reviewer acknowledgement pythagorashttp://www.pythagoras.org.za acknowledgement to reviewers 59 the quality of the articles in pythagoras crucially depends on the expertise and commitment of our peer reviewers. reviewing is an important part of scholarly work, making a substantial contribution to the field. reviewers’ comments serve two purposes, guided by two inter-dependent objectives: • pythagoras wishes to publish only original papers of the highest possible quality, making a meaningful contribution to the field. reviewers advise the editor on the scholarly merits of the manuscript to help him evaluate the manuscript and to decide whether or not to publish it. reviewers are encouraged to reject a manuscript if it is scientifically flawed, merely sets out observations with no analysis, provides no new insights, or is of insufficient interest to warrant publication. • pythagoras is committed to support authors in the mathematics education community. reviewers help the author to improve the quality of their manuscript. reviewers are encouraged to write their comments in a constructive and supportive manner and to be sufficiently detailed to enable the author to improve the paper and make the changes that may eventually lead to acceptance. the following summary of outcomes of the reviewing process in 2014 shows that our reviewers do well in achieving both objectives: we sincerely thank the following people who have reviewed these manuscripts for pythagoras in 2014. we very much appreciate their time, expertise and support of pythagoras amidst pressures of work. ana p. lombard andrew talmadge aneshkumar maharaj anita campbell anna crowe anna posthuma ann-sofi röj-lindberg ansie harding anthony essien belinda huntley bruce brown calisto munongi cally kuhne carol bohlmann cheryl reeves in an effort to facilitate the selection of appropriate peer reviewers for manuscripts for pythagoras, we ask that you take a moment to update your electronic portfolio on www. http://pythagoras.org.za, allowing us better access to your areas of interest and expertise, in order to match reviewers with submitted manuscripts. if you would like to become a reviewer, please visit the pythagoras website and register as a reviewer. to access your details on the website, follow these steps: 1. log into pythagoras online at http://www. pythagoras.org.za 2. in your ‘user home’ select ‘edit my profile’ under the heading ‘my account’ and insert all relevant details, bio statement and reviewing interest. it is good practice as a reviewer to update your personal details regularly to ensure contact with you throughout your professional term as reviewer to pythagoras. please do not hesitate to contact me if you require assistance in performing this task. rochelle flint submissions@pythagoras. org.za tel: +27 21 975 2602 fax: +27 21 975 4635 page 1 of 2 craig pournara cyril julie david andrich deonarain brijlall dirk wessels duncan samson elizabeth burroughs erica spangenberg hamsa venkat hennie boshoff janine hechter joseph dhlamini karin brodie kerstin jordaan lyn webb 1. accepted after one round of review, with ‘minor’ changes as specified by reviewers and editor. 2. accepted after two or more rounds of review, with major changes specified by reviewers and editor. 3. includes one case where the authors did not resubmit after required to make major changes. 4. all submissions undergo a preliminary review by the editor team to ascertain if it falls within the aims and scope of pythagoras and is of an acceptable standard. no. manuscripts processed in 2014 (outcome complete) 21 accepted without changes 0 (00.0%) accepted with minor changes (to the satisfaction of the editor)1 5 (23.8%) accepted after major revisions (re-submit, then re-review)2 5 (23.8%) rejected after review not acceptable to be published in pythagoras3 6 (28.6%) rejected without review not acceptable to be published in pythagoras4 5 (23.8%) http://www.pythagoras.org.za http://www.pythagoras.org.za reviewer acknowledgement pythagorashttp://www.pythagoras.org.za page 2 of 2 if you would like to become a reviewer, please visit the pythagoras website and register as a reviewer. manare setati mandisa lebitso mdutshekelwa ndlovu melih turgut michael de villiers michael murray nancy chitera nelis vermeulen nick taylor paul mokilane piera biccard rajendran govender sarah bansilal segun adeyefa shaheeda jaffer sharon mcauliffe sheena rughubar-reddy sibawu siyepu tim dunne toni beardon tracy craig verona leendertz werner olivier zain davis zalman usiskin zingiswa jojo reviewers (continued): 60 pyth 36_2 reviewer acknolegdement.indd reviewer acknowledgement pythagorashttp://www.pythagoras.org.za page 1 of 2 reviewing is an important part of scholarly work, making a substantial contribution to the field. reviewers’ comments serve two purposes, guided by two inter-dependent objectives: • pythagoras wishes to publish only original papers of the highest possible quality, making a meaningful contribution to the field. reviewers advise the editor on the scholarly merits of the manuscript to help him evaluate the manuscript and to decide whether or not to publish it. reviewers are encouraged to reject a manuscript if it is scientifically flawed, merely sets out observations with no analysis, provides no new insights, or is of insufficient interest to warrant publication. • pythagoras is committed to support authors in the mathematics education community. reviewers help the author to improve the quality of their manuscript. reviewers are encouraged to write their comments in a constructive and supportive manner and to be sufficiently detailed to enable the author to improve the paper and make the changes that may eventually lead to acceptance. the following summary of outcomes of the reviewing process in 2015 shows that our reviewers do well in achieving both objectives: we sincerely thank the following people who have reviewed these manuscripts for pythagoras in 2015. we very much appreciate their time, expertise and support of pythagoras amidst pressures of work. in an effort to facilitate the selection of appropriate peer reviewers for manuscripts for pythagoras, we ask that you take a moment to update your electronic portfolio on http:// www.pythagoras.org.za, allowing us better access to your areas of interest and expertise, in order to match reviewers with submitted manuscripts. if you would like to become a reviewer, please visit the pythagoras website and register as a reviewer. to access your details on the website, follow these steps: 1. log into pythagoras online at http://www. pythagoras.org.za 2. in your ‘user home’ select ‘edit my profile’ under the heading ‘my account’ and insert all relevant details, bio statement and reviewing interest. it is good practice as a reviewer to update your personal details regularly to ensure contact with you throughout your professional term as reviewer to pythagoras please do not hesitate to contact me if you require assistance in performing this task. duncan hooker submissions@pythagoras. org.za tel: +27 21 975 2602 fax: +27 21 975 4635 anelize van biljon anilkumar krishnannair anita campbell anthony essien antonia makina belinda huntley benadette ainemani bruce brown carol macdonald caroline long clement dlamini connie skelton david mogari david mtetwa david reid debbie stott deonarain brijlall dirk wessels doug clarke duncan mhakure edgar guacaneme elmarie meyer elsa lombard elspeth khembo erna lampen eugenia vomvoridi-ivanovic eunice moru faaiz gierdien gary powell hamsa venkat the quality of the articles in pythagoras and the credibility and reputation of our journal crucially depend on the expertise and commitment of our peer reviewers. acknowledgement to reviewers no. manuscripts processed in 2015 (outcome complete) 26 accepted without changes 0 (0%) accepted with minor changes (to the satisfaction of the editor)1 10 (38.5%) accepted after major revisions (re-submit, then re-review)2 2 (7.7%) rejected after review not acceptable to be published in pythagoras3 9 (34.6%) rejected without review not acceptable to be published in pythagoras4 5 (19.2%) 1. accepted after one round of review, with ‘minor’ changes as specified by reviewers and editor. 2. accepted after two or more rounds of review, with ‘major’ changes specified by reviewers and editor. 3. includes two cases where authors did not resubmit after required to make major changes. 4. all submissions undergo a preliminary review by the editor (and associate editors) to ascertain if it falls within the aims and scope of pythagoras and is of an acceptable standard. includes two cases where authors did not resubmit after extensive feedback prior to reviewing. reviewer acknowledgement pythagorashttp://www.pythagoras.org.za page 2 of 2 helena miranda ingrid sapire jacob jaftha janine hechter jayaluxmi naidoo jogy alex johan meyer johann engelbrecht joseph dhlamini jurie conradie kakoma luneta kerryn vollmer kosie smit lindiwe tshabalala lizelle fletcher lorna holtman lyn webb marc schäfer mark jacobs mdutshekelwa ndlovu michael de villiers michael mhlolo michael murray neil eddy niren naidoo pam lloyd paul mokilane paula ensor piera biccard radley mahlobo rajendran govender richard alexander sally hobden sarah bansilal satsope maoto sharon mcauliffe sheena rughubar-reddy sibawu siyepu stanley adendorff tim dunne ursula hoadley vasuthavan govender willy mwakapenda yip cheung chan zain davis zwelethemba mpono if you would like to become a reviewer, please visit the pythagoras website and register as a reviewer. reviewers (continued): reviewer acknowledgement open accesshttp://www.pythagoras.org.za page 1 of 2 reviewing is an important part of scholarly work, making a substantial contribution to the field. reviewers’ comments serve two purposes, guided by two inter-dependent objectives: • pythagoras wishes to publish only original papers of the highest possible quality, making a meaningful contribution to the field. reviewers advise the editor on the scholarly merits of the manuscript to help him evaluate the manuscript and to decide whether or not to publish it. reviewers are encouraged to reject a manuscript if it is scientifically flawed, merely sets out observations with no analysis, provides no new insights, or is of insufficient interest to warrant publication. • pythagoras is committed to support authors in the mathematics education community. reviewers help the author to improve the quality of their manuscript. reviewers are encouraged to write their comments in a constructive and supportive manner and to be sufficiently detailed to enable the author to improve the paper and make the changes that may eventually lead to acceptance. the following summary of outcomes of the reviewing process in 2017 shows that our reviewers do well in achieving both objectives: we sincerely thank the following people who have reviewed these manuscripts for pythagoras in 2017. we very much appreciate their time, expertise and support of pythagoras amidst pressures of work. in an effort to facilitate the selection of appropriate peer reviewers for manuscripts for pythagoras, we ask that you take a moment to update your electronic portfolio on http:// www.pythagoras.org.za, allowing us better access to your areas of interest and expertise, in order to match reviewers with submitted manuscripts. if you would like to become a reviewer, please visit the pythagoras website and register as a reviewer. to access your details on the website, follow these steps: 1. log into pythagoras online at http://www. pythagoras.org.za 2. in your ‘user home’ select ‘edit my profile’ under the heading ‘my account’ and insert all relevant details, bio statement and reviewing interest. it is good practice as a reviewer to update your personal details regularly to ensure contact with you throughout your professional term as reviewer to pythagoras please do not hesitate to contact us if you require assistance in performing this task. publisher: publishing@aosis.co.za tel: +27 21 975 2602 fax: +27 21 975 4635 the quality of the articles in pythagoras and the credibility and reputation of our journal crucially depend on the expertise and commitment of our peer reviewers. no. manuscripts processed in 2017 (outcome complete) 25 accepted without changes 0 (0%) accepted with minor changes (to the satisfaction of the editor)1 5 (20%) accepted after major revisions (re-submit, then re-review)2 5 (20%) rejected after review − not acceptable to be published in pythagoras3 6 (24%) rejected without review − not acceptable to be published in pythagoras4 9 (36%) no. articles currently in review 11 anass bayaga andile mji andrew maffessanti andrew stevens anelize van biljon anita campbell ansie harding anthea roberts anthony essien antonia makina barbara postuma benadette ainemani benita nel bhekumusa khuzwayo bruce brown carol bohlmann carol macdonald caroline long cheryl reeves clement dlamini craig pournara cyril julie david mogari dirk wessels duncan mhakure erica spangenberg erna lampen eunice moru helen sidiropoulos helena miranda ingrid sapire jaya naidoo jenefer golding jogy alex johan hugo kakoma luneta kathryn mellor l.m. kaino leila goosen lekwa mokwana leonor camargo lindiwe tshabalala lizelle fletcher lyn webb margot berger maria trigueros marie joubert marthie van der walt acknowledgement to reviewers 1. accepted after one round of review, with ‘minor’ changes as specified by reviewers and editor 2. accepted after two or more rounds of review, with ‘major’ changes specified by reviewers and editor 3. includes one case where the authors did not resubmit after required to make major changes 4. all submissions undergo a preliminary review by the editorial team to ascertain if it falls within the aims and scope of pythagoras, is of sufficient interest to our readers, offers substantially new knowledge, and is of sufficient quality to be sent for review. we have this year received a flood of 18 submissions from indonesia, russia and south america that we unfortunately had to reject mainly because the authors’ english language constructs simply could not carry the mathematics education concepts they were trying to communicate, and fixing it seemed impossible, despite our commitment to support authors and despite our professional language editing services. we do not include these 18 articles in our statistics, because that would distort the acceptance rate. http://www.pythagoras.org.za http://www.pythagoras.org.za http://www.pythagoras.org.za http://www.pythagoras.org.za http://www.pythagoras.org.za reviewer acknowledgement open accesshttp://www.pythagoras.org.za page 2 of 2 mdu ndlovu michael mhlolo michael murray nosisi feza nyna amin olatunde osiyemi odette umugiraneza pam lloyd patrick barmby piera biccard samuel khoza sarah bansilal satsope maoto shaheeda jaffer sibawu siyepu suela kacerja tulsi morar vasuthavan govender verona leendertz wajeeh daher willy mwakapenda yael shalem yip-cheung chan zingiswa jojo zulkardi zulkardi http://www.pythagoras.org.za a site-based model for professional development in mathematics at the elementary school level denise s. mewborn university of georgia email: dmewborn@coe.uga.edu patricia d. huberty university of georgia email: pattihuberty@charter.net teaching is a profession in which one must continue to learn because there is no one right way to teach, and one does not “master” teaching. students change, curricula change, and schools change; thus there is always a need to grow as teachers. heaton (1994, 2000), an experienced teacher who engaged in a yearlong self-study as she tried to change her teaching practice to be consistent with current calls for reform, characterized teaching as “inherently under construction … and continuous invention” (p. 341). she noted that teaching is very situation-specific so teachers must constantly tinker with their practice. cobb, wood and yackel (1990) asserted that a teacher’s classroom is a “learning laboratory” (p. 131). in response to requests from local teachers, we developed a professional development model based on the notions of a classroom as a learning laboratory and teachers as learners. our overarching goal was to help teachers reconsider the fundamentals of both what it means for children to learn mathematics, and for them to teach mathematics. our sub-goal was to improve student achievement in mathematics by enhancing teachers’ classroom practices to enable them to meet the learning needs of all students. in this paper we present a description and evaluation of a model of professional development that we have been using with elementary school mathematics teachers for several years. we describe the origin of the model and its connections to both teachers’ needs and the literature-base on teacher development, the implementation of the model, and the results we have seen in both student achievement and teachers’ practices. the professional development projects described in this manuscript were supported by funding from the eisenhower program for improving mathematics and science instruction and the teacher quality grants program. conceptual basis for the model our work in professional development was born out of needs identified by teachers and administrators in local schools. concerned about overall student achievement and disparities in teachers’ comfort levels with and support for teaching mathematics versus teaching literacy, the leadership team from an elementary school approached us and asked how we might help them address their concerns. as we collaborated on the design of the professional development project, teachers brought to the table their prior experiences with professional development and their understanding of the needs facing their students and fellow teachers. we brought to the table our past experiences with professional development and our knowledge of the research literature on effective practices in professional development. together, we crafted a workable model that has since been implemented in five schools. the teachers’ prior experiences with professional development suggested three major themes. first, mathematics staff development that spans grades prekindergarten through fifth grade is too general to be useful. the content demands and student needs in place value, geometry, data analysis and other topics are very different across the grades spanned in an elementary school. second, staff development must be directly linked to teachers’ classroom practices. teachers told us they did not want to attend a oneweek “crash course” in the summer or an after school workshop that resulted in a notebook of activities that they would have to sift through in order to find what applied in their classrooms. they wanted sustained staff development. third, the teachers wanted the staff development activities to be site-based so that the staff 2 pythagoras 59, june, 2004, pp. 2 7 denise mewborn, patricia huberty developers understood their students, their curriculum, and their school structures. although the research base on effective professional development is in its infancy, the teachers’ concerns resonated with common messages in the literature. these messages include the importance of focusing on mathematics content, student thinking, and curriculum (gearhart et al., 1999), situating staff development in the context of teachers’ practices (fennema et al., 1996, kazemi & franke, 2000; schifter, 1998), providing sustained staff development (garet, porter, desimone, birman, & yoon, 2001; schifter, 1998), and fostering collegial relationships through staff development (garet et al., 2001; silver, smith, & nelson, 1995; stein, silver, & smith, 1998). description of the model our professional development model has three main components. the first and most important is classroom-based support for individual teachers and/or teams of teachers. the second is quarterly cross-gradelevel staff development sessions, and the third is monthly grade-level meetings. two additional components of the project include support for the school’s math committee and support for parent involvement in students’ mathematics learning. this manuscript describes only the classroom-based support aspect of the project. staff the project staff consists of a university-based mathematics educator (first author), a school-based mathematics specialist (second author), and a university-based mathematician. the mathematics educator serves as the project coordinator to mediate university and school paperwork and politics. the mathematics educator also serves as a resource and support for the mathematics specialist. the schoolbased mathematics specialist is an experienced classroom teacher with a strong background in mathematics and mathematics education. the specialist is responsible for the bulk of the professional development activities described in this manuscript. the university-based mathematician is a person with an interest in school mathematics who serves as a sounding board when mathematical issues arise. across the five schools in which this model of professional development has been employed, there have been two different mathematics educators, three different mathematics specialists, and three different mathematicians. contexts the five schools in which we have implemented the model are in the same school district in the town where our university is located. the elementary schools are pre-k or k through 5th grade schools (ages 4 through 11). the district contains 13 elementary schools and serves over 11,000 students in elementary, middle and high school, with 62% of these students living in poverty as defined by the federal government1. the demographics of individual schools vary widely, however. the demographics of the five schools in which we have employed this professional development model are summarized in table 1. implementation. the focus of the classroom-based support is to prepare teachers to effectively use strategies known to enhance student achievement: hands-on learning, group work, class discussions, and the use of questions requiring higher-order thinking in both instruction and assessment (milken family foundation, 2000). the support occurs primarily in the form of planningteaching-debriefing cycles with individual teachers or teams of teachers at the same grade level. the goal of working with a team of teachers is to promote a “community of learners” among the staff so that mathematics teaching becomes a publicly shared responsibility (garet et al., 2001). a typical planningteaching-debriefing cycle with one grade level includes one or two planning periods in which teachers and the mathematics specialist discuss the upcoming sequence of lessons. this discussion includes an assessment of students’ readiness for the topic (including a discussion of common misconceptions students have and known weaknesses in students’ achievement in this area), a review of the topics that have already been covered and in what manner, a summary of teachers’ goals for these lessons, and an exploration of possible manipulatives, representations, and examples to be used in the lessons. between planning sessions the mathematics specialist and the teachers spend time doing individual 1 students are identified as living in poverty by a federal standard based on household income. students living in poverty are eligible to receive a hot meal at lunch at a reduced price or for free, depending on their family’s income. this is commonly referred to as the “free and reduced price lunch rate” in the united states and is synonymous with the poverty rate. 3 professional development in mathematics at the elementary school level planning and gathering resources. during the next planning session they share their plans and resources with the team to gather suggestions for modification. then, the lessons are implemented. the implementation occurs in a variety of ways–with each teacher teaching the lessons in her own classroom with supp teach math the m teach do a math supp skills type pract the p spec durin spec discu how appr poss emph sessi of expla writt hom inten all s strate have of the model because it enables teachers to see individual students through new eyes. this model promotes active learning on the part of teachers (garet et al., 2001) because they are “actively engaged in meaningful discussion, planning, and practice” (garet et al., 2001, p. 925). 4 school a school b school c school d school e number of students in k-5 430 400 525 400 430 % of students living in poverty 82% 47% 56% 67% 81% race/ethnicity of students: white black hispanic asian multi-racial 15 55 25 2 5 42 36 7 24 4 33 58 2 4 4 28 55 12 2 3 12 75 9 2 2 % of 4th grade students who failed to meet mathematics standards on criterionreferenced test in 2000-2001 49% 24% 45% 38% 68% table 1: school demographics ort from the mathematics specialist, with one er teaching the lessons with support from the ematics specialist as other teachers observe, with athematics specialist teaching some lessons and ers conducting others. the goal is for teachers to s much of the teaching as possible with the ematics specialist and other teachers serving in orting roles to increase the teachers’ pedagogical and comfort so that they are able to continue this of instruction when the project is finished. in ice, however, we find that during the first year of roject teachers are more comfortable with the ialist doing most of the teaching. several times g the instructional sequence the mathematics ialist and the teachers hold debriefing sessions to ss evidence of students’ mathematical thinking, the instruction facilitated student learning, the opriateness of the materials being used, and the ible next steps after these lessons. there is an asis on examining student work during these ons. the mathematics specialist leads discussions students’ verbal comments, questions, and nations and promotes examination of students’ en work (produced during the lesson or as ework). this focus on students’ thinking is ded to draw teachers’ attention to the learning of tudents and to help them focus on instructional gies that are meaningful for their students. we found that this is a particularly powerful aspect impact of the model across the five schools we have seen a variety of changes in teachers’ classroom practices in mathematics. for some teachers this change is in one particular content area, such as fractions, in which they now feel more confident about teaching the material because they understand it themselves. other teachers feel better about motivating students to learn mathematics through the use of children’s literature, manipulatives and group work. others have changed their approach to classroom discourse because they have seen the benefits of having students talk and listen to one another about mathematics. some teachers have found a connection between their longheld views about teaching and learning in other subjects (such as literacy or social studies) and their emerging views about mathematics. in schools where administrators have been actively involved in observing instruction throughout the project, we have seen a change in their expectations. many administrators have gained an appreciation for the ways that mathematics instruction might be conducted and for the value of discourse in the classroom. in one school, the professional development process facilitated a school-wide examination of the practice of ability grouping students for mathematics instruction. teachers in the fourth and fifth grades in this school believed that students needed to be denise mewborn, patricia huberty grouped with same-ability peers so that mathematics instruction could be provided on an appropriate level. in general, this arrangement led to a predominant focus on lower-level skills with the lowest-achieving students and an accelerated pace with a modicum of problem solving with the highest-achieving group of students. when the mathematics specialist encouraged the fourth-grade teachers to engage their students in more problem solving tasks, the teachers expressed considerable skepticism that their lowest-achieving students were capable of problem solving. the mathematics specialist therefore spent a week in each classroom and presented the same four problem solvin partici of stu lesson the problems using a variety of strategies. as a result of these lessons and the associated discussions, the teachers began to examine their perceptions of what students could do and the role of problem solving in learning mathematics. they also began to reconsider both the mathematics content and the teaching strategies to which they exposed their students. over the last two years, the entire school (including parents) has begun a discussion of how to best meet the needs of the academically diverse student body, and ability grouping has come into question. as a result of these discussions, the school has implemented an inclusion model for mathematics instruction in grades k, 1 and 2. rather than separating children by ability, the teachers who are certified for special education and gifted education go into the general education classroom during mathematics instruction rather than pulling children out of the classroom for specialized instruction. this experiment is ongoing so we do not yet have test data or anecdotal data from parents and teachers. however, we believe that it is a sign of progress that teachers and parents are questioning the value of ability-grouping and are taking steps to understand it better. test scores, while they cannot be solely attributed to professional development, provide some evidence school 2000 2001 2002 2003 school a 51% 52% 56% 61% school b 81% 76% 68% 73% school c 73% 55% 66% 72% school d 59% 62% 46% 65% school e 50% 33% 24% 58% table 2: test scores for participating schools g tasks to each group. all three teachers pated in each lesson, working with small groups dents while the mathematics specialist led the . students in all three groups successfully solved of improved student achievement. the data in table 2 show the percentage of fourth-grade students who met or exceeded standards on a state-mandated criterionreferenced test. we administered a teacher survey in please mark the extent to which you agree or disagree with the following statements: strongly disagree disagree agree strongly agree the kind of teaching advocated by the project is helping my students reach higher levels of mathematics achievement. 1 4 22 17 the type of mathematics teaching advocated by the project is very different from what i have been doing. 1 15 22 7 i value the kind of mathematics teaching advocated by the project. 1 15 28 the project provided me with knowledge, skills, or resources that are useful to me in the classroom. 2 17 26 the project helped me pay closer attention to particular things i was doing in the mathematics classroom. 3 16 26 the project led me to think about an aspect of my mathematics teaching in a different way. 1 21 23 the project led me to try new things in the classroom. 1 20 24 i feel more confident about teaching mathematics now. 4 27 13 table 3: teacher survey 5 professional development in mathematics at the elementary school level schools c, d, and e in order to gauge whether professional development was having an impact on teachers beyond supplying them with new mathematical tasks and materials. the survey items in table 3 were taken from the teacher questionnaire developed for the study of instructional improvement conducted by the consortium for policy research in education at the university of michigan. we selected items from this existing survey because we thought these items tapped the kinds of deeper teacher learning we were trying to foster. overall, we believe that our informal and formal assessments of the professional development project show that we are meeting our goals. critique of the professional development model this staff development model has advantages and disadvantages. we believe the advantages outweigh the disadvantages, but it is important to articulate both. the advantages of the model primarily correspond to the characteristics of effective staff development identified by the teachers and in the literature. all staff development is conducted entirely at the school site and involves all teachers in the building who have responsibility for mathematics instruction, which facilitates school-wide, within grade level, and cross grade level dialogue about mathematics instruction, curriculum, and assessment. by conducting staff development activities in the building where teachers work, we are able to see first-hand their teaching environment and work with their students. teachers do not have to be away from the building during the school day or drive to another location at the end of a long school day for staff development. furthermore, they do not have to endure staff development that “won’t work with my students” because we do work with their students. the “grain size” of our approach is both an advantage and a disadvantage. the grain size of most professional development is too large (e.g. geometry for k-5 teachers). our grain size is one classroom at a time and one topic at a time. using this approach, we do not “cover” every topic with every teacher. however, we believe that the impact is far greater when we help teachers think differently about the entire enterprise of teaching mathematics via specific lessons on particular topics. by taking this approach, we work to build content knowledge as well as pedagogical content knowledge in the mathematics that teachers teach (borko and putnam, 1995) rather than simply focusing on building isolated skill sets (national commission on mathematics and science teaching, 2000; renyi, 1996). a substantial disadvantage of our model is that it requires a significant commitment of resources. the mathematics specialist is essentially an additional teacher in the building, but she does not contribute to reducing the student-teacher ratio. we have been fortunate to have external funding for our work thus far, but any replication or expansion of this model is expensive. in addition, the human resources required for this model are considerable. the mathematics specialist must be someone whose content and pedagogical content knowledge span the mathematics curriculum of pre-kindergarten through fifth grade. in addition, the specialist must be able to work well with teachers as peers in a non-evaluative, coaching relationship. it does not necessarily follow that someone who is a good teacher of mathematics to children is a good professional developer. thus, it takes a person with fairly specialized knowledge and skills to serve as a mathematics specialist. we do not have structures in which to prepare such people, so it is rather an ad hoc creation. as with most models of professional development, this one fails to address problems of sustainability. while it is plausible that changes in a particular teacher’s practice will be sustained over the long term, this model does not account for teacher turnover at the school level. this model does not necessarily lead to the development of a professional community of learners in the school to support teachers who are new to the school as they develop teaching practices that are consistent with those of the rest of the school. conclusion as a field of inquiry, mathematics education suffers from a lack of research on professional development and teacher learning. what we have is a collection of “cases” much like the one we have provided here. most accounts are descriptive, and a few contain some data about effectiveness. we must guard against viewing these accounts as simply a collection of stories. we need to mine these stories by conducting cross-case analyses in order to develop theoretical frameworks about professional development that account for teachers’ knowledge, teachers’ practice, and student learning. as cooney (1994) note: “if we are to move beyond collecting interesting stories, theoretical perspectives need to be developed that allow us to see how those stories begin to tell a larger 6 denise mewborn, patricia huberty story” (p. 627). the development of theoretical frameworks will help us design future professional development projects, conduct research on professional development, and evaluate the effectiveness of such projects. thus, it is critically important for mathematics educators to craft professional development programmes that reflect best practices, to share these programmes widely, and to research the effectiveness of such program so that we can move toward effective means for supporting teachers as they engage in the continual journey that is teaching. references borko, h., & putnam, r. t., 1995, “expanding a teacher's knowledge base: a cognitive psychological perspective on professional development” in t. r. guskey, (ed.), professional development in education: new paradigms and practices: pp. 35-65. new york: teachers college press cobb, p., wood, t., & yackel, e., 1990, “classrooms as learning environments for teachers and researchers” in r. davis, c. maher, & n. noddings (eds.), constructivist views on the teaching and learning of mathematics (jrme monograph 4, pp. 125-146) reston, va: national council of teachers of mathematics cooney, t. j., 1994, “research and teacher education: in search of common ground” in journal for research in mathematics education, 25(6), pp. 608-636 fennema, e., carpenter, t. p., franke, m. l., levi, l., jacobs, v. r., & empson, s. b., 1996, “a longitudinal study of learning to use children’s thinking in mathematics instruction” in journal for research in mathematics education 27, pp. 403-434 garet, m.s., porter, a.c., desimone, l., birman, b.f., & yoon, k.s., 2001, “what makes professional development effective? results from a national sample of teachers” in american educational research journal, 38(4), pp. 915-945 gearhart, m., saxe, g. b., seltzer, m. schlackman, j., ching, c. c., nasir, n., fall, r., bennett, t., rhine, s., & sloan, t. f., 1999, “opportunities to learn fractions in elementary mathematics classrooms” in journal for research in mathematics education, 30, pp. 286-315 heaton, r. m., 2000, teaching mathematics to the new standards: relearning the dance. new york: teachers college press. heaton, r. m., 1994, “creating and studying a practice of teaching elementary mathematics for understanding” (doctoral dissertation, michigan state university, 1994), dissertation abstracts international, 55-07a: 1860 kazemi, e., & franke, m. l., 2000, “understanding teacher learning as changing participation in communities of practice” in m. l. fernández (ed.) proceedings of the twentysecond annual meeting of the north american chapter of the international groups for the psychology of mathematics education: pp. 561566, columbus, oh: eric clearinghouse for science, mathematics, and environmental education milken family foundation, 2000, how teaching matters: bringing the classroom back into discussions of teacher quality, princeton, nj: ets. available at: www.ets.org/research/pic national commission on mathematics and science teaching (ncmst), 2000, before it's too late: a report to the nation from the national commission on mathematics and science teaching for the 21st century, jessup, md: national commission on mathematics and science teaching renyi, j., 1996, teachers take charge of their learning: transforming professional development for student success, washington, dc: national foundation for the improvement of education schifter, d., 1998, “learning mathematics for teaching: from a teachers’ seminar to the classroom.” in journal of mathematics teacher education, 1, pp. 55-87 silver, e. a., smith, m. s., & nelson, b. s., 1995, “the quasar project: equity concerns meet mathematics education reform in the middle school” in w. g. secada, e. fennema, & l. b. adajian (eds.), new directions for equity in mathematics education: pp. 9-56 new york: cambridge university press and national council of teachers of mathematics stein, m. k., silver, e. a., & smith, m. s., 1998), “mathematics reform and teacher development from the community of practice perspective” in j. greeno & s. goldman (eds.), thinking practices: a symposium on mathematics and science learning: pp. 17-52, hillsdale, nj: erlbaum 7 http://www.ets.org/research/pic description of the model staff contexts implementation. impact of the model critique of the professional development model conclusion references article information author: susan nieuwoudt1 affiliation: 1faculty of educational sciences, north-west university, potchefstroom campus, south africa correspondence to: susan nieuwoudt email: susan.nieuwoudt@nwu.ac.za postal address: po box 20189, noordbrug 2522, south africa dates: received: 21 aug. 2014 accepted: 29 sept. 2015 published: 30 nov. 2015 how to cite this article: nieuwoudt, s. (2015). developing a model for problem-solving in a grade 4 mathematics classroom. pythagoras, 36(2), art. #275, 7 pages. http://dx.doi.org/10.4102/pythagoras.v36i2.275 copyright notice: © 2015. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. developing a model for problem-solving in a grade 4 mathematics classroom in this original research... open access • abstract • introduction and orientation • conceptual and theoretical framework • empirical investigation    • aim of the investigation    • research method    • analysis and discussion of the findings       • problem 2       • problem 3       • problem 4       • problem 7 • conclusions • acknowledgements    • competing interests    • author's contribution • references abstract top ↑ the teaching of problem-solving through the development of a problem-solving model was investigated in a grade 4 mathematics classroom. learners completed a questionnaire regarding their knowledge of mathematical problem-solving, their attitudes towards problem-solving, as well as their experiences in solving problems. learners’ responses revealed overall negative beliefs towards problem-solving as well as a lack of knowledge about what problem-solving in mathematics entails. the teacher then involved the learners in a structured learning programme where they worked in cooperative groups of six on different kinds of mathematical problems to solve. the groups regularly engaged in discussions about the different strategies they were using to solve a specific problem and eventually succeeded in formulating a generic problem-solving model they could call their own. the model was effectively used by the learners to solve various mathematical problems, reflecting their levels of cognitive development to a certain extent. introduction and orientation top ↑ problem-solving has to be the primary goal of the teaching and learning of mathematics, giving each learner the opportunity to engage in problem-solving activities (nctm, 2000). learners not only learn mathematics while solving problems, but also develop problem-solving skills and strategies while doing mathematics (lesh & zawojewski, 2007; schoenfeld, 1992, 2013). the identification and solving of problems using critical and creative thinking, working in groups and recognising that problem-solving contexts do not exist in isolation are some of the general aims set for education and training in south africa (department of basic education, 2011). moreover, problem-solving is also part of every content area in the south african intermediate and senior phases mathematics curricula. south african grade 8 learners performed poorly in the trends in international mathematics and science study (timss) (howie, 2004; reddy, 2006). the timss evaluated, among others, acquired mathematics knowledge as well as the use of logical thinking while solving problems (heideman, 1999). brenner, herman, ho and zimmer (1999) attribute the outstanding performance of learners from singapore, korea and japan to effective teaching and learning of problem-solving skills at school. south african mathematics learners’ poor results, in contrast, seemed to relate to learners’ inadequate mathematics knowledge and skills, especially problem-solving skills, and to poor mathematics teaching and learning (howie, 2004; reddy, 2006). this is a reflection of the situation not only in grade 8 mathematics classrooms but also in grade 4, where learners encounter problem-solving for the first time in a more formal (structured) way than before. although issues related to problem-solving in mathematics have been widely researched at secondary school level, little is known about problem-solving strategies at grade 4 level. the aim of this article is to report on the process by which grade 4 mathematics learners develop a problem-solving model while solving problems. more specifically, the following research question was addressed: how can problem-solving be taught in a grade 4 mathematics classroom? conceptual and theoretical framework top ↑ the research reported in this article was executed from a social-constructivist perspective regarding the learning of mathematics. learners construct their own mathematical knowledge by connecting mathematical facts, procedures and ideas (hiebert & grouws, 2007). understanding or meaningful learning involves not only internal or mental representations of individual learners, but also social and cultural aspects. the development of mathematical concepts and mathematics learners’ problem-solving abilities is highly interdependent and socially constructed (lesh & zawojewski, 2007). therefore, the teaching of mathematics through problem-solving provides opportunities for learners to gain understanding and attain higher levels of achievement (rigelman, 2007). problem-solving refers to a mathematical situation that poses a mathematical question to which the solution is not immediately accessible to the solver, because they do not have a way to relate the data to a solution (callejo & vila, 2009). for the purpose of this article, mathematical problem-solving refers to a person's efforts to solve a problem that they have not encountered before. through solving a given problem, a person should learn some mathematics. perspectives on problem-solving vary from a more traditional approach to a models and modelling approach. traditionally, problem-solving involved the following steps: mastering the prerequisite mathematical ideas and skills, practising the newly mastered ideas and skills in solving word problems, learning general problem-solving processes and, finally, applying the learned ideas and skills to solve real-life problems. lesh and zawojewski (2007) view problem-solving as modelling: in response to a real-life problem situation, the problem solver will engage in mathematical thinking as they produce or develop a sensible solution for the problem. this suggests that people learn mathematics through problem-solving and that they learn problem-solving through doing mathematics. for schoenfeld (2013), solving problems is part of the doing and sense-making of mathematics. in doing mathematics, learners investigate, make conjectures and use problem-solving strategies to verify those conjectures. most current problem-solving models have adapted polya's four-phase model of understanding the problem, devising a plan, carrying out the plan and looking back (polya, 1973). lester (1985) added meta-cognitive behaviour to polya's model. schoenfeld (1992) included managerial processes in the teaching of problem-solving, to be discussed with learners while they are solving problems. fernandez, hadaway and wilson (1994) introduced a dynamic and cyclic interpretation of the model, including meta-cognitive processes (self-monitoring, self-regulating and self-assessment). according to this model, a learner starts solving a problem by engaging in thought to understand a given problem, then moves into the planning stage. after some time spent on making a plan, the learner's self-monitoring of understanding creates the need to understand the problem better and the learner returns to the understanding-the-problem stage. in their research on the nature of problem-solving behaviour, lester and kehle (2003) come to the conclusion that the knowledge of good problem-solvers not only exceeds the knowledge of poor problem-solvers, but also is more connected. good problem-solvers pay more attention to the structural features of problems, while poor problem-solvers’ attention is focused on surface features. furthermore, good problem-solvers are better users of meta-cognition during problem-solving. schoenfeld (1992) refers to meta-cognition as the ability that enables problem-solvers to break down a problem into sub-problems, solving the sub-problems and eventually solving the original problem. wilson and clark (2004) report on students’ use of meta-cognitive language to describe how they go through a meta-cognitive cycle (awareness, evaluation, regulation, evaluation) during problem-solving activities. primary school mathematics learners are not mathematical problem-solvers by nature; therefore, they have to be taught problem-solving skills and strategies (mccormick, miller & pressley, 1989; lesh & zawojewski, 2007). this can be done by using problem-based mathematics lessons (van de walle, karp & bay-williams, 2013). these lessons consist of three parts, namely a ‘before’ part when pre-knowledge is assessed and the problem is presented to the learners, the ‘during’ part when the learners attempt to solve the problem and the ‘after’ part when the learners discuss and reflect on their solutions. problem-solving can be successfully executed in small groups (mcleod, 1993). the interaction between the teacher and learners as well as among the learners working together in small groups can improve the quality of the teaching and learning of mathematics (berry & nyman, 2002). rather than working on their own, learners in groups have more opportunities to participate in problem-solving activities, discover problem-solving strategies for themselves and report back to other groups than when working on their own (cangelosi, 2003). the understanding of the role of beliefs and dispositions in problem-solving has not changed much since schoenfeld's work in 1992 (callejo & vila, 2009; lesh & zawojewski, 2007). certain beliefs with respect to mathematical problem-solving sometimes have negative influences on learners’ mathematical thinking, such as: mathematics problems have only one correct answer; there is only one correct way to solve a mathematics problem; only a few learners understand mathematics – other learners are supposed to memorise and apply what they have learnt without understanding; learners who have understood the mathematics they learnt will be able to solve any problem in five minutes or less (schoenfeld, 1992). a supportive problem-solving environment can change learners’ dispositions towards problem-solving (yudariah, yusof & tall, 1999) from negative to positive. middleton, lesh and heger  (2003) conducted problem-solving sessions among learners where they had to solve mathematical problems in small groups. during the sessions learners not only shared their mathematical thinking processes while collaborating in groups, but also revealed their beliefs and dispositions with respect to the mathematics dealt with in a specific problem. although there is a strong relationship between learners’ approaches to problem-solving and their belief systems, it is difficult to determine a causal relationship between specific beliefs and problem-solving activities (callejo & vila, 2009). learners’ beliefs regarding the level of effort required to solve a mathematics problem, as well as their self-confidence in mathematics problem-solving, influence the learners’ involvement in solving a given problem. from the above arguments it should be clear that school mathematics can be taught through problem-solving and that learners learn mathematics while solving problems. therefore, the development and use of a problem-solving model has the potential to assist learners in the learning of mathematics. empirical investigation top ↑ aim of the investigation this article reports on one aspect of a broader study (graaff, 2005), namely an empirical investigation into teaching problem-solving through developing a problem-solving model in a grade 4 mathematics classroom. research method a qualitative research method by means of a case study was employed in a grade 4 mathematics classroom in an urban school in gauteng, south africa. a purposive sample of one grade 4 mathematics class was chosen from the three grade 4 classes (the population) taught by the participating teacher. the gauteng department of education, as well as the school, granted written permission for the research. parents of all the participating learners granted informed consent. we guaranteed that learners’ identities would not be revealed. the class was divided into six small groups of six learners each. these learners’ performance in mathematical problem-solving was studied for a period of eight months. for the duration of the investigation, all mathematics topics were taught through problem-solving. initially (before they had to solve any mathematical problems), a questionnaire regarding their experiences in problem-solving, their attitudes towards problem-solving, their efforts at problem-solving and their knowledge of solving problems was completed by the class group. learners from one small group (the investigative group, from now on referred to as ig or group a) were also interviewed with respect to their beliefs about mathematics, problem-solving and group work. the small groups, labelled a−f, were each given different kinds of problems to solve. the teacher used problem-based mathematics lessons (van de walle et al., 2013) to teach mathematics to the grade 4 learners. while solving a problem, group members had to design a problem-solving ‘model’ (see table 1), indicating step by step how a learner should go about solving the specific problem. after a problem had been solved, different groups from the class had the opportunity to illustrate and explain their problem-solving models. table 1: summary of different groups’ efforts towards a problem-solving model. after the grade 4 class, with the assistance of their teacher, had developed their problem-solving model (see figure 4), the model was used to solve more mathematics problems. analysis and discussion of the findings learners’ responses to the questionnaire revealed negative attitudes and beliefs towards problem-solving that could be attributed to a lack of exposure to problem-solving in previous grades. when confronted with questions about solving problems, learners showed little confidence in answering these questions: can you solve a mathematics problem (like the example)? do you understand what is asked in the problem? where would you start solving the problem? do you have a plan in mind (to help you to solve the problem)? this is consistent with the findings of callejo and vila (2009:123) that even high school mathematics learners are reluctant to solve unknown problems because of negative beliefs towards problem-solving. from the interviews with the ig learners (group a) the following became evident: although the group members did not regard mathematical problems as difficult as such, and were not afraid to attempt solving a problem, three learners admitted that they did not have an idea where to start to solve a given problem. whereas only one group member read the problem more than once, trying to understand the problem, the other two learners used trial and error to solve the problem, but did not show any indication that they had tested their solution. one must conclude that they had a lack of knowledge about what problem-solving in mathematics entails. as has been indicated, the initial data gathering about learners’ attitudes towards and knowledge about problem-solving was followed by the implementation of a problem-solving approach in the classroom. i now provide a few brief illustrating examples of group discussions during the problem-solving sessions as a background to the summary of the findings in table 1. problem 2 nine children attended a birthday party. on one of the plates on the table were 24 cocktail sausages, among other food. the birthday girl wanted everyone to have the same number of sausages. how many sausages did each child get to eat? two of the members of group b read the problem to the other group members. although luke read fluently, carin struggled to pronounce some of the words. james drew 24 cocktail sausages and nine learners beneath the sausages (see figure 1). ockert: let's distribute the sausages among the learners. figure 1: group b's drawing for problem 2. [james drew lines between the sausages and the nine learners. adeli distributed the next nine sausages, while the others kept on counting.] group members: 1, 2, 3, 4, 5, 6, 7, 8, 9. ockert: we divide each sausage into nine parts [referring to the remaining five sausages.] [adeli wrote down the answer as 215, clearly not the correct answer.] (graaff, 2005, p. 70) problem 3 a wall is built by laying 17 rows of bricks, using 69 bricks for each row. how many bricks are used? while one learner from group a was reading the problem to the other members of the group, another learner started writing down ‘69’ 17 times, indicating the rows of bricks in the wall. another learner knew she had to multiply 69 by 17, but could not find the answer. a learner from group c thought that adding the sum of 17 × 9 and 17 × 6 would give the correct answer. problem 4 a string of beads is made by using three red beads for every five blue beads. how many red beads are there in a string containing 60 blue beads? one of the members of group a, jaco, read the problem to the others. two of the girls in the group did not understand what ‘three red beads for every five blue beads’ meant. another girl, ronel (with encouragement from the teacher) tried to explain it to the others: ronel (to jaco): draw a big circle, put three red beads on the circle. [jaco drew the circle with the 3 beads on the circle (see figure 2).] figure 2: group a's drawing for problem 4. ronel: jaco, you have to draw five blue beads on the circle. we have to complete the circle using 50 blue beads. jane: no, 60 blue beads. ronel: oh yes, 60. jaco: what now? ronel: well, just continue the same way. now you draw five blue beads and three red beads. [jaco continued like this until there were a total of 60 blue beads. the group members then counted the red beads.] group members: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36. ronel: there are 36 red beads on the string. (graaff, 2005, p. 68) problem 7 a girl has two skirts and three blouses that can be mixed and matched. how many outfits can she put together? amanda (a member of group e) took a pencil and drew the three blouses and two skirts, showing the different combinations in figure 3a. ben did not know the word ‘blouse’ and amanda explained (to the other boys in the group) that a blouse is a girl's shirt. jake showed amanda that there are more ways to combine the skirts and blouses (see figure 3b). although the other group members agreed with jake's solution, carol drew her own picture, in order to assure herself that jake had drawn all the possible combinations. figure 3: group e's drawings for problem 7. from the solutions to problems 1 to 7 (see table 1), as well as other problems solved during the investigation, the following became clear: some learners experienced difficulties in understanding a given problem. most learners realised they had to do something (make a plan) to solve a given problem. learners used different problem-solving strategies (draw a picture, do a calculation, act it out, etc.) to solve a problem. learners were not able to solve some of the problems because their calculations were wrong. learners did not always check solutions to given problems. not all the stages of the problem-solving models used by polya (1973), schoenfeld (1992, 2013), fernandez et al. (1994), and others referred to earlier in the article, are reflected in table 1. this result is in line with the views of mccormick et al. (1989) and lesh and zawojewski (2007) that learners are not problem-solvers by nature, and that to become successful problem-solvers, learners have to be supported in discovering problem-solving strategies. in addition, the social interaction between learners during the solving of the respective problems assisted them in exploring and constructing ‘new’ mathematical knowledge. each group had the opportunity to display and explain their model to the other groups. during the class discussions each group tried to convince the other groups that their model could be used to solve any problem in grade 4 successfully. the teacher asked the learners the following questions: teacher:can your group's model for problem-solving be used to solve a mathematics problem? learner 1: yes. teacher: will you always get the right answer when you use this model? learner 2: sometimes, but not always. teacher: how would we know that the answer to the problem is wrong? learner 1: when you mark it wrong. learner 2: when the different groups’ answers are not the same. learner 3: when we haven’t answered the question. teacher: yes! how do we know that we haven’t answered the question? learner 4: at the end, after we have done everything. (graaff, 2005, p. 79) the teacher then asked the grade 4 learners how they would guide other learners when (1) they didn’t understand a problem, (2) they had to make another plan when their plans to solve a problem did not work out and (3) how would they (the learners) know that their solution to a specific problem was correct. learners reworked their models, resulting in the problem-solving model illustrated in figure 4. this cyclical model closely resembles the mathematical problem-solving ‘method’ originally initiated by polya (1973), and adapted by fernandez et al. (1994) and others. figure 4: the problem solving model compiled by the grade 4 learners. when the groups applied the developed model to solve some other mathematical problems, the initial observations of the teacher were confirmed, namely that learners with a well-developed number sense solved problems with more ease than those with a weak number sense, that learners’ ability to perform the basic operations correctly enabled them to solve problems and that learners needed basic mathematical knowledge and skills to solve problems. conclusions top ↑ grade 4 learners, assisted by their mathematics teacher, were able to compile a problem-solving model while trying to solve novel mathematics problems. although grade 4 learners were able to use the compiled model to solve other mathematical problems, the success of the teaching of mathematics through problem-solving depends on more than one factor. for problem-solving to be effective in a primary school mathematics classroom, the mathematics teacher has to plan thoroughly for teaching, involve the learners actively in the learning-teaching activities and play a crucial role as facilitator by teaching problem-solving with the aid of a guideline such as a problem-solving model. mathematics teachers need to understand the role of problem-solving in learners’ everyday lives, as well as the importance of problem-solving in the mathematics classroom. by incorporating problem-solving in their classrooms, teachers will enable learners not only to attain one of the general aims of the south african curriculum, namely to identify and solve problems and make decisions using critical and creative thinking, but also to attain a specific aim for school mathematics, namely to apply mathematics to solve problems, using acquired knowledge and skills. acknowledgements top ↑ this article is partially based on the research by magda graaff (graaff, 2005) in one of her grade 4 mathematics classrooms in a primary school in randfontein, gauteng, south africa. competing interests the author declare that she has no financial or personal relationship(s) that may have inappropriately influenced her in writing this article. author's contribution s.n. 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(1999). changing attitudes to university mathematics through problem solving. educational studies in mathematics, 37, 67–82. http://dx.doi.org/10.1023/a:1003456104875 microsoft word 13-21 pilay & pilay equipartitioning points.doc pythagoras, 71, 13-21 (july 2010) 13 equipartitioning and balancing points of polygons    shunmugam pillay & poobhalan pillay  school of mathematical sciences  university of kwazulu‐natal  pillaysc@ukzn.ac.za & pillayph@ukzn.ac.za    the centre of mass g of a triangle has the property that the rays to the vertices from g  sweep out triangles having equal areas. we show that such points, termed equipartitioning  points in this paper, need not exist in other polygons. a necessary and sufficient condition  for a quadrilateral to have an equipartitioning point is that one of its diagonals bisects the  other.  the  general  theorem,  namely,  necessary  and  sufficient  conditions  for  equipartitioning points for arbitrary polygons to exist, is also stated and proved. when this  happens, they are in general, distinct from the centre of mass. in parallelograms, and only  in them, do the two points coincide.  this paper provides a specific scenario that can be utilised to stimulate learners into investigating a geometrical problem through the questioning process. the investigation leads to conjectures that are verifiable with the aid of computer software like the geometer’s sketchpad, and then the formal proofs verifying the conjectures are made. traditionally the function of proof has been seen to be to verify or justify a mathematical statement. it is well known that proofs, especially in geometry, are a major problem for learners. for most learners mathematical statements (theorems) seem obvious and they see no need for proofs. often, they do not appreciate a need for proofs and get frustrated when asked to prove a given theorem or “to prove” a statement from a textbook or a teacher (de villiers, 2003, p. 5). to stimulate genuine interest in proof, de villiers (1997), chazan (1990) and others have argued that it requires that students are inducted early into the art of problem posing, allowing sufficient opportunity for exploration, conjecturing, refuting, reformulating and explaining. movshovitz-hadar (1988) has similarly suggested stimulating presentations of results that solicit the surprise and curiosity of students so that they are susceptible to responsive proofs, which leave them with “an appreciation of the invention, along with a feeling of becoming wiser” (p. 15). pólya emphasizes that the students’ interest and motivation must stem from mathematics itself. it is the duty of the teacher to select tasks for the students that are: not too difficult and not too easy, natural and interesting and appropriate to their knowledge. mathematics should also be used to develop problem solving skills and gain confidence and hence students “may experience the tension and enjoy the triumph of discovery” (1945, p. v). further the reform movement in mathematics education requires the teacher to provide tasks that are stimulating, relevant and foster growth in knowledge. this can be achieved by helping students connect mathematical ideas and concepts to real life situations (national council of teachers of mathematics, 1991). to this end, the problem in figure 1 can be posed and investigated in a grade 11 class. such an investigation may proceed along the following lines. equipartitioning and balancing points of polygons 14 mr baker makes biscuits that are in the shape of polygons. he has templates of triangles, quadrilaterals, pentagons and so on, but although all of them are convex, most of them have random dimensions. he finds a point in his triangle template that has the property that the three triangles formed by joining the point to the three vertices all have the same area. this pleases him, for he can then sell each piece separately at the same price. he calls such points, if they can be found, equipartitioning points. investigate the following. 1. how does he locate the equipartitioning point inside the triangle and how can he be certain that the point is the correct one? how many such points are there? 2. can he do the same with a quadrilateral? that is, can all quadrilaterals be divided into four equal triangles in the manner described? 3. justify your answer. if it cannot be done, for which quadrilaterals would mr baker be able to locate at least one equipartitioning point? 4. investigate the same question for convex polygons having five or more sides. 5. mr baker is aware that all polygons have balancing points, also called centres of mass, namely, points at which the mass of the polygon may be presumed to be concentrated. (here he regards the polygon as a plate having uniform density). would the balancing point of a polygon be the point mr baker is looking for? figure 1: the problem the investigation the square, rectangle and parallelogram learners should have no difficulty in identifying the equipartitioning points in these shapes as the points of intersection of the diagonals. finally they should explain why such points work. the triangle 1. using appropriate software, locate such points for a predetermined number of triangles. also ascertain how many such points there are. 2. use the points obtained to conjecture a rule by which these points may be located without using technology. 3. give a formal proof of your assertions in 1 and 2. we believe that with a little help maybe, all learners would come to the conclusion that every triangle has exactly one equipartitioning point, and that it is situated at the point of intersection of (any) two of its medians. the uniqueness of the equipartitioning point also can be used to argue that for any triangle all three medians go through the same point, namely, it’s equipartitioning point. this leads to a secondary conjecture. the kite use appropriate software again to determine whether kites have equipartitioning points. make a conjecture and give a formal proof of your claim. the general quadrilateral do the same for trapeziums and cyclic quadrilaterals. what do you conclude? which quadrilaterals have equipartitioning points? are these the only ones? shunmugam pillay & poobhalan pillay 15 centres of mass each learner could then be asked to plot the equipartitioning point of stiff triangular boards, and then try to support the triangle with a finger held up vertically under the board, at the point where the equipartitioning is located. if the construction is accurate, the board will balance perfectly. it is known that all polygons possess such balancing points. the terms centres of mass, centre of gravity and centroid are also used. hence, for triangles, the equipartitioning point coincides with the centre of mass. formal statements and proofs formal definition let a1a2…an be a polygon. an interior point q of the polygon is called an equipartitioning point of a1a2…an if the n triangles qa1a2, qa2a3, …,qan-1an, qana1 all have the same area. in figure 2 we have that dc bd adcarea abdarea  . hence if ad is a median, bd = dc, so area abd = area adc. that is, a median bisects the area of a triangle. figure 2 theorem 1 a triangle has a unique equipartitioning point, namely, the point of concurrency of its medians. proof we first prove that the point g of concurrency of the medians of a triangle is an equipartitioning point. figure 3 the symbols a, b, c in figure 3 refer to the areas of the triangles. now 2a + b = 2c + b and a + 2b = a + 2c lead to 2a = 2b = 2c. so g is an equipartitioning point. equipartitioning and balancing points of polygons 16 conversely, let q be any equipartitioning point of  abc produce bq to meet ac at d. let a, x and y be the areas as indicated in figure 4. since q is an equipartitioning point, a = x + y. figure 4 now y a x a qd bq  so x = y, hence ad = dc, proving that q lies on the median through b. by symmetry, q also lies on the median through a. hence, q is the point of concurrency g of the medians. we have shown that g is the only equipartitioning point. the case for a quadrilateral the following is of independent interest. theorem 2 a diagonal of a quadrilateral (not necessarily convex) bisects the area of a quadrilateral  the diagonal bisects the other diagonal (see coxeter & greitzer, 1967, pp. 54-55). proof suppose one of the diagonals bisects the other. refer to figure 5. let the area of  ade = a and the area of  abe = b. then it is an easy matter to see that the same diagonal bisects the area as well. figure 5 conversely, let abcd be a quadrilateral, not necessarily convex, such that db bisects the area of abcd. draw the altitudes ae and cf of  abd and  dbc as shown in figure 6. since the area of  adb is equal to the area of  cdb, we have 2 1 db.ae= 2 1 db.cf, so ae = cf. let us first assume that e and f are distinct points. since ae // fc, aecf is a parallelogram. hence the midpoint of diagonal ac is on fe (and hence on db or db produced), proving that one of the diagonals bisects the other. shunmugam pillay & poobhalan pillay 17 if e = f, aec is a straight line, db bisects ac and is perpendicular to it. (in this case abcd is a kite). this completes the proof. figure 6 whereas all polygons have centres of gravity, not all polygons have equipartitioning points, as the case for triangles seems to suggest. the next result provides a necessary and sufficient condition for a quadrilateral to have an equipartitioning point. the convex and non-convex cases are treated separately. theorem 3 a quadrilateral has an equipartitioning point q  one of its diagonals bisects the other, and then q is the midpoint of the first diagonal (see gilbert, krusemeyer, & larson, 1993). proof we consider two cases, one where the quadrilateral is convex, and the other where it is not convex. case 1 let abcd be a convex quadrilateral such that ac bisects bd (see figure 7). figure 7 let q be the midpoint of ac and let a, b and c be the indicated areas. then c = a + b, proving that the midpoint q of diagonal ac is an equipartitioning point. conversely, suppose a convex quadrilateral abcd has an equipartitioning point q (see figure 8). then diagonals qb and qd bisect the areas of quadrilaterals abcq and aqcd respectively. hence from theorem 1, both qb and dq contain the midpoint m of ac. equipartitioning and balancing points of polygons 18 figure 8 suppose that q is different from m. the straight line through q and m contains d and b, hence dqb is the diagonal db and it bisects ac. but triangles adq and abq have the same area, so dq = qb . we have shown that in the case where q and m are distinct, diagonal db bisects diagonal ac and the equipartitioning point q is the midpoint of db. now suppose the equipartitioning point q = m, the midpoint of ac. the diagonal ac then bisects the area of abcd. from theorem 2, diagonal ac bisects diagonal db and q (= m) is the midpoint of ac. so the theorem is proved for convex quadrilaterals. case 2 let abcd be a quadrilateral with bcd > 180° (see figure 9). we shall prove that abcd has an equipartitioning point q  ac (produced) bisects db, and then q is the midpoint of ac. figure 9 one direction is straightforward; if ac produced bisects the diagonal bd, then it is easy to see that if q is the midpoint of ac, then the areas of triangles qab, qbc, qcd, and qda are all equal. suppose then abcd has an equipartitioning point q in its interior. the areas of triangles qab, qbc, qcd and qda are all equal. then diagonals aq and qc bisect the areas of (convex) quadrilaterals abqd and bqdc respectively. from theorem 2, the midpoint m of bd lies on aq and qc produced. since m is in the exterior of abcd, a, q, c and m are collinear. then bq bisects the area  abc so q is the midpoint of ac. also ac bisects bd, so we are done. shunmugam pillay & poobhalan pillay 19 remarks learners may now be asked: 1. explain why parallelograms and kites always have equipartitioning points. 2. prove that a trapezium has an equipartitioning point if, and only if, it is a parallelogram. the general case we are now in a position to generalise to an arbitrary polygon. theorem 4 let a1a2…an be a polygon. for any three adjacent vertices x, y, z of the polygon, let y' be the midpoint of xz. then a1a2…an has an equipartitioning point q  the n lines pp', as p runs through the n vertices, are concurrent, and then the point of concurrency is the equipartitioning point of a1a2…an. figure 10 proof suppose the polygon (refer to figure 10) has an equipartitioning point q. let x, y, z be adjacent vertices. from theorem 2 q lies on yy' (possibly produced). it follows that q lies on each of the n lines pp', as p runs through the n vertices, so the n lines are concurrent. conversely, if q lies on each such yy' (possibly produced), it is easy to see that for any three adjacent vertices x, y, z, triangles qxy and qyz have equal area, so q is an equipartitioning point, proving the theorem. remarks learners may be asked to verify the following: 1. in the special case where the polygon is a triangle, the three lines mentioned above are just the medians, and, since the medians of a triangle are always concurrent, all triangles have equipartitioning points. 2. in the case of a quadrilateral the concurrency of the four lines above is equivalent to the statement that one of the diagonals bisects the other, and then the point of concurrency is the midpoint of the bisecting diagonal. balancing points in contrast to equipartitioning points, balancing points (also called centres of mass, centres of gravity or centroids) of all plane regions, and in particular, of all polygons, always exist. the following result (wales, 2010) provides a method to locate balancing points of polygonal plane regions. it should be mentioned that the centre of mass or centroid of equal point masses placed at the vertices of a polygon, in general, do not coincide with the centre of mass of a polygonal plane region. equipartitioning and balancing points of polygons 20 theorem 5 let r be a plane region bounded by a polygon (see figure 11). suppose r is subdivided into two regions r1 and r2 having areas a1 and a2 respectively. let g1 and g2 be the centres of mass of r1 and r2. then the centre of mass g of r is a point in the interior of the line segment g1g2 satisfying the condition a1.g1g = a2.g2g. in particular, if the two areas are equal, then g is the midpoint of g1g2. figure 11 the next theorem classifies those plane quadrilaterals having equipartitioning points coinciding with their centre of mass. theorem 6 let abcd be a quadrilateral that has an equipartitioning point q. then q is the centre of mass of abcd if, and only if, abcd is a parallelogram. proof let abcd be a parallelogram whose diagonals ac and bd intersect at m. figure 12 the centre of mass g1 of  abd lies on am with g1m = 3 1 am. the centre of mass g2 of  cbd lies on cm with g2m = 3 1 cm. since am = cm, m is the midpoint of g1g2. since  abd and  cbd have equal areas, g, the centre of mass of abcd, is the midpoint of g1g2, by theorem 5. that is, g = m. since the diagonals partition a parallelogram into four equal areas, m = q, the equipartitioning point. so q = g is the centre of mass of abcd. conversely, let quadrilateral abcd have equipartitioning point q, and further, suppose q = g, the centre of mass of abcd. let the diagonals intersect at m. by theorem 4, we may assume am = cm and bq = dq. suppose aqc is not a straight line. let g1 and g2 be the centres of mass  cbd and  abd respectively. then g1 lies on cq and g2 lies on aq and g1qg2 is a triangle. but g1gg2 is a straight line, contradicting q = g. hence aqc is a straight line. it follows that ac and bd bisect each other. hence abcd is a parallelogram. shunmugam pillay & poobhalan pillay 21 the parallelogram technique the following construction provides a simple technique to find the centre of mass of an arbitrary plane quadrilateral. trisect the four sides and complete the parallelogram determined by these eight points (see figure 13). the diagonals of the parallelogram meet at the centre of mass of the plane quadrilateral. figure 13 discussion we trust that this paper will encourage teachers and learners to investigate different partitionings of polygons into parts of equal area, because it is a very rich topic with some challenging, still unsolved problems (see for example croft, falconer, & guy, 1991). references chazan, d. (1990). quasi-empirical views of mathematics and mathematics teaching. interchange, 21(1), 14-23. doi: 10.1007/bf01809606 coxeter, h. s. m., & greitzer, s. l. (1967). geometry revisited. washington, dc: mathematical association of america. croft, h. t., falconer, k. j., & guy, r. k. (1991). unsolved problems in geometry. newyork: springerverlag. de villiers, m. (1997). the role of proof in investigative, computer-based geometry: some personal reflections. in j. king & d. schattschneider, geometry turned on: dynamic software in learning, teaching, and research (pp. 15-24). washington, dc: mathematical association of america. de villiers, m. (2003). rethinking proof with sketchpad. emeryville: key curriculum press. gilbert, g. t., krusemeyer, m., & larson, l. c. (1993). the wohascum county problem book. dolciani mathematical expositions, 14(10), 68-70, washington, dc: mathematical association of america. movshovitz-hadar, n. (1988). stimulating presentations of theorems followed by responsive proofs. for the learning of mathematics, 8(2), 12-19,30. pólya, g. (1945). how to solve it. a new aspect of mathematical method. princeton, nj: princeton university press. national council of teachers of mathematics. (1991). professional standards for teaching mathematics. reston, va: national council of teachers of mathematics. wales, j. 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/hrv (za stvaranje adobe pdf dokumenata najpogodnijih za visokokvalitetni ispis prije tiskanja koristite ove postavke. stvoreni pdf dokumenti mogu se otvoriti acrobat i adobe reader 5.0 i kasnijim verzijama.) /hun 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/pdfxoutputintentprofileselector /documentcmyk /preserveediting true /untaggedcmykhandling /leaveuntagged /untaggedrgbhandling /usedocumentprofile /usedocumentbleed false >> ] >> setdistillerparams << /hwresolution [2400 2400] /pagesize [612.000 792.000] >> setpagedevice microsoft word 28-35 essien.doc 28 pythagoras, 69, 28-35 (july 2009) an analysis of the introduction of the equal sign  in three grade 1 textbooks    anthony a. essien  marang centre for maths and science education  university of the witwatersrand  anthony.essien@wits.ac.za      this paper attempts to analyse how the concept of the equal sign is introduced to learners  in  grade  1  textbooks  in  use  in  south  africa.  in  doing  this,  three  grade  1  textbooks  (learner’s book and their accompanying teacher’s guide) were analysed  in terms of the  first  appearance  of  the  equal  sign.  analysis  reveals  that  the  equal  sign  is  introduced  operationally and  its  introduction  is subsumed  in  the  introduction of  the addition and  subtraction signs. the author argues for the equal sign to be accorded equal status and  attention as the addition and subtraction symbols by both the curriculum and grade 1  textbooks and also makes an argument for the introduction of the equal sign first (using  appropriate pictorial representations and artefacts) before the introduction of the plus and  minus signs.  research has long recognised that learners tend to misunderstand the equal sign as an operator, that is, as a symbol inviting them to “do something”, to “find the answer”, rather than as a relational symbol signifying equivalence or quantitative sameness. investigations carried out in the early 1980s up to the present day have shown that the equal sign is understood by many learners at primary and early secondary levels as either a do-something symbol (that is, as an automatic invitation to write the answer), and/or a unidirectional symbol (asquith, stephens, knuth, & alibali, 2007; behr, erlwanger, & nichols, 1980; carpenter, franke, & levi, 2003; demonty & vlassis, 1999; essien & setati, 2006; falkner, levi, & carpenter, 1999; herscovics & kieran, 1980; kieran, 1981; knuth, stephens, mcneil, & alibali, 2006; liebenberg, sasman, & olivier, 1999; mcneil, grandau, knuth, alibali, stephens, hattikudur, & krill, 2006; prediger, 2010; sáenz-ludlow & walgamuth, 1998; stacey & macgregor, 1997). in my previous study (essien & setati, 2006), where i explored the understanding of the equal sign among grade 8 and 9 learners in a south african context, it emerged that learners in these grades also conceive of the equal sign as a do-something or unidirectional symbol. research has attributed such narrow understanding of the equal sign to, amongst others, the use of calculators, direct verbal to written translation of mathematical sentences, the use of metaphors to explain, for example why a + b cannot be simplified further, the use of contexts that do not denote an equivalent relation (e.g., drink + drive = death as one sees in some billboards in nigeria), and the way the equal sign is introduced to learners in early grades. the present analysis of textbooks, therefore, arose out of the need to explore learners’ first encounter with the concept of the equal sign. according to barbe, bosch, espinoza and gascon (2005), one of the moments necessary for the successful completion of didactic processes (or processes of study) involving a concept is the moment of the first encounter. knowledge on how to organise the learners’ first encounter with a particular concept is crucial to the type of internalisation that occurs in the learners. this paper reports on an analysis of three grade 1 textbooks (both learner’s book and teacher’s guide) – maths for all 1 (schools development unit (sdu), 2003a, 2003b), classroom mathematics grade 1 (jenkins et al., 2003a, 2003b) and successful numeracy grade 1 (chantler, hoffmann, & stephanou, 2008; smith, chantler, hoffmann, & stephanou, 2008) in order to examine how the equal sign is introduced to learners in these textbooks. it must be noted, therefore, that the present analysis is not aimed anthony essien 29 at a comparative analysis of the three textbooks; neither is it aimed at recommending one textbook over the others. it is rather aimed at an in-depth examination of how the equal sign is first encountered in textbooks (and the curriculum), and thereafter to propose one way of introducing the equal sign. grade 1 textbooks were chosen because it is at this grade that the equal sign is introduced to learners. the three textbooks were chosen for a number of reasons. first, because they are books that are commonly used by learners and especially teachers in grade 1. second, the books are written for the outcomes based education (obe) – the framework for the revised national curriculum statement (rncs) (department of education (doe), 2002). third, the textbooks, to a large extent, reflect the manner in which teachers generally introduce the concept of the equal sign as revealed by research (essien & setati, 2006; kieran, 1981; 1992; mcneil et al., 2006; behr et al, 1980). finally, the three textbooks each have an accompanying teacher’s guide. this is critical to the present analysis because, one presupposition could be that learner books are usually written with the learners in mind, while the guidelines for the teacher and didactical underpinnings are provided in the teacher’s guide. as such, it was quite logical that both learner book/activity book and the accompanying teacher’s guide became the objects of analysis. 1 the overall argument in this paper is that because the textbooks introduce the equal sign within the context of addition and subtraction, they (the textbooks) entrench the understanding of the equal sign as an operational rather than a relational symbol. in the light of this observation, i propose a different way of introducing the equal sign to learners (especially in terms of grade 1 content sequencing) that, in my opinion, would enable an understanding or grasp of the significance of the equal sign as a relational symbol. i also argue for the equal sign to be given the same status and attention (as the addition and subtraction signs) by both textbooks and the curriculum. the equal sign in the rncs the rncs does not foreground the importance of the equal sign in grade 1 school mathematics. neither does the mathematics curriculum for the foundation phase emphasise the importance of teaching/introducing the equal sign to learners. the notion of the equal sign is, however, embedded in learning outcome 1, which requires of learners to be able to “recognise, describe and represent numbers and their relationships, and to count…” (doe, 2002, p. 6) and to be able to order and compare collections of objects using the words “more”, “less” and “equal” (doe, 2002, p. 14). the curriculum also expects learners to know how to perform calculations using the appropriate symbols to solve problems involving addition and subtraction, etc. the national council of teachers of mathematics (nctm) principles and standards for school mathematics, unlike the rncs, makes a particular allusion to the equal sign symbol by referring to it as “an important algebraic concept that students must encounter and begin to understand in the lower grades (nctm, 2000, p. 94). the principles and standards also notes that the common learners’ understanding of the equal sign at this stage (foundation phase) should be more accurate than the limited understanding of the equal sign as signifying “the answer is coming”. learners need to understand that the equal sign “indicates a relationship – that the quantities on each side are equivalent” (nctm, 2000, p. 94). in the principles and standards, unlike the rncs, therefore, we see an explicit recognition of the necessity for direct and explicit instruction as far as the equal sign is concerned. the rncs seems to take for granted the fact that the equal sign, (unlike the plus sign or minus sign), needs to be highlighted explicitly in texts and pedagogy in order for learners to develop a more sophisticated understanding of the equal sign. yet as the above mentioned research has shown, the concept of “equal” is a complex and difficult one for learners to comprehend and the misconceptions that many learners from grades 1-9 posses attest to this fact. 1 it must be noted that there is a dearth of grade 1 learner’s books. in my visits to many primary schools in the johannesburg area, i observed that what is available and used in most schools are rather the learner’s workbooks/activity books (rather than learner’s book). it was therefore expedient to examine some of these learner workbooks/activity books. in the three textbooks under consideration, one (successful numeracy) is a learner workbook and the other two are learner books. an analysis of the introduction of the equal sign 30 critical analysis of the equal sign in the textbooks introduction of the equal sign in the three textbooks maths for all is a mathematics textbook series used widely in south african schools. written in the light of outcomes-based education, math for all attempts to integrate the activities used in the book to other learning areas, and to learners’ daily activities in the home, school, etc. activities used in the textbook are also such that encourage learners to work in a range of ways – talking, writing, singing, listening to stories, playing games, drawing, collecting, sorting, etc. the learner book, at the bottom of each page, provides the assessment focus and instructs teachers as to what they should ask learners to do. in maths for all the introduction of the equal sign is preceded by the introduction of the counting (number) system using various diagrams and strategies. the first appearance of the equal sign occurs with the introduction of addition. learners are asked to put one counter next to another and to say how many counters there are altogether. the teacher is instructed to place an item on the desk, and then place another one next to it and to ask the learners how many items there are altogether. the idea is to show, for example, that “one and one makes two” and to show the learners that this is written as 1 + 1 = 2 (see sdu, 2003b, p. 12). pictorial representations of items to be added, with the placeholders after the equal sign, like the one shown in figure 1, are also used in the introduction of the plus sign to learners (for the actual image, see sdu, 2003a, p. 17): figure 1: introduction of the equal sign in maths for all the equal sign is also used in the introduction of the minus sign, using representations such as insects, eggs, counters, fruits, etc. after contrasting addition and subtraction, the placeholder is used with the equal sign in an exercise which the learners are to do (see sdu, 2003a, p. 29). in the teacher’s guide for maths for all 1, there is a detailed description of how the teacher should introduce the addition sign with the use of counters and other objects (balls, trees, etc) to demonstrate the combination process. the same process is used to introduce the subtraction symbol. the teacher’s guide stresses the importance of understanding the value of a number and the correct use of the language of operations (such as add, plus, subtract, take away, etc). nowhere in the teacher’s guide are there any explicit instructions on how the teacher ought to introduce the equal sign. in fact, the textbook takes for granted that the learners would automatically know what the equal sign means when placed between objects or when used with a placeholder. even when the commutative property of addition is introduced later in the chapter (through placeholders) there is no mention of the equal sign as signifying equivalence relations. the learner is left to believe that equal sign means “makes” as in “one plus one makes two” used in the earlier introduction of the addition symbol. classroom mathematics is a mathematics textbook series that adheres to the principles of obe inasmuch as it advocates integration between learning areas and uses real life contexts in the introduction of several mathematics concepts. the first appearance of the equal sign in classroom mathematics also occurs during the introduction of addition after the constructs of “more” and “less” have been introduced to learners. learners are given some exercises to complete. anthony essien 31 in the first exercise involving robots, learners are given an addition problem on the left-hand-side of the equal sign (example, 6 + 5 =) and three options in form of traffic lights (example, 10, 11, 15) on the righthand-side of the equal sign. learners are expected to colour in the correct answers with red, orange or green. in the second exercise, the learners physically see the objects (dots) and how the addition of the objects is written mathematically, as illustrated in figure 2 (for the original, see jenkins et al., 2003a, pp. 41-43). 3 + 1 = 4 2 + 4 = figure 2: introduction of the equal sign in classroom mathematics the emphasis here, according to the teacher’s guide, is to teach learners how to write number sentences. learners are then given many drilling questions with placeholders and diagrams to reinforce the concept of addition. the same process is used to introduce the subtraction symbol. learners are given a set of questions, first without diagrams, and then with diagrams and asked to write the correct number in the placeholders (see p. 65). in this book, it is interesting to note that the symbols are used first, and only later are the learners provided with pictorial representations. like maths for all, nowhere in the classroom mathematics learner’s book are there any explicit pictorial diagrams aimed at enabling learners see and understand the significance of the equal sign in the addition or subtraction process. in the teacher’s guide, however, teachers are urged to show learners that “is equal to” means “the same as”. the teacher’s guide also urges teachers to draw pictures of, say, 1 + 1 = 2 and to explain this to the learners (jenkins et al., 2003b, p. 38). there is also an elaborate explanation and activities around subtraction and its introduction in the teacher’s guide, but nothing on the equal sign at this stage. in successful numeracy, the equal sign appears first in an activity where learners are asked to count how many objects are in the picture, and how many are left after crossing out some of the items (see figure 3 for an example of this). in so doing, the book introduces the concept of subtraction. count: how many?  cross out 3 5 – 3 =  figure 3: introduction of the equal sign in successful numeracy after this, just like maths for all, pictorial diagrams and the accompanying mathematical sentences are used to introduce the addition sign (using placeholders) much like in figure 1 above for maths for all. like classroom mathematics, while there are several activities showing learners and teachers what to do/how to introduce the minus and plus symbols, the learner’s workbook of successful numeracy has no an analysis of the introduction of the equal sign 32 activities aimed at explicitly entrenching in learners the significance of the equal sign. the teacher’s guide, however, urges teachers to introduce the equal sign, without explicitly saying how teachers are to accomplish this (smith et al., 2008, p. 39). the teacher’s guide later encourages teachers to ask learners to match pictures of shape with their real objects like: a rectangle = a table or a chalkboard; a square = a window plane or a chair seat. as i will explain later, placing the equal sign between objects only lead to misconceptions about the significance of the equal sign. practical implications of using the textbooks research has long recognised the implications of an operational rather than a relational understanding of the equal sign in algebra and other branches of mathematics. it is not the aim of this paper to restate such implications as a product of using the three textbooks under consideration. it is, however, pertinent to observe that when one looks at the diagrams above from the textbooks, it is easy to see why learners would understand the concept of addition and subtraction better than they would understand the concept of the equal sign. from the pictorial representations in the learner books, it is difficult for learners to grasp the concept of the equal sign (as signifying an equivalent relation) from the mathematics mathematical representations (both pictorial or symbolic representations) that are used in the three textbooks. i argue that a cursory explanation of the equal sign as “the same as” or “makes” is highly unlikely to firmly entrench the meaning of the equal sign in learners. rather, learners, because they are introduced to the equal sign in the way the textbooks do (and in the context of addition and subtraction without explicit attention to the equal sign per se), would always tend to conceive of the equal sign as simply a tool for writing the answer. in figures 1, 2, and 3, learners may well see that the equal sign as used in the texts plays a role in the rightand left-hand relation of the question. but it would be difficult for learners to know that this relation is that of quantitative sameness between objects on the left-handside and objects on the right-hand-side. recommendations the rncs textbooks are written in accordance with what the curriculum stipulates as knowledge that learners need to acquire. the first obvious recommendation as far as the curriculum is concerned is for it to give more attention to the concept of the equal sign. as research by knuth et al. (2006) has shown, understanding the equal sign does matter as the limited notion of what the equal sign signifies is a major stumbling block in the learning of algebra in later grades. they also contend that the lack of explicit focus on the equal sign in curricula can only contribute to inadequate understanding of the equal sign since they (curricula) form the foundation for textbooks and teachers. given the importance of the equal sign in mathematics and the fact that the equal sign is one of the most used, if not the most used, notation in mathematics, and more importantly, given the fact that the equal sign is a symbol that is very easily misunderstood by learners, i argue that the equal sign needs to be accorded the same status as the plus and minus signs in the rncs. textbooks the preoperational stage (roughly from age two to age six or seven) is the second stage of the piagetian genetic epistemology. most learners in grade 1 fall into this category. at this stage, according to piaget, the child learns to manipulate his environment symbolically through inner representations, or thoughts, about the external world. also, during this stage, the child learns to represent objects by words and to manipulate the words mentally, just as he/she earlier manipulated the physical objects themselves (inhelder & piaget, 1958). that said, it is commendable that the textbooks use symbols and representations to mediate, albeit, implicitly the mathematical understanding of the equal sign. to this effect, future grade 1 mathematics textbooks should be encouraged to use pictorial representation of object familiar to the context of learners of that age bracket to introduce the concept of the equal sign. care must be taken, however, to avoid using the equal sign between two objects (e.g., a window = a rectangle) as this anthony essien 33 does not represent a relationship of equality between numbers and therefore, does not focus on the significance of the equal sign. in the textbooks too, a common feature is that the equal sign is introduced alongside the addition and subtraction signs with elaborate and articulated ways of introducing the plus and minus signs. the minus and plus signs, it can be argued, are an invitation to do something since there cannot be an addition or subtraction sign in a question (in the foundation phase) that does not require the learners to compute. this is probably why the learners also take the equal sign as a command to do something since all three signs are introduced simultaneously. hence there is a need for textbooks to emphasise on the correct use and understanding of the concept of the equal sign and to also advise teachers to teach it (the equal sign) explicitly. in all three textbooks, there were elaborate pictorial representations to illustrate the meaning of the plus and minus signs in the learner’s book. there is none of such to illustrate the significance of the equal sign. there are, in fact, no activities targeting the introduction of the equal sign. how can texts begin to use placeholders if learners have no clear definition and understanding of the equal sign? even in two of the three books where there is an explicit mention of the equal sign, the teacher’s guides only say, “teach the equal sign”, or get learners to use the equal sign in number sentences. the equal sign does not, thus, enjoy equal status as the plus and minus signs. yet, the equal sign is as important as these other two signs. i propose that of the three signs (plus, minus and equal signs), the equal sign should be introduced first. one way of doing this is to introduce the equal sign after the learners have been taught the constructs of “less” and “more” which precede the concepts of plus and minus both in the curriculum and in the textbooks under consideration. the learners would thus understand when a quantity is “more”, “less” or “equal”, as shown in figure 4. figure 4: a suggested approach to introduce the equal sign after this, the teacher/textbook needs to continue explaining the role of the equal sign while introducing the addition sign. the equal sign should not be taught formally only when dealing with addition, but also reinforced when the concept of subtraction is introduced, and further revisited with the teaching of the concepts of “heavy” and “light” which are the next constructs in the curriculum and in the textbooks i analysed. an analysis of the introduction of the equal sign 34 to do this, the scale balance such as the ones in figure 5 can be used. 2 first, the constructs of “more than” and “less than” and “equal to” are revised, by putting objects (oranges for example) on the scale balance. then, when the constructs of “heavier than” and “lighter than” are introduced to learners using the scale balance, learners’ knowledge of the significance of the equal sign should also be reinforced. figure 5: different scaling systems which can be used to introduce the equal sign an important point to consider is that not only should the teacher’s guide contain instruction on how to teach the equal sign, the learner’s book must also have activities around the role or significance of the equal sign. as falkner et al. (1999, p. 233) argue, “teachers [and i would add textbooks] should…be concerned about children’s conceptions of equality as soon as symbols for representing number operations are introduced”. learners must not only be taught how to use the equal sign, but more importantly, they should be taught the significance of the equal sign in a mathematical sentence. conclusion in this paper, i have argued that introducing the equal sign within the context of addition and subtraction, as do the three textbooks maths for all, classroom mathematics and successful numeracy, has far reaching implications for the understanding of the equal sign. i have also shown that the textbooks and the curriculum have not given equal attention and status to the equal sign compared to the addition and subtraction signs. giving equal status would entail that (1) the curriculum makes explicit allusion to the equal sign and provide the assessment criteria for how the teacher would know when learners have understood the significance of the equal sign; (2) textbooks provide a detailed description of how the equal sign should be introduced (especially in the teacher’s guide); (3) textbooks use placeholders, pictures, numbers and mathematical sentences to demonstrate equality (in learner books); and (4) textbooks provide drilling exercises that reinforce the concept of the equal sign. i have also proposed one way of introducing the equal sign to learners arguing that it should be first introduced when the learners are being taught the concepts of “more” and “less”, before the introduction of the addition and minus symbols. i maintain that if learners first encounter the equal sign in this manner, they are more likely to develop a relational interpretation of the equal sign. acknowledgements this paper is based on a research report supervised by prof. mamokgethi setati. i am grateful to her for her insights. i am also grateful to prof. jill adler for her initial comments on the original manuscript. 2 lyon’s (2003) critique of the use of the scale balance to introduce the equal sign deals with the limitation of this pedagogic strategy in showing that sin30° = 0,5 or that √2 = 1,4142… i would argue that the use of the scale balance is only a means to an end – of entrenching the significance of the equal sign as denoting quantitative sameness (and in later grades, as denoting identity). anthony essien 35 references asquith, p., stephens, a., knuth, e., & alibali, m. 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(1997). building foundations for algebra. mathematics teaching in the middle school, 2, 252-260. impact of a distance mathematics course description and impact of a distance mathematics course for grade 10 to 12 teachers gerrit stolsa, alwyn olivierb and diane graysonc a,cuniversity of pretoria and buniversity of stellenbosch email: agerrit.stols@up.ac.za, baio@sun.ac.za and c dgrayson@absamail.co.za this paper explores the impact of a one-year in-service distance education mathematics course that follows a problem-solving approach. the course aims to develop teachers’ mathematical thinking skills. results show that the course improved the teachers’ confidence, problem-solving skills and teaching skills. results also show that the course on mathematical thinking skills helps to improve teachers’ understanding of mathematics content. introduction and background the fact that there is a crisis in mathematics education is common knowledge and a cause for concern. for the past five years only approximately 20,000 grade 12 learners (4%) a year have passed mathematics on higher grade (hg) in the senior certificate examinations (department of education, 2001; kahn, 2004). given that a pass in hg mathematics is a prerequisite for entry into science-based studies at university, there are serious implications for the country’s ability to produce enough scientifically skilled professionals. it was with these points in mind that a distancebased course for grade 10-12 teachers was developed by the centre for the improvement of mathematics, science and technology education (cimste) at the university of south africa (unisa). the course was designed and written by the second author in 2001 and 2002 and implemented by the first author for the first time in 2002. the focus of the course is on problemsolving, in the context of algebra. in the planning phase of a course, an important question is what the course should look like in order to have the maximum impact possible on teachers and their learners. what kind of mathematical skills and knowledge do teachers need in order to teach effectively? it is impossible to answer such questions before we have decided what we want from the learners. we naturally want the learners to be able to do mathematics, but then the question arises: what is mathematics and how do you learn it? it is therefore important, when designing a course, to start with a view of the nature of mathematics and the learning of mathematics. dossey, mccrone, giordano and weir (2002) believe that there is no general agreement on the question: the wide variety of its [mathematics’] applications in our society is easy to list. but, the nature of mathematics itself is hard to capture. this results from a lack of consensus, even among mathematicians, as to what constitutes ‘mathematics’ and what ‘doing mathematics’ means. (2002: 4) in reality, most professional mathematicians spend little thought on the fundamental nature of their subject. what we want to know is: which skills do we value as mathematical skills? this can change according to changes in our society. confrey and lachance explain why: these skills (computational skills) allowed students to secure jobs and to become informed citizens in an industrial society. however, with advances in technology, such computational skills are no longer as important. instead, students need to develop critical-thinking skills to interpret data appropriately and to use technology to solve more complex problems. thus, changes in our society have led to a change in what we value in mathematical skills. (2000: 232) we currently support the following view expressed in the south african national curriculum statement on grade 10-12 mathematics (department of education, 2003): mathematics enables creative and logical reasoning about problems in the physical and social world and in the context of mathematics itself. it is a distinctly human activity practised by all cultures. knowledge in the mathematical sciences is constructed through the establishment of descriptive, numerical and symbolic relationships. mathematics is based on 32 pythagoras 65, june, 2007, pp. 32-38 gerrit stols, alwyn olivier, and diane grayson observing patterns which with rigorous logical thinking, leads to theories of abstract relations. mathematical problemsolving enables us to understand the world and make use of that understanding in our daily lives. mathematics is developed and contested over time by social interaction through both language and symbols. (2003:7) in terms of this statement, we want the learners to: • solve problems, using creative and logical reasoning. • find relationships and express them symbolically. • observe patterns to find relationships and prove them. to expand on this, in the words of the education development centre (2000), “students need to be thinkers: pattern hunters, experimenters, describers, tinkerers, inventors, visualisers, conjecturers, guessers and seekers of reasoned argument and proof.” we can describe justifying, explaining, analysing, generalising and defining as mathematical thinking. we believe the best way to help learners to develop the ways of thinking that are characteristic of mathematics is through problem-solving (education development centre, 2000: xiv). according to resnick (1989), theory and research show that we develop habits and skills of interpretation and meaning construction though a process of socialisation or enculturation rather than through instruction: …becoming a good mathematical prob lem-solver – becoming a good thinker in any domain – may be as much a matter of acquiring the habits and dispositions of interpretation and sense-making as of acquiring any particular set of skills, strategies, or knowledge. if this is so, we may do well to conceive of mathematics education less as an instructional process (in the traditional sense of teaching specific, well-defined skills or items of knowledge), than as a socialisation process. in this conception, people develop points of view and behaviour patterns associated with gender roles, ethnic and familial cultures, and other socially defined traits. when we describe the processes by which children are socialised into these patterns of thought, affect, and action, we describe long-term patterns of interaction and engagement in a social environment. (1989: 58) this view of enculturation highlights the importance of perspective and point of view as core aspects of knowledge. the case can be made that a fundamental component of thinking mathematically is having a mathematical point of view, or having a mathematical attitude of mind – seeing the world in the way mathematicians do. the focus of our course is therefore on developing a mathematical attitude of mind and the way to do it is to immerse participants in a typical mathematical culture. theoretical context one of the most important factors influencing learner performance is the teacher. in a synthesis of research related to the president’s education initiative, taylor and vinjevold (1999) indicate that there are problems with teachers’ knowledge and skills and, in consequence, with their teaching approaches: … reform initiatives aimed at revitalising teacher education and classroom practices must not only create a new ideological orientation consonant with the goals of the new south africa. they also need to get to grips with what is likely to be a far more intractable problem: the massive upgrading and scaffolding of teachers’ conceptual knowledge and skills. …the fundamental mechanism for its propagation [the vicious cycle of rote learning] is the lack of conceptual knowledge, reading skills and spirit of enquiry amongst teachers. (1999: 160) what is needed if mathematics teachers are to become more effective are professional development opportunities to strengthen their conceptual knowledge and problem-solving skills. it comes as no surprise that a focus of the new south african further education and training (fet) curriculum is on helping learners develop problem-solving skills because problem-solving is central to the constructivist-based teaching of mathematics. a problem-centred learning approach is based on the acceptance that learners construct their own mathematical knowledge. the difficulty that arises, however, is that the teachers are not trained to teach problem-solving and did not experience the power of a problem-centred teaching approach themselves. the way in which teachers have been taught themselves plays an important role in the way they think about teaching. it is therefore important in a course for the training of teachers to 33 description and impact of a distance mathematics course for grade 10 to 12 teachers allow those teachers to experience problem-solving first hand. in a report, the national commission on teaching and america's future (1996:20) discuss the fact that teacher preparation and professional development programmes must consciously examine the expectations embodied in new curriculum frame-works. the report also comments on the need for these programmes to develop strategies that help teachers learn to teach in these much more demanding ways. teachers’ programmes, according to loucks-horsely, hewson, love, and stiles (1998: 36), must be organised around problem-solving and must be directly related to teachers' work with their students. this paper discusses the impact of a problemsolving course on a group of 27 teachers, in terms of both their mathematical understanding and their attitudes towards mathematics and the teaching of mathematics. description of the course the purpose of the unisa (university of south africa) mathematics for teachers course, for which the students receive 12 credits (an average teacher will take about 120 hours to complete the course successfully), was to improve the teachers’ mathematical thinking skills and pedagogical content knowledge, mainly in the context of problems. the duration of the course is one year and is offered by means of distance teaching. unisa is a distance education institution, a factor which poses challenges in itself. to try to overcome some of the problems presented by a distance course (e.g. sharing solutions through discussions with others), we held three workshops of two hours each. we encouraged the teachers to form peer groups by explaining the advantages of such groups to them and sending them a list of the telephone numbers and addresses of all the students enrolled for the course. the central focus of the course is on problemsolving, that is, non-routine problem-solving – either through illustrations of the process of problem-solving, or through students’ own engagement with problems. in order to encourage the teachers to reflect regularly on the mathematics that they had learnt and to think of ways to introduce their new-found knowledge in the classroom, we asked them to keep and submit a journal and encouraged them to update their journals regularly – at least once a week. as part of the course, teachers were required to complete four assignments, which we marked and returned to them. examples of problems from the study guide the examples below are taken from the course study guide. they will give the reader a better idea of what we mean when we ask learners to find relationships and express them symbolically and observe patterns to find relationships and prove them. example 1: short cut a) develop a short method to calculate 17 + 16 + 15 + 14 + … + 3 + 2 + 1. b) use your method to calculate 1 + 2 + 3 + 4 + 5 + … + 99 + 100. c) generalise! d) can you use your method to calculate 2 + 4 + 6 + 8 + 10 + … + 98 + 100? e) can you use your method to calculate 1 + 3 + 5 + 7 + 9 + … + 97 + 99? example 2: consecutive numbers some numbers can be written as the sum of two or more consecutive whole numbers. for example: 13 = 6 + 7 14 = 2 + 3 + 4 + 5 15 = 7 + 8 = 1 + 2 + 3 + 4 + 5 some numbers cannot be written as the sum of consecutive whole numbers. investigate: which numbers can and which numbers cannot be written as the sum of consecutive whole numbers. try to develop a general theory or method or formula that will enable you to: a) immediately decide if any given number can be written as the sum of consecutive numbers. b) easily write the number as the sum of consecutive numbers. example 3: generalise a) find the value of (1 – 1 4 )(1 – 1 9 )(1 – 1 16 )(1 – 1 25 )(1 – 1 36 ) … (1 – 1 10000 ) b) generalise. example 4: petrol price in january the petrol price is increased by 10%. then, in february the petrol price was reduced by 10%. john says that the petrol price is now the same as it was before the first increase. is this correct? explain! impact of the course on teachers in 2002, twenty-seven teachers of grades 10 to 12 mathematics enrolled for the mathematics for teachers course. the majority of the teachers were 34 gerrit stols, alwyn olivier, and diane grayson teaching at rural black schools. the reflective journals provided us with documentation of a continuous cycle of inquiry. apart from the journal, various data were gathered in order to assess the impact of the course. at both the beginning and the end of the course the teachers completed a questionnaire and a test. the questionnaire asked teachers to indicate their level of confidence about teaching various topics in the grades 10, 11 and 12 syllabi using a 5-point likert scale. the test comprised questions similar to those on the grade 12 examinations and included questions with a range of levels of cognitive demand (see appendix a). the same questionnaire and test were administered at both the beginning and the end of the course. further information was obtained from teachers’ assignments, journals, evaluations completed at the end of workshops and an end-of-year evaluation. effects on teachers’ content knowledge at the end of the course we asked the teachers to evaluate the course by completing a questionnaire. we divided the free response questions into categories. thirteen teachers completed and returned the questionnaire/survey. the respondents gave feedback on the impact of the course on their content knowledge and teaching practice as follows: 1) has the course improved your subject content knowledge? yes: 13 (100%). no: 0. 2) in what ways has the course improved your subject content knowledge? improved problem-solving skills: 6 (46%). improved mathematical thinking skills: 3 (23%). no response: 4 (31%). 3) in what ways has the course failed to improve your content knowledge? none: 11 (85%). 4) has the course improved your teaching practice? yes: 13 (100%). no: 0. 5) in what ways has the course improved your teaching practice? more confidence: 1 (8%). learners are more interested: 1 (8%). improved learners’ class attendance: 1 (8%). improved teachers’ problem-solving skills: 3 (23%). improved teaching skills: 3 (23%). the preand post-tests showed that there was a 13.6% improvement in the teachers’ ability to answer grade 12 exam-type higher grade questions (see figure 1). the average mark was 32.4% for the pre-test and 46% for the post-test. the improvement in the teachers’ ability to answer grade 12 exam-type higher grade questions was unexpected, because the course did not explicitly focus on the content knowledge that was tested in the preand post-test. some of the comments that the teachers made in the journals were: i am discovering something new every day in my learning. my knowledge horizon is expanding although i have been a teacher for more that 15 years. i have discovered ways of attempting a problem if you don’t have a clue of what to do. it has helped me to upgrade my mathematical insight. effects on classroom practice anecdotally, the course had an impact on the classroom practice of the teachers. some of the comments they made were: the weeks’ work will affect my classroom practice in that i will simply be moving away from the traditional way of teaching. i will make sure that the learners learn through social interaction and reflection, no longer through practice and repetition. i will present tasks and problems that will lead the learners to inventing mathematics. this piece of work will enrich the knowledge of my learners as far as arithmetic sequence is concerned. this is very exciting, as they will have a deeper understanding of the concept. the learners are more interested in mathematics, understand it more quickly and attend their classes better. all of these exercises empower you to teach certain topics… it is true that experience is the best teacher. effects on teachers’ attitudes towards mathematics from the feedback it was evident that all the respondents believed that the course had improved their teaching practice. according to them they are more confident than before about teaching mathematics. one of the teachers wrote: one is empowered to teach in the classroom without fear. the students 35 description and impact of a distance mathematics course for grade 10 to 12 teachers understand me more clearly than before i registered for this course. the course also changed the way that some of the teachers think about mathematics: it was very challenging and has changed my mindset about mathematics so that i now realise that mathematics is about discovery and about being able to make conjectures and to reason. effects on teachers’ reflective skills initially the teachers were very negative about the weekly journal entries, but their attitude changed as they progressed. in the beginning one of the teachers wrote: “i think that the journal exhausts our study time”. later in the year, the same teacher wrote: i am now developing a positive attitude about this journal. i can see that the questions in the journals help us to link what we have learnt in the study guide with the activities in the classroom situation. twenty of the 27 teachers were from the same geographical area, which contributed to the fact that 77% of the teachers worked with peers. the journals and the fact of working with peers helped the teachers to be more reflective. the journal gave the researchers some insight into what was going on in the teachers’ minds during the problemsolving course. in the beginning of the course the teachers expressed their frustrations: this was too demanding and thought provoking. one had to recognise patterns which were not always easy to find. the assignment is really challenging. i didn’t expect this. i am thinking of stopping this course. as i am working through this unit, i am getting frustrated … i see the problems for the first time. some of the teachers complained about the study guide. they wanted examples, followed by similar problems to solve. they looked for examples in other books: as i was working through this unit, i was frustrated. i was frustrated to see myself reading so many books and not finding the exact book that would provide me with relevant information. but the frustration soon changed into joy. the same teacher wrote some time later: my frustration ended in excitement. even though i was frustrated, at the end of each and every question i had gained something exciting. one teacher wrote in her journal: “i am really frustrated. i wish i could get the answer.” two days later she wrote: at last i got the answer and i am so happy. no amount of money can buy my happiness. possible impact of the course on the learners the study only determined the impact of the course on the teachers. although we attended some of the teachers’ classes, we could not determine the impact of the course on their classroom practice. instead of trying to determine the direct impact of the course on the learners, we decided to divide the investigation into two parts. the first question we asked was: is there a correlation between the teachers’ knowledge and their learners’ knowledge? the second question was whether this course improved the teachers’ knowledge. what we know is that there is a strong correlation between the teachers’ knowledge and their learners’ knowledge. a previous study undertaken by stols (2003: 246-250) with the same teachers revealed that the correlation between these teachers’ ability to answer exam-type questions and the learners’ ability to answer similar examtype questions is very strong (the pearson correlation coefficient is 0.8008; this correlation is statistically significant, because of the low p-value of 0.0005). therefore, improving the teachers’ ability to answer exam-type questions will also improve their learners’ ability (see figure 2). in the results, we mentioned that the teachers’ ability to answer mathematical thinking course teachers’ ability to answer exam-type questions improved by 13,6% figure 1. improvement in the teachers’ ability to answer exam-type questions 36 gerrit stols, alwyn olivier, and diane grayson exam-type questions improved by 13,6%. it is therefore reasonable to believe that this problemsolving course in mathematical thinking could help the learners as well. this belief was confirmed by the results for one school. the pass rate of the grade 12 learners in mathematics higher grade in this school was 51.4% in 2001 and it improved to 94.4% in 2002 (that is the year in which the mathematics teacher at the school was enrolled for this course). the teacher believes that the improvement in the results was due his participation in this course. discussion the course improved the confidence, problemsolving skills and teaching skills of the teachers. one of the teachers who did the course wrote in his journal: i believe that creativity is there in each and every mind. it only needs to be activated in order to make it useful in other situations. the course also improved the teachers’ content knowledge because it helped them to help themselves to master content. it is clear from the comments by the teachers that this course (problem-solving approach) helped the teachers to improve their metacognitive skills. a possible reason is that teachers have become more reflective because of the journals they kept and another possible reason is that a problem-solving approach may help teachers to enhance their metacognitive skills. routine exercises on the other hand may engender a false impression of success and understanding because it is possible to experience success without understanding. we would conclude that teacher training programmes that are organised around problem solving and that are directly related to teachers’ work with their students are a powerful means of helping teachers who are preparing to teach mathematics. this approach will help teachers to become more reflective and will develop their metacognitive skills. it will help them to help themselves in future. conclusion to prepare teachers to help learners learn mathematics is not easy. traditionally, the focus of teacher training programmes was on the upgrading of content knowledge and on ways of explaining the new knowledge to learners. but schoenfeld (1994) states that teachers need more than that: “the danger in this kind of ‘content inventory’ point of view comes from what it leaves out: the critically important point is that mathematical thinking consists of a lot more than knowing facts, theorems, techniques, etc.” it remains important, however, that the teachers should at least know the mathematics they teach. yet, considering the requirements outlined above, it is clear that they must know more than that. ball (2003) explains the knowledge that teachers need as follows: teaching requires justifying, explaining, analysing errors, generalising, and defining. it requires knowing ideas and procedures in detail, and knowing them well enough to represent and explain them skilfully in more than one way. this is mathematics. the failure to appreciate that this is substantial mathematical work does teachers – and the improvement of teaching – a disservice. (2003: 4) this course helps teachers deepen their content knowledge and their pedagogical content knowledge, improve their problem-solving skills, and develop their metacognitive skills so that they can continue to learn in future without relying on a structured course. because of the strong correlation between teachers’ knowledge and learners’ knowledge, the course will eventually make a difference in the classroom. teachers inset course learners relationship? correlation: 0,8008 (p-value: 0,0005) figure 2. the relationship between teachers’ and learners’ abilities 37 description and impact of a distance mathematics course for grade 10 to 12 teachers mathematical problem-solving (pp. 32-60). reston, va: national council of teachers of mathematics. acknowledgements this paper was made possible by a grant from the carnegie corporation of new york. the statements made and views expressed are solely those of the author. schoenfeld, a.h. (1994). what do we know about mathematics curricula? journal of mathematics behaviour, 13, 55-80. stols, g.h. (2003). the correlation between teachers’ and their learners’ mathematical knowledge in rural schools. in p. bongile, m. dlamini, b. dlamini, & v. kelly. (eds.), proceedings of the 11th annual saarmste conference (pp. 246-250). cape town: university of cape town. references ball, d.l. (2003). mathematics in the 21st century: what mathematical knowledge is needed for teaching mathematics? retrieved november 28, 2003, from http://www.ed.gov/ inits/mathscience/ball.html confrey, j. & lachance, a. (2000). transformative teaching experiments through conjecture-driven research design. in a.e. kelly. (ed.), handbook of research design in mathematics and science education (pp. 231-266). new jersey: lawrence erlbaum associates. taylor, n. & vinjevold, p. (1999). getting learning right. johannesburg: joint education trust. department of education. (2001). national strategy for mathematics, science and technology education to address the problem in mathematics, science and technology education. pretoria: department of education. appendix a: preand post-test please answer the following questions without obtaining help from anyone. 1 ; 1 6 ≠≤ − xx x 1. solve for x: 2. for which values of k are the roots of k 12 2 2 =++ + xx x with x ≠ –1 real? department of education. (2003). national curriculum statements grades 10-12 (general): mathematics. pretoria: department of education. 3. solve for x: 9 x + 3x = 27(3x + 1) 4. a house contractor has subdivided a farm into 100 building lots. he has designed two types of homes for these lots: colonial and ranch style. a colonial home requires r300 000 of capital and yields a profit of r40 000 when sold. a ranch-style house requires r400 000 of capital and yields an r80 000 profit. if he has r36 million of capital on hand, how many of each type should he build for maximum profit? will any of the lots be vacant? dossey, j., mccrone, s., giordano, f. & weir, m.d. (2002). mathematics methods and modelling for today's mathematics classroom: a contemporary approach to teaching grades 712. canada: brooks & cole. education development centre (2000). connected geometry: teachers guide. chicago: everyday learning. kahn, m. (2004). for whom the second bell tolls: disparities in performance in senior certificate mathematics and physical science. perspectives in education, 30(4), 149-156. 5. if ax2 + bx + c = 0 and a + b + c = 0 find the numerical value of x. loucks-horsely, s., hewson, p., love, n. & stiles, k. (1998). designing professional development for teachers of science and mathematics. thousand oaks, ca: corwin press. 6. show that a = 0 if ba2ba2 ba2aba2a2 2 2 − −−+ = xx x national commission on teaching and america's future (1996). what matters most: teaching for america's future. summary report. new york: national commission on teaching and america's future. 7. solve for x and y: ( ) 0142 =⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −− y yx 8. is the following statement true: ( ) 122121 −− +=+ aaaa ? why? olivier, a. (2000). study guide: mathematics for teachers i. pretoria: unisa publishers. 9. is the following statement true: 2(4x + 4–x) = 8x + 8–x ? why? resnick, l. (1989). treating mathematics as an illstructured discipline. in r. charles & e. silver (eds.), the teaching and assessing of 10. solve for x: (x 2 + 2x)2 − x2 − 2x − 6 = 0 38 << /ascii85encodepages false /allowtransparency false /autopositionepsfiles true /autorotatepages /all /binding /left /calgrayprofile (dot gain 20%) /calrgbprofile (srgb iec61966-2.1) /calcmykprofile (u.s. web coated \050swop\051 v2) /srgbprofile (srgb iec61966-2.1) /cannotembedfontpolicy /warning /compatibilitylevel 1.4 /compressobjects /tags /compresspages true /convertimagestoindexed true /passthroughjpegimages true /createjdffile false /createjobticket false /defaultrenderingintent /default /detectblends true /colorconversionstrategy /leavecolorunchanged /dothumbnails false /embedallfonts true /embedjoboptions true /dscreportinglevel 0 /syntheticboldness 1.00 /emitdscwarnings false /endpage -1 /imagememory 1048576 /lockdistillerparams false /maxsubsetpct 100 /optimize true /opm 1 /parsedsccomments true 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<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> >> >> setdistillerparams << /hwresolution [2400 2400] /pagesize [612.000 792.000] >> setpagedevice 63 p14-21 berger layout final 14 pythagoras 63, june 2006, pp. 14-21 making mathematical meaning: from preconcepts to pseudoconcepts to concepts margot berger school of mathematics, university of witwatersrand mberger@maths.wits.ac.za i argue that vygotsky’s theory of concept formation (1934/1986) is a powerful framework within which to explore how an individual at university level constructs a new mathematical concept. in particular i argue that this theory can be used to explain how idiosyncratic usages of mathematical signs by students (particularly when just introduced to a new mathematical object) get transformed into mathematically acceptable and personally meaningful usages. related to this, i argue that this theory is able to bridge the divide between an individual’s mathematical knowledge and the body of socially sanctioned mathematical knowledge. i also demonstrate an application of the theory to an analysis of a student’s activities with a ‘new’ mathematical object. introduction the issue of how an individual makes personal meaning of a ‘new’ mathematical object is a fundamental issue in mathematics education. in particular, at many universities the student is frequently introduced to a new mathematical object through a definition1. from this definition, the learner is expected to construct the properties of the object (tall, 1995). in many instances neither diagrams nor exemplars of the mathematical object are presented alongside the definition; initial access to the mathematical object is through the various signs (such as words and symbols) of the definition. in this paper i examine how a student at university level makes meaning of a mathematical object presented in the form of a definition. i argue that vygotsky’s theory of preconceptual and pseudoconceptual thinking (1934/1986) provides an appropriate structure within which to explore how a student constructs concepts which are both personally and culturally meaningful. in particular, i show that vygotsky’s theory can be used to explain how idiosyncratic usages by learners of ‘new’ (to the student) mathematical signs get transformed into mathematically acceptable usages. i illustrate the theory through an example in which we see how a university student makes meaning of a mathematical object introduced through a definition. my argument revolves around vygotsky’s idea of the functional use of a sign. in terms of this 1 the introduction of new mathematical objects through definitions is common practice in many south african universities and in certain british universities such as warwick (alcock and simpson, 2001). notion, and as i argue later, the learner’s use of the mathematical signs in activity and communication is a necessary first step in the appropriation of mathematical meaning. this usage precedes an understanding of the mathematical object signified by the mathematical sign. this argument contradicts the position most mathematics educators take, often implicitly, that understanding and the construction of mathematical meaning needs to occur prior to the use of the symbol. although my focus is on how a student at university level makes meaning of a new mathematical object presented in the form of a definition, my arguments also relate to school level mathematics. understanding the extreme case of a mathematical object introduced through a definition provides a window into what is happening when a learner encounters a new mathematical object, no matter the academic level and no matter that it may be introduced through exemplars and/or with diagrams (as is common practice in many south african high schools). background several mathematics education researchers have considered how an individual, at university level, constructs a mathematical concept and some have developed significant theories in response. the most influential of these theories focus on the transformation of a process into an object (for example, tall, 1995; dubinsky, 1991; czarnocha et al., 1999). according to tall et al. (2000), the idea of a process-object duality originated in the 1950s in the work of piaget who spoke of how “actions and margot berger 15 operations become thematized objects of thought or assimilation” (cited in tall et al., 2000: 1). in adopting a neo-piagetian perspective, these researchers and their various followers successfully extend piaget’s work regarding elementary mathematics to advanced mathematical thinking. for example, czarnocha et al. (1999) theorise that in order to understand a mathematical concept, the learner needs to move between different stages. she has to manipulate previously constructed objects to form actions. “actions are then interiorised to form processes which are then encapsulated to form objects” (1999: 98). processes and objects are then organised in schemas. dubinsky (1991) names this theory apos (actions, process, object, schema). but much of this process-object theory does not resonate with what i see in my (university) mathematics classroom. for example, it does not help me explain or describe what is happening when a learner fumbles around with ‘new’ mathematical signs making what appear to be arbitrary connections between these new signs and other apparently unrelated signs. similarly, it does not explain how these incoherent-seeming activities can lead to usages of mathematical signs that are acceptable to professional members of the mathematical world and personally meaningful to the learner. i maintain that the central drawback of these neo-piagetian theories is that they are rooted in a framework in which conceptual understanding is regarded as deriving largely from interiorised actions; the crucial role of language (or signs) and the role of social regulation and the social constitution of the body of mathematical knowledge is not integrated into the theoretical framework. given that mathematics learning is by its very nature a social activity, mediated and constituted by language, signs and tools (for example, textbooks), a concentration on interiorised actions is problematic. indeed, meaning, thinking and reasoning need to be seen as products of social activity (lerman, 2000: 23), not just the characteristics of a decontextualised individual. in this regard, a framework is required in which the link between an individual’s construction of a concept and socially sanctioned knowledge (existing in the community of mathematicians and in reified form in textbooks) is foregrounded. furthermore, given that mathematics can be regarded as the “quintessential study of abstract sign systems” (ernest, 1997) and mathematics education as “the study of how persons come to master and use these systems” (ibid.), a framework which postulates semiotic mediation as the mechanism of learning, seems apposite. i claim that vygotsky’s much-neglected notions of preconceptual and pseudoconceptual thinking, allied with a notion of the functional usage of a sign (1934/1986), provide theoretical tools with which the researcher or teacher can make sense of what is happening when a student attempts to construct a new mathematical concept. vygotsky’s theory of concept formation although vygotskian theory has been applied extensively in mathematics education, most of the research has focused on the mathematical activities of a group of learners or a dyad rather than the individual (van der veer and valsiner, 1994). indeed, van der veer and valsiner claim that the use of vygotsky in the west has been highly selective. they argue that “the focus on the individual developing person which vygotsky clearly had … has been persistently overlooked” (1994: 6; italics in original). in this paper, i focus on the activities of an individual. my focus on the individual is motivated by the situation at many universities (south africa and worldwide) in which much learning occurs in the context of an individual sitting with a textbook. (of course this does not deny that learning may also take place in other settings such as lecture rooms or discussions with other students). it is important to note that a focus on the individual (possibly with a textbook or lecture notes) does not contradict the fundamental vygotskian notion that “social relations or relations among people genetically underlie all higher functions and their relationships” (vygotsky, 1981: 163). after all, a situation consisting of a learner with a text is necessarily social; the textbook or exercises have been written by an expert (and can be regarded as a reification of the expert’s ideas); also the text may have been prescribed by the lecturer with pedagogic intent. thus a focus on the individual does not in any way undermine the significance of the social. functional use of the sign my understanding of how a student constructs a new mathematical concept is based on vygotsky’s theory of how a child learns the meaning of a new word. in this regard it is important to note that vygotsky regards a word as embodying a concept. he postulates that the child uses the ‘new’ word making mathematical meaning: from preconcepts to pseudoconcepts to concepts 16 for communication purposes before that child has a fully developed understanding of that word: words take over the function of concepts and may serve as means of communication long before they reach the level of concepts characteristic of fully developed thought (uznadze, cited in vygotsky, 1934/1986: 101). the use of a word or sign to refer to an object (real or virtual) prior to ‘full’ understanding resonates with my sense of how an undergraduate student makes a new mathematical object meaningful to herself. in practice, the student starts communicating with peers, with lecturers or the potential other (when writing) using the signs of the new mathematical object (symbols and words) before she has full comprehension of the mathematical object. it is this usage of the mathematical signs, with the accompanying communication, that gives initial access to the new object. secondly vygotsky argues that the child does not spontaneously develop concepts independent of their meaning in the social world: he does not choose the meaning of his words… the meaning of the words is given to him in his conversations with adults (vygotsky, 1934/1986: 122). thus the social world, with its already established definitions (as given in dictionaries or books) of different words, determines the way in which the child’s understanding of the object needs to develop. analogously, i argue that in mathematics, a student is expected to construct a concept whose use and meaning is compatible with its use in the mathematics community. to do this, that student needs to use the mathematical signs in communication with more knowledgeable others (including using textbooks). through this usage, which is socially regulated (via the interaction with a text or others), the meaning of a concept can evolve for the learner in a way that is compatible with its culturally accepted usage (berger, 2004a). this functional use argument is reminiscent of dörfler’s (2000) thesis that in order to appropriate a new mathematical object, the mathematics student has to be willing to adopt an attitude whereby he participates in the discourse of mathematics as−−−−if the discourse is meaningful and coherent, even if he does not experience it as such. it is also supported by pimm’s (1987) argument about the importance of learners’ mathematical talk, no matter the impreciseness of this talk. once something is expressed, however haltingly and incompletely, then questions can be asked about the current formulation in order to encourage greater refinement, precision and clarity. (ibid.: 31) semiotic mediation vygotsky (1978) regarded all higher human mental functions as products of mediated activity. the role of the mediator is played by a psychological tool or sign, such as words, graphs, algebra symbols, or a physical tool. these forms of mediation, which are themselves products of the socio-historical context, do not just facilitate activity; they define and shape inner processes. according to vygotsky, action mediated by signs is the fundamental mechanism which links the external social world to internal human mental processes and he argues that it is by mastering semiotically mediated processes and categories in social interaction that human consciousness is formed in the individual (wertsch and stone, 1985: 166). allied to this, concept formation, as discussed above, is only possible because the word or mathematical object can be expressed and communicated via a word or sign whose meaning is already established in the social world. in mathematics, the same mathematical signs mediate two processes: the development of a mathematical concept in the individual and that individual’s interaction with the already codified and socially sanctioned mathematical world (radford, 2000). in this way, the individual’s mathematical knowledge is both cognitively and socially constituted. this dual role of a mathematical sign by a learner before ‘full’ understanding is not well recognised by the mathematics education community; indeed, its manifestations in the form of activities such as manipulations, imitations and associations are often not appreciated by mathematics educators.2 for example, certain proponents of the calculus reform movement believe that mathematical skills are somehow separate from understandings. as ganter (2001: 23) puts it: what is “being debated are which basic mathematical computations and skills should go along with conceptual understanding” (my italics), as if conceptual understandings were somehow 2 an exception to this is sfard (2000) whose theory of templatedriven use implies use of symbols before understanding of mathematical objects. margot berger 17 independent of computations and skills. seldon and seldon (2001) describe, without any hint of misgiving, how textbooks used by the ace teaching cycle which is based on apos theory, contain no template solutions to problems. they do not even seem to consider that imitation may play an important role in learning. vygotsky’s theory, that usages of the sign are a necessary, but not sufficient, part of concept formation, manages to provide a link between certain types of mathematical activities (including those activities frequently ignored or regarded pejoratively by mathematics educators) and the formation of concepts. for this reason, it has great implications for desirable pedagogical practices.3 different preconceptual stages vygotsky elaborated his theory by detailing the stages in the formation of a concept. he claimed that the formation of a concept entails the learner moving through different preconceptual stages (heaps, complexes and potential concepts). according to kozulin (1990: 159), vygotsky’s position was that these preconceptual types of representation are retained by older children and adults, who quite often revert to these more ‘primitive’ forms depending on their interpretation of a given task and on their chosen strategy for solution. it is in this latter sense that i maintain that university mathematics students use preconceptual (that is, heap and complex) thinking when dealing with new ideas. most importantly, this movement is not linear. indeed the learner may move back and forth between the different stages. during the syncretic heap stage, the child groups together objects or ideas which are objectively unrelated. this grouping takes place according to chance, circumstance or subjective impressions in the child’s mind. for example, a learner is using heap thinking if she interprets, say, the meaning of a mathematical sign based on the layout of the page. in the complex thinking stage, the learner’s activities are driven by non-logical activities such as template-matching, associations, imitation, manipulations, etc. as a result, complex thinking 3 it is important to stress that, in terms of the above theory and in contrast to the back-to-basics position, adequate use of a mathematical sign is not sufficient evidence of a student’s understanding of the relevant mathematical concept. often manifests as an idiosyncratic usage of mathematical signs. an example of complex thinking using association is as follows: on first encountering the derivative, f ′′′′ (x), of a function f(x), many learners associate the properties of f ′′′′ (x) with the properties of f(x). accordingly, these learners assume that since f(x) is continuous, so is f ′′′′ (x). clearly this is not logical; indeed it is mathematically incorrect. an example of activity guided by complex thinking using template-matching (sfard, 2000) is when the student uses the template 1, a a a = ∈ � to argue that 0 1 0 = , which is, of course incorrect. however the argument that, say, 3 1 3 = , is correct (and may also be based on templatematching). but my point right now is not how the student uses the signs but rather that she uses the signs. through this use, the student gains access to the ‘new’ mathematical object and is able to communicate (to better or worse effect) about it. and, as i have just argued, it is this communication with more knowledgeable others which enables the development of a personally meaningful concept whose use is congruent with its use by the wider mathematical community. my observations of undergraduate students over the years ties in very well with the idea that preconceptual thinking is a necessary part of successful mathematical concept construction. with regard to potential concepts, vygotsky (1934/1986: 135) argues that complex thinking creates the basis for later generalisations in that the learner classifies different objects into groups (or complexes) on the basis of particular characteristics. however, classification would not be possible without abstraction of these particular characteristics. thus the learner engages in abstractions concurrently with complex thinking. vygotsky calls the formation that results from the grouping together of objects on the basis of a single attribute or a set of attributes, a ‘potential concept’. abstractions are inherent in the construction of any mathematical concept and so potential concepts in the vygotskian sense abound in mathematical thinking. but the abstraction of attributes is so profoundly intertwined with the formation of complexes in advanced mathematical thinking that it is impossible to distinguish making mathematical meaning: from preconcepts to pseudoconcepts to concepts 18 potential concepts from mathematical complexes. for this reason i suggest that the potential concept is not a particularly useful or appropriate category of analysis, particularly in the advanced mathematical domain. vygotsky distinguishes between five different types of complexes. i will not go into detail here about the different forms and indicators of complex thinking in the mathematical domain. (for such an elaboration, see berger, 2004b, 2004c.) suffice to note that complex thinking manifests as a non-logical usage of signs, as discussed above. however, i will elaborate on the pseudoconcept since this construct provides a bridge between preconceptual (i.e. complex) and conceptual thinking. the pseudoconcept: a bridge between the individual and the social at this juncture, we need to establish what vygotsky meant by conceptual thinking. in conceptual thinking, the links between properties and aspects of the concept and between different concepts are logical and the ideas form part of a culturally-recognised and consistent system of hierarchical knowledge. this differs from complex thinking where non-logical thinking predominates. but how does the transition from complexes to concepts take place? according to vygotsky, it is through the use of pseudoconcepts. pseudoconcepts have a dual nature. that is, they resemble true concepts in their appearance, but the thinking behind the pseudoconcept is still complex in character. that is, with complex thinking, the student is still using association, template-matching or imitation and they may even hold contradictory ideas about the one concept. but, importantly, the learner is able to use the pseudoconcept in communication and activities as if it were a true concept. the use of pseudoconcepts is ubiquitous in mathematics and is analogous to the use by a child of a word in conversation with an adult before the child fully understands the meaning of that word. pseudoconcepts occur whenever a student uses a particular mathematical object in a way that coincides with the use of a genuine concept, even though the student has not fully constructed that concept for herself. for example, a student may use the definition of the derivative of a function to compute the derivative of the function before she ‘understands’ the nature of the derivative or its properties. vygotsky (1934/1986) argues that the use of pseudoconcepts enables children to communicate effectively with adults and that this communication is necessary for the transformation of the complex into a genuine concept. verbal communication with adults (…) become[s] a powerful factor in the development of the child’s concepts. the transition from thinking in complexes to thinking in concepts passes unnoticed by the child because his pseudoconcepts already coincide in content with adult concepts (vygotsky, 1934/1986: 123). thus the pseudoconcept functions as the bridge between concepts whose meaning is more or less fixed and constant in the social world (such as that body of knowledge we call mathematics) and the learner’s need to make and shape these concepts so that they become personally meaningful. in this way, the pseudoconcept can be regarded as a link between the individual and the social. furthermore, the notion of the pseudoconcept is entirely consistent with the functional use of a sign. the pseudoconcept can be used to explain how the student is able to use mathematical signs (in algorithms, definitions, theorems, problem-solving, and so on) in effective ways that are commensurate with that of the mathematical community even though the student may not fully ‘understand’ the mathematical object. the hope is that through appropriate use and social interventions, the pseudoconcept will get transformed into a concept. brief demonstration i will use the above theory to explain how a firstyear mathematics major student at a south african university moves from an idiosyncratic usage of signs (using, i claim, preconceptual thinking) to a conceptual (or perhaps pseudoconceptual) usage of signs. the activity took place during an interview which i conducted, video-taped and later transcribed and analysed in 2002 (berger, 2002). john had been given the following definition which he has not seen before, although he is familiar with the definite integral and the notion of a limit. margot berger 19 definition of an improper integral with an infinite integration limit if f is continuous on the interval [a, ∞), then ∞ →∞ =∫ ∫( ) lim ( ) b b a a f x dx f x dx if →∞ ∫lim ( ) b b a f x dx exists, we say that the improper integral converges. otherwise the improper integral diverges. this is followed by several questions each of which is presented on its own, in order (for example, john has not seen question 4 when he first encounters, say, question 1). 1. (a) can you make up an example of an improper integral with an infinite integration limit? 1. (b) can you make up an example of a convergent improper integral with an infinite integration limit? m 4. determine whether ∞ ∫ 3 1 dx x converges or diverges. john’s response, in part, to question 1(a) is to generate a string of signifiers: ∞ ∫ 0 f ( x )dx = →∞ ∫ 2 2 0 lim f ( x )dx = →∞ ∫ 2 2 0 lim xdx = ∫ 2 0 xdx clearly what john has written is objectively meaningless and inconsistent. but the point is that john is using the new mathematical signs in mathematical activities (incoherent as they are to the outsider). in response to question 1(b), he writes: →∞ ∫ 2 2 0 lim ( )f x dx = ∞ →∞ ∫2 0 lim xdx = ∞ ∫ 0 xdx again his response appears incoherent and confused. but once more john is using the ‘new’ (to him) signs in mathematical activities. i suggest that notions of complex thinking can help the educator understand what is happening. specifically, i suggest that john’s response to both question 1(a) and 1(b) is dominated by complex thinking. in question 1(a) he has manipulated the template of an improper integral so that it eventually has the form of a definite integral (i.e. ∫ 2 0 xdx ), a form with which he is familiar. in question 1(b), he manipulates this further to get back to the template of an improper integral (albeit it does not converge). the point is: by using various signs in mathematical activities (a functional usage involving template-matching, associations and manipulations primarily) john is able to engage with the mathematical object on first contact, albeit in an idiosyncratic fashion. in this way, john gains a point of entry into mathematical activities with the object before he ‘knows’ that object. the question now is: how does john move from this (objectively) incoherent usage to a usage which is both personally satisfying and mathematically acceptable? i suggest that the answer lies in john’s imitation of the improper integral sign. that is, john is finally able to appropriate the socially-sanctioned usage of the improper integral sign through interaction with the mathematics textbook (a resource comprising socially sanctified mathematics). specifically, it is only after john has seen exemplars in the textbook of improper integrals and their evaluation, that he starts to use the improper integral in a way that is consonant with its definition. indeed, after seeing textbook exemplars, he is able to answer question 4 in a coherent fashion. that is, he writes ∞ →∞ =∫ ∫3 3 1 1 lim b b dx dx x x = →∞ −    2 1 2 lim b b x = →∞ − −  2 2 lim 2 b b = −2. and he states that this integral is convergent. although john has integrated ∫ 3 dx x incorrectly, ( − =∫ 3 2 1 2 dx x x ), his response is coherent; also he uses correct procedure and appropriate notation. this is a much improved response compared to his response to question 1. furthermore, john tells me that the examples are useful to him and that he is no longer confused. this contrasts with earlier statements that he is very confused about notions of convergence and divergence and the improper integral. my contention is: it is john’s functional use of the improper integral sign (initially association, template-matching and manipulations and then imitation) that enables him to move from activity dominated by complex thinking to conceptual (possibly pseudoconceptual) activity. allied to this, he is able to move from a confused notion of the improper integral (by his own admission) to a personally meaningful usage (again, in terms of his own assessment). making mathematical meaning: from preconcepts to pseudoconcepts to concepts 20 conclusion in this paper, i have argued that vygotsky’s theory of preconceptual and pseudoconceptual thinking (1934/1986) provides an appropriate framework within which to explore how students construct concepts which are both personally and culturally meaningful. in particular, i have argued that the notion of the functional usage of a sign (that is, the use of a mathematical sign prior to full understanding of the mathematical object that it signifies) together with the construct of the pseudoconcept, can be used to explain how the divide between an individual’s initial mathematical activities and a socially sanctioned mathematical definition is bridged. related to this, i have argued that idiosyncratic mathematical activities can be regarded as manifestations of complex thinking. with social regulation, these complexes can be transformed into pseudoconcepts and (through further activities and further social regulation) concepts. given that so many activities that constitute the functional usage of a sign (for example, manipulations, imitations, associations and template-matching) are dismissed or ignored by many mathematics educators, i suggest the need for research which explores the relationships between these different usages of signs and meaning-making. similarly, research that illuminates the bridges between personal and socially sanctified usages of mathematical signs, and that explicates the transformations from complexes to pseudoconcepts to concepts, is required. references alcock, l. & simpson, a. (2001). the warwick analysis project: practice and theory. in d. holton (ed.), the teaching and learning of mathematics at university level: an icmi study (pp 99-111). dordrecht: kluwer academic publishers. berger, m. (2002). the appropriation of mathematical objects by undergraduate mathematics students: a study. unpublished doctoral dissertation, university of the witwatersrand, johannesburg. berger, m. (2004a). the functional use of a mathematical sign. educational studies in mathematics, 55, 81-102. berger, m. (2004b). heaps, complexes and concepts (part 1). for the learning of mathematics, 24 (2), 2-6. berger, m. (2004c). heaps, complexes and concepts (part 2). for the learning of mathematics, 24 (3), 11-17. czarnocha, b., dubinsky, e., prabhu, v. and vidakovic, d. (1999). one theoretical perspective in undergraduate mathematics education research. in 0. zaslavsky (ed.), proceedings of the 23rd conference of the international group for the psychology of mathematics education, (vol. 1, pp 95-110). haifa, israel: pme. dörfler, w. (2000). means for meaning. in p. cobb, e. yackel, and k. mcclain (eds.), symbolizing and communicating in mathematics classrooms: perspectives on discourse, tools, and instructional design (pp 99-132). mahwah, nj: lawrence erlbaum. dubinsky, e. (1991). reflective abstraction in advanced mathematical thinking. in d. tall (ed.), advanced mathematical thinking (pp 95123). dordrecht: kluwer academic publishers. ernest, p. (1997). introduction: semiotics, mathematics and mathematics education. philosophy of mathematics education, journal 10. retrieved december 9, 2000, from http://www.ex.ac.uk/~pernest/pome10/art1.htm ganter, s.l. (2001). changing calculus: a report on evaluation efforts and national impact from 1988-1998, maa notes #56. washington dc: mathematical association of america. kozulin, a. (1990). vygotsky’s psychology: a biography of ideas. cambridge, ma: harvard university press. lerman, s. (2000). the social turn in mathematics education research. in j. boaler (ed.), multiple perspectives on mathematics teaching and learning (pp 19-44). westport, ct: ablex publishing. pimm, d. (1987). speaking mathematically: communication in mathematics classrooms. london: routledge & kegan. radford, l. (2000). signs and meanings in students’ emergent algebraic thinking: a semiotic analysis. educational studies in mathematics 42, 237-268. seldon, a. and seldon, j. (2001). tertiary mathematics education research and its future. in d. holton (ed.), the teaching and learning of mathematics at university level: an icmi study (pp 237-254). dordrecht: kluwer academic publishers. sfard, a. (2000). symbolizing mathematical reality into being − or how mathematical discourse and mathematical objects create each other. in p. cobb, e. yackel, and k. mcclain (eds.), margot berger 21 symbolizing and communicating in mathematics classrooms: perspectives on discourse, tools, and instructional design (pp 37-98). mahwah, nj: lawrence erlbaum. tall, d.o. (1995). cognitive growth in elementary and advanced mathematical thinking. in l. meira and d. carraher (eds.), proceedings of the 19th conference of the international group for the psychology of mathematics education, (vol. 1, pp 61-75). recife, brazil: pme. tall, d.o., thomas, m., davis, g., gray, e., and simpson, a. (2000). what is the object of the encapsulation of a process? journal of mathematical behavior, 18 (2), 1-19. van der veer, r. and valsiner, j. (1994). reading vygotsky: from fascination to construction. in r. van der veer and j. valsiner (eds. and trans.), the vygotsky reader (pp 1-9). oxford: blackwell publishers. vygotsky, l.s. (1981). the genesis of higher mental functions. in j.v. wertsch (ed.), the concept of activity in soviet psychology (pp144-188). armonk, ny: m.e. sharpe. vygotsky, l. s. (1934/1986). thought and language, a. kozulin (ed. and trans.), cambridge, ma: mit press. wertsch, j.v. and stone, c.a. (1985). the concept of internalisation in vygotsky's account of the genesis of higher mental functions. in j.v. wertsch (ed.), culture, communication and cognition (pp 162-179). new york: cambridge university press. during the three years which i spent at cambridge my time was wasted, as far as academical studies were concerned, as completely as at edinburgh and at school. i attempted mathematics, and even went during the summer of 1828 with a private tutor (a very dull one) to barmouth, but i got on very slowly. the work was repugnant to me, chiefly from my not being able to see any meaning in the early steps in algebra. this impatience was very foolish, and in after years i have deeply regretted that i did not proceed far enough at least to understand something of the great leading principles of mathematics; for men thus endowed seem to have an extra sense. charles darwin (autobiography) designing mathematical-technological activities designing mathematical-technological activities for teachers using the technology acceptance model gerrit stols university of pretoria email: gerrit.stols@up.ac.za the objective of this study was the design and development of mathematical technological activities. the guiding model in developing these activities was the technology acceptance model (tam). the two main issues to address, according to the model, are the “perceived ease of use” and the “perceived usefulness” of the software. the activities were used, refined and tested in a professional development programme which consisted of five hands-on laboratory sessions, lasting 2 hours each, spaced over a 3month period. the study found that the approach of letting teachers experience technology while engaged in authentic mathematical tasks promotes the perceived usefulness of geometer’s sketchpad (gsp) by the teachers. at the end of the series of workshops, 91% of the teachers indicated in the summative evaluation that these workshops helped them to understand geometry better. this finding was supported by the 21.5% increase in the pre-test and post-test average score. background & theoretical framework computers play an important role in the daily lives of learners and teachers. although the power of computers and software is increasing, we are not seeing the impact of this in our classrooms. even when computers are available, teachers do not use them as expected (vrasidas & glass, 2005). although the use of technology does not guarantee effective learning, some aspects of technology make it easier to create learning environments that are in line with the principles of learning (bransford, brown & cocking, 1999). for example, it is possible to make complicated subjects like geometry more accessible with the aid of technology by using new approaches to visualisation and analysis through animation, and other techniques that tap the multiple intelligences of the learners (bransford et al., 1999). according to bransford et al., the new technologies can help learners to visualise difficult-to-understand concepts and help to create an active problem-solving environment. but despite all the possible advantages, teachers in general do not make use of these technologies. salomon (2005) believes that what is needed is better teacher preparation. wenglinsky (1998: 33) also emphasises the importance of preparing teachers to use technology by stating: “[t]he study suggests that federal and state policymakers should redouble their efforts to ensure that teachers are properly trained to use computers.” teacher preparation and professional development opportunities play a crucial role in helping teachers to use technology for these reform-oriented purposes (willis & mehlinger, 1996). but teacher preparation, according to salomon (2005: xvii), is a long process that requires first-hand learning by experience, active doing and deep reflection. this implies the development of activities for teachers. the question, however, is what kind of activities can be used in teacher preparation programmes that will increase the possibility that the teachers themselves will use technology in their own classrooms. a model that can guide us in this process is the technology acceptance model (tam). fred davis and richard bagozzi developed the technology acceptance model (tam) as an attempt to explain factors that influence computer users’ decisions about how and when they will use it. tam is a general model that predicts and explains user acceptance of information technology systems. tam postulates that an individual’s attitude towards technology and behavioural intention to use it is determined by the perceived usefulness and perceived ease of use of that specific technology (davis, 1989; venkatesh & davis, 2000). according to this model, there is a strong causal link between intention and actual behaviour. sheppard, hartwick and warsaw (1988) found a correlation of 0.54 in their study between behavioural intention and actual use. in the context of teachers’ professional development, “perceived usefulness” is about the extent to which a teacher believes that the use of technology – in this case geometer’s sketchpad (gsp) – will enhance his or her own understanding or will enhance the teaching/learning process. “perceived 10 pythagoras 65, june, 2007, pp. 10-17 gerrit stols ease of use” is about teachers’ belief that using the specific technology geometer’s sketchpad will be free of effort. hu, clark and ma (2003) used the technology acceptance model (tam) to examine the use of powerpoint by school teachers. they found that there is a significant influence path from job relevance to perceived usefulness and finally user acceptance. although perceived ease of use of technology did not directly affect user acceptance, it did via perceived usefulness. this implies that a teacher will not use a technology just because it is easy to use, but perceived difficulty may prevent him or her from using a technology. we can therefore conclude that the tam may also be used in the context of teachers’ technological professional development. designing principles of the activities the target population of this study was in-service teachers of grade 11 and 12 south african mathematics. in order to address the grade 11 and 12 teachers’ perceptions of the usefulness of geometer’s sketchpad (gsp) in school geometry, the south african school-geometry curriculum was used as a basis for the development of the activities. vrasidas and glass (2005) expressed the view that teachers should not be taught out of context. according to these authors the best way to prepare teachers is to use technology while working on meaningful tasks. the activities must be of such a nature that teachers experience technology while engaged in authentic mathematical tasks. garofalo, drier, harper, timmerman, and shockey (2000) devised a set of guidelines to be used when developing mathematical activities and materials. two of the guidelines will help to increase the perceived usefulness of the technology by the teachers: worthwhile mathematics must be addressed, and the activities must take advantage of technology. it is, however, important at this point to remember that the introduction of technology will not in itself improve the quality of education. a lot depends on how we use it. wenglinsky (1998) examined data obtained from the 1996 national assessment of educational progress in an attempt to determine the relationship between computer use and students’ achievements in mathematics. this research emphasises the importance of professional development, and of the development of higher order thinking skills: professional development and higherorder thinking are both positively related to academic achievement: students with teachers who have had such professional development show higher levels of achievement, as do those who are taught higher-order skills with computers (wenglinsky, 1998: 28-29). the aim was therefore to develop higher order thinking skills in the context of high school geometry problems. this is in line with findings reported in literature. for example, it has been argued that the most successful technology-based projects used content and instructional approaches compatible with the local curriculum (zhao, pugh, sheldon & byers, 2002). further, wenglinsky (1988) has also pointed out that the use of technology must focus on the development of higher order thinking skills. while working on the activities the teachers must experience the usefulness of the technology for themselves. choosing software one of the reasons why teachers are reluctant to use technology for instruction is that they believe it will take more time (vrasidas and glass, 2005). teachers indicated that they do not have the time to learn how to use computers themselves and then to pass that knowledge on to the learners. vrasidas and glass (2005: 3) state, “in 1999 the most frequently reported obstacles to the use of computers and the internet in the classroom by public school teachers were lack of released time for teachers to learn how to use computers”. to address this problem it is important that the technology used for professional development is easy to use and does not take too much time and effort to master. this is also important from the perspective of tam because the behavioural intention of teachers to use technology is determined by the perceived usefulness and perceived ease of use. technological applications have their own advantages and constraints and therefore some software is better suited to particular parts of the curriculum than other software programs. it is important that the software should fit the planned activities. vessey and galletta (1991) explain that cognitive fit is a “cost benefit characteristic that suggests that, for most effective and efficient problem-solving to occur, the problem representation and any tools or aids should all support the strategies (methods or processes) required to perform that task”. according to russell, bebell, o’dwyer, and o’connor (2003), it seems prudent to focus teacher preparation on a specific technology, in depth, rather than adopt a more general approach. the instructional objective with graphing software, 11 designing mathematical-technological activities for teachers using the technology acceptance model according to wong (1998), is to develop and reinforce concepts, to rectify common errors, to check graphical solutions, to solve equations graphically, to test conjectures through problemposing, to encourage users to become metacognitive, to help users to acquire information technology skills and to enhance the desire to learn. geometer's sketchpad (gsp) can help enhance some of the essential skills for teaching mathematics. sanders (1998) comments that the appropriate use of gsp can enhance mathematics teaching, improve conceptual development and visualisation, and create opportunities for creative thinking. in this study gsp was used because it is an easy program to learn, but also a powerful tool for the development of geometrical understanding. the software prompts the user to undertake mathematical investigations and can be used for exploration, discovery and problem-solving. gsp was originally designed for teaching geometry in secondary schools. it includes the classical ruler and compass constructions. the program allows teachers and learners to work quickly through numerous examples (by dragging) and enables them to discover patterns, to explore and to test conjectures by constructing their own sketches. the latest version of gsp has a number of trigonometric and algebraic features, which enable teachers and learners to find relationships between symbolic and graphic representations. designed activities the activities start with a high level of guidance on how to use gsp in order to help the teachers to perceive gsp as an easy program to use. mishra and koehler’s (2006) approach to the introduction of technology was followed. they believed that the teachers should learn about technology as and when necessary to complete their tasks. while working through the activities using gsp, they also learned how to use the software. it was therefore important to choose activities that would cover all the basic functions of gsp. although the aim of the series of workshops was the development of higher order thinking, the activities start with basic discovery in order to help the teachers to learn how to use the basic functions of the software. in summary, the activities have been designed in such a way that the level of mathematical thinking increases (from basic discoveries to advanced problem-solving) while the technology guidance gradually decreases (see figure 1). examples of activities in the first activity the mathematical thinking level is low (basic discovery) but there is a high level of technological guidance. activity 1 construct a cyclic quadrilateral abcd. measure the sum of the opposite interior angles. drag points a, b, c and d. does the sum stay the same? you can do the activity using the following steps: step 1: click on the gsp icon . the following screen will appear (see figure 2). step 2: select the compass tool and construct a circle. step 3: select the point tool and construct four points on the circumference of the circle. step 4: select the text tool and click on the four constructed points to label them a, b, c and d. step 5: select the straight-edge tool and construct the four segments ab, bc, cd and da (see figure 3). step 6: select the selection arrow tool and mark points a, b and c. (the points will be highlighted.) select “measure” on the menu toolbar (see figure 4). click ”angle” and the measure of ∠abc will appear: m∠abc = 97.46°. measure ∠bcd, ∠cda and ∠bad in the same way. step 7: to calculate the sum of the opposite angles, click on the “measure” toolbar and choose “calculate” from the toolbar. the following will appear: “click: m∠abc, +, m∠adc, ok”. the sum of ∠abc and ∠adc will appear. step 8: drag the different points. does the measure of the angles stay the same? what about the sum of the interior opposite angles? activity 3 use gsp to convince learners that the line joining the mid-points of two sides of a triangle is parallel to the third side and equal to half the third side. how do you determine if two lines are parallel? the following activity was included because it requires and helps develop a high level of technological skills, while it still only requires a low level of mathematical thinking skills 12 gerrit stols technology guidance level of mathematical thinking figure 1. model for designing activities toolbox menu figure 2. gsp interface b c d a figure 3. the result of step 5 figure 4. menu toolbar 13 designing mathematical-technological activities for teachers using the technology acceptance model (knowledge). it is an advanced discovery activity. the purpose of the activity was to enable the teachers to use as many different functions and features of gsp as possible. it helped the author to assess the teachers’ competence in the use of the different functions of gsp. the group discussed the definitions of the incentre, circumcentre, orthocentre and centroid of a triangle before they started with the activity. activity 9 three of the four points of concurrency associated with triangles are the incentre, circumcentre, orthocentre and centroid. three of them are on the same line. the line is called the euler line. the euler line is named after leonard euler, who proved that these three points of concurrency are collinear. construct the euler line of a triangle. which point of concurrency is not on the euler line? use the drag test to see if this is always true. the successful completion of activity 9 indicates the ability to use the software. it was therefore important to allow enough time for all the teachers to complete this activity successfully. after this the focus shifts to the development of mathematical thinking skills. for example, activities 10 and 12 require more mathematical thinking skills than computer skills: activity 10 what is the sum of the measures of the 100 interior angles of a 100-gon (a triangle is a 3-gon)? how did you solve the problem? activity 12 if you have a piece of land that is a quadrilateral, what kind of quadrilateral will be formed if you take the midpoint of each of its four sides and join these midpoints? use gsp to prove your belief. explain mathematically why this will always be true. to increase the level of the perceived usefulness of gsp, the activities stimulated the teachers to think and reflect in terms of their own classroom practice and to reflect on the value of using gsp to convince their learners of the truth of the geometry theorems. this was, for example, the aim of activities 17 and 30. it is important to note that the discovery of a conjecture, although convincing because of the drag-test, does not replace formal mathematical proofs. activity 17 use gsp to convince the learners of the truth of the following grade 10 theorems: • triangles on the same base (or on equal bases) and between the same parallel lines are equal in area. • parallelograms on the same base (or on equal bases) and on the same side of the base are equal in area if they lie between the same parallels. activity 30 a) use the following school geometry theorems and use gsp to convince the learners that they are true (that is inductive reasoning). b) prove these theorems using deductive reasoning: if a line is drawn parallel from one side of a triangle, it will divide the other two sides proportionally. if a line divides two sides of a triangle in the same proportion, then that line is parallel to the third side. the most difficult activity in terms of mathematical thinking according to the teachers was activity 36. solving this activity requires doing mathematics and using higher order reasoning. gsp could only help the teachers to understand the problem, but did not help them in solving it. activity 36 divide the area of ∆abc (see figure 5) into three equal parts by drawing two lines parallel to bc. what is the ratio of ak:kl:lb? explain mathematically why the ratio divides the area of ∆abc into three equal parts. extension problems were also given to challenge the more able teachers. they were asked to develop their own geometry and analytical geometry questions using gsp. in addition to the activities, the workshop manual contains reflective a b c k l figure 5. triangle abc 14 gerrit stols questions to help the teachers to reflect on the usefulness of gsp in their own teaching when they reach the end of each workshop. methodology data for the study were drawn from a series of five two-hour workshops that were conducted by the author over a period of three months. the participants were 15 teachers from previously disadvantaged schools in the pretoria region in south africa. all of them were grade 11 and 12 geometry teachers. all the teachers had taught mathematics in previously disadvantaged rural schools. the local department of education selected the participants. of the 15 original participants, 12 completed the series of five workshops. the reason for non-completion of the series of workshops by three participants was an inadequate level of computer literacy. in the first questionnaire, the teachers had to indicate their level of computer literacy. the following data were collected on the computer background of the 15 teachers: • 2 of the 15 teachers have an e-mail address. • 14 of the 15 teachers have computers at school. • 8 of the 15 teachers have computers at home. • 9 of the 15 have computer labs at school. • 1 of the 15 has access to the internet at home. • 7 of the 15 use word processors for school work. one teacher indicated that he was not computer literate; there were also three beginner computer users, nine novice users and two expert users. the non-computer literate teacher and two of the three beginner users failed to complete the program. this suggests that non-computer literate and beginner users first need to complete a general computer skills training session before they can enrol for such a program. this essentially entails being able to use a mouse, open files, save files, copy and paste, etc. data collection procedure the teachers completed questionnaires during and after each workshop and were asked to keep a reflective journal. the reflective journals provided us with documentation of a continuous cycle of enquiry. in addition to the journals, data were gathered to assess the impact of the workshops. at both the beginning and the end of each series of workshops, the teachers completed the same test. the test comprised questions similar to the grade 12 second paper examination questions. further information was obtained from teachers’ journal entries, workshop evaluations completed at the end of each workshop, a summative evaluation questionnaire and interviews. two of the teachers were interviewed after the last three workshops. a pilot study was conducted with one teacher – a fully computer literate teacher. the purpose of the pilot study was threefold, namely to: • follow the change and growth in the teacher’s conceptual development and understanding of mathematics. • develop and refine training materials (teachers' worksheets and notes). • determine the effective duration and nature of a full basic course for gsp teachers. the developed and refined materials from the pilot study were used in a workshop for 15 teachers with different levels of computer literacy. a series of five two-hour workshops were conducted. results although 10 out of the 15 teachers are positive about geometry, they find it difficult to teach geometry. when asked to write about their experience of teaching geometry and their beliefs about learning geometry, they had the following to say: • 7 out of 15: teaching geometry is a nightmare /difficult/hell. • 5 out of 15: learners have negative attitudes. • 1 out of 15: learners find it difficult. • 1 out of 15: learners must learn theorems and practice geometry. • 1 out of 15: learners like to experience things. the feedback indicated that the respondents were confident about teaching geometry after attending the workshops. the teachers agree (with 9 of the 12 strongly agreeing) that workshops helped to increase their levels of confidence. some of the feedback included the following: i will try to open afternoon classes in the computer laboratory for my pupils so that i can give them more attention to improve their geometry. learners are more enthusiastic about maths. they work more problems than usual. the learners will be very positive in doing geometry unlike the situation that we are having at our schools where they are negative. at the end of the series of workshops, 11 of the 12 teachers indicated in the summative evaluation that the workshops helped them to understand geometry better. the most useful part was using geometer's sketchpad (gsp) to design riders, 15 designing mathematical-technological activities for teachers using the technology acceptance model which was the purpose of the fifth workshop. in this regard, feedback included statements such as: i had a problem before but now i think i will approach it (geometry) with confidence. after this workshop i think i have gained confidence in geometry teaching and i’m having a positive attitude. when i came here i had a very negative attitude as learners did not really understand geometry, but now i am positive that i will also change the learners’ attitude toward geometry. although the teachers were not allowed to use gsp during the preand post-test the teachers’ pretest and post-test results showed a 21.5% improvement. the same tests were used. the average marks were 54.8% for the pre-test and 76.3% for the post-test. the teachers’ ability to answer the questions improved in all the questions. eighty-three percent of the teachers find that gsp is an easy program to use. all the teachers believe that the use of gsp in schools will help learners, and 11 of the 12 believe that it is possible to use gsp in schools. the same 11 teachers also believe that learners will be able to learn to use the software. in the summative evaluation 9 out of 12 of the teachers believed that it is not possible to teach geometry successfully without using technology. discussion, implications and limitations looking at the results from the perspective of the tam, the teachers perceived geometer’s sketchpad (gsp) to be extremely useful. the approach of integrating the development of computer skills with geometric discovery has proved to be successful. future workshops must include more group discussions on pedagogical issues. the focus of these workshops was on conceptual development, problem-solving and mastering the software. from the videos and interviews it is clear that the teachers demonstrate conceptual understanding. they interpret mathematical principles in a problem and translate those ideas into a coherent mathematical representation using the important facts of the problem. teachers show good conceptual understanding of the geometry and choose appropriate representations. from the perspective of the tam the two main factors that determine an individual’s intention to use a technology are “perceived usefulness” and “perceived ease of use”. given the strong causal link between intention and actual behaviour we can conclude therefore that these teachers would use gsp in their classrooms to teach geometry if it was possible for them to do so. the teachers who attend these workshops are very positive and want to learn more. the responses and evaluations after each workshop show that the teachers want more and longer workshops. all of them agree (with 9 out of 12 indicating that they strongly agree) that they would gladly attend similar workshops in future. gsp often requires teachers to rethink their teaching practice. in the south african context, the perceived usefulness was dangerously high. the implication of the fact that 9 out of 12 of the teachers believe that it is not possible to teach geometry successfully without using gsp is that they have a valid excuse for not teaching geometry effectively in their own schools because the teachers who attended this series of workshops were all from previously disadvantaged schools and not all of the schools have computer labs. sixty percent of the teachers indicated that they have computer labs at their schools, with an average of 17.6 computers per lab. this figure may look impressive for a third world country, but the computers are normally old computers. the majority of these schools can also not afford to buy a site licence for gsp. one way to address the problem in future is to spend time with the teachers after the computer sessions to assist them in developing similar pen and paper or folding activities that do not require the use of a computer to give them a more realistic view of the use of technology. more research is needed to determine the impact of a high “perceived ease of use” of technology on the part of teachers who do not have, or cannot afford, the technology. without knowing the full picture, using technology could do more harm than good. a possible future teaching sequence could, for example, consist of manual explorations, followed by computer explorations and then manual explorations again. references bransford, j., brown, a. & cocking, r. (1999). how people learn. washington, dc: national academy press. davis, f.d. (1989). perceived usefulness, perceived ease of use, and user acceptance of information technology. mis quarterly, 13(3), 319-340. 16 gerrit stols venkatesh, v. & davis, f.d. (2000). a theoretical extension of the technology acceptance model: four longitudinal field studies. management science, 46(2), 186-204. garofalo, j., drier, h., harper, s., timmerman, m.a. & shockey, t. (2000). promoting appropriate uses of technology in mathematics teacher preparation. contemporary issues in technology and teacher education, 1(1), 6688. vessey, i. & galletta, d. (1991). cognitive fit: an empirical study of information acquisition. information systems research, 2(1), 63-86. hu, p.j., clark, t.h. & ma, w.w. (2003). examining technology acceptance by school teachers: a longitudinal study. information management, 41(2), 227-241. vrasidas, c. & glass, g.v. (eds.) (2005). preparing teachers to teach with technology: current perspectives on applied information technologies. greenwich: information age publishers. mishra, p. & koehler, m. j. (2006). technological pedagogical content knowledge: a framework for teacher knowledge. teachers college record, 108(6), 1017–1054. wenglinsky, h. (1998). does it compute? the relationship between educational technology and student achievement in mathematics. princeton: educational testing service. russell, m., bebell, d., o’dwyer, l. & o’connor, k. (2003). examining teacher technology use: implications for preservice and inservice teacher preparation. journal of teacher education, 54(4), 297-310. willis, j. & mehlinger, h. (1996). information technology and teacher education. in j. sikula, t.j. butter & e. guyton (eds.), handbook on research in teacher education (pp 978-1029). new york: simon & schuster macmillan. salomon, g. (2005). preface. in c. vrasidas & g.v. glass (eds.), preparing teachers to teach with technology: current perspectives on applied information technologies (pp. xvxviii). greenwich: information age publishers. wong, k.y. (1998). graphing software: computers for mathematics instruction (cmi) project. retrieved june 25, 2004, from http://academic.sun.ac.za/mathed/software/grap hmat.pdf sanders, c.v. (1998). geometric constructions: visualizing and understanding geometry. mathematics teacher, 91(7), 554-556. zhao, y., pugh, k., sheldon, s., & byers, j. (2002). conditions for classroom technology innovations: executive summary. teachers college record, 104(3), 482-515. sheppard, b.h., hartwick, j. & warsaw, p.r. (1988). the theory of reasoned action: a metaanalysis of past research with recommendation for modifications and future research. journal of consumer research, 15, 325-343. “who wishes correctly to learn the ways to measure surfaces and to divide them, must necessarily thoroughly understand the general theorems of geometry and arithmetic, on which the teaching of measurement ... rests. if he has completely mastered these ideas, he ... can never deviate from the truth.” abraham bar hiyya 17 << /ascii85encodepages false /allowtransparency false /autopositionepsfiles true /autorotatepages /all /binding /left /calgrayprofile (dot gain 20%) /calrgbprofile (srgb iec61966-2.1) /calcmykprofile (u.s. web coated \050swop\051 v2) /srgbprofile (srgb iec61966-2.1) /cannotembedfontpolicy /warning /compatibilitylevel 1.4 /compressobjects /tags /compresspages true /convertimagestoindexed true /passthroughjpegimages true /createjdffile false /createjobticket false /defaultrenderingintent /default 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<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> >> >> setdistillerparams << /hwresolution [2400 2400] /pagesize [612.000 792.000] >> setpagedevice 66 p76-84 venkat final 76 pythagoras 66, december, 2007, pp. 76-84 mathematical literacy – mathematics and/or literacy: what is being sought? hamsa venkat marang centre for mathematics and science education, university of the witwatersrand email: hamsa.venkatakrishnan@wits.ac.za introduction mathematical literacy was introduced as an alternative option to mathematics in the further education and training (fet) phase (grades 1012, learners generally aged 15-18) in south africa in january 2006. as a new subject in the fet phase, and with aims that differ somewhat from the notion of mathematical literacy that figured within the mathematical literacy / mathematics / mathematical sciences (mlmms) learning area in the general education and training (get) phase, teachers are faced with implementing a subject which does not have established aims, understandings and practices associated with it. in this paper i use empirical data to present two agendas – mathematics and literacy – that two teachers i collaborate with are drawing upon in different ways, according to their interpretations of putting policy into practice. these empirical data are located within a trajectory of research in mathematics education in south africa that has pointed to problems when attempts have been made to link mathematics with other agendas – the need for relevance being a prime example. many of these critiques have raised the issue of the ‘displacement’ of mathematics when other agendas are brought into the arena. in contrast, i will argue here that the empirical data from mathematical literacy lessons point to ways in which ‘tweaking’ the flow of questions and activities can allow for a productive integration of mathematical and literacy-oriented agendas, with each agenda working to bolster, rather than distract from, the other. the mathematical literacy curriculum statement defines the subject in the following terms: mathematical literacy provides learners with an awareness and understanding of the role that mathematics plays in the modern world. mathematical literacy is a subject driven by life-related applications of mathematics. it enables learners to develop the ability and confidence to think numerically and spatially in order to interpret and critically analyse everyday situations and to solve problems. (department of education, 2003a: 9) this definition, with its emphasis on enabling learners to act in ways that involve awareness of the roles and uses of mathematics, and an inclination and ability to think mathematically and use mathematics in real-life, shares much in common with contemporary notions of ‘literacy’. gee’s (2001) discourse-based definition of literacy relates to “the mastery of or fluent control over a secondary discourse”. ‘secondary’ discourses are “ways of being in the world” that are purposively promoted or pursued, often within the school context. they differ from ‘primary’ discourses, gee’s term for discourses that are naturally acquired through a person’s initial ways of understanding and interacting in the world. this definition of literacy emphasises behaviour and identity, and whilst knowledge is implicated within both the mathematical literacy definition and gee’s definition of literacy, both notions of literacy emphatically involve more than acquisition of content. walsh (1991) also places emphasis on understanding and interpreting the world within her description of literacy as: a creative activity through which learners can begin to analyse and interpret their own lived experiences, make connections between these experiences and those of others, and, in the process, extend both consciousness and understanding. (1991: 6) the links between the curricular definition of mathematical literacy and these broader understandings of how literacy is comprised are unsurprising, given the overt acknowledgement of gee’s notion of literacy within the organisation for economic cooperation and development / pisa notion of mathematical literacy (organisation for economic cooperation and development, 2003), which in turn has been acknowledged as an important source of ideas about how the mathematical literacy subject area has been constituted in south africa (department of hamsa venkat 77 education, 2007). the sense that mathematical literacy involves acquiring the mathematically literate discourses required for constructive and critical participation in, and interpretation of, the practices related to the future life roles of learners − a self-managing person, a contributing worker, a participating citizen (department of education, 2003a: 10) − is therefore foregrounded in the definition and preamble to the curriculum specification within this document. this agenda differs in important ways from the agenda driving mathematics as a subject in the fet phase. mathematics is defined in its fet curriculum statement as follows: mathematics enables creative and logical reasoning about problems in the physical and social world and in the context of mathematics itself. it is a distinctly human activity practised by all cultures. knowledge in the mathematical sciences is constructed through the establishment of descriptive, numerical and symbolic relationships. mathematics is based on observing patterns; with rigorous logical thinking, this leads to theories of abstract relations. mathematical problem solving enables us to understand the world and make use of that understanding in our daily lives. mathematics is developed and contested over time through both language and symbols by social interaction and is thus open to change. (department of education, 2003b: 9) whilst the notion of mathematics as a subject with applications in real-life is mentioned, emphasis is laid on abstract rather than concrete concepts, on intra-mathematical connections rather than mathematics-real-world connections, on rigour and logic rather than interpretation and critique, and on knowledge itself, as well as applications of knowledge. i am not suggesting here that mathematics curricula do not require literacy in walsh’s sense of analysing and interpreting situations. i am however pointing to the extensive body of evidence that suggests that mathematical learning is often restricted to the acquisition of content with little room for sense-making and interpretation of problems (mukhopadhyay & greer, 2001; schoenfeld, 1985). many critiques have argued for a more literate agenda within mathematics, such as kilpatrick, swafford, & findell (2001), in order to facilitate flexible conceptual understanding and the willingness to make sense of situations alongside procedural fluency. in this paper, the mathematical agenda refers therefore to the more restricted (but widely prevalent) emphasis on mathematical content and procedures. critiques of the mathematical literacy curriculum statement, for example, amesa (2003), have pointed out that the curriculum specification (detailed largely in terms of mathematical content) conflicts with, and runs the risk of derailing, the broader literacy-focused agenda in which acquisition of content forms just one part. thus, within the mathematical literacy curricular documentation there are tensions between whether mathematical literacy should focus more on furthering a mathematical agenda in terms of extending the learning and understanding of mathematical content, or a literacy agenda that suggests the need to develop learners’ willingness and ability to use mathematical thinking to analyse, interpret and solve problems in increasingly complex contexts. in spite of this conflict between the tone of the policy texts and the nature of the curriculum specification, the overall signals tend to state that the two agendas – developing mathematics and developing literacy – need to be balanced and can, and should, be integrated: the challenge for you as the teacher is to use situations or contexts to reveal the underlying mathematics while simultaneously using the mathematics to make sense of the situations or contexts, and in so doing develop in your students the habits or attributes of a mathematically literate person. (department of education, 2006: 4) this debate on whether a mathematical agenda can and should be brought together with the literacy agenda forms the focus of this paper. i consider the ways in which the introduction of mathematical literacy in the fet phase, with aims that differ from those associated with mathematics, might provide openings for the linking of the ‘mathematical’ and ‘literacy’ agendas. in order to consider this question, i detail some of the positions that have been taken up in previous south african debates on whether mathematics can be linked with other agendas. a key antecedent with associated literature was the attempt to shift towards a more ‘relevant’ mathematics curriculum within curriculum 2005 in the general education and training phase. much of this writing (though not all) pointed to the incommensurability of the ‘mathematics’ and ‘relevance’ agendas, providing examples of the ‘hollow’ use of ‘relevant’ problems. however, this debate was located within a curriculum that was firmly rooted in a mathematical literacy – mathematics and/or literacy: what is being sought? 78 mathematical agenda, so part of my aim is to consider the ways in which mathematical literacy teaching may change the terms of this debate. in the body of this paper i present two excerpts of mathematical literacy teaching drawn from research exploring the implementation of mathematical literacy in schools, in which i have been involved through the marang centre at wits university. the first excerpt shows a teacher foregrounding a more mathematical agenda; the second shows a teacher foregrounding a more literacy-focused agenda, but with the mathematical features somewhat backgrounded. this format may suggest that my aim is to add evidence into the ‘incommensurability’ claim mentioned earlier within the contours of mathematical literacy. this is not the case. in the analysis that follows the two excerpts, i provide evidence, from teachers’ comments and from other classroom observations, to argue that these excerpts are better interpreted as ‘near misses’ rather than ‘incommensurability’ in terms of integrating the mathematical and literacy-based agendas, and that the contrasts of foreground reflect differing understandings about what mathematical literacy is about. greeno (1998) developed the notion of ‘attunements’ – “regular patterns of an individual’s participation” – to environmental ‘affordances’ and constraints, which reflect historical routines of practice based on these understandings to explore interactions in teaching and learning environments. the two excerpts in this study detail contrasting ‘attunements’ relating to what should be aimed for in mathematical literacy, with understandings that draw upon educators’ priorities – or deficiencies – in the context of their prior experiences of mathematics teaching. a ‘near miss’ perspective, rather than simply highlighting an absence of mathematics, allows me to highlight what is achieved in class in relation to teacher’s aims. i also identify gaps in terms of what might be possible with better integration of the two agendas. both educators whose teaching is presented in this paper saw the need for mathematics and literacy to be developed, and during the course of 2006, i observed some lessons in which integration of the two goals was closer to being attained and others in which one or the other agenda took precedence. i conclude that educators should aim to improve awareness of the twin agendas at work, and improve the flow of questions and activities. in this way, both of the approaches presented in the excerpts can better achieve the kinds of integration suggested in the definition of mathematical literacy and in other related policy documents. integrating agendas – historical antecedents questions were asked regarding the role and function of mathematical knowledge and skills in relation to their integration with other agendas, in the context of a shift to a more ‘relevant’ curriculum 2005. in that debate, some argued that the mathematics and relevance agendas were focused on fundamentally different goals, and hence, could not be reconciled – the ‘incommensurability’ claim (davis, 2003). others, while tentatively supporting the shift to mathematics teaching and learning that highlighted the links between mathematics and the real world, pointed out the dangers of either agenda taking precedence (sethole, 2003), or more pessimistically, of the mathematics becoming ‘lost’ (adler, pournara, & graven, 2000) – an issue that is considered later in this paper in relation to the literacy agenda. whilst not related directly to the introduction of curriculum 2005, barnes’ (2005) research on realistic mathematics education (rme) advocated the use of a relevant, experientially real context as a starting point for the process of mathematisation, and highlighted their role in helping learners make sense of a situation. although rme is closest to the literacy agenda within mathematical literacy, it needs to be remembered that this approach is firmly focused on a mathematical agenda in which contexts are viewed largely as vehicles for progressive mathematisation. this differs from the emphasis of the mathematical literacy policy documents on inculcating attitudes and behaviours that quip learners for participation in future life roles. the south african literature thus provides a mixed message regarding the useful integration of mathematics with other agendas, and alerts the reader to the mathematical emphasis of previous efforts. research outline and data sources the two excerpts of teaching presented in the following section were drawn from a longitudinal study of policy implementation in one inner-city johannesburg school in which three mathematics educators taught the three grade 10 mathematical literacy classes. the research involved weekly observations in all three classes during the course of 2006. field notes were taken during each observation, records were kept of informal hamsa venkat 79 conversations with the educators, and copies of tasks and materials given to learners were collected. semi-structured interviews were conducted with these educators in the fourth semester; these were tape-recorded and transcribed. additional comments are incorporated from a focus group interview that was conducted in the fourth semester which included the two main teachers. the excerpts presented in this paper, both taken from lessons that took place in october 2006, present examples of the foregrounding of a mathematical agenda by one educator, and an emphasis on literacy – over mathematical thinking – by the other. whilst neither excerpt can be described as ‘typical’ of the educators’ teaching, their differing agendas were more evident in their reflections on the policy. their understandings of and priorities in implementing mathematical literacy are considered in relation to an analysis of their different approaches to implementation. pseudonyms are used throughout. excerpt 1 charles naughton, an educator with 25 years’ teaching experience in mathematics and science at fet level, had been at the school for nine years. he had decided to do a lesson on the use of percentages, because learners had struggled to understand the topic in the previous lesson. his class consisted of 24 learners, 12 male and 12 female, all black. charles had, at several points in the course of the year, taken time to go back over aspects of relatively basic mathematics that had emerged as problematic within lessons. for example, i observed lessons on understanding concepts related to decimals, and a lesson working on the meaning of ‘per’ across a range of problems (for example, kilometres per hour). as learners entered, charles wrote a title on the board: ‘all the ways of using percentages’. he then said to the class ‘make up an example of turning a fraction into a percentage, an everyday example’. faced with silence, he added ‘look around the class. think of something that might be different here’. a boy responded ‘boys and girls’. charles, standing at the board, nodded and said ‘yes’. he asked the class how many girls were in the group. learners began looking around and counting. initially, the answers called out most insistently were 11 girls, 12 boys, and he wrote on the board: g = 11 b = 12 a girl then called out that there were actually 12 girls in the group. charles responded: ‘no, that’s too easy. let’s say that there are 10 girls and 12 boys.’ changing the numbers written on the board accordingly, he then went on to ask the class what fraction of the class were girls and what fraction were boys, and, following their answers, wrote down on the board: fraction of g = 22 10 fraction of b = 22 12 he then asked, “how do we change these to a percentage? for example, if the proportion stayed the same, but the numbers changed, what percentage would be girls?” after 30 seconds, a learner who was using a calculator volunteered an answer. charles wrote the learner’s method and answer on the board: g = 22 10 x 1 100 = 45,4 = 45% excerpt 2 freddy dube is the head of mathematics, and elected to teach mathematical literacy in grade 10. he attended provincial mathematical literacy training in 2005 and 2006, and has completed an honours course which included mathematical literacy modules. he has taught mathematics for nearly 10 years, across grades 8-12. his class consisted of 26 learners, 14 boys and 12 girls, all black. across the year, freddy had drawn from a range of different resources to provide or design ‘real-life’ problems. for example, he used newspaper articles about the effects of heavy rainfall on south african dam levels, bank pamphlets about loan repayment rates, and a mathematical literacy textbook to work out floor areas on house plans. freddy settled the class to silence and then switched on the overhead projector. he showed the class a transparency (appendix 1) detailing body mass index (bmi) information – which he had photocopied from the teacher guide (department of education: 2006, 25) – and then wrote on the board: bmi = body mass (kg) height2 (m2) bmi = body mass (kg) x 10 000 height2 (cm2) mathematical literacy – mathematics and/or literacy: what is being sought? 80 freddy explained that bmi is commonly used to judge whether a person is overor underweight, and then went through the text and figures on adult bmi on the transparency. he added that there were separate bmi graphs for boys and girls. twenty minutes into the lesson, he gave out a worksheet with the transparency information on one side (appendix 1), while on the other side were the two bmi formulae which he had written on the board, together with examples of how to use them and some tasks that he had devised (figure 1). asking the class to focus on the first task (calculate your teacher’s bmi), he wrote his own weight and height on the board as follows: mass = 67kg height = 1,7m freddy then asked the class to calculate his bmi using the first formula on the worksheet. not all learners in the class had a calculator, but most could observe one being used at their table. the class began to calculate his bmi, and after looking over individuals’ work, freddy wrote on the board: bmi = 67kg (1,7m)2 he asked the class for answers. ‘39,4’ was the first response, and he wrote ‘ = 39,4 kg/m2’ underneath. he then asked the class what advice they would give him based on this figure. individual learners began scanning the information given on the bmi ranges. some started laughing, saying, “sir, you’re obese”. others called out “use bio-slim” and “not eating”. their laughter was due in part to the fact that freddy is very slim. at this point, freddy commented, ‘well, you can’t always tell about someone’s bmi just by looking at them’, and noted too that there were categories of people that fall outside bmi scales, such as pregnant women. as the class began their own discussions around this, he asked them to work in groups and calculate each of their bmis. as they did this, one learner pointed out that the answer given for freddy’s bmi was incorrect. he asked the class to recalculate, and as they worked, freddy suggested that those working with scientific calculators could use the exponent key, and those without would have to do the multiplication manually. he wrote on the board: (1,7m) 2 = 1,7m x 1,7m = 2,89 m2 bmi = 67 kg 2,89 m2 = 23,2 kg/m2 answers were contributed by the learners at task 1 1) determine whether your ml teacher is overweight or not. 2) what advice can you give him/her? 3) determine whether your group members are overweight or not, including yourself. 4) what advice can you give them if they are overweight? task 2 (data collection) 1) collect weight and height of at least 15 adults/ young adults. 2) you can start collecting data from school figure 1. worksheet with formulae and classroom tasks. hamsa venkat 81 each stage underlined above, and freddy concluded that he fell within the ‘healthy weight range’. he then asked the class to proceed with the subsequent tasks on the worksheet. analysis both excerpts provide examples of teacher decision-making within the flow of classroom activity. stein, smith, henningsen and silver (2000) have stressed that looking at tasks alone is insufficient, since the nature and demand of tasks are frequently altered significantly during implementation: as mathematical tasks are enacted in classroom settings, they become intertwined with the goals, intentions, actions, and interactions of teachers and students. (2000: 24) while stein et al. were concerned with changes in the demands of tasks, my interest is in how the actions and interactions detailed above reveal the goals and intentions of the two teachers. both excerpts share a largely educator-led format, in which learner contributions were both expected and encouraged – this latter feature was reflected across almost all the lessons i observed with both teachers throughout the year. in addition, the pace in both excerpts was unhurried, an aspect that both educators found possible within their mathematical literacy teaching, in contrast to the content-laden, time-restricted pressures that constrain their mathematics teaching. both educators try to set up their task in terms of a situated problem. in charles’ case, learners are asked to calculate percentages based on a feature in their classroom, and are cued further to include the concept of ‘difference’. freddy’s task is located in the context of bmi, with relevant information presented in text, graphs and equations. it is interesting to note that charles begins by highlighting ‘percentages’ on the board, and then suggests a context within which percentage can ‘occur’, whereas freddy first introduces the bmi concept, followed by the use of graphs and equations within this context. this contrast in ‘attunements’ – to mathematics in charles’ case, and to an emphasis on interpreting answers in the context of the situated problem in freddy’s case – persists across the excerpts and more widely across their teaching. charles – excerpt 1 charles’ decision-making within the lesson flow (excerpt 1) suggests that his aim is to focus on checking, revising and teaching the procedure for turning a fraction into a percentage. thus, when ‘real-life’ provides him with equal numbers of boys and girls in his class – a scenario which opens up the possibility for learners to guess, remember or understand that this would be equivalent to 50% – without recourse to the procedure – he rejects these data in favour of ‘made-up’, unequal numbers producing fractions that require the procedure if they are to be converted into percentages. the procedure is asked for, carried out and checked, and passing mention is made of proportionality as a rather contrived motivation for why percentage might be useful to work out, although this is not dwelt upon. charles’ concern with a mathematical agenda, with mathematical concepts and connections, has been voiced frequently during the course of the year: i keep on wanting to help them achieve the actual mathematics. this point came across in his reference to a task involving cell phone tariffs during the focus group interview, where he felt that the ‘realistic’ complexity of the numbers involved in the call rates tended to obscure the mathematics: i’ve also felt all the way along that using numbers like 28,76 cents per call or something was not necessary at this early stage. we should still be keeping things simple so they see how it works rather than get confused with those actual numbers, you know, like on the price of a cell phone call. the literacy agenda here is not being dismissed but it is being deferred because it is considered too complex for the present, and a distraction that prevents learners from seeing the underlying (mathematical) structure of the situation. freddy – excerpt 2 freddy’s awareness and use of ideas from a wide range of sources, including the policy documents, is evidenced when he draws from an exemplar unit in the mathematical literacy teacher guide document (department of education, 2006). in his opening exercise, freddy draws attention to the text and context of bmi, going through the contents of the overhead, and at times adding further explanation. the formulae written on the board are introduced, but not worked through – he writes down his height and weight and asks the class to use the formulas to work out and interpret his bmi. when an answer is provided, freddy’s attention shifts directly into asking the class to interpret and understand this answer in context – an activity that most learners participate in actively mathematical literacy – mathematics and/or literacy: what is being sought? 82 and succeed in doing. however, neither the structure nor the calculation procedure is interrogated at this stage. this suggests either an assumption that the answer offered is correct, or an ‘attunement’ which is much more focused on encouraging learners to interpret answers in relation to the information that is provided on the worksheet. when asked on other occasions about his views on mathematical literacy, freddy has stressed the importance of interpreting and understanding the context: in maths literacy, we try by all means when we are teaching, we try to involve some real life situations so that learners can see the meaning of maths literacy. and we should actually – it does not actually force them to remember the rules and laws that they've learnt from their previous grades, such as grade nine. they just come up with their own ways of solving problems. it is interesting to note here that learners in the class do pick up on the fact that the answer they have got doesn’t connect well with ‘reality’ – freddy’s size does not indicate obesity – and offer an alternative answer. a need to make sense of the situation, to make the mathematics connect to the context, is clearly present within the classroom in spite of initial lack of attention to the mathematical aspects. general discussion several avenues are available in both of these excerpts to ‘add in’ and integrate the backgrounded agenda. the question in this paper addresses the way in which a more integrated approach might affect both the educator’s existing agenda and the backgrounded agenda. in charles’ case, the initial answer of equal numbers of girls and boys could have been followed up. charles’ assumption that someone in the class would have been able to understand that 12/24 was equivalent to 50% was undoubtedly true, but what were the alternatives? a lesson with 10 girls and 12 boys could have been the situation in a previous lesson, for example, and the relevant gender fractions of the class could have been compared. the question could have been asked, “today, 50% of the class is made up of girls. what percentage of yesterday’s class were girls?” this would have opened up avenues for comparing 12/24 with 10/22, for comparing 10/22 with 11/22, and for comparing 11/22 with 12/24. in terms of sense-making, this opens up opportunities to reason and understand that 10/22 must be less than 50%, to estimate that the answer is probably only a little less than 50%, and also opens up a motivation for why the concepts and procedures related to percentages might be useful in this context as a tool of comparison. the procedure can then be checked, revised and taught, as occurred within the lesson. such questions are not significantly different from those posed by charles, but they are guided by the twin aims of promoting mathematical learning and a sense of literacy in terms of making sense of the situation. similarly, in freddy’s case, if some of his alertness to the need to understand the meaning and consequences of numbers in the context of bmi had been focused on encouraging the class to either understand the structure of the formula, and/or to estimate the likely answer before calculating and to checking the answer given, sense-making would have been better supported. furthermore, some discussion about the structure and procedures associated with the bmi formula could have promoted a linkage to concepts relating to concentrations and density – and ratio and proportion, which are likely to feature in other mathematical literacy lessons. it would also have allowed for discussion about the historical development and use of the bmi concept and its implications. once again, these possible adjustments do not represent major changes to the flow of the lesson in both cases. more importantly, the broadening of the scope of questions, whilst allowing an alternative agenda to figure, works also to support the educator’s foregrounded agenda. conclusions in discussing the differences between literacy and language, gee (2001) emphasises that literacy involves more than a mastery of the language’s grammar – its associated rules, procedures and syntax. by way of parallel, mathematics education is replete with evidence suggesting that an emphasis on the rules, procedures and syntax is predominant in a range of countries (schoenfeld, 1985), and has led to widespread concerns about learners’ ability to use and apply mathematics to make sense of situations (smith, 2004; steen, 2001). much of the south african literature related to debates around relevance suggests that linking mathematics with other agendas is problematic. adler et al. (2000) present a number of vignettes with varying degrees of linking, and conclude that whilst integration remains desirable, the aims and conditions of mathematics teaching make it difficult, if not impossible to achieve: hamsa venkat 83 this all suggests that the promises of the new curriculum might be unfounded and that what might be desirable at the level of policy and advocacy might not be feasible at a practical or a theoretical level. (2000: 12) adler et al.’s suggestions for supporting teachers in acquiring the requisite skills for more relevant mathematics teaching include “more realistic time frames” and the increased use of “integrated mathematical tasks”. the mathematical literacy policy documents advocate the common use of such tasks, and the comments of both teachers indicates that a less congested curriculum has provided them with more time and space than was available in their mathematics teaching. in this paper i have attempted to use the empirical data to demonstrate that the mathematics and literacy agendas are not incompatible. i would argue that the excerpts point to a need for improved awareness of the different ways of thinking about the agendas that underlie mathematical literacy – and mathematics – teaching. this awareness can then be followed up with practical solutions for extending teaching repertoires to integrate mathematics and literacy agendas in ways that support rather than detract from either agenda in isolation. the evidence presented here further suggests that the implementation of mathematical literacy in south africa may provide fertile ground for locating a mathematical agenda within the broader notions of sense-making and interpreting the world that constitute the notions of literacy considered in this article. references adler, j., pournara, c. & graven, m. (2000). integration within and across mathematics. pythagoras, 53, 2-13. amesa. (2003). amesa submission to the department of education on the national curriculum statement grades 10-12 (schools) and in particular on the mathematics and mathematical literacy subject statements. diepriver: amesa. barnes, h. (2005). the theory of rme as a theoretical framework for teaching low attainers in mathematics. pythagoras, 61, 42-57. davis, z. (2003). free associations: from canned laughter to relevant mathematics to the tamagochi. pythagoras, 58, 2-6. department of education. (2003a). national curriculum statement grades 10-12 (general): mathematical literacy. pretoria: government printer. department of education. (2003b). national curriculum statement grades 10-12 (general): mathematics. pretoria: government printer. department of education. (2006). national curriculum statement grades 10-12, teacher guide, mathematical literacy. pretoria: government printer. department of education. (2007). national curriculum statement grades 10-12 (general): subject assessment guidelines, mathematical literacy. pretoria: government printer. gee, j. p. (2001). literacy, discourse, and linguistics: introduction and what is literacy? in e. cushman, e. r. kintgen, b. m. kroll & m. rose (eds.), literacy a critical sourcebook (pp 525-544). boston: bedford st martins. greeno, j. (1998). the situativity of knowing, learning and research. american psychologist, 55(1), 5-26. kilpatrick, j., swafford, j. & findell, b. (2001). adding it up: helping children learn mathematics. washington, dc: national academy press. mukhopadhyay, s. & greer, b. (2001). modeling with purpose: mathematics as a critical tool. in b. atweh, h. forgasz & b. nebres (eds.), sociocultural research on mathematics education: an international perspective. mahwah, new jersey: lawrence erlbaum associates. organisation for economic cooperation and development (oecd). (2003). the pisa 2003 assessment framework − mathematics, reading, science and problem-solving knowledge and skills. paris: oecd. schoenfeld, a.h. (1985). mathematical problem solving. new york: academic press. sethole, g. (2003). meaningful contexts and dead mock reality: experiences in black and white mathematics classrooms. paper presentation, conference of the learner’s perspective study, international research team, december 1-3. university of melbourne. smith, a. (2004). making mathematics count: the report of professor adrian smith's inquiry into post-14 mathematics education. london: department for education and science. mathematical literacy – mathematics and/or literacy: what is being sought? 84 steen, l.a. (2001). the case for quantitative literacy. in national council on education and the disciplines & l. a. steen (eds.), mathematics and democracy (pp 1-22). washington d.c.: the mathematical association of america. stein, m.k., smith, m.s., henningsen, m.a. & silver, e.a. (2000). implementing standardsbased instruction: a casebook for professional development. new york: teachers college press. walsh, c.e. (1991). literacy as praxis: a framework and an introduction. in c.e. walsh (ed.), literacy as praxis: culture, language and pedagogy (pp 1-22). new jersey: ablex publishing corporation. appendix 1. overhead transparency detailing body mass index (bmi) information (photocopied by freddy from the teacher guide, department of education: 2006, 25). watson 20 pythagoras 60, december, 2004, pp. 20-27 chanting as mathematical enculturation anne watson university of oxford email: anne.watson@educational-studies.oxford.ac.uk introducing assessment unison response is a major teaching strategy in mathematics throughout the world. the spoken word is the dominant source of mathematical stimulus, the dominant shaper of the mathematical environment in which school learners construct mathematical meaning. the teachers’ utterances are intended to be expert, and the creation of learner responses is intended to involve learning of some kind. this paper is intended to offer theoretical considerations, based on experience and observation of interactional routines in classrooms rather than on imagination or thought-experiment, which may lead towards a more developed theory of the roles and effectiveness of unison responses in mathematical learning. the word ‘unison’ is used to describe situations in which learners say the same thing at the same time; often this has rhythmic, semi-conscious qualities in which case it can be described as ‘chanting’. i use the word ‘chorus’ interchangeably with ‘unison’ although it also has greek roots relating to the provision of narrative during the action of a play. ‘routine’ is used to describe patterns of interaction which have become habitual through regular use; these may or may not involve unison elements. examples here are two contrasting examples of chorused response. example 1 the teacher showed learners how to multiply two binomial expressions in brackets to make a trinomial. as the teacher wrote the terms of the product she recited phrases loudly and slowly, leaving pauses where all learners were expected to insert words in unison. teacher: the first multiplied by the first gives the learners: first term teacher: and the first multiplied by the second gives the some learners: second term teacher: and the second multiplied by the first gives the some learners: third term teacher: and the last by the last gives you the learners: last term i cannot be sure that all learners were answering for the first and last terms, but it sounded like strong unison response. there was a noticeable difference between the loud, confident, bright sound of the first and last responses and the less secure, lower tones of the middle two responses. no other ways to think about this equivalence were offered in this lesson, no diagrams and no numerical examples. i was able to see the work learners did on their exercises after these interactions. for many learners the order in which terms were multiplied was not a problem: they knew what they needed to do but not necessarily how to do it. i had an impression that even many of those who did not join in the middle part of the chorus knew which pairs of terms to multiply to get the middle terms. reasons for getting the wrong answer were more likely to be errors in multiplying, especially where there were negative signs, or attempts to combine unlike terms in the product. this latter error made me wonder if they understood the word ‘term’ in algebra. confusion between long-term, short-term, school term and algebraic term is an example of difficulties with ambiguity in the mathematical register (pimm, 1987; tobias, 2003). i also found myself asking ‘what do you know about minus signs?’ and they would immediately say ‘two minuses make a plus’ but could not necessarily apply this. the exercise required several manipulative subprocedures which learners had apparently shown competence in performing in the past, but which were not being brought into play automatically where it was appropriate. example 2 learners were asked to count down from 100 in steps of 1 10 9 . to an observer it was clear that they were used to being asked to do such things, because they immediately became quiet, some of anne watson 21 them closed their eyes, others looked down at their desks as if what they needed was written there, others mouthed the first two answers to each other silently. after several seconds of thought the teacher signalled them to begin chanting answers in unison. why not try it yourself before you read on? as they proceeded some smiled and the speed increased until the rhythm they had created suddenly broke down. if you have tried this yourself you will know when and why this happens. this cycle of speeding up rhythmically and then having to slow down and rethink happened a few times (how many times?) and eventually the teacher stopped them (they were into negative numbers by this time), and asked them to report on where they found it difficult and why. some of their reports were about relationships within the structure of numbers; others were about the process of chanting in unison. the teacher then asked them to spend a few minutes working on fraction questions which arose for them about the exercise, and they did so, discussing their answers and calculations and observations with each other in small clusters. some of them were trying the same kind of exercise with different constant differences. in this kind of classroom, where learners can talk openly about their chanting experiences, it is common to hear remarks such as: i wasn’t sure whether i was saying things with the others or a split second afterwards i kept getting lost and used the others to get back in i couldn’t think quickly enough when things got hard, but it didn’t matter because i could use other people’s thinking i was amazed at how quickly i could do this but now i want to check whether we were right chanting has not guaranteed learning, and the particular answers are not important in themselves, but the experience has given learners something to explore. critics of over-reliance on chanting, while recognising the need to provide ways to memorise mathematics, point to a model of passive learning which is implied and a lack of understanding which would result. in the second example this is clearly not the case. although there are sequences of apparent passivity, the story is much more complex than a simple distinction between apparent passivity and activity. oral traditions mathematics is not a subject which has a strong oral tradition. it is passed on mainly through text, teachers and the experience of working on mathematical questions and problems at every level. teachers and textbook writers mediate knowledge through verbal, graphical and symbolic routines. at its most mundane it is learnt by repeated exercising in order to become fluent with techniques; the learner has to remember what to do. there are ways to support this recall by learning facts and instructions by rote, but it is a fallacy to thus equate rote-learning with an oral tradition. yet mathematics does, through the way it is taught, have moments of oral universality. in the english-speaking world, everyone has heard ‘times the top by the top and the bottom by the bottom’, or ‘the square on the hypotenuse … etc.’, but these fulfil the role of universally known pneumonics rather than cultural tradition. an oral tradition stimulates intellectual and social development: proverbs and riddles are used to develop reasoning power; stories about behaviour contain information about cultural norms and expectations and provide help to resolve dilemmas; word games 'strengthen' memory; rhymes encourage counting and the use of number words (reagan, 1995). memory of the stories permits their recall; remembered words can be brought into conscious use when appropriate to inform future action. “a purely oral tradition knows no division between recollecting and doing” (illich and saunders, 1988: 15). memorised phrases provide the raw material for thought and discussion. why bring ‘oral tradition’ into the discussion of mathematics classrooms? because there are features of the mathematical canon which could parallel all of the oral genres described above by reagan, but we cannot go far in that direction without the need for written symbolism rather than just the spoken word. what this alignment shows is that mathematics does have its own rich and complex culture, and while learners need to acquire props to give them access to the culture, learning is the experience of using the props to resolve problems rather than their acquisition. what i see in classrooms, however, is reliance on oral interactions for a variety of low level reasons which fail to engage learners with the most important part of oral enculturation: the use of shared oral experience to promote thought, chanting as mathematical enculturation 22 discussion, resolution and the development of a mathematical culture. while so many classrooms in the world lack textbooks, it is naïve of mathematics educators to reject rote-learning, that is learning text so that it can be recited word-for-word, as an educational tool since at the very least it gives access to material for later consideration. reghi et al (1991) compared the effects of rotelearning among asian learners and conceptual learning among australian learners, testing both groups for their understanding. their results revealed that a combination of rote-learning and thoughtful repetition of the learnt words, used by the asian learners in their study, leads to deeper understanding than ‘reading for understanding’ alone. it seems that several oral repetitions of the text, each with a different focus on the content, help develop deep, structural knowledge rather than just mechanistic recall (marton and booth, 1997). what is puzzling here is why rote-learning should be done at all by literate learners who can always have access to written text. it seems to be seen by learners to resolve a perceived need to acquire surface knowledge which can then be used easily in tests, brought into use in new contexts where needed, and worked on in future to develop deeper knowledge. the relationship between repetition and rotelearning is therefore two-way. repetition of a thoughtful, attentive kind is required if rotelearning is to lead to deep learning, but repetition of words, possibly at a more superficial level, is also the mechanism for learning by rote. recall of the songs and playground chants of childhood is easy, and their acquisition was often effortlessly achieved by listening to and joining in with others. repetition, particularly when there is no access to books, is therefore an important cultural requirement for learning, both for acquisition of the words and for returning to them again and again to develop deeper understanding. in the passage above i am guilty of not distinguishing between rote-learning and chanting. the chanting i have heard in classrooms is sometimes a mechanism for rote-learning, but not always, as i shall show. similarly, it is not a mechanism which ensures understanding. unison responses in a variety of classrooms in this section i am going to describe many examples of the use of unison responses in classrooms and examine the learning opportunities they offer learners. several of these occurred during enquiry into interactional routines in south african classrooms. as an outsider i observed nineteen secondary mathematics lessons, containing between 40 and 50 learners each, taught by thirteen teachers in four urban schools. classes ranged from year 8 to year 12. two of the schools were under-resourced and had some teachers who voluntarily described themselves as having very low morale. one of the schools had a more stable history, with learners from a wide range of social backgrounds and well-qualified teachers. these three were township schools. the fourth school was formerly european mono-cultural and was well-resourced. the sample was completely opportunistic. in addition, the willingness of teachers to have me in their classrooms suggests that they were confident practitioners showing me their best practice. i had field notes of all the interactional routines which included the expectation of whole-class unison response used in these lessons. i analysed them to identify possible purposes, to conjecture about the learning opportunities they offered to learners, and to imagine ways in which they might fail in their purpose. my claim is not that these were the actual purposes, the actual learning or the actual failures. rather it is to discuss the issues which were raised for an informed observer, with professional and academic perspectives, which might then frame future research. the possible purposes i identified were: recall of words or concepts instructions reasoning routines commentary participation these categories are not exclusive, and there may be other ways to categorise such routines, but they did encompass everything i saw. i see them as tools for further thought, rather than a definitive frame for future work. recall of words or concepts teachers use unison routines both to generate, and to remind learners of mathematical facts. for example, the teacher says: a square must have four equal the sentence must end with a noun, so choice is limited and most learners say "sides" immediately. the emphasis seems to be on the syntactic structure rather than the four-ness or equality involved in squares. compare this to another version in which the teacher says: a square must have and learners say: anne watson 23 four equal sides response in the second case was less united and several learners were silent as they were not being prompted to recite a well-known phrase. possibly some learners were silent because they did not know which of several possible answers would be deemed correct. more likely they were waiting for clues or cues about what to say. the second response requires some knowledge of concepts; the first requires a limited level of classroom norms and linguistic cadence, knowledge of words would be enough. another familiar example is the recitation of “vertically opposite angles are equal”. it is hard to understand ‘vertically’ in this context, which relates to ‘vertex’. without this latin root or some other contact with a latinate language, how can ‘vertically’ make sense for learners? the normal meanings of ‘opposite’, such as ‘living opposite me’ or ‘the opposite team’, even ‘opposite angles of a quadrilateral’, do not quite contain the meaning of ‘opposite’ in this phrase. we ought to assume that ‘angle’ is understood, although if learners believe, as many do, that the size of an angle is related to the length of the lines delineating it, this will be a confusing statement. finally the word ‘equal’ does not give you an instruction, it is a statement of a relationship. an alternative version of this could be the instruction ‘equate vertically opposite angles’. what is a learner to do with this phrase? it does not tell you what to do and is full of confusing words. taking time to develop this understanding of the phrase is important, as zevenbergen (2000) and tobias (2003) suggest, but even knowledge of what it means and how it can be used is no good if you cannot find the vertically opposite angles in a diagram because they don’t look like the ones in the teacher’s diagram. knowing the words is only a tiny part of understanding what the words are about; understanding includes knowing how and when to use the property. the first kind of routine was observed frequently, but seems unrelated to conceptual understanding. the second kind was seen less frequently but seems to invite more engagement with meaning as well as words. instructions teachers use unison response in gap-filling routines to instil, through repetition in several situations throughout their school experience, rules about how to do mathematics. for example, a teacher says: what we do to one side we do to the and the learners finish the phrase by saying: other in this case the whole phrase has not been said by the learners, they have merely finished a linguistic structure with an obvious response. in another example, the teacher says: what do we do with two minuses? and learners answer: two minuses make a plus this is intended to provide an inner monologue to tell learners what to do, but it is well-known that such a monologue is often applied in inappropriate circumstances, such as "-2-3 = +5". in the algebra example at the start of this paper, we saw that learners do not necessarily recall it when appropriate. these phrases are merely tools, and, without explicit work on how they can be used in contexts, they can have an effect on mathematics like random hammering. imagine a teacher then asking learners to create questions in which the phrase might be useful, and offering examples of where other learners have misapplied the rule. in this case the remembered rule becomes attached to an exploration of where it is, and where it is not, useful. memorised phrases, as in profound oral traditions, are used as the raw material for contemplation and deepening understanding, but in many mathematics lessons such phrases can be seen as a meaningless instruction or as an endpoint in themselves. reasoning routines unison responses can appear to engage learners in sequences of mathematical reasoning. for example: teacher says: if it isn't positive it must be learners respond: negative the teacher models how she hopes learners will think, but this line of reasoning is mathematically oversimplistic; it ignores zero. chorused phrases often have a symmetry which is interrupted by special cases! to use this reasoning routine the pupil needs to know when she should consider signs and hence bring the routine into play. as with instructional routines, learners need to know when and how to use them. also, in this case, the learners have only supplied the missing linguistic opposite and may not connect this gap-filling activity with mathematical meaning at all. a more complex example of this was demonstrated with a class which was working on a problem involving angles in a circle. on the board was a ‘toolkit’ of diagrams illustrating angles in chanting as mathematical enculturation 24 the same segment (called ‘the butterfly’), a cyclic quadrilateral, and the alternate segment theorem. the teacher points to the diagram given with the problem and says: does this look like a butterfly? learners say: no does this have any external angles? learners say: no so it must be about cyclic quadrilateral learners say: yes again the teacher is offering a model for reasoning, a way to go about answering examination questions. there is no guarantee that learners are making the active choices which the teacher intends through just responding with "no" when it seems sensible to do so, as in ritual speech (pimm, 1987: 73); learners recognise a linguistic routine and give a well-known response. just as in the previous example it is interesting to look for what is missing here. there is no ‘angle in a semicircle’ tool offered, nor the possibility that the circle is a red herring and the question might be resolved using angle properties of general quadrilaterals alone. however, as a model of gathering what you know and relating it to what you are given and what you want (mason et al, 1982) it provides a general heuristic which the teacher uses frequently and hopes the learners will adopt. thus it could be providing scaffolding for learners to become more able to engage in a mathematical reasoning routine. what seems just as likely, however, is that learners are merely picking up clues from the teacher’s intonation and the knowledge that he will go through all the unhelpful possibilities available, one at a time, coming to the correct one last. this, of course, is not a useful reasoning routine because it depends on knowing the outcome! commentary teachers often ask learners to join in a commentary as they work through an example on the board. learners who provide their own commentary while reading through worked examples appear to learn with more understanding than those who merely look for a template or pattern into which they can fit different numbers (anthony, 1994), so the aim of developing a commentary is well-founded. (it also, incidentally, coincides with the use of ‘chorus’ in greek plays to mean the use of narrative to elaborate on the action.) but in many cases i saw of this, learners were only invited to comment on mundane steps. in one case, the teacher was demonstrating algebraic substitution and did not invite any participation until she got to "seven from four..." to which learners responded “you can’t”. this was a minor aspect of the whole piece of work (and one could question whether “you can’t” is a correct or helpful mathematical answer at secondary level). another example concerned the calculation of the area of a shape made up of rectangles; learners were not invited to respond until the teacher got to “three times four is …?” when they were invited to complete the story by saying “twelve”. i saw no examples of learners being invited to join in the commentaries on central conceptual or reasoning aspects of a worked example in unison, and it is hard to imagine how this would work, since learners make their own sense of what they see and there is no reason to suppose that unison response would be at all possible. nevertheless, where there were learnt routines or arithmetical aspects to the work, learners sometimes joined in together to say what was being written or performed. participation some of the examples given above require so little from learners in terms of mathematical engagement that i began to wonder if the main function is social participation rather than enculturation into mathematical activity. seen as social participation some of the examples above make more sense. with quite an advanced class, finding the coordinates of a point of intersection, a teacher says: this bracket gives us the x-coordinate of one of the points and this bracket gives us the xcoordinate of the and the learners respond: other the learners are not involved in the reasoning; they merely follow familiar word rhythms to supply the end of a sentence, as they have done in every mathematics lesson. they would rarely have to reproduce this bit of reasoning for themselves as it is specific to a very small subset of problems they may meet. the likelihood that when they do need it they will think to use it as a recalled routine is very small, yet the unison response gives the impression of mathematical engagement and provides the teacher with feedback from which she can draw confidence. unison response has, as i have tried to show, tremendous potential to engage learners in cognitive activity, but seems often to be reduced to mundane or social uses and even its potential for supporting the learning of useful text by rote is underexploited. anne watson 25 chanting for engagement and awareness the research above is reported more conventionally in watson (2002), but my thinking has developed since that publication. awareness of subtle differences in oral routines led me to ‘collect’ more examples. in particular, as in example 2 at the start of the paper, i looked for uses of chanting which went beyond mundane aspects of mathematics and which provided the learners with some raw material for future work. by looking at these i saw that unison responses have a strong social purpose in that they give learners a role to play in the mathematics lesson, but they go beyond this. they also enculturate learners into relating words and mathematical structures and into participating in other mathematical practices, rather than just classroom practices. they also use voice as a mechanism for generating physical engagement in mathematics, sometimes supplanting mental engagement. chanting to relate words and mathematical structure the example of counting-back given at the start of the paper uses a relationship between number structure and rhythm to generate knowledge of, and interest in, fractions. an easier and much more common example of this is the chanting of the eleven times table. because of the rhythm, we could all say that ‘tum elevens are tumty-tum; plonk elevens are plonky-plonk’ even though these make no mathematical sense, but we could also say that ‘twelve elevens are twelvty-twelve’ and indeed they are. the rhythm mimics the numerical structure and the relationship between ‘twelvtytwelve’ and ‘132’ is worthy of mathematical exploration. if all that happens is that the teacher stops the chanting at ‘nine elevens’ a valuable opportunity to learn about number has been lost. in a video, dave hewitt teaches a lesson on ‘doing and undoing’ as a metaphor for solving linear equations (open university, 1992). at one stage he sets up a unison response similar to this: teacher: if i added 5, i subtract learners: 5 teacher: if i added 19.2, i subtract learners: 19.2 teacher: if i added c, i subtract learners: c teacher: if i added ‘three-pi-squared’ i subtract learners: three-pi-squared teacher: if i added alpha, i subtract learners: alpha in this lesson there are other sequences in which subtraction as ‘undoing’ adding is emphasised; the point of this sequence is partly to enculturate learners into the range of possible mathematical objects they may encounter, so that their experience of constant terms in equations is not restricted to small integers. this kind of reminder is repeated in other lessons every now and then. the rhythm emphasises that they can all be treated in the same way as more familiar numbers and reminds learners of the oppositional pairing of addition and subtraction. the intense repetition over a short period of time may be more effective in instilling the oppositional pairing than irregular repetition spread thinly throughout several lessons. chanting for multiple sensory contact with mathematics in a lesson about expanding brackets, a jamaican teacher, ceva macpherson, wants learners to meet the expansion of (a+b)2 and to at least be aware that it is not a 2 + b2 , a common error. but the lesson achieves more than this; it provides an image which, for those who can work mentally with spatial images, gives access to what the expansion really is. the lesson starts with manipulation of shapes and description of the results in terms of conservation of area, building up to: thus, as long as learners know the area of rectangles, and are happy with conservation of area, and understand that it is area they are finding, they construct the algebraic representations for themselves. they share their ideas and the class reaches agreement about the equivalence. but the lesson is not left there. they then chant and dance, to the jamaican anthem ‘one love, one heart’, that ‘a plus b squared … is a squared plus b squared plus 2a times b’ rhythmically many many times. choice of an unfamiliar tune would give more to remember, not less. the choice of the anthem is a a b b chanting as mathematical enculturation 26 important as it is known and loved by everyone, it is a common heritage and binds them socially and culturally. at the end of this experience it is less likely that anyone will omit the final term after the a 2 + b2 because they would recall that the chant continued in a certain way. they may not recall what the missing item is, but recalling the chanting itself might trigger a return to the spatial memory to reconstruct it. chanting is used to develop memory of mathematics which has already been understood through other senses; visual, oral, musical, verbal and physical experience combine with the symbolic representations. in these examples of effective chanting, a move from listening, thinking and consciously repeating to the automatic, rhythmic, mantra-like states of chanting has been deliberately invoked in order to engage learners subconsciously with mathematics. chanting as talking mathematics raj varma’s lessons in the uk are carefullyconstructed journeys through complex structures of mathematics, with a focus on understanding. in a lesson on multiplicative reasoning he spends a great deal of time getting learners to distinguish between fractions which are greater than one and those which are less than one. it is important to take time over this because many learners will have images of fractions as parts of pizzas rather than numbers of any size, and he intends eventually that they should be able to predict whether multiplying by a particular fraction will increase or decrease a quantity. his lessons include unison gap-filling routines and the development of rhythms of words which relate to structure, such as i have already described. but he also gets whole classes of learners to repeat phrases which have no rhythm to them, such as: ‘if the numerator is greater than the denominator, the number is greater than 1: if the denominator is greater than the numerator, the number is less than 1’. learners cannot just open and close their mouths in the right place, the sentence is too complicated for that. guessing what the words will be is hard, there are too many distinctions to make. indeed the phrase draws attention to the fact that there are distinctions to be made between numerator and denominator, and between greater and less than 1. rhythm is not a big feature of this repeated phrase; to do it correctly you have to be engaged with meaning to a certain extent and get the distinctions right. this is not the kind of unison response which aims for subconscious automatisation of mathematical knowledge; it aims to promote conscious awareness of complexity. if nothing else, those engaging in this lesson might later recall that there is something you have to think about when multiplying by fractions, that fractions can be seen as ‘numbers’, that the words ‘numerator’ and ‘denominator’ are worth knowing better, and that lexical density is a feature of mathematics (halliday & martin, 1993). learners gain familiarity with the mathematical register by connecting technical terms loosely with partly understood concepts in the same way as someone who learns a song in a new language might later understand its meaning and, indeed, refine their knowledge of the words. in subsequent interactions in the classroom raj asks learners to repeat the phrases individually when offering solutions or methods, modelling the kind of thinking he hopes they will all do silently in future. thus he supports learners in linking what they said in unison to what they might think on their own. he claims, and his learners and colleagues corroborate this, that saying things in your own voice helps ‘fix’ them, and the complexity of what he asks them to repeat forces them to think more about what they are saying. in such lessons, unison response is used for particular purposes within a more complex teaching situation, not as a method of transmitting knowledge. it might be helpful to distinguish between chanting, which implies automatisation of speech so that subconscious relationships can be made, and unison response in general, which includes chanting but also allows the highly conscious activities generated by raj. conclusion and a research agenda pimm (1987) says that there is … a deep-rooted belief on the part of many teachers that there is a power in someone saying things aloud, and therefore it is better for the learners to say the central part for themselves, rather than merely hear it expressed by the teacher (p. 54). in the examples above we have seen several manifestations of this belief, not all of which appear to benefit learning. a superficial understanding of the role of recitation in learning is demonstrated in some routines, separating unison response from its place within oral traditions by detaching it from meaning, and failing to make use of the learnt words for critical examination, discussion and revisiting in different contexts. furthermore, a sense that hearing one’s own voice anne watson 27 saying something makes it more likely to be retained can lead to emphasising retention of verbal sounds as the only way to accrue factual and procedural knowledge. in their desire to be understood, teachers can too easily identify chorus responses as feedback giving evidence of learning. more helpfully, they may believe that learned responses provide scaffolding for learners to develop appropriate inner monologues which are versions of the speech patterns offered by experts. brodie (1989) points out that this process may be especially prevalent where learners are not learning in their first language. only in raj varma’s lessons, when learners are invited to say things individually, is any attention paid to those who do not fulfil the response ‘correctly’; in other cases it seems to be assumed that being immersed in the chorus will eventually result, through enculturation, in full participation and correct recall. there is very little which has been written about the use of unison response in the learning of mathematics, and in the professional arena it is common to reject it as an approach. however, it does not make sense to dismiss unison response as a poor teaching strategy simply because it is strongly associated with underachieving educational settings.1 some learners do succeed in mathematics through being taught in typically unison classrooms. further analysis from linguistic, psychological, educational and sociocultural perspectives would be beneficial, as would exploration of the following questions: how can learners be helped to apply learnt phrases and ritual speech patterns meaningfully in their work? how can learners be helped to discriminate between appropriate and inappropriate applications of learnt phrases? how much use, and what kinds, of chorus response are purposeful? how have successful learners from classrooms where chorusing is a main teaching mode used their experience to achieve mathematical understanding? what ways can be found, particularly in underresourced 1 recently it has been widely promoted in the uk through official channels, including the distribution to all school mathematics departments of videos of raj varma’s teaching, and david hewitt’s teaching to teacher educators of all subjects. resourced classrooms, to relate learnt phrases to other representations of their mathematical meanings? references anthony, g., 1994, “the role of the worked example in learning mathematics” in jones, a. et al., eds, same papers, hamilton, nz: university of waikato brodie, k., 1989, “learning mathematics in a second language”, educational review 41(1), pp. 39-53 halliday, m. & martin, j., 1993, writing science: literacy and discursive power, london: falmer illich, i. & saunders, b., 1988, the alphabetization of the popular mind, london: marion boyars marton, f. & booth, s., 1997, learning and awareness, new jersey: lawrence erlbaum mason, j., burton l. & stacey k., 1982, thinking mathematically, london: addison wesley open university, 1992, em236 learning and teaching mathematics. milton keynes: open university pimm, d., 1987, speaking mathematically: communication in the classroom, london: routledge reagan, t., 1995, “language and skills or oracy in traditional african education”, journal of research and development in education 28(2), pp. 106-112 reghi, m., astilla, e. & watkins, d., 1991, “theasian-as-a-rote-learner stereotype: myth or reality?”, educational psychology 11(1), pp. 21-34 tobias, b., 2003, “do you speak mathematics?”, pythagoras 58, pp. 21-26 watson, a., 2002, “use of unison responses in mathematics classrooms”, in winter, j. and pope, s., eds, research in mathematics education volume 4: papers of the british society for research into learning mathematics, pp. 35-49, london: british society for research into learning mathematics zevenbergen, r., 2000, “‘cracking the code’ of mathematics classrooms: school success as a function of linguistic, social and cultural background”, in boaler, j., ed., multiple perspectives on mathematics teaching and learning, pp. 201-224, london: ablex microsoft word 64 front cover final.doc 62 pythagoras 64, december, 2006, pp. 62-69 mathematical literacy: myths, further inclusions and exclusions cyril julie school of science and mathematics education, university of the western cape cjulie@uwc.ac.za mathematical literacy as a new subject in the final three years (grades 10 to 12) of schooling in south africa is discussed. the discussion is driven by debates and deliberations around the introduction of mathematical literacy as they emerged in the public domain and constructs emanating from a diverse field of literature related to both mathematics and mathematical literacy. in particular the national curriculum statement grades 10 to 12: mathematical literacy is held in focus and somewhat mirrored against critical mathematical literacy. it is concluded that there should be guarding against mathematical literacy degenerating into a 21st century form of arithmetic. introduction mathematical literacy has been instituted as a specific subject in south africa for the last three years of schooling, the further education and training (fet) phase (grades10, 11 and 12). all learners not selecting mathematics will have to study mathematical literacy as a compulsory subject in the fet phase. this decision was reached after much discussion and debate on issues around the compulsory nature of forms of mathematical studies for all learners in the fet phase. some of these issues were differentiation in mathematical studies, the mathematics needed for entry into tertiary studies and the possibility that learners in schools in particularly low socioeconomic environments will be doomed to mathematical studies of some inferior nature. notwithstanding these debates and differences of opinion there is agreement that a mathematically literate populace is needed and schools should produce such graduates. the debates and deliberations referred to above were conducted in various forums mostly by academics from tertiary institutions. in this paper some of the constructs emanating from these debates and deliberations and related constructs emanating from the literature specifically linked to what the goals of mathematical literacy should be are discussed in relation to the national curriculum statement (grades 10-12 – general): mathematical literacy (department of education, 2006a). the constructs are: the teachability of mathematical literacy, the lack of a recreational component in mathematical literacy, mathematical literacy and an action component, the dilemma of contexts for mathematical literacy and the resilience of people’s qualitative improvised strategies to resolve quantitative dilemmas. a note on mathematical literacy mathematical literacy is variously defined. these definitions can be deemed as on a continuum. at the one end of the continuum, mathematical literacy is viewed as entry into mathematics and at the other end as a means to interact with mathematical installations in society. exemplary of “mathematical literacy is viewed as entry into mathematics” definition is that given by kilpatrick (2001). he introduces the term “mathematical proficiency, defining it in terms of five interwoven strands to be developed in concert.” and according to him “mathematical literacy…fits very well [with] mathematical proficiency” (kilpatrick, 2001: 106-107, italics in original). the five strands are: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning and productive reasoning. in his explanations of these strands, mathematical literacy as entry into mathematics comes through clearly. for example, adaptive reasoning is the “capacity for logical thought and for reflection on, explanation of, and justification of mathematical arguments” and productive disposition “includes the student’s habitual inclination to see mathematics as a sensible, useful, and worthwhile subject to be learned, coupled with a belief in the value of diligent work and in one’s own efficacy as a doer of mathematics” (kilpatrick, 2001: 107). the other end of the definition continuum is driven by skovsmose’s critical mathematics education paradigm (skovsmose, 1994). he coined the term mathemacy and explains it in an enquiring way as follows: cyril julie 63 as the notion of critical education developed, literacy was seen as a competency which enabled students to see and reinterpret part of (their) reality and to react to this reality…the essential question for critical mathematics education, then, is whether or not it is possible to develop a competency, mathemacy, which has a potential similar to that of literacy and which may help students to reinterpret their reality and to pursue a different reality? (skovsmose (in collaboration with nielsen and powell), 1995: 7-8). critical mathematical literacy has a focus on citizenship and an interest in mathematical models structuring social life. there is a specific interest in the critique of such models (see for example julie, 1998a and gellert, 2004). skovsmose (2004) provides a kind of technique to identify the “mathematics in action” regulating societal matters. julie (2006) argues that in essence this “mathematics in action” is actually “mathematics in and for action”. in this paper the definition of mathematical literacy as in the region of the critical mathematical literacy guided the interrogation of the south african national curriculum statement (ncs) (grades 10-12 – general): mathematical literacy (department of education, 2006a) and some issues surrounding the implementation of this subject. of the principles underpinning the ncs (grades 10-12 – general): mathematical literacy, “social transformation” and “human rights, inclusivity, environmental and social justice” (department of education, 2006a: 1) indicate some support for critical mathematical literacy. this is further buttressed by the definition of mathematical literacy given as: mathematical literacy is a subject driven by life-related applications of mathematics. it enables learners to develop the ability and confidence to think numerically and spatially in order to interpret and critically analyse everyday situations and to solve problems. (department of education, 2006a: 9) however, as will be discussed below, there is still some way to go for the realisation of a critical mathematical literacy education at least at the level of the intended curriculum. notwithstanding this intention, any curriculum has to be taught in schools and hence this issue is discussed in relation to the teaching of mathematical literacy in the next section. the teachability of mathematical literacy it is a widely held opinion that mathematical literacy is more difficult to teach than the “normal” school-going mathematics. at a meeting, organised by higher education south africa (hesa), to discuss the mathematics and mathematical literacy in the revised curriculum statement for the further education and training band this issue was raised. one of the participants related how this difficulty manifested itself when a highly adept and well-versed teacher experienced this difficulty when teaching mathematical literacy to a group of preservice teachers. the teacher also referred to similar experiences when discussing mathematical literacy at a regional meeting of the association for mathematics education of south africa. although no firm evidence about the difficulty of teaching of mathematical literacy was offered at the two meetings, the general noddings and “oohs” and “aahs” of the audience gave the impression of a general agreement that mathematical literacy is indeed difficult to teach. is mathematical literacy really that difficult to teach? is it teachable at all? in order to understand this it is necessary to delve into notions of teaching and the development of teachable products. teaching is a deliberative act in which a knowledgeable person relays her/his understanding of a body of declared and established knowledge to someone not as yet knowledgeable about this body of declared and established knowledge.1 this relaying includes revealing and providing opportunities for engagement with the domain’s navigational technologies and the necessary skills for the construction of artifacts deemed belonging to the domain. it is a social activity having a distinctly historically-developed-and-changing identity in distinct settings. teaching a group of grade 1 learners, although there are overlaps, has a different identity from that of teaching a group of grade 9 learners. within this process of relaying, values are implicitly relayed and appropriated. there are, amongst others, the values of mathematics per se i.e. of “the nature of mathematics”, the values the teacher holds with respect to the goals of teaching and the values the learners bring to and appropriate in the classroom. the implicit relaying of values is in many 1 the presentation of teaching as “relay” does not exclude teaching underpinned and informed by one or other form of constructivism as a theory of learning. in fact, most constructivist-inspired teaching has elements of the expository teaching and vice versa. exposition is more implicit in constructivist-inspired teaching and the other way round in expository. the difference thus is one of degree and not of kind. mathematical literacy: myths, further inclusions and exclusions 64 instances in the form of “anecdotal mentioning” in statements such as “mathematics has always played a role in war and the money spent on it can be better used for feeding the poor of the world” when teaching quadratic theory and doing an applications-like problem on the distance traveled by a projectile. or through “admonishing statements” such as “can you see that if we just follow the applications of the definitions and the theorems we logically come to the desired conclusion. in a lot of issues in real life, it is also like this – do the things logically from the start and a defensible conclusion will follow” after, say, demonstrating the method for dealing with a geometry rider. both these utterances are not what are being explicitly taught. they emerge if and when the teacher finds it opportune to make these utterances and so brings to bear his/her value positions on the situation at hand – expenditure on warfare versus expenditure on feeding the poor and destitute or the power of the mathematical-like rule-following in a closed system as a desirable form of logical reasoning. teaching is decidedly different from what happens when a member of a knowledge-making community informs and conveys her/his frontier knowledge products to the affiliated knowledgemaking community. in this instance he/she is not relaying declared and established knowledge. rather, she/he is bringing to the affiliated community her/his knowledge constructs at the frontiers of a field in a discipline. this knowledge is not-yet declared, established and vetted by the community. frontier knowledge is not relayed but rather co-explored by a community of practitioners in order to make it part of the body of declared and established knowledge of the domain. it is not a given that after declaration and establishment a body of knowledge is immediately teachable. to become teachable, especially as school knowledge, it has to and does go through a series of transpositions from its declarative and established forms (artique, 1994: 28). these are didactical transpositions and it is through mathematical didactical analysis that the subject matter is made amenable for school teaching. mathematical literacy, in its variety of presented definitions, as a domain of declared and established knowledge needs to proceed through a similar set of didactical analyses to make it teachable. if now the claim is made that it is more difficult to teach mathematical literacy than other school-going mathematics then it seems wise to search for this “difficulty” at the level of the processes of didactical analysis of mathematical literacy from its definitions to school-teachable knowledge. the difficulty, i contend, has to do with some inadequacies of the didactical analysis process. these inadequacies can have various sources. two of the most important sources are: (1) lack of epistemic dependence on experts wittmann (1998) defines the core task of mathematics education, as a field of study, as the development of learning resources which lead to productive and meaningful learning. he argues that mathematics educators play a crucial role in this regard and that one of their tasks is the didactical analysis of subject matter to design school learning resources. this process, wittman insists, is not the primary task of teachers because they do not have the time to do this. this “not having the time” to engage in the act of didactical analysis is also the case with “subject advisors” within departments of education. wittman’s proposal is essentially a call for the epistemic dependence on experts which hardwig (1985) presents as having “…good reasons for believing a proposition if one has good reasons to believe that others have good reasons [and] that…because the layperson is the epistemic inferior of the expert (in matters which the expert is expert), rationality sometimes consists in refusing to think for oneself.” didactical analysis by mathematics educators as experts should render an elementarised and integrity-preserving version of the body of knowledge which is in principle teachable for the level for which this body of knowledge is elementarised. the experts elementarise this body of knowledge through theorisation and thought-experimentation which leads to hypothetical learning trajectories. this does not guarantee that the hypothesised trajectory will work in the real classroom situation. this leads to the second inadequacy in the process followed with respect to mathematical literacy. (2) lack of experimentation with hypothetical teaching trajectories there is emerging realisation that teaching products should be designed and developed along the same lines as engineering products are developed (burkhardt and schoenfeld, 2003). such a process, it is argued, will increase the likelihood that the distance between the intended and implemented is decreased. teachable products should thus be obtained through a process of research and development with real classrooms as phenomenal domains and sites of experimental implementation. this process has not, as yet, been followed for mathematical literacy in south cyril julie 65 africa. it appears that the expectation is that teacher inservice providers will develop learning resources for courses that will be offered to teachers. these materials should then form the basis for classroom learning resources. the problem with such an approach is that it does not provide an empirical base of learner handling of subject matter. there is emerging acceptance that learner handling of subject matter should form an integral part of the desirable knowledge that teachers should be exposed to. the integration of learner handling of school-going mathematics and the school-going mathematical content is increasingly gaining acceptance as a particular kind of mathematics – mathematical knowledge for teaching (ball and bass, 2000). design research conducted with real classrooms as phenomenal domains will provide such knowledge of learner behaviour and such knowledge is virtually nonexistent for school-going mathematical literacy. it should not now be concluded that a research and development approach will render teacher-proof teaching products. teachers always amend and adapt products to fit the contextual demands they are faced with. however, having exemplary teaching products informed by the reality of classrooms will provide teachers with a more considered base from which to make their decisions, amendments and adaptations. the seeming difficulty of teaching mathematical literacy is thus more a system-level problem than a teaching problem. despite the issue of the teachability of mathematical literacy, there are others that need highlighting in order for the ncs (grades 10-12 –general): mathematical literacy to move towards the expressed intentions. lack of a recreational component in mathematical literacy the second chapter of the ncs (grades 10-12 – general): mathematical literacy (department of education, 2006a) is a discussion on the purpose, scope, educational and career links and the learning outcomes. conspicuous in this discussion is the absence of reference to mathematics as recreation. recreational mathematics is a fairly well-established activity. it is commonplace to find books and articles of a mathematical recreational nature. the many books by, for example, gardner (1981) and others attest to this. if one takes the view of a well-rounded and educated person also being someone who participates in some form of recreational activity then recreation through mathematics cannot per se be excluded from the arsenal of recreational pursuits such a person can engage in. one definition of mathematical literacy is provided by the program for international student assessment (pisa) (oecd, 2000). this definition is in terms of the desirable outcomes and is stated as “to make well-founded judgments and to use and engage with mathematics in ways that meet the needs of that individual’s life as a constructive, concerned and reflective citizen” (oecd, 2000: 21). this definition privileges citizenship and awareness-building of a participatory democracy. even if it can be argued that remotely “needs” is inclusive and that anything can fill this “needs” slot, it is unlikely that “needs” would be beyond what will appear in mathematical literacy examinations. thus “recreational” mathematics will be suppressed or even disallowed and excluded. lack of an action component in the discussion on the purpose of mathematical literacy attention is given to citizenship. it is stated: to be a participating citizen in a developing democracy, it is essential that the adolescent and adult have acquired a critical stance with regard to mathematical arguments presented in the media and other platforms. the concerned citizen needs to be aware that statistics can often be used to support opposing arguments, for example, for or against the use of an ecologically sensitive stretch of land for mining purposes. in the information age, the power of numbers and mathematical ways of thinking often shape policy. unless citizens appreciate this, they will not be in a position to use their vote appropriately. (department of education, 2006a: 10) this points in the direction of the exclusion of an action component. mathematical literacy is embedded in the applications of mathematics and mathematical modelling. mathematical modelling and the applications of mathematics are essentially about the development of some mathematical technology to realise a certain objective. this objective is an action objective. for example, if a mathematical model is developed to predict the supply and demand of teachers then the action objective is to plan in such a way for the education of teachers so that the demand will be met. although it can be argued that the ncs (grades mathematical literacy: myths, further inclusions and exclusions 66 10-12 – general): mathematical literacy does include an action component, it is rather a removed futuristic one. so, for instance, dealing with investments and their growth has limited, if any, immediate action value for the majority of learners who will be following the mathematical literacy curriculum. there are, however, instances of engagement with issues in mathematical literacy where learners should be sensitised to the possibilities of action due to the outcome of the analysis of the situation by mathematical means. in his inaugural address as professor of applied mathematics at the university of cape town, ellis (1976: 17) highlights this notion of action and asserts that if the results from a mathematical analysis illuminate profound injustice of a politically-instituted practice then non-action is meaningless. after analysing the growth of pass law arrests and finding its exponential growth pattern, he asserts …the only true basis of freedom is a realistic vision of the alternative possibilities before us. mathematical studies can sometimes help us in understanding what alternative possibilities are. but such an understanding is quite valueless unless it affects our actions. an understanding of the courses of any social wrong, which does not lead to some corrective action to right that wrong, is meaningless. (ellis, 1976: 17) thus through mathematical literacy it is not only an analytical competence that needs to be developed but also an action competence. this competence is viewed as making decisions individually or collectively and embarking on action, based on the decisions, to address the issue at hand (jensen and schnack, 1997: 168-169). the dilemma of contexts in mathematical literacy the ncs (grades 10-12 – general): mathematical literacy, as is the case for other definitions of mathematical literacy, gives much attention to the desirable contexts to be used. which contexts to use in mathematical literacy is a complex issue. in her statement that “organizing a lawn mowing business in math class is neither real lawn practice nor real school practice” (lave, 1988: 20), lave brings forth the dilemma of context and this is repeated by muller when he states that contextually-driven mathematics “is neither ‘real’ mathematics nor recognizably ‘real life’.” (2000: 67). muller (2000), drawing heavily on dowling’s (1998) work, essentially argues against the use of contexts emanating from the life-world of learners from marginalised and poverty-infested backgrounds since according to him this will not provide such learners with the necessary “article of universally recognized cultural capital such as a school diploma [which] confers symbolic power on the holder” (muller, 2000: 62). the mathematical knowledge, preferred by muller, that should be distributed for the attainment of such a “school diploma” is that which is an elementarised version of pure mathematics. a careful study of the ncs (grades 10-12 – general): mathematical literacy (teacher guide) (department of education, 2006b) indicates that this is subtly the case for mathematical literacy at the suggested implemented curriculum level. it is not only the ‘unreality’ of contexts that is an issue. pollak (1969) drew attention to context problem types and identified “five problem types that are normally included under the rubric ‘application of mathematics’.” he further “discards three of these [contexts problem types] as inappropriate since they do not take into account the complexity of the world outside of mathematics or their reality is fairly unreal” (in julie, 1998b: 294). according to julie (1998) pollak suggest two desired problem contexts. the first is closed context problems – problems for which the model of a contextual situation is essentially provided and the learners explore the given mathematical model. the other is the open type context problem – “problems which emulate the…way applied mathematicians would receive a problem, in all its complexity, from a commissioning agency” (julie, 1998b: 294). busse and kaiser (2003) draw attention to the changes a context undergoes when encountered by a learner. they distinguish between objective figurative context – “the real description of the real scenario” and the subjective figurative context – “the interpretation of the objective figurative contexts” (busse and kaiser, 2003: 4). their proposition is that it should not be unquestionably accepted that there is a one-to-one mapping between the contexts as presented in a task and contexts as interpreted by learners. the issue of learning transfer and the use of context is discussed by boaler (1993) and she suggests “that contexts may be useful in relation transfer even though contexts as generally used are not useful” (boaler, 1993: 13, italics in original). skovsmose (1998) addresses the question of using contexts emanating from the immediate environment of the learner. he argues that in their cyril julie 67 daily trade practices people do not view the work they are doing as essentially mathematical. they are simply exercising their skills and competences dealing with the job at hand. thus, skovsmose argues, a tailor is doing tailoring and not the mathematics of mapping a two-dimensional cloth in the plane to a three-dimensional body in space. true there is nothing wrong with this argument but it cannot serve as reason for prohibiting someone else from addressing such an issue from a mathematical angle. in fact, there are many instances, as skovsmose admits in his tailoring example, where the mathematical description of some human activity served as inspiration for the development of some form of mathematics. in addition to the argument about the nonmathematical attention people involved in activities and practices, other than visibly mathematical work, give to their work, skovsmose also advances the argument that learners are generally not interested in dealing with activities from their backgrounds. he proposes that contexts dealing with possible futures of learners should be foregrounded as an alternative. using piloting as an example to be foregrounded, skovsmose realises the problematic he is confronted with. it is this: his father was a tailor and he would have been averse to deal with the mathematics of tailoring in his school mathematics classes. but he was (and presumably still is) not interested in piloting. the crux of the question of desirable contexts to be used in mathematical literacy is a major dilemma facing mass schooling and education, and can be captured as “what are the contexts and situations within which school-learning activities should be embedded so that they will appeal to all learners to be constructively engaged in the learning to be fostered by these contextuallyembedded activities?” the ncs (grades 10-12 – general): mathematical literacy (teacher guide) (department of education, 2006b) indicates that not much attention has been given to this issue. in most instances contexts emanated from curriculum and learning resource designers’ perceptions on what preferred contexts should be. the resilience of the mixture of qualitative and quantitative procedures to resolve everyday quantitative dilemmas a feature standing out in mathematical literacy for school-going is that school-goers should develop competence to use mathematics in their daily lives. the manifestation of this desire is that issues and situations such as shopping, baking, cooking and everyday financial calculations are used as contexts for mathematical treatment. whilst not arguing against this there is not clarity for which issues in their daily lives ordinary people actually need mathematical calculations and procedures. i argue that those situations in which people use mathematical ways in some instinctive way should not be colonised for classroom use and teaching, no matter how much mathematical elegance might be extracted from such activities. meeting a self-employed, self-appointed and nonunionised porter at the airport i got involved in a discussion with her about her work and the amount of money she earns. the conversation went as follows (in the primary language of the porter and myself, of course). cj: hoeveel maak jy nou op ’n dag? p: as’t goed gaan da ga ek hystoe met about ’n honerd, one fifty ’n dag. dit depend oek of ek dollars of ponne kry, nie rante nie. die americans en die anne mense gie altyd dollars ma die engelse mense gie ponne, daai dik coins. cj: hoe exchange jy dit na rante? p: ek ga sommer hie na absa da binne en dan change hulle’t vi my. cj: hoe wiet djy dat hulle gie vir jou die regte amount? p: hulle sal my nie rob nie. hulle ken al vi my. maar ek kyk oek na rie bod. nou is ’n dollar ampe tien rand. as ek drie dollars het dan moet hulle vi my ampe dêtig rant gie. die bank moet mos altyd iets kry vi hulle expenses. soe hou ek trek. cj: jou kinners oppie skool. wat sal djy laaik hulle moet leer van dié goed? p: niee, hulle moet lee wat die teachers sê hulle moet lee. in summary she told me that she earned about one hundred to one hundred and fifty rand per day. she gets her “tips” from tourists in dollars or pounds and reads on the display board at the bank at the airport what the exchange rates are. she then does an approximation and works in some additive and qualitative way by anchoring on some whole number. this is not a strange way of working. people, regardless of levels of education, do such qualitative calculations all the time and my contention is that no amount of schooling will change this. as already stated, to change ordinary commonsense ways to mathematical ways and hope that people will apply such in their day-to-day mathematical-like activity is folly. people do not mathematical literacy: myths, further inclusions and exclusions 68 walk around with abstract mathematical models in their heads. when confronted with a mathematicallike dilemma they resort to qualitative ways of dealing with these dilemmas and utilise all sorts of contextually-driven procedures to resolve these dilemmas. the social history of mathematics indicates clearly that mathematics has both a qualitative and quantitative dimension with the quantitative one having been accorded the status of pure mathematics. kaplan (1999) in his social history of zero masterfully illustrates the acceptance of zero qualitatively before its acceptance as a quantity. this gives credence to the rallying assertion of realistic mathematics education (gravemeijer, 1994) that “reality is the source and domain of application of mathematics”. this might now mean that the common-sense qualitative ways that people use to resolve their day-to-day quantitative-like dilemmas should be incorporated in mathematical literacy. this will require that learners be exposed to many such common-sense qualitative ways of resolving dayto-day quantitative dilemmas so that generalisable patterns across contexts can be observed. this moves us into the domain of contexts as vehicles for the development of mathematical concepts, procedures, relationships between concepts and justifications for such relationships. this is antithetical to the objective of mathematical literacy defined in the negative as not working at the frontiers of mathematical knowledge production but using existing mathematical knowledge to read, analyse, build alternatives and act in the world. conclusion mathematical literacy is here to stay. globilisation, the availability of technology allowing for the access to knowledge in an unprecedented way and the world-wide thrust for regular testing of the state of mathematical literacy in a country through international comparative studies such as timms contribute favourably to the maintenance of mathematical literacy. however, as pointed out above, mathematical literacy is fraught with myths, omissions and unwarranted ambitions. mathematical literacy should, i contend, contribute towards the development of a “scientific temper” (russell, 2001) which he argues is important for the maintenance of a culture of nondomination and non-discrimination. a mathematical literacy not contributing to this will be futile and be nothing more than a 21st century version of arithmetic which, in south africa at least, was devised to maintain discrimination 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(2000). reclaiming knowledge: social theory, curriculum and education policy. london: routledge/falmer. organisation for economic co-operation and development. (2000). measuring student knowledge and skills. paris: oecd pollack, henry. (1969). how can we teach the applications of mathematics? educational studies in mathematics, 2, 393-403. russell, bertrand. (2001). the scientific outlook. london: routledge skovsmose, ole. (1994). toward a critical philosophy of mathematics education. dordrecht: kluwer academic publishers. skovsmose, ole (in collaboration with lene nielsen and arthur powell). (1995). critical mathematics education. research report r-952023, aalborg university: department of mathematics and computer science. skovsmose, ole. (1998). critical mathematics education. in c. alsina, j.m. alvarez, m. niss, a. perez, l. rico, and a. sfard (eds.), proccedings of the 8th international congress on mathematics education (pp. 413-425). seville: s.a.e.m thales. skovsmose, ole. (2004). mathematics in action: a challenge for social theorising. philosophy of mathematics education journal. retrieved may, 15, 2005 from http://www.ex.ac.uk/ ~pernest/pome18/contents.htm wittman, erich, c.h. (1998). mathematics education as a ‘design science’. in a. sierspinska and j. kilpatrick (eds.), mathematics education as a research domain: a search for identity. dordrecht: kluwer academic publishers. a habit of basing convictions upon evidence, and of giving to them only that degree or certainty which the evidence warrants, would, if it became general, cure most of the ills from which the world suffers. – bertrand russell article information authors: herbert b. khuzwayo1 sarah bansilal2 affiliations: 1department of mathematics, science and technology education, university of zululand, south africa2department of mathematics education, school of education, university of kwazulu-natal, south africa correspondence to: sarah bansilal postal address: 8 zeeman place, malvern 4093, south africa dates: received: 14 mar. 2012 accepted: 04 oct. 2012 published: 14 dec. 2012 how to cite this article: khuzwayo, h.b., & bansilal, s. (2012). granting learners an authentic voice in the mathematics classroom for the benefit of both the teacher and the learner. pythagoras, 33(2), art. #163, 7 pages. http://dx.doi.org/10.4102/ pythagoras.v33i2.163 copyright notice: © 2012. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. granting learners an authentic voice in the mathematics classroom for the benefit of both the teacher and the learner in this original research... open access • abstract • introduction • discussion    • providing opportunities for learners’ voices, even if they are dissenting    • building, not breaking    • learning to listen, listening to learn    • listening to understand    • listening to know: another perspective of mathematical knowledge for teaching • concluding remarks • acknowledgements    • competing interests    • authors’ contributions • references abstract top ↑ an important aspect of democratic education is the value it places on providing spaces for disagreement and argument as opportunities for learning. this is unlike an ‘occupation of the minds’ philosophy, which denies people the right to see alternatives. in this article we explore one aspect of this area of democratic education: the issue of providing opportunities for learners’ voices. we acknowledge the importance of this, even if the voices are dissenting; such dissent is important for teachers to learn more about the learners. we subsequently look at the kind of listening that a teacher can do in order to learn, and consider some cases from literature about teachers who struggle to listen and what happens when they learn to listen to their learners. finally, we argue that a perspective aligned with preparing learners to contribute to a democratic society advises a rethink of the construct of mathematical knowledge for teaching. by learning how to listen in a respectful manner and as part of a negotiation and co-evolution of shared understanding, teachers can deepen and shift their understanding of mathematics, their understanding of learners and their understanding of their own learning. introduction top ↑ on the critical nature of mathematics, skovsmose (2004) states that ‘mathematics education could mean empowerment, but also suppression. it could mean inclusion but also exclusion and discrimination’ (p. 2). although he does not endorse claims about the existence of intrinsic connections between mathematics and democratic values, or claims that mathematics education can serve anti-democratic interests, skovsmose does suggest that ‘mathematics must be reflected on and criticized in its variety of forms of action’ (p. 2). in this article, in addition to looking at the teachers’ role in empowering learners by creating spaces for their voices, we reflect on how this process can also empower teachers by providing opportunities to deepen their own mathematical knowledge for teaching.khuzwayo (1998) argues that a dominant feature in mathematics education in south africa during the apartheid years was the attention paid to the ‘occupation of our minds’ (p. 219). he states that redress in mathematics education in south africa will require serious commitment to ending this occupation. an occupation perspective denies people the opportunity to see alternatives. this is played out in the process of the development of a curriculum, the style of textbooks, the notion of success and the kind of teaching that is encouraged in classrooms. he states that ‘we have been blinded so that we are unable to see the alternatives in both our teaching and learning of mathematics’ (khuzwayo, 1998, p. 221). another characteristic of the occupation of the minds philosophy is its focus on obedience to authority and passiveness; it discourages curiosity and independence. fasheh (1996) emphasises that ‘ending the occupation of our minds is a personal task, its continuation depends solely on our acceptance of it. so is its termination’ (pp. 25–26). in this article, we look at one of the implications of such a commitment. we argue that a concern with development and democracy in mathematics education requires an interrogation of the kind of classroom that is needed to initiate a disruption of the occupation. of course such a disruption will require attention to many facets and in this article we focus on one issue: how teachers benefit when they grant learners a real voice in the classroom. in preparing learners to participate in and contribute to a democratic society, we will need to give them opportunities to experience such processes. it is increasingly important, particularly in south africa, for people to realise that democracy involves responsibility. in order to avoid a situation where ‘citizens and public officials can use democratic principles to destroy democracy’ (guttmann, 1987, p. 4), citizens need to see that democracy grants all people the right to voice their views. this aspect, where contestation is not only allowed but actively sought, is a valuable component of democracy. as guttman argues: the most distinctive feature of a democratic theory of education is that it makes a democratic virtue of our inevitable disagreement over educational problems in a way much more likely to increase our understanding of education and of each other … (p. 11) this supports the notion that contestation is an important part of democratic processes and must be valued and supported, not reduced or suppressed. thus a democratic theory of education presupposes a classroom where argument and disagreement are actively encouraged. particularly in mathematics, contestation offers opportunities to increase our understanding of the subject, our understanding of learners and our understanding of teachers. in a traditional mathematics context, however, a classroom is seen as the domain of the teacher: they set the agenda for the content, the opportunities for mediation of the content and the actual practices of mediation. how does this picture change when learners are granted a voice in the classroom − not a superficial voice, but an authentic voice that can influence what happens, why it happens and how it happens in the classroom? one implication of creating space for learners’ voices in a mathematics classroom is that as well as contributing to the learners’ experiences of democracy and their own learning, this process also contributes to the teacher’s mathematical knowledge for teaching. current conceptions of the notion of mathematical knowledge for teaching (ball, hill & rowan, 2005) suggest that interactions with learners contribute to developing one component of a teacher’s mathematical knowledge for teaching (knowledge of learners and content). this limits the critical role learners can play in the development of all domains of mathematical knowledge for teaching. it also undermines the importance of acknowledging and listening to multiple potential meanings of voice from various democratic classroom settings (where the voice of the learner is considered to be important). this appears to be in contrast to issues raised by studies such as the learner’s perspective study (see clarke, 2002). the findings of that study raised awareness of the role of learners as a legitimate voice to take into consideration when thinking about teachers’ knowledge. by taking seriously the learners’ voices we can enrich our attempts to develop a notion of mathematical knowledge for teaching by focusing on interacting with, observing and reflecting on the activity of the learners. mathematical knowledge for teaching has received much attention in recent times (adler & davis, 2006; adler, pournara, taylor, thorne & moletsane, 2009; ball & bass, 2000; ball, thames & phelps, 2008; ball et al., 2005; perressini, borko, romagnano, knuth & willis, 2004). ball et al. (2005) define four domains of mathematical knowledge for teaching: common content knowledge (mathematical knowledge of the school curriculum), specialised content knowledge (mathematical knowledge teachers make use of in teaching), knowledge of students and content (knowledge that combines knowledge of content and learners – in this domain, teachers need to be able to anticipate learner errors and common misconceptions, interpret learners’ incomplete thinking and predict what learners are likely to do with specific tasks and what they will find interesting or challenging), and knowledge of teaching and content (knowledge about instructional sequencing of particular content, and about salient examples for highlighting mathematical issues). in these definitions the role of the learners in contributing to the development of the knowledge is relegated to the sidelines, just in terms of predicting what they would do in a particular situation. let us look at the following statement more closely: [mathematical knowledge for teaching] allow[s] teachers to assess their students’ work, recognizing both the sources of student errors and their understanding of mathematics being taught. they also can appreciate and nurture the creative suggestions of talented students. (conference board of the mathematical sciences, 2001, p. 3) this statement suggests that teachers can only ‘appreciate’ and listen to some good suggestions of ‘talented’ students. it backgrounds the fact that interactions with learners can deepen teachers’ knowledge in the different domains. in this article we first consider the issue of providing opportunities for learners’ voices. we then acknowledge that, even if the voices are dissenting, such dissent is important for teachers to learn more about the learners. we subsequently look at the kind of listening that a teacher can do in order to learn more, and consider some cases from literature about teachers who struggle to listen and what happens when they learn to listen to their learners. finally, we argue that a potential benefit of preparing learners to contribute to a democratic society is the enhancement of the teacher’s mathematical knowledge for teaching. discussion top ↑ providing opportunities for learners’ voices, even if they are dissenting how do we create authentic opportunities for learners’ voices, so that we learn about the learners, what they value and what they want? vithal (1999) has shown that project work can be used as a site for learners to play a meaningful role in their own learning. she comments that: during project work, as democracy comes to have a place in a mathematics classroom, authority gets rearranged, and in turn, the kinds of authority influences the forms of democratic life possible within a classroom or school. (p. 29) in vithal’s (1999) study about project work in a primary mathematics classroom, it was found that the concerns driving the learners were different from those of the teacher: while the teacher wanted to raise more general societal issues of differences and inequalities between pupils’ background such as the parents’ different incomes, … the pupils focused on the specific issue of their school fees and their needs and concerns at school. (p. 29) although these differing drives may be a potential source of conflict, such situations present opportunities for teachers to learn more about the concerns that drive the learners. it is understandable that a learner and a teacher do not share the same perspective on classroom issues, but it may be surprising to hear learners voice their displeasure about certain practices, such as in assessment. many teachers are burdened by large teaching and marking loads, and consequently may sometimes feel that they are doing their learners a favour when they mark assessments and return them timeously, expecting the learners to appreciate the effort it took to get the marking done in time. however, in a study by bansilal, james and naidoo (2010), grade 9 learners expressed strong views about what they expected from their teachers in terms of responses to learners’ work. they expected the teacher to provide meaningful feedback to their work, to diagnose their errors and to show them how they could close the gap. they expected the teacher to provide feedback that would improve their understanding. clearly, learners have their own opinions about the teacher’s responsibilities towards the learners’ understanding. these findings confirm that teachers could enhance their mathematical knowledge for teaching by addressing learners’ expectations. common advice to novice teachers is that they should walk around the classroom and offer advice to learners whilst learners work on their pre-assigned tasks. there is an assumption that all learners welcome feedback from their teachers. however, this may be a simplistic assumption by teachers and teacher educators. naidoo’s (2007) study identified a learner who preferred to ‘call a teacher … when you are stuck … it’s better … instead of the teacher coming and bothering you all the time’ (p. 72). the learner wrote in her journal that she hated it when she was in the middle of something and the teacher wanted to mark her work. she felt that the teacher’s unsolicited feedback interrupted her concentration. her remarks convey the sentiment that the teacher’s presence was not always welcome when she was trying to work on her own. this is contrary to what teachers commonly believe about their offering of comments to learners – that it is always helpful and welcome. acknowledging such unexpected reactions can help teachers to develop a more nuanced understanding of their learners’ needs. building, not breaking the intention of the teacher when listening to learners should be to build and not break a learner’s confidence. the study by bansilal et al. (2010) revealed that learners did not welcome derogatory comments about their abilities from their teachers, since they viewed these as personal. the study highlighted the role that the teacher plays in building or breaking a learner’s self-confidence. it was found that shy learners were not confident enough to approach the teacher because, in their experience, there was a chance of their being insulted by the teacher. one learner said that she did not want to hear the words ‘you are so stupid’. the learners felt belittled and lost their confidence when faced with teachers’ negative comments. on a similar note, moodley’s (2008) study on south african learners’ self-efficacy beliefs about mathematics reported that most learners (from her sample of 32 grade 11 mathematics learners) felt that their mathematics teachers displayed a negative attitude towards them. ninety-one per cent of her sample indicated that the teacher ignored them when they asked questions and 93% indicated that the teacher made them feel silly when they asked questions in the mathematics classroom. some of the learners’ comments were: ‘he tries to be funny but he doesn’t know that he actually embarrasses and hurts people’; ‘you know you afraid to ask questions. maybe the teacher will make you feel stupid’ and ‘i hate being looked down upon’ (pp. 57–59). these comments support the fact that when teachers make negative comments to learners, the learners are embarrassed and feel belittled. these unpleasant experiences discourage the learners from seeking help from their teachers.similarly, young (2000) explains that the ‘most powerful and potentially dangerous dimensions of students’ feelings about feedback is the extent it impacts on themselves as people’ (p. 414). in his study, derogatory verbal comments were experienced as being ‘absolutely annihilating’ for the learner in the learning experience. learning to listen, listening to learn the studies cited above highlight the fact that learners have their own expectations of the teacher, which may be different from what the teacher assumes. granting a learner a voice presupposes a willingness on the part of a teacher to listen. effective teaching by listening depends on learners trusting, respecting and valuing the teacher. maoto and wallace (2006) suggest that to ‘be alert to learners’ learning requires a teacher who listens to learners’ thinking rather than simply listens for the right answer’ (p. 67). maoto and wallace detailed the hesitant progression of gerty, a teacher from the limpopo province in south africa, in moving from always wanting to tell, to learning how to listen. as gerty struggled to develop an awareness of her learners’ understanding by examining ideas underlying their confusion, gerty developed a deeper understanding of the learners’ understanding of factorisation and other concepts. maoto and wallace provide an insightful interrogation of gerty’s interactions with one learner who did not use the equal sign appropriately, but wrote (p. 63): gerty, by listening to the learner, was able to separate his misuse of the equal sign notation, and his subsequent erroneous representation of factorisation, from his understanding of the process of factorisation. the learner’s response reveals that he understands the three demands inherent in the process of factorisation of expressions (identifying a common factor to the different terms, extracting the factor and identifying the remaining factor of each term). however, the learner is unable to use the mathematical notations appropriately to represent these steps. by listening to the learner about his unorthodox ways of representing the factorisation process, gerty enhanced her own specialised as well as her common content knowledge (ball et al., 2005) in the area of factorisation and use of algebraic notation.gerty sought ways to design or adapt activities to meet her learners’ needs by scrutinising textbooks, which further deepened her understanding in the domains mentioned. she used the learners’ misconceptions to guide her remediation, which led her to examine mathematical ideas more closely, further enhancing her knowledge. thus her engagement with her learners’ understandings and struggles prompted her to refine her goals and extend her own understanding of what it means to teach for her learners’ understanding. maoto and wallace (2006) note that learning to listen is a key aspect of teaching for understanding and they emphasise that it is necessary to spot the appropriate moments to tell, to know when to clear up a controversy, or to decide when it is good to probe. learning to listen can be seen as a catalyst for teacher learning, but what kinds of listening must a teacher engage in? davis (1997) distinguishes between three types of listening: • evaluative (listening for and checking) • interpretive (listening to and recasting) • hermeneutic (listening between the words and actions). listening for a predetermined answer and checking that what is said is what is expected means that the listener may misunderstand or fail to appreciate what is being expressed. it is listening that involves having a ‘correct’ answer in mind. in this situation, the listener is not really interested in what the other person is saying, beyond its match (or mismatch) to the expected answer. it also reinforces a culture of test and check for learners. listening to and recasting in their own terms (interpretive listening) can be valuable for a teacher. listening to is characterised by deliberative attending, and suggests trying to understand the sense that the learner is making of the mathematics and using that as a guide for further direction (davis, 1997). however, this deliberative attending may sometimes lead the listener to overlook or misinterpret what is being articulated. hermeneutic listening (davis, 1997), on the other hand, is about negotiation, respectfulness, engagment and co-evolution. it is messy and may appear unstructured but it can be a transformative experience (as in the case of gerty from the maoto and wallace, 2006, study) if one is willing to interrogate the taken-for-granted assumptions that frame our perceptions and actions. listening to understand thomson and thomson (1996) remark that ‘how one teaches a subject is influenced greatly by the many ways one understands it’ (p. 16). we expand on that: how one teaches should be influenced greatly by what one knows of the many ways the learners understand it. henderson (1996) suggests that a teacher needs to be in touch with the different ways learners understand particular concepts:i believe mathematics is a natural and deep part of human experience and that experiences of meaning in mathematics are accessible to everyone … it is necessary to bring the power back to the meaning. (p. xx) henderson (1996) reminds us that a proof that is normally conceived is not the goal of mathematics, but just a means to an end. the goal is understanding, that is experiencing meaning in mathematics; without understanding one can never be satisfied. henderson found in a course that he had taught for over 20 years that approximately 30% of his learners show him a meaning that he has never seen before, and which changes his own meaning. he argues that the different meanings people bring with them thus require different answers to the ‘why’ question. for example, when asked to construct a straight line, the different meanings held by different people require that different conditions be satisfied in order to justify whether a figure is a straight line. henderson concludes that one can learn much about mathematics from those who have different meanings, who have different ‘why’ questions and who consequently require different proofs (p. xxiv). in his study on a teacher’s progression in learning to listen, davis (1997) found that a teacher, wendy, initially engaged mostly in evaluative listening. he noted one occasion when a potentially powerful learning opportunity was not picked up as wendy apparently sought to stay the planned course (p. 360). however, when wendy began to develop her own ideas about creating improved opportunities for learning for her learners, ‘her modes of questioning and attending to student articulations … changed dramatically’ (davis, 1997, p. 369). davis describes her changed manner of listening as interpretative listening. as their researcher-teacher partnership evolved, there seemed to be a further change in wendy’s mode of attending to her learners’ actions. this progression to hermeneutic listening was enabled by the ‘presence of an interested colleague [davis] who was willing to offer support and assistance and able to disrupt some of what was being taken-for-granted’ (p. 371). he also attributes her transition to the fact that wendy had time to think, having taken off a couple of months for maternity leave. davis (1997) also argues that: the important distinguishing characteristic between conventional and constructivism-informed teaching is not to be found in the way the teacher speaks or structures her lesson (i.e., in the visible) but in the manner in which he or she listens (i.e., in the invisible). (p. 364) the ‘participatory, transformative competency of hermeneutic listening’ enabled the teacher to enter as a participant into the learning of the classroom community. in the study, this learning to listen took time, deep reflection and the presence of a concerned colleague. in their study, peressini and knuth (1998) describe how a teacher, george, struggled to accept a different solution method (to a task based on combinations) offered by a group of learners. although he displayed flexibility and strove to listen to the learners and to make sense of what they were saying, it took the intervention of a pre-service teacher to convince george by explaining the group’s solution process. these authors comment that the ‘change in discourse was an “ah-ha” moment for george as he finally exclaimed that he “got it”’ (peressini & knuth, 1998, p. 120). initially, george was puzzled by how the learners’ ‘wrong process’ resulted in the same value that he obtained, but he struggled to see why this was so, until the intervention of the pre-service teacher. the student teacher helped george to see that the learners used a sample space consisting of sets with 10 elements amounting to a total of () outcomes; this was different from his sample space of sets of 4 elements amounting to a total of () outcomes, because they reasoned about the selection in a different manner. the learners’ solution was , whilst george’s solution was (pp. 118–125). the differences in approach served to provide a meaningful example of the identity in the context of choosing winning tickets in a raffle (note that this generalisation is the authors’ interpretation and does not appear in peressini & knuth, 1998). in this case, george reached a different level of understanding of the use of combinations in the solution of the task as well as a different level of understanding of his learners when he was able to negotiate and permit a co-evolution of a shared meaning. thompson and thompson (1994) examine the reflexive relations between the different ‘ways of knowing’ (p. 281) the ideas of speed and rate of a teacher, bill, and a learner, ann. in their study, they analyse the sources of the eventual dysfunctional communication and conclude that bill had encapsulated his deep understanding of rate and proportionality within his language for numbers, operations and procedures; this undermined his effort to help ann’s understanding. her conceptualisation of the situation was as a measurement task. this led to a language mismatch because, whilst the teacher spoke in terms of calculations, the learner was thinking of measurement; this led to a breakdown in communication and the learner was reduced to tears. thomson and thomson comment that bill’s: language of arithmetic served him well as a personal representational system, or as a system for communicating with other competent quantitative reasoners … yet … [it] served him poorly when trying to communicate with children … who had not constructed the meanings and images that [he] had constructed. (p. 301) the authors contend that sometimes ‘nothing can be gained … if the teachers’ attention actually contributes to the child’s difficulty’ (thomson & thomson, 1994, p. 301). this suggests that when teachers are unable to participate in the kind of hermeneutic listening that allows for negotiation, co-evolution and shared meaning, the teachers’ interventions can sometimes be more harmful than helpful, as was the situation in the case of bill and ann. the cases presented in this section illustrate that listening to learn is a challenging process because the teacher has to accept that the learners’ different perspectives are legitimate and the teacher has to try to answer the ‘why’ question arising from the learners’ different meaning of the situation. often, mismatches that occur are never exposed because the learners’ perspectives are not brought forward in a typical classroom. listening to know: another perspective of mathematical knowledge for teaching the discussion so far has revealed that, for a teacher, learning to listen can be a powerful learning tool for their own learning. current discussions about mathematical knowledge for teaching sometimes tend to be silent on the contribution that can be made by learners in developing teachers’ mathematical knowledge for teaching. the conception and definition of mathematical knowledge for teaching needs to be broadened to encompass the role learners can play in its development. how does mathematical knowledge for teaching change if we take the learner as the starting point? it may not significantly change the descriptions of the constructs encompassed by ball et al. (2005), but it will change our understanding of the development and purpose of mathematical knowledge for teaching and how it is used in a mathematics classroom that seeks to view learners as the starting point. the study by maoto and wallace (2006) showed that gerty’s engagement with her learners’ understandings led to a deepening of her own mathematical knowledge for teaching. davis’s (1997) study about wendy showed that her participation in the learning of her learners was enabled by the development of her hermeneutic listening. this development was facilitated by time to reflect and by the presence of a supportive colleague. peressini and knuth’s (1998) study on george revealed that his ability to listen and to understand his learners’ alternative solution strategy was facilitated by the intervention of a student teacher. this acceptance of the different proof deepened his own understanding of the equivalence of the two approaches and strengthened his understanding of the ways in which combinations could be applied to such problems. thompson and thompson (1994) revealed that bill’s strong content knowledge did not help him to cross the divide between him and his learner. he was unable to understand the learner’s different way of understanding the concept of speed, which resulted in an impasse. these different cases demonstrate that the classroom can be a powerful site for the development of a teacher’s mathematical knowledge. however, the development can only be enabled if the teacher is willing to grant the learner a voice and can engage in the kind of hermeneutic listening that allows participants to seek a shared meaning embedded in negotiation, respectfulness and co-evolution, despite the fact that this may be messy and appear unstructured. furthermore, such learning is enhanced by the availability of interested colleagues who are willing to help the teacher disrupt the familiar ways of knowing and doing mathematics. thus, by foregrounding the learners’ role in the development of mathematical knowledge for teaching, our actual understanding of teaching mathematics for understanding is deepened and shifted. concluding remarks top ↑ in this article we argued that it is a democratic virtue to permit disagreement and contestation, for it is these processes that can lead to ending the occupation of our minds in education. the purpose of this article was to consider some of the implications of granting learners a voice and listening to them. in addition to helping learners experience democratic ways of handling dissent and disagreement, encouraging learners to participate more fully in classroom activities will also be a learning experience for the teacher. vithal’s (1999) study was presented as an example of using project work as a vehicle for increasing the learners’ participation in decision-making in their school. we noted that an important aspect of granting learners a voice is that they may not support the teacher’s priorities. this disagreement may be seen as a valuable opportunity for teachers to learn more about their learners. it may be that their expectations are very different from the teacher’s, and in some cases are even disapproving of the teacher’s actions. results from two studies support our contention that a teacher’s role in listening to their learners is to build and not to break the learners’ confidence. we then turned our attention to the act of learning to listen, and how such a focus on listening could bring about meaningful change in the teacher’s own learning. by presenting various cases from existing literature, this article showed that when teachers engage in hermeneutic listening, their own mathematical knowledge for teaching is deepened and strengthened. bill was an example of a teacher who, because of his inability to listen hermeneutically, was unable to appreciate the different way of understanding held by his learner and consequently was unable to meet her learning needs. thompson and thompson (1994) comment that in such cases of mismatched instruction, the teachers’ intervention may cause more harm. thus this article has argued that allowing learners an authentic voice can contribute to their experience of how democratic processes work when there are dissenting voices. furthermore, by drawing on other research results, we have shown that a classroom in which the teacher is able to listen and which encourages learners to speak can be a crucial site for the development of mathematical knowledge of teachers. by learning how to listen in a respectful manner and as part of a negotiation and co-evolution of shared understanding, teachers can deepen and shift their understanding of mathematics, their understanding of learners and their understanding of their own learning. khuzwayo (1998) argues that redress in mathematics education in south africa requires commitment to end the ‘occupation of our minds’. this article has unpacked this process a little by focusing on the transformations that take place when learners’ voices are taken seriously. when teachers provide their learners with the opportunity to have their voices heard, they initiate a disruption to the occupation of learners’ minds by encouraging independent thought and discouraging passive acceptance of authority. during this process of considering alternatives proposed by their learners, teachers deepen their own understanding of the procedures and concepts that they are teaching, as links and relationships between the concepts and procedures are interrogated. for example, we saw how gerty had to go beyond her own understanding of an equal sign as denoting equivalence to acknowledge that her learner used it to represent the steps in the problem. she was able to see that her learner understood the factorisation procedure, when she was prepared to engage in hermeneutic listening to overcome the notational misconception. george experienced a leap in understanding of the concept of combinations when he forced himself to consider whether the learners’ alternate modelling of a problem was correct, although different from his own model, thus seeking equivalence between two different representations of the situation. henderson maintains that by seeking to understand the different meanings held by learners in his class, his own meaning of the concepts has changed. these learning experiences of the teachers recounted here did not take place in a lecture hall or from a textbook, but were a consequence of their attempts to understand their learners’ ways of thinking. it was their learning to listen that facilitated deep shifts in their mathematical knowledge for teaching and provided examples of how teachers can also contribute to their own ‘ending the occupation’ of their minds. acknowledgements top ↑ competing interests we declare that we have no institutional and/or personal or financial relationships which may have inappropriately influenced us in writing this article. authors’ 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(2000). i might as well give up. journal of further higher education, 24(3), 409–418. http://dx.doi.org/10.1080/030987700750022325 cross-cultural peer teaching the experience of cross-cultural peer teaching for a group of mathematics learners tracey d fox, natasha b vos and johanna l geldenhuys nelson mandela metropolitan university email: johanna.geldenhuys@nmmu.ac.za despite the post-1994 government’s efforts to put the necessary legislation in place and to work hard to reform the education system in south africa and improve standards, inequalities still exist in many schools. instead of focusing on the barriers to learning in schools, this paper, within the framework of the asset-based approach, describes the experiences of learners involved in a cross-cultural peer teaching initiative between a privileged private school and a township school in port elizabeth. the aim of the project was to explore the possible advantages of cross-cultural peer tutoring of certain sections of the new mathematics curriculum for both the tutors and tutees, especially to see whether the township learners’ understanding of the learning content could be improved. both quantitative and qualitative research methods were used to collect the data. the results showed that the township learners’ understanding of the mathematic topics dealt with during the peer teaching session was enhanced and that both groups gained from the cross-cultural peer teaching interaction. introduction every year the learners of a privileged private school in port elizabeth engage in a community service program. a group of them decided that they would like to begin an outreach program in order to make a difference in the community. a student teacher posted at this school joined the learners and they decided to initiate a peer teaching program at a township school. the learners felt that they could use assets such as a good knowledge of their school subjects, supportive parents and teachers, financial means and transport, to help their less-privileged counterparts. the student teacher was able to link them up with another student teacher, who was placed at a disadvantaged school in zwide, port elizabeth, where circumstances were very much the same as a decade ago and remedial intervention was drastically needed, especially in the learning area of mathematics. since mathematics was the school subject method of one of the student teachers and a learning area of all the learners, it was the logical choice for peer tutoring. the student teachers were also required to do a research project as part of their research methodology module and this peer tutoring program led to the inspiration for a research project to evaluate the effectiveness of cross-cultural peer teaching as a means of overcoming some of the many challenges faced by the mathematics learners of the township school. the hope was not only to enhance the disadvantaged learners’ mathematical performance, but also to develop social interaction and positive attitudes and values in learners from both schools. the aim of the research project was therefore to explore the possible advantages of cross-cultural peer teaching of some sections of the new mathematics curriculum for both the tutors and tutees, and especially to see whether the latter came to a better understanding of the learning content. brief overview of peer teaching peer teaching refers to the concept of learners teaching other learners. while this takes place in a school environment, it could be either formal or informal as long as it is directed, planned or facilitated by a teacher (wagner, 1982: 5). peer teaching has been ongoing in various forms of education for hundreds of years and was evident in ancient roman education as well as in the practices of early judaism (wagner, 1982: 7-25; topping, 1988: 12; goodlad, 1998: 2). peer teaching as a model of instruction became redundant in the nineteenth century when teaching developed as an organised profession but there was a resurgence during the 1960s. a growing interest in improving the standards of achievement in american schools led to a desire to focus on “individualised instruction” which was lacking in the teachercentred style (topping, 1988: 16; goodlad, 1998: 2). during the 1970s, particularly in britain, there was an increase in peer-assisted learning. many research projects were set up to ascertain the effectiveness of peer teaching in assisting pythagoras 65, june, 2007, pp. 45-52 45 the experience of cross-cultural peer teaching for a group of mathematics learners education. the pimlico connection, founded by sinclair goodlad in 1975 and the cambridge stimulus project set up in 1987 are amongst the biggest research projects on the subject. from these, over 180 projects have sprung up in the uk (beardon, 1995: 106). the findings of these early research projects have been very positive. there was a reported gain in cognitive development as well as improvement in self-concepts, social skills and communication skills for both the peer learner and the peer helper (beardon, 1995: 104-105; goodlad, 1998: 5; hill & topping, 1995: 142-145; potter, 1995: 126; rutherford & hoffmeyer, 1995: 233). while not all the research has been conclusive, and in a small percentage no significant gains were recorded, the majority of the research is in favour of peer teaching as a supplement to formal education and a major resource that should be utilised in schools (topping, 1988: 72-85, 218). according to wagner (1982: 218-224), the most frequently reported advantages of peer teaching are: • peer teaching can develop a deep bond of friendship between the helper and the learner, which facilitates in the development of social skills and selfesteem. • peer helpers are often effective in teaching children who do not respond well to adults. • more individualised attention is possible. • the helper may increase his own understanding as well as confidence and self-esteem. • additional motivation for learning may come through peer teaching. • peer helpers may be more patient with a slow learner than teachers. • peer teaching reinforces prior knowledge and peer helpers may reorganise knowledge more effectively to increase understanding. definition of terms for clarity and consistency the following terms will be used in this research report: • the concept cross-cultural refers to dealing with two or more different cultures. in this investigation it is the white and xhosa cultures. • the terms disadvantaged school and township school are used interchangeably in this report to distinguish the schools located in the so-called township areas where communities still face poverty and other social disadvantages, caused mainly by the segregation during apartheid. • the peer helper is defined as the learner from the privileged school who supports and aids in the instruction of a learner from the disadvantaged school. other terms for this role include: tutor, mentor and peer teacher. • the peer learner is defined as the learner from the disadvantaged school who requires remedial teaching and is given assistance by the peer helper. another term for this role is tutee. south african context over the past 13 years there have been many political, social and economic changes in postapartheid south africa with considerable accompanying change in the education system, including the introduction of an outcomes-based curriculum. with all the reviews and amendments to the curriculum the stage has been reached where effective implementation needs to be seen in schools. although changes were driven by the government’s will to remedy past injustices in education provision, these have not necessarily resulted in major changes at the classroom level (vandeyar & killen, 2007: 11). apartheid has unfortunately left behind some particularly damaging legacies due to its policy of racial segregation which promoted a sense of superiority of whites (sorensen & gregory, 1998: 179; furstenberg, 1995: 79). bantu education reinforced the policy of black inferiority and denied the potential of thousands of learners, focusing more specifically on developing a subordinate labour force. limited funding was provided for black education and this led to characteristics such as poor school maintenance and poorly trained and under-qualified teachers in township schools (nelson mandela foundation, 2005: 81). since 1994, there has been a complete overhaul of the education system, in order to right this injustice and bring equality for all. despite the work done by government to reform the education system in south africa and improve standards in schools, research by the human science research council and the eastern province council indicates 46 tracey d fox, natasha b vos and johanna l geldenhuys that the “majority of children in rural poor communities are receiving less than their right in a democratic south africa” (nelson mandela foundation, 2005: vii). while the study referred to here was conducted in the more rural areas of south africa, personal experience in township schools in the port elizabeth area indicates that the findings apply to these schools as well. these township schools can still be classified as disadvantaged as they face enormous challenges and barriers to learning, which limit the implementation of the new outcomes-based curriculum. vandeyar and killen (2007: 103) point out that the major limiting factor with regard to curriculum changes in south africa between 1996 and 2000 was the legacy of the poor and often inadequate pre-service and in-service training programs. it is, however not our intention to focus on these learning barriers, but rather on intervention that may minimise them. within the framework of the asset-based approach, the capacities, knowledge, skills and social resources of people and communities are utilised to the benefit of all parties involved. it is about recognising and appreciating all people, including learners, for what they know and are able to do. potential can be converted to opportunities. people and relationships could therefore serve as an asset, as can particular knowledge and expertise, time, facilities, resources, services and financial means. it is clear that cross-cultural peer teaching complies with the characteristics of the asset-based approach. the asset-based approach is relationship driven and the shared experience and knowledge is a process of growth. relationships need to be built between individuals through a process of facilitation, based on the strengths and talents of the individuals involved (ebersöhn & eloff, 2006: 461-462; bouwer, 2005: 51-52). problem formulation and research objectives having identified a drastic need for remediation in township schools and with the altruistic efforts of the private school learners, the question arose as to whether cross-cultural peer teaching could be effective as a means of raising the performance of grade 9 learners in mathematics in township schools. recognising the advantages of individualised attention and a good level of mathematics education as characterised in a private school, it was felt that learners from this school could make available their “assets” to the learners from a township school to the mutual advantage of everybody. of particular interest was the potential effect that peer helpers might have in acting as role models with a positive attitude towards learning. the problem can be encapsulated by means of the following research questions: • is cross-cultural peer teaching effective in assisting the township learners’ understanding of certain sections of the mathematics curriculum? • what can both the peer tutors and tutees gain from the cross-cultural peer teaching relationship? the primary objective was to determine whether peer teaching would be an effective way of supplementing the disadvantaged learners’ understanding of certain sections of the new mathematics curriculum in order to improve their performance in mathematics. a secondary objective was to explore and describe the peer teaching experience of both the peer learners and the peer helpers. research method mixed method research was applied which requires that both the quantitative research paradigm and the qualitative research paradigm be used within a stage or across two of the stages in the research process (johnson & christensen, 2004: 49). quantitative research in the quantitative phase of the project experimental research was used. the sample here was firstly, an experimental group of 15 learners, both male and female, which were randomly selected from two grade 9 mathematics classes in a large township school with a total of 315 grade 9 learners. secondly, a control group of 15 grade 9 learners was selected in the same manner. no information of the learners’ academic performance was previously known and they were asked to volunteer to participate in this research project. remedial classes were administered to the experimental group on a friday afternoon for an hour in the form of peer teaching intervention. the tutor sample consisted of eleven grade 9 to 12 volunteer learners from a private school. these learners were fully informed of the research project and were eager to see whether their actions could make a difference. the peer teaching sessions were opened up by a teacher who introduced the topic on the chalk board. the peer helpers would then disperse themselves among the learners, in a ratio 47 the experience of cross-cultural peer teaching for a group of mathematics learners of one peer helper to approximately two peer learners. in these groups they would attempt to solve problems related to the topic covered, thereby practising their skills. this structure enabled learners to ask questions and receive individual attention. four topics of the grade 8 mathematics syllabus were revised in the tutoring sessions, namely ratios, percentages, angles and expanding pairs of brackets. there were a number of reasons for this selection of content. firstly, many of the peer helpers were only in grade 9 themselves, and would not be able to assist learners beyond this level. secondly, for the study to be valid, the control group must have had equal opportunity to achieve on this level. finally, it was recognised that many learners struggle in higher grades due to gaps in their knowledge of foundational concepts. due to the poor culture of learning in the township school, a lack of commitment and erratic attendance by many of the participants in the experimental group were identified. in order to conduct valid research there was a need to maintain consistent attendance and they were required to commit to full attendance for a period of three weeks. this was far below the initial intended time-frame. however, due to time constraints of the full school calendar and external factors, as well as the student teachers’ university commitments, this became necessary. thereafter a test dealing with the four topics that were covered in the tutoring sessions was written by both the experimental and control groups. the test papers from both the experimental and control group were mixed in order to prevent bias. they were then marked objectively and test scores were compared using microsoft excel. graphs were plotted to represent the findings. qualitative research the qualitative method included firstly, observation by the researchers and an independent teacher from the private school throughout the peer helping sessions. field notes as described by greeff (2002: 317) were made of the participants’ comments throughout the program. secondly, three sets of interviews were conducted. initially, informal interviews were conducted with the grade 9 mathematics teachers at the township school. the teachers were asked for their perspectives on the challenges of teaching the new curriculum. the opinions gathered were studied to establish benefits and recommendations. next, semi-formal interviews, containing both closed and open-ended questions, were conducted with the 15 learners from the experimental sample in groups of threes after they had written the tests. interviews, as opposed to questionnaires, were selected due to the fact that many learners struggle to express their opinions in english, and shy away from writing. group interviews, as explained by struwig and stead (2001: 99), ensured that the learners were more relaxed and did not feel intimidated. additionally, it also allowed learners to assist each other in expressing their feelings. the last set of interviews was conducted with the peer helpers after the completion of the outreach program. six of the 11 the peer helpers were interviewed in pairs. responses of both the peer learners and peer helpers were analysed according to key questions to ascertain shared beliefs and to establish any trends. ethical measures all learners were treated with the utmost respect and participation was voluntary. the research objectives and procedures were explained to the learners in advance. before the participants gave their written consent, learners were requested to commit themselves to attend all remedial classes, to write the test, and to give feedback in an interview. they were assured of confidentiality and therefore the names of learners and schools are not mentioned in this report. trustworthiness in trying to answer the research question the researchers made use of various approaches and obtained data quantitatively through a written test and qualitatively through observations and interviews with both the peer helpers and the peer learners. by implementing triangulation the researchers found consistency in their results, thereby ensuring trustworthiness. objectivity was maintained throughout the investigation. this was particularly important when gathering and analysing data through both tests and interviews. in designing the written test, the researchers focused on content of the grade 8 syllabus to ensure that the control group was not disadvantaged. there was no bias in the selection of the sample, as it was random and voluntary. the learners’ academic performance prior to the research was not known to the researchers; and neither the experimental group, nor the control group could be labelled cleverer than the other. the duration of the experimental intervention was restricted due to the limiting factors in a real 48 tracey d fox, natasha b vos and johanna l geldenhuys life school environment. these include sports, staff meetings, pubic holidays and the like. this can be seen in a positive light, as the recommendations are not theoretical but can then be applied to every school context. the credibility of the research lies in the fact that the information gathered through the interview process comes from the learners’ authentic experiences. it can also be seen as highly dependable because learners from both schools had similar responses within the interview context. in addition, triangulation confirmed dependability. finally, a literature control of previous research findings, both internationally and nationally, was undertaken to recontextualise the results of this research. results and discussion the findings were unanimous that cross-cultural peer teaching is highly effective in raising the standard of understanding in the mathematics curriculum. the results from the different research instruments will now be addressed separately. written test the results of the written test which consisted of problems on ratios, percentages, angles and expanding pairs of brackets, showed that the experimental group exposed to peer tutoring, performed significantly better than the control group that did not participate in the tutoring sessions. there was a dramatic increase in the peer learners’ skills and understanding of the topics covered. the mean percentage scored (average) for the written test of the control group was 21.74% and for the experimental group it was 40.58%. the research group scored 18.4% higher than the control group. this is a dramatic difference and even on its own would be a strong argument for peer teaching to facilitate improvement in mathematics performance. however, it must also be stated that despite this performance there is still a great need for remedial attention – as the average mark of 40.58% should not be the desired pass mark. what is even more appalling is that the subject content tested is content that all learners in grade 9 should be familiar with in order to cope with the mathematics syllabus in the higher grades. from figure 1 it can be seen that learners in the 0 10 20 30 40 50 60 70 80 90 100 0 5 10 15 20 25 30 students listed 1-30 te st s co re s as p er ce nt ag es control group research group linear (research group) linear (control group) figure 1. individual test scores of learners in the control group and research group 49 the experience of cross-cultural peer teaching for a group of mathematics learners experimental group obtained far higher test scores than their counterparts, with the highest mark of 95% recorded. those with extremely low test scores did appear in both groups, but the majority of learners in the research group passed with marks from 30% to 50%. this is in comparison to the majority of learners in the control group who scored below 30%. observation through observation it was noted that the peer learners’ attitude to mathematics was positively influenced by the constructive role models that the private school learners provided. there was also a notable increase in their levels of confidence as evident from their greater participation in asking questions and giving comments. other results from the observation are incorporated in the results of the interviews. teacher interviews the teacher interviews revealed that a major challenge of the new curriculum is the pace. in this township school mathematical concepts are explained in the learners’ second language and the teachers’ experience is that they progress at a slower pace. the individualised attention of peer teaching could assist with this. this result is confirmed by the research of sentson (1994: 109) who has found that since comprehension is poor, learners suffer with low marks not because they do not have the skills but because many cannot fully understand the english, primarily the written words. engelbrecht, green, naicker and engelbrecht (1999: 75) are in agreement that research has shown that peer tutoring is beneficial to both partners in reading and mathematics. in questioning teachers at the township school, it was determined that one of the definite challenges of teaching mathematics is to cope with the large number of learners. this results in limited individual attention. peer teaching, by its very nature, can help to overcome this barrier to learning. another significant point mentioned by the teachers is the non-existence of a culture of learning in most township schools. this is characterised by a lack of enthusiasm, apparent laziness of learners, inadequate attendance and limited task and homework completion. teachers recognised that performance in mathematics could improve if learners were stimulated to practise solving problems. not only does extra-mural peer teaching afford learners the opportunity to practise solutions, but the peer helpers also role-modelled positive attitudes and enthusiasm for the subject. other researchers, for example hill and topping (1995: 136) and wagner (1982: 83), found that peer teaching raises the aspirations and motivation of the peer learners to strive for university exemption. interviews with the peer learners and peer helpers the learners from both schools responded very favourably to the peer teaching. their responses to the closed questions in the interview is summarised in table 1. from table 1 it can be seen that for both the peer learners and the peer helpers the experience was 100% enjoyable and they could see a future in peer helping – particularly in benefiting from understanding the learning content. with regard to the participants’ increased confidence in mathematics, the peer helpers showed that a striking 67% had noticed that they themselves had grown in confidence in this learning area. even more significant is the fact that peer learners’ response peer helpers’ response topics of interview questions % yes % no % yes % no enjoyment / increased enthusiasm 100 0 100 0 increased confidence in maths 80 20 67 33 forming cross-cultural relationships 87 13 50 50 belief in sustainability of peer teaching & helping 100 0 100 0 table 1. summary of peer learners’ and helpers’ responses to interviews 50 tracey d fox, natasha b vos and johanna l geldenhuys low-achieving learners are more effective as peer helpers. goodlad and hirst (1990: 6) ascribed this to the fact that these helpers have “more empathy and tend to have better insight into the learner’s difficulties.” this offers a great opportunity for learners who are struggling and lack confidence to gain a better self-concept and improve their ability through helping other learners. furthermore, one of the greatest advantages of peer teaching is that learners feel comfortable asking questions of learners their own age. eighty percent of the peer learners felt that their confidence in mathematics had increased as a result of their involvement with peer tutoring. large class sizes and lack of discipline cause some teachers to be impatient and intolerant of questions asked. many learners interviewed commented that they were afraid to ask questions during class time because the teacher often shouted or called them stupid. another great advantage of peer teaching is that learners could be motivated by seeing their peers mastering concepts; as one learner put it “here’s a person my own age who understands, maybe i can too!” adolescents are far more aware of building relationships within their peer group and relate far better to their peers at this stage of their lives, than to adults. therefore, it has been observed that they respond favourably and more openly to peer tuition. this was mentioned in interviews with the peer helpers and learners, and was also mentioned as a significant factor in a similar research project of goodlad and hirst (1990: 9). the following quotation of wagner (1982: 221) also confirms this finding: “it has long been obvious that children learn from their peers, but a more significant observation is that children learn more from teaching other children”. both groups reported a considerable improvement in self-esteem. this finding is also reported in other literature on this topic (ehly & larsen, 1980: 11; goodlad & hirst, 1990: 8; pastoll, 1992). with regard to forming cross-cultural relationships, it was observed that learners were growing in understanding and appreciation of one another. the social awareness that developed by peer helpers who were stretched out of their comfort zone, was dramatic and 50% said that they can relate cross-culturally with ease. the peer learners expressed enjoyment in meeting learners from other schools and 87% of them were of the opinion that they can relate cross-culturally quite comfortably. this cross-cultural interaction has brought with it benefits of understanding and sensitivity towards people from different cultural, racial and socio-economic groups. this is extremely valuable in overcoming the negative lingering effects of apartheid. beardon’s (1995: 107) research has similarly revealed significant gains in the development of peer learners’ social, problem-solving and communication skills. this fact may account for the large difference (37%) between the responses of the tutors and tutees in this category. the peer helpers and peer learners responded very favourably to the peer teaching and 100% of them believed in the sustainability of peer tutoring. this is in line with the views of ebersöhn and eloff (2006: 462) that the information-sharing trend of the asset-based approach broadens mutual knowledge and deepens the sustainability of programs. recommendations in evaluating the gains achieved by this research project it can be recommended that peer teaching be implemented in all schools across south africa – particularly between advantaged and disadvantaged schools. especially with the introduction of inclusive education in south africa, and a greater focus on learner-centred education, peer teaching offers a major resource to teachers (goodlad & hirst, 1990: 20; harper, maheady & mallette, 1994: 230). peer teaching can take place either within the formal classroom context or as part of an extramural program. it is further recommended that the duration of the peer teaching sessions be extended to at least two hours a week. a greater number of hours would result in higher academic performance and deeper, more meaningful relationships developing between learners. this program should also be ongoing and should not be limited to only a few weeks of the year before examinations or the like. it is recommended that the peer helpers be given training before instituting a program such as this and that they are provided with the necessary materials for each tutoring session. due to the cross-cultural language barrier, another recommendation could be that peer helping should only take place between learners of the same cultural group. however, although some learners did struggle to relate, the social gains of breaking down barriers and developing understanding between racial and socio-economic groups far outweigh the language difficulty. 51 the experience of cross-cultural peer teaching for a group of mathematics learners harper, g.f., maheady, l. & mallette, b. (1994). in j.s. thousand, r.a. villa & a.i. nevin (eds.), creativity and collaborative learning: a practical guide to empowering students and teachers. baltimore: paul h. brookes. conclusion peer-teaching offers teachers a great opportunity in overcoming the challenges of the demanding new mathematics curriculum where comprehension is a barrier and understanding is slow due to language capacity. the individual attention of peer teaching allows for concepts to be explained repeatedly and for learners to ask questions when they don’t understand. from this study it can be concluded that peer teaching is an extremely valuable asset and benefits both the peer learner and the peer helper, as well as the community. there is a huge potential for peer teaching to be developed further into opportunities for learning and understanding in south african schools. hill, s. & topping, k. (1995). cognitive and transferable gains for student tutors. in s. goodlad (ed.), students as teachers and mentors (pp 135-154). london: kogan page. johnson, b. & christensen, l. (2004). educational research: quantitative, qualitative and mixed approaches. boston: pearson. nelson mandela foundation. (2005). emerging voices: a report on education in south african rural communities. pretoria: hsrc press. pastoll, g. (1992). tutorials that work. cape town: arrow publishers cc. references potter, j. (1995). new directions in student tutoring: the uk experience. in s. goodlad (ed.), students as teachers and mentors (pp 121-134). london: kogan page. beardon, t. (1995). peer assisted learning in raising standards. in s. goodlad (ed.), students as teachers and mentors (pp 104-120). london: kogan page. rutherford, m. & hofmeyer, r. (1995). student tutoring in developing countries: practice and possibilities. in s. goodlad (ed.), students as teachers and mentors (pp 233-246). london: kogan page. bouwer, c. (2005). identification and assessment of barriers to learning. in e. landsberg (ed.), addressing barriers to learning (pp 45-60). pretoria: van schaik publishers. ebersöhn, l. & eloff, i. (2006). identifying assetbased trends in sustainable programmes which support vulnerable children. south african journal of education, 26(3), 457-472. sentson, c. (1994). the effect of language of presentation on pupils’ performance in a mathematics test. south african journal of education 14(3), 109-115. ehly, s.w. & larsen, s.c. (1980). peer tutoring for individualized instruction. boston: allyn & bacon. sorensen, l. & gregory, i. (1998). ‘i understand more than i understood’: exploring the possibilities of using students as tutors in south african township schools. in s. goodlad (ed.), mentoring and tutoring by students (pp 176190). london: kogan page. engelbrecht, p., green, l., naicker, s. & engelbrecht, l. (1999). inclusive education in action in south africa. pretoria: van schaik publishers. struwig, f.w. & stead, g.b. (2001). planning, designing and reporting research. cape town: pearson education south africa. furstenberg, l. (1995). investigating support systems in science teaching – a south african perspective. in s. goodlad (ed.) students as teachers and mentors (pp 78-85). london: kogan page. topping, k. (1988). the peer tutoring handbook: promoting co-operative learning. cambridge: brookline books. goodlad, s. & hirst, b. (1990). explorations in peer tutoring. england: basil blackwell. vandeyar, s. & killen, r. (2007). educators’ conceptions and practice of classroom assessment in post-apartheid south africa. south african journal of education 27(1), 101115. goodlad, s. (ed.) (1998). mentoring and tutoring by students. london: kogan page. greeff, m. (2002). information collection: interviewing. in a.s. de vos (ed.), research at grass roots (pp 291-320). pretoria: van schaik publishers. wagner, l. (1982). peer teaching: historical perspectives. connecticut: greenwood press. 52 understanding is not a private possession to be protected from theft, but rather a capacity to be developed through the free exchange of ideas. m. s. wiske << /ascii85encodepages false /allowtransparency false /autopositionepsfiles true /autorotatepages /all /binding /left /calgrayprofile (dot gain 20%) /calrgbprofile (srgb iec61966-2.1) /calcmykprofile (u.s. web coated \050swop\051 v2) /srgbprofile (srgb iec61966-2.1) /cannotembedfontpolicy /warning /compatibilitylevel 1.4 /compressobjects /tags /compresspages true /convertimagestoindexed true /passthroughjpegimages true /createjdffile false /createjobticket false /defaultrenderingintent /default /detectblends true /colorconversionstrategy /leavecolorunchanged /dothumbnails false /embedallfonts true /embedjoboptions true /dscreportinglevel 0 /syntheticboldness 1.00 /emitdscwarnings false /endpage -1 /imagememory 1048576 /lockdistillerparams false 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<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> >> >> setdistillerparams << /hwresolution [2400 2400] /pagesize [612.000 792.000] >> setpagedevice abstract introduction methodology analysis and results discussion conclusions acknowledgements references about the author(s) jogymol alex faculty of educational sciences, walter sisulu university, south africa kuttikkattu j. mammen faculty of education, university of fort hare, south africa citation alex, j.k., & mammen, k.j. (2018). students’ understanding of geometry terminology through the lens of van hiele theory. pythagoras, 39(1), a376. https://doi.org/10.4102/pythagoras.v39i1.376 original research students’ understanding of geometry terminology through the lens of van hiele theory jogymol alex, kuttikkattu j. mammen received: 15 may 2017; accepted: 05 sept. 2018; published: 18 oct. 2018 copyright: © 2018. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract after a long six-year lapse, the curriculum and assessment policy statement introduced in 2012 included geometry as part of the south african grade 12 mathematics paper 2. the first cohort of matriculation students wrote paper 2 in 2014. this article reports on the understanding of geometry terminology with which a group of 154 first-year mathematics education students entered a rural south african university in 2015; 126 volunteered to be part of the study. responses to a 60-item multiple-choice questionnaire (30 verbally presented and 30 visually presented items) in geometry terminology provided the data for the study. a concept’s verbal description should be associated with its correct visual image. van hiele theory provided the lens for the study. an overall percentage mean score of 64% obtained in the test indicated that the majority of the students had a fairly good knowledge of basic geometry terminology. the students obtained a percentage mean score of 68% on visually presented items against that of 59% on verbally presented items implying a lower level thinking as per van hiele theory. the findings of this study imply a combination approach using visual and verbal representations to enhance conceptual understanding in geometry. this has to be complemented and supplemented through scaffolding to fill student teachers’ content gap. introduction internationally, identifying the challenges in the preparation of mathematics teachers is a growing field of research as it is one of the most urgent problems faced by those who wish to improve student learning. for example, the cross-national study on the preparation of middle school mathematics teachers by schmidt (2013) and the review of 26 studies on prospective teachers’ content knowledge in geometry and measurement by browning, edson, kimani and aslan-tutak (2014) are some of the studies that show that the international mathematics education community is trying to address some of the issues in pre-service teacher education. teachers are the determining factors of successful educational change within an education and training system so that advancement of teachers through education is a method to access optimal and successful educational changes (mostafa, javad, & reza, 2017). the south african education system is confronted with the under-preparedness of teachers particularly in the teaching of mathematics in rural areas (aldridge, fraser, & ntuli, 2009). this can be attributed to the ongoing curriculum changes in south africa since its democratic inception in 1994. the education community has seen many policy revisions, modifications and reformations such as curriculum 2005 in 1998, the national curriculum statement in 2001 and the revised national curriculum statement in 2002 (king, 2003). the revised national curriculum statement came in to effect in the further education and training band (fet) in 2006, where euclidean geometry was excluded from the compulsory mathematics curriculum component (see alex & mammen, 2014). later on, a further revision of the curriculum called curriculum and assessment policy statement came into effect in the fet phase in grade 10 in 2012 and the first cohort of matriculation students wrote the mathematics paper 2 with euclidean geometry as a compulsory topic in 2014. the year 2013 was the very last year when the students wrote geometry as part of the optional paper 3 for mathematics in grade 12. for that reason, in this study, the year of passing grade 12 was taken as a factor contributing to the performance in geometry. the purpose of this study was to investigate the understanding of geometry terminology with which pre-service student teachers entered a rural university in 2015. insight of this nature is important for effective remedial teaching measures. conceptual understanding in school mathematics and geometry in south africa, one of the aims of teaching mathematics is to develop an understanding of spatial concepts and relationships (department of education, 2003). the idea of re-conceptualising the approach to geometry teaching and learning was placed in the foreground of the introduction of curriculum 2005 in 1998 by the south african national ministry (king, 2003). one of the aims as stipulated in the national curriculum statement grades r–12 is that teachers of mathematics need to produce learners who are able to communicate effectively using visual, symbolic or language skills in various modes (department of basic education, 2011). learning with understanding has been problematic, especially in the domain of mathematics (stylianides & stylianides, 2007). learning with understanding has increasingly received attention from educators and psychologists, and has progressively been elevated to one of the most important goals for all students. in order to prepare mathematically literate citizens for the 21st century, classrooms need to be restructured so that mathematics can be learned with understanding (carpenter & lehrer, 1999). french (2004) emphasises that students’ general mathematical competencies have been closely linked to their geometric understanding. according to couto and vale (2014), the development of geometrical thought is an important auxiliary to solving problems in students’ daily lives. it is important to develop the skills to see, analyse and think about the spatial objects and their images within the child (battista, 2007). this shows the importance of geometry in the overall mastery of mathematics, and further explains why geometry assumes a dominant place in the school mathematics curricula of many countries (atebe, 2008). in order to meaningfully teach mathematics in general and geometry in particular, developing each student’s conceptual understanding is important. concept is an element of understanding and knowledge (öksüz, 2010). mathematics education should include appropriate emphasis on the teaching of conceptual understanding of mathematics. according to kilpatrick, swafford and findell (2001), a significant indicator of conceptual understanding is the ability to represent mathematical situations in different ways and knowing how different representations can be useful for different purposes. suh (2007) stressed the use of representations to foster conceptual understanding. according to cunningham and roberts (2010), when in the process of trying to recall a concept, it is not usually the concept definition that comes to a student’s mind but the prior experiences with diagrams, attributes and examples associated with the concept. conceptual knowledge of geometrical concepts goes further than the skills required to manipulate geometric shapes (luneta, 2015). it is crucial that future teachers know the basic concepts well in order to understand complex concepts (couto & vale, 2014). according to brown, cooney and jones as cited in cunningham and roberts (2010), pre-service elementary teachers did not possess the level of mathematical understanding that was necessary to teach at the level recommended by the national council of teachers of mathematics (nctm). the situation is similar in south africa. for example, while bennie (1998) reported lack of conceptual understanding among teachers, atebe and schäfer (2010) reported on the lack of conceptual understanding in mathematics in general and geometry in particular among school learners. as such, there was a need for additional research which warranted this study. identifying students’ prior knowledge, before commencing teaching had been considered as a good measure to pitch the learning and teaching at an appropriate level to enhance learning. teaching and learning of geometry terminology understanding geometry is an important mathematical skill since the world in which we live is ‘inherently geometric’ (clements & battista, 1992, p. 420). geometry focuses on the development and application of spatial concepts through which children learn to represent and make sense of the world (thompson, 2003). geometry is an essential part of the south african mathematics curriculum (alex & mammen, 2016). according to genz (2006), evidence from a variety of sources makes it clear that students at junior school level are not learning geometry concepts appropriately in order to prepare them for success in their high school geometry course. ‘the most basic type of knowledge in any particular field is its terminology’ (bloom, 1956, p. 63). de villiers, as cited by feza and webb (2005), suggests that acquisition of technical terminology is the key to success in learning geometry. students need to acquire the correct technical terms and be able to use them correctly to communicate their ideas about concepts in geometry (atebe & schäfer, 2010, p. 54). sherard (1981) states that our basic speaking and writing vocabularies are rich in many geometric terms, such as point, line, angle, parallel, perpendicular, plane, circle, square, triangle, and rectangle and this geometric terminology helps us to communicate our ideas to others in precise forms. geometry is slotted as an important school subject because it provides perspectives for developing students’ deductive reasoning abilities and the acquisition of spatial awareness (nctm, 1989). jones (2002) suggests that geometry helps the students to develop the skills of visualisation, critical thinking, intuition, perspective, problem-solving, conjecturing, deductive reasoning, logical argument and proof. in the literature, spatial sense, spatial perception, spatial insight, spatial visualisation and spatial orientation have been used for reference to spatial skills in geometric thinking (bennie, 1998). according to clements and battista (1992, p. 444), ‘spatial ability is important in students’ construction and use of non-geometrical’ concepts. spatial ability plays a very important part in the development of geometric concepts and their representations (nickson, 2000). the nctm draft standards 2000 document suggests that mathematics instruction programmes should pay attention to geometry and spatial sense so that students, among other things, use visualisation and spatial reasoning to solve problems both within and outside of mathematics (mathematics learning and teaching initiative [malati], 1997). correct terminology of concepts is necessary to avoid misconceptions and confusion. the pre-service training period of teachers is ideal to ensure grounding in correct geometric terminology. couto and vale (2014) state that in portugal, the mathematics programme for basic teacher education stresses several factors such as: the visualisation and comprehension of properties of geometrical figures, the understanding of how important these are for the development of the student’s spatial awareness and the introduction of the study of geometrical transformations from early years, and how it progressively widens. according to gal and linchevski (2010), in the professional development of teachers, one should examine ways to incorporate theories of visual perception as well as their use in analysing difficulties. furthermore, as ndlovu, wessels and de villiers (2013) point out, the quality of teachers determines the quality of an education system. van hiele theory on learners’ understanding of geometry how children develop their understanding of geometry and spatial sense has been an area of research over the past 60 years (alex, 2012). during 1997, malati tried to re-conceptualise the teaching and learning of geometry in south african schools (bennie, 1998). for that re-conceptualisation to happen and to propose changes to the curriculum, malati felt that a model to understand the geometric thinking of learners would be needed (king, 2003). the group found that the van hiele model of geometric thinking would be a framework to understand the geometric thinking of learners. the theory of geometrical thinking proposed by van hiele in the 1950s suggested five sequential and discrete levels of thought a learner passes through, namely visualisation (recognising and naming figures), analysis (describing the attributes of shapes), informal deduction (classifying and generalising by attributes), deduction (developing proofs using axioms and definitions) and rigor (working in various geometrical systems) (alex, 2012). the van hiele levels ‘explain the understanding of spatial ideas and how one thinks about them’ (luneta, 2015, p. 11). the levels are hierarchical and each level is characterised by its own language and vocabulary (van hiele, 1986). the first two levels (visualisation and analysis) are particularly important for the discussion in this article. in the former, the learners reason about basic geometric concepts such as simple shapes, primarily by means of visual considerations of the concept as a whole without explicit regard to properties of its components (burger & shaughnessy, 1986). in the latter, learners begin to identify properties of shapes and learn to use appropriate vocabulary related to properties (teppo, 1991). van hiele theory has much influence on the argument of cunningham and roberts (2010) that in the process of trying to recall a concept, prior experiences with diagrams, attributes and examples associated with the concept come to a student’s mind before the concept’s definition. a learner operating at analysis level will be able to master the visual and verbal attributes better than a learner operating at visualisation level. according to van hiele (1986), teaching of geometry is central to the development of logical thinking, a key element of mathematical understanding. van hiele asserted that teachers have a crucial role in the process of teaching and learning (couto & vale, 2014). the more a teacher knows about the way students learn, the more effective that individual will be in nurturing mathematical understanding (swafford, jones, & thornton, 1997). the van hiele theory aims to explain how children learn geometrical concepts (lim, 2011). according to shulman (1987), teaching necessarily begins with a teacher’s pedagogical content knowledge. according to rossouw and smith (1997), the rich base developed from research on van hiele levels and how students learn geometry is an important source of understanding teachers’ pedagogical content knowledge of geometry teaching. the van hiele theory with this significant pedagogical implication in geometry thinking levels forms the basis of this study. research questions the research questions investigated in this study are: (1) what was the conceptual understanding of basic geometrical terminology of the sample of 2015 entry level mathematics education students in the rural university in relation to the curriculum followed by the pre-2014 matriculants and matriculants of 2014? (2) was there a statistical difference in the overall performance of students in matching the verbal description with their correct visual images? and (3) was there a statistical difference in the performance of the students with respect to the different concepts tested? methodology this research adopted a positivist paradigm and a quantitative approach. this case study design mainly focused on matching the verbal description with their correct visual images in the geometrical concepts and terminology and as indicated earlier was a retrospective study. instrument euclidean geometry in south african schools is usually the figures in the plane (atebe, 2008). the instrument used was a 60-item multiple choice questionnaire with four options from which the students were expected to choose the best answer. the questionnaire was constructed by atebe (2008) and the researchers adopted it with permission. the aim was to explore students’ understanding of some key technical terms frequently encountered in the teaching and learning of school geometry (atebe, 2008). atebe used the split-half method to check the reliability of the test construction and the spearman-brown reliability coefficient (r) calculated for the test was 0.87; thus, the test was found to be reliable as the sample of atebe and the sample for this study were similar in nature. the items mainly tested the terminology associated with concepts about lines, angles, triangles, quadrilaterals and circles. there were two conceptually identical but structurally different sets of items for each of the 30 selected terminologies, one in a verbal form (with no diagrams), and the other in a visual form (with diagrams). in other words, the multiple choice questionnaire consisted of 30 verbally presented and 30 visually presented items. all the items in the questionnaire were then juggled around so that items in the homologous (i.e. identical) pair of items were separated far away from the other. the purpose of the identical pair was to determine whether a student who chose the correct verbal description of a geometric concept also chose the correct visual representation and vice versa. for example, in figure 1, item 1 (verbal) and item 10 (visual) form a homologous pair used in the questionnaire. figure 1: a homologous pair. the instrument had two sections, the first one to gather biographic data and the second one consisted of the 60-item multiple choice questionnaire. the lead researcher administered the instrument during the first two-hour mathematics education lecture. analysis and results general information of the students who participated in the study the mean average age of the students who participated in the study was 22 years. out of the 126 students who participated in the study, who were enrolled for the year 2015 in year 1, 88 (70%) passed matric before 2014 (pre-2014 matriculants) and 38 (30%) were matriculants of 2014. ethical considerations this article reports on a study that was conducted in a historically disadvantaged rural university in the eastern cape. to conduct the study, permission was sought from the head of the department of mathematics, natural and consumer sciences education. a permission letter was obtained and approved by the faculty of educational sciences ethics committee. entry level mathematics education students were informed of the purpose of the test and a request for voluntary participation was made. out of the 154 students who enrolled for the course for the year 2015, 126 (86 male students and 40 female students) voluntarily took part in the study. it was agreed that anonymity and confidentiality of the data would be guaranteed. this information was also printed on the general information of the instrument and there was space for participants to sign for informed consent. there was no reward for participation. performance of entry level mathematics education students in the geometry terminology test the scoring of the terminology section was calculated using microsoft excel 2013. each item was awarded 1 mark and the total was 60 marks. for each student, the total was then converted to a percentage. the general performance of the students was calculated in terms of the overall percentage mean score as shown in figure 2. figure 2: performance of entry level mathematics education students in the geometry terminology test. for research question (1), an overall percentage mean score of 64% obtained in the test indicated that the majority of the students (64%) in this study had a fairly good knowledge of basic geometric terminology. the study further aimed to determine the students’ ability in visually presented and verbally presented terminology items. the students obtained a percentage mean score of 68% on visually presented items against a percentage mean score of 59% on verbally presented terminology items. this meant that the students’ performance was better in dealing with visually presented terminology items than the verbally presented items for the same concept. to answer research question (2), a further analysis was also done to find out the performance of students in the multiple choice questionnaire in relation to pre-2014 matriculants and the matriculants of 2014. table 1 depicts the results. table 1: performance of students in the multiple choice questionnaire in relation to pre-2014 matriculants and the matriculants of 2014. from table 1 it can be noted that p < 0.05 and that the obtained t-value 6.748214 is significant, as |t| > 1.96 at α = 0.05 for a two-tailed independent samples test. this shows that there is a significant statistical difference in the overall performance of the students in favour of matriculants of 2014. from table 2 it can be noted that p < 0.05 and that the obtained t-value 6.260318 is significant, as |t| > 1.96 at α = 0.05 for a two-tailed independent samples test. this shows that there is a significant statistical difference in the performance in visually presented items in favour of matriculants of 2014. table 2: performance of students in the visually presented items in relation to pre-2014 matriculants of and the matriculants of 2014. from table 3 it can be noted that p < 0.05 and that the obtained t-value 5.932766 is significant, as |t| > 1.96 at α = 0.5 for a two-tailed independent samples test. this shows that there is a significant statistical difference in the verbally presented items in favour of matriculants of 2014. table 3: performance of students in the verbally presented items in relation to pre-2014 matriculants of and the matriculants of 2014. from table 4 it can be noted that p < 0.05 and that the obtained t-value 4.137707405 is significant, as |t| > 1.96 at α = 0.5 for a one-tailed independent samples test. this shows that there is a significant statistical difference in the performance of the students in favour of visually presented items. table 4: performance of pre-2014 matriculants in relation to the verbally presented and visually presented items in the test. from table 5 it can be noted that p < 0.05 and that the obtained t-value 2.319806649 is significant, as |t| > 1.96 at α = 0.5 for a one-tailed independent samples test. this shows that there is a significant statistical difference in the performance of the students in favour of visually presented items. table 5: performance of matriculants of 2014 in relation to the verbally presented and visually presented items in the test. students’ knowledge of different concepts in geometry to address research question (3), students’ mean scores in the terminology test were calculated separately for items on geometric terminology associated with the concepts in three categories: lines, circles, and triangles and quadrilaterals. table 6 shows how the different concepts were asked in the terminology test. table 6: the different concepts tested in the terminology test. the results were analysed using microsoft excel 2013 and are shown in figure 3. figure 3: performance of entry level mathematics education students in the multiple choice questionnaire according to concepts. figure 4: performance of entry level mathematics education students in the multiple choice questionnaire according to concepts. it was found that the students performed better in the terminology associated with lines (70%) followed by circles (64%) and the terminology in triangles and quadrilaterals were the worst performed (52%). it was found that the matriculants of 2014 performed better in the terminology associated with lines (83%) followed by circles (75%) and the terminology of triangles and quadrilaterals (63%) than the pre-2014 matriculants who scored 64%, 60% and 48% respectively. from table 7 it can be noted that p < 0.05 and that the obtained t-value 6.131306938 is significant, as |t| > 1.96 at α = 0.5 for a two-tailed independent samples test. this shows that there is a significant statistical difference in the performance of the students in the concept of lines in favour of matriculants of 2014. table 7: performance of students in the line concepts in relation to pre-2014 matriculants and matriculants of 2014. from table 8 it can be noted that p < 0.05 and that the obtained t-value 4.943971704 is significant, as |t| > 1.96 at α = 0.5 for a two-tailed independent samples test. this shows that there is a significant statistical difference in the performance of the students in the circles concepts in favour of matriculants of 2014. table 8: performance of students in the circles concepts in relation to pre-2014 matriculants and matriculants of 2014. from table 9 it can be noted that p < 0.05 and that the obtained t-value 4.738459571 is significant, as |t| > 1.96 at α = 0.5 for a two-tailed independent samples test. this shows that there is a significant statistical difference in the performance of the students in triangles and quadrilateral concepts in favour of matriculants of 2014. table 9: performance of students in the triangles and quadrilaterals concepts in relation to pre-2014 matriculants and matriculants of 2014. it is noted from tables 1–5 and tables 7–9 that the performance of the students who wrote geometry as part of their matriculation examination in the year 2014 outperformed the students who passed earlier than 2014 as the percentages for verbally presented items, visually presented items and all the concepts were higher than the students who passed matric before 2014. discussion this study found that students’ performance was better in dealing with visually presented terminology items than verbally presented ones. this could lead to the conclusion that the students in the study, although high school graduates, could probably be operating at lower van hiele levels of geometric thinking. according to couto and vale (2014), the development of geometrical thought is an important ancillary to solving problems in students’ daily lives and the attainment of these ideas depends greatly upon the teacher and their knowledge. the performance of the student teachers of this study raises a concern in this respect and they are to be trained at university entry level in order to reach the expected levels of the van hiele theory. in support of the van hiele theory, couto and vale (2014) state that geometrical thought is gradually developed in students starting with recognition of figures, moving on to differentiation up to the emergence of deductive reasoning. an earlier study by alex and mammen (2014) on the van hiele theory of geometrical thinking found that the majority of south african grade 10 learners were at the visualisation level. one of the characteristics of visualisation level thinkers is that they reason about basic geometric concepts such as simple shapes, primarily by means of visual consideration of the concept as a whole without explicit regard to properties of its components. other south african studies such as those of de villiers and njisane (1987), siyepu (2005), atebe (2008) and luneta (2015) indicated that high school learners in general, and more especially grade 12 learners, were functioning below the levels that were expected of them, that is, they were at the concrete (visualisation) level rather than at the abstract level in geometry. the results from this study were inconsistent with the study by bozkurt and koç (2012) on first-year elementary mathematics teacher education students where the majority of the participants (32%) could not provide a definition of a prism (word skills). also, the results reported here were inconsistent with those of couto and vale (2014) on pre-service teachers in portugal. their study showed a weak performance in the test on issues addressing elementary knowledge of geometry where there were only 34% correct answers in knowledge and understanding of concepts and mathematical knowledge. the overall percentage mean score of 64% in the present study was inconsistent with atebe (2008) in that the 64% was better than the mean score (47.85%) of the south african subsample of learners in his study. the results from the present study were also inconsistent with the study by cunningham and roberts (2010) where 23 elementary pre-service teachers were assessed on their ability to answer questions involving geometry concepts and a weak or limited understanding of certain concepts was reported. the results, however, were confirmation of the analyses by luneta (2015) of 1000 grade 12 scripts from 2012 in south africa which followed the national curriculum statement, where most of the students made conceptual errors in questions in geometry. the data show that the matriculants of 2014 outperformed, with a significant statistical difference, the pre–2014 matriculants. the present study revealed a very significant gap in the performance of the pre-service student teachers in geometry at the university entry level in favour of students who came through a compulsory geometry curriculum. this might also be due to the constraints in the secondary school mathematics curriculum originating from curriculum reforms. this is in support of the inference put forward by wilburne and long (2010) that many pre-service teachers find that they never get the opportunity to really study the mathematics curriculum in depth but are expected to know and teach it with meaning in their student teaching and beginning of teaching experiences. conclusions this study investigated the knowledge of basic geometric terminology with which pre-service student teachers enter the rural university. even though it was found that the majority of the students had a fairly good knowledge of the geometric terminology, the students performed better in dealing with visually presented terminology items than verbally presented ones. this raised a concern that the majority of students were operating at the visualisation level of van hiele’s geometrical thinking. the study revealed that the matrics of 2014 performed better in all aspects tested than pre-2014 matriculants. it can be concluded that curriculum constraints due to the ongoing changes in the school mathematics curriculum might have adversely affected students’ performance in geometry. the study also gives insight into the quality of students received by universities for teacher education courses which reflects the quality of geometry learning in our schools. visual and verbal representations in geometry should complement and supplement each other to enhance conceptual understanding. the use of multiple representations carefully built into the geometry curriculum will ensure that students meaningfully understand the concepts they are learning. pre-service students and their educators need to adopt a combination approach since visual representations enhance spatial understanding and verbal representations promote mathematical terminology and mathematical language development besides general vocabulary and language development. the curriculum of the universities should include more opportunities for mathematics education students to familiarise themselves with school geometry content so as to allow them to teach it with understanding and meaning-making to learners in their careers as future teachers. acknowledgements the authors gratefully acknowledge marc schäfer for giving permission to use the test constructed by humphrey atebe. competing interests the authors declare that they have no financial or personal relationships that may have inappropriately influenced them in writing this article. authors’ contributions j.a. conducted the research and wrote the manuscript. k.m. made conceptual contributions and provided critical revision, guidance, editing and final approval of the document. references aldridge, j., fraser, b., & ntuli, s. 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(2010). secondary pre-service teachers’ content knowledge for state assessments: implications for mathematics education programs. iumpst: the journal. vol. 1 (content knowledge), january, 1–13. retrieved from https://files.eric.ed.gov/fulltext/ej872484.pdf article information authors: joseph m. furner1 cheryl kenney2 affiliations: 1department of teaching and learning, college of education, florida atlantic university, united states2limestone creek elementary, florida, united states correspondence to: joseph furner email: jfurner@fau.edu postal address: florida atlantic university, college of education, department of teaching and learning, 5353 parkside drive, ec 207d, jupiter, florida 33458, united states dates: received: 21 oct. 2011 accepted: 09 nov. 2011 published: 02 dec. 2011 how to cite this article: furner, j.m., & kenney, c. (2011). counting on frank: using bibliotherapy in mathematics teaching to prevent de-geniusing. pythagoras, 32(2), art. #133, 7 pages. http://dx.doi.org/10.4102/pythagoras.v32i2.133 copyright notice: © 2011. the authors. licensee: aosis openjournals. this work is licensed under the creative commons attribution license. issn: 1012-2346 (print) issn: 2223-7895 (online) counting on frank: using bibliotherapy in mathematics teaching to prevent de-geniusing in this original research... open access • abstract • introduction • mathematically gifted students • the ‘dumbing down’ phenomenon of the gifted and talented • using bibliotherapy in the classroom • summary • acknowledgements    • competing interests    • authors’ contributions • references • appendix    • a sample bibliotherapy lesson plan featuring counting on frank    • menu of themes or key concepts    • discussion questions    • possible activities    • interactive mathematics websites that may appeal to the mathematically gifted abstract (back to top) today the understanding of mathematics is critical in an increasingly technological age. teachers must play an important role to ensure that all students display confidence in their ability to do mathematics. often gifted students of mathematics can be made to feel bad by their peers just because they know mathematics and things come easily to them. children’s and adolescent literature has now been recognised as a means of teaching mathematics to students through the use of stories to make the mathematics concepts relevant and meaningful. literature can also be used as a form of therapy to reach students who may be frustrated with children picking on them for being good at mathematics. story and picture books such as counting on frank, math curse and a gebra named al are now available to use in the classroom as forms of bibliotherapy in helping students come to terms with issues relating to mathematics that haunt them. in this article we discuss the phenomenon of dumbing down by the gifted population to fit in with their peers. we propose using reading and discussion (bibliotherapy) to aid in preventing de-geniusing of mathematically gifted students. introduction (back to top) everyone is born a genius, but the process of living de-geniuses them. (r. buckminster fuller 1855–1983)today many young people in our schools who excel in the area of mathematics are often labeled or stigmatised by their peers as being nerds or know-it-alls. teachers must ensure that all students develop confidence in their ability to do mathematics − and gifted youngsters are no exception. mathematical understanding is critical in an increasingly technological society. it is critical in our globally competitive world that most countries now produce more workers in the science, technology, engineering and mathematics fields. teachers must make sure we nurture our strong mathematics, science, and technology students so that they pursue careers in these fields and do not shy away from them due to being worried about being considered ‘nerdy’. gifted students of mathematics are often made to feel bad by their peers because they excel in mathematics and other academic disciplines come easy for them. young people today are influenced by hollywood stars, musicians, and professional sports players. ‘being a mathematician’ is not always looked upon as being as glamorous or as exciting as being a national football league player. within the past decade or two, children’s and adolescent literature has been recognised as a means of teaching mathematics concepts to students through the use of stories to make mathematics ideas relevant and meaningful. literature can also be used as a form of therapy (bibliotherapy) to reach students who may be frustrated with being taunted or teased for excelling in mathematics. story and picture books such as counting on frank (clement, 1991), math curse (scieszka & smith, 1995) and a gebra named al (isdell, 1993) are available to use in the classroom as forms of bibliotherapy to help students come to terms with issues that haunt them just because they are good at mathematics. in this article we propose using reading and discussion (bibliotherapy) to aid in preventing the de-genuising of mathematically gifted youngsters. rozalski, stewart and miller (2010) have found that by using carefully selected thematic books teachers can use literature to reach young people who are experiencing difficult situations. the following dialogue is related by one of the authors as a parent’s interaction with her son, since it relates to why a gifted mathematics student may resort to avoiding showing their real knowledge of the subject matter because of peer pressure, and the de-geniusing that occurs amongst our gifted and talented students: whilst in the kitchen preparing dinner early one evening, my 11–year-old son, kevin came to me with some exciting news. ’guess what mom?! guess what mr d said today?’ ‘what?’ i replied enthusiastically. i knew mr d was kevin’s sixth grade maths teacher who held kevin in high regard as a talented maths student. ‘mr d said i should be in algebra. he wants me to start working on algebra.com.’ ’that’s because you are so smart in maths, kevin. i think that it is a wonderful idea.’ ’yeah, mr d says i am so smart in maths i should be in gifted, but my fcat [florida comprehensive assessment test] score in fifth grade was off by one point, so i can’t. he makes me help all the other kids.’ ’when you have a talent or gift, you should always share that with others,’ i said. ’yeah, but they bug me too much. they’re always asking me for the answers and to help them when they don’t want to do the work, so now i just pretend i don’t know how to do the problems even when i do.’ i can’t tell you the shock and dismay i felt when i heard those words − ‘pretend i don’t know’ − come out the mouth of my child. i have always encouraged my sons to be proud of their talents; to recognise their limitations but give their best in whatever they do. now i was hearing that my son was deliberately ‘dumbing down’ to adapt to an uncomfortable situation with his peers. it went against everything i thought i had taught them about selfconfidence, pride and acceptance of who they are. when the shock of his statement wore off and i had some time for thoughtful consideration, i realised that in as much as his high mathematical intellect is a gift that we celebrate, for my son this ‘gift’ is also a burden. ‘dumbing down’ is a coping mechanism that many gifted children use as a way to fit in socially with their peers. they don’t want to be seen as different, even in the most positive regard. both teachers and parents interact with youngsters on a daily basis and see young people ‘shut down’ due to peer pressure or societal stigmas like being a ‘nerd’ just because mathematics comes easy for them. dyer (2004) claims that many people are easily influenced by their peers and society pressures. as humans we allow others’ influences to impact our egos and success in life. hence ‘de-geniusing’ can happen to anyone, as in the case of kevin above and many gifted children around the world. it is a real phenomenon that is happening to our youngsters in schools today, who ‘dumb down’ in order to fit in with their peers and society as a whole. renzulli (2008, 2011a, 2011b) has found that teachers need to do more for the gifted to better reach them and foster an atmosphere where they can excel in academics without dumbing down the curriculum or short changing some students who have great potential. this article provides a review of the literature as it relates to both the gifted mathematics student and the use of bibliotherapy, and will make an argument for using bibliotherapy to help in the prevention of de-geniusing of mathematically gifted children. a sample lesson plan employing a children’s literature picture book will be shared, as well as how mathematics teachers can work towards emphasising the importance of children being proud of their know-how of mathematics, so as not to dumb down or minimise their true gifts and talents. floyd and hébert (2010) contend that using picture book biographies can help to nurture the talents of gifted youngsters. mathematically gifted students (back to top) many mathematically gifted students tend to display several characteristics that identify them as talented in this sphere, such as: • being interested in numerical analysis • having memory retention for storing main features of problems and solutions • able to appreciate parsimony, simplicity, or economy in solutions • reasons effectively and efficiently • solves problems intuitively using insight • ability to reverse steps in the mental process • organises data and experiments to discover patterns and relationships • exhibits flexibility in solving problems (johnsen, 2004; vantassel-baska & little, 2011). plucker (2008) and johnsen (2004) synthesised cognitive characteristics that gifted students frequently exhibit, particular to affective characteristics like having a sense of humour, being highly sensitive, taking other perspectives and being empathetic and often perfectionists. schools need to provide professional development for teachers so that they can establish positive situations for gifts and talents to emerge so that individuals who are proud of their talents do not dumb down their gifts due to societal and peer pressures. there is a myth that gifted children are better adjusted, more popular and happier than average children. the reality is that more frequently the opposite is true (renzulli, 2011b). for most gifted youngsters childhood is more pleasurable because they derive joy from challenge and reward from work. at the same time, it is a childhood that is more painful, more isolated and more stressful, because these gifted children often do not fit in with their peers socially and have high expectations for themselves, as do others (freedman & jensen, 1999). also, since gifted youngsters’ intellectual capabilities are so strong, they have less need to develop their emotional intelligence. this underdeveloped emotional intelligence is a major pitfall in social interactions with peers. parents and teachers need to nurture and guide gifted children to becoming well-balanced individuals, and using bibliotherapy to address de-geniusing is one possibility. vantassel-baska and little (2011) also believe that we need to foster such support for gifted students whilst also providing a strong emphasis on content knowledge, helping students to appreciate their gifts and excel in mathematics. renzulli (2011a) and diezmann and english (2001) believe that enrichment for mathematically gifted students in the elementary school needs to extend beyond puzzles or busy work and should support the development of mathematical power through a differentiated curriculum. although mathematically gifted children are characterised by the quality of their reasoning abilities, they require appropriate and challenging learning experiences to facilitate their cognitive development. enrichment that consists of ‘busy work’ or irrelevant topics has limited academic value for gifted students. although a student might be gifted, he or she still needs appropriate teacher support in dealing with challenging tasks that extend mathematical understanding as well as affective needs like being accepted by their peers in spite of their giftedness. enrichment programmes play a key role in equipping these students with the foundational knowledge and skills to actively pursue their interests (renzulli, 2011a, 2011b). in addition, children’s self-reflection of their learning can empower them in their work as mathematicians. diezmann and watters (2000) contend that mathematical tasks that facilitate learning should be commensurate with the capability of the learner. for gifted students this requires flexibility in the nature of tasks and appropriate support from others. tasks of sufficient difficulty need to be carefully chosen or existing classroom tasks need to be adapted, that is, ‘problematised’. once the task is appropriately challenging, the teacher needs to provide support for the student. the need for support should be viewed positively rather than negatively, because the more complex task provides an opportunity for mathematical learning not provided by an easier task. furthermore, the teacher provides feedback to the student, highlights successful strategies and acknowledges the student’s capability. gifted peers may also provide support and feedback in a positive way for young gifted students to be accepting of their true mathematical gifts, whereas other peers may make fun of their mathematical knowledge. appropriate time allocation for tasks is also an important consideration. mathematically gifted students achieve mastery faster and generally have more lengthy concentration spans than non-gifted students. however, engaging in challenging tasks is time-consuming. time is also required for the incubation of ideas, which is associated with insight into challenging problems. thus, an effective goal should be that gifted students do fewer and more complex tasks over a longer period of time (diezmann & watters, 2000). this may also be why many advocate gifted mathematics students being placed in ability groupings. it is important that schools and parents help to raise well-rounded gifted young people who accept their gifts and do not minimise them. parents can impart valuable lessons to their gifted children whilst helping them achieve a balanced life. it is fine for gifted children to spend a lot of time pursuing their interests, but even the most advanced children should participate in family life, chores, playing with siblings and community involvement. a well-balanced life equals a well-rounded, happy young person. the ‘dumbing down’ phenomenon of the gifted and talented (back to top) under-achievement in exceptionally gifted students has been documented in the literature. fears of envy or retaliation and peer pressure can lead precocious intellects to resort to various forms of disengagement and withdrawal to avoid the emotional pain of such conflict (grobman, 2006). most students want to fit in, as do the gifted, and they may mask their talents in order to do so. hoover-schultz (2005) contends that at first glance ‘gifted under-achievement’ seems like an oxymoron; it can also be an educational enigma. the loss to society can be tragic when students do not achieve their full potential due to peer pressure; minor neurological problems may also make under-achievers behave this way. we must support all students also, or especially, the gifted so that they elevate themselves to their full potential. cross (1997) raised this issue repeatedly in her research about under-achieving gifted students, students who know that they are different, that they show intelligence and are outspoken, and then people tend to isolate them and put labels or expectations on them. dumbing down can actually be thought of as a ‘coping’ strategy for these students in order to fit in (cross, 1997, 2002). cross believes that one of the most detrimental coping mechanisms academically gifted students employ is under-achievement. according to cross (1997) we must provide effective guidance to the gifted students so that they can chart a new course for not lowering their achievement just to fit in, whether with peers, society or competitive occurrences. teachers need to teach their gifted students strategies so that they will not employ these severe patterns of coping behaviours that include trying to blend in with non-gifted students, under-achievement, and even suicidal behaviour (cross, 1997). gifted programmes certainly have many benefits. for example, a study by cross, stewart and coleman (2003) showed that co-participants in the study described feeling more accepted in the gifted magnet school than in previous non-magnet schools, as they did not have to change who they were or dumb down their intellect to fit in. whilst it my not be possible for all gifted student to attend a gifted magnet school, there are certainly benefits to this so that students do not have to become de-geniused. plucker (2008) and cross (2002) believe that we must challenge many of the myths about the social and emotional developmental of gifted students, provide appropriate counselling and create learning environments where students with gifts and talents can thrive. in this article we suggest the use of bibliotherapy as a form of counselling to reach these gifted students, so that they will not dumb down to fit in with classmates and society. bain, choate and bliss (2006) found in their empirical study that 77% of their participants felt that gifted students are more likely to have problems with social relationships compared to the general population. this leads to them behaving in a way to fit in better with all their peers at school. young people who are gifted are often placed front and centre in their school and community domains, as examples of the best individuals society can offer. doing this adds erroneous pressure upon these youngsters, which is often unfair and leads to problems later on in life. many gifted students are often de-geniused. we must help gifted students to create plans that develop their talents to an optimal level. we must talk to them and work with them; they need to know that using coping strategies to dumb down their intelligence is really hurtful to them and ultimately to our society. using bibliotherapy in the classroom (back to top) affective factors play an important role in learning (mcleod, 1992), and teachers need practical classroom strategies to address the feelings of their students who are suffering from mathematics anxiety.teachers need to create supportive environments in which their students feel comfortable in expressing how they feel about their mathematical experiences. one approach in helping young people to express themselves comfortably is through bibliotherapy. bibliotherapy is reading of selected literature to produce affective change and promote personality growth and development (abdullah, 2002; betzalel & shechtman, 2010; doll & doll, 1997; forgan, 2002, 2003; furner, 2004; jeon, 1992; heath, sheen, leavy, young & money, 2005; jack & ronan, 2008; lenkowsky, 1987; reis & renzulli, 2004; rozalski, stewart & miller, 2010; sridhar & vaughn, 2000; sullivan & strang, 2003). bibliotherapy can be used to help young people understand themselves and cope with problems by providing literature relevant to their personal situations and developmental needs (betzalel & shechtman, 2010). hébert and kent (2000) advocate the use of young adult literature for gifted teenagers to address social and emotional concerns. teachers using this approach hold a fundamental belief that reading will influence thinking and behaviour, and that through guided discussions selected readings can be focused on the specific needs of students. the bibliotherapy process is fairly easy to understand and implement. the therapeutic experience whilst reading a book happens each time we pick up a good book and say ‘this character is very much like me. i can relate to this person’. this interaction is known as identification, and the more we have in common with people we meet in our reading, the closer the identification process will be. that identification produces a sense of tension relief, or ‘catharsis’, an emotional feeling that tells us we are not alone in facing our problems. as we enjoy the book, we learn vicariously through the characters in the book. we gain new ways of looking at troublesome issues we face and insight evolves. with this new insight, changed behaviour may occur as real-life situations similar to those experienced in the books are confronted (furner, 2004). this may be the case with kevin, presented in our case scenario earlier. when reading counting on frank he may relate to the character and emulate his attitudes or methods for coping with his feelings as well as using the strategies presented for being proud of his mathematic prowess. the three most recognised stages of bibliotherapy are identification, catharsis, and insight (forgan, 2002; halsted, 1994; jack & ronan, 2008); however, another that is less mentioned in the literature yet especially interesting for teachers working with mathematics-anxious students is the concept of universalisation (slavson, 1950), or the recognition that our problems are not unique. through universalisation we realise that we, as sensitive individuals, ‘are in this together’. thatcher and fletcher (2008) have found that not all teachers realise or see the value in using literature or bibliotherapy for addressing students’ problems, nor how to use such a process. books can help teachers guide the emotional development of their students far more than intellectual discussion, because stories directly affect human emotions (forgan, 2002, 2003; furner, 2004; rozalski, stewart & miller, 2010). a skillful author can help young people connect with others who have similar problems. if books or short stories can touch young people emotionally, these students may be much more receptive to ideas presented by the author than if they are presented to them in a lecture by a concerned teacher. students who are unable to talk about their anxieties often can identify with characters in books strongly enough to experience the catharsis and acquire some important insights (halsted, 1994; heath, moulton, dyches, prater & brown, 2011). for bibliotherapy to be successful, a meaningful follow-up discussion is required (forgan, 2002; furner, 2004). to simply read a good book with an entire class is not bibliotherapy. it is very important that young people not only read books, but also become involved in discussions, counselling and follow-up techniques such as role-playing, creative problem solving, relaxation with music, art activities and journal writing (furner, 2004; forgan, 2002, 2003; hébert, 1991, 1995; hébert & furner, 1997). when presented in this way, bibliotherapy can be enjoyable whilst providing a time for solid introspection for young people. it is important that sensitive mathematics teachers help students recognise that their abilities may differ from those of their peers, acknowledge that they may also have areas of weakness and assist them in developing self-esteem by becoming satisfied with who they are as individuals (ableser, 2008). teachers using bibliotherapy may be successful in doing so by making connections to the students’ attitudes and feelings with the characters in story books (regan & page, 2008). leininger, dyches, prater and heath (2010), burke (2009), and kurtts and gavigan (2008) have all found that there are a wide range of literature and picture books useful in conducting bibliotherapy to address a wide range of conditions and societal factors impacting young people today. through this counselling approach students come to understand that their avoidance or dumbing down was a learned behaviour; they were not born with this feeling, and they can be taught to overcome it by consistently implementing their self-monitoring strategies to overcome the social stigmas that peers put on them about being good at mathematics. since bibliotherapy is one avenue for students to discuss feelings about problems with others who share similar issues, the use of guided reading (forgan, 2002) could naturally become one component of systematic desensitisation proposed by educational and psychological experts. the book counting on frank by rod clement (1991) depicts the story of a middle school boy’s gift of mathematics. he goes through life knowing many things, counting, comparing and knowing many number facts that all come easily to him. the book portrays him as somewhat of a nerd or geek. the book presents the reader with a variety of mathematics problems in a humorous yet realistic fashion as they relate to numbers, counting, and mathematics facts. the lesson plan (see appendix) designed for teachers at the middle and high school levels provides a variety of follow-up activities for use during or after reading counting on frank. these activities are appropriate for infusing affective teaching into a mathematics curriculum over the course of a semester or even an entire academic year. the activities were designed with the realisation that teachers are often under pressure to teach many mathematical concepts and skills in an academic year. forgan (2003), in his book entitled teaching problem solving through children’s literature, offers many children’s books to use in bibliotherapy sessions to help students overcome a variety of problems confronting them. integrating activities like bibliotherapy and discussion whilst teaching the content consistently throughout a semester or an academic year would allow a teacher to address the feelings of their students as they learn mathematics. such activities become an affective strand incorporated throughout the mathematics curriculum. teachers may consider working with the school counsellor and use such an activity at least once a month, or whenever they detect that their gifted students are avoiding showing their true expertise in mathematics with their peers. it is important for teachers to do these recommended activities with their mathematically gifted youngsters, to help them gain the mathematics confidence whilst preventing de-geniusing. summary (back to top) there are many students in schools around the world like kevin or the boy in the book counting on frank, who are made to feel like geeks or who dumb down their mathematics excellence in an effort to be accepted by their peers. ‘de-geniusing’ is a common phenomenon amongst gifted and talented students. teachers need to take the time in their teaching to address the affective needs of these students, since many may be made to feel like nerds just because they are good at mathematics. this may also be relevant in other disciplines like the sciences and technologies. to address this issue, teachers need to be trained in how to use bibliotherapy to assist students in accepting their giftedness in mathematics. such strategies involve teachers taking time to discuss with their students how they are feeling about learning mathematics. reading children’s or adolescent literature may be one approach that mathematics teachers can use to reach students who may feel bad due to peers making fun of them because of their mathematical knowledge. bibliotherapy is a therapeutic discussion-generating technique which offers caring teachers appropriate affective strategies for dealing with preventing ‘de-geniusing’ of the mathematically gifted in classrooms, so that students achieve success truly reflective of their true aptitude. it is important to note that as part of the national council of teachers of mathematics standards (1989, 2000), teachers are responsible for assessing students’ disposition toward mathematics. during the bibliotherapy session teachers should also do the mathematics with the students as discussed in the adolescent literature book. bibliotherapy is not just a ‘warm and fuzzy’ approach. it is a serious form of psychological counselling and should be done to help students so to prevent this form of ‘de-geniusing’ in mathematics or any other academic discipline. bibliotherapy then can serve as a sensitive and non-intrusive way to help students solve problems and cope with issues in their personal life, so that they will not affect their academic potential. teachers need to take the time to discuss problems with students and employ techniques for students to share and discuss concerns which are often not brought out due to shame, fear, guilt or worries about fitting in. kevin, like many of our young gifted kids today, dumbed down his true aptitude for mathematics so that their peers would not make fun of him or use him to get the ‘answers’. our mathematically gifted students should be proud of their giftedness in mathematics, and need to learn coping strategies for dealing with the peer pressures that come with a discipline that is not as glamorous as being a movie or rock star or a professional sports player. we need to help young people to be proud of their mathematical giftedness. acknowledgements (back to top) competing interests the authors declare that they have no financial or personal relationship(s) which may have inappropriately influenced them in writing this article. authors’ contributions j.f. was the project leader, and c.k. made conceptual contributions and researched relevant literature. j.f. and c.k. prepared the sample student narrative based on c.k.’s personal interactions. j.f. wrote the manuscript with edits and contributions from c.k. references (back to top) abdullah, m.h. (2002). bibliotherapy. eric digest. bloomington, in: eric clearinghouse on reading, english, and communication. (eric document reproduction service no. ed470712). available from http://www.eric.ed.gov/pdfs/ed470712.pdfableser, j. (2008). authentic literacy experiences to teach and support young children during stressful times. young children, 63, 74–79. bain, s.k., choate, s.m., & bliss, s.l. (2006). perceptions of developmental, social, and emotional issues in giftedness: are they realistic? roeper review, 29, 41−48. http://dx.doi.org/10.1080/02783190609554383 betzalel, n., & shechtman, z. (2010). bibliotherapy treatment for children with adjustment difficulties: a comparison of affective and cognitive bibliotherapy. journal of creativity in mental health, 5(4), 426−439. http://dx.doi.org/10.1080/15401383.2010.527816 burke, a. (2009). gifted and grieving: why it is critical to offer differential support to gifted kids during times of loss. gifted child today, 32(4), 30−37. clement, r. (1991). counting on frank. milwaukee, wi: gareth stevens publishing. cross, t.l. (1997). psychological and social aspects of educating gifted students. peabody journal of education, 72(3/4), 180−200. http://dx.doi.org/10.1080/0161956x.1997.9681873 cross, t.l. (2002). competing with myths about the social and emotional development of gifted students. gifted child today, 25(3), 44−45. cross, t., stewart, r.a, & coleman, l.j. (2003). phenomenology and its implications for gifted studies research: investigating the lebenwelt of academically gifted students attending an elementary magnet school. journal for education of the gifted, 26, 201−220. diezmann, c.m., & english, l.d. 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(2011). content-based curriculum for high-ability learners (2nd edn.). waco, tx: prufrock press inc. appendix (back to top) a sample bibliotherapy lesson plan featuring counting on frank counting on frank by rod clement depicts the story of a middle school boy’s gift of mathematics. he goes through life knowing many things, counting, comparing, and knowing many number facts that all come easily to him. this lesson plan provides a variety of follow-up activities for use during or after reading counting on frank. integrating such activities into teaching allows a teacher to address the feelings of their students as they learn mathematics. the book offers a series of questions at the end of the book called here’s a chance to use your brain! which many young gifted students may enjoy investigating. figure 1: book cover of counting on frank by rod clement. menu of themes or key concepts • everyone has a unique way of looking at the world. • the mathematically gifted should not be embarrassed because they are good at mathematics. • mathematics is used in everyday life events. • some gifted and talented students are embarrassed by their mathematics abilities and tend to diminish their skills or talents. • mathematics can be easy, fun and rewarding. • there are many misconceptions about mathematics. discussion questions • some people say that mathematics is used in everyday life. can you give some examples? • how do you feel when adults say mathematics surrounds us? • do you think people learn mathematics in different ways? why? how? • how do you learn mathematics best? • how do you feel when an adult tells you to use your brain? why? • what do you think is the most frightening thing about mathematics? what is the best thing about mathematics? • describe how your mathematics teacher helps you to use your brain. how can you do this on your own? • do you share your feelings about mathematics with other people? • most people think that the ability to do mathematics in a necessary life skill. how does that belief make you feel about your mathematics ability? do you agree or disagree? • should all smart students be expected to love mathematics? why or why not? • what advice would you give to younger students about learning and enjoying mathematics? • what are some ways you can be proud of your mathematics skills? possible activities • the book concludes with four pages of mathematics problems directly related to the story that give students a chance to use mathematics and numbers in fun ways. the teacher can encourage the students to explore and solve these problems. • students can write their own word problems relating to everyday life events for classmates to solve. • students can write a reflective essay about the importance of mathematics in everyday life and how this affects their own beliefs about their mathematics skills and abilities. • have students write a poem about their mathematical ability and allow time to share with the class. • have students create a picture collage of examples of mathematics use in everyday life. • have students keep a mathematics journal to record their feelings about mathematics and share times when others made them feel like a nerd or bad for being good at it. • have students create a ‘counting’ mathematics song and allow them to perform in front of classmates. • group students into teams to play maths jeopardy. students have to work collaboratively to solve word problems. • invite professional community members to come in to discuss how they use mathematics in their business. • allow students to investigate mathematicians through time and identify some who may have been gifted and have them report back their contributions to mathematics. • student may want to review the book and notice the character and how he is dressed and portrayed. does the book portray him as a ‘nerd’? how would you change this portrayal? interactive mathematics websites that may appeal to the mathematically gifted www.funbrain.com www.coolmath4kids.com www.math.com www.brainpop.com http://nrich.maths.org a dynamic approach to quadrilateral definitions rajendran govender university of the north email: rajengovender@absamail.co.za michael de villiers university of durban-westville email: profmd@mweb.co.za this study examined 18 prospective secondary mathematics teachers' understanding of the nature of definitions, and their use of the dynamic geometry software sketchpad to improve not only their understanding of definitions but also their ability to define geometric concepts themselves. results indicated that the evaluation of definitions by accurate construction and measurement enabled students to achieve a better understanding of necessary and sufficient conditions, as well as the ability to more readily find counter-examples, and to recognise and improve on uneconomical definitions. introduction research by linchevsky, vinner & karsenty (1992), among others, on definitions in mathematics, has indicated that many student teachers do not understand that definitions in geometry have to be economical (contain no superfluous information) and that they are arbitrary (in the sense that several alternative definitions may exist). it is plausible to conjecture that this is probably due to their past school experiences where definitions were largely supplied to them directly. it would appear that in order to increase future teachers' understanding of geometric definitions, and of the concepts to which they relate, it is essential to engage them at some stage in the process of defining geometric concepts. the research reported here concentrated mainly on student teachers’ (prospective mathematics teachers) understanding of the nature of definitions and the development of their ability to evaluate and formulate definitions in a sketchpad context (see govender, 2002). sketchpad was used to expose the students to the process of defining as a creative activity in which students can be fully involved, rather than perpetuating the view of definition as an imposed body of knowledge immune to any change or development. the following research questions were addressed: • what prior understanding of the nature of definitions do student teachers have before being engaged in a process of formulating some definitions for themselves? • to what extent does the student teachers' understanding of the nature of definitions change while involved in a process of evaluating definitions by means of construction, measurement and dragging within a sketchpad context? • how competent are the student teachers in evaluating other definitions after being engaged in the preceding process? definitions because student teachers often meet mathematics structured only as in condensed formal mathematics textbooks, their learning takes place within this structure. the textbooks used in schools give concisely expressed definitions, and this has an effect both on how our students view definitions and how teachers teach. however, this structured approach can easily lead to a common but false perception that there is only one (correct) definition for each defined object in mathematics. the fact that several different (correct) definitions may exist for a particular concept is seldom addressed in such a structured, pre-packaged approach. 34 pythagoras 58, june, 2004, pp. 34-45 mailto:rajengovender@absamail.co.za mailto:profmd@mweb.co.za rajendran govender, michael de villiers a further misconception that can easily develop from a traditionally axiomatically structured approach is that mathematics always starts with definitions, which can give the false impression that definitions of mathematical objects are given a priori in nature. in such a structured approach, students are not brought to realise that definitions do not exist independently of human experience in some "ideal" platonistic world, so that all we can do is to "discover" them. the fact that definitions are not discoveries, but human "inventions" for the main purpose of accurate mathematical communication, is therefore not addressed. fortunately, the national curriculum statement grades 10-12 (schools) lays a foundation for the use of technology such as dynamic geometry software for exploration and investigation. in particular, the grade 10 learners ought to explore necessary and sufficient conditions for the various quadrilaterals and investigate ways of defining various polygons. furthermore, according to the new fet curriculum, grade 10 learners ought to realise that definitions are not absolute but fixed on the basis of principles that will result in conciseness and efficiency. definitions, axioms and preceding theorems thus serve as starting points for deductive arguments in the expansion of the axiomatic system. for the quadrilaterals, decisions as to which system of definitions to use can depend on either a partition or inclusiveness approach, although the latter is favoured in the interests of efficiency. learners need to be made aware of these factors in determining definitions. definitions are important in mathematics, but they ought to evolve naturally from previous knowledge, models or real experiences that the child can relate to. otherwise, confusion reigns. mathematical definitions are generally very concise, contain mathematical terms, and require an immediate synthesis of the information if understanding is to result. however, although research confirms that most children cannot operate on an abstract or formal level until junior high school, we find many textbooks and teachers presenting formal definitions to children before they reach junior high school. in fact, learners’ spatial thinking needs to be relatively mature (welldeveloped) before they are able to decide what the definition of a rhombus (for example) should be. indeed, according to the van hiele theory, they need to be at least at van hiele level 3 (see burger & shaughnessy, 1986; de villiers, 1997). the van hiele theory the van hiele model has important implications for the teaching of geometry. it can be used firstly to guide students to achieve a more sophisticated level of geometric thinking, and secondly to assess students' abilities. the most obvious characteristic of the theory is the distinction of five discrete thought levels in the development of students' understanding of geometry. the levels, labelled "visualization (recognition)," "analysis," "informal deduction (ordering)," "deduction" and "rigor" describe characteristics of the thinking process. assisted by appropriate instructional experiences, the model asserts that the learner moves sequentially from the initial, or basic level (visualization), where space is simply observed and the properties of figures are not explicitly recognised, through the sequence listed above to the highest level (rigor), which is concerned with formal abstract aspects of deduction. few students are exposed to, or reach, the latter level. according to this model, the learner cannot achieve a specific level of thinking without having passed through previous levels. according to the van hiele theory, the main reason the traditional geometry curriculum fails is that it is presented at a higher level than those at which students are operating; in other words, students cannot understand the teacher nor can the teacher understand why they cannot understand. the general characteristics of the first four levels, the ones commonly displayed by secondary students and most relevant to this study, are given: level 1: recognition: students visually recognise figures by their global appearance. they recognise triangles, squares, parallelograms and so forth by their shape, but they do not explicitly identify the properties of these figures. level 2: analysis: students start analyzing the properties of figures and learn the appropriate technical terminology for describing them, but they do not interrelate figures or properties of figures. level 3: ordering: students logically order the properties of figures by short chains of deductions and understand the relationships between figures (for example, class inclusions). level 4: deduction: students start developing longer sequences of statements and begin to understand the 35 a dynamic approach to quadrilateral definitions significance of deduction and the role of axioms, theorems and proofs. the van hiele theory is a useful framework for interpreting and analysing student teachers’ levels of understanding for the following reasons: • it seeks to explain not only why students have trouble in learning but also what could be done to remove these stumbling blocks. • the differences between the levels can be projected in terms of the objects and structure of thought at each level (see fuys et al., 1986). • each level is characterised by a series of specific actions/performances. for example, by using task-based interviews burger and shaughnessy (1986) identified more fully what students do at the first four levels. due to space limitations, we shall only discuss level 3. level 3 formulate economical, correct definitions for figures. are able to transform incomplete definitions into complete definitions and more spontaneously accept and use definitions for new concepts. accept different equivalent definitions for the same concept. classify figures hierarchically; for example, quadrilaterals. explicitly use the logical form if.... then to formulate and handle conjectures, and implicitly use logical rules such as modus ponens. are uncertain and lack understanding regarding the functions of axioms, definitions and proof. • students attempting definitions of concepts would be influenced by their level of understanding. for example, students who are still at van hiele level 1 tend to give visual definitions, e.g. a rectangle which looks like this (draws or identifies a quadrilateral with all angles 90 degrees and two long and two short sides), while those students at van hiele level 2 tend to give correct, uneconomical definitions (eg. a rectangle is a quadrilateral with opposite sides parallel and equal, all angles 90 degrees, equal diagonals, half–turnsymmetry, two axes of symmetry through opposite sides, two long and two short sides, and so on.). students at van hiele level 3 tend to give correct, economical definitions, e.g. a rectangle is a quadrilateral with two axes of symmetry through opposite sides. (compare de villiers, 1997: 15-17). research design the target population the target population of this study was the 18 mathematics education 3 students at umbumbulu college of education (uce), which is located approximately 15km south of the durban international airport, in the kwa makutha township. uce is a three year teacher college which gives students a professional teacher’s diploma. the college prepares its students for teaching at both the junior and senior secondary level. hence, their mathematical preparation is less than what a regular mathematics student gets at a regular university. on the other hand, there is a lot of emphasis on didactics and pedagogy at the teacher training college. characteristics of the subjects (sample) [backgound of students] quadrilaterals like the parallelogram, square, rectangle, rhombus and kite are taught in all schools in south africa. all the students had done mathematics up to matric level at school. many of them had attempted standard grade mathematics and obtained poor passes. a few had attempted higher grade mathematics and also performed poorly (see table 3.1). at college the students in the mathematics method 1 and 2 courses revisited many of the topics from the junior secondary and senior secondary mathematics school syllabi. their first and second year mathematics education courses included many of the learning theories and other associated topics such as mathematical thinking; cooperative learning; investigative mathematics; obe; and assessment. in the final year of study a great deal of time was allocated to calculus. these students were expected to be conversant with most of the aspects associated with quadrilaterals, in particular the properties of rhombuses, the definition of a rhombus and the associated theorems. all students were black and had only attended schools for blacks in their previous classes. there were 10 males and 8 females in the given sample. table 1 shows: 36 rajendran govender, michael de villiers • the gender distribution in the sample • age of the respective students • the students’ mathematics symbols in the senior certificate examination and the grades on which the subject was written (higher grade or standard grade) • the students’ second year college exam mathematics content results. the overall academic performance for each student in the senior certificate examination is poor. the overall academic performance of most of the students in the second year college exam is clearly below average. the students were preparing themselves to write their final college exit exam. they were all willing participants since they all wanted to become computer literate and also wanted to understand the topic better. the topic is not in the college syllabus and the researcher promised the subjects that he would make them computer literate and issue a certificate of participation to each student. many of the students were computer illiterate. during the afternoons and during the mathematics method period the students were exposed to a basic computer literacy programme. the students had no prior exposure to sketchpad but through organised planning they were brought together in the afternoons in order to familiarise them with the general use and application of the computer software – sketchpad. furthermore, they had not been previously exposed to the process of defining. before engaging them in defining rhombi, the students had to be given some knowledge of the properties of rhombuses and associated theorems. this was well within the capabilities of the students. name age gender mathematics higher grade mathematics standard grade second year final second year supplementary buzani 22 f f 40% 51% ephraim 23 m f 42% 39% innocent 21 m f 43% 44% jwara 20 m f 58% khethu 20 m e 54% letha 21 m e 53% mathunzi 23 m e 59% mthembu 23 f e 61% ntombi 22 f f 56% owen 21 m e 63% petros 23 m f 40% 47% sanele 22 m e 50% siboniso 21 m e 59% sifiso 23 m e 62% siyand 23 m f 43% 50% thandiwe 23 f e 52% xolis;e 23 f e 55% fikile 23 f f 51% table 1: participants in the study research approach the developmental research approach of the freudenthal institute at utrecht university in the netherlands was used (gravemeijer, 1994). basically this approach involves the designing, assessment and consequent redesign of curricula for learning and teaching mathematics. use of activities from de villiers (1999) was made in this study, as well as the associated theoretical framework in relation to different functions of proof in mathematics. 37 a dynamic approach to quadrilateral definitions the van hiele theory of learning geometry also provided a useful conceptual framework for interpreting and analysing the student teachers' levels of understanding (see burger & shaughnessy, 1986). data collection both qualitative and quantitative data were collected between february and may 2000 from 18 students, largely by means of one-to-one task-based interviews. the rhombus was chosen for the task-based activity. the researcher chose this quadrilateral because in his experience as a teacher he had found that learners at schools have a lot of difficulty in understanding this concept, and in meaningfully applying the knowledge gained about a rhombus. they also have difficulty in "connecting" the knowledge gained to other quadrilaterals. the collection of data was of a dual nature, i.e. there were written responses to some questions and verbal responses to others. in addition, there were follow up interviews to some of the written responses during the task-based interview. all interviews were tape recorded and transcribed. bell (1995) is of the view that for the sake of accuracy a tape recorder is a vital device, especially in interviews. the collection of data was divided into three sessions as follows: • ascertaining the student teachers' prior understanding of definitions; • engaging them dynamically in the process of evaluating different given descriptions (definitions) for a rhombus; and • assessing their own ability in evaluating other, new definitions for a rhombus. clarification of some terminology this section is included in this paper because it contains definitions intended to be helpful in the analysis. the arbitrariness of mathematical definitions the arbitrariness of mathematical definitions refers to the existence (or choice) of different, alternative but correct definitions for the same concept. for instance, when defining a trapezoid one can define it as a quadrilateral having at least one pair of opposite sides parallel. on the other hand, one can define it, if one wishes, as a quadrilateral having exactly one pair of opposite sides that are parallel. if you choose the first definition, a parallelogram is also a trapezoid. if you choose the second one, it is not. if the concept that definitions are arbitrary is well understood the above fact will not cause confusion, otherwise it might cause a great deal (see vinner, 1991). necessary and sufficient conditions for a condition in a given description (definition) to be sufficient, it must contain enough information (properties) to ensure that not only do we obtain the elements of the set we want to define, but only those elements (and not any others). however, normally we want to use as little information as possible, i.e. only as much as is really necessary. correct definitions a description (definition) which contains conditions (properties) that are sufficient is said to be correct. in a correct definition, all the conditions may be necessary or some of the conditions may be unnecessary, i.e. it is possible to have unnecessary conditions in correct definitions. incorrect definitions a definition is incorrect if it contains an incorrect property or if it contains insufficient properties. incomplete definitions a definition is incomplete if it contains necessary but insufficient properties. so an incomplete definition is also an incorrect one. economical and uneconomical definitions a correct definition can be either economical or uneconomical. an economical definition has only necessary and sufficient properties. it contains no superfluous information. on the other hand, an uneconomical definition has sufficient, but some unnecessary properties. in other words, it contains more information than necessary (redundant information). rationale for use of the words economical and uneconomical it is generally accepted that definitions in mathematics should be minimal (economical). by this we mean that definitions should not contain parts which can be mathematically inferred from other parts of the definitions. for instance if one decides to define a 38 rajendran govender, michael de villiers rectangle in euclidean geometry by means of its angles it is preferable to define it as a quadrilateral with three right angles and not as a quadrilateral having 4 right angles. this is because in euclidean geometry, if a quadrilateral has 3 right angles one can prove that its fourth angle is also a right angle. so when formally defining the meaning of a term it is customary to give only the minimum required to understand the term. this minimality principle is a crucial structural element of mathematics organised as a deductive system. in fact, it shapes the way in which mathematics progresses when it is presented deductively, namely after the definition, theorems which give you additional information about the concept are formulated and proved. linchevsky et al (1992:54) made the following point in regard to definitions being minimal: “if mathematical definitions were not minimal we would have to prove their consistency. for instance, if you define an equilateral triangle as a triangle whose sides are congruent and all its angles are congruent then you have to proceed by showing that these two properties can "live together." the most appropriate way of doing it in this case is to show that if all sides of a given triangle are congruent then its angles are also congruent. therefore, what is the point of defining an equilateral triangle by both its sides and its angles if you should prove exactly the same theorems you would have to prove when going the minimal way? being minimal is being economical”. however, there are a few cases in geometry where definitions are not minimal. a familiar one, perhaps, is the way in which some textbooks define congruent triangles. for example, “congruent triangles are triangles which have corresponding sides congruent and corresponding angles congruent” (gonin et al, 1974). we know, though, that it is sufficient to require less than that for two triangles to be congruent. this is expressed by each of the four congruence axioms. data analysis and findings the analysis and findings will be presented in the context of each of the critical questions. student teachers' prior understanding of definitions two of the tasks used are given and discussed below. t foll h defi leth xoli . t info rhom nec app h defi mat nto t suff or t t g (n task 1 how would you describe what a rhombus is, over the phone, to someone who is not yet acquainted with it? he students’ responses were classified into the owing categories ere are some examples of incomplete or wrong nitions: a: i would say a rhombus is a four-sided figure. it has both pairs of opposite sides equal. it also has both pairs of opposite sides parallel. sle: a rhombus is a parallelogram with four sides. both opposite sides are equal and both opposite angles are equal and opposite sides parallel hese descriptions clearly contain too little rmation to guarantee the construction of a bus. these student teachers seem to view essary conditions as sufficient conditions and ear to be operating only at van hiele level 1. ere are some examples of typical uneconomical nitions: hunzi : i would tell him/her that it is a four sided, figure with all sides equal. if you join the opposite angles, the angle at the centre where the lines bisect each other will be a right angle and diagonals bisect the angles of the rhombus. mbi: a rhombus is a four-sided figure, with diagonals bisecting each other at right angles and with each pair of opposite sides equal and parallel. hese student teachers provided more than the icient information required to construct a rhombus o deduce the other properties from it, and therefore ask 1 incomplete or wrong definitions uneconomical definitions economical definitions roup =18) 4 13 1 39 a dynamic approach to quadrilateral definitions would appear to be operating at van hiele level 2 (compare burger & shaughnessy, 1986). the following student teacher was the only one to provide a correct, economical definition: sifiso: i would say a rhombus is a closed four sided figure with all its sides equal. their selection of incomplete descriptions (definitions) makes it clear that these student teachers don’t understand that a description (definition) must contain necessary and sufficient properties. whilst many students chose various options to describe a rhombus, none of the students chose the full complement of the correct descriptions, namely c, d, f, g. this therefore suggested that the students did not have a well-developed ability to evaluate definitions. since this definition does not contain any redundant or superfluous information, it would appear that the student teacher is operating at van hiele level 3. co na on ma ch co sel • • • eight students selected: (b) a rhombus is any quadrilateral with perpendicular diagonals. a single student teacher, owen, chose only one description. the others, by choosing more than one, seemed to have at least an intuitive understanding of the arbitrary nature of definitions. summary • the majority of the student teachers appeared to have an intuitive understanding of the arbitrary nature of definitions. • several student teachers understood a definition (description) of a given figure to be a list of properties of that given figure, which in fact is an uneconomical way to describe a figure (thus suggesting van hiele level 2 understanding). • some student teachers incompletely define (describe) figures by viewing necessary conditions as sufficient conditions. constructively evaluating different definitions for a rhombus the student teachers were next given a number of 40 task 2 which of the following descriptions do you think you would be able to use? circle these descriptions. a. a rhombus is any quadrilateral with opposite sides parallel. b. a rhombus is any quadrilateral with perpendicular diagonals. c. a rhombus is any quadrilateral with two perpendicular axes of symmetry (each through a pair of opposite angles). d. a rhombus is any quadrilateral with perpendicular, bisecting diagonals. e. a rhombus is any quadrilateral with two pairs of adjacent sides equal. f. a rhombus is any quadrilateral with all sides equal. g. a rhombus is any quadrilateral with one pair of adjacent sides equal, and opposite sides parallel. none of the student teachers selected the full mplement of correct descriptions (definitions) mely c, d, f and g. however, three students selected ly correct descriptions (definitions). for example, thunzi chose d and f, owen chose g whilst sanele ose d and f. the remaining 15 students chose at least one rrect description (definition), although they also ected incomplete descriptions. for example: twelve students selected: (a) a rhombus is any quadrilateral with opposite sides parallel. eleven students selected: (e) a rhombus is any quadrilateral with two pairs of adjacent sides equal. prerecorded sketchpad scripts from de villiers (1999), and asked the following main questions while playing the scripts: a. does the script construct a correct rhombus? b. which description in the table matches the script? c. drag the figure. does it always remain a rhombus? d. is the given information sufficient for the construction of a rhombus? if yes/no, why? e. is all the given information necessary for the construction of a rhombus? f. is the description correct? if yes/no, why? g. is the description economical or uneconomical? why? rajendran govender, michael de villiers although initially the student teachers required some guidance in constructing the required givens (prerequisite points) for each script, they quickly became independent as they progressed through the 7 scripts. working through each script provided good learning opportunities for the student teachers to check whether the conditions for each script were sufficient to produce a rhombus. due to space limitations, we shall here only discuss the script rhombus 7. an example of an on-screen sketch produced by this script is shown in figure 1 (though appearance may vary depending on the relative positions of the pre-requisite points a and b). when the construction was finished, the researcher firstly questioned the student teachers about the displayed lengths and gradients on the screen. the student teachers showed understanding of the displayed measurements. upon asking the students whether the script constructed a correct rhombus, all responded that it was correct, apparently judging purely from a visual perspective. only upon encouragement did the students check out their claim by measuring the sides to see if they were really all equal. furthermore, all 18 student teachers matched the script rhombus 7 correctly to description g. the student teachers were then requested to drag the figure task 3 one way of testing a description is to construct a figure complying with the description to see if it really gives the desired figure (we will use sketchpad). open the scripts rhombus 1.gss (windows), and go on to check the descriptions a-g. construct the appropriate givens required for each script, and click on the step button repeatedly to make each script construct its figure. when the construction is finished, match each script with a description in the table. drag the figure to see if it always remains a rhombus. in the table below, cross out the script names of any scripts that construct figures that are not always rhombuses. script description (a-g) rhombus 1 rhombus 2 rhombus 3 rhombus 4 rhombus 5 rhombus 6 rhombus 7 on sketchpad and observe whether it always remained a rhombus. after the student teachers had dragged the figure around numerous times, they were confident figure 1: example of an on-screen sketch a b c d m ac = 3.715 cm m ab = 3.715 cm slope ac = 0.442 slope db = 0.442 slope ab = -1.291 slope cd = -1.291 41 a dynamic approach to quadrilateral definitions that this figure always remained a rhombus. the student teachers also indicated that the given information was sufficient as well as necessary. some selected extracts from typical responses are given below: researcher: i now want you to look for a description in the list that fits in with rhombus 7? xolisile: g. researcher: ok, right, g is correct. i want you to focus on the definition or description there and i want you to tell me if the given information is sufficient for the construction of a rhombus? xolisile: yes. researcher: why? xolisile: the script always constructs a rhombus. if you drag it you always get a rhombus. researcher: ok. is the information necessary for the construction of a rhombus? xolisile: yes, it is necessary. all the student teachers identified it is a correct, economical definition because it contains necessary and sufficient conditions, which is indicative of van hiele level 3 thinking. researcher: is it an economical definition? xolisile: yes. researcher: why? xolisile: it is necessary and sufficient. researcher: would you say that the definition is a correct definition of a rhombus? mathunzi: ja, i would say it is a correct definition of a rhombus. researcher: would you say it is a correct economical definition? mathunzi: ja, it is economical. researcher: why do you say economical? mathunzi: aaaah. it is sufficient and necessary. d, des tha ide nam summary the student teachers’ responses to task 4, in comparison to their earlier responses to task 2, clearly suggest the following as a result of being involved with the process of constructive evaluation of definitions in a dynamic geometry environment: • the student teachers appear to have developed a deeper understanding of the arbitrary nature of definitions. • they showed improved ability to select correct alternative definitions for a rhombus. • the scripts and the use of dynamic dragging provided the student teachers with the opportunity to check whether the conditions in the given definitions were sufficient for the accurate construction of a rhombus. • as a consequence, the student teachers exhibited a better understanding of when conditions are: • necessary and sufficient • necessary but not sufficient • sufficient but not necessary (and therefore also some ability to distinguish between economical and uneconomical definitions) student teachers' competency to assess other definitions after the preceding process task 5, consisting of 2 items, was utilised to determine how competent the students were in evaluating other possible definitions for rhombi after the preceding learning experience. it was given to the students immediately after all the interviews were completed. this task comprised only written responses which were done by all the students at one sitting, though all the students had access to sketchpad whilst answering these questions. a discussion of one of the items from task 5 is provided below. 42 task 4 list the descriptions from a-g that you think best describes a rhombus seventeen out of 18 students listed all four (i.e. c, f and g) correct descriptions as the ones that best cribe the rhombus in task 2. only one student, ndiwe, did not choose all four, but managed to ntify three out of the four correct descriptions, ely c, d and f. t desc task 5 consider the following definition: “a rhombus is any quadrilateral with all sides equal, opposite sides parallel, and perpendicular and bisecting diagonals”. a. what are your comments about this definition? b. is it a good definition? c. if not, how would you change it? his item gives a correct but uneconomical ription of a rhombus and therefore was designed rajendran govender, michael de villiers to test the students’ ability to recognise that the description is correct, but uneconomical (i.e. contains more information than necessary). fourteen out of 18 of the respondents recognised that this description was uneconomical (sufficient but has unnecessary properties). this item was also implicitly intended to evaluate the students’ understanding of a “good” definition and it was encouraging to note that all fourteen students indicated that the description was not a good definition. the following are typical examples: jwara's response: a. correct definition. but it is sufficient and unnecessary. b. no. fikile's response: a. this is a correct uneconomical definition because it includes unnecessary properties and can be made economical in a number of ways . b. no, it is not a good definition. sifiso's response: a. this definition is correct but it also contains unnecessary information therefore it is uneconomical. b. no, because it is uneconomical. however, not all fourteen who identified it as an uneconomical definition gave clear responses. indeed some showed either a measure of confusion or difficulty in clear formulation. extracts from these four responses are now presented and discussed individually. mthembu's response: a. the definition is correct but some of the information is not there (uneconomical). b. no. though mthembu says it is uneconomical, he says “some of the information is not there”. this may suggest that he is actually operating at van hiele level 2, and might prefer more properties (rather than less) in the definition, rather than simply not understanding the meaning of the term uneconomical. letha's response: a. the definition is uneconomical for a rhombus. it is insufficient but necessary. b. no. similarly, letha stated that it was uneconomical, but then contradicts himself by claiming that the information is insufficient. it is difficult to ascertain whether the difficulty is conceptual or just a matter of a minor confusion of correct mathematical terminology. siyande’s response: a. uneconomical because the definition is insufficient. b. no. just like letha there is a contradiction in this statement as a definition cannot be (correct) economical, as well as insufficient. it appears that siyande might have been reasoning as follows: if the properties are more than sufficient then it means that it is “not sufficient” which he then equivalently matched with the word “insufficient”. however, without further interviewing it is impossible to determine whether the problem is conceptual, linguistic or a mixture of both. here are some examples of the four responses that did not identify the definition as uneconomical: innocent: a. this is a good definition. in fact it is a correct economical definition as it contains necessary and sufficient conditions. b. yes. ephraim: a. i think this definition is necessary and sufficient because it include all the properties of a rhombus. b. yes. ntuli: a. economical correct, definition contains necessary & insufficient conditions. b. no. (ntuli may have thought it was not a good definition because he personally preferred another correct definition) item 5(c) was intended to assess whether the student teachers could provide a shorter, correct definition of a rhombus. nine of the students came up with shorter, correct economical descriptions by 43 a dynamic approach to quadrilateral definitions leaving out some properties. six of these students chose to define the rhombus in terms of its sides (a fairly standard textbook definition) as follows: a rhombus is any quadrilateral with all sides equal. two students preferred to define a rhombus as a parallelogram with one pair of adjacent sides equal (which is also used in some textbooks). for example: a rhombus is any quadrilateral with one pair of adjacent sides equal, and opposite sides parallel. however, sboniso defined a rhombus just in terms of its diagonals (a seldom used definition in textbooks). for example: a rhombus is any quadrilateral with perpendicular bisecting diagonals. six students came up with different shorter versions, which were either still uneconomical, or insufficient, or contained an incorrect property. for example: uneconomical: mthembu for example gave a definition that was correct, but still uneconomical: a rhombus is any quadrilateral with both pairs of opposite sides equal and parallel and with diagonals bisect each other perpendicularly. note that the definition would have been economical (and a non-standard definition) had mthembu left out the condition that the diagonals bisected each other. ntombi’s response: a rhombus is any quadrilateral with all sides equal, opposite sides parallel and perpendicular, bisecting diagonals, forming two axes of symmetry. ntombi seemed to only list all the properties of a rhombus she knew, making it even more uneconomical. her reasoning appears to be at van hiele level 2. incomplete (insufficient): petros gave a definition that was insufficient. for example: a rhombus is any quadrilateral with opposite sides parallel. thandiwe also gave a definition that was insufficient. for example: a rhombus is any quadrilateral with all pairs of opposite sides parallel and bisecting diagonals. however, it was encouraging to note that half the student teachers presented correct economical definitions in 5(c), which is reflective of van hiele level 3 thinking. summary • the majority of the student teachers were able to identify (realise) the description in task 5 as correct but uneconomical, and half of them were able to change it to a correct economical description of a rhombus in 5(c). • it would appear that this improved ability to evaluate and improve a given definition could in some measure be attributed to the earlier learning activities, i.e. the earlier construction, measurement and dragging activities with sketchpad. concluding remarks the use of construction and measurement to evaluate the correctness of geometric statements (conjectures) before proofs are done is of course common practice among mathematical researchers. as a teaching approach it is also not new. for example, a similar approach was used effectively in the useme teaching experiment during 1977/78 (see human & nel et al, 1989). similarly, smith (1940) reported marked improvement in pupils' understanding of "if-then" statements by letting them first make constructions to evaluate geometric statements. in his research he found that it enabled pupils to learn to clearly distinguish between the "given condition(s)" and the "conclusion(s)", and laid the conceptual groundwork for an improved understanding of the eventual deductive proof. however, this study is markedly different in that it took place within the context of dynamic geometry, where a geometric configuration can be continuously dragged into different shapes to check for invariance. ideally, students should test geometric statements by making their own constructions within sketchpad. however, since this requires a rather high level of technical knowledge of the software, it was decided to provide them with ready-made scripts that they could play through step by step and observe as the figure was gradually constructed. as the scripts are dependent on the arbitrary construction and positioning of the "given points", they sometimes produce crossed quadrilaterals which was a little confusing to some students. accordingly, in the revised version of this activity (see de villiers, in press) use will instead be made of the "hide/show" button facility of sketchpad to produce figures step by step, ensuring that they all initially appear to be a rhombus. only upon further dragging would students 44 rajendran govender, michael de villiers then be able to ascertain whether it always remains a rhombus, and therefore whether the conditions are really sufficient. it should also be noted that since the dynamic geometry software provided conviction to all the student teachers, the role of the eventual deductive proofs (i.e. to prove the sufficiency of the definitions) was conceptualized as that of systematization rather than that of verification. although it was not a main focus of this study, the issue of hierarchical vs. partition definitions for a rhombus arose quite a few times while interviewing (or in discussion with) the student teachers. however, the dynamic nature of the rhombi constructed in sketchpad seemed to make the acceptance of the hierarchical classification of a square as a special rhombus far easier than in a traditional non-dynamic environment, as the student teachers could easily drag the constructed rhombus until it became a square. this is, however, a matter for further research. acknowledgement this research was partially funded by a national research foundation (nrf), pretoria, south africa grant from the spatial orientation & spatial insight (sosi) project, coordinated by prof. dirk wessels (unisa), dr. hercules nieuwoudt (puche) and prof. michael de villiers (udw). the opinions and findings do not necessarily reflect the views of the nrf. references bell, j., 1995, doing your research project. buckingham: open university press burger, w.f. & shaughnessy, m., 1986, “characterizing the van hiele levels of development in geometry” in journal for research in mathematics education, 17(1), pp. 31-48 de villiers, m., 1997, “the future of secondary school geometry” in pythagoras, 44, pp. 37-54 de villiers, m., 1998, “to teach definitions in geometry or teach to define?” in olivier, a & newstead, k. (eds.) proceedings of 22nd pmeconference, university of stellenbosch, 12-17 july 1998, vol. 2, pp. 248-255 de villiers, m., 1999, rethinking proof with geometer's sketchpad, key curriculum press, usa de villiers, m. (in press), rethinking proof with geometer's sketchpad 4, key curriculum press, usa gonin, a.a., archer, i.j.m., slabber, g.p.l., la rey nel, g.d.e., 1974, “congruent triangles” in modern graded mathematics for standard 8 (second edition), elsies river: national book printers limited govender, r., 2002, student teachers' understanding & development of their ability to evaluate & formulate definitions in a sketchpad context, unpublished master's thesis, university of durban-westville gravemeijer, k., 1994, “educational development and educational research” in journal for research in mathematics education, 25 (5): pp. 443-471 human, p. g. & nel, j. h. in co-operation with m. d. de villiers, t. p. dreyer and s. f. g. wessels., (eds.), 1989, useme teaching experiment. appendix a: alternative geometry curriculum material. rumeus curriculum material series no. 11, university of stellenbosch linchevsky, l., vinner, s., karsenty, r., 1992, “to be or not to be minimal? student teachers views about definitions in geometry” in geeslin, w., graham, k. (eds.), proceedings of the sixteenth international conference for the psychology of mathematics education , vol. 2, pp. 48 55. durham, usa department of education, 2002, national curriculum statement grades 10 12 (schools): mathematics (draft), pretoria: department of education smith, r. r., 1940, “three major difficulties in the learning of demonstrative geometry” in the mathematics teacher, 33, pp. 99-134, 150-178 vinner, s., 1991, “the role of definition in the teaching and learning of mathematics” in d. tall (ed.), advanced mathematical thinking, kluwer academic publishers, pp. 65-81 45 a dynamic approach to quadrilateral definitions introduction definitions the van hiele theory level 1: recognition: students visually recognise figures by level 2: analysis: students start analyzing the properties o level 3: ordering: students logically order the properties o level 4: deduction: students start developing longer sequenc level 3 research design the target population research approach data collection clarification of some terminology this section is included in this paper because it contains d the arbitrariness of mathematical definitions necessary and sufficient conditions correct definitions incorrect definitions incomplete definitions economical and uneconomical definitions rationale for use of the words economical and uneconomical data analysis and findings student teachers' prior understanding of definitions task 1 summary constructively evaluating different definitions for a rhombu student teachers' competency to assess other definitions aft uneconomical: mthembu for example gave a definition that was incomplete (insufficient): petros gave a definition that was summary concluding remarks acknowledgement references article information authors: piera biccard1 dirk wessels1 affiliations: 1department of curriculum studies, research unit for mathematics education, stellenbosch university, south africa correspondence to: piera biccard email: biccardwps@gmail.com postal address: private bag x1, matieland 7602, south africa dates: received: 31 mar. 2015 accepted: 01 sept. 2015 published: 07 dec. 2015 how to cite this article: biccard, p., & wessels, d. (2015). student mathematical activity as a springboard to developing teacher didactisation practices. pythagoras, 36(2), art. #294, 9 pages. http://dx.doi.org/10.4102/pythagoras.v36i2.294 copyright notice: © 2015. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. student mathematical activity as a springboard to developing teacher didactisation practices in this original research... open access • abstract • introduction • didactisation and mathematical work of teaching framework • active students as a theoretical construct in teaching • the study    • participants and research design    • validity and reliability • the results • conclusion • acknowledgements    • competing interests    • authors’ contributions • references abstract top ↑ this article is part of a larger study on teacher development. the main study investigated teacher development within primary school mathematics teachers’ classrooms to determine if teaching practices could be enhanced through a didactisation-based programme. it sought to develop teachers within their own environments and classrooms. design research (both designing the conditions for change and studying the results of those conditions) enabled the researchers to design a programme that was congruent with teachers’ own needs and experiences. the programme ran for a period of a year with regular contact between the teachers and the researcher conducting the programme (the first author). the programme set out nine didactisation practices: active students, differentiation, mathematisation, vertically aligned lessons, accessing student thinking and ideas, probing student thinking and ideas, connecting student ideas, assessing students and reflecting on practice. one practice, student activity, is the focus of this article. it was found that by initiating discussion and cognitive conflict in teachers by using modelling problems, and further allowing teachers to observe pupils working in groups with modelling problems, teachers were starting to incorporate the didactisation practices within their own classrooms. this article documents specifically the fundamental role of student mathematical activity and the importance of improving student mathematical experiences, both for teacher development and for student mathematical learning. the study may be valuable in structuring and planning further effective teacher development programmes. introduction top ↑ mathematics teaching and learning is a priority in many countries around the world. south africa is no different and the latest national assessments indicate mathematics education in south africa is in dire straits, with the grade 9 average for mathematics being 14% (department of basic education, 2013). although the situation is complex, and many factors contribute to this, mathematics teachers need to be developed so that a change in performance can be realised. in this article, a teacher development programme is described that formed part of a larger study on didactisation practices (biccard, 2013). the focus of this professional development programme was the didactisation practices of primary school mathematics teachers (grade 5 and grade 6). these practices were filtered from treffers’s (1987) term ‘didactisation’ and wilson and heid’s (2010) ‘mathematical work of teaching’. design research allowed an integrated, iterative research design between the researcher and five teacher participants. teacher change in the study was catalysed through teacher exposure to mathematical modelling tasks and being involved in in-depth reflective discussions. the focus of this article is on how student activity (as one of the didactisation practices) was optimised by teachers and how this effected a change within the classroom culture and within teachers’ own development of didactisation practices. traditional mathematics classrooms are characterised by low student-initiated mathematical activity. lessons often follow a predictable teacher explanation session followed by routine procedures completed by students. mathematics presented to students in these classrooms is largely deductive: a general rule is taught upfront and practised by means of various examples in what freudenthal (1991) describes as an ‘anti-didactical inversion’. there is a need to create a more constructive type of classroom, where students are active participants in their own mathematical learning: classrooms where student ideas are stimulated, and teacher-student as well as student-student discussions are rich ground for mathematical thinking. we need classrooms where the type of mathematical thinking required of students is more inductive, more personal and more meaningful. the programme envisaged teachers encouraging students to become more mathematically active in their lessons. how, and to what extent, this could happen was an important question in the design of the programme. the programme sought to encourage teachers to incorporate more opportunities for their students to: think and talk about what they were doing, arrange the learning process and be involved in constructing their own meaning. the effect of increasing student mathematical activity in mathematics classrooms needed to be gauged and documented. mathematical activity where students are exploring, conjecturing and making connections is more difficult to incorporate in lessons than simply solving mechanical problems. the effect of broadening student mathematical activity experiences on both the nature of student learning and teacher development needed to be studied. it was hoped that the classrooms would change from being based on teacher explanations only, to being more learner centred and more problem centred. the spin-off of making classrooms more learner participatory is that the role of both the students and the teachers had to be reconsidered. it was through mediating these changes that teachers would reflect on and develop their practices. a number of questions are scrutinised in this article. can teachers change the level of student mathematical activity in their classrooms? what is the nature of this change? how does this affect the nature of mathematical learning in these classrooms? how would an increase in student activity assist teachers in developing other didactisation practices? the level, type and nature of student activity is the focus of this article. didactisation and mathematical work of teaching framework top ↑ realistic mathematics education (rme) provides a basis for the theoretical framework of the article. rme is a teaching and learning theory that is underpinned by the work of freudenthal, treffers, gravemeijer and others. the principles of guided re-invention, self-developed models and didactical phenomenology (gravemeijer, 1994) were incorporated into the study. at the heart of rme theory is the concept of mathematisation and how important it is for students to re-invent mathematisation. treffers (1987) defines didactisation as ‘the essence of didactical action which makes mathematisation possible’ (p. 58). furthermore, treffers outlined four components of didactisation from a rme perspective. they are: active students – what mathematics are the students working through or dealing with? is it procedures only or do they have to engage with the material and make sense of it? are there different ways of approaching the problem or are they simply repeating a procedure? is there a need to discuss what they are working on? is there more to discuss than only the solution? differentiation – does the problem lend itself to a variety of approaches? can students with different understanding deal with and access the problem? mathematisation – this is at the heart of a mathematics lesson. what mathematics are students learning and how are they learning it? can they bridge from a contextual problem to mathematics? how can they shorten their methods by using more elegant solution paths? vertically aligned lessons – this refers to the teaching and learning trajectory a teacher has set for students, whether it is a short-term trajectory through a particular section of work or a long-term yearly trajectory. in terms of mathematics, a progressive mathematisation is sought where there is a building of mathematical structures or a deeper, more complex understanding of surface features of a problem. essentially, does the teaching trajectory allow for scaffolding or reflecting on student informal ideas to more formal mathematics? these components formed an integral part of the didactisation practices envisaged for teacher development in the study. they encapsulate essential features of mathematics classrooms where student involvement and student understanding is central. understanding mathematisation is fundamental to theorising about teacher actions and teacher decision-making that may lead to significant learning in a mathematics classroom. freudenthal (1993, p. 72) formulates a description of mathematisation as turning ‘mathematically underdeveloped matter into more distinct mathematics’ while he stipulates that mathematisation is translated into ‘reinventing’ (freudenthal, 1991, p. 67). mathematisation is therefore closely tied to student activity and student thinking and raising student understanding of mathematics to more abstract levels. gravemeijer (1994) sets out four level-raising features of mathematisation: generalising – looking for analogies classifying and structuring. certainty – reflecting, justifying, proving (using a systematic approach, elaborating and testing conjectures). exactness – modelling, symbolising and defining. brevity – symbolising and schematising (developing standard procedures and notations) (p. 82). the level of thinking in mathematics is raised from looking for analogies and structuring to reflecting and proving and finally to symbolising and schematising. this level raising, according to gravemeijer specifically involves generalising and formalising. formalising embraces modelling, symbolising, schematising and defining while generalising refers to a ‘construction of connections’ and not an application of general knowledge (gravemeijer, 1994, p. 83). teacher development programmes within mathematics education would benefit from keeping these descriptions close to their aims and focus. teacher development needs to be centred on student mathematising and how teachers and students together can raise the level of thinking through solving contextual problems. level raising in mathematics classrooms presupposes mathematical activity on the part of the students. teacher development should consider teachers and students as active reflective participants within their own classrooms as suggested by fosnot, dolk, zolkower, hersch and seignoret (2006): we engaged in-service teachers in experiences that involved action, reflection and conversation within the context or learning/teaching. we took the perspective that teachers need to construct new gestalts, new visions of mathematics teaching and learning. to do this they need to be learners in an environment where mathematics is taught as mathematising, where learning is seen as constructing in terms of professional development of teachers. (p. 7) to further understand and guide the actions and decisions of mathematics teachers, a framework that focused on the work of teachers was also needed to understand the very complex, messy domain of mathematics classrooms. wilson and heid (2010) provided a proficiency framework to describe teacher knowledge, actions and skills. as part of the framework, they described the mathematical work of teaching (p. 6). this aspect of their framework reminds us that teaching comprises essential components that can be developed in all teachers. these components serve as good starting points in teacher development programmes since they focus on specific teacher actions and decisions-making ideas. the mathematical work of teaching framework comprised the following teacher actions: probe mathematical ideas access and understand the mathematical thinking of learners know and use the curriculum assess the mathematical knowledge of learners reflect on the mathematics of practice. a proficiency framework was preferred to a knowledge-only framework since it allowed closer access to identifying and understanding authentic mathematics teacher decisions that translate into actions in classrooms. the mathematical work of teaching framework made it possible for the researcher to prepare for the practical aspect of the teacher development programme and to bridge the day-to-day teacher actions with the theoretical aspects of teaching. on reflection of the above components, treffers’s vertically aligned lessons and wilson and heid’s ‘know and use the curriculum’ were considered to be of a similar nature so that they are grouped together. furthermore a teacher’s proficiency or competence to connect student ideas and thinking is vital to effective mathematics teaching and learning so it was included as a didactisation practice (biccard, 2013). this resulted in nine didactisation practices drawn from treffers and wilson and heid. those from treffers relate to the what of effective mathematics teaching and those from wilson and heid relate to the how of effective teaching (biccard, 2013, p. 49). the nine didactisation practices incorporated in the main study are: student activity differentiation mathematisation vertically aligned lessons access student thinking and ideas probe student thinking and ideas connect student ideas assess student thinking reflect on practice. as the professional development programme progressed, it was found that an improvement in student activity was the first to develop. furthermore, it was found that other didactisation practices are stimulated by, and could further be developed through, meaningful student activity and teacher reflection on this activity. this resulted in a hierarchy of didactisation practices with student activity at its base (biccard, 2013, p. 278; reproduced as figure 2). meaningful teacher development that results in effective classroom implementation of reform ideas requires firstly that students are mathematically active. active students as a theoretical construct in teaching top ↑ starting with a basic definition of activity allows one to place it within the mathematics learning domain. according to the concise oxford english dictionary activity includes the definition: ‘the condition in which things are happening or being done’. this means that activity can be graded by the conditions under which the activity takes place. to extend this would mean that the activity needs to be judiciously placed and incorporated by the teacher within a lesson to maximise its effect in learning. it would also mean that the conditions in a lesson are guided by a teacher’s decisions, actions, beliefs and aims. to change the conditions under which an activity takes place, the teacher’s decisions before and during that lesson would need to be examined and developed. further reflection after a lesson would refine these ideas. in a traditional setting mathematical learning takes place under conditions where teachers explain set methods and students repeat these procedures. however in problem-centred environments the teachers present students with contextual problems. through interacting with the problem and socially developing meaning with each other, students construct mathematical concepts. the conditions under which teaching and learning take place in these settings is different and so for teachers and students to adapt to a reformed setting is significant. brousseau (1997) defines the term devolution of a problem as the students accepting responsibility for solving a problem (p. 30). he explains that students have to adapt to this new role or milieu which constitutes an adidactical situation. a meaningful learning situation is one in which the conditions in which things happen are different from conditions where students only answer drill-type questions. hmelo-silver (2004) confirms the role of responsibility in making students more active in lessons. in traditional classrooms, teachers assume almost all the responsibility of the mathematical work to be done; their students are largely passive observers of their teacher taking part in all the ‘action’. handing over the activity of doing mathematics to the students implies handing over responsibility to the students. in traditional classrooms teachers only hand over repeating the procedure they have just shown, but the responsibility of coming up with the mathematical procedure or method is the teacher’s. the traditional teacher believes that students are not able to come up with significant ways of thinking or working mathematically; they have to be shown. allocating the responsibility of thinking mathematically to the students may be accompanied by feelings of loss of control by the teacher and a feeling of being overwhelmed by the sense of responsibility by the students. this is in line with stigler and hiebert’s (1998) formulation that teachers do not want students to struggle with mathematics; teachers see their role as having to make things as easy as possible for students (p. 3). the roles of both the teacher and the students are modified under these changing conditions. when teachers learn to trust the mathematical thinking of their students and to value the meaningful learning that takes place through problem-solving, teachers will change their goals for mathematics lessons. when teachers place students in mathematically active roles in their classrooms, not only is mathematical thinking enhanced but teachers develop as a result too. skemp (1986) explains that traditional mathematics teaching is dominated by verbal-algebraic imagery and not visual imagery since the verbal-algebraic is easier to communicate and is analytical, sequential and logical while visual imagery is integrative, simultaneous and intuitive (p. 104). the reliance on verbal definitions affects students’ meaningful learning in traditional classrooms since, ‘concepts of a higher order than those which people already have cannot be communicated to them by a definition’ (skemp, 1986, p. 25). however, traditional teaching that is definition based without the supporting student interaction and activity is still a major focus in mathematics classrooms today. student learning and a change of teacher practices may be enhanced by moving from a verbal and static approach towards a more dynamic and active approach. treffers (1987) advocated that students also work interactively with each other. he linked the role of meaningful student activity to mathematisation (p. 249). he maintained that interactive learning would allow students to either shorten their own methods or to become aware of the advantages and disadvantages of their own ideas and methods. this vertical mathematisation is ultimately the goal of mathematics teaching and learning. biccard (2013) analysed pegg and tall’s (2005) fundamental cycle of concept construction and found that in the comparison of concept construction by several authors, the term ‘process’ was critical in vertical mathematisation. the term process was accepted as performing a composite cognitive activity (biccard, 2013, p. 59); when students reflected on the effect of their activities or actions, this assisted in building concepts or procepts (pegg & tall, 2005, p. 473). furthermore, freudenthal’s (1991) analysis of the van hiele levels in learning processes reminds us that the student’s activities on a lower level become the objects of analysis on a higher level (p. 98). it would therefore appear that part of the conditions under which student activity becomes meaningful in mathematical learning is that of reflecting on their activity and the effect of their actions. classrooms that are teacher-directed and teacher-presented allow very little student discussion or deep reflection. vygotsky (1978) specifically forges the link between speech and action (p. 25). he proposes that speech and action are part of the same psychological function and that the more demanding the activity, the more important the role speech plays in solving the problem. traditional classrooms that are largely silent may pose problems for students in understanding significant mathematical concepts. it also allows less insight into the mathematisation process. students may also struggle with real-world application problems. lesh and doerr (2003) describe modelling and model-eliciting activities. the tasks are rich, real-world context-based and require groups of students to solve them. the tasks are designed so that students have to formulate a situational model of the problem as well as a generalisable model for similar situations. lesh and doerr explain that the models students construct are ‘sharable, manipulatable, modifiable, and reusable tools for constructing, describing, predicting or controlling mathematically significant systems’. modelling is the type of task that changes the conditions under which mathematical activity takes place. it creates an innovative learning environment where students use mathematics towards a significant end. this is also true of teacher learning through professional development programmes. modelling tasks were used during the teacher development programme in the main study to stimulate paradigm shifts within practising teachers and to catapult teacher thinking about the nature of mathematical activities. the first step in developing teachers’ didactisation practices would be to help them to understand the importance of mathematically meaningful student activities whereon they could build their didactisation proficiencies. the professional development programme designed in this study changed the conditions under which teachers learn by presenting them with modelling problems; firstly, they solved the problems as a group and, secondly, they observed groups of students solving the same problem. this led to teachers reflecting on these activities and making different decisions for their lessons in ways that each teacher decided according to their own goals and resources. biccard (2013, p. 291) concludes that modelling tasks became a knowledge resource for teachers. she further documents (p. 189) that modelling problems made teachers stop and consider their beliefs and orientations about mathematics teaching. their observation of students solving modelling problems became rich ground for discussing the nature of different mathematical problems in classrooms. other studies (schorr & lesh, 2003; stillman, 2010) have also shown that modelling tasks have a profound influence on teacher thinking and that teachers themselves experience discomfort (ng, 2010, p. 142) with mathematical tasks of a different nature to traditional tasks. the study top ↑ participants and research design the study involved five primary school mathematics teachers. they were teaching grade 5 and grade 6 students (aged 10–12 years). the five teachers volunteered to be part of the programme, which lasted about 9 months over a period of one year. teachers were briefed on the intentions and aims of the study and they signed consent documentation. they understood that all references were anonymous and they could withdraw at any time. the study implemented the principles of design research. this means that the researcher had to create an innovative learning environment for the teachers and study the conditions that made the environment conducive to professional development and, similarly, study the effect of the professional development. cobb, confrey, disessa, lehrer and schauble (2003, p. 9) outlined five features of design research (also known as design experiments): design experiments develop local theories about learning processes and how to support those learning processes. design experiments are highly interventionist in nature. design experiments are both prospective and reflective. they include a hypothetical learning trajectory as well as a retrospective analysis. design experiments have an iterative design. the theory generated by design experiments is relatively humble. bakker (2004) explains that there are three phases in design research: a planning and preparation phase, a teaching experiment and a retrospective analysis. these phases are iterated so that the participants can become active members of the design and so that the researcher can make changes to the research design so that the research question can be answered or more robust analyses can be made. design research becomes part of the learning landscape it studies and as such requires ‘robust designs – ones that produce impressive results, not only under ideal conditions, but also under severe but realistic constraints’ (walker, 2006, p. 13). the ‘situated nature of the retrospective analysis is the strength of the methodology’ (cobb et al., 2003) since results are presented in the context from which they emanated so that the reader can decide on their significance. the main study included the following sequence of activities in the teacher development programme that was designed: an observation of each of the volunteer teachers in practice. the didactisation principles were gauged using a variety of instruments. the pedagogy scale, use of context scale and mathematical content scale as presented by fosnot et al. (2006, p. 10) were used, as well as categories set out by timms (2006; see table 3), pea (1993; see table 4) and stigler and hiebert (1998; see figure 1). teachers also filled in lesson reflection sheets after each session. a number of contact sessions with all five teachers. during the first, fourth and seventh contact sessions, teachers worked through a modelling problem as a group. as concluded by biccard (2013, p. 290), modelling tasks were a vehicle that developed teachers’ didactisation practices by creating a scaffold for teachers to think about their own actions. a ‘fishbowl’ session, in which teachers observed a small group of students engaging in the same modelling task the teachers had completed in the previous session. reflection session and reflection instruments at various stages. figure 1: promotion of inductive reasoning. these activities were repeated over three cycles. the researcher also provided various resources for teachers. during the third contact session teachers were given two groups of tasks: one group were of a problem-solving nature, while the other was traditional numbers-only problems. they were asked to match the number-only problem to its contextual partner problem. teachers were then asked to discuss the nature and value of each type of problem in classrooms. through this discussion teachers were assisted in changing traditional problems into more context-based and collaborative problems for their classrooms. teachers also contrasted the differences between the different types of problems. it was not a requirement that teachers were to present any particular (modelling or problem-based) lesson in any of the observed lessons. it was also clear to teachers that they did not have to present any particular content during observed lessons. teachers were to continue covering their curriculum during observed visits. teachers were left in control of ‘filtering’ aspects of the professional development programme through to their own classrooms and their own practices. part of the programme involved resourcing teachers with the type of problems that could lend themselves to more active students in classrooms. stigler and hiebert (2004) present three broad ideas on how to improve classroom teaching of mathematics based on 10 years of trends in international mathematics and science study (timss) video studies (p. 16). their first suggestion is to focus on the details of teaching and not teachers. by this they mean that the focus should be on improving teaching methods, specifically the way in which the teacher and students interact. according to them, this can be ‘more powerful than the curriculum materials that teachers use’. their second suggestion is to become aware of cultural routines that are prevalent in classrooms within the same country and their third suggestion is to build a knowledge base for the teaching profession. teachers need access to examples of alternative teaching methods that they can integrate into their own teaching (stigler & hiebert, 2004, p. 12). validity and reliability since design research is qualitative in nature, the validity and reliability of the study was hinged around mckenney, nieveen and van den akker’s (2006) three concepts of rigour, relevance and collaboration. in terms of rigour, the main study (biccard, 2013, p. 152) explicated a number of the mckenney et al.’s guidelines such as: setting out a conceptual framework, triangulating data sources and providing a context-rich description. for relevance the study took place within the natural setting of teaching and learning while collaboration took place between the researcher and teachers in that their ideas guided the sessions and resources for future sessions. furthermore, a number of the strategies proposed by mcmillan and schumacher (2006) were included in the main study (biccard 2013, p. 153) such as prolonged fieldwork, multi-method strategies, verbatim accounts, mechanically recorded data and a participant researcher. the results top ↑ ‘student activity’ was one of nine didactisation practices that were described and explicated in the professional development programme of the main study. the didactisation principles were gauged by the researcher during observation of lessons. the first observation lesson took place before the professional development programme had started. teachers were asked to present a typical lesson. the second and third observation lessons took place after three (and six) professional development contact sessions while the final lesson observation took place at the very end of the programme (one year after the beginning of the programme). although the programme took place over a year, actual contact was maintained with the teachers for a period of nine months since research is not allowed at schools during the final term of the school year when pupils and teachers are preparing for the final examinations. this article will only focus on student activity during the four lessons each of the five teachers presented. a brief summary of the student activities for each lesson is outlined in table 1. the full versions of the lessons are described in biccard (2013). table 1: student activity per lesson. the main shift in the cycles of student activity in these results is from teacher doing to student doing. another shift is from individual work to pair work or group work. the type of activities moved from single solution to multi-step solutions and multi-approach solutions. the teachers were ‘devolving’ the responsibility of the mathematical work to their students as the lessons progressed over the year-long period. the type of activities changed from teacher explaining ideas and concepts to students encountering and engaging with the ideas and concepts. this change in the activity level, and the types of activities students were involved in, was gauged as summarised in table 2. table 2: occurrence of student activities over five lessons. the five lessons were observed and the activities students were involved with were recorded and are presented in table 2. activities such as reading, writing, organising, physical activity (cutting, constructing, etc.), anticipating and modelling were more evident in latter lesson observations. these activities raised the mathematical load the students were involved in. as the students became more actively involved in constructing meaning, the role of the teachers was also gauged through the middle part of a lesson. it is titled the teacher’s probing role; the three descriptive categories were taken from timms (2006): the teacher as supervisor of the activity, in which the teacher answers questions or clarifies if students ask. the teacher as director or manager, in which the teacher initiates discussion and controls the topic, but allows or invites input. the teacher as facilitator, in which the teacher sets up a structure, interacts with students and students interact with each other and the materials (p. 4). the findings are summarised in table 3. table 3: teacher probing of student ideas. the teachers’ roles changed as they incorporated more mathematical activity for their students and as they varied the activities that students were involved in. the change in teacher role and student role is reciprocal and is influenced by the landscape of mathematical activities in the classroom. as the conditions and responsibilities changed due to students being more active in the lessons, so the type of mathematical thinking involved changed (see figure 1). furthermore, teacher thinking about mathematics teaching also developed. the use of materials also changed across the lessons. from a distributed cognition point of view (pea in 2007, p. 13), the distribution of intelligence is across a system that comprises the individual, tools and the social context in which learning takes place. pea (1993) also maintains that tools, artefacts and external representations mediate a problem situation. table 4 displays the tools and materials that were incorporated into the four lessons as well as the social context of the lesson. underneath each teacher’s lesson description the coding of +m or + s is used to indicate if the materials (m) or social context (s) were improved from the baseline lesson. table 4: change in the use of materials and social context per lesson. as can be seen in table 4, teachers moved from individual work to pair work and group work. pair work was most common. it may have allowed teachers more control over the class as they ventured into changing the social context of their classrooms. teachers also started using large sheets of paper for students to work on as well as introducing physical items for the students to use. this was necessitated by teacher planning for a wider variety of activities in their classrooms. a further look at the +m and +s coding may allow us to deduce that a change in materials seems to be accompanied by a change in social setting, but a change in the social setting does not have to be accompanied by a change to the materials used in the lesson. another deduction could be that teachers may find it easier to change the social setting of their lessons than change the materials required to extend activity within the lesson. this may be affected by resources that are available or curriculum coverage pressures. distributed cognition assists in understanding that abstraction as a result of reflection can be a reflection on social, mental or physical activity. a further aspect of development within these mathematics classrooms can be dissected using stigler and hiebert’s (1998) analysis of timss video lessons where mathematical thinking can be thought of in two broad terms: inductive thinking and deductive thinking. typical traditional lessons involve mostly deductive thinking where the teacher presents a method and students use the method on a number of similar problems while with inductive reasoning students move from solving problems to generalising and abstracting mathematical ideas from the problems. stigler and hiebert outline three differentiating student mathematical activities on an inductive-deductive thinking continuum: practising a routine procedure, applying procedures in new situations and inventing new procedures or analysing new situations. the 20 lessons in the main study were analysed for this article using these three constructs to understand the role of student activity in the quality of student thinking. the resulting figure 1 highlights that student activity did improve during the professional development programme on didactisation practices and the quality of mathematical thinking moved towards more inductive type thinking. although lessons showed shifts along an inductive-deductive continuum, it is important to state that mathematics lessons may include a judicious mix of the two. traditional lessons tend to be exclusively deductive in nature. it is evident in this study that teachers are trying new ways of teaching mathematics. the changing roles of both students and teachers mediated by increased student activity and ameliorated by the social interaction through activities raised the level of teacher professional activity in the classrooms. the results of this study show that teachers incorporated more active learning activities in their own way and with the content that they were required to teach. the results suggest that smaller problem-based activities are the first step in changing activity levels in classrooms while modelling problems were starting to come to the fore at the end of the programme. teachers’ didactisation practices in the main study showed signs of development through the programme. after careful consideration of each teacher’s personal development, the following hierarchy was formulated to show which didactisation practices developed first. as a result of increased student activity, the other practices were made visible to teachers and could evolve and develop. effective professional development of mathematics teachers should therefore have a solid base of how and why student activity contributes to both abstract and generalisable thinking in students and changing of teaching practices in teachers. figure 2 shows the foundational construct that student activity played figure 2: didactisation practices hierarchy. in this teacher development programme. the hierarchy also shows that the more abstract thinking required by teachers is in the form of generalisations needed for connecting student ideas and vertically aligning lessons. the increasing difficulty in these practices is as a result of abstraction and generalisation on the teacher’s part. conclusion top ↑ teacher professional development programmes for both inservice and pre-service teachers would benefit from focusing on the level and conditions of student activity. this formed a cornerstone in this article. although the professional development programme in the main study focused on all nine didactisation practices, student activity appears to have resonated most with the teachers. it was only once they incorporated student activity into their own teaching in their own ways and to suit their own needs, did the other didactisation practices develop. it also provided teachers with their own experiences to share and build upon. by focusing on increasing student mathematical activity in the classroom teachers were able to mediate the challenge of the changing roles for themselves and their students. an evolution of changing social contexts for the lessons took place. students were given more responsibility in the latter lessons as they were given different activities to complete. students were also doing more talking in the latter lessons. it was however found that tying together and connecting different student ideas and integrating different ways of thinking about concepts was still challenging for teachers by the end of the programme (biccard, 2013, p. 272). this suggests the need for prolonged studies of this nature or studies that focus exclusively on student activity or how teachers can be developed to more skilfully connect student ideas that result from this activity. student mathematical activity was a key result of the main study. extending the range and depth of student mathematical activity in classrooms may be pivotal in realising teacher change and improved mathematical performance of students. acknowledgements top ↑ competing interests the authors declare that they have no financial or personal relationships that may have inappropriately influenced them in writing this article. authors’ contributions p.b. 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(2010). framework for mathematical proficiency for teaching. athens, ga: center for proficiency in teaching mathematics at the universtiy of georgia. article information author: barbara posthuma1 affiliation: 1faculty of education sciences, north-west university, south africa correspondence to: barbara posthuma postal address: private bag x1290, potchefstroom 2520, south africa dates: received: 17 nov. 2011 accepted: 21 oct. 2012 published: 19 nov. 2012 how to cite this article: posthuma, a.b.(2012). mathematics teachers’ reflective practice within the context of adapted lesson study. pythagoras, 33(3), art. #140, 9 pages. http://dx.doi.org/10.4102/ pythagoras.v33i3.140 copyright notice: © 2012. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. mathematics teachers’ reflective practice within the context of adapted lesson study in this original research... open access • abstract • introduction • problem statement • conceptual theoretical framework    • symbolic interaction theory    • social reconstructionist theory    • mathematics teachers’ reflective practice    • lesson study       • research on lesson study and mathematics teaching       • lesson study in the south african context • empirical investigation    • research design       • participants       • procedure       • data collection       • data analysis       • trustworthiness       • ethical considerations • findings    • findings from the analyses of the lesson plans    • findings of lesson study group reflection after observing lessons    • findings of final group reflection on the influence of lesson study on teaching mathematics • discussion    • limitations of the study    • further research • conclusion • acknowledgements    • competing interests • references abstract top ↑ there seems to be paucity of research in south africa on mathematics teachers’ reflective practice. in order to study this phenomenon, the context of lesson study (in an adapted form) was introduced to five mathematics teachers in a rural school in the free state. the purpose was to investigate their reflective practice whilst they collaboratively planned mathematics lessons and reflected on the teaching of the lessons. data were obtained through interviews, video-recorded lesson observations, field notes taken during the lesson study group meetings and document analyses (lesson plans and reflective writings). the adapted lesson study context provided a safe space for teachers to reflect on their teaching and they reported an increase in self-knowledge and finding new ways of teaching mathematics to learners. this finding has some potential value for planning professional learning programmes in which teachers are encouraged to talk about their classroom experiences, share their joys and challenges with one another and strive to build a community of reflective practitioners to enhance their learners’ understanding of mathematics. introduction top ↑ the ability to reflect on practice is considered a necessity for effective instruction (sowder, 2007). when teachers reflect on their classroom practice they carefully consider the problems in their own teaching and think about how those problems are related to their learners’ understanding of concepts. they are aware of the consequences of their teaching and how their own assumptions or beliefs can influence their teaching. however, day (1999) argues that other teachers are needed in the reflective process in order to achieve deep levels of reflection. it seems that systematic investigation of practice with the help of a critical colleague enhances the reflective process. teachers may for instance find it beneficial to come together in groups or teams to discuss their teaching in a supportive atmosphere (farrell, 2004). this view mirrors pollard’s (2002) finding that the value of engaging in reflective activity is almost always enhanced if carried out in association with other colleagues. york-barr, sommers, ghere and montie (2006) concur with pollard and maintain that reflecting on practice with another person has the potential to greatly enrich understanding and support improvements in practice. they believe that reflecting with a colleague can assist in gaining awareness of some of the fixed assumptions a teacher might have and as such help the teacher to view events from another perspective (york-barr et al., 2006). lesson study is a professional development model that provides an opportunity for teachers to reflect collaboratively on their planning and teaching of a lesson. murata (2011, p. 10), for instance, claims that lesson study is centred around teachers’ interests, is learner focused, provides opportunities for teachers to be researchers, provides plenty of time and opportunities for teachers to reflect on their teaching practice and learner learning, and is collaborative. problem statement top ↑ in spite of what has been said in the introduction, there seems to be paucity of research in south africa on mathematics teachers’ reflective practice, which strengthened the rationale to research this issue. furthermore, lesson study has not been used widely as a professional development model in south africa, despite evidence that the lesson study structure enables teachers to engage in significant professional development with a minimum of resources and offer possibilities for rural teachers to use the resources they already have to improve the teaching and learning of mathematics (taylor, anderson, meyer, wagner & west, 2005). based on evidence regarding the importance of teachers reflecting collaboratively on their teaching, lesson study (in an adapted form) was used as the context of an investigation of the nature of mathematics teachers’ reflective practice (posthuma, 2011). this article reports on how the adapted lesson study context influenced mathematics teachers’ reflections before, during and after teaching a lesson. the article is structured as follows. firstly, a conceptual theoretical framework is outlined. this is followed by an exposition of the research method that was applied. the findings are listed thereafter, followed by a discussion against the backdrop of the conceptual theoretical framework. the article concludes with suggestions for further research. conceptual theoretical framework top ↑ the main concepts that this research addressed were mathematics teachers’ reflective practice and lesson study (in an adapted form). in this section these concepts are linked theoretically to establish a conceptual theoretical framework for the study that was done. two theories, symbolic interaction and social reconstructionism, informed the conceptualisation of the study. symbolic interaction theory symbolic interaction theory examines the ways in which people make sense of their life situations and the ways in which they go about their activities, in conjunction with others, on a day-to-day basis (prus, 1996). as a theoretical framework, it suited the interpretive nature of this study which is centrally concerned with meanings teachers construct whilst reflecting collaboratively on their planning and teaching of a lesson. the four basic propositions of symbolic interaction are: • individuals act and interact within larger networks of individuals and groups that have an impact on them • human beings are active in shaping their own behaviour • individuals can engage in thought and change their behaviour as they interact with others • and to understand human conduct requires a study of their covert behaviour (blumer, 1969). this final proposition has major methodological implications, namely that the procedures used should allow sympathetic introspection as a part of the methodology (pedro, 2001). the adapted lesson study context was explored not only against the backdrop of symbolic interaction, but also within the milieu of the social reconstructionist theory, discussed in the following section. social reconstructionist theory zeichner and tabachnick (1991) outlined four major theoretical traditions in the reflective teaching literature. each of these is concerned with thoughts and practices connected to particular educational aims and values. these traditions are the academic, social efficiency, developmentalist and social reconstructionist traditions. the latter views schools and teachers as agencies of change for the creation of a more just and humane society. this tradition has three central strands. the first encourages teachers to focus their attention both inwardly at their own practice and outwardly at the social conditions wherein these practices are situated (carr & kemmis, 1986). the second advocates a democratic and emancipatory force as teachers focus their reflections on issues related to inequality and injustice, in the context of their own classroom, school and society. the third demonstrates a commitment to reflection as a communal activity and seeks to create communities of learning where teachers can support and sustain one another (zeichner & tabachnik, 1991, p. 9). according to mewborn (1999), ‘reflection and action together are seen as a bridge across the chasm between educational theory and practice’ (p. 317). lesson study formed the practice part of this study where the teachers tried to construct meaning about their classroom practice socially through their interaction with one another. in the adapted lesson study process the teachers were constantly reflecting on their practice and both theories (theory of symbolic interactionism and social constructionist theory) embrace the social interaction and reflection concepts. mathematics teachers’ reflective practice thoughts on reflection and reflective practice have evolved over many decades, if not centuries, through carefully constructed theory and research applications (york-barr et al., 2006). dewey is frequently recognised as the eminent 20th-century influence on reflection in education (ottesen, 2007; pollard, 2002; rodgers, 2002; york-barr et al., 2006; zeichner & liston, 1996). for dewey (1933), true reflective practice takes place only when the individual is faced with a real problem that they need to resolve and seek to resolve in a rational manner. the seminal work of schön (1983) has also inspired a renewed interest in reflective practice in the field of education (lee & tan, 2004; valli, 1997). although schӧn clearly relates reflection to action through using the terms ‘reflection-in-action’ and ‘reflection-on-action’, other researchers seem to view reflection as a special form of thought (artzt, armour-thomas & curcio, 2008; sparks-langer & colton, 1991). for the purpose of this investigation, reflection was considered to involve questioning the effectiveness of practice before (reflection-for-practice), during (reflection-in-practice) and after the act of teaching (reflection-on-practice). research indicates that the content of teachers’ reflection in their classrooms relates to their learners’ prior knowledge (ward & mccotter, 2004), their own instructional strategies (lee, 2005; ward & mccotter, 2004), discipline (lee, 2005), the teacher-learner relationship (lee, 2005), mathematical content (mckeny, 2006) and disturbing aspects of teaching experiences (ottesen, 2007). the quality of teachers’ reflection can be measured at different levels as identified by a number of researchers (e.g. lee, 2005; valli, 1992; van manen, 1977). for the purpose of this study the teachers’ quality of reflection was rated according to lee’s (2005) levels of reflection. according to lee, a teacher who reflects on level 1 (r1: ‘recall level’) is mainly concerned with mastery and/or application of technical means for achieving given educational ends, and includes a simple description of observation or a focus on behaviours or skills from past experience. reflection on level 2 (r2: ‘rationalisation level’) is directed at an interpretive understanding of the meanings of educational experiences and choices of action within a particular social and institutional context (lee, 2005). reflection on level 3 (r3: ‘reflective level’) links classroom practice to the broader arena of political, moral and ethical forces (lee, 2005) and this type of reflection is considered to be critical reflection (jay & johnson, 2002). in order to reflect on a critical level about the teaching and learning of mathematics, the teacher has to question moral, ethical, and other types of normative criteria related directly and indirectly to the classroom (van manen, 1977). within the social reconstructionist tradition, teaching and teacher education are important determinants for the creation of a more just and humane society. this perspective prioritises critical reflection in terms of how teachers’ actions influence the stability within both the school and society (zwodiak-myers, 2009). lesson study lesson study is a collaboration-based teacher professional development approach that originated in japan (fernandez & yoshida, 2004). according to takahashi, watanabe and yoshida (2006), lesson study has played an important role in professional development in japan since the public education system was introduced more than a hundred years ago. one reason for the popularity of lesson study in japan might be that:lesson study provides japanese teachers with opportunities to make sense of educational ideas within their practices; change their perspectives on teaching and learning; learn to see their practices from a learner’s perspective; and enjoy collaborative support among colleagues. (takahashi, watanabe & yoshida, 2006, p. 201) the lesson study process is cyclical and has the following basic stages (murata, 2011, p. 6): 1. consider goals for student learning and development, plan a research lesson based on these goals. 2. observe the research lesson and collect data on student learning and development. 3. use these data to reflect on the lesson and on instruction more broadly. 4. if desired, revise and re-teach the research lesson to a new group of students. research has shown that lesson study impacts on teachers’ understanding of learner thinking, it enhances teachers’ content knowledge and awareness of new approaches to teaching, it helps teachers to connect their practices to school goals and broader goals, and it creates a demand for improved instruction and allows competing views to be heard during the reflection-stage of the lesson study cycle (lewis, 2000). friedman (2005) argues that the habit of personally reflecting on one’s teaching that occurs during the lesson study process is a habit that remains with teachers long after the lesson study process is over. research on lesson study and mathematics teaching meyer and wilkerson (2011) investigated the impact of lesson study on teachers’ knowledge for teaching mathematics and found that lesson study provided opportunities for improved teacher knowledge in three of the five groups that participated. they conclude that more research is needed to examine the impact of lesson study on teachers’ knowledge.olson, white and sparrow (2011) studied the influence of lesson study on mathematics teachers’ pedagogy and found that three of the five teachers who completed the lesson study process changed their pedagogies over 18 months. before lesson study, the three teachers maintained control of the classroom discourse and frequently asked ‘why?’ without knowing what to do with the information that they gained. after completing the lesson study process, these three teachers asked purposeful questions and used the learners’ responses to guide instructional decisions. lesson study in the south african context in south africa, a school-based in-service education intervention programme, modelled along the lines of the japanese lesson study, was launched in 2000 in mpumalanga (jita, maree & ndlalane, 2006). according to jita et al. (2006), the lesson study approach has established a system in which teachers have grown accustomed to relying on one another, coaching, leading discussions and exploring alternative solutions to problems experienced in their teaching of mathematics. a study by coe, carl and frick (2010), in a rural primary school in the western cape, sought to determine the value that a group of teachers placed on the process of lesson study as a model for their own learning and instructional improvement. their findings highlight the following benefits of lesson study: 1. lesson study offers an effective strategy to bring teachers out of isolation, allowing them to experience meaningful collaboration with fellow teachers. 2. the process of lesson study is embedded within the classroom context by setting goals and then planning instruction with the purpose of moving the learners closer to the goals. a connection between the content of the research lesson and the remainder of the curriculum is established. furthermore, lesson study provides an opportunity to observe the learners during the research lesson. the post-lesson discussion is also valuable in terms of validating and developing the perceptions of learners in relation to the prescribed goal. 3. lesson study can be a catalyst for transforming new instructional strategies into routine classroom practice. 4. continuous support (by fellow teachers) is embedded within the model of lesson study. the lesson study process involves reflection on the planning and teaching of a lesson as a communal activity, and hence fits into the framework of the social reconstructionist tradition. teacher reflection within the lesson study model of professional development focuses on social conditions of practice (a group of teachers trying to make sense of the teaching and learning of mathematics) in a democratic way. the conceptual theoretical framework (discussed previously) links mathematics teaching to teachers’ reflective practice, within the context of adapted lesson study. the lesson study cycles of planning, teaching and evaluation correspond with teachers’ reflection-for-practice, reflection-in-practice and reflection-on-practice. the theory of symbolic interaction and the social reconstructionist tradition both embrace critical reflection and social interaction and therefore together form the backdrop of this study. empirical investigation top ↑ research design a case study design suitable for a qualitative approach was chosen. according to cohen, manion and morrison (2005) a case study provides a unique example of real people in real situations. case studies offer a multi-perspective analysis in which the researcher considers not just the voice and perspective of one or two participants in a situation, but also the views of other relevant groups and the interaction between them (nieuwenhuis, 2010). participants the participants in this study were five mathematics teachers from one school in the thabo mofutsanyana district in the free state. the criteria for selection included the factors of convenience, access and willingness to participate. meetings with the teachers took place in the teachers’ school environment.the biographical information of the participants is provided in table 1. pseudonyms are used to protect the identities of the participants. table 1: biographical information of participants. table 1 indicates that all the participants were experienced teachers with basic teaching qualifications and, except for dianne, had all been teaching for 14 years or longer. the pass rate for mathematics at the rural township school was 19% in 2010. the medium of instruction at the school was english. setati, adler, reed and bapoo (2002) argue that in the remote rural areas of south africa where access to english outside the classroom is severely limited, the classroom context is more appropriately described as a learning environment in which english is a foreign language. this notion has implications for the present study, where the participants had to reflect on their practice through the medium of english. in addition, they all taught mathematics to learners who did not have textbooks. procedure the data-gathering procedure followed is outlined in table 2. table 2: procedure for data gathering. although the group provided feedback on how to improve each of the lesson plans, the final lesson plan for each teaching phase remained the responsibility of the teacher who would teach the lesson. this is a deviation from the japanese lesson study, although the same procedure was followed as suggested by lewis (2000): the research lesson should be observed by other teachers; the research lesson should be planned collaboratively; there should be an overarching goal for the research lesson; and the research lesson should be recorded and discussed. (pp. 4–6) the lesson study cycle is illustrated in figure 1. figure 1: lesson study cycle. data collection the lesson study cycle involves three phases: • planning • teaching • evaluation (reflection). data were collected to coincide with each of these phases. during the planning phase, data were collected through document analyses (lesson plans); during the teaching phase, data were collected through lesson observation (video recorded so that the lesson study group could observe the lesson the same afternoon); during the reflection or evaluation phase, data were collected through semi-structured interviews, reflective writings and field notes. a final group interview with all the participants was conducted after the last lesson study cycle to establish how the reflective processes of lesson study influenced their classroom practice. data analysis the data gathered during the lesson study cycles were analysed during and after the data-gathering process, in line with creswell’s (2003, p. 18) qualitative case study approach in which the researcher collects open-ended, emerging data with the primary intent of developing themes from the data. all interviews were transcribed verbatim and the computer software program atlas.ti 6 (atlas.ti scientific software development, 2010) was used to assist with the data management, coding, categorisation, abstracting and conceptualising stages of the analysis. trustworthiness the qualitative data collected were in the form of observations, interviews and document analyses. the observations and interviews were electronically recorded and transcribed. participants had the opportunity to review these transcriptions at the end of the data collection period to ensure accuracy and provide additional research data. data from multiple sources were used to help verify findings: data collected through interviews were verified, for instance, with information gathered from the observations and the document analyses. ethical considerations permission to access a school in the thabo mofutsanyana district was obtained from the free state education department. potential participants were verbally briefed and presented with details of the research in writing. each teacher signed a letter of consent. participation was voluntary and the teachers had the option to withdraw at any stage. they were assured that all data would be kept confidential and pseudonyms would be used in the report. in addition, the principles of respect for personal autonomy, benevolence and justice guided the research. findings top ↑ findings from the analyses of the lesson plans the first lesson was planned collaboratively by the group (the ‘research lesson’) and subsequent lesson plans were refined by the lesson study group and presented by the different teachers. the participants used a lesson plan template that made provision for reflection on their learners’ current understanding of the concepts. the template also made provision for reflection after the lesson was taught. the lesson plans were analysed to establish whether the teachers reflected on their learners’ thinking and understanding of the concepts to be taught, whether provision was made for learners who might have struggled with the concepts or learners who needed to be challenged, and whether there was any evidence of linking the current lesson to the lesson study goal determined by the lesson study group. the analysis of the lesson plans revealed no reflection on learners’ current mathematical knowledge or on how to address any misconceptions that might have occurred when teaching the concepts. there was also no evidence in these lesson plans of teaching mathematical concepts with a long-term goal in mind. for example, during the post-observation interview mary admitted that she had not planned the lesson with her learners in mind: ‘you know, i did not now think of them … grasping gradually …’ and dianne said: ‘… previously we did not spend much time on lesson plans.’ in addition, the level of the content the participants selected to teach differed. for example, the content that the lesson study group selected for mary’s grade 8 class (the first lesson plan) was on a very low cognitive level (the examples used were x + 3 = 7, x – 10 = 3 and 32 + x = 34) in contrast to the content that morgan selected for his grade 10 class (which was arranged from basic examples to more complicated examples, including fractional algebraic equations such as ). the selection of tasks on a low cognitive level might reveal the individual teacher’s perception of their learners’ ability to cope with the mathematics content. findings of lesson study group reflection after observing lessons the lesson study group observed the video recording of each lesson and discussed their observations. according to taylor et al. (2005), observing a lesson enables teachers to shift their thinking from a teaching focus to a learning focus whilst puzzling over their learners’ mathematical thinking. as observers, they are free to focus on the actual work the learners are doing, as well as the learners’ thought processes. during the lesson observations the focus was on the teacher’s presentation as well as on the learners’ content knowledge. mary’s learners struggled to transpose the constant term to the other side of the equation (using the additive inverse) and as a result the volume of work done during the 45-minute period was very low. it seemed to me that mary did not understand her learners’ thinking and failed to encourage them to clearly explain and justify their reasoning. for example, she asked one learner to come to the board to solve the linear equation 32 + x = 34. the learner struggled and mary did not help her, but instead told her to ask learners from the class to help, which resulted in chaos. she moved between the desks to mark the learners’ work and if a learner wrote a wrong answer, she sent the learner to the board to look at the examples, not helping the learner to understand the method or asking them to explain their solution method to her. according to warfield, wood and lehman (2005) such teachers do not reflect deeply about their learners’ mathematics or about their own teaching. however, morgan allowed learners to solve the equations using their own methods and expected them to explain and justify their reasoning and to listen to and question the reasoning of other learners. in this way he actively involved all learners in the lesson: i invite the learners to come and show me what they are doing on the board, so that i can exactly know what they know and what they don’t know … the lesson study group reflected critically on mary’s lesson and expressed concern about treating the learners fairly and catering for all learners’ needs (reflecting on lee’s r3: reflective level). during the group reflection, morgan said: we should cater for all learners … sometimes you plan a worksheet for your class, and after one or two examples you see they don’t understand, and then only a few sums are done ... the lesson study group seemed to reflect more openly and talk more freely as the lesson study process continued. they constantly reflected on their learners’ lack of understanding of basic concepts, for example, learners struggling to add 3x to 6x. table 3 summarises the content of reflection, as well as the level of the group reflections on each lesson. table 3: summary of content and level of reflection during the lesson study group reflections. from table 3, it appears that the participants reflected on their teaching styles, on their learners’ understanding of concepts, and on matters such as class size, textbooks, class arrangement and language. within the framework of the symbolic interaction theory, the teachers were making sense of their own and each other’s classroom practice collaboratively. during the group reflections on the lessons they observed, the lesson study group reflected critically (r3) on mary and vicky’s lessons, considering the implications of each teacher’s actions on their learners’ understanding of mathematics. the video recordings revealed mary’s impatience with the learners’ inability to solve basic linear equations as well as vicky’s impatience with her learners’ lack of basic mathematical knowledge (such as the product of multiplying two negative numbers will be a positive number) and vicky admitted: ‘ai, ... i lose my temper sometimes … (laughs).’ morgan replied: ‘we need to be very patient with the learners. when we teach them, we need to make sure you motivate them.’ against the backdrop of the social reconstructionist theory, the teachers who participated in this study reflected communally, supported each other during the feedback sessions, but also critically considered the effects of their own and their colleagues’ classroom practice on their learners’ mathematical growth and well-being. findings of final group reflection on the influence of lesson study on teaching mathematics all the participants reported that they had gained personally and professionally from being part of the lesson study group. a summary of their reflections appears in figure 2. figure 2: influence of lesson study on teaching as reported by the participants. the benefits of lesson study reported by the participants are summarised in figure 2, a circular diagram that illustrates the influence of the lesson study cycle on the participants’ reflective journeys. they reported that they had improved their lesson planning as a result of the lesson study group planning sessions. they were more confident about their teaching after seeing themselves on video. they expressed a deeper awareness of their learners’ needs. they learned from watching their fellow participants on video to change their teaching to become more learner-centred, and they felt as if they were doing self-research by being part of this research study. against the framework of the symbolic interaction theory, these teachers reflected collaboratively and tried to make sense of their classroom practice and their teaching of mathematics. they reported positively on the lesson study process, but whether they will adapt their classroom behaviour as revealed by the lesson observations remains to be researched. discussion top ↑ lesson study as reported by this research supports lewis’s (2000) view that lessons planned in the lesson study context impact on teachers’ individual professional development and their view of learners (they ‘learn to see children’). new content and approaches are acquired, competing views of teaching emerge and a demand for improvement is created. it also confirms hix’s (2008) contention that the sum of planning collaboratively, anticipating learner responses, creating evaluation questions for observers, observing the public teaching, and discussing and reflecting on the observations are beneficial to teachers’ reflective practice. the results of this study mirror friedman’s (2005) report that the major advantage of lesson study for teachers is the collaboration factor. taylor et al. (2005) report on the following benefits of the lesson study professional development model: • an effective detailed lesson plan achieves the goal of more effective learning by learners • the lesson study model provides a highly motivated structure for planning and teaching a lesson • reflecting and thinking in the company of other teachers allow for sharing, interacting questioning assumptions, and reassessing common practices • observing a lesson enables a shift in thinking from a teaching focus to a learning focus • focusing on learner thinking provides opportunities for feedback to support changes in teaching mathematics • lesson study transforms working relationships and conversations between teachers. as expected in terms of the symbolic interaction theory, the adapted lesson study provided a context for the teachers to derive meaning about their teaching actions through social interaction with their fellow teachers. however, they were also actively shaping their own behaviour and reflecting on how to change their teaching actions to improve their learners’ understanding of mathematics concepts. in line with the social reconstructionist tradition, which prioritises reflection-in-practice and -on-practice, the teacher becomes an agent of change to improve social conditions of practice as a result of critical reflection. this issue is especially applicable in south african society, where most teachers teach in rural areas, using a language that is not their mother tongue, in schools with limited resources. limitations of the study the five participants were of the same cultural and language group, and taught at the same school. a more diverse sample that excludes ethnic and geographical biases could arguably have cast more light on the research problem.another limitation of this research study pertains to the researcher’s inability to speak the home language of the participants. language emerged as a contextual factor that possibly influences participants’ reflective practice (posthuma, 2011). although allowance was made for an interpreter during the last group reflection, the participants’ verbal and written reflections showed that they struggled to express themselves in english during the lesson study group meetings. one has to allow for the fact that their reflections might have been misinterpreted. further research it would be worthwhile to undertake a follow-up study with the participants of this study to understand the long-term effects of reflective processes in a lesson study context. furthermore, there might be other contextual factors that influence mathematics teachers’ reflective practice that need to be researched further, for example, gender, personality characteristics and culture. the reason gender is considered to be a possible influence on the reflective process is because the male participants in this study were very reluctant to write about their reflections in a reflective diary. personality characteristics might also play a role in teachers’ reflective practice, based on dewey’s (1933) three attitudes that he considered to be integral to reflection, namely, open-mindedness, responsibility and wholeheartedness. reflection might also be culturally bound. for example, lee and tan (2004) investigated student teachers’ reflective practice in malaysia and found that their private reflections were on a deeper level than their public reflections. an intercultural study, for example, comparing south african mathematics teachers’ reflective practice to that of a different culture, will provide for a more comprehensive body of knowledge on reflective practice.the results of this study could furthermore be used in the planning of future continuing professional teacher development programmes. the positive feedback of the participants on the lesson study process suggests that lesson study should be initiated in other settings. further research should explore lesson study as a model in south africa for successful continuing professional teacher development and teacher education programmes. conclusion top ↑ the teachers who participated in this study were teaching mathematics to learners in a rural area with few or no resources, in a language that was not their mother tongue. the symbolic interaction theory provided a backdrop for the adapted lesson study as a way for these mathematics teachers to derive meaning about their teaching actions through social interaction with their fellow teachers. the adapted lesson study experience allowed them an escape from their isolated practices, providing them an opportunity to reflect on their problems cooperatively in a safe space. however, they were also actively shaping their own behaviour and some were critically reflecting on how to change their teaching actions to improve their learners’ understanding of mathematics concepts. the social reconstructionist approach prioritises reflection-in-practice and reflection-on-practice that enables the teacher to become an agent of change to improve social conditions of practice. this issue is especially applicable in south african society where teachers’ histories need to be taken into account when planning and implementing professional development programmes. acknowledgements top ↑ i would like to acknowledge the support of my supervisor, prof. kobus maree and co-supervisor, dr gerrit stols in the completion of this research. in addition, i thank prof. hannes van der walt from north-west university as well as sanpad (south africa netherlands research programme on alternatives in development) for supporting me in writing this article. competing interests i declare that i have no financial or personal relationship(s) which may have inappropriately influenced the writing of this article. references top ↑ artzt, a.f., armour-thomas, e., & curcio, f.r. 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(2009). an analysis of the concept reflective practice and an investigation into the development of student teachers’ reflective practice within the context of action research. unpublished doctoral dissertation. brunel university, middlesex, united kingdom. article information authors: aarifah gardee1 karin brodie1 affiliations: 1school of education, university of the witwatersrand, south africa correspondence to: aarifah gardee email: aarifahgardee@gmail.com postal address: school of education, wits university, 27 saint andrews road, parktown 2193, south africa dates: received: 23 mar. 2015 accepted: 01 oct. 2015 published: 27 nov. 2015 how to cite this article: gardee, a., & brodie, k. (2015). a teacher's engagement with learner errors in her grade 9 mathematics classroom. pythagoras, 36(2), art. #293, 9 pages. http://dx.doi.org/10.4102/pythagoras.v36i2.293 copyright notice: © 2015. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. a teacher's engagement with learner errors in her grade 9 mathematics classroom in this original research... open access • abstract • introduction • slips, errors and misconceptions • how teachers deal with misconceptions • research design and methodology • analysis and discussion of results    • mistakes that occurred in the classroom    • errors dealt with by the teacher    • how the teacher dealt with the errors    • changes over time in dealing with errors • conclusions • acknowledgements    • competing interests    • authors’ contributions • references • footnotes abstract top ↑ how errors are dealt with in a mathematics classroom is important as it can either support or deny learner access to mathematical knowledge. this study examines how a teacher, who participated in a professional development programme that focused on learner errors, engaged with mathematical errors in her grade 9 classroom. data were collected over a two-year period in the form of videotapes and were analysed qualitatively. our findings illustrate that this teacher dealt with four types of mistakes: slips, errors derived from misconceptions, language-related errors and errors that occurred from the incorrect usage of the calculator. she dealt with these by correcting, probing or embracing them. we found that, over time, this teacher dealt with more errors and corrected and embraced errors in different ways. we recommend that teachers use their professional knowledge to decide when, why and how it is appropriate to correct, probe or embrace errors in light of their knowledge of the content and their learners. introduction top ↑ errors play a central role in the mathematics classroom as they are a reflection of the manner in which learners reason and they illuminate the processes through which learners attempt to construct their own knowledge (olivier, 1989). errors can be used by teachers to provide learners with epistemological access to mathematics and contribute to developing learners’ conceptual understanding (brodie, 2013). therefore, the manner in which a teacher deals with learner errors is crucial, as it can either enhance or limit learners’ understanding of mathematics. methodologies for remediating errors are not always satisfactory, especially when additional work or re-explaining of ideas are used as remedies (borasi, 1987). while much research has been done on the nature of learners’ errors and their underlying misconceptions (hansen, 2011; nesher, 1987; olivier, 1989) and how teachers might deal with errors (borasi, 1994; swan, 2001), very little work has illustrated how teachers actually deal with errors in their mathematics classrooms (heinze & reiss, 2007). the purpose of this study is to investigate the type of errors a teacher chooses to deal with in her mathematics classroom and the manner in which she deals with the errors. since the data from this study were obtained over a two-year period, we also decided to investigate if there were any shifts in the teacher's practice in dealing with errors over time. this research is part of a larger project called the data-informed practice improvement project (dipip). dipip works with mathematics teachers in professional learning communities so that they may develop their understanding of the importance of mathematical errors in the classroom and the learners’ reasoning behind their errors and collectively strategise how to deal with them (brodie, 2013). the first and second sections of this article draw on the literature to illuminate the kinds of errors that might occur in the classroom and the ways in which teachers might deal with errors. in the third section, we explain the research design and methodology used to analyse our data, which is then followed by an analysis and discussion of our results in the fourth section. finally, we suggest recommendations based on our findings. slips, errors and misconceptions top ↑ there are many reasons why learners may not obtain the correct solution to a mathematical problem. these reasons may include, but are not limited to, carelessness, a lack of knowledge of the mathematical concepts or the learners not understanding what is required of them in a mathematical task (swan, 2001). terms like ‘misconceptions’, ‘errors’ and ‘mistakes’ are often used interchangeably to describe any solution that is a deviation from the expected result. however, these three terms are not synonymous and refer to different kinds of mistakes. the first is what olivier (1989, p. 12) calls ‘slips’, which are mistakes made as a result of carelessness and which are easily rectified when pointed out. slips are not symptoms of conceptual misunderstandings. slips are common; we all make them as learners and doers of mathematics. slips are sporadic. errors, however, are systematic. they occur on a regular basis and are pervasive and persistent, often across contexts. errors occur at a deeper conceptual level than slips, so correcting errors is usually not enough to address these conceptual misunderstandings. the underlying conceptual framework that causes the errors is called a misconception (nesher, 1987). nesher argues that misconceptions lead to a cluster of errors, which are not sporadic. misconceptions generate errors. but how are misconceptions generated? the theory of constructivism proposes that we actively construct knowledge by using our prior knowledge as a foundation to build new knowledge. the processes of assimilation and accommodation enable us to restructure our existing schemas to develop our conceptual knowledge (hatano, 1996). assimilation occurs when new knowledge is ‘amalgamated’ into existing schema (hatano, 1996, p. 202). accommodation occurs when new knowledge is in conflict with existing schema and reorganisation of the schema needs to occur to incorporate the new knowledge. assimilation and accommodation working together lead to the reconstruction of knowledge (hatano, 1996), which means that learners are not only actively engaging in constructing knowledge, but they are also reorganising their knowledge into more powerful schema. the process of accommodating new knowledge is more challenging than assimilating knowledge into existing schema. by attempting to assimilate knowledge that we should accommodate, we tend to ‘overgeneralise’ new knowledge based on prior correct knowledge (olivier, 1989). we apply knowledge that is correct in one domain to another in which it no longer works (smith, disessa & roschelle, 1993). this is why errors are not random; they have some grounding in learners’ correct prior knowledge. a constructivist framework suggests that errors are sensible and reasonable to learners and that they illuminate important aspects of learners’ reasoning, both valid and not valid. dipip focuses on working with teachers to understand the reasoning behind learner errors and to build on this reasoning to develop new mathematical concepts (brodie, 2013, 2014). how teachers deal with misconceptions top ↑ the word ‘error’ in the education system tends to have negative connotations. summative assessments used widely in schools perpetuate the misconception that making errors is punishable through the system of deducting marks for wrong performances (nesher, 1987). treating errors as problems may disrupt learners’ confidence in their previously learned correct knowledge (nesher, 1987). ingram, baldry and pitt (2013) argue that although teachers may not explicitly tell the learners that making errors is problematic, the manner in which teachers deal with errors, by avoiding opportunities for learners to make and discuss mistakes in the classroom, implicitly suggests that errors are problematic (heinze & reiss, 2007). hansen (2011) argues that teachers need to treat errors sensitively and productively, as errors can be used as tools, not only to motivate learners but also to assist them in developing their conceptual knowledge by learning from their errors. teachers may also regard errors as a failure on their part. this is reflected in brodie’s (2014) research, wherein teachers blamed the learners or themselves for the errors made in class. much of the research on errors and misconceptions argues that errors are a normal part of the learning process (borasi, 1987; brodie, 2013, 2014; heinze & reiss, 2007; ingram et al., 2013; nesher, 1987; smith et al., 1993). even experienced mathematicians make errors and in so doing create new knowledge (borasi, 1994). in classrooms, errors make for points of engagement with learners’ current knowledge (brodie, 2014). this notion of errors gives us a way to help teachers see learners as reasoning and reasonable thinkers and the practice of mathematics as reasoned and reasonable (ball & bass, 2003). if teachers search for ways to understand why learners may have made errors, they may come to value learners’ thinking and find ways to engage their current knowledge in order to create new knowledge. an important issue for teachers’ thinking about errors relates to the role and responsibility of teachers in producing errors. errors are seldom taught directly by teachers and yet all learners, even ‘strong’ learners, develop them at some point (brodie, 2014). however, teachers sometimes exacerbate errors through taken-for-granted use of language and concepts (brodie, 2014) and, at another level, through not making errors public and dealing with them (brodie, 2014; ingram et al., 2013). research on teachers’ dealing with learner errors in mathematics is limited, but two authors have developed frameworks for this purpose. peng and luo (2009) identify four kinds of error analysis that teachers can use to engage with learners’ written texts: identify, interpret, evaluate and remediate. in two case studies they report on, the teachers were able to identify the learners’ errors but struggled to interpret them appropriately. they were therefore not able to appropriately evaluate or remediate the errors. in a study with 45 pre-service teachers, prediger (2010) asked them to analyse a learner's error and their analyses suggests four characteristics necessary for diagnostic competence of learner errors: interest in learner thinking, interpretive attitude of understanding the learner's thinking from their perspective, general knowledge of learning processes and domain-specific mathematical knowledge. most of the pre-service teachers in prediger's study showed an interest in understanding the learners’ errors but those who did not have an interpretive attitude were likely to make suggestions for remediation that were confusing or that re-taught what the learner already knew, rather than pinpointed the source of the error. pre-service teachers who showed an interpretive attitude with some general knowledge of learning were able to partially understand how the learners might be thinking but were not able to activate the mathematical knowledge that they needed to fully understand and work with the error. only pre-service teachers who activated all four levels of competence were able to make appropriate interventions. while peng and luo (2009) and prediger (2010) use the notion of remediation of learner errors, borasi (1994) argues that there is a difference between diagnosing and remediating errors, with the aim of eliminating them, and using them as ‘springboards for inquiry’ where errors become part and parcel of mathematical development and knowledge creation (see also lakatos, 1976). based on the above, brodie (2013) suggests a framework for analysing how teachers interact with learner errors. teachers can avoid, correct, probe or embrace errors. teachers may avoid or ignore errors because they are insecure about their content knowledge, they may not regard errors as important tools for learning, they may not want to shame learners or they fear that errors may be ‘contagious’ (swan, 2001, p. 151). teachers often correct errors, thereby making the correct knowledge accessible to the learners. correcting errors suggests that teachers have identified and evaluated the errors rather than interpreted the errors from the learners’ perspectives. probing errors involves teachers attempting to understand how errors make sense to learners, usually by asking learners ‘probing questions or pressing questions’ to gain access to learner thinking (brodie & shalem, 2011, p. 431). by asking such questions, teachers support learners to develop reasoning and learners learn to explain their thinking and justify their ideas. embracing errors is where teachers use errors constructively to generate new knowledge for the learner who has made the error and for other learners (brodie, 2013); that is, they use errors as tools to enhance epistemological access. we use this framework to analyse how the teacher in this study dealt with the errors her learners produced. research design and methodology top ↑ the teacher whose lessons we analysed participates in a professional learning community organised by dipip on an ongoing basis. this teacher is one of about 40 teachers who are part of this project and was selected for this study because she seemed to have a range of strategies in working with learner errors. the data are in the form of videotapes, which were taken before the project started as a baseline and have been collected at various points during the project over two years (2012–2013). for the purpose of this study, we analysed nine videotaped lessons in a grade 9 class. each of the nine lessons was categorised by the dipip project as either individually planned or jointly planned lessons. individually planned lessons involved the teacher teaching a lesson planned by herself as part of her daily routine. the jointly planned lessons were planned in collaboration with other teachers in the professional learning community and aimed to deal with the possible learner errors that might emerge during the lessons1. in each year (year 1 and year 2) there was a set of individually planned and jointly planned lessons. table 1 provides an overview of the types of lessons. table 1: overview of the types of lessons. the first step in data analysis was to watch each of the videos. the first author made notes on the time that an error was seen in the video, the nature of the error and the manner in which the teacher dealt with the error. to further ensure the validity of our results, she re-watched the video and documented excerpts illustrating how the teacher dealt with the error and the possible reasons that the learners provided for making such errors, if they were expressed. the first author then discussed her coding system and checked her interpretations with the second author and all disputes were resolved by discussion. thereafter, we arranged the data into a table. we documented the error, our thoughts on the error, the manner in which the teacher dealt with the error and our thoughts on the teacher's approach to dealing with the error in light of the literature. initially, we intended to classify errors using the two categories, slips and errors stemming from misconceptions, extracted from our literature review. however, when watching the lessons, we realised that there were two additional types of errors which are conceptual in nature; that is, they are not slips, but are not derived from misconceptions. these conceptual errors were language-related errors and errors derived from the incorrect usage of the calculator, which we included in our framework. we then classified each of the errors in our table of results as either being a slip, an error derived from a misconception, a language-related error or an error derived from incorrect usage of the calculator. the manner in which the teacher dealt with the error was classified as correcting the error, probing into the error or embracing the error, as discussed above. we acknowledge that the practices of one teacher cannot be generalised to other teachers in the project, or to other teachers more generally. however, this analysis has enabled us to test and refine our categories for analysis and we intend to analyse the shifts among other teachers in the future. the analysis of one teacher is useful in that it enables us to discuss in detail the different categories and how she developed her practices over time. analysis and discussion of results top ↑ mistakes that occurred in the classroom there were four categories of mistakes that occurred in in the classroom: slips, errors derived from misconceptions, language-related errors and errors derived from incorrect usage of the calculator. to illustrate the nature of each category of mistakes we provide an example of each category below. an example of a slip occurred when the teacher asked learners how many times 2 goes into 36. a learner responded by saying 13. this learner could have treated 36 as 26. at a grade 9 level, this error can be attributed to carelessness and can be easily corrected by checking the calculation. at this level, it is unlikely that the mistake indicates a conceptual misunderstanding; hence, it can be classified as a slip. an example of an error derived from a misconception occurred when the teacher asked the learners to add to . a learner gave an incorrect answer of one and a half. since the teacher did not probe the error, we thought of possible methods to get the answer. the first possible error is that the learner may have added the numerator and the denominator separately: [eqn 1] this error is evidence of a misconception because the learner overgeneralised the addition of whole numbers to the addition of fractions. the learner could have also added the numerator and denominator based on an overgeneralisation of multiplication and division of fractions. much research suggests that misconceptions are a result of prior correct knowledge interfering with new knowledge. however, new learning, such as the multiplication and division of fractions can also interfere with prior correct learning (olivier, 1989). the second error in this answer is that the learner divided 18 by 12 instead of 12 by 18 to simplify the fraction. this learner probably assumed that 12 divided by 18 is equivalent to 18 divided by 12, a misconception that can maybe be attributed to the overgeneralisation of the commutative properties of addition and multiplication of numbers to division. an example of a language-related error occurred when the teacher asked the learners for a definition of the word ‘expression’. a learner stated that an expression refers to ‘making a number bigger’, which suggests that this learner confused the word ‘expression’ with ‘expansion’. despite their similar pronunciations, these concepts refer to different mathematical objects or processes and have different spellings (adams, 2003). we classified this as a language-related error and not a slip because such errors are reasonable and sensible and usually occur where learners do not fully grasp the concepts. this learner did not fully grasp the concept of an expression in mathematics and used it interchangeably with the word expansion. language-related errors are likely to occur across learners, rather than be idiosyncratic, once again highlighting the reasonableness of such errors. an example of an error due to the incorrect usage of the calculator occurred when the teacher asked the learners how to represent −1x in the expression −3x2 − 1x. a learner stated that negative one is like zero. when probed by the teacher, the learner said that she used a calculator to obtain the answer of zero. there is a tendency for variables in a scientific calculator on computer mode to represent numbers saved in its memory. if there is no number saved within the memory, variables are equated with zero. hence, by typing an expression into a scientific calculator, learners may get incorrect answers. in classifying errors linked to the incorrect usage of the calculator, it was established in the classroom by both the teacher and the learners that a calculator was used to get the incorrect answers. these errors are conceptual because they relate to not understanding how the calculator works and are likely to be repeated and systematic in nature. table 2 illustrates the frequency of each of the four kinds of mistakes across the lessons that were analysed. there were a total of 69 mistakes made across the four lesson categories. table 2: types of errors that occurred per lesson category. most of the errors that were made across these four lesson categories arose from misconceptions. this indicates that the majority of the incorrect answers made across the analysed lessons were derived from an overgeneralisation of correct prior knowledge. nesher (1987) argues that misconceptions give rise to a cluster of errors. this means that from one misconception, there can be many errors of a similar nature in each lesson category. this could possibly be the reason why the frequency of errors due to misconceptions is significantly higher than the other conceptual errors and slips. errors dealt with by the teacher of the 69 mistakes that were made across the four categories of lessons, 45 were dealt with by the teacher, while 24 were not. some of these 24 mistakes were ignored by the teacher: she did not acknowledge or engage with the learners’ responses and, in some cases, she may not have heard them. others were ignored because they were shouted out, a deliberate strategy of this teacher. table 3 illustrates the number and the nature of the mistakes that were dealt with by the teacher and should be read in conjunction with table 2. while table 2 shows that there were a total of seven slips that occurred during the lessons, table 3 shows that one slip was dealt with by the teacher. similarly, of the 55 errors derived from misconceptions (table 2), 36 were dealt with by the teacher (table 3) and all of the five language-related errors and two incorrect usage of the calculator were dealt with (table 2 and table 3). figure 1 illustrates the number of mistakes in each category dealt with by the teacher (table 3) in relation to the number of mistakes made in each category (table 2). figure 1 illustrates that the teacher dealt with all the language-related errors as well as all the errors derived from the incorrect usage of the calculator. she further dealt with 65% of errors that were derived from misconceptions. table 2 and table 3 illustrate that most of the misconceptions that were not dealt with occurred during the first individually planned and jointly planned lessons. it is possible that the teacher may have benefited from her participation in her professional learning community as she dealt with more errors during the last two individually planned and jointly planned lessons. one of the seven slips that occurred during the lessons was dealt with by the teacher, which could be due to the fact that slips are not as serious as errors because they are sporadic and due to carelessness. table 3: number of different kinds of mistakes dealt with by the teacher. figure 1: number of mistakes dealt with by the teacher. how the teacher dealt with the errors we categorised the manner in which the teacher dealt with errors using brodie’ s (2013) three categories: correcting errors, probing errors and embracing errors. there was one slip that was probed by the teacher. we have included it in the category of probing errors despite slips not being classed as errors. examples of correcting, probing and embracing errors from the lessons are provided to show the reader how we coded our data. the first example is an excerpt where the teacher corrected an error. in this excerpt, the teacher asked the learners to share 12 sweets according to the ratio 1:2:3. she asked different learners to answer how many sweets will be represented by 1, 2 and 3 in the ratio: teacher: what is two parts of the twelve sweets? learner a: four. teacher: that is four sweets. and now, i’m coming to this side, learner b? what is three parts of twelve sweets? learner b: three. teacher: is three. alright, i want people on this side to help learner b. because learner b is sitting on this side. so what is three parts of twelve sweets, learner c? learner c: six. teacher: six. learner b's response to how many sweets are represented by 3 in the ratio was incorrect. the teacher then asked another learner what the correct answer was and it was established to be 6. despite addressing the error, the teacher did not get to the bottom of why this error was made as she had another learner correct learner b. we note here that even if the teacher gets another learner to correct the error, it still counts as correcting the error because the underlying conceptual issues are not dealt with. borasi (1987) argues that this manner of correcting errors is ineffective as any learning that occurs may be temporary. this is because errors are evidence of conceptual misunderstanding and simply correcting them does not mean that the conceptual basis of this error is corrected. this is emphasised by brodie (2013) who argues that the manner of correcting errors may not contribute to supporting learners’ access to mathematical knowledge. the second example is an excerpt of the teacher first probing an error and then embracing the error. in this excerpt, the teacher asked the learners to find n + m + p if n + m= 11. this is a very long excerpt. we have divided it into three smaller parts to make the discussion that follows easier to understand. part a: teacher: if nplus m is equal to eleven. then what is n plus m plus p? lift up your hand. if n plus m is equal to eleven. then what is n plus m plus p? learner h? learner h: it might be fifteen and a half. teacher: it is fifteen and a half. and how did you get fifteen and a half? learner h: because n is equal to five and a half and m is equal to five and a half. so p is five and a half again. teacher: right. how do you … why do you get five and a half? learners: sixteen and a half. teacher: where do you get five and a half? learner h: because five and a half plus five and a half is equal to eleven. teacher: is there anything that tells us that should be solved like that? learners: [inaudible] teacher: it's your mind set. who else has got another answer for that sum? because he, learner h, what did he … did he divide that eleven by two. and why did he divide by two? learner h: because half of eleven is five and a half. teacher: because half of eleven is five and a half. so in other words, when you look at n and you look at m, what do you see there? look at n and look at m. because we are trying to find out why did you think of dividing that eleven by two? where did you get that from? learner h: because mam, half of eleven is five and a half. and if you equal five and a half and five and a half, it gives you eleven. part b: teacher: right. learner i, do you want to say something? learner i: mam, i think its eighteen mam, because m plus n is equal to eleven. so m can be five, ncan be six. so if it's n plus m plus p, then m can be five, ncan be six and p can be seven. teacher: why did you say m can be five, ncan be six. what informs you of that? what in that sum? what is m and n? how do we call m and nin that sum? learners: variables. teacher: they are? learners:[shout out] variables, unknown numbers. teacher: they are variables. what is a variable? learners:[inaudible] teacher: sssshhhh … learner j? learner j: that is a letter from the alphabet that represents a number. part c: teacher: that is a letter from the alphabet that represents a number. right, so when learner i said that m can be represented by five, ncan be represented by six. so when we add both of those, it could be that m is two and nis nine. or m is seven and nis … what would n be, if mis seven? what would n be? learner k: four. teacher: it would be … four. alright. so we agree that m and n are variables there and they are representing numbers. so, they can be any number. now why do you say p is seven? where do you get that from? where do you get that from because we have just said that m and n are letters of the alphabet which are representing a number. learner: [inaudible] teacher: ok, he says that because we say m is five and n is six, so automatically p will be seven. but if i say to you mis two and nis nine. what would p be? learner l: [shout out] ten. learner m: [put up his hand] mam, the answer is eleven plus p. teacher: the answer is eleven plus p. in part a of the excerpt, the teacher constantly probed learner h's error. she asked him how he got the answer of 15½, why he got 5½ for n, whether there was any information in the problem that made him think that problem could be solved this way and continued to probe his error. the teacher responded to this error by interacting and engaging with the error to access the learner's reasoning and to support him in reflecting on his solution process. in part b, the teacher used the same approach when dealing with learner i's error. in both cases, the learners’ reasoning clearly made sense to them, but would not make sense to a mathematician and did not make sense to the teacher. this is because the learners were assigning values to the variables which were not necessary according to the question. by examining part a and part b, it is evident that the teacher uses the teaching strategy of questioning to probe the errors, for example by asking learners to justify their answers in a discussion as to why they thought the question should be solved in the manner they suggested. what makes this category of dealing with errors different from embracing errors is that she did not use these learner justifications to promote epistemological access. in part c, after obtaining a definition of variables from the learners, the teacher began to problematise the question. using the definition that variables represent numbers, the teacher substituted different values that add up to 11 in place of mand n to show that these variables do not represent unique numbers. finally, one of the learners gave the correct answer, which was explained later in the lesson. throughout this excerpt, the teacher did not tell the learners at any point that they were wrong; instead she used questions to support the learners’ understanding that the variables m and n can represent any two numbers whose sum is 11. the above excerpt illustrates that this teacher did not only tolerate errors, but used them for epistemological purposes. the learners had learned previously that there can be a finite set of values for variables in an equation and an infinite number in an expression. here, the teacher further supported learners in developing a conceptual understanding of variables in equations and expressions through conversation. the manner in which the teacher dealt with the error enabled learners to perceive that their errors are reasonable and are an integral part of learning mathematics (brodie, 2013). this excerpt indicates how errors can be embraced ‘as a point of contact with learners’ thinking and as points of conversation, which can generate discussions about mathematical ideas’ (brodie, 2013, p. 8) and contribute to the enrichment and development of mathematical knowledge. what is interesting about all the errors that were embraced across the lessons is that the conversations were all lengthy and required a large amount of time. hansen (2011) argues that dealing with misconceptions to enable learning is very time consuming. the exploration of mathematical ideas is encouraged by brodie (2007), who argues that these conversations are important as they can foster an increase in genuine learner thinking. table 4 illustrates the number of errors that were corrected, probed or embraced by the teacher across the four categories of lessons. table 4 shows that the teacher corrected more errors in comparison to the number of errors she probed or embraced, which is a finding that might be expected of most teachers (brodie, 2013). however, in the case of this teacher, 21 errors were either probed or embraced, which is similar to the number corrected, suggesting that this teacher was engaging errors in a way that is different from many teachers. this is unexpected and may be attributable to the work that the teacher did with her professional learning community in dipip. in order to investigate this suggestion further, we looked at shifts in how the teacher dealt with errors over time. table 4: the manner in which the teacher dealt with the errors. changes over time in dealing with errors there was a change in the manner in which the teacher dealt with errors across the four categories of lessons which took place over a two-year period. table 4 shows that the teacher corrected and probed more errors over time but the number of errors embraced remained relatively constant. table 5 shows the number and percentage of errors dealt with by the teacher per lesson category. table 5: number and percentage of errors dealt with by the teacher per lesson category. table 5 illustrates that there was an increase in the percentage of errors dealt with by the teacher across the lessons: from approximately 32% and 65% to 86% and 100%, although the 100% was of a small number of errors. table 4 shows that the teacher dealt with more errors by correcting and probing them; however, the number of errors embraced remained relatively constant over time. what our quantitative analysis does not show is whether the teacher dealt with errors differently over time from how she dealt with them initially in the first categories of lessons. despite the quantity of embraced errors being the same across the lesson categories, the manner in which the teacher embraced errors was less time consuming in comparison to how she embraced errors in the first individually planned category of lessons2. in addition, the teacher tended to elicit more incorrect responses from a single question which was corrected with a single response in the second individually planned and jointly planned lessons. that is why there are more errors that were corrected, especially in the second individually planned lessons and the second jointly planned lesson. we now describe how the teacher's manner of dealing with errors changed compared to how she dealt with them initially. in terms of correcting errors in the first individually planned and jointly planned sets of lessons, the teacher responded to errors using the initiation-response-evaluation cycle: the teacher asked a question, a learner responded incorrectly and an evaluation of the error followed (brodie, jina & modau, 2009). however, instead of telling the learner they were incorrect, the teacher asked other learners to evaluate the incorrect answer, which was established to be wrong. thereafter, the teacher asked another learner to give the correct answer. this method of evaluating errors after they were made shifted in the second individually planned lessons. instead of correcting or getting other learners to correct errors immediately after they were made, the teacher usually asked the learners if there were any other answers, which elicited more errors. often a number of other errors arose and the teacher faced the dilemma of which errors to follow up. if she chose to correct the errors, she did not just tell learners they were wrong, but she explained to the learners why they were wrong. this method of correcting errors was prevalent in the manner in which the teacher dealt with eight of the ten errors corrected in the second individually planned lessons and four of the six errors corrected in the second jointly planned lesson. this illustrates a progression on the part of the teacher in correcting errors from providing the correct answer to a single error, to eliciting a cluster of errors, correcting them and explaining the correct answer. in terms of the teacher's shift when embracing errors, we showed that initially the process of embracing errors was time consuming. in later lessons, the teacher managed to embrace errors, but in less time. the teacher initially addressed the errors by probing learners as to whether they agreed or disagreed with an incorrect solution. after she probed learner thinking, she suggested a pathway to follow and simplified her questions to obtain the solution. this simplification of questions could be a reason that she embraced errors in less time than in the first sets of individually planned and jointly planned lessons. we were careful to check whether the simplification of questions led to what stein, smith, henningsen and silver (2000) call task decline, because if there was too great a decline in the cognitive demands of the task it would not count as embracing errors. we classified this manner of dealing with errors as embracing errors when the teacher used the error to generate new knowledge about particular concepts. conclusions top ↑ in this article, we have shown that the teacher dealt with four types of mistakes, namely slips, errors derived from misconceptions, language-related errors and errors derived from the incorrect usage of the calculator. we categorised the manner in which she dealt with the errors in three categories, namely correcting errors, probing errors and embracing errors, and showed how each of these approaches to dealing with errors provides different forms of access to knowledge. we found that most of the mistakes made throughout the four categories of lessons were conceptual in nature and that the teacher probed and embraced almost as many errors as she corrected. we also showed that the percentage of errors dealt with by the teacher across the lessons increased: from 32% and 65% to 86% and 100%, although the 100% was of a small number of errors. the shifts in the teacher's practice could possibly be due to the influence of her participation in her professional learning community, which supported her to engage with learner errors; however, confirming this conjecture requires further research. we also argued that the manner in which the teacher corrected and embraced errors changed over time. the teacher managed to elicit and correct more errors and she managed to embrace errors in less time. good teaching requires using learner errors constructively in class on the basis of teachers’ professional knowledge and judgements. embracing errors, as we have illustrated, has the potential to allow learners to develop a rich understanding of concepts. it is preferable that teachers embrace errors rather than correcting or probing errors, which provide learners with limited access to knowledge in comparison to the access afforded to learners when teachers embrace errors. however, we do not argue that teachers should always embrace errors, because as hansen (2011) and our findings suggest, embracing errors may be extremely time consuming. with the demands of the curriculum, it would be difficult for teachers to constantly embrace errors. we do think however, that embracing errors may be less time consuming than re-teaching and re-explaining ideas, which are not conducive to eradicating misconceptions (borasi, 1987). teachers should be aware of the benefits and limitations of correcting, probing and embracing errors. using their professional knowledge, teachers should decide when and why it is appropriate to correct, to probe and to embrace errors in light of their knowledge of the content and their learners. for example, it might not make much sense to embrace a slip. probing or correcting slips would be a more suitable method of dealing with the mistake. in probing and embracing errors, teachers are likely to develop their learners’ mathematical proficiency and reasoning skills, help them become aware of their own errors and develop a sense of agency in relation to their mathematical learning. acknowledgements top ↑ funding for the dipip project was provided by the gauteng education and development trust and the national research foundation. competing interests the authors declare that they have no financial or personal relationship(s) that may have inappropriately influenced them in writing this article. authors’ contributions a.g. (university of the witwatersrand) and k.b. (university of the witwatersrand) worked together on this article in discussing, drafting and refining the article. the article is based on a.g.'s research project, of which k.b. was the supervisor. references top ↑ adams, t.l. (2003). reading mathematics: more than words can say. the reading teacher, 56(8), 786–795. ball, d.l., & bass, h. (2003). making mathematics reasonable in school. in j. kilpatrick, w.g. martin, & d.e. schifter (eds.), a research companion to principles and standards for school mathematics (pp. 27–44). reston, va: nctm. http://dx.doi.org/10.1111/j.1949-8594.2003.tb18153.x borasi, r. (1987). exploring mathematics through the analysis of errors. for the learning of mathematics, 7(3), 2–8. available from http://www.jstor.org/stable/40247900 borasi, r. (1994). capitalizing on errors as ‘springboards for inquiry’: a teaching experiment. journal for research in mathematics education, 25(2), 166–208. available from http://www.jstor.org/stable/749507 brodie, k. (2007). teaching with conversations: beginnings and endings. for the learning of mathematics, 27(1), 17–22. available from http://www.jstor.org/stable/40248555 brodie, k. (2013). the power of professional learning communities. education as change, 17(1), 5–18. http://dx.doi.org/10.1080/16823206.2013.773929 brodie, k. (2014). learning about learner errors in professional learning communities. educational studies in mathematics, 85, 221–239. http://dx.doi.org/10.1007/s10649-013-9507-1 brodie, k., jina, z., & modau, s. (2009). challenges in implementing the new mathematics curriculum in grade 10: a case study. african journal of research in mathematics, science and technology education, 31(1), 19–32. http://dx.doi.org/10.1080/10288457.2009.10740648 brodie, k., & shalem, y. (2011). accountability conversations: mathematics teachers’ learning through challenge and solidarity. journal of mathematics teacher education, 14(6), 419–439. http://dx.doi.org/10.1007/s10857-011-9178-8 hansen, a. (2011). children’s errors in mathematics: understanding common misconceptions in primary school. (2nd edn.). exeter: learning matters ltd. hatano, g. (1996). a conception of knowledge acquisition and its implications for mathematics education. in p. steffe, p. nesher, p. cobb, g. goldin, & b. greer (eds.), theories of mathematical learning (pp. 197–217). mahwah, nj: lawrence erlbaum. heinze, a., & reiss, k. (2007). mistake handling activities in the mathematics classroom: effects of an in-service teacher training on students’ performance in geometry. in j.-h. woo, h.-c. lew, k.-s. park, & d.-y. seo (eds.), proceedings of the 31st conference of the international group for the psychology of mathematics education (vol. 3, pp. 9–16). seoul: pme. ingram, j., baldry, f., & pitt, a. (2013). the influence of how teachers interactionally manage mathematical mistakes on the mathematics that students experience. in b. ubuz, ç. haser, & m. mariotti (eds.), proceedings of the 8th congress of the european society of research in mathematics education (pp. 1487–1495). ankara: european society of research in mathematics education. available from http://cerme8.metu.edu.tr/wgpapers/wg9/wg9_ingram.pdf lakatos, i. (1976). proofs and refutations. cambridge, uk: cambridge university press. http://dx.doi.org/10.1017/cbo9781139171472 nesher, p. (1987). towards an instructional theory: the role of students’ misconceptions. for the learning of mathematics, 7(3), 33–39. available from http://flm-journal.org/articles/4582e5c6877c0cbf1cff62aba1aa0b.pdf olivier, a. (1989). handling pupils’ misconceptions. pythagoras, 21, 10–19. peng, a., & luo, z. (2009). a framework for examining mathematics teacher knowledge as used in error analysis. for the learning of mathematics, 29(3), 22–25. prediger, s. (2010). how to develop mathematics-for-teaching and for understanding: the case of meanings of the equal sign. journal for mathematics teacher education, 13(1), 73–93. http://dx.doi.org/10.1007/s10857-009-9119-y smith, j.p., disessa, a.a., & roschelle, j. (1993). misconceptions reconceived: a constructivist analysis of knowledge in transition. journal of the learning sciences, 3(2), 115–163. http://dx.doi.org/10.1207/s15327809jls0302_1 stein, m., smith, m., henningsen, m., & silver, e. (2000). implementing standard-based mathematics instruction: a casebook for professional development. new york, ny: teachers college press. swan, m. (2001). dealing with misconceptions in mathematics. in p. gates (ed.), issues in teaching mathematics (pp. 147–165). london: falmer press. footnotes top ↑ 1. see brodie and shalem (2011) for a description of the dipip activities and how the jointly planned lessons arose out of previous error analysis. 2. the manner in which the teacher probed errors remained constant over time. this is because probing errors merely required the teacher to access justifications for errors. microsoft word 32-42 long.docx 32 pythagoras, 70, 32-42 (december 2009) from whole number to real number:  applying rasch measurement  to investigate threshold concepts  caroline long  the centre for evaluation and assessment, university of pretoria  caroline.long@up.ac.za      the  developments  in  mathematics  that  take  place  in  grades  7  to  9  constitute  critical nodes in a learner’s schooling. one of the major transitions to be made is  from  whole  numbers  to  real  numbers,  which  involves  the  understanding  of  rational  (and  irrational)  numbers,  and  concepts  such  as  ratio,  proportion  and  percent.  i  hypothesise  that  ratio  is  a  threshold  concept  that  provides    the  conceptual gateway to higher order concepts. the research problem is to describe  the learning challenges and provide an array of insights and strategies that will  inform teaching. the theory of conceptual fields provides the framework for this  research  (vergnaud,  1988).  the  rasch  measurement  model  (rasch,  1960/1980)  articulates  the  qualitative  and  quantitative  aspects  of  the  research.  this  paper  provides an overview of the broader study and reports on an aspect of the data  analysis.  from whole number to real number usiskin (2005) highlighted the critical mathematical developments that take place during the “transition years”, most notably the transition from whole number1 to real number. in a detailed mathematical analysis of the unfolding development of number systems from whole numbers to fractional numbers, fractional numbers to rational numbers, and rational numbers to real numbers, skemp (1971) notes that there are both smooth assimilations and difficult accommodations that take place in this unfolding. each new system retains elements of the previous system, but introduces new notation, new meanings to operations, and contributes additional rules. in each new [number] system there are sub-sets which are isomorphic2 with earlier systems. this [isomorphism] allows us to move freely from one number system to another, and also to mix systems provided that each one is operated according to its own methods. the overall result is a conceptual system of enormous power and flexibility (skemp, 1971, p. 226). it is this structure of number systems, where aspects of the system, for example the fraction notation used for both fraction and ratio, deceptively alike and yet subtly different, that is the rubicon for most learners of mathematics as they make the transitions from whole numbers to rational numbers and then to real numbers. the assimilation into existing schemas and accommodating existing schemas to incorporate new 1 the term whole number is used by usiskin (2005). in the sense it is used here the term whole number can be used interchangeably with natural number. 2 two number systems are isomorphic if 1) there is a mapping of one into the other that puts them into one-to-one correspondence. 2) under this mapping, sums and products are preserved. caroline long 33 concepts is a challenge for most learners. this accommodation is particularly challenging in the topic of percent. as parker and leinhardt (1995) note, percent is a dense language, where the changing meanings are not always accommodated into existing schema. in the chinese curriculum, the term percent ratio is used (cai & sun, 2002). this term indicates the close relationship between percent use and ratio. there is also however, a part whole use of percent which is closer to fraction use (parker & leinhardt, 1995). challenges of major proportion in this domain are not new to the mathematics community. one of the major challenges confronted by the mathematicians of antiquity (340 b.c.) was the discovery that not all numbers could be represented by an integer over (divided by) an integer, of the type b a . this critical understanding confronting the mathematicians was that there is no magnitude however small that will divide into both the side of a square and its diagonal. similarly, there is no magnitude that will divide an integral number of times into both the diameter and the circumference of a circle. the startling fact that numbers like 2 could not be written as a ratio of two whole numbers and the equally startling fact that two line segments exist for which there is no common measure caused great consternation in the pythagorean community in as far back as 340 b.c. (eves, 1990). the historical account of how eudoxus resolved the crisis through formulating a definition of ratio, based on magnitudes that were independent of whether numbers were commensurable or incommensurable, is a telling account of how mathematics developed in response to a question that concerned the mathematical community (eves, 1990, p. 150). the concept of an irrational number is a threshold concept3 that is difficult to learn, but once grasped provides the conceptual gateway to higher mathematics. this critical mathematical node has caused and still causes a measure of consternation among teachers and learners of mathematics today. according to eves, “the incommensurable case was relegated to an appendix [in some textbooks], to be covered at the teacher’s discretion, and sometimes it was omitted entirely, as being beyond the rigor of the course” (1980, p. 57). the omission of this crisis, and its resolution, from the high school topic of rational (and irrational numbers) is a tragedy, or at least a flaw, in that it leaves a conceptual gap in the understanding of irrational numbers. irrational numbers are often introduced as non-terminating, non-repeating decimals, whereas the missing link is that the ratio of two incommensurable magnitudes will result in neither recurring nor terminating decimals. my initial fascination with the concept of incommensurability, and irrational numbers (long, 2006a), was temporarily replaced by a need to understand why the concept of ratio, and the consequent conceptualisation of rational numbers, a necessary precursor to the understanding of irrational numbers, is difficult to learn. essentially the need to explore the topics of ratio and the related topics of fraction, proportion, and percent, for the purposes of teaching and learning ratio, proportion and percent has directed this research study. a further goal, through careful scaffolding of these pivotal concepts, is to contribute to an understanding of rational number, a threshold concept on the path to real number. are ratio, proportion and percent difficult concepts? usiskin (2005) describes one of the conceptual difficulties in the move from whole number to real number as the complexity of rational and real number concepts. for example ‘fraction’ has three different meanings, namely fraction as a number representing a quotient, as a number indicating a position between 0 and 1 and as a number that is not an integer. there are also different notations, fraction, decimal and percent, for the same number (smith, 2002). percent has a part whole meaning in addition to a ratio meaning depending on the context (parker & leinhardt, 1995). according to parker and leinhardt, teachers and learners use the different meanings embedded in each particular problem situation intuitively but the conceptual differences are not necessarily made explicit. the complexity of the real number system and indeed much of school mathematics provides a challenge for researchers in that “a single concept does not refer to only one type of situation, and a single situation cannot be analysed with only one concept” (vergnaud, 1988, p. 141). another complexity is that “a 3 the term “threshold concept” and its meaning is attributed to meyer and land (2005). from whole number to real number 34 single concept develops not in isolation but in relationship with other concepts, through several kinds of problems and with the help of several wordings and symbolisms” (vergnaud, 1988, p.141). in order to encompass the complexity it is therefore important to study conceptual fields. a conceptual field is defined as a set of situations, the mastering of which requires mastery of several concepts of different natures (vergnaud, 1988, p.142). it is broad enough to accommodate the complexity of related concepts and processes that are required to solve a “bulk” of problem situations, but not too unwieldy that it cannot provide a manageable research domain. the multiplicative conceptual field4 is conceptualised as “all situations that can be analysed as simple and multiple proportion problems and for which one usually needs to multiply or divide. several kinds of concepts are tied to those situations in addition to the thinking required to master [the problems]” (vergnaud, 1988, p. 141). concepts include multiplication and division, fraction, ratio, rate, rational number, linear and non-linear functions, vector spaces and dimensional analysis. the development of proficiency in this field begins in the early grades and continues through high school and further. vergnaud (1988) notes that the study of cognitive development can only meaningfully take place in problem contexts in that a concept is not a true concept unless it is operationalisable. the complexity of a problem depends “on the structure of the problem, on the context domain, on the numerical characteristics of the data, and on the presentation” (vergnaud, 1988, p. 143). however, as noted by vergnaud (1988) the impact of these factors depend on the cognitive level of the student. vergnaud, asserts that while the origins of mathematical concepts lie in practical problems the power of mathematics lies in the process of transforming these intuitive and implicit knowledge first used in the solving of problems or a class of problems, into explicit and generalisable knowledge that can be applied to more than one situation. the necessity is therefore to describing mathematical concepts and cognitive processes in the mathematical language which makes explicit hierarchical progression (1988). timss 1999 and 2003 results a secondary analysis of the trends in international mathematics and science study (timss) 1999 and 2003 results investigating only the items which fell into the domain ratio, proportion and percent, showed that in general south african grade 8 learners do not have an operational understanding of this domain; they are not able to solve problems in which these concepts are embedded. this predicament is not unique to south africa as learning the concepts in this domain has provided challenges to learners in many countries (hart, 1981; parker & leinhardt, 1995; vamvakoussi & vosniadou, 2007). the secondary analysis was extended to the contextual data provided by the timss study concerning both the intended and the implemented curriculum. a comparison of the south african curriculum with the timss framework indicated insufficient attention to content detail (long, 2006b, mullis et al., 2003). this omission may be explained by the south african policy context, where it is expected that textbooks interpret the curriculum and provide teachers with additional support, beyond what is stated in the curriculum documents. according to teacher reports concerning the implemented curriculum, the topics relating to ratio were in some cases only taught to the “more able” children (long, 2006b; mullis et al., 2003; reddy, 2006). it is well known that the south african education system has a large tail of inefficient and dysfunctional schooling (taylor, 2007), making it difficult to formulate policy or strategies for intervention based on national aggregated data alone. therefore, in order to investigate the challenges confronting mathematics teaching and learning, selected timss items were administered to learners at two well functioning schools at grades 7, 8 and 9 level, comprising a sample of 330 learners. subsequently interviews were conducted with selected learners for the purposes of further investigating the cognitive development of learners at these grades (long, 2008). 4 the additive conceptual field includes the set of situations for which additive structures are required and sufficient. the algebraic conceptual field builds on the additive and multiplicative conceptual fields. caroline long 35 assessment instrument for ratio, proportion and percent items the requirement for designing an assessment instrument is firstly to have a clear idea of the domain to be tested, and to define the construct of interest (wright & stone, 1979). this requirement is satisfied, more or less, in class tests and the matriculation examination, because experienced teachers have implicit knowledge of the curriculum and the kinds of questions that should be asked. in the case of this study the construct of interest is explicitly constructed as the multiplicative conceptual field, comprising the elements, fractions, ratio, proportion and percent. a second requirement for any instrument is to contain items which will elicit reliable information about the domain to be tested. this precondition necessitated developing an instrument5 comprising items that are realisations of the construct and that would accurately target the population to be tested in terms of difficulty level. this requirement was assisted by having data available on the items from the timss 2003 study so that the difficulty level of items for the south african grade 8 population could be established (long, 2008). the third and fourth requirements, according to wright and stone (1979), are to demonstrate that the items when taken by a suitable population are consistent with expectations and that the patterns of learner responses are consistent with expectations. finding unexpected responses in either the items or the learners requires an investigation of the items used to measure the construct and an investigation of the learner profile (including curricula and instructional experiences) on the test as a whole and consequently, if required, a deeper investigation of the construct being measured. in summary, any test functions as a test of the learner, a test of the test and the test designer, and in addition is a test of the defined construct being tested. in all, 36 items6 from the timss released items were identified as located in the multiplicative conceptual field; these items constituted the research instrument. some of the items had been categorised as belonging to the domain ratio, proportion and percent; however other items also exhibiting multiplicative structures had been categorised as geometry, measurement, data handling (probability), and algebra (patterns). as a proxy for a pilot test, the difficulty levels of the items were checked against the timss 2003 south african results. in order to include some items that could provide information at the lower levels of proficiency, it was decided to include six grade 47 items. the rash measurement model the prior empirical requirement of the rasch model (rasch, 1960/1980), concurring with wright and stone (1979), is to have an explicit understanding of the latent trait, that is, in the case of this research study proficiency in solving problems requiring the concepts fractions, ratio, proportion and percent, and to construct and refine an instrument made up of items that operationalise this trait. a feature of the rasch model permits the discovery and amplification of item anomalies, which are inconsistent with the general expectations of the instrument. the test instrument used in this study proved to fit the requirement of the rasch model of having a clearly defined domain. the researcher is required to further investigate (post-hoc) any particular item shown to be misfitting, and if necessary eliminate the item, while at the same time identify a plausible explanation of the item misfit in terms of its own characteristics. one item, on the topic of probability did not discriminate between learners of greater proficiency and learners of less proficiency, as measured on the test as a whole. on further investigation it was inferred that probability had not been adequately taught and therefore this item was equally likely to be guessed by both high and low proficiency groups of learners, or to elicit an intuitive response, by both groups of learners. 5 the development of an instrument implies the development of a measure that makes the outcome meaningful. 6 one item was rejected making only 35 items. 7 timss 2003 conducted mathematics testing at grades 8 and 4 only from whole number to real number 36 likewise data from learners who for some or other reason do not perform as expected can be eliminated from the analysis, on the basis of explicit reasoned arguments, for the purposes of refining the instrument and establishing probabilistic estimates of item difficulty and learner ability on the same scale. if for example some learners had not taken the test seriously and guessed all the way through, this anomaly would be picked up and for the purposes of developing the scale, these learners would be eliminated. the learners eliminated for the purposes of developing the scale, could still subsequently be allocated a location on the scale for other purposes. the rasch model, by locating both items and individual learners on the same scale, provides an answer to the severe criticism of applications of statistics in some psychological research that statistical methods can only provide information on groups of individuals8. this class of rasch models enables “individualcentred statistical techniques” where each individual is characterised separately (rasch, 1960/1980). it is also possible to develop a profile of individual learners. the clear articulation of qualitative and quantitative information is provided by this model. interpretation of the rasch model the results of the assessment were captured and analysed using rumm software (andrich, sheridan, & luo, 2007). the resulting person–item distribution map shown in figure 1 indicates the difficulty of the items on the right, from high difficulty at the top to low difficult at the bottom. on the left the learners are ranked from high proficiency as measured by this test, at the top, to low proficiency at the bottom. the item mean is set at zero and difficulty levels established through ratio comparisons. the location of learners is calculated in relation to the items and in relation to each other9. for the purposes of this paper, it suffices to say that a learner at a particular point on the scale, for example at -1.359, 50% of the learners can be expected to have item 4 correct, less than 50% of them can be expected to have any specific item higher that item 4 on the scale correct, and more than 50% of them can be expected to have any specific item lower that item 4, for example item 20, correct. the spread of items and learners along the whole scale indicates that the test functions well. information on the learners at all levels can be obtained. the lower level of the scale provides information on learners with lower proficiency. even though five items were added from timss grade 4, there were some learners for whom even these items had little chance of eliciting a correct response. data analysis percent for the purposes of this paper, i have limited the discussion to four items that included the concept of percent and that are located at different levels of difficulty (see figure 1). from the perspective of teaching and learning, we ask, “what makes these items more or less difficult, and therefore easier or harder to learn?” vergnaud (1988), drawing from empirical work on elements of the multiplicative conceptual field, reports that the complexity of items depends on the context, the mathematical structure, the presentation (or notation) and the number range used for the problem. the analysis of the items presented here draws on these four constructs. in addition to locating item difficulty, additional information is provided by distracter analyses. the reasons for learners choosing specific distracters are inferred. these inferences were explored through subsequent interviews at a subsequent phase of the project. the analysis presented here however will remain at the level of a conceptual analysis of items, with empirical support provided by the rasch model. the four items represented increasing levels of difficulty and by assumption complexity. table 1 presents the problem description, the context, the mathematical structure and the type of notation used. 8 the details of this story may be read in the forward by benjamin wright in rasch (1960/1980). 9 the probabilistic process is explained in andrich and marais (2008). caroline long 37 location persons items[locations] o = 2 persons o | 7.0 | | | 34 | | 6.0 | | | | 5.038 | 26 5.0 | | | | o | 4.0 o | | | | 27 | 3.0 o | | o | o | oo | 35 2.0 | 30 31 ooo | o | oo | oo | 1.0 o | 28 0.937 ooooo | 7 ooooo | 32 oooooooooo | 10 29 ooooooo | 0.0 ooooo | 18 -0.060 ooooooooooooo | 9 8 15 oooooooooooo | 33 16 oooooooooooo | 5 25 14 oooooooooo | -1.0 ooooooooooooooooooo | 3 21 ooooooooooo | 23 -1.359 oooooo | 4 24 ooooooooo | 20 6 12 11 ooooooooo | -2.0 ooooooooo | 22 oooo | ooooo | 2 17 o | 13 1 ooooo | -3.0 | | o | | | -4.0 | figure 1: person-item location distribution 26. a computer club had 40 members, and 60% of the members were girls. later, 10 boys joined the club. what percentage of members now are girls? 7. when a new highway is built, the average time it takes a bus to travel from one town to another is reduced from 25 minutes to 20 minutes. what is the percent decrease in time taken to travel between the two towns? 8. a shop increased its price by 20%. what is the new price of an item which previously sold for 800 zeds? 4. at a play, 25 3 of the people in the audience were children. what percent of the audience was this? from whole number to real number 38 table 1: percent items description context mathematical structure notation 4. at a play, 25 3 of the people in the audience were children. what percent of the audience was this? everyday context part-whole 25 3 children (3 out of 25) 10025 3 25 3  x x bx ax b a  fraction notation, percent notation 8. a shop increased its price by 20%. what is the new price of an item which previously sold for 800 zeds? financial context price increase 800 + 20% of 800 100% of 800 + 20% of 800 = 120% of 800 percent notation 7. when a new highway is built, the average time it takes a bus to travel from one town to another is reduced from 25 minutes to 20 minutes. what is the percent decrease in time taken to travel between the two towns? measurement (time) percentage ratio 25 – 20 25 5 converted to percent bx ax b a  percent notation 26. a computer club had 40 members, and 60% of the members were girls. later, 10 boys joined the club. what percentage of members now are girls? everyday part-whole percentage ratio 60% of 40 = 24, 40 + 10 = 50 50 24 convert 250 224   bx ax b a  percent notation item 4 (location -1.359), a relatively easy item was set in an everyday context that is familiar to most learners. the mathematical structure can be described as a part-whole or fraction-related use of percent. the equivalence concept can be used to convert from a fraction to a decimal or percent. this fraction use of percent is the first use encountered in the curriculum, and it is hypothesised by some research studies (parker & leinhardt, 1995), that learners remain at this fraction meaning of percent notation, and therefore do not progress to a ratio understanding of percent. the use of fraction notation in item 4 makes the item more difficult than for example items 13, 1, 2, 17 and 22 (see figure 1) in which natural language is used. the next four items, 20, 6, 12, and 11 exhibiting fraction notation, were according to the empirical results found to be more difficult than the easiest five items expressed in natural language. while fraction notation is the item 4 problem, learners are required to operationalise the concept of percent and work with percent notation. the setting for item 8 (location -0.060) is a financial situation, which learners may have encountered in a shopping experience10. the mathematical structure of the problem requires a ratio understanding of percent, and an understanding of percent notation. the context for the problem in item 7 (location 0.937), involves a change in time and a percent decrease. the mathematical construction entails picturing the two time periods (related to the distances travelled), finding the difference, and then finding the ratio of the difference to the original time. this ratio (in fraction form) is then transformed into a percentage. the increase in conceptual difficulty from item 8 to item 7 is substantial. this is reflected by the difference between these items on the scale (see figure 1). the context for item 26 (location 5.038) involves a calculation, a change in time and a further calculation. the context can be described as an everyday context, but the problem construction involves two steps with two different uses of percent. the first step is to find 60% of 40 members (the referent), which is 24 (the number of girls). the referent whole, 40, then changes (10 boys are added, making 50). the next requirement is to convert the 24 (girls) out of 50 (members) into a percentage. this problem involves switching between fraction and percent notations. 10 it is however the mathematical gaze which transforms the shopping experience into a mathematical task, rather than the act of shopping (dowling, 1998). caroline long 39 three of the items (items 4, 8, and 7) were in multiple choice formats for which 4 or 5 response options were provided. item 26 was presented in a constructed response format. three of the four items (items 4, 8 and 26) comprised the equivalence relation, bx ax b a  . this underlying structure can be regarded as the explicit and generalisable knowledge that can be applied to more than one situation, the power of mathematics that is acquired through the process of transforming intuitive and implicit knowledge that is first encountered in many different situations. making explicit the mathematical language when describing mathematical concepts and cognitive processes, gives form to the hierarchical progression (1988). both item 8 and item 26 formed part of the interview phase; detailed information on learner responses has been reported in long (2008). for the present the discussion will be restricted to the information obtained from the test and from the distracter analyses. item 8 is given as an example. analysis of item 8 the rumm software used to conduct the analysis provides a graph (see figure 2) showing the choice of options for learners of different proficiency levels. for the purposes of analysis the sample was divided into five quintile groups with approximately 63 learners in each group. the mean of each quintile group is calculated on the performance of learners in that ordinal group on the test as a whole. on the horizontal axis, the person locations are exhibited. the mean of each group is marked on the horizontal axis by pointers at 2.09 (lowest quintile); -1.19; -0.66; -0.02; and 1.23 (highest quintile). on the vertical axis the probability of attaining a correct response is exhibited. for example, a learner located at 1.23 (the highest quintile) on the horizontal scale, the frequency of selecting the correct response c (see line marked 3) is 72%. the frequency of selecting d (line marked 4), 16%, of selecting b (see line 2), 7%, and of selecting a (see line 1), 5%. figure 2: item 8 distracter analyses for quintile groups for learners at the low end of the scale, at –2.09, the frequency of selecting the correct option c is 11% (see line 3). the frequency of selecting option d (4) is 57%. this option was also the most likely choice for the second lowest quintile group. the selection of this distracter was due to a part-whole understanding of percentage, possibly influenced by the presence of 1000 as one of the options. the learners at the lower end of the scale, it is inferred, have remained at the early cognitive stage of learning percentage, where fractions, decimals and percent notation are used interchangeably. from whole number to real number 40 this disaggregated information shown in figure 2, provides the researcher, and teacher, with relevant information. the single statistic, that 60% of the class got this item correct, is replaced with more useful information about different subgroups, which range from 72% (the highest quintile), to 11% (the lowest quintile), respectively of learners selecting the correct response. the additional information about what kinds of conceptual errors are made is also useful. the information on the distracters as well as inferred reasons is shown in table 2, in which the information is provided in numerical form. table 2: inferred mathematical procedure for responses by quintile group mean locations of 5 quintile groups choice inferred mathematical procedure -2.09 -1.19 -0.66 -0.02 1.23 c. 960 zeds percentage increase 11% 24% 20% 62% 72% d. 1000 zeds part whole 800 to 1000 increase represents 20% of final amount 57% 45% 40% 17% 16% b. 900 zeds estimate a number greater than 800 21% 24% 26% 16% 7% a. 640 zeds decrease by 20% rather than increase misreading 11% 7% 6% 5% 5% after option c and d, distracter b elicited the most responses. this preponderance may have been due to the fact that learners estimated an answer greater than 800. distracter a may have been selected on account of carelessness. conclusion the transitions from whole number to real number, requires an understanding of concepts such as fraction and ratio, elements of the multiplicative conceptual field. an operational understanding attained through working with these concepts provides the basis for an understanding of rational number. the shift from the “conceptualization of the ratios of integers from relations between numbers to numbers [themselves]” is necessary for understanding the threshold concept rational number (vamvakoussi & vosniadou, 2007, p. 266, my emphasis). this understanding of rational numbers is a necessary pre-curser to understanding irrational numbers, and therefore real numbers, which provides the conceptual gateway to higher order concepts. parker and leinhardt (1995) remind us that percent is a complex construct where the underlying referent is not always made explicit. percent builds on fractions and ratio; in some ways functioning like fractions and decimals, but in many ways differently, for example, one cannot add percentages unless the referent populations are equivalent or equal. it is tempting to think that access to the historical account of discovering irrational numbers and the eudoxian resolution of the crisis confronting mathematicians would enable understanding of incommensurability11. the answer emerging from this research study is that this historical insight may contribute to conceptual understanding, but the true concept cannot be attained without some cognitive engagement on the part of the learner assisted by carefully constructed situations that provide a context, with which learners feel familiar but which demand the development of further mathematical understanding and the associated development of cognitive processes. 11 the introduction via decimal numbers, invented centuries later than the discovery of irrational numbers, circumvents the very crisis and potential understanding of irrational numbers and incommensurability, these concepts being the precursors to the understanding of non-terminating and non-repeating decimal numbers. caroline long 41 in an appeal for greater regard for the complexity underlying the language of percent, parker and leinhardt (1995) propose that attention be given to the underlying mathematical structure of percent and the complex relationship between percent and the related concepts of ratio, proportion, fractions and functions. vergnaud (1988) proposes that research into the learning and teaching of mathematics requires both a mathematical view and a psychological view, and that conceptual analysis of the mathematical situations, and observation of cognitive schemes that are engaged in the solving of these situations, be described using mathematical terminology. this approach enables the teacher and researcher to locate the learner on a mathematical path. it has been acknowledged that educational measurement as it manifests in testing and statistics may be a rather blunt instrument for the fine qualitative distinctions required to understand teaching and learning. the rasch measurement model however, is sensitive to qualitative nuances and indeed requires subject experts to engage at all levels of the assessment process, from the conceptualisation of the construct to the analysis and interpretation of the data. in this study the rasch model and the aligned software have enabled finer degrees of understanding of this complex field. references andrich, d., & marais, i. 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(1971). the psychology of learning mathematics. harmondsworth: penguin. from whole number to real number 42 smith iii, j. p. (2002). the development of students' knowledge of fractions and ratios. in b. litwiller & g. bright (eds.), making sense of fractions, ratio and proportions (pp. 3-17). reston, va: the national council of teachers of mathematics. taylor, n. (2007). equity, efficiency and the development of south african schools. in t. townsend (ed.), international handbook of school effectiveness and improvement (pp. 523-540). new york: springer. usiskin, z. (2005). the importance of the transition years, grades 7-10, in school mathematics. ucsmp newsletter. spring 2005. vamvakoussi, x., & vosniadou, s. 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/hrv (za stvaranje adobe pdf dokumenata najpogodnijih za visokokvalitetni ispis prije tiskanja koristite ove postavke. stvoreni pdf dokumenti mogu se otvoriti acrobat i adobe reader 5.0 i kasnijim verzijama.) /hun 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/pdfxoutputintentprofileselector /documentcmyk /preserveediting true /untaggedcmykhandling /leaveuntagged /untaggedrgbhandling /usedocumentprofile /usedocumentbleed false >> ] >> setdistillerparams << /hwresolution [2400 2400] /pagesize [612.000 792.000] >> setpagedevice microsoft word 17-27 bansilal.docx pythagoras, 69, 17-27 (july 2009) 17 the use of real life contexts in the cta:  some unintended consequences  sarah bansilal  university of kwazulu‐natal  bansilal@ukzn.ac.za      the common tasks for assessment (cta) was a new assessment programme that was  introduced in 2002 in south africa for all grade 9 learners. the purpose of this paper is to  articulate  some  concerns  around  the  use  of  contextualised  assessment  activities  in  the  cta. the study reported here was carried out in 2003. data for the study was generated  from  lesson  observations  and  interviews  with  the  participant  teachers  and  groups  of  learners. it is argued that although the intentions behind the design of the cta are well  meaning  and  noble,  there  are  in  fact  some  learners  who  may  be  unintentionally  disadvantaged by the design of the cta which uses an extended context as a source for  all the assessment tasks. in this paper two unintended consequences of using  ‘real life’  contexts  are  identified  and  the  implications  of  these  are  discussed,  by  linking  the  observations to research carried out in the uk and the usa.   the purpose of this paper is to explore some unintended consequences arising from the use of real life contexts in the design of the externally mandated national assessment programme in the form of the common tasks for assessment (cta). i present two vignettes of two grade 9 mathematics classrooms drawn from a study of the classroom practices of two mathematics teachers who were participants in a study in 2003 which explored the mathematics teachers’ professional knowledge by looking at how and why the teachers mediated the cta in the ways that they did (bansilal, 2006). the abovementioned study was carried out on four participant teachers, and their grade 9 learners drawn from three different schools. in this paper, i focus on some unintended consequences arising from the use of these “real life” contexts in the assessment tasks, which have implications for the validity of these tasks as assessment tools. firstly, the issue of contexts which are out of certain learners’ experiences is highlighted. in the first vignette, we are introduced to a teacher, who in compensating for her learners lack of knowledge of pizzas, simplifies the actual assessment task. her actions seem to be within her prescribed role of helping her learners to succeed (department of education (doe), 2002), but her intervention results in a simpler task. in the second vignette we are introduced to a learner, sipho, who displays a tendency to respond to the task based on his everyday experiences. the discussion is then focused on how learners could be disadvantaged by their propensity to base their responses on their everyday experiences instead of the mathematical considerations that are required by the task designers. these observations are linked to some concerns raised by cooper and colleagues (cooper & dunne, 1998; cooper & harries, 2003) about certain learners being disadvantaged by the use of such contextualised tasks in national assessment programmes in the uk as well as the work of boaler (2003) with respect to the stanford achievement test (sat)-9 assessment programme in the usa. the organisation of the paper is as follows: in the background section, i first present some background surrounding the cta programme before discussing some research around the use of “real life” contexts in assessment tasks. i then briefly discuss the design of the study, and thereafter two vignettes are presented. each vignette is followed by a reflective commentary which identifies and clarifies the main points raised. thereafter i discuss the implications of the findings by relating the identified points to relevant research and i end off the paper by identifying areas for further research. the use of real life contexts in the cta 18 background as part of the broader curriculum reform process occurring in south africa, a new assessment programme at the grade 9 exit level of the get phase was introduced in 2002. the common tasks for assessment (cta) was completely different in approach, form and substance from the senior certificate examination which is written at the end of the 12 years of schooling. the document: draft framework for the development of the cta’s (doe, 2002) was circulated in 2002 and it is this document that i use to trace some guiding principles and the rationale behind the design of the cta. initially there were two parts to the cta, section a which focused on the assessment of performance-based competencies and section b which focused on “competencies that could be assessed with paper and pencil form of assessment” (doe, 2002, p. 6). however the section b of the cta did not appear in the years 2004 to 2006, but reappeared in 2007. this paper is based on the section a of the cta programme in 2003. the cta was intended to be “used as an external summative assessment instrument” (doe, 2002, p. 4). the framework document articulates strong concerns that learners should not be disadvantaged by their abilities, language skills and learning styles. for example some recommendations are that the tasks should “accommodate the diversity of learning styles and abilities” (p. 12), “consider the diverse learning styles and learning needs … [and] determine how adaptations could be built in to address barriers to learning” (p. 14) and “should avoid all biases, e.g., language, urban and rural contexts, gender, disability, etc.” (p. 12). these excerpts convey a strong recommendation that learners with different learning styles must be accommodated and that the tasks should not have language or context biases which may lead to certain learners being disadvantaged by their backgrounds. an interesting and unusual suggestion (for summative assessment tasks) was that the teacher could play an active role during the assessment programme in order to help learners overcome their barriers. one of the stipulations was that the cta tasks should be structured “so that teachers can help learners succeed” (doe, 2002, p. 12). the document contains references to a teacher’s manual which should be used to guide the teacher. the framework document states that the manual should “specify clearly what the teacher is expected to do in each activity” (p. 8). however, none of the participant teachers had seen such a manual. it is possible that the manual had been reduced to the marking memorandum, which all the teachers had received. in the absence of detailed guidance on the interventions that were permissible, teachers exercised their own judgments and intervened in different ways, ranging from not providing any hints at all, to working out certain tasks in the class (bansilal, 2006). it will be seen from the discussion that this combination of the use of “real life” contexts with the stipulation that teachers were allowed to intervene, produced a particular set of threats to valid assessment. the department of education was explicit that the cta should be designed as performance based assessment tasks; the framework document states that the “..[cta] will target outcomes not best assessed in a pencil and paper exercise, … should be based on performance-based assessment” (p. 9). the document explains performance based assessment: performance–based assessment permits learners to show what they can do in a real life situation, i.e. this is a type of assessment that emphasises the learners’ ability to use or transfer their knowledge, understanding and skills into action. … performance-based assessment provides a systemic way of evaluating those reasoning skills and outcomes … that cannot be assessed adequately by a paper and pencil test. (doe, 2002, p. 9) it is emphasised once again (on p. 12) that the tasks “should be grounded in real life contexts”. the above excerpts demonstrate the doe’s intention to move the assessment away from pencil and paper tests and conveys the view that assessment tasks set within real life contexts are able to accommodate the diverse learning styles and learning needs of the south african learner population. however, i will argue that in fact certain learners are disadvantaged by their tendency to base their responses on their everyday experiences. looking at mathematics in “real life” or “real world” contexts is a common refrain heard by teachers, learners and parents throughout the world. moschkovich (2002) explains that real world when used to describe curriculum assessments, or mathematical activity can refer to activities in which students might sarah bansilal 19 engage during the course of their present daily lives or to future activities in which students might engage as adults at work. this emphasis on “real life” contexts is a common trend in mathematics curricula globally. cooper and dunne (1998) note that in recent years, in the uk, there has been an increasing emphasis on understanding, investigation and the application of mathematics in realistic settings, as opposed to the emphasis in the 1960s of abstract algebraic approaches. moschkovich (2002) states that curriculum and teaching standards, in the usa, for the mathematics classroom have made calls for engaging students in real world mathematics rather than mathematics in isolation. a powerful use of contexts is in the mediation of the teaching of new mathematical concepts. the real world can be a source of analogies that can be used to make the mathematical ideas more relevant and more meaningful for learners. one well-known theory of learning is that proposed by the realistic mathematics education school of thought which has its roots in freudenthal’s interpretation of mathematics as a human activity. within this philosophy, learners are encouraged to learn mathematics by mathematising subject matter from real contexts and their own mathematical activity, rather than presenting mathematics to them as a ready-made system with general applicability (gravemeijer, 1994). real situations may include contextual problems or mathematically authentic contexts for learners where they experience the problem presented as relevant and real. examples of such contexts can be found in freudenthal, (1983), linchevski and williams (1996) and william(1997). boaler (1993) wrote: the move away from the teaching of mathematics as a complete series of abstract calculations has also been supported by a number of different assertions regarding the enhancement of mathematical understanding. the abstractness of mathematics is synonymous for many with a cold, detached remote body of knowledge….[an image that] may be broken down by the use of contexts which are more subjective and personal also improving the ability of students to interpret events around them. (p. 2) however, the simplistic assumption that the use of contexts provides meaning and improves the understanding of mathematical concepts is challenged by certain researchers. arcavi (2002) cautions us that familiarity with a context does not necessarily make the understanding of the mathematics easier. he uses the example of f(x) = x to show that a mathematical idea may sometimes be easier to use and conceptualise in a de-contextualised environment than in a familiar context. evans (1995) criticises the simplistic notion that giving real world contexts for mathematical concepts provides meaning for learners – a meaning that supposedly exists in some absolute sense and is illustrated by or modelled in that real world context. in a similar vein, william (1997) notes that sometimes contexts are used which bear little or no relation to the mathematics being taught and which serve primarily to legitimate the subject matter being taught (“mathematics looking for somewhere to happen”). boaler (1993) states that “misconceptions prevail such as the belief that mathematics in an ‘everyday’ context is easier than its abstract equivalent” and notes that the degree to which the context of a task affects students’ performance has been widely underestimated. boaler’s (2003) assertion that teaching situations in which students are learning are very different from standardised assessments in which they are being tested, has direct reference to this paper. i am not taking issue with the use of realistic mathematics contexts to enhance understanding of various mathematical concepts – my focus is on the use of realistic contexts in the mathematics cta programme. in particular, i am concerned about the ways in which certain learners could be disadvantaged by some unintended consequences brought about by, the design of the tasks and/or assumptions about learners experiences. cooper has written extensively with various colleagues (cooper, 1998; cooper & dunne, 1998; cooper, dunne & rodgers, 1997; cooper & harries, 2003) about the use of these real life contexts in national assessment tasks in the uk. cooper and his colleagues’ studies alert us to the danger of certain learners’ abilities being underestimated by national assessment tasks set within real life contexts. the use of these “realistic” test items may present the danger of learners relying on their everyday knowledge instead of their mathematics knowledge. when learners get confused about which knowledge to draw upon, it may not mean that they do not know the mathematics, but it may just mean that they did not realise what they were being asked for. cooper and dunne (1998) used bernstein’s theory to demonstrate how questions the use of real life contexts in the cta 20 that were set in everyday contexts in national mathematics tests in the u.k disadvantaged working-class children. cooper and dunne (1998) drew upon data from primary schools to show why certain “realistic” mathematics test items were associated with the under-estimation of children’s actually existing capacities in mathematics and that the extent of such under-estimation varied by social class. the study the data presented in this paper was drawn from a larger study designed to explore the professional knowledge of four grade 9 mathematics teachers in the context of mediating the 2003 cta (bansilal, 2006). the methodology employed was a case study approach. cohen, manion and morrison (2000) maintain that case studies provide a unique example of real people in real situations, enabling readers to understand the events more clearly than simply presenting them with abstract theories or principles. in order to understand the teachers’ actions, the case study allowed me to focus on the particularity of each teacher’s context. data for the larger study was generated from classroom observations, semi-structured and informal interviews and my field notes. during the analysis stage of the research study mentioned above, certain compelling aspects related to learners’ experiences of the real life contexts used in the cta, came to the fore. it is my purpose, in this paper to raise some of these issues related to the learners’ experiences of the cta. in this paper i use my field notes, journal entries, and lesson and interview transcripts to construct two vignettes  a party without pizzas and sipho and the weather in durban. the vignettes are constructed using polkinghorne’s (1995) narrative analysis technique where the data elements are configured into a story that unites and gives meaning to the data. accompanying each vignette is a reflective commentary, on issues pertaining to the central research question: what are some unintended consequences of using real life contexts in the cta? this question is then explored in further detail in the discussion section of the paper using polkinghorne’s (1995) analysis of narratives strategy applied to the two vignettes. the focus in this paper is not on the teachers’ pedagogic styles or repertoires. vignette 1: a party without pizzas the setting for this vignette is vanitha who is teaching in an ex-hod school in a durban south township, where most of her learners are african. in this lesson, vanitha was involved in preparing her learners to respond to a task whose context was out of her learners’ experience. the task is provided below. task three: activity 2 (pairs) at a restaurant at the waterfront in cape town, tourists have a choice of different pizzas: base toppings thick pineapple regular salami tuna mushroom if a tourist buys a pizza with three toppings, how many combinations are possible? (use any systematic counting method that you have learnt.) i was surprised when vanitha started her lesson off by inviting her learners to a diwali party instead of discussing the cta task. however i soon realised that she was using the context of party with drinks and snacks to introduce the meaning of combinations. the following excerpt from her lesson illustrates her strategy of using a similar task that was set within a different context (of providing refreshments) to mediate the cta activity. instead of asking them for combinations of pizza toppings and bases, she asked sarah bansilal 21 her learners to choose combinations of snacks and drinks (t = teacher, and s = learner): t: i think you have been very good so far so what i am going to do is invite all of you to my house. would you be able to come? s: yes t: ja [meaning yes], but now i am only going to serve 2 types of cool drinks right because i don't have a lot of money. i could only serve coke to you or fanta. is that okay? now listen carefully you can't have both, you can either choose coke or you can choose fanta. i am also going to give you some eats. i am going to give you some eats. it is going to be limited, very limited. so these are the things that i am offering to you. so you got a variety there. can you see? i have got almond biscuits, ginger biscuits and peppermint biscuits. you know what is that? i have some date rolls, i don't know if you like that. date rolls, it’s biscuits but it got dates in it. how many of you like that? s: (inaudible) t: and i have made a fruit cake as well. okay you can see we got to do a lot of eating on diwali day. there's coke and fanta and 5 kinds of things that you can eat…… while talking to the class, vanitha put up this list on the board with the drinks on one side and the biscuits on another side: coke fanta almond biscuits a ginger biscuits g peppermint biscuits p date rolls d fruit cake f she continued: t: but you can only choose 3. you can eat only 3. okay so what are the combinations, the choice you can choose? first you got to decide whether you got to have coke or fanta. now if you decide to have coke, you can have 3 types of biscuits. whatever it is. okay now lets look at all the combinations. if you have fanta you can also have 3 types of eats that's there. any 3 combinations. all right now look at this.[points to her list] you have gotten 5 different kinds of things here. how many combinations can you have? … what are the possible combinations that you can have? what are the possible combinations that you can have? let’s use some abbreviations here. [as vanitha called out the abbreviations she wrote these letters next to the item it represented] almond biscuits, what you call it? a, and ginger biscuits g, and peppermint biscuits what will we call it? s: p she then led her learners to list the possible combinations in the next 20 minutes. thereafter vanitha linked the ways in which they generated the list to the pizza problem: t: so you know how to work with these combinations. in your activity you will read it, it has nothing to do with these combinations, it has to do with pizza … you will have to list your combinations in this manner and the question is you have to find out how many combinations can you have? after handing out the tasks to the learner she continued in a similar vein: t: ok let’s look at activity 2 now. you are at a restaurant and you are looking at a pizza, you are going to buy different pizzas. right you can either have pizza’s with a thick base or you could order a pizza with a regular base. and how many toppings are there? s: 4 the learners then completed the task during the lesson, while vanitha walked around and emphasised the connections between the two tasks. later on, in an informal conversation conducted after the lesson, vanitha explained that she had suspected that many of her learners had not experienced eating pizza and she had therefore asked the class the the use of real life contexts in the cta 22 previous day, how many learners had eaten pizza before. there were six learners who had indicated that they knew what pizza was and vanitha feared that the terms base, toppings, mushroom, salami and tuna, would be a barrier that would disconnect them from engaging with the mathematics of the task which involved carrying out the relatively simple mathematical procedure of listing the various combinations. this reasoning led her to design a similar task which was set within a context that was familiar to all her learners. this lesson foregrounds the question: what happens when a context is out of the learners’ experiences? in this situation vanitha intervened by working through a similar question set within a known context. in order for the learners to complete the task vanitha needed to help them understand the concept of combinations. by using a familiar context (of cool drinks and snacks), vanitha wanted to help her learners to understand the procedures involved in generating a list of all possible combinations of elements from two given sets. the learners were then able to apply the procedure they used in the party context to the pizza context because they understood that they needed to choose between the bases firstly and then a second choice between the various toppings. at this stage they needed to just distinguish between the sets from which they could make the choices from. once they understood the role played by the toppings and bases in the procedure, they were no longer intimidated by the unfamiliar words — mushroom, salami and tuna — because they knew that the terms were part of the set from which the second choice of four elements had to be chosen. however vanitha’s intervention simplified the cta task by reducing it to an application of a familiar operation to be performed on two sets. the consequences of the context being out of learners’ experiences is explored further in another paper by the author (bansilal, 2008), where issues around the fairness and validity of the cta are interrogated, by using learners’ responses to the pizza activity discussed in this vignette. vignette 2: sipho and the weather in durban the following activity was under discussion in thami’s class, in a school in a township in the durban south region. the school is a high achieving school and boasts an excellent pass rate in the matriculation examination. in the cta lessons thami usually went over as many as three or even four cta activities in one lesson, leaving the learners to complete the activities in their study time. these study times were periods in the morning before school officially started or in the afternoons after the school day had ended during which the learners worked unsupervised in their classes. in this lesson thami’s discussion is centred around the activity (below) that requires learners to interpret two graphs on temperature and rainfall and thereby decide which times would be suitable for repairs. task 1, activity 4 (pairs) work4u intends to do the repairs in the dry season when day temperatures are as high as possible. on the next page you will find two graphs with information about the temperatures and rainfall on robben island. use these graphs to help work4u decide during which months they can do the work. give reasons for your choice. t: now let’s compare these 2 graphs. now you notice here all right according now to the 2 graphs, you notice that on graph number one, what type of graph is that one you see there you have the horizontal axes? horizontal axes is what you have. horizontal axes is what you have. on your first graph what you have? s: months. t: months okay. this is january, february up to december then thereafter you see on that one you have on axes. what type of axes is that one? that is? s: inaudible t: okay thereafter if you are looking at the first graph you see, how many months? s: twelve. sarah bansilal 23 t: twelve months. okay so when i look at the graph now you find that when the months where the temperature is high so among those twelve months, when will be the temperatures be high? s: january, february, december t: yes, those three thereafter you have such now according to our explanation here, they said that this company wants to do the repairs when the temperature is high. so the temperatures are high on those months that you said. so that is how you answer that one. so those months that you have said, looking at your graph. okay thereafter they also want to do, they say repairs when the city is dry. okay when is the city dry? tell me when is it dry? when there is too much rainfall or when there is no rainfall? where there is no rainfall. when do you think the city is dry? sipho: july. when thami posed a question about when the city is dry, sipho’s answer of july was related to the weather patterns in durban. durban usually has wet summers (in december) and dry winters (in july). the rainfall pattern represented in the graph given, was different  in the graph the rainfall in the month of july is third highest. the graph represents the rainfall pattern of robben island, which is in cape town, which was out of most of the learners’ experiences. cape town has dry summers and wet winters which is reflected in the graph. in this instance thami identified that the learners’ everyday knowledge contradicted the context that was used in the cta problem. so in order to clarify this issue thami asked them to disregard the durban experience and drew their attention to the graphs: t: that’s according to our city here in durban, so now in your class refer to graph number 2 there. okay in your graph 2 there is the highest rainfall in which month? where is the highest rainfall? sipho: june it is important to note that when thami directed sipho (and others) to look at the graph, sipho’s response that june had the highest rainfall was correct. thami acknowledged that it was correct and then told the learners precisely how they needed to use the information from the graphs. t: okay but now they are saying here this company here, this work4u wants to do repairs when the season is dry. so you are saying is the season is dry when there is no rainfall. okay so you look at that [graph] and now decide amongst those tropical months you see there. you see on this month which has a small rainfall. okay so you combine now, combine because i actually detailed for you there. they only said that this company does repair in a dry season and when the temperatures are high. then you combine now the answer you got in number one with the one you have from number 2 [referring to the graphs in the task]. okay so you decide which month, it means that month you see, the one that overlaps combine both number 1 and number 2. after allowing the learners some time to consider the problem, thami then moved on to a further activity based on the same two graphs that is given below: task 1, activity 5 (individually) macy works for work 4u and knows that her brother camille, who is in grade 9, is very interested in the analysis of statistical data. she prepared the following questions for him based on the information in the graphs on the next page. please help camille to answer the following questions. 5.1 5.2 5.3 5.4 during which months do you think the weather is more favourable for tourism? give reasons for your answer. an excerpt from thami’s discussion is given below: t: now in 5.4 during which month do you think the weather is more favourable to tourist. when do we have a lot of tourists visiting this part and all the surrounding area? s: in summer. t: in summer. okay, but now in your case you need to look at why is it in summer. see if you can tell me. can you support your statement? why in summer? the use of real life contexts in the cta 24 sipho: because it is the holidays. t: its the holidays right. it's the festive season. so in our case we refer to our graph. okay we know visitors come when it is summer. we have quite a number of people coming along to the beach. right okay so in your case there you cannot have the tourist in the winter especially in robben island. you can't say you have tourist in winter when it is ice, you can't, so decide on your own why you are saying that. unfortunately the teacher closed the discussion at this point on this issue and asked the learners to continue with the task, without probing the learner further to see whether he understood the issue or not. this excerpt again shows sipho’s tendency to answer questions based on his everyday knowledge and experiences of the times that tourists visited durban. when thami asked the class why tourists visited durban in summer, siphos’ response was because it was holidays, and not the expected answer which should have been because the weather was warmer and dryer. durban has mild weather throughout the year. so in sipho’s experience the weather was not a determining factor in the times tourist chose to visit. more tourists visited durban during the summer because it was holidays, probably because it is the longest holidays in the year. other possible reasons for the higher number of visitors could be because of the longer days or because it coincided with the festive season. for the case of cape town the weather in winter is a deterrent to tourists, because of the low temperatures and high rainfall. learners were expected to infer this from the graphs showing the low temperatures and high rainfall in july. many of these learners had not been to cape town or other cold places and did not have the experience of the icy and wet winter described by the graphs. in durban weather is hardly ever a deterrent  it is very common to have beach festivals during the winter holidays, because the weather in july (average 25 °c) is not a deterrent to tourists. in any case, durban has high rainfall in summer and that is still not a deterrent. thami tried to direct their attention to the extremities of the cape town weather by saying: “…you cannot have the tourist in the winter especially in robben island. you can't say you have tourist in winter when it is ice...” in this instance thami had identified that his learners’ everyday experience was not helpful in answering the question in the cta problem. he pointed out that the learners needed to infer their answers from the graph of temperatures in cape town and not from their everyday experience of tourism patterns in durban. it turns out that sipho was very positive about the cta because he saw it as offering opportunities for participation. during an interview with a group of learners from the class, sipho commented: “i think the cta’s are, they created them to make us pass, they are good …. a lot of people are participating now, they are succeeding”. sipho’s responses indicate that he felt that their participation levels were higher in the cta, than in the usual assessments. it is important to note that the interview was conducted after the learners completed the cta tasks, but before they received their results. these learners did not see their tasks after submitting them  they just received their scores. had sipho received feedback on his responses to the cta tasks, he would not have been so positive. sipho’s responses in the lesson described above, illustrates a difficulty associated with test items drawn from real life contexts  learners can get confused about which knowledge they should use. cooper and dunne (1998) point out the likelihood for confusion around what relevant knowledge should be brought to bear when answering test items set within “real life” contexts. they showed that there was a group of children in their study who had the mathematical competence but did not always demonstrate it without being induced to by the interviewer. this was because many children initially gave responses which they saw as appropriate for the setting, but was not appropriate or comprehensive enough in the context of the mathematics test. however on being probed during their interviews they were able to produce the response considered mathematically appropriate for the setting. in fact sipho was able to interpret the information from the graph correctly, when probed by thami, but his initial response was to relate the situation to his durban experience. sipho, in one lesson, twice based his initial response on his everyday experiences and not on the graphs provided. furthermore, sipho’s comment that they are succeeding, implies that he considered that the learners’ various responses were acceptable for the cta tasks. this reasoning was perhaps because the questions in the tasks convey the impression that different responses would be considered, unlike the usual mathematics assessment task, which usually needed a fixed procedure with a fixed final result. sarah bansilal 25 however, the reality is that even with these “real life” tasks, not all responses are considered as legitimate to the task designers. as seen in the two excerpts of classroom interaction, had sipho responded based on his everyday experiences, his response would have been considered incorrect. cooper and harries (2003) has shown that many children import a variety of extra-school realistic considerations into their solutions even though the test designers intend them not to do so. faced with some realistically contextualised items children may introduce more than they are required or expected to do. in this episode we see that the only realistic consideration that was allowed was the assumption that tourists would prefer visiting places in warm and dry weather (which is questionable anyway, since there are many who may prefer colder weather for reasons of their own). the consideration that tourist visit in summer because it was the festive holidays, was one that was not recognised by the task designers. yet this was a reason that sipho had offered, based on his own experiences in durban. however his consideration that there was a link between holidays and tourists was one which was not appropriate for the task, because the legitimate answers had be drawn from an interpretation of the two graphs. sipho’s comments during the interview also demonstrate that he felt that these extraschool realistic considerations were acceptable resulting in his (erroneous) perception of having performed favourably in the cta task. discussion the move by south african education authorities to the use of contextualised assessment activities, presents many unintended consequences, some of which pose a threat to the validity and reliability of the cta programme. the first vignette was intended to draw our attention to the possibility that for some learners the context used is out of their experience. in the vignette provided, the teacher was able to address this problem by first introducing the mathematics by using a more familiar context. thereafter the learners applied the procedure to the unfamiliar context, with the teacher helping them to link the elements of the procedure to the previous task. this intervention by the teacher effectively simplified the task and reduced it to a rote application of a known procedure to a simplified context. this has implications for the validity of the cta task as a summative assessment tool. what about those learners for whom the context was out of their experience but whose teachers did not intervene to help them understand the mathematical elements or the context? there is also the possibility that the context was out of their experience of some teachers? both the teachers and the learners would have been disadvantaged, and the fairness of the cta as an assessment tool is implicated. (see bansilal (2008) for some descriptions of responses from learners who did not understand what a pizza was). when a mathematics task utilises a context that is out of the learners’ experience, the learner has nothing to draw upon from her/his background to understand the questions. consequently the context itself will block the learners from understanding what is required from them. so an appropriate question to pose would be: whose “real life” contexts are being privileged in these assessment tasks which draw on these “real life” contexts? it is important to recognise that my comments are based on the use of the contexts in externally mandated national assessment tasks, and not a typical classroom episode, where new contexts can be used to extend learners’ understanding of the world around them. in a normal classroom setting their inexperience of the middle class lifestyle may not pose a barrier to their progress in understanding because of the opportunities that are available for discussion and clarification, provided of course that the teacher understands the context. the second vignette was intended to draw our attention to the fact that certain learners tend to rely on their everyday experience when answering tasks set within real life contexts. could there be certain learners who are more likely than others to be disadvantaged by their tendency to respond based on their experiences? cooper and dunne’s (1998) study concerning the differences between the approach favoured by working and service class children has reference here. these authors show that working class children seem to be more predisposed than service class children to initially employ their everyday knowledge in answering mathematics test items and that this can lead to the under-estimation of their actual capacities with respect to the demands of the mathematics school curriculum. in fact an analysis of the responses to one test item showed that working class children were almost twice as likely as service class children to refer only to their everyday knowledge in answering the researcher’s query. this finding the use of real life contexts in the cta 26 is of concern because this study has revealed that some learners have a predisposition to employ their everyday knowledge, and their mathematical competence may not be revealed in such a situation. learners may not provide the expected answer because they were confused about what was being asked for and not because they did not know the correct answer. in addition, within a national assessment task, they may not be given the opportunity to explain their responses. boaler’s work (2003) concerning the sat-9 assessment in the usa has reference here. she described the work done in a project school, railside high, which was situated in a low income area where students had access to few financial resources both in school and at home. it was found that the railside learners did much worse on the sat-9 test items than in the district tests. she argues that the sat-9 questions used long sentences and words unknown to many students new to the country (such as spool, cable crew, wired). furthermore, she claims that the sat-9 question assessed many things  confidence in the face of unfamiliar answers, context knowledge, and language  but none of these were indicators of mathematics knowledge. boaler states that there was a high correlation between socioeconomic status and sat–9 success and that that the evidence collected from students at railside suggests that the low performance of students in the sat-9 is related less to mathematical understanding than it is to language, context interpretation (which relies heavily on language) and test-taking skills. she states: “using such tests … as a supposed tool to increase the performance of underachieving students, particularly those from low-income and ethnic minority homes does not therefore seem to be a wise decision on the part of californian policy makers” (2003, p. 6). boaler’s comments should influence us to pause here in south africa and interrogate the preferred style of the cta which is supposed to be designed in ways that promote inclusion, accommodate diversity, identify and address barriers to learning thus helping learners to succeed. however these two vignettes demonstrate that the opposite may in fact be true. the contexts used are sometimes out of the learners experiences. in this scenario, teachers would need to intervene to help learners access the context. however the teachers’ interventions must be pedagogically sound without lowering the demand of the tasks. if teachers do not intervene in meaningful ways, the learners will be disadvantaged. the second vignette illustrates that certain learners tend to rely on their everyday experience by relating the task to extra–mathematical considerations which are not considered as legitimate by the task designers. learners who have a tendency to rely on their everyday knowledge when answering these “real life” tasks are disadvantaged. the work done by cooper and his colleagues in the uk shows that service class learners seem to be more predisposed than middle class learners to rely on their everyday experiences and are therefore more likely to be disadvantaged by these task. so contrary to the declarations of the intentions of fairness and the stipulation that learners should not be disadvantaged by the design of the tasks, many learners may be disadvantaged, albeit unintentionally. concluding remarks it is commendable that the doe in trying to find fairer systems of assessment has made a shift towards using alternate kinds of assessment strategies. however it is crucial that these large scale assessment programmes be researched and trialed with groups of learners from diverse backgrounds to find out whether the strategies are an improvement on the pencil and paper tasks or indeed, whether they are more disadvantageous to certain learners (as suggested by the work of cooper in the uk and boaler in the us). it is also necessary for research to be undertaken to find out more about what support learners need in order to ensure their success and in the same vein, what type of support teachers need in order to help their learners succeed. it does not make any sense to move to alternate ways of assessment which by their very nature, demand more support in order to be successful, without a commitment to changing the levels of support available to teachers. i end this paper by calling for more research around several issues which have been identified in this paper. firstly, there is a need to find out whether the assessment tasks set within an extended context can be successful at assessing students’ mathematical knowledge and skills at the grade 9 level. a second question directed more generally at the use of contextualised items is: what support must be provided to sarah bansilal 27 teachers to help them understand their role in the assessment process? and a compelling fourth question in the light of south africa’s rapidly changing class system is: which learners are more disadvantaged than others by the use of these “real life” contexts in externally mandated assessment tasks? references arcavi a. (2002). the everyday and the academic in mathematics. in m. e. brenner & j. n. moschovich (eds.), everyday and academic mathematics in the classroom (pp. 12-29). reston, va: the national council of teachers of mathematics. bansilal, s. (2006). south african grade 9 mathematics teachers, professional expertise: a situative perspective. (unpublished doctoral dissertation). curtin university of technology, perth, australia. bansilal, s. (2008). assessing the validity of the grade 9 mathematics cta. paper presented at the 5th conference of the association of commonwealth examinations and accreditation bodies. umalusi, pretoria. boaler, j. (1993). the role of contexts in the mathematics classroom: do they make mathematics more real? for the learning of mathematics, 13(2) 12-17. boaler, j. (2003). when learning no longer matters – standardized testing and the creation of inequality. phi delta kappan, 84 (7), 502-506. cohen, l., manion, l., & morrison, k. 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(1997). social class, gender, item type and performance in national tests of primary school mathematics: some research evidence from england. paper presented at the annual meeting of the american educational research association, chicago. department of education (2002): a draft framework for the development of common tasks for assessment (cta). pretoria: national department of education. evans, j. t. (1995). adults and numeracy. (unpublished doctoral dissertation). university of london, london. freudenthal, h. (1983). didactical phenomenology of mathematical structures. dordrecht: reidel. gravemeijer, k. p. e. (1994). developing realistic mathematics education. utrecht: freudenthal institute. linchevski, l., & williams, j. s. (1996). situated intuition, concrete manipulations and mathematical concepts: the case of integers, in l. puig & a. gutierrez (eds.), proceedings of the twentieth conference of the international group for the psychology of mathematics education (vol. 3, pp. 265-272). valencia: university of valencia. moschkovich, j. n. (2002). bringing together workplace and academic mathematical practices during classroom assessments. in m. e. brenner & j. n. moschovich (eds.), everyday and academic mathematics in the classroom (pp. 93-110). reston, va: the national council of teachers of mathematics: polkinghorne, d. e. (1995). narrative configuration in qualitative analysis. in j. a. hatch & r. wisniewski (eds.), life history as narrative (pp. 5-23). london: falmer press. william, d. (1997). relevance as macguffin in mathematics education. paper presented at the british educational research association conference, york. abstract introduction and problem statement conceptual framework theoretical framework empirical investigation findings discussion conclusion and future directions acknowledgements references about the author(s) divan jagals school for professional studies in education, faculty of education sciences, north-west university, south africa martha van der walt school for science and technology education, faculty of education sciences, north-west university, south africa citation jagals, d., & van der walt, m. (2018). metacognitive awareness and visualisation in the imagination: the case of the invisible circles. pythagoras, 39(1), a396. https://doi.org/10.4102/pythagoras.v39i1.396 original research metacognitive awareness and visualisation in the imagination: the case of the invisible circles divan jagals, martha van der walt received: 14 sept. 2017; accepted: 02 july 2018; published: 13 aug. 2018 copyright: © 2018. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract awareness of one’s own strengths and weaknesses during visualisation is often initiated by the imagination – the faculty for intuitively visualising and modelling an object. towards exploring the role of metacognitive awareness and imagination in facilitating visualisation in solving a mathematics task, four secondary schools in the north west province of south africa were selected for instrumental case studies. understanding how mathematical objects are modelled in the mind may explain the transfer of the mathematical ideas between metacognitive awareness and the rigour of the imaginer’s mental images. from each school, a top achiever in mathematics was invited to an individual interview (n = 4) and was video-recorded while solving a mathematics word problem. participants also had to identify metacognitive statements from a sample of statement cards (n = 15) which provided them the necessary vocabulary to express their thinking during the interview. during their attempts, participants were asked questions about what they were thinking, what they did and why they did what they had done. analysis with a priori coding suggests the three types of imagination consistent with the metacognitive awareness and visualisation include initiating, conceiving and transformative imaginations. these results indicate the tenets by which metacognitive awareness and visualisation are conceptually related with the imagination as a faculty of self-directedness. based on these findings, a renewed understanding of the role of metacognition and imagination in mathematics tasks is revealed and discussed in terms of the tenets of metacognitive awareness and imagination. these tenets advance the rational debate about mathematics to promote a more imaginative mathematics. introduction and problem statement there are not many references to imagination in metacognition research, particularly in the context of mathematics education. there are, however, publications that illuminate the importance of imagination in fields related to mathematics education. some examples of these fields include physical science and technology education (e.g. nemirovsky & ferrara, 2009). as an example of this application, imagine looking up at a water wheel and noticing how it accelerates, or imagine yourself turning a wheel on a bicycle with your hand. can you sense its acceleration, direction and force with the energy applied? the scarcity of accepting the applicability of imagination in the field of mathematics could be because imagination received a bad reputation in western philosophy where some scholars believed that imagination is inferior to reason (e.g. spinoza’s naturalistic theory of imagination) while others (who agree with hume’s theory of imagination) argue that imagination constrains metaphysics and makes way for scientific reasoning. we hold on to wenger’s (1998) view of imagination, stating: imagination is ‘a process of expanding our self by transcending our time and space and creating new images of the world and ourselves’ (p. 176). both these schools of thought, however, imply that the understanding of imagination in metacognition and mathematics education literature have, thus, become vague. as a result, a predominant objectivist perspective on school mathematics is held, one that upholds mathematics as a ‘rational structure to reality, independent of the beliefs of any particular people’ (murray, 2013, p. 387). visualisation and imagination are both terms that preserve the spatial layout of an image. the ability to focus your attention on to an object, to imagine being closer or further away from it, and to perceive the greater detail (e.g. at the centre of the object compared to elsewhere) are some examples of the powerful capacity that facilitates the metacognitive skill of visualisation (gilbert, 2005). however, research by fried (2014), for example, shows that students prefer algorithmic mathematical problems, emphasising routine memorised procedures, over non-routine problems that require some form of visualisation. in this regard, reddy et al. (2016) show that, for grade 9 advanced international benchmark levels, learners are expected to apply and reason in a variety of problem situations, to solve linear equations and make generalisations, all of which requires metacognition (jagals & van der walt, 2016) and visualisation. visualisation is therefore not only about seeing, but is a process and product of reflecting, thinking deeply and creating illustrations or mental models as meta-representations of what is imagined. in classrooms where routine procedures dominate problem-solving activities, a typical ‘talk and chalk’, and ‘drill and practice’ pedagogy reigns, and might explain why south africa falls short in the trends in science and mathematics study when compared to the mathematics performance of other countries (reddy et al., 2016). we therefore argue that, in non-routine problems, the power of metacognition and imagination can make the mathematical concepts visible, tangible and represented in a more accessible form through visualisation. learners who develop this imaginative power can become more aware of their own understandings, in the sense that they will be able to see the invisible ideas in mathematics, and be equipped to solve problems of a similar nature. such a holistic view of mathematics problem-solving emphasises a different and more sufficient approach to teaching and learning of visualisation. if we wish to understand the conceptual linkage between imagination and metacognitive awareness, then the reductionist view of mathematics must be revisited to bring to mind and imagine what is unseen about the problem or its approach. a holistic view of mathematics aspires to a more imaginative approach to teaching and learning mathematics. in the sections that follow, the inclusion of the imagination as part of the holistic view of mathematics education is explored in terms of the role metacognition and imagination play to facilitate visualisation about a euclidean geometry task. in contrast to the rational view of mathematics, we suspected that learners who follow such a holistic approach would more likely be able to become aware of their own strengths and weaknesses during problem solving. as part of a larger project (jagals, 2013), the study reported in this article explored the metacognitive awareness and imagination that emerged from learners’ mathematics problem-solving experiences. specifically, the study sought to answer the question: what role does metacognitive awareness and imagination play in facilitating visualisation in solving a mathematics task? the concepts of metacognitive awareness and imagination were aligned against the theoretical framework of embodied, situated and distributed cognition to explore learners’ awareness and imagination of their visualisation processes. the conceptual and theoretical framework on which the study was based and the empirical design of the study that was done follows. the results informed the role of imagination in metacognitive awareness during a mathematics visualisation task. in essence, the findings portray the results of four instrumental case studies and a discussion in view of the theoretical framework. this is concluded by some guidelines for classroom practice and recommendations for future consideration. conceptual framework imagination can serve as an indispensable tool for unlocking and discovering reality, and the mathematical ideas that surround reality’s hidden structures. without knowing how to ‘see’ a geometric point, or without awareness of the fact that a geometrical plane has no thickness, or that a series of points about the same axis can form a circle rather than a straight line, a personal mistrust in one’s thinking and awareness about reality (schoenfeld, 2013), and mathematics for that matter, can develop. in euclidean circle geometry, imagination can therefore serve as a medium for scientific visualisation, as galileo used for his thought experiments, and may help in understanding how mathematical models are formed, thus encouraging reason. take for example a situation where learners reflect on problem-solving processes. they tend to look at and sometimes away from their written or sketched work. they, then, do not only think metacognitively about the process, but also bring to mind and imagine what is unseen about the problem or its approach. imagination and circle geometry the circle is probably one of the oldest figures of mathematics (kasner & newman, 2001). it has characteristics of a non-straight line and is often regarded as a polygon with an infinite number of sides. thus, a circle is a limit of the inscribed polygons. there are, in fact, some interesting generalisations of the circle when viewed this way. for example, greek mathematicians such as apollonius posed the classical circle problem: given three fixed circles, find a circle that touches them all. teachers and learners could, through visualisation, try to methodologically fit a circle to touch all circles, starting with the smallest possible radius and using trial and error. the link between visualisation and imagination could offer both teachers and learners a more constructive approach to this problem. imagination, in light of wenger’s (1998) definition, could facilitate metacognitive thinking about the experience, to encourage reasoning towards a new knowledge of reality, or at least of circles. as this example suggests, creative problem-solvers engage in metacognitive thinking about strategies and monitor and evaluate their attempts. during this process the faculty of the imagination is employed to facilitate the visualisation of the problem-solving process. the rationale behind this argument is that mathematical models (in the case of apollonius’s circles) are generated through problem-solving experiences and critical thinking which have become a cornerstone in visualisation. visualisation then serves as a metacognitive skill (gilbert, 2005). yet, problem-solving skills require metacognitive awareness, which, in turn, predicts imagination (liang, hsu & chang, 2013). learning opportunities that are, therefore, framed around exploring, questioning, understanding and imaginative experiences are exemplary of the role metacognitive awareness plays in the imagination. theoretically, franklin and graesser (1999) clarify that precision is needed to explain what it means when a mathematical image is brought to mind. the image conjured up about apollonius’s circles (mentioned above) preceded any explanation about radius, foci, area, circumference or diameter. yet, the image conjured up by the mention of ‘circles’ is an implicit theory with particular visual and spatial content. further examples of such an implicit theory of circles include wheel of fortune, circle of life, ring of fire, sphere of philosophy, to name but four. the theory behind these images is that they raise awareness of circle properties (or at least knowledge of its shape). however, when confronted with mathematical ideas, such as the properties of a circle, a learner will typically distinguish between three stages of this imaginative experience. franklin and graesser explain these stages as metacognitive awareness of (1) a conjured image, (2) a line drawing not necessarily labelled and (3) attached symbols that serve as a map or a plan with some symbolised spatial relation. imagination as visualisation of metacognitive awareness visualisation, also called the representational view of the mind (makina, 2010), integrates the mental processes made up of visual imagery, visual memory, visual processing, visual relationships, visual attention and visual imagination (p. 25). the aim, then, of teaching mathematics, with these visual functions in mind, is to support learners in the construction of mental representations of mathematical phenomena such that they develop an awareness of the underlying notions or concepts that the mathematical ideas develop from. such an understanding not only brings to mind the imagined visual object (such as a circle), but also creates awareness of the knowledge of the person, task and strategies in order to provide a mental space in which these representations can be regulated. mainly, visualisation should be considered as a fundamental aspect of mathematics learning and understanding (makina, 2010) as learners learn to demonstrate their thinking and, thus, become metacognitively aware of the experiences this thinking involves. such experiences could be reading for a particular purpose to select important words or phrases, describing them, providing proof of the knowledge they are accepting, examining or recognising. as a result, these representations can act as ways in which thinking and reasoning can be visualised, and acted out, or embodied, to extend the knowledge of one’s own understanding, the problem or task and the strategies to solve a particular problem. imagination can therefore serve as a crucial faculty of the metacognitive awareness of one’s own and others’ mathematical understanding. the awareness, then, transposes to other higher order faculties, such as metacognitively planning, monitoring and evaluating one’s attempts in solving a mathematics task, and assists in directing one’s learning and problem-solving attempts. theoretical framework the cognitive processes related to the expression of metacognitive awareness are embodied, situated and distributed in nature. embodied, situated and distributed cognition the theory of embodied, situated and distributed cognition is applied as a lens to explore the role of metacognitive awareness, imagination and visualisation. this theory suggests we belong within a reality through actual engagement with that reality, and in the activities we engage in we develop the power of our imagination (murray, 2011), for instance solving a mathematics task. in terms of visualisation, this means that learners might imagine themselves drawing a circle as part of an activity, becoming aware of an (often similar) experience in order to gain knowledge (or cognition) from that experience needed to solve the problem or create a mathematical model. according to theorists of imagination (murray, 2013; yueh, jiang & liang, 2014) and embodied, situated and distributed cognition (clark, 1998; nemirovsky & ferrara, 2009), cognition becomes embodied when the mental mathematical imaginations (e.g. of a circle) are expressed physically, that is, outside of the mind. this can be through body poise, facial expressions, gestures, utterances or any other motor activity, like drawing a circle, which resembles some form of mathematical intuition. this embodiment of mathematics can be harnessed in situated specific contexts. o’connor and aardema (2005), for example, explain that cognition is imagined interaction with the imaginary world. it is in this world that senses, acts and ideas are shaped to visualise mental mathematical models. there are, however, some doubts as to the applicability of situated cognition to imagination (jansen, 2013). reasons for this include the offline nature of metacognition and imagination where the object or model is absent, and needs to be conjured up in the mind, and therefore does not define any particular situation. as an example of the situatedness, jagals and van der walt (2016) developed a mathematics task, based on an example by fortunato and hecht (1991), during which the image of circles needs to be imagined and then visualised in the context situation of the problem. then there is also the argument that situatedness deals with the conscious rather than the imagination which requires a phenomenal consciousness (awareness about a particular phenomenon). imagination therefore deals with the paradigmatic of higher cognitive levels, beyond awareness, towards intuition. it therefore seems that imagination also associates with the distribution of cognition. in this sense, knowledge of circles’ (mathematical) properties can be applied in calculations involving glass mirrors or sacrobosco’s sphere, thereby distributing the knowledge from one domain to another. in another thread, an argument can be made that a circle, as a mathematical object, can be used to show the definitions and logic, thoughts and other non-mathematical ideas (as in the case of venn diagrams), and therefore plays a role, in terms of its imagined image in the metacognitive awareness. in developing an empirically categorised imagination scale liang et al. (2013) identified initiating, conceiving and transforming as three types of imagination. in this study, however, to explore the role of metacognitive awareness and imagination in facilitating visualisation in solving mathematics tasks, these types of imagination are aligned against the theoretical framework of embodied, situated and distributed cognition (clark, 1998). the theoretical framework thereby serves to inform a more conciliatory image of the role of metacognitive awareness and imagination in facilitating visualisation and the consequences for metacognition and mathematics education research. tenets of metacognitive awareness and imagination metacognitive awareness refers to reflecting on understanding and regulating knowledge (schraw & moshman, 1995). when the mathematical model is not immediately tangible and accessible (like the conjured image of a circle) it can be connected with through the power of the imagination (murray, 2011). this connection, according to wenger (1998) ‘needs an opening. it needs the willingness, freedom, energy, and time to expose ourselves to the exotic, move around’ (p. 185). when learners engage and experiment with reality, they develop the power of imagination. wenger’s comments indicate a strong parallel between imagination and metacognitive awareness. initiating imagination involves the imaginative capability to explore and produce new, unfamiliar and unique ideas (lin, hsu & liang, 2014). learners who reflect deeply on themselves are more metacognitively aware. perhaps this explains why new and novel ideas are often generated in solitude (lin et al., 2014). conceiving imagination is the imaginative capability to understand mathematical ideas through personal intuition and using one’s senses (lin et al., 2014). typically, individuals who have a strong sense of conceiving imagination are capable to understand and conceptualise mathematical ideas through rehearsed concentration (and thereby memorise a formula or strategy) and follow logical steps or apply appropriate strategies through metacognitive awareness of the knowledge of these strategies. transforming imagination is the imaginative capability to identify a pattern, and to transform it into an abstract idea to distribute and apply the conceptualised understanding across different situations (lin et al., 2014). individuals who have a strong transforming imagination therefore tend to reflect on mathematical ideas from past experiences, become aware of these knowledge and strategies, and mimic the route of memorised strategies they employed in different situations. they can also transform one abstract idea into another by manipulating the abstract pattern, typically as a learner will manipulate the formula for the area of a circle (a = πr2) to calculate the circle’s radius. in other words, the role of metacognition and the imagination becomes powerful in an environment which provides learners with personal mathematical independence. when reflecting on a problem-solving experience the imaginer can become metacognitively aware of the imaginings and can express these imaginings in the form of a drawing or written steps (embodying their cognition) as strategies are applied (nemirovsky & ferrara, 2009). utterances such as drawings or written steps allow us to explore the processes whereby learners express their imaginations during visualisation. to do this, these imaginations through metacognitive awareness were explored across a collective case study based on a mathematics word problem in the case of the invisible circle. empirical investigation in order to explore the role of metacognitive awareness and imagination in facilitating visualisation in solving a mathematics task, a predominantly qualitative approach with a collective case study design was followed. an interpretivist perspective allowed us to explore the role of metacognition and imagination to facilitate visualisation, by means of the tenets of metacognitive awareness. the tenets show how the conceptual and theoretical frameworks of this study are intertwined and the empirical investigation that follows explores the significance of these tenets further. research design four instrumental case studies were conducted to explore the role of metacognition and imagination, to facilitate visualisation, towards embedded, situated and distributed cognition during visualisation of elementary euclidean circle geometry. each case study facilitated an understanding of the construction of metacognitive awareness on three levels, to be triangulated. sampling of the participants four participants were purposively and conveniently invited to take part in an individual interview and included the top achievers from grade 8 and grade 9 in the senior phase having mathematics as a compulsory school subject. since this was not a comparative study, we wanted to know how academically strong learners think and do mathematics and therefore asked teachers to identify participants who achieved 70% or higher during the previous year for mathematics. in addition to this criterion, teachers were asked to identify those top achievers who would be willing and comfortable to communicate about their thinking. according to literature, learners who perform well in mathematics can provide more accurate information about their thinking and problem-solving behaviour. if learners were identified with poor academic achievement and who could not explain their thinking and reasoning, then it would have limited the opportunity to collect data regarding metacognitive awareness. teachers at the three participating schools identified these participants mainly because of their achievement in mathematics at the end of the previous school year. to enhance the trustworthiness of this small case study, one main criterion for sampling these learners was that their teachers had to identify them as learners who were not too shy to provide information about themselves, particularly about their thinking. the assumption was that having a high achievement in mathematics suggests they had the necessary knowledge and understanding of mathematical ideas to solve open ended and non-routine mathematics word problems. learner a, a 15-year-old girl, has a seemingly quiet nature. she had an average of 70% for mathematics at the end of grade 8 and took 9 min and 20 s to solve the given word problem. learner b was 15 years old and from an all-girls school. she smiles a lot and seemed to enjoy the discussion and questions in the interviews. she averaged 80% for mathematics at the end of grade 7 and solved the word problem in 16 min and 58 s. learner c was 14 years old. she seemed more interested in the study than were the other participants. she asked questions about why the study was done and whether she was allowed to ask questions during the interview. this learner had obtained an average of 76% at the end of grade 7 and solved the word problem in the shortest amount of time, only 4 min. learner d was a 13-year-old boy, the youngest, and from an all-boys school. his teachers considered him a top student, achieving 96% at the end of grade 8. he took 9 min and 30 s to solve the problem, almost equal to learner a. research instruments the four instrumental case studies, each instrumental towards the collective case study, were conducted using: (1) a word problem based on the area of circles to transgress the mathematical ideas from invisible to visible, (2) metacognitive statement cards to elicit metacognitive awareness of the mathematical ideas underpinning the visualisation of circles, (3) observations of the utterances and gestures that pose the imaginative capabilities of the participants during visualisation, and (4) a collection of the images of participants’ own conjured mathematical models of the invisible circles on which the word problem was based. a brief outline on each of these instruments now follows. the task of the invisible circles the idea of the task was initiated by jagals (2013), published for the first time in jagals and van der walt (2016), and read as follows: suppose a circle’s diameter is 20 cm. this is also four times the radius of a second circle. calculate the area of a third circle if the third circle’s radius is half of the second one. (p. 158) the task was inspired by a similar activity by fortunato and hecht (1991) and adheres to the characteristics of a rich mathematical task identified by boston and wolf (2006), including the requirement of multiple approaches and representation, engagement, curiosity and creativity and has the potential to relate and extend to other tasks. participants had no time restriction to complete the task and were allowed to use a non-scientific calculator. no images of circles were given as this allowed the participants to exhibit their own visualisation of the problem, and to elicit the mathematical ideas from invisible to visible. after the problem was solved, the participants were interviewed individually. metacognitive statement cards before solving the problem, each participant received a set of 15 metacognitive statement cards, taken from the idea by wilson (2002) and van der walt, maree and ellis (2008), on which metacognitive awareness capabilities were spelt out. participants also received blank cards in case they wanted to write some thoughts that were not already stated. no participant made use of the blank cards. participants were asked to focus on their thoughts while solving a mathematics task. the statements had a bearing on the embodied, situated and distributed cognitive processes identified by clark (1998). examples of the metacognitive statements include: i drew an image to better understand the question (embodied), i thought about something that i had done in the past that was helpful (situated), i thought about a different way to solve the problem (distributed). the procedure with the metacognitive statement cards followed after the invisible circle problem was solved and served as a starting point before interview questions were asked. after the task was solved, participants had to reflect on what they did and indicated which thoughts represented on the cards were thoughts that they were thinking. participants had to reflect on the problem-solving experience and select only the cards with the statements that they are aware of and can agree to have used. these cards were then placed in the order they had been used and numbered consecutively to obtain a detailed account of the cognitive processes followed. the metacognitive statement cards were collected to ensure that the best possible data of participants’ reflections and reporting on their experiences could be obtained. observations of the utterances and gestures that pose imaginative capabilities careful observation was used to note any utterances and gestures or movements with hands and arms that can indicate the embodiment of mathematical ideas. mainly, gestures of a rest position, stroke, pointing, preparation or movement about the work place, retraction and movement of hands, or movement of the lips and other relevant body language which represented the embodiment of cognition of mathematical ideas served as criteria for noting a gesture or utterance. images of participants’ own conjured mathematical models since the word problem was meant to elicit imaginative capabilities, it was anticipated that participants will conjure images of the elements of a circles (e.g. radius, chord, circumference and diameter) which, at the end, served as a mathematical model for the word problem. each participant’s conjured images served as evidence of the situated and distributed cognition instigated by the information provided in the word problem. trustworthiness issues participants continuously referred back to the video recordings and reflected on their actions and the statements they made. in this sense we consider their statements as valid and trustworthy. the metacognitive statement cards also validated interview responses. to ensure trustworthiness the guidelines by elliott, fischer and rennie (1999) were followed. these included: checking the interpretations of participants’ utterances with the participants and cross-checking the findings with the conceptual-theoretical framework of this study. in line with curtin and fossey (2007), we also offer a thick description of the empirical investigation and findings, and employ within-method triangulation where selection of metacognitive statement cards, the individual interviews on the problem-solving experience, observations, and images serve as different methods within the same methodological approach. ethical considerations permission and ethical clearance was sought and obtained from the department of basic education, and the university’s ethics committee (reference nwu-00043-11-a2) in which the initiated study was proposed; learners’ parents and school principals also gave consent. participants were informed about the aim of the investigation and could have withdrawn at any moment, although none did. participants’ identities were protected by using pseudonyms (e.g. learner a). data collection procedures data were collected for each instrumental case study through four research instruments. first, inductive analysis was conducted regarding the selection of metacognitive statement cards after problem solving. a priori analysis was then conducted on the transcriptions at interview level with a focus on what the learners were thinking, and why they did what they had done. this was followed by identifying the utterances or gestures that relate to imaginative capabilities as observed through video playback. together with these recordings, we also collected particular ‘screen shots’ of the learners’ mathematical models to represent the circle images of learners’ imaginations, to showcase how they have conjured and imagined distributed mathematical ideas regarding the elements of a circle. together, the following instrumental cases were chosen because they are thought to be instrumentally useful towards the aim of this investigation. data processing procedures each statement card represented a metacognitive awareness statement relating to either embodied, situated or distributed cognition. the video recording was played back to identify the metacognitive statements that were used during particular moments where mathematical ideas were expressed through gestures and utterances. imaginative capabilities and metacognitive awareness were identified by visible (observable) regulatory actions such as starting the next step, getting an answer on the calculator or rereading the question. interviews were first transcribed verbatim, then entered into the computer as a word document and saved using pseudonyms for the participants (learners a, b, c and d). second, a priori codes were identified based on the conceptual-theoretical framework and categorised in the transcriptions. we read and reread the transcriptions and labelled related sentences or words according to the a priori codes. particular sentences, words or phrases were then cross tabulated according to the a priori codes and were subsequently compared to determine if there was a pattern or structure in participants’ visualisation, metacognitive awareness or imaginative capabilities. codes represented in the transcriptions (in terms of responses and phrases) were summarised and then tabled in four comparing columns. the columns included the identified codes for each participant. patterns emerging from the data were compared between participants’ overall visualisation. care was taken to identify which metacognitive awareness and imaginative capabilities the participants exhibited. similar patterns for the different themes were joined together and comparisons were made between all participants and relevant themes. findings following is a narrative account on each of the cases’ findings, flowing from the four instrumental case studies. in each case, tenets of metacognitive awareness (by franklin & graesser, 1999), imagination (by liang et al., 2013; lin et al., 2014) as well as the embodied, situated and distributed cognitive processes (by clark, 1998) were identified. the main findings of the instrumental cases follow. instrumental case 1 – learner a learner a solved the mathematics problem in 9 min and 20 s. she drew three circles as illustrated in figure 1, and labelled them in the centers as a, b and c. she circled the middle circle, labelled a, again and reread the question before making a gesture by pointing with her pen from one side of circle a to the other, as if drawing a line through a, and in doing so embodied the mathematical idea of the circle’s diameter. figure 1: learner a’s mathematical model for the three circles. the image that learner a conjured portrays her mathematical model for the three circles. even though she labelled them clearly, she did not attach any symbols to suggest possible mathematical ideas about each circle. she read and reread the problem again, stopped and then drew her own version of what she had read through her conceiving imaginative capability. she also applied appropriate strategies by rereading the question, concentrating more and reading longer, while looking up at her written work and drawings. she paused looking at her work and then looked away from her work, staring in the air. she pointed with a pen towards her written and sketched work while reading the word problem. after every sketch or step, she paused and scanned her page from top to bottom. a seemingly major challenge was when she became aware that she ‘couldn’t remember the formula’ to calculate the area of a circle (jagals, 2013, p. 127). trying three different strategies, she reflected on her knowledge and practices from experiences. the three strategies included: trying half of the diameter, then drawing a big circle with a shaded area, and eventually writing the word area on a piece of paper and underlining it. this provided the situated cognition through which she then remembered the formula and wrote it down. she explains that ‘i think that’s why i underlined the word area. … i then remembered the formula for the area’ (p. 127). almost immediately after getting the answer the learner made another attempt to distribute her cognition by using the formula known to calculate the circumference of a circle. she compared the answers and where deemed necessary tried another way to solve the problem. she identified the formula as a pattern and adapted it to calculate the area. she then wrote her final answer for the area of circle c as 78,5 cm2 (jagals, 2013, p. 127). instrumental case 2 – learner b learner b solved the problem in 16 min and 58 s. unlike learner a, this participant first read the whole problem three times before drawing the circles as illustrated in figure 2a. she seemed to have a calm and relaxed approach, doing what she does slowly and double-checking her work regularly. she first drew circle a and then labelled it. then she drew circles b and c and labelled them. figure 2: (a) learner b’s mathematical model for the three circles, (b) learner b’s mathematical formula to calculate the area of a circle and (c) learner b’s mathematical formula to calculate the area of a circle after changes (made to figure 2b). after drawing each circle, she read the question again and attached symbols to the circles by using information. she appeared to read some parts more and longer, stopped, read again. when the video recording was played back, the researcher asked at this stage what the learner was doing, and she commented: ‘i was putting like into a picture format, yes, like individual, like a, b, c’ (jagals, 2013, p. 133). she moved closer to the question on the desk. she read it, pointing at some parts of the question and looking up at her drawings, then back to the question. she moved the calculator slightly out of the way and began writing a =. she paused while looking at circle c and then finished the first step with a substituted value for π. when the researcher asked the learner why she was pausing, she said: i wanted to write my formula so i went back to check what are we doing, what area, perimeter or volume, so then i paused to double check; okay what is the area, so i specifically went to look for, if it’s area. (jagals, 2013, p. 133) this learner’s completed visual formula is depicted in figure 2b. she paused, sat back in her chair and then erased the exponent (2) as well as the radius (r) in the formula (refer to figure 2b), wrote times r, hesitated (paused) and placed the square at π as illustrated in figure 2c. at this stage, the learner commented: ‘i was very uncertain there’. she paused again, pen between the lips, and after some time corrected the formula. after substituting the value for π and the radius into the second step and using a calculator to get the answer, the learner evaluated her answer. she recalculated using other strategies and became aware that π was wrongly substituted. she corrected this and wrote the final answer as 19,625 cm2 (jagals, 2013, p. 134). instrumental case 3 – learner c learner c solved the problem effortlessly and, almost, mechanically, in 4 min. she read the question only partly, concentrating on some particular aspects at a given moment. the steps were taken without doubting the formula, units or substituted values – unlike learner a or learner b. after reading the question, she drew three circles of similar sizes as depicted in figure 3. figure 3: learner c’s mathematical model for the three circles. without showing lines to picture radius or diameter, she wrote down the given and mentally calculated information next to the matching circles. the learner calculated or deduced the information for circle b and circle c mentally, not showing any written work for her conclusions about the diameter or radii. she read information every time before she wrote something, thus breaking the question up into smaller manageable parts. while referring to the video recording afterwards, the learner had difficulty describing what she had done. she kept her answers short and sounded uncertain. the dialogue between the learner and researcher explains this: researcher: why did you read the question again? learner c: i was looking at this [pointing with her finger towards her solution]. researcher: what were you doing there? (jagals, 2013, p. 141) her response had a futile motive. she commented on a particular step: ‘i was writing and then i closed the pen and then i wanted to work out the sum’. learner c was not aware of what she had done nor could she provide clear reasons for doing what she did. she monitored her work less often than the other participants did, and only reread the question once. she also did not evaluate her answer. she did, however, explain that ‘it was bothering me that there were no labels’ suggesting that she was metacognitively aware of the imaginative capability that the problem requires of her. instrumental case 4 – learner d learner d started drawing the circles almost immediately after being given the word problem as illustrated in figure 4. although all four participants drew the three circles and filled in information about each circle, learner d only included information for circle a. he also did not make use of algorithms or so-called steps; instead he conjured a picture, wrote descriptive words as labels and did mental calculations. figure 4: learner d’s mathematical model for the three circles. circles b and c were just drawn and labelled but no symbols were attached inside or outside these circles. he claims that: ‘i drew it, what they said it is, i drew it on. so if it is a big circle, i draw a big circle if it is a small circle i draw a small circle’. after drawing the circles, he started calculating the area by first making sure he understood the given information, which was not written routinely. he wrote the formula for the area of a circle and continued to substitute the values for π as well as the predetermined mentally calculated value for the radius of circle c. learner d seemed surprised when he wrote down the answer as shown on the calculator’s screen. this fraction led him to solve the problem, manually, by doing long division. discussion the purpose of this research was to explore the role of metacognitive awareness and imagination in facilitating visualisation in solving a mathematics task. references to metacognitive awareness in mathematics education research often fail to acknowledge the imagination as a key role player in the cognitive processes during visualisation. some studies have shown that teachers’ metacognition is not adequate to model metacognitive awareness and undertones the vagueness in our understanding of the imagination as a faculty of self-directed learning, as it is rarely promoted in mathematics classrooms (van der walt et al., 2008). the typical objectivist perspective on mathematics is, perhaps, the result of this understanding, and impacts on teachers’ approaches to, and education philosophy of, mathematics. in this regard we offer here an understanding of the conceptual linkage between the imagination and metacognitive awareness, to advance this debate about the rational view of mathematics, and to promote a greater imaginative mathematics. we therefore contemplated the visualisation activities learners engage in can prompt their imaginative capabilities. originating from this investigation, and in line with the model by lin et al. (2014), learners expressed three types of imaginative capabilities as initiating, conceiving and transformative imaginations. to understand the cognitive processes embedded in these imaginations we aligned lin et al.’s (2014) model against the embodied, situated and distributed cognition, as the conceptual-theoretical framework borrowed from clark (1998), and produced the notion that the imagination becomes a powerful construct in the cognitive processes during visualisation. it provides learners with the opportunity to foster metacognitive awareness. becoming metacognitively aware of the imaginings of mathematical ideas not only creates awareness of the underlying mathematical models, of elementary euclidean circle geometry, for example, but suggests learners can express these imaginings in the form of a drawing or written steps and, in doing so, depict and become aware of their cognitive processes. the result is that initiating, conceiving and transformative imaginings can become embodied, situated and distributed in the visualisation of elementary euclidean circle geometry. moreover, this understanding hints at the imagination as a faculty of self-directed learning and a, philosophically speaking, underlying construct of the elements of a circle in circle theory. the four instrumental case studies collectively reflect this theoretical stance and support the role which the tenets of metacognitive awareness and visualisation, through embodied, situated and distributed cognition in the imagination, plays in facilitating visualisation in solving a mathematics task. to elucidate these findings, table 1 reflects the tenets of metacognitive awareness and visualisation in the imagination. table 1: summary of findings that reflect the tenets of metacognitive awareness and imagination. based on these tenets, it seems reasonable to suggest that metacognitive awareness and imaginative capabilities can serve as guiding principles embedded in visualisation tasks. to do so, the tenets need to be incorporated in the development of appropriate activities (such as open-ended non-routine word problems) to elicit the necessary mental images that will conjure metacognitive awareness. each of these underlying tenets seems to advance, collectively, the rational debate about mathematics, promoting a more imaginative mathematics. in respect of the role of metacognitive awareness of the learners, all cases were joined and overlapping with franklin and graesser (1999)’s stages as metacognitive awareness of mental images. in this sense, each learner conjured an image, drew lines that were not necessarily labelled and attached symbols with spatial relation to their images. these images served as the expressions of their embodied mathematical ideas of the three circles, represented as a mathematical model with circle properties. the possibility that one had more or less metacognitive awareness of the mathematical idea of a circle was not evident. instead, each case raised awareness of a deliberate conscious attempt to decipher the task and, within the perimeters of elementary circle geometry properties, conjure a circle based on what they have read and reread, although they were not instructed to do so. the conjured image was embodied, situated and then distributed even though the task did not expect learners to draw an image. their visualisation, then, appeared obvious and evident, even if they did not, in all cases, attach symbols to the image as franklin and graesser (1999) proposed. this obviousness of conjuring an image based on the metacognitive awareness of the mathematical ideas captured in the task reflects a radical constraint to the view of mathematics as a rational structure to reality. it is this independent belief, of what the task requires, that separates these learners’ attempts from one another. it is therefore not the imaginative capability, of what the mathematical model looks like, or what strategies they used to calculate the area, that distinguishes their metacognitive awareness from one another; perhaps this as well, but the fact is that each participant portrayed a conjured image, a formula and a series of steps (be it correct or not) which reflects in all cases the tenets of embodied, situated and distributed cognition. in so doing, the mathematical models that these learners made explicit show that the role of metacognitive awareness was largely emphasised by the mathematical ideas that the task required (e.g. learner b illustrated this in figure 2 as ratio of proportion of the elements of the circles). what the ground is for this intuitive mathematical idea remains in the imagination. conclusion and future directions visualisation research has recently received increasing attention with a focus on metacognitive awareness and visualisations, including imagination. since imagination is a scarce topic in metacognition research, a recent emphasis in the south african school mathematics curriculum prompts mental images, predictions and visualising of thoughts and decisions to promote self-directed learning as imaginative states of mind. the conceptual link, therefore, between these imaginative states of mind and the cognitive structures that draw on mathematical ideas seem to relate to a demand for exploring the conceptual understanding of the role of metacognitive awareness and imagination in facilitating visualisation in solving a mathematics task. the significance of this study is the close connection between metacognition and imagination as portrayed through the tenets of metacognitive awareness to illustrate the role it plays during problem-solving tasks. educators need to design learning environments that support learners’ metacognitive development and encourage them to engage their imaginations in the learning process. in mainstream mathematics, the holder of a rational view will need to reflect on the cognitive demands of the problem and the associated tenets of the imaginative capabilities to solve circle geometry problems that are non-routine and open-ended in nature. the beholder of a more imaginative and holistic view, however, should reflect on at least two kinds of imagination: what do i imagine myself knowing? and how do i imagine this? even in this sense, metacognitive awareness of oneself and of the mathematical ideas that govern reality needs further contemplating. acknowledgements the authors acknowledge funding from the national research foundation. views expressed are not necessarily those of the national research foundation. competing interests the authors declare that no significant competing financial, professional or personal interests might have influenced the performance or presentation of the work described in this manuscript. authors’ contributions d.j. conceived of the presented idea, developed the theory and performed data collection and analysis. m.v.d.w. verified the analytical methods, and the tenets that emerged from them, and supervised the findings of this work. both authors discussed the findings and contributed to the final manuscript. references clark, a. 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(2014). how does human aggregate moderate the effect of inspiration through action on the imagination of engineering majors? international journal of engineering education, 30(1), 128–135. microsoft word 3-15 mogari et al.doc pythagoras, 70, 3-15 (december 2009) 3 lesotho’s students’ achievement in mathematics  and their teachers’ background  and professional development    david mogari 1  jeanne kriek 2  gerrit stols 3  ogbonnaya ugorji iheanachor 4    1, 2, 4 institute for science and technology education, university of south africa  mogarld@unisa.ac.za; kriekj@unisa.ac.za; ogbonui@unisa.ac.za  3 dept. of science mathematics and technology education, university of pretoria  gerrit.stols@up.ac.za      the  study  sought  to  obtain  information  on  the  relationship,  if  any,  between  students’  achievement  and  teacher  background  as  well  as  between  students’  achievement  and  professional development. the respective correlation coefficient showed that there was a  significant relationship between students’ achievement and teacher background and an  insignificant relationship between students’ achievement and the extent of professional  development. of the three components of teacher background (viz., qualification, subject  major  and  teaching  experience),  teacher  qualification  showed  the  strongest  significant  relationship with students’ achievement, followed by level of subject major. but teaching  experience displayed a curvilinear significant relationship with students’ achievement. it  was concluded  that quality qualifications and deep subject content knowledge  tend  to  make teachers more effective while teachers improve and develop when they are exposed  to quality professional development programmes.  the study primarily seeks to obtain information about the relationship, if any, between students’ achievement in mathematics and their teachers’ background (refers to qualifications, mathematics major and teaching experience) and professional development, respectively. the study is necessitated by, seemingly, continual poor results produced in mathematics in lesotho. the examinations report compiled by the examination council of lesotho (ecol) indicates that, in the past five years, less than 12% of the candidates were able to score up to 50% in the mathematics examination of the cambridge oversees school certificate (ecol, 2005). the report also shows that in the junior certificate (jc) 2004 mathematics examination the average students’ performance was f+. for 2005, 4% of the students scored grades a and b while 13% scored grades c and d (ecol, 2006). for the situation to improve the focus has to be on teachers. they are the ones who daily interact with learners in the class. they also have to prepare and plan lessons they are to teach by knowing, arrange and organise the subject matter knowledge, prepare and organise the necessary teaching-learning resources, and select and identify appropriate teaching strategies for their lessons. in addition teachers have to be ‘confident in their ability and skills to guide and facilitate meaningful learning’ (onwu & mogari, 2004, p. 162), to be able to communicate knowledge and develop complex thinking and problem-solving skills to students (louckshorsley, hewson, love, & stiles, 1998) as well as being able to inspire students to learn. lesotho’s students’ achievement and their teachers’ background and professional development 4 teachers, therefore, need relevant education and training to adequately prepare them to aptly handle and manage the teaching-learning tasks they are expected to carry out in class. the education and training provided to teachers should not only focus on familiarising them with various instructional models. but it should also put emphasis on deepening their understanding of the mathematical content, their interpretations of the mathematical content in the context of facilitating meaningful learning, their knowledge of learners’ conceptions and learning difficulties (shulman, 1986). in south africa, for example, a teacher development programme is essentially required to make a teacher be a learning mediator; interpreter and designer of learning programmes and materials; leader, administrator and manager; scholar, researcher and life-long learner; role model and moral being; assessor; and subject specialist (department of education, 2000, p. 13-14). it is hoped that if teachers can display all the seven competencies, they will be in a position to teach effectively and thus facilitate meaningful learning. it can, thus, be argued that the success of students with their studies is at the core of the goals of any education system. possibly, it is for this reason that teachers are expected to continually upgrade and update both their content and classroom practice knowledge. hence, the current study, among others, seeks to establish whether there is any relationship between students’ achievement and the amount of professional development available to teachers. from the point of view of ensuring students’ success in their studies, the common practice has been to look for teachers with appropriate qualifications. the assumption being that, such qualifications can provide teachers with relevant mathematical content knowledge and classroom practice knowledge and skills. the main focus should not only be on what the qualifications can provide, but should also be on the quality of education. hence, mechanisms, processes and procedures have been put in place, in countries such as south africa, to assure and promote the quality of education and training offered in higher education institutions. each institution is expected to formulate a quality management system to promote and assure quality in its core activities of teaching and learning, research and community engagement. institutions would then have their quality management systems audited with a view to identifying their strengths and weaknesses (higher education quality committee, 2007a). in addition, programmes by the higher education institutions offered are reviewed to establish whether their provision complies with the set standards (higher education quality committee, 2007b). as part of the quality management system, institutions are required to determine the appropriateness and relevance of the qualifications they offer by, among others, at reasonable intervals, conduct tracer studies and employer surveys. therefore, it is argued that the mathematics teachers with a qualification offered within the purview of such an institutional quality management system have the necessary expertise to do well in their profession. that is, such teachers have been afforded, during pre-service or in-service training, a mathematical content knowledge sufficient enough to enable them manage accordingly the cognitive demands and challenges posed by the content they teach. in terms of classroom practice training, teachers have been exposed to the requisite instructional strategies and techniques to make learning meaningful and fun to students. it is for this reason some information is sought about the relationship, if any, between students’ achievement and teacher’s background. rationale for the study the students’ poor achievement in mathematics has become an issue of global concern. for many years, teachers, researchers and other interested parties have raised debates about which school variables have any association with students’ achievement (darling-hammond, 2000; reynoid & farrell, 1996). to this end a number of research studies have focused on a wide array of factors presumed to be associated with students’ achievements in mathematics. for instance, some of the studies focused on teacher qualifications (e.g., darling-hammond, 2000; rice, 2003; wenglinsky, 2000), some others on teacher subject major (e.g., wilson & floden, 2003), some others on teacher teaching experience (e.g., betts, zau, & rice, 2003), and some others on teacher professional development (e.g., franke, 2002; kennedy, 1998; varella, 2000). teachers’ background the need to improve students’ achievement in mathematics in lesotho is extremely critical. however, the factors that actually are related to students’ achievement in mathematics in the said country, seemingly, david mogari, jeanne kriek, gerrit stols & ogbonnaya iheanachor 5 have not been identified by any empirical study and so are not well understood. it is for this reason the present study is conducted in lesotho with a view to providing some knowledge about the relationships between students’ achievement in mathematics and factors such as teachers’ qualifications, subject major, years of experience and teacher professional development. teacher qualification, subject major and years of experience can be considered to constitute teacher background hence they are treated as its composite variables in the present study. but the significant role these factors tend to play in the teaching and learning of mathematics has led to the possible relationships between students’ achievement and each of them also being considered. teachers’ qualifications there is strong evidence supporting the need for teachers to have rich mathematical content knowledge and deep understanding (brown & borko, 1992, p. 209). it is thought that these teacher traits can be related to students’ achievement (collias, pajak, & rigden, 2000; sanders & rivers, 1996). therefore, it is argued that students with less exposure to qualified teachers seem far less likely of achieving academic success than those with more. a number of studies have examined the ways in which teachers’ highest qualifications are related to students’ achievement and many of these studies found that teachers’ highest qualifications correspond positively with students’ achievement. for instance, betts, zau and rice (2003) found that teachers’ highest degree correlates positively with students’ achievement. rice (2003) found that when teachers have an advanced degree in their teaching subjects it will have a positive impact on the students’ achievements. greenwald, hedges and laine (1996) conducted a meta-analysis of studies that examined the relationship between school resources and student achievement; and found a significant positive relationship between teachers’ qualification (measured as having a master’s degree or not having a master’s degree) and students’ achievement. goldhaber and brewer (1996) indicated that an advanced degree that was specific in the subject taught was associated with higher students’ achievement. on the contrary, there are studies that present opposing results. for example, greenberg, rhodes, ye and stancavage (2004) and wenglinsky (2000) found that postgraduate qualifications at masters or higher level were not significantly related to students’ achievement. further study is therefore necessary to shed more information on the relationship, if any, between teachers’ qualifications and students’ achievement in mathematics. teachers’ subject majors the importance of the relationship between teacher subject major and student achievement have repeatedly been acknowledged by leading education groups such as the education trust, the education leaders council, and the national commission on teaching and america's future despite being characterised by their diversity and commitment (thomas & raechelle, 2000). several other studies have shown a positive connection between teachers’ subject majors and higher students’ achievement in mathematics. for example, wilson and floden (2003) established that students of mathematics teachers with mathematics degrees as majors tend to demonstrate higher academic achievement in mathematics. but, wilson and folden assert that there is a limit at which further mathematics knowledge no longer helps the teacher. goldhaber and brewer (1996) found that teachers having a major in their subject area are the most reliable predictor of students’ achievement in mathematics and science. similarly, darlinghammond (2000) reported in a review of a study of high school students’ performance in mathematics and science that a teacher having a major in his/her teaching subject was the most reliable predictor of students’ achievement scores in mathematics and science. also, wenglinsky (2002) and greenberg et al. (2004) indicated that teachers with mathematics major correlated with higher students’ achievement in mathematics. hill, rowan and ball (2005) found that teachers’ specialised mathematical knowledge was significantly related to student achievement. however, a few other researchers reported inconsistent results about the relationship between teachers’ subject majors and students’ achievement. for example, ingvarson, beavis, bishop, peck and elsworth (2004) found that a number of studies on the relationship between teachers’ subject majors and student’s achievement in mathematics suggest complex and inconsistent results. martin, mullis, gregory, hoyle and shen (2000) and wenglinsky (2000) also lesotho’s students’ achievement and their teachers’ background and professional development 6 discovered that mathematics major could not be associated with teacher effectiveness that is linked to meaningful learning which in turn leads students’ success. perhaps, there is a need to explore more the issue of relationship, if any, between students’ success and teachers’ subject major. teachers’ experience a number of studies found that teachers’ years of experience positively correlate with students’ achievement. for example betts, zau and rice (2003) reported that teachers’ experience significantly correlates with students’ achievement in mathematics. a report by the centre for public education (2005) showed that there was a positive correlation between teaching experience and higher students’ achievement. in fact, teachers with more than five years teaching experience were found to be the more effective than those inexperienced ones. greenwald et al. (1996) in their meta-analysis of data from 60 studies indicated that teachers’ years of teaching experience positively correlates with students’ achievement. similar results were also found by rivkin, hanushek and kain (2005). their study showed that students of experienced teachers achieved better than those taught by novice teachers. it was reported in darling-hammond (2000) that teaching experience is related to students’ achievement even though the relationship curvilinear. darling-hammond established that mathematics students taught by teachers with less than five years experience had lower levels of achievement. in particular, the achievement of students tends to increase as teachers spend more years teaching. strangely though, there was no significant difference between the achievement of these students and those taught by teachers with more than five years of experience. the reason for this somewhat weird observation, as darling-hammond (2000) explains, could be that there is a tendency to be complacent by teachers after some years of teaching as result teacher effectiveness deteriorates. another possible reason advanced by darling-hammond is that for some reason teacher’s enthusiasm fizzles out and this leads to low morale. contrary to these findings, a few studies like hanushek (1997), martin et al. (2000) and wenglinsky (2002) revealed that the number of years in teaching is not associated with students’ achievement. these findings could be attributed to the teachers’ high level of preparedness as a result of good quality pre-service education and training obtained. the current study intends to contribute more on the issue of relationship between students’ achievement and teaching experience. professional development another aspect considered critical so far improving student achievement in mathematics is concerned, is teachers’ professional development. loucks-horsley et al. (1998) refers to teachers’ professional development as the opportunities offered to practising teachers to develop new knowledge, skills, approaches and dispositions to improve their effectiveness in their classrooms. in other words, it is advancement/enhancement of teachers’ knowledge of the students, the subject matter, teaching practices, and education-related legislation professional development programmes could include formal and informal means of helping teachers to not only learn new skills but also develop insight into pedagogy and their own practice, as well as exploring new or advanced understanding of content and resources. professional development, for example, can take place through workshops, cluster meetings, formal presentations by the more knowledgeable persons, further studies and self-evaluation of one’s own practice. however, for some reasons, there have been concerns raised in some quarters about the ineffectiveness of teachers’ professional development offered. ball, lubienski and mewborn (2001) reckon teachers’ professional development is intellectually superficial, disconnected from deep issues of curriculum and learning, fragmented, and non-cumulative. little and mclaughlin (1993) argued that professional development programmes only update teachers’ knowledge instead of providing an opportunity for sustained learning on issues pertaining to curriculum, students or teaching. on the contrary, varella (2000) and franke (2002) indicate that teachers’ professional development has positive effects on students’ achievement on condition it happens over a considerable time. what is also important for a professional development programme to be effective is what it seeks to achieve or is meant for. for example, the study by carpenter, fennema, peterson, chiang and loef (1989) show that professional development rooted in subject matter, focused on students learning and on effective ways to gauge david mogari, jeanne kriek, gerrit stols & ogbonnaya iheanachor 7 learning impacted significantly positive on students’ achievement. kennedy (1998) who reviewed 10 research studies on the impact of teachers’ professional development programmes on students’ achievement also came up with similar findings. kennedy found that teachers’ professional development improved students’ achievement when it focused on strengthening teachers’ content knowledge and related instructional practices; how students learn; and ways to help students understand subject knowledge. therefore, the focus and purpose of a teachers’ professional development programme are of utmost importance. if teachers can benefit from a professional development programme mainly because it focused on and addressed their specific needs, can this change be related to an improvement in student achievement? it is this that the current study seeks to examine. methodology research design the study followed a co-relational research design to determine the relationships, if any, between students’ achievement and teacher background and professional development, respectively. students’ achievement was also related with each component of teacher background (viz., teachers’ subject majors, qualifications, and teaching experience). sample a convenient sample of form c (grade 10) mathematics teachers was derived from 54 secondary schools in the maseru district, lesotho. of these schools, 40 (75%) are owned by the missions, 6 (10%) are owned by the government, 4 (7,5%) are owned by the communities and 4 (7,5%) are owned by private individuals or organisations. of the 53 teachers that availed themselves for the study, 6 came from government schools, 6 were from community schools, 37 descended from mission schools and 4 were from private schools. a stratified random sample of 40 teachers was selected from the initial 53 to ensure that each type of school was proportional represented in the final sample. students who took part in the study were those taught by these teachers. instrumentation a self report questionnaire called mathematics teaching opinionate scale (matos) was used. this is a modified version of a self report survey questionnaire developed by horizon research incorporated (2001) in the united states to gather in-depth information from teachers. the questionnaire was modified by only including sections that elicited detailed information relevant to the present study. the content validity was tested by involving experts in the field of mathematics education. its reliability was determined using cronbach’s alpha coefficient and yielded values of 0,76 for teachers’ background and 0,79 for professional development. the questionnaire was also pre-tested with 13 form c mathematics teachers who were not part of the current study. a form c mathematics examination question paper was administered to students during the end of the academic year. this is an examination conducted by the examination council of lesotho (ecol). mathematics question papers are jointly set by mathematics teachers and examiners. they are then content validated by the subject officers, specialists and the subject team members drawn from ecol and the national curriculum development centre. the reliability of the 2006 form c mathematics question paper was determined by k-r 21 formula which gave a value of 0,92. data collection the matos self report was administered to the participating teachers and data on mathematics students’ achievement was obtained from ecol 2006 form c examination results. lesotho’s students’ achievement and their teachers’ background and professional development 8 results students’ achievement and teachers’ background table 1 shows that 65% of the teachers have been teaching for more than 10 years and 80% of the teachers have a minimum of a bachelor’s degree. furthermore, 52,5% of the teachers have majored in mathematics or mathematics education and this implies that there is considerable number of the mathematics teachers who, arguably, could be deemed not to have enough mathematics knowledge and skills. table 1: teachers’ demographic information (n = 40) teaching experience 0 – 5 years 6 – 10 years 11 –15 years 16 –20 years over 20 years % of teachers 20,0 15,0 32,5 12,5 20,0 qualification certificate diploma bachelors masters doctorate 5,0 15,0 67,5 12,5 0,0 mathematics/mathematics education major yes no 52,5 47,5 correlation analysis was then used to determine whether there was any relationship between students’ achievement in mathematics and each component of teachers’ background (viz., teaching experience, qualifications and subject majors). there results thereof are presented in table 2. table 2: correlations between students’ achievement and each component of teacher’s background (n = 40) variables correlation coefficients teaching experience 0,393*(pearson r) qualifications 0,547**(point biseral rpt bis) mathematics or mathematics education majors 0,467*(biseral rbis) *significant at p < 0,05, ** significant at p < 0,01 according to table 2 there were significant positive relationships between students’ achievement and each of the components of teachers’ background. on the issue of a curvilinear relationship between students’ achievement and teacher experience raised in darling-hammond (2000) and hawkins, et al. (1998), it was determined whether the teachers’ years of experience greater than five years and that greater than ten years were respectively related to the students’ achievement. table 3 shows that there was a significant relationship between students’ achievement and teachers’ years of experience greater than five years. but, there was no statistically significant relationship between students’ mathematics achievement and teachers’ experience of more than ten years. in fact the data show that the trend of relationship between students’ achievement and teacher experience seems to level off round about ten years. david mogari, jeanne kriek, gerrit stols & ogbonnaya iheanachor 9 table 3: correlations between students’ achievement in mathematics and teaching experience variables r teaching experience > 5 years 0,416* teaching experience > 10 years 0,313 *significant at p < 0 ,05 regression analysis was used to examine the relationship between the dependent variable (student achievement) and the independent variables (teachers’ qualifications, subject majors and years of teaching experience). it allows for the determination of the variance between the dependent variable and the independent variables as well as to determine the independent variables that are statistically significant predictors of students’ achievement in mathematics. table 4 shows the regression analysis results involving students’ achievement in mathematics as the criterion variable (dependent variable) and the three independent variables (teachers’ qualifications, subject majors and years of teaching experience). table 4: relationship between the criterion variable (achievement) and the three independent variables (regression analysis) (n = 40) model summary r r square f p 0,600 0,360 4,321 0,015 model unstandardised coefficients standardised coefficients t p β std. error beta constant teaching experience qualifications subject majors 0,450 0,159 0,771 0,348 1,070 0,155 0,434 0,417 0,188 0,373 0,176 0,420 1,026 1,778 0,835 0,678 0,316 0,089 0,412 table 4 indicates that the three statistically significant predictors accounted for 36 percent of the students’ achievement in mathematics (r2 = 0,36). teaching experience (β = 0,16, p < 0,5), teachers’ qualifications (β = 0,77, p < 0,5) and subject majors (β = 0,35, p < 0,5) demonstrated significant relationships on students’ achievement in mathematics. the coefficients of the model indicate that the three regressors can be ranked in order to quantify their relationship with the dependent variable by starting with teachers’ qualifications (0,77), subject major (0,35) and teaching experience (0,16). in other words, in the context of teachers’ background, teachers’ qualifications accounted for 77% variation in students’ achievement in mathematics, while 35% and 16% can be attributed to teachers’ subject majors and teaching experience, respectively. students’ achievement and professional development in terms of professional development, the following aspects were looked at: time spent by teachers on professional development in the last three years, the frequency of various forms of professional development programme and the activity carried out during a professional development programme. each of these aspects was then correlated with students’ achievement. lesotho’s students’ achievement and their teachers’ background and professional development 10 time spent on professional development the duration of teachers’ participation in professional development in the last three years is shown in table 5. the table shows that only 20 percent of the teachers have spent 35 or more hours in professional development in the last three years. table 5: duration of professional development in the last three years (n = 40) time percentage of teachers none less than 6 hours 6-15 hours 16-35 hours more than 35 hours 22,5 17,5 25,0 15,0 20,0 the frequency of various forms of professional development programmes table 6 presents the various professional development programmes the teachers have taken part in during the last three years. table 6: teachers’ participation in professional development programmes (n = 40) programme % of teachers taken a formal college/university mathematics course 22,5 taken a formal college/university course in the teaching of mathematics 25,0 observed other teachers teaching mathematics as part of your own professional development (formal or informal) 70,0 met with a local group of teachers to study/discuss mathematics teaching issues on a regular basis 65,0 collaborated on mathematics teaching issues with a group of teachers at a distance using telecommunications 32,5 served as a mentor and/or peer coach in mathematics teaching, as part of a formal arrangement that is recognised or supported by the school or district 32,5 attended a workshop on mathematics teaching 52,5 attended a mathematics teacher association meeting 37,5 from table 6, it is evident that observing other teachers teaching mathematics either formally or informally was the most commonly reported form of professional development. meeting with a local group of teachers to study or discuss mathematics teaching issues on a regular basis was the second most frequently used professional development programme while attending a workshop focused on mathematics teaching was the third. correlation analysis was used to find the relationship between students’ achievement in mathematics and the respective indices of the three most popular forms of professional development programmes, namely, observing other teachers, meeting to study or discuss mathematics teaching and attending workshops on mathematics teaching. table 7 shows that there was no significant relationship. david mogari, jeanne kriek, gerrit stols & ogbonnaya iheanachor 11 table 7: correlations between students’ achievement in mathematics and variables defining professional development programmes (n = 40) variables rbis observing other teachers 0,05 meeting to study or discuss maths teaching 0,10 attending workshop 0,27 the activity carried out during a professional development programme according to carpenter et al. (1989) professional development programmes are successful if it is rooted in subject matter and focused on student learning. table 8 indicates that only 5 percent of the teachers reported that their professional development largely emphasised deepening their mathematics content knowledge, while 12,5 percent reported that their professional development activities largely emphasised understanding student thinking in mathematics. table 8: emphasis of teachers’ professional development activities (n = 40) professional development activity percentage of teachers no response not at all slightly moderately largely to a great extent deepening my own mathematics content knowledge 7,5 5,0 30,0 52,5 5,0 0 understanding student thinking in mathematics 7,5 5,0 27,5 47,5 12,5 0 learning how to use inquiry/ investigation-oriented teaching strategies 7,5 5,0 10,0 62,5 15,0 0 learning how to use technology in mathematics instruction 7,5 20,0 25,0 45,0 2,5 0 learning how to assess student learning in mathematics 7,5 5,0 12,5 62,5 12,5 0 learning how to teach mathematics in a class that includes special needs students 7,5 27,5 25,0 12,5 10,0 17,5 the teachers were also asked to indicate how much emphasis was placed on the various professional development activities they participated in the past three years. table 9 shows the correlation between emphasis on the professional development activities and students’ achievement. the table shows that each of the professional development activities that emphasise deepening teachers’ mathematics content knowledge, understanding students thinking in mathematics and learning how to assess student learning in mathematics correlate positively but insignificantly with students’ achievement in mathematics. table 9: correlation between students’ achievement and professional development activities variables rbis deepening my own mathematics content knowledge 0,318 understanding student thinking in mathematics 0,353 learning how to use inquiry/investigation-oriented teaching strategies -0,224 learning how to use technology in mathematics instruction -0,047 learning how to assess student learning in mathematics 0,125 *p < 0,05, **p < 0,01 (n = 40) lesotho’s students’ achievement and their teachers’ background and professional development 12 table 9 shows that professional development activities that emphasised learning how to use inquiry/investigation-oriented teaching strategies and learning how to use technology in mathematics instruction to a great extend have negative insignificant relationship with students’ achievement in mathematics. students’ achievement correlated with teachers’ background and professional development the respective components of the two main variables (viz., teachers’ background and professional development) were then combined and each of the variables was correlated with students’ achievement. according to table 10 teachers’ background has a positive significant relationship with students’ achievement while professional development has positive but insignificant relationship with students’ achievement. table 10: correlation between students’ achievement in mathematics and combined indices of teachers’ background, and professional development variables rbis teachers’ background 0,552** professional development 0,209 *significant at p < 0,05, **significant at p < 0,01 to further verify the results in table 10, multiple regression analysis of the combined variables with students’ achievement was carried out using spss. the results of multiple regression analysis presented in table 11 do confirm that teachers’ background correlate significantly with students’ achievement while there is no significant relationship between students’ achievement and the extent of professional development. table 11: combined effects of the indices of teachers’ background, professional development sum of squares df mean square f sig. teachers’ background regression residual total 15,134 26,853 41,998 3 23 26 5,045 1,168 4,321 0,015 professional development regression residual total 9,657 31,953 41,609 5 20 25 1,931 1,598 1,209 0,341 discussions the purpose of the study was to determine if there is any relationship between students’ achievement teachers’ background and professional development, respectively. the results of the study show that there was a significant positive relationship between students’ achievement and teachers’ background. the results are consonant with prior findings by goldhaber and brewer (1996), betts et al. (2003), darlinghammound (2000), wilson and floden (2003). this implies that an improvement in teachers’ background can be connected with an improvement in students’ achievement. for a country like lesotho, which has been having a problem of poor performance in mathematics of the years (ecol, 2005; 2006), david mogari, jeanne kriek, gerrit stols & ogbonnaya iheanachor 13 there might be a need to prioritise the improvement of mathematics teachers’ background if the students’ achievement is to be improved. similar results were also obtained between students’ achievement and each of the components of teachers’ background, namely, teachers’ qualifications, teachers’ subject majors, and teachers’ years of experience. of the three, teachers’ qualification correlated most significantly with students’ achievement followed by teachers’ subject major and lastly teaching experience. in other words, students whose teachers have higher qualifications are likely to perform better in mathematics than students whose teachers have lower qualifications. this finding confirms those by greenwald et al. (1996), goldhaber and brewer (1996), betts et al. (2003) and rice (2003). it can be argued that the strong connection existing between teachers’ qualification and students’ achievement implies that as teachers acquire additional qualifications their knowledge, skills and attitude tend to improve. in turn the effectiveness of teachers in the classroom can translate into better learning for students (kriek, 2005). in addition, the regression analysis results in table 2 showed that teachers’ qualifications are the greatest predictor of students’ achievement in mathematics in lesotho. however, the results also showed that 20% of teachers do not have a degree and 47,5% do not have mathematics major (see table 1). it may very well be that this unpleasant state of affairs is connected with the prevalent high rate of students’ poor achievement in mathematics in lesotho. thus, a significant relationship was obtained in this study between students’ achievement and teachers’ mathematics major. this further supports the earlier findings of goldhabler and brewer (1996), wenglinsky (2002), wilson and floden (2003), and greenberg et al. (2004). the results seems to imply that teachers without mathematics major are not considerably knowledgeable in mathematics content and this tends to affect the quality of the teachers’ pedagogical content knowledge which is essential in facilitating meaningful learning. the result showed that there was a significant relationship between students’ achievement in mathematics and teachers’ years of experience, even though the correlation coefficient obtained was the lowest as compared to those of the other two components of teachers’ background. nevertheless the results do affirm those by greenwald et al. (1996), hawkins et al. (1998) and rivkin et al. (2005). what also emerged from the results is the issue of curvilinear relationship between students’ achievement and teachers’ experience. the trend of the relationship tends to level off when teachers reach ten years of service. it may very well be that the teachers’ effectiveness fizzles out with time mainly because most teachers (≥80% according to table 5) have had less than 35 hours of professional development over three years. this is limited to have any significant effect on teacher improvement (garet, porter, desimone, birman, & yoon, 2001). furthermore, the popular forms of professional development for the teachers in the present study (in order) are watching their peers teaching; interacting with peers regularly and attending workshop on mathematics teaching (see table 6). it can be argued that even though teachers are encouraged to discuss and share their daily classroom experiences among themselves and observe each other teaching, there has to be “an expert figure” to provide corrective feedback and suggestions (onwu & mogari, 2004). therefore, it may be that no new ideas and skills are fed from the external source into the communities of teachers instead ideas and experiences are continually being recycled among the participating teachers. this, then, compromises the quality of professional development programme teachers attended. hence, as teachers acquire more years of teaching there is little improvement in their students’ achievement. in conclusion the study presented yet further information on the relationship between students’ achievement and teachers’ background as well as between students’ achievement and teachers’ extent of professional development. the study showed that the former relationship was positive and significant while the latter was just positive. this implies that the quality of qualifications teachers are exposed to is closely related to how their students achieve. the qualifications should provide teachers with the necessary amount of subject content and skills to become effective in their classrooms. furthermore, for teachers of mathematics to be competent enough they should acquire the highest possible amount of mathematics in their qualifications. in terms of the insignificant relationship between students’ achievement and the extent of professional development, the issue of quality seems yet again to be at the centre. if teachers are not subjected to a quality professional development programme they tend not to improve and develop (the professional lesotho’s students’ achievement and their teachers’ background and professional development 14 affairs department, 1999). this means that the teachers’ classroom practice fails to improve regardless of the time they spent teaching. it may be argued that the teachers’ knowledge of relevant instructional strategies, knowledge of appropriately representing mathematics content for teaching, knowledge of students’ conceptions and students’ learning difficulties, (see shulman, 1986), cannot be improved through a poorly structured and badly planned professional development programme. it then tends to follow that the effectiveness of any teachers’ professional development programme is essential for the improvement of students’ learning. since the current study only focused on the relationship, possible studies might be necessary to determine the effects of teacher background (as defined in the current study) and the extent of professional development on students’ achievement. references ball, d. l., lubienski, s. t., & mewborn, d. s. 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can be opened with acrobat and adobe reader 5.0 and later.) >> /namespace [ (adobe) (common) (1.0) ] /othernamespaces [ << /asreaderspreads false /cropimagestoframes true /errorcontrol /warnandcontinue /flattenerignorespreadoverrides false /includeguidesgrids false /includenonprinting false /includeslug false /namespace [ (adobe) (indesign) (4.0) ] /omitplacedbitmaps false /omitplacedeps false /omitplacedpdf false /simulateoverprint /legacy >> << /addbleedmarks false /addcolorbars false /addcropmarks false /addpageinfo false /addregmarks false /convertcolors /converttocmyk /destinationprofilename () /destinationprofileselector /documentcmyk /downsample16bitimages true /flattenerpreset << /presetselector /mediumresolution >> /formelements false /generatestructure false /includebookmarks false /includehyperlinks false /includeinteractive false /includelayers false /includeprofiles false /multimediahandling /useobjectsettings /namespace [ (adobe) (creativesuite) (2.0) ] /pdfxoutputintentprofileselector /documentcmyk /preserveediting true /untaggedcmykhandling /leaveuntagged /untaggedrgbhandling /usedocumentprofile /usedocumentbleed false >> ] >> setdistillerparams << /hwresolution [2400 2400] /pagesize [612.000 792.000] >> setpagedevice abstract introduction literature review and theoretical perspectives research design and methodology learners’ performance on the concepts in secondary maths and science items discussion conclusions and implications acknowledgements references footnotes about the author(s) craig pournara school of education, university of witwatersrand, johannesburg, south africa jeremy hodgen school of education, university of nottingham, united kingdom yvonne sanders school of education, university of witwatersrand, johannesburg, south africa jill adler school of education, university of witwatersrand, johannesburg, south africa citation pournara, c., hodgen, j., sanders, y., & adler, j. (2016). learners’ errors in secondary algebra: insights from tracking a cohort from grade 9 to grade 11 on a diagnostic algebra test. pythagoras, 37(1), a334. http://dx.doi.org/10.4102/pythagoras.v37i1.334 original research learners’ errors in secondary algebra: insights from tracking a cohort from grade 9 to grade 11 on a diagnostic algebra test craig pournara, jeremy hodgen, yvonne sanders, jill adler received: 04 apr. 2016; accepted: 02 sept. 2016; published: 31 oct. 2016 copyright: © 2016. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract it is well known that learner performance in mathematics in south africa is poor. however, less is known about what learners actually do and the extent to which this changes as they move through secondary school mathematics. in this study a cohort of 250 learners was tracked from grade 9 to grade 11 to investigate changes in their performance on a diagnostic algebra test drawn from the well-known concepts in secondary maths and science (csms) tests. although the csms tests were initially developed for year 8 and year 9 learners in the uk, a rasch analysis on the grade 11 results showed that the test performed adequately for older learners in sa. error analysis revealed that learners make a wide variety of errors even on simple algebra items. typical errors include conjoining, difficulties with negatives and brackets and a tendency to evaluate expressions rather than leaving them in the required open form. there is substantial evidence of curriculum impact in learners’ responses such as the inappropriate application of the addition law of exponents and the distributive law. although such errors dissipate in the higher grades, this happens later than expected. while many learner responses do not appear to be sensible initially, interview data reveals that there is frequently an underlying logic related to mathematics that has been previously learned. introduction this article has its origins in the discourse of poor performance and failure in school mathematics in south africa and other developing countries. that said, not all learners in south africa are failing mathematics but the vast majority are not coping (spaull, 2013). attempts to seek and gather evidence of improved learner performance are often thwarted by a lack of longitudinal data or instruments that are not sufficiently sensitive to detect change at lower performance levels. furthermore, in developing countries where these problems are most acute, there is seldom adequate funding to develop more appropriate and sensitive measures. in our project we seek to shift the discourse from ‘learners can’t do’ to ‘what learners can do is …’. this requires sensitive diagnostic instruments that enable us to measure change over time, even when learners’ performance is poor. however, there are no adequate local diagnostic instruments available for this purpose. consequently, we drew on a selection of algebra items from the concepts in secondary maths and science (csms) tests (hart & johnson, 1983) and tracked the performance of a group of 250 learners on these items across their grade 9, 10 and 11 years of schooling. this has enabled us to investigate the kinds of algebraic errors learners make and how these change over time. our findings show that many learners in grade 11 still have difficulty with aspects of basic algebra. this reinforces the comments made on the performance of grade 12 learners on the national senior certificate paper 1 each year. for example, in 2012 the examiners commented: many of the errors made in answering this paper have their origins in poor understanding of the basics and foundational competencies taught in the earlier grades. for example, algebraic manipulation, factorisation, solution of equations and inequalities. (department of basic education, 2012, p. 12) in this article we draw on quantitative and qualitative data to illuminate the kinds of errors that learners make in relation to basic algebra. thus we go beyond merely noting that learners make errors to provide insights into what errors are being made and the extent to which these decrease or persist over time. in addition, we provide evidence of the kinds of rational descriptions that learners provide of their thinking as they work on algebraic tasks. we pay particular attention to errors of conjoining, difficulties with negatives and brackets and learners’ tendency to evaluate expressions rather than leaving them in the required open form. our analysis also shows evidence of curriculum impact, that is, how learners apply new procedures and laws they have been taught, but in inappropriate ways. typically, this changes over time as learners become more familiar with the new procedures and when they should be applied but the cycle continues as they learn further procedures and then apply these inappropriately. this finding gave rise to more substantial concerns regarding the use of the csms items outside the uk and with older learners. based on a simple rasch analysis of the grade 11 data we show that the items performed sufficiently well for the purposes of this study although the nature of the errors is clearly impacted by learners’ exposure to more mathematics as they move from grade 9 to grade 11. literature review and theoretical perspectives internationally, there has been much interest in learners’ errors in mathematics for the past 30 years. the early work drew strongly on constructivist perspectives (e.g. ben-zeev, 1998; borasi, 1994; olivier, 1989; radatz, 1979), others adopted a socio-cultural perspective (e.g. ryan & williams, 2007) and more recent work comes from a discursive approach (e.g. brodie & berger, 2010). in south africa, there has been a resurgence in the focus on learner errors and error analysis in mathematics in recent years (brodie, 2014; brodie & berger, 2010; herholdt & sapire, 2014; makonye & luneta, 2014; shalem, sapire, & sorto, 2014). this is not surprising given the poor performance in mathematics in sa and the associated focus on mathematics professional development. the work continues to reflect a range of theoretical perspectives to describe and explain errors. we work from a sociocultural perspective but draw on some aspects of discursive approaches (sfard, 2008). in agreement with all the work referred to above, we consider learners’ errors to be rational attempts to make sense of mathematics. we acknowledge that, for the most part, errors are systematic, persistent and resistant to ‘fixing’ through instruction. research on the transition from arithmetic to algebra has been ongoing since the 1970s (e.g. collis, 1978) and much has been learned about learners’ difficulties in making the transition. following the csms study, the increasing competence and confidence in algebra and multiplicative structures (iccams) study used the csms algebra items to investigate changes in learners’ understanding of algebra 30 years later. it was found that more learners were achieving low marks, fewer were scoring very high marks and there was little change in the kinds of errors learners made (hodgen, brown, coe & küchemann, 2012). linked to the csms study, küchemann (1981) identified six ways in which learners interpret letters; these ideas have influenced much of the work that has followed. one interpretation is that letters are abbreviations for the names of objects. asquith, stephens, knuth, and alibali (2007) have shown that this error tends to disappear by the 10th grade in the united states. many have reported the persistent tendency for learners to think that a letter stands for a single, specific number (asquith et al., 2007; booth, 1984; collis, 1975). christou and vosniadou (2012) have shown that 10th grade learners in greece tend to substitute natural numbers in algebraic expressions even when they recognise that letters can stand for any number. there is also a great deal of evidence which shows that prior knowledge of arithmetic has a negative impact on the learning of algebra. this is typically manifest in errors relating to ‘lack of closure’ where learners do not accept expressions such as a + 2 as final answers. (e.g. booth, 1984; christou & vosniadou, 2012; collis, 1978; macgregor & stacey, 1997). despite all that has been learned, algebra remains a substantial obstacle for many learners of secondary mathematics internationally as borne out in the performance of sa learners on the csms items 45 years after they were first developed. the design and content of the concepts in secondary maths and science algebra test the algebra items used in this study were first developed and administered to a nationally representative sample of english learners in the 1970s as part of the csms study. the design of the items was strongly informed by piagetian ideas that were dominant at the time and were intentionally designed to be recognisably connected to the uk mathematics curriculum of the time, but required learners to: use methods ‘which were not obviously ‘rules’ (hart & johnson, 1983, p. 2), avoid ‘excessive computation’ (p. 22) and make use of simple numbers in non-routine problems. the csms algebra test focuses on generalised arithmetic, the use of symbols to denote numbers and letters as variables (collis, 1975; küchemann, 1981). consider the following item: ‘if e + f = 8, e + f + g = …’. this was designed to test whether learners would accept the ‘lack of closure’ (collis, 1975) of the expression 8 + g, that is, to see it as an entity in its own right (sfard, 1991). without this, learners tend to see the expression as an instruction to do something and give numerical responses (such as 9 or 12) or the ‘compressed’ response 8g. research design and methodology over the period 2010 to 2014 our project worked in 10 schools in the johannesburg area in south africa. six of the schools are in so-called townships where there are only basic resources for teaching and learning. the other four schools are low-fee schools located in suburban areas. here too resources are limited although these schools are typically better equipped than those in the townships. in all schools, english is the language of instruction but almost no learners speak english as their main language. initial selection of sample we set out to track a cohort of learners from grade 9 to grade 11 (approximately 15 to 17 years old) across all project schools, regardless of whether they progressed to the next grade at the end of the year and regardless of whether they chose mathematics or mathematical literacy1 in grade 10 or grade 11. in 2011 we selected a sample of approximately 1500 grade 9 learners with the sole criterion being that their mathematics teachers in that year had been participating in the professional development activities offered by the project. while we anticipated some attrition over the three-year period, the attrition rate was far greater than expected and in 2013 we ended up with a tracked cohort of only 250 learners. there are many reasons for this attrition, including learners moving schools, teachers not administering the test to learners who had failed or who were taking mathematical literacy and the withdrawal of one school from the project. the limitations of this sample are that it became a small, opportunistic, non-random sample with a smaller proportion of weaker learners than the intended sample. nevertheless, it provides a unique snapshot of learners’ performance in algebra across the transition from grade 9 to grade 11. the test instruments and coding the test instruments did not consist only of csms items. we designed three tests, one for each grade, containing curriculum-related algebra and functions items and a selection of csms algebra items which were common to all three tests. we focus here only on the csms items, which have been validated in the uk in both studies. in selecting the items, we chose only those clusters of items that had been ‘levelled’ in the csms hierarchy (see hart, 1981, for more details). however, we excluded some of the more difficult items, partly because of the length of the test and partly because we anticipated that very few learners would cope with them. the order of the items differed from the uk test. in 2010 the csms items were piloted with grade 8 and grade 10 learners in schools similar to the project schools. the results of this pilot indicated that the items and the associated coding scheme were appropriate in the sa context. the three full tests were then piloted in project schools with different cohorts prior to administering to the tracked cohort. the tests were administered in october each year (i.e. near the end of the school year). in 2011 and 2012 they were administered by project team members. in 2013 they were administered by team members in some schools and by teachers in other schools. we deliberately wanted to include teachers in the research process but, with hindsight, this was not wise because some teachers did not follow the directions given regarding who was to write the test, which ultimately impacted on the size of the tracked cohort. the test codes were adapted from the csms codes. in the main, we used the most prevalent csms codes and then added additional codes to capture the wide range of responses that would otherwise have merely been coded as ‘other’. coding was carried out by trained research assistants and moderated by senior members of the project team. interview data in 2012 and 2013, following the analysis of pilot data and the test scripts, several small studies were conducted by postgraduate students (honours level) who were supervised by senior members of the project team. these studies involved an analysis of scripts and one-on-one task-based interviews where each postgraduate student interviewed learners from project schools on questions similar to selected test items in order to investigate learners’ thinking in relation to introductory algebra. we draw on these studies to illustrate our claims about learners’ thinking and strategies in relation to their errors. we explicitly acknowledge the postgraduate students as authors of the work reported here. however we have re-transcribed and re-analysed all interview extracts that are included in this article. ethical issues ethical clearance was obtained from the university (2010ece60c) and from the gauteng department of education for this research. informed consent was obtained from schools, parents and learners for both the tests and the interviews. names of learners and schools have been kept confidential and we have not reported separately on particular schools. pseudonyms have been used in all learner transcripts. permission was obtained from the postgraduate students to use their data as part of the wider project research. reliability and validity as noted above, the validity of the items for the sa context was confirmed through piloting. the reliability of the coding was increased through a moderation process: all learner interviews were re-transcribed and they were then re-analysed by two of the authors to confirm the initial interpretation. since the csms items had been developed for year 8 and year 9 learners in the uk, it was necessary to validate it for older learners in sa. we therefore conducted a simple rasch analysis, using winsteps (version 3.73)2, to assess the validity of the scale for older learners. we chose the grade 11 data since these learners had performed better on the test, thus enabling better discrimination of items at the higher and lower ranges of difficulty. we used three fit statistics to judge the quality of the selected csms items as an instrument for measuring learners’ understanding of algebra in the sa context: infit mean square (mnsq), outfit mnsq and item-scale correlation. since the test is a low stakes diagnostic test, we took a relatively liberal approach and so values of infit and outfit mnsqs higher than 2.0 or lower than 0.5 were regarded as a cause for possible concern (linacre, 2002). for the item-scale correlation, the discrimination index should ideally be above 0.4. a discrimination index below 0.2 is regarded as low and items with such a score may be candidates for deletion (wu & adams, 2007). there were four items with problematic fit statistics: q2, q5.1, q9.1 and q10.1. however, none of these questions is considered to be a threat to the scale. learners’ performance on the concepts in secondary maths and science items a summary of learners’ performance on the items across the three years is provided in table 1. the figures indicate the item facility (i.e. the percentage of learners who answered each question correctly). this is followed by a brief discussion of the results. table 1: percentage of correct responses to each item per grade. the general performance of the learners over the three years is low but this was not unexpected. there is a general increase from grade 9 to grade 11, although there are six items where performance dipped from grade 9 to grade 10 and two items where performance dropped from grade 10 to grade 11. several of these are discussed in later sections. the four items that grade 11 learners found most difficult were q2, q8.4, q10.4 and q11. more than 16% did not attempt the latter three items in grade 11. by contrast fewer than 1% of learners omitted q2 in grade 11 although only 1% answered it correctly. it is surprising that q1.5, q1.6 and q1.9 were among the most difficult items for learners at grade 11 level since these look similar to questions found in grade 8 and grade 9 textbooks in sa. reflecting on question 1 as a whole, it is noticeable that the performance on questions with brackets or negatives is well below the other items for all three years. for example q1.3 and q1.5 are very similar except for the operation in the bracket. yet the performance on these two items is considerably different, particularly in grade 9 and grade 10. other errors include inappropriate application of the addition law of exponents and the distributive law. these errors are discussed in more detail in the next section. the poor performance on q1.2 appears to be related to learners’ expectation that they should do something with the unlike terms to produce an answer. the slight drop in performance on q3.1 from grade 9 to grade 11 is surprising, as is the small size of the gain on q4.1, q8.1 and q8.2. one possible explanation is that asking simple arithmetic questions in the context of an algebra test confuses learners. alternatively, given that the first part of our test consisted of curriculum-level questions, some learners might have expected that these questions were more complex than intended. with regard to question 5, there is a noticeable contrast in performance between q5.2 and q5.3 in grade 9 and grade 10, although this was expected. it is also worth noting that responses to q5.2 were unstable with many learners moving from a correct response in one year to an incorrect response the following year. for example, in grade 9, 80 learners gave the correct answer but of these 44% got it incorrect in grade 10. similarly in grade 10, 100 learners gave the correct answer but of these, 30% got it incorrect in grade 11. it may be that this item is more error-prone than initially appreciated. if one merely glances at the question it is quite easy to focus on the increase on the left side and assume the same on the right side whereas it requires paying more careful attention to the relationships to recognise that such reasoning is faulty. question 6 and question 7 involved substitution. learners had particular difficulty with q6.2. given the prevalence of premature closure in their responses across the test, it is possible that many learners could not make sense of the equation b + 2 = 2b since for them b + 2 can be written as 2b and so they did not see the statement as one of mathematical equivalence in the way that was intended. question 9 and question 10 required algebraic expressions for the perimeter of polygons. while performance improved systematically on q10.2, performance on q10.3 dropped in grade 10 and was still poorer than q10.2 in grade 11. the higher number of errors is likely due to the combination of letters and numbers in q10.3 with 23% of grade 11 learners making conjoining errors and hence expressing the perimeter as 18u. question 11 was the only csms item selected that focused explicitly on letter as object. not surprisingly many learners treated the letters as objects with at least 35% of learners each year stating that 4c + 3b stands for 4 cakes and 3 buns. common learner errors while learners’ poor performance is reason for concern, an analysis of their errors provides insight into their reasoning and shows evidence of rational albeit incorrect strategies. these insights enable us to focus on what learners can do rather than on what they can’t do. in this section we focus on key errors and illustrate these with examples from questions 1, 3, 4 and 5. we also include interview data from the postgraduate students’ research projects. the most common errors involve conjoining and premature closure, negatives and subtraction, multiplication and indices, the equality relationship and evaluating letters rather than accepting an open expression as a final answer. we also provide evidence that learners change their strategies depending on particular details of a question. errors involving conjoining and premature closure errors involving conjoining or premature closure occur in the responses to many items as shown in table 2. while the actual number of conjoining errors decreases by grade 11, it still constitutes a large proportion of the errors on several questions. table 2: percentage of responses showing conjoining error. in q1.2 it could be argued that the nature of the question prompts learners to do something and so a reasonable response is to conjoin 2a and 5b. the interview data provides insight into how learners produce conjoined answers. mashazi (2012) asked lizwe in grade 9 to ‘add 5 to 3x’ and to ‘multiply 3x + 1 by 5’: lizwe: they said add 5 to 3x so i said 5 plus 3 equal to 8 then after that i took the x and put it next to 8 to get 8x. (p. 20) lizwe: they say multiply 3x + 1 by 5 so i first took the three and one and i added it because 3 plus 1 is 4 then i put the x and times by 5 and got 20x (p. 23) when asked to ‘multiply t + 2 by 3’ shenaaz, also grade 9, adopted the same strategy (govender, 2012): shenaaz: then [question] 2.2 was t + 2. so i said, 2 times 3. and isn’t 2 times 3 which is 6. so then after i put the 6 and then i put the t. govender: okay, how did you get the 6? shenaaz: okay, i said 2 times 3 which is 6. then i brought down the t. cos you can’t multiply the t by something else. it’s gotta come after you know. the learners ‘simplify’ the binomial by attending first to the numbers, then appending the letter and then multiplying by the constant. their use of language shows explicitly that when they conjoin unlike terms, they ignore the letter and then append it as one writes units in a measurement problem. both learners use ‘put’ to explain how the letter comes to be placed next to the number and shenaaz also says ‘i brought down the t’ illustrating that the letter is attended to last. the error of attending first to numbers is persistent in terms of the proportion of errors, as illustrated in table 3. the numbers in the questions are either added or multiplied. in some cases the letter is conjoined (e.g. 7n) while in others it is not (e.g. n + 9). table 3: percentage of errors showing focus on numbers. in the sections that follow we continue to point out errors involving conjoining although they are no longer in the foreground. it is worth noting at this point that when learners conjoin, their talk typically reflects an operational view of the expression they are dealing with. this is evidenced in treating subtraction as ‘take away’ and in counting the numbers of things they are working with, for example the numbers of b’s in an expression. errors involving subtraction and negatives learners’ errors in simplifying algebraic expressions suggest they may not be paying attention to signs and operations. this is more visible when the question involves subtraction and negatives and it appears that learners are focusing mainly on letters and numbers. for example in q1.5 more than 10% of grade 9 and grade 10 learners gave answers of a ± 2b and one of the most common errors in q1.6 was 4a − b. such responses come from collecting terms and ignoring the syntax of the expression. kalidheen (2012) provides evidence of grade 11 learners who continue to struggle with algebraic syntax involving negatives. for example, she asked simon to respond to the following question: subtract 2b from 8, 2b − b, 2b − a and 2b − 2a. simon: it’s going to be 8 minus 2b and the answer is going to be 8 minus 2b. the answer will still be 8 mam … cos… er here we have 8. we have numbers only and not letters and this side on the right we have two b’s so there is no way we can take it out from 8 while there’s no b’s. kalidheen: so you’ve got subtract 2b from 8 and your answer is 8. right? simon: still 8. kalidheen: so you left out the 2b. why? simon: cos, the 2b mam … cos … it is what, it is what, it had to be subtracted from 8 and we couldn’t. kalidheen: ok, so what happens if i make this 2b minus b (i.e. 2b − b)? what will happen then? simon : 2b minus b? no mam it’s, two b’s minus one b so it’s gonna be b. … kalidheen: and then, if i had … ok let’s say i change this to 2b minus a. simon: 2b minus a? mam you couldn’t … it can’t be … it can’t be done cos there’s no, there’s no a’s here, it’s only b’s. kalidheen: ok, so, what would happen here? what will the answer be? simon: no mam, it will be left like this (i.e. 2b − a). simon provides the correct answers when both terms contain letters. however, when only one term contains a letter, he eliminates the term because it cannot be subtracted from 8 ‘while there’s no b’s’. yet he does not eliminate the term in a when dealing with 2b − a. he is one of many learners who uses a different strategy depending on whether both terms contain letters. his responses also show that he is working with a partitioning structure of subtraction, that is, as ‘take away’ (haylock, 2006). such a view of subtraction is inadequate for algebra but it goes some way to explaining why learners might claim incorrectly that 8 − 2b = 8 but also claim correctly that 2b − a cannot be done ‘cos there’s no a’s here [to take away]’. in contrast, other interviews show evidence of learners paying explicit attention to signs and operations yet over-generalising methods from equation solving when working with expressions (e.g. gumpo, 2011; mashazi, 2012). mashazi (2012, p. 21) asked themba (grade 9) to explain his solution to: ‘simplify 3x + 2 + x’: themba: they said simplify. the first thing that i did, i grouped the like terms then i got the answer. mashazi: how did you group your like terms? themba: they said 3x + 2 + x then i said eh, when x comes between 3x and 2 it changes the sign to negative x, then i said 3x − x + 2, then i got the answer for 3x − x which is 2x and i left the 2 there, then i said 2x + 2 which gave me 4x. mashazi: let us go back to where you added the like terms. you said when x moves closer to 3x, it changes the sign. why? themba: because that is how mam taught me that when x eh, when the equation moves to the other side it changes the sign. mashazi: oh! when it is an equation. is this an equation? themba: yes, ma’am. mashazi: why do you say it is an equation? themba: because, ma’am, it has the variables. themba is confident in his selection and use of a strategy, albeit an inappropriate one. he changes the sign if a term moves to another position in the expression and his justification reveals some confusion between an equation and an expression. it also reflects inappropriate criteria for defining an equation and incorrect description of the objects he is working with, such a referring to a term as an equation. errors related to multiplication and indices there is much evidence of learners applying the distributive law and the addition law of exponents inappropriately across various items in question 1, particularly in grade 10. such errors account for the dip in the percentage of correct responses in q1.1 and q1.3 in grade 10. by grade 11 these errors were less frequent and this partially explains the substantial increase in the percentage of correct responses in items involving brackets. errors involving the addition law for exponents, that is, a m.a n = a m+n, are best illustrated by q1.1, q1.4 and q1.5 as shown in table 4. in q1.1, 7a 2 is by far the most common error each year. the increase in the number of learners making this error in grade 10 may suggest that learners’ knowledge of algebraic simplification is unstable. however, the increase in grade 10 may also suggest that learners expected a more complex question given that the first part of the test contained more difficult curriculum items. table 4: percentage of incorrect responses involving exponential laws. many learners treated the presence of brackets as a signal to multiply, not paying attention to the operations adjacent to the brackets. consequently there is much evidence of the incorrect application of the distributive law in questions involving brackets. this is clearly illustrated in q1.5 and q1.6 as shown in table 5. the percentage of learners giving these responses grows substantially from grade 9 to grade 10 and then drops off in grade 11. table 5: percentage of incorrect responses involving distributive law. of all the csms test items, question 1 was most similar in form to sa curriculum items in algebra. consequently, learners saw opportunity to use the algebraic laws and procedures they had been taught and this may account for the inappropriate use of such curriculum knowledge. exponential laws are introduced informally in grade 9, then formalised for integer exponents in grade 10 and extended for rational exponents in grade 11. the increased attention given to the laws in grade 10 may explain the spike in this error. the drop in grade 11 may reflect that this knowledge is now more stable. the trend in q1.4 and q1.5 is similar. the distributive law is first introduced in grade 8 and reinforced in grade 9 for more complex algebraic expressions. it is therefore surprising that the misapplication of the law appears more frequently in grade 10 and not in grade 9 as was the case with the exponential law. the reduction in errors involving the distributive law suggests that learners are more familiar with the correct application of the law in grade 11. in q1.9 there is evidence of both the addition law of exponents and some attempt at the distributive law. it appears that that many learners treated the question as the product of binomials: (a + b)(a−b) the substantial increase in these errors from grade 9 to grade 10 is further evidence of curriculum effects since the factorising and expansion related to the difference of two squares is given more attention in grade 10. errors related to evaluation of letters when learners are not yet able to deal with a lack of closure in algebraic expressions, a common strategy is to evaluate the letters (küchemann, 1981). item q5.3 was deliberately designed to test whether learners would accept the lack of closure inherent in the expression 8 + g. as expected, many learners evaluated the letters with errors involving evaluating, accounting for 51% of all responses in grade 9. this dropped to 31% in grade 11. however, of all the errors committed in grade 11, evaluating of letters constituted 52%. there were two common strategies for evaluating: equal splits: splitting the known quantity equally depending on the number of letters on the left and then giving the new letter the same value as the others, for example e = f = 4 so g also has a value of 4 and hence e + f + g = 12. assigning a value of 1: the new letter is given a value of 1, presumably because its value is unknown and hence can be any value, so the simplest value is chosen, giving e + f + g = 8 + 1 = 9. both strategies are reported in other studies that have made use of the csms items (macgregor & stacey, 1997; oldenburg, hodgen & küchemann, 2013). we consider the equal splits strategy to be a more sophisticated strategy because it indicates that learners are looking for patterns in the relationship rather than merely adding one. equal splits is the most common strategy constituting 29%, 31% and 33% of the errors each year respectively. it is also a persistent strategy across the three years. for example, of the 62 grade 10 learners who used equal splits, 47% had also used it in grade 9 and 37% continued to use it in grade 11. another noticeable trend was the move from the add 1 strategy to the equal splits strategy. for example, in grade 9, 46 learners used add 1. of these, 35% used the same strategy in grade 10 but 27% changed to equal splits. in grade 10, 46 used add 1. of these, 24% moved to equal splits and 26% gave the correct answer in grade 11. other less common strategies included assigning consecutive whole numbers, which typically occurred when the constant was odd (du plessis, 2012), and solving for e. in the interview extracts we gain insight into learners’ reasoning about q5.3 or similar items from the honours projects. thabo, a grade 9 learner interviewed by brown (2012, p. 95), argues confidently for an answer of 12 to ‘if e + f = 8, then e + f + g = …’: thabo: if you can take a look at e plus f equals to 8, it actually tells us that this e is 4 and this f is 4. that’s why they got 8. and right here we’ll say e plus f plus g equals to, ah, then one must say 4 plus 4 plus 4 is, 4 plus 4 plus 4, i think i’ll, i will actually say 4 times 3 which is 12. that’s why i got this 12. thabo says ‘it actually tell us’ indicating that he perceives the value of e and f to be given, and is not considering any other values for the letters. by contrast lindo (grade 11), in madosi (2012, p. 25), is very clear that she guessed her answer to ‘if a + b = 6, then a + b + c = …’: madosi: so how did you get 3? lindo: i just guessed the 3. i just thought this 6, they are two variables, meaning i add two numbers to get 6. madosi: ok. lindo: 3 plus 3, and here also i said 3 because there are 3 numbers, i also used the same number that i thought they used here. in both cases learners ‘know’ that they need to obtain values for the letters and both make use of an equal splits strategy. thabo assumes the values are implied in the question and recognises that he can multiply to obtain the sum. lindo guesses a value of 3 but in another section of the interview acknowledges that she could have chosen different values. another strategy, which was more common among grade 11 learners, involved treating the expression as an equation and solving for the unknown. this is illustrated in du plessis (2012), where bokang solves for e in ‘if c + d = 6 then c + d + e = …’ (see figure 1): bokang: i thought, i thought eh here neh, since c well, eh, c plus d is 6, i thought we were looking for e only. then if we are looking for e only, then which means here was 3 and here was 3. then you add this and this and you get 6. then when you go to the other side then e is minus 6. in both cases learners ‘know’ that they need to obtain values for the letters and both make use of an equal splits strategy. thabo assumes the values are implied in the question and recognises that he can multiply to obtain the sum. lindo guesses a value of 3 but in another section of the interview acknowledges that she could have chosen different values. another strategy, which was more common among grade 11 learners, involved treating the expression as an equation and solving for the unknown. this is illustrated in du plessis (2012), where bokang solves for e in ‘if c + d = 6 then c + d + e = …’ (see figure 1): bokang: i thought, i thought eh here neh, since c well, eh, c plus d is 6, i thought we were looking for e only. then if we are looking for e only, then which means here was 3 and here was 3. then you add this and this and you get 6. then when you go to the other side then e is minus 6. figure 1: bokang’s response to ‘if c + d = 6, then c + d + e = …’ bokang interprets the question to mean that he needs to find the value of e. he assigns equal splits to c and d and then treats the expression as an equation, assuming zero on the right side and then transposing the 6. his comment ‘i thought we were looking for e only’ suggests he views this item as an equation to be solved and there is only one unknown. while he still evaluates the letter, his approach is more sophisticated than merely assigning equal splits or adding one. choosing strategies based on particular features of the item there is evidence across several interview items that learners change their strategies depending on the specifics of the question. for example, du plessis (2012) shows that learners used different strategies in responding to a variation of q5.3 depending on the choice of letters (consecutive versus non-consecutive) and constants (odd or even). earlier we provided evidence of how simon reasoned differently when subtracting a constant compared with subtracting a term involving a letter. in the example below musa (grade 11) reasons differently depending on the operation and the coefficients (kalidheen, 2012, p. 48): musa: the first question says simplify where possible. the equation is 2a plus 3b plus c (2a + 3b + c). in this case you would have to add…, you would have to add all of them together but we have 2a plus 3b plus c so add the like terms together which is 2a plus 3b which will give us 5ab plus this c, ja , plus … ja, ja, so the final answer will be 5ab plus c. … kalidheen: which ones are the like terms? musa: 2a plus 3b. they are like terms in, because a and b have terms in front of them and c does not. kalidheen: when you say term, what do you mean? musa: i mean the number. when dealing with subtraction, musa reasons differently. he is given 2b − a and says the answer is b (kalidheen, 2012, p. 48): musa: 2b minus a, 2b minus, ok, 2b minus a, which it stands for 1 right? so if a stands for 1, so just, you would be left with b. yes, you would be left with b because, a because a in maths, if we use it as an x, x stands for actually one, so if we actually took out the a-factor which is in this case, er the one, so it means you must take out the one from the 2b. when dealing with subtraction, musa reasons differently. he is given 2b − a and says the answer is b (kalidheen, 2012, p. 48): musa: 2b minus a, 2b minus, ok, 2b minus a, which it stands for 1 right? so if a stands for 1, so just, you would be left with b. yes, you would be left with b because, a because a in maths, if we use it as an x, x stands for actually one, so if we actually took out the a-factor which is in this case, er the one, so it means you must take out the one from the 2b. musa’s strategy for adding terms is to identify like terms although he does so based on an incorrect definition. when subtracting, he makes no mention of like terms. instead he assigns 1 to a and then focuses on the numbers to ‘take out one from the 2b’. here again we see a view of subtraction as take away. the different strategies for addition and subtraction suggest that he does not see similarities between items that add algebraic terms and those that subtract them. discussion the quantitative and qualitative analyses provide complementary insights into learners’ errors in basic algebra. from the quantitative evidence we see that conjoining errors become less frequent by grade 11 but they are still persistent. another persistent error is the focus on numbers and the lack of attention to operations. learners’ talk confirms that they are separating numbers from letters when dealing with expressions. errors related to the learning of new procedures appear to depend on when the new procedures are taught. for example the inappropriate application of the distributive law increased in grade 10 but dropped off by grade 11. although we cannot be certain, it appears that by grade 11 learners are familiar with when the distributive law can be applied which suggests they may be paying more attention to the operations and not simply to the presence of brackets. learners’ performance on items involving subtraction and negatives improved substantially from grade 10 to grade 11 but was still below performance on items without brackets and negatives. further research needs to focus on learners’ reasoning when dealing with subtraction and negatives both in number work and algebra. the interviews show evidence of learners talking of subtraction as ‘take away’, even when dealing with algebraic terms. this restricted view of subtraction may be a consequence of limited exposure to other subtraction structures such as reduction, comparison and inverse-of-addition (haylock, 2006) in number work. thus, despite evidence that arithmetic is an obstacle to learning algebra, there is need for placing greater attention on relationships between numeric quantities particularly in numeric subtraction scenarios. there is widespread evidence in the tests and interviews of learners evaluating letters rather than accepting expressions such as 8 + g as a final answer. the most common evaluation strategy was equal splits and it was persistent up to grade 11. this suggests that learners assume the letters must have values which may be a curriculum effect of the emphasis on equation solving. we have shown instances were learners are certain that the letter has a particular fixed value and other instances where learners recognise that the letter could take on several different values. these are both steps towards the notion of letter as variable but learners need to develop this more sophisticated interpretation of letters earlier in order to cope with algebra and function. the interview data provides evidence of learners incorrectly naming mathematical objects, for example referring to like terms as those with coefficients other than one, regardless of the letter. in other work emanating from the project we have argued for paying greater attention to the words learners use to name or refer to the mathematical symbols, expressions, graphs and for more opportunities for learners to talk mathematics (adler & ronda, 2015; adler & venkat, 2014). one of the most alarming observations each year was the high percentage of errors that were coded ‘other’, meaning that there was insufficient trend to add a particular code for the error. although the diversity of such errors reduced by grade 11, the apparent idiosyncrasy of the errors remains a cause for concern. however, the interviews across all postgraduate projects confirmed that learners are frequently able to justify their strategies and answers with some rational connection to the mathematics they have learned. this highlights the need to gain deeper insight into learners’ reasoning as they work with basic algebra and hence for teachers to provide more opportunity for learners to explain their strategies verbally. conclusions and implications in this article we have taken a first step in identifying and describing the algebraic errors that learners make as they move from grade 9 to grade 11. we have also provided some evidence of their reasoning in relation to these errors. this enables us to see what learners can and do do as they progress to higher grades, thus going beyond the general calls from grade 12 examiners for greater attention on the ‘basics’ of algebra. since this research provides insights into learners’ algebraic errors and their related thinking, it also provides a starting point to address the errors though attention to the underlying erroneous thinking rather than attempting to ‘fix’ the errors through reteaching. explanations for learners’ errors depend on theoretical orientations but there is agreement across perspectives that learners need to move from operational to structural ways of thinking about the symbols and their relationships. this will require deliberate instruction with tasks that push learners towards a structural view. furthermore, increased attention to learners’ ways of speaking about algebra will provide greater insight into their thinking. there is no ‘quick fix’ to deal with learners’ errors in algebra, particularly when one sees how some errors persist deep into senior secondary mathematics. however, the insights provided by the research reported here give some indication of useful starting points such as specific attention to the meaning of brackets and a stronger focus on negatives and different views of subtraction. acknowledgements this work is based on the research of the wits maths connect secondary project at the university of the witwatersrand, supported by the firstrand foundation mathematics education chairs initiative and the department of science and technology and administered by the national research foundation. any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the institutions named above. the authors wish to thank vasen pillay for the vital role he played as data manager in the project. competing interests the authors declare that they have no financial or personal relationship(s) that may have inappropriately influenced them in writing this article. authors’ contributions c.p. conducted data analysis and wrote the majority of the article; j.h. did rasch analysis, wrote sections of the article relating to csms design and the validation test for sa; y.s. re-transcribed interview data, checked accuracy of transcripts and conducted data analysis; j.a. was project leader and provided critical comments on drafts of the article. references adler, j., & ronda, e. 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(2007). applying the rasch model to psycho-social measurement: a practical approach. melbourne, australia: educational measurement solutions. footnotes 1. in south africa, all students are required to take mathematics for all 12 years of schooling. typically they will choose between mathematics or mathematical literacy at the end of grade 9. many learners change from mathematics to mathematical literacy in grade 10 or grade 11 if they are not coping with mathematics. 2. following linacre (2015), this analysis was re-performed after the data was cleaned to recode the most unexpected responses as missing. article information authors: satsope maoto1 kwena masha1 affiliations: 1department of mathematics, science and technology education, school of education, university of limpopo, south africa correspondence to: satsope maoto email: satsope.maoto@ul.ac.za postal address: private bag x1106, sovenga 0727, south africa dates: received: 01 mar. 2015 accepted: 11 sept. 2015 published: 20 nov. 2015 how to cite this article: maoto, s., & masha, k. (2015). towards an understanding of students’ thinking in learning new and unfamiliar concepts: focus on the factorial function. pythagoras, 36(2), art. #288, 10 pages. http://dx.doi.org/10.4102/pythagoras.v36i2.288 copyright notice: © 2015. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. towards an understanding of students’ thinking in learning new and unfamiliar concepts: focus on the factorial function in this original research... open access • abstract • introduction and research questions • theoretical and conceptual framework • research methodology and methods    • ethical considerations    • quality criteria • findings and discussion    • practical experiences, representations and counting or enumeration strategies       • response 1: 5 ways       • explanation 1:       • explanation 2:       • response 2: 20 ways       • response 3: 23 ways       • response 4: 25 ways       • explanation 1:       • explanation 2:       • reflecting on the students’ challenges in working from practical engagement    • textual experiences, interpretations and applications       • response 1: 120 ways       • explanation 2:       • explanation 3:       • explanation 4:       • response 2: 24 ways       • explanation 1:       • explanation 2:       • reflecting on textual experiences, interpretations and applications    • reflections on both practical and textual experiences in a pedagogical context       • insights into the depth and quality of students’ understanding of their solutions       • complex nature of pre-service teachers’ learning • conclusion    • challenges associated with engaging in a self-directed learning activity    • ways in which pedagogical attributes affect learning • acknowledgements    • competing interests    • authors’ contributions • references abstract top ↑ this study used participant observation to explore students’ thinking when learning the concept of factorial functions. first-year university students undertaking a mathematics methodology course were asked to find the number of ways in which five people could sit around a circular table with five seats. using grounded theory as a qualitative research strategy, we analysed student responses and written reflections according to the sequence of their experiential realities: practical and textual experiences. this was followed by an analysis of their reflections on both experiences in a pedagogical context. we found that the way basic mathematics operations are learned impacts on the student’s ability to experience components of new problems as familiar. consequently, they encounter these problems as new and unfamiliar. at the same time we found that engagement with practical experience does allow for the emergence of representations that have the potential to be used as foundations for learning new and unfamiliar concepts. the blending of practical, textual and teaching experiences provoked students’ thinking and ultimately their understanding of a given new and unfamiliar mathematics concept. introduction and research questions top ↑ this article reports on an exploration of first-year university students’ (pre-service teachers) thinking in solving a factorial function problem. the role of the teacher in the activity was limited to presenting the situation and introducing some resources at different stages of the process. the reason for this was twofold: to introduce students to a different teaching approach in which learning is at the foreground and to introduce them to ‘pedagogical content knowledge’ (shulman, 1986, p. 9). two research questions emerged during the study, namely: what are the challenges associated with engaging in a self-directed learning activity of a new and unfamiliar mathematics concept? in what ways do pedagogical attributes affect learning of a new and unfamiliar mathematics concept? the concept of the factorial function is associated with permutations, combinations, stirling numbers and number theory (bhargava, 2000). the factorial of a positive integer , denoted by , is the product of all positive integers less than or equal to . symbolically, the function is defined by: in the context of mathematics the definition and symbolic representations appear simplistic. learners know how to multiply integers and rules of permutations and combinations can be easily followed. when the situation manifests itself in a real-world context, learners often encounter challenges, in particular with how one perceives continuous multiplication in a real-world context. malisani and spagnolo (2009) attribute some of these challenges to the introduction of the concept of variables in the teaching of mathematics. they argue that ‘the concept of variable is used with different meanings in different situations’ (p. 21) and often unconsciously. this is a core problem in the growth of mathematics concepts. skemp (1978) vividly demonstrates this in his conception of instrumental and relational understanding. pirie and kieren (1994) explore the same idea in their notion of growth in mathematical understanding. the phenomenon is adequately addressed by tall (2008) in his characterisation of cognitive development through three worlds of mathematics. lack of systematic development from one area of concept development and representation to another, as discussed by these three authors, often results in learning problems. elia and spyrou (2006, p. 257) explicitly state that ‘the understanding of functions does not appear to be easy’. this is particularly so with the concept of factorial function as it requires the blending of different knowledge structures. functions, especially in their symbolic form, are generalisations of relationships. some studies, such as ones by amit and neria (2008) and tarlow (2008), have reported that the ability to generalise and represent relationships requires higher order thinking skills, such as visualisation, holistic thinking, flexibility, reasoning and abstraction. pure mathematics at university level tends to be taught within a formal framework of axiomatic systems and mathematical proof (tall, 2008). development of an understanding of mathematics concepts requires students to link ideas together for themselves and to take some responsibility for their own learning (maoto & wallace, 2006). thus, the role of a teacher shifts to a facilitative one to allow students time to develop mathematical ideas rather than to impose those ideas onto them. most research in relation to the teaching and learning of the factorial function is covered in comprehensive studies that focus on combinatorics (halani, 2013; lockwood, 2011). complex thinking processes are involved in learning combinatorics, including factorial functions. lockwood (2011) presents two perspectives of thinking that underpin learning combinatorics: the process-oriented perspective and the set-oriented perspective. meanwhile, halani (2013) classifies the thinking that is involved in learning the factorial function into eight different categories: addition, union, standard odometer, wacky odometer, generalised odometer, deletion, equivalence classes and ratio. other researchers (bintz & moore, 2003; braithwaite & goldstone, 2013) focus on the teaching approaches that facilitate the learning of factorial functions. bintz and moore (2003) use specifically constructed stories that they read to their middle school learners to introduce factorials. in the context of university students, braithwaite and goldstone (2013) support an approach where what they term grounded representations precede formal representations. the purpose of this literature review was not to identify gaps with respect to locating the study in literature. that would have defeated the grounded theory approach that we have adopted (bitsch, 2005). instead, we highlighted literature within which we could explain our emergent understandings of students’ learning of factorial function. theoretical and conceptual framework top ↑ our emerging conceptualisation of the framework that best accounts for how the study unfolded and was processed and reported upon is informed by three theories: constructivism as a referent for teaching (tobin & tippins, 1993), theory of reflective practice (schön, 1983) and grounded theory (glaser & strauss, 1967). constructivism, when used as a referent (tobin & tippins, 1993), demands that students’ experiential reality provides a foundation upon which their knowing is constructed. growth with regard to understanding of concepts is facilitated through a scaffolding process (vygotsky, 1962). schön’s (1983) idea of reflection-in-action is helpful with regard to how and when the knowledgeable other (vygotsky, 1962) can provide timeous scaffolding. the origins of grounded theory lie in glaser and strauss’s (1967) response to the dominant role theoretical frameworks have played in research, especially in the context of qualitative studies. according to charmaz (2008), grounded theory ‘takes a systematic inductive, comparative, and interactive approach to inquiry’ (p. 156). through this process ‘not only are the surprising data emergent, but the researcher’s theoretical treatment of them is also emergent’ (p. 157). our conceptualisation of the interrelationships of the three theories is captured in figure 1. figure 1: the emergent conceptual framework. in this model, the teaching and learning process unfolds first. the emerging data are analysed with the purpose of facilitating and supporting learning. the process is enriched by reflection-in-action. where necessary, intervention in the process comes in the form of scaffolding. the interplay between emerging data, the results of reflection-in-action and scaffolding activities continues until the outcomes of learning are achieved. subsequently, grounded theory allows us to process the whole learning experience anew. the process is now guided by the need to understand what happened, focusing on the research outcomes. research questions emerge and are pursued, guided by reflection-on-action (schön, 1983). the process allows for the interrogation of the value of the questions that emerged during the facilitation of learning. theoretical sampling, constant comparative analysis and emergence of categories or themes become realisable. research methodology and methods top ↑ this study was situated within the first author‘s own classroom, which consisted of 89 first-year university students who were registered for a mathematics methodology module. data gathered in a setting where one could actually talk directly to participants, and see them behave and act within their context and ultimately reflect on their own learning, is a major characteristic of qualitative research (creswell, 2007). following the qualitative research strategy of grounded theory, data collection, analysis and interpretation proceeded interdependently and iteratively and was not influenced by the literature (bitsch, 2005). data were mainly collected through submission of written responses, while participant observation during normal classroom interactions provided insights into the students’ responses. all 89 students worked in self-selected groups with a minimum of five members. the question given to them was: ‘in how many ways can five people sit around a circular table? investigate and thereafter explain how you will teach this to grade 9 learners’. through the provision of scaffolding (vygotsky, 1962), the students were expected to outline the key elements required to bring about an understanding of the activity and to clearly explain and justify their responses. on completion of their written responses, the students had to reflect on their use of mathematical strategies, mathematical processes and mathematical content, as well as reflect on the low and high moments they experienced. a day after their first submission they were given reference material which highlighted, in limited detail, linear and circular permutations, indicating symbolic generalisations of the two related functions. the intention was to scaffold the understanding and offer them an opportunity to attach meaning to, and to improve on, their initial explanations and justifications. it was also intended for the reference material to provoke their thinking towards generating functions for their different emerging sequence of numbers. the written responses of the groups were submitted in three stages. the first set of responses were submitted immediately after practical experiences, the second set was submitted immediately after textual experiences and the last set came after students’ reflections on both experiences in a pedagogical context. during whole-class discussions, guided by their submitted responses at each stage, individual groups were asked to confirm the different interpretations that existed within and across the groups. it was this alternating between data collection and analysis that influenced how we organised and interpreted their findings. the back and forth experiences and reflective interpretations by both students and us provided a rich platform to trace students’ thinking towards an understanding of the problem. to back up the findings and discussions we used the exact final responses of the students without disclosing their identities, as agreed upon. in between the students’ quotes, we highlighted emerging lines of investigations and ideas that influenced students’ conclusions. ethical considerations this qualitative study complied with all ethical requirements of the university. approval was obtained from the department in which the study was located, the students and the relevant university structures beyond the department. the nature and purpose of the study were declared, inclusive of potential audiences and substantive foci. erickson (1998, p. 1161) writes: ‘consent that is genuinely informed and without coercion reduces the risk of social harm because it affirms the dignity and respects the agency of those who will be involved in the study’. we agreed on anonymity of the participants, hence no individual identities were divulged during this study. the students themselves were beneficiaries of the results of this study. they were inducted into the dynamics of what it means to learn mathematics with a view to teaching young learners. quality criteria the prolonged engagement (semester), persistent observation, ongoing probing during a number of whole-class discussions, peer debriefing and member checks provided enough opportunity to hear the students’ voices, which contributed to establishing credibility of this study (bitsch, 2005; guba & lincoln, 1989). recursive submissions from the students and sufficient descriptive data added to both confirmability and transferability of this study (guba & lincoln, 1989). findings and discussion top ↑ the practical interaction by the students with the task presented them with a unique learning experience, including counting of actual seating positions, engagement with some of the theories of counting presented in the handouts and the need to approach the task from the perspective of a teacher. each of these stages provided us with a glimpse into the thinking that students applied as they learned mathematics in the context of becoming teachers. in order to preserve the order through which the experiences unfolded, this section is presented in the three stages as they emerged: (1) practical experiences, representations and counting or enumeration strategies, (2) textual experiences, interpretations and applications and (3) reflections on both practical and textual experiences in a pedagogical context. practical experiences, representations and counting or enumeration strategies the students’ engagement with the task involved talking about the task and using sketches on paper to determine the solution. some students created the actual space in which seating would take place and exchanged seats, while other students recorded or counted the number of possible seating permutations. lack of systematisation by students on how the activity was unfolding was apparent in the initial stage. however, over time, strategies became evident. four different answers emerged, namely (1) 5 ways, (2) 20 ways, (3) 23 ways and (4) 25 ways in which five people could sit around a circular table. in almost all instances it was evident that the students considered the representation to be linear rather than circular. each of these four different responses is examined in order of complexity. response 1: 5 ways two groups of students gave an answer of five possible permutations, with the following explanations: explanation 1: firstly, each occupies a seat around the table. then each moves to the next chair while the other one from right hand side moves to the chair left by the other. the one on the right hand side moves to make space for the other. at the end all five people would have occupied five chairs in different positions around that circular table. explanation 2: there are 5 ways: letters of the alphabets represent names of people the two strategies are clearly distinct and yet similar. in explanation 1 the focus is on each of the five people having an opportunity to occupy each of the five seats. the order in which they are seated remained static. however, it is still important that all of them are seated at the same time, though in different seats. practical observations were used to arrive at the number of times this was possible. explanation 2 reflected a different strategy, where the students used the last five letters of the alphabet to represent people. this offered some systematisation of the process which was then used to visualise how five people could change their seating arrangements. the first and the last letters of the alphabet were systematically exchanged, leading to a total of five possible permutations of the seating arrangement. the relative positions of letters remained the same. v would always be next to and to the left of w, except when one of them is the last and the other is the first letter in the sequence. in the context of a circular table, the order of these letters remained the same. this makes the explanation similar to the first. the difference between the two strategies is with regard to their potential in scaffolding towards the concept of factorial. the use of letters in explanation 2 allows for better manipulation. response 2: 20 ways the explanation of one group that arrived at this conclusion was: if the first person occupied a position, the others can sit around 4 remaining positions. we say 4 ways for 1 person if stationary, therefore 5 people each with 4 ways: 5 × 4 to get 20 ways. the strategy was to fix the first of the five positions while varying the remaining four; halani (2013) calls that standard odometer thinking. this seating permutation could be achieved in exactly five different ways as there were five people. once that was done, the remaining four positions produced four possible alternatives. while not explained by the students, this strategy of fixing of positions is similar to and enriches explanation 2 in response 1. this new thinking is an important aspect of the concept of factorial. furthermore, the idea of expressing the numerical representation of fixing and varying seating positions through multiplication makes this an invaluable development in working towards the concept of factorial. response 3: 23 ways this response showed a two pronged strategy in which two different representations were used to arrive at 23 combinations. however, a closer review of the processes followed indicated that this group made an error and, in fact, the calculation leads to a total of 20 instead of 23 permutations. the explanation was: 1st position 5 candidates; 2nd position 4 candidates; 3rd position 3 candidates; 4th position 2 candidates; 5th position 1 candidate; added to the sequential positioning of 5 equals to 23 ways. let’s say the candidates are named a; b; c; d; e sequential positioning equals 5 ways the 1st is now last and so on 1st position: 7 ways; 2nd position: 5 ways; 3rd position: 3 ways; 4th position: 2 ways; 5th position: 1 way therefore 5 + 7 + 5 + 3 + 2 + 1 = 23 ways the first strategy used considers the possibilities for each of the five seating positions. while this is still a feature of halani’s (2013) odometer thinking, these positions were counted independently. the students did not consider the strings of seating arrangements that could arise from the approach. the subsequent possibilities of occupying a seat were not taken as linked to the previous possibilities. it is the absence of this connection that, we believe, makes the strategy ineffective. the second strategy was to use letters of the alphabet to represent different strings of seating arrangements. the sequence of the letters was maintained, unless subsequent letters were at the opposite ends of the string. the two strategies yielded 15 and 5 ways respectively leading to a total of 20 permutations. the 7 ways for the first position and the 5 for the second appear to be mistakes in the students’ work. initially, the numbers were correctly given as 5 and 4. in terms of the ultimate purpose of engaging with the concept of factorial, the two strategies can be scaffolded for a successful count. the first strategy was, however, closer to the numerical and symbolic representation of the concept of factorial than the pictorial representation. response 4: 25 ways two groups arrived at this conclusion using two distinct processes. explanation 1: each person shifted 5 times; 5 × 5 = 25. by changing positions of 5 people until the first one occupies the first position again. learner a sits in 5 times learner b sits in 5 times learner c sits in 5 times learner d sits in 5 times learner e sits in 5 times 5 + 5 + 5 + 5 + 5 = 25 explanation 2: according to statistics there is only one way 5 people can sit around a circular table. the first person can occupy any of the five positions. the same could happen to the remaining four people. in this manner all five people will occupy 5 different positions, which means 5 people, 5 positions each. we multiply 5 by 5 to obtain 25 (5 × 5 = 25) ways. explanation 1 suggests that the learners were counted separately and not in their strings of seating arrangements. each of the five learners had an opportunity to occupy any of the five seats, translating into a total of 25 possibilities. the potential to scaffold this into the concept of factorial is very low as there are numerous issues (structure, permutations) that need to be addressed. explanation 2 deploys an external tool: statistics. the explanation did not make it clear whether all the seats were occupied simultaneously or one at a time. reflecting on the students’ challenges in working from practical engagement the students’ responses to the assigned task reveal two key challenges associated with dealing with mathematical concepts encountered in real-life situations. the first challenge is that students struggle with representing the problem in a way that makes it accessible mathematically. each group had no problem in realising that they were dealing with a counting activity as reflected in the four conclusions presented. initially they thought that role-playing the situation would enable them to observe and better understand the given factorial function problem. lack of systematisation made this option less feasible, leading to its abandonment. acting out as a form of direct representation proved ineffective as people proved hard to manipulate (as learning aids). this led to the use of letters of alphabet as symbolic representations of people. letters proved easier to manipulate, as evident in almost all the responses. however, the choice of letters imposed yet another restriction. in all the cases, either the first or the last five subsequent letters were chosen and the order was rigidly adhered to and maintained. the order of taking the first letter to the last resulted in a different seating arrangement only if one deals with linear arrangement. in a circular arrangement, one ends up with the same seating arrangement. the second challenge was the problem of how multiplication manifests itself in real-life situations. in real life, multiplication is encountered as repeated addition of the same quantity. thus we do not look for multiplication per se but for the same quantity added a number of times. textual experiences, interpretations and applications at this stage, students were given a handout explaining the factorial concept and differences between linear and circular arrangements. furthermore, they were advised to make use of internet and library services to find more information. this exposure alerted students to a different way of learning mathematics – the use of literature (bintz & moore, 2003) rather than a prescribed textbook. this offered students an opportunity to review their earlier submissions. the second set of students’ written responses followed this intervention. two different answers were arrived at: 120 and 24 permutations. response 1: 120 ways seven groups arrived at 120 seating permutations. their procedures, interpretations and justifications fall into three different categories. the first category comprises students who used colours. explanation 1: use of colours to represent the five people. it is a probability problem, which needs possible outcomes. …we first took 2 people and sit them opposite to each other and not allowing them to change positions. for example we name our 5 people, green, blue, red, yellow and white then sit them such that red is opposite green. make green and red not moving and let the others change positions. for example: red, green, blue, white, yellow red, green, blue, yellow, white red, green, yellow, blue, white … there are six possible arrangements. if we fix the position of red and replace green, we also find another six possibilities. if we continue by not altering the positions of red and changing the ones that sit opposite red, we found 24 outcomes (6 × 4). if we change the position of red and replace it by each of the other colours at a time, we will have another 24 arrangements different to each colour and we multiply the 24 different arrangement by the number of colours we found a total of 120 arrangement (5 × 24). initially, the students classified the activity as a probability problem. however, that did not influence what followed. instead, the students developed a systematic way of representing the different arrangements. the technique involved fixing some positions while exchanging the others. this improved the clarity of their observations and they thus appeared to have extended standard odometer thinking to wacky odometer thinking (halani, 2013). ultimately, they arrived at 24 arrangements for each colour occupying the first position resulting in 5 × 24 = 120 different possible combinations. the use of colours resolved the rigidity of linearisation inherent in the use of letters. the second category comprised students who used the factorial formula without justifying its link to the problem at hand. the distinction between linearity and circularity was not addressed. explanation 2: say that people are seated as person a; b; c; d and e. such an order can be represented in a linear list as abcde. with the use of this notation, we can count the number of possible lists. there are: 5 possible choices for the first spot (a, b, c, d or e) 4 choices for the second 3 choices for the third 2 choices for the fourth 1 choice for the fifth spot. that is, there 5× 4× 3× 2× 1=120 ways. the list can be written as: abcde; abced; abecd; abdec; ebdca; edbca; etc. therefore, there are 120 ways. the five people keep changing seats one at a time in such a way that none of them repeats to have the same neighbour. a string of five letters was used as a way of representing people. however, the rigidity of the order of the letters was rescinded. having explored the possibilities of occupying each of the positions, the group then used multiplication to arrive at 120. the transition from choices for each seat to multiplication is not clear. this transition makes their explanation similar to those in which a formula was used. the third category comprised explanations that provided some link between representation using letters and factorial function. the students first made use of formulae to calculate the number of permutations, they then use actual seating, represented by letters, to confirm their findings. explanation 3: 5!=5× 4× 3× 2× 1=120 therefore, there are 120 ways since these people are to sit in different ways, this implies repetition is not allowed and the order at which these people are arranged is important. the number of permutations of ׳n׳ different people taken at a time is as follows: first mark the sitting positions as 1; 2; 3; 4; 5. note that the positions must be at equal distance from each other to avoid conflict. name the five people as a; b; c; d and e. sit one person at a time. since there are five people, the first person, suppose is ‘a’ can sit in 5 ways. the second person ‘b’ from the four remaining can sit in ‘4’ ways as the fifth position is occupied by ‘a’. the third person ‘c’ will sit in 3 ways and the fourth will sit in 2 ways. finally the fifth person ‘e’ will sit in only one way. … this can be: abcde abecd abdce acedb aebdc acdeb acdbe abedc adbec aecdb adebc adbce acbde adcbe aedbc aebcd aebcd acbed adceb aedcb abdec abced acebd adecb this arrangement gives 24 positions if a is fixed in the 1st position. another 24 when a is fixed in the 2nd position and so on resulting in 24+24+24+24+24=120 ways explanation 4: there are 120 ways 1!=1 2!=2× 1=2 3!=3× 2× 1=6 4!= 4× 3× 2× 1=24 5!= 5× 4× 3× 2× 1=120 for 5 people we have 5!=5× 4× 3× 2× 1=120 ∴ 120 ways we indicated person number 1 as a; 2 as b; 3 as c; 4 as d and 5 as e no. 1 2 3 4 5 1 abcde bacde cabde dabce eabcd 2 abced baced cabed dabec eabdc 3 abdce badce cadbe dacbe eacbd ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 24 aedcb bedca cedba decba edcba each person can sit in 24 ways and there are 5 people. ∴ 24× 5=120 in explanation 3, the permutation strategy is used. the strategy was confirmed by showing what all arrangements that begin with letter a would look like. the point was made by the students that, since there are five different letters, a conclusion of 5 × 24 = 120 makes sense. in explanation 4, the notion of the factorial function is extrapolated from a simple situation to the one where there are five people. this was followed by the actual representation of those ways using letters. while conclusions are simultaneously drawn from two contexts, the link between those contexts remains obscure. the notion of a factorial as a product of all positive integers equal to and less than the given number is not matched by the representations that students used. two systems that are not necessarily aligned have now taken hold in students’ engagement with the task. response 2: 24 ways in this case groups resorted to strong algorithmic procedures where numbers were processed without any justification given for the action taken. in all instances, the first part of the strategy was to calculate the number of possible arrangements, assuming that they were in a linear form. a total of 120 permutations was arrived at. the second part involved division by 5, arguing that the circular arrangement warranted division. the rationale for division by 5 was not explained. however, two different categories could still be observed: (1) abrupt transition without justifying the transition from linear to circular and (2) a fairly appropriate justification that showed only language as a challenge, not the understanding of the concept. in the first category, students resorted to strict adherence to the formula found in the text. the linear arrangement was used to find the total number of possible outcomes. division was then used to arrive at the arrangements for a circular setting. a typical response is seen in explanation 1. explanation 1: five people have 5 factorials in a linear permutation. the linear permutation is found when multiplying the factorials of five: 5× 4× 3× 2× 1=120. to convert it to circular permutation you have to divide the linear permutation by the number of people, 120÷ 5=24 the formula can be defined as the rationale for division by the number of seats as they convert from the linear to circular permutations was not given. the way the information was presented suggests that the action was undertaken because the formula suggested it. in the second category, students started the same way as in the first. however, the impact of a circular arrangement on the distinction of these arrangements is recognised and acknowledged. explanation 2: a round table differs with a normal table in that it has no ends. therefore, you can arbitrarily select seat #1. you can fix one person in place (or, equivalently, rotate the table like a carousel so that that person always winds up in the same place). this will avoid multiple equivalent scenarios in which the people are seated in the same order, but have just shuffled a seat or two over (these don’t count as different arrangements). then you have free rein to arrange the other four people, so that’s 4!=24 arrangements. alternatively, you can figure out that there are 5!=120 arrangements overall. however, each unique arrangement is actually repeated five times because there are five seats around the table, there are 5 different versions of every possible seating arrangement. (for instance, abcde, bcdea, cdeab, deabc, eabcd are all equivalent). so this means you must divide by 5: . the case for repeating arrangements was adequately made and demonstrated in the response. the conclusion that ‘each different representation is actually represented 5 times’ is critical to the ultimate formula for calculating the circular arrangement. the only uncertainty was the connection between the repeating arrangements and the use of division to address that. can one say that the students had consciously used the division by 5 to counter the five repetitions? more explanation would have provided a better perspective in this regard. later on in this article we show how these kinds of gaps translate into challenges when the solutions are to be explained in a pedagogic situation. reflecting on textual experiences, interpretations and applications the students’ engagement with text reveals two insights into their learning, especially in mathematics: the authoritative role that text plays and the gap between real life and mathematical representations. authoritative role of text in students’ learning: upon receiving the text, the students focused on how a number of arrangements could be calculated given the number of objects. since the text first addressed arrangements in a linear setting and later those in a circular setting, students used exactly the same approach in responding to their assigned task. in particular, the focus was on the formulae given in the text. this appeared to have obstructed their ‘mental flexibility in switching from one solution method to another’ (amit & neria, 2008, p. 124). for the majority of students, identification of the formula was an end in itself. we saw this in students who arrived at the response of 120. immediately after identifying the formula for n! they plugged it in and stopped. nothing much was done in terms of how well the formula represented the task at hand. the same challenge was observed with those students who arrived at an answer of 24. while this group could clearly differentiate between linear and circular arrangements, it is the transition between the two that reflects the authoritative power of the text. the formula was simply accepted as appropriate for circular arrangements without much effort to justify it. the problem with this approach is that it relinquishes the responsibility for learning to the text and, by implication, to the author. it is the author’s responsibility to explain why a particular idea works and how it works in reality. the gap between real life and mathematical representations: how does division manifest itself in the real world and can it be observed? this seems to be a major obstacle in how the students moved from linear to circular arrangements. what is in the real world? saying the statement is about 120 divided by 5 does not say much more than how the statement reads. however, if one sees 120 divided into fives (groups of 5) or 120 divided into five groups, or repeated subtraction of 5 from 120, then one has more chance of observing the phenomenon in reality. limited use of varied verbal representations, among other things, denies our students the richness of observing and engaging with mathematical objects in real life. as argued by elia and spyrou (2006), mathematics teaching in schools focuses on the use of algebraic representations of functions, thus hindering the application of functions in other representational modes. reflections on both practical and textual experiences in a pedagogical context this section analyses students’ responses to the question of how they would teach the factorial function to grade 9 learners. the students were asked to reflect on their own experiences in solving the problem at hand. this new perspective prompted varied responses that (1) gave us insights into the depth and quality of students’ understanding of their solutions and (2) revealed complex nature of pre-service teachers’ learning. insights into the depth and quality of students’ understanding of their solutions the students’ struggles with the problem evolved from managing their observations and making sense of text or seeing connections with real life, to expressing their newly constructed knowledge at a basic level that would make grade 9 learners understand the problem. almost all the groups encountered a challenge regarding how to systematically observe and record the different seating arrangements, for example: we decided to do it practically by sitting in a group of five people whereby we were changing our sitting positions. but along the practical route, we got confused and dropped it. our frustration is that we are failing to arrange those five people around the table and how will they circulate on that particular table. we find it difficult to determine whether those five people will exchange their seats or just move from one seat to another because that table is circular. it is this basic challenge of systematisation that later translated into a bigger problem when students were asked to think about how they would teach the concept to a grade 9 class. the enactment, as they exchanged seats, was disorderly, making it difficult to keep track of which arrangement had been enacted and which had not. the reflections suggest that, at that stage, the students had limited understanding of the problem. however, the introduction of symbolic representation of people through the use of letters and colours allowed the arrangements to be manipulated on paper. still, the power of the manipulation on paper was not utilised to understand the actual arrangements in reality. instead, the students were satisfied to arrive at the required number of arrangements. the continuous interaction between mathematical and real-life representations plays a pivotal role in deepening knowledge construction. it allows one to value the emerging constructs and its absence often leads to frustrations as is evident in the following reflections: we were frustrated more when someone had the answer but couldn’t explain to us how it could be done by illustrating practically so, by seating on a circular table and changing our seating positions now and again until we get the suitable answer that we had by using the permutation formula. my frustration is that i didn’t understand the formula of permutations which i have obtained from internet because if i use this formula to calculate i get 120 ways but if i try to do it practically it becomes impossible. i don’t understand where the formula comes from and how they derived it. knowledge, by its nature, enriches one’s sense of seeing. if new knowledge does not enhance one’s broadened vision then there is a challenge. the students’ discovery of the formula in this instance led to frustration instead of a sigh of relief – the ‘aha’ moment. their inability to relate the textual and the practical experiences was apparent. crusius (1991, p. 38), using gadamer’s fusion of horizon, would regard this as a failure of ‘an event of truth, a revealing-concealing that goes beyond the spontaneous, unscrutinized projections of preunderstanding’. the problem provides insights into the students’ enculturation (bishop, 1991) in problem-solving. we argue that the challenges encountered are not necessarily limited to the current task but reflect general challenges the students face in dealing with authentic problems in mathematics. it is more of a skill problem than a knowledge one. the way mathematics is traditionally taught in schools does not encourage the ‘seeing’: i don’t know why we were multiplying instead of adding. … we didn’t know whether we had to add or multiply, but the formula illustrated that we should multiply but still we were more confused because we could not find a suitable answer. in real-life situations, multiplication manifests itself as repeated addition of the same quantity. in the students’ eyes it is addition that is observed. how that translates into multiplication frustrated them. in the absence of reciprocal translation skills between the two contexts, concepts could be playing themselves out without students noticing. complex nature of pre-service teachers’ learning learning for the sake of solving a mathematical problem for oneself is different from learning with the purpose of explaining the concept to others. in a context where that ‘other’ involves learners with varying mathematical needs, the demand becomes one where one has to think of a multiplicity of representations at that elementary level. learning mathematics for others is aligned to shulman’s (1986, p. 9) pedagogical content knowledge, which includes ‘the ways of representing and formulating the subject that make it comprehensible to others’. for this to happen one must be fairly grounded in the concept. the students’ reflections in explaining how they would teach factorial function to grade 9 learners revealed varying degrees of discomfort with their own understanding. from the first set of reflections it was clear that their understanding of the concept lacked depth, for example: we could not find a suitable answer and we would not know how to explain it to grade 9s as we don’t know how to explain it between each other. we were not happy the time when we were not able to find the solution … and we were still confused on what to … teach to grade 9 learners. it was difficult to reach the answer because we were applying many ways in fact it was really confusing. with pre-service teacher’s learning, it is important not just to find a solution, but to explain it to oneself or peers, even before thinking of explaining it to others. while in two instances it was obvious that students still had to attend to their own needs, in the next set of reflections it became evident that resolving the problem at their own level was not enough, for example: we used the formula which is n!. then we used our calculators to find the answer which is 120. we then tried to make it easy to a grade 9 learner to understand it. after we had submitted our assignment, the lecturer gave us pamphlets that explain permutation better and they made us to understand better than before. but still the challenge was how we are going to teach that to the grade 9 learners. initially, students went about the task using their own shallow understanding of the problem. the latter part of their reflections was more revealing. the pamphlets helped them understand the problem better than before. however, the challenge of teaching grade 9 learners became more evident. this means that, while the students were now in a position to deal with the problem at their own level, they could not visualise how the same could be done at a lower level. conclusion top ↑ in this article we pursued first-year university students’ thinking in solving a real-world problem in a context of minimal teacher interference. students were exposed to three interactive phases, namely practical experience, textual experience and reflection-on-teaching experience. during practical experience, which braithwaite and goldstone (2013) call grounded representations, students were offered an opportunity to see more, to independently manipulate, interpret, reflect in and on action, and to construct visual representations of the problem at hand. during textual experiences, called formal representations by braithwaite and goldstone, they engaged with new symbols, calculations and representations. during reflection-on-teaching experience the students were forced to rethink what their new knowledge meant at a conceptually lower level. drawing from these three interactive phases, we organise our conclusion guided by our research questions. challenges associated with engaging in a self-directed learning activity all the groups experienced varying needs for systematisation in engaging with the problem at hand. this is in line with the findings of english (2005) and melusova and sunderlík (2014). having failed to act out different seating arrangements, symbols were employed, resulting in improved representations. the inherent order of the symbols unfortunately added limitations to representing different arrangements. while manipulatives do facilitate learning (abramovich & pieper, 1995), we observed that their inherent features sometimes limit their benefits. the nature of prior learning also plays a significant role in engaging with combinatorics thinking (lockwood, 2011; melusova & vidermanov, 2015) and we have similar findings. while in real life the students could observe that they were expected to add, the textual experience was encountered in the form of multiplication and division. the two experiential domains could not be reconciled. the students’ prior learning of multiplication and addition was not always reconciliatory especially in the context of real-life experience. we thus conclude that the nature of prior learning, its richness with respect to its representations, is necessary in learning new and unfamiliar mathematics concepts. in this way prior learning facilitates folding back (pirie & kieren, 1994). all the responses, no matter how inconceivable they might have been, offered opportunities for construction of the concept of a factorial function. the mathematical representations used, the struggles with counting different seating arrangements and so on gave us primitive forms of the concept to work with. that is, no matter how unfamiliar the concept is, there is always an opportunity to find a possible baseline to work from, especially when a real-life experience forms part of the engagement. ways in which pedagogical attributes affect learning the question that asked students to reflect on how they would teach the factorial function to grade 9 learners evoked different orientations to those they used when engaging with the problem. the solutions that were considered adequate no longer held when subjected to the intensity of rethinking them in the context of grade 9 learners’ capacity to learn. eventually, it is the viability of their newly found knowledge that is questioned. we observed that the textual experience allowed the students to adopt a formal representation of the factorial function without establishing images of the concept as it manifests itself in the experiential practical world. the knowledge that learners need seems to reside in this gap and the students have limited access to it. knowledge for teaching requires co-evolution along with content knowledge. it is hard to learn mathematics content first and only thereafter think of how to teach that content at lower levels, or at the same level for that matter. the rich, meaningful and conscious evolution of concept formation allows students multiple perspectives from which they can engage with learners at various stages of learning. any struggle a student encounters in figuring out how to help learners learn a particular concept is a sign that there are gaps in the student’s learning of the concept itself. mathematics content knowledge that is approached from a pedagogical context is process-rich and multi-representational, and so poised to serve learning of new and unfamiliar concepts. in this orientation, pre-service teachers’ learning is more engaging than is traditional mainstream mathematics content learning. melusova and sunderlík (2014) arrived at the same conclusion when they investigated pre-service teachers’ problem-solving processes in combinatorics. we observed that when the students had to solve the problem for themselves, they hurried through some aspects of their learning, focusing more on the actual solution. they were more product-driven than process-focused. that way, learning outcomes were limited in their usefulness. we are, therefore, persuaded that pedagogical attributes, when meaningfully incorporated into the learning of new and unfamiliar mathematics concepts, enrich learning and orient knowledge to productively service future learning. acknowledgements top ↑ competing interests the authors declare that they have no financial or personal relationships that may have inappropriately influenced them in writing this article. authors’ contributions s.m. 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(1962). thought and language. cambridge, ma: mit press. microsoft word 53-64 atebe schafer.docx pythagoras, 71, 53-64 (july 2010) 53 beyond teaching language: towards terminological primacy  in learners’ geometric conceptualisation    humphrey uyouyo atebe & marc schäfer  education department  rhodes university  humphrey.atebe@wits.ac.za & m.schafer@ru.ac.za  this paper reports on a specific aspect of a broader geometry conceptualisation study that  sought  to explore and explicate  learners’ knowledge of basic geometric  terminology  in  selected nigerian and south african high schools. it is framed by the notion that students’  acquisition of the correct terminology in school geometry is important for their success in  the subject. the original study  further aimed  to determine  the relationship  that might  exist between a learner’s ability in verbal geometry terminology tasks and his/her ability  in visual geometry terminology tasks. a total of 144 learners (72 each from south africa  and  nigeria)  were  selected  for  the  study,  using  both  the  stratified  and  the  fish‐bowl  sampling techniques. a questionnaire consisting of a sixty‐item multiple‐choice objective  test provided the data for the study. an overall percentage mean score of 44,17% obtained  in  the  test  indicated  that  learners  in  this study had only a  limited knowledge of basic  geometric  terminology.  the  nigerian  subsample  in  the  study  had  a  weaker  understanding  of  basic  geometric  terminology  than  their  south  african  counterparts.  importantly, there were high positive correlations between participants’ ability in verbal  geometry terminology tasks and their ability in visual geometry terminology tasks. these  results are consistent with those of several earlier studies, and provide a reasonably firm  basis for certain recommendations to be made.      language is undoubtedly an essential tool in communication, and perhaps geometry stresses the use of language more than any other part of a mathematics curriculum (ashfield & prestage, 2006; hoffer, 1981). research on language, particularly, in africa where most formal education is conducted in a teaching language (such as english or french) usually alien to learners and teachers, has over the years focussed on learners’ difficulties with the teaching language and the need to make them proficient in it (oyoo, 2009). the assumption here, at least in the main, is that once learners have attained an acceptable level of proficiency in the teaching language (semantic understanding), then their understanding of the subject matter (conceptual/mathematical understanding) will naturally follow. a critical analysis of findings from studies on the educative efficacy of teaching languages, however, indicates that learners who have achieved a commendable level of “proficiency in the language of teaching have often been found unable to follow classroom discussions [even] with ‘good’ teachers” of the subject (von glaserfield, as cited in oyoo, 2009, p. 197). the problem of comprehension tends to arise when both the learner and the teacher know the meaning of a word at a basic semantic level and therefore assumes that the word’s technical or conceptual meaning is equally shared. this suggests that during teaching, the teacher uses certain words in a sense peculiar to the subject that is not precisely understood by the learners. while learners’ proficiency in the teaching language is important for social interaction in the classroom, learning mathematics generally, and geometry specifically, involves more towards terminological primacy in learners’ geometric conceptualisation 54 than just social interaction in the classroom. learners need to acquire the correct technical terms and be able to use them clearly to communicate their ideas about concepts in geometry. premised upon this statement, the present study was undertaken to explore learners’ knowledge of basic terminology in plane geometry in selected nigerian and south african high schools. the authors acknowledge that language proficiency amongst learners cannot be assumed to be constant within schools and across schools. nevertheless, for the purposes of this paper we assumed a common basic level of proficiency in english, the language of teaching in the participating schools. rationale and significance the point that has been stressed by the south african national curriculum statement concerns the need to promote learners’ “competence in mathematical process skills such as investigating, generalising and proving” (department of education, 2003, p. 9), a competence vital for their overall mastery of mathematics. an emerging argument from research is that learners’ ability to classify geometrical objects in terms of their properties facilitates their acquisition of these mathematical process skills (roberts, 2010), and few would argue that knowledge of the correct technical language plays an important role in the classification process. as indicated, this study is underpinned by the notion that students’ acquisition of the correct terminology in school geometry is important for their success in the subject. although it has long been recognised that “the most basic type of knowledge in any particular field is its terminology” (bloom, 1956, p. 63), it is only recently that this notion has begun to attract empirical attention in geometry education. recent research has indicated that a lack of language competency impedes progress in geometric understanding (feza & webb, 2005). de villiers (as cited in feza & webb, 2005, p. 45), for example, stresses the point that “success in geometry [indeed] involves acquisition of the technical terminology”. all too often, however, students lack the appropriate vocabulary to express the distinguishing properties of a figure or compare shapes in an orderly manner (renne, 2004). the learners’ repertoire of geometric terminology that we explore in this study is not synonymous with and does not replicate that investigated in any earlier studies: it draws upon their findings but breaks new ground in the field of enquiry. students’ difficulty with, and misconceptions in geometry can, in part, be linked to their “lack of exposure to proper [geometric] vocabulary” (oberdof & taylor-cox, 1999, p. 340) in the subject (mji & makgato, 2006). it is thus hoped that this study may be able to indicate to teachers in nigeria and south africa (and elsewhere) possible reasons for the misunderstandings and difficulties that plague discussions of geometry in the classroom. insight of this nature is important for remedial teaching design and delivery. furthermore, the determination and subsequent analysis of the correlations between a student’s ability in verbal geometry terminology tasks and his/her ability in visual geometry terminology tasks, as exemplified by students’ scores in the test instrument used in this study, are of particular significance to both prospective and practising teachers. while it is important to teach the terminology associated with a given content area in geometry, it is equally important for teachers to ensure that a learner who can give a correct verbal description of a geometric concept also has the correct concept image associated with that concept, and vice versa. this is vital to assessment of learners’ conceptual understanding in this learning area (hiebert & lefevre, 1986). this paper reports on a specific aspect of a broader study (atebe, 2008), and extends a previous preliminary report (atebe & schafer, 2010) by offering much richer data, in particular a concept-byconcept analysis of learners’ performance, grade-level analysis and interpretations. for more detail concerning the current study and its significance, refer to atebe (2008). theory and related research in 1957, a husband-and-wife team of dutch mathematics educators, pierre van hiele and dina van hielegeldof, completed companion doctoral dissertations at the university of utrecht that focussed on the difficulties that school children experience in geometry and how they can best be assisted through teaching. the van hieles asserted that students pass through several levels of reasoning in their development of geometric ideas (clements, 2004; pegg, 1995). they developed a theory of levels of humphrey atebe & marc schäfer 55 thought in geometry, now known as the van hiele theory, which suggests that students pass through a sequence of five hierarchical levels of geometric thinking as they progress from merely recognising geometric shapes to being able to construct a formal geometric proof (atebe & schafer, 2008; clements, 2004; teppo, 1991; van hiele, 1986, 1999). the theory also offers a model of teaching to promote learners’ levels of understanding in geometry (fuys, geddes, & tischler, 1988; van hiele, 1986). this implies that there are at least two aspects to the van hiele theory: the thought levels and the learning phases. the van hiele levels are: recognition, description, ordering, deduction and rigour (see hoffer, 1981 for details), while the van hiele learning phases are: information, guided orientation, explicitation, free orientation, and integration (see atebe, 2008 for details). various aspects of the van hiele theory have been explored in previous publications. for example, usiskin (1982) and atebe (2008) offer comprehensive reports on the van hiele theory as a whole; atebe and schafer (2008) explicate the thought levels in some detail, and atebe and schafer (2009) focus in depth on the teaching phases of the van hiele theory. in this paper, therefore, relatively cursory treatment is accorded the van hiele thought levels and learning phases. nevertheless, since the properties of the van hiele levels – the very aspect of the van hiele theory with reference to which the results of this study were analysed and interpreted – have hitherto received little scholarly attention, they remain the main focus of this literature review. properties of the van hiele levels the van hieles made certain observations about the nature of the levels of thinking in geometry and their relationship with geometry classroom teaching. in order to understand how an individual student progresses from one van hiele level to the next, early van hiele researchers like usiskin (1982), fuys et al. (1988) and of course, the van hieles themselves, identified the following properties as pertaining to the levels: 1) fixed sequence/hierarchy, 2) linguistic character, 3) adjacency, 4) discontinuity, 5) retention and 6) ascendancy/progress. only property 2 – the linguistic character of the van hiele levels – is elaborated on in this paper (see usiskin, 1982; atebe, 2008 for discussion of the others). linguistic character of the van hiele levels an important consequence of the van hiele theory is its pedagogical relevance to teaching in geometry. each of the levels is characterised by its own linguistic symbols, terminology and network of relations. people reasoning at different levels speak different languages and the same term is interpreted differently. in a classroom situation, for example, one might find the teacher, the texts, and the students functioning at different levels and hence using different linguistic symbols or networks of relations. consequently, the students and the teacher do not understand each other (mason, 1998; van hiele, 1986). the mismatch between teaching and students’ cognitive levels in geometry is caused largely by teachers’ failure to deliver teaching to the pupils in a language that is terminologically appropriate to the students’ thinking levels (oyoo, 2009; van hiele, 1986). van hiele (1986) sees this property as the most critical in the learning process: in education, teachers often give their students unsolvable problems. they use the language of the third level and the pupils often are not even able to use the language of the second level. sometimes the pupils have not even formed a language of the first level that accompanies the visual structure. (p. 90) the result of such teaching is that the learners are obliged to imitate, but without understanding, the action structure of the teacher. van hiele (1986) therefore proposes that “a teacher beginning the teaching of geometry should address himself to the pupils in a language they understand” (p. 45). by this van hiele means that teachers should use level-appropriate terminology, symbols, and general language in their geometry teaching practices. in line with this view, the third of the van hiele learning phase descriptors as formulated by atebe (2008, pp. 72, 286) recommends that “teachers [should] ask questions that seek to clarify [their] students’ imprecise terminology and gradually introduce formal mathematical terms”. hoffer (1981, p. 12) similarly suggests that precise terminology should “be thrust on students” early in their geometry course in order to remediate their use of imprecise geometric terminology, which is often attributable to misconceptions that they hold. towards terminological primacy in learners’ geometric conceptualisation 56 given the foregoing theoretical perspectives on geometric terminology and students’ acquisition of geometric ideas, the determination of the repertoire of geometric terminology at the disposal of high school learners (in nigeria and south africa) for possible remedial teaching design and delivery cannot be taken for granted. the teaching of basic geometric terminology should be given pride of place in the teaching of high school geometry. the research goals this study sought to explore and explicate high school learners’ knowledge of basic technical terms in geometry in selected nigerian and south african schools. the study further sought to determine what relationship might exist between a learner’s ability in verbal geometry terminology tasks and his/her ability in visual geometry terminology tasks. method the research design is that of a collective case study (stake, 2000), focusing on a total of 144 mathematics learners drawn from two high schools in nigeria and south africa – one school from each country. as a research approach, a case study typically investigates the “particularity and uniqueness of a single case, coming to understand its activity within important circumstances” (stake, 1995, p. xi). however, stenhouse (as cited in schäfer, 2003, p. 49) points out that in the case of “educational case studies”, collective case study approaches – such as that employed in this study – are increasingly being used. the sample and sampling procedures the sample comprised a total of 144 nigerian and south african high school learners with a mean age of 16,6 years. of the 144 learners, 72 were drawn from a state-owned high school in ojo local education district in lagos state, nigeria, and the other 72 from a ‘township’ high school in makana educational district in eastern cape province, south africa. in each of these schools, 24 learners from each of grades 10, 11 and 12 were selected for the study using stratified and fish-bowl sampling techniques. given the limitation of involving only one school from each of the participating countries, the choice of which schools to study was a critical factor. although geographical accessibility and proximity, functionality, and gender composition (single-gender or co-educational) were some of the important factors that influenced our choice of school, the primary criterion was their easy accessibility to the majority of nigerian and south african learners because of the relatively low cost of attending them. for more comprehensive information concerning the sample and the sampling procedures, see atebe (2008). collection of data data were collected for this study mainly through the construction and administration of a structured questionnaire which was issued to the learners in the form of a pen-and-paper test. the questionnaire consisted of a sixty-item multiple-choice objective test referred to as the terminology in plane geometry test (tpgt). each question was followed by a list of four options (or foils) from which the learners were expected to choose the correct answer. choppin (1988, p. 354) asserts that the objective test, as a means of data collection in educational research, is structured in such a way that “the testees must choose their answers from a specific list of alternatives rather than creating them for themselves”. anderson (1990) similarly states that distinct choices in a questionnaire eliminate possible ambiguity in the responses of the research participants and facilitate a very precise form of data analysis. for the tpgt, two conceptually identical but structurally different sets of questions were constructed for each terminology. that is, for every question that was presented in a verbal form (i.e. without diagrams), there was a corresponding one presented in a visual form (i.e. presented diagrammatically). all the items in the test were then juggled, so that each member of every homologous (i.e. identical) pair of questions was sepa numbere with the whether visual (o below, e question its visua question for the t were exa and 3) li arated from t ed items) an goal of this a student w or conceptual exemplify a p n 5: what i a. a b. a c. a d. a al image hom n 24: in the a. a b. d c. c d. o tpgt, a tota amined. the nes and angl f circle radius chord diameter tangent arc sector cyclic quad concentric the other. he d a correspo study, the p who can give l) image asso pair of homo is the name o a tangent a radius an arc a diameter mologous pair diagram, o aob db cod ob al of 30 term e concepts to les. figure 1 figure 1: co drilateral c circles ence, the tp onding 30 vi purpose of th a correct ve ociated with logous quest of the chord r (or counterp is the centre ms pertaining o which thes presents the oncepts and t triangles a equilateral isosceles tr scalene tria right-angle similar tria altitude of area of a ri number of number of diagonals o lines of sym pgt consiste isually repre he homologo erbal descrip the concept tions used in that passes t rpart) is: e of the circle to three diff e terms relat e terminology their associa and quadrila triangle riangle angle ed triangle angle f a triangle ight-angled tr f sides in a tria f sides in a qu of a quadrilat ymmetry ed of 30 verb esented ones us pair of qu ption of a ge , and vice ve n the tpgt. through the c e. which of t ferent but int te are: 1) cir y examined. ated terminol aterals riangle angle uad. teral humphre bally represe (the even-n uestions in th ometric conc ersa. questio centre of a ci he following errelated con rcles, 2) trian logy in the tp lines and an acute angle right angle obtuse angle reflex angle alternate ang vertically op complement supplementa correspondin parallel lines perpendicula ey atebe & m ented questio numbered ite his test was t cept also ha ons 5 and 24 ircle? g is a diamete ncepts in pla ngles and qu tpgt ngles e e gles pposite angle tary angles ary angles ng angles s ar lines marc schäfer 57 ons (the oddems). in line to determine s the correct , reproduced er? ane geometry uadrilaterals, s r 7 e e t d y , towards terminological primacy in learners’ geometric conceptualisation 58 administration and grading of the tpgt we personally administered the tpgt to all 144 learners (three learners were absent in nigeria on the day of this test, though), with the assistance of the mathematics teachers from the participating schools. with regard to test grading, scoring of students’ responses was done using scantools for windows version 2.2. the percentage score was calculated for each student and an item analysis of students’ responses was done using microsoft excel (see atebe, 2008 for details). results and discussion information about students’ knowledge of basic geometric terminology is acquired through analysis and interpretation of participants’ performance in the tpgt. overall participants’ performance in the tpgt students’ general performance in the tpgt was described in terms of the overall participants’ percentage mean score obtained in this test. table 1 summarises participants’ performance in the tpgt. table 1: percentage mean score of all participants in the tpgt school n mean score std deviation min score max score ns sas 69 72 40,49 47,85 16,78 13,82 17 27 90 87 note: ns = nigerian subsample; sas = south african subsample as is evident from table 1, the percentage average score obtained by learners from the nigerian subsample in the tpgt was 40,49%, and that of the learners from the south african subsample, 47,85%. a simple calculation from table 1 indicates that the percentage mean score obtained by all the participating learners in the tpgt was 44,17%. given that the items that made up the tpgt were largely van hiele level 1 in nature, and that the tpgt as a whole was a straightforward test of learners’ knowledge of the simplest and most common geometric terminology frequently encountered in junior and high school geometry, this rather low percentage mean score is an indication that this cohort of high school learners had a low level of terminology knowledge in this learning area. that is, learners in this study had a weak understanding of basic terminology associated with high school geometry. performance of nigerian and south african learners in the tpgt the mean score of learners from nigeria was compared with that of learners from south africa. the aim was to determine how nigerian high school children compare with their south african peers in the tpgt (see atebe, 2008 for the basis for this comparison). the results are summarised in table 2. table 2: school percentage means for learners in the tpgt school n mean score sd t-value df p-value ns 69 40,49 16,78 -2,85 139 0,0051* sas 72 47,85 13,82 note: ns = nigerian subsample; sas = south african subsample *p < 0,05 table 2 indicates that the mean score of the south african learners on the tpgt was higher than the mean score obtained by the nigerian learners. a test of significance revealed that the difference between humphrey atebe & marc schäfer 59 the means of the nigerian and the south african learners in the tpgt was significant in favour of the south african learners. what this result shows is that, on average, participants from south africa performed significantly better than their nigerian peers in the tpgt; or in other words, that the nigerian subsample in this study had a somewhat weaker understanding of basic geometric terminology than its south african counterpart. grade level performance in the tpgt grade level analysis of learners’ performance in the tpgt focused on the relative performance of grade 10, 11 and 12 learners in the nigerian and south african subsamples. these results are represented on figure 2. 34 42 4545 41 58 0 10 20 30 40 50 60 10 11 12 m ea n s co re ( % ) grade level ns sas figure 2: grade level performance of learners in the tpgt figure 2 reveals a marginal progressive increase in performance along the grade levels for the nigerian subsample. for the nigerian participants, the percentage mean score (45%) obtained by grade 12 learners was marginally higher than that of grade 11 learners (42%), which was in turn marginally higher than the mean score of the grade 10 learners. given these little differences in the mean scores of learners from across grades 10–12 of the nigerian subsample, it could be hypothesised that the nigerian high school learners in this study add only a little to their repertoire of geometric terminology as they progress from grade 10 through 12. an interesting revelation in figure 2 about the grade level performance of south african participants is that the grade 10 learners outperformed the grade 11 learners in the tpgt. as the figure illustrates, the mean score (45%) of south african grade 10 learners was marginally greater than the mean score (41%) obtained by grade 11 learners. south african grade 12 learners, however, obtained a higher mean score (58%) than both grade 10 and 11 learners. what these results show is that the south african grade 11 learners involved in this study had a weaker understanding of basic geometric terminology than their peers in grades 10 and 12. that grade 10 learners from the south african subsample outperformed their grade 11 peers in the tpgt turned out not to be by chance, since for all other tests used in the larger study (atebe, 2008), the grade 10 south african learners consistently obtained higher mean scores than the grade 11 learners. why this should have been so was not investigated in the current study. the grade level analysis of learners’ performance in the tpgt further indicated that south african learners, with the exception of grade 11, obtained higher mean scores than their nigerian peers. in fact, the nigerian grade 12 learners obtained a mean score (45%) equal to that of the south african grade 10 learners. in grade 11, the mean score of the nigerian learners (42%) was marginally greater than that of their south african counterparts, which was 41%. towards terminological primacy in learners’ geometric conceptualisation 60 grade level comparison of mean scores in the tpgt further analysis was done to determine whether or not the differences in the mean scores of the nigerian and the south african participants in the tpgt reported in the preceding section at each grade level were significant. the results, which are represented in table 3, indicated the following: there was a statistically significant difference in the mean score of nigerian grade 10 learners and south african grade 10 learners in favour of the latter. that is, south african grade 10 learners performed significantly better than their nigerian peers on the tpgt. the test of significance also revealed that although nigerian grade 11 learners obtained a marginally higher mean score on the tpgt than their south african counterparts, the difference in the mean scores of these two groups was not statistically significant. this means that nigerian grade 11 learners did not achieve significantly better results than their south african grade 11 counterparts in the tpgt. table 3: grade level mean scores in the tpgt grade nigerian subsample south african subsample t-value df p-value n mean sd n mean sd 10 11 12 21 24 24 33,62 41,67 45,33 10,15 19,51 17,13 24 24 24 44,63 40,50 58,42 7,22 8,16 16,94 -4,23 0,27 -2,66 43 46 46 0,0001*** 0,7882 0,0107* note: *p < 0,05; ***p < 0,001 the t-test further revealed that there was a significant difference between the mean score of nigerian grade 12 learners and that of their south african peers in favour of south african learners. that is, south african grade 12 learners performed significantly better than nigerian grade 12 learners in the tpgt. these results further buttress our earlier claim that on average, south african learners involved in this study have a better knowledge of basic geometric terminology than their nigerian counterparts. mean scores in the tpgt of all participants by gender as figure 3 illustrates, this study identified a gender difference in performance in the tpgt in favour of male learners. on average, male learners obtained higher scores, with a mean score of 48%, than female learners, who obtained a mean score of 41%. 48 41 0 10 20 30 40 50 male female m ea n s co re ( % ) gender figure 3: gender difference in mean scores in tpgt a test of significance conducted indicated that the difference between the male and female mean scores was statistically significant, as shown in table 4. humphrey atebe & marc schäfer 61 table 4: mean scores in the tpgt by gender gender n mean score sd t-value df p-value male 68 48,04 16,49 -2,84 139 0,0053** female 73 40,71 14,19 note: **p < 0,01 these results were consistent with those of usiskin’s (1982, p. 84) study, in which he reported that in the comparative and similar entering geometry test, the mean score of american high school male learners was significantly greater than that of their female counterparts. the results were also consistent with those of barnard and cronjé’s (1996, p. 1) study in which “differential gender performance was in favour of most males” in south africa in a 20-item multiple-choice euclidean geometry test. see atebe (2008) for gender differences determined for each of the participating schools. mean scores of all participants in the tpgt by concept for the analysis in this section, students’ mean scores in the tpgt were calculated separately for items on geometric terminology associated with the concepts of circles, triangles and quadrilaterals, and lines and angles. the results are as shown on figure 4. 40 45 47 0 15 30 45 60 circles triangles and quadrilaterals lines and angles m ea n s co re ( % ) geometric concepts figure 4: mean scores of learners in the tpgt by concept as figure 4 illustrates, the learners participating in this study showed a rather weak knowledge of geometric terminology across all three geometric concepts according to which items in the tpgt were drawn up. in relative terms, however, these learners demonstrated a better knowledge of terms in geometry associated with the concept of lines and angles than those associated with circles, or with triangles and quadrilaterals. the higher mean score of 47% indicated that these learners were more comfortable when dealing with geometric terminology associated with lines and angles than that associated with the concepts of circles, and triangles and quadrilaterals, for which they obtained mean scores of 40% and 45%, respectively. these results were found to be partly consistent with those of kouba, brown, carpenter, lindquist, silver and swafford (as cited in clements & battista, 1992, p. 421), who reported that in america, “students’ performance at identifying common geometric figures, such as parallel lines…[was] acceptable”, but that students’ knowledge of some basic geometric terms associated with the concept of the circle was deficient. towards terminological primacy in learners’ geometric conceptualisation 62 correlation between verbal and visual abilities of learners in the tpgt the correlation between students’ ability in verbal and visual geometry terminology tasks was determined. by correlating students’ scores from the verbally presented items in the tpgt, correlation coefficients were obtained separately for the nigerian and south african subsamples. both correlations were found to be significant, as indicated in table 5. table 5: correlation coefficients for the tpgt by school school verbally presented items visually presented items n r-value p-value mean sd mean sd ns sas 11,16 12,68 5,12 4,61 13,13 15,99 5,45 4,58 69 72 0,83 0,63 0,00001*** 0,00001*** note: ns = nigerian subsample, sas = south african subsample ***p < 0,001 as is evident in table 5, the values of the correlation coefficients calculated for both the nigerian and south african subsamples are positive. this means that a student who correctly answered a verbally presented question in the tpgt also answered its visually presented identical counterpart correctly, and vice versa. that the values of the coefficients are fairly large does not necessarily indicate that the learners have an impressive grasp of geometric terminology. the coefficients only give information about the level of consistency with which participants responded to homologous pairs of questions in the tpgt. the correlation coefficient calculated for the nigerian subsample (0,83) was greater than that calculated for the south african subsample (0,63).this again does not imply superiority in performance on the part of the nigerian subsample over that of the south african. what it does mean is that nigerian learners were more consistent in their responses to the items in the tpgt than their south african peers. that is, more nigerians passed or failed identical pairs of questions in the tpgt than south africans. the higher mean score obtained by the south african subsample in the tpgt (reported earlier) compared with that of the nigerian participants supports this disclaimer. even with the interpretation given in the preceding paragraph, there could be many other dimensions to what these correlation coefficients tell us about students’ conceptual understanding of geometric terminology. that the correlation coefficient for the nigerian subsample was greater than that of the south african participants means that more nigerian learners than south africans who knew the correct verbal description of a geometric concept also had the correct visual/concept image associated with the concept. in terms of the conceptual understanding of basic terminology in geometry, this would mean that south african participants were less conceptually grounded than their nigerian peers; it could be that the sas learners engaged in random guessing that resulted in the correlation coefficient being lower than that of the nigerian participants. another interpretation, and perhaps one more tenable, is that south african learners had a more comprehensive understanding of basic geometric terminology than their nigerian counterparts and hence obtained a higher mean score in the tpgt. but this understanding is less conceptual, as the sas learners tended to understand the terminology better only in one form of presentation, namely the visual form. simple calculations from table 5 reveal that there is a wider difference between the mean scores of south african learners for the verbally presented items and the visually presented items than there is for learners from the nigerian subsample. this indicates that the south african learners were less successful with geometry terminology tasks that were presented in verbal form than the nigerian participants. to conclude, table 5 further indicates that learners from both the nigerian and the south african subsamples had a better understanding of geometric terminology presented in terms of visual tasks than those that were presented in verbal form, and hence obtained higher mean scores in the former than in the latter. humphrey atebe & marc schäfer 63 conclusion this study has investigated learners’ knowledge of basic geometric terminology in selected nigerian and south african high schools. the motivation was the theoretical standpoint that students’ acquisition of the correct terminology is important for their success in the subject (van hiele, 1986). analysis of participants’ responses to the tpgt showed that learners in this study had a limited and arguably inadequate knowledge of basic geometric terminology, since they were only able to obtain an overall percentage mean score of 44,17% in this test even when the tpgt dealt with the simplest of geometry facts and concepts. learners from the south african subsample performed significantly better than their nigerian peers in the tpgt: there was a statistically significant difference between the mean score of south african learners (47,85%) and that of the nigerian subsample (40,49%) in favour of the former. given the lower mean score, learners from the nigerian subsample in this study were considered to have a weaker understanding of basic geometric terminology than their south african counterparts. in this study, differential gender performance was in favour of the male learners in respect of the basic geometric terminology test. there were high positive correlations between participants’ ability in verbal geometry terminology tasks and their ability in visual geometry terminology tasks. for the participants in this study, learners who knew the correct verbal description of a geometric concept also had the correct visual image associated with the concept, and vice versa. learners’ mean score for the visual geometry terminology tasks was, however, greater than their mean score for the verbal geometry terminology tasks comprising the tpgt. this would mean that learners are more comfortable dealing with visual geometry terminology tasks than verbal geometry terminology tasks. recommendation given the results of this study, it is important that the teaching of basic geometric terminology be foregrounded in school mathematics. this could assist the teacher in detecting misconceptions among learners and enable him/her to correct them early in classroom interaction. it is also recommended that teachers make use wherever possible of authentic examples or graphic representations thereof during geometry teaching, since learners tend to have a better understanding of geometry terminology tasks that are presented in visual form than those that are presented in verbal form. acknowledgement the sponsorship of this research by the firstrand foundation mathematics education chair at rhodes university is hereby acknowledged. opinions expressed and conclusions arrived at, are those of the authors and are not necessarily to be attributed to rhodes university 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can be opened with acrobat and adobe reader 5.0 and later.) >> /namespace [ (adobe) (common) (1.0) ] /othernamespaces [ << /asreaderspreads false /cropimagestoframes true /errorcontrol /warnandcontinue /flattenerignorespreadoverrides false /includeguidesgrids false /includenonprinting false /includeslug false /namespace [ (adobe) (indesign) (4.0) ] /omitplacedbitmaps false /omitplacedeps false /omitplacedpdf false /simulateoverprint /legacy >> << /addbleedmarks false /addcolorbars false /addcropmarks false /addpageinfo false /addregmarks false /convertcolors /converttocmyk /destinationprofilename () /destinationprofileselector /documentcmyk /downsample16bitimages true /flattenerpreset << /presetselector /mediumresolution >> /formelements false /generatestructure false /includebookmarks false /includehyperlinks false /includeinteractive false /includelayers false /includeprofiles false /multimediahandling /useobjectsettings /namespace [ (adobe) (creativesuite) (2.0) ] /pdfxoutputintentprofileselector /documentcmyk /preserveediting true /untaggedcmykhandling /leaveuntagged /untaggedrgbhandling /usedocumentprofile /usedocumentbleed false >> ] >> setdistillerparams << /hwresolution [2400 2400] /pagesize [612.000 792.000] >> setpagedevice article information author: wajeeh daher1,2 affiliations: 1department of educational sciences, an-najah national university, palestine2department of mathematics education, al-qasemi academic college of education, israel correspondence to: wajeeh daher postal address: po box 8861, nazareth 16000, israel dates: received: 09 feb. 2012 accepted: 16 nov. 2012 published: 10 dec. 2012 how to cite this article: daher, w. (2012). student teachers’ perceptions of democracy in the mathematics classroom: freedom, equality and dialogue. pythagoras, 33(2), art. #158, 11 pages. http://dx.doi.org/10.4102/ pythagoras.v33i2.158 copyright notice: © 2012. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. student teachers’ perceptions of democracy in the mathematics classroom: freedom, equality and dialogue in this original research... open access • abstract • introduction • literature review    • democratic education    • perceptions of teachers    • pedagogy and didactics       • pedagogic aspects of democracy in the mathematics classroom versus didactic issues • the research framework • methodology    • the research setting and participants    • data production and analysis       • validity and reliability of the research tools and procedure       • ethical considerations • findings    • student teachers’ perceptions of freedom    • student teachers’ perceptions of equality    • student teachers’ perceptions of dialogue • discussion    • pedagogic (un)democratic acts in the mathematics classroom    • didactic (un)democratic acts in the mathematics classroom • conclusions • acknowledgements    • competing interests • references abstract top ↑ this article studies student teachers’ perceptions of the pedagogic and didactic aspects of teaching and learning mathematics in a democratic classroom. it is concerned primarily with issues of democracy in the mathematics classroom, specifically freedom, equality and dialogue. the research was conducted in two mathematics teacher education classes, where students were in their third year of study to major in mathematics. to find these students’ perceptions of democracy in the mathematics classroom the first two stages of the constant comparison method were followed to arrive at categories of democratic and undemocratic acts. the participants in the research emphasised that instructors should refrain from giving some students more time or opportunities to express themselves or act in the mathematics classroom than other students, because this would make them feel unequal and possibly make them unwilling to participate further in the mathematics classroom. the participants also emphasised that instructors should not exert their power to stop the flow of students’ actions in the mathematics classroom, because this would trouble them and make them lose control of their actions. further, the participants mentioned that instructors would do better to connect to students’ ways of doing mathematics, especially of defining mathematical terms, so that students appreciate the correct ways of doing mathematics and defining its terms. introduction top ↑ cristillo (2009) found that palestinian universities continue to emphasise teacher-centred teaching approaches and assessment in spite of faculty exposure to student-centred approaches and the increased use of information technology by students. similar findings are found in other reports, for example world bank (2008), which describes educational reform in the middle east and north africa. to summarise this situation, the report (p. 88) says that there is little evidence of a significant shift in educational practices away from a traditional model of pedagogy, in which, despite pedagogical reforms characterised by student-centred learning, competency-based curricula and focus on critical thinking, the main student activities in classrooms continue to be copying from the blackboard, writing and listening to the teachers. this educational scene should be seen in light of the fact that palestinian university campuses are places where students experience democratic politics, an experience which is transferred to society at large (abu lughod, 2000). this fact prompted cristillo (2010) to suggest practices of student-centred classrooms, in which discussion, debate, collaborative problem solving and critical thinking prevail, as practices that contribute to palestine’s success in local, regional and global economies, as well as in the training of student teachers for democratic politics. the above situation emphasises the importance of and need to emphasise democratic issues in palestinian university settings in general and particularly in mathematics pedagogic training courses in palestinian universities, as well as the importance of analysing student teachers’ perceptions as a consequence of this emphasis. this is also emphasised by bailey and murray (2009), who cite reports from the palestinian ministry of education and higher education, unesco and the world bank to conclude that higher education institutions in palestine do not train pre-service teachers in an adequate way, and that this training does not suit the specific demands of the palestinian education system. it is our conviction that educating future teachers to respect democratic teaching will help to change positively the educational scene in palestinian schools, and as a result the palestinian people, especially in the time of occupation. the current research examines mathematics student teachers’ perceptions of (un)democratic acts in the mathematics classroom after they treat this issue in the context of their pedagogic training course. specifically, it focuses on the students’ perceptions of freedom, equality and dialogue. literature review top ↑ in this review we will consider the educational constructs with which this research is concerned: democratic education, teachers’ perceptions and pedagogic versus didactic aspects of teaching. democratic education issues of democracy in the mathematics classroom have drawn the attention of researchers for at least three decades, during which time different aspects of these issues have been studied:• the right to equal access to mathematical ideas (allen, 2011; ellis & malloy, 2009; moses & cobb, 2001) • authority in the mathematics classroom (amit & fried, 2005; povey, 1997; skemp, 1979) • promoting equality in the mathematics classroom (croom, 1997) • promoting democracy in the mathematics classroom (allen, 2011; ellis & malloy, 2009) • diversity of curriculum and classroom (ellis & malloy, 2009) • revisiting old ideas in new ways (ellis & malloy, 2009) • dialogue in the classroom (ball, goffney & bass, 2005; hannaford, 1998; skovsmose, 1998) • proving in the mathematics classroom (almeida, 2010; skovsmose, 1998) • engaging in ethnomathematics (ball et al., 2005; skovsmose, 1998) • controlling the flow of discourse (barner, 1998; zhang, 2005). these different issues related to democracy in the mathematics classroom are described in more detail below. firstly, the general issues are described, where these issues could be related to various democratic classroom topics. secondly, the democratic issues that are of particular interest in this research are described: freedom, equality and dialogue. two issues that were studied regarding promoting democracy in the mathematics classroom are the beliefs of the educational agents that influence students’ learning and the qualities that encourage democratic practices in the classroom. allen (2011) says that promoting democracy in the mathematics classroom of the 21st century involves different agents (teachers, students, administrators and parents) who should hold the following beliefs as a condition for this promotion: • all students are capable of learning powerful mathematics • mathematics was and is still invented by humans • students can and should help to design their mathematical learning experiences • thinking mathematically means solving problems to which we do not know the answer • successful education requires meaningful relationships between students and teachers. ellis and malloy (2009) say that the literature on democratic education identifies the following qualities as encouraging a democratic classroom: • a problem solving curriculum, in which the problems are related to students’ lives and society, and where students have access to information that helps them to solve the problems in diverse ways. • inclusivity and rights: students should have access to mathematical ideas from multiple perspectives and have diverse experiences and approaches in solving problems. • equal participation in decisions, which are arrived at as a result of open, persuasive and negotiable discussions of mathematical and social issues and ideas. these discussions help students to create, clarify, and re-evaluate their own ideas and understand the ideas of others. • equal encouragement for success through encouraging the development of habits of drawing conclusions and critically evaluating implications from mathematical data for personal and social action. ellis and malloy (2009) also elaborate on diversity in the mathematics classroom, considering democratic education a process of collaborative reconstruction of a curriculum that is inclusive of diversity on the part of both the teacher and the students. such diversity is exemplified in, amongst other things, the ways in which mathematics is taught and learned. diversity exists not only inside the classroom, but also between different classrooms based on the needs, preferences and experiences of the students and teacher. relating mathematical problems to students’ lives is a matter of engaging students in ethnomathematics activities. skovsmose (1998) described ethnomathematics as the study of mathematics presented in many forms in ‘traditional’ societies, in society routines, techniques and handicrafts, and in all kinds of ordinary life. further, ball et al. (2005) say that teachers should design contexts that are rooted in the broader and more diverse experiences and cultures of their students, as well as in other cultures. this, they say, is ‘crucial for developing the understanding and appreciation of diverse traditions, values, and contributions’ (p. 4). such understanding and appreciation of diverse traditions, values and contributions are expected to help to maintain democratic social values and practices. the diversity in the mathematics classroom described above can be related to freedom in the classroom. this is emphasised by the australian-proposed charter of academic freedoms, as reported by gelber (2009), which mandates that students be provided with diverse scholarly viewpoints. in other words, the charter considers diversity of viewpoints as a prerequisite for encouraging freedom in the classroom. freedom in the mathematics classroom is also related to encouraging students to explore mathematical ideas (sgroi, 1995) and the freedom to start with the easiest questions when given a classroom task (susuwele-banda, 2005). two issues that were studied regarding the right to equal access to mathematical ideas are how access to mathematical ideas (1) influences students’ behaviour as citizens in a democratic society and (2) is influenced by societal perceptions and practices. moses and cobb (2001) argue that access to mathematics, especially algebra and advanced mathematics, is a civil right as important as the right to vote, that is, they argue that access to mathematics is a very important democratic principle. citing examples from the literature, ellis and malloy (2009) describe societal perceptions and practices that are not democratic and which cause students not to have equal access to mathematical ideas: perceptions of ability by the teacher and society, cultural discontinuity in learning and instruction, tracking, poverty and school finance, and low expectations from teachers, parents or society. regarding promoting equality in the mathematics classroom, croom (1997) mentions the following practices: • accommodating the learning styles of culturally diverse students. • organising development activities that enhance teachers’ knowledge of different mathematical world views and cultural perspectives. • ensuring that minority and female students have an equal opportunity to learn substantive mathematics. • promoting the achievement of limited-language-proficiency students. • devising strategies that form partnerships with families. • affirming the richness and strength of cultural diversity. it could be claimed that croom’s suggested practices to promote equality in the mathematics classroom are primarily cultural and social, and fit a community that consists largely of immigrants or of culturally diverse students. other researchers talk about the relation of mathematics to maintaining a democratic society: the theme of authority is one of the features of mathematics and is related to equality. some researchers describe how the use of mathematics helps to achieve authority, for example porter (1995, as reported in wagner & herbel-eisenmann, 2011) points out that bureaucrats in democracies obscure their authority by mathematical means, quantifying them without seeming to make decisions. wagner and herbel-eisenmann (2011) cite such political practices to argue that promoting authority for students makes the teaching and learning of mathematics an essential condition to give students power in society. dialogue is an important practice in the mathematics classroom. ball et al. (2005) say that teachers should listen closely to students’ ideas; skovsmose (1998) elaborates more on this issue, stating that ‘[i]t is important to make possible an interaction in the classroom which supports dialogue and negotiation’ (p. 200), in order to maintain democracy in the mathematics classroom. some researchers relate dialogue practices in the mathematics classroom to the students’ proving processes. almeida (2010) regards proving actions in the classroom as having a democratic flavour when they are accompanied by interrogation, convincing and agreement actions. almeida emphasises that teachers’ explanatory arguments should be open to students’ scrutiny and debate and, at the same time, pupils’ sense of argumentation, reasoning and reasonableness should be regarded as legitimate. skovsmose (1998) also emphasises the importance of respecting students’ sense of argumentation, saying that the teacher ‘should be aware of the student’s good reasons in order to escape the paradigm of classroom absolutism’ (p. 200). dialogue is also related to the equality and authority issue in terms of controlling the flow of discourse. zhang (2005) considers this control a symptom of exerting power in a discourse. in fact, zhang considers power a relationship with the others in a discursive practice. this consideration can be used as a tool for analysing power in the mathematics classroom dialogue: where, in a traditional classroom, the teacher controls the discourse and thus the power, in the collaborative classroom, the power is distributed amongst the students too, with some at times having more power than others. zhang further mentions mathematics knowledge, social assignation, gender, class, race, and religion as factors that influence an individual’s power in the mathematics classroom. barner (1998) mentions various ways in which students act to control or influence the flow of group discourse, and thus exert power on the group activity: initiating a negotiating event, initiating off-task talk, and rejecting or ignoring off-task talk by continuing the negotiating event or by initiating a new one. dialogue is not only related to equality and authority in the mathematics classroom, but also to freedom. for example, geoghegan, petriwskyj, bower and geoghegan (2003) argue that to foster children’s ability to express unique ideas teachers need to allow them some freedom to move in the classroom with autonomous flexibility and to interact with their peers in informal ways. the above descriptions of the various democratic issues in the mathematics classroom not only highlight the uniqueness of each issue, but also their inter-dependence: the realisation of one assures or leads to the realisation of the others. perceptions of teachers teacher’s perceptions are an important element that influence the teacher’s teaching method and their behaviour in the classroom. this means teachers’ perceptions are an element that influences students’ learning as well. research shows what the impact of teachers’ perceptions, expectations and opinions on their behaviour in the classroom and on their students’ learning can be. trouilloud, sarrazin, martinek and guillet (2002), for example, report that students’ achievements validated their teacher’s expectations of them. several researchers have examined the influence of perceptions on behaviour, saying that one’s perceptions of the self and the reality in which one lives influence one’s behaviour. perceptions are not a passive process of absorption of facts and impressions, but a compound and active process which is influenced by various variables, such as past experiences, expectations and emotions, which design our perception’s domain (bar-al & noymeyer, 1996). chanal, sarrazin, guay and boiché (2009) say that the theories concerned with perceptions of the self suggest that perceptions students have about their ability in a specific subject are not side effects, but an important cognitive state that influences students’ behaviour and academic achievement. this importance of the students’ perceptions explains why it is necessary to examine student teachers’ perception of democratic acts in the mathematics classroom, so as to evaluate their future teaching practices in the classroom and examine how they prepare their students for democratic life, and independent and critical thinking. pedagogy and didactics two educational aspects with which this research is concerned are pedagogy and didactics. andrews (2007) differentiates between the two by saying that pedagogy is concerned with the curriculum in both broad and narrow forms, and the underlying systemic aims and objectives of education. in addition, pedagogy transcends subject boundaries and acknowledges general theories of teaching and learning. for andrews, pedagogy includes didactics, which is concerned with the strategies and warranted approaches to subject teaching and learning. these strategies and approaches may vary from one subject to another. in addition, didactics acknowledges theories of teaching and learning but from the subject-specific perspective. shulman (1987) described pedagogic knowledge as the knowledge of pedagogic principles and techniques that are concerned with efficient teaching, and which is not concerned with a specific subject. shulman described pedagogic content knowledge as the knowledge of the teaching methods of a specific subject. in the current research we will treat the pedagogic and didactic aspects of democratic acts in a mathematics pedagogic training course in these terms, where the didactic aspect is the pedagogic content knowledge, according to shulman. these two aspects should be attended to in the mathematics classroom, in which each complements the other in general and as regards democracy in particular. the didactic aspect is not largely attended to in the literature on democracy in the mathematics classroom. pedagogic aspects of democracy in the mathematics classroom versus didactic issues some of the previous issues and sub-issues regarding democracy in the mathematics classroom can be related to the political aspect of teaching and learning mathematics, whilst others can be related to the cultural, social, pedagogic and didactic aspects of teaching and learning mathematics. for example, some sub-issues of the right to equal access to mathematical ideas can relate to the political aspect, such as poverty and school finance; the sub-issues of societal perceptions of ability, cultural discontinuity in learning and instruction can relate to the pedagogic aspect of teaching and learning. on the other hand, the equality sub-issue of devising strategies that form partnerships with families can relate to the social aspect of learning and teaching mathematics in the mathematics classrooms; the sub-issue of affirming the richness and strength of cultural diversity can relate to the cultural aspect. regarding the didactic aspect of teaching and learning mathematics, the problem-solving curriculum sub-issue of relating problems to students’ lives and society can relate to the didactic aspect of teaching and learning mathematics because this sub-issue influences teachers’ methods of teaching mathematics. the current research is concerned with both the pedagogic and the didactic issues of teaching mathematics so can be considered to embrace a continuum of researches in these two fields. the research framework top ↑ this research focuses on the pedagogic and didactic aspects of teaching and learning mathematics in a democratic classroom. it is concerned primarily with some of the previously described issues of democracy in the mathematics classroom, specifically freedom, equality and dialogue. these issues are important issues of democracy as well as of teaching and learning mathematics (alrø & johnsen-høines, 2010; alrø & skovsmose, 2002; national council of teachers of mathematics [nctm], 2000; vithal, 2000a, 2000b; wilson & lloyd, 1996). freedom can be viewed as the freedom to choose and act as a means to control learning and equality as ‘the fair and equal distribution of mathematical knowledge and other educational opportunities for all members of society’ (vithal, 1999, p. 27). dialogue is viewed as the verbal interactions amongst learners and between them and their teacher; this interaction could be about their experience of learning and about forms of classroom practice (boylan, 2009).various researchers have been involved with democracy in the mathematics classrooms, but most of these were concerned with global pedagogic issues, for example providing equal access and attainment for all students (allen, 2011). the current research attempts to examine complementary issues: pedagogic and didactic issues of democracy inside the mathematics classroom, significantly concerned with the pedagogic and didactic issues of teaching and learning mathematics. the current research sheds light on the pedagogic and didactic democratic acts in the mathematics classroom, as perceived by student mathematics teachers. this will direct student teachers and teacher educators to the methods and behaviours that encourage democracy in the mathematics classroom and, as a result, encourage constructivist and social constructivist teaching and active learning of mathematics. the research question being addressed is: what pedagogic and didactic acts do student mathematics teachers perceive as democratic in a mathematics classroom (a university pedagogic training classroom)? more specifically, what pedagogic and didactic acts do student mathematics teachers perceive as: 1. promoting or lessening freedom 2. promoting equality or inequality 3. encouraging or discouraging dialogue in a mathematics classroom? methodology top ↑ the research setting and participants the research was conducted in two mathematics pedagogic training classes at a large university in palestine. the student teachers were in their third year of study, majoring in mathematics. one class was held in the second semester of the 2010–2011 academic year and the second class was held in the second semester of the 2011–2012 academic year. the first class included 26 student teachers; the second included 41 student teachers. the mean age of the participants was 21.4 years with a standard deviation of 1.2. the mathematics pedagogic training course addressed various issues of teaching mathematics in the classroom, including the issue of democracy, which was raised near the end of the course. the assignments in the course included exams, a reflective portfolio, preparing a unit of three or more lessons on a mathematical topic and microteaching in front of the class. the microteaching was assigned from the beginning of the semester, 15−20 minutes for each student. the instructor had two strategies: interfering during the microteaching if there was a critical remark to give, for example correcting a wrong mathematical definition that the student who did the microteaching gave, and discussing the microteaching with the whole class at the end of the microteaching session. the portfolio included writing reflections on the course’s lectures, on the school training and any other mathematics related work. data production and analysis at the end of each course, the student teachers were asked to give their evaluation of the course outline with respect to the democratic aspects of freedom, equality and dialogue. they had to write down six acts which occurred or were discussed during the course: one that supported freedom, one that hindered freedom, one that promoted equality, one that hindered equality, one that promoted dialogue and one that hindered dialogue. the participants’ evaluations of the democratic or undemocratic aspects of the pedagogic training course constituted the data in this research. in order to evaluate the democratic aspects of the course’s outline, the student teachers were asked to give their evaluations on a voluntary basis. in the first class, 24 of the 26 student teachers participated; in the second class, 38 out of 41 participated. the first two stages of the constant comparison method (glaser & strauss, 1967) were followed to arrive at categories of democratic and undemocratic acts (acts that support freedom or hinder it, acts that promote equality or hinder it and acts that promote dialogue or hinder it). these stages were: • categorising data: putting together data expressions or sentences that imply a category of democracy, for example, putting together all expressions or sentences that imply a freedom category, for example, the freedom to express opinion. • comparing data: comparing expressions or sentences within each previously built category. this gave rise to sub-categories. for example, in the category ‘the freedom to express opinion’, comparing expressions or sentences in students’ answers differentiates between expressions and sentences that imply expressing opinion and those that imply another category of freedom. validity and reliability of the research tools and procedure the validity of the research analysing procedure was guaranteed by ensuring theoretical saturation by continuing the analysis of the participants’ answers until (1) no new category or theme regarding freedom, equality and dialogue appeared, and (2) every category was well developed in terms of its properties and dimensions demonstrating variation (strauss & corbin, 1998, p. 212). lincoln and guba (1985) say that no validity exists without reliability, so the ensuring of validity also ensures reliability. this means that following theoretical saturation maintains not only the validity of the research procedure but also its reliability. two experienced coders (one of them the author) coded the participants’ texts, searching for categories in the participants’ perceptions of the three aspects of democracy in the mathematics classroom, freedom, equality and dialogue. the agreement between the coders (cohen’s kappa coefficient) was 0.86 for freedom, 0.79 for equality and 0.85 for dialogue. ethical considerations the participants were asked to give their evaluations of the pedagogic training class regarding democracy aspects on a voluntary basis. further, they were directed not to write their names on the assignment. the goal was for them to describe anonymously and thus openly their perceptions of the democratic atmosphere in the mathematics pedagogic training classes. the participants were directed to write six incidents that occurred during the course’s lectures. the goal was to make sure that the participants considered both sides of the democratic phenomenon in the classroom, and not just the positive side. these steps were performed to ensure ethical treatment of the researcher–participant relationship (ramos, 1989). the university gave its consent for this research to be conducted in the mathematics pedagogic training course, on the condition that the student teachers agreed to participate in it. the participating student teachers gave their permission for their answers to be available for this research and agreed that their exact sentences could be used for the research goals. findings top ↑ the findings on student teachers’ perceptions of the three democratic issues will be presented in the following order: the freedom issue, the equality issue and the dialogue issue. for every issue, four tables will be built, describing categories related to: acts promoting the pedagogic aspect of the issue, acts lessening the pedagogic aspect of the issue, acts promoting the didactic aspect of the issue and acts hindering the didactic aspect of the issue. the pedagogic categories are related to general acts, that is, acts which could happen in any classroom and not only in the mathematics classroom. the didactic categories are related to acts that happen specifically in the mathematics classroom. student teachers’ perceptions of freedom the acts described by the participants as promoting or lessening freedom in the mathematics pedagogic training course can be divided into pedagogic acts and didactic acts. these two types of acts can be further divided into subcategories. these subcategories, with appropriate examples, are described in tables 1–4. table 1: pedagogic acts promoting freedom. table 2: pedagogic acts lessening freedom. table 3: didactic acts promoting freedom. table 4: didactic acts lessening freedom. table 1 shows that the participants talked primarily about three pedagogic acts that promote freedom in the classroom: the freedom to express oneself, the freedom to decide and the freedom to act. table 2 points at the importance that the participants gave to having freedom to act and communicate. table 3 shows that the freedom to express oneself and the freedom to decide are not only important pedagogic acts that assist students’ freedom, but also didactic acts that lead to the same result: the students having learning freedom. finally, table 4 points to the freedom of students to ask questions as another important block of didactic acts that promote or hinder students’ freedom in the mathematics classroom. student teachers’ perceptions of equality categories of equality prevalence or absence are divided into subcategories with appropriate examples in tables 5–8. table 5: pedagogic acts promoting equality. table 6: pedagogic acts promoting inequality. table 7: didactic acts promoting equality. table 8: didactic acts promoting inequality. table 5 and table 7 show that the participants perceived the instructor–student relationship as important to pedagogic and didactic equality in the mathematics classroom. table 8 shows that the participants were concerned with the teaching processes that influence their learning processes, particularly those processes related to the teaching methods and teacher’s examples. table 6 shows that it was important for the participants that the instructor does not perform discriminating acts between them that could be related to their achievement. student teachers’ perceptions of dialogue categories of dialogue encouragement or discouragement are divided into subcategories with appropriate examples in tables 9–12. table 9: pedagogic acts that encourage dialogue. table 10: pedagogic acts discouraging dialogue. table 11: didactic acts encouraging dialogue. table 12: didactic acts discouraging dialogue. table 9 shows that the participants perceived the pedagogic acts that are related to expressing oneself and students’ roles as important to dialogue in the classroom. in addition, the possibility and potentiality of mathematical discussions and finished or unfinished talk also caught the attention of the participants, not only for encouraging discussions but also for discouraging them, as table 10 and table 12 show. interestingly, as table 11 shows, the prevalence of dialogue in the mathematics classroom is related to justification and use of natural language. discussion top ↑ pedagogic (un)democratic acts in the mathematics classroom the research findings, as can be seen in table 1, show that the student teachers considered self-expression to be an important aspect of freedom. the rest of the aspects mentioned by the participants as acts of freedom in the mathematics classroom, the freedom to act, to decide and to criticise, indicate the importance of self-realisation, which is a required learning outcome (isman, altinay & altinay, 2004). the importance that the participants gave to self-expression as a self-realisation aspect could be due to different causes, namely, (1) the importance of students’ self-realisation to their activity in the mathematics classroom, (2) the course outline, where the students were not satisfied with the freedom issues in the course, and (3) the national and political conditions of the participants, which could hinder the advancement of their self-expression and self-realisation. these findings agree with kesici (2008) who reported that the school teachers participating in his study pointed out that teachers would better allow school students to express their ideas, which promotes their self-realisation. the participants in the current research, as table 2 shows, also pointed out the importance of having the freedom to act and communicate, but it seems that some of them did not understand the limits of educational freedom: they considered the instructor’s criticism of their microteaching as negatively influencing this teaching. the participants were new to teaching, so they were not sure of themselves as teachers, and thus considered any criticism or interruption as a threat to their teaching. instructors should take into consideration their students’ knowledge and experience and state their feedback on students’ action in a way that the students accept, so that they benefit from this feedback. in addition, a dialogue should be held between the students and the instructor to discuss the role of critiquing in the democratic mathematics classroom, where this critiquing is an indicator of a healthy democratic classroom (macmath, 2008). an alternative explanation could be that the course outline failed to convince the students of the prevalence of freedom in it, so they expressed their perception of the lack of it. the second issue of democracy in which this research is interested is equality, which has been researched educationally from various angles; one aspect that has been researched extensively is the gender aspect (canadian international development agency 2010; united states agency for international development [usaid], 2008). here, as table 5 shows, the participants were concerned with being not discriminated against because of their gender and specifically not discriminated against in their course grades. students’ role in teaching and learning, as table 5 shows, caught the attention of the participants too as an issue related to equality in the mathematics classroom: their ability to present their experiences in the classroom made them feel equal with the instructor. the student’s role in the instructor–student relationship is considered by various researchers to be a component of democracy in the elementary and secondary classroom (see e.g. davis, 2010; larrivee, 2002). in this research, as table 5 shows, the student teachers were content to exchange their role with the role of their instructor, probably because this exchange empowered them and supported their self-esteem. talking about inequality in the classroom, as perceived by the participating students, usaid (2008) four types of equality in education can help us. these types are: equality of access, equality in the learning process, equality of educational outcomes and equality of external results. here, as table 6 shows, the participants were concerned with inequality between students in the learning process (time allocated to each of them in microteaching or discussion) and in the educational outcomes (the absence of clarity in the assignments evaluation criteria). the third issue of democracy with which this research is concerned is the dialogue issue. the participants in this research mentioned four different communicational acts as democratic: discussing, asking, arguing and listening. these acts imply the importance of open dialogue and open discussion in the classroom in order to foster a democratic climate (harwood & hahn, 1990). further, the issue of democracy limit is of relevance here. this is about the situations in which teachers have to end an educational action in the classroom to start another for whatever reason. sometimes, students do not understand their teacher’s decision and consider it undemocratic. morrison (2008) reminds, whilst talking about her experience with student teachers, that students may consider some classroom acts as undemocratic if they mistake positive freedom for negative freedom; in our case, the student teachers did not accept the necessity of putting an end to a classroom dialogue, probably thinking that the dialogue should continue until all issues related to it were settled. the pedagogic dialogue aspects of the mathematics classroom that the participants perceived were similar to those pointed out by other researchers. for example, poduska (1996) says that teachers implementing a democratic pedagogy should not only encourage open dialogue, but also encourage critical student feedback on aspects of the school. didactic (un)democratic acts in the mathematics classroom the participants in the research, as table 3 shows, admired their ability to suggest at the beginning of the course what the parts of a mathematical lesson should be and the components of each part. they considered this ability a democratic act; this agrees with allen (2011), who says that to promote democracy in the mathematics classroom students should help in designing their mathematical learning experiences. the participants, as table 3 shows, were also concerned with problem solving and their ability to suggest and use solution methods other than those suggested by the teacher. the ability of students to suggest and use their own solution methods is emphasised in previous research as a way to advance students’ mathematical thinking (fraivillig, 2001) and improve their achievement (grouws & cebulla 2000). this concern with problem solving has different aspects, freedom being just one of them, but the results imply that freedom related to problem solving in the mathematics classroom is supposed to contribute to students’ learning and achievement, as well as their feeling of didactic freedom. previous researchers (akinbobola & afolabi, 2010; kesici, 2008) have pointed at student-centred and guided-discovery teaching methods as methods that should prevail in the democratic classroom, for they encourage learners to be autonomous and responsible (akinbobola & afolabi, 2010). it seems that the students who have not practised teaching in schools were not impressed by these teaching methods, for they had not seen their practical benefits, so they took into their consideration mainly their didactic ability to act and decide. on the other hand, the participants were concerned also with the ability of the students to ask questions in the mathematics classroom, probably because asking questions allows students to verify their understanding and be more autonomous as learners and less dependent on the teacher (anthony & walshaw, 2007). to discuss the participants’ perceptions of didactic equality in the mathematics classroom, let us stay within the framework of usaid’s (2008) categorisation of equality types. the participants were concerned with three of these types when talking about didactic acts that promote equality in the mathematics classroom: equality of access to problem solving and solution methods; equality in the learning process, including the instructor as one of the learners; and equality in the educational outcomes that include the instructor’s feedback and the students’ discussion about their experiences. the findings regarding the participants’ perceptions of didactic equality also agree with kesici (2008), who pointed out that providing equality in the classroom should include both the learning process and classroom activities.  talking about didactic acts that promote equality in the mathematics classroom, as table 7 shows, the participants talked about the need to provide different solution methods in the democratic mathematics classroom. this agrees with ellis and malloy (2009) who said that students in the democratic classroom should have access to mathematical ideas from multiple perspectives and have diverse experiences and approaches in solving problems. here, the equality is a consequence of the recommended educational environment and not a procedural one. it can be said that the participants were concerned with teaching processes that influence learning processes. the diversity of mathematics teachers’ methods emphasised by the participants has also been attended to in the literature. for example, one of the principles set out by the national council of teachers of mathematics (nctm, 2000) is the equity principle which states that all students must have the opportunity to study and the appropriate support to learn mathematics. moreover, klassen (2008) suggests practical adaptations and instructional strategies that could address the diversity of students in the mathematics classroom. here the participants suggested that teachers should implement diverse teaching methods and give diverse examples to reach all students in the mathematics classroom. the participating student teachers, as table 11 shows, were aware of different acts in the mathematics class that encourage dialogue and at the same time result in better learning: expressing oneself, inquiring about ideas, listening to ideas, building ideas and suggesting topics for discussion. these class acts point at effective communication between the teacher and the students and encourage positive learning processes (kibler, rush & sweeney, 1985). a healthy communication established between the teacher and the students not only increases students’ educational success, but also allows a safe environment for them (kohn, 1997). thus, open class dialogues not only point at a democratic class, but also contribute to students’ learning processes and outcomes. regarding didactic acts that do not promote dialogue in the mathematics classroom, as table 12 shows, some participants said that by not accepting their ways of defining mathematical terms the instructor discouraged their participation in discussions about these terms. this perception made it difficult for the instructor to develop and build on students’ ways of doing mathematics. schifter (2001) says that being sensitive to students’ conceptual issues is critical for building on and developing students’ thinking, so it can be said that the instructor, by not connecting to the students’ definitions, made them reluctant to participate in further class discussions. the teacher’s interruptions of students’ talk also made it difficult for the teacher to build on and develop students’ thinking if the students took the interruptions to be an act of authority. conclusions top ↑ allen (2011) describes the classroom where democratic mathematics education cannot take place: the students learn the same way, work toward a single best solution, minimise interaction and teamwork, and focus on the mathematical ends or answers rather than the means or processes. the participants in this research pointed at some of the opposite of these acts as didactic democratic acts, as they perceived them in the frame of a pedagogic training course. some of these acts are related to didactic freedom in the mathematics classroom, for example encouraging students to suggest their own ways of solving mathematical problems and freedom of self-expression. other acts are related to didactic equality in the mathematics classroom, for example allowing students to have different ways to learn new mathematical topics, so that they have equal opportunities to learn and understand mathematics. the rest of the acts are related to didactic dialogue in the mathematics classroom, for example collaborative discussions on mathematics education issues amongst students to arrive at new mathematical concepts, justification of claims and communication using natural language. the students’ perceptions of democratic didactic acts, as described above, agree with ellis and malloy (2009) about the democratic mathematics classroom, but the participants also pointed to a need for freedom of students to suggest their own ways of solving mathematical problems, probably because this advances their mathematical thinking and achievement (fraivillig, 2001; grouws & cebulla, 2000). these acts, in addition to being democratic (vithal, 1999, as cited in ellis and malloy, 2009), would result in a better environment for students’ learning of mathematics (staples & bartlo, 2010).didactic acts that the students identified as undemocratic were: freedom acts, such as withholding the freedom to decide or to ask question (see e.g. olaye, 2008); equality acts, such as not giving diverse examples; and dialogue acts, such as not connecting to students’ ways of doing mathematics, and interrupting students’ work or discussions. the identification of these acts emphasises the importance and positive influence of specific practices in the classroom in general and in the mathematics classroom in particular. the results also extend the repertoire of these practices to include connecting to students’ ways of doing mathematics. regarding pedagogic democratic acts in the mathematics classroom, the participants in the research identified the following acts as democratic: freedom acts, such as the freedom to criticise, to express opinion, to decide and to act; equality acts, such as exchanging roles with the teacher; and dialogue acts, such as exchanging ideas. these pedagogic acts, implemented in the democratic classroom, would positively influence the learning outcomes (isman, altinay & altinay, 2004). the participants identified the following pedagogic acts as undemocratic: freedom acts, such as hindering, interrupting, prohibiting, putting limits or criticising students’ acts; equality acts, such as discriminating between students and performing unclear acts that obscure the equality in grade assignment; and dialogue acts, such as ending class discussions. implementation of the mentioned democratic acts and avoidance of the undemocratic acts would not only result in a better environment for students’ learning of mathematics, but they would also help to make students better citizens (skovsmose, 1998). the findings of the research imply that instructors should try not to give certain students more time or opportunities to express themselves or act in the mathematics classroom than other students. such discrimination makes the other students feel unequal and possibly discourages them from further engaging in the class (povey, 2010). the findings also suggest that instructors should not exert their power to stop the flow of students’ actions in the mathematics classroom, because this troubles students and causes them to lose their control of their actions, especially when teachers’ power is not perceived positively (botas, 2004). instructors should try to connect to students’ ways of doing mathematics, especially their ways of defining mathematical terms and giving solutions to mathematical problems, because otherwise students do not appreciate the correct ways of doing mathematics and defining its terms. further, this connecting to students’ ways of doing mathematics helps them to develop their thinking (schifter, 2001). specifically within the palestinian context, it is hoped that more emphasis will be put on democracy in classrooms, especially mathematics classrooms, where students gain a powerful knowledge for their future life. this emphasis will provide students with better foregrounds – interpretations and conceptualisations of their future, their possibilities, and their life conditions given the social, cultural, economic and political environment in which they live (skovsmose & valero, 2005) – which, in turn, will positively influence what the students do and want to do, by providing them with resources and reasons to get involved in their learning and society as acting persons. this is especially important in palestine, where the israeli occupation still prevails and negatively influences the foregrounds of students. acknowledgements top ↑ competing interests the author declares that he has no political, cultural or personal relationships which may have inappropriately influenced him 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(2005). analyzing the power relationships in mathematics classroom. journal of the korea society of mathematical education series, 9(2), 115−124. abstract introduction literature review and theoretical framework study methodology findings of the study conclusion acknowledgements references about the author(s) liveness mwale department of mathematics, science & business education, tshwane university of technology, south africa willy mwakapenda department of mathematics, science & business education, tshwane university of technology, south africa citation mwale, l., & mwakapenda, w. (2018). ‘eighteen hands high’: a narrative reading of animal farm from a mathematical perspective. pythagoras, 39(1), a403. https://doi.org/10.4102/pythagoras.v39i1.403 original research ‘eighteen hands high’: a narrative reading of animal farm from a mathematical perspective liveness mwale, willy mwakapenda received: 30 oct. 2017; accepted: 05 sept. 2018; published: 15 nov. 2018 copyright: © 2018. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract this article addresses the interconnection between two education practices: reading and mathematics. these are two common aspects of schooling. learners engage with these critical practices, regularly. there hardly goes a day in the life of schoolgoing children without them engaging in these two practices in one form or another. however, the point of this article is to examine how these two practices come together in activities of learning mathematics, and seeing mathematical ideas within the context of reading texts that may be considered non-mathematical. the question we address is: to what extent are high school mathematics learners able to see mathematics in non-mathematical reading texts? we examine this question based on an analysis of research project data collected from south african high school learners’ interactions with animal farm, one of the 2015 grade 10–12 south african english home language literature books. the learners were drawn from five schools in three provinces, namely limpopo, gauteng and eastern cape. in the first phase of data collection, 430 grade 10–12 learners were purposively sampled to participate in the study. in the second and third phases of the study, 100 out of the 430 learners were purposively sampled. these were learners who had read animal farm before the time of data collection. based on our analysis, we argue that high school mathematics learning and teaching do not adequately prepare learners to be able to see mathematics in spaces that may be isolated from traditional environments in which learners learn mathematics. introduction in his book animal farm, george orwell (1945) writes: boxer was an enormous beast, nearly eighteen hands high, and as strong as any two ordinary horses put together. (p. 2) there were fifteen men with half a dozen guns between them. (p. 63) food stuffs had increased by 200 per cent, 300 per cent, or 500 per cent. (p. 57) these excerpts from animal farm are examples of the many instances in which mathematical ideas, especially those connected to number, are used in the book. the author used his ‘common’ knowledge of mathematics and his familiar language to present the story of animal farm using mathematical ideas explicitly or implicitly. it is possible that the author’s intentions were not to present mathematics or mathematical ideas, but because some storylines needed the use of mathematical language, he could not do so without using mathematics. this article emerges from a study that assessed learners’ abilities to interpret what they read and in particular, to ‘see’ mathematical aspects in the book animal farm. the study sought to find out learners’ abilities to read mathematically since mathematics is a specialised language that requires a specialised domain of practice. animal farm was one of the english home language literature books for high school learners in grades 10–12 in south africa in the 2015 academic year. according to the department of basic education (2014), other novels for english home language were the great gatsby (fitzgerald, 2008) and pride and prejudice (austen, 2008). learners were presented with excerpts from animal farm such as the ones quoted above. they were required to identify the mathematics part of the excerpts and to interpret what the excerpts meant. in the first excerpt, the mathematics part is ‘eighteen hands high’. according to conversion rates one adult hand is approximately 0.1016 m long. therefore, boxer’s height in metres was approximately 1.83 m. it was important for learners to understand this mathematical aspect in order to make sense of the extract. without this understanding, the statement: ‘eighteen hands high’, does not make sense as one reads it in the printed media. understanding what one is reading and how one needs to read is a critical skill required in relation to learning and achievement in education generally and mathematics education specifically. results from several international educational achievement tests such as trends in international mathematics and science study (timss), progress in the international reading literacy study (pirls) and southern and eastern africa consortium for monitoring education quality (sacmeq) and local standardised tests – national school effectiveness study (nses) and annual national assessments (anas) – show that the vast majority of south african learners in all the grades are significantly below where they should be in terms of the curriculum and, more generally, have not reached a host of normal literacy and numeracy milestones (spaull, 2011). national averages of 30% – 35% on tests of mathematics and languages are the norm for tests calibrated to measure grade-appropriate performance as a 50% score, and can be seen in both the nses and the ana evaluations (taylor, 2011). the ability to read and comprehend mathematical ideas in non-mathematical texts is important in learning mathematics especially at grades 10–12 (further education and training, fet) level. stanic and kilpatrick (1989) pointed out that non-mathematical texts have non-routine mathematical problems and non-routine problem-solving skills. non-routine problem-solving is characterised as a higher level skill to be acquired after the skill of solving routine problems which in turn is to be acquired after learners learn basic mathematical concepts and skills. the study from which this article emerges analysed fet learners’ readabilities (abilities to read) of mathematical texts in a non-mathematics specialised book, animal farm. the study attempted to address two problems affecting mathematics learning in south africa. firstly, in terms of the problem of poor performance in mathematics and literacy tests, we addressed the question: how much are learners able to see, identify and comprehend mathematical ideas in non-mathematical texts? secondly, are learners at fet level able to extend their routine mathematical knowledge to non-routine mathematical knowledge in non-mathematical texts? literature review and theoretical framework reading and mathematics are regarded as basic activities in schools. however, research findings lament learners’ inadequate attainment of these basics. learners’ attain basics that enable them to solve well-defined, ready-made, classroom problems but are unprepared to deal with ill-structured situations that they face in real life contexts. learners are found ill-prepared to deal with real-life situations. educators have begun to argue that the traditional sense of basic skills as algorithms for acquiring stable bodies of facts has created rote thinkers. if, instead, students are to become critical thinkers, then a new view of the ‘basics’ must be forged (borasi & siegel, 2001). learners do not seem to acquire the benefits of the basics of education. what is reading? in the conceptualisation of the study, several meanings of reading were explored. the seminal work of lee (1969) provided two categories of answers to the question ‘what is reading?’ she defined reading as ‘translating symbols into sound, saying words, getting meaning from the printed page’. she also defined it as: bringing personal meaning to the printed page, reacting to the ideas, evaluating the author’s recorded thoughts, gaining increased understanding through experiencing the recorded understandings of another. (p. 403) lee’s first definition involves the mechanics of reading, the superficial (not by any means unnecessary) phases of reading. the second definition involves more personal, more true aspects of reading by giving meaning to the printed words in the readers’ own understanding and experiences. the first definition assumes that the meaning of what one is reading is in the printed words while the second definition assumes that the printed words have no meaning in themselves, but that the reader gives meaning to the words. lee’s second definition is among others more personal, transformative, political, radical and progressive. allen (2000) defined reading as a process of moving between texts, moving out from the independent text into a network of textual relations. meaning becomes something that exists between a text and all the other texts to which it refers and relates. the text becomes the inter-text. allen’s definition contends that the written text does not have meaning of its own. it depends on other texts to which it refers and relates to provide meaning to the reader. the reader makes sense of the text in an interrelated way. this definition of reading shows the relationship that exists between and among texts in bringing meaning to the reader. the definition is in line with lee’s first definition of reading as ‘getting meaning from the printed page’. however, allen goes a step further to explain the interrelatedness of texts in bringing meaning to the reader. otherwise, it is difficult for the reader to make sense of the texts without what he calls inter-text. educational research has pinpointed a number of factors that contribute to underachievement and underlying many of these factors is reading ability. differences in the sociocultural backgrounds and home backgrounds of certain groups of and individual students have been shown by a large body of research to affect the ways in which students learn (pretorius, 2002; pretorius & ribbens, 2005). research has shown that a literate home background in which parents read a great deal themselves and also read to their young children, and in which there is a significant collection of varied books which children have opportunities to read themselves, contributes significantly to the level of achievement in later life. following on from this, a number of studies have shown that the amount of leisure reading in which individuals engage is directly related to their reading achievement (evans, 2002). most sub-saharan countries suffer from a scarcity of materials for students to read on their own, materials such as storybooks or informational texts. school libraries, classroom libraries and community libraries are rare (unesco, 2007). put simply, there are few opportunities for students to engage in the reading of extended texts that would allow them to practise their reading skills and engage in the kinds of choices around reading that promote independence (macro international, 2008). studies on the development of reading have shown that variations in children’s reading skills are associated with large differences in reading experience. children at the 80th percentile in reading level were estimated to average more than 20 times as much reading per day as children at the 20th percentile. many african education systems including south africa do not provide adequate reading opportunities to learners. learners have insufficient access to reading materials. this affects both their skill and will to read and develop into mature and effective readers to be able to cope with advanced academic work at the secondary school level. for example, according to sailors et al. (2014), in malawi, the core curriculum textbooks related to literacy instruction in the primary grades are the language textbooks for chichewa (the local language) and english texts only. there are no reading textbooks, nor is there a curriculum area called reading. chall, jacobs and baldwin (1990) observed that reading is a very powerful learning tool, probably the most important and empowering skill that learners need in a learning context. reading ability in the middle school years is crucial in later academic success. ‘reading science and social studies texts becomes an almost impossible task for students who cannot read at this level’. if schools wish to improve the overall academic performance of their learners, they should change their assumptions about what is important, and provide opportunities for their learners to develop their reading skills. reading mathematically adams (2003) explained that mathematics is a language that people use to communicate, to solve problems, to engage in recreation and to create works of art and mechanical tools. she defined mathematics reading as reading words, numerals and symbols in order to uncover the message of and about mathematics. included also in reading mathematics is the reading of graphics, diagrams and illustrations. adams identified three areas of focus in reading mathematics: words, numerals and symbols. words in mathematics may or may not have the same meaning as in everyday language or outside the discipline. for example, a ruler in mathematics means an instrument for measuring length while in everyday life it means someone in power or authority. examples of words that have the same meaning in mathematics and everyday life include dozen and time. words in mathematics are also used in context form to communicate ideas. for example, consider the statement: ‘write any two numbers whose product is 50’. the benefits of reading to learn mathematics include: (1) contributing to better learning and understanding of mathematical content, (2) developing new learning strategies useful in new learning situations and (3) developing a deeper understanding of mathematics as a discipline. reading to learn mathematics may be able to play a role in bringing about much-needed reform of the mathematics curriculum and lead to a reconceptualisation of the role of the traditional ‘basics’ in educating students as critical thinkers. a review of the research on the reading process shows how the concept of reading as a transaction contributes to the attainment of these goals for mathematics instruction (borasi & rose, 1989). reading fluency is a significant variable in secondary students’ reading and overall academic development. for both reading and mathematics, children’s performance at the end of elementary school is an important predictor of their ultimate educational success. if they have not mastered certain basic skills, they can expect problems throughout their schooling and later. research on reading indicates that all but a very small number of children can learn to read proficiently, though they may learn at different rates and may require different amounts and types of instructional support. furthermore, experiences in pre-kindergarten and the early elementary grades serve as a crucial foundation for students’ emerging proficiency. similar observations can be made for mathematics (kilpatrick, swafford, & findell, 2001). theoretical framework the study from which this article emerged was informed by dowling’s (1998) theory of mathematical practice. the theory of mathematical practice gives theoretical tools to approach and analyse school mathematical knowledge and how language affects the understanding of this knowledge. it gives a distinction between practice that is highly organised at the level of language and practice that generally lacks systematic organisation in language. dowling has thus identified four domains of mathematical practice: esoteric, expressive, descriptive and public domains to distinguish how mathematics content and expression of the content are used in mathematics practice. dowling uses the concept of classification to produce a model for analysing different types of mathematics statements in pedagogic texts and providing a language for describing relationships between school mathematics and other domains of practice. dowling’s work involves analysis of the relationship between mathematical and extra-mathematical knowledge, contents, discourse and practices. the argument in this regard is that academic (generally) and mathematical (specifically) activities are incommensurate with everyday activities and that academic mathematical knowledge cannot be used as a theory for facilitating adequate or appropriate understanding of everyday practices (north & christiansen, 2015). ensor and galant (2005) pointed out that the power of the theory of mathematical practice is threefold. firstly, it allows us to classify the content and mode of expression of school mathematics with respect to other subjects in the curriculum. secondly, it illuminates the task of apprenticeship, which is to move novices from the public (or other domains) into the esoteric domain. thirdly, it allows us to discuss the articulation between school mathematics and out-of-school practices. domains of mathematical practice dowling (1998) identified four domains of mathematical practice: esoteric, expressive, descriptive and public domains. these domains are characterised by the degree of specialisation of mathematical content and mode of expression employed in the messages through which the practices of the activity are transmitted (see figure 1). figure 1: dowling’s domains of mathematical practice. figure 1 shows two axes: the content axis and the expression axis. both axes have strong and weak classifications. classification here refers to the boundary strength between mathematics as specialised knowledge and practices, and everyday knowledge and practices (bernstein, 1971). along the content axis, the esoteric and expressive domains of mathematical practice have strong classification while the descriptive and public domains have weak classification. what this means is that the esoteric and expressive domains have a strong degree of specialisation of mathematical content which can unambiguously be identified as mathematics (north & christiansen, 2015). the descriptive and public domains on the other hand have a weak degree of mathematics content such that mathematics appears to be something other than mathematics, that is, ‘mathematics parading as something other than itself’ (dowling, 2013, p. 331). along the expression axis, the esoteric and descriptive domains have strong classification while expressive and public domains have weak classification. what this means is that the esoteric and descriptive domains use a strong mode of expression through which the message of mathematics is transmitted. the expressive and public domains use a weak mode of expression to transmit the message of mathematics. several researchers have used dowling’s domains of mathematical practice to analyse and discuss different issues in mathematics education. dowling developed the theory to help him to construct the sociological language of description for the analysis of school mathematics textbooks. this helped him to make empirical claims about the texts themselves and the wider practice of school mathematics in england and wales. north and christiansen (2015) used the domains of mathematical practice to analyse the examination papers for mathematical literacy in south africa. parker (2006) used dowling’s domains of mathematical practice to analyse the national curriculum statements for mathematics. this theoretical framework has also been used to analyse other mathematical texts (dowling, 1994, 1995), mathematics classrooms (coombe & davis, 1995; johnstone, 1993) and to recontextualise pedagogic practices in initial teacher education and initial teaching appointments. although the domains of mathematical practice have mainly been used to analyse mathematics textbooks and examination papers, this study used the domains in analysing learners’ understanding of mathematics in contexts other than the mathematics textbook. the domains were used to analyse learners’ access of mathematics outside mathematics textbooks and the mathematics classroom. this is important because it shows the connection between mathematics and everyday activities and practices. two important aspects were analysed and discussed using the four domains of mathematical practice. these are learners’ mathematical content knowledge in animal farm and learners’ understanding (and access) of mathematical aspects (content) in animal farm. the study used excerpts from animal farm (that are connected to the concept of number) as pedagogic texts for learners to interpret. ensor and galant (2005) explain that dowling’s model is valuable as it allows us to discuss the articulation between school mathematics and out-of-school practices. north and christiansen (2015) argue that dowling’s work involves analysis of the relationship between mathematical and extra-mathematical knowledge, content, discourse and practices. a central argument in this regard is that mathematical activities are incommensurate with everyday activities and practices. teaching and learning of mathematics can start with the non-mathematical content as a vehicle towards accessing the specialised mathematical content. conversely, teaching and learning of specialised mathematics content should enable learners to make connections between mathematics and out-of-school or everyday experiences. study methodology the study collected both qualitative and quantitative data, most of which were qualitative. data were collected from grades 10–12 learners from five public schools in three provinces in south africa using three different questionnaires in three phases. most learners from these schools were from a predominantly low socio-economic background. four out of the five schools had school feeding programmes. the school without a feeding scheme at the time of the data collection was in gauteng and had learners with similar characteristics to the ones from the other provinces. the data were collected towards the end of the 2015 academic year. the schools were purposively sampled as they were participating in a community engagement programme based at a tertiary institution. the community engagement programme involved providing in-service training to mathematics and science teachers and supporting the teaching of grade 10–12 learners in mathematics and physical science. the study purposively sampled grades 10–12 learners who were studying mathematics during the period of data collection. phases of the data collection in the first phase of data collection, a questionnaire was administered to 430 grades 10–12 learners. they provided information on the reading resources that were (made) available to them. they also reported on their favourite reading materials. the learners also provided information on how their reading materials were connected to mathematics. the second phase of data collection started with a document analysis of animal farm. animal farm was one of the books in the english home language curriculum in 2015 for the fet phase in south africa and was identified as one of the learners’ favourite reading materials in phase one. we read animal farm several times (document analysis) and identified mathematical aspects from the book. the mathematical aspects showed the connection that the book has with mathematics. a questionnaire, developed based on the connection between mathematics and animal farm, was administered to some learners who participated in the first phase of data collection. we purposively sampled 100 learners who had read animal farm by the time of data collection. the data were collected from 100 grades 10–12 learners who had read animal farm. in the third phase of data collection, a questionnaire was administered to 83 out of the 100 learners who participated in the second phase of data collection. these were the learners who were available during the day of data collection. the questionnaire was also based on animal farm but was different from the previous one. in this phase, deliberate efforts were made by the researchers to extract statements in animal farm that had connections with the concept of number. learners were asked to identify mathematics from the extracts and to explain the meanings of the extracts. the data collected from the three phases were entered into an excel spreadsheet and analysed. ethical considerations several ethical considerations were undertaken when conducting the study from which this article was taken (mwale, 2017). ethical clearances were granted from the department of education district office in march 2009 (reference tzimbizo/232009) to establish the community engagement project. learners who participated in this study were sampled from the project schools. further consent was granted by the same office to conduct research from the project schools. at school level, consent was granted by the principals, parents of participating learners and the learners. parents and learners signed consent forms to accept their participation in the study. they were, however, free to withdraw from the study anytime they felt compelled to do so. participating learners were assured of confidentiality of the findings of the study and that their identities would not be revealed at any point. in analysing learners’ responses below we use the notation l1, l2, … for learner 1, learner 2, …. the ethics committee of the university also gave approval for data collection in the sampled schools in june 2014. findings of the study in phase one of data collection, we found that the reading materials available to secondary school learners were mostly school textbooks in science, mathematics and english. these books were made available to learners by their schools. the other reading materials that were not made available by the schools were magazines, newspapers, storybooks and the internet but were read by few learners. for the reading materials provided by the schools, most learners read english books. they included novels, drama, short stories and poetry books. most learners also indicated that their favourite reading materials were english textbooks. in phase 2 of data collection, most learners were unable to indicate how the stories they read were connected to mathematics. the common response from those who read english books was that their books were storybooks and were not connected to mathematics in any way. similarly, learners were unable to show the connections between animal farm and mathematics in phase 2 of data collection. since learners were still unable to see or identify mathematical aspects from animal farm in the second phase of data collection, deliberate efforts were made in the third phase of data collection to extract statements connected to the concept of number and ask learners questions based on these aspects. learners were given an example to show them how the concept of number is implied in the statement. results from this phase showed that most learners were now able to see and identify mathematical aspects from animal farm especially from extracts that had explicit connection to the concept of number. learners only saw the connections when the extracts were isolated from the storyline. however, most learners were unable to interpret the extracts. they tended to give the literal meaning of the extracts that did not make sense. we therefore concluded that learners had a weak ‘mathematical gaze’ and comprehension skills to enable them to see and comprehend what they read. learners’ perspectives of mathematics from animal farm the questionnaire that the learners responded to had 10 excerpts, connected to the concept of number. the excerpts were taken from chapter 1 of animal farm. in choosing these excerpts, we ensured that they had number words or an application of the concept of number which could be identified by the learners. we ensured that the excerpts had enough words to convey the mathematical meaning they carried. as such, leaving out some words from the sentences that were extracted would still bring out the mathematical aspects captured in the sentence. is there mathematics in the excerpt? learners were asked to indicate if there was any mathematics in the given excerpts. table 1 shows the results. table 1: is there mathematics in the excerpt? table 1 shows the percentage and number of learners who were able to (yes), unable to (no) or did not respond to the question: ‘is there mathematics in the excerpt?’ there were 100 learners who were asked to respond to this question. the results show that most learners were able to ‘see’ mathematics in excerpts 1–4. the percentages of those who were able to see mathematics in these excerpts ranged from 66% to 78%. these excerpts had an explicit connection to the concept of number. for extracts that had an implicit connection to the concept of number, the percentages were lower. none of the extracts had 100% yes responses from the learners. this shows that some learners did not see mathematics in the given excerpts, even though some of them had number words such as eighteen, two and a thousand. the table shows that some learners did not provide any response to the excerpts. the number of non-respondents was large in excerpts 5–10. although it was not made known to us why many learners left the excerpts unanswered, non-response is likely to have arisen from learners’ uncertainty about possible connections between the excerpts and mathematics. the ‘mathematics part’ of the excerpts apart from identifying mathematics in the excerpts, learners were asked to identify the mathematics part of the excerpts. this helped the researchers to know if learners were indeed able to recognise or see mathematics in the excerpts. the data analysis showed that most learners were able to identify mathematics as it appeared in the excerpts. table 2 is an example of learners’ responses in relation to identification of the mathematics part of the excerpt ‘boxer was as strong as any two ordinary horses put together’. table 2: mathematics part of ‘boxer was as strong as any two ordinary horses put together’. table 2 shows aspects that learners identified as the mathematics parts of the excerpt ‘boxer was as strong as two ordinary horses put together’. in terms of number words, the expected response was ‘two’. sixty-three (63) out of 75 learners (i.e. 84%) were able to identify two as the mathematics part. other learners indicated that the mathematics part was ‘addition’. this response appeared to have been based on the learners’ reading of the phrase ‘put together’. although ‘put together’ is used in mathematics to mean addition, it did not carry the same meaning in this case. the following were some of the reasons why some learners thought that there was no mathematics in the excerpts: no numbers or numerical values in the statement. no mathematical language. because there is no such topic in math. none of the words are found in mathematics content. meaning of the excerpt after identifying the mathematics part of the excerpt, learners were asked to give the meaning of the excerpt to show their understanding of the excerpt. this is in line with lee’s definition of reading: being able to interpret, analyse and evaluate the author’s thoughts. table 3 is an example of learners’ responses on the meaning of the excerpt ‘boxer was eighteen hands high and strong’. table 3: learners’ responses to ‘boxer was eighteen hands high and strong’. table 3 shows learners’ responses upon being asked to provide the meaning of the excerpt ‘boxer was eighteen hands high and strong’. in this case the number word ‘eighteen’ is being used to show the size of the horse called boxer. we had expected learners to explain the meaning of the excerpt in relation to the height of boxer who was 18 hands high. in analysing learners’ responses, it is evident that most learners did not understand and appreciate the author’s use of ‘hands’ as a unit of measurement. learners l2, l4 and l5, for example, provided the literal meaning of the word hands, which does not make sense. learners l7 and l9 used their knowledge of number to explain the meaning of the excerpt as ‘the size of someone who is too tall and strong’. none of their responses recognised the word hands according to the author’s idea. findings such as these were evident in the analysis of responses to several other excerpts given to learners. interpretation of findings the analysis presented in this article shows that although most learners were able to see mathematics in the excerpts, they were unable to articulate the mathematical meaning of the excerpts. their meanings reflected simple paraphrases of the excerpts or literal meanings that did not make sense. for example, the following are some of the learners’ interpretations of the excerpt ‘boxer was as strong as two ordinary horses put together’: adding, adding two horses. adding two horses that are ordinary to make one strong horse. adding two horses with similar strength. boxer = 2 horses. two horses were put together. (mwale, 2017, p. 174) this observation raised several questions such as: did learners understand what they read: the story in general and the extracts in particular? why did their interpretations not connect to the story, and to mathematics if they now saw mathematics in the extracts? for example, the following are some of the learners’ interpretations of the excerpt ‘boxer was eighteen hands high and strong’: raising hands to count up to eighteen. he was helping or giving a hand to others. eighteen people raise their hands to measure someone. (mwale, 2017, p. 173) what do these findings imply for the learning of mathematics? in analysing the excerpts with respect to learners’ interpretations we found that the excerpts from animal farm can be characterised by low levels of institutionalised ways of expression and mathematical content. according to dowling (1998) they fall under the public domain: ‘the universe of mathematical statements which are unambiguously mathematical, either in terms of the content that they refer to, or in the language which is used to do this’ (p. 135). therefore, the excerpts could not be easily identified as mathematical in nature because of their weak mathematical contents and the weak mathematical language of expression used. as is the case with the public domain, they followed no set rules or specialised technical or mathematical language. animal farm is a storybook and therefore the language used is in most cases different from a mathematics textbook. a storybook and a mathematics textbook have what dowling (1992) describes as distinct regions of practice and have fundamentally different modes of practice. dowling (1992) describes mathematics as an example of a practice that is heavily dominated by explicit principles. in contrast, animal farm uses metaphors with implicit, implied and hidden meanings of words including mathematical aspects. for example, words such as ‘hour’, ‘eighteen’, ‘first’, ‘whole’ and ‘five times’ in the excerpts above have been used in metaphorical ways such that although they are mathematical in nature, they carry a different meaning in the excerpts. the metaphorical demands of the excerpts may have affected learners’ interpretations. secondly, orwell, the author of animal farm, used mathematical terms in his familiar, local and everyday mode of expression which is different from that of the participants of this study. the author was a european and this study was carried out in south africa. dowling (1996) refers to this difference as ‘language that is not within the readers’ everyday domain of participation’. for instance, expressions such as: ‘lose an hour’s sleep’, ‘eighteen hands high’, ‘two ordinary horses put together’, ‘the whole farm was asleep’, ‘first rate intelligence’, and ‘five times in succession’ may not be the learners’ everyday domain of participation. therefore, the text sent different messages to the learners other than what the author intended it to be. the purpose of the public domain in mathematics books is to help learners to access the esoteric domain. although the excerpts can be termed as public, they are not meant to help the learners to access the esoteric domain. the mathematical aspects in the excerpts serve a different purpose from the mathematical aspects in the mathematics practical problems. they appear as the authors’ way of communicating mathematical aspects in his familiar, everyday language and not necessarily expressing mathematics in everyday, familiar language. harel and kaput (1991) explain that the to-and-fro movement between mathematical and everyday representations in such a case contrasts with descriptions of the one-way movement from everyday to mathematical meanings in mathematics. morgan (1998) and sfard (2008) argue that the activity of explaining using practical terms differs from the deductive reasoning based on mathematical definitions and theorems that characterises academic and advanced mathematics discourses. therefore, the excerpts from animal farm could not easily send a message of mathematics to the learners because they were not meant to send such a message. in addition to language issues that affected learners’ abilities to see and interpret mathematical aspects from animal farm, the study also unveiled learners’ inadequate mathematical knowledge and comprehension (reading) skills. adequate mathematical knowledge could enable them to see mathematics from animal farm as identified by the researchers. dowling (1998) argued that gaining mastery of esoteric domain mathematics equips one with a mathematical ‘gaze’ with which one can look out upon the world and ‘see’ mathematics in it. this gaze results in the incorporation of aspects of everyday settings into mathematics learning. this raises the question of whether secondary school learners are attaining the esoteric domain of mathematical practice to enable them to acquire the mathematical gaze. dowling (1996) pointed out that the other domains of mathematical practice do not adequately prepare learners to see and do mathematics like the esoteric domain. he argued that school mathematics is driven by the esoteric domain with its intents, purposes, self-referential and closed. school mathematics might recruit a whole range of non-mathematical texts and contexts for pedagogic purposes, but this ‘use’ of mathematics is by and large confined to the school. the recontextualised mathematics is not the core of school mathematical knowledge. it is what teachers and textbooks use to help learners to access subject matter knowledge and it is used in schools only. north and christiansen (2015) reported that the predominant use of public domain inhibits mathematical understanding and affords only a limited degree of lifelong preparation. the implication for mathematics books that use the public domain more than necessary is that learners may not be able to access the mathematical knowledge they are meant to learn. in his empirical studies of school textbooks, dowling (1998) identified practical problems that obscure the specialised mathematical knowledge and thus close opportunities for participation in school mathematics. this finding has an implication on solving word or practical problems in mathematics. le roux and adler (2015) explain that solving a practical problem involves first recognising the problem as a particular type of mathematics problem, or genre, and then recognising the mathematical-practical boundary and the specialised mathematical knowledge that casts a gaze on the practical. can learners without a mathematical gaze see mathematics in word problems? and if they cannot see mathematics, can they solve word problems? another critical issue that this study has highlighted is learners’ limited comprehension skills. the study showed that learners were unable to interpret what they read. this finding has an implication for mathematics learning and for learning in general. do learners understand what they read or learn? in mathematics, learners’ inability to interpret what they read is clearly shown in word problems. a number of studies have highlighted several challenges that learners face in solving word problems. dossey (2000) pointed out that the linguistic form of word problem texts affects students’ efficacy in solving word problems. for bilingual learners, another known factor related to the problem of comprehension is their proficiency in the language in which the problem is stated. bernado (1999) reported that the ability to understand word problems is affected by internal variables or factors related to the student’s prerequisite knowledge and information processing skills related to the use of knowledge. le roux and adler (2015) investigated how linguistic features of practical problems represent mathematics, act textually by relating texts and identifying people. they reported that a text may be included or excluded or given significance in representing objects and activities in the practical problem. depending on how the learner understands and interprets the text, it can diverge from what it is expected to represent and in so doing produce a ‘hybrid’ text. implications for mathematics learning the south african fet mathematics curriculum covers 10 main content areas: functions, number patterns, sequences and series, finance, growth and decay, algebra, differential calculus, probability, euclidean geometry and measurement, analytical geometry, trigonometry and statistics (department of basic education, 2011). these are the non-arbitrary, esoteric domain content that learners are supposed to acquire by the end of three years of fet. according to the department of basic education (2011), one of the specific aims of the mathematics curriculum is ‘to be able to use contextual problems relating to health, social, economic, cultural, scientific, political and environmental issues whenever possible’ (p. 13). this objective provides an example of the connections between the mathematical and non-mathematical contexts that the curriculum aims at promoting. in this study we analysed learners’ perspectives of connections between mathematics and their english literature book, animal farm. the assumption was that they had acquired adequate subject matter knowledge and skills to enable them make connections, see, identify and interpret mathematics from what they read based on the intended curriculum. however, the analysis in this study showed otherwise. mwakapenda (2008) argued that connections are central to the way the discipline of mathematics, its learning outcomes and assessment standards are conceptualised. the notions of representation and integration are found to be key aspects in understanding connections in mathematics, making links within itself and other disciplines. mathematics is not about reasoning for its own sake; it is concerned with processes that are connected to activities and problems of the social, physical and mathematical worlds involving human practices in all cultures. however, connections in mathematics cannot be easily made by someone who lacks the conceptual understanding of mathematical concepts and procedures and relations between concepts and procedures (kilpatrick et al., 2001). donoghue (2001) proposed that a strong mathematics background helps students to make connections among mathematical ideas and also between mathematics and other content learning areas. this raises the question of whether secondary school learners are achieving the necessary conceptual understanding to enable them to make connections between mathematics and what they read. can they make connections between mathematics and ‘contextual problems relating to health, social, economic, cultural, scientific, political and environmental issues whenever possible?’ (department of basic education, 2011, p. 13). this has an implication on the way learners are taught mathematics especially to acquire conceptual understanding. the learners’ inability to make connections between mathematics and what they read showed their lack of understanding of mathematical concepts including the basic number system. with this deficiency, how do we expect them to learn the more complex ten topics outlined for the fet mathematics curriculum? boaler (2013) showed that when mathematics is opened up broader mathematics is taught, mathematics that includes problem-solving, reasoning, representing ideas in multiple forms and asking questions, improving learners’ performance and encouraging them to study mathematics at advanced levels. mathematical problems that demand connection making and creativity are more engaging for learners of all levels. when all aspects of mathematics are encouraged, rather than procedural fluency alone, many more learners participate in the learning process, leading to higher achievement. boaler referred to this opening and broadening of the mathematics taught in classrooms as mathematical democratisation. narrow mathematics teaching combined with low and stereotyped expectations for learners produce low achievement in mathematics. in this article, we have examined the interconnection between two education practices: reading and mathematics. in particular, we have examined how these two practices come together in activities of learning mathematics and seeing mathematical ideas within the context of reading texts that may be considered non-mathematical, as in the case of animal farm. we have presented our analysis of learners’ understanding of mathematical ideas by looking at the context of the reading materials (animal farm) available to them. considered in this way, our study is at the interconnection of the domain of mathematics and the domain of language (everyday ordinary public language). several studies have emphasised the importance of recognising this problematic interface in mathematics teaching and learning (barwell, 2014; moschkovich, 2008; planas & civil, 2013; setati, 2008). our analysis has shown that learners were not able to see mathematics embedded in animal farm. that is, they were largely unable to see mathematics in spaces that may be isolated from traditional environments in which they learnt mathematics. we argue that the reason for this finding is due to the fact that mathematics learning and language learning occur as separate practices. in particular, we contend that this finding is due largely to the inability of mathematics teaching to draw on the non-mathematical domain, the domain that is more easily available and accessible to learners. our argument concurs with contextualisation perspectives advanced by lakoff and nuñez (2000) in relation to the interface between mathematical cognition and everyday reality. lakoff and nuñez have explored the question ‘how much of mathematical understanding makes use of the same kinds of conceptual mechanisms that are used in the understanding of ordinary non-mathematical domains?’ (p. 28). our finding suggests that meaningful mathematics teaching and learning need to connect with and make use of everyday understandings that are located in spaces and domains that may be considered non-mathematical. conclusion in this article, we have addressed the interconnection between two education practices: reading and mathematics. we have reviewed the literature related to reading and reading mathematically. we have examined how these two practices come together in activities of learning mathematics and seeing mathematical ideas within the context of animal farm, a textbook that may be considered non-mathematical. we explored the extent to which grades 10–12 mathematics learners were able to see mathematics in animal farm. the analysis showed that learners were able to see mathematics in animal farm only when they were given excerpts that had mathematics words and applications related to the concept of number. we have observed that animal farm falls into the public domain of knowledge. we have argued that high school mathematics knowledge (esoteric domain) is largely inconsistent with the public domain knowledge. the learners’ inability to see mathematics in animal farm points to the fact that school mathematics learning and teaching do not necessarily prepare learners to be able to see mathematics in spaces such as animal farm. ability to see mathematics in such spaces requires a recontextualisation of mathematics knowledge beyond what may be accessible based on exposure to environments in which learners 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(2007). education for all by 2015: will we make it? paris: oxford university press. microsoft word 64 front cover final.doc 14 pythagoras 64, december, 2006, pp. 14-28 mathematical literacy in south africa and functional mathematics in england: a consideration of overlaps and contrasts hamsa venkatakrishnan and mellony graven marang centre, school of education, university of the witwatersrand venkatakrishnanh@educ.wits.ac.za and gravenm@educ.wits.ac.za introduction the call for students with a flexible range of quantitative skills that can be applied in a diverse range of contexts has gained considerable ground internationally over the last decade or so (kilpatrick et al., 2001; steen et al., 1990). lyn steen, a key american proponent of what he terms ‘quantitative literacy’, suggests that in everyday life and work situations, the knowledge and skills involved are related to, but different from, those associated with traditional mathematics learning: whereas the mathematics curriculum has historically focused on school-based knowledge, quantitative literacy involves mathematics acting in the world. typical numeracy challenges involve real data and uncertain procedures but require primarily elementary mathematics. in contrast, typical school mathematics problems involve simplified numbers and straightforward procedures but require sophisticated abstract concepts. the test of numeracy, as of any literacy, is whether a person naturally uses appropriate skills in many different contexts. (steen, 2001: 6) a mathematical literacy programme, aiming to address this area, and offered as an alternative to the mathematics programme, has recently (january 2006) been introduced in grade 10 classes (learners generally aged 15) across south africa. this programme is intended to run across the further education and training (fet) phase – grades 1012, and will lead to a mathematical literacy qualification that is viewed as suitable for learners who wish to proceed onto courses at higher education level that do not have a significant mathematical content, or into vocational courses or employment. mathematical literacy qualifications at fet level have been available within the adult learning sector since 2001, but this paper focuses primarily on the current school-based introduction of mathematical literacy, incorporating the more general aspects of critiques that have been undertaken of the programme on offer in the adult sector. in this paper, we consider the policy context of problems and aims within which mathematical literacy was proposed and discussed, before being formalised within policy texts, and compare this trajectory with the notion of ‘functional mathematics’ which is currently being discussed in england, with small-scale trials of curriculum and assessment models for the course in the 14-19 age range scheduled for september 2006. mathematical literacy has been defined within the south african national curriculum statement for the fet phase in the following terms: mathematical literacy provides learners with an awareness and understanding of the role that mathematics plays in the modern world. mathematical literacy is a subject driven by life-related applications of mathematics. it enables learners to develop the ability and confidence to think numerically and spatially in order to interpret and critically analyse everyday situations and to solve problems. (department of education (doe), 2003a: 9) functional mathematics in england is currently defined thus: • each individual has sufficient understanding of a range of mathematical concepts and is able to know how and when to use them. for example, they will have the confidence and capability to use maths to solve problems embedded in increasingly complex settings and to use a range of tools, including ict as appropriate. • in life and work, each individual will develop the analytical and reasoning skills to draw conclusions, justify how they are reached and identify errors or inconsistencies. they will also be able to validate and interpret results, to judge hamsa venkatakrishnan and mellony graven 15 the limits of their validity and use them effectively and efficiently. (qca, 2005: 2) clearly, there are overlaps in these definitions – particularly in their common stress on applying mathematics across a range of contexts, and the focus on developing both an understanding of concepts and a willingness to bring application skills to bear when solving ‘realistic’ problems. thus, a comparison of the aims and current structuring of the two policies rests upon some degree of shared vision for what they wish to put into place within the field of mathematics education. there has been criticism in south africa and england that policy studies have tended to focus on policy implementation, whilst leaving the policy itself outside the frame of discussion (in england, ball, 1997; vally, 2003 in south africa). in order to overcome this criticism, and to facilitate an analysis of the historical backcloths to the introduction of policy in the two countries, we use the first two phases of what ball (1994) has termed a ‘policy trajectory’. policies are viewed here in three contexts: the ‘context of influence’, in which the concerns and discourses that lead to the constitution of texts associated with a policy are considered; the ‘context of policy text production’ – the texts themselves and the discourses surrounding their introduction; and the ‘context of practice’, in which the interpretations of policy into practice and the effects of these interpretations are considered. this analytic tool highlights the dynamic nature of policy, showing how policy comes to take particular forms, and stresses the ongoing, dialectical and contested nature of debates around issues and goals. the enactment phase – the ‘context of practice’ has just begun in south africa; in england, the ‘context of influence’ is currently feeding in to the development of policy texts for functional mathematics. given the comparative nature of this exercise, we summarise in table 1 (see next page), the way in which the schooling and qualifications structures in the two countries’ senior phases are linked within their respective national qualifications frameworks (nqf), and locate mathematical literacy/functional mathematics within these frameworks. we then briefly detail key elements of the landscape of mathematics education in these phases in both countries, including some of the problems that the new courses are intended to address. south africa approximately 75% of learners in south africa currently continue their schooling into the fet phase (doe, 2005b). this phase in south africa consists of grades 10-12, with learners generally aged between 15 and 18. the introduction of mathematical literacy alongside mathematics makes a mathematically-orientated course compulsory for all these learners. this aims to remedy a situation in which just over 40% of all the end of fet phase senior certificate candidates nationally took no mathematics courses at all in the fet phase, another 50% approximately were entered for the standard grade (sg) mathematics examination, and under 9% were entered for the higher grade (hg) examination – the course and examination pass needed for entry to higher education courses with a significant mathematical component (all data from 2003, taken from perry, 2004). an analysis of trends across the previous decade or so showed that whilst enrolment for sg mathematics had increased significantly, pass rates had dropped, and that whilst hg pass rates had improved, enrolment at this level had plummeted. the pass rate for mathematics at sg or hg of just under 50% compares poorly with the general pass rate for the senior certificate of 73%. england the 14-19 age range in england encompasses two phases of education – key stage 4 and the postcompulsory 16-19 phase. key stage 4 covers years 10 and 11 (learners aged 14-16) and represents the final phase of compulsory schooling, culminating with most students taking general certificate of secondary education (gcse) examinations in a range of subjects. gcse mathematics is currently three-tiered into higher, intermediate and foundation levels, with overlapping grade boundaries. a ‘c’ grade in mathematics at gcse is generally required for entry into higher education. in 2003, 94% of the age cohort was entered for the gcse exam, with 48% of this cohort gaining at least a ‘c’ grade (data taken from the dfes statistical bulletin, cited in smith, 2004: 59). the proportions of students continuing in fulltime education in the post-compulsory phase has risen significantly over the last thirty or so years (ofsted, 2005), with over two thirds of all 17year olds now remaining in full-time education or training. these students can select from a range of academic or vocational courses. particular concerns have been expressed though about the low proportions choosing to take mathematics 16 courses in this age range in england (mathematics is not compulsory in england beyond the age of 16), in comparison to other european countries (tomlinson et al., 2004), and the particularly low proportions (6.5% of the age cohort in 2002) taking mathematics at advanced level (smith, 2004) – the qualification needed for entry to mathematically based disciplines in higher education. given these backcloths, we now consider the policy trajectory in the two countries in terms of the context of influence. the context of policy text production is analysed in south africa only, as england has not yet moved into this stage. within this exploration, we look at the combinations of spheres of influence, concerns and goals that led to the perceived need for programmes focused on quantitative literacy, and the ways in which these have become enshrined within south africa’s policy texts. table 1: schooling and qualification structures senior phase in south africa and england south africa nqf school year ave age of learners maths literacy (ml) notes 1 grade 9 last year of compulsory schooling 15 end of general education and training (get) phase notions of ml are strongly incorporated into mathematics in this phase 2 grade 10 16 3 grade 11 17 4 grade 12 18 fet ml focuses on these years and is offered alongside fet maths. end of fet phase – senior certificate exam taken england nqf school year ave age of learners func maths (fm) notes entry 1 gcse maths grades d g 2 years 10 & 11 year 11 last year of compulsory schooling 14-16 fm is to be offered across all these levels & will be a compulsory part of gcse maths gcse maths grades a* c 3 year 12 & 13 16-19 fm is offered as a separate option at this level a range of mathematical courses currently available including the traditional ‘a’level maths, but mathematical courses are not compulsory in this phase hamsa venkatakrishnan and mellony graven 17 south africa context of influence in the process of attempting to dismantle the machinery of apartheid within education, the african national congress (anc) party’s early policy documents stressed the need for both access and redress, coupled with the need for economic growth (anc, 1994). also evident within these priorities was an emphasis on the disciplines of science, mathematics and technology as critical to achieving these goals, with existing curricula described as “academic, outmoded and overloaded”, and calls for science and mathematics education to be: transformed from a focus on abstract themes and principles to a focus on the concrete application of theory to practice. it must ensure that students and workers engage with technology through linking the teaching of science and mathematics to the life experiences of the individual and the community. (anc, 1994: 84) several aspects pertinent to the development of mathematical literacy are highlighted in these quotations – the need for mathematics to be linked more closely to real-life, for mathematics to shift away from a focus on the abstract towards the concrete, and away from an overriding focus on content, and the need to integrate education (for learners) and training (for workers). underpinning the school-based sector in particular, there was an emphasis on a rounded, liberal education which emphasised the need for active citizenship and political critique (christie, 1999). through the 1990s, an extensive array of governmental white papers and education and training acts put policies into place, which, to varying degrees, focused on the goals outlined in the quote above. the south african qualifications authority (saqa) was established in 1995 to oversee the integration of formal school and vocational qualifications within a national qualifications framework (nqf). an eight-level model was developed in which level 1 certification was equivalent to the end of the general education and training (get) phase certificate, level 4 was equivalent to fet certificate level, levels 2 and 3 were broadly defined as running parallel to grades 10 and 11 in schools, and higher levels ranged across from graduate to doctoral level qualifications. concurrently, the curriculum development working group within the department of education (doe) produced a discussion document (curriculum framework council, 1995) aiming to translate into practice the anc desire to replace the ‘outmoded’ curriculum. this document introduced the notion of replacing traditional subjects with ‘integrated learning areas’; mathematics became a part initially of the ‘numeracy and mathematics’ learning area, which later became mathematical literacy, mathematics and mathematical sciences, or mlmms. mathematical literacy, on paper at least, was ‘born’ here. saqa adopted the notion of learning areas, including mlmms, and also took on board the adult training led shift to using ‘competences’ to define the curriculum. once again this was driven by a desire to move away from an overemphasis on content, and defined ‘outcomes’ that the curriculum should produce for learners; some highly generic – e.g. “participating in civil society and democratic processes through understanding and engaging with a range of interlocking systems”, and others related to specific areas of study. this approach of ‘outcomes-based education’ subsequently became a key feature of curriculum 2005 (c2005), which was introduced in 1997, and is currently in place across the get phase (grades 0-9). within mathematics, implementation of c2005 was seen primarily in the transition to theme-based, contextualised learning, and group work (graven, 2002). however, the highly sought integration between education and vocational training did not flow through into the department of education’s curriculum for schools. the reason for this lay with saqa’s preferred approach to structuring the curriculum. led by the concerns of adult education (greenstein, 2003) and based on selections of modular units with associated ‘unit standards’, this model was rejected by the department of education for schooling. the department chose instead to retain the format of holistic learning programmes. this ‘disconnect’ plays into the current situation in south africa where there are two distinct mathematical literacy programmes – the saqa unit standard-based programme which is used in the adult learning sector, and the department of education’s mathematical literacy programme of work detailed within their fet curricular statement which is focused on the school sector. both programmes are based on very similar mathematical content, and lead to fet level certification, but their associated structures of learning follow historical differences in their patterns of work – modular block training models based on a selection of unit standards to make up the necessary 16 credit minimum in the adult 18 training sector, and grade-based holistic programmes of study in schools. there were other more indirect pressures for change in the existing provision of mathematics in the fet phase. it was argued that there was a mismatch between the process-oriented programmes of study within c2005 in the get phase and the much more traditional content oriented programmes in the fet phase (chisholm et al., 2000), and that the transition between these phases was particularly problematic in subjects relying on hierarchical development of concepts, mathematics being one of these (taylor & vinjevold, 1999). there was also criticism of the end of fet phase senior certificate exam, as failing to meet the needs of learners, universities or employers (doe, 1998), and calls in the same report for schools to provide alternative pathways integrating more vocationally-oriented programmes in the fet years. additionally, the government pointed to schools’ abusing the hg/sg curriculum differentiation model by entering many more candidates for the sg band in order to secure better pass rates for themselves – bodies representing the higher education sector were particularly concerned that this abuse further reduced the pool of learners studying higher level mathematics, and consequently, restricted many learners’ access to mathematical disciplines at graduate level (sauvca/ ctp, 2003). the fact that the numbers entered for and passing mathematics were significantly lower than corresponding rates for other subjects continued to raise concerns (adler et al., 2000), which were further intensified by the widely publicised poor performance of south african learners on the international timms-r tests in 1999 (howie, 2001). there was, thus, a combination of factors through the 1990s which came to exert significant pressure for change in the existing provision for mathematics in the fet phase – pressures which culminated with saqa’s 1998 proposition that future qualifications awarded within fet equivalent bands (i.e. nqf levels 2-4) should contain a minimum of 16 credits in mathematics or mathematical literacy (i.e. approximately 13% of the credits required for certification at this level). this model of alternatives shifts the discussion into how these two options are ‘different’ from each other and to how they relate to each other – taken up within the next section in which we discuss the policy texts detailing curricular content and assessment models for mathematical literacy, the ‘context of policy text production’ in south africa. context of policy text production as noted earlier, there are two key sources of documentation on the curriculum and assessment of mathematical literacy in south africa – the saqa ‘unit standards’ for mathematical literacy and the department of education’s mathematical literacy curriculum statement, which details learning outcomes and assessment standards for school learners in grades 10-12 (department of education, 2003a). whilst our focus is primarily on the latter, we draw in more general critiques of saqa’s unit standards in order to gain a better understanding of the nature of curricula associated with mathematical literacy in south africa. the saqa proposition to make 16 credits in mathematics/mathematical literacy compulsory was taken up urgently within the adult training sector with hurried attempts made to put together unit standards in mathematical literacy in order to ensure that people enrolled on these programmes could gain certification (laridon, 2006). hallendorf’s (2003) and brombacher’s (2006) historical sections on the development of these unit standards note that given the time constraints for publication, this curriculum did use the existing fet mathematics curriculum as a starting point, and attempted to extract the ‘fundamental’ mathematics whilst removing the ‘pure mathematics’. the association for mathematics education in south africa (amesa) expressed concerns at the time that this approach would tend to work against the goals that mathematical literacy was expected to deliver: we are concerned that mathematical literacy should not be a ‘watered down’ academic mathematics but rather mathematics with a different emphasis. if the purpose of the fetc (fet certificate) is among other things to benefit society then the mathematics needed by the learner is not necessarily more (in terms of knowledge) than that covered at the get level, but rather the mathematical thinking skills – habits of mind – to be able to apply that learning in various contexts. and further that: we do not see mathematical literacy as necessarily developing a lot of new mathematical knowledge, but rather being about using and applying get mathematics in more sophisticated contexts and at higher levels of analysis and understanding. (amesa, 2001, see hamsa venkatakrishnan and mellony graven 19 http://academic.sun.ac.za/mathed/ames a/fetcomment.htm) a number of issues are flagged up here – that mathematical literacy’s predominant focus should not be to further learners’ mathematical ‘content’ learning, that the emphasis should be on applications, and that development in mathematical literacy should be understood in terms of learners’ ability and willingness to solve problems in increasingly complex contexts. the moves to restrict the further learning of ‘pure mathematics’ within mathematical literacy, alongside the simultaneous move to withdraw curricular differentiation along the sg/hg lines which existed previously, were criticised though in higher education-based submissions. sauvca/ctp, the body representing university vice-chancellors and technikon principals, argued for the retention of sg mathematics alongside mathematics/mathematical literacy, stating their concerns thus: we fear that the introduction of the new mathematics subject coupled with the alternative of mathematical literacy will see a stampede from mathematics to mathematical literacy. this has two disastrous consequences. first, far too few learners will take mathematics; secondly, those with only mathematical literacy will probably be effectively denied access to a crucial range of higher degree opportunities in science, engineering, the health sciences and commerce. (sauvca/ ctp, 2003: 1) criticisms in this vein, alongside the need to ensure that higher education recognised mathematical literacy as a valid prerequisite course for entrance to graduate study in nonmathematical disciplines, put pressure on curriculum developers to retain content that went beyond the scope of the get curriculum, in spite of amesa’s concerns that this would make the subject less appropriate for lower attaining learners and their needs in relation to mathematical literacy. thus content relating to use of the basic trigonometric ratios and the sine and cosine rules, for example, forms part of the department of education’s fet schools’ syllabus for mathematical literacy. in contrast again, hallendorf’s (2003) report, compiling feedback from a range of education and employers’ groups about saqa’s unit standards, having researched a number of international studies attempting to define and describe what constitutes mathematical literacy/quantitative literacy/numeracy, concluded that the standards in the level 2-4 range were: inappropriate in terms of the emerging definition of mathematical literacy, and expressed needs. (hallendorf, 2003: 29) the contestation of the mathematical literacy policy by different lobby groups is clearly evident here, with some parties seeking actively for programmes that would ‘re-form’ mathematics in ways that would work to broaden access and application skills, and others largely concerned with increasing the proportions capable of working with traditional mathematics. the ongoing nature of this contestation is evident in the recent draft report produced by brombacher (2006), reviewing, after consultation, saqa’s mathematical literacy unit standards in the level 2-4 range. his conclusions are expressed thus: the unit standards are a) too mathematical in nature to develop the attributes of mathematical literacy and b) too irrelevant for the qualifications they are meant to be developing fundamental skills for. (brombacher, 2006: 14) in structural terms, saqa locates mathematical literacy within the ‘fundamental’ mathematics subfield. all learners in the adult and school sectors aiming for qualifications in the level 2-4 range have to take either mathematics or mathematical literacy in order to be certified. the department of education’s decision to offer school learners the choice between mathematics and mathematical literacy has been criticised by proponents of mathematical literacy. the department’s argument here is that essentially, mathematical literacy forms a subset of mathematics and that mathematical literacy learning can be assumed within the learning of mathematics. thus, the department of education’s curriculum statement for mathematics states: mathematics is being used increasingly as a tool for solving problems related to modern society. the financial aspects of dealing with daily life are informed by mathematical considerations. mathematical ways of thinking are often evident in the workplace. the learning outcomes and assessment standards in mathematics are designed to allow all learners passing through this band to develop into citizens who are able to deal with the mathematics that impinges on the society they live in and on their daily lives. (doe, 2003b: 11) 20 this point though has been disputed by wider evidence in the field of mathematics education (schoenfeld, 1985), and conflicts also with the view that the nature and aims of mathematical literacy are very different from those associated with mathematics (brombacher, 2006). the department of education’s mathematical literacy curriculum (doe, 2003a) also keeps in place the basic structure of the mathematics curriculum. thus, whilst the definition given in the opening section and the section detailing the purposes of introducing mathematical literacy (for developing the “self-managing person”, the “contributing worker” and the “participating citizen” (doe, 2003a: 9-10)) stress the subject as useful in a broad, future-life oriented sense, the operationalisation of these aims into curriculum is done through learning outcomes in four contentoriented areas: “number and operations in context”, “functional relationships”, “shape, space and measurement” and “data handling”. this runs largely parallel to the learning outcomes for mathematics: “number and number relationships”, “functions and algebra”, “shape, space and measurement” and “data handling and probability”. amesa (2003) surmised that the reason for the use of a content-led approach based on these areas was for the sake of “portability and mobility” between mathematics and mathematical literacy, but cautioned against this: the two subjects are so dissimilar in philosophy and purpose that such portability and mobility should not be a consideration. (amesa, 2003: 4) amesa (2003) follows the arguments made by steen (2001) that curricula framed in terms of mathematical skills have tended to lead to teaching and learning focused on acquiring procedures, as opposed to developing mathematical attitudes. in this critique, amesa advocates a ‘behaviour’defined curriculum, more focused on the kinds of actions and attitudes that are helpful when faced with a problem context that can be ‘mathematised’, and as such, more in line with steen’s delineation of ‘expressions’ of quantitative literacy. the examples provided within the assessment standards in the curriculum statement, which break down learning outcomes across the three grades of the fet phase, do emphasise the use of ‘real’ problems, but the format tends to stress their use as useful ‘vehicles’ upon which mathematical content can be carried and then foregrounded (cooper & dunne, 2000; sethole, 2003). this view is reinforced within the department of education mathematical literacy curriculum statement’s “content and contexts” section (doe, 2003: 3843) in which bald lists of content for each grade and within each learning outcome are detailed, followed by a few paragraphs about incorporating these into appropriate contexts in order to facilitate learning. an alternative view to viewing contexts as a preamble to the mathematics is the one that stresses contexts as locations for application of prior mathematical learning. both of these views though, are explicitly disputed in department of education assessment guidelines documentation for mathematical literacy, which states: the emphasis in learning should be on enabling learners to develop mathematical knowledge while dealing with issues, rather than on applying mathematics after “learning the basics”. mathematical literacy should be imbedded in applications and extracted from problems in a variety of contexts. the most noticeable change in approach to the teaching and learning of mathematics in mathematical literacy is the delaying of formal methods (algorithms) in favour of extended opportunities to engage with mathematics in diverse contexts. learners will often meet problem situations in their adult lives for which there are no ready-made formulas or procedures to provide solutions. (doe, 2005: 14) overall, therefore, it would appear that there are mixed messages within the department of education’s documentation for mathematical literacy. whether educators will give more emphasis to context-specific problem solving using mathematics, or to the mathematics involved in solving contextual problems remains unclear at this stage. given the extensive evidence of assessment driving the emphases within the taught curriculum in ‘high stakes’ exams (wiley & yoon, 1995), the nature of the assessments that will be used within mathematical literacy are likely to be critical – in particular, the format, contexts and nature of tasks that are to be developed for the end of the fet phase external examination which has currently been given 75% of the overall weighting. we move on now to contrast this trajectory with the initial phase – the ‘context of influence’ for functional mathematics in england. hamsa venkatakrishnan and mellony graven 21 england context of influence policy intervention in education in england has been a common feature of the schooling landscape over the last twenty or so years, with intense focus on mathematics education over the last decade. policy initiatives over this last decade have affected all phases of education – the national numeracy strategy was introduced in primary schools (years reception to 6, students aged 5-11) in 1999, followed by its extension into secondary schools (years 7-11, students aged 11-16) initially as part of the key stage 3 in 2001, and latterly within the secondary strategy. the structure and sequencing of content of advanced level (‘a’level) mathematics (taken in the post-compulsory phase, students aged 16-18) was also changed in 2000. outside the arena of schooling, acute problems with low levels of basic numeracy across the adult population in england were also identified in the late 1990s (moser, 1997), detailing too, some of the consequences of this for individuals in their capacity as consumers and citizens, and for their employability. from a labour perspective, this position was set against the presence of serious skills shortages, a situation highlighted again in a recent skills white paper (department for education and skills (dfes), 2005). a number of influential reports were commissioned by the labour government in the first five years of the new century, looking at various aspects of the supply and demand chain for people with mathematical skills and qualifications. focusing largely on the upper end of the qualification range, roberts’ report (roberts, 2002) identified the mismatch between supply and demand of graduates in mathematics, physics and engineering sciences, and went on to detail the low proportions taking mathematics at ‘a’ level and the further haemorrhage of numbers choosing to go on to mathematics at degree level. also, in relation to the question of supply and demand, the report noted employers’ concerns about shortcomings within this relatively highly qualified minority: there are mismatches between the skills of graduates and postgraduates and the skills required by employers (for example, many have difficulty in applying their technical knowledge in a practical environment and are seen to lack strong transferable skills). (roberts, 2002: 2) turning back to the education and training sector, two inquiries were directed to look at education and training provision within the 14-19 age range. the smith report (smith, 2004) focused specifically on mathematics provision, whilst the tomlinson report (tomlinson, 2004) looked at provision more generally across this range. both reports noted the need to improve engagement with mathematics and motivation to learn in the compulsory phase (14-16 age range), in a context where disaffection and lack of engagement have been cited as significant and increasing problems for secondary schools (ofsted, 2004), in order to raise participation rates in mathematics and more generally in the post-compulsory phase (16-19). they also pointed to the lack of flexibility in curricular provision which compounded the problems reported above. as in the south african context, tomlinson’s recommendations in particular, argued for a much closer integration of academic and vocational programmes, but again similarly, the formats recommended to achieve integration were largely rejected by the department for education and skills (dfes, 2005). a key recommendation within the tomlinson report was the need to secure a ‘core’ programme of learning comprising functional mathematics, functional literacy and communication and ict. whilst the content of these subjects was not defined in this report, it was argued that the following criteria needed to be satisfied: [the content] should be based on a common understanding of what learners need to develop in each subject, including both knowledge and capacity to apply it. it must: • equip young people with the knowledge and skills in each subject that they will need to progress and succeed in learning, he, employment and adult life. that means that it can only be determined in consultation with end-users, including he, employers and community groups; • encourage progression to at least level 2 [equivalent to gcse grade c or above on england’s nqf], as young people move through the diploma framework, with opportunities and encouragement to progress to level 3 [equivalent to ‘a’level standard]; and • encourage the extended study of these subjects as part of main learning. (tomlinson, 2004: 31, brackets added to fill in background detail). in england also then, the notion of a mathematical course built on the likely needs of various aspects of students’ future lives, and the need to emphasise application skills within this course, was being 22 suggested here. importantly, and in contrast to the south african situation, the tomlinson report argued for functional mathematics qualifications to be part of the ‘core’ programme, i.e. compulsory for all learners across the 14-19 phase and available as qualifications at entry to level 3 on england’s nqf. however, this was only accepted by government up to level 2 (age 16) qualifications. other aspects of the argument for offering functional mathematics across levels were to provide a common recognised qualification across the school and adult learning sectors which opened up access to further mathematical learning in the form of functional mathematics at the next level or mathematics courses at the same level, and also the desire to rationalise the often poorly understood range of mathematical courses available to students and adults in the post-16 age range in particular. this provision of a common curriculum and assessment structure for functional mathematics across all sectors contrasts with the continued presence of two different structures for the school and adult learning sectors in south africa. the tomlinson report also proposed that all the core components including functional mathematics assessment ought to adopt a ‘mastery model’, (a proposal accepted by the department for education and skills), explaining the rationale for this in the following terms: we would propose that they should be developed using a ‘mastery’ model, unlike assessment of gcses which allows high performance in one aspect of a subject to compensate for lower performance in another. this would mean that to attain in core learning, young people would need to command good knowledge and skills in every aspect of the component. it would make this assessment a more significant hurdle, but it would ensure that all young people are adequately equipped across the range of functional mathematical, literacy and communication and ict skills. (tomlinson, 2004: para. 164) this assessment model diverges significantly from that used for mathematical literacy within south african schools, where worries about lack of adequate staffing and of teaching expertise as well as the need to establish the course, have resulted in the ‘temporary’ acceptance of relatively low thresholds (30%) for a pass (doe, 2005a), alongside the decision to delay the teaching and assessment of some of the standards to a later date (brombacher, 2005; laridon, 2004). additionally, and in line with the emphasis given to integrating technology, the department for education and skills has provisionally accepted the recommendation that functional mathematics assessment should, in part at least, be available, in a computer-based format. the smith inquiry, running in parallel to tomlinson’s working group, focused on pedagogy, curriculum and assessment in the post14 age range in mathematics education, with the central aim of increasing the numbers taking mathematics across all levels. the lack of specialist mathematics teachers was raised in their findings, and, as in south africa, the lack of provision for ongoing professional development was flagged. whilst functional mathematics was not a specific focus of this report, the findings criticised the existing provision of mathematics courses and qualifications in the 14+ age range: it is clear that the overwhelming majority of respondents to the inquiry no longer regard current mathematics curricula, assessment and qualifications as fit for purpose. (smith, 2004: para. 0.25) thus, the smith inquiry agrees with the tomlinson proposal of ‘progressive pathways’ in mathematics, providing a structure within which more learners could be supported to reach level 2 and level 3 qualifications. the need to incorporate greater emphasis on technology, modelling and applications within mathematics is noted in both of these reports. hoyles et al.’s (2002) research investigating the mathematics used by adults in a variety of work situations highlighted both these aspects, and noted again, employers’ concerns that these aspects did not seem to be emphasised sufficiently within the existing provision for mathematics. the authors also stressed the view that mathematics curricula focusing on the ‘basics’ – essentially arithmetical skills – would not produce the much more flexible kinds of reasoning skills that were a common feature of the workplaces that were researched. this provides a summary of the concerns that led to the current definition of functional mathematics, detailed at the start of this paper. two groups – king’s college london/edexcel and leeds university – are currently working on projects focused on recommending new mathematics curriculum and assessment pathway structures for the 14-19 age range. these structures will incorporate the government’s decision to introduce a compulsory functional mathematics hamsa venkatakrishnan and mellony graven 23 core at entry level and levels 1 and 2, with a functional mathematics option available at level 3, as part of a broader move to try and increase the numbers at all levels taking mathematical courses. in the next section, we extend the discussion of the key overlaps and contrasts that figure within the current conceptualisation and structuring of the two courses. discussion of overlaps and contrasts some aspects of the overlaps and contrasts between mathematical literacy and functional mathematics have been introduced in the last section – amongst these, the fact that mathematical literacy is offered as an alternative to mathematics in south africa’s fet, in contrast to its structuring as a compulsory ‘hurdle’ for achieving a mathematics gcse grade ‘c’ in england. also, differences in the assessment models for the two courses were also introduced above. the aims of mathematical literacy and functional mathematics suggested overlaps in the conceptualisation of what these courses are designed to achieve for learners – improved outcomes for employers and employment, a more active citizenship role, and a more confident ability to participate in everyday life – and are being sought in both countries. a more detailed analysis of differences is now undertaken through the use of ball’s (1994) notion of the ‘essential circuits’ of education. building on bernstein’s (1971) delineation of the ‘basic message systems’ of education, ball suggests that the transmission of messages about education policy comes through four key channels, or ‘essential circuits’: the organisational models proposed, curriculum, pedagogy and assessment. we use these four aspects to compare and contrast the structures associated with functional mathematics and mathematical literacy (abbreviated to fm and ml in the following sections), noting two things: firstly, that there is limited detail currently within the policy texts on the pedagogical aspects of implementing these kinds of programmes; and, secondly, that our analyses of functional mathematics are, as stated previously, based currently on proposed, rather than actual, structures. organisational models as stated before, the introduction of ml in south africa makes a mathematical course compulsory for all learners in the fet phase. within schools, this effectively makes mathematical work compulsory for learners up to the age of 18. in england in contrast, the department for education and skills has decided that fm should be a compulsory component up to level 2 only, although it may be opted for at level 3 also. this retains the preexisting status quo, in which mathematical courses were compulsory to the age of 16. the south african organisational model for ml as an alternative to mathematics reflects a notion that it is somehow ‘different’ from mathematics. proponents of ml strongly advocate this view, but as noted earlier, warn of the dangers prevalent in this kind of curricular differentiation: mathematical literacy is different from mathematics not in level or complexity but rather in kind and purpose. (brombacher, 2006: 10) however, the suggestion in the curricular statement for the fet mathematics programme (doe, 2003b) that the assessment standards for mathematics are sufficient to meet the needs of mathematical literacy, points to a view of ml as a subset of mathematics. when combined with the fact that ml has been developed in part to provide access to mathematical courses for learners who previously fell outside the net of sg mathematics, this tends to reinforce the notion of ml in the former categories – lower level and less complex. in england, the positioning of fm as a compulsory ‘hurdle’ towards the achievement of the ‘iconic’ mathematics grade ‘c’ produces a somewhat different conceptualisation. fm in this model is clearly overtly viewed as a subset of mathematics, but a subset that will lead to a recognised ‘stand-alone’ fm qualification, with clear openings for progression to either mathematics qualifications at that level, or higher level fm qualifications. its role as a ‘hurdle’ for gcse confers ‘status’, but the extent to which the requirement for ‘mastery’ will interfere with this status remains an issue that is being grappled with (acme, 2005) – a small or narrow fm component makes mastery easier to achieve, but reduces its status, (and runs the risk of reducing fm to ‘basic skills’), whilst a larger, broader component confers status but makes mastery harder to attain and thus threatens the establishment of fm and pass rates at gsce. the need for ‘progressive pathways’ in mathematical courses is given much greater emphasis in england than in south africa, where the need to establish access to mathematics in the 24 fet phase has taken priority. one of the criticisms of existing provision in england was that there were too many mathematical ‘dead-ends’, militating against ongoing participation and engagement with the subject (smith, 2004). in contrast to the south african context then, where advocates of ml have argued against the need for ‘articulation and portability’ between ml and mathematics as a fundamental requirement of ml (amesa, 2003), discussions around fm in england are located within the premise that such articulation is important. the agreement that a single set of curriculum and assessment standards for fm should be developed for use across all phases, in contrast to the ongoing presence of two parallel sets of standards in south africa, further attests to this emphasis. curriculum we pointed out earlier that whilst the mathematics curriculum in south africa had been ‘pruned’ significantly in order to produce the ml curriculum, the latter curriculum continued to raise concerns that it was ‘overloaded’ (amesa, 2003), and still contained some relatively high level mathematical content. the ‘hurdle’ and ‘mastery’ requirements in england, in addition to evidence that approaches based explicitly on modelling in increasingly complex contexts are often more difficult than traditional approaches – as noted within the construction of the ‘competency clusters’ used in the international mathematical literacy tests used in pisa assessments (oecd, 2003) – are likely to result in a lower level of mathematical content being selected for the fm curriculum for level 2 (the end of the compulsory phase for fm in england) than is currently within the ml curriculum. (it needs to be noted here though, particularly within the school-based context, that fm at level 2 on england’s nqf is aimed predominantly at 16-year olds, whilst ml at level 4 on south africa’s nqf is aimed predominantly at 18-year olds.) pedagogy the shift to a modelling-based approach within both ml and fm has been acknowledged as likely to be problematic for significant numbers of teachers in both countries. amesa (2003: 6) comments on the south african situation thus: current teachers, in the main, lack the capacity both to connect their mathematics to real contexts and struggle to see the internal connections between mathematical concepts. reflecting similar concerns, the king’s college london/edexcel team in england have added the criterion of “improving classroom practice” to their list of aims for fm (acme, 2005a). assessment the contrast between the ‘mastery’ requirement for fm in england and the relatively low threshold set for at least the initial years of implementation of ml in south africa has already been introduced. basic outlines of the assessment format that will be used for ml have already been detailed (doe, 2005a) – a combination of continuous assessment tasks spread across the course and terminal external examinations at the end of the fet phase, with a 25%/75% split in the allocation of marks to the two respective components. current guidelines given in the document cited above suggest that the examination component will be made up of two papers, one focused on shorter questions based on ‘knowing’ and ‘routine applications’, and the other with more extended questions emphasising ‘applications’ and ‘reasoning and reflecting’ (the pisa taxonomy based on ‘competency clusters’ and timss’s delineation of ‘cognitive domains’ are cited as sources for this particular structure). assessment models are still under discussion in england, but their emphasis on integrating computer use into some aspects of the assessment of fm does not figure within the south african context. this is understandable given the extensive evidence of the lack of infrastructure, although a more aggressive approach to providing technology within the implementation of ml as part of the strategy for redress was advocated by amesa (2003). in table 2 below, we provide a brief summary of this discussion of overlaps and contrasts. 25 south africa: ml england: fm to increase the numbers of students taking mathematical courses at all levels aims more overt political rhetoric in terms of broadening access and improving civic citizenship. need for articulation between ml and mathematics is not highlighted. more overt emphasis on society’s need for more people with quantitative literacy skills. to rationalise provision and provide coherent routes for progression. need for articulation and portability between fm and other mathematical courses stressed. use of relevant, realistic and increasingly complex contexts; interpreting texts; modelling situations; problem-solving emphases within course integrating the use of information and communication technology within fm learning and teaching. ml makes mathematical studies compulsory for all fet phase learners (i.e. in the post compulsory phase, to age 18/lev 2-4). fm leaves mathematical courses compulsory up to age 16 (lev 2) but optional thereafter (lev 3). ml offered as an alternative to mathematics in the fet phase. fm compulsory for all learners taking gcse mathematics; ‘hurdle’ to achieving a grade ‘c’. course organisation ml taught separately to mathematics. fm teaching integrated within mathematics teaching at level 2. fm offered as stand alone course at level 3. ml viewed primarily as a way of providing mathematical access to school learners who previously did no maths courses in the fet phase. less emphasis given to the need for ml to articulate with mathematics course in fet, or to further provision. fm viewed as fitting within a mathematical ‘pathway’. thus the need for a common curriculum across school and adult education, so that learners can progress onto further mathematical courses with fm qualifications at particular levels. emphasis on integrating content and context curriculum more high level mathematical content. current discussions emphasise lower level mathematical content in ‘complex’ contexts. use of technology emphasised. assessment models relatively low pass thresholds (30%) to be accepted in the initial years with only a selection of the curriculum initially being assessed. need for ‘mastery’ and associated high pass thresholds table 2: summary of overlaps and contrasts in south africa’s mathematical literacy and england’s functional mathematics 26 concluding remarks this analysis indicates finer-grained differences in emphases between the two policies which go beyond the overt structures and take in aspects of the historical problems which the initiatives were intended to address. the overlaps in concern are striking given the very different historical, social and economic contexts in the two countries. further overlaps and contrasts are likely to emerge as english policy makers negotiate the ‘context of policy text production’ and south african teachers grapple with the ‘context of practice’, and the affordances and constraints that impact on their enactments of policy. we have begun to research the ‘context of practice’ of mathematical literacy in south africa working with teachers in a small sample of schools. brombacher (2006) argues that mathematical literacy programmes across the world essentially develop out of two pressures, which he describes in the following terms: there is pressure to provide greater access to mathematics for more people – the democratisation of mathematics. on the other hand there is an imperative that more people should be able to use mathematics in order to more effectively participate in and contribute to the twenty-first century world in which we live – mathematics for democracy. (brombacher 2006: 2, original emphases) english policy makers are currently discussing structures that will address these aims; south african teachers are working within their policy makers’ structures with their grade 10 mathematical literacy classes. in doing so, they are faced with the challenges presented by some of the mixed messages in the policy texts that we have discussed in this paper – for example, are mathematical literacy and functional mathematics ‘basic maths’ or a ‘different maths’? one of the teachers that we are working with had asked his class for their thoughts on how they perceived mathematical literacy. overwhelmingly, he said, their responses consisted of lines such as: it’s maths for the people who can’t do maths. slowly, he is trying to overcome this perception and engage learners in a more constructive way with mathematical content and contexts, and thus use the opening provided for the first goal, the democratisation of mathematics, within the introduction of mathematical literacy, to move towards the achievement of the second goal, mathematics for democracy, through engendering the disposition to ‘mathematise’ in a variety of meaningful contexts. acknowledgement special thanks to professor paul laridon for a lengthy informal interview detailing the historical background to mathematical literacy in south africa. references acme. 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(2001). the case for quantitative literacy. in national council on education and the disciplines & l. a. steen (eds.), mathematics and democracy (pp. 1-22). washington d.c.: the mathematical association of america. 28 steen, l. a. mathematical sciences education board., & national research council (u.s.). (1990). on the shoulders of giants : new approaches to numeracy. washington, d.c.: national academy press. taylor, n., & vinjevold, p. (1999). getting learning right, report of the president's education initiative research project. johannesburg: joint education trust. tomlinson, m., with the working group on 14-19 reform, & great britain. department for education and skills. (2004). 14-19 curriculum and qualifications reform: final report of the working group on 14-19 reform. annesley: dfes publications. vally, s. (2003). reassessing policy and reviewing implementation: a maligned or misaligned system? in l. chisholm & s. motala & s. vally (eds.), south african education policy review (pp. 697-743). sandown: heinemann. wiley, d. e., & yoon, b. (1995). teacher reports on opportunity to learn: analysis of the 1993 california learning assessment system (clas). education evaluation and policy analysis, 17(3), 355-370. a habit of basing convictions upon evidence, and of giving to them only that degree or certainty which the evidence warrants, would, if it became general, cure most of the ills from which the world suffers. – in g. simmons microsoft word 64 front cover final.doc 70 errata: pythagoras 63, june, 2006, p. 19 in pythagoras 63, margot berger’s article making mathematical meaning: from preconcepts to pseudoconcepts to concepts contained errors. an electronic gremlin changed some of the mathematical signs to boxes. the relevant page, along with introductory text from the previous page of the original article, is reprinted below with apologies to the author. brief demonstration i will use the above theory to explain how a firstyear mathematics major student at a south african university moves from an idiosyncratic usage of signs (using, i claim, preconceptual thinking) to a conceptual (or perhaps pseudoconceptual) usage of signs. the activity took place during an interview which i conducted, video-taped and later transcribed and analysed in 2002 (berger, 2002). john had been given the following definition which he has not seen before, although he is familiar with the definite integral and the notion of a limit. definition of an improper integral with an infinite integration limit if f is continuous on the interval [a, ∞), then ∞ →∞ =∫ ∫( ) lim ( ) b b a a f x dx f x dx if →∞ ∫lim ( ) b b a f x dx exists, we say that the improper integral converges. otherwise the improper integral diverges. this is followed by several questions each of which is presented on its own, in order (for example, john has not seen question 4 when he first encounters, say, question 1). 1. (a) can you make up an example of an improper integral with an infinite integration limit? 1. (b) can you make up an example of a convergent improper integral with an infinite integration limit? m 4. determine whether ∞ ∫ 3 1 dx x converges or diverges. john’s response, in part, to question 1(a) is to generate a string of signifiers: ∞ ∫ 0 f ( x )dx = →∞ ∫ 2 2 0 lim f ( x )dx = →∞ ∫ 2 2 0 lim xdx = ∫ 2 0 xdx clearly what john has written is objectively meaningless and inconsistent. but the point is that john is using the new mathematical signs in mathematical activities (incoherent as they are to the outsider). in response to question 1(b), he writes: →∞ ∫ 2 2 0 lim ( )f x dx = ∞ →∞∫2 0 lim xdx = ∞ ∫ 0 xdx again his response appears incoherent and confused. but once more john is using the ‘new’ (to him) signs in mathematical activities. i suggest that notions of complex thinking can help the educator understand what is happening. specifically, i suggest that john’s response to both question 1(a) and 1(b) is dominated by complex thinking. in question 1(a) he has manipulated the template of an improper integral so that it eventually has the form of a definite integral (i.e. ∫ 2 0 xdx ), a form with which he is familiar. in question 1(b), he manipulates this further to get back to the template of an improper integral (albeit it does not converge). the point is: by using various signs in mathematical activities (a functional usage involving template-matching, associations and manipulations primarily) john is able to engage with the mathematical object on first contact, albeit in an idiosyncratic fashion. in this way, john gains a point of entry into mathematical activities with the object before he ‘knows’ that object. the question now is: how does john move from this (objectively) incoherent usage to a usage which is both personally satisfying and mathematically acceptable? i suggest that the answer lies in john’s imitation of the improper integral sign. that is, john is finally able to appropriate the socially-sanctioned usage of the improper integral sign through interaction with the mathematics textbook (a resource comprising socially sanctified mathematics). specifically, it is only after john has seen exemplars in the textbook of improper integrals and their evaluation, that he starts to use the improper integral in a way that is consonant with its definition. indeed, after seeing textbook margot berger (errata continued) 71 exemplars, he is able to answer question 4 in a coherent fashion. that is, he writes ∞ →∞ =∫ ∫3 3 1 1 lim b b dx dx x x = →∞ −⎡ ⎤ ⎢ ⎥⎣ ⎦ 2 1 2 lim b b x = →∞ −⎡ ⎤ −⎢ ⎥⎣ ⎦ 2 2 lim 2 b b = −2. and he states that this integral is convergent. although john has integrated ∫ 3 dx x incorrectly, ( − =∫ 3 2 1 2 dx x x ), his response is coherent; also he uses correct procedure and appropriate notation. this is a much improved response compared to his response to question 1. furthermore, john tells me that the examples are useful to him and that he is no longer confused. this contrasts with earlier statements that he is very confused about notions of convergence and divergence and the improper integral. my contention is: it is john’s functional use of the improper integral sign (initially association, template-matching and manipulations and then imitation) that enables him to move from activity dominated by complex thinking to conceptual (possibly pseudoconceptual) activity. allied to this, he is able to move from a confused notion of the improper integral (by his own admission) to a personally meaningful usage (again, in terms of his own assessment). reviewer acknowledgement pythagorashttp://www.pythagoras.org.za acknowledgement to reviewers the quality of the articles in pythagoras crucially depends on the expertise and commitment of our peer reviewers. reviewing is an important part of scholarly work, making a substantial contribution to the field. reviewers’ comments serve two purposes, guided by two inter-dependent objectives: • pythagoras wishes to publish only original papers of the highest possible quality, making a meaningful contribution to the field. reviewers advise the editor on the scholarly merits of the manuscript to help him evaluate the manuscript and to decide whether or not to publish it. reviewers are encouraged to reject a manuscript if it is scientifically flawed, merely sets out observations with no analysis, provides no new insights, or is of insufficient interest to warrant publication. • pythagoras is committed to support authors in the mathematics education community. reviewers help the author to improve the quality of their manuscript. reviewers are encouraged to write their comments in a constructive and supportive manner and to be sufficiently detailed to enable the author to improve the paper and make the changes that may eventually lead to acceptance. the following summary of outcomes of the reviewing process in 2011 shows that our reviewers do well in achieving both objectives: we sincerely thank the following people who have reviewed for pythagoras in 2011. we very much appreciate their time, expertise and support of pythagoras amidst pressures of work. andile mji anthony essien belinda huntley benadette ainemani caroline long clement dlamini craig pournara dirk wessels dorit patkin erna lampen faaiz gierdien gary sharp gerrit stols hamsa venkat helena miranda helena wessels hennie boshoff hugh glover humphrey atebe jacques du plessis janine hechter jenny campbell jill adler johan meyer john malone kerryn vollmer lindiwe tshabalala luckson kaino lyn webb lynn bowie maggie verster marc north marc schäfer margot berger mark jacobs mellony graven michael de villiers nelis vermeulen paula ensor percy sepeng piera biccard piet human retha van niekerk sharon mcauliffe sizwe mabizela stanley adendorff verena nolan vimolan mudaly willy mwakapenda zane davis zonia jooste in an effort to facilitate the selection of appropriate peer reviewers for manuscripts for pythagoras, we ask that you take a moment to update your electronic portfolio on www. pythagoras.org.za, allowing us better access to your areas of interest and expertise, in order to match reviewers with submitted manuscripts. if you would like to become a reviewer, please visit the pythagoras website and register as a reviewer. to access your details on the website, follow these steps: 1. log into pythagoras online at http://www. pythagoras.org.za 2. in your ‘user home’ select ‘edit my profile’ under the heading ‘my account’ and insert all relevant details, bio statement and reviewing interest. 3. it is good practice as a reviewer to update your personal details regularly to ensure contact with you throughout your professional term as reviewer to pythagoras. please do not hesitate to contact me if you require assistance in performing this task. rochelle flint submissions@pythagoras. org.za tel: +27 (0)21 975 2602 fax: +27 (0)21 975 4635 page 1 of 1 no. manuscripts processed 27 accept without changes 0 (0.0%) accept with minor changes (to the satisfaction of the editor) 8 (29.6%) accept after major revisions (re-submit, then re-review) 4 (14.8%) reject – the article is not acceptable to be published in pythagoras 15 (55.6%) abstract introduction literature review framework the study findings discussion conclusion acknowledgements references footnotes about the author(s) sarah bansilal school of education, university of kwazulu-natal, south africa citation bansilal, s. (2017). the application of the percentage change calculation in the context of inflation in mathematical literacy. pythagoras, 38(1), a314. https://doi.org/10.4102/pythagoras.v38i1.314 original research the application of the percentage change calculation in the context of inflation in mathematical literacy sarah bansilal received: 21 sept. 2016; accepted: 01 may 2017; published: 31 july 2017 copyright: © 2017. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract the school subject mathematical literacy requires application of mathematics procedures in various contextual settings, but not much is known about the ways in which students engage with contextual settings such as inflation. this qualitative study was conducted with in-service mathematical literacy teachers in south africa with the purpose of exploring the extent to which the teachers recognised the contextual constraints involved in applying the percentage change calculation to the inflation context. the written responses of the 406 mathematical literacy teachers were scrutinised to identify their interpretations of the contextual constraints involved in applying the percentage change procedure to the context of inflation. the item required the application of two successive percentage change operations (corresponding to the inflation rates for the 2 years). of the 406 responses that were analysed, 260 (65%) were unable to take account of all the contextual constraints. there were 108 teachers who reduced the procedure to a one-step calculation while 64 teachers interpreted the context as a percentage decrease scenario. a large number of teachers (162) struggled with the interpretation of the role of the year, k, in the relationship between the quantities. the findings indicate that engagement with and understanding of the concept of inflation is dependent on a synthesis of the contextual constraints into the mathematical procedures. this article provides some insights into the struggles with making sense of the contextual nature of inflation which is an area that has received little attention in mathematics education studies. the teachers’ struggles likely mirror learners’ struggles and hence the research applies in a similar way to learners. introduction globally there has been much interest in how learners at school could be better prepared to participate meaningfully in situations they encounter out of school. this out-of-school world has increasingly become influenced by mathematics, and is described by orrill (2001, p. xiv) as being ‘awash with numbers’ and drenched with data. the increasing emphasis on the application of mathematics in ‘realistic’ settings is reflected in calls for curricula to focus on applications of mathematics to real-world situations rather than mathematics in isolation (cooper & dunne, 2004; de lange 2003; department of basic education [dbe], 2011a; moschkovich, 2002; organisation for economic co-operation and development, 2010; steen 2003). steen described the mathematics in the workplace as requiring a sophisticated use of elementary mathematics while in a classroom it is mostly an elementary use of sophisticated mathematics. south african education authorities, in trying to make a shift towards mathematics in real-life settings, introduced the school subject mathematical literacy which is compulsory for learners who are not studying mathematics in grades 10–12. mathematical literacy is described as follows in curriculum documents (dbe, 2011a): the competencies developed through mathematical literacy allow individuals to make sense of, participate in and contribute to the twenty-first century world – a world characterised by numbers, numerically based arguments and data represented and misrepresented in a number of different ways. (p. 8) the mathematical literacy curriculum includes topics such as inflation, taxation, tariff systems, health issues, household bills, exchange rates, maps and scale drawing, among many others (dbe, 2011a). of interest to this study is the concept of inflation which has traditionally not been taught in a mathematics class, but is a context in which the elementary mathematics concept of percentage increase and decrease can be applied. on its own, the percentage change procedure is used in the early grades by children as young as 11 years old (dbe, 2011b). however, when used to make interpretations about the consequences of the inflation rate figures, the procedure is no longer as simple. in this study, i try to unpack some of the complexities involved in applying the percentage change procedure to the context of inflation; i draw upon a framework used by zazkis, dautermann and dubinsky (1996) to disentangle some of the contextual constraints. this is done to better understand how knowing in mathematics and knowing in the context of inflation develop together. the participants in the study were 406 practising mathematical literacy teachers who were enrolled in an in-service programme at a university in kwazulu-natal, south africa. the research question underpinning this study is: what are the teachers’ interpretations of the implicit contextual constraints when carrying out calculations based on the concept of inflation? it is hoped that this study about teachers’ use of the percentage change calculation in inflation can add to our knowledge about the conceptual understanding of mathematical literacy concepts. a further contribution is the use of theorems-in-action in the analysis which can provide valuable insight into teaching and learning. an improved understanding of the types of problems students could experience when faced with applying percentage changes successively, especially in the context of inflation, can help teachers plan how to mitigate these. literature review a central concept underlying inflation calculations is that of percentage, on which there have been numerous studies. parker and leinhardt (1995), in their exhaustive review of seven decades of research into the concept of percentage, noted that the domain of percentage is covered in grades 6 through 8. consequently, most research about students’ understanding of percentage has been carried out with young learners and has been concerned with ‘how students perform, what specific errors students make and why students might make these errors’ (parker & leinhardt, 1995, p. 423). parker and leinhardt suggested that the richness of the percentage topic has been left unexplored because instructional emphasis has been placed on procedures. the concept of percentage forms part of the mathematical ideas encompassed by multiplicative structures (vergnaud, 2009) and is encountered in financial mathematics concepts, proportional relationships, rates and exponential growth, among many others. percentage with its practical applications is widely used in the marketplace, in newspapers, in magazines, in the evening news, and in everyday commerce (parker & leinhardt, 1995). despite its apparent simplicity, there are many misconceptions and incorrect applications of the concept. people often make mistakes in evaluating the result of successive percentage changes, for example a reporter ‘mistakenly judged a 5% wage increase to be one-fourth, when it actually was one-fifth, of a preceding 20% decrease’ in wages (chen & rao, 2007, p. 327). there is often also confusion between the multiplicative meaning and an additive interpretation (chen & rao, 2007; parker & leinhardt, 1995) the concise, abstract language of percentages often uses misleading additive terminology with a multiplicative meaning. a change in the monthly inflation rate from 5% to 4% could be reported statistically as a decrease of one percentage point in the rate (using percentages as additive numbers) or as a 20% decrease in inflation (one point on five points using percentages as referent quantities to create a new percentage). because of the dominance of operations with whole numbers, people may mistakenly apply a whole number strategy and add or subtract the individual percentages directly (chen & rao, 2007). parker and leinhardt (1995) note that by reporting data in percentage form it is possible to disguise or distort information about referent quantities (the original quantities on which the percentage was based). when data is reported in percentage form, information about the referent quantities may be easily distorted or misrepresented. when corporate executives receive a 5% salary increase, it may sound insignificant until the person’s salary is revealed as r32 million. on the other hand, a 200% percent increase in a domestic worker’s wage may seem very large even though it may mean that the wage has increased from r1000 to r3000. studies about inflation have emanated mainly from economic perspectives focusing on people’s everyday meaning and perceptions of inflation (leiser & drori, 2005), inflation credibility (degree of acceptance of official inflation figures; rossouw & padayachee, 2007), prediction and memory of rates of inflation (kemp, 1991) and the psychology of inflation (batchelor, 1986). in their study among economics students at a south african university rossouw and joubert (2005) found a higher degree of acceptance of the consumer price index as an accurate indication of inflation than other indices. rossouw and padayachee (2007) found a great difference in inflation credibility among students in different faculties. the study found that students who studied in the economics field displayed the highest credibility, confirming that knowledge and information enhance the credibility of inflation figures. the authors argued that dissemination of information improves the understanding of inflation. one of their recommendations was that policy initiatives must be supported by means of communication strategies aimed at improving the general level of understanding of inflation and its measurement (rossouw & padayachee, 2007). an important issue that needs to be explored is the extent to which ordinary people understand the inflation figures released each month by stats sa. it is even more crucial that mathematical literacy teachers who teach the concept of inflation develop a deep understanding of the mathematical calculation behind the inflation rate figures reported in the press, to ensure that they can facilitate a sound understanding of the concept of inflation with their mathematical literacy learners. in south africa there are various inflation indices such as the consumer price index, the consumer price index excluding mortgage and the producer price index. in this study the focus was on the consumer price index which is taken as the average (weighted mean) cost of the ‘shopping basket’ of goods and services for a typical south african household. the typical south african basket contains about 1500 different consumer goods and services, and is based on many processes before average cost of the ‘basket’ is worked out. price movements on the goods in the basket are measured and the consumer price index is compiled using the price movements per product and their relative weight in the basket (lehohla, 2011). framework in order to better understand the linking of the mathematics and contextual perspectives, i draw upon a framework used by zazkis et al. (1996) that they used to describe the merging of the visual and analytic perspectives when faced with a problem. the visualisation/analysis model of zazkis et al. (1996) specified the two elements – visualisation and analysis – as two interacting modes of thought. the authors have tried to disentangle acts of analysis and visualisation to represent how the two interacting perspectives of the object may move together until they form a new totality. initially each act in a particular mode is separate. for example, the thinking may begin with an act of visualisation, v1, which could entail the learner looking at some ‘picture’ and constructing some mental actions or processes on it. the next step is an act of analysis, a1, which consists of some kind of coordination of the objects and processes constructed in step v1. as the movement between the v and a is repeated, the perspectives move closer together creating a spiral effect as illustrated in figure 1. figure 1: the visualisation/analysis model. as the movement between the two modes proceeds, the analytic and visual perspectives move closer together and at some stage become inseparable, with each one supporting the other and together forming a totality. in this study, we can distinguish two domains, the mathematical and the contextual settings. an understanding of the inflation rate figures requires continuous transitions between the mathematical and contextual settings. these shifts result in a series of re-interpretations of the mathematics signifiers, which are directed by the constraints of the contextual setting. the signifiers (or variables) in this setting are elaborated in equations 1 to 4 and tables 1 to 3. these transitions can be viewed as similar to the movement between the two perspectives encompassed in the visualisation/analysis model. in this instance, as in the visualisation/analysis model, the reflection on the mathematics quantities deepens the reflection on the contextual quantities and vice versa and the movements between the two become progressively closer. at first the passage from one to the other may represent a major mental effort, but gradually, the two kinds of thought become more interrelated, fusing and forming a new totality. table 1: response showing a one-step solution. table 2: response of t18. table 3: response of t35. what follows now is an attempt to disentangle the substance of the shifts between the two perspectives by examining the transformation of the arithmetic procedure used to find the percentage by which a quantity has changed into one that is used to quantify the percentage change in the cost of a basket of goods over a certain period. the basic procedure that measures the percentage change of a quantity is given in equation 1, where p0 is the initial value, p1 is the final value and c is the percentage change in the quantity expressed as a decimal. this can be extended to capture the percentage change in costs of goods and services over a fixed period of 1 year. the inflation rate calculation process is represented as follows: here, p1 represents the current average price level, p0 the price level a year ago, and the rate of inflation (i1) during the past year is calculated by equation 2, where the symbol i1 is used in place of c. this relationship between the average price levels and the calculated value of the inflation rate gives meaning to the figures reported in the media each month as the ‘monthly inflation rate’ figures for the consumer price index (bansilal, 2011). usually, inflation rate figures are reported on a monthly basis. the ‘monthly inflation rate’ refers to the year-on-year rate calculated on a monthly basis and does not represent a month-on-month rate. for example, if i1 is the monthly inflation rate for march 2013, this means that p0 and p1 represent the price levels in march 2012 and march 2013 respectively. when calculating the inflation rate, the various signifiers take on further contextual constraints that are elaborated in box 1. it can be seen that equation 2 has been constrained by conditions taken on by the signifiers (variables), which relate to the relationship between the variables, as detailed in box 1. box 1: equation 2, showing the calculation of the inflation rate. box 1 captures the transition of c (percentage change in a quantity) as described by equation 1 in the mathematics domain to i1 as described by equation 2 embodied in the context of inflation. for the purposes of this discussion, in order to understand the additional contextual constraint, equation 2 is now re-written to include the year, k, over which the measurement is made, and this is shown in box 2. box 2: equation 3, showing the constraint made by the year. furthermore, there are 12 measurements, done on a monthly basis. thus if m is the month in which the measurement was made, ik can then be expressed as , where m = 1, 2, 3, 4, … 12. hence, equation 3 can be expressed even more generally as shown in box 3. box 3: equation 4, showing the constraint of the month of measurement. each contextual constraint, as in the visualisation/analysis model of zazkis et al. (1996) requires a synthesis or coordination with the mathematical operations leading to progressively new interpretations of the signifiers and equation. finally, the new object described by equation 4 is encapsulated as an object (bansilal, 2011). the study the participants in the study were a group of 406 kwazulu-natal in-service mathematical literacy teachers who were enrolled in a programme designed to retrain practising teachers in the school subject mathematical literacy. the 2-year part-time programme comprised modules that focused on applications in different fields of mathematics. inflation was one of the many concepts that were covered and the unit on inflation was covered over two 2-h sessions. during this time the teachers were introduced to the concept and they discussed ways in which the concept of inflation appeared and was used in everyday situations. the teachers also spent time in groups working with calculations based on inflation figures. data for this qualitative study were generated from the teachers’ responses to items set as part of an assessment in the programme. the research question explored in the study was: what are the teachers’ interpretations of the implicit contextual constraints when carrying out calculations based on the concept of inflation? the main task presented a hypothetical item whose price was tracked over a particular period of 3 years by assuming that the price increases of the item were in line with the inflation rate figures at that time. during the particular period chosen, the inflation rate figures fluctuated with no steady pattern, and were presented in a graph. figure 2 contains the task (question) and an accepted solution. figure 2: task with suggested solution. note that the figures for the monthly inflation rates were provided in a graph containing all the monthly figures over a period of 4 years. differences in reading of the values of the points on the graph were taken into consideration during the analysis process. the analysis procedures were guided by the notion of ‘theorems-in-action’, which vergnaud (1998) used in the context of the multiplicative conceptual field to describe the mathematical relations considered by students when they choose a concept to solve the given problem. these theorems are implicitly held by students and may not be articulated and may be correct or incorrect. during the analysis, each written response was first expressed as an equation or ‘theorem–in–action’ based on the various signifiers (ik, pk) and the operations carried out on them. the next layer of analysis involved grouping the various equations into a smaller number of categories according to the contextual constraints that were recognised or not recognised by the theorem-in action underpinning the response of each of the 406 teachers. an important reliability check when conducting qualitative research is to ensure traceability so that anyone can retrace the process. cohen, manion and morrison (2011, p. 202) note that an audit trail ‘enables the research to address the issue of confirmability of results’. the transcriptions of each of the teachers’ responses were recorded in an excel worksheet. each response was translated into the underlying theorem-in-action and recorded on the same sheet. the next stage involved the coding of these theorems-in-action. the coding was revisited by the author at a later stage to check for consistency and accuracy. findings in this section, the teachers’ calculations are presented in eight different categories which are then used to analyse the contextual conditions recognised by the teachers. note that in representing the responses of teachers as examples of particular categories, the representations are reproduced verbatim. the responses are then presented in the form of an equation representing the theorem-in-action. responses were coded according to the order in which the teachers appeared in a list. the code t26, for example, shows that it is the response of teacher number 26. category 1: responses aligned to equation of the 406 teachers, 133 produced calculations similar to that appearing in figure 1 and a further 13 teachers made slight calculation errors. hence, 35% of the group were able to present a calculation based on equation 5 followed by equation 6. category 2: responses containing a one-step calculation using the inflation rate i2 a large number of teachers (108) did a one-step calculation using a different version of equation 6 that used the inflation rate i2, instead of the rate i1. an example of such a response (t29) is reproduced in table 1. hence, 27% of the teachers produced a response that can be represented by equation 7. the inflation rate used was that for march 2005 (i2). here i2 was taken as the percentage by which p0 (price of item in year 2003) increased 2 years later to p2 (price of item in year 2005), displaying a misconception that the rate i2 measures the percentage change in the price of goods over 2 years, that is instead of (equation 2). category 3: use of the compound interest formula some teachers (nine) used a similar approach to those in category 2 but the calculation took into account the 2-year period between 2003 and 2005, and assumed that the inflation rate was constant over the 2-year period. these teachers used the compound interest formula to calculate the cost. t18’s response is reflected in table 2. in this approach, the annual inflation rate has been assumed to be the same over the 2-year period of calculation. these teachers used a compounding formula, which is sometimes used in illustrative predictions when the rate is averaged over a few years. however, in this calculation the assumption is that the inflation rate was constant for both years. category 4: responses containing two separate calculations another approach, based on yet a different combination of equation 5 and equation 6 was presented by 12 teachers. these teachers produced a response consisting of two separate calculations, the first one using the march 2003 inflation rate (i0), and the second disconnected calculation being the same as that produced by the people in category 1 using equation 7. one such response by t35 is reflected in table 3. as is seen, from the theorem-in-action, there is no connection between the two calculations and the answers are presented as two separate results. these 12 teachers’ responses do not show recognition of the recursive nature of the price calculation, namely that the changes in the price of the item over consecutive measurement periods are cumulative with the subsequent inflation rate one acting on the changed price. equation 9 takes i0 as the percentage change of p0 to p1, instead of it being i1. equation 7 was also identified in category 2, and reflects a misconception that the rate i2 measures the percentage change in the price of goods over 2 years, from p0 to p2. category 5: use of the inflation rate i0 a similar theorem-in-action to those of category 4 was evident in the responses of 21 teachers who also produced a response of the form equation 10, with some producing a one-step calculation based on equation 9 and others using a two-step or even three-step calculation using equation 9 in the first step. table 4 summarises the various responses of these 21 teachers. table 4: responses that used the inflation rate i0. table 4 shows variations in the ways in which equation 9 was used, with 11 teachers producing a one-step calculation (only equation 9) while a further five did a two-step calculation (equation 9 followed by equation 10). five teachers produced a three-step solution using the three inflation rates i0, i1 and i2, incorrectly assuming that there are three possible year-on-year inflation rate calculations over a 2-year period. category 6: use of other rates there were five teachers who produced a one-step calculation using a rate different from i0, i1 or i2. one of these teachers used the vat (value added tax) rate (table 5). table 5: response illustrating other rates. category 7: calculations assuming a decrease in price many teachers (68) felt strongly that the car would be cheaper in 2005 than it was in 2003, and their theorems-in-action encompassed a percentage decrease operation on p0 as shown in table 6. in fact, 29 teachers used a percentage decrease calculation, that is, a subtraction form of equation 7, while others used different variations of the percentage decrease calculation. table 6: responses which assumed a decrease in price. there was a large number (19) who first found the difference between i0 and i2 and used the result in the percentage decrease calculation denoted as equation 21. the teachers’ responses in this group contradicted the order of the difference calculation in equation 2, because the sign of the inflation rates in their calculation is negative. so equation 18 from table 6 for example, can be expressed as , showing that the order condition discussed in box 2 of the framework has been contradicted. the responses that assumed a decrease in price are presented in table 6. hence the responses of 68 teachers revealed that they believed so strongly that the price should have decreased over that period, that they disregarded the constraint of the order of the difference in the calculation. category 8: other responses responses that could not be identified with the seven preceding categories were placed in the category ‘other responses’ and these are described in this section. table 7 presents two equations and the others are described thereafter. table 7: responses taken as ‘other’. there were three teachers who took the inflation rate as a month-on-month rate, as shown by t405 in table 7. some responses were based on a direct proportion calculation (five people) that set up various incorrect proportions as shown by t4 in table 7. there were 11 teachers whose responses did not specify details of the calculations, and 18 teachers who did not produce any response. discussion the findings indicate that all except 34 teachers included a percentage increase or decrease calculation showing that 92% of the group did not have problems with using the percentage change formulas represented by the decontextualised formula of equation 1. however, it was in identifying the quantities anchored in the context, and interpreting the realisations of contextual constraints, that most difficulties emerged. the representation of each teacher’s response in terms of a theorem-in-action enabled the identification of the conditions that were not recognised by each teacher. as explained in the framework, the simplistic inflation rate calculation is given by equation 2: however, a more comprehensive representation of the relationship between the various contextual quantities is given by equation 4: the analysis revealed that the extent of the teachers’ engagement in the problem varied widely. it was found that some contextual conditions were more easily grasped than others. the discussion is arranged around the various contextual constraints embedded in the inflation rate calculation process. constraint of the base quantity one constraint was on the base quantity in calculating the percentage change. the initial base quantity of p0 from equation 2 has attained a realisation into pk−1 as the base for the percentage change comparison as shown by: responses in category 5 which were represented using equation 12 and equation 13 contradicted this condition of what the base or referent quantity should be in each percentage increase calculation. for example, in equation 12, p2 = p1 + (i1 · p0) or equivalently, , which means taking the difference in the price from year 1 to year 2 as a percentage of year 0. these teachers have not demonstrated an understanding of the meaning behind the inflation rate figures: that the inflation rate figure is the measure of the percentage by which the cost of the basket in the previous year has increased. parker and leinhardt (1995) noted that the presentation of data in percentage form sometimes makes it difficult for the referent quantities (the original quantities on which the percent was based) to be discerned. this can often lead to a distortion or misinterpretation of information, which may have been the situation here. constraint of the year of measurement closely related to the condition on the base or referent quantity in the calculation was the associated constraint of the year of measurement of the quantities, as described in equation 3, which shows the relationship between quantities ik, and pk and pk−1, where pk is the result, but only when the operation of adding a percentage change is carried out on pk−1 using ik as the rate. this struggle to recognise the constraining condition of the years on the variables pk and ik was evident in the responses shown in categories 2, 4, 5 and 6 and exemplified in equations 7, 9, 10, 11, 12, 13 and 14. the theorems-in-actions underpinning these equations do not correctly reflect the relationship that is described by equation 3. in fact, non-recognition of this condition constituted the biggest hurdle in this task, with the responses of 162 teachers overlooking this constraint. the constraint of the year required a simultaneous consideration of two quantities, which constitutes areas of conceptual difficulty in other contexts such as distance, speed and velocity. it required reasoning about the relationship between the cost variables pk and pk−1 and the inflation rate variable, ik. in this case, there is a changing relationship between and within the variables ik and pk and it was necessary to recognise these different relationships. tasar (2010) noted a similar difficulty (in the context of distance, speed and acceleration) in understanding the different ways of decrease or increase (e.g. as a constant rate of increase, increasing rate of increase, and a decreasing rate of increase). tasar reflected that a fall in inflation rate is an example of a decreasing rate of increase, that is, the prices are increasing by a smaller percentage, contrary to the misunderstanding that it indicates a fall in prices. unpacking these relationships requires what oerhtman, carlson and thompson (2008) describe as covariational reasoning1. this refers to the ability to form an image of two varying quantities and coordinating their changes in relation to each other, which seems to be difficult to achieve (oerhtman et al., 2008; thompson, 1994). for a more detailed discussion of this covariational relationship, the reader is referred to bansilal (2017). constraint on the order of the difference another constraint on the procedure arises from the order of the calculation of the change in the quantity. the original difference between p1 and p0 in equation 1 has attained a realisation into pk – pk−1, as shown in equation 3, where the order is important, and the sign of the difference denotes specific meaning. this order of the difference constraint constituted a problem for many teachers whose responses were described in category 7, and exemplified in equations 15–21. the non-recognition of the constraint is related to the misconception that a decrease in the inflation rate should be associated with a decrease in prices. it is not surprising that many teachers worked with depreciation, since a typical scenario in dealing with car purchases is one that considers the value of the purchased vehicle after a certain time period. this involves depreciation, and this is often found in textbook tasks but, more importantly, affects teachers in their daily lives. it is possible that the teachers did not fully understand the context or pay enough attention to the question and assumed that the question asked for the value of the same car after 2 years. these teachers went as far as subtracting the two rates and putting the difference as the net effect of inflation over the 2 years, to ensure that the cost 2 years later, was lower than the cost in the first year. in subtracting the two rates, the teachers have mistakenly applied a whole number strategy to the percentage values by subtracting the individual percentages directly, as if they were whole numbers (chen & rao, 2007). a percentage actually is a comparison between two referent quantities, and cannot make sense unless one knows what the referent quantities are (parker & leinhardt, 1995). the subtraction of two percentage values has no meaning because the referents in each case are different, and the result of the difference is thus not appropriate for this context of application. the referent quantities in any percentage relationship is very important and one cannot make any conclusion about the relationship without considering them because they are a fundamental property of percentage. constraint of the recursive nature of the inflation calculation there were some responses which ignored the recursive nature of the calculations: that is, the cumulative effect of increasing (or decreasing!) prices to the cost of the basket. the rate ik is the percentage change in prices from year k−1 to year k and i(k+1) is the percentage change in prices from year k to year k+1. the variable pk can also be calculated as follows: hence the cumulative effect of the successive percentage increases or decreases in the cost of the basket is factored into the signifier pk. the neglect of this constraint was identified in categories 4 and 8. the application of the percentage increases or decreases is not complicated (last year’s price plus the increase caused by inflation). however, it is the recognition that the cumulative effect of inflation is enabled by the repeated application of the percentage change formula that seems to be the issue. thompson (1994) alluded to the difficulty of recognising this recursive nature of the inflation calculation. he reported that when given a question based on a graph of yearly inflation figures, many mathematics secondary teachers looked at ‘inflation as increase in price per year (i.e., rate of increase with respect to time), which translates into slope, instead of as percentage change, which is a recurrence relationship’ (thompson 1994, p. 21). constraint of the month the condition of m, where m represents the month for which the inflation rate is being reported, is shown in equation 4. thus, the quantities ik, pk and p(k−1) attain a further realisation into respectively, where k is the year and m is the month of measurement. the month was provided in the task, making it very difficult to ignore this condition; however, there were three people in category 8 who misused the annual rate. they assumed it was a monthly rate by multiplying the monthly rate by 24. understanding the inflation rate the analysis via theorems-in-action identified a variety of ways in which the various quantities (or signifiers) were combined and the ways in which the relationships between the quantities were constructed. the number of different equations that were uncovered was at least 24, each of which revealed a different understanding of the relationship between the signifiers. the choice of combinations of the signifiers and the operations on them are likely to be arbitrary, such as t4, if the understanding of the relationships between the contextual quantities is unsound. clearly it is difficult to represent a situation without a clear understanding of the context. the various ways of carrying out operations on the different quantities, demonstrate the teachers’ struggles in trying to combine the mathematics and the contextual domains. the visualisation/analysis model, according to zazkis et al. (1996) results ultimately in a refinement of the thinking strategies and a more sophisticated reflective analysis. in the case of the question in this research study the thinking strategy remains the same: last year’s price + increase caused by inflation. the increased sophistication in the reflective analysis emerges in the recognition that the cumulative effect of inflation is enabled by the repeated application of the percentage change formula. the visual representation of the visualisation/analysis model with its developing spirals shows how the reflections can increase in sophistication and, as each constraint is recognised, it leads to the selection and appropriate interpretation of the relevant quantities. conclusion in this article, the written responses of 406 mathematical literacy teachers were scrutinised to identify their interpretations of the contextual constraints involved in applying the percentage change procedure to the context of inflation. the study provided a specific and detailed analysis of the complexities that can arise in the use of rates and percentage for working with inflation. of the 406 responses that were analysed, 65% of them were unable to take account of all the contextual constraints. some of the constraints were more easily taken up than others. for example, most teachers realised that the month of measurement was constant, while most struggled with interpreting the role of the year, k, in the relationship between the quantities. the study illustrates the complex interaction between mathematics and context that arise when applying mathematical tools to concepts in other fields. the manner in which particular elements of the context of application expand and adapt the mathematical tool as it is interpreted, as well as the manner in which the use of the mathematical tool forces the precise identification of elements of the context are both important and evident in the data of this study. the understanding of the relationship between the contextual quantities shifts and is shifted by the understanding of the mathematical relationships. developing our understanding of this interpretation process is useful for deepening the teaching and learning of the application of mathematics. an implication of this study is that in order to deal with complex applications, one must have a sophisticated understanding of the mathematics procedures, even for a procedure as elementary as the percentage change calculation. this resonates with steen’s (2003) call for the development of sophisticated understanding of elementary mathematics rather than elementary understanding of sophisticated mathematics. this study has illustrated that one of the complexities related to attaining a sophisticated understanding of elementary mathematics is that of understanding the constraints of the context in which the elementary mathematics is being applied. the study has also raised issues about whether the in-service programme provided adequate support for the teachers to engage with the contextual settings. most teachers did not develop a robust understanding of the notion of inflation, suggesting that the purpose and design of such programme must be carefully reconsidered. acknowledgements competing interests the author declares that she has no financial or personal relationships that may have inappropriately influenced her in writing this article. references bansilal, s. 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(1996). coordinating visual and analytical strategies. a study of students’ understanding of permutation and symmetry groups. journal for research in mathematics education, 27(4), 435–457. https://doi.org/10.2307/749876 footnotes 1. a detailed discussion of the covariational reasoning with respect to the changes in the inflation rates and the changes in the prices of goods appears in bansilal (2017). abstract introduction general theoretical frame part 1: how to be a student of school geometry part 2: how to draw a square drawing versus construction part 3: a square in drag as concrete universal concluding remarks acknowledgements references footnote about the author(s) zain davis school of education, university of cape town, south africa citation davis, z. (2016). a square in drag as concrete universal; or, hegel as a sketchpad programmer. pythagoras, 37(1), a322. http://dx.doi.org/10.4102/pythagoras.v37i1.322 original research a square in drag as concrete universal; or, hegel as a sketchpad programmer zain davis received: 16 dec. 2015; accepted: 05 sept. 2016; published: 17 nov. 2016 copyright: © 2016. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract in this article, by way of an analysis of a case of mathematics teacher training, i explore the general idea of pedagogic expectation of an alignment of pedagogic identities and specific realisations of mathematics in pedagogic contexts. the particular case analysed has a constructivist orientation, but the analytic resources brought to bear in the analysis can be used more generally for the description and analysis of pedagogic situations. the analysis is framed chiefly by the philosophical work of georg hegel alongside basil bernstein’s sociological discussion of evaluation in pedagogic contexts. the argument proceeds in three inter-related parts, the first of which produces an analytic description of the discursive production of the desired pedagogic subject–in this case, a teacher/student of geometry–in which i show how explication and abbreviation are used discursively in an attempt to construct the desired teacher/student–that is, a particular pedagogic identity. the second part of the argument describes the discursive production of mathematics content in a manner intended to align content with the desired teacher/student and introduces the notion of a regulative orientation in order to grasp the differences in the mathematical work of students. the third part is a synthesis of parts one and two, showing how pedagogic identity and mathematics contents are brought together as correlative effects of each other. introduction these days it is commonplace for official elaborations of national curricula to make very noticeable pronouncements on the nature of the school learner and the teacher and not only on the contents of school subjects. curriculum-specific expectations about teachers and their learners are not something new. the main difference from the past is that ideas of who the learner and teacher ought to be are made a great deal more explicit now than was the case when social arrangements were a lot more stable and the life trajectories of individuals could be predicted and mapped out with a reasonable degree of confidence. many of the effects on mathematics education of the social, political and economic transformations that irrupted over the course of the 20th century, spilling over into the present one, are drawn together and focused in the various realisations of pedagogic constructivism (see dowling, 2002, p. 36ff.), that is, the use of pedagogic principles derived from philosophical positions on epistemology arguing that all knowledge held by individuals is necessarily constructed by the knower. the sticky fingerprints of pedagogic constructivism can be detected on curriculum statements, school texts and, in more complicated ways, on pedagogic practices across the globe. the conceptions of the mathematics teacher, of the student and of mathematics that derive from a constructivist frame, or even from resistances to such framing today, are rather different from the conceptions of teachers, students and school mathematics that circulated before the emergence of pedagogic constructivism. today we have no choice but to engage with the far-reaching effects of pedagogic constructivism on mathematics education and on the training of mathematics teachers. this article is an instance of such in the context of teacher education. one of the important pedagogic problems that we have to confront as trainers of mathematics teachers–and one which arises as an effect of the insistent hegemonic grip of pedagogic constructivism on contemporary mathematics education–is the very real potential for teachers and students to become unmoored from mathematics and drift into unproductive, incoherent solipsism as they attempt to develop skills that are intended to have the ultimate effect of encouraging and sustaining productive reflections on mathematics. piaget’s (1995) sophisticated arguments elaborating his concerns about the negative effects of egocentric and sociocentric thought on learning, along with his cautions on the dangers of educational arrangements that tend to authorise knowledge statements gerontocratically, seem to arrive in the pedagogic constructivist universe in a curiously simplistic and fractured form. the constructivist propositions on the impossibility of the transmission of knowledge from one mind to another are often recontextualised to pedagogic constructivism as propositions declaring the impossibility of teaching, with the expectation that the teacher no longer teaches. that is, the impossibility of the direct transmission of knowledge is transmuted into a prohibition against teaching. where that does happen, such impossibility is often restricted only to the teacher, not to students, because it is usually accepted (implicitly or explicitly) that students are able to learn from each other, typically via some form of group-based activity, and without the pedagogic evaluations of the teacher. the teacher who would adopt pedagogic constructivism as the central organising framework for teaching and learning school mathematics is confronted with the rather delicate task of acknowledging the student as always-already knowledgeable–but must nevertheless have the latter arrive at the realisation that they are, in fact, also ignorant–and of valuing the everyday experience of the student, yet simultaneously getting them to realise that mathematics contents are grounded in a necessity derived from mathematics rather than from either everyday experience or some social mechanism (like voting, for example). the socrates of plato’s dialogues is the exemplary pedagogue in this regard, skilfully engaging his confident interlocutor (usually a beautiful young man) in a manner that gives impetus to the youth’s journey from a confidence grounded in the apparent certainty of experience to an acknowledgement of his profound ignorance, which plato saw as a necessary precondition for access to knowledge rather than remaining mired in mere opinion (see bloom, 1991). so, how is a teacher to be trained to become a pedagogic constructivist? it appears that a favoured way of training teachers to that end is to embody the central propositions that are to structure pedagogic practice in an exemplary form of pedagogic engagement that models the practice for the neophyte. it often turns out to be the case that the trainee adopts the position of a school learner while the trainer adopts the position of a pedagogic constructivist school teacher. the case discussed in this article is an instance of pedagogic constructivist mathematics teacher education that proceeds along the lines just sketched, revealing in the process a series of interesting features of the approach that are worth considering in some detail. however, to arrive at a more precise framing of the central issues, some or other set of theoretical resources needs to be recruited to enable us to reflect on the immediacy of the initial encounter with the empirical record of the case. it is to that task that i now turn. general theoretical frame in his discussion of the pedagogic device, bernstein (1996, p. 50) arrives at a position claiming that evaluation is the key to pedagogic practice. with his use of the term evaluation bernstein is referring to instructor-learner interactions that include pedagogic exchanges between instructors and learners, as well as engagements with problems, tests, projects, examinations and so forth. it is in that broad sense that the term is used here. for bernstein, pedagogic practice is necessarily saturated with evaluative acts that are continually performed by pedagogic subjects and so routinely generates evaluative judgements of the knowledge claims and statements made by instructors and learners. schooling can be viewed as a context in which the encounters between selected fields of knowledge and learners are staged, with pedagogic evaluation functioning as the mechanism mediating the encounter between a field of knowledge and the learner. pedagogic evaluation marks out what are to be taken as legitimate and illegitimate learner responses to the recurrent demands on them to produce utterances–written, spoken and gestural–in pedagogic situations. what pedagogic evaluation thus inserts into the pedagogic situation is a demand for and assessment of what ought to be the content of instructor and learner activity. in other words, the staging of an encounter between the learner and a field of knowledge necessarily produces a moralising of learner and instructor. what the ought of pedagogic evaluation proposes is a correlation of a pedagogic identity with particular realisations of content. bernstein (1990, p. 183) announces a proposition on this specific feature of what he terms pedagogic discourse in a somewhat clumsy way, as the embedding of an instructional discourse in a regulative discourse, where the latter is a discourse of social order and is dominant. dowling (2009, pp. 81–83) has detailed a series of problems with bernstein’s formulation of the proposition, which the interested reader can review for themselves. the central proposition that i wish to exemplify and explore in this article is the following: the ought of the pedagogic situation is internally split between some idea, or expectation, of a specific pedagogic identity on the one hand and particular realisations of content on the other, with the pedagogic identity believed to be correlated to particular realisations of content. the particular pedagogic identities and their imagined correlative content could be thought of in very precise terms or in fairly vague terms in different pedagogic contexts. in either case, some recognition of what the learner ought to be like and what their presentation of content ought to entail is always present. in my exemplification of the central proposition of this article i focus on a specific case of the training of in-service teachers in a geometer’s sketchpad environment as part of a mathematics education course on the teaching of school geometry, offered as a module on a bachelor of science honours (bsc hons) programme at a south african university.1 of course, a serious drawback of case studies is the difficulty of producing generalisable propositions about the phenomena that are perceived to be present. however, the great utility of case studies resides in the opportunities they offer for the production of hypotheses fuelled by the detailed study of the internal relationships and processes found to be inherent to identified phenomena. such hypotheses then become available for exploration across cases and for larger samples, possibly in transformed states. the geometry module of concern here made up the first part of a two-module bsc hons course on geometry and was run over seven four-hour sessions. the students were in-service secondary school teachers who were required to teach some geometry as part of the national school mathematics curriculum. all of the module sessions took place in a computer laboratory, where each teacher had individual access to a computer terminal. the design of the geometry module was underpinned by principles of pedagogic constructivism alongside van hiele’s theory of the development of geometrical thinking (van hiele, 1986) and included the use of sketchpad environments to develop the dispositions towards mathematics deemed appropriate by the lecturers. a video recording of each session was produced and made available for analysis in the archive of information on the module. in addition to the video recordings, the archive contained a transcript of an interview with the lecturers of the two modules that made up the course, a course outline for each of the modules, the course material that was handed to students during each session and a copy of a geometry test that teachers sat at the start of the first module. the test was designed to enable the lecturer to read the teachers’ responses to the test items in terms of van hiele’s theory. the test was not referred to by the course lecturers other than in the first session of the first module. this article is not primarily concerned with either the general problems of pedagogic identity or with the problems of teaching and learning school geometry. what is of primary concern here is the attempted constitution of specific pedagogic identities in pedagogic situations and the attempted alignment of such identities with particular relations of content. more specifically, the pedagogic and discursive mechanisms by which such identity and content constitution and correlation are fashioned are of key interest. the rest of the article unfolds as three inter-related parts, the first of which explores the construction of the student as lacking, while the second focuses on the elaboration of mathematics content. part three produces a synthesis of parts one and two, showing how pedagogic identity and mathematics contents are brought together as correlative effects of each other. in each of the distinct parts of the article some additional methodological resources will be required to develop the argument a bit further, but will be introduced as needed rather than spelt out in a separate section. part 1: how to be a student of school geometry the lecturer for the first module of the course was a mathematics educator who was situated in the education department of the university, while the lecturer for the remaining module was a mathematician from the mathematics department. the mathematics educator saw her task as one of preparing the students to be disposed towards mathematics in a manner that would facilitate their explorations of the mathematics they would encounter in the second module, ‘supporting what [lecturer 2] saw as aims’, as she put it during the interview. in her introduction to the course she wrote that her part of the course would ‘focus on getting to know geometer’s sketchpad as a tool to investigate geometrically’, as well as giving students the ‘opportunity to reflect on the development of spatial knowledge and pedagogy that supports such development’. the mathematician, who was the more vocal of the two, drew a distinction between learning mathematics as a mere series of techniques for solving various classes of problems and learning something about what mathematics is. ‘i have taught for many years and i have students who do well based on learning techniques, but they learn very little’, she complained during the interview. ‘they learn very little about what mathematics is about, and a lot of people struggle very hard in that learning of very little’, she continued, characterising her module as one that would ‘allow the students to see things differently, to appreciate things differently and to definitely make connections between diverse things that they never used to connect’. lecturer 1 explained that she saw the need ‘to build in a bit of geometric thinking and geometric development’ as the means for realising her goal of supporting lecturer 2. in his clarification of hegel’s dialectic, mccumber (1993) developed a number of ideas that prove to be very productive for constructing descriptions of pedagogic exchanges between teachers and their students. while mccumber’s purposes are philosophical and so rather different from mine, the descriptive resources he developed are admirably suited to describing the flow of pedagogic discourse. this is, perhaps, not so surprising given that the form taken by the philosophical system developed by hegel in his logic (hegel, 1975) was, in part, a pedagogical solution to having been set the impossible task of preparing a couple of philosophy courses in under two weeks in december 1808 (pinkard, 2000, p. 322), as well as having to teach students who had no training in philosophy and who had great trouble in engaging with his phenomenology (hegel, 1807/1977). in his re-presentation of the inner workings of hegel’s dialectic, mccumber (1993, pp. 130–43) proposes to achieve his ends by targeting the dynamic ‘gestures’ inherent to the dialectics of thought. the ‘gestures’ detailed by mccumber function equally well as resources for describing and analysing the flow of discourse in pedagogic situations. following mccumber, i propose that one way of characterising discourse in a pared-down form is as the gathering together of terms, or groups of terms, that are related to other terms or groups of terms. this may appear to be a rather impoverished reduction of discourse as it unfolds in the interactions between teachers and their students, but the apparatus that emerges around the idea will be seen to be surprisingly productive in capturing the dynamic flow of such exchanges. since this discussion is conditioned by bernstein’s theorisation of pedagogy, pedagogic exchanges are to be understood as instances of pedagogic evaluation. consequently, the elements that i borrow from mccumber will be treated as a set of resources contributing to the operationalising of the description of pedagogic evaluation in pedagogic situations. i will not reference mccumber on every occasion that i use the terms that he uses to construct his description of hegel’s dialectic, but i will italicise my first references to the terms. recall that my purposes are different from those of mccumber, so that my use of his terms change their meanings from those intended by mccumber somewhat. i need to introduce a shorthand notation to render the resources concisely. the terms that are referred to shall be referred to as markers and indicated using the notation mi when appropriate. the use of the term ‘markers’ is appropriate for at least two reasons. firstly, it corresponds to hegel’s use of the german term merkmal and, secondly, it corresponds to what the french refer to as le trait signifiant, the signifying feature (žižek, 1994, p. 47). it is especially the latter connection that is methodologically suggestive here because the flow and punctuations of pedagogic discourse are indissolubly bound up with the signifying features that emerge in pedagogic exchanges between teachers and their students, and the discursive punctuations tell us a good deal about the operation of pedagogic evaluation. an aggregation of markers is indicated by the use of the symbol ‘ · ’ positioned between each successive pair of markers, as in: m1 · m2 · m3. any collection of singular markers, say mj, mk and ml, can be aggregated by conjunction to produce mj · mk · ml. given any aggregation, say m1 · m2 · m3, a disjunction may be produced by selecting out one or more terms, as in m3 or m1 · m2 or m1 · m3 and so forth. central to the flow of discourse are the gestures of abbreviation and explication, both of which relate groups of terms. abbreviation effects a substitution of a group of markers by one or more markers, but always with fewer markers than the original group: (ma · mt · … · mr) mm indicates that mm abbreviates the aggregate ma · mt · … mr. in general, it tends to be the case that abbreviation substitutes a single marker for an aggregate of markers. explication substitutes an aggregate of markers for a smaller aggregate of markers, or for a single marker: the expression mk⇐ (mσ · mη · … · mπ) indicates that mσ · mη · … · mπ is an explication of mk. the production of expressions of the type described here should in no way be seen as constituting a calculus with rules of transformation enabling one to move from one expression to the next. the expressions are ‘bare bones’ symbolisation of features of pedagogic exchanges that could be described in ordinary language. the great utility of the expressions is their ease of use in summarising the features i am interested in, enabling the production of very compact global summaries of general features of entire lectures or lessons. one of the important differences between my use of this notation and mccumber’s, which is a consequence of dealing with actual pedagogic exchanges between a lecturer and her students, is that where mccumber uses finite series of marker subscripts that follow the sequence on natural numbers (1, 2, …, n), i cannot do so because the temporal unfolding of pedagogic texts is not such that everything pertaining to a particular abbreviation or explication emerges in strict temporal succession. therefore, while mccumber (1993, p. 132), for example, indicates explication as mn+1 ⇐ (mn-x · … · mn), where n > x, i use a series of subscripts that do not imply an order (e.g. mk ⇐ (mσ · mη · … · mπ)) when describing gestures in general terms to avoid suggesting a necessary strict temporal succession of the markers in any expression that is intended to refer to pedagogic exchanges. that said, i do, nevertheless, wish to capture the temporal emergence of markers. to that end, as they emerge, each marker is given a natural number subscript strictly observing the order of the natural numbers, so that the nth marker to emerge is given the subscript n. when a marker is introduced, by whatever mechanism, the presentation of the marker as the focus for further work is referred to as immediation. as part of her general strategy for dealing with students’ responses, the lecturer would listen to them carefully, repeat what they said in a modified form, but usually seeming to preserve the sense of their statement, and then write down one or more words on the chalkboard to summarise the exchange. in fact, we might interpret the lecturer’s strategy as one that effectively produces a series of markers. in the first session of the course, after spending around six minutes on introductions and administrative business, the lecturer kicked off the action by soliciting from her students their associations with school geometry: ‘the first question is: if you think of school geometry, what comes to mind? affective words, content, everything. anything.’ in different terms, as an opening gambit the lecturer requested an explication of the term geometry from her students. i shall indicate geometry by the marker m1, so m1: geometry is the first marker to be immediated. the students reacted to the lecturer’s request by offering the terms listed as follows: m8: hard to teach m6: reasoning m3: theorems m2: parallelograms m9: hard to learn m7: diagrams m5: proofs m4: angles m10: only clever people m11: coordinate geometry as different students made statements or offered various terms, the lecturer wrote single words or short phrases up on the chalkboard, placing the expressions in different columns, the organising principles of which were not announced. rather than substitute the lecturer’s words and phrases with terms of my making, i have chosen to keep her expressions and treat them as markers. the marker indicators alongside each term were, of course, not part of what the lecturer wrote up on the chalkboard and have been inserted here to indicate the sequence in which the students suggested the terms. in each of the figures that i use for the purpose of presenting the terms written up on the chalkboard by the lecturer i will preserve the relative spatial positioning of the terms as far as is possible. what emerges as an initial explication of m1: geometry is constructed by treating the students collectively as a communal pedagogic subject, constituting terms from the students’ individual offerings and aggregating them as though they derived from a single individual. the lecturer’s particular arrangement of markers in distinct columns, as shown below, indicates an implicit abbreviation of selections of markers, effecting a reorganisation of markers. the lecturer’s abbreviations (viz. m12, m13, m14 and m16) are indicated in bold at the head of each column. a belated offering of a marker by a student, indicated as m15, was inserted by the lecturer as she revealed her abbreviations of students’ markers and written between m4 and m11. m16: affective m14: products m13: processes m12: contents m8: hard to teach m6: reasoning m3: theorems m2: parallelograms m9: hard to learn m7: diagrams m5: proofs m4: angles m10: only clever people m15: diagrams given m11: coordinate geometry the production of multiple abbreviations of the students’ explications by the lecturer can be viewed retroactively as produced by a series of disjunctions performed on the aggregate m2 · m3 · … · m11 · m15 to generate the sub-aggregates m2 · m4 · m11 · m15, m3 · m5, m6 · m7 and m8 · m9 · m10. the sub-aggregates are regulated by the implicit markers m12: contents, m13: processes, m14: products and m16: affective as they are used to effect the abbreviations (m2 · m4 · m11 · m15) ⇒ m12, (m3 · m5) ⇒ m13, (m6 · m7) ⇒ m14 and (m8 · m9 · m10) ⇒ m16. a fundamental purpose of the use of the series of abbreviations by the lecturer is revealed to be one concerned with generating material that enables her to focus attention on the affective. she performed a disjunction on the aggregate of category markers, m12 · m13 · m14 · m16, selecting out m16 for further elaboration: l: and here you have some affective factors. can anybody say something positive about geometry? ss: [laughter] l: what was positive? what can you think of as positive from geometry? your experience of geometry. a student offered a few suggestions that are included under the category of the affective by the lecturer, extending the explication of m16: affective to include m17: can reason things out and m18: learning. m16: affective m8: hard to teach m9: hard to learn m10: only clever people m17: learning m18: can reason things out before moving on, this is an appropriate point at which to introduce a couple of mccumber’s gestures: reflection and counter-reflection. mccumber (1993, p. 132) stipulates that markers that have been conjoined by both ⇒ and ⇐ are to be considered as having been caught up in the transition to reflection, for which he uses the connective ⇑. mccumber’s point appears to be that the movement from abbreviation to explication for a given marker has the effect of shifting attention to reflection because it gives us pause: subsequent to abbreviation and explication we are caught in a situation that has us move from ‘something done to something thought about’ (p. 134; italics in the original). as an example, consider m16. having introduced m16: affective as an abbreviation of the aggregate m8 · m9 · m10, viz., (m8 · m9 · m10) ⇒ m16, the immediate additional attention given to m16 entailed a further explication of m16, viz., m16 ⇐ (m17 · m18), which has the consequence of generating m16 ⇐(m8 · m9 · m10 · m17 · m18) as a fuller explication. considering m16 as an abbreviation now, we have (m8 · m9 · m10 · m17 · m18) ⇒ m16. we thus have two ways of reading m16. first, as an abbreviation: the properties marked by m8, m9, m10, m17 and m18 are collectively to be called ‘affective’. next, as the subject of a series of explications: ‘affective’ has the series of effective properties marked by m8, m9, m10, m17 and m18. stated in this way, as though m8, m9, m10, m17 and m18 were necessary properties of m16, we immediately feel the conceptual strain of encountering the contingent (i.e. m8, m9, m10, m17 and m18) appearing in the guise of the necessary. the impossibility thus produced forces on us the gesture of reflection: ‘something thought about’. to indicate that we are dealing with an abbreviation or explication of m16 under the aspect of reflection, we write (m8 ⇑m9 ⇑m10 ⇑m17 ⇑m18) ⇒ m16 to represent abbreviation and m16 ⇐(m8 ⇑m9 ⇑m10 ⇑m17 ⇑m18) to represent explication, from which we can write (m8 ⇑m9 ⇑m10 ⇑m17 ⇑m18) ⇑m16 as representing reflection, tout court. the important hegelian point is that, in general, the precipitation of an impossibility out of the construction of a coincidence of abbreviation and explication engenders the gesture of reflection. to symbolise counter-reflection mccumber stipulates that an expression indicating reflection, like (m8 ⇑m9 ⇑m10 ⇑m17 ⇑m18) ⇑m16, permits us to write the aggregate [(m8 ⇑m9 ⇑m10 ⇑m17 ⇑m18) ⇒m16] · [m16 ⇐(m8 ⇑m9 ⇑m10 ⇑m17 ⇑m18)] from which we are free to focus on either of the more complex markers or their constituents. the lecturer quizzed a student on the teaching of geometry, using his responses to focus in on an explication of m9: hard to learn. l: mm, mm. and teaching geometry? s: teaching geometry. that is what the kids have a problem with doing. they … in other words, that they prefer you having something that’s already mapped out for you. in other words, you start at a point and you end. and the steps in-between they know how to get there. but geometry is a little bit more … you have to search. you have to play around with things. and extract from what you’re given. you have to extract the solution. l: do you think that’s what makes it hard for them? as the students responded to the lecturer’s question she listed their responses in a manner that amounts to explications of both m8 and m9. the linking of the student’s response to m9: hard to learn effects a disjunction on the aggregate m8 · m9 · m10 · m17 · m18, resulting in an immediation on m9. the lecturer focused briefly on getting the students to explicate m8: hard to teach in a similar fashion, once again using disjunction on m8 · m9 · m10 · m17 · m18 (to immediate m8), but then shifted the emphasis to explicating m9: hard to learn, producing a fairly extended series of explications. as usual, the indexes of markers indicate the order in which they emerged and the spatial positioning of the markers indicates their locations relative to other markers. m8: hard to teach m9: hard to learn m21: foundations not in place m20: open-ended, creativity needed, needs problem-solving skills m23: textbook bound m25: visualisation: reading of diagrams m27: no practical link m28: not important, for example, for commerce m19: manoeuvre: search to get to the endpoint of a problem m26: apply what is learnt m22: perception: only clever people m24: children required to work above their developmental level the lecturer brought the explications of m8 and m9 to a close and then effected a disjunction on the aggregate (m21 · m23) to select out and immediate m21: foundations not in place. l: lots of interesting things that you bring up here. the first that i want to explore with you further, you said they find it difficult because the foundation is not in place, because in the lower grades they were textbook bound. what would you say is a proper foundation for geometry at the levels at which you teach it, which i take is high school level? right? what do you see as a proper foundation? what would you want? as she spoke she cleared a section of the chalkboard and wrote the heading ‘foundation’, effectively generating a new marker off m21: foundations not in place, which i shall refer to as m29: foundation. this move is one that contracts m21 to produce a form that the lecturer deems to be more appropriate to the further development of her teaching. mccumber does not have a term for such a gesture, so i’ll name it contraction, and we can say that the lecturer performed a contraction on the marker m21: foundations not in place to produce the marker m29: foundation. in addition, i use the symbol ⊂ as a connective to indicate contraction, as in m29 ⊂ m21. the students generated a series of explications in response to m29: foundation: m29: foundation m30: practical exploration m31: measurement m32: introduce geometry in the same way as it developed historically m33: basic axioms m34: use and understanding measurement instruments. length vs size of angle. units (vocabulary: big vs long) the lecturer then effected a disjunction on (m20 · m25 · m27 · m28 · m19 · m26 · m22 · m24), immediating m24: children required to work above their developmental level. l: the other thing i want to pick up with you is … children are asked to work above their developmental level. how do we gauge their developmental level? what gives you an idea that they are not ready for it and at another time, when? what are the characteristics of that readiness from your experience? say we think of a grade 8 learner coming in to your class, and you need to do geometry with them. what is it, typically, that you find they can’t do, which you think they are just not ready for? the lecturer’s questioning of the students implicitly effected a contraction to produce the marker m35: developmental level from m24 (i.e., m35 ⊂ m24), with m35 becoming the object of explication, the result of which is m35 ⇐ (m36 · m37). m35: developmental level m36: statement/reason; given/asked m37: link together → build a logical argument with m29 and m35 the discussion is seen to home in on a pair of twin lacks, one of which is associated with teaching and the other with learning. the message is that teaching is responsible for setting in place adequate foundations for students to be working at an appropriate developmental level, but that both teaching and students are lacking. it is at this point that the lecturer introduced a shift from the realm of opinion and experience to knowledge as it is realised in theory, by asking the students to redescribe the lack in school students in theoretical terms. l: i’m sure that you’ve done some learning theory. things like that. last year and this year. have you? can you link any of that knowledge, of theories of learning, of theories of development, to this thing of ‘they are not ready’? the students were not all that forthcoming on the matter, but that was not an issue for the lecturer because her aim had been to set in place the conditions for her to insert the van hiele theory of the development of geometrical thinking, thus producing a relation that functioned like an abbreviation, viz., ((m36 · m37) · (m30 · m31 · m32 · m33 · m34)) ⇒ van hiele. by means of using what students would have experienced as a record of their own thoughts on school geometry, they had been taken from their experiences to a point where they would have to confront themselves as lacking with respect to both pedagogy and mathematics. van hiele testing was presented as a benign, scientific measure of the state of geometrical thinking, as the lecturer attempted to downplay the element of lack: l: what we are going to do now ties in with developmental levels. i’m going to give each of you a copy of the van hiele geometry test, or assessment instrument. now, this instrument is used all over the world for in-service teachers and for children at all kinds of levels to ascertain their geometric thought levels. so this is not something you can fail or pass. it simply throws out a level of thinking, and we’re going to work with this van hiele level of geometric thinking a lot. we’re going to relate what we do on sketchpad back to this. we’re going to try and ascertain our own levels of thinking. we are going to become aware of where we are aiming to get to and what the characteristics are of a level we need to be on to understand our school geometry. twenty-eight minutes into the lecture the lecturer handed out what she referred to as a van hiele test for students to work on individually. she collected their answer sheets and the test booklets about 36 minutes later. after the students had a break, the lecturer assured them that the developmental levels revealed by a van hiele test are not age dependent but related to previous experience: l: interestingly enough, i’ve worked a lot in adult education, and we find the exact same developmental path, from young, young children–i’m talking from three years and on–or with adults that haven’t been schooled before. so it seems to be an inherent development of geometrical thought that’s not so much dependent on age, but on what came before. okay? at this point it would be helpful to present an overview of the flow of the pedagogic exchanges between the lecturer and her students. figure 1 shows a network representing the flow of explications and abbreviations that emerged during the lecture. starting from an invitation to students to explicate geometry, the pedagogic exchanges are regulated by the lecturer, who uses carefully selected disjunctions to steer the discussion in the direction of explications of the affective, from which she finds material enabling her to pick up on teaching and learning with respect to failure, thereby registering the existence of a pair of lacks that are to be unified by considering them in relation to the van hiele theory of the development of geometrical thinking. figure 1: network showing the flow of abbreviations and explications from m1 to the insertion of van hiele. in slightly different (hegelian) terms, we see that the lecture unfolds by starting with the actuality of the phenomenal experience of the students and then focuses in on affective issues, and it is only in relation to the affective that we see the formal structure of reflection emerging in the pedagogic exchanges, and precisely around the twin issues of a lack in teaching and a lack in learning. while the references to the lacks in teaching and learning connect deeply with teachers’ experiences, this is also the moment when the actuality of their immediate experiences is suspended by the production of the twin contractions that immediate the markers of foundation and developmental level, m29 and m35. what these might be pushes pedagogic discourse to confront the openness of possibility, which is exactly what reflection generates. the introduction of the van hiele test and the appeal to that theory of the development of geometrical thought closes down the openness and inserts a new necessity into the discourse, where van hiele is intended to describe and explain everything—successes and failures. the lecturer’s general pedagogic strategy of inviting her students to offer explications of various terms means that she constantly has to deal with the contingent: she cannot know in advance what will emerge from her students’ responses. we have also seen that she uses disjunctions and explications very skilfully to keep the discourse flowing in the general direction that she prefers. what this tells us is (1) that the contingent emerges at each of the moments of actuality, possibility and necessity in the discursive flow and (2) that the lecturer’s disjunctions, abbreviations and invitations to explicate are constituents of pedagogic evaluation. as already mentioned, the flow of pedagogic exchanges produces two inter-related splits, one internal to knowledge, the other internal to the pedagogic subject. the former is registered in the indirectly voiced distinction between knowledge deriving from pedagogic experience versus knowledge deriving from the van hiele theory (opinion versus knowledge, in platonic terms), the latter in the distinction between the pedagogic subject’s imagined geometrical skill (as registered in grade level, for example) versus their determined van hiele level. so, by the conclusion of the first session of the course, the students had been taken from the certainty of their phenomenal experiences as teachers of geometry to the point of encountering themselves as lacking in both knowledge of geometry as well as in the teaching of geometry, thus setting the stage for their engagement with geometry in the sessions to follow. part 2: how to draw a square the sketchpad environment is, amongst other things, an environment that exploits the visual representation of mathematical content as a central resource for the teaching and learning of school mathematics. each of the students had a personal copy of sketchpad, including a manual and a copy of the geometer’s sketchpad learning guide (gslg; chanan, 2001) and each had access to their own computer workstation in the computer laboratory used for the duration of the module. at the start of session 2 the lecturer drew the students’ attention to a task in the gslg that required them to construct a square: l: now you will see on page sixteen constructing a square is the task there. my purpose for taking you to this task is, apart from what you will learn in the outcomes that they have there. i talk you through this thing. say do this, do this, do this. and you will always be dependent on me. right? i want you to follow the text and try out, following the text, what they want you to do. it will take you, i think, around about half an hour to come up with a constructed sketch of the square, and after that we’re going to discuss why it had to be along these lines. the problem of constructing a square in the task on page 16 of the gslg is explicitly used as a vehicle for introducing the sketchpad neophyte to the commands and functions available in the sketchpad environment. the students responded to the task in three different ways. firstly, there were those who attempted to produce a four-sided figure that looked more or less like a square and then proceeded to drag the vertices of their drawings to produce an approximation to a square. the images in figure 2 show the work of two such students. next were the students who produced squares by measuring the lengths of the sides and the angles between adjacent sides of the quadrilaterals they drew. such students used the lengths and angle measures to adjust their quadrilaterals so that the definition of a square was satisfied. a third group of students followed the instructions listed in the gslg and were able to produce the constructions as illustrated in the text (see figure 3 for an example). we shall refer to the three methods used to produce a square as method 1, method 2 and method 3, respectively. figure 2: attempted imagistic approximations to a square. method 1 shows an attempt at using pictorial similarity as the chief resource for producing a square: if it looks like a square then it is a square. the students using method 1 would, of course, have to know something about what a square looks like and so would more than likely know something about the definition of a square. however, it is the image of the square that appears to be primary for such students, so that the formal definition of the square functions as a description of the image and is therefore secondary to that image. in other words, what can be seen enjoys a much more substantial reality that that which is discursively rendered. the students employing method 2 use the definition of a square, so that the image of the square is a representation of a square as discursively defined. here the definition of the square is central to the reality of what a square is. however, the students using this method imagine that accurate empirical measurement is required to realise a square, so that empirical measure is central to the realisation of the square. method 3 enables the students to produce a construction that is more stable than that generated by method 2. a square produced by method 3 is not destroyed by dragging any of its vertices, unlike one produced using method 2. method 3 is, in a sense, indifferent to empirical measure in that the square is, in the first instance, produced by fixing its elements in a system of formally defined determining relations. here the particular empirical measures of angles and sides of any square so produced are effects of the formally defined relations between sides and angles. further, the image of the square, that is, what a square looks like, is also an effect of those formally defined relations. the different emphases of the three methods discussed above can be read as co-present in particular instances of the production of a square. for example, the student using method 2 starts by drawing a roughly square-like quadrilateral (image) and then adjusts the positions of its vertices by using measures of angles and lines (empirical measure) so that the relations between the angles and lines conform to the formal definition of a square (definition). similarly, we would expect that the student starting from formally defined relations (definition) as in method 3 checks that the square they produced looked like a square (image) and that they might even measure angles or lengths (empirical measure) if the image suggests that a quadrilateral other than a square has been produced. at the very least, the students using method 1 appear to appeal to a proto-definition of a square: the software forces them to select and draw the four vertices and four line segments (definition) that are needed to be arranged in a particular configuration that is visually recognisable as square-like (image), the latter requiring some sense of measure to realise (empirical measure). what is of interest is that three methods show three different emphases. method 1 is dominated by the image, method 2 by empirical measure and method 3 by formally defined relations. this statement needs to be understood by focusing on what the primary resource is and what is secondary in regulating the activity of producing a square in each of the methods, where what is secondary is positioned as an effect of that which is primary. central to each method is a particular type of regulation of mathematical activity and corresponding to each regulative type we have a primary means for evaluating the existence of the square that can be framed in the form of a series of questions: (1) does it look like a square? (2) are its sides of equal length and each of its angles 90 degrees? (3) does its construction satisfy the formal relations entailed in the definition of a square? as before, in each of the three questions we can hear the echo of the other questions, but what is important to attend to is the determination of what i shall refer to as a regulative orientation, indexed by taking a particular question as primary. a regulative orientation can be understood as colouring the student’s mathematical activity in a particular way, producing an over-determining effect on that activity. we might describe the primary regulation of the mathematical activity of the student in more general terms as exhibiting iconic, empirical or formal orientations. describing the activity of students in general terms permits us to describe empirical instances of mathematical activity beyond the case under consideration here. to that end, we need to express the primary and secondary resources that figure in regulative orientations in general terms as well. there are three general types of resources: images, empirical tests and formally defined relations between mathematical objects. in the particular instance discussed here, the empirical test was one of measuring angles and lengths, but empirical tests can take many different forms. school learners, for example, are often asked to check the validity of their algebraic transformations of expressions by substituting actual numbers for the variables or unknowns, or teachers might substitute numbers into expressions in an attempt to convince learners of the validity of a statement. now it might be argued with some justification that an appeal to an image is also a sort of empirical test–and i would agree. an important difference between the appeal to an image and measuring or substituting numbers into expressions is that the former entails a very weak form of validity, usually requiring more stable empirical testing or appealing to formal definitions or propositions to convince oneself and others of the truth of a claim grounded in the iconic. in fact, as concerns the mathematical validity of claims, it is apparently the case that the claims generated from the different regulative orientations can be arranged in a hierarchical sequence starting from the mathematically least valid: firstly, claims grounded in the iconic, then those that are empirically grounded and, finally, those grounded formally. this is not to deny that pedagogic experience has often shown us that formally grounded claims might well be the least convincing to the school mathematics student. table 1 summarises significant features of the analytical language developed from a consideration of session 2 of the geometry module to this point. i shall now return to session 2 to develop the discussion a bit further. drawing versus construction once most of the students managed to produce the square as required by the gslg task, the lecturer marked out the differences between what she referred to as drawing and construction: l: when we started out you made some drawings with those tools. right? you can draw a circle; you can draw a line; you can draw a square with the lines. do you agree? what makes those drawings different from the construction of the square that you did? what’s the difference between just taking the line and drawing the square to the construction of the square? s: i suppose that when you’re drawing the square you start moving the points and it does not necessarily stay a square. but when it’s constructed and you start moving the points around it will remain a square, no matter what angle you rotate it through, and no matter how close you bring the points, or how far apart they are. l: did you observe that? if you just draw a square, then you can even mark the points, or whatever. if you just draw, the thing remains free. right? you can move those lines into any other configuration that you want to, which remains free. it isn’t defined as a square. it just looks like a square. with me … whereas, when you constructed the square, you sat with a defined object, which you could manipulate on the screen. you could take any of the points and move it around. yes, it shrunk, or, you dilated it. or, you can turn it, but if you watch the measurements, you still let your angle remain ninety degrees, and however the side lengths change, they remain equal to each other. do you agree? the lecturer, following the gslg, set drawing and construction in opposition. she associated drawing with the simple reproduction of images of geometrical objects and construction with the formal definitions of those objects. for the lecturer, drawing was grounded in the iconic and construction in the formal. the gslg refers to a particular test to be used to check whether or not a geometrical object was produced by straight-edge and compass construction, viz., the drag test. as its name suggests, the test requires the student to drag some element of a figure, usually a point, to check whether the figure is stable. if the figure does remain stable, then the student can assume that the relations between its constitutive elements remain true to the formal definition of the particular geometrical the figure it depicts. if, for example, the square was not produced by the equivalent of straight-edge and compass construction, then the relations between its sides and angles would be altered by the drag test, destroying the square (see figure 3). so, even if a square is produced in strict accordance with its definition, but not by straight-edge and compass construction, it is fated to be destroyed by the drag test. within the sketchpad environment the strong distinction being made in this particular instance is between geometrical objects produced by euclidean methods and those not so produced. for the lecturer, the use of the formal definition of objects is associated only with ‘construction’. over the seven lectures that make up the module, the lecturer used a number of oppositions that she saw as related to the drawing-construction opposition: spatial thought vs geometric thought, free vs defined, and remembering vs visualising. figure 3: attempt at the construction of a square as illustrated in the gslg. the drawing-construction opposition (see figure 4) used by the lecturer (and the gslg) also suggests related descriptive categories other than drawing and construction, but which remain hidden in the lecturer’s interaction with her students. we can define two descriptive categories that are neither drawing nor construction by considering the contradictories of construction and drawing in relation to the opposition drawing-construction: firstly, we have a class of squares that are reasonably accurate representations of squares within the sketchpad environment, obtained by means of empirical testing, but which collapse when subjected to the drag test; secondly, there is also a class of objects intended as squares but which do not even look like squares (see figure 2 for an example). figure 4: ‘drawing’ vs. ‘construction’ of a square in sketchpad. by using the semiotic schema proposed by greimas (1968), which he derived from the aristotelian square of oppositions, i can expand the lecturer’s descriptive schema from a simple opposition to a more complex schema of four inter-related categories, a first rendition of which is shown in figure 5. the categories are drawing, construction, not-drawing and not-construction, where drawing vs construction is the primary opposition of contraries as constituted by the gslg and the lecturer. the oppositions drawing vs not-drawing and construction vs not-construction are oppositions of contradictories derived from the primary contrary opposition. the opposition not-construction vs not-drawing is also one of contrariness and those categories entail the sub-contraries. the relations between drawing and not-construction on the one hand and construction and not-drawing on the other are relations of entailment: drawing is entailed in the category of not-construction and construction in that of not-drawing. figure 5: greimassian semiotic square derived from the drawing-construction opposition. i can now describe the categories derived from the drawing-construction opposition in terms of the categories of regulative orientation developed earlier (refer to table 1). in the context of the sketchpad environment, what are considered drawings as opposed to constructions are those sketches produced under iconic and empirical regulative orientations. the empirical orientation is one that entails a degree of iconicity, which suggests that not-construction be associated with an iconic regulative orientation and drawing with an empirical regulative orientation. further reflection on the category not-drawing suggests that the categories of regulative orientation need to be rethought. one the one hand, not-drawing corresponds to objects that are produced with, at best, a partial description of the object in mind. within the expanded schema it might be interpreted as the only category that does not appeal to a definition adequate to the purpose of producing an image of a square. on the other hand, as all mathematics teachers are aware, there exist notational resources, like particular symbols for indicating right angles and equal lengths, that can be attached to any suitable rough sketch to mark it as a square, an example of which is shown in figure 6. figure 6: a pictorial description of a square. the class of such squares would not function as legitimate responses to the demands of the task in this instance: such a representation of a square is really functioning as a pictorial description, in much the same way as a definition presented in written or spoken language, rather than a euclidean construction of a square. given that construction has to be understood as entailed in not-drawing, and thinking about a formal regulative orientation as constituted by both algorithmic and propositional orientations, with the former entailed in the latter, construction can be associated with an algorithmic regulative orientation and not-drawing with a propositional regulative orientation. such an arrangement makes good sense since the construction of the square emerges from the constitution of an appropriate method that is derived from euclidean propositions and relations between them. what has thus been produced, summarised in table 2, is a revision of the regulative orientations that were presented in table 1. table 1: summary of analytical terms and their relations. table 2: summary of revised analytical terms and their relations. the diagram of the semiotic square derived from the drawing-construction opposition can now be elaborated a bit further by mapping the revised regulative orientation categories on to the series drawing, construction, not-construction, not-drawing, as in figure 7. figure 7: revised regulative orientations mapped onto the construction-drawing opposition. the iconic and propositional regulative orientations, which now stand in opposition by being associated with the sub-contrary opposition not-construction vs not-drawing, bring the latter opposition under the aspect of the sensible-intelligible and opinion-knowledge oppositions central to the platonic universe. part 3: a square in drag as concrete universal the manner in which the lecturer uses the drawing-construction opposition reveals something interesting about the position of the empirical in the privileged pedagogy of the module. recall that the lecturer asserted the following in support of construction: when you constructed the square, you sat with a defined object, which you could manipulate on the screen. you could take any of the points and move it around. yes, it shrunk, or, you dilated it. or, you can turn it, but if you watch the measurements, you still let your angle remain ninety degrees, and however the side lengths change, they remain equal to each other. (my emphasis) the straight-edge and compass construction of the square demonstrated in the gslg determines the square as such, so that the construction is itself the proof that the figure is indeed a square. the lecturer and her students, however, never used the construction to prove that the figure they constructed was a square. instead, they used measure as an empirical test of the truth of the square as square. even in her discussions with individual students the lecturer repeatedly called for and encouraged them to perform empirical tests using measure to check their constructions. we see that, on the one hand, squares produced by empirical testing fail the drag test and are thereby marked as non-legitimate constructions; on the other hand, empirical testing is considered sufficient to validate the equivalent of a straight-edge and compass construction of the square. in other words, empirical testing comes to be aligned with both drawing and construction and, in that way, with both the individual and the universal. it would therefore appear that an empirical orientation is used in an attempt to assert the universal and that we have a so-called inductive pedagogy in play. in her written introduction to the course the lecturer had informed students that they would ‘get opportunities to engage in hypothesising, testing by experiment and proving in the context of synthetic geometry; transformations and analytic geometry’, but there was very little evidence of students engaging with proof over the seven sessions. what the session reveals is the general structure of the inductive pedagogies widely championed in the preparation of teachers for the teaching of school mathematics today. the structuring matrix of the pedagogy is of the form particular-singular-universal, where the particular is given as a particular task and the singular is constituted by the contingent activity of the student. that is, students are confronted with some particular situation from which a problem emerges (construct a square) and in response to which a series consisting of singular student reactions is produced (individual squares) to mediate between the particular and the universal (square-as-such). here an attempt is made at effecting a shift from the particular to the universal by establishing a pair of links of the form: particular-singular/singular-universal. hegel (1816/1969, p. 690) describes induction as the syllogism of experience–of the subjective taking together of the individuals into the genus and of the conjoining of the genus with a universal determinateness because this latter is found in all the individuals. (italics in the original) hegel does, however, warn of the problems of proceeding inductively as he sets about describing the central features of induction, from which we can recognise that mathematics as we know it could not have been generated inductively. that is, mathematical systems would collapse if they were constructed solely on series of inductive conclusions that remain problematical. [induction] is … essentially a subjective syllogism. the middle terms are the individuals in their immediacy; the subjective taking together of them into the genus by means of allness is an external reflection. on account of the persistent immediacy of the individuals and their consequent externality, the universality is only completeness, or rather remains a problem. in induction, therefore, the progress into the spurious infinite once more makes its appearance; individuality is supposed to be posited as identical with universality, but since the individuals are no less posited as immediate, that unity remains only a perennial ought-to-be; it is a unity of likeness; those which are supposed to be identical are, at the same time, supposed not to be so. it is only when the a, b, c, d, e are carried on to infinity that they constitute the genus and give the completed experience. the conclusion of induction thus remains problematical. (hegel, 1816/1969, pp. 690–691; italics in the original) we should note that the pedagogic modality privileged by the lecturer is only apparently inductive. firstly, and most obviously, the pedagogic modality has as its object the reproduction of mathematics, not the production of new mathematics. this is not to deny that the student might experience the reproduced mathematics as new. secondly, the organisation of the pedagogic text is such that the tasks the student is confronted with already have encoded into them, in a condensed form, the conceptual work that went into the production of the mathematics content. that is, tasks are always-already structured from the position of knowledge of the content that is to be acquired. thirdly, it follows that the aforementioned structuring of tasks provides an implicit guarantee that the finally arrived at content will indeed be legitimate mathematics content. in other words, an elaborate game is being played in which the student agrees to proceed as though they are a producer of mathematical knowledge who employs inductive reasoning, while the mathematical truth of the results of such activity are guaranteed in advance of their production. hegel (1816/1969, p. 690) points out that induction in expressing that perception in order to become experience ought to be carried on to infinity, presupposes that the genus is in and for itself united with its determinateness. therefore, strictly speaking, it rather presupposes its conclusion as something immediate. … an experience that rests on induction is accepted as valid although the perception is admittedly incomplete; but the assumption that no contradictory instance of that experience can arise is only possible if the experience is true in and for itself. thus the syllogism by induction, though indeed based on an immediacy, is not based on that immediacy on which it is supposed to be based, on the merely affirmative [seiende] immediacy of individuality, but on the immediacy which is in and for itself, the universal immediacy. (italics in the original) therefore, for induction to apparently operate as a reliable syllogism, it must be based on an immediacy which is always-already the universal immediacy. the relation between the individual realisation of an object and its recognition as a legitimate instance of a universal notion is a problem of perennial concern to teachers. the question i now address is that of how sketchpad is used to assert the universal in the case under discussion here. i have already briefly discussed how the drag test focuses attention on the universal defining features of the square. the students’ initial attempts at producing squares in the sketchpad environment represent a contingent series of phenomenal experiences and constitute a form of reflection that hegel refers to as positing reflection. every square that is offered by a student is different from every other square offered by the student and by other students and so the series of phenomenally generated squares constitute a series of individuals. the series of squares produced by the students is a subset of the infinite class of squares that it is possible to produce in the sketchpad environment by the range of methods used by the students. the exceptional element of the class of such squares that has the effect of halting the infinite flow of squares at the level of phenomenal experience is the constructed square. the latter is an interesting object. on the one hand it is merely one among the set of possible phenomenally experienced squares; on the other hand, it is square-as-such, standing for the entire class of squares. in the expression of genus-species relations, the constructed square is genus and, at the same time, species. that is, of the class of squares, the constructed square is the exceptional element that completes the infinite series by being it and so, paradoxically, being the final element. each phenomenally experienced square, including the exceptional element, finds its ground in the exceptional element, so that the genus finds itself amongst its own species. once we have moved from an enumeration of the series of phenomenally experienced squares to the production, and addition, of the exceptional square that is both genus and species (square-as-such), then we have also moved from positing to external reflection. in hegelian terms, the exceptional element–which is the square that remains a square when subjected to the drag test–achieves the status of concrete universal. that is, an individual through which shines the universal notion of square. the use of the drawing-construction opposition encoded into sketchpad fitted nicely with the lecturer’s pedagogy, in which her engagement with students had them shift from the apparent certainty of their individual experiences to confronting themselves as lacking and (hopefully) on to a deeper engagement with the content of a lecture. recall that the student as lacking is not asserted, or presumed, but has to be produced in the course of the flow of pedagogic exchanges between lecturer and students, or between text and students. what might be considered failure–the production of drawn rather than constructed squares by many students–is thus a legitimate element of the pedagogy modelled by the lecturer. the production of the series of squares is simultaneously the production of markers of another series–namely, the van hiele levels of geometrical thought–which is not immediately apparent to students but which is pointed out and alluded to at various moments in lectures. the production of a drawn square is not simply wrong. rather, its presence is ultimately used to legitimate the deployment of the series of van hiele levels to read geometrical activity. the infinite class of phenomenally produced squares, of which the students’ particular series is an instance of positing reflection, is thus closed in a second way by means of the introduction of the van hiele levels. the shift from positing to external reflection is therefore repeated, but this time with respect to the lack that emerges as indexed by the individual squares that are produced. what we have is the production of a second exceptional element, viz., the van hiele level. using lacan’s (1966/2006) distinction between the ‘subject of the enunciated’ and the ‘subject of the enunciation’, one might argue that the two series generated by the pedagogic exchanges–the first concerning the production of squares and the second the production of van hiele level markers–have as their respective targets the subjects of the enunciated (squares) and of the enunciated (van hiele levels), the latter being the place from which the enunciation is presumed to be made. the subject of the enunciation is, precisely, the student-as-subject. a concern with the subject of the enunciation is, ultimately, a moral concern, its incessant question directed at the student being: ‘who are you?’ recall that, in his attempt at clarifying the structure of pedagogic discourse, bernstein (1996) described pedagogic discourse as an instructional discourse embedded in a regulative discourse, with the regulative discourse being a moral discourse. the analysis of the shift from positing to external reflection with respect to two series (squares-as-squares and squares-as-levels) certainly resonates with bernstein’s proposition. concluding remarks the structure of session 2, in which the focus was on the production of a square in the sketchpad environment, was repeated in the remaining five sessions. firstly, some particular was introduced to the students who were then required to investigate it in a manner that promoted the production of a series of individuals. from the latter, conjectures were generated which were subsequently tested and ‘verified’ by some empirical procedure–usually measurement–which was taken to establish the truth of the matter. simultaneously, by means of the drag test, sketchpad designers seek to encourage and train users in the construction of geometrical objects that are exemplifications of the universal ideas pertinent to the particular geometrical objects of concern. however, there was inevitably a retreat into the empirical as measurement remained the preferred test of validity, so that the empirical and iconic regulative orientations emerged as dominant in the activity of students. it was interesting to see that students spent inordinately long periods of time on tasks that they ought to have been able to complete in matters of minutes, being teachers of geometry. this was the case even in the later sessions, by which time they were very familiar with the sketchpad interface. the appeal to van hiele had the potential to influence students to develop ways of working with geometry that would privilege algorithmic and propositional regulative orientations, but the retreat to the empirical appeared to counteract that potential, so that van hiele ended up as a resource for registering the student as lacking rather than for assisting them in developing their geometrical thinking. in different terms, van hiele ended up as a resource for moralising the student. acknowledgements the production of data for this article was funded by the national research foundation under grant number fa2006031800003. any opinions, findings and conclusions or recommendations expressed in this material are those of the author and do not reflect the views of the national research foundation. this publication is based on research that has been supported in part by the university of cape town’s research committee (urc). competing interests the author declares that he has no financial or personal relationships that may have inappropriately influenced him writing this article. references bernstein, b. (1990). class, codes and control. volume 4: the structuring of pedagogic discourse. london: routledge & kegan paul. bernstein, b. (1996). pedagogy, symbolic control and identity. theory, research, critique. london: routledge & kegan paul. bloom, a.d. (ed.). (1991). the republic of plato. new york, ny: basic books. chanan, s. (2001). the geometer’s sketchpad, version 4: learning guide. emeryville, ca: key curriculum press. dowling, p. (2002). the sociology of mathematics education: mathematical myths/pedagogic texts. london: routledge. dowling, p. (2009). sociology as method: departures from the forensics of culture, text and knowledge. rotterdam: sense publishers. greimas, a.j. (1968). the interaction of semiotic constraints. yale french studies, 41, 86–105. hegel, g.w.f. (1969). science of logic. (a.v. miller, trans.). amherst, ny: humanity books/george allen & unwin. (original work published 1816) hegel, g.w.f. (1975). hegel’s logic: being part one of the encyclopaedia of the philosophical sciences. oxford: clarendon press. hegel, g.w.f. (1977). phenomenology of spirit. (a.v. miller, trans.). oxford: oxford university press. (original work published 1807) lacan, j. (2006). the subversion of the subject and the dialectic of desire in the freudian unconscious. écrits: the first complete edition in english. (b. fink, trans.). new york, ny: norton. (original work published 1966) mccumber, j. (1993). the company of words: hegel, language, and systematic philosophy. evanston, il: northwestern university press. piaget, j. (1995). egocentric thought and sociocentric thought. sociological studies. (l. smith, ed.). london: routledge. pinkard, t. (2000). hegel: a biography. cambridge: cambridge university press. van hiele, p.m. (1986). structure and insight: a theory of mathematics education. orlando, fl: academic press. žižek, s. (1994). the metastases of enjoyment: six essays on woman and causality. london: verso. footnote 1. written permission to videotape lectures and to audiotape interviews for the purposes of research, and to use those records for analysis and reporting on at conferences and in academic publications, was obtained from the participants in the study. microsoft word 3-16 huntley et al.doc pythagoras, 69, 3-16 (july 2009) 3 can multiple choice questions be successfully used as an  assessment format in undergraduate mathematics?    belinda huntley; johann engelbrecht; ansie harding  university of pretoria  huntley@sjc.co.za; johann.engelbrecht@up.ac.za; ansie.harding@up.ac.za      in this study we propose a taxonomy for assessment in mathematics, which we call the  assessment component taxonomy, to identify those components of mathematics that can be  successfully  assessed  using  alternative  assessment  formats.  based  on  the  literature  on  assessment  models  and  taxonomies  in  mathematics,  this  taxonomy  consists  of  seven  mathematics assessment components, hierarchically ordered by cognitive level, as well as  the nature of the mathematical tasks associated with each component. using a model that  we developed earlier for measuring the quality of mathematics test items, we investigate  which  of  the  assessment  components  can  be  successfully  assessed  in  the  provided  response  question  (prq)  format,  in  particular  multiple  choice  questions  (mcqs),  and  which  can  be  better  assessed  in  the  constructed  response  question  (crq)  format.  the  results of this study show that mcqs can be constructed to evaluate higher order levels of  thinking  and  learning.  the  conclusion  is  that  mcqs  can  be  successfully  used  as  an  assessment  format  in  undergraduate  mathematics,  more  so  in  some  assessment  components  than  in  others.  the  inclusion  of  the  prq  assessment  format  in  all  seven  assessment components can reduce the large marking loads, associated with continuous  assessment practices in undergraduate mathematics, without compromising the validity  of the assessment.  in south africa, as in the rest of the world, the changes in society and technology have imposed pressures on academics to review current assessment approaches. changes in education assessment are currently being called for, both within the fields of measurement and evaluation as well as in specific academic disciplines such as mathematics. geyser (2004) summarises the paradigm shift that is currently under way in tertiary education as follows: the main shift in focus can be summarized as a shift away from assessment as an add-on experience at the end of learning, to assessment that encourages and supports deep learning. it is now important to distinguish between learning for assessment and learning from assessment as two complementary purposes of assessment. (p. 90) mathematics at tertiary level remains conservative in its use of alternative formats of assessment. as goals for mathematics education change to broader and more ambitious objectives (nctm, 1989), such as developing mathematical thinkers who can apply their knowledge to solving real problems, a mismatch is revealed between traditional assessment and the desired student outcomes. it is no longer appropriate to assess student mathematical knowledge using general assessment taxonomies, because these taxonomies are not pertinent to mathematics and do not identify those levels of mathematics that can be assessed using alternative formats of assessment. with this background, we propose a taxonomy of mathematics, which we call the assessment component taxonomy, to identify those components of mathematics that can be successfully assessed using alternative assessment formats. using a model that we developed earlier (huntley, engelbrecht & mcq in undergraduate mathematics 4 harding, 2008; 2009) for measuring the quality of mathematical test items, we investigate which of the assessment components can be successfully assessed in the provided response question (prq) format, in particular multiple choice questions (mcqs), and which can be better assessed in the constructed response question (crq) format where students have to construct and supply their own responses (engelbrecht & harding, 2003). mathematics assessment models an assessment model emerges from the different aspects of assessment: what we want to have happen to students in a mathematics course, different methods and purposes for assessment, along with some additional dimensions. the first dimension of this framework is what to assess, which may be broken down into: concepts, skills, applications, attitudes and beliefs. niss (1993) uses the term assessment mode to indicate a set of items in an assessment model that could be implemented in mathematics education. these items include the following: o the subject of assessment, i.e. who is assessed o the objects of assessment, i.e. what is assessed o the items of assessment, i.e. what kinds of output are assessed o the occasions of assessment, i.e. when does assessment take place o the procedures and circumstances of assessment, i.e. what happens, and who is expected to do what o the judging and recording in assessment, i.e. what is emphasised and what is recorded o the reporting of assessment outcomes, i.e. what is reported, to whom. for the purpose of this study, the focus will be on the objects of assessment in the niss model outlined above, i.e. types of mathematical content (including methods, internal and external relations) and which types of student ability to deal with that content. this varies greatly with the place, the teaching level and the curriculum, but the predominant content objects assessed seem to be the following: (a) mathematical facts, which include definitions, theorems, formulae, certain specific proofs and historical and biographical data. (b) standard methods and techniques for obtaining mathematical results. these include qualitative or quantitative conclusions, solutions to problems and display of results. (c) standard applications which include familiar, characteristic types of mathematical situations which can be treated by using well-defined mathematical tools. to a lesser extent, objects of assessment also include: (d) heuristic and methods of proof as ways of generating mathematical results in non-routine contexts. (e) problem solving of non-familiar, open-ended, complex problems. (f) modelling of open-ended, real mathematical situations belonging to other subjects, using whatever mathematical tools at one’s disposal. in mathematics, we rarely encounter (g) exploration and hypothesis generation as objects of assessment. with regards to the students’ ability to be assessed, the first three content objects require knowledge of facts, mastery of standard methods and techniques and performance of standard applications of mathematics, all in typical, familiar situations. as we proceed towards the content objects in the higher levels of the niss assessment model, the level of the students’ abilities to be assessed also increase in terms of cognitive difficulty. in the proof, problemsolving, modelling and hypothesis objects, students are assessed according to their abilities to activate or even create methods of proof; to solve open-ended, complex problems; to perform mathematical modelling of open-ended real situations and to explore situations and generate hypotheses. in the niss assessment model, objects (a) – (g) and the corresponding students’ abilities are widely considered to be essential representations of what mathematics and mathematical activity are really about. the first three objects in the list emphasise routine, low-level features of mathematical work, whereas the belinda huntley, johann engelbrecht & ansie harding 5 remaining objects are cognitively more demanding. objects (a), (b) and (c) are fundamental instances of mathematical knowledge, insight and capability. current assessment models in mathematics education are often restricted to dealing only with these first three objects. one of the reasons for this is that methods of assessment for assessing objects (a), (b) and (c) are easier to devise. in addition, the traditional assessment methods meet the requirement of validity and reliability in that there is no room for different assessors to seriously disagree on the judgement of a product or process performed by a given student. it is far more difficult to devise tools for assessing objects (d) – (g). inclusion of these higherlevel objects into assessment models would bring new dimensions of validity into the assessment of mathematics. webb and romberg (1992) argue that if we assess only objects (a), (b) and (c) and continue to leave objects (d) – (g) outside the scope of assessment, we not only restrict ourselves to assessing a limited set of aspects of mathematics, but also contribute to actually creating a distorted and wrong impression of what mathematics really is (niss, 1993). assessment taxonomies according to the world book dictionary (1990), a taxonomy is any classification or arrangement. taxonomies are used to ensure that examinations contain a mix of questions to test skills and concepts. a leader in the use of a taxonomy for test construction and standardization was ralph w. tyler, the “father of educational evaluation” (romberg, 1992, p. 19) who in 1931 reported on his efforts to construct achievement tests for various university courses. the next step was taken by benjamin bloom (1956), who organised the objectives into a taxonomy (dedicated to tyler) that attempted to reflect the distinctions teachers make and to fit all school subjects. in bloom’s taxonomy of educational objectives, objectives were separated by domain (cognitive, affective and psychomotor), related to educational behaviours, and arranged in hierarchical order from simple to complex. bloom’s taxonomy has often been seen as fitting mathematics especially poorly (romberg, zarinnia & collis, 1990). it is quite good for structuring assessment tasks, but freeman and lewis (1998) suggest that bloom’s taxonomy is not helpful in identifying which levels of learning are involved. as ormell (1974) noted in a strong critique of the taxonomy, bloom’s categories of behaviour “are extremely amorphous in relation to mathematics. they cut across the natural grain of the subject, and to try to implement them – at least at the level of the upper school – is a continuous exercise in arbitrary choice” (p. 7). since its publication, variants of bloom’s taxonomy for the cognitive domain have helped provide frameworks for the construction and analysis of many mathematics achievement tests (begle & wilson, 1970; romberg et al., 1990). attacking behaviourism as the bane of school mathematics, eisenberg (1975) criticised the merit of a task-analysis approach to curricula, because it essentially equates training with education, missing the heart and essence of mathematics. expressing concern over the validity of learning hierarchies, he argued for a re-evaluation of the objectives of school mathematics. the goal of mathematics, at whatever level, is to teach students to think, to make them comfortable with problem solving, to help them question and formulate hypotheses, investigate and simply tinker with mathematics. in other words, the focus is turned inward to cognitive mechanism. smith et al. (1996) propose a modification of bloom’s taxonomy called the math taxonomy (mathematical assessment task hierarchy) for the structuring of assessment tasks. the categories in the taxonomy are summarised in table 1. table 1: math taxonomy (adapted from smith et al., 1996) group a group b group c factual knowledge information transfer justifying and interpreting comprehension applications in new situations implication, conjectures and comparisons routine use of procedures evaluation mcq in undergraduate mathematics 6 in the math taxonomy, the categories of mathematics learning provide a schema through which the nature of examination questions in mathematics can be evaluated to ensure that there is a mix of questions that will enable students to show the quality of their learning at several levels. it is possible to use this taxonomy to classify a set of tasks ordered by the nature of the activity required to complete each task successfully, rather than in terms of difficulty. activities that need only a surface approach to learning appear at one end, while those requiring a deeper approach appear at the other end. previous studies have shown that many students enter tertiary institutions with a surface approach to learning mathematics (ball et al., 1998) and that this affects their results at university. there are many ways to encourage a shift to deep learning, including assessment, learning experiences, teaching methods and attitudinal changes. the math taxonomy addresses the issue of assessment and was developed to encourage a deep approach to learning. it transforms the notion that learning is related to what we as educators do to students, to how students understand a specific learning domain, how they perceive their learning situation and how they respond to this perception within examination conditions. recently, work on how the development of knowledge and understanding in a subject area occurs has led to changes in our view of assessing knowledge and understanding. for example, in biggs’ (1991) solo taxonomy (structure of the observed learning outcome), he proposed that as students work with unfamiliar material their understanding grows through five stages of ascending structural complexity. in the interests of higher quality tertiary education, a deep approach to learning mathematics is to be valued over a surface approach (smith et al., 1996). students entering university with a surface approach to learning should be encouraged to progress to a deep approach. studies have shown (ball et al., 1998), that students who are able to adopt a deep approach to study tended to achieve at a higher level after a year of university study. mathematics assessment components based on the literature on assessment models and taxonomies in mathematics (bloom, 1956; niss, 1993; smith et al., 1996), we argued that for purposes of this study it was necessary to adapt the reviewed taxonomies in order to address the issue of assessing the cognitive level of difficulty of mathematical tasks, as well as the cognitive skills associated with each level. with this background, we propose a taxonomy of mathematics, which we call the assessment component taxonomy, to identify those components of mathematics that can be successfully assessed using alternative assessment formats such as mcqs. this taxonomy consists of a set of seven items, hereafter referred to as the mathematics assessment components. this set of seven mathematics components was ordered by the cognitive level, as well as the nature of the mathematical tasks associated with each component. this mathematics assessment component taxonomy is particularly useful for structuring assessment tasks in the mathematical context. the proposed set of seven mathematics assessment components are summarised below: (1) technical (2) disciplinary (3) conceptual (4) logical (5) modelling (6) problem solving (7) consolidation in this proposed set of seven mathematics assessment components, questions involving manipulation and calculation would be regarded as technical. those that rely on memory and recall of knowledge and facts would fall under the disciplinary component. assessment components (1) and (2) include questions based on mathematical facts and standard methods and techniques. the conceptual component (3) involves comprehension skills with algebraic, verbal, numerical and visual (graphical) questions linked to standard applications. the assessment components (4), (5) and (6) correspond to the logical ordering of proofs, modelling with translating words into mathematical symbols and problem solving involving word problems and finding mathematical methods to come to the solution. assessment component (7), belinda huntley, johann engelbrecht & ansie harding 7 consolidation, includes the processes of synthesis (bringing together of different topics in a single question), analysis (breaking up of a question into different topics) and evaluation requiring exploration and the generation of hypothesis. using bloom’s taxonomy (bloom, 1956), and the math taxonomy (smith et al., 1996), the proposed mathematics assessment components can be classified according to the cognitive level of difficulty of the tasks as shown in table 2. table 2: mathematics assessment component taxonomy and cognitive level of difficulty mathematics assessment components cognitive level of difficulty 1. technical 2. disciplinary lower order / group a 3. conceptual 4. logical middle order / group b 5. modelling 6. problem solving 7. consolidation higher order / group c table 3 summarises the proposed mathematics assessment components and the corresponding cognitive skills required within each component. based on the literature on assessment, the necessary cognitive skills required by students to complete the mathematical tasks within each mathematics assessment component were identified. table 3: mathematics assessment component taxonomy and cognitive skills mathematics assessment components cognitive skills 1. technical o manipulation o calculation 2. disciplinary o recall (memory) o knowledge (facts) 3. conceptual comprehension: o algebraic o verbal o numerical o visual (graphical) 4. logical o ordering o proofs 5. modelling translating words into mathematical symbols 6. problem solving identifying and applying a mathematical method to arrive at a solution 7. consolidation o analysis o synthesis o evaluation question examples in assessment components in the following discussion, question examples within two of the mathematics assessment components have been identified according to table 3. these items, one mcq and one crq, were selected from the tests and examinations in the first year mathematics major course (math109) at the university of the witwatersrand, johannesburg. the classification of the question according to one of the assessment mcq in undergraduate mathematics 8 components was validated by a team of lecturers (experts) involved in teaching the first year mathematics major course at the university of the witwatersrand. in addition, the examiner of each test or examination was asked to analyse the question paper by indicating which assessment component best represented each question. in this way, the examiner could also verify that there was a sufficient spread of questions across assessment components, and in particular, that there was not an over-emphasis on questions in the technical and disciplinary components. this exercise of indicating the assessment component next to each question also assisted the moderator and external examiner to check that the range of questions included all seven mathematics assessment components, from those tasks requiring lower-order cognitive skills to those requiring higher-order cognitive skills. example 1: conceptual mathematics assessment component assessment component 3: conceptual (mcq) state why the mean value theorem does not apply to the function defined by 2 2 ( ) ( 1) f x x   on the interval [ 3, 0] . a. ( 3) (0)f f  b. f is not continuous c. f is not continuous at 3x   and 0x  d. both a and b e. none of the above math109 june 2006, section a: mcq, question 7 in the conceptual question (example 1), the student is required to apply his/her knowledge of the mean value theorem to a new, unfamiliar situation which requires that the student selects the best verbal reason why the mean value theorem does not apply to the function f and the interval given in the question. this question requires a comprehension of all the hypotheses of the mean value theorem and tests the students’ understanding of a situation where one of the hypotheses to the theorem fails. example 2: problem solving assessment component assessment component 6: problem solving (crq) this question deals with the statement 3 3 3( ) : ( 1) ( 2)p n n n n    is divisible by 9 , for all , 2n n  (1.1) show that the statement is true for 2n  . (1.2) use pascal’s triangle to expand and then simplify 3( 3)k  . (1.3) hence, assuming that ( )p k is true for 2k  with k  , prove that ( 1)p k  is true. (1.4) based on the above results, justify what you can conclude about the statement ( )p n . math109 june 2006, section b: algebra. question 1 in the problem solving crq (example 2), the students are required to use the principle of mathematical induction to prove that the statement ( )p n is true for all natural numbers 2n  . the crq has been subdivided into smaller sub-questions involving different cognitive skills to assist the student with the method of solving using mathematical induction. in sub-question (1.1), the students need to establish truth for 2n  by actually testing whether the statement ( )p n is true for 2n  . hence (1.1) assesses within the technical mathematics assessment component. sub-question (1.2) involves a numerical belinda huntley, johann engelbrecht & ansie harding 9 calculation, the result of which will be used in the proof by induction. hence (1.2) also assesses within the technical assessment component. in sub-question (1.3), students are required to complete the proof by induction, by assuming the inductive hypothesis that ( )p k is true for 2,k k  , and proving that ( 1)p k  is true. since sub-question (1.3) requires the cognitive skills of identifying and applying the principle of mathematical induction to arrive at a solution, (1.3) assesses within the problem solving mathematics assessment component. sub-question (1.4) concludes the proof by requiring the students to justify that both of the conditions of the principle hold, and therefore by the principle of induction ( )p n is true for every 2,n n  . hence (1.4), requiring no more than a simple manipulation, assesses within the technical assessment component. this problem solving crq illustrates that often those questions involving higher order cognitive skills subsume the lower order cognitive skills. the quality index (qi) model the quality index (qi) model, developed by huntley et al. (2008, 2009), can be used both to quantify and visualize the quality and nature of a mathematics question. three parameters, namely discrimination index, confidence index and expert opinion were used to develop the qi model to quantify the quality of a question. in addition, a fourth parameter, namely the level of difficulty, was used to qualitatively contribute to the characteristics of a question. in order to graphically represent the qualities and characteristics of a question, 3-axes radar charts are constructed, where each of the three measuring criteria are represented as one of the three arms of the radar plot. in this model, no distinction is made between the relative importance of the three criteria in their contribution to the overall quality of a question. the quality index (qi) is defined to be the area of the triangle in the radar chart. for the qi model, the smaller the qi value of the radar plot i.e. the closer the qi value is to zero, the better the quality of the question. to visualize the difficulty level of the test item, a corresponding shading of the radar plot was chosen to represent the six difficulty levels: very easy; easy; moderately easy; moderately difficult; difficult; very difficult. the shading for the easy items is a lighter shade of grey and for the more difficult items, a darker shade of grey is used, with black representing very difficult items. briefly, the qi model can be used both to quantify and visualise how good or how poor the quality of a mathematics question is. the following three features of the radar plots could assist us to visualise the quality and the difficulty of the item: o the shape of the radar plot; o the area of the radar plot; o the shading of the radar plot. further discussion of the application of the qi model and radar charts will be presented in the component analysis of this study. research questions the quality index model (qi), briefly outlined in this study, was used in order to address the following research question: can we successfully use mcqs as an assessment format in undergraduate mathematics? in order to answer the research question, the following sub-question was formulated: which of the mathematics assessment components can be successfully assessed using the prq assessment format and which of the mathematics assessment components can be successfully assessed using the crq assessment format? mcq in undergraduate mathematics 10 in order to address the research sub-question, the qi model was used to identify those components that can be successfully assessed using the provided response format, in particular mcqs, and which can be better assessed in the constructed response question format. to assist with this process, we used the proposed mathematics assessment component taxonomy. research design response data from 14 different mathematics tests, both crq and prq format, written between august 2004 and june 2006 were collected. the study was set in the context of a first-year level mathematics major course at the university of the witwatersrand, johannesburg. in total, 207 test items were analysed in this study. the rasch model, as a statistical tool, was used in the quantitative data analysis of this study. it is a probabilistic model by which linear measures are created to be used in subsequent parametric tests (rasch, 1980). in comparison to traditional analysis techniques, the rasch model can be used (1) to analyse and improve a test instrument, and (2) to generate linear (interval strength) learner scores, thus meeting the assumptions of parametric statistical tests such a t-tests and anova (birnbaum, 1968). the rasch model focuses on the interaction of a person with an item rather than upon the total test score. one of the basic assumptions of the rasch model is that a relatively stable latent trait underlies test results (boone & rogan, 2005). for this reason, the model is also sometimes called the latent trait model. a feature of traditional test theory is that many of the statistics depends on the assumption that the true scores of people are normally distributed (andrich, 1988). an important advantage of the rasch latent trait model is that no assumptions need to be made about this distribution, and indeed, the distribution of abilities may be studied empirically. it was for this reason that the rasch model was chosen above other traditional statistical procedures for the quantitative research methodology of this study. component analysis using the qi model, a component analysis of selected questions, both mcqs and crqs, in the seven different mathematics assessment components was conducted. two such examples of the component analysis will now be illustrated. example 1 is a crq item within the disciplinary component and example 2 is a mcq item within the conceptual component. for each item, the question is followed by a radar plot and a table summarising the quality parameters of the test item, i.e. item difficulty; discrimination; confidence index; expert opinion and the final quality index. each of the axes of the radar plots are labelled with the corresponding values for discrimination, confidence index and expert opinion. the quality index (qi) is displayed alongside the radar plot. the shading of the radar plot corresponds to one of the six item difficulty levels as outlined in the brief discussion of the qi model in this study. the black shading of the radar plot in example 1 indicates a very difficult item while the lighter grey shading of the radar plot in example 2 indicates an item of moderate difficulty. the comments briefly summarise the difficulty level, the three measuring criteria and the overall quality of the item. all these parameters were used to define the qi model developed earlier by huntley (2008). belinda huntley, johann engelbrecht & ansie harding 11 example 1: disciplinary component c363b prove, using the intermediate value theorem, that there is a number exactly 1 more than its cube. crq, calculus, march 2006, q3b c363b comment assessment component disciplinary prq/crq crq item difficulty 3.94 very difficult discrimination 0.295 discriminates well confidence index 0.274 small deviation from expected confidence level expert opinion 0.574 large deviation from expected performance quality index 0.177 good quality crq example 2: consolidation component a45ma4 if f is an odd function and g is an even function then a. f g is an even function b. f g is an odd function c. f is a one-to-one function d. g is a one-to-one function prq, algebra, march 2005, tut test a, q4 mcq in undergraduate mathematics 12 a45ma4 comment assessment component consolidation prq/crq prq item difficulty 1.11 moderately difficult discrimination 0.275 discriminates well confidence index 0.698 large deviation from expected confidence level expert opinion 0.296 small deviation from expected performance quality index 0.207 good quality prq research results and discussion in the presentation of the results, a comparison of test items, both prqs and crqs, within each assessment component will be made. table 4 summarises the quality of the item, both prqs and crqs, within each assessment component. within each component the number of good and poor quality items are given, both for the prq and crq formats. the numbers are also given as percentages of the total number of items. technical in the technical assessment component, there is a higher percentage (73%) of good prqs than good crqs (41%). 73% good prqs compared to 41% good crqs shows us that prqs are more successful than crqs as an assessment format in the technical component. there is also a much higher percentage (73%) of good prqs than poor prqs (27%). crqs, however, are not that successful in this component, with the results showing 59% poor crqs compared to 41% good crqs. the conclusion is that the technical assessment component lends itself better to prqs than to crqs. disciplinary in this study, the disciplinary component is the assessment component with the most items (58), of which 34 were crqs and 24 were prqs. in this component it is interesting to note that the percentages of good prqs (50%) and good crqs (47%) are almost equal. in addition, there is no difference between the good prqs (50%) and the poor prqs (50%), with very little difference between the good crqs (47%) and poor crqs (53%). prqs and crqs can be considered as equally successful assessment formats in the disciplinary component. belinda huntley, johann engelbrecht & ansie harding 13 table 4: component analysis – trends component no. of prqs no. of crqs total no. of items good quality items poor quality items good prqs good crqs poor prqs poor crqs technical 11 22 33 17 (52%) 16 (48%) 8 (73%) 9 (41%) 3 (27%) 13 (59%) disciplinary 24 34 58 28 (48%) 30 (52%) 12 (50%) 16 (47%) 12 (50%) 18 (53%) conceptual 26 30 56 28 (50%) 28 (50%) 14 (54%) 14 (47%) 12 (46%) 16 (53%) logical 7 6 13 5 (39%) 8 (61%) 1 (14%) 4 (67%) 6 (86%) 2 (33%) modelling 3 10 13 8 (62%) 5 (38%) 2 (67%) 6 (60%) 1 (33%) 4 (40%) problem solving 7 4 11 6 (55%) 5 (45%) 4 (57%) 2 (50%) 3 (43%) 2 (50%) consolidation 16 7 23 12 (52%) 11 (48%) 7 (44%) 5 (71%) 9 (56%) 2 (29%) conceptual the conceptual component also contained many items (56), with an almost equal number of prqs and crqs (26 prqs versus 30 crqs). 50% of the items are of good quality and 50% are of poor quality. in this component, there is no clear trend that prqs are better than crqs or vice versa. there is a slight leaning towards good prq assessment (47% good crqs compared to 54% good prqs). therefore, in the conceptual assessment component, prqs could be used as successfully as crqs as a format of assessment. logical in this study, it is interesting to note that the majority of questions within the logical component were of a poor quality mainly due to the large percentage of poor prqs. there are noticeably more good quality crqs (67%) than good quality prqs (14%), and noticeably more poor quality prqs (86%) than poor quality crqs (33%). a very high percentage of the prqs (86%) in the logical component were of a poor quality. the conclusion is that the logical assessment component lends itself better to crqs than to prqs. modelling in the modelling component, very few prqs were used as assessment items in comparison to crqs, 3 prqs versus 10 crqs, probably because it is difficult to set prqs in this component. despite the small number of prqs, it was encouraging to note that the good prqs (67%) far outweighed the poor prqs (33%). so in terms of quality, the prqs were highly successful in the modelling component. there are also more good crqs (60%) than poor crqs (40%). it appears that although more difficult to set in the modelling component, prqs could be used as successfully in the modelling assessment component as crqs. problem solving although the problem solving component had the least number of items (11), it is interesting to note that there are more prqs (7) than crqs (4). there is a slightly higher percentage (57%) of good prqs than good crqs (50%). although the sample is too small to make definite conclusions, there is no reason to disregard the use of prqs in this assessment component. in fact, prqs seem to be slightly more mcq in undergraduate mathematics 14 successful than crqs, and the conclusion is that prq assessment format can add value to the assessment of the problem solving component. consolidation it was somewhat surprising to note that corresponding to the highest level of conceptual difficulty, the consolidation component displayed an unusually higher proportion of prqs (16) to crqs (7). this supports the earlier claim that prqs are not only appropriate for testing lower level cognitive skills (haladyna, 1999; thorndike, 1997; williams, 2006). in the consolidation component there is a significant higher percentage (71%) of good crqs than good prqs (44%). in addition, there is a higher percentage of poor prqs (56%) than good prqs (44%). the high percentage of good crqs (71%) in comparison to poor crqs (29%) indicates that the consolidation assessment component lends itself better to crqs than to prqs. conclusion the mathematics assessment component taxonomy, proposed by the authors in this study, is hierarchical in nature, with cognitive skills that need a surface approach to learning at one end, while those requiring a deeper approach appear at the other end of the taxonomy. the results of this study have shown that it is not necessary to restrict mcqs to the lower cognitive tasks requiring a surface approach. the prq assessment format can, and does add value to the assessment of those components involving higher cognitive skills requiring a deeper approach to learning. according to smith et al. (1996), many students enter tertiary institutions with a surface approach to learning mathematics and this affects their results at university. the results of this research study have addressed the research question of whether we can successfully use mcqs as an assessment format in undergraduate mathematics and the mathematics assessment component taxonomy was proposed to encourage a deep approach to learning. in certain assessment components, mcqs are more difficult to set than crqs, but this should not deter the assessor from including the prq assessment format within these assessment components. as the discussion of the results has shown, good quality mcqs can be set within most of the assessment components in the taxonomy which do promote a deeper approach to learning. results of this study (huntley, 2008) show that the more cognitively demanding conceptual and problem solving assessment components are better for crqs. traditional assessment formats such as the crq assessment format have in many cases been responsible for hindering or slowing down curriculum reform (webb & romberg, 1992). the prq assessment format can successfully assess in a valid and reliable way, the knowledge, insights, abilities and skills related to the understanding and mastering of mathematics in its essential aspects. as shown by the results of this study, mcqs can provide assistance to the learner in monitoring and improving his/her acquisition of mathematical insight and power, while also improving their confidence levels. furthermore, mcqs can assist the educator to improve his/her teaching, guidance, supervision and counselling, while also saving time. the prq assessment format can reduce marking loads for mathematical educators, without compromising the value of instruction in any way. inclusion of the prq assessment format into the higher cognitive levels would bring new dimensions of validity into the assessment of mathematics. table 5 presents a comparison of the success of mcqs and crqs in the mathematics assessment components. as table 5 illustrates, the enlightening conclusion is that there are only two components where crqs outperform mcqs, namely the logical and consolidation assessment components. in two other components, mcqs are observed to slightly outperform crqs, namely the conceptual and problem solving assessment components. the mcqs outperform the crqs substantially in the technical and modelling assessment components. in one component there is no observable difference, the disciplinary assessment component. in addressing the research question formulated as “can we successfully use mcqs as an assessment format in undergraduate mathematics?”, this paper has addressed the sub-question formulated as “which of the mathematics assessment components can be successfully assessed using the prq assessment belinda huntley, johann engelbrecht & ansie harding 15 table 5: comparison of the success of mcqs and crqs in the mathematics assessment components mathematics assessment component comparison of success technical mcqs can be used successfully disciplinary no difference conceptual mcqs can be used successfully logical crqs more successful modelling mcqs can be used successfully problem solving mcqs can be used successfully consolidation crqs more successful format and which of the mathematics assessment components can be successfully assessed using the crq assessment format?”. to address the sub-question, the mathematics assessment component taxonomy was proposed. in terms of the mathematics assessment components, it was noted that certain assessment components lend themselves better to mcqs than to crqs. in particular, the prq format proved to be more successful in the technical, conceptual, modelling and problem solving assessment components, with very little difference in the disciplinary component, thus representing a range of assessment levels from the lower cognitive levels to the higher cognitive levels. although crqs proved to be more successful than mcqs in the logical and consolidation assessment components, mcqs can add value to the assessment of these higher cognitive component levels. greater care is needed when setting mcqs in the logical and consolidation assessment components. the inclusion of the prq format in all seven assessment components can reduce marking loads for mathematics educators, without compromising the validity of the assessment. the prq assessment format can successfully assess in a valid and reliable way. the results have shown that mcqs can improve students’ acquisition of mathematical insight and knowledge. the prq assessment format (including mcqs) can be used as successfully as the crq assessment format to assess undergraduate mathematics. references andrich, d. 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(1990). a new world view of assessment in mathematics. in g. kulm (ed.), assessing higher order thinking in mathematics (pp. 21-38). washington, dc: american association for the advancement of science. romberg, t. a. (1992). mathematics assessment and evaluation. imperatives for mathematics educators. albany: state university of new york press. smith, g. h., wood, l. n., crawford, k., coupland, m., ball, g., & stephenson, b. (1996). constructing mathematical examinations to assess a range of knowledge and skills. international journal of mathematical education in science and technology, 27(1), 65-77. thorndike, r. m. (1997). measurement and evaluation in psychology and education (6th ed.). upper saddle river, nj: prentice-hall. webb, n., & romberg, t.a. (1992) implications of the nctm standards for mathematics assessment. in t.a. romberg (ed.), mathematics assessment and evaluation: imperatives for mathematics educators (pp. 3760). albany: state university of new york press. williams, j. b. (2006). assertion-reason multiple-choice testing as a tool for deep learning: a qualitative analysis. assessment and evaluation in higher education, 31(3), 287-301. doi:10.1080/02602930500352857 world book dictionary (1990). chicago: world book inc. book review research for educational change book title: research for educational change: transforming researchers' insights into improvement in mathematics teaching and learning authors: jill adler and anna sfard (eds.) isbn: 978-1-138-18732 publisher: routledge, london, 2017, £88.00* *book price at time of review review title: research for educational change reviewer: mellony graven1 affiliation: 1department of education, rhodes university, south africa corresponding author: mellony graven, m.graven@ru.ac.za how to cite this book review: graven, m. (2017). research for educational change. pythagoras, 38(1), a380. https://doi.org/10.4102/pythagoras.v38i1.380 copyright notice: © 2017. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. introductory comments it is not often i read an entire academic book almost in one sitting as was the case with this book. i found the insights captured compelling and thought provoking – pushing me to read on while constantly reflecting on the central question posed at the start of the introduction: ‘how can educational research fulfil its commitment to educational practice?’ the book focuses on the difficult challenge of how to turn research into educational improvement, so that research-generated insights can impact mathematics teaching in classrooms particularly in contexts where teaching and learning are challenged by social injustice and poverty. to do this, the book adopts a quite novel structure well-suited to the aim. it creatively structures the chapters into two main parts, each focused on a different discourse – though these discourses engage with predominantly the same data set. the first discourse is that of researchers speaking to one another (part 2) and the second discourse is that of researchers speaking directly to practitioners and teachers (part 3). part 1 and part 4 provide introductory and reflective comment for these parts. because of these two discourses the authors: hope that the book, whether in its entirety or in specific parts, will be of use to a wide range of agents – researchers, teachers, teacher-researchers, policy makers and curriculum developers. (p. 5) while i agree that this book is relevant to these agents i found the book particularly useful for secondary teachers and teacher educators because it provides powerful illumination of how theory can and does inform practice in concrete ways – thus addressing the challenge of many pre-service and in-service teacher conceptions of theory being largely irrelevant to the day-to-day practice of teaching. the book emerges from the work of the south african wits maths connect secondary (wmcs) project, which professor jill adler leads. in this project researchers (including national and international research collaborators such as anna sfard) and teachers collaborate to ‘try to make a difference by proposing specific changes in specific practices’ (p. 2). the project is argued to be special in two key ways: first, the task of translating research into practice is actually performed here, not just discussed. second, the project – being done in south africa, a country whose education system is now widely recognized as failing the majority of learners – tackles the question of how to turn research into a lever for practice in the context of learning and teaching hindered by poverty, oppression and social injustice. (p. 2, italics in original) while the wmcs project provides the broad inspiration for the book the chapters focus on one lesson (recorded and transcribed) of one participating mathematics teacher referred to as mr t. the lesson on quadratic equations and inequalities is with a grade 11 class of students in a township fee-paying school in gauteng (a highly urban province of south africa). mr t’s school, as expanded on in chapter 2, is considered a relatively typical school sharing multiple resource challenges with what shalem and hoadley (2009) term ‘schools for the poor’, in which the majority of south african secondary teachers teach. mr t is part of the wmcs professional development project. it is in this learning context of participation in this project – that brings researchers and teachers together to search for sustainable ways forward to the challenges in secondary mathematics education in south africa – that mr t agrees to have his lesson recorded and analysed from a broad range of research perspectives. the researchers (who include the authors of the chapters herein) then analyse this data from various perspectives while maintaining a focus on the key aim of translating research into practice. the introductory chapter by jill adler and vassen pillay provides a ‘panoramic view’ of the broader south african context while chapter 2 sets the scene for the school, mr t and the lesson data that forms the basis for the chapters in parts 2 and 3. chapter 1 provides an excellent overview of the current state of secondary mathematics education and mathematics education research in south africa that makes excellent reading for those interested in understanding the many challenges teachers and learners face, in resource constrained contexts, as well as the opportunities such contexts open up for researchers. as the authors point out, this type of setting is still quite rare for mathematics education research, which tends to be conducted in more affluent environments. the book then offers ‘another attempt at correcting this imbalance’ (p. 6). two discourses: researchers to researchers; researchers to practitioners part 2 and part 3 are the heart of the book. in part 2 researchers analyse mr t’s lesson data from a range of different perspectives using a discourse of researchers speaking to one another in a way that highlights insight into the teaching and learning process. so anna sfard analyses the lesson from her commognitive perspective, jill adler and erlinda ronda from their mathematics discourse in instruction framework developed as a key part of the wmcs project, anthony essien analyses the dialogic and argumentation structures in the lesson, while finally kate le roux bring a discourse analysis to bear on the lesson. the mathematics discourse in instruction perspective is particularly interesting in that it is a framework that emerges from the wmcs project and is a framework that is used, alongside lesson study design, to engage with teachers on their practices. in this respect it provides a powerful boundary object (as elaborated by hamsa venkat in chapter 12) for researchers and teachers, like mr t and ms h. ms h’s teaching data is brought into part 3 by adler and ronda when translating ‘research insights to teaching a lesson’. as setati points out in her afterword, mathematics discourse in instruction ‘enables adler and ronda to point to strengths and weaknesses in mr t’s lesson, avoid some deficit language, and also have productive conversations with teachers that inform action’ (p. 211). in part 3 of the book the discourse shifts from speaking to fellow researchers to speaking directly to education practitioners and in particular to teachers. this is where ‘the translation from research to practice takes place’ (p. 5) and it is this part that i believe will be particularly of use for teacher educators. the chapters here seek to communicate, as clearly as possible and in a practitioner-oriented discourse, the implications of the insights gathered from careful lesson analysis (reported in part 2) for impacting on practice. each of the researchers thus here takes up the innovative task of bridging discourses and addressing different practitioners (teachers, curriculum designers, professional developers and even concerned newspaper readers). the innovation of this exercise and the change in discourse is most clearly visible in the innovative format of letters to mr t provided in chapters 7 and 9 by anna sfard and audrey msimanga (who collaborated with anthony essien on his analysis of dialogic and argumentation structures of mr t’s lesson in chapter 5) respectively. this novel approach deserves some elaboration not only because it provides a powerful means of shifting researcher to researcher discourse to researcher to practitioner discourse, but also because it illuminates tensions in navigating the deficit trap, as referred to in part 4, and which i elaborate on in the final section of this review. in anna sfard’s eloquently written letter to mr t in chapter 7 titled ‘teaching mathematics as an exploratory activity: a letter to the teacher’ she shares with mr t insights gained from her analysis in chapter 3 that drew on her discursive commoginitive perspective and in particular her notion of ritual and explorative participation (sfard, 2008). in the letter she explains the difference between two modes of doing mathematics, namely the ritualised and explorative and argues for ‘the superiority of explorative mathematics’ (p. 133). this assumption of superiority, and the validity of the assumption, is dealt with in chapter 3 where her analysis of mr t’s lesson reveals that ‘the learners’ mathematical discourse was ritualized rather than explorative’ (p. 59). in her concluding remark in that chapter she reflects: whether i want it or not, this report on what i saw in mr t’s classroom will probably be perceived as judgmental and overly negative, whereas my right to judge is likely to be questioned on the basis of my not being sufficiently acquainted with the goals, ways of life, and backgrounds of the participants of the study, and above all, not cognizant of the past deprivations and the present needs of the wider community. (p. 60) sfard continues in chapter 3 to argue in ‘defence of the legitimacy and potential value’ (p. 60) of her analysis, pointing to both her expertise in mathematics education and her longer-term involvement as a collaborator in the wmcs project. in her letter to mr t she foregrounds her outsider status to the practice she is commenting on – beginning her letter with ‘i am no longer a professional mathematics teacher myself’ – then continues to outline the expertise she believes she brings. she argues that her many years of research experience have enabled her the privileged position of viewing and analysing lessons and transcripts multiple times, ‘with all the time in the world to ask questions and test possible answers’ (p. 123) and this enabled her to: perfect my ways of constructing interpretations and then, after trial, improve my tools as an interpreter even further. all this made me aware of things that usually escape the attention of teachers, who are too busy with moment-to-moment decision making to notice. (p. 123) acknowledging that this is a ‘luxury not many teachers can enjoy’ (p. 123), she then continues to explain ways in which mr t could change his practice to enable more explorative discourse and participation in his class. in her analysis she highlights that what learners are offered is what you get back from them. the letter communicates profound insights that maintain deep respect for the teacher and gratitude for the opportunity to engage with the lesson data. the final summarising remark is captured in the form of simple advice: attend to how you talk and make it clear that mathematics is not about symbols but rather about mathematical objects for which the symbols are but mere ‘avatars’ (p. 132). this letter, and msimanga’s, provide powerful examples of how researchers might shift their discourse and style of writing, in order to speak more directly to teachers in ways that are respectful of their practice and acknowledge the inevitable differences between what teachers are able to notice and reflect on in the moment-to-moment and day-to-day practice of teaching and what researchers are able to notice when reviewing lesson data multiple times. furthermore, the letter aptly captures the tension in navigating respect for teacher practices while simultaneously pointing to ‘deficiencies’. this tension is powerfully elaborated on by einat heyd-metzuyanim in chapter 11 who provides a critique of the way in which the book set out, as captured in the introduction, to avoid deficit interpretations that point to what is not present even while noting ‘we do not wish to disown these critical descriptions’ (p. 3). this is elaborated on in the next section as it relates to powerful meta-level learning offered by the book. meta-level learning emerging from the chapters the three chapters in the final part, titled beyond school: some meta level learning, and mamokgethi phakeng’s afterword provide powerful reflections on what the book contributes. einat heyd-metzuyanim, hamsa venkat and nuria planas each highlight what is made visible in the dialogue between discourses. each brings a different but powerful perspective to what is offered by the book and each author compares and contrasts across the chapters and the discourses to ‘pull the different threads together’ (p. 5). these chapters and the afterword jointly illuminate that while the book provides insights for how research might better impact teaching and how researcher and practitioner discourses might enter into productive dialogue, it does not provide easy ways to do this. rather it reveals that in the process many tensions emerge. in particular, the intention to avoid imposing deficit discourses on mr t’s practice is raised as a critical challenge as bringing a research perspective onto teaching inevitably focuses on what is present and absent in terms of what is relevant (placed in focus) for each perspective. einat heyd-metzuyanim’s chapter titled ‘a meta level reflection on dialogue between discourses’ powerfully captures this tension. her chapter i believe makes key reading for all researchers navigating the ethical tension of deficit discourses present in their research writing. she points out that in each of the chapters in part 2 deficits in the lesson are highlighted in relation to the actions of the teacher from the perspective taken – i.e. dominance of ritual over explorative, lack of exploratory talk, instruction that is authoritative rather than dialogic and texts provided to learners lacking opportunity for meaningful engagement with mathematical concepts. she asks: how come most of what is said in these chapters refers to what cannot be seen in mr t’s classroom, what isn’t there and not what is? even when there is talk of what is, such talk is in relation to what isn’t. (p. 176, italics in original) einat thus refers to ‘the trap of the “deficit talk” about mathematics instruction’ (p. 176). she however engages further, urging us to consider ways to minimise this trap. this for me is one of the most exciting contributions of the book as this ‘trap of deficit discourse’ has weighed heavily on me since the start of my research with and on (after setati, 2005) teachers in the mid-1990s. while i have worked against falling into this trap i have not had a frame for articulating and conceptualising how we might escape the trap. einat argues that viewing school mathematics according to the ‘ring model’ of the mathematics community is at the heart of the problem. that is, the model – with mathematicians in the core ring embedded in a ring of mathematics teacher educators embedded in the outer most ring of school mathematics teachers – necessarily leads to highlight the deficits of any of the outer rings over the innermost ring. thus, by definition, school mathematics will often be ritual compared to the ‘hard core’ of the mathematical community (p. 177). she continues to point out that if we were to see in school mathematics not what we wish to see, we would admit that ‘school mathematics is first and foremost about grades and measures of achieving them (namely tests)’ (p. 178) and so school mathematics is more about identities than about mathematical objects. einat thus calls on researchers to rather seek to better understand the constraints, which are surprisingly similar across continents, and to view teachers as experts and central participants in their own communities. she instead argues for an overlapping model of professional communities rather than the ring model where the three communities of mathematics teacher educators, mathematicians and school mathematics teachers are each separate but with areas of overlap. in this respect building opportunities for rich dialogue and engagement between communities in these overlapping spaces is important to enabling movement beyond the deficit trap. indeed, even while struggling with such tensions the book makes a significant start towards illuminating possibilities for meaningful dialogue and engagement between communities. as hamsa venkat notes in chapter 12, the mathematics teaching framework that adler and ronda present in chapter 8, and which emerges from the wmcs project, provides a useful boundary object ‘commonly recognizable across the mathematics education research and mathematics teaching community, while providing spaces for development of activities within both of these communities in dialectical ways’ (p. 190). concluding remark i highly recommend this book for mathematics education researchers, teachers and teacher educators interested in increasing the impact of research on teaching and learning, particularly in contexts of social inequality and economic disadvantage, and to those with an ethical commitment to navigating emergent tensions such as the production of deficit narratives of teacher practices. references shalem, y., & hoadley, u. (2009). the dual economy of schooling and teacher morale in south africa. international studies in sociology of education, 19(2), 119–134. https://doi.org/10.1080/09620210903257224 sfard, a. (2008). thinking as communicating. new york, ny: cambridge university press. setati, m. (2005). researching teaching and learning in school from “with” or “on” teachers to “with” and “on” teachers: conversations. perspectives in education, 23, 91–102. available from http://hdl.handle.net/10520/ejc87300 statistical education of prospective engineers focus on the statistical education of prospective engineers in south africa temesgen zewotir and delia north university of kwazulu-natal email: northd@ukzn.ac.za the paper deals with the teaching of statistics to engineering students at tertiary level in south africa. a number of suggestions are made in order to improve the statistical education of engineering students, thus potentially enabling future prospective engineers to optimise the power of statistics in their profession. though the focus here is on suggesting ways to improve the statistical education of engineers at tertiary level, current changes in the school curriculum are eluded to, as this adds another dimension to early statistical education of future engineers. introduction statistics has been described as the science of learning from data. it includes everything from planning for the collection of data and data management, to end-product activities such as the drawing of conclusions from data and the presentation of results. as is the case in many scientific professions, the engineering profession relies on numerical measurements to make decisions in the face of uncertainty. whenever there is uncertainty or prediction involved, then statistics, with probability theory as major building block, plays a significant role. this has lead to a great demand for familiarity with basic statistical techniques and inference procedures in the workplace. in particular, with the advances in technology and the associated increased ability to produce and process large masses of numeric readings, data handling and statistical techniques play an ever increasing role. it is thus very pleasing to note that the new curriculum currently being phased in at schools in south africa (department of education, 2003), includes data-handling throughout the various levels of schooling. this is in direct contrast to what had been the case prior to the adoption of this new school curriculum. as is the case all over the world, statistics courses forms an essential part of all engineering programmes at tertiary institutions in south africa. these courses typically deal with descriptive data handling procedures, probability theory, common univariate distributions, bivariate distributions, estimation of parameters, tests of hypothesis and regression analysis. the course tends to dwell more on theory and less on applications of statistics, thus fostering an inwardly focused approach where theory plays the dominant role, followed by a few techniques, with the hope that the value of the subject will speak for itself. it is argued here that the underlying purpose is implicit rather than explicit (mclean, 2000). this is the traditional approach to teaching statistics (see, for example, moore, 1993; bazargan, 2002; north & zewotir, 2006). this setting creates a common criticism that undergraduate engineering statistics courses are too academic in focus, excessively theoretical, and divorced from real problems that appear in the engineering industry. engineering students generally attend statistics modules separately from non-engineering students studying statistics, thus giving the perfect opportunity for engineering-specific examples and applications of statistics to be used in such a module, yet it generally does not happen. bearing in mind that south african scholars will in future be leaving school with basic statistical literacy skills, it is essential that the teaching of statistics to engineers needs to be modernised so that full use can be made of the higher level of statistical proficiency that will be present upon entering the tertiary institution. the problem outlined above and need for change in statistical education of engineering students however, is not exclusive to south africa. several educators have described the need for specific changes in statistics education for engineers (see, for example, box, 1990; bisgaard, 1991; hogg, 1994; higgins, 1999; disney, bendell & mccollin, 1999; acosta, 2000; vardeman, 2002; and references therein). what is surprising, however, is that the discussion forum and research in this regard is limited in south africa. moreover, the general unpleasant perception about statistics amongst engineering students in south africa is labelled and attributed to poor mathematical 18 pythagoras 65, june, 2007, pp. 18-23 temesgen zewotir and delia north background of the formerly disadvantaged racial groups (see, for example, blinnaut & venter, 2002; de wet, 2002; steffens, 1998). though the disparate schooling system of apartheid, and its legacy, has its own impact in the south african learners’ overall performance, it would be naïve to associate everything which is deficient with apartheid schooling. students are only able to enrol in the engineering faculty if they meet the basic entry requirements as set by the faculty; in fact, only highly competent matriculants (regardless of their race) will qualify to enter the faculty and thus become prospective engineers. an overview of the current engineering statistics in south africa bear in mind that at most south african universities and technikons, engineering students’ first encounter with statistics courses are at third year level. the initial stages of any engineering programme, by necessity, include a vast amount of calculus and numerical analysis. as mathematics is an essential key tool for statistics teaching and learning, it would seem reasonable to assume that teaching statistics to such a group of engineering students should be free of problems caused by poor mathematical preparation and one should thus find such a course to have a high pass rate. unfortunately, this expectation is far from reality as engineering students find statistics courses difficult with a resulting poor pass rate in such courses. despite the effort that instructors of engineering statistics devote to the course, many students experience anxiety when they are required to take statistics courses, as these courses are rumoured to be difficult to pass. cruise, cash and bolton (1985) argued that anxious students’ image of statistics is generally not a very positive one, with the resulting failure rate of such students being an indicator of the negative effect that the anxiety has on their chances of passing. with this in mind, it is important to examine the failure rate of a typical statistics module to third year engineers at a south african university. as an illustration, we used the failure rate of the same course over a number of years. note that this module is an essential part of the engineering programme and thus has to be passed prior to them graduating with a degree in engineering. the number of passes and failures from 1997-2005 academic years is reflected in table 1. using the cochran-armitage trend test (margolin, 1988; agresti, 2002) we analysed the pattern of the failure rate of this module over the last nine years to see if any significant trend developed over this period. the cochran-armitage statistic (z=11.2100 and p<0.0001) provides strong evidence of a positive trend. this shows an increasing failure rate for students taking engineering statistics courses for the period 1997 to 2005. this is not just by chance, in fact, it is a statistically significant trend. as is usual when a monotonic effect is observed, the linear logit model (margolin 1988; agresti 2002) was fitted with logit(πt)=α+βt where πt is the failure rate at time t=1,2,…,9; t=1 indicates the academic year 1997, and t=9 is the academic year 2005. the results are reflected in table 2. the estimated multiplicative effect of a unit increase in academic year on the odds of fails is exp(0.2137)= 1.238. deviance and pearson chi-square divided by the degrees of freedom are used to detect overdispersion or under-dispersion in the logistic regression. values greater than 1 indicate overdispersion, that is, the true variance of the failure rate is greater than what it should be under the given model. if this happens the resulting estimates are consistent, however, estimates of the variance are not. it can result in spuriously small standard errors of the estimates (barron, 1992). this inconsistent variance estimate invalidates any hypothesis testing. the most common and most widely implemented approach to remedy this is the use of “quasi-likelihood” through the introduction of a scale term into the variance equation. this approach has the advantage that it inflates the variance of each of the observations by a like amount, so that the estimated values will be the same – just the associated standard errors will be year passed failed failure rate 1997 174 37 0.17536 1998 234 22 0.08594 1999 149 34 0.18579 2000 191 31 0.13964 2001 119 89 0.42788 2002 216 40 0.15625 2003 163 64 0.28194 2004 138 161 0.53846 2005 228 131 0.3649 table 1. the number of passes and fails in engineering statistics at the ukzn 19 focus on the statistical education of prospective engineers in south africa inflated. logistic regression with quasi-likelihood over-dispersion is implemented in a wide variety of statistical packages, including sas. statistical hypothesis tests or confidence intervals using this adjusted fit provide valid inference (allison, 1999). the values of pearson chi-square and deviance divided by the degrees of freedom are significantly larger than 1. this evidence of over-dispersion indicates inadequate fit of the logit model. nevertheless, limited inference can be made from the fit. this limited inference is only about the estimates of the parameters as they are consistent. accordingly, the estimate of the logistic regression coefficient shows an increasing failure rate pattern. we refitted the model by adjusting for overdispersion. the result is presented in table 3. as noted earlier, the adjustment does not change the parameter estimates. the values of pearson chisquare and deviance divided by the number of degrees of freedom are close to 1. all the statistical tests, namely, the likelihood ratio, the score and wald tests show that the failure rate increases over the academic years. on the average, the failure rate in year (t+1) is exp(0.2137)= 1.238 times year t failure rate. in other words, on the average, failure rate increases 23.8% a year. figure 1 displays the observed and logit model fitted values. the plots show the increasing pattern of failure rate. the results from two logistic regression model fits assure the existence of positive trend of failure rate. in the first model there is no allowance for over-dispersion, in the second the quasi-likelihood approach to overdispersion is employed. all the analyses support our call for revisiting the current offering of engineering statistics at tertiary level in south africa. we know that poor mathematical preparation cannot be the problem, as discussed above, yet there is strong evidence of increased failure rates in engineering statistics courses. parameter estimate standard error wald chi-square p value intercept -2.1969 0.1282 293.4256 < 0.0001 year 0.2137 0.0196 119.4811 < 0.0001 testing global null hypothesis: β=0 test chi-square df p-value likelihood ratio 130.5489 1 < 0.0001 score 125.6644 1 < 0.0001 wald 119.4811 1 < 0.0001 goodness of fit criteria value df value/df p-value deviance 117.6822 7 16.8117 < 0.0001 pearson 119.2952 7 17.04217 < 0.0001 table 2. logistic regression analysis result parameter estimate standard error wald chi-square p value intercept -2.1969 0.5258 17.46 < 0.0001 year 0.2137 0.0802 7.11 0.0077 testing global null hypothesis: β=0 test chi-square df p-value likelihood ratio 7.7654 1 0.0053 score 7.4548 1 0.0068 wald 7.1070 1 0.0077 goodness of fit criteria value df value/df p-value deviance 6.9989 7 0.9998 0.428994 pearson 7.0948 7 1.0135 0.419077 table 3. the quasi-likelihood logistic regression analysis result 20 temesgen zewotir and delia north according to the engineering council of sa records, between 1998 and 2004 50,570 people enrolled at south african universities for engineering courses and 8,900 graduated. this a graduation rate of 17.5 percent across all engineering disciplines. the graduation rate for engineers is even lower at universities of technology. between 1998 and 2004 there were 139,820 enrolments and 14,250 graduates – a rate of 10 percent across all disciplines (south african migration project, 2007). this is further echoed by boroughs (2007) who states that the work environment in south africa is continually improving for black engineers, as affirmative action opens up more opportunities, but engineering educators note that the supply of engineering graduates is shrinking. the reasons for an inability to succeed can be discussed by considering the curriculum, including what happens in individual courses. steffens (1998) remarked that statistics syllabi in south africa have traditionally been very theoretical and have deliberately shied away from official (“birthand-death”) statistics. he also noted that a more balanced attitude has lately become popular internationally. we thus argue that the key to solving the problem of increasing failure rates amongst students in the engineering statistics courses may lie in examining the nature of the material in such a course. the overall goal must be to deliver a product which is relevant to the needs of future engineers and to structure the course in such a way as to maximise the possibility of motivating students about the need for statistics in their profession. this will go a long way towards replacing anxiety and negativity with recognition of the relevance of statistics to their future careers. clearly the professionals are not interested in the logic of statistical analysis, but will get motivated when learning statistical methods through handson experience related to solving problems in their discipline. problem-solving approach for engineering students there is a growing body of literature providing suggestions and discussions strongly favouring the teaching of statistical concepts through a practical approach (cobb, 1993; forte, 1995; rossman, 1995; moore, 1997; schaeffer, 1998; moore, 2000, gelman & nolan, 2002; north & zewotir, 2006), rather than the traditional mathematical approach. the focus of this approach is to promote general classroom activities and discussions on substantive application issues relevant to the students’ field of study, so that the student may discover statistical principles and the relevance thereof, rather than being able to prove mathematically why the principles hold. it is thus directed around a problem-solving approach, i.e. data to be collected as the result of a problem/question/statement to be analysed. it is very pleasing to note that this is the approach that has been outlined in the new national curriculum statement (department of education, 2003), where a problem-solving approach has been taken throughout the datahandling sections. the added advantage of taking the problem-solving approach to curriculum development at tertiary level as opposed to only at school level, however, is that it lends itself to being discipline-specific and can thus be far more 1997 1998 1999 2000 2001 2002 2003 2004 2005 0.0 0.1 0.2 0.3 0.4 0.5 0.6 1997 1998 1999 2000 2001 2002 2003 2004 2005 0.0 0.1 0.2 0.3 0.4 0.5 0.6 figure 1 observed and logistic regression predicted failure rates observed predicted f ai lu re r at e year figure 1. observed and logistic regression predicted failure rates 21 focus on the statistical education of prospective engineers in south africa effective in motivating future engineers as to the power of statistics in their future careers. we believe that the content of an introductory statistics course for engineers should be determined by the types of problems that engineers are most likely to encounter. further, we believe that the topics defined by solving such problems should be introduced in a manner that is similar to how these problems would be encountered in practice rather than being presented in a fashion that is determined for mathematical convenience. we are thus in favour of introductory engineering statistics courses being driven by problems rather than by techniques, with applied problems, rather than mathematical derivations, forming the basis for such a course. in decades gone by, large data sets were avoided in class as the computational power was a serious time constraint; however, with recent technological advances there is now no need to teach in the classic way. in order to most effectively modernise a statistics course to engineers, one must start by initiating discussions between the school of statistics and the engineering faculty as a ‘buy-in’ from both these parties will be necessary prior to achieving the outcomes mentioned above. the next step would be to consult with the customer to ascertain the exact nature of the desired product in order to be certain of relevance. when deciding on the appropriate material for an introductory statistics course to engineers, one might obtain information from several companies that employ large numbers of engineers as their input is vital when redesigning such a course. also, the information obtained when performing numeric readings in engineering experiments in other courses is a valuable source of appropriate data capturing opportunities for an introductory statistics course. above all, the data used must be seen to be collected in order to solve a problem and the student ideally needs to be part of the data capturing process in order for the statistical process to achieve maximum appreciation by the student (moore, 2000). above all, we believe that an introductory course in statistics for engineers must be considered in conjunction with their entire curriculum. no matter how good an introductory course in statistics might be, if students are not asked to use this material in any subsequent courses, they will soon forget it and most probably question why they were required to take the course in the first place. thus, we propose to enlarge our area of concern from just an introductory course in statistics to how the concepts from this course can be utilised, reinforced, and enhanced in subsequent engineering courses. the statistical concepts obtained should be an integral part of all laboratory experiences in subsequent courses. all of this necessitates a true collaborative effort between the engineering faculty and the statistics lecturers as they will need to work together in order to determine where statistical techniques can be used and which techniques are most appropriate in other modules not lectured by the statisticians. conclusion the type of introductory statistics course we are proposing will evolve as we gain information from industry and the engineering labs. the effectiveness of the course will increase as statistical content is added to the engineering labs and students are required to use statistical methods in their subsequent engineering curricula. there is no doubt that engineering students will become more motivated about learning statistics if they see the relevance thereof in subsequent modules in their programme, and ultimately, the power of statistics in the field of engineering will be appreciated by them. the core of the effort would be to develop a revised laboratory program for engineers in order to highlight the benefits to be gained by appropriate utilisation of statistical techniques. in the spirit of continuous improvement and of designing quality into a product rather than trying to address problems after the manufacturing stage, one can emphasise not being satisfied with simply dealing with variability in existing experiments, but rather designing new experiments that better emphasise the engineering issues. as a start, however, it is essential to ensure that statisticians fully understand the engineering experiments as they are currently being conducted. references acosta, f.m. 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(2002). providing ‘real’ context in statistical quality control courses for engineers. proceedings of the sixth international conference on teaching of statistics. retrieved july 18, 2005, from http://www.stat. auckland.ac.nz/~iase/publications/1/5e2_vard.p df forte, j.a. 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<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> >> >> setdistillerparams << /hwresolution [2400 2400] /pagesize [612.000 792.000] >> setpagedevice mwakapenda 28 pythagoras 60, december, 2004, pp. 28-35 understanding student understanding in mathematics willy mwakapenda school of education, university of the witwatersrand email: mwakapendaw@educ.wits.ac.za introduction understanding is one of the most important traits associated with the attainment of educational goals. however, nickerson (1985) observes that although the concept of understanding is a fundamental one for education, “what it means to understand is a disarmingly simple question to ask but one that is likely to be anything but simple to answer” (p. 215). a significant concern in school mathematics is learner understanding of mathematical concepts. kilpatrick, swafford and findell (2001) have described conceptual understanding as a critical component of mathematical proficiency that “is necessary for anyone to learn mathematics successfully” (p. 116). in particular, kilpatrick et al (2001) have argued that: students with a conceptual understanding know more than isolated facts and methods. they understand why a mathematical idea is important and the kinds of contexts in which it is useful. they have organised their knowledge into a coherent whole, which enables them to learn new ideas by connecting those ideas to what they already know. (p. 118) vygotsky (1962) makes the following more theoretical observation: concepts do not lie in the child’s mind like peas in a bag, without any bonds between them. if that were the case, no intellectual operation requiring coordination of thoughts would be possible, nor any general conception of the world. not even separate concepts as such could exist; their very nature presupposes a system. (p. 110-111) the revised national curriculum statement grade r-9 (department of education, 2002) observes that “mathematical ideas and concepts build on one another to create a coherent structure”. accordingly, the teaching and learning of mathematics is supposed to enable learners to “develop deep conceptual understandings in order to make sense of mathematics” (p. 5, emphasis added). it therefore suggests that finding out how learners have organised their knowledge of mathematical concepts might be a way of establishing how they understand those concepts. a strategy known as concept mapping has been associated with exploring learner understanding in terms of how they make links between concepts. there has been a growing interest in the use of concept mapping in teaching and research across various fields in education (novak, 1998). a concept map (see, for example, figure 1 below) is a graphical tool for representing concept relationships. in a concept map, lines are drawn between pairs of concepts to denote relationships between concepts. linking words on the lines indicate how pairs of concepts are related (kennedy & mcnaught, 1997). concept mapping has frequently been used as a teaching tool to help students “learn more meaningfully” and form a “conceptual understanding of the subject” (novak, 1990: 943). the potential of concept mapping to make a knowledge discipline more “conceptually transparent” (novak, 1998: 162) has been particularly recommended. in a concept map, “the network of propositions interlinking a group of concepts tells us much about the meaning of the concept from the perspective of the map makers” (roth & roychoudhury, 1992: 357). interrelationships between concepts are considered to be an important attribute of knowledge. interrelationships represent an essential feature of a learner’s thinking, understanding and meaningmaking in a particular learning area. the nature of concept mapping as described above makes it a useful tool for assessing and researching learner understanding. however, while there has been widespread use of concept mapping to investigate learner understanding in science education (chemistry, biology and physics), there is a marked under-utilisation of concept mapping as a tool in mathematics education research. raymond’s (1997) study is one of the few that have used concept mapping to explore mathematical knowledge and understanding in a qualitative way. willy mwakapenda 29 in the literature on concept mapping, there seems to be a taken-for-granted assumption that a concept map represents a totality of an individual’s understanding (williams, 1998). however, just as knowledge is argued to be a construction resulting from a dynamic relationship between prior and new knowledge, so is a concept map a product: a construction that is not “independent of its author” (henderson, 1991: 160). more usefully, concept maps need to be seen as representing “contents of consciousness” that need to be “inspected, edited, and shared with others” (eisner, 1993: 6). the potential of concept mapping as a tool in mathematics education research is explored in a broad study by mwakapenda and adler (2002), from which this paper emerges. the study sets out to answer the question: what do the concept maps and follow-up interviews indicate about students’ understanding of specific mathematical concepts? the resulting article explores learners’ understanding of specific mathematical concepts in the south african (senior phase) curriculum. the article highlights the usefulness of concept mapping in researching understanding, and shows, in particular, that students’ understanding of concepts is highly related to the broader context in which they learn mathematics, an aspect that has been rarely explored in studies that have used concept mapping as a research technique. moreover, an issue that does not seem to have had focused attention in such studies relates to the question of how one is able to determine when something has been “understood”. by sharing insights into aspects of learning experiences related to specific mathematical concepts, the study reported in this article provides a stimulus for discussing questions related to the complex issue of what constitutes understanding. study design the study involved first-year students from the university of the witwatersrand. there were three groups: students with at least a 60% pass on the higher grade matriculation (grade 12) mathematics examination who had enrolled for a mathematics major; students with at least a 60% pass on the standard grade matriculation mathematics examination who were enrolled at the college of science (an access college); and students who did not obtain a 60% pass on the standard grade matriculation mathematics examination who had enrolled in a foundation mathematics course. the study therefore involved three quite different groups of mathematics students in terms of previous school performance and their current university enrolment. participation in the research was purely voluntary. the study had intended to involve thirty students, ten from each group. a total of twenty-two students volunteered to participate in the research. an important aspect of the study’s methodological approach concerned a reflection on concept mapping itself as an activity in school mathematics. although concept mapping has been described as a teaching tool, particularly in the sciences, none of the students involved in this study had any experience of concept mapping in their learning of science and mathematics. a critical component of the implementation of the study involved introducing these students to the nature and activity of concept mapping (mwakapenda, 2001). students were then asked to construct a concept map which they would use to show a friend how the following 16 concepts were related: ratio, parallel, function, tangent, infinity, perpendicular, inverse, zero, equation, limit, absolute value, similar, gradient, angle, variable, bisector. why were these particular concepts selected? there is little doubt that zero, ratio and angle can be regarded as elementary and basic concepts in school mathematics. the mathematical concepts used in this investigation were drawn, in consultation with mathematics education specialists at wits university, from a study of the curriculum, textbooks and assessment items being used for senior secondary mathematics. these concepts were selected because they were considered by mathematics education specialists to be key mathematical concepts. these were included in the task because of their prevalence and significance in the south african school curriculum. these concepts cut across the algebraic, numerical and geometric settings of secondary school mathematics. the aim was to see how students would link concepts across topics and settings that are often fragmented in the way they are presented in the curriculum. the number of concepts to be mapped in the task was relatively large. because students would have been familiar with these key concepts, having encountered them over and again in their five years of secondary mathematics study, it would be interesting to see the ways in which they had understood possible connections between them. although the students’ familiarity with the concepts could be assumed, the connecting of concepts in the form of a concept map was not common practice in school mathematics activity, hence the need for an extensive introduction on concept-mapping understanding student understanding in mathematics 30 activity. this is described fully in mwakapenda (2001). reflective interviews were then conducted with students on their completed concept maps. the aim was to probe students’ understanding. the interviews presented an opportunity for students to explain and elaborate on the meanings represented in the links, and to allow them to provide appropriate examples to illustrate these links. this made it possible for the researcher to gain qualitative insights into the various dimensions of understanding which the students may have developed in their learning of school mathematics, and what these might mean for mathematics education and practice. the dimensions of understanding being explored here are closely linked to the notion of the “functional access” students may have to mathematical knowledge (lawson & chinnappan, 2000: 34). this level of access involves “not only having knowledge but also doing something with it” (nickerson, 1985: 234). data analysis in many concept mapping studies, the analysis of concept maps is predominantly quantitative and proceeds by scoring various aspects of student figure 1: angie’s concept map figure 1: angie’s concept map willy mwakapenda 31 maps such as the presence and accuracy of hierarchy levels, propositions, links and crosslinks, and specific examples provided to illustrate links (ruiz-primo & shavelson, 1996). in this study, i identified the number of concept links and examples related to these links. novak and musonda (1991: 127) have argued that “any map scoring procedure reduces some of the richness and detail of information contained in a concept map”. therefore, in the greater part of this study, students’ maps, and their elaboration of these in the interviews, were analysed in terms of the organisational principles (prawat, 1989) which students seem to have used in constructing the maps. for example, the maps were examined to determine whether students considered certain concepts as central in developing links between concepts. the central concepts (e.g. angle or gradient) that students used were identified. the meanings students associated with these concepts were examined. students’ descriptions of their maps were then analysed to examine the “completeness” (nickerson, 1985) of connections made. as well as providing insights into the meanings and nature of links between concepts, the analysis also raised questions about students’ understanding of specific concepts. an analysis of one student’s concept map detailed analyses and findings related to students’ concept maps from the broad study have been presented elsewhere (see mwakapenda, 2004; mwakapenda & adler, 2003). in this article, the analysis focuses on the map drawn by angie (pseudonym), a student from the foundation mathematics group. in the discussion that will follow, some implications arising from the analysis of the data from angie are described and discussed in the context of the broad study involving other students. figure 1 shows angie’s concept map. in relation to the given task, and in quantitative terms, we can see from figure 1 that angie used 14 out of 16 given concepts. we can also see that the map has two parts: the top part consisting of seven (mainly geometric) concepts with angle as a central concept, and the bottom part consisting of eight (largely algebraic) concepts. seen in this way, the top part can be described as a more centralised assemblage of concepts. qualitatively, in comparison with the top part, the bottom part of the concept map is not organised around any central concept. it can be described as a linear assemblage of concepts. however, the fact that angie drew a concept map shows that she saw that there were links between various concepts and she was able to display these links as can be seen in figure 1. it is also observed that the concept of tangent was used twice while two concepts: “similar” and “absolute value”, were not used at all in the map. this suggests that angie did not immediately see how these two concepts were linked to other concepts presented in the task. the appearance of the concept “tangent” in both parts of the concept map suggests a link between the geometric and algebraic concepts, an aspect that angie did not seem to have noticed. in a follow-up interview, angie said the following in an attempt to describe and clarify the links she had displayed in her concept map (wm = interviewer). angie: i looked at these words. okay. then i said, what can i do with these? then i said i am going to concentrate on one word, which is angles. wm: you said you should concentrate on angles. how did you decide? angie: i thought there are so many things that can be linked to angle from these given words. so i said angles can be perpendicular [draws a sketch], can be a bisector here cut into two equal parts [draws a sketch]. we can form a tangent [draws a sketch]. angle can form a tangent, can be parallel [draws a sketch], angles can be placed as variables [points at angle x in a triangle]. and angles can be zero. as can be seen, angie was able to see that a number of concepts, six in this case, “could” be linked to angles. however, stating and describing the conceptual linkages that she saw appears to have presented problems for angie. for example, what does it mean when she says: “angles can be perpendicular” and “angle[s] can be parallel”? in the context of school geometry, it may be more adequate to say that lines (rather than angles) can be perpendicular or parallel. nevertheless, the links angie describes are revealing when compared with the corresponding “sketches” (henderson, 1991) shown on her concept map in the top part of figure 1. the sketches (drawn as a consequence of probing) in the top part shows that angie associated angle with the following contexts: parallel lines and a transversal, an unknown angle (i.e. variables) in a triangle, perpendicular lines, and a tangent to a circle. what is revealed here is that angie is able to draw sketches to describe links between angle and other concepts. what can also be seen in the interview excerpt above is angie’s apparent struggle to appropriately understanding student understanding in mathematics 32 verbalise these links. also, the sketches angie drew did not seem to have helped her express these links more adequately. there is some inconsistency between the sketches she drew and the statements she makes about the sketches in figure 1. on their own, the sketches describe a representation of mathematical knowledge that may be said to be more identifiable with the context of school geometry. angie said the following when she was asked to explain the links she had made between the concepts in the bottom part of the concept map. angie: limit can be used as equations having an inverse. wm: can you give an example? what you mean by “limit can be used as equations”? angie: having an inverse? wm: yah. angie: [long pause] ... i can’t … wm: would you like to think about it more? angie: … limit … [struggles to give examples] … well i just can’t. i don’t remember exactly… like this is, it’s a long time i have done this. angie seems to be able to see some links between the more “algebraic” concepts in the bottom part of figure 1. for example, she sees “limit” and “equations” as concepts that are connected in some way, although it is not clear what she means by “limit can be used as equations”. describing links between algebraic concepts and providing examples to illustrate and explain these links was not easy for angie. the inability to give appropriate examples appears to be due to her not having been able to “remember exactly” what these could be. given that these concepts were taught while in high school, it can be suggested that angie found it easier to remember the “contexts” or mathematical situations in which the geometry concepts were learnt and what some of the organising features of the geometry were. angle appears to be an organising feature for the geometry concepts in angie’s understanding in relation to the given task. a question that arises here and one that needs further exploration is: what could be the organising features of “algebra”? to what extent can we say that angie was unable to remember or think about the organising features of the algebraic concepts given in the concept mapping task? in the above analysis, one is able to see how angie explained her difficulty in describing links she was able to see between concepts. her explanation concerned the fact that she could not remember what these links were, given that “it was a long time” since she had learnt these concepts. the point is that being able to remember concepts and how they may have been used in school seems to have played an important role in angie’s ability to see links between concepts. discussion the analysis has indicated a lack of “fluency” (williams, 1998: 414) in angie’s articulation of the perceived connections between given concepts. this lack of fluency suggests that there are specific ways of expressing mathematics that angie (as well as other students in her foundation mathematics group) seems not to have adequately developed while in school. this is confirmed by the following remark made by a student in angie’s group: you know, in maths, we are taught to do maths. you know, to discuss maths, maths is not expressed in that way. at school we are taught to work maths on a paper. sometimes it’s even difficult to understand a teacher … when he talks. but it would be far better if you write something down. i cannot be with you in maths but when you write something there i will understand. when you write it down rather than expressing it… we cannot express maths like some other subjects… you can talk about psychology, what you are discussing, unlike maths. we can see from above that lack of expertise in expressing mathematics and inadequate understanding of mathematical language and mathematics itself are likely to constrain students’ abilities to describe conceptual links between mathematical concepts. expertise in expressing mathematics and how this is enabled by concept mapping as a pedagogical tool needs further exploration. the above analysis underscores the central issue that concepts are not seen as entities on their own. the analysis suggests that, apart from being linked to other concepts, concepts are linked to contexts (represented by sketches or diagrams) associated with the learning and experiences of school mathematics. angie found it easier to remember contexts in which particular concepts (e.g. angle) were learned than to describe ways in which such concepts are related. the link between concepts and contexts is important since it supports the widely acknowledged view that knowledge willy mwakapenda 33 cannot be separated from the situations in which it is learned and used. the individual’s participation in the production of this knowledge is also critical (lave, 1991). what then can be said about the use of concept mapping as a tool for exploring students’ understanding of mathematics? in particular, what does the concept map and follow-up interview indicate about students’ understanding of specific mathematical concepts? proponents of concept mapping assume that knowledge within a content domain is organised around central concepts and that to be knowledgeable in that domain, students need to be able to display a highly integrated structure of concepts (mckeown & beck, 1990). based on this assumption, the analysis presented here suggests that angie displayed a partial integration of knowledge of concepts. describing the links in this knowledge was, however, problematic. for angie as well as other students involved in the study, it may be more accurate to characterise her mathematical connections as “representations” (novak & gowin, 1984: 40) of what they know and how they came to know about specific concepts. the above analysis indicates that it is possible to access some insight into students’ understanding of mathematical concepts by examining the connections they make in a concept map. the ability to make connections between concepts is an important aspect of understanding the specific concepts concerned. what can be said about the completeness of the connections students made? it is possible that, although students participating in this study may have been taught specific concepts, they may not have understood well enough to be able to communicate the knowledge gained. this result is not unusual given the culture of many classrooms where mathematics is typically taught as a disconnected set of facts and rules, unrelated to each other and to other knowledge disciplines. it is widely acknowledged that mathematics instruction often prevents meaningful learning and does not give students an opportunity to understand mathematical concepts and to critically and freely reflect on relationships between ideas (boaler, 1997). roth and roychoudhury (1992) have pointed out that even though instruction may attempt to show connections, textbooks and teachers can never provide all possible connections. besides, no matter how many formulations there are and how explicit they are, students will always have to construct their own ways of expressing the relationship between pairs of concepts. (p. 547) concept mapping is therefore a useful tool for exploring and documenting these connections and for promoting understanding of their conceptual meanings in mathematics. however, as indicated in the research design, interviews need to be set up to augment information from students’ concept maps. the interview provides space for obtaining a more informed account of students’ understanding of mathematics and why they connect mathematical concepts in the way they do. this is to emphasise the point that knowledge cannot be divorced from the individuals involved in the production of knowledge (lerman, 1998). a critical aspect concerning these connections relates to the issue about what we can learn by looking at the concept map. it is noted here that depending on the perspective one takes, not much can be learnt about student knowledge by focusing on the concept map only. concept mapping originated from practitioners in the cognitive science field, a field that does not seem to openly problematise the claim that a concept map provides a totality of an individual’s knowledge. allchin (2002) argues that concept maps are “inherently selective. they can only represent selectively, based on the mapmaker’s purpose” (p. 146, emphasis in original). while a map is a model of reality, one needs to understand the map’s context in order to appropriately interpret how it represents that reality. the map externalises only a part of an individual’s thoughts (roth & roychoudhury, 1992). it is possible to get a fuller picture of a student’s learning if we consider a concept map as a product that depends highly on the contexts of its production: the individuals who produced it, and who they are and where they come from: their learning histories, and the mathematical opportunities and resources to which they have access. a concept map is therefore not considered as an end product that represents a totality of an individual’s knowledge. rather, it is perceived as a working representation of what students seem to currently know and have experienced. in this regard, nickerson (1985) suggests that “one’s understanding must depend on the amount of knowledge one has about the concepts involved”, and that “the degree to which one understands [the concepts] must depend on the richness of the conceptual context in which the [concepts] can be interpreted” (p. 217). in acknowledging the context-dependent nature of understanding, nickerson then makes the key point that “one’s understanding of something should probably not be understanding student understanding in mathematics 34 thought of as right or wrong, but rather as … more or less complete” (p. 220). concept mapping is therefore seen as a vehicle for entering into a dialogue with students about mathematical knowledge and key concepts in school mathematics. the findings illustrate that to know something (e.g. a mathematical concept) is not to know it as an entity having a life of its own, but it is to know it in relation to something else: its context. the concept map provides a partial representation of this knowledge. the ability to establish meaning from this knowledge is likely to be hindered by students’ inability to talk clearly about concepts. concept mapping gives students the space to talk about concepts. it creates an opportunity for students to clarify what they have learnt about mathematical concepts, and, in the process, it identifies for students and educators what further learning and relearning needs to be sought. concept mapping is emerging as a reflective tool in mathematics learning in south african classrooms (see, for example, van rensburg et. al, 2001: 21, 34, 49). however, there is a need to articulate and reflect on the epistemological assumptions guiding the appropriation of concept mapping as a tool in the developing context of mathematics education in south africa. for mathematics education research generally, the activity of concept mapping opens up possibilities for gaining insights into what a learner knows and understands and the form that this understanding takes. references: allchin, d., 2002, “the concept map is not the territory”, studies in science education 37, pp. 143-148 boaler, j., 1997, experiencing school mathematics: teaching styles, sex and setting, buckingham: open university press department of education, 2002, revised national curriculum statement grades r-9 (schools): mathematics, pretoria: department of education eisner, e.w., 1993, “forms of understanding and the future of education research”, educational researcher 22(7), pp. 5-11 henderson, k., 1991. “flexible sketches and inflexible data bases: visual communication, conscription devices, and boundary objects in design engineering”, science, technology & human values 16(4), pp. 448-473 kennedy, d., & mcnaught, c., 1997, “use of concept mapping in the design of learning tools for interactive multimedia”, journal of interactive learning research, 8(3-4), pp. 389406 kilpatrick, j., swafford, j. & findell, b., eds., 2001, adding it up: helping children learn mathematics, washington dc: national research council lave, j., 1991, “situating learning in communities of practice”, in resnick, l., levine, j. & teasley, s., eds., perspectives on socially shared cognition, pp. 63–82, washington dc: american psychological association lawson, m., & chinnappan, m., 2000, “knowledge connectedness in geometry problem solving”, journal for research in mathematics education 31(1), pp 26-43 lerman, s., 1998, “a moment in the zoom of a lens: towards a discursive psychology of mathematics teaching and learning”, in olivier, a. & newstead, k., eds., proceedings of the 22nd international conference for the psychology of mathematics education, vol. 1, pp. 67-81, stellenbosch: university of stellenbosch mckeown, m., & beck, i., 1990, “the assessment and characterisation of young learners' knowledge of a topic in history”, american educational research journal 27(4), pp. 688-726 mwakapenda, w., 2001, “‘quadrilateral equations’ are easy to solve: findings from a concept mapping task with first-year university students”, pythagoras 54, pp. 33-41 mwakapenda, w., 2004, “examining understanding in mathematics: a perspective from concept mapping”, in vithal, r., adler , j. & keitel, c., eds., mathematics education research in south africa: perspectives, practices and possibilities, pretoria: human sciences research council mwakapenda, w. & adler, j., 2002, “do i still remember?’: using concept mapping to explore student understanding of key concepts in secondary mathematics”, in malcolm, c. & lubisi, c., eds., proceedings of the 10th annual conference of the southern african association for research in mathematics science and technology education, part ii, pp. 60-67, durban: university of natal mwakapenda, w. & adler, j., 2003, “using concept mapping to explore student understanding and experiences of school mathematics”, african journal of research in willy mwakapenda 35 mathematics, science and technology education 7, pp. 51-62 nickerson, r.s., 1985, “understanding understanding”, american journal of education 93, pp. 201-239 novak, j. d., 1990, “concept mapping: a useful tool for science education”, journal of research in science teaching 27(10), pp. 937949 novak, j. d., 1998, learning, creating, and using knowledge: concept maps as facilitative tools in schools and corporations, mahwah, n.j.: lawrence erlbaum associates novak, j. & gowin, d. b., 1984, learning how to learn, cambridge: cambridge university press novak, j. d. & musonda, d., 1991, “a twelve-year longitudinal study of science concept learning”, american educational research journal 28(1), pp. 117-153 prawat, r. s., 1989, “promoting access to knowledge, strategy, and disposition in students: a research synthesis”, review of educational research 59(1), pp. 1-41 raymond, a., 1997, “the use of concept mapping in qualitative research: a multiple case study in mathematics education”, focus on learning problems in mathematics 19(3), pp. 1-28 roth, w.r. & roychoudhury, a., 1992, “the social construction of scientific concepts or the concept map as conscription device and tool for social thinking in high school science”, science education 76(5), pp. 531-557 ruiz-primo, m. & shavelson, r., 1996, “problems and issues in the use of concept maps in science assessment”, journal of research in science teaching 33(6), pp. 569600 van rensburg, m., bierman a., de kock, h, kwakwa, b., le roux, p., lourens, r., malan, f., molefe, k. & kambule, t., 2001, mathematics: the outcomes way, cape town: kagiso education vygotsky, l. s., 1962, thought and language, new york: john wiley williams, c., 1998, “using concept maps to access conceptual knowledge of function”, journal for research in mathematics education 29, pp. 414-421 "if you want mathematics to be meaningful, you must resign of certainty. if you want certainty, get rid of meaning. you cannot have both." imre lakatos, “proof and refutations” (quoted by amy j. hackenberg, one of pythagoras’ reviewers, from the university of georgia) microsoft word 43-56 hamsa et al.docx pythagoras, 70, 43-56 (december 2009) 43 critiquing the mathematical literacy assessment taxonomy:  where is the reasoning and the problem solving?    hamsa venkat 1  mellony graven 2  erna lampen 1  patricia nalube 1    1 marang centre for mathematics and science education, wits university  hamsa.venkatakrishnan@wits.ac.za; christine.lampen@wits.ac.za; patricia.nalube@wits.ac.za  2 rhodes university  m.graven@ru.ac.za      in this paper we consider the ways in which the mathematical literacy (ml) assessment  taxonomy provides spaces for the problem solving and reasoning identified as critical to  mathematical  literacy  competence.  we  do  this  through  an  analysis  of  the  taxonomy  structure within which mathematical literacy competences are assessed. we argue that  shortcomings in this structure in relation to the support and development of reasoning  and problem solving feed through into the kinds of questions that are asked within the  assessment  of  mathematical  literacy.  some  of  these  shortcomings  are  exemplified  through the questions that appeared in the 2008 mathematical literacy examinations. we  conclude the paper with a brief discussion of the implications of this taxonomy structure  for the development of the reasoning and problem‐solving competences that align with  curricular  aims.  this  paper  refers  to  the  assessment  taxonomy  in  the  mathematical  literacy curriculum statement (deparment of education (doe), 2007).  mathematical literacy was introduced as a new subject in the post-compulsory further education and training (fet) curriculum in 2006. its introduction made a mathematically-oriented subject – either mathematics or mathematical literacy – compulsory for all fet learners. the curriculum statement for mathematical literacy defines the subject in the following terms: mathematical literacy provides learners with an awareness and understanding of the role that mathematics plays in the modern world. mathematical literacy is a subject driven by life-related applications of mathematics. it enables learners to develop the ability and confidence to think numerically and spatially in order to interpret and critically analyse everyday situations and to solve problems. (doe, 2003, p. 9) this definition alongside the broader description of the new subject’s aims in this document places emphasis on the need to develop life-oriented competences for a range of everyday situations in which mathematical reasoning and mathematical tools can be brought to bear productively to aid informed decision making and problem solving. situational reasoning relating to the identification and selection of salient features of the context is therefore required alongside and integrated with mathematical reasoning. critiquing the mathematical literacy assessment taxonomy 44 within this definition, we highlight two features that the broader literature suggests are central to the notion of mathematical literacy1 that is promoted in the south african rhetoric: firstly, the need for the reasoning that is implicated within the need to “think numerically and spatially” and to “interpret” and “critically analyse everyday situations”; and secondly, the need for “problem solving”. both of these aspects have significant bodies of literature associated with them in the field of mathematics education – some focused on the teaching and learning of mathematics, and some specific to discussions of mathematical literacy. olkin and schoenfeld (1994), focusing on mathematics, describe problem solving in terms of “confronting a novel situation and trying to make sense of it” (p. 43). steen, a leading advocate of what he terms “quantitative literacy” comments that centrally what quantitatively literate citizenship requires is “a predisposition to look at the world through mathematical eyes … and to approach complex problems with confidence in the value of careful reasoning” (steen, 2001, p. 2). in both of these quotations – across mathematics and mathematical literacy – reasoning and problem solving are seen to involve complex problems/novel situations. further, steen’s viewpoint suggests that “complex problems” are essential to developing mathematical literacy as a life-oriented competence. in this paper our focus is on exploring the extent to which reasoning and problem solving as described in the literature (halmos, 1975; polya, 1962; steen, 2001) and in the rhetoric of international parallels to ml (e.g., functional mathematics in england) figure within the assessment of ml in south africa. the results of the first ml examinations within the new fet national senior certificate (taken in 2008 in which 79% of the cohort attained a ‘pass’ or above (≥ 30%)) were met with some scepticism in public commentaries (e.g., jansen, 2009). earlier in 2008, academic critiques had already raised concerns that the exemplar papers for ml did not appear to align well with the taxonomy against which assessments were supposed to be designed, with an overrepresentation of questions focused on the lower levels (prince, frith, & burgoyne, 2008). this taxonomy, provided in the subject assessment guidelines (sag) document (doe, 2007), is outlined in the following terms, with guideline percentages given for the distribution of marks across its levels in ml assessments: level 1: knowing (30% of marks) level 2: applying routine procedures in familiar contexts (30%) level 3: applying multi-step procedures in a variety of contexts (20%) level 4: reasoning and reflecting (20%) prior analyses have considered the 2008 ml examination papers in relation to this taxonomy (umalusi, 2009), and pointed to an over-representation of lower level questions in paper 2 specifically. paper 2, according to the specification in the sag document, should focus predominantly on level 3 and level 4, with a small proportion of marks allocated to level 2 whilst paper 1 focuses solely on levels 1 and 2. given these analyses, our aim in this paper is to consider more specifically whether the taxonomy provides sufficient openings for developing what the literature base identified above tells us are key competences for becoming mathematically literate citizens – competences that are also highlighted in the rhetoric of the south african ml curriculum statement. in order to facilitate our consideration of this issue, we have structured this paper as follows. we begin with a brief review of literature that point to reasoning and problem solving as critical to the notion of mathematical literacy. we then go on to consider the taxonomy structure in some detail, and locate the areas and levels within which reasoning and problem solving are assessed, and the ways in which other competences figure alongside these. we argue that shortcomings in this structure in relation to the support and development of reasoning and problem solving feed through into the kinds of questions that can be asked within the assessment of ml. these shortcomings are exemplified through the questions that appeared in the 2008 ml examinations. we conclude the paper with a brief discussion of the implications 1 we use the capitalised “mathematical literacy” to refer to the south african subject specification and its enactments, and the small “mathematical literacy” to refer to a more generalised notion of a life-related competence that has a significant literature base associated with it in the field of mathematics education. hamsa venkat, mellony graven, erna lampen & patricia nalube 45 of this taxonomy structure for the development of the reasoning and problem-solving competences that align with curricular aims. specifically, we relate this discussion to how we interpret students’ performance in the 2008 ml examination. essential competences for mathematical literacy one of the difficulties in trying to identify essential competences for mathematical literacy is the fact that a range of interpretations exist as to what mathematical literacy actually is. in her analysis of a range of perspectives on mathematical literacy, jablonka notes that even the notion of essential competences – “transferable methodological and process skills” (2003, p. 79) in her terms – is only possible within particular conceptions of what mathematical literacy is about. she argues that the centrality of problem solving and reasoning are emphasised in the view of mathematical literacy embodied in the definition and problems set within the international comparison oriented program for international student assessment (pisa) tests for example (organisation for economic cooperation and development (oecd), 2003). she points out the imperatives for standardisation across cultures and measurement of competence within these tests, and argues that these imperatives take precedence over the need for genuine relevance. of interest in the south african ml context is the explicit acknowledgement of the oecd conception of mathematical literacy and its associated view of development in mathematical literacy competence within the development of the south african ml taxonomy. this suggests that reasoning and problem-solving development will be central to the assessment framework. further insights on the nature of the problem solving needed for mathematical literacy is provided within the documentation associated with the introduction of functional mathematics as a new subject in england. the definition of functional mathematics in england overlaps in significant ways with the south african definition of ml, containing phrases such as these: functional mathematics requires learners to be able to use mathematics in ways that make them effective and involved as citizens, able to operate confidently in life and to work in a wide range of contexts. ( qualifications and curriculum development agency, 2007, p. 19) problem solving is stressed as a key feature of being able to use mathematics in functional ways, with some detail provided on the ways in which problems should be presented in order to aid the development of a functional competence: it is important that learners are not told, at the time a problem is set, which of the mathematical tools they have at their disposal will actually be needed. selecting the right tools is a core aspect of becoming functional in mathematics. (functional skills support programme, 2007, pp. 22-23) and further: it is very important for learners to experience the need to decide for themselves whether a problem can be addressed using mathematics, what mathematics might help, and how the problem should be set out mathematically. (functional skills support programme, 2007, p. 24) the importance of the lack of “obvious-ness” of solution routes in order for genuine problem solving to occur has also been repeatedly emphasised within the mathematics education literature: ...to have a problem means: to search consciously for some action appropriate to attain a clearly conceived, but not immediately attainable, aim. to solve a problem means to find such action. (polya, 1962, p. 117) a further feature of this kind of problem solving in which the solution route is not immediately obvious is the importance of problem-posing, of asking questions that are appropriate to the context of the problem: the hardest part of answering questions is to ask them. […] the chief problem is likely to be “what is the problem?”. find the right question to ask, and you’re a long way toward solving the problem you’re working on. (halmos, 1975, pp. 466-467) critiquing the mathematical literacy assessment taxonomy 46 we argue that, both in mathematics and in mathematical literacy, the processes of problem-posing and problem-solving require reasoning in a range of ways – in asking the right questions, in choosing the mathematical tools that might assist in answering them, in selecting the most appropriate ways of representing information using these tools, in working systematically, and in analysing and interpreting results in context (with contexts usually drawn intra-mathematically in mathematics and from real life situations in ml). an important feature of this view of reasoning is the way in which it permeates the problem-solving process, rather than following the problem-solving process. we return to this point when we consider the south african mathematical literacy taxonomy in the following section. in the english functional mathematics specification, four features of “level differentiation” are outlined for describing problem demand in the sphere of contextualised problems: o the complexity of the situation o the familiarity to the learner of the situation or problem o the technical demand of the mathematics required o the independence of the learner in tackling the situation or problem. (functional skills support programme, 2007, p. 25) this model of complexity differs from the south african ml taxonomy in important ways. in the next section, we consider the outline and specification of this taxonomy in some detail. the mathematical literacy taxonomy the headings of the four levels of the ml taxonomy, presented in the introduction, suggest two overt strands that vary across the levels: a mathematical strand evident within the reference to “knowing” (mathematical facts) at level 1 and moving to the use of increasingly complex (or at least lengthy) procedures (levels 2 and 3); and a contextual strand that moves from “familiar” contexts (level 2) to a “variety of contexts” (level 3). combining content (in terms of facts and procedures) and context oriented complexity within a single hierarchy appears to suggest that both these aspects become more complex together. this contrasts with the view presented in functional mathematics in the last section where the categories suggest that these can vary independently of each other, with “technical demand of the mathematics” providing one avenue for making problems more demanding, and “complexity of the situation” providing another avenue. a further feature at the outline level is that recall of fact and engagement with mathematical procedures is followed by “reasoning and reflecting” at level 4. in the section above and in earlier writing (venkat, graven, lampen, nalube, & chitera, 2009), we have commented that this “deferral” of reasoning – construed as following the “doing” of mathematics – is problematic from the perspective of the literature outlined on what it means to be mathematically literate. the subject assessment guidelines document provides descriptions of each of the taxonomy levels in figure 1 (doe, 2007, pp. 27-28). given our focus on reasoning and problem solving as central competences that need to be developed for mathematical literacy, it was instructive to start by looking for where these features occurred within the assessment taxonomy. in the terms outlined in our literature review, problem solving appears to figure only at level 3 (amp1). whilst there is reference to problem solving at level 2 (arp1), the emphases on the “obvious-ness” of the procedure that is needed and the immediate availability of all the required information would tend to disqualify this reference in the terms outlined in the literature on problem solving. problem-posing, in the ways referred to within our overview of the literature, figures only at level 4. hamsa venkat, mellony graven, erna lampen & patricia nalube 47 figure 1: description of the mathematical literacy assessment taxonomy levels given then that problem solving (using well-known procedures) makes an appearance only at level 3, with reasoning deferred to level 4, we then began to look across the levels at the ways in which complexity appeared to be conceptualised within the taxonomy. we were able to discern four threads relating to mathematical development within the structure, with other aspects appearing in much more localised ways within particular levels. the developmental threads we identified are as follows: o calculation/procedure o algebraic competence o dealing with tabulated information o knowledge and competence in data representation in the following sections we discuss each of these developmental threads, and then go on to identify aspects that appear in more “singular”, non-developmental ways within the taxonomy structure. 2 abbreviation of level headings and numbering of aspects within levels added by us to facilitate reference. level 1: knowing (k)2 level 2: applying routine procedures in familiar contexts (arp) level 3: applying multi-step procedures in a variety of contexts (amp) level 4: reasoning and reflecting (rr) tasks at the knowing level of the ml taxonomy require learners to: k1 calculate using the basic operations including: o algorithms for +, -, ×, and ÷; o appropriate rounding of numbers; o estimation; o calculating a percentage of a given amount; and o measurement k2 know and use appropriate vocabulary such as equation, formula, bar graph, pie chart, cartesian plane, table of values, mean, median and mode. k3 know and use formulae such as the area of a rectangle, a triangle and a circle where each of the required dimensions is readily available. k4 read information directly from a table (e.g. the time that bus number 1234 departs from the terminal). tasks at the applying routine procedures in familiar contexts level of the ml taxonomy require learners to: arp1 perform well-known procedures in familiar contexts. learners know what procedure is required from the way the problem is posed. all of the information required to solve the problem is immediately available to the student. arp2 solve equations by means of trial and improvement or algebraic processes. arp3 draw data graphs for provided data. arp4 draw algebraic graphs for given equations. arp5 measure dimensions such as length, weight and time using appropriate measuring instruments sensitive to levels of accuracy. tasks at the applying multistep procedures in a variety of contexts level of the ml taxonomy require learners to: amp1 solve problems using well-known procedures. the required procedure is, however, not immediately obvious from the way the problem is posed. learners will have to decide on the most appropriate procedure to solve the solution to the question and may have to perform one or more preliminary calculations before determining a solution. amp2 select the most appropriate data from options in a table of values to solve a problem. amp3 decide on the best way to represent data to create a particular impression. tasks at the reasoning and reflecting level of the ml taxonomy require learners to: rr1 pose and answer questions about what mathematics they require to solve a problem and then to select and use that mathematical content. rr2 interpret the solution they determine to a problem in the context of the problem and where necessary to adjust the mathematical solution to make sense in the context. rr3 critique solutions to problems and statements about situations made by others. rr4 generalise patterns observed in situations, make predictions based on these patterns and/or other evidence and determine conditions that will lead to desired outcomes. critiquing the mathematical literacy assessment taxonomy 48 calculation/procedure across the four levels of the taxonomy, there is recurring reference to calculations and procedures. at the most basic level, the four taxonomy labels indicate that development within this thread proceeds along the lines of knowledge/recall of procedure (k), application of routine procedures (arp), application of multistep procedures (amp), identifying the appropriate questions to ask in order to identify procedures to solve problems using mathematics as a tool (rr). whilst this labeling indicates a rather mathematically-based progression across levels, closer reading of the descriptions indicates that such mathematically-oriented progression is intertwined with the following three sub-threads: o the degree of “known-ness” of the procedure to be applied o the degree of “immediate availability” of the information required to use the procedure to solve the problem o the degree of “obvious-ness” around what procedure to use we now discuss each sub-thread in turn. degree of “known-ness” of the procedure to be applied the descriptions across the levels point to a progression from “basic operations” to “well-known procedures” to “identification of” an appropriate mathematical procedure, which may also be well-known or even basic. the listing of basic arithmetical operations under k1 suggests that a school mathematics curriculum hierarchy of procedure may underlie the way progression within this sub-thread is interpreted. as such, there is an overlap between this sub-thread and the notion of the “technical demand of the mathematics” that is presented as one variable within problem demand in functional mathematics. degree of “immediate availability” of the information required to use the procedure to solve the problem. the descriptions under arp1 and amp1 allude to this as a developmental sub-thread of application of procedures. the lack of any allusion to contextualisation under k1 suggests that required information may be directly given within decontextualised questions – a suggestion confirmed by the presence of such questions in paper 1 in the 2008 ml examination, in spite of advice to teachers to avoid this practice in the sag: “when teaching and assessing mathematical literacy, teachers should avoid teaching and assessing mathematical content in the absence of context” (doe, 2007, p7). degree of “obvious-ness” around what procedure to use comments within the descriptions for arp1 and amp1 suggest that this factor represents a key discriminator between level 2 and level 3 questions. when combined with the lack of contextualisation in some level 1 questions, this sub-thread relates to the degree of explication of specific procedures in the statement of the question. this suggests that at levels 1 and 2, direct pointers towards the required mathematics are acceptable, resulting in heavily scaffolded questions in the examination papers that emphasise and test calculation skills, thus vitiating the assessment of problem solving. several interesting issues are raised within the calculation/procedure developmental thread and the subthreads within it in the context of the south african ml curriculum. first there is the mathematical orientation of the progression – a feature that appears to contradict the curriculum specification within which overt mathematical progression is limited, and explicitly acknowledged: for mathematical literacy, the assessment standards do indicate progression from grade to grade. however, this progression is not markedly evident in some of the assessment standards. the complexity of the situation to be addressed in context, through using the mathematical knowledge and ways of thought available to the learner, is where the extent of the progression needs to be ensured. (doe, 2003, p. 38) the framing of progression in terms of mathematical procedure within the taxonomy is somewhat problematic. we acknowledge and accept that this progression is both a necessary and useful part of the frame of mathematical literacy development – that recognising and unpacking mathematical progression within the context of problem solving provides both rationales for mathematical development and a range of increasingly sophisticated tools for making sense of situations. however, to allow mathematical hamsa venkat, mellony graven, erna lampen & patricia nalube 49 progression to dominate the assessment of a curriculum that is oriented towards quantitative and mathematical reasoning for life-situations appears akin to the “tail wagging the dog” and runs the risk of diminishing the emphasis on understanding everyday contexts that is central to the curriculum rhetoric. secondly, the progression appears to move from an emphasis on basic arithmetical procedure into directed straightforward procedures, and then onto less directed and more complex procedures. the calculating thread therefore appears to build in procedural complexity alongside a “degree of explication” strand, with procedures getting more complex and diminishing explication of the procedure to use occurring together. this suggests that these two features cannot be varied independently. this view tends to contradict steen’s position that quantitative literacy more often involves “real data and uncertain procedures but require[s] primarily elementary mathematics” (steen, 2001, p. 6). here, steen’s view suggests that a diminishing degree of explication – again related to the degree of scaffolding of specific procedures ought to be an overarching development thread in order to meet the goals of mathematical literacy, with limited shifts towards more complex procedures. the shift from a directed emphasis on basic calculation towards the selection and use of mathematical tools within a problem-solving frame correlates with what has been presented by some writers as a potential model of how sophistication builds up in mathematical literacy. maguire and o’donoghue (2003) present an “organising framework” for numeracy within which numeracy builds from a “formative phase” with an emphasis on arithmetic skills, into a “mathematical phase” in which the mathematics in everyday situations is made explicit, and then into an “integrative phase” where mathematics, communication and culture have to be synthesised in context. this view though, tends to be contradicted by findings in mathematics education located within a situated learning perspective (boaler, 1997; scribner, 1984). both these studies suggest that people can become good (efficient and effective) at activities that they practise – designing efficient distribution schedules in scribner’s case and non-routine, mathematics problems in boaler’s case. given that the taxonomy allows for 60% of the marks in ml assessments, even at grade 12 level, to come from basic arithmetic and simple calculation without involving reasoning or problem solving in context, our concern is that there simply is not enough pressure to infuse ml learning with these latter features. instead, the mathematical calculation thread allows for procedural mathematics to dominate in ways that often work against the aims of the curriculum. algebraic competence this developmental thread appears somewhat incongruous when considered alongside the advocacy in policy documents for work in ml to be led by the need to “engage with contexts rather than applying mathematics already learned to the context” (doe, 2003, p. 42) – given that the emphasis appears to be on mathematically focused progression located within one particular domain of mathematics – algebra. once again, whilst there are stronger indications here of an algebraically focused interpretation of progression, i.e. knowledge and use of formulae (k3) to solving equations algebraically (arp2) or drawing algebraic graphs for equations (rather than situations) (arp4) to generalising patterns in context (rr4), statements incorporate the degree of immediacy of information needed to apply algebraic tools as part of the increasing complexity of problems (e.g., k3). dealing with tabulated information information presented in tabular form curiously seems to receive special attention within the taxonomy, rather than a more general competence in dealing with numerical information from a variety of sources and representations (including graphs and written text) that is often highlighted in literature dealing with the nature of mathematical or quantitative literacy (e.g., steen, 2001). progression within this developmental thread appears to be delineated primarily in terms of the degree of interpretation needed to “select” appropriate information from a table in order to answer a question or solve a problem. thus, at the knowing level, the information needed can be found “directly” from the question in the table provided (k4), whilst a higher level competence in this strand would require some interpretation in order to select the “most appropriate data from options in a table” (amp2). the notion of immediacy is therefore present within this thread as well. critiquing the mathematical literacy assessment taxonomy 50 knowledge and competence in data representation the hierarchy presented in this developmental thread appears to suggest that low level competence is evident in knowledge and use of some basic representations (bar charts, introduced in the foundation phase content mathematics curriculum), are mentioned specifically – k2). this proceeds into an unspecified drawing of “data graphs for provided data” (arp3), and then into a more critical engagement with the kinds of data presentation that might support specific positions or arguments (amp3). critiquing provided data presentations can potentially fall within the remit of rr3, but is not indicated as the final progression of this thread. within this thread too therefore, the “production” of mathematics prevails at levels 1 and 2, with critical engagement appearing only at level 3, implying that reasoning will not be assessed in the reading of tables or the drawing of graphs. non-progressing descriptors this identification of developmental threads and some cross-cutting sub-threads leaves some aspects of the specification of the taxonomy as yet untouched. the “missing” descriptors appeared to us to stand in much more isolated ways within particular levels without a sense of progression attached to the skill or competence presented. examples of this include the emphasis on measurement which appears in arp5. whilst this descriptor can be interpreted as following on from “given” dimensions which feature in k3, the activity of measuring appears at this point only and appears to qualify as a routine procedure regardless of the context of the actual measurement. similarly, the need for “one or more preliminary calculations before determining a solution” (amp1) appears to push a question into the third level irrespective of the actual calculations required. the practical consequence is that the stringing together of routine calculations is construed as indicative of a high level of mathematical literacy. the key location for non-progressing descriptors though occurs within the “reasoning and reflecting” level. problem posing and interpretation or adjustment of answers in context only figure at this level, with lower levels providing few, if any, handles for the development of this kind of reasoning and critique capacity. as stated earlier, we have expressed our concern that the taxonomy appears in this way to separate the “doing” of ml from the “reasoning” required for ml (venkat et al., 2009). one consequence of this separation is that lower level reasoning skills are rendered invisible, for example in question 2.2.2 in example 1 (see further on), which requires candidates to visualise where a maximum height occurs as the arms of a wind turbine circulate. developing such visualisation skills is seen as an important component of reasoning for both mathematical literacy (de lange, 1999) and for mathematics (presmeg, 1986). a second consequence is that, given that reasoning and reflecting is understood to “follow” the calculating of answers, reasoning questions tend to focus narrowly on rather limited interpretation and commentary on previously calculated answers. question 2.4.3 in paper 2, asking candidates to comment on whether a payment option involving a deposit and monthly instalments or one involving taking out a loan for an outright purchase is preferable, following step by step calculations for the previous steps provides a good example of this kind of limited interpretation demand. essentially in this view, reasoning is reduced to a “reflection on prior calculation in context” – calculations that are so extensively scaffolded that the need for reasoning about the chain of steps required to make an informed decision is effectively removed. in summary, two aspects occur recurrently in this analysis of the taxonomy – firstly, a tendency towards procedural orientations to progression in several of the threads, and secondly, the notion that the degree of “immediacy” of information availability and/or “explication” of the required mathematical tools provides a sub-thread contributing to mathematical progression. key problems, in our view, relate firstly to the lack of a developmental thread across taxonomy levels related to openings for reasoning and problem solving, compounded by the interpretation of reasoning as a “deferred” activity understood in rather limited ways. secondly, the emphasis on mathematically-oriented progression is problematic in assessing a curriculum that contains a limited degree of mathematical development in its specification. this tends to result in a somewhat arbitrary breakdown of procedures into taxonomy levels. this breakdown is made much more explicit in the taxonomy version provided in the examination guidelines document which ml teachers are encouraged to use, in which, for example, calculating a hamsa venkat, mellony graven, erna lampen & patricia nalube 51 percentage of a given quantity qualifies for a knowing (k) level competence, whilst calculating a percentage increase or decrease of a given quantity falls with the applying routine procedures (arp) level. anomalies result from this kind of breakdown in mathematical terms – a percentage reduction problem (e.g., to calculate the effect of reducing r400 by 12%), can be calculated as 0,88  r400 for a one-stage solution or as a two-stage solution by calculating the 12% amount and subtracting. the first route in procedural terms carries exactly the same technical demand as a level 1 question, but some mathematical reasoning about the structure underlies this calculation. the second route makes the procedure ‘longer’ – and the emphasis on moving from “routine procedures” to “multi-step procedures” in the level headings of the taxonomy suggests that it is this “length of procedure” orientation that prevails in the taxonomy, rather than the reasoning underlying the more efficient procedure. as such, the kind of reasoning that underlies the more efficient procedure in the first solution tends to remain invisible in the taxonomy frame. in turn, this invisibility makes it unlikely that such reasoning will be encouraged in ml classrooms. in the next section, we compare three questions from the 2008 ml examinations which exemplify the problems we have identified in this section. the first example is from paper 1 which aims to assess ml at levels 1 and 2, while the other examples are from paper 2, aimed at assessing ml at levels 3 and 4. questions from the 2008 mathematical literacy examinations example 1 – paper 1, question 2.2 2.2  critiquing the mathematical literacy assessment taxonomy 52 commentary we note that within the second diagram given, a mathematical representation of the wind turbine is provided to candidates. this provision has several consequences. firstly, the need to engage with the context through making sense of the photo representation of the wind turbine, and connecting this representation to the textual information given beneath it is diminished. the need to select the salient features of the context in order to create a mathematical model is therefore completely removed. further, the reasoning load required to answer question 2.2.1 to 2.2.4 is also consequently reduced, although as we have pointed out, low-level visualising is still required to “imagine” the location of a blade to get the maximum height above ground (in spite of the fact that the taxonomy renders this invisible). we argue that the lack of openings for problem representation, reasoning and problem solving in the formulation of questions such as this one follow from the structure of the taxonomy. paper 1 is structured so that it contains only questions at levels 1 and 2 of the taxonomy, and our analysis above indicates that neither reasoning nor problem solving are represented at these levels. essentially therefore, given the mathematical model and the formulae needed for the circumference and area of a circle, the question becomes a test of the candidate’s ability to calculate – again, in line with the orientation of the taxonomy, but much less in line with the spirit of the curriculum. hamsa venkat, mellony graven, erna lampen & patricia nalube 53 example 2 – paper 2, question 2.4 2.4 thandi decides to buy a dishwasher based upon the advertisement below. 2.4.1 suppose thandi decides to buy the dishwasher using the instalment option. (a) what is the balance owing after paying the deposit? (b) calculate the total cost of the dishwasher. 2.4.2 suppose thandi takes a loan from abc bank for the full cash price of the dishwasher. she is charged interest of 18% p.a. compounded monthly and agrees to repay the loan over two years in equal monthly instalments. use the formula a = p(1 + i)n to calculate the total amount to be paid back, where: a = total amount to be paid back p = loan amount i = monthly interest rate n = number of months over which the loan will be taken 2.4.3 which method of payment would you advise thandi to choose? give a reason for your answer. commentary essentially this paper 2 question asks candidates to calculate and then compare across the three options available (with sub questions directing each calculation), and to then select one and provide some kind of justification for it. questions such as 2.4.3, which asks candidates to look over a prior sequence of well structured calculations, and to justify a selection in some way from these appears to be the most common way in which “reasoning and reflecting” questions are incorporated into the ml examination. none of these skills appears to us to merit the conferral of a high level of reasoning in relation to mathematical literacy. however, given the lack of a developmental thread related to reasoning, and the scaffolding away of a more genuine orientation to problem solving, such questions are by default classified at level 4 within the taxonomy. critiquing the mathematical literacy assessment taxonomy 54 example 3 – paper 2, question 4.2 commentary this question is also drawn from paper 2, which should be primarily constituted by tasks at levels 3 and 4 of the taxonomy. the phrasing of the initial context in question 4.2 draws students into the reading of a relatively complex text, yet with all information explicitly provided. much of the remaining complexity is removed by the structure of the sub-questions in question 4.2.1. we point out here the step by step nature of the scaffolding provided in question 4.2.1 in order to get to a value for how much lebo has left in hand after expenses each month. as in the previous example, problem representation skills are reduced significantly through both the scaffolding and the provision of a table for collation, and each part of the question requires relatively directed and relatively basic calculation skills. the provision of annexure c containing a table formatted to allow each of the expenses to be listed in rands and cents in a sensible order, completely removes the level 3 requirement to “decide on the best way to represent data to create a particular impression.” as with the questions in examples 1 and 2, we note the potential of this scenario to open up avenues for complex problem solving, but in the over-scaffolding that follows, much of the potential for decisionmaking and interpretation is removed. the difficulty essentially, therefore, lies not in the choice of problem contexts, but in the design of the questions within them, which seem to be designed to mimic the taxonomy in their unfolding. conclusions widespread evidence of assessment driving teaching (clarke, 1996) indicates that the taxonomy upon which ml assessments are designed has implications for teaching. our analysis of the taxonomy reveals that an emphasis on problem solving only really comes into play at level 3 of the taxonomy and yet the hamsa venkat, mellony graven, erna lampen & patricia nalube 55 literature and curriculum documents suggest that this is the crux of the aim of ml. the emphasis on routine calculations is marked overtly at levels 1 and 2 – which by definition then, fall outside the realm of problem solving. thus immediately visible from this taxonomy is the way in which the key aims of ml as given in the definition are located primarily in levels 3 and 4. indeed the definition of ml is that it is a subject driven by life related applications which enables learners to think numerically and spatially “in order to interpret [level 3] and critically analyse [level 4] everyday situations and to solve problems [level 3]” (doe, 2003, p. 9, emphasis and levels added). in relation to the specification that paper 1 only includes questions at level 1 and level 2 of the taxonomy we have to question the validity of such a paper in assessing the aims of ml as given by the definition. as we have pointed out, some aspects of the literature point to difficulties with interpreting the basic calculation focus of levels 1 and 2 as an appropriate lead in to service the problem solving and reasoning demands of levels 3 and 4. furthermore if across the two papers 60% of the marks can be obtained without any problem solving, mathematical interpretation and reasoning, then again the validity of what a ‘pass’ in this subject means in relation to the curriculum statement is called into question. in this respect the very high pass rate in mathematical literacy is unsurprising. perhaps what it indicates is that 79% of all learners who wrote the examination were able to perform/demonstrate basic mathematical operations, calculations/ skills and apply them to some extent in familiar situations where the necessary procedure is relatively obvious from the information given (which is immediately available). perhaps the proportion of learners who received more than 60% for the examination is a better reflection of those who were able to demonstrate some competence in meeting what the curriculum rhetoric suggests as requirements to be mathematically literate. references boaler, j. (1997). experiencing school mathematics: teaching styles, sex and setting. buckingham: open university press. clarke, d. (1996). assessment. in a. j. bishop, k. clements, c. keitel, j. kilpatrick, & c. laborde (eds.), international handbook on mathematics education (pp. 327-370). dordrecht: kluwer academic publisher. de lange, j. (1999). framework for classroom assessment in mathematics. utrecht: freudenthal institute & the national center for improving student learning and achievement in mathematics and science. available from http://www.fi.uu.nl/catch/products/framework/de_lange_frameworkfinal.pdf. functional skills support programme. (2007). resources to support the pilot of functional skills: teaching and learning functional mathematics. london: crown copyright. halmos, p. (1975). the problem of learning to teach. american mathematical monthly, 82(5), 466-476. jablonka, e. (2003). mathematical literacy. in a. j. bishop, m. a. clements, c. keitel, j. kilpatrick, & f. k. s. leung (eds.), second international handbook of mathematics education (pp. 75-102). dordrecht: kluwer academic publisher. jansen, j. (2009, january 4). old school: new system produces the same results. the sunday tribune, p. 20. maguire, t., & o’donoghue, j. (2003). numeracy concept sophistication – an organizing framework, a useful thinking tool. in j. maaß & w. schlöglmann (eds.), proceedings of the 10th international conference on adults learning mathematics (pp. 154-161). linz, austria: alm and johannes kepler universität. department of education. (2003). national curriculum statement grades 10-12. (general): mathematical literacy. pretoria: department of education. department of education. (2007). national curriculum statement grades 10-12 (general): subject assessment guidelines. mathematical literacy. pretoria: department of education. organisation for economic cooperation and development. (2003). the pisa 2003 assessment framework mathematics, reading, science and problem-solving knowledge and skills. paris: organisation for economic cooperation and development. olkin, i., & schoenfeld, a. h. (1994). a discussion of bruce reznick's chapter. in a. h. schoenfeld (ed.), mathematical thinking and problem solving (pp. 39-51). hillsdale, nj: lawrence erlbaum associates. polya, g. (1962). mathematical discovery: on understanding, learning and teaching problem-solving. new york: wiley. presmeg, n. (1986). visualisation in high school mathematics. for the learning of mathematics, 6(3), 42-46. critiquing the mathematical literacy assessment taxonomy 56 prince, r., frith, v., & burgoyne, n. (2008, august). mathematical literacy  grade 11 and 12 exemplars. paper presented at the south african mathematics society workshop. pretoria. qualifications and curriculum development agency (2007). functional skills standards. london: qualifications and curriculum development agency. available at http://orderline.qcda.gov.uk/gempdf/ 1847215955.pdf. scribner, s. (1984). studying working intelligence. in b. rogoff & j. lave (eds.), everyday cognition: its development in social context. cambridge, ma: harvard university press. steen, l. a. (2001). the case for quantitative literacy. in l. a. steen (eds.), mathematics and democracy. the case for quantitative literacy (pp. 1-22). washington, dc: the mathematical association of america. available from http://www.maa.org/ql/mathanddemocracy.html. umalusi. (2009). 2008 maintaining standards report. from nated 550 to the new national curriculum: maintaining standards in 2008. pretoria: umalusi. venkat, h., graven, m., lampen, e., nalube, p., & chitera, n. 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/hrv (za stvaranje adobe pdf dokumenata najpogodnijih za visokokvalitetni ispis prije tiskanja koristite ove postavke. stvoreni pdf dokumenti mogu se otvoriti acrobat i adobe reader 5.0 i kasnijim verzijama.) /hun 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can be opened with acrobat and adobe reader 5.0 and later.) >> /namespace [ (adobe) (common) (1.0) ] /othernamespaces [ << /asreaderspreads false /cropimagestoframes true /errorcontrol /warnandcontinue /flattenerignorespreadoverrides false /includeguidesgrids false /includenonprinting false /includeslug false /namespace [ (adobe) (indesign) (4.0) ] /omitplacedbitmaps false /omitplacedeps false /omitplacedpdf false /simulateoverprint /legacy >> << /addbleedmarks false /addcolorbars false /addcropmarks false /addpageinfo false /addregmarks false /convertcolors /converttocmyk /destinationprofilename () /destinationprofileselector /documentcmyk /downsample16bitimages true /flattenerpreset << /presetselector /mediumresolution >> /formelements false /generatestructure false /includebookmarks false /includehyperlinks false /includeinteractive false /includelayers false /includeprofiles false /multimediahandling /useobjectsettings /namespace [ (adobe) (creativesuite) (2.0) ] /pdfxoutputintentprofileselector /documentcmyk /preserveediting true /untaggedcmykhandling /leaveuntagged /untaggedrgbhandling /usedocumentprofile /usedocumentbleed false >> ] >> setdistillerparams << /hwresolution [2400 2400] /pagesize [612.000 792.000] >> setpagedevice abstract introduction research design (participants, measures and models) methodology findings conclusion and implications acknowledgements references about the author(s) jeremiah maseko department of childhood education, faculty of education, university of johannesburg, johannesburg, south africa kakoma luneta department of childhood education, faculty of education, university of johannesburg, johannesburg, south africa caroline long department of childhood education, faculty of education, university of johannesburg, johannesburg, south africa citation maseko, j., luneta, k., & long, c. (2019). towards validation of a rational number instrument: an application of rasch measurement theory. pythagoras, 40(1), a441. https://doi.org/10.4102/pythagoras.v40i1.441 original research towards validation of a rational number instrument: an application of rasch measurement theory jeremiah maseko, kakoma luneta, caroline long received: 01 aug. 2018; accepted: 23 sept. 2019; published: 05 dec. 2019 copyright: © 2019. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract the rational number knowledge of student teachers, in particular the equivalence of fractions, decimals, and percentages, and their comparison and ordering, is the focus of this article. an instrument comprising multiple choice, short answer and constructed response formats was designed to test conceptual and procedural understanding. application of the rasch model enables verification of whether the test content was consistent with the construct under investigation. the validation process was enabled by making explicit the expected responses according to the model versus actual responses by the students. the article shows where the rasch model highlighted items that were consistent with the model and those that were not. insights into both the construct and the instrument were gained. the test items showed good fit to the model; however, response dependency and high residual correlation within sets of items was detected. strategies for resolving these issues are discussed in this article. we sought to answer the research question: to what extent does this test instrument provide valid information that can be used to inform teaching and learning of fractions? we were able to conclude that a refined instrument applied to first-year students at university provides useful information that can inform the teaching and learning of rational number concepts, a concept that runs through mathematics curricula from primary to university. previously, most research on rational number concept has been conducted on young learners at school. keywords: fractions; equivalence; decimals; rasch model; teacher education; percentage; compare; conversion. introduction venkat and spaull (2015) reported that 79% of 401 south african grade 6 mathematics teachers showed proficiency of content knowledge below grade 6–7 level in a southern and east african consortium for monitoring educational quality (sacmeq) 2007 mathematics teacher test. universities recruit and receive students from some of these school where these teachers are teaching. in the previous years of teaching first-year students in the mathematics module in the foundation phase teacher development programme, we noticed that each cohort of prospective teachers come with knowledge bases that are at different levels. these classes, of students’ with varied mathematics knowledge, are difficult to teach unless you have some idea of their conceptual and procedural gaps. this varied knowledge base is greatly magnified in the domain of rational numbers in which they are expected to be knowledgeable and confident in order to teach and lay a good foundation in future teaching. an instrument, functioning as a diagnostic and baseline test for the 2015 first-year foundation phase cohort, was constructed at the university level in the fractions-decimals-percentages triad. this instrument aimed at gauging the level of students’ cognitive understanding of rational numbers as well as evaluating the validity of the instrument that was used to elicit their mathematical cognition. all the participants admitted into the foundation phase teacher training programme were tested on 93 items comprising multiple choice, short answer and constructed response formats. that elicited both conceptual and procedural understanding. application of the rasch model enabled a finer analysis of the test construct, the individual item and person measures, and the overall test functioning through making explicit the expected responses according to the model versus the actual responses by the students. in addition, the test as a whole was investigated for properties that are requirements of valid measurement such as the local independence where each item functions independently of each of the other items. this article reports on how students displayed gaps in their rational number knowledge base but focuses primarily on the validation of the instrument. the following questions are answered: to what extent does the test provide valid measures of student proficiency? how might the test be improved for greater efficiency of administration, and greater validity for estimating student proficiency? the aims of the immediate analyses were to: evaluate the assessment tool in terms of fit to the model, both item and person fit, thereby checking whether the tool was appropriate for this student cohort. provide detailed descriptions of selected items in relation to the students taking the test. the validity and reliability of the assessment tool were analysed through the rasch model incorporating both the dichotomous and partial credit model using rasch unidimensional measurment models (rumm) software (see andrich, sheridan, & luo, 2013). the processes of analysis and refinement, and the final outcome of this cycle are described. as this test was used as a preliminary diagnostic instrument, we regard ongoing cycles of refinement as pertinent in the interests of informing the teaching of mathematics on fractions-decimals-percentages to preservice cohorts of teachers in our programme. literature review in an attempt to clarify the assessment, how it was conducted and its purpose, we provide the justification for the exercise. it was critical to ensure that certain conditions were satisfied in order to safeguard the effectiveness of the assessment as well as the validity of the test items. stiggins and chappuis (2005) explained that assessment must be guided by a clear purpose and it must accurately reflect the learning expectations. wiliam (2011) affirms that a method of assessment must be capable of reflecting the intended target and also act as a tool for gauging teaching proficiency. these were the core intentions of the assessment in this research, and therefore the validation of the test as a whole, and the validation of independent items was critical and appropriate. the learning and teaching of rational number concepts is particularly complex. the representation of a fraction as = 0.24 has a meaning different to whole numbers 6 and 25. the numbers 6 and 25 are called local values while together, as a single entity, yielding 0.24, they constitute a global value and have a different meaning and value from 6 and 25 represented separately (gabriel, szucs, & content, 2013; sangwin, 2007). these authors found that it was not a simple process for either learners or adults to cross the bridge from whole numbers to fractions (global value form). vamvakoussi and vosniadou (2007, 2010) identify two distinguishing features of rational numbers. these are firstly that for each point on a number line, for example , there are infinite representations, that is, etc. secondly, between any two points on the number line there are infinitely many numbers. given this complexity, the operations on rational numbers, for example addition, subtraction, multiplication and division, require procedures that may previously have been learned when working with natural numbers but that now appear to generate misconceptions and associated errors (harvey, 2011; pantziara & philippou, 2012; shalem, smith & sorto, 2014). in fact the operations on rational numbers are somewhat distinct, and require additional conceptual understanding together with the associated procedures. besides the features mentioned previously, there are different representational systems for rational numbers, namely common fractions, decimal fractions and percentages. while there is equivalence across the three systems within the triad fractions-decimals-percentages, this equivalence is not obvious at face value unless the student has understood the organising principles of each system. for instance, the denominator of a percentage representation is always 100, for common fractions the choice of denominator is infinite, while for decimal fractions, the denominator is 1 (one). the apparent simplicity of the percentage because of its everyday use belies the complexity of this ‘privileged proportion’ (parker & leinhardt, 1995, p. 421). for example, an additive difference between two percentages, may be confused with a ratio difference. hiebert and lefevre (1986, p. 3–4) define conceptual knowledge as ‘knowledge that is rich in relationship, that can be thought of as a connected web of knowledge, a network in which linking relationships are as prominent as the discrete pieces of information’. such knowledge is described as that which is interconnected through relationships at various levels of abstraction. conceptual knowledge is essential for learners to have conceptual understanding as in its absence they will indulge ineffectively in problem solving and follow wrong procedures to solve them. conceptual knowledge plays a more important role, although interactively the two facets support a solid knowledge foundation. stacey et al. (2001) found that preservice primary school teachers had problems understanding the size of decimals in relation to zero including limited awareness on the misconception that ‘shorter is larger’ among learners. ryan and williams (2007) also highlight and explain misconceptions and associated errors on adding and subtraction fractions, working with decimals, and the meaning of place value that are commonly committed by learners, such as having problems with zero when subtracting smaller from larger digits. huang, liu and lin (2009) report that preservice teachers in taiwan displayed better fraction knowledge of procedures but lacked conceptual knowledge because of the way they had received this knowledge themselves. they recommended that these preservice teachers need more opportunities to construct their conceptual knowledge before they graduate. pesek, gray and golding (1997) believe that clear understanding of rational numbers is one of the most foundational sections in the primary school curriculum and yet, presently, is one of the least understood by both teachers and learners. identifying mathematical competence levels of incoming preservice teachers provides an opportunity for the timely remediation of at-risk students. the conceptual complexities that generate misconceptions and associated errors emerge from lack of conceptual understanding (ryan & williams, 2007; charalambous & pitta-pantazi, 2005, 2007). research shows that in most cases both teachers and learners appear to have instrumental understanding of fractions, but do not really know why the procedures are used (post, harel, behr, & lesh, 1991). students tend to develop conceptual schemes and information processing capacities to master fractions, decimals and percentage concepts individually but they also need to understand the commonalities between the different representations in their interaction with each other (kieren, 1980). the educational aim however is for these students to have a balanced ability to follow a procedure with conceptual or relational understanding as the two facets interactively support a solid knowledge foundation (zhou, 2011). assessment and measurement the rich theorising of and research into rational numbers provides the theoretical base for the assessment instrument, which therefore meets the requirement for measurement to define clearly what is to be tested (wright & stone, 1979, 1999). the next requirement is to outline the interrelationships between component parts of the construct; in the case of this study, the interrelationships between fraction, decimal and percentage representations. the third stage is the construction and selection of items that will operationalise the construct, keeping in mind its complexity, and which will provide the teacher with evidence of misconceptions that would need to be addressed in class. a final phase is the post hoc verification of the functioning of the test as a whole and of the individual items. research design (participants, measures and models) the primary study (maseko, 2019) investigated the extent to which the 2015 cohort had mastered and retained their procedural and conceptual knowledge from their school level mathematics. this prior study reports on the level of relational understanding in the triad of concepts fractions-decimals-percentages of the first-year foundation phase student teachers entering the education programme. this article reports on the appropriateness of the instrument designed to test the students’ levels of understanding and conceptual knowledge as they entered the teacher education programme. the assessment tool was administered to the whole population of students that were admitted into the foundation phase teacher training programme (n = 117). the test comprised 93 items that were designed to elicit prior knowledge at the beginning of the academic year. the main research study comprised five conceptual categories that facilitated the analyses. the categories are understanding rational number concepts: definitions and conversions (14 items); manipulating symbols (operations) (17 items); comparing and sequencing rational numbers (15 items); alternate forms of rational number representation (35 items); as well as solving mathematical word problems with rational number elements (12 items). the items were drawn from selected projects, for example ‘the rational number project’ (cramer, behr, post, & lesh, 2009), and other such literature, and then adapted to post secondary school level. the items were primarily informed by the conceptual categories above, and could be identified according to the following requirements: the items demanded a demonstration of procedural as well as conceptual understanding. the items included fraction, decimal and percentage representations. items were generated with the specific purpose of evoking misconceptions. the items were comprehensive, covering most concepts and sub-concepts within the three representational systems – fractions, decimal fractions and percentages. the format of the test item types included multiple choice items, short answer, as well as extended response items. the reason for such a comprehensive selection of items was that the lecturers needed to identify the many difficulties and misconceptions the students could bring into their first semester mathematics class. a range of difficulty that would include learners of current low proficiency, and high proficiency, was also required. also, at the time of setting the items, the instructors were not sure from which categories the difficulties would emerge. the rasch model was applied in this study in order to either confirm or challenge the theoretical base, to check the validity of the instrument, and to measure the students’ cognition of rational number concepts. the hypothesis was that the assessment tool would function according to measurement principles. the rasch model provided information of where the item functioning and student responses were unexpected. possible explanations could then be inferred, and presented, as well as provide some indications for the refinement of the test instrument. methodology there are other theories developed that can be used to validate and authenticate tests, such as classical test theory (ctt) (treagust, chittleborough, & mamiala, 2002). rasch measurement theory (rmt) is generally used when measurement principles are considered, as the rasch model adheres to measurement principles as conceptualised in the physical sciences (rasch, 1960/1980). the application of the rasch model is premised on the particular strength implicit in the model: that both item and person parameters are aligned on the same scale (wei, liu, & jia, 2014). by considering both the validity and reliability of the test items, and of the person responses, the rasch model identifies aspects for further improvement as well as signs of biases (if any) (smith, 2004; bond & fox, 2007, 2015; long, debba, & bansilal, 2014; bansilal, 2015). ethical considerations this study has been cleared by the university of johannesburg ethics committee, with the ethical clearance number sem 1 2018-021. findings the first analysis showed the test instrument to have a sound conceptual base and to be well targeted to the cohort, with a range of items, such that the students of current lower proficiency could answer a set of questions with relative ease, while students of high proficiency would experience some challenging items. table 1 shows summary statistics of the rasch analysis. in this model, the item mean is set at zero, with items of greater and lesser difficulty calibrated against the mean. person proficiency is then estimated against the item difficulty. the item standard deviation was 1.6302. the person mean location is estimated to be −0.4238 logits, and the person standard deviation is 0.9686, which shows fairly good targeting and spread. the person separation index of 0.9114 shows that the assessment tool was able to differentiate well between students’ proficiencies and that the power of fit was excellent, in essence a high reliability. table 1: summary statistics of fractions, decimals and percentages. as observed in the person-item map figure 1, a range of items from easy to difficult was achieved, and the test is well targeted. figure 1: rasch model – person-item original map. easier items are located at the lower end of the map (item 65 and item 66), while the difficult items are located at the higher end (item 27 and item 28). similarly, learners of high proficiency are located higher on the map, 2.903 and 1.733, while learners of low proficiency are located at −2.159 and −2.143. the mathematical structure of the rasch model is such that where a person’s proficiency location is aligned with an item difficulty location, an individual of that proficiency level has a 50% probability of answering an item of that difficulty level correctly (rasch, 1960/1980). from the model one is able to predict how a student in a particular location will perform against an item: at, below or above their location on the scale. individual item analysis the individual items when constructed were initially reviewed by the lecturers. the application of the rasch model provided empirical output calibrating a relative location and giving the probability that a person located at a certain proficiency location will get the item correct within the instrument. item 63 (43. fraction form of 0.21) at position −0.646 is shown on the category probability curve (figure 2). aligned with item 63 are seven students (each represented by an ×, as shown on figure 1). from their overall performance on the test as a whole, these students are estimated to have a 50% chance of answering item 63 correctly. each of the items shown by the category probability curves can be represented to show the item’s unique characteristics in relation to the student cohort as a whole. figure 2: item 63 – category probability curve. in figure 2, depicting item 63, the horizontal axis shows the student locations from −5 to +5. the vertical axis indicates the probability of getting a correct response. the item difficulty is calibrated at −0.646 (the dotted line shows the meeting point of the two curves). as stated previously, the seven students located at this point will have a 50% probability of answering the question correctly. students located above −0.646 will have a greater than 50% probability of answering this question correctly. students located below −0.646 will have a less than 50% chance of answering this question correctly. the light grey curve indicates the probability, according to the model, of a correct response. inversely, the solid black curve shows the probability of getting an incorrect answer. both curves plot either an increased or decreased probability of a correct response from a particular location of both a question item as well as a person responding to that item. when an item is difficult or easy for the students, the curves show a shift of the meeting point away from the zero position (0) on the x-axis. two items, item 58, a relatively difficult item with an item location of about +3 logits (see figure 3), and item 39, a relatively easy item, with an item location of about −3, are presented (see figure 4). figure 3: item 58 – difficult category probability curve. figure 4: item 39 – easy category probability curve. very few students are to the right of position +3, implying that it was only students located at +3, or higher, that had a greater than 50% probability of answering the item correctly. item 39 (figure 4) had a 50% or greater probability of being answered correctly even by students with relatively low proficiency. all those to the right of location −3 had a greater than 50% chance of providing the correct answer. in this next discussion we compare two students, one located at +3 and another located at −3, on item 63 (figure 5). the student located at +3 has a 97% chance of answering this item correctly; however, the student located at −3 has about a 4% of answering the item correctly. the model predicted this outcome, which is not to say that the low proficiency student could not answer a very difficult item correctly, but that this outcome was highly unlikely. figure 5: item 63 – category probability curve. in summary, applying the rasch model to a data set is essentially testing a hypothesis that invariant measurement has been achieved. where there are anomalies, the researcher is required to investigate the threat to valid measurement. the model enables the researchers to identify the items that did not contribute to the information being sought or those items that were deemed faulty in some respect. likewise, where students’ responses to the question were unexpected the researchers were also alerted. the rasch model is to some extent premised on the guttman pattern, which postulates that in addition to some difficult questions, a person of greater proficiency should answer all the items correctly that a person of lower proficiency answers correctly. likewise, easier items should be answered correctly by low proficiency learners, and also by moderate proficiency and higher proficiency learners. while a strict guttman pattern is not possible in practice, the principle is a good one (dunne, long, craig, & venter, 2012). we briefly report on six students against four questions close enough to their locations to illustrate the relationship of person proficiency to item difficulty as seen through the guttman pattern model. the student of low proficiency (a, location −2.159) struggled with the range of items that included the easiest of the items. the other student categorised as of low proficiency (b, location −2.143), offered no response to these particular items. from the person-item map, we would expect students at these locations to have a 50% chance of answering correctly, meaning that if there were 100 students at that location approximately 50 could have answered the items correctly. of the two students in the moderate category, one of the students (c, location 0.003) did not attempt the easiest item (location −2.234) (missing response), while the other student (d, location 0.029) answered this item far below his location correctly. the next two items which were above the two students’ locations were either not answered or answered incorrectly. the two students located in the high proficiency category are located at 1.733 logits (e) and 2.903 logits (f), more than a logit apart. we therefore deal with them separately. student e answered the easiest item correctly and this was to be expected; however, the next easiest item was answered incorrectly. in theory the student should have had a greater than 50% correct response. the difficulty of the third item is aligned with the proficiency of learner e. in theory learner e has a 50% chance of answering item 57 correctly. item 58 has a greater difficulty by a large margin. one would expect the student to perhaps get this incorrect. student f (location 2.903) answered three items correctly but was not able to answer item 58 (location 2.903) correctly. according to the model the student had a 50% probability of answering this item correctly, as it is located at the same point on the scale. in the case of the most difficult item, item 58, the requirement was to make decisions on converting the existing form before comparing and sorting the elements in ascending order. the cognitive demand required the students to connect their knowledge and make decisions in the process of working out the solution. table 2: six students, low, moderate and high proficiency vs performance on four items. problematic items it was noted in the first analysis that there were two items that did not function as expected. these two items were removed from this analysis, although for future testing they may be refined. one multiple choice item was removed due to an error. the second item, question 8a, was revised as shown below and was reserved for the next cycle. item 88 (question 8a) was found to be a misfit as the grammatical representation of the mathematical idea is confusing. the original and possible revised versions are briefly discussed below. original question: tell if the fraction on the left is less or greater than or equal to the fraction on the right. use < or > or = for each case to make the statement true. 1.5 150% the responses to item 8a produced the distribution displayed in figure 6. figure 6: item 88 (8a) – item characteristic curve. the black dots represent the means of the 5 class intervals into which the students were divided. the allocation to class intervals is decided by the researcher. the black dots representing students’ mean responses did not follow the expected pattern according to the model. the expectation is that students of lower ability will be less likely to answer an item correctly than those of higher ability. the analysis revealed that learners of lower proficiency (four × marks left of 0 logits) on the test as a whole performed relatively higher than the students of higher proficiency (one × mark at about 1 logits). this anomaly was investigated, and it was found that the grammar and length of the instructions appeared to have interfered with the understanding of the question. for the next three items in question 8 the instructions did not seem to mislead the students. when the instructions were revised and reduced to ‘use < or > or = for each case to make the statement true’, the whole question seemed clearer. local independence a further check on the validity of the test required an investigation of local independence. in any test, one expects that each item would contribute some information to the test construct (andrich & kreiner, 2010). there may be cases of construct irrelevance, where items do not contribute to the construct, and may be testing other dimensions, or construct underrepresentation, where the construct is not fully represented (messick, 1989). on the other hand, there may be cases where there is response dependency, where answering a second item correctly is dependent on answering the previous item correctly. another threat to validity of the construct is where there are too many items targeting one aspect of the construct, for example five items asking for similar knowledge. in such a case the student who knows the concept is unduly advantaged, while a student who does not know the concept is unduly disadvantaged. high residual correlations between items can be resolved by forming a subset, essentially a super-item, where the two items contribute to the score (andrich & kreiner, 2010). in this instrument analysis, we checked the residual correlations of the items and found high correlations, both positively correlated sets of items and negative correlations across some items. the implications of such a threat to local independence is that there are many items contributing the same information, as in a high positive correlation, and those with a negative correlation are ‘pulling in the other direction’. a resolution of this threat is to remove the items that seem to test the same thing or create subtests of items that are highly correlated, by investigating both the item context and the statistics it conveys. in a second round, eight items were removed due to redundancy. in order to resolve response dependency, 18 subtests were created. these subtests were then checked for ordered or disordered thresholds. for illustrative purposes four sets of items are discussed. question 6: item 6a ‘draw a representation of fraction , and item 6b ‘explain the meaning of the following fraction: , were subsumed into a subtest. the subtest was structured in such a way that instead of having two items that were highly correlated, there was one partial credit item, for which the student could obtain a 0 for none correct, a 1 for one of the two questions correct, or a 2 for fully correct. on investigating the subtest, question 6 (combined a and b), which required students to both draw a representation and explain the meaning of , the now partial credit item, it was observed that the common response was either none correct, or both correct. the middle category for which one mark awarded was almost redundant. the solution was to re-score the item as a dichotomous item and the resulting category probabilistic curve to show an improved scoring (figure 7b). figure 7: (a) subtest 3 (question 6), (b) subtest 3 (question 6) – re-scored. question 27 required the students to provide the fraction and percentage form for 0.75 as individual responses, but the correct answer depended on whether the student knew how to perform the conversions to both forms of fractions from the decimal form, that is, fraction and percentage form. this was the second set of items observed to be highly correlated and was subsumed into a subtest. for this subtest (see figure 8) it was found that the category probability curves functioned appropriately. the three categories, 0, 1 and 2, corresponded to both incorrect, one correct and two correct. the group of students of middle proficiency were most likely to obtain 1 mark for being proficient in converting a decimal fraction to either a common fraction or a percentage, whereas the higher proficiency group obtained the full 2 marks, meaning that they were proficient in both conversions of the item. figure 8: subtest 12 (question 27a and 27b). the next subtest was created by subsuming four items into one set. the four sections of the question asked similar questions, which were to convert from an improper fraction to a mixed fraction. these four items – 11a = ; 11b = ; 11c = and 11d = – appear to be testing only one skill because the distribution showed that students either answered all four items correctly or answered none correctly. the resulting category probability curve is shown in figure 9a. there may be a case here for rescoring, 0, 1 or 2 (see figure 9b). figure 9: (a) subtest 5 (question 11) – original score category probability curve, (b) subtest 5 (question 11) – re-scored category probability curve. the final subtest was made up of four different question items, where the requirement was to order a combination of the fractions-decimals-percentages representations in ascending or descending order (see figure 10). figure 10: final subtest. here it appeared that although these items were highly correlated, they increased in complexity. this subset functioned as expected in that the categories mark increase in proficiency with a clearer differentiated distribution of the curves (figure 11). figure 11: subtest 17 (questions 39–42) – re-scored category probability curve. as exhibited in the examples above, the investigation of specific subsets, from both a conceptual perspective and a statistical perspective, was conducted in order to ascertain which items could reasonably be subsumed into subtests. the subtests that functioned as expected were retained, but for those whose categories were for some conceptual reason not functioning according to measurement principles, the rescoring of the subtests items was implemented. the process reported in this article works together with a qualitative investigation that was done in the main study, and also formed part of improving the functioning of the instrument (dunne et al., 2012; maseko, 2019). the outcome, after this final analysis, was a test with 50 items, including both dichotomous and polytomous items, 22 of which were multiple choice format and 28 constructed response format. figure 12 appears more compact in the distribution of both the test item difficulty locations and students’ proficiency levels. the easiest of the items (st031) is by far the easiest and there is some distance from the next easiest items by almost 2.5 logits points. there were two items that were at difficulty levels where no one had a 50%, or greater, chance of answering correctly. figure 12: revised person-item map. table 3 shows the refined test person mean; a mean of −0.4172 in the initial analysis, moved closer to zero, 0.099, implying that by resolving some of the test issues, the targeting of the test to the learners was found to be better. the item standard deviation in the initial analysis was rather large at 1.6302, but after the refinement of the test it moved closer to 1, at 1.4933. the person standard deviation after refinement was somewhat smaller, implying that the range of proficiency was narrower. table 3: a comparison of the initial and final analyses. conclusion and implications as stated in the introduction, this article forms part of a larger study into the student understanding of rational number, fractions, decimals and percent. the purpose of the investigation was to gather information about the cohort entering the foundation phase teacher development on working with rational numbers, especially fractions-decimals-percentage. this article reported on how the instrument was functioning to assess their knowledge level of work done at school. the assessment tool covered understanding rational number concepts, manipulating symbols (operations), comparing and sequencing rational numbers, alternate forms of rational number representation, as well as solving mathematical word problems with rational number elements. it is clear that the number of items does not impact the quality of the test. beyond a certain amount, some of the items might be redundant. one has to check if the test instrument as a whole is fit for purpose. beyond the total score obtained by each student in the test, the rasch model indicates a position on a unidimensional scale where the student’s proficiency level is differentiated. the power and usefulness of the rasch model is that it supports the professional judgement of the subject expert in making decisions about the validity of items (smith & smith, 2004). the rasch model was applied in this study in order to confirm or challenge the theoretical base, to check the validity of the instrument and to quantify the students’ cognition of rational number concepts. the application of the rasch measurement model enabled checking whether the test content was consistent with the construct under investigation, and supported expectations of a sharper understanding of these students in terms of proficiency level within a set of items in the test. the outcome showed the data to fit the model, the person separation index was high, and the target was appropriate, thereby confirming the theoretical work that supported the design of the test. acknowledgements this work is based on a doctoral study for which the university of johannesburg has provided financial support. the opinions, findings, conclusions and recommendations expressed in this manuscript are those of the authors and do not necessarily reflect the views of the university of johannesburg. competing interests the authors declare that they have no personal relationships that may have inappropriately influenced the writing of this manuscript. authors’ contributions this study is based on the submitted doctoral research of j.m. the conceptualisation of this manuscript was done by all three authors. k.l. provided input on the literature and methodology. c.l. assisted with the rasch analysis. the final product received input from all three authors. funding information this research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors. data availability statement data sharing is not applicable to this article as no new data were created or analysed in this study. disclaimer the opinions, findings, conclusions and recommendations expressed in this manuscript are those of the authors and do not necessarily reflect the views of the 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(1979). the measurement model. in b.d. wright & m.h. stone (eds.), best test design (pp. 1–17). chicago, il: mesa press. wright, b.d., & stone, m.h. (1999). measurement essentials. wilmington, de: wide range, inc. zhou, z. (2011). the clinical interview in mathematics assessment and intervention: the case of fractions. in m.a. bray & t.j. kehle (eds.), the oxford handbook of school psychology (pp. 351–368). oxford: oxford university press. https://doi.org/10.1093/oxfordhb/9780195369809.013.0131 abstract introduction location in the curriculum what are area and perimeter? relationship between area and perimeter: mysterious connection misconceptions about area and perimeter theoretical framework methodology reliability and trustworthiness findings area is length multiplied by breadth overgeneralisation same a – same b discussions and conclusion summary of results implications acknowledgements references about the author(s) france m. machaba mathematics education department, university of south africa, south africa citation machaba, f.m. (2016). the concepts of area and perimeter: insights and misconceptions of grade 10 learners. pythagoras, 37(1), a304. http://dx.doi.org/10.4102/pythagoras.v37i1.304 original research the concepts of area and perimeter: insights and misconceptions of grade 10 learners france m. machaba received: 27 may 2015; accepted: 23 mar. 2016; published: 31 may 2016 copyright: © 2016. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract this article focuses on learners’ understanding and their descriptions of the concepts of area and perimeter, how learners solve problems involving area and perimeter and the relationship between them and misconceptions, and the causes of these misconceptions as revealed by learners when solving these problems. a written test was administered to 30 learners and clinical interviews were conducted with three of these learners, selected based on their responses in the test. this article shows that learners lack a conceptual understanding of area and they do not know what a perimeter is. learners also hold misconceptions about the relationship between area and perimeter. it appears that inadequate prior knowledge of area and perimeter is the root cause of these misconceptions. this article provides suggestions on how to deal with the concepts of area and perimeter. introduction research in the field of mathematics education, locally and internationally, often reveals poor understanding of the concepts of area and perimeter (gough, 2004; helen & monicca, 2005; tirosh & stavy, 1999). it was found that the concepts of area and perimeter are a continual source of confusion for learners. van de walle, karp and bay-williams (2014) suggest that it is perhaps because both area and perimeter involve measurements, or because students are taught formulae for both concepts at about the same time, that they tend to get the formulae confused. the confusion between these two concepts results in learners developing misconceptions. this article is based on the study by machaba (2005), which posed the following questions: how do grade 10 learners describe the concepts of area and perimeter? how do grade 10 learners solve problems involving area and perimeter and the relationship between them? what misconceptions are evident? what might be at the root of these misconceptions? in this article, i argue that learners lack conceptual understanding of the concept of area and they do not know what a perimeter is. it further appears that inadequate prior knowledge of area and perimeter is the root cause of these misconceptions. firstly, the location of this study in the curriculum will be discussed. i will then explain and describe the concepts of area and perimeter and the thought processes around area and perimeter, drawing from a variety of publications. after describing the theoretical framework of the study, the methodology used and the analysis of the data, the conclusion of the study once again focuses on the research questions and the answers yielded by an analysis of the data. finally, i will make practical suggestions on how educators and textbook writers or curriculum designers can improve learners’ ability to deal with the concepts of area and perimeter. location in the curriculum the national curriculum statement for mathematics, grade 7–9, determines that learners should be able to describe and represent the characteristics of and relationships between 2-d figures and 3-d objects in a variety of orientations and positions (department of education, 2003). this means that learners should be able to apply their knowledge of area and perimeter appropriately in various situations. the gauteng institute for curriculum development (1999, p. 46) in their progress map argue that learners should develop an understanding of the relationship between area and perimeter and use appropriate methods to solve problems involving area and perimeter. for example, learners should notice that all rectangles with the same perimeter will not necessarily have the same area and, where necessary, learners should be able to develop general formulae to calculate the area and perimeter. this means that learners should develop a conceptual understanding of area and perimeter through measuring areas and perimeters in a real-world context. they should work concretely (fitting, cutting, folding, matching and counting) to develop an understanding of the concept of area and should simultaneously develop the appropriate language to explain area. they should compare different regular and irregular figures and explain and justify which would be more suitable for measuring area (for example, it is easier to measure the area of a square than that of a circle, because no open spaces are left unmeasured when measuring the area of a square). they should also devise methods to measure the areas of regular and irregular figures (gauteng institute for curriculum development, 1999, p. 45). what are area and perimeter? dickson (1989, p. 79) defines area as ‘the amount of surface of a region’, and perimeter as ‘the distance around the region’. she argues that these definitions are not adequately covered in the lower grades, when learners merely learn to define area as the product of length and breadth (a = l × b), which is completely divorced from the idea of covering surface. nunes, light, mason and allerton (1994) concur that children’s success in understanding area is not independent of the resources they are given to represent area during problem-solving. learners need objects or resources like bricks and cuttings which they can fit, fold, match and count (gauteng institute for curriculum development, 1999), so that they can work concretely to develop a conceptual understanding of area and perimeter. this implies that for learners to understand the concept of area and perimeter, the formula length × breadth does not suffice. relationship between area and perimeter: mysterious connection the mathematics education literature reports that many learners and even adults adhere to the view that figures with the same perimeter must have the same area (e.g. outhred & mitchelmore 1996; tirosh & stavy, 1999; tsamir & mandel, 2000; kidman, 2001; helen & monicca, 2005; van de walle et al. 2014). these studies interpreted learners’ responses as resulting from a misunderstanding of the concepts of area and perimeter, that is, figures with the same perimeter must have the same area and vice versa. furthermore, tirosh and stavy (1999) found that for a significant percentage of learners a predictable relationship between area and perimeter is that when the area of a figure decreases or increases, the perimeter will also decrease or increase. learners may not realise that it is possible to have many rectangles with the same area, but different perimeters. if the perimeter is the same in a set of rectangles, then the area of those rectangles does not have to be the same. for example, rectangles with the same area can have many different perimeters. for example, a 3 × 4 and a 2 × 6 rectangle have the area of 12 square units, but their perimeters are 14 units and 16 units respectively. rectangles with the same perimeter can have many different areas. for example, a 3 × 4 and a 2 × 5 rectangle both have a perimeter of 14 units, but their areas are 12 square units and 10 square units respectively. on the other hand, learners can also establish a relationship between area and perimeter that rectangles with the same area have dimensions that are factors of the fixed area. when the difference between the dimensions of a rectangle with a fixed area is the smallest, you will have the smallest perimeter. when the difference between the dimensions of a rectangle with a fixed area is the largest, you will have the largest perimeter. given a fixed perimeter, the rectangle with the largest area will be the one with the dimensions that are closest together (a square). given a fixed perimeter, the rectangle with the smallest area will be the one with the dimensions farthest apart. the literature, as stated above, indicates that there are many misconceptions for learners and adults about the complex relationship between area and perimeter. learners need to have experiences in which they are manipulating the spaces that they are measuring, to construct deep understanding. because of this, it is important to use a variety of manipulatives to develop the concepts. if these are not used, learners would view the relationship between area and perimeter as the result of the application of the intuitive rule ‘same a – same b’ (same perimeter – same area; same area – same perimeter). this kind of mysterious connection between perimeter and area is further discussed below. misconceptions about area and perimeter olivier (1989, p. 12) defines misconceptions as ‘errors or wrong answers that are systematic in that they are applied regularly in the same circumstances’. he says they are symptoms of underlying conceptual structures that are causes of error. smith, disessa and roschelle (1993, p. 1) argue that misconceptions are flawed ideas that are strongly held by a student and which interfere with learning. clement, in confrey (1990, p. 18), defines misconception as ‘conceptual stumbling blocks, inconsistent semi-autonomous schemes, and cognitive processes responsible for errors in problem solving’. these definitions imply that misconceptions are a part of learning and that they are inevitable, but, if noticed, they can be dealt with appropriately. when learners construct knowledge, reconstructing and reorganising their prior knowledge and aligning it with their new knowledge, misconceptions are likely to arise. the point here is that misconceptions are not mistakes that can easily be corrected by telling the learners that they are wrong. learners need experiences that will enable them to reorganise their thinking. olivier (1989, p. 12) states that ‘if we want to account for pupils’ misconceptions, we must look at pupils’ current schemas and how they interact with each other, with instruction and with experience’. this means that misconceptions are a function of how learners construct knowledge. this is also supported by smith et al. (1993), who argue that misconceptions arise from prior learning, either in the classroom (especially for mathematics) or from learners’ interactions with the physical and social world. olivier (1989) believes that the source of misconceptions is mostly an overgeneralisation of previous knowledge (knowledge that was correct in an earlier domain) to an extended domain (where it is not valid). this means that learners who have been taught the formulae for the areas of a square and a rectangle in their early stages of learning, overgeneralise these formulae, extending them to other figures such as triangles and irregular figures. in other words, they use the formula length times breadth for all figures. dickson (1989) asserts that most children responded to the word ‘area’ by saying ‘length times breadth’, irrespective of the figure being considered. another common misconception has been researched by tsamir and mandel (2000). they argue that if learners are presented with a square where one side is lengthened and the other shortened by the same amount, these learners claim that both the perimeter and the area of the created rectangle are equal to the perimeter and area of the original square. they claim that learners are using a rule called ‘same a – same b’, a rule based on their intuition. learners’ justifications of their answers were based on the intuition that if the same number was transferred from one place to another, there would be no change in the perimeter or the area of the figure formed. this type of reasoning has been explained by piaget’s theory about the use of compensation to attain conservation. kidman (2001) and zacharos (2006) concur that the learners intuitively had a perception of area, where doubling the length of the sides of a rectangular region can be seen as doubling the area. these support research findings by outhred and mitchelmore (1996), who argue that learners believe that when the sides of a square are doubled, so is its area. i was interested to see if the result of the current research study would produce the same misconception of ‘same a – same b’ as has been explained above. nunes et al. (1994, p. 256) indicate that the concept of area is ‘prone to misconceptions, is difficult to teach, and that misconceptions are retained even in the upper middle school age range’. they cite the most common misconception of the relationship between the area and perimeter, which is that the area delimited by the perimeter remains the same, even though the figure and delimited surface have changed. this was evident when limiting an area with a string fixed at the corner and gradually changing the surface by moving the tacks that held the other three corners. they said children tend to think that the area remains constant, despite the visible changes in the surface. they argue that another cause of these misconceptions is that learners are taught or learn a procedure or formula for the area of the rectangle, rather than forming a solid understanding of the relationship between the side lengths and the area of the figure. in my study i was interested to see if learners in south africa do the same as reported by dickson (1989) and tsamir and mandel (2000). do learners in south africa overgeneralise? do they apply the ‘same a – same b’ intuitive rule? and if so, what reasons do they give for their conceptions? theoretical framework this study is informed by piaget’s theory of constructivism, which is about learners assimilating new learning into their existing schema. constructivism holds that learning occurs efficiently and effectively when new knowledge is linked to existing or prior knowledge (hatano, 1996). kramer (1996, p. 6) states that one of the important contributions of piaget’s ideas is the notion that ‘learners build or construct new knowledge or skills based on what they know or can do’. this means that learners cannot learn mathematics effectively in isolation from what they are already familiar with. this theory is supported by mogari (1998), when he says learners do not enter classrooms with blank minds. instead, they bring with them ideas, conceptions and experiences about mathematical principles, practices and concepts. the theory of constructivism also emphasises the fact that the construction of knowledge is dependent not only on what the child already knows, but also on what they have to know. skemp (1976, p. 20) describes this kind of understanding as ‘instrumental understanding’ and contrasts it with ‘relational understanding’. instrumental understanding is manifested when learners know rules and formulae and have the ability to use them without reason, not knowing where those rules and formulae come from. for example, many learners know that the formula to calculate the area of a rectangle is ‘length multiplied by breadth’, but they do not know why this is so. he argues that learners should develop a relational understanding of area and perimeter. in other words, learners should know both ‘what’ to do and ‘why’ when dealing with problems that involve area and perimeter (skemp, 1976, p. 20). this also implies that learners should be able to associate or relate the concepts of area and perimeter with other mathematical concepts and their everyday life experiences. for example, skemp mentions in his research that he asked his learners the question: ‘what is the area of a field 20 cm by 15 yards?’ the reply was: ‘300 square centimetres’. when asked why not 300 yards, his learners replied that ‘area is always measured in square centimetres’ (p. 23). the integrated networks of connections between ideas referred to as cognitive schemas are the product of constructing knowledge and the tools with which additional new knowledge can be constructed (skemp, 1976). thus, van de walle (2007) defines understanding as the quality and quantity of connections that an idea has with existing ideas. understanding depends on the existence of appropriate ideas and the creation of new connections. understanding at this rich and strongly interconnected end of the continuum can be referred to as ‘relational understanding’, while that at the other end of the continuum, where the ideas are completely isolated, can be referred to as ‘instrumental understanding’ (skemp, 1976; van de walle et al. 2014). knowledge learned by rote, where ideas are nearly always isolated and disconnected, is almost always understood instrumentally. when learners come by knowledge through self-discovery, it has more meaning because it facilitates the process of building cognitive structures (constructing a network of connected ideas) and because the recall of information (concepts and procedures) is far easier than when knowledge is given (van de walle et al. 2014). relational understanding suggests that when a learner solves a mathematical problem, they arrive at a concept which is in fact produced from a schema (a network of connected ideas). this process produces the kind of learner who is independent, able to think, able to express ideas and to solve problems, representing a shift to a learner-centeredness, that is, learners as knowledge developers and users, rather than storage systems and performers. methodology the research methodology used in this study was qualitative and was organised around a written test administered to 30 learners as well as a clinical interview carried out with a sample of six learners. the interviews were conducted after all the learners had written the test of six questions (figure 1), which was based on previous studies such as those discussed above. in the interview i probed my participants to explain their interpretations, experiences and insights with regard to each of their responses to the test items and their reasoning when it comes to the concepts of area and perimeter. figure 1: the written test on area and perimeter. grade 10 learners were chosen from a secondary school in soshanguve. the reason why i chose this school was because i was not teaching in this school. this assisted to minimise researcher bias that might emerge from familiarity with the learners and with the school concerned. below is the written test that was given to learners to write on the concept of area and perimeter. rationale for the choice of questions question 1 this question tests learners’ conceptual understanding, without using formulae, and their ability to express these ideas or concepts in words. question 2 this question was also aimed at testing learners’ understanding of the concepts of area and perimeter, without being given numbers or measurements. it tests whether learners are able to count square units to determine the area of a figure. it also tests whether learners know that calculating the area of a rectangle by multiplying the number of square centimetres in a row by the number of rows is the same as multiplying the number of square centimetres in the length by the number of centimetres in the breadth. question 3 this question tests the learners’ ability to calculate the perimeter and area of an irregular figure when given measurements. question 4 this question tests learners’ understanding of the concept of area and whether they can measure the area of an irregular figure. question 5 this question tests learners’ understanding of the relationship between area and perimeter and whether they apply the intuitive ‘same a – same b’ rule, finding that when increasing the length of two opposite sides of a square by a given factor and reducing the length of the other two remaining sides by the same factor, the perimeter and the area will remain the same. question 6 this question tests learners’ understanding of the relationship between area and perimeter and their application of the intuitive ‘same a – same b’ rule, finding that when increasing or multiplying the length of two opposite sides of a square by a given factor and reducing or dividing the length of the other two remaining sides by the same factor, the perimeter and the area will remain the same. to reiterate, the purpose of the study was to investigate the insights and misconceptions that some grade 10 learners in one school in soshanguve have with regard to the concepts of area and perimeter. a written test was administered to 30 learners in a classroom and an interview was conducted with a selected six of these learners. however, for this article i report on the results of only three of the learners with whom the interviews were conducted, because only from them had i obtained saturated data, so i was forced to omit the data obtained from the other three learners. the results of the interviews and the written test will be reported simultaneously. ethical considerations permission to conduct the research was granted by the gauteng department of education, the district and the school where this research was conducted. as this study was not conducted at my school, i wrote a letter to the principal of the school, describing the required grade (grade 10) and the purpose and the rationale of the study. grade 10 learners were informed of the study so that they could decide whether or not to participate in the study. learners who agreed to participate in the study were guaranteed anonymity and confidentiality. at both school and individual levels, participants’ anonymity and confidentiality were maintained by use of pseudonyms (e.g. l1, l2 and l3). learners were informed that their real names would not be used in the study and whatever they said would be kept confidential. i developed a rapport with them so that they would not perceive me as an evaluator or judge, that is, as somebody who wanted to detect their learning flaws or faults that could be used to determine their promotion. rather, i intended to be perceived as one who was interested in how they think and reason mathematically. participants were informed that they would be provided with the report of the study. reliability and trustworthiness data were collected by me using both the instrument (test) and interviews. to ensure reliability of the instrument i initially collected pilot data and then tested an instrument to see if it would be interpreted in the same way. since some of the questions had not yet been used in any research before, i thought it would be imperative to find out whether the test items were appropriate and tested my critical questions through piloting. i involved five grade 10 learners in piloting. i gave test items to each and they spent 45 minutes on average answering the questions. i marked their test and chose two learners for an interview. the selection of the two was based on how they had answered the test items. i selected one whose performance was good in the test and one who performed poorly. furthermore, i presented the test at conferences and postgraduate meetings, where it underwent rigorous peer reviewing before taking its final form. because i interviewed few learners, one cannot generalise the findings beyond the studied cases. this is the nature of case studies. however, consistent with the objective of the study, the findings could provide principles for dealing with learners’ misconceptions. table 1 shows the learners’ performance in each of the questions and the numbers and percentages of correct, partially correct and incorrect answers given by all 30 learners who wrote the test. table 1: summary of findings (n = 30). findings the findings as summarised in table 1 reflect poor performance in almost all questions, with the exception of question 2b, which 63% of learners answered correctly. it would seem that most of the learners had not yet come across the mathematical concept of perimeter. this is evident from the fact that none of the learners could define perimeter. it was therefore imperative to find out during the interview why learners were unable to define area and perimeter without using the formulae. it is evident from table 1 that 67% of the learners answered question 1a incorrectly, while 90% failed to answer question 1b correctly. the answers to the rest of the questions were also interesting, with more than 50% incorrect answers. i hoped to discover the reasons for this during the interviews. of course, the table does not show how the learners solved the problems, or how during the interviewing process some were able to obtain the correct solutions. the discussion that follows is an analysis of the responses of the three selected learners to each of the questions in the test and during the interview. i will refer to the three learners that were interviewed as l1, l2 and l3, without implying through the labelling the order in which they were interviewed. the researcher will be referred to as the ‘interviewer’ in the transcript. the major finding was that the learners held the same misconceptions that had been identified by other researchers. this claim is based on both the test responses and the follow-up interviews. area is length multiplied by breadth the learners cited the ingrained, formalised method of multiplying length by breadth to get the area. this indicates a lack of conceptual understanding of area as a surface and perimeter as the distance around the edge of the figure. the learners described both the area and the perimeter in terms of a formula. for example, with regard to question 1a and question 1b, l1 responded as follows: l1: it is the answer that you get after multiplying both the length and the breadth. interviewer: why did you define by using the formula, when you had been forbidden to use the formula? l1: mmm. … i use the formula because there was no other way i can define the area without using the formula. l1: perimeter is the sign of showing that it is a cm, km or m. interviewer: what do you mean? can you say more on that? l1: i mean like in a ruler [showing me a ruler], these are millimetres and centimetres, these are perimeters. similarly, l2 responded as follows: l2: [perimeter] is the size of an object. interviewer: have you ever heard about the word perimeter in your life? l2: no. overgeneralisation the learners overgeneralised when moving from working with rectangles to working with non-rectangles. they thought that the formula a = l × b could be applied to non-rectangles. for example, the learners responded as follows to question 5: l1: no, the leaf does not have an area because there is no length and breadth. l3: i do not think the leaf has an area, because the leaf is not a rectangle and does not have length and breadth. same a – same b the learners claimed that when the length of two opposite sides of a square were increased by a given number of centimetres and the length of the other two sides were decreased by the same number of centimetres, both the perimeter and the area would remain the same. similarly so if the length of two sides is multiplied by a certain factor and the other sides are divided by the same factor. the response to questions 6 was as follows: interviewer: you said your answer would be ‘equal to’ in your solution of 6(a) and 6(b). can you give a reason why you said so? l1: sir, i think if you lengthened these two sides of the square, nee! … by 6 nee!! … mmm … and shortened the other two sides by 6 again [talking and demonstrating with her hands and fingers] is like you did nothing, is like you add 6 subtract 6 is zero, so that is why i say the answer is equal. interviewer: which means, what you are saying is the perimeter of the rectangle will be equal to the perimeter of square? l1: yes, sir. interviewer: what about the area? l1: the area of the square will also be equal to the area of rectangle. interviewer: why are you saying so? l1: because, sir, you add 6 and subtract 6. it is clear that l1 uses the intuitive ‘same a – same b’ rule when solving the problem. l2 did likewise: l2: no, i think the perimeters of the two diagrams are equal, sir. l2: because the two opposite sides of the square are increased by 6 cm and the other two have been also decreased by 6 cm. interviewer: what about the area of the two diagrams? l2: i think, are also equal because of the 6 cm, which was added and subtracted. the application of the intuitive ‘same a – same b’ rule confirms findings by (tsamir & mandel, 2000). in their findings, they reported that in the same mathematical problem learners correctly claimed that the perimeters of the square and the rectangle are equal when adding and subtracting 6 cm to opposite sides of the square. this type of reasoning has been explained by piaget’s theory about the use of compensation to attain conservation. tirosh and stavy (1999) also viewed these responses as an instance of the intuitive ‘same a – same b’ rule. this rule is an instance of overgeneralising and arriving at erroneous conclusions such as that if the perimeter of the original square is equal to the perimeter of the created rectangle, the areas of these figures must be equal too. discussions and conclusion in this section, i shall return to my research questions and answer them on the basis of the analysis of my data. i will treat each question as a subheading of this section when stating my findings. i shall state explicitly what i have found. lastly, i will discuss the implications of my findings and reflect on them. the critical questions that guided this study were: how do grade 10 learners describe the concepts of area and perimeter? how do grade 10 learners solve problems involving area and perimeter and the relationship between them? what misconceptions are evident when learners are solving these problems? what might be the cause of these misconceptions? how do learners describe the concept of area? learners had problems defining the concept of area without using the formula a = l × b. most learners were unable to define area as the amount of a surface of a region, with the exception of l2 and l3, who used the everyday notion of area as being a ‘place’ or a ‘space’. in terms of skemp (1976), these learners do not have relational understanding (knowing what to do and why), but an instrumental understanding (doing something without understanding) of the concept of area. they were unable to build on their existing knowledge (schema) of the area as a surface which they should have possessed to create a new knowledge of the formula of an area. this means that learners do not have a conceptual understanding of area. it appears that they do not know where the formula a = l × b comes from or why a = l × b. kilpatrick, swafford and findell (2001, p. 119) say that a significant indicator of conceptual understanding is being able to represent mathematical situations in different ways and knowing how different representations can be useful for different purposes. in this study, with the exception of l2 and l3 who used the everyday notion of area as a ‘place’ or a ‘space’, learners could not, apart from the formula a = l × b, give any different representations or use methods like the square centimetre grid. it appeared to me that these learners do not know that the formula a = l × b generalises an arithmetic pattern (usiskin, 1998) and is derived from somewhere. this was evident to me when none of the learners mentioned the word grid or something similar. dickson (1989) also found a lack of conceptual understanding of area in her research, when she indicated that some of her learners defined area as a = l × b, yet regarded it as completely divorced from the idea of covering surface. how do learners describe the concept of perimeter? i found that learners do not know what the concept ‘perimeter’ entails. none of the learners could correctly define perimeter. one of the learners defined perimeter as units, for example km, m, cm, while others defined perimeter as ‘the length and breadth’. dickson (1989) defines perimeter as the distance encompassing a region. none of the learners defined perimeter in this way. i can only assume that the lack of knowledge about perimeter is because it is not an everyday notion as area is. how do learners solve problems involving area and perimeter and the relationship between them? in this study learners were able to calculate area when given measurements, but were unable to determine the area when measurements were not given on the figure. they did not know that the area could be determined through counting square centimetres. their failure to make a connection between the figure with square centimetres and the one with measurements leads us to conclude that they also lack a conceptual understanding of area. the lack of the integrated network of connections between ideas (cognitive schema) was the product of being unable to construct new knowledge based on existing knowledge. learners who have difficulty translating a concept from one representation to another have difficulty solving problems and understanding computation (van de walle et al. 2014). this is where the idea of smith et al. (1993) comes into play, namely that making connections between multiple representations helps to develop meaning. all three learners responded by saying that an irregular figure (such as that of a leaf) does not have an area, because it does not have length and breadth. this implies that these learners only know the procedure or formula for calculating area fluently, accurately and efficiently when given numbers to substitute, which skemp (1976) refers to as instrumental understanding. kilpatrick et al. (2001) maintain that learning procedures without connections to understanding, meaning or concepts is merely memorisation and at the lowest level of cognitive demand. learners were unable to calculate perimeter, which shows that they do not have an understanding about the concept of perimeter. in their research report, kilpatrick et al. (2001) say that when learners have acquired conceptual understanding in an area of mathematics, they see the connections among concepts and procedure and (that) some facts are consequences of others. this report confirms that learners lack conceptual understanding (relational understanding) of area and perimeter and the connections between them. what misconceptions are evident when learners are solving problems related to area and perimeter? the first misconception that these learners displayed was overgeneralisation. they thought that the formula a = l × b, which is used for rectangles, could be used for non-rectangles as well. this was evident when l1 multiplied 10 cm by 9 cm to get an area in question 4. dickson (1989) found that most children responded to the word ‘area’ by saying ‘length times breadth’, irrespective of the figures being considered (rectangles or non-rectangles). it was interesting to see that one learner showed a working knowledge of subdividing the l-shape in question 4 into rectangles to enable him to apply the ‘length times breadth’ formalisation. the second misconception that was found was the application of the intuitive ‘same a – same b’ rule. learners claimed that when increasing the lengths or adding a certain figure to the lengths of two opposite sides of a square and reducing the lengths or subtracting the same figure from the lengths of the other two remaining sides, the perimeter and the area would remain the same. this finding resonates with the findings of tsamir and mandel (2000). learners also claimed that when increasing or multiplying the size of two opposite sides of a square by a given factor and reducing or dividing the size of the other two remaining sides by the same factor, the perimeter and the area would remain the same. all the learners (l1, l2 and l3) therefore correctly claimed that the perimeters of the square and the rectangle would be equal when adding and subtracting 6 cm to opposite sides of the square (question 5). they also correctly claimed that the area of the rectangle remained the same when multiplying and dividing opposite sides of the square by equal factors. however, in each instance they extended their claims to the other concept as well (to the area in the addition and subtraction of a certain length of the sides and to the perimeter in the multiplication and division of the lengths of the sides). tsamir and mandel (2000) confirm that correct answers found with an intuitive rule therefore do not necessarily reflect students’ understanding of the concepts. all the learners (l1, l2 and l3) wrongly assumed that when two opposite sides of a square are lengthened by 6 cm and the other two sides are shortened by the same number (6 cm), then the area of the given square would be equal to that of the resulting rectangle. they also wrongly assumed that when two opposite sides of a square are multiplied by 6 cm and the other two sides are divided by 6 cm, the given square and the created rectangle would have the same perimeters. the justifications of their claims were in line with the application of the intuitive ‘same a – same b’ rule. learners also wrongly concluded that if the perimeter of the original square in question 5a is equal to the perimeter of the created rectangle, the perimeter of the original square in question 6a would also be equal to that of created rectangle and that if the area of the original square in question 6b is equal to the created rectangle, the area of the original square in question 5b would be equal to that of the created rectangle. this erroneous conclusion is another form of overgeneralisation which indicates a lack of knowledge of the two concepts: perimeter and area. tirosh and starvy (1999) suggest two ways in which the intuitive rule ‘same a – same b’ are formed; they indicate that: (1) it may be one of a small set of universal, innate primitives and (2) it is an overgeneralisation from successful experiences. often, both in everyday life and in school situations, the rule ‘same a – same b’ is in fact applicable (e.g. ‘same heights of juice in two identical cups – same amount to drink’, ‘same number of candies – same price’). it is reasonable to assume that children generalise such experiences into a universal maxim: ‘same a – same b’. what might be causing these misconceptions? it was evident from their incorrect definitions of area and perimeter that all three of the interviewed learners lacked prior knowledge of area and perimeter and that they had no conceptual understanding of perimeter as a distance. furthermore, none of them mentioned the square centimetre grid in their discussion with the interviewer. learners could not add new knowledge to the existing knowledge by making sense of what is already inside their heads. they could not organise, structure and restructure their experience in the light of available schemes of thought (van de walle et al. 2014). dickson (1989, p. 79) says that ‘the confusion between area (the amount of surface) and perimeter (the distance round a region) is nearly always due to inadequate preparation in the early stages’, when the learners are given the formulae a = l × b and p = 2(l + b) without adequate explanation of the concepts. this is also confirmed by olivier (1989) and smith et al. (1993), who argue that prior knowledge or existing knowledge may contribute to the development of misconceptions in the cognitive structure of the learner. the misconceptions created by prior learning were evident when learners applied the formula a = l × b to non-rectangles. it was further evidenced that misconceptions resulted from structures (a = l × b) that apply appropriate in one domain (a rectangle) being overgeneralised to another domain (non-rectangles). summary of results in the light of the above findings i can say that in dealing with the concepts of area and perimeter, learners have the following problems: they lack conceptual understanding of area as a surface. this became evident when they described area as length multiplied by breadth. they overgeneralise, in other words, they assume that the formula a = l × b, as applied to rectangles, can also be applied to non-rectangles. they use the intuitive ‘same a – same b’ rule when dealing with area and perimeter and therefore obtain only partially correct (false positive) results. they believe that when the size of two opposite sides of a square are increased by a given factor and then the size of the other two remaining sides is decreased by the same factor, the perimeter and the area would remain the same. implications this study, like other similar studies (e.g. dickson, 1989; tirosh & stavy, 1999), suggests how teachers, textbook writers and curriculum designers could improve learners’ understanding of the concepts of area and perimeter. teachers should drastically change their approach in teaching these concepts, by shifting the emphasis away from teaching the formulae for area and perimeter to making use of activities that would develop learners’ conceptual understanding of area and perimeter. they should ensure that learners develop a conceptual understanding of area and perimeter by using square centimetre grids and by letting the learners do fitting, cutting, folding and counting activities, instead of teaching them formulae like a = l × b and p = 2(l + b), procedures, rules and principles, without explaining where they come from and why they are that way. teachers should distinguish between the concepts of area and perimeter, yet emphasise the relationship between them, so that learners will not see them as isolated concepts. in their progress map, the gauteng institute for curriculum development (1999) indicate that learners should develop an understanding of the relationship between area and perimeter and use efficient methods appropriately to calculate and solve problems that involve area and perimeter. if teachers and textbook authors could, in the lower grades, emphasise the fact that the area is the size of a surface and the perimeter is the size of the edge of a figure, many misconceptions could be avoided. the extent of such misconceptions was evident when l2 and l3 responded in question 5 that a leaf does not have an area, because there is no length and breadth. teachers should also be aware of the role that the intuitive rule plays in the concepts that learners form. in other words, when designing problems, teachers should consider whether they might elicit the use of an intuitive rule or counter it. this also implies that teachers should not be satisfied with the correct answers alone, but probe further to be certain that the learners are not just applying the intuitive ‘same a – same b’ rule (olivier, 1989; tsamir & mandel 2000). acknowledgements competing interests the author declares that he does not have financial or personal relationship(s) that may have inappropriately influenced him in writing this article. references confrey, j. 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(2006). prevailing educational practices for area measuring and student failure in measuring areas. journal of mathematical behavior, 25, 224–339. http://dx.doi.org/10.1016/j.jmathb.2006.09.003 world view theory and the conceptualisation of space in mathematics education (an abridged version of this paper was presented at the saarmste conference, cape town in january 2004) marc schäfer rhodes university email: m.schafer@ru.ac.za introduction the cornerstone of current education trends that recognise prior knowledge as fundamental to the learning process, is the notion that beliefs and experiences that learners bring to the classroom influence their learning experiences in the classroom (cobern, gibson and underwood, 1999). world view research in science education has proved an important tool in unravelling learners’ beliefs and perceptions in an attempt to better understand what experiences they bring to a classroom situation. cobern (1991, 1993, 1994, 1996, 1997) has successfully spearheaded much of the world view research in that field, with particular emphasis on learners’ perceptions and understanding of nature. very little, if any, world view research has been conducted in a mathematics education context. this paper reports on a phd study that specifically employed a world view research strategy in exploring grade 11 learners’ perceptions of space in an attempt to shed light on their broader spatial conceptualisation. before describing the methodological model which framed the world view research of the study, this paper will briefly discuss the three fundamental theoretical concepts that underpinned this research: spatial conceptualisation, space, and world view. traditionally, comment on and analysis of learners’ spatial conceptualization was framed around performance in pen-and-paper tests. it is, however, the assertion of this research that performance in pen-and-paper tests provides only part of a more comprehensive picture. spatial conceptualisation is seen as a complex and rich blend of spatial performance ‘measured’ in pen-andpaper tests, and a personal world view of space. the theoretical framework spatial conceptualization it is a common perception that spatial understanding is fundamental to the understanding and “appreciation of our inherently geometric world” (nctm, 1989) in general, and to mathematical cognition in particular. battista et al. (1982) recognised this by suggesting that spatial thinking is an important aspect of mathematical performance. much research has focused on the relationship between spatial understanding and mathematics achievement (bishop, 1980). educators and policymakers the world over have intuitively recognised the role of spatial thinking and most mathematics curricula incorporate aspects of developing spatial thinking, usually in the form of formal geometry. there is, however, little consensus on the concept of space, even while most western mathematics curricula are firmly rooted in a euclidean paradigm which sees space as a measurable and rational system. euclidean space is seen as ordered, where shapes are measurable and positioned in a definite manner (euclid, 1956). there is a sense of geometry – “the branch of mathematics concerned with the properties and relations of points, lines, surfaces and solids; the relative arrangement of objects or parts” (concise oxford dictionary, 1995). newtonian space is consistent with euclidean space in terms of its sense of order and tangibility. it affirms the “reality of space” (gardner, 1999) and maintains that space exists independently of the subject’s awareness. kantian space, on the other hand, suggests that “space (and time) are not features of absolute reality but only forms of sensibility, elements of our subjective cognitive constitution, and that everything that has spatial properties – all the objects of our experience – are mere appearances as opposed to things in themselves” (gardner, 1999). 8 pythagoras 59, june, 2004, pp. 8-17 mailto:m.schafer@ru.ac.za marc schäfer this implies that space should not be conceptualised in terms of objective features only. in kant’s concept of space, the space is independent of its content. this means that our subjective understandings and experiences form an integral part of our overall perception of space. kant suggests that the order we find in nature is the order that exists in our minds, an order which is embedded in or reflects our own structure of mind (stumpf, 1994; want and klimowski, 1996). there is a tendency in current thinking to embrace a broader view of geometry. the post-1994 curriculum for mathematics in south africa, for example, sees space and shape within a context of social experiences. one of the specific outcomes of the mathematics curriculum suggests that learners need to be able to “describe and represent experiences with shape, space, time and motion, using all available senses” (south africa, 1997). this apparent shift is consistent with a global epistemological paradigm move towards recognising that cognition is an active and complex process of social interaction. there has been a plethora of research literature on spatial conceptualization and its importance in the cognitive development of spatial capacity. much of the literature, however, is in agreement that there is little consensus in the terminology and definitions used. there also appears to be little consensus in the theoretical frameworks underpinning spatial development. nothing much has changed since battista’s et al. (1982:332) comment that “the role that spatial thinking plays in mathematical performance has not been adequately described”, or bishop’s (1980) lament that there exist inconsistencies in terminology and methodology of research in the general discourse of spatial ability. for the sake of ensuring clarity and seeking consistency in the use of terminology from the literature (bishop, 1980), this research project was shaped around the following fundamental concepts of spatial conceptualisation: spatial capacity: the all-encompassing concept which embraces spatial visualisation, orientation, perception and ability. in this research spatial capacity refers in particular to that aspect of spatial conceptualisation which is measurable in a pen-andpaper and/or hands-on activity test. spatial visualisation: the ability to manipulate mentally, rotate, twist, or invert a pictorially or physically presented stimulus object. the underlying ability in spatial visualisation appears to be connected to movement, transformation and manipulation. it is dynamic and involves motion. spatial orientation: the ability to recognise the identity of an object when it is seen from different angles. it is the ability to make sense of spatial orientations of an object relative to different positions of itself or of other objects. spatial conceptualisation: the fundamental concept that ultimately incorporates spatial capacity and perceptions of space. world view as funk (2001) suggests, the meaning of the term world view seems self-evident. the german translation weltanschauung implies a perception of the world, “a conception of the world” or a “particular philosophy of life” (concise oxford dictionary, 1995). the 19th century german philosopher wilhelm dilthey was especially interested in world views. “he was fascinated by the familiar, yet vexing, fact that world views vary so widely and conflict so sharply even when they are based on the reasoned arguments of philosophers” (rickman, 1979:47). according to dilthey (rickman, 1976:141) “the formation of world views is determined by the will to stabilize the conception of the world, the evaluation of life and the guidance of the will”. world views are dynamic and develop under varied conditions. dilthey (rickman, 1976:139) asserts that: …climate, race and nationality, determined by history, and the development of states, the temporal delimitation into epochs and ages in which nations cooperate, combine to produce the special conditions which influence the rise of differences in world views. dilthey’s views need to be seen in context with his own epistemological assumptions. he assumes that “we know the world through our feelings and strivings as well as through our sense impressions and thinking” (dilthey, cited in rickman, 1976:15). the epistemological underpinning of this study has elements of dilthey’s assumption and is rooted in the idea that an individual’s cognition is a complex process which is informed by world views and presuppositions. funk (2001) makes an interesting observation with which i strongly identify. he suggests that an individual’s world view may not always be explicit. few people take time to thoroughly think out, much less articulate, their world view. when exploring 9 conceptualisation of space in mathematics education and discussing their world views the participants of this research project intimated that they had never really thought and talked about their presuppositions and conceptions of the world around them. holmes’ work on dilthey, as cited in cobern (1991), suggests that our world view, or weltanschauung, initially rests on a more fundamental implicit world picture, or weltbild, which develops in the context of the world in which we live, or lebensbild. this concept of world view has influenced current world view theorists and has occupied an important place in anthropology, but according to kearney (1984) no comprehensive model had been formulated prior to his logicostructural model which will be discussed later. he refers to world view as a “culturally organised macrothought: those dynamically interrelated basic cognitive assumptions of a people that determine much of their behaviour and decision making” (kearney, 1984:1). cobern (1991:19), a scienceeducation researcher who has embraced the notion of world-view theory in science-education research, takes kearney’s definition a step further and asserts that “a world view inclines one to a particular way of thinking”. in cobern’s work (1991), which rests heavily on kearney’s (1984) theoretical framework, world-view research focuses on students’ presupposition about their world, that is, their epistemological macrostructures. one of the central themes of this research is cobern’s (1991:20) observation that “knowing more about students’ world views should help researchers come to a better understanding of conceptual change by providing a more complete understanding of conceptual structure. it should enable educators to better understand students’ attitudes and achievement in general”. kearney’s (1984:2) unique theoretical worldview framework is based on the assumption that a world view is a “dynamic, more or less internally consistent system which demonstrates logical and structural regularities”. he suggests that the structural composition of a world view consists of seven universals, or cognitive categories (cobern, 1991): self, the other (non-self), relationship, classification, causality, space and time. kearney refers to his model as logico-structural integration (kearney, 1984) because he asserts that the worldview categories are filled with “logically consistent presuppositions about reality” (cobern, 1991:39). as slay (2000) notes in reviewing these universals, they can serve as an effective framework for analysis of a world view. although there is a recognition that the universals above need to be seen holistically, my research specifically focused on the category space and, in conjunction with exploring issues of spatial capacity, investigated presuppositions about space held by 32 grade 11 students. it is my assertion that spatial conceptualisation is a complex blend of spatial capacity and world view of space. as it is beyond the scope of this paper to consider all of kearney’s universals, i will dwell only on the space universal. for details regarding the other six universals refer to cobern (1991), slay (2000) and schäfer (2003). world view in the context of this research the key theoretical issues that are particularly relevant to this study and specifically frame this research in order to achieve part of the objectives are: • the systematic application of the logicostructural model as proposed by kearney (1984) to explore world views of space; • the creation and exploration of world-view profiles to illustrate students’ presupposition about space. space space means many things to many people and is key to this study. in kearney’s (1984) logico-structural model, space is the seventh universal, although he links this very closely to time. he theorises that as things are located in space, they are also inextricably located in time. for the purpose and scope of this study, however, space and time have been segregated. when looking at the world-view aspect of space, as wide a definition for space as possible is sought. in kearney’s work (1984:92) he observed that space is used to refer “to many different concepts, ranging from an easily measurable geographical space to more metaphorical usages such as psychological, life and mathematical space”. in his anthropological dealing with world-view aspects of space, kearney (1984) was mainly concerned with the relationship between the environmental space of a people and their images of it. in my attempt to ground this research in the literature, my readings led me along a fascinating path of interesting interpretations of space. this journey facilitated transient visits to gestalt psychology, ancient greek philosophy, mathematics and modern western philosophy. as it is beyond the scope of this short paper to refer to these visits in detail, i provide only a very short and superficial overview. for further details, refer to schäfer (2003). 10 marc schäfer absolute and relative space: among theorists in gestalt psychology, koffka (1935) distinguished between geographical environment (absolute space) and the behavioural environment (relative space). he held that the geographical environment is ‘stimulus providing’ and that the behavioural environment depends upon the geographical environment and the organism itself. mathematical space, physical space and psychological space: downs and stea (1973) cite lewin (another gestalt psychologist), who stressed the relationship of and distinctions among mathematical space, physical space and psychological life space. these concepts of space resemble those of koffka, who collapsed mathematical and physical space into absolute space and referred to lewin’s psychological space as relative space. in greek philosophy plato and aristotle explored the notion of space in terms of its physical characteristics and boundaries. plato viewed space as a receptacle or vessel for objects (caygill, 1995), whereas aristotle suggested that space is “the boundary of the containing body at which it is in contact with the contained” (aristotle cited in mckeon, 1941:31). one of the most influential geometers who speculated with mathematical space, euclid, acquired his early education in plato’s academy (hollingdale, 1994) and was taught by geometers from that school of thought (euclid, 1956). in his work, the elements, regarded as one of the most influential texts of mathematics (hollingdale, 1994), euclid employed a precise, innovative, rigorous and logical methodology (using assumptions, postulates and theorems) to describe and prove geometrical concepts (mlodinow, 2001), which at the time informed the frames of reference for spatial understanding. descartes, in his quest to describe the universe in terms of definable co-ordinates or positions, leaned towards the platonic position, with the identification of space with “extension in length, breadth, and depth” (descartes cited in caygill, 1995:368). building on the work of galileo, descartes was one of the first advocates of geometrical physics. sorrell (1987) suggests that descartes’ physics was constructed out of mathematical facts about material things, from facts about size, shape, composition and speed. descartes claimed that “he was in the habit of turning all problems into geometry” (davis and hersch, 1986). descartes asserted that mathematics (geometry in particular), which was based on direct observation, is fundamental to understanding the universe. the result of this reductionist approach was a very heavily structured and grid-like outlook on space. indeed descartes is credited as the founder of the cartesian plane, the foundation for coordinate geometry, a system which facilitates the solving of algebraic problems through applying geometric principles. in his book la geometrie he suggests that “any problem in geometry can easily be reduced to such terms that a knowledge of the lengths of certain straight lines is sufficient for its construction” (descartes, 1925:2) newton distinguished between absolute space and relative space. in his view, absolute space is “without relation to anything external, remains always similar and immovable” (newton cited in caygill, 1995:368) – it is the space of god. relative space, on the other hand, “is some movable dimension or measure of absolute spaces; which our senses determine by its position to bodies; and which is commonly taken for immovable space” (newton cited in caygill (1995:368). according to gardner (1999), newton’s view is of space as an absolutely real, self-subsistent ‘container’ which would exist even if no physical objects were contained within it. leibniz refutes both descartes’ and newton’s views that space is in some sense substantial. he argues that space is relative (caygill, 1995). leibniz suggests that the universe is made up of monads, which have no shape or size. a monad is a point, not a mathematical or a physical point but a metaphysically existent point. whereas descartes is arguing for a rational and material space, leibniz clearly proposes a space consisting of non-corporeal forms. leibniz, however, says that there must be some relation between all the monads which make up the universe, some explanation for their orderly actions which leibniz refers to as pre-established harmony (stumpf, 1994). leibnizian view is of space as a logical construction out of relations between objects (gardner, 1999). it can be conceived that in newton’s absolutist model of space, “the universe could shift its position in space and could have been created at a different time from that at which it actually came into existence” (gardner, 1999:71). leibniz’s relational view of space, however, “grants the plain possibility of empty space and empty time”. locke avoids the material/non-material debate by arguing that space is a simple concept based entirely on our senses of sight and touch. space therefore is “a simple idea which is modified into measures of distance and into figures” (caygill, 1995:368). kant’s view on space changed and evolved over time. he initially identified with the leibnizian definition of space as “the objective relation of 11 conceptualisation of space in mathematics education substances” (caygill, 1995:369). with the emergence of a revised definition of metaphysics, which was no longer equated with the science of substantial forces but was identified as the science of the limits of human cognition, kant turned to a more subjective understanding of space. he asserted that space is an intuition (gardner, 1999). more specifically, kant speaks of space as a priori intuition which means that it is not derived from experience. this is to say that a perception of space is pure and cannot be derived from either sensibility or understanding. “space cannot be derived from outer experience” (caygill, 1995:372). personal space: personal space refers to “an area with invisible boundaries surrounding a person’s body into which intruders may not come” (sommer, 1969:26). it is not spherical in shape, according to sommer (1969), as some people are able to tolerate closer presence of strangers at their sides than directly in front. further, the extent of this space changes from individual to individual as some people like to be close enough for warmth and comradeship whereas others like to be far enough to avoid space invasion. the invasion of personal space is an intrusion into a person’s self-boundaries. these may be physical or they may be psychological. i may be made uncomfortable by an individual’s physical closeness; or i may feel uneasy at an individual’s intrusion into my relationship space with another person. when exploring issues of space with my sample the notion of personal space surfaced on numerous occasions. space in the context of this research: one of the central aims of this research was to construct individual world-view profiles of space of the participating pupils. as i sought as wide a perception of space as possible i consciously tried not to let the above definitions and understandings dictate my interview approach. i therefore steered clear of providing a single dogmatic definition of space. as it happened, it was fascinating to experience how many of the above ideas emerged throughout the conversations. many of the subjects took a platonic stance and saw space as something empty into which objects could be placed. others identified with the personal dimension of space and articulated how they treasured their own space bubble. others again adopted a more kantian perspective and equated space with their feelings – as something that feels good and beautiful. many saw space merely in its physical and planetary (cosmic) form while others took the newtonian and cartesian stance and saw space as an ordered, absolute and definable system. the different ideas and positions on space were not mutually exclusive – there was a lot of overlap and diffusion of different ideas. an interpretive methodology the fundamental research question that this research aimed to answer was whether an understanding of a world-view-theory approach contributed towards the exploration of an individual’s spatial conceptualisation. to answer the question the study pursued and was structured around the following objectives: 1. to explore spatial capacity (spatial visualization and orientation skills) using both pen-and-paper test items and hands-on activities; 2. to explore spatial conceptualization by engaging in world-view profiles of space using a logicostructural approach; 3. to investigate consistencies and relationships between spatial capacity, world views and aspects of mathematics performance of grade 11 learners; and 4. to show that spatial conceptualisation is a rich and complex blend of spatial capacity and world view. this paper only reports on the issue of worldview profiles. the study was grounded in an interpretivistnaturalistic paradigm as it was concerned with the individual and was aimed at understanding the subjective world of the individual (schwandt, 1994; lincoln and guba, 1985; denzin and lincoln, 1994; cohen and manion, 1994; cohen, manion and morisson, 2000), and made use of multiple sources and types of data (le compte, millroy and preissle, 1993) to ensure credibility (lincoln and guba, 1985). this research comprised a multi-sited case study (stake, 2000) involving a group of 32 grade 11 learners from five different secondary schools in the eastern cape, a province of south africa, from a diverse range of cultural and socio-economic backgrounds. several techniques for data collection were employed. they included the use of questionnaires, administering a pen-and-paper and a hands-on activity test, and engaging in conversations. the interview was used in this research as the dominant technique to tease out the participants’ own view of space. as i felt that the concept of space could be very abstract, complex and possibly inaccessible for many of them and consequently a very difficult topic for conversation, i modelled my 12 marc schäfer strategy on cobern’s (1993) and slay’s (2000) world-view research in which they made use of a framework of themes around which conversations and interviews could be structured. i was reluctant to over-structure my interviews and so stifle spontaneous conversation and insight. nevertheless, i thought that some enabling framework which remained consistent for all the 32 interviews would facilitate the eventual analysis more effectively than having to analyse 32 totally open-ended conversations. in retrospect this proved to be an important aspect of the methodological design, serving as a frame of reference for analysing the conversations and building meaningful concept maps. throughout the interviews and conversations i consciously encouraged the participants to talk as freely and generally as they could. i avoided coercing them into a discursive domain that could be described specifically as “scientific” or “mathematical.” the framework originally devised by cobern (1993:935) relied on a set of “bipolar descriptive codes”. these bipolar codes (also sometimes called vector pairs), representing the opposing ends of a continuum, were used to explore students’ conceptualisation of and beliefs about nature. the set of bipolar codes, also referred to as vector pairs, that cobern used were: naturalism and religion, chaos and order, mystery and knowledge, function and purpose, mundane and special, and science and no science. in addition to a few extra vector pairs, i made use of cobern’s (1993) model above (see table 1), and for each of the pairs i devised some guiding questions that would frame the structure of the interviews. naturalism – religion chaos – order mystery – knowledge function – purpose mundane – special mathematical – non-mathematical (cartesian – non-cartesian) finite – infinite tangible – non-tangible internal – external table 1: framework for interviewing using bipolar codes. citing jones, cobern (1993:939) suggests that “bipolar coding involves selecting two related codes that together distinguish a range of beliefs with respect to one presupposition”. for example, as described below, people could for example believe space to be orderly, chaotic, or something in between the two extremes. the nine descriptive pairs of bipolar codes as used in this study, are discussed below. naturalism and religion naturalism, as used in this study, refers to the belief that “material or physical causation provides the basis” (cobern, 1993:939) for understanding space. it fundamentally rules out theistic involvement in space. religion, however, is the opposite – it asserts the involvement of the supernatural in space. chaos and order chaos implies that space is changeable, random and unpredictable, whereas order implies the opposite – space is predictable, based on rules and principles. mystery and knowledge this vector pair describes “the extent to which one believes [space] to be fathomable” (cobern, 1993:942). those who find space mysterious are clearly more impressed with what is not known than what is, whereas those who find space knowable have significant material understanding of space or events in space (cobern, 1993). function and purpose a function explanation of space refers to a “teleonmic or a structure/function explanation of space” whereas purpose explanations refer to “transcendent purpose or cosmic teleology” (cobern, 1993:944). mundane and special this vector pair seeks to deconstruct whether space is perceived as something “beyond the ordinary” (cobern 1993:946) or something mundane and prosaic. mathematical and non-mathematical as this study is situated in a mathematical context there was an interest in the extent to which mathematics informed a participant’s view of space. finite and infinite this vector pair seeks to describe the extent to which participants view space as an infinite concept (infinitely large or infinitely small) or whether they see space in terms of defined parameters. tangible and non-tangible a ‘tangible’ explanation of space is one where space is perceived as something that one can touch and 13 conceptualisation of space in mathematics education see, whereas a ‘non-tangible’ perception of space is characterised by mystery and abstractions. internal and external this vector pair seeks to explore the extent to which space is perceived as a phenomenon “out there” or something which is internalised and part of the self. the interview transcripts then formed the basis for the construction of world-view profiles in the form of concept maps. figure 1 provides an example of one participant’s concept map. the transcripts and concept maps were reviewed and edited by the participants (member-checked) before they were finalised and analysed. the bi-polar codes assisted in clustering themes. the concept maps were effective and useful in establishing meaningful overviews of the individual world views of the participants. they gave an interesting glimpse into the participants’ perception of space and illustrated the richness and complexities of their perceptions. the world-view profiles provided fascinating insight into what the participants thought of space. as, however, the study aimed to provide as comprehensive a picture as possible of the sample’s conceptualisation of space, i considered it necessary to also establish some sense of how the participants thought of space. i therefore space was created by god who is invincible i think figure 1: a concept map of the conversation of one participant of endless green grass and then my small room at the hostel, freedom … created the world into that space he just breathed wonderful space…he created space for everything it’s not the same everywhere… you can like different spaces everyone else’s space was also created by him you can feel him around his space there are no predetermined rules…space could be like a cloud or space could still be in the clouds space does not have a beginning…it’s big and diverseyet it is not chaotic, and not unorderly it carries millions of oxygen atoms and hydrogen in the air. it is so important space is beautiful because it belongs to me it is mysterious because it always remains hidden from everything i also find it mysterious that with all the technology, so many things remain unexplained it has a purpose – to change the world and for us to enjoy people in prison, bad people, don’t have space because they invaded other people’s space – they violated that space sometimes i could have more space than i do…i want to grow vertically and not horizontally everything rests on it earth is insignificant to all of space…it’s just floating around but you cannot see space – but you can see things in space and there’s a lot of space in my mind – one of the biggest spaces you cannot touch space, but space touches you 14 marc schäfer investigated beyond the content level and explored some of the higher-order thinking processes that the participants were involved in and engaged in a metalevel analysis. the extent to which the participants’ perception of space related to their spatial conceptualisation in terms of spatial capacity was, however, nigh impossible to establish. a deeper level of analysis was therefore required. the participants’ dialogue therefore evolved into a vehicle through which a meta-analysis of higherorder thinking was conducted. the world-view profiles of space thus emerged as important tools for exploring meta-level thinking. a process of meta-analysis through the engagement of a consensual validation team consisting of three expert researchers and i, a methodological model was developed which facilitated a process of meta-analysis. the concept of validation was grounded in the notion that “validation is not just seen as part of a final product control process or verification, but rather a continuous process of credibility, growth and understanding” (schäfer, 1999). through a process of discussion and seeking consensus amongst the team, five meta-themes (criteria) that characterised the participants’ thinking skills in this study were identified. they were: • the capacity to abstract (abstraction); • the capacity of insight (insightful); • the capacity to deal with complex concepts and issues (complexity); • the capacity to critically engage (critical); • the capacity to be imaginative (imagination). for detailed definitions and explications of the above criteria refer to schäfer (2003). it was felt inappropriate to attach a quantity for each participant’s “performance” in terms of the criteria. instead, each participant was positioned on a continuum in each of the meta-criteria. thus the notion of a “meta-profile” developed (schäfer, 2003). the idea of a “meta-star” developed whereby each arm of the star represented each participant’s emphasis on the five meta-criteria. for example, a longer “abstraction” arm than “critical” arm would suggest a better capacity to abstract than to critically engage. figure 2 shows an example of one of the participants’ meta-stars. the respective lengths of the arms of each metastar were arrived at through consensus amongst the members of the validation team. this process involved numerous rounds of analyses and discussion. the collection of meta-stars then formed the basis for a meta-narrative which complemented and supplemented the content-narrative based on the various world-view profiles. the meta-stars could also be grouped into numerous clusters such as schools and gender for interesting comparisons and insight. it needs to be emphasised that the metastars describe individual meta-profiles relative to the themes and criteria of this study. figure 2: the meta-star of one of the participants abstraction imagination criticalinsightful complexity conclusion the general world view of space of the sample was characterised by rich, complex and multifaceted ideas. space was seen as: • having a strong religious element; • relatively orderly, peaceful and underpinned by design; • somewhat obscure and mysterious; • special and beautiful; • something which has direction; • making us insignificant and small; • having no beginning and no end; • something which we cannot touch, yet feel; • something that is visible in the sense that we can see that there is nothing there. the overall perception of space of the sample leant towards a newtonian division of absolute and relative space. in terms of magnitude it regarded space as infinite thereby refuting the platonic position which asserts that space has length, depth and breadth. in terms of the cartesian reductionist and grid-like outlook on space, this sample preferred to view space as mysterious, infinite and somewhat 15 conceptualisation of space in mathematics education obscure. it often referred to space in kantian terms and related to space in terms of subjective feelings. on a meta-level it was found that generally the sample engaged at a relatively low level of critical engagement. interestingly, the girls were rated at a higher level than their male counterparts for their capacity to abstract and be complex, whereas the males rated higher for their capacity to show insight, be imaginative and be critical. kearney’s (1984) logico-structuralist model is not without its problems. although the framework of bipolar codes was meant to enable the establishment of world-view profiles, it can be argued that the framework was too rigid, prescriptive and linear in its approach and did not allow for deviation and flexibility. it can also be argued that the bipolar codes in themselves were too limiting and restrictive. for example, in the first bipolar code, the assumption was made that the antithesis of naturalism is religion. although the notion of a bipolar continuum provided for compromise and a softening of mutual exclusivity, it did not challenge the fundamental assumption of bipolarity. the same can be said for the other codes. by the same token it can be reasoned that the strength of the logico-structuralist approach lies in its structure and rigour. world view is a complex concept to describe and explore, and kearney’s model provided a workable framework and point of reference for researching world views. in terms of a more global and comprehensive world view, it could be argued that this study focused too narrowly on the space universal at the expense of the other six universals. this study was particularly concerned with the presuppositions of space in conjunction with spatial capacity, and hence focused only on the space universal. it is thus acknowledged that the world-view claims made in this study pertain only to the understanding of space and consequently reflects only a small aspect of a more global world view. the relevance of this study to mathematics education needs to be seen in the context of recognising prior knowledge, beliefs, presuppositions and experiences. the notion of a world view is fundamental to the unravelling of prior beliefs and it is hoped that the qualitative methodological model described in this paper will contribute to further world view studies. references battista, m.t., talsma, g. and wheatley, g.h., 1982, “the importance of spatial visualization and cognitive development for geometry learning in pre-service elementary teachers” in journal for research in mathematics education, 13(5), pp. 332 – 340 bishop, a.j., 1980, “spatial abilities and mathematics education: a review” in educational studies in mathematics, 11, pp. 257 – 269 caygill, h., 1995, a kant dictionary, 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(eds.), image and environment, pp.1 – 7, london: edward arnold 16 marc schäfer euclid, 1956, the thirteen books of the elements: volume i, new york: dover publications. funk, k., 2001, what is a worldview?, at http://engr.oregonstate.edu/~funkk/personal/worl dview.html gardner, s., 1999, kant and the critique of pure reason, london: routledge. hollingdale, s., 1994, makers of mathematics, london: penguin books kearney, m., 1984, world view, california: chandler & sharp publishers koffka, k., 1935, principles of gestalt psychology, new york: harcourt-brace le compte, m.d., millroy, w.l. and preissle, j., 1992, the handbook of qualitative research in education, london: academic press lincoln, y.s. and guba, e.g., 1985, naturalistic inquiry, london: sage mckeon, r., 1941, the basic works of aristotle, new york: random house mlodinow, l., 2001, euclid’s window: the story of geometry from parallel lines to hyperspace, london: allen lane, the penguin press nctm, 1989, curriculum and evaluation standards for school mathematics, reston, va.: the council rickman, h.p., 1979, wilhelm dilthey: pioneer of the human studies, london: paul elek rickman, h.p., 1976, dilthey: selected writings, cambridge: cambridge university press schäfer, m., 1999, designing and making a difference: an exploration of technology education for rural school teachers, unpublished master’s thesis, rhodes university, grahamstown, south africa schäfer, m., 2003, the impact of learners’ spatial capacity and world view on their spatial conceptualization: a case study, unpublished doctoral thesis, curtin university of technology, perth, australia also available at http://adt.curtin.edu.au/theses/browse/by_author/ s.html schwandt, t.a., 1994, “constructivist, interpretivist approaches to human enquiry” in denzin, n.k. and lincoln, y.s., eds., handbook of qualitative research (2nd ed.), pp. 118 – 137, london: sage publications sommer, r., 1969, personal space: the behavioural basis of design, englewood cliffs, nj.: prentice-hall sorrel, t., 1987, descartes: a very short introduction,. oxford: oxford university press stake, r.e., 2000, “case studies” in denzin, n.k. and lincoln, y.s. (eds.), handbook of qualitative research (2nd ed.), pp. 435 – 454, london: sage publications slay, j., 2000, culture and conceptualization of nature: an interpretive analysis of australian and chinese perspectives, unpublished doctoral thesis, curtin university of technology, perth, western australia south africa (republic), 1997, senior phase (grades 7 to 9). policy document, pretoria: government printer stumpf, s.e., 1994, philosophy: history and problem (5th ed.), new york: mcgraw-hill want, c. and klimowski, a., 1996, introducing kant, cambridge: icon books 17 (an abridged version of this paper was presented at the saar introduction the theoretical framework spatial conceptualization world view world view in the context of this research space absolute and relative space: among theorists in gestalt psyc mathematical space, physical space and psychological space: personal space: personal space refers to “an area with invis space in the context of this research: one of the central ai an interpretive methodology naturalism and religion chaos and order mystery and knowledge mundane and special a process of meta-analysis it was felt inappropriate to attach a quantity for each part conclusion references 63 p22-29 mbekwa layout final 22 pythagoras 63, june 2006, pp. 22-29 teachers’ views on mathematical literacy and on their experiences as students of the course monde mbekwa school of science and mathematics education, university of the western cape mmbekwa@uwc.ac.za this paper reports on a study undertaken at the university of the western cape with a class of 32 inservice teachers who had completed six months of an advanced certificate in education (ace) course in mathematical literacy in 2004. the teachers completed an evaluation questionnaire, which asked them about their common sense conception of mathematical literacy and their reflections on their student experiences in the first six months of the course at uwc. the study has found that some of these students’ understanding of mathematical literacy corresponds to contradictory conceptions of mathematical literacy, which appear in the literature. some expected that the course would be a watered-down version or easy mathematics whilst some viewed it as mathematics with applications in everyday life. whilst some of the teachers perceived the course content to be “difficult”, it is important to note that almost all the teachers, except those who dropped out, have completed and passed the course, which indicates that it was a worthwhile endeavour. introduction i am a member of a three-person team that taught an advanced certificate in education (ace) course in mathematical literacy in 2004 at the university of the western cape, south africa. participants in the course were 32 in-service teachers out of an initial cohort of 140 course participants at four higher education institutions in the western cape. the 140 teachers were selected by the western cape education department and registered, in equal groups of 35, at the four higher education institutions in the western cape, namely the universities of the western cape (uwc), cape town (uct), stellenbosch (us) and the newly established cape peninsula university of technology (cput). the last mentioned institution was born out of the amalgamation of the two former technikons in the western cape, the peninsula technikon (pentech) and the cape technikon (cape tech). these four universities had tendered as a consortium and won the right to offer the advanced certificate in education in mathematical literacy to in-service teachers from western cape schools. the major aim of the ace in mathematical literacy is to prepare teachers to teach the new subject “mathematical literacy”, introduced in the further education and training (fet) phase (grades 10 to 12) in all south african secondary schools in 2006. the course commenced in june 2004 and was scheduled to be completed in 2006, the year in which the new subject, mathematical literacy, was going to be introduced into south african high schools. the fact that only 140 teachers in the western cape started the course, and that it was scheduled for completion in the year in which mathematical literacy was planned to be introduced, eloquently illustrates the lack of readiness of the department of education to implement its own policy innovations. the national department of education declared that, as from 2006, mathematical literacy would be a compulsory subject for all those learners in the fet band who do not take mathematics as a subject. in anticipation of this, the education and training of teachers in mathematical literacy would be an imperative, considering the vast numbers of learners who are not doing mathematics as a subject at school. clearly, it makes sense to train and educate teachers who are not teaching mathematics, in addition to teachers of mathematics, so as to have an adequate number of teachers to teach mathematical literacy to the huge numbers of learners who would not be taking mathematics in 2006. it is illuminative of the enormity of the task at hand to note the report of the national minister of education, naledi pandor (2004), on the final results of the senior certificate at the end of 2004. she reported that, of the 471 080 learners who wrote the grade 12 examinations in 2004, 283 744 or 57.2% wrote mathematics. this implies that 42,8% did not write mathematics, which is close to the number of those who wrote mathematics. monde mbekwa 23 one can imagine what the situation would be like if these figures are extended to all learners who would not be doing mathematics in the fet phase, and consequently, would have to do mathematical literacy. it stands to reason that it will be beyond the capacity of the present core of qualified mathematics teachers to cope with the total population of learners at school. hence the recruitment of tens of thousands more teachers would be imperative to cope with this new demand. most of these teachers, who need training, will have to be recruited from nonmathematical disciplines because it would be impossible for the existing mathematics teachers to teach both mathematics and mathematical literacy. it might even be necessary to recruit from beyond the borders of south africa. it is in the context of this new reality and the need to ensure the availability of teachers who are qualified to teach the new subject that the national department of education is embarking on this retraining exercise. the introduction of this new subject in the south african schooling system forms part of the momentum of curriculum reform propelled by the introduction of c2005 in the 1990s and its subsequent revision through the revised curriculum statement in the 2000s. because most of the teachers who will be required to teach mathematical literacy are not practising mathematics teachers, they will need both subject content knowledge and pedagogical content knowledge. it is obvious that these teachers will also need to have an understanding of mathematical concepts, and knowledge of mathematical algorithms and problem solving skills. the challenge for teacher educators would be to ensure that teachers develop these skills and knowledge. as adler (2003: 4) has put it: there is little contention that teachers need to know the subject matter they are teaching, and moreover, that they need to know how to present this clearly to learners. the nature of the course at uwc in 2004 the university of the western cape (uwc) registered 35 students on 20 june 2004, to commence studies at the beginning of the second semester in july 2004. it was planned for these students to complete the course in four semesters, implying that the first group would finish in june of 2006 and thus graduate in the uwc case in september 2006. there were 35 students who registered in june 2004 but by the end of the year three had left the course. hence at the commencement of examinations in november, 33 had finished the semester, having been joined by one student who registered late. the curriculum designed at uwc for the certificate comprises the following components: mathematical literacy for teaching, mathematics education, computer studies and hiv/aids education and counselling. the computer studies and hiv/aids modules were added to all ace courses offered by uwc so that students could complete the ten modules required to qualify for the ace course. the first component of the course, mathematical literacy for teaching, comprises four modules and the other components, two modules each to make up a total of ten modules. the decision taken by the lecturers, especially for the mathematical literacy for teaching component, was to deal with one learning outcome of the national curriculum statement per semester. the learning outcomes as indicated in the mathematical literacy curriculum, are the following: (1) number and operations in context (2) functional relationships (3) space, shape and measurement and (4) data handling. the focus of this paper is on the uwc module mathematical literacy for teaching 113, which deals with the first learning outcome of mathematical literacy, namely number and operations in context as stated in the national curriculum statement. the national curriculum statement for mathematical literacy explains the focus of this outcome as follows: the focus of this learning outcome is on the investigation and solution of problems that require a sound understanding of numbers and their use in calculations, especially in financial contexts, ranging from personal to international issues…. learners should develop sound estimation and mental calculation skills and a facility in using equivalent forms to simplify calculations. proper conceptual understanding will be required to enable learners to use calculators appropriately and effectively. (doe, 2003: 11) the curriculum designed at uwc for this learning outcome includes the following topics: integers; fractions; ratio, rate and proportion; formula construction and substitution in formulae; exponents, scientific notation and compound interest. after the completion of the module, i wanted to gauge the students’ common sense teachers’ views on mathematical literacy and on their experiences as students of the course 24 understanding of mathematical literacy, and also to find out how they experienced the course. the research question the main aim of this study was to gauge students’ common sense understanding of the notion of mathematical literacy, which they are expected to teach at the end of their study, and their motivations and perceptions about the course. this study aimed at responding to the following research questions: • what are teachers’ motivations for registering for the ace in mathematical literacy? • what is the teachers’ common sense understanding of mathematical literacy? • what are the teachers’ views of their experiences as students of mathematical literacy? • how do the teachers evaluate the course? literature review the south african national department of education in its national curriculum statement (2003: 10) conceives mathematical literacy as being: … driven by life-related applications of mathematics. it enables learners to develop the ability and confidence to think numerically and spatially in order to interpret and critically analyse everyday situations and to solve problems. the international programme committee for icmi study 14 (2002) condenses the understanding of mathematical literacy cited above when it states that mathematical literacy belongs to the domain of “applications and modelling of mathematics”. this means that mathematical literacy implies modelling, which refers to the translation of reality to mathematical models and “application” implies the use of mathematical models to solve real life problems. one can observe the dual mathematical implication “modelling” ⇒ moving from reality→ to mathematics and “applications”⇒ moving from mathematics→ to reality. in articulating the purpose of mathematical literacy, the ncs states that mathematical literacy “will develop the use of basic mathematical skills in critically analysing situations and critically solving everyday problems.” (doe, 2003: 10). this understanding of mathematical literacy characterises it as having an important application or functional use in real life. hence it is not viewed as being mathematics for its own sake but as mathematics having relevance in contextual problem solving (romberg, 2001; oecd, 2001). this is also what jablonka (2003) refers to as mathematical literacy for developing human capital. whilst there is a paucity of literature on mathematical literacy and also little consensus on the definition of mathematical literacy, all definitions of mathematical literacy provided in the literature highlight the functional dimension of mathematics by focusing on its application in the lives of citizens. in addition to the idea of mathematical literacy having a critical application dimension, it is also seen as having an important role in encouraging people to engage in and understand mathematics, implying that mathematics does not only have an aesthetic value but also a use value. in this sense then, one can view mathematical literacy as mathematics with concrete and practical value in day-to-day existence. it can also be understood as literacy in the same way as literacy in a language, which is indispensable in a world in which language is a tool for sense-making and social intercourse. in this sense, a mathematically literate person is one who understands the language of mathematics, with its nuances and its applications, as a communication tool. the view of mathematical literacy as mathematics with a utility value is borne out by pisa (the programme for international student assessment), which views mathematical literacy as: …the capacity to identify, understand and engage in mathematics, and to make well founded judgements about the role that mathematics plays in an individual’s current and future private life, occupational life, social life with peers and relatives, and life as a constructive, concerned and reflective citizen. (oecd, 2001: 22) hence pisa proposes that mathematical literacy should be seen not only as a tool for solving life problems but also as a tool to understand a mathematised world. lastly, the mathematical council of the alberta teachers’ association views mathematical literacy as: • connecting mathematics to the real world • using mathematics appropriately in a variety of contexts monde mbekwa 25 • communicating using the richness of mathematics • synthesising, analysing, and evaluating the mathematical thinking of others • appreciating the utility and elegance of mathematics • understanding and being conscious of what has been learned mathematically (mcata, 2002: 2) the above conception of mathematical literacy emphasises both its contextual connectedness and its utility value as an instrument to appreciate the elegance of mathematics. methodology the sample and data collection at the beginning of the 2005 academic year, 32 questionnaires were issued to the teachers who had registered for the ace course in mathematical literacy in 2004. students took the questionnaires away to complete at home. of these, 20 questionnaires were completed and returned. eight questions appear in the questionnaire ranging from why the students applied to do the course, what their understanding of mathematical teacher teaching experience in years subjects taught in 2004 grades taught highest qualification in maths 1 9 technology/science 8, 9 matric 2 15 technology/needlework 11, 12 matric 3 12 maths/life orientation 8, 10 de iii1 4 12 maths/history 8 de iii 5 18 economics/accounting 9,10,11,12 matric2 6 21 biology 11, 12 ptd iii3 7 18 technical subjects 10,11,12 n44 8 16 ems 8, 9 9 15 geography 11 12 10 8 metal work/technical drawing 8, 9, 10, 11, 12 math 1 11 14 accounting 10, 11 grade 9 12 14 ems/economics/bus. econ. 8, 10, 11, 12 matric 13 15 needle work/technology/travel & tourism 8, 10, 11 & 12 college yr 1. 14 11 geography/english 9, 10 matric 15 15 history/arts& culture 9, 12 matric 16 12 maths/science 9 hde iv5 17 17 maths/science 8 hde iv 18 20 maths 9, 10 diploma in education 19 19 accounting/ems 8, 9, 10, 11, 12 math 1 20 15 afrikaans 9, 10, 11, 12 matric table 1: profiles of questionnaire respondents 1 de 111 is a teaching qualification: a three-year post grade 12 teaching qualification. 2 matric is grade 12. 3 ptd is a primary teachers’ diploma: a three-year post grade 12 teaching qualification. 4 n4 is a one-year post grade 12 qualification obtained at a technical college. 5 hde is a four-year post grade 12 teaching qualification formerly obtained at a university. 26 literacy was, their expectations of the course, and experiences during mathematical literacy classes to what their attitude was towards the examination and their course assignments. the profiles of the teachers who responded to the questionnaire appear in table 1 on the preceding page. from this data, one can see that these are experienced teachers with the lowest experience of teaching being eight years. one can also see that of the six (30%) teachers who have a teaching qualification in mathematics, there are five teachers (25%) who taught mathematics in 2004. there were 17 teachers with matric or grade12 mathematics, a percentage of more that 85%. of these, two had a first year university mathematics qualification. only one teacher indicated that he or she had grade nine as the highest qualification in mathematics. from this data, one could then expect that more than 95% of the teachers, very close to the total sample, had a reasonable familiarity with high school mathematical concepts. data analysis and findings i analysed all the responses to the questions and clustered the teachers’ responses according to the questions they responded to, and then identified trends in their responses. here, i report on the teachers’ responses to questions relating to reasons for registering for the course, the teachers’ common sense understanding of mathematical literacy, their understanding of the course content and their attitude towards examinations and the formative assessment tasks. i am reporting on responses to these questions because they are closely tied to the research questions. motivations for applying on the question why students chose to apply for the course, seven out of the 20 students (i.e. 35%) saw this as an opportunity to be able to teach mathematical literacy. they answered that they joined the course: “to teach mathematical literacy.” four (20%) saw this as an opportunity to teach mathematics, e.g. “ek wil graag meer betrokke raak by die gee van wiskunde.” [i want to be more involved in the teaching of mathematics.] some (25%) saw it as an opportunity to improve themselves or to broaden their mathematical knowledge, e.g “i had to retrain myself in a new field. mathematics literacy seemed to be a fresh difference to what i normally do.” the rest of the students (15%) saw this as an opportunity to secure their jobs, e.g. “there was an opportunity to study. to secure my job in the mathematics department.” common sense understanding of mathematical literacy responses to the question on the teachers’ common sense conception of mathematical literacy can be categorised into two areas: (1) the “functionalist” view, which regards mathematical literacy as that type of mathematics that finds application in people’s lives. as some teachers put it: “mathematics is part of every person’s life. people must handle their own finances and do calculations. when you do that you are mathematically literate.” about 30% of the respondents have this view of mathematical literacy, which coincides with the view that mathematical literacy has to do with contextual problems in people’s daily lives. (2) another view, which teachers hold, is that mathematical literacy is a simplified or an easier version of the mathematics that learners do at school. as they put it: “basic knowledge of mathematics” or “an easy version of mathematics. just to help learners in the real world.” the latter response is a combination of what i regard as a functional understanding of mathematical literacy and the alternative understanding that mathematical literacy is no different from school-going mathematics except that it is easier than mathematics. i regard this conception of mathematical literacy as a lower level conception of mathematical literacy. many mathematics educators agree that mathematical literacy is not a watered down version of mathematics and that it demands the same rigour that is exercised in the pursuit of an understanding of formal mathematics. madison (2005) underscores the seriousness with which mathematical literacy (which he refers to as quantitative literacy) ought to be dealt when he states: there can be no doubt that ql (quantitative literacy) is difficult… the difficulty of ql, however, is rooted in its sophisticated uses of elementary mathematics and their immersion in extraneous, varied and possibly confusing terminology. using mathematics in multiple and unpredictable contexts requires both an monde mbekwa 27 understanding of mathematical concepts and practice at retrieving and applying them. often contexts are replete with the language of science, statistics, economics or engineering. relevant information may be ambiguous or hidden. sorting all this out, modelling with mathematics or statistics, doing the mathematics and interpreting the results is challenging indeed. (madison, 2005: np) brombacher (2005) undergirds the view that mathematical literacy is not easy mathematics when he writes in the cape times: what i really worry about is the possibility that mathematical literacy may be interpreted as the new mathematics standard grade. let me be emphatic about this. it is not! mathematical literacy is a different kind of mathematics, not a different, lower level of mathematics. mathematical literacy will be at least as demanding as mathematics to teach and certainly as challenging for pupils to learn. understanding of course content the other questions in the questionnaire revolved around students’ understanding of the course content and their experience of learning mathematics literacy. teachers who were positive about their classroom experience and understanding of the course constituted 52% of respondents, while 48% did not understand the course and found it difficult, e.g. “dit is moeilik vir ons wat nie wiskunde verstaan nie” [it is difficult for us who do not understand mathematics] or “mathematical literacy for teaching is a bit more difficult to understand.” another example of the teachers’ dissatisfaction with the course, which i found interesting and contradictory, is that one teacher felt uncomfortable with the fact that the course was not “school based”. he or she states, seemingly after consulting those teachers who teach mathematics: “die wiskunde wat ons hier doen, sê die wiskunde opvoeders is nie skool gebaseerde nie. hulle sê dit is baie moeilik.” [the mathematics teachers say that the mathematics that we do here is not school based. it is very difficult.] an example of a response i regard as positive is the following: “relevant. besides the fact that after a long day of seven periods. teaching. loaded administration duties, 54km of travelling, the classes are enrichfull [sic] and enjoyable.” i also regard the following as a positive response: “ek geniet die klasse en leer elke keer iets nuut. sommige van die werk is moeilik, maar met oefeninge kom ek by. wiskunde word gebied deur metodes wat ek nog nie gesien en dis alles baie nuut vir my.” [i enjoy the classes and learn something new each time. part of the work is difficult but with exercises i manage. mathematics is presented through methods which i have not seen before and everything is very new to me.] evaluation of the course one question related to the students’ attitude towards the assignments and the final examination. all students were positive about the assignments because the assignments “give(s) you more time to reflect on what you were doing in class”. six of the teachers (30%) were concerned that they did not have the time to do the assignments because of the pressure of being full-time teachers. on the examination paper only four students (20%) felt that the paper was fair and not difficult. the rest felt that the paper was difficult and that the lecturers had misled them with regard to the scope of the examination paper that they were given. one student put it this way: “standard of paper was too high – reminded me of old regime where papers were set to ‘catch you out’– very stressful & too little time. we were misled with the scope.” whilst generally students felt that the examination itself was difficult, they did not have the view that the course was difficult as evidenced by nearly 50% who saw the course as fair and enjoyable. in the examination, there was one little misunderstanding about the time allocated for the examination paper but all students passed the final examination paper, after only five students who did not obtain the requisite pass mark qualified for re-examination. reflections on the findings reasons for registering for the course from the teachers’ responses to the question why they registered for the course on mathematical literacy, i identified two types of reasons, namely, pedagogical and strategic reasons. pedagogical reasons are reasons given by the teachers that have to do with their view of their teaching responsibilities and the furtherance of their professional and qualification status in the subject. strategic reasons are reasons that have to do with job security. from reading their responses i deduced that strategic reasons were also disguised as pedagogical reasons. reflecting on the reasons articulated by these teachers on why they chose to do the course, one teachers’ views on mathematical literacy and on their experiences as students of the course 28 finds that some of these teachers had a strategic objective in joining the course, which did not necessarily coincide with the education department’s vision for requesting them to register for the course. they felt that if they obtained a qualification in mathematical literacy, it would make them key teachers in the department and consequently could not easily be declared in excess and hence would not be redeployable. this would achieve the objective articulated so bluntly by 15% of the teachers that they joined the course to secure their jobs in the department. these teachers have a particular understanding of the education department’s policy of rationalisation that certain categories of teachers like mathematics teachers cannot easily be redeployed. more than a third of the teachers saw themselves fulfilling the vision of the department of education, in other words, responding to the needs of the department to address a curriculum development objective. a quarter of the teachers had the view that their engagement with the course would enable them to teach mathematics, in other words, they had the perception that mathematical literacy prepares them to teach mathematics. from this one can deduce that these teachers did not see any distinction between mathematics and mathematical literacy. at present there are debates about the compatibility or incompatibility of mathematics and mathematical literacy (see madison, http://www.aacu.org). madison (2005: np) proposes a solution to this dilemma by suggesting an integration of what he terms “formal mathematics and ql (quantitative literacy) mathematics through more contextual teaching, thereby making mathematics more apparently relevant to contemporary society”. hence these teachers, although they might not be aware of the debates, are spot on in assuming that by being students of mathematical literacy, they would gain enough mathematical knowledge and pedagogical content knowledge to enable them to teach more than mathematical literacy. the challenge is for teacher educators to be aware that, to teach mathematical literacy, one also needs to use mathematical tools and simultaneously acquaint teachers with this knowledge. there is also an advocated position that mathematical literacy and mathematics can be integrated. madison (2005: np) argues for the need for this integration and states: [these] pedagogical changes, admittedly more difficult, include increased extraction of abstractions from examples, better understanding of effective contextual teaching practices and more attention to research results about how people learn. one can infer that the 25% of respondents who state that they registered for the course to teach mathematics form part of those teachers who do not teach mathematics at their schools because it is highly unlikely that those who already teach mathematics would see their participation in the mathematical literacy course as a springboard to teach the mathematics they already teach. these teachers saw their participation in the course as an opportunity to gain enough subject content knowledge in mathematical literacy to make it possible for them to teach mathematics. the implication of this perception is that the content of mathematical literacy is the same as the content of mathematics or, as argued above, it is possible to integrate the two. others decided to satisfy a personal need to improve their qualifications by enrolling for a course when an opportunity presented itself. if one looks at the profile of teachers in terms of their mathematics qualifications, one finds that the vast majority (i.e. 85%) have passed grade 12 mathematics although the majority (75%) do not teach mathematics. one can thus understand that the time period that has passed since they did grade 12 can explain their perception of the course as difficult. nevertheless, there is almost a 50/50 split between those who regard the course as difficult and those who regard it as manageable and hence it is also understandable that an equal number of teachers do not regard the course as difficult. it is informative to note that 30% of these teachers are qualified mathematics teachers even though not all of them teach the subject. the statement that the subject matter of the mathematical literacy class is not school based is strange, coming from teachers who teach under the new curriculum 2005 of the south african national education department, which professes to encourage creativity in teachers as “the designers of the curriculum” (doe, 2000: 13) in its norms and standards document for educators. perhaps, their perception that the course in mathematical literacy is not school based might come from the fact that very few textbooks on mathematical literacy have been written and these teachers have not yet come across those textbooks which have already been written. their comments might also be a consequence of their not being used to the kind of creativity that the lecturers exercised in coming up with contextual examples in the teachers’ class activity and written work. monde mbekwa 29 common sense understanding of mathematical literacy generally what has been gauged from the responses of these teachers is that the majority of the respondents have an understanding of mathematical literacy as an easier version of mathematics. this is understandable because mathematical literacy as a school subject is still in its infancy and mathematics educators still have no clear-cut answers on what mathematical literacy is or what it should be. nevertheless, there is some agreement that mathematical literacy has to do with real life application of mathematics. the question that one can pose is – if a citizen can read his or her municipal account and can decide on the basis of the reading on the water or electricity meter that the municipality has overcharged him or her or that when the weather forecaster indicates that the probability of rain is 30%, that it does not mean that it will drizzle but that there is a small chance of rain given the prevailing weather conditions – does it mean that the citizen is mathematically literate? the teachers’ conception of mathematical literacy then represents spontaneous views not tempered by interrogation of the literature on the subject – a literature which is still seeking answers. coincidentally this conception of mathematical literacy, which these teachers have, concurs with some ideas on mathematical literacy that are found in the literature. experiencing and evaluating the course the issue of the alleged difficulty of the subject is also an issue, which raises more questions than answers because it has not been established what the teachers mean by difficult or what aspects of the course they found difficult. is it the content which they found difficult or is it the activity or process approach whereby they were required to investigate scenarios that would lead them to understand contextually embedded mathematical concepts? conclusion at the time of finalising this paper, 29 of the initial 35 teachers who registered for the course had completed their final assessments for the course and it is highly likely, judging from their previous academic performance, that all of them will graduate with the ace: mathematical literacy qualification in september 2006. from this study one can consider the exercise of taking these teachers through this course over two years and the high retention rate of 83% at one institution as evidence of the success of this project, albeit on a small scale. references adler, j. (2003). global and local challenges of teacher development. in j. adler & y. reed (eds.), challenges of teacher development: an investigation of take-up in south africa. pretoria: van schaik. brombacher, a. (2005, march 1). mathematical literacy vital for life today. cape town: cape times. department of education. (2003). national curriculum statement: grades 10 – 12 (general): mathematical literacy. pretoria: department of education. department of education. (2000). norms and standards for educators. government gazette no. 20844. pretoria: department of education. international programme committee for icmi study 14. (2002). study 14: applications and modelling in mathematics education – discussion document. educational studies in mathematics, 51, 149-171. jablonka, e. (2003). mathematical literacy. in a. bishop, c. clements, j. keitel, j. kilpatrick & f.k.s. leung (eds.), second international handbook of mathematics education (pp 103142). dordrecht: kluwer academic publishers. madison, b. l. two mathematics: ever the twain shall meet? retrieved may 2, 2005, from http://www.aacu.org. mathematical council of the alberta teachers’ association. mathematical literacy…an idea to talk about. retrieved may 2, 2005, from www.mathteachers.ab.ca/mcata. organisation for economic co-operation and development (oecd). (ed). (2001). knowledge and skills for life. first results from the oecd programme for international student assessment (pisa) (2000). paris: oecd. pandor, naledi (2004). the release of senior certificate examination results by the minister of education. retrieved may 2, 2005, from www.education.gov.za. romberg, t. a (2001). mathematical literacy: what does it mean for school mathematics (pp 5-8). wisconsin school news. available at: http://www.wcer.wisc.edu. retrieved may 7, 2005. abstract introduction research methodology results conclusions acknowledgements references about the author(s) michael murray school of mathematics, statistics and computer science, university of kwazulu-natal, south africa citation murray, m. (2017). how does the grade obtained at school for english and mathematics affect the probability of graduation at a university? pythagoras, 38(1), a335. https://doi.org/10.4102/pythagoras.v38i1.335 original research how does the grade obtained at school for english and mathematics affect the probability of graduation at a university? michael murray received: 22 apr. 2016; accepted: 30 may 2017; published: 25 aug. 2017 copyright: © 2017. the author licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract over half of all students enrolling at a particular university in kwazulu-natal fail to complete a degree. this article aims to determine to what extent the marks they obtain for english and mathematics at school impact on their probability of graduation at this university. in addressing this problem, other student specific factors associated with their gender, race and the type of school they attended need also to be properly accounted for. to provide answers for this study, the performance of 24 392 students enrolling at the university over the period 2004 to 2012 was followed until they graduated or dropped out from their studies. a structural equation model was fitted because it allows one to separate a direct effect from that of an indirect effect. gender, race and school background were found to be very significant with men, black africans and students coming from a less privileged school background having a smaller probability associated with eventually graduating from this university. men tend to perform better than women in mathematics, with women performing better men in english. more importantly, however, a single percentage point increase in one’s mark for english increases the probability associated with graduating from this university far more than would be the case if their mathematics mark were to increase by a single percentage point. in light of these mediated results, perhaps this university should be directing their efforts more towards improving the english (rather than mathematical) literacy of students entering the university. introduction prior to 2008 students completing their school-leaving grade 12 examinations in south africa could take subjects at a higher or a standard grade level. in 2008 all subjects were collapsed into a single grade with mathematical literacy being offered as an alternative for students who would have preferred to take mathematics at a standard grade level prior to 2008. entry into a particular programme at a specific university in kwazulu-natal in south africa (hereafter the university) depends on an appropriate level of marks (called a matric point score) being recorded for these school-leaving examinations. this article seeks to determine to what extent the marks that are being recorded for english and mathematics impact on students’ probability of graduation at this university. factors associated specifically with the student such as their gender, race and age and the type of school they attended may also have an impact on the probability associated with graduation from the university. factors that are specific to the university such as their college of enrolment and whether they have been given residence-based accommodation need also to be controlled for when attempting to produce an appropriate model for predicting graduation at the university. by making use of a structural equation based methodology, this article allows each one of the above factors to have a direct effect on the probability associated with graduation; some of these factors (namely race, gender and type of school) also exert an indirect effect on graduation that is being mediated by the marks that students are managing to obtain in english and mathematics for their school-leaving examinations. literature review support for some of the factors that we will be including in our structural equation based prediction model come from a variety of international and locally based studies. naylor and smith (2005) found that gender and school type play a very significant role in determining how students perform at a uk university. tomul and savasci (2012) found that school type was a very strong predictor for determining academic success at a university. surprisingly, family background variables such as parents’ educational status, income and fathers’ occupation were found not to be very strong predictors in their particular study. caldas and bankston (2004) also obtained similar results, noting that the type of dwelling in which the family lives and the number of siblings in the family were also not very strong predictors of student achievement. chiu (2007), in an extensive study covering 41 countries, found that students who lived with two parents, lived without grandparents and lived with fewer siblings (especially older ones) did better in their studies. in a united kingdom based study, thiele, singleton, pope and stanistreet (2016) found that living in a more deprived area adversely affected student performance. race was also found to be an important factor with asian and black students not performing as well as white students. they also found that female students performed better than male ones in the more language-based disciplines, a result which will also be borne out in our study. in the united states, coleman et al. (1996) found that the type of school being attended did not always have a significant effect on the academic success of a student at a university. instead, they found that academic performance was being determined primarily by the living conditions that exist in one’s home and the relationships that one has with one’s family. in south africa, however, apartheid has created a dysfunctional society with people from similar racial and socioeconomic backgrounds being forced to live together in townships and rural settlements that are often very poorly resourced. many of these homes will have absentee fathers who are being forced to seek employment far away from their family’s place of residence. wealthier schools will be able to appoint better qualified teachers. added to this mix is a powerful teacher’s union whose policies attempt to entrench the job security of teachers in the less wealthy schools irrespective of whether they can teach their subjects or not. in a recent study, murray (2016) found this to be an important factor affecting performance at the university. consequently, effects that have been observed in other international studies may not be true given a south african context. bohlmann and pretorius (2008) found that the marks that south african students obtain for important school subjects such as mathematics and english do not necessarily serve as very strong predictors for determining whether a student will be able to succeed in their studies at a university level. spaull (2011) and van der berg (2008) found that south african learners coming from a poor socioeconomic background, when given the opportunity to attend a better resourced school, are often able to significantly improve their mathematical and reading skills. complicating matters, however, is the fact that teaching in most of these township schools is being done in one’s mother tongue with a switchover to english only taking place later on in one’s primary school education. because students entering their high school years, are now being taught important concepts in another language, the lack of meaning that they are now experiencing with the texts that they are having to read is being `overcome’ with a culture of rote learning. bird and welford (1995) have highlighted this problem in south africa, suggesting that this culture of rote learning is one of the main causes for the high failure rates that are being recorded at most of the universities in south africa. studies by feast (2002), howie (2003), van dyk, zybrands, cillie and coetzee (2009) and johnstone and selepeng (2001) have also all found, at both a tertiary and higher education level, that differences between an institution’s language of teaching and a student’s home language negatively affect a student’s ability to successfully complete a course or degree. by allowing race, gender and school background to impact directly on the probability being associated with graduation at this university, and then also to impact indirectly on this probability through a mechanism that is being mediated by the marks obtained in english and mathematics school-leaving examinations, questions along the following lines can now hopefully be appropriately answered: do women perform better than men at university? do women perform better than men even after an appropriate adjustment has been made for other predictive factors associated with race, school background, college of enrolment and the receipt of residence-based accommodation? how do women who have done well in english at school compare with those who have not done well in english with respect to their performance at university? research methodology the data that have been used for this study comes from a university situated in kwazulu-natal. a total of 24 392 students enrolled for a degree at this university over the period 2004 to 2012. for each student, the mark (expressed as a percentage) that they obtained for mathematics and english in their school-leaving examinations was collected together with some of the other more important predictor variables that have been highlighted in the previous section of this article. university specific variables indicating whether or not they received some form of residence-based accommodation while studying at university were also included together with the college in which they chose to study. table 1 contains a listing of all the predictor variables that were collected. national senior certificate refers to a new single grade-based exam that was introduced in 2008 to replace the standard and higher grade examinations that were in existence before that date. permission to use this data set was given by the teaching and learning office at the university. table 1: variables that were included in the analysis. the results in table 2 indicate that students enrolling for a degree at this university are performed better in english than mathematics when writing their school-leaving examinations. because a national body adjusts the marks that are obtained for each subject according to a set of targeted guidelines, one cannot interpret this result as necessarily implying that mathematics is a more difficult subject to take than english at a matric level. nor can it be argued that the knowledge base that they will need to succeed with their studies at this university is better for english because the average marks being recorded for english are higher than those being recorded for mathematics. table 2: performance measures for english and mathematics over the period 2004–2012. school quintile all public schools in south africa are given a ranking based on the level of poverty that exists within the community in which the school is located. this ranking takes into account the average level of income that is earned by someone in the area surrounding that school, the unemployment rate and the level of education within that community. schools falling in the bottom 20% of this ranking (i.e. the poorest schools) are then classified as being quintile 1 schools. schools falling within the top 20% of this ranking are said to be quintile 5 schools. in table 3, t-tests comparing the performance of students in english and mathematics at a quintile 5 school with those coming from a less privileged quintile 1–4 school indicate unsurprisingly that students in the better resourced quintile 5 schools perform significantly better than students coming from the less privileged quintile 1–4 schools. this result may be a reflection of better teaching of these subjects at the quintile 5 schools. it could also reflect a greater level of `spoon-feeding’ and rote learning, which may not necessarily translate into a higher probability of being able to graduate at a university when faced with the extra workload pressures that exist in a university environment. interestingly enough, the results in table 3 suggest that the gap in performance between the quintile 5 and quintile 1–4 schools is much greater in english than mathematics. a greater prevalence of mother tongue based instruction in the poorer quintile 1–4 schools may be a reason for this. also, the teaching of mathematics lends itself more easily to the application of a `correct’ formula without a basic understanding of why this formula is necessary. consequently, extra marks can be more easily obtained in mathematics than would be the case when having to write for example an english essay. table 3: two sample test results for comparing the mathematics and english results of quintile 5 schools with quintile 1–4 schools (assuming unequal variances in the grouped populations). choice of response variable a 0/1 variable indicating whether a student has eventually been able to graduate from this university has been used as a response variable y for this article. students who were still busy with their studies when the study period ended were deleted from the data set. the results given in table 4 indicate that more than half of the students who enroll for a degree at this university fail to complete their degree. we want to determine what sort of effect the variables listed in table 1 have on a student’s prospect of eventually graduating. given that each one of these students would have taken mathematics and english as a subject at school, by allowing the effect of race, gender and school background (on graduation) to be mediated through the type of mark they recorded for mathematics and english, we also want to determine whether the university should be looking at offering an extra course in english or mathematics (or both) to help students bridge the gap that exists between what they actually know and what they need to know in order to improve their chances of graduating at this university. table 4: observed graduation rates at the university over the period 2004–2012. results a logistic regression based analysis table 5 contains graduation rates for each one of the variables listed in table 1. focusing on gender one can see that women have more chance of graduating with a degree from this university. black africans do not perform as well as non-black africans, students coming from a quintile 5 background do better than students coming from a less privileged school background. students enrolled in the college of agriculture, engineering and science do not do as well as students enrolled in one of the other colleges. table 5: a demographic breakdown of graduation rates at this university. all the results given in table 5 however represent marginal effects. fitting a logistic model to all the covariates that appear in this table will allow each effect to be estimated after an appropriate adjustment for all the other variable effects have been made. an interaction term denoting whether as a black african child they attended a quintile 5 school has also been included in the results that appear in table 6. table 6: parameter estimates associated with covariates affecting the probability of graduation at the university. parameter estimates that are significantly positive in value indicate covariates that will help to increase the probability associated with graduating from the university. for example, the parameter estimate –0.237 associated with being male implies that men have a smaller probability associated with eventually graduating than women. in terms of an odds ratio, the above estimate implies that the probability associated with a man graduating is 0.789 times the probability associated with a woman graduating; i.e. . using a 5% level of significance, the results in table 6 suggest that male and black african students have significantly less chance of graduating than female and non-black students. those who wrote the national senior certificate did not performing as well as those who finished matric before 2008. obtaining a higher mark for mathematics and english can be associated with an increased probability of eventually graduating. focusing on college of enrolment it is important to note that students in the college of management studies and law form a baseline college from which to make a comparison. thus, students enrolled in the college of humanities or the college of health sciences have a higher probability associated with graduating when compared with a student enrolldc in the college of management studies and law. students in the college of agriculture, engineering and science have a smaller probability associated with graduating when compared with a student enrolldc in the college of management studies and law. a structural equation model based analysis in this section, we will focus on the marks that students obtained for mathematics and english when writing their school-leaving examinations. given that these subjects are compulsory for all school leavers we want to identify whether obtaining a particular mark for mathematics improves one’s probability of graduating from this university more than obtaining the same mark for english. at the same time, we want to separate the total effect that race, gender and school background have on graduation into an indirect effect that is mediated by the type of mark obtained at school for mathematics and english and a direct effect representing the residual effect of these three factors on graduation that results after having controlled for the mediation effects of mathematics and english. by doing this, we can now address other issues that may be of interest such as whether the university should look at offering an extra course in english or mathematics (or both) to help students bridge the gap between what they actually know and what they need to know in order to improve their chances of graduating at this university. structural equation based models allow one to estimate direct and indirect effects for a given problem. referring to the diagram given in figure 1, the observed variables are represented by rectangular boxes with an arrow between boxes representing a hypothesised pathway between these variables whose ‘value’ we want to estimate. the arrow linking the box labelled black african, male and quintile 5 school to the box labelled graduated represents a set of direct path-based effects for race, gender and school background that we want to estimate. the arrows linking the box labelled black african, male and quintile 5 school to the box labelled graduated via a pathway that passes through the box labelled mathematics and english represents a set of indirect effects associated with race, gender and school background whose estimated effects on graduation we also want to estimate. estimation of these path-based coefficients is done using a maximum likelihood method with the fit of the model structure being based on a comparison of the observed variance covariance matrix and the variance covariance matrix that is implied by the path model in figure 1. figure 1: a structural equation model for graduation with direct and indirect mediated effects. path-based estimates associated with the fitting of a logistic model to that part of the path model in figure 1 that affects graduation are given in table 7. odds ratios have been included to help with the interpretation of direct effects, noting that these are effects that are observed after an appropriate adjustment for the effects of other variables in the table have been made. path-based estimates for the indirect effects that are mediated through mathematics and english are given in table 8. table 7: direct effect estimates associated with graduation for the model structure given in figure 1. table 8: indirect effect estimates that are mediated through mathematics and english. focusing on the direct effect estimates that are given in table 7, noting that these estimates include an appropriate adjustment for the mediation effects of english and mathematics that form part of table 8, coming from a more privileged school background increases the probability of graduating at this university. male and black african students have a smaller probability of graduating and staying in residence improves one’s probability of graduating from this university. at a college level, students enrolled in the college of agriculture, engineering and science have less chance of graduating than students enrolled in the college of law and management studies. students enrolled in the college of humanities or the college of health sciences however have more chance of graduating than students enrolled in the college of law and management studies. the strongest direct effect of all on graduation is the effect recorded by students who wrote the national senior certificate examinations. these students significantly underperformed when compared with someone who wrote the higher and lower grade examinations that were in existence before 2008. focusing on the path-based estimates that appear in table 8 linking english and mathematics to the probability of graduation, not unsurprisingly having a higher mark for either subject improves one’s probability of eventually graduating from this university. consequently, offering appropriate bridging courses in english or mathematics for students enrolling at this university should help to improve graduation rates at this institution. more importantly, however, the estimated effect for english (0.021) is more than double that for mathematics (0.009) implying that a single percentage increase in one’s mark for english increases the probability of graduating much more than a single percentage increase in one’s mark for mathematics. in light of these mediated results, should a choice between running a bridging course in mathematics or a course in english be necessary, more attention and resources should be directed towards improving the english (rather than mathematical) literacy of students entering this institution. gender focusing on gender and its mediated effect on graduation based on the type of mark recorded for mathematics and english, the results given in table 9 indicate clearly that men tend to do better than women in mathematics with the opposite being true for english. the path-based estimates that appear in table 8 linking male to english and mathematics respectively reinforce this fact. in particular, after an appropriate adjustment for race and school background has been made, men score a mark for mathematics that is on average 3.7% higher than that for women. women, on the other hand, do better in the languages, scoring a mark for english that is on average 3.4% higher than that for men. with these results in hand, the university may want to `consider’ introducing a bridging course in mathematics for women and a bridging course in english for men enrolling at this university. table 9: two sample test results comparing the mathematics and english results of men with women (assuming unequal variances in the grouped populations). race focusing on race and its mediated effect on graduation based on the type of mark recorded for mathematics and english, the results given in table 10 indicate for both subjects that black african students significantly underperform when being compared with non-black african students. the path-based estimates in table 8 linking black african to english and mathematics reinforce this fact. table 10: two sample test results comparing the mathematics and english results of black african with non-black african students (assuming unequal variances in the grouped populations). school background focusing on school background, the inclusion of an interaction term (quintile 5 × black african) that denotes a black african who attended a quintile 5 school allows one to isolate the direct and mediated effect on graduation of this cohort from that of other non-black african students who also attended a more privileged quintile 5 school. the results in table 7 indicate that attending a quintile 5 school increases the probability of being able to graduate at this university. conclusions in this article, we have been concerned primarily with constructing an appropriate prediction model for graduation at the university. by making use of a structural equation model we have been able to model not only the direct effects associated with graduation but also the indirect effects that have been appropriately mediated by the marks that students obtain for english and mathematics in their school-leaving examinations. focusing on the direct effects that were observed, gender and race play a significant role with men having a smaller probability of graduating (when compared to women) and black african students not doing as well as their non-black african counterparts. students coming from a more privileged school background have a greater probability of eventually graduating from the university. focusing on the indirect effects, a strong gender bias emerges with men tending to perform better in mathematics than women and women better than men in english. more importantly, however, a single percentage increase in one’s mark for english serves to increase the probability of graduating from the university by an amount which far exceeds what would be the case with a single percentage point increase in one’s mathematics mark. in light of these mediated results, when it comes to holding a discussion around bridging and the type of courses that should be developed for struggling students, more attention should be given towards improving the english (rather than mathematical) literacy of students entering this institution. acknowledgements competing interests the author declares that he has no financial or personal relationships that may have inappropriately influenced him in writing this article. references bird, e., & welford, g. 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(1966). equality of educational opportunity. washington, dc: u.s. government printing office. feast, v. (2002). the impact of ielts scores on performance at university. international education journal, 3(4), 70–85. available from https://ehlt.flinders.edu.au/education/iej/articles/v3n4/feast/paper.pdf howie, s.j. (2003). language and other background factors affecting secondary pupil’s performance in mathematics in south africa. african journal of research in science, mathematics and technology education, 7, 1–20. available from http://hdl.handle.net/10520/ejc92785 johnstone, a., & selepeng, d. (2001). a language problem revisited. chemistry education: research and practice in europe, 2(1), 19–29. available from http://www.chem.uoi.gr/cerp/2001_february/pdf/05johnstone.pdf murray, m. (2016). does poor quality schooling and/or teacher quality hurt black south african students enrolling for a degree at the university of kwazulu-natal? plos one, 11(4), 1–11. https://doi.org/10.1371/journal.pone.0153091 naylor, r., & smith, j. (2005). schooling effects on subsequent university performance: evidence for the uk university population. economics of education review, 24(5), 549–562. https://doi.org/10.1016/j.econedurev.2004.07.016 spaull, n. (2011). primary school performance in botswana, mozambique, namibia and south africa. sacmeq working paper. available from http://www.sacmeq.org/sites/default/files/sacmeq/publications/08_comparison_final_18oct2011.pdf thiele, t., singleton, a., pope, d., & stanistreet, d. (2016). predicting students’ academic performance based on school and socio-demographic characteristics. studies in higher education, 41(8), 1424–1446. https://doi.org/10.1080/03075079.2014.974528 tomul, e., & savasci, h.s. (2012). socioeconomic determinants of academic achievement. educational assessment, evaluation and accountability, 24(3), 175–187. https://doi.org/10.1007/s11092-012-9149-3 van der berg, s. (2008). how effective are poor schools? poverty and educational outcomes in south africa. studies in educational evaluation, 34(3), 145–154. https://doi.org/10.1016/j.stueduc.2008.07.005 van dyk, t., zybrands, h., cillie, k., & coetzee, m. (2009). on being reflective practitioners: the evaluation of a writing module for first-year students in the health sciences. southern african linguistics and applied language studies, 27(3), 333–344. https://doi.org/10.2989/salals.2009.27.3.10.944 article information author: erna lampen1,2 affiliations: 1school of education, university of the witwatersrand, south africa 2research unit for mathematics education, stellenbosch university, south africa correspondence to: erna lampen email: ernalampen@sun.ac.za postal address: po box 455, stellenbosch 7599, south africa dates: received: 12 nov. 2014 accepted: 20 apr. 2015 published: 24 june 2015 how to cite this article: lampen, e. (2015). teacher narratives in making sense of the statistical mean algorithm. pythagoras, 36(1), art. #281, 12 pages. http://dx.doi.org/10.4102/pythagoras.v36i1.281 copyright notice: © 2015. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. teacher narratives in making sense of the statistical mean algorithm in this original research... open access • abstract • introduction • problem statement • the research setting       • ethical issues • the research design • mean and average in validated discourses    • dictionary definitions of mean and average    • statistics education research: understanding average and mean       • everyday meanings of average       • didactical meanings of the statistical mean    • historical discourses: from the arithmetic mean to the statistical mean       • the arithmetic mean       • the mean of a distribution • conceptualising the relationship between average, arithmetic mean and statistical mean for instructional discourse    • framing the discussion of the meaning of the mean • results: narratives about mean and average    • meaning 1: mean is average    • meaning 2: average gives a general picture    • meaning 3: average is middle    • meaning 4: average is most    • meaning 5: the mean is a value to compare to    • meaning 6: far from the mean is not average    • meaning 7: mean is a constant and a norm • discussion    • evening out as a process to derive the mean algorithm    • structural differences between the arithmetic mean and the statistical mean       • the statistical mean as a norm in relation to the mean algorithm • conclusion • acknowledgements    • competing interests • references • footnotes abstract top ↑ teaching statistics meaningfully at school level requires that mathematics teachers conduct classroom discussions in ways that give statistical meaning to mathematical concepts and enable learners to develop integrated statistical thinking. key to statistical discourse are narratives about variation within and between distributions of measurements and comparison of varying measurements to a central anchoring value. teachers who understand the concepts and tools of statistics in an isolated and processual way cannot teach in such a connected way. teachers’ discourses about the mean tend to be particularly processual and lead to limited understanding of the statistical mean as measure of centre in order to compare variation within data sets. in this article i report on findings from an analysis of discussions about the statistical mean by a group of teachers. the findings suggest that discourses for instruction in statistics should explicitly differentiate between the everyday ‘average’ and the statistical mean, and explain the meaning of the arithmetic algorithm for the mean. i propose a narrative that logically explains the mean algorithm in order to establish the mean as an origin in a measurement of variation discourse. introduction top ↑ this article explores the knowledge needed by teachers to enable meaningful mathematical discourse in instruction (venkat & adler, 2012) for the statistical mean. in venkat and adler’s (2012) work, mathematical discourse in instruction comprises inter alia the explanations and discussions a teacher creates between the stated problem, the initial object, transformations of the object and applications of the result. establishing coherence between these aspects is the learning task and enabling the construction of coherence by learners through tasks and discussions is the most important role of a teacher. whilst mathematical discourse in instruction can be understood narrowly as a discourse that aims for local, micro-level coherence from one step of a transformation to another, ending when the problem at hand is solved, the mathematical discourse in instruction that i advocate builds on and is framed by a conceptual orientation (thompson, philipp, thompson & boyd, 1994) and aims at constructing meaning for statistical procedures that have the statistical horizon in mind, to paraphrase ball (1993). thompson’s (2013, p. 61) rendition of piaget and garcia's (1991) notion of meaning – ‘meaning comes from an assimilation's implications for further action’ – motivates for a conception of the mean that includes a rationale for its use in more advanced statistical processes such as calculating the standard deviation and linear regression. thompson calls for research on teachers’ mathematical meaning for teaching in recognition that developing mathematical meanings for teaching requires deep reflection on connections and organisations between mathematical objects and processes in relation to the larger mathematical project: that of providing opportunities for learning to think mathematically. in particular, this article wants to promote deep reflection on the connection between the use of the statistical mean as a central value for a data set and the mathematical procedure to calculate the mean. such knowledge of the uses of mathematical procedures to create statistical tools is specialised content knowledge (ball, thames & phelps, 2008) that will help teachers to conduct classroom discussions that promote statistical reasoning. problem statement top ↑ at school level statistics is usually taught by mathematics teachers, whose studies may not have included courses in statistics. hence, the instructional discourse of statistics tends to be restricted and mostly aimed at instruction for performing well-defined mathematical procedures, such as calculating the mean when it is asked for explicitly. in contrast, statistical thinking ‘involves “big ideas” that underlie statistical investigations’ (ben-zvi & garfield, 2004, p. 7). big ideas that have been made explicit in statistics education literature include the ideas of variation and distribution, where measures like the mean and median act as representative values and summaries of distributions. the statistical mean derives its meaning from mappings between practical, everyday discourse about varying observations and mathematical discourse in which the algorithm for the arithmetic mean is understood to effect equal sharing. evident from statistics textbooks, the mean is at most reported as ‘the average’ in a context without any further attempt at explicating the meaning of average. on the one hand, the problem is that average has many contextual meanings that do not all map onto the statistical mean (watson, 2006). on the other hand, the mathematical algorithm is adopted as the way to obtain the average, rather than logically explained. teachers who are not aware that different meanings can be assigned to average in context may treat average and mean as synonyms in classroom discussions and fail to provide opportunities to shift classroom discourse from purely informal understandings of average towards statistically literate discourse. teachers who cannot logically explain the mean algorithm may fail to explain why it yields a statistically representative number and why the mean is an important statistic in more advanced procedures. although there is a substantial amount of research about teachers’ and learners’ explanations of average and mean (shaugnessy, 2007), an aspect that has not been researched explicitly is the conflation of the arithmetic mean and the statistical mean in teachers’ discourses for instruction. i use the term arithmetic mean to refer to the mathematical structure of the mean algorithm and the use of the mean in other than statistical contexts. for example, the calculation of the gradient between two points, and division as equal sharing in typical school tasks, use the mean algorithm without viewing the resultant number as a measure of central tendency for a data set. in this article i explore the following research question: how do high school mathematics teachers reason about the relationships between average and mean and the structure of the mean algorithm? i analyse a group of high school teachers’ discussion of the meaning of the statistical mean in relation to data contexts and the algorithm. i show that the teachers’ narratives about the mean shift from limited contextual enactments of ‘average’ and ‘middle’ to using the mean as a norm to compare data values to. cognitive conflict about the interpretation of the equal values obtained by fair sharing, whilst the measured values were variable, enabled discursive shift towards statistical discourse. the findings have implications for teachers’ potential instructional discourses and suggest a need for an object definition of the statistical mean that takes account of the structure of the mean algorithm. the research setting top ↑ the discussion that provides the data for this article took place in the third session of a semester course in introductory statistics for high school teachers. the course formed part of an honours degree in mathematics education. i was the lecturer of the course and engaged the teachers as students in deep discussions of data contexts, engaging with and contrasting everyday reasoning with statistical reasoning in such contexts. twelve students were enrolled in the course. i arranged the students into three groups of four and video-recorded the discussions of two of the groups. i constituted the groups in a way that would reflect the language complexities of classroom discourse in south africa, but also provide the best possible chance of promoting discussion. i mainly controlled for power issues related to age, gender and previous knowledge of statistics. group 1 comprised mature students who are experienced mathematics teachers, evenly divided according to gender and previous knowledge of statistics. two students (kh and rk) had taken statistics as an undergraduate course. only one student (kh) had english as a first language. group 2 comprised young students, with little or no teaching experience. in this group only one student was male, but gender power issues amongst the younger students were unproblematic. two students (sds and gg) had english as their first language and three (sds, nm and mm) had recently done a statistics course in their b.ed. programme. in total, five of the eight students in the video-recorded groups had done statistics courses prior to this course and five of them were teaching statistics at grade 10 level at the time of the research. the third group was not included in the study as a separate group, although the contributions of these students were included in analysis of whole class discussions. i decided not to include the last group since they were least balanced in terms of my criteria. the discussions were transcribed from the video tapes and analysed together with the students’ written work. ethical issues i studied the group and classroom discussions during the course as part of my doctoral research. ethical clearance for the study was duly obtained from the ethics committee of the relevant university's school of education. after a contact session during which information about my research was provided and the conditions for consent were negotiated with the students, they gave informed consent that their recorded discussions and their written work may be used as research data and disseminated in scholarly conferences and publications. the conditions for consent were anonymity in the wider dissemination of the research and ensuring that their withholding consent would not influence their participation in the course or their assessments. the research design top ↑ for this case study i undertook discourse analysis of three sessions of the course in order to investigate emergent statistical reasoning. i used sfard’s (2008) theory of commognition to inform the analysis of the uses of words and other symbols in different discourses. key to commognition is the notion of thinking as communication and of learning as a process of shifting discourses. this theory allowed me to interrogate the participants’ every day and informal statistical reasoning about the meaning of the mean, rather than discount it as idiosyncratic. in order to analyse shifts in discourses, the target discourses must be defined and operationalised. i conceptualised everyday discourse, informal statistics discourse and literate statistics discourse as follows: everyday discourse about average and mean refers to concrete objects and observations of similarity amongst objects and bases arguments on practical considerations in context and personal experience and opinion. informal statistics discourse about average or mean comprises narratives that informally explore and compare measurements of variable attributes to derive an informal value of central tendency, related to an informal measure of spread. literate statistics discourse distinguishes between average as a contextual observation and mean as an abstract measure of central tendency of a data set, and relates the mean as a measure of central tendency to standard deviation as a measure of spread. these operational definitions of the discourses guided my data analysis. for example, everyday discourse was coded if a participant referred to a person as being ‘average’, or ‘the average one’, without recourse to measurements. sfard (2008, p. 57) calls such flattened discourse instances of ontological collapse, in which a construct like the mean is treated as if it belongs to the world of direct observation. informal statistics discourse was coded when participants indicated, through words, inscriptions or gestures, that average or mean is a position on a continuum which serves to facilitate informal comparison of many objects. such objectification of average from being a property of an object to being a position on an informal scale indicates a shift to informal statistics discourse. the participants in my study did not provide narratives that could be coded as literate statistics discourse. such discourse would, for example, refer to the need for a set of data, a formal calculation of the mean and a contextual interpretation of the number obtained. commognitive research requires in-depth analysis of the uses of words and discursive patterns in extended discussions. words are concepts and the ways in which participants elaborate on word uses through other words or representations like gestures allow the researcher to make conjectures about participants’ discourses and hence understanding of concepts. mean and average in validated discourses top ↑ the word usage of the participants in my research is not independent of culturally validated uses in different discourses. hence, i begin by contrasting the meanings of average and mean as they are used in three discourses: everyday discourse evident from dictionaries, statistics discourse used in subject dictionaries and mathematics discourse as evident from the historical emergence of the arithmetic mean. then i discuss literature about discourse on average and mean that emerge in teaching and learning situations. dictionary definitions of mean and average a study of dictionary entries under ‘average’ and ‘mean’ reveals an opaque and circular relationship between the two terms. in table 1 i compare the definitions of average from a dictionary of everyday usage: the merriam-webster online dictionary (merriam-webster, 2015), and a statistics dictionary: collins dictionary of statistics (porkess, 2004). table 1: comparison of definitions of average in everyday and statistics discourses. a comparison of the everyday and statistics definitions of average in table 1 indicates that average as being typical or representative of a group is a shared meaning in the two discourses. however, in everyday discourse average is ‘an estimation or approximation to an arithmetic mean’ whilst in statistics discourse average may refer to ‘any (or none) of mean, mode, median and midrange’. hence, the statistics point of view acknowledges that the term average derives meaning mainly from context and the everyday perspective acknowledges that what is average may be approximately the same as the value calculated by the mean algorithm. a second observation is that in both discourses average is implicitly utilised as a point for comparison. in the examples provided for average as typical or representative (see entries numbered 1 in table 1), objects are described in comparison to average as ‘above average’ or ‘is average’. this use of average is not made explicit, yet i will argue later that the mean as a logical point to which to compare other measurements is a crucial narrative in a discourse about variation. in table 2, in everyday discourse the term ‘mean’ is explained as a middle position (though not necessarily a number) between extremes and as a calculated value that falls within a range of values. similarly, in statistics discourse mean is defined as a measurement of average, with the vague concession that there are different ways to measure average appropriately. table 2: comparison of definitions of mean in everyday and statistics discourses. the definitions of ‘mean’ in the merriam-webster online dictionary (merriam-webster, 2015) emphasise the ‘laws’ for calculating the arithmetic mean or the expected value and refrain from explicit contextual examples; these therefore belong to a more abstract discourse than the definitions of ‘average’. this analysis and comparison of the sanctioned meanings of average and mean reveals a disjunct that begs explanation: intuitively and informally average is representative and serves as a point of comparison, yet these meanings are not carried over in the definitions of the mean. in particular, it is problematic for instructional discourses that porkess’s (2004, p. 14) statistical definition of the arithmetic mean as a ‘measure of an average value’ fails to explain why the calculation of the arithmetic mean is a measure of an average value or how it manages to be a middle, typical or representative value. research about understanding of the statistical mean in teaching and learning situations indicates that the conflation of average and mean is problematic for teaching, since it leaves the ontologies of the mean and the average unexplained. a teacher who needs to answer the question ‘what is the statistical mean?’ may invoke the calculation procedure to imply ‘the mean is what is does’, but, as the statistics education literature reports, the process-definition is open to varied interpretations. statistics education research: understanding average and mean in-depth interviews as well as large-scale studies that have researched the meanings learners and teachers assign to the mean provide wider context for the meanings of average and mean, which are reflected in dictionaries. it also illuminates the potential for confusion in statistics classrooms: literally, participants in a classroom discussion may not be talking about the same thing when they refer to average or to mean. everyday meanings of average various meanings of average in everyday discourse are described in statistics education literature. both teachers and learners routinely elaborate the meaning of ‘average’ as ‘middle’. in turn, ‘middle’ is understood in more than one way: sometimes middle is determined by active ordering of measurements of some attribute, where after the middle position between minimum and maximum is assigned to ‘average’. this meaning of average can be mapped on the statistical median or on the midrange. sometimes, middle is achieved by excluding extreme values so that middle refers to an interval of similar values rather than a single value. this meaning of average can be a precursor of a measure of spread of similar values, rather than a measure of central tendency (konold & pollatzek, 2004; makar & mcphee, 2009; mokros & russell, 1995; watson & moritz, 2000). average is also explained as ‘typical’ in everyday discourse. when data are available, ‘typical’ tends to be associated with the most frequent observation (konold & pollatzek, 2004) but also with a reasonable range of values (makar & mcphee, 2009). in these meanings the confusion between average as a single value or a range of values is evident: average as the ‘most frequent’ observation can be mapped onto the statistical mode rather than the statistical mean, whilst a ‘reasonable range’ indicates early notions of spread of near-similar data points. the complexity does not end here. everyday meanings of average do not depend on the comparison of numerical values. interpretations of average are often based on qualitative judgments of what is experienced as ‘not extreme’. hence, a person can be described as average in appearance, based on a qualitative judgement of appearance that lies between extremes, for example the extremes of ugly and attractive. ‘average’ in context may be so tightly associated with normative contextual descriptions that it is associated with adjectives like good, bad (to score an ‘average’ mark is good or bad, depending on the value of the average mark), low, high, cheap or expensive, rather than reflecting a relationship between overt or covert measurements of an attribute of a collection of objects (lampen, 2013). these everyday meanings of average held by teachers and learners suggest that simply explaining the number obtained by the mean calculation as the average does not provide access to statistical discourse. indeed, the equal sharing meaning suggested by the mean algorithm is not associated with average by people who do not know the algorithm (mokros & russell, 1995): in many everyday contexts where observations are not equal, the mean as an equal share makes little sense. didactical meanings of the statistical mean attempts to unpack the mean didactically as a statistical object have led to descriptive definitions such as an equal share, true value, signal in noise, balance point or representative value (konold & pollatzek, 2004). in these definitions the mean refers to a distribution of data, abstracted from a collection of contextual measurements. studies of meanings assigned to the mean have not specifically asked participants to explain what they understand by these descriptions; rather the descriptions have been used by researchers to categorise ways in which participants interpret graphs and data sets. only rarely have learners or teachers without formal statistical background responded in these statistically descriptive categories (groth & bergner, 2006; watson & moritz, 1999) and there is consensus that such abstract meanings of the mean are difficult to develop (konold & pollatzek, 2004; watson & moritz, 2000). makar and confrey (2004) concur that the statistical relationship between a distribution as an object and the mean as a measure of the object is opaque, whilst mokros and russell (1995) draw attention to the disjunct between understanding the process of measuring the distribution and the mean as an object when they say ‘the mathematical relationship [of the mean algorithm and the uses of the statistical mean] itself remains opaque’ (p. 22). cortina, saldanha and thompson (1999) propose a conceptualisation of the statistical mean that consciously measures variation and yields an object: students need to create the mean as an adjustment on the measure of group performance … as one runs through the contribution of cases to the mean of the group. (p. 2) however, in their conceptualisation, the mean as an object is a multiplicative concept that serves as a measurement of group performance, hence it foregrounds the relationship: historical discourses: from the arithmetic mean to the statistical mean historically the concept of the mean can be traced back to estimation in order to solve practical, measurement-related problems and the geometric construction of different means in mathematics, namely the harmonic, geometric and arithmetic means. statistical use of the mean can only be traced back to the 19th century (bakker, 2004). in this section i draw on research about the historical development of the mean algorithm to show that the arithmetic mean and the statistical mean are different concepts, despite having the same algorithm. the difference lies in the discourses in which they are used. the arithmetic mean bakker (2004) describes two different calculation procedures that were historical precursors of the mean algorithm, even if these processes were not named with terms related to average or mean. the historical enacted algorithms provide insight into the uses and therefore the concepts that have underpinned the concept of average. the first procedure uses one representative value multiplicatively to estimate a large total number. bakker (2004) gives two examples. in the first example1 the number of leaves on a twig was multiplied by the number of twigs on the tree to estimate the number of leaves on the tree. in the second example, the thickness of a brick was estimated and multiplied by the number of layers of bricks in a wall in order to estimate the height of the wall2. in these early historical examples the term average does not appear; instead the method or process of calculating some practical quantity was described in words. the goal was to determine a direct measurement for a physical object. bakker interprets the relevance of these examples as incorporating notions of the arithmetic mean in relation to the statistical concept of representativeness (the number of leaves on one twig is representative of the number of leaves on all the other twigs). the totals in the examples were calculated according to the algorithm: structurally, ‘a representative object’ represents the mean and its value can be calculated by a simple transformation of the relationship above. it is important to note that in this historical use of finding a total number of objects the mean was not an unknown or hypothetical value. it was the smallest component unit (a brick in a wall or leaves on a twig) that could be used to access measurements of larger, composite objects (rows of bricks and walls or leaves on a tree). hence, there is no intuitive conceptual step to ‘creating’ the arithmetic mean by equal sharing. in practice, bricks are made to a standard size whilst the heights of walls vary; it does not make practical sense to ask how wide a brick must be to build a wall of a given height with a given number of rows. the geometric concepts of arithmetic, geometric and harmonic means existed long before the statistical concept of mean and were studied in pythagoras's time (around 500 bc). in ancient greece, where these concepts were mathematically formalised, lengths were constructed with the use of compasses and straight edges and treated as concrete objects (to the extent that numerical discourse on square root lengths was problematic). bakker (2004, p. 56) cites the theorem of pappus in which the arithmetic mean, the geometric mean and the harmonic mean of two line segments were indicated in a single construction (see figure  1). the construction placed the two line segments ab and bc as extensions of each other, so that the combined length was a + c and formed the diameter of a circle. hence, the arithmetic mean was half of the diameter (the total length), which is the radius. figure 1: theorem of pappus: od is the arithmetic mean of ab and bc. through the construction of pappus (ca. 320 ad) the arithmetic mean existed as an object with a measurable length. the formula that was used to calculate b as the average or middle length of two lengths a and c was: in this equation it is clear that the mean length (b) is between the two lengths it has to average. expressed in words, b is the length between a and c such that the difference between the lengths of a and b is the same as the difference between the lengths of b and c. however, reasoning about the lengths of geometrically constructed line segments as in pappus's theorem does not lead to the mean algorithm, since the radius of a circle is always half the diameter, and not an nth part. only in the 16th century, and possibly enabled by the development of the decimal system, was the arithmetic mean generalised to more than two cases (bakker, 2004). bakker draws attention to the historical process, since about 700 bc, of averaging the value of cargo losses at sea, so that such losses could be shared equally between merchants and shippers. this meaning of average is reflected in the following definition of average as a transitive verb: to divide among a number, according to a given proportion; as, to average a loss (merriam-webster, 1913). according to bakker, it is unclear how average in this sense came to signify the arithmetic mean and when and how the shift from the concept of the arithmetic mean to the statistical concept of representative value or balance point of a data set occurred. such loose ends in overlapping discourses about average and mean are problematic in teaching for statistical reasoning. the mean of a distribution the use of mean in a discourse on variation, hence statistical discourse, developed quite recently in the history of mathematics. until about the 19th century the calculation of the mean was used to find a ‘real’ value, a measurement of a physical object (e.g. the diameter of the moon or the number of leaves on a tree). bakker (2004) dates the first use of the mean as ‘the representative value for an aspect of a population’ around 1835, when the belgian statistician quetelet invented the term l’homme moyen, the average man. this use of the mean as a representative value rather than a ‘real’ value, as in astronomy, was an important, yet difficult step in the development of variation discourse on the mean. fifty years after quetelet's invention, charles peirce, mathematician and philosopher, wrote in 1877 how problematic it was to map continuity of measurement onto situations where measurements are in discrete units, in order to report averages like ‘there are in the united states 10.7 inhabitants per square mile’ or to talk of ‘the average man’. according to bakker peirce preferred ‘most men’ instead of ‘the average man’ (p. 61). conceptualising the relationship between average, arithmetic mean and statistical mean for instructional discourse top ↑ i now report on the meanings of the statistical mean that emerged in a discussion of the mean algorithm by a group of high school teachers, after which i reflect on connections between their narratives about the mean and average, and their understanding of the meaning of the division step in the mean algorithm; finally, i consider possibilities for integrated discourse for instruction of the mean as a statistical concept. framing the discussion of the meaning of the mean prior to the discussion of the meaning of the mean, the students had studied real data of samples of prices of used cars and drawn various graphs of the data with the aid of fathom™ in order to investigate shapes of distributions and to estimate measurements that could reasonably serve to represent and summarise central tendency and spread. they had also compared calculated values of the mean and the median to their estimations on graphs. furthermore, the sensitivity of the mean to extreme values had been explored empirically and discussed as a reason for representing and comparing skewed data sets by the median rather than the mean. hence, all the students knew how to find the median and how to calculate the mean. i introduced the following prompt for the discussion of the meaning of the mean algorithm: ‘what is the logic or common sense behind using the mean as a measure of centre?’ the aim of the discussion as a learning task was to engage the students in analysing the meanings of average and mean, and in constructing a logical connection between the syntax of the mean algorithm and the role of the mean as a statistical measure of centre. in my analysis of the discussions i looked for ‘seed concepts’ that could be used in discourses for instruction to develop statistical reasoning about the mean. in particular, i wanted to understand if and how the participants considered the enacted meanings of addition (putting together) and division (sharing or grouping) in their explanations of the mean algorithm. it transpired that their discourse maps well onto everyday discourses such as those evident from the dictionary entries. the students too explained mean as average and average as mean with ‘middle’ as the predominant spatial image. they were at a loss to give meaning to the mean algorithm, yet they developed a generative narrative of the mean as a norm or a value to which to compare measurements. this narrative holds the key to a new object definition of the mean. i will now report on seven meanings that emerged during group and whole class discussion of the meaning of the mean algorithm. the excerpts are provided in chronological order and provide the opportunity to describe discursive shifts in the discussion. in order to establish confidence in the credibility of my own interpretive narratives (and hence the validity of my research) i provide extended transcripts of the discussions (sfard, 2012, p. 8). full transcripts of the discussions are available in lampen (2013). results: narratives about mean and average top ↑ meaning 1: mean is average throughout the group and class discussions the students explained the mean as the ‘average’ in contexts in which they imagined the mean could be used. the excerpt in box 1 is an example. the numbered turns provide a chronological order for the students’ utterances. at first glance it appears that the students are treating mean and average simply as synonyms, yet in turn 10 and turn 15 kh's utterances suggest a primary ontological position for average. the students seem to share the common sense meaning of average that they believe ‘people’ have. the discussion about the mean as an object (‘the mean is …’) stops here. the ontological collapse in this narrative prevents the students from further reasoning. the requirement to further unpack the meaning of average seems ridiculous: the mean is ‘just’ the average as if the average was self-evident and no further explanation is needed. box 1: mean is average. meaning 2: average gives a general picture in the excerpt in box 2 the discussion shifts to why the mean is used as a measure of centre. the discussion is based on references to imagined contexts of real objects: that of a class of ‘kids’ of different heights and cars with different prices. through its conflation with average the mean provides ‘an impression’ and ‘a general picture’ of a situation. in this narrative the mean provides one with a bird's eye view in which the differences between the imagined objects recede and the similarities remain. box 2: average gives a general picture. intertwined with the impression narrative in box 2, a narrative about mean-as-middle develops. in contrast with the impersonal ‘it gives …’ (box 2, turn 18 and turn 23), the ‘middle’ narrative in box 3 draws the observer into the context: ‘you have to order it’; ‘you take the middle value’ and ‘then you know’; ‘exactly half are above that height and exactly half are below’. in the excerpt in box 3 the use of middle in relation to average and median raises conflict. box 3: average is middle. meaning 3: average is middle in the excerpt in box 3 rk, who is the leading discussant, first describes average as a value in the middle of some interval where objects (kids) would converge if compared by a measurement like height (turn 20). in turn 23 rk insists that this average as a middle value gives a general impression of the situation. kh (turn 26) initiates a discussion about middle as being representative and the procedure to find the middle value. she queries the assertion that average is the only middle value through her reference to the median. rk's leading narrative about the mean as a ‘middle value’ is within everyday discourse in which physical examples and imagined contexts are used to give weight to the argument. kh's narrative, on the other hand, is anchored in statistical discourse, drawing on the procedural definition of the median. the students seem to have control over the median: they are certain they find the middle when they calculate the median position, whilst there is no such agency in their narrative about the mean. since the logic by which mean becomes middle is not clear, the students are unable to resolve the conflict around the meaning of the mean-as-middle, and rk and kh (turn 44 and turn 45) retreat to the initial realisations of mean as ’the general picture’ and ‘an impression’ of what is going on in a situation in which it is used. an underlying problem is that the objects that support the reasoning at this stage are a concrete, although imagined, collection of ‘kids’. the mean does not have anything more to say about this collection; average is adequate. with no recourse to logical reasoning about the syntax of the mean algorithm in relation to average and average-is-middle, there is no opportunity to develop more abstract statistical narratives about the mean. as i mentioned before, the students knew how to calculate the mean and how to find the median; hence, their confusion between mean and median cannot simply be ascribed to lack of algorithmic knowledge. meaning 4: average is most in the excerpt in box 3, turn 20, rk pointed out that the mean is such that ‘generally … you find kids around that’, and is therefore a centre within an interval. in the excerpt in box 4 (turn 49 to turn 50), another property of average is realised in everyday discourse, namely that average describes an interval that captures most objects. box 4: average is most. in turn 49 gk agrees with the narrative that the mean as the average gives a general picture of some aspect of a context. she then realises her understanding of the use of the mean algorithm. the result of ‘add[ing] up the total and dividing it by the number’ is realised as a frequency of occurrence ‘how often you can get it’. with her verbal realisation of average as most, gk gestures grouping together of objects within brackets. in turn 49 (box 4) gk strengthens the realisation of average as a place rather than a measurement or a property of an object: ‘most of the learners are here … in a certain average’. utterances of ‘most’ are interpreted in the statistics education literature as unrepresentative modal understandings of the mean (mokros & russell, 1995), but i interpret gk's combined verbal and gestural realisations as ‘most will be around the mean, because they are average’ (see also rk's utterance in box 3, turn 20). gk does not refer to a measurement that occurs most often (the mode), but to the majority of cases that were grouped together as ‘average’. rk does not explicitly take up the notion of average as an interval; on the contrary, his emphasis on ‘general’ together with a sweep of the hand (box 4, turn 50) supports replacement of many measures by one. at this stage in the discussion the student teachers do not have access to narratives that unpack the meaning of the mean; instead, their narratives compare uses of the statistical mean with the everyday, self-evident notion of average. figure 2 summarises the available narratives that relate mean to average in context. figure 2: three narratives about the mean as the average in everyday discourse. the ontology of the mean – what the mean is – is completely realised in intuitive everyday understanding of average in which similarity and extremity are observed properties of objects. the epistemology of the mean is similarly intuitive and practical: we come to know what the mean is through its uses in everyday contexts. hence, both ontology and epistemology of the mean in these teachers’ narratives are intuitive and restricted to everyday discourse. the meanings they assign to the mean as average are reflected in the dictionary definitions i mentioned earlier. the problem is that even the definitions in the statistics dictionary do not provide a way out of the conundrum of the conflation of mean and average. in the ensuing discussion the conflation of mean and average is gradually resolved. by comparing measurements to the mean, the mean is useful to determine what is not average. meaning 5: the mean is a value to compare to in order to focus the discussion on the syntax of the mean algorithm, i led the student teachers to think about the division step as equal sharing and then challenged: ‘what does it help you to pretend they are all the same? they are not the same!’ (in reference to the sample of car prices that was used in the group discussion). the students haltingly started to compare a state in which all the cars were hypothetically assigned the same price and the actual state of variable prices. in the excerpt in box 5 rk replaces vague impressions of mean as average and middle by a narrative about the mean as a calculated number that is in the middle of the average values and a value that anchors the actual values mathematically: if the mean is known, the actual values can be found by addition or subtraction. this understanding can be related to the definition of the mean as a measurement of average in the statistics dictionaries (see table 2) and stimulates the abstraction of the mean from average. box 5: the mean is a value to compare to. meaning 6: far from the mean is not average concurrent with the discussion of the first group reported so far, the second group of four students that were video-recorded raises the distance of a point from the mean as a means to judge in context whether an object is average or not. in turn 269 (box 6) nm talks about her learners’ marks and in turn 273 gg talks about prices of used cars; the implication of the discussion is that distance from the calculated mean holds qualitative information about the object: a mark far from the mean may be judged (turn 270) as good or bad, whilst a price that differs by r60 000 from the mean is ‘way out of the average’ and presumably too expensive in comparison to the rest. equal sharing is the enacted concept that is related to the mean as a point of comparison. these narratives about distance from the calculated mean indicate a further shift in discourse from every day to informal statistics discourse as it allows the meaning of the mean as a ‘constant’ or a ‘norm’ to emerge. box 6: far from the mean is not average. meaning 7: mean is a constant and a norm the discussion of the meaning of the mean algorithm closes with tentative object definitions of the mean as a constant amidst variable measurements and as a norm. the accompanying procedure is that of levelling out variable measurements. in the excerpt in box 7 rk (turn 144) tentatively realises the mean as some constant value compared to the variable measures in a data set. this realisation signals a crucial shift in his discourse: without the mean, we are aware of relative variation amongst actual measurements; with the mean we become aware of deviation from a single hypothetical measurement. rk interprets this ‘constant’ as an approximation to the actual values in context. rk's choice of the term constant was meaningful. the merriam-webster online dictionary (merriam-webster, 2015) defines the noun ‘constant’ as follows: ‘a number that has a fixed value in a given situation or universally or that is characteristic of some substance or instrument’. box 7: mean is a constant and a norm. sds's explanation (turn 232 and turn 247) of the result of evening out as norm supports the shift in the discourse from intuitive awareness of variation in context to comparing measurements to a fixed number. in these attempts to define the mean as an object, the position of the mean (in the ‘middle’) is not mentioned. levelling out and fair sharing emerge as process meanings of the division step. figure 3 provides a summary of the narratives of the meaning of the mean algorithm. figure 3: informal statistical narratives on the meaning of the mean algorithm. in the discussion of the meaning of the mean algorithm, the mean emerged as a hypothetical, abstract object that serves as an objective point of comparison amongst measurements. hence, the conflation of average and mean is resolved and the students’ narratives now belong to informal statistical discourse. discussion top ↑ the meanings of the mean and average that emerged in my study support findings in the literature that the mean algorithm is badly understood by teachers. the tendency to accept the mean as a ready-made formula to assign a number to a variety of everyday meanings of average is pervasive and persistent. the reported discussion suggests that, unless teachers consciously work to separate the meanings of the calculated mean and the contextual average, their discourses for instruction will be limited to everyday, experiential meanings. from the students’ discussion i identified two seed narratives for developing connections between average, the mean algorithm and the statistical mean. the students’ narratives presented the mean as an evening-out process and the mean as an object, namely a norm to compare to. i propose that these two narratives are conceptual process-object counterparts that can be developed to logically relate the arithmetic mean to the statistical mean. in the rest of the discussion i will illustrate a possible discourse for instruction towards this integration. evening out as a process to derive the mean algorithm evening out is reported in the literature as an intuitive process to find a mean value (bakker, 2004). in the absence of data, evening out is used even by young learners when they can draw on case-value bar graphs. a case-value bar graph represents specific cases and their measurement values as bars with different lengths. in accompanying discourses for instruction teachers view the task as completed when the evening out of bars is achieved, but the process is not abstracted in relation to the mean algorithm. furthermore, narratives about evening-out processes refer to the bars (case values) and not to the differences between the bars. yet, evening-out processes are based on redistributing differences between bar lengths. i will illustrate how attention to the evening out of differences can be productively used in a measurement of variation discourse that shifts to the statistical mean. the bars in a case-value bar graph can be ordered from small to large to support a narrative about ordered evening out. the process is illustrated in figure 4. figure 4: evening out differences between ordered measurements. as a narrative the algorithm proceeds as follows: even out the difference between the smallest and the second smallest measurement by taking away half of the difference between the measurements and adding it to the smallest measurement. then the difference between the largest measurement and the two equalled measurements is shared equally amongst all three bars to achieve the mean measurement. this process can be extended to any number of measurements. modelling the evening-out action closely, the algebraic process yields a mathematical narrative about the algorithm for the statistical mean, as shown in figure 5. figure 5: algebraic derivation of the algorithm for the statistical mean. structural differences between the arithmetic mean and the statistical mean the evening-out process to derive the statistical mean can be described as a first-divide-then-redistribute process, since in this enacted narrative division happens first and is effected on a single measurement at a time. each bar is divided according to the proportion required to even out bars that are shorter. in this example, in the first step the difference between the shortest bar and the second shortest bar is halved, whilst in the second step, the difference between the length of the evened bars and the remaining long bar is divided into thirds. the redistribution between the bars is additive. consequently, there is a disjunct between the mathematical structure of the mean algorithm (where division is the final action) and the meaning derived from the evening-out process. the disjunct demands a statistical redefinition of the object that is constructed by evening out. the object definition of the mean as a ‘fair share’ is not compatible with the process of sequential sharing between two measures at a time. an object definition based on the narratives that emerged about the mean as a norm in my research is the following: the mean is an origin of zero variation for the purpose of measuring variation. the statistical mean as a norm in relation to the mean algorithm statistics education literature abounds with reports of learners’ inappropriate comparison of distributions according to a contextually meaningful measure, rather than a statistical measure of central tendency (bakker & gravemeijer, 2004; ben-zvi & arcavi, 2001; konold & pollatzek, 2004). various explanations are given for such non-statistical comparison, such as students’ perceived roles in the task context (bakker, 2004), their level of knowledge of the context (pfannkuch, 2011) and local rather than global conception of distributions (ben-zvi & arcavi, 2001). in addition, i argue that comparison to the mean is not logically motivated in a measurement of variation discourse. measurement of variation raises the questions of where to measure from, that is, what value shall act as the ‘zero’ or ‘origin’, and what the unit is that shall be iterated. the answers to these questions do not lie in discourse about average in context, but fundamentally engage with the arithmetic mean as a statistical model. the evened-out value acts a standard of zero variation amongst varying measures in a data set. just as any measurement tool has a zero value from which deviations are quantified, so the mean is the origin for measuring variation in a data set. the standard deviation, also based on the concept of a mean, can then be developed as the unit of measurement of variation. conclusion top ↑ in addition to reflecting on the connections between statistical concepts, a teacher who wishes to teach statistics as a cycle of enquiry (wild & pfannkuch, 1999) needs to reflect deeply on the connections amongst three discourses: the everyday discourse in the real-world context in which the enquiry takes place, the how to and why discourses about the applications of the statistical concepts that are to be developed through this enquiry and the why discourse that logically motivates the mathematical tools that are used in statistics. the last discourse is neglected in statistics education research and hence in the education of mathematics teachers who teach statistics at school. in this article i have argued that the teachers in my study could initially not create a narrative about the mean as a statistical object. their explanations conflated mean with vague and varied ideas about average and middle in imagined situations. through focused discussion of the mathematical structure of the mean algorithm they were able to construct narratives about the statistical mean as a constant and a norm or standard to which actual data can be compared. such understanding of the statistical mean is a big idea in a discourse in which statistics is the science of measuring variation. averaging in the sense of calculating a mean pervades the structure of more complicated statistical models. therefore, for discussions of the mean to be statistical rather than informal the mean must be used with conscious consideration of variation and, most importantly, the endeavour to measure variation. the implication of this study for teachers’ statistical discourses for instruction is twofold: instructional discourse must consciously strive to separate the meanings of average in context and the statistical mean. the intuitive understanding of the mean as the middle value of an interval of average (not extreme) values in a data set should be taken up in a deviation discourse, which raises the need to measure variation. hence, i draw the attention of teachers to another big idea, namely that statistics is concerned with the measurement of variation, rather than merely the description of variation. without instructional discourses that consciously differentiate between average and mean, meaningful integration discourses about these concepts are not possible. the object conception of the mean as a norm or a standard has the potential to construct clear narratives of the difference between the statistical mean and the arithmetic mean. in arithmetic narratives the mean is understood as a fair share, whilst in statistical narratives the mean is the origin or zero variation value from which variation is measured. i showed how intuitively accessible evening-out procedures can be ordered and used to derive the mean algebraically. the conception of the mean as a norm or standard is thus rich in connections to intuitive reasoning as well as formal statistical reasoning. further classroom-based research is needed to understand how teachers develop instructional discourses about measurement of variation and the mean as an origin 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(1999). statistical thinking in empirical enquiry. international statistical review, 67(3), 223–248. http://dx.doi.org/10.2307/1403705 footnotes top ↑ 1.an ancient indian story reported by hacking (1975). 2.from the history of the peloponnesian war (431–404 bc). see bakker (2004). article information authors: verona leendertz1 a. seugnet blignaut1 suria ellis2 hercules d. nieuwoudt3 affiliations: 1faculty of economic sciences and information technology, north-west university, vaal triangle campus, south africa 2statistical consultation services, north-west university, potchefstroom campus, south africa 3school of natural science and technology for education, north-west university, potchefstroom campus, south africa correspondence to: seugnet blignaut email: seugnet.blignaut@nwu.ac.za postal address: post box 1174, vanderbijlpark 9100, south africa dates: received: 10 apr. 2015 accepted: 09 sept. 2015 published: 27 nov. 2015 republished: 17 dec. 2015 how to cite this article: leendertz v., blignaut a.s, ellis s., & nieuwoudt, h.d. (2015). the development, validation and standardisation of a questionnaire for ict professional development of mathematics teachers. pythagoras, 36(2), art. #297, 11 pages. http://dx.doi.org/10.4102/pythagoras.v36i2.297 note: this article has been republished with several corrections. copyright notice: © 2015. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. the development, validation and standardisation of a questionnaire for ict professional development of mathematics teachers in this original research... open access • abstract • introduction • research design and methodology • fourteen stages of questionnaire development    • stage 1: aim of the questionnaire    • stage 2: select the questionnaire    • stage 3: link with the research question    • stage 4: link with key aspects    • stage 5: clarify information to address key aspects    • stage 6: determine the target population    • stage 7: compile the questions and the metrics    • stage 8: create the questionnaire       • parts a and b: personal and demographical information       • parts c–f: governance, school environment, information and communication technology, professional development and open and distance learning       • part g: professional development models    • stage 9: determine the data collection strategy    • stage 10: piloting the questionnaire    • stage 11: prepare for data collection    • stage 12: collect data    • stage 13: analyse the data       • parts a and b: descriptive statistics       • factor analysis       • factor analysis: part c       • factor analysis: part d       • factor analysis: part e       • factor analysis: part f       • part g: structural equation modelling    • stage 14: report the findings • conclusions • acknowledgements    • competing interests    • authors’ contributions • references • footnote abstract top ↑ this article reports on the development of a custom-made questionnaire. the questionnaire was developed with the aim to compile guidelines for the professional development (pd) of mathematics teachers for the pedagogical use of information and communication technology (ict) integration in teaching and learning. during the standardisation and validation of the questionnaire it was distributed to 179 schools and 300 teachers in eight educational management districts in the western cape, south africa. the extracted factors had a reliability level higher than 0.8, which indicates the items in the questionnaire are significant to address the research problem and the questionnaire is valid for ict pd. introduction top ↑ information and communication technology (ict) is increasingly becoming embedded in teaching and learning; it is thus imperative that teachers integrate the underpinning concepts and skills of teaching with technology into sound classroom pedagogy. cassim (2010) conducted a secondary analysis on the south african data set of the second information technology in education study (sites 2006) (pelgrum & law, 2008) to determine the extent to which technological pedagogical and content knowledge (tpack) (mishra & koehler, 2006) contribute towards grade 8 mathematics teaching in south african schools. results from this secondary data analysis revealed that the pedagogical use of ict in mathematics teaching and learning is neither comparable to other education systems (leendertz, blignaut, els, nieuwoudt & ellis, 2013), nor on par with the goals stipulated in the white paper on e-education (department of education [doe], 2004). these results indicated that there is a dire need to address the continuous professional development (pd) of mathematics teachers’ ict pedagogy, with a focus on: (1) transforming ict mathematics education at system, school and teacher levels (cassim, 2010) and (2) providing continuous pd to improve teachers’ pedagogical use of ict in teaching and learning in schools (department of basic education [dbe], 2012). there are no guidelines for continuous pd of mathematics teachers on the pedagogical use of ict during face-to-face teaching, in a blended environment or in open distance learning (odl). the dbe aims for system-wide integration of icts into teaching and learning (doe, 2004). there are many facets that have to be addressed before south africa can coherently compete in the global society while adhering to the quality targets envisaged in the e-education policy (doe, 2004). the researchers searched the internet and significant databases for a suitable research instrument to address the research problem, but could not find a relevant questionnaire to use or adapt. they decided to compile an objective custom-made questionnaire in order to contribute to the discipline a usable instrument to measure the professional developmental needs of teachers relating to their pedagogical use of ict for mathematics education. this article describes the compilation, validation and standardisation of an instrument for pd of mathematics teachers towards ict integration. research design and methodology top ↑ the question that guided this research was: what are the processes to develop, validate and standardise a custom-made instrument for pd of mathematics teachers towards ict integration? this question is complex and relates to the two separate social realities of objectivist and subjectivist assumptions (burrell & morgan, 1979). the subjective method (adjustable exploration) focuses on the ongoing attainment and internalisation of existing knowledge and skills rooted in the known activity (engelström, 2004) and the objective method (radical exploration) broadens the horizon of the current knowledge of the phenomenon through construction of new knowledge (table 1). phase i is an interpretivist phase to describe and understand the world from the perspective of those who are directly involved in the social processes (denzin & lincoln, 1994) of continuous teacher pd; and phase ii is a structuralist phase rooted in a materialist view of the natural and social world that advocates sociology of radical change from an objectivist standpoint. this tactic therefore demands a multimode methodological approach which (1) enables the integration of different theoretical stances in order to answer the research question, (2) provides access to exploratory and confirmatory methods and (3) fosters the processes and outcomes of a multimode analysis (cohen, manion & morrison, 2011). table 1: two phases of the multi-mode research. the research therefore comprised a fully multimode equal status design of the qualitative and quantitative phases (leech & onwuegbuzie, 2007), which resulted in description of the data, constructive explanation of the data and hypotheses of the outcomes (bryman, 2006). by combining the two approaches, the researchers could merge findings in order to develop new theory presented as guidelines for ict pd of mathematics teachers (sechrest & sidana, 1995). table 1 indicates the paradigm characteristics (ontology, epistemology, human nature and methodology) of the two phases. phase i entails the inductive analysis of the documents selected through systematic literature review – the systematic review and recording of a body of previously documented knowledge on the phenomenon of the pd of mathematics teachers for the pedagogical use of ict in odl. phase ii includes the distribution of a custom-made questionnaire to compile guidelines for the pd of mathematics teachers towards the pedagogical use of ict in odl. survey research comprises the use of survey interviews or questionnaires. a prerequisite for the design of a good survey is deciding what is to be measured (fowler, 2009). generic features of questionnaires include: (1) collecting numerical data on a single data collection strategy over a large geographical area, (2) pursuing a large target population, (3) supplying explanatory and inferential information, (4) calculating frequencies, (5) standardising information (all participants complete the same questionnaire), (6) making generalisations and (7) detecting patterns according to the responses of the target population. surveys can be either exploratory or confirmatory. while confirmatory relates to the use of an existing model, or the testing of a causal relationship or a hypothesis, exploratory relates to statements, models or patterns in the data which are consequently investigated through correlations, factor analysis, regression analysis or stepwise regression analysis (cohen et al., 2011). surveys generally collect trustworthy data relating to a large number of variables (mcmillan & schumacher, 2001) and are widely used in the social sciences to collect data on target populations relating to specific phenomena at a particular point in time (fowler, 2009; neuman, 2011). surveys are often used to determine the levels of knowledge, plan intervention programmes, evaluate curricula and create intervention strategies. surveys are designed with the aim of producing statistics about the target population (fowler, 2009) and to generate precise, dependable and valid data, much thought and meticulous planning are required to select appropriate questions to address the research question (neuman, 2011). for survey design and development, specific preliminary factors should be considered: (1) the motivation for the enquiry, (2) the target population of the research, (3) the availability of funding for the data collection and (4) the data collection strategy (cohen et al., 2011). the researchers, to no avail, searched for an existing questionnaire which could be used or modified to measure the envisaged variables of this study. choosing a suitable questionnaire is a complex process; even more aspects come into play when designing and developing a survey (creswell, 2012). the researchers therefore embarked on constructing a custom-made questionnaire to determine the attitudes, opinions, behaviours, characteristics and pd requirements in terms of the pedagogical use of ict in odl. fourteen stages of questionnaire development top ↑ the researchers followed the 14 stages of cohen et al. (2011) for questionnaire design and development. table 2 provides a succinct summary of the 14 stages. table 2: stages of questionnaire development. stage 1: aim of the questionnaire a questionnaire aims to gather data from a large number of respondents on the same phenomena with the intention of describing the nature of the current circumstances or to determine whether relationship exists between specific events (cohen et al., 2011). the aim of this questionnaire was to (1) measure the professional developmental needs of teachers relating to their pedagogical use of ict for mathematics education, (2) obtain a holistic view of the demographical outline of mathematics teachers in the western cape education department (wced), (3) enquire about the current pd status of the teachers regarding ict integration and (4) use the results to develop guidelines for the professional development of mathematics teachers towards the pedagogical use of ict in odl. stage 2: select the questionnaire mainly two types of questionnaires exist: cross-sectional (obtain data on present trends, attitudes and beliefs of the participants; creswell, 2012) and longitudinal (collect data on a particular group over an extended period). this study compiled a cross-sectional questionnaire to gain the opinions of mathematics teachers about the provision of ict resources, ict trends, ict teaching and learning conditions at schools and their pd needs towards ict integration in teaching and learning. the data collection (phase ii of the study) took place as a single data collection and a cross-sectional survey was the most viable method to collect the exploratory and explanatory data (neuman, 2011). this type of survey is relatively quick to conduct, moderately economical to administer and holds a strong probability of participation. the research collected data from a specific group of participants, subjected the data to inferential statistics and provided data for retrospective and prospective enquiry (cohen et al., 2011) on: (1) the governance of ict, (2) the ict resource provision status of schools of different quintiles in the wced, (3) the extent to which the school environment is conducive to ict integration, (4) the current practices in ict within mathematics classrooms, (5) the pd activities previously initiated and conducted within the wced and (6) the pd needs and preferable modes for future pd towards ict integration. stage 3: link with the research question the design of a questionnaire involves the translation of the research question and objectives into information embedded in the questions. it is therefore essential that the research question should be linked to the rationale of the research. the question for this study was: what are the guidelines for the pd of mathematics teachers towards the pedagogical use of ict in odl? the survey questions were aimed at addressing this particular research question. stage 4: link with key aspects four key variables (governance, school environment, pd and odl) emerged as themes from the document analysis from phase i (adjustable exploration). each of these variables represents a section of the questionnaire with multiple sub-questions; two variables (pd and odl) were grouped into a single section with multiple sub-questions (neuman, 2011). additionally, the analysis required personal information from the teachers as respondents, as well as demographical information from the participating schools, in order to compare whether there were significant differences between schools in terms of context, quintile1 and socio-economic status (cohen et al., 2011). stage 5: clarify information to address key aspects in order to address the key aspects in the research, certain categories of questions should be posed. neuman (2011) proposes that the survey should include a list of category questions that relate to attitudes, beliefs or opinions, behaviour, characteristics, expectations, self-classification and knowledge. the researchers grouped these questions in parts c, d, e and f of the questionnaire according to the identified themes. stage 6: determine the target population sampling is an integral part in the survey approach (cohen et al., 2011). funds available for collecting data, time available for the data collection and access to the respondents are aspects that the researchers have to take into consideration at the onset of the research before samples of participants are selected (cohen et al., 2011). the five key factors that influence sampling are: (1) sample size, (2) representation and restrictions of the sample, (3) means of contact with the sample, (4) sample selection strategy and (5) the research methodology. as soon as these aspects have been addressed, the sample selection can be performed. the target population is a specified large group of many subjects from which a researcher draws a sample (neuman, 2011). a sample is a set of units a researcher chooses from the large group and generalises to a particular population (neuman, 2011). sample selection is a critical component of research which is guided by the type of analysis being performed. sampling in quantitative research is when a researcher selects units and regards them as representative of the total population. the features of the sample should emphasise key elements in a complex social world in order to give clarity about and insight and understanding into the concerns in the social world (neuman, 2011). it is therefore important that the researchers make sampling decisions during the initial planning of the research project (cohen et al., 2011; neuman, 2011). the systematic literature analysis revealed various ict projects in south africa since the launch of the e-education policy (doe, 2004). western cape teachers received extensive ict training through the khanya project (wced, 2011) and therefore the western cape was identified as the most viable target population. the mathematics teachers of the wced were therefore the target population for this study. the researchers considered five key factors when selecting the sample: (1) the sample size, (2) the symbolisation and limitation of the sample, (3) access to the sample, (4) the use of a sample strategy and (5) the conduct of research. there is no clear-cut method to select an appropriate sample (neuman, 2011). it is an intricate process which depends on: (1) the aim of the study, (2) the characteristics of the population under study, (3) the level of accuracy needed from the data, (4) the response rate estimated from the data collection, (5) the number of variables included in the study and (6) the research methodology used for the study (cohen et al., 2011). for quantitative research a large sample is beneficial as larger samples contribute towards increased reliability and the use of sophisticated statistical procedures demands a large number of observations. the researchers applied the eight stages of planning a sample strategy according to cohen et al. (2011). table 3 provides a summary of the planning stages of the sample selection procedure of this research. table 3: stages in sample selection. the wced is divided into eight education districts: four rural districts, which correspond to one or more municipalities, and four urban or metro districts located within the city of cape town. each educational management district has a district director, a circuit team with a circuit team manager who coordinates the tasks of the curriculum advisors, special needs education professionals, institutional management and governance planning and a school governance and management team (wced, 2013). stage 7: compile the questions and the metrics babbie (2010) and mcmillan and schumacher (2001) propose that a researcher formulate questions and constructs to: (1) operationalise the variables of the research, (2) collect data for analysis and interpretation, (3) ensure maximum response and (4) extract data for analysis to address the research problem. this formulation procedure should take into account: general question format. the arrangement of the questions in the questionnaire is of equal importance to the nature and wording of the questions asked. jumbled and shortly phrased questions confuse respondents and can lead to respondents not replying to questionnaire items (babbie, 2010). the researchers planned the layout of the questions in the questionnaire to optimally use space and yet simultaneously contribute towards readability. the researchers consulted a language expert to assist with the wording of the questions. usable format for respondents. the researchers provided tick boxes to capture responses easily. for parts c–f of the questionnaire, the researchers used a five-point likert scale (1 = strongly agree; 2 = agree; 3 = strongly disagree; 4 = disagree; 5 = do not know). ordering of items in the questionnaire. the researchers placed the demographic data at the beginning of the questionnaire (parts a and b) to provide an easy start and put the respondents at ease. the more complex questions on attitudes, attributes, opportunities, self-classification and knowledge followed in parts c– g. questionnaire instructions. a cover letter introduced the background and the rationale for the research, followed by basic instructions on how to complete the questionnaire. pretesting of the questionnaire. to ensure that the questions were clear and unambiguous, the researchers approached mathematics teachers to assess the readability and layout of the questionnaire. data processing. a data processing pane on the questionnaire was created: (1) to assist the researchers during the capturing and checking of the data and (2) for respondents to see which data would be used for the analysis (babbie, 2010; mcmillan & schumacher, 2001). table 4 provides a synopsis of the seven guidelines used during the compilation of the questionnaire. table 4: seven guidelines to formulate questions. stage 8: create the questionnaire the researchers compiled the questionnaire according to the seven identified parts. parts a and b: personal and demographical information part a requested personal information of the respondents: (1) gender, (2) age, (3) home language, (4) language of instruction, (5) years of teaching experience in the various grades, (6) highest qualification or highest professional qualification and (7) subject specialisation. part b comprised demographical information relating to the: (1) school district, (2) nearest town or city, (3) geographical location (rural or urban area), (4) quintile of the school, (5) number of mathematics classes per grade, (6) number of mathematics classes teachers teach, (7) availability of computers for teaching, learning and administration and (8) access to the internet for administration, teaching and learning. for some questions, the respondents had to select the appropriate option or fill in the applicable response. parts c–f: governance, school environment, information and communication technology, professional development and open and distance learning parts c–f comprised complex questions on attitudes, attributes, opportunities, self-classification and knowledge relating to the research. part g relates to pd models. parts c–f posed questionnaire items relating to the four themes (governance; school environment; odl; pd). the selected quotations from the qualitative analysis illustrate the underlying constructs. a closed form method of a five-point likert scale (1 = strongly agree, 2 = agree, 3 = strongly disagree, 4 = agree, 5 = do not know) pinpoints respondents’ inputs. table 5 provides an example of the compilation of the questions and illustrates the relationship between the literature constructs as derived from the literature review and the individual questions. table 5: description of compilation of questions for parts c–f. part c of the questionnaire addressed the governance of ict implementation at provincial (wced), district and school level. the majority of the items in this section relate to the objectives of the three-phase ict integration plan as stipulated in the white paper on e-education (doe, 2004). aspects relating to the school environment were grouped as one part of the questionnaire with various sub-questions (part d). the questions relate to aspects regarding the extent to which mathematics teachers: (1) create a classroom environment where they utilise the internet and ict applications, (2) download resources, (3) develop lessons with the help of ict, (4) stimulate their learners to be creative, (5) use the vast array of resources to improve their academic performance, (6) achieve the assessment standards and (7) have a positive attitude towards the use of ict for teaching and learning of mathematics. in part f, the researchers aimed to: (1) confront issues about pd of mathematics teachers, (2) gain insight into the opinions of mathematics teachers on future pd opportunities, (3) learn about the context in which pd activities should occur and (4) assess the modes in which pd should take place. part g: professional development models the final part of the questionnaire (part g) included four pd models conceptualised using the multiple pd models and frameworks identified in the literature review. the researchers adapted a variety of pd models, tested in other education systems across the world by the expert researchers in this particular field, to suit the background and context of south african schools. the four models included the best practices for pd that could work in the south african context. stage 9: determine the data collection strategy the data collection of a survey can be conducted via a postal interviews, personal interview, telephone, and internet-based surveys. each of these strategies has its own strengths and weaknesses (cohen et al., 2011). the researchers administered the questionnaire by personally delivering it to respondents and again collecting it from their individual sampled schools. stage 10: piloting the questionnaire the validation of the questionnaire before distribution to the participants is critical. two components have to be piloted: (1) the instructions to the respondents and (2) the questionnaire itself (mcmillan & schumacher, 2001). after compilation of the initial questionnaire, it was formally peer reviewed by five mathematics lecturers at the unit for open distance learning at the authors’ university. subsequently the researchers piloted the questionnaire with: (1) two mathematics teachers from a quintile 1 school, (2) two mathematics teachers from a quintile 3 school and (3) one mathematics teacher from a quintile 5 school. this contributed towards the validity and reliability of the questions and ensured the comprehensiveness of the questionnaire (cohen et al., 2011; strydom, 2005). stage 11: prepare for data collection before data collection the researchers obtained ethical clearance from the university's ethics committee and permission from the wced to distribute the questionnaire in the eight education districts. ethical clearance ensures that the respondents are protected from harm and that the researcher ensures the respondents of confidentiality, anonymity and the non-traceability of their participation in the research. the researchers planned the data collection in accordance with the schedule of the district offices and contacted the relevant parties (subject advisors, schools) via email and followed up by telephone. stage 12: collect data the first author confirmed the appointments for the data collection sessions and visits to schools telephonically and travelled in 37 days a total of 9287 km in order to collect the data. this was a considerable task in terms of man-hours and cost. the researchers collected data from farm, semi-urban, urban, former model c and independent schools across the districts. stage 13: analyse the data various statistical procedures (descriptive statistics, factor analysis and structural equation modelling) were followed to analyse the collected data. parts a and b: descriptive statistics the biographical information was presented with frequencies and percentages. table 6 outlines the summary of the biographical information. both men (45%) and women (55%) were well represented, from the eight education districts in the western cape, with a good distribution in age, teaching experience, language, qualification, quintile and type of school. table 6: descriptive statistics for parts a and b. only 7% of the mathematics teachers held a nqf level five qualification; 14% of the mathematics teachers had postgraduate degrees (table 6). the majority (84%) specialised in mathematics teaching and 16% had no formal mathematics qualification. only 5% of the participating schools in the western cape did not have computer laboratories. few mathematics teachers (8%) rated their computer literacy as poor; in general they had adequate access to computers, interactive whiteboards and the internet either at home or school. however, only 11% had experience of participating in some sort of online course. factor analysis the researchers applied construct validity through factor analysis to group the items of the questionnaire meaningfully. using construct validity validates the extent to which the questionnaire used in this research for data collection corresponds with the theoretical context (cohen et al., 2011; neuman, 2011). after factor analysis had been conducted on parts c–f, the researchers performed a reliability test using cronbach's alpha, applying an acceptable reliability level of 0.7, to test whether the extracted frequencies were significant and if the items within the questionnaire were valid. this study used representative reliability: (1) to make ensure that the data and findings were suitable, predictable, reliable and replicable and (2) to minimise the external sources of variation in the data (excluding acute answers from the data analysis) (cohen et al., 2011; neuman, 2011). for the alpha coefficient the following categories applied to this factor analysis: >0.90: very highly reliable 0.80–0.90: highly reliable 0.70–0.79: reliable 0.60–0.69: marginally reliable =0.60: low reliability the questionnaire included subscales; therefore the reliability of each set of factors was calculated individually. the factors were extracted according to kaiser's criteria and 10 of the 11 extracted factors had a reliability level higher than 0.8, which indicates the items in the questionnaire are significant to address the research problem. table 7 illustrates the pattern matrix of the correlation coefficient between the factors in parts c–f. variables with factor loadings of 0.3 or lower were extracted from the tables and variables with more than one factor loading were grouped according to best interpretability. each factor was scrutinised, evaluated and named according to a theme best suited to the factor. the barlett's test of sphericity showed a significance (p) of less than 0.0001 for the factor analysis from parts c–f. table 7: pattern matrix of factor analysis for parts c–f. factor analysis: part c four factors were extracted with the factor analysis. the kaiser meyer olkin (kmo) measure of 0.912 indicated adequate data for factor analysis with a total variance of 59%. communalities varied from 40% for factor 1 and 50% for factor 2 to 55% for factor 3 and 59% for factor 4. the factors showed a high reliability of 0.95 for factor 1, 0.85 for factor 2, 0.81 for factor 3 and 0.88 for factor 4. the four factors were named: responsibility of dbe) (factor 1), responsibility of management (factor 2), responsibility of teaching and learning (factor 3) and policy initiatives (factor 4). the mean of 3.26 for responsibility of dbe indicates that mathematics teachers in general think that it is the responsibility of dbe is to administer ict integration at schools in the wced, ensure that intervention strategies are in place for phase iii of the e-education policy and provide pd towards ict integration. mathematics teachers did not all regard ict integration to be the responsibility of management (mean 2.92). some mathematics teachers were of the opinion that the school management team should have an ict strategic plan for: (1) providing access to ict tools for teaching and learning, (2) creating a timetable for admission to ict facilities and (3) affording time to prepare ict integration lessons. the mean of 3.24 for policy initiatives indicates that mathematics teachers in the wced agree that previous and current ict initiatives do not address their needs for ict integration in mathematics teaching and learning (table 7). factor analysis: part d two factors were extracted with a kmo measure of 0.906 indicating adequate data for factor analysis with a total variance of 59%. communalities varied from 53% for factor 1 to 61% for factor 2. the factors showed a marginal reliability of 0.60 for factor 1 and 0.71 for factor 2. factor 1 relates to technology knowledge (tk), and factor 2 corresponds with technological pedagogical and content knowledge (tpack). a mean of 3.19 for tk indicates that quite a few teachers regard tk as more important than tpack. they believe that if they understand ict, they can (1) use it in their daily lives, (2) employ it in their work environment and (3) identify when ict can complement other methods to achieve the learning outcomes for mathematics. the mean of 2.71 for tpack indicates that some mathematics teachers regard ict pedagogical skills to impart knowledge and skills as less important (table 7). factor analysis: part e two factors were extracted with the kmo measure of 0.893 indicating adequate data for factor analysis with a total variance of 55%. communalities varied from 46% for factor 1 to 55% for factor 2. the factors showed a high reliability of 0.92 for factor 1 and a reliability of 0.78 for factor 2. factor 1 relates to contributors to social professional identity (spi) and factor 2 corresponds with ict and social professional development (spd). a mean of 3.26 for contributors to spi indicates that holistic growth through self-awareness and constructive socialisation is critical to many mathematics teachers. mathematics teachers should have internet access so that they can create a network with other mathematics teachers to discuss their uncertainties and share their best practices. a mean of 2.62 for ict and spd indicates that mathematics teachers regard their holistic growth through social interaction as more important than the extent to which they use ict to add to their spd (see table 7). factor analysis: part f three factors were extracted with the kmo measure of 0.927 indicating adequate data for factor analysis with a total variance of 55%. communalities varied from 50% for factor 1 to 57% for factor 2 and 61% for factor 3. the factors showed a high reliability: of 0.92 for factor 1, 0.87 for factor 2 and 0.87 for factor 3. factor 1 relates to teachers’ expectations for pd, factor 2 corresponds with building a spi and factor 3 relates to pd models and frameworks. means of 3.19 for teachers’ expectations for pd, 3.23 for building a spi and 3.26 for pd models and frameworks indicate that mathematics teachers in the wced regard all these as important during pd. mathematics teachers expect the dbe, provincial departments and schools to work together to develop an ict strategic plan, which includes a pd model that is versatile, so that they can develop at their own pace, perceptive to address subject-specialised training and insightful to their developmental needs (table 7). part g: structural equation modelling the aim of the structural equation model was to determine whether there were statistically significant relationships between the four themes. table 8 indicates the statistically significant relationship between governance and school environment (p < 0.05) and between school environment and odl (p < 0.05). the standardised regression weight for governance and school environment was 0.61 and for odl and school environment was 0.97. although not statistically significant, the standardised regression weight for odl and pd was –0.90 and for school environment and pd was 1.07. the standardised regression coefficients indicated that when correlations between variables were taken into account, governance had a positive influence on school environment and odl, and school environment had a positive influence on odl and pd. however, odl had a negative influence on pd (see table 8). table 8: standardised regression weights and correlations between governance, school environment, open distance learning and professional development. figure 1 illustrates the structural equation model from governance, school environment, odl and pd. the comparative fit index (cfi), also known as the bentler comparative fit statistic, compares the fit of a target model to the fit of an independent model. the goodness-of-fit measures for the model comprised: chi-square test statistic divided by its degrees of freedom (cmin/df) value of 3.40, which was less than 5, which indicated a good fit. cfi value of 0.88, which indicated an acceptable overall fit (mueller, 1996). figure 1: structural equation model from governance, school environment, open distance learning and professional development. root mean square error of approximation measures the differences in the observed values in the model. a value of 0.11 was not smaller than 0.1 for unacceptable fit (blunch, 2008). the structural equation model illustrates the validation of how the four themes were supportive towards achieving the object of the research. even though odl has a negative influence on pd; odl is a new mode of service delivery in terms of pd in the current education system in south africa. when teachers become acquainted with pd through odl, and the benefits thereof, their perceptions and attitudes towards odl as a method of pd may change. stage 14: report the findings the first author wrote a full report on the findings from the statistical analysis with clear guidelines for the pd of mathematics teachers for the pedagogical use of ict in odl. additionally the research: (1) developed a model for identity essentials for pd for mathematics teachers in south africa and (2) compiled strategies on how to conduct systematic literature and quantise literature data through exploratory factor analysis. the researchers are in the process of: (1) distributing the questionnaire in finland to do a comparison between pd needs of mathematics teachers in the two contexts and (2) constructing national guidelines for pd for ict integration. conclusions top ↑ with the 14 stages of questionnaire development, the research developed, validated and standardised an instrument for pd of mathematics teachers for the pedagogical use of ict, which (1) enables each context to assess the pd requirements, (2) gives access to the developmental needs of mathematics teachers to allow for the creation of context-specific pd programmes and (3) can consequently be used by other researchers to compare the contexts of other south african provinces, as well as to explore and describe pd needs in diverse contexts. the factor analysis extracted factors with high reliability which accentuates the value of the analysis. from the results it became clear that fundamentally ict integration and implementation initiatives start with the dbe. before developing future pd which aligns with the continuous teacher pd management system and phase iii of the e-education policy, the dbe should appoint a dbe panel comprising national and provincial ict coordinators. once all the groundwork has been done they can plan future ict initiatives, develop pd models that suit the south african school context and supply the provincial education departments and schools with guidelines for pd. the dbe should invest in the provision of ict equipment and human capital, reinstate the laptop initiative for teachers and supply schools with networked computer facilities so that they can explore online platforms for pd. acknowledgements top ↑ this work is based on research conducted for a phd study. the research is in part funded by the national research foundation of south africa. any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and therefore the nrf does not accept any liability in regard thereto. this publication is based on research that has been supported in part by the university of cape town's research committee. competing interests the authors declare that they have no financial or personal relationships that may have inappropriately influenced them in writing this article. authors’ contributions v.l. 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(2011). khanya: summary of the project. available from http://www.khanya.co.za/projectinfo/?catid=32 western cape education department. (2013). a brief introduction to the educational districts. available from http://wced.pgwc.gov.za/branchidc/districts/districts.html footnote top ↑ 1. a quintile is a cut-off point to receive a subsidy from the government for the provision of resources. abstract introduction brief description of the underlying project and the research question high-stakes examinations examination-driven teaching as underpinning of the project research design data analysis results and discussion conclusion acknowledgements references about the author(s) onyumbe okitowamba school of science and mathematics education, university of the western cape, south africa cyril julie school of science and mathematics education, university of the western cape, south africa monde mbekwa school of science and mathematics education, university of the western cape, south africa citation okitowamba, o., julie, c., & mbekwa, m. (2018). the effects of examination-driven teaching on mathematics achievement in grade 10 school-based high-stakes examinations. pythagoras, 39(1), a377. https://doi.org/10.4102/pythagoras.v39i1.377 original research the effects of examination-driven teaching on mathematics achievement in grade 10 school-based high-stakes examinations onyumbe okitowamba, cyril julie, monde mbekwa received: 16 may 2017; accepted: 13 mar. 2018; published: 28 june 2018 copyright: © 2018. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract various efforts are underway to improve achievement in high-stakes examinations in school mathematics. this article reports on one such initiative which focuses on the development of quality teaching of school mathematics by embedding it within an examination-driven emphasis. a quantitative approach was used to analyse the performance of grade 10 learners in three consecutive end-of-year school-based examinations set by the initiative. results indicate a trend in a positive direction over the three-year period. nevertheless, there was a discernible decrease between the first and second administration of the examinations. it is concluded that examination-driven teaching holds a promise for enhancing achievement in high-stakes school mathematics examinations if sensibly and sensitively implemented. introduction underachievement in school mathematics is a concern in most countries in the world. watson and de geest (2012) sketch the situation of underachievement in mathematics by drawing attention to ‘identifiable groups of students, such as those with different language backgrounds and those from lower socioeconomic rankings [who] underachieve in national and international tests’ (p. 213). regarding the situation in south africa, reddy and janse van rensburg (2011) highlight two relevant characteristics in the south african education system. the first is that the national average mathematics achievement score for different grade levels across the schooling system is similar and stable, around 30% to 40% at different grades. the second is that there is a high differentiation of the educational performance of students from various socio-economic backgrounds. by stating that ‘methods of approaching this issue range from macro-changes in policy, curriculum and assessment to institutional change, provision of extra teaching and micro-advice about inclusive teaching in classrooms’, watson and de geest (2012, p. 213) draw attention to the efforts embarked on to address the issue in the united kingdom. the reference to ‘provision of extra teaching’ is akin to interventions, such as additional classes offered by universities and ngos to selected learners in grades 10 to 12 and extra tuition and vacation tutoring schools offered by the provincial education departments to improve achievement outcomes in the national senior certificate (nsc) mathematics examinations (reddy, berkowitz, & mji, 2005). the seriousness with which countries take the improvement of achievement in mathematics of learners from low socio-economic and historically disadvantaged sectors of a country’s demographic makeup is evident in current reform initiatives in school mathematics. for example, in australia a desktop review was conducted to identify ‘gaps in current pedagogical approaches and learning resources for the teaching of mathematics to inform the mathematics by inquiry initiative’ (australian academy of science, 2015). one issue that had to be addressed in this initiative in australia was linked to the teaching of socio-economic and historically disadvantaged groups. the commission given to the australian academy of science by the australian government’s department of education and training was specifically stated as ‘which pedagogical approaches have been shown to work with specific groups under-represented in advanced mathematics at senior secondary level (girls, indigenous, disadvantaged students)?’ (australian academy of science, 2015, p. 17). it is also now commonplace in research reports that there is an explicit disaggregation of results along the lines of gender, socio-economic status and, where relevant, language diversity of participating cohorts in the research. the australian situation is different from the south african one, since in south africa much effort is invested in the improvement of achievement in mathematics of low socio-economic and historically disadvantaged groups. the popularisation of a programme of teaching adopted to enhance achievement in marginalised groups in a high-stakes mathematics examinations is vividly portrayed in the 1988 film stand and deliver, depicting jaime escalante’s work with disadvantaged latino-american students in east los angeles. in the movie the producers obviously used their creative licence to render a fictionalised account of the real situation. however, it is widely (see, for example, jesness, n.d.) reported that at the start of escalante’s programme only two of the five students who wrote the advanced placement calculus examination passed. the pass rate steadily increased and in 1982 eighteen students passed. the film focuses primarily on the 1982 cohort of students. escalante’s methods of teaching and ways of working with students are described in escalante and dirmann (1990). in southern africa there is a paucity of research related to efforts to enhance the achievement in mathematics in high-stakes examinations of students from low socio-economic environments. this does not imply that such projects and efforts do not exist. many projects report on the impact of their initiatives to improve achievement in high-stakes school mathematics (see reddy et al., 2005). what is not visible in the reports of these projects and efforts are issues such as the underlying pedagogical and theoretical underpinnings of these projects. in addition to project reports there are some research-based projects on learner achievement in high-stakes end-of-year mathematics and the professional development of teachers. mogari, kriek, stols and iheanachor (2009), for instance, report on such a study. although the teachers in the study reported the professional development activities with which they were involved, there is no clear indication of the theoretical underpinnings. this article reports on a classroom-based project to improve achievement in high-stakes examinations. the mentioned underpinnings of the project and results of learner achievement over three years are presented. brief description of the underlying project and the research question the project, the local evidence-driven improvement of mathematics teaching and learning initiative, has as part of its aims the increase in the number of learners taking mathematics as an examination subject for the nsc examination, an increase in the pass rates and an improvement in the quality of the passes in the participating schools. the project developed an intentional teaching model (julie, 2013) for guiding instructional practices in classrooms. mathematics teachers from secondary schools in low socio-economic areas – bellville south, bishop lavis, bonteheuwel, elsies river, gugulethu, heideveld, kleinvlei, langa, manenberg, mfuleni and strand – in the cape peninsula participate in the project. the project focuses on the development of high-quality mathematics teaching to improve achievement in mathematics. a belief underlying the project is that improvement of teaching can lead to an enhancement of achievement in high-stakes examinations. generally the project operates by offering workshops and institutes attended by participating teachers. workshops are conducted after school and are usually of approximately two hours duration. two to three workshops are held per term for the first three terms of the school year. institutes are extended and residential gatherings held normally from a friday afternoon to sunday lunchtime. two institutes per year were held for the three years, 2012 to 2014. overall the teachers were engaged in 64 hours of continuing professional development activities for the three years for which results in the high-stakes school-based mathematics examination were tracked. the content of the professional development activities focused on pedagogical issues such as analysis of lesson excerpts, discussions around dilemmas teachers face in their teaching, searching for ways to address these dilemmas and the design of lessons. another feature of the content of the continuing professional development is that in most of the meetings teachers worked on mathematical problems with the aim of developing their mathematicalness – flexible ways of dealing with mathematics. the mathematics of the tasks is explored and discussed. the ways teachers worked with the tasks and the facilitation are then discussed in relation to how teachers can engage learners in doing mathematics. an example of a dilemma teachers face that was raised by teachers is that of learners not really doing homework. the purposes of homework were then discussed. one of the purposes offered was consolidation of completed work. this was connected in the discussions to the issue of forgetting. the outcome of the deliberations around the issue led to the development of a strategy for which the term ‘spiral revision’ was coined. basically this consists of learners being presented with two to three exercises of previously covered work which they have to complete in class. this has to be done 3–4 periods per week in about 7–10 minutes before dealing with the lesson for the day. ‘spiral revision’ is the project’s version of ‘distributed practice’ (see, for example, johnson & smith, 1987; seabrook, brown & solity, 2005; smith & rothkopf, 1984) which meta-analysis of meta-analytic studies found as one of the aspects that contributed towards enhancing achievement (hattie, 2009). the other purposes of homework were not addressed and teachers generally used their own ways of dealing with these purposes. other pedagogical aspects engaged with during the workshops and institutes were clarity to both teachers and learners of the intentions or goals of a lesson, the use of feedback and provision of opportunities to work with different problem types. these are also aspects which hattie’s (2009) meta-analytic work showed had moderate to high effect sizes related to achievement. the objective of the project, as stated above, is the improvement of achievement in high-stakes examinations. high-stakes examinations, which are discussed below, thus played a structuring role within which the above pedagogical aspects were dealt with as shown by using the example of quadratic inequalities. this brought the issue of examination-driven teaching into the picture. the research question being reported on in this article is: does an examination-driven teaching strategy improve achievement in high-stakes school-based end-of-year summative mathematics examinations in grade 10? high-stakes examinations as is evident from the research question, the notion of a high-stakes examination is one of the constructs of importance in this article. various notions of high-stakes examinations exist. these are normally linked to the purposes of the examinations. howie (2012, p. 82) classifies three kinds of assessments around high-stakes examinations – ‘classroom assessment, system assessments [and] public examinations’ – with their purposes, frequencies, test cohorts and subject area coverage. she does not include school-based end-of-year summative assessments in her classification of assessments. in a recent survey on assessment in mathematics suurtamm et al. (2016), include the last-mentioned assessments and view it as ‘increasingly play[ing] a prominent role in the lives of students and teachers as graduation or grade promotion often depend on students’ test results’ (p. 4). in this article, a high-stakes examination is one that has direct consequences, positive or negative, for the examinees. particularly for grade 10 learners, the school-based end-of-year mathematics examination has consequences such as promotion to grade 11 or not and the right to continue taking mathematics as an examination subject for the nsc examination. non-continuation with mathematics up to grade 12 is a major issue. for the research reported here, of the 403 learners in the five participating schools who wrote the 2012 project-designed examination in grade 10 only 280 proceeded to write the 2014 nsc mathematics. this is an instance of the decrease in taking mathematics from grade 10 to 12 in a cohort of learners. adler and pillay (2017) indicate that in one of their project schools only 22% of learners who took mathematics in grade 10 proceeded with mathematics as an examination subject 3 years later. various reasons for learners’ non-continuation with mathematics up to grade 12 are offered. some of these are: failing mathematics in grade 10 but being promoted to grade 11 due to fulfilling the promotion rules and dropping mathematics for mathematical literacy and failing grade 11 (adler & pillay, 2017). the importance of taking mathematics, and being successful, as an nsc examination subject–for access to tertiary studies, job opportunities, etc.–attests to the high-stakes nature of the school-based end-of-year mathematics examination in grade 10. for school-based end-of-year summative assessments the focus is on learners. as indicated above, success or not in these examinations has consequences for them. at a very basic level success (and the level of the success) or failure on these assessments decides whether or not learners will be able to proceed from grade 10 onwards to be awarded a certificate of worth that they can use after their completion of schooling. the nsc in south africa is such a certificate. the consequence of having at least this certificate is that it greatly enhances the chances of school-leavers to obtain employment and access to further studies. in this regard statistics south africa (2016, p. xiv) reports that ‘those without matric constituted more than 58% of the unemployed among the black africans and coloured population’. matric refers to the exit level grade (or grade 12) of the south african non-compulsory schooling system. success in grade 10 is the entry point for learners in their journey of pursuit to obtain this valued certificate. in this article, high-stakes examinations are viewed as those that allow learners to progress from one grade to another and in particular to the exit level, grade 12. as mentioned, the high-stakes examination is the school-based summative examination learners write at the end of the school year, grade 10 for the purposes of this study. this examination is normally internally constructed and marked. to ensure quality, the head of the mathematics department of the school normally moderates both the construction and the marking of the examination. further quality assurance and consistency across schools are ensured through a process of external moderation by the mathematics curriculum advisors of the department of education (jacobs, mhakure, fray, holtman, & julie, 2014). regarding the administration of the end-of-year school-based examination, similar security procedures such as the preparation, printing and release of the examination on the day it is written, as is the case for the nsc examinations, are followed. the staff of the school does invigilation during the writing, normally with the subject teachers not invigilating their own examination. school-based summative end-of-year examinations thus follow the processes and procedures that approximate those of the high-stakes nsc examinations. thus, in terms of purpose and the entire set of processes and procedures, the end-of-year grade 10 examination is a high-stakes examination. examination-driven teaching as underpinning of the project according to julie (2013), examination-driven teaching is normally viewed as ‘teaching the content of previous examinations and anticipated questions that might crop up in an upcoming examination of the subject’ (p. 1). examination-driven teaching is a controversial issue. debates about it abound. opponents of examination-driven teaching argue that it leads to the fragmentation of knowledge, the restriction to low-level content, the fostering of the loss of disciplinary coherence, mitigation against flexible knowing, curriculum contraction, deskilling of teachers and the inhibition of making sound instructional decisions due to the predominant psychometric paradigm underlying high-stakes examinations (davis & martin, 2006; shepard & dougherty, 1991; van den heuvel-panhuizen & becker, 2003). davis and martin (2006) also draw attention to examination-driven teaching being more often adopted as a preferred instructional approach to address low performance of learners from low socio-economic environments. proponents of examination-driven teaching, on the other hand, draw attention to its advantages for improving achievement outcomes. these include clarity of instructional goals, cost-effectiveness, motivation and examination assistance for learners by providing clarity on the kinds of problems they can expect to encounter in a high-stakes examinations and the feedback that examinations-driven teaching provides to teachers for instructional decision-making (popham, 1987; shepard & dougherty, 1991). notwithstanding the debates about examination-driven teaching, there are considered positions about the structuring roles examinations exert on instructional practices. one such position is that examinations play a major role in the constitution of legitimate and valued school mathematics knowledge. bishop, hart, lerman and nunes (1993, p. 11) contend that ‘examinations operationalise the significant components of the intended mathematics curriculum, so they tend to determine the implemented curriculum.’ according to julie (2013) ‘the intended and interpreted curricula provide only boundaries of content to be dealt with but the implemented curriculum is heavily driven by the examined curriculum’ (p. 6) and the examined curriculum eventually drives what is taught regardless of what the aims of the curriculum are. the recognition of the structuring effects of the examined curriculum provides a strong argument that in order for teaching to comply with meaningful learning, examinations must be changed (burkhardt & pollak, 2006; van den heuvel-panhuizen & becker, 2003). julie (2013) presents an argument that examination-driven teaching can contribute towards meaningful learning. he states that examination-like questions and mathematics problems that learners are exposed to during classroom teaching can be changed to questions that elicit process skills and develop critical and conceptual thinking skills. swan and burkhardt (2012, p. 5), concurring with julie, state that if items that require that learners demonstrate their ability of critical and conceptual understanding are included in high-stakes examinations then ‘teachers who teach to the test [can] deliver a rich and balanced curriculum’. a careful study of the nsc mathematics examination indicates that such questions are part of the examination. admittedly, the number and variety of such items need to be increased in assessments in all grades in the further education and training band. to realise this increase, attention must be given to the percentages of items prescribed to be at certain levels of cognitive demand in the mathematics curriculum documents. in south africa there is currently an emergence of the use of large-scale systemic assessments to structure continuing professional development initiatives for mathematics teachers. shalem, sapire, and huntley (2013) worked with teachers to do curriculum mapping of large-scale assessments which led to teachers reflecting on their instructional practices with respect to the content taught and the level of cognitive demand that is focused on. the data informed practice improvement project of brodie (2013) focused on teachers engaging with misconceptions and errors resulting from learners’ responses to examination items. this was followed by teachers designing lessons based on the analysis of the examination items. the lessons were implemented in their classrooms and teachers reflected on the efficacy of their implementation to address the identified errors and misconceptions. in the project of interest in this article, examinations are used in a similar way to those described in the foregoing paragraph. learners’ responses in examinations are used to reflect on difficulties learners display in examinations, design of activities to address such difficulties and backward mapping from the high-stakes nsc mathematics examination to provide focus for teaching in lower grades. for example, in the curriculum and assessment policy statement document (department of basic education, 2011, p. 13) the content related to quadratic inequalities is given as solve ‘quadratic inequalities in one variable and interpret the solution graphically’. this has to be dealt with in grade 11 and manifests itself as ‘solve for x: (x + 1)(4 − x) > 0’ as a level 1 (lowest level) question in the nsc mathematics examination. in grade 10 it is prescribed that quadratic graphs of the form y = af(x) + q, where f(x) = x2, should be dealt with but the solution of quadratic inequalities is not. this is understandable given the restriction, as stated in the aforementioned sentence, for graphs of quadratic functions in grade 10. in the project, learners are exposed to solving quadratic inequalities under the topics dealing with the real number system. the graphs of quadratic functions, without specifying the defining expression, are given and learners have to solve quadratic inequalities with a generically specified defining expression as given figure 1. figure 1: task on quadratic inequalities when dealing with the real number system. tasks of the nature given in figure 1 are done during the first week of the first term when learners have not yet dealt with the quadratic functions and their graphs. this is followed up when quadratic functions of the form indicated for grade 10 are taught and items related to quadratic inequalities are included in the year-end examination as indicated in figure 2. figure 2: examination item on the quadratic function (the quadratic inequality item is 6.1.4). it is the contention of the project that if learners start engaging with questions that they will encounter in the ultimate nsc mathematics examination as early as grade 10 then they will have high levels of fluency to deal with the cognate problems in the nsc mathematics examination. in this section we presented an indication of how examination-driven teaching is conceived and implemented as underpinning in the project. the next section discusses the research design. research design a quantitative design was adopted in this study because learners’ scores are used to describe the phenomenon being investigated. the study is a trend study where results of the same phenomenon are tracked over a period. it is different from a tracer study which follows the results of the same cohort over a period. the trend of the overall mathematics scores in the end-of-year summative school-based grade 10 examination over three years – 2012 to 2014 – was thus investigated. trend studies are appropriate in situations of curriculum stability. the trends in international mathematics and science study (timss) project does trend studies (martin, mullis, & chrostowski, 2004). an important requirement of trend studies is that the items used in the assessment instrument should be similar in kind and degree. end-of-year summative school-based grade 10 examinations are such and thus appropriate as instruments for use in the research reported here. two methods, rasch modelling and other statistical methods, were used. these methods are described in the section on analysis procedures. sample and sampling procedure the sample was an opportunistic sample of five schools whose teachers were involved in the continuing professional development initiative. ten schools were initially involved in the project. after the first year of implementation, the participation of one of the schools was terminated due to unsatisfactory participation in project activities. not all the schools wrote the project-set common examinations for the reporting period. the reasons for this are: (1) the timing and availability of the common question papers in that some schools had their examination timetables ready before common agreed examination dates could be negotiated, (2) the standard of question papers was deemed too high in terms of their cognitive demand according the judgement of the teachers of their learners’ cognitive levels and (3) the grade 10 learners of one school were not available in 2013 and 2014 because they had gone to another school following a prior arrangement. it needs to be borne in mind that teachers’ participation is voluntary and so the decision to write the common examination or not rests with the schools. voluntary participation and the right to withdraw from research activities or part of it are important ethical principles in a research project involving human participants. this was made clear to teachers at the start of the continuing professional development initiative. this resulted in five schools who wrote the project common examination for the three years. a possible threat emanating from working with samples over different years is that the characteristics of the cohorts of participants might change and have a confounding effect. major confounders are normally race, gender, age, class size and school type. regarding gender, although the names of the learners appeared on the scripts, the difficulty of using names as a signifier for gender is highly problematic. ‘cyril’, for example, can either be male or female. furthermore, in a school-based examination learners do not indicate their gender. gender dimensions were thus not included. other confounding factors that might be linked to the contexts of the schools might have changed. however, the nature of schools in south africa is such that the enrolments are reasonably stable with regard to socio-economic status and demographic composition. our own observations during classroom support visits revealed no observable change along these lines. in line with common practice for school-based end-of-year summative mathematics examinations, the examinations are governed by the assessment guidelines as described in the curriculum and assessment policy statement. the curriculum and assessment policy statement document describes modalities such as the topics and their weightings to be covered and percentage of marks to be allocated to the different levels of cognitive demand. the school-based end-of-year summative mathematics examination comprises two papers of 2 hours duration each. the first paper deals with the topics (their weightings given between brackets): algebra and equations (and inequalities) (30 ± 3), patterns and sequences (15 ± 3), finance and growth (10 ± 3), functions and graphs (30 ± 3) and probability (15 ± 3). the topics dealt with in the second paper are: statistics (15 ± 3), analytical geometry (15 ± 3), trigonometry (40 ± 3) and euclidean geometry and measurement (30 ± 3). the examinations adhered to these guidelines and were thus similar in kind and degree. to protect anonymity the schools are named a, b, c, d and e. table 1 gives the number of learners in the different schools for the period 2012 to 2014. table 1: number of learners per school, per year and per paper. it can be observed from table 1 that a small number of learners who wrote the first paper did not write the second one. in this study, the scores for learners who missed a paper were treated as missing data. thus the total number of learners was taken as 403 for 2012, 381 for 2013 and 406 for 2014. the presented results are thus not representative of all the schools participating in the local evidence-driven improvement of mathematics teaching and learning initiative project or of grade 10s in the western cape province or in south africa. therefore, to generalise about the outcomes for the entire province or for the country requires careful consideration if the results are to be more broadly applied. it also needs to be borne in mind that there are many interventions, for which information was not gathered for the participating schools, addressing the low performance of learners in school mathematics. thus, the results may be confounded by influences of such interventions. however, it is well known that many interventions at grade 10 level focus on selected learners with potential and short-term teacher initiatives focus on the enhancement of subject matter knowledge. the focus of the underlying project as referred to above was on the development of quality teaching. data and data collection the data were the scores learners obtained in the end-of-year grade 10 mathematics examinations. these scores comprise 75% of the total mark of 200 that is awarded for mathematics. the other 25% is compiled from tasks, tests and the mid-year examination. participating mathematics educators, mathematicians, mathematics teachers and mathematics curriculum advisors set the examinations. the mathematics educators and the mathematics curriculum advisors firstly designed draft items. these were discussed with teachers at workshops to ensure that there was fairness with regard to the topics that were covered in their teaching. upon reaching consensus, the examination papers and the memoranda of marking were moderated by the participating mathematicians. figure 2 is an example of an item of the examination. the project staff designed the final versions of the examination papers and electronic versions were dispatched to schools for them to put in a format as required by the schools. for example, most schools follow the format where the cover page of their examination papers must have the school’s emblem on it. in order to prevent leakage of the examination papers, the school management teams were approached to timetable the examinations for the same date and time. the five schools agreed to this request. as is normal for school-based end-of-year examinations the responsible mathematics teachers of the schools marked the scripts. except for two schools, the same teacher taught grade 10 for the three years. for the one school where this was not the case, the school uses the strategy of a teacher ‘taking the learners through’ from grade 10 to 12. this school had two teachers involved and they both attended all the project activities. the other school changed the teacher responsible for teaching mathematics in grade 10 in 2012 due to the responsible teacher being on maternity leave for the first half of 2013. the other teacher taught grade 10 for 2013 and 2014 and also attended all project activities. to ensure consistency of marking across the five schools, a -day common marking session was held in 2012. this was not repeated for 2013 and 2014 since the same teachers who were involved in the 2012 administration were those for 2013 and 2014. it was assumed that the teachers would mark the scripts according to the procedures applied for the 2012 examination. further, the project had access to the scripts to record the marks and no observable deviations from agreed-upon marking procedures developed in 2012 were found in 2013 and 2014. the marked scripts were collected from the schools once all the administrative procedures that schools are required to do were completed. the score for each item – the sub-sections of a question – for each learner from a school was captured. therefore, the only data recorded were the scores as reflected on the scripts of the learners. after the collection of the common examination scripts, the data were checked, cleaned and coded as described in okitowamba (2015). ethical considerations the university’s research ethics committee cleared the project of which this particular study is a part with the ethics registration number 11/9/33. the project was also approved by the western cape education department through a memorandum of understanding between the university and the western cape education department. in order to maintain anonymity, the names of learners were not used and not recorded in the data files. the scripts were assigned numbers for purposes of checking during the data cleaning phase. the names of schools were also anonymised as indicated in table 1. data analysis rasch procedures there is a vast body of literature in which rasch measurement theory is broadly explained, from its origin to its applications (andrich, 1978; bond & fox, 2010; dunne, long, craig, & venter, 2012; griffin, 2007; long, 2011; rasch, 1960/1980; wilson, 2005; wright & stone, 1979). an important use of rasch analysis is the computation of ‘“measures” that can … be used with parametric statistical tests’ (boon, staver & yale, 2014, p. 3). furthermore, application of rasch procedures provides a solution ‘of measuring changes across time in achievement … at the same grade level over several years’ (scantlebury, boone, kahle, & fraser, 2001, pp. 649–650). similar to the study reported here, rasch measurement was used to compare different cohorts of students in australia and to detect improvement in students’ mathematics achievement in lower secondary schools over time (afrassa & keeves, 1999). this method fits the purpose of the investigation as it can help to detect whether the trend in achievement for different cohorts of learners is in a positive or negative direction. a more in-depth discussion regarding the implementation of rasch analysis for this research is given by okitowamba (2015). the software winsteps 3.4.1 (linacre, 2008) was used for rasch measurement. given that the scores were not dichotomous and for these examinations partial scores are awarded, the procedures for rasch partial credit model for polytomous data were applied. reliability and validity concerning reliability, rasch measurement provides person reliability index and item reliability indexes. these indices were specifically calculated from the raw data for this article and are presented in table 2. table 2: reliability index by year. the acceptable range for reliability coefficients of an instrument is that it should be greater than or equal to 0.70 (mcmillan & schumacher, 2010). the indexes in table 2 indicate that reliabilities were within the acceptable range for the three cohorts of learners in this study. the near equality of these coefficients is also indicative of the examinations functioning in a similar fashion for the three years. regarding validity, content validity was assured through the construction of the test by the set of ‘experts’ – the teachers, mathematics educators, mathematicians and the mathematics subject advisor. the examinations were administered under normal conditions for examinations during the end-of-year examination period and followed the same processes and procedures for such examinations. there were no deviations from the way end-of-year examinations are conducted and administered. this implies that the ecological validity – ’the degree to which an assessment of events, activities, participation, or environments reflects everyday life expectations’ (crist, 2015, p. 2) – of the study was high. reasonably high ecological validity of studies is, according to black and wiliam (1998), an important determinant for acceptance by teachers of ideas and notions emanating from research. in the rasch analysis construct validity focuses on the idea that the recorded performances are reflections of a single underlying construct [and] fit indices help the investigator to ascertain whether the assumption of unidimensionality holds up empirically. (bond & fox, 2010) for a test to satisfy unidimensionality only the items within the range 0.5 and 1.5 are deemed productive for measurement (linacre, 2008, p. 227). the ranges for the infit and outfit statistics for 2013 were 0.85 to 1.14 and 0.62 to 1.28 respectively. thus a condition for construct validity satisfied. this was also the case for the 2014 test where the range for the infit statistic was 0.23 to 1.44 and 0.29 to 1.4 for the outfit one. it was not the case for the 2012 test–infit statistic range 0.37 to 1.97 and outfit statistic range 0.26 to 1.75. for the 2012 examination, four items fell outside the range and were not included in the analysis. these items were spread over the two papers. in total they accounted for 2.5% of the total marks. they were at the ‘easy’ end of the person-item map. it was deemed that the contribution to the total score of the four items was negligible in the sense that it would not affect the analysis detrimentally. the items were excluded from the analysis rendering all three complying with a condition for construct validity. the 2012 examination had 74 items (405 persons), the 2013 one 82 items (82 persons) and there were 62 items (407 persons) for the 2014 examination. fit statistics provided by rasch analysis is a quality control mechanism to determine whether data holds the assumption of unidimensionality (each item contributes to measure only one construct that represents one attribute or ability at a time). the data used thus measured what it was intended to measure (construct validity). other statistical analysis tests statistical significance between the scores of 2012 and 2013, 2013 and 2014 and 2012 and 2014 was determined using t-tests. the software spss version 23 was used. ‘effect sizes’ were calculated to assess the effectiveness of the intervention. according to coe (2002, p. 1): ‘effect size’ is simply a way of quantifying the size of the difference between two groups [and] … particularly valuable for quantifying the effectiveness of a particular intervention. … it allows us to move beyond the simplistic, ‘does it work or not?’ to the far more sophisticated, ‘how well does it work in a range of contexts?’ different formulae exist for determining effect size (see, for example, cohen, 1988; kerby, 2014; mcgraw & wong, 1992). in this study, the one that was used by afrassa and keeves (1999, p. 4) was utilised because of the similarity of this research and the study reported here. the formula is: where = estimated mean for group one (year x) = estimated mean for group two (year x + 1) s1 = standard deviation of the mean of group one s2 = standard deviation of the mean of group two since this study mirrors the one by afrassa and keeves (1999), the interpretation of the effect sizes given by them (p. 4) are adopted. namely: es < 0.20: the size of effect is trivial 0.20 ≤ es < 0.5: the size of effect is small 0.50 ≤ es < 0.80: the size of effect is medium es ≥ 0.80: the effect size is large there are other interpretations for acceptability of effects sizes. hattie (2009), for example, accepts an effect size as useful if it exceeds 0.4. results and discussion the results of the analysis are presented in this section. they are given for three periods: 2012 to 2013, 2013 to 2014 and 2012 to 2014. table 3, table 4 and table 5 present the outcomes of the various statistical analyses. table 3: mean scores of the five schools for the three cohorts. table 4: t-test for mean difference for cohorts 2012 and 2013. table 5: the effect sizes for different periods. table 4 indicates that the differences were significant for the three periods. for the period 2012 to 2013 the mean score declined. it is conjectured that the introduction of the curriculum and assessment policy statement in grade 10 in 2012 contributed to the low average score in 2012. furthermore, regarding the contribution of the project, it required teachers to use teaching strategies with which they were not fully conversant. given that teaching is a habit, it is widely accepted that the appropriation of other strategies takes time. for the period 2013 to 2014, the mathematics mean scores improved from 22.32% to 35.42% which was a significant increase. contributors to this improvement are obviously the increased familiarity with the curriculum and an emerging implementation of some of the strategies linked to examination-driven teaching (julie, 2016). the trend also reveals a significant positive increase for the period 2012 to 2014. figure 3 presents the trends graphically. figure 3: trend of learners’ mathematics performance over time. the calculated effect sizes are given in table 5. table 5 shows that the effect from 2012 and 2013 was trivial. as is argued by julie (2016) this decrease follows the pattern of any improvement initiative where there is an initial deterioration before improvement. the effect size from 2013 to 2014 was large and between 2012 and 2014 it was medium. taking the average effect of 0.40 as a crude indicator of an overall effect then it is small but within hattie’s (2009) cut-off for a reasonable effect. conclusion in light of the study outcomes reported above, there are positive indicators that the trend of mathematics achievement in school-based end-of-year summative examinations moves in a positive direction over time when an examination-driven teaching strategy is employed. however, this improvement is not necessarily immediate and the nature of examination-driven teaching must be carefully considered and crafted to counter a minimalist view of teaching to the test. examination-driven teaching is definitely not the only contributor since enhanced achievement in high-stakes examinations is closely related to socio-economic status and ultimately the desirable achievement results that are sought will only materialise if socio-economic inequalities are substantially reduced. acknowledgements competing interests the authors declare that they have no financial or personal relationships that may have inappropriately influenced them in writing this article. authors’ contributions o.o. conducted the research, did the statistical analysis and wrote the draft of the manuscript. c.j. was the project leader, conceptualised the project, assisted with the data collection and analysis and contributed to the writing of the manuscript. m.m. contributed towards the discussion and conclusions of the research and did final editing. funding this research is supported by the national research foundation under grant number 77941. any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views the national research foundation of south africa. references adler, j., & pillay, v. 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(1979). the measurement model. in b.d. wright & m.h. stone (eds.), best test design (pp. 1–17). chicago, il: mesa press. 66 p85-96 forgasz & murimo final pythagoras 66, december, 2007, pp. 85-96 85 depictions of females and males in mozambican and victorian (australia) primary mathematics textbooks adelino evaristo murimo and helen forgasz monash university, australia email: mailto:ademurimo@yahoo.com.au and helen.forgasz@education.monash.edu.au the depiction of females and males in mozambican primary mathematics textbooks for grades 6 and 7 were examined, and comparisons made with victorian (australia) textbooks for years 5 and 6. it was found that mathematics learning was portrayed as a male domain in the mozambican textbooks, reflecting what used to be the case with australian texts of the 1970s, although there are some differences between the two countries that may be culturally based. the victorian textbooks depicted mathematics learning as a domain for all children, and in the majority of categories examined, females and males were distributed fairly evenly. compared to earlier reported findings, there was a general improvement in the portrayal of females in victorian textbooks. introduction and background to study in mozambique, activities aimed at eliminating gender disparities in education became popular in the 1990s through unicef intervention projects soon after the country ratified the declaration on education for all (efa) (bernard & cabral, 2002). since then, there have been three main objectives forming the focus of unicef projects: to increase the enrolment rates of girls in the early school grades, to ensure that enrolled girls do not drop-out, and to ensure that girls do not fail the grade they are attending (guz, 2000; bernard & cabral, 2002). however, despite the efforts of the government and non-governmental organisations, a national evaluation report carried out by unicef indicated that although the school participation of girls and boys in grades 1 to 5 tended to be similar, there were fewer girls than boys in grades 6 and 7, and there were differences across the provinces (guz, 2000). moreover, fewer than 40% of mozambican children complete primary education and there is a gender gap in completion rate of over 20%, with fewer girls completing primary schooling than boys. unicef therefore questioned whether mozambique was going to achieve one of the key purposes of the united nations millennium development goals (mdg), the completion of primary education for all by 2015. to date, official accounts of the gender disparities in mozambican schools have been attributed to poverty in the majority of families, and to schools which are unsafe for girls because of harassment by adult males (guz, 2000; bernard & cabral, 2002; unicef, 2006). it has been claimed that some parents tend to keep their daughters at home so that they can help with farming, fetching water, and even income generation. the long walking distances to schools and the lack of piped water and separate facilities in schools have also been identified as causes for gender disparities in education. as a consequence, the affirmative actions of government and nongovernmental organisations to promote girls’ education have concentrated mostly on subsidiary issues, such as money given to girls to support them at school. little has been done to look into the curriculum, that is, to question to what extent the curriculum is ‘girl-friendly’. in mozambique, gender equity in education has been discussed mainly in terms of equal participation of girls and boys in the classroom (guz, 2000; bernard & cabral, 2002; unicef, 2006). although this is an important starting point, one of the other critical dimensions of gender equity in education is equal educational outcomes (fennema, 1990). equal educational outcomes demand that girls and boys must both experience good quality teaching. in fact, fennema (ibid.) linked the problem of gender equity in mathematics education to certain teaching approaches valued by teachers: achieving equivalent outcomes in mathematics education for males and females may require that teachers actually treat the sexes differently. it could be that the most effective teaching for males is different from that of females. for example, there is some evidence that classes organised to provide competition result in somewhat better learning for males, while classrooms that encourage cooperation depictions of females and males in mozambican and victorian (australia) primary mathematics textbooks 86 are better for females…[italics added] (fennema, 1990: 5-6). fennema (ibid.) also argued that there were reasons other than economic ones leading to gender disparities in mathematics learning. she pointed out that females contributed to mathematics knowledge in the past, but that societal attitudes inhibited such contributions. she suggested that girls should be given opportunities to experience and to contribute to mathematics knowledge just as boys do. it was fennema’s (1990) arguments that motivated the analysis of gender differences in mozambican textbooks and the comparison with victorian (australia) textbooks that are reported in this article. the inclusion of textbooks from victoria was purposeful because efforts to bring about gender equity in textbooks in australia date back to the 1970s (australian college of education, 1977; clarkson, 1993; forgasz, 1996; mckimmie, 2002). also, gender issues have been discussed in teacher education programs in australia and there is general acceptance that females and minority groups should be treated fairly in school materials (clarkson, 1993). the study reported here sought to answer the following questions: 1) are the photos and drawings illustrating females and males distributed with the same frequency in the mathematics textbooks? 2) among the photos and drawings illustrating people, how often do females and males appear together? 3) do females and males share equal active (physical) and passive (non-physical) roles? in their active and passive roles, are females and males equally mathematically cognitively engaged? 4) are females and males mentioned with the same frequency in mathematical problems? 5) how are selected mathematics educational values (accessibility and ‘specialism’ – defined on page 89) evident in the textbooks? are there differences along gender lines? 6) what are the important differences in the findings in the textbooks of mozambique and victoria? for a better understanding of the differences and similarities in the data obtained in this study, an overview of the mozambican system of education is described briefly in the section that follows. mozambican system of education due to important differences between mozambique and victoria in terms of language used for school instruction, curriculum and culture, the mozambican system of education is described in order to assist in understanding the findings from this study. the language of instruction in mozambique is portuguese, a legacy of over 500 years of western and colonial experience. as in many other african countries, the mathematics curriculum in mozambique is a curriculum transplantation, that is, the content reflects western values (gerdes, 1988). despite the vast numbers of people entering schools soon after independence in many former western colonies, including mozambique, the mathematics curriculum retained its western roots. the mozambican ministry of education and culture is currently attempting to restructure the curriculum by considering issues such as native languages, gender and cultural values (guz, 2000). the formal education system used in mozambique is a ‘5 + 2 + 3 + 2’ model which means children study five years at lower primary level, two years at upper primary, three years at lower secondary, and two years at upper secondary level. they graduate from one level to the next level by sitting for national examinations. portuguese and mathematics are compulsory subjects for all students up to lower secondary level. in order to move from one level to the next, students must be competent in both subjects as these are considered the core subjects in further studies (ministério de educação e cultura, 2003). at the upper secondary education level, mathematics is optional. mathematics teaching in mozambican schools in a typical mozambican primary school mathematics classroom, the furniture is organised in rows of individual chairs and desks, usually facing the blackboard where teachers have their tables nearby. since 2003, every primary school child has received a free textbook which must be returned to the school at the end of the year to be reused by another child. mathematics lessons usually begin with the teacher speaking to the whole class, giving instructions about activities in the textbook. students then complete the activities individually. even in cases where children are seated in desks in pairs, or when there are no desks and they sit on the floor, students are working individually. usually the teacher will call for silence as students adelino evaristo murimo and helen forgasz 87 work on the activities; any queries related to the activities are directed to the teacher. in mozambican classrooms the value of competition in mathematics learning is emphasised more than cooperative work, and students with difficulties are usually ignored (luís, 2004). prior research gender issues and mathematics learning gender equity in mathematics learning has been defined according to three dimensions: equity as equal opportunity to learn mathematics, equity as equal educational treatment, and equity as equal educational outcomes (fennema, 1990). the dimension referred to as equal opportunity to learn mathematics, focuses on the equal right to participate in mathematics learning for both females and males. equity as equal educational treatment is concerned with fair teaching in the mathematics classroom (and in textbooks), as females and males experience it through teacherstudent (and textbook-student) interactions. the third dimension, equity as equal educational outcomes, attempts to ensure that: at the end of schooling, there should be no differences in what females and males have learned, nor should there be any gender differences in how students feel about themselves as learners of mathematics. males and females should be equally willing to pursue mathematics-related careers and should be equally able to learn new mathematics as it is required (fennema, 1990: 5). the reasons for gender differences in mathematics learning are multifaceted and there are many explanations for them (for example, leder, 1990; 1992; leder, forgasz & solar, 1996). leder (1990; 1992) reviewed research on gender differences in mathematics learning and identified the main variables commonly used to explain why, on average, females’ achievement levels were lower than males’, and why females were less likely to be enrolled in challenging mathematics courses – as summarised in table 1. as shown in table 1, gender differences in mathematics learning have been explained using two main categories of variables: learner-related and environmental variables. the learner-related variables are directly associated with the learner, while the environmental variables are external influences on the learner. each category of variable is further divided into sub-categories. the learnerrelated variables are divided into cognitive development (spatial ability and verbal ability) and the personal beliefs. the internal belief system comprises a range of affective variables (table 1) which were found to influence participation and achievement in mathematics, with males generally cognitive development spatial ability verbal ability confidence usefulness of mathematics sex-role congruency learner variables beliefs motivation fear of success attributional style learned helplessness mastery orientation performance following failure society law, media, peers, cultural expectations home parents, siblings, socioeconomic status environmental variables school teachers, organization, curriculum, textbooks, assessment, peers table 1. two categories of variables commonly used to explain gender differences in mathematics learning (adapted from leder, 1990: 15). depictions of females and males in mozambican and victorian (australia) primary mathematics textbooks 88 holding more functional (likely to succeed) beliefs than females (leder, 1990). leder’s (ibid.) second category of variables comprised environmental variables which were divided into three subcategories: society, home and school (table 1). based on an extensive literature review, koehler (1990) found that learning mathematics among males and females was strongly influenced by the educational environment including, for example, textbooks that were well known for their stereotyped remarks, sex bias, and the presentation of mathematics as a male domain (more suited to males than to females). in classrooms, teachers were found to spend more time with boys than with girls, with boys asking and being asked more questions than girls. boys were also criticised and praised more often than girls. textbooks, gender and mathematics learning forgasz (1996) investigated gender equity in an australian mathematics textbook and produced a checklist of five equity issues (table 2). it was claimed that in order to be considered gender equitable, textbooks and other educational materials needed to meet the five criteria (ibid.). appropriate depictions of females and males in mathematics textbooks are important because pictures and text that depict gender stereotyped roles and occupations and gender-biased language can send implicit messages to females and males that they incorporate into their understandings of life around them, and their place in it. genderbiased text and images can contribute to or reinforce students’ beliefs about gender role expectations. when the materials portray stereotyped images, students might develop inaccurate perceptions of their real potential and the potential of others (stretmatter, 1994). values of accessibility and ‘specialism’ in mathematics learning bishop (1996: 19) defined values in mathematics education as “the deep affective qualities which education aims to foster through the school subject of mathematics”. according to bishop (1988) and bishop, seah and chin (2003), mathematics teachers promote three categories of values in classrooms: general educational, mathematical, and mathematics educational. general educational values do not have a mathematical component; however, teachers use a mathematical activity to discuss issues related to environmental protection or behaviour, for example. mathematical values are related to the nature of mathematics itself, particularly the way mathematics knowledge was developed in different cultures over time. mathematics educational values are related to the nature of teaching and learning mathematics as a school subject. although the three categories of values are important for research, the focus in the study reported here was only on mathematics educational values. seah (1999) characterised mathematics educational values in the following terms: the norms and practices of doing mathematics as advocated by the mathematics teacher or textbook reflect values which are both mathematical and educational. such values include encouraging or expecting pupils to display their workings of a problemsolving exercise in detail, as well as to double-check answers for accuracy before submission. (1999: 18) according to seah (ibid.), there are many categories of mathematics educational values to be found. however, in his study of textbook materials, seah focused on five values that were most likely to be found in the classroom or in textbooks. these 1 equal numbers of males and females 2 males and females sharing active (‘doer’) and passive (helper/observer) roles 3 a balance of contextual settings of interest to males and females (not necessarily gender-neutral activities) 4 the settings, occupations and professions illustrated or mentioned are not gender-stereotyped 5 the images of people and their names clearly reflect a multicultural profile table 2. a checklist for a gender-equitable mathematics textbook (adapted from forgasz, 1996: 2). adelino evaristo murimo and helen forgasz 89 values, and their complementary pairs, are listed in table 3. as can be seen in table 3, each mathematics educational value has a complementary value. accessibility and ‘specialism’ were selected as the focus of the study reported here because they contain text features that can be analysed along gender lines. text features of accessibility can convey embedded messages to students that everyone can find a place in mathematics learning (bishop, 1988; seah, 1999) rather than mathematics as a field for only a certain group of people (for example, males). text features of ‘specialism’ convey the implicit message that in order to do well in mathematics, one needs to possess special knowledge and skills, such as appropriate vocabulary and formulae. in fact, accessibility is one of the principles that bishop (1988) suggested for the development of an enculturation curriculum aimed at reaching all children: mathematics education should be for all… the curriculum content must not be beyond the intellectual capabilities of the children, nor must the material examples, situations, and phenomena–to–be– explained, be exclusive to any one group in society. (1988: 96) according to seah (1990), accessibility in mathematics learning is promoted in text materials when the following text features are evident: workers, labourers, females, and minorities are depicted using mathematical ideas in their daily life; students are invited to work collaboratively, and share information; the textbook writers convey concern by checking affective variables such as student confidence, attitude, and self-evaluation. ‘specialism’ is the complement of accessibility, with text features including the following: mathematical concepts are presented with specialist vocabulary, often introduced through formal definitions; mathematics is presented as a product of gifted human actions (seah, 1999). the instruments used in this study to analyse the selected mathematics textbooks were developed out of the gender equity checklist suggested by forgasz (1996) and the text features conveying accessibility and ‘specialism’ suggested by seah (1999). in the next section, a brief overview of the assumptions about the social world that informed the study is presented. thereafter, the research methods used to select the textbooks, and the methods adopted to assess the validity and reliability of data will be discussed. assumptions about the social world the study was informed by three paradigms, namely, positivism, interpretivism and constructionism, each of which reflects assumptions about the social world (terre blanche & durrheim, 1999). underpinning the positivist paradigm is the belief that there is an objective reality and that this reality can be accessed through observation that is directly experienced, measured, and verified by independent researchers. the aim of the positivist paradigm is to verify and describe in a rigorous way what people have already taken hold of or suspected (terre blanche & durrheim, 1999). the interpretive point of view is concerned with the meanings that people attach to existent facts. in adopting an interpretive stance, the intention is to find out and describe how social phenomena are experienced by the people involved in those situations (terre blanche & durrheim, 1999). the basis of the social constructionist paradigm is a belief that there are multiple and valid realities. hence, constructionist researchers are interested in understanding how such realities, or ‘human constructs’, appear among a particular group of people. in other words, constructionism is interested in understanding how people determine what is normal and abnormal in particular contexts (terre blanche & durrheim, 1999). terre blanche and durrheim (1999) used a photograph in a magazine to illustrate how the positivist, interpretivist and constructionist paradigms have three quite distinct readings of the same scene. in this study the three paradigms were blended “because different paradigms exist values complementary values instrumental understanding relational understanding formalistic view of mathematics learning activist view of mathematics learning ‘specialism’ accessibility theoretical relevance process product table 3. mathematics educational values (adapted from seah, 1999: 125-126). depictions of females and males in mozambican and victorian (australia) primary mathematics textbooks 90 simultaneously, it is possible for the same researcher to draw on more than one paradigm, depending on the kind of work they are doing” (terre blanche & durrheim, 1999: 7). the positivist position was of value to the study because it was important to know observable facts such as the number of females and males portrayed in the textbook materials. likewise, the interpretive position was required to interpret the hidden messages that readers may gain from the textbooks, for example, it was important to discern whether mathematics might be portrayed as a ‘male domain’, ‘female domain’, or a ‘domain for all, females and males’. the constructionist perspective was equally valuable because of sociopolitical differences between mozambique and australia, for example, culture, context, language, and curriculum. methods selection of textbooks and grades mozambican primary mathematics textbooks for the grades 6 and 7 were selected for analysis because they were recently issued in mozambique and there has been no study of them. the analysis of gender issues in these textbooks was particularly important as gender differences in primary education in mozambique, as discussed earlier, are relatively large in these grades. nelson mathematics textbooks were selected in victoria, australia, because they were first introduced in the same year as the mozambican textbooks (2003). also, nelson mathematics is one of the most popular titles in victoria. it should be noted that with respect to curriculum content, grades 6 and 7 in mozambique correspond to years 5 and 6 in victoria, respectively. data collection, validity and reliability of data content analysis is a non-reactive research method aimed at gathering and analysing the content of a text. the content of a text refers to words, meanings, pictures, symbols, ideas, themes, or any other explicit or implicit messages that can be communicated through a text (neuman, 2003). in this study the four selected textbooks were thoroughly analysed page by page. in all pictures and drawings with gender-identifiable and countable people, the unit of analysis was a person. the number of pages in the textbooks varied between 100 and 120. the wording of every written problem was also analysed. the illustrated characters or photographed individuals and their actions were thus analysed and described in order to infer the implicit messages that they might convey to the reader. crowded images, such as people in a train or market, and images showing only a small part of the human body (a hand, for example) were excluded from the analyses as it was difficult to identify their gender. the validity and reliability of the subjective data collected were determined by recoding the same data from the texts over time (from january to june 2006). as the meanings of the visual material are culture-bound, the interpretation depends on the researchers’ knowledge of the context (neuman, 2003). the instruments to collect data were validated by running a two-hour seminar involving mathematics and science students and lecturers at monash university, victoria, australia. the participants represented different backgrounds. they were asked to identify the roles of the people illustrated in a number of randomly selected pages of the textbooks, and were asked to select, from a list provided, the values promoted by the textbook authors. during the seminar the selections were shared and discussion yielded three important outcomes that influenced the course of this study: 1) by comparing the findings obtained by the seminar participants with those of the firstnamed author of this article (undertaken a week earlier), the data gathering instruments were validated. 2) the data gathering instruments were improved by being able to describe more rigorously the features of people/characters portrayed in the textbooks that are engaged in active or passive roles. 3) criteria were established to identify whether the textbook people/characters were mathematically cognitively engaged or not. results the first question of the study sought to identify whether the photos and drawings illustrating females and males in the various textbooks were distributed with the same frequency throughout, in order to meet one of the criteria for gender equity put forward by forgasz (1996). the numbers of females and males depicted, as well as the frequency ratio of females to males, are shown in table 4. as can be seen in table 4, there were differences between the textbooks of mozambique and victoria in terms of the frequency ratios. the female to male ratios in the mozambican textbooks were lower than 1.00:1 – indicating that males adelino evaristo murimo and helen forgasz 91 were illustrated more often than females. the victorian year 5 textbook illustrated equal numbers of females and males, whereas the year 6 textbook illustrated more females than males. these results indicate that the mozambican texts were more consistent with pattern of male and female depictions found in textbooks from earlier times in australia in which more males than females were portrayed (australian college of education, 1977). in the textbooks of the mid-1980s in victoria there was an attempt to portray females and males more equally, but it was found that there were still more males than females (clarkson, 1993). the present study suggests that victorian textbook writers and publishers seem to be responding positively to the need for gender inclusivity (forgasz, 1996). the second question of the study sought to determine how often females and males appeared together in the same photos or drawings. this question was important as the illustration of females and males in the same drawing could be used to reinforce gender-stereotypes, for example, by depicting males as the ‘doers’ and females as the ‘helpers’ or ‘observers’ (forgasz, 1996). it was found that in the victorian textbooks females and males were rarely illustrated together whereas in the mozambican textbooks there was a trend to depict females and males together in about 27% of the images, with boys measuring and building, and girls standing, observing and playing. the third question of the study was concerned with comparisons in the numbers of females and males in active roles (physical activities) and passive roles (non-physical activities). forgasz (1996) associated active roles with doing something and passive roles with helping or observing. based on these readings, active and passive roles were inferred from the actions undertaken by the characters illustrated in the textbooks. table 5 presents a summary of the data (frequencies and female to male ratios) on the roles played by females and males. the objective for the calculation of the female to male ratios in active and passive roles was to verify whether, as in the past, there were more males than females playing active roles and more females than males in passive roles (australian college of education, 1977; clarkson, 1993; mckimmie, 2002). as can be seen in table 5 the female to male ratios for the mozambican textbooks are lower than 1.00:1 in all categories. this means that more males than females were portrayed in active roles, as was also found in earlier times in australia (australian college of education, 1977) and by clarkson (1993). more males than females were also found in passive roles in mozambican textbooks, which differs from past analyses (australian college of education, 1977; clarkson, 1993; mckimmie, 2002). the illustrations in the victorian textbooks were more varied. female to male ratios revealed that: • year 6 textbook: females and males shared equal numbers of passive roles (ratios of 1.00:1) and, in contrast with earlier textbooks (australian college of education, 1977; clarkson, 1993), more girls than boys (ratio 1.60:1) were portrayed in active roles. • year 5 textbook: fewer females than males were depicted in active roles (ratio 0.77:1), and more females than males were portrayed in passive roles (ratio 1.43:1). the year 5 textbook region level females males f:m ratio grade 6 16 21 0.76 : 1 mozambique grade 7 10 19 0.52 : 1 year 5 20 20 1.00 : 1 victoria year 6 18 15 1.20 : 1 table 4. distribution frequency of females and males in photos and drawings. active roles passive roles region females males f:m ratio females males f:m ratio grade 6 7 9 0.78:1 9 12 0.75:1 mozambique grade 7 4 8 0.50:1 6 11 0.55:1 year 5 10 13 0.77:1 10 7 1.43:1 victoria year 6 8 5 1.60:1 10 10 1.00:1 table 5. distribution of active and passive roles between females and males. depictions of females and males in mozambican and victorian (australia) primary mathematics textbooks 92 depictions were consistent with earlier findings. (australian college of education, 1977, mckimmie, 2002) forgasz (1996) stated that a textbook reflects gender equity when females and males share equal passive and active roles. this was not always the case in the four textbooks, although the differences in numbers were not always great. an attempt was also made to identify, through inference, whether females and males were equally mathematically engaged within their active and passive roles. in this study, to be mathematically cognitively engaged was defined as “thinking about mathematical concepts”; not mathematically cognitively engaged was defined as “thinking about something not mathematically-related” or “not thinking at all”. it was found that in the mozambican textbooks, females and males were not equally mathematically cognitively engaged within their active and passive roles because in both cases, more males were depicted as mathematically cognitively engaged. this finding is consistent with earlier analyses of victorian textbooks in which females and males were also not equally mathematically cognitively engaged in active and passive roles (australian college of education, 1977; forgasz, 1996). however, in both active and passive roles there were slightly more females than males depicted as mathematically cognitively engaged in the mozambican textbooks. this finding is consistent with earlier analyses in the australian context (clarkson, 1993; forgasz, 1996; mckimmie, 2002) in which there was a trend towards depicting females in more positive roles, mathematically, than in earlier times. the present study has shown that the numbers of females have now surpassed that of males with respect to the depiction of mathematically cognitively active roles. one of forgasz’ (1996) criteria for gender equity is that textbooks should present a balance of contextual settings of interest to males and females. while this criterion was not specifically examined in the present study, the genders of the people mentioned in the worded problems were categorised using ‘manifest coding’ (neuman, 2003). manifest coding refers to the coding of textual features such as the pronoun ‘she’ for a female, or the name ‘paul’ for a male. all the words in the mathematical problems presented in the texts were analysed. it was found that in the mozambican texts, males were mentioned twice as frequently as females. in contrast, in the victorian texts, females and males were mentioned in equal numbers. an important difference between the mozambican and the victorian texts was the predominance of gender-neutral references in the victorian texts. in contrast to portuguese (the language used in mozambique), many aspects of the english language are not gender-specific. in using english, the victorian textbook writers have this characteristic available to them in their attempts to be gender inclusive. the depictions of males and females in the textbooks from mozambique more strongly reflected traditional, gender-stereotyped roles, with respect to mathematics learning, than did the victorian textbooks. whether conscious or unconscious, the message likely to be conveyed to students working with the mozambican textbooks is that mathematics is a field of study more appropriate for males than for females – that mathematics is a male domain. the penultimate research issue investigated in this study was to identify text features of accessibility and ‘specialism’ in mathematics learning and to examine them for gender differences. the use of specialist vocabulary, formulae, and the names or illustrations of male mathematicians were considered text features of specialism as they convey the message that mathematics is a product of gifted people, particularly those who understand the jargon and symbols involved, and is not for ordinary people (seah, 1999). in contrast, when there were images and contexts depicting workers, labourers, women, and minorities applying mathematical concepts to solve problems in daily life, or when students were invited to pose problems, to work cooperatively and share information, and when affective variables such as attitudes, confidence and selfevaluation were verified, then the text features were considered to be valuing accessibility in mathematics learning, as they communicate implicit messages that every child can find a place in the mathematics learning process. table 6 illustrates the number of times that females and males (and gender identifiable people) were depicted in the text features valuing specialism in the selected textbooks. adelino evaristo murimo and helen forgasz 93 as can be seen in table 6, in the mozambican textbooks there were gender differences favouring males, as male names and faces were associated with mathematics specialist vocabulary two to three times as often as female names and faces. in the victorian textbooks the gender differences in numbers were not great, due to the use of gender inclusive language; most of the problems and examples were often attached to people whose gender was unidentifiable. the mozambican textbooks acknowledged the contributions to mathematics of three male mathematicians, two of whom were john venn and rene descartes; the victorian textbooks did not refer to any mathematicians. table 7 illustrates the number of times females and males were depicted in the text features valuing accessibility in mathematics learning in the selected textbooks. it is evident that the text materials in the two regions promote accessibility to mathematics learning to some extent. however, the aspects of the textbooks for which accessibility was promoted were culture-bound. text features favouring accessibility in the mozambican textbooks were demonstrated through the use of the local context, and mathematics was operationalised almost equally by females and males. there were thus no gender differences for this text feature, as females and males were almost equally portrayed using mathematics to solve their daily life problems. the value of accessibility in mathematics learning in the victorian textbooks was promoted through invitations to the reader to pose problems, to engage in collaborative work and information sharing, and through drawing attention to affective variables such as attitudes, confidence, and selfassessments. as shown in table 7, there were no gender differences in the promotion of accessibility as the numbers of females and males were small, and there was a high proportion of gender neutral language as reported earlier in this paper. with respect to the value of ‘specialism’, the results suggest that the messages conveyed in the mozambican textbooks reinforce a perception that mathematics is an exclusive and specialised subject predominantly for males, while the victorian textbooks imply that mathematics is a gender inclusive discipline. with respect to the complementary value of accessibility, the mozambican textbooks convey the message that the mathematics of everyday situations is a gender inclusive mozambique victoria grade 6 grade 7 year 5 year 6 f m gu f m gu f m gu f m gu use of specialist vocabulary or a formula 9 27 13 22 1 6 7 9 11 9 13 mathematics knowledge: a product of gifted human action 2 1 f: females m: males gu: gender unidentifiable table 6. gender distribution among text features valuing ‘specialism’ in mathematics learning. mozambique victoria grade 6 grade 7 year 5 year 6 f m gu f m gu f m gu f m gu context: workers, labourers, females and minorities apply mathematical ideas to solve daily life problems. 17 13 5 4 1 2 1 1 students pose problems, work cooperatively and share information 15 2 1 11 affective variables (attitude, confidence, self-evaluation, etc) are checked 5 5 26 1 4 27 table 7. gender distribution among text features valuing accessibility in mathematics learning. depictions of females and males in mozambican and victorian (australia) primary mathematics textbooks 94 activity. the victorian textbooks focused on different aspects of accessibility, namely, its pedagogical and affective dimensions. again, the victorian textbooks were found to be gender inclusive due mainly to the use of the gender neutral aspects of the english language. conclusions and recommendations clarkson (1993), forgasz (1996), and mckimmie (2002) reported improvements in the depictions of females in victorian textbooks. this study has noted further improvements, although still not ideal. forgasz’ (1996) recommendations to ensure that mathematics textbooks are gender equitable were generally evident in the victorian textbooks examined; however, further improvements are needed. for example, the textbook writers and publishers should ensure that males are not poorly depicted in attempts to depict females in more positive roles. not all the criteria suggested by forgasz (1996) were examined in this study. since australia is a multicultural country, there is a need to investigate whether the characters illustrated, their names, and the contexts in which they are portrayed reflect the country’s multicultural diversity. while females were portrayed in the mozambican textbooks, the content reflected what used to be the case with australian textbooks in the 1970s (australian college of education, 1977), although there are small differences that are likely to be culturally based. these findings provide evidence of gender bias in favour of males, with images of males outnumbering those of females in photos and drawings. more males than females were also depicted in active (physical) roles, and as being mathematically cognitively engaged. worded problems also mentioned more males than females. the mathematics educational value of ‘specialism’ was attached to male names and faces through the use of specialist vocabulary and formulae. consequently, there is a risk that mozambican children will perceive mathematics as an exclusive subject that is more appropriate for males than females. in portraying the mathematics education value of accessibility through everyday life contexts, this dimension of mathematics – also referred to as numeracy (australian association of mathematics teachers, 1998) – is likely to be seen as equally valuable for males and females. since victorian textbooks generally reflected gender equity, there is much that the mozambican ministry of education and culture can learn, particularly as gender equity is now one of the education policy goals. to ensure gender equity in mozambican textbooks something needs to be done urgently. based on the findings of this study, it is recommended that the mozambican ministry of education and culture should elaborate guidelines on gender equity to orient the writers of textbooks and other instructional materials. the guidelines should also be available for teachers to align their classroom practices with gender equitable textbook materials. it is acknowledged that the discussion of gender equity in mathematics textbooks does not, perhaps, answer all problems of gender differences in mathematics learning in mozambique. however, the findings of this study are important because they open up discussion around gender issues in mathematics education in the country. leder (1990) described many variables associated with gender differences in mathematics education. these variables should be investigated in the context of mozambique and elsewhere in order to identify the contribution of each variable to the observed gender differences favouring males in educational outcomes. certainly, one of the most important variables that requires urgent investigation in mozambique is the influence of various cultural dimensions and economic issues on the educational outcomes of girls and boys in the context of high poverty rates faced by the majority of mozambican families (unicef, 2006). it is also important to bear in mind that previous research reviewed for this study was carried out in english, a language that has a gender neutral facility which, it was argued, made it easier for the victorian textbook writers to be gender inclusive. this conclusion generates additional questions. what else may be hidden within the structure of various languages that is worthy of investigation? how do the students’ mother languages (bantu languages) affect the way in which they interpret gender issues portrayed in textbooks and other school materials? the findings of this study cannot be generalised to other textbooks and regions. however, the general popularity of the textbooks in mozambique and victoria that were examined should not be disregarded. finally, the findings of this study are of significance. mozambican textbook writers and publishers portrayed mathematics learning as a male domain whereas the victorian textbook writers and publishers attempted to portray mathematics learning as a domain for all children, adelino evaristo murimo and helen forgasz 95 female and male. as gender equity in education is one of the goals in mozambique advocated in educational policies, mozambique should ensure that mathematics texts are not biased in favour of one particular group. both females and males should be depicted positively in textbooks in order to support all children in reaching their full potential in mathematics learning. references australian association of mathematics teachers (aamt). (1998). policy on numeracy education in school. occasional publication. adelaide: aamt. australian college of education. (1977). sex bias in mathematics texts. tasmanian chapter newsletter (november 11), 31, 6. bernard, a. & cabral, z. (2002). gender and education in mozambique: analysis of results, lessons and recommendations. retrieved january 12, 2006, from: http://www.unicef.org/ evaldatabase/index_18996.html bishop, a.j. (1988). mathematical enculturation: a cultural perspective on mathematics education. dordrecht, the netherlands: kluwer academic publishers. bishop, a.j. (1996). how should mathematics teaching in modern societies relate to cultural values? some preliminary questions. paper presented at the seventh southeast asian conference on mathematics education, hanoi, vietnam. bishop, a.j., seah, w.t. & chin, c. (2003). values in mathematics teaching – the hidden persuaders? in a.j. bishop, m.a. clements, c. keitel, j. kilpatrick & f.k.s. leung (eds.), second international handbook of mathematics education, vol.2 (pp 717-765). dordrecht, the netherlands: kluwer academic publishers. clarkson, p. (1993). gender, ethnicity and textbooks. the australian mathematics teacher, 49(2), 14-16. fennema, e. (1990). justice, equity, and mathematics education. in e. fennema & g.c. leder (eds.), mathematics and gender (pp 1-9). new york: teachers college press. forgasz, h. (1996). equity and the selection of textbooks: an analysed example and a checklist. vinculum, 33(4), 6-8. gerdes, p. (1988). on culture, geometrical thinking and mathematics education. in a.j. bishop (ed.), mathematics education and culture (pp 137-162). dordrecht, the netherlands: kluwer academic publishers. guz, a. (2000). education for all: the year 2000 assessment report of mozambique. retrieved january, 20, 2006, from http://www2.unesco. org/wef/countryreports/mozambique/contents.ht ml koehler, m.s. (1990). classrooms, teachers, and gender differences in mathematics. in e. fennema & g.c. leder (eds.). mathematics and gender (pp 128-148). new york: teachers college press. leder, g.c. (1990). gender differences in mathematics: an overview. in e. fennema & g.c. leder (eds.), mathematics and gender (pp 10-26). new york: teachers college press. leder, g.c. (1992). mathematics and gender: changing perspectives. in d.a. grouws (ed.), handbook of research on mathematics teaching and learning (pp 597-622), new york: macmillan. leder, g.c., forgasz, h.j., & solar, c. (1996). research and intervention programs in mathematics education: a gendered issue. in a. bishop, k. clements, c. keitel, j. kilpatrick, & c. laborde (eds.), international handbook of mathematics education, part 2 (pp 945-985). dordrecht, netherlands: kluwer. luís, m.c. (2004). por uim currículo de formação do professor de matemática na perspectiva de construção do comhecimento. unpublished doctoral thesis, pontifícia universidade católica de são pauco (puc), brazil. mckimmie, t. (2002). gender in textbooks. vinculum, 39 (4), 18-23. ministério de educação e cultura. (2003). plano estratégico da educação. mozambique. retrieved january 11, 2006, from: http://www. mined.gov.mz/ neuman, w.l. (2003). social research methods: quantitative and qualitative approaches (5th ed.). boston: allyn & bacon. seah, w.t. (1999). the portrayal and relative emphasis of mathematical and mathematics educational values in victoria and singapore lower secondary mathematics textbooks. unpublished masters thesis, monash university, victoria, australia. stretmatter, j. (1994). towards gender equity in the classroom. every teachers’ beliefs and practices. usa: state university of new york press. depictions of females and males in mozambican and victorian (australia) primary mathematics textbooks 96 terre blanche, m. & durrheim, k. (1999). histories of the present: social science research in context. in m. terre blanche & k. durrheim (eds.). research in practice: applied methods for the social sciences (pp 1-16). cape town: cape town press. unicef (2006). mozambique: the children primary school years. retrieved january 11, 2006 from: http://www.unicef.org/mozambique /children_1594.html “research on gender and mathematics has provided a powerful scientific discourse during the past three decades. the entire educational communitycomposed of practitioners, researchers, and policymakersneed to continue to engage in this discourse about and to explore ways to deepen our understanding of what equity is and how it can be achieved. it is in discourse about philosophical questions as well as research questions that our understanding of gender and mathematics will grow.” elizabeth fennema article information authors: temesgen zewotir1 delia north1 affiliations: 1school of statistics and actuarial science, university of kwazulu-natal, south africa correspondence to: delia north email: northd@ukzn.ac.za postal address: private bag x54001, durban 4000, south africa dates: received: 03 apr. 2011 accepted: 01 nov. 2011 published: 28 nov. 2011 how to cite this article: zewotir, t., & north, d. (2011). opportunities and challenges for statistics education in south africa. pythagoras, 32(2), art. #28, 5 pages. http://dx.doi.org/10.4102/ pythagoras.v32i2.28 copyright notice: © 2011. the authors. licensee: aosis openjournals. this work is licensed under the creative commons attribution license. issn: 1012-2346 (print) issn: 2223-7895 (online) opportunities and challenges for statistics education in south africa in this original research... open access • abstract • introduction • statistics education at school level • statistics and statistics education at university level • professional association • conclusion • acknowledgements    • competing interests    • authors’ contributions • references • footnote abstract (back to top) the south african educational system is in a state of transformation as the government embarks on a process of grappling with legacies of the past, whilst balancing risks and opportunities for the future. accordingly, a new school curriculum with outcomes-based education as the fundamental building block was introduced along a sliding scale, starting in 1997. this curriculum, with a vast statistics content, has the potential to change the face of statistics education in south africa, as statistics had previously been virtually absent from the school syllabus. this article highlights the challenges to and opportunities for optimising the teaching of statistics across all education levels in south africa. introduction (back to top) in this information-driven society it is vital that all citizens are able to orient themselves in a world where evidence-based decision making is likely to call for data collection, organisational, analytical, interpretation and communication skills. accordingly, statistics education is facing a critical global challenge to meet with an ever-increasing need to disseminate more data, accurately, in shorter times and in forms desired by users for further analysis (wallman, 1993). in south africa the training of statisticians and raising of levels of statistical literacy have to be planned carefully to make the most effective use of the limited training resources available. this article assesses the challenges and opportunities for statistics education in south africa. statistics education at school level (back to top) it has been well documented that statistical concepts should be introduced into the school curriculum (e.g. franklin et al., 2005; gal, 2002; wild & p fannkuch, 1999).this has resulted in mathematics curricula for primary and secondary schools being reformed to include statistics (north & scheiber, 2011). accordingly, in 1995 the minister of education announced the introduction of a new curriculum with outcomes-based education (obe) as the fundamental building block (botha, 2002; department of education, 1997). this curriculum was intended to catapult south africa into the 21st century (chisholm et al., 2000). this curriculum was launched in march 1997 and became known as curriculum 2005 ( department of education, 1997). implementation of curriculum 2005 was along a sliding scale, starting with grade 1 in 1998. this curriculum was subsequently revised in 2002 and renamed the revised national curriculum statement for grades r–9, and implemented along a sliding scale starting in 2004. the national curriculum statement (ncs) was subsequently defined for grades 10–12 (department of education, 2002, 2003) and was implemented for grade 10 in 2006, grade 11 in 2007 and grade 12 in 2008. the first year that the grade 12 students had received the full benefit of the new curriculum and wrote the first national senior certificate school-leaving examination based on the new ncs was thus 2008. in the new curriculum emphasis is specifically placed on shifting from the traditional aims and objectives approach to obe. this paradigm shift is seen as a prerequisite for achievement of the vision of an internationally competitive country. outcomes-based curriculum development starts with the formulation of the purposes of learning, which are then used as the criteria for further curriculum development and assessment. one of the specific outcomes identified in the school mathematics curriculum is the use of data from various contexts to make informed judgements (steffens & fletcher, 1999; north & zewotir, 2006a). this serves the ever-increasing need to understand data and to translate them into usable knowledge in the technological age of rapid information expansion. recognition of the cross-curricular need for statistics as an anticipated outcome resulted in the assessment criteria of the ncs, including the collection of data (using methods such as interviews and sampling), application of statistical tools and communication and critical evaluation of findings. the new obe system requires that each learner either does mathematics or mathematical literacy in each school year – a major shift from what had been the case prior to adoption of the new curriculum. it was previously possible to complete schooling without doing any form of mathematics in the last three grades of the schooling system. to get a grasp of the scale of the change, one needs to examine the number of school leavers with some form of mathematics in 2007 (when the old system was in place) compared to 2008 (first graduation year of learners under the ncs). a huge ‘wave’ came through in 2008, when a total of 592 017 students registered for the final school examination in mathematics or mathematical literacy, compared to a total of 347 570 in the previous year (appel, 2008). statistics (data handling) forms an integral new part of the mathematics syllabus and is also present in the mathematical literacy syllabus of the ncs, adding to the dilemma of training many in-service mathematics teachers who have generally had little or no training in statistics. the dilemma is thus two-fold – training more mathematics teachers, and ensuring that existing teachers can cope with the vast new statistics content of the ncs (north & scheiber, 2008). the statistics component of the ncs was initially developed by the department of education without any input from the south african statistical association (sasa), the mouthpiece of the majority of professional and practicing statisticians in south africa. training in statistics for mathematics teachers for the ncs was carried out by department of education subject advisors, themselves products of the previous curriculum and thus also not having the required statistics knowledge (north & scheiber, 2008). this prompted intervention from sasa to ensure that the potential successful introduction of statistics, as specified in the ncs, was achieved. sasa is accordingly actively involved in the teaching of statistics in schools via its education committee, a subcommittee with the specific brief of furthering statistics education at tertiary and preparatory (school) level. in 1998 contact was made with the association for mathematics education of south africa (amesa),the association to which school mathematics teachers belong, with a view to closer collaboration with the intention of including school teachers in some of the initiatives of the sixth international conference on teaching statistics (north & zewotir, 2006a). this occasion marked the beginning of a closer bond between sasa and amesa and has resulted in sasa members taking an active part in subsequent congresses and meetings of amesa, as well as contributing to the amesa newsletter. the sasa education committee initiated an awareness of the dilemma of statistics education of school teachers by giving various talks at local conferences, holding workshops and helping with teacher training. however, it was only when statistics south africa (stats sa), the national statistics office, launched the maths4stats campaign (stats sa, n.d.) that the human capacity and finances were available to address the dilemma on a national basis. the maths4stats project addresses the dilemma by aiming to provide a roll-out plan for statistics training for roughly 10 000 mathematics teachers (grades 10–12) from 28 000 schools. the objective of themaths4stats campaign was to create a specialised body of teachers with a passion for mathematics, and to instil a love for and interest in mathematics and statistics in teachers and learners. details of the project can be found in north and scheiber (2008) and stats sa (n.d.). the long-term aim of this project is to strengthen expertise in statistics at all levels so that ordinary people have trust in the information they receive from stats sa (lehohla, 2002). through a focus on in-service teachers, the scope of statistics education is being broadened considerably at school level. the issue of statistics training of pre-service teachers is addressed by universities and teacher training centres, as they are starting to incorporate statistics education courses into their formal teacher training courses (wessels, 2008). teachers are thus being prepared to teach statistics by enabling pre-service teachers to develop the statistical literacy, thinking and reasoning abilities called for in the ncs. there is a need for teachers to have the knowledge to teach the elements that are essential for basic statistical thinking, that is, an appreciation of uncertainty and data variability and their impact on decision making (sylwester, 1993). this is an attempt to make precise and explicit what the data have to say about the problem of interest (mallows, 1998). by providing exposure to and instruction in this type of thinking, we can hasten development of the way of approaching problems and methodology for problem solving which will be called on in later years (chance, 2002). statistics and statistics education at university level (back to top) statistics syllabi were introduced at south african universities in the 1930s and had a theoretical focus, deliberately shying away from applied statistics (steffens, 1998). statistics historically started at second-year level (in a three-year programme leading to a bachelor’s degree in statistics), as one needed a high level of mathematics in order to follow the calculus-based approach to teaching statistics. historically, south african universities thus produced statisticians that were not ideally suited to any occupation other than in academia, since the statistics training was geared towards furthering the discipline in a theoretical way (de wet, 1998). it must, however, be noted that a few individuals did manage to excel in the application of statistics through their own self-interest rather than through skills acquired from their tertiary-level statistics training. for instance, dr herbert sischel (1915–1995), a world-renowned statistician from the university of the witwatersrand, made great advances in both theoretical and applied statistics. in recent times a more balanced view of theory and applications has become apparent in statistics training at south african universities (de wet, 1998; steffens, 1998). currently statistics departments typically offer a three-year programme leading to a bachelor’s degree in statistics, a one-year honours degree, oneor two-year masters programmes and four-year phd programmes. masters and phd programmes are mainly by research dissertation only. the current structure of the statistics courses in the various south african universities is very similar. two courses (one per semester) at first-year level, two course sat second-year level and four courses at third-year level are generally required for the bsc programme. the two courses (one per semester) in the first year offer an introductory approach to the theory, principles and applications of statistics. the two courses in second year mainly deal with distribution theories, estimation procedures and inference. the third-year courses are a mix of methods and applications with a theoretical basis. relatively intermediate-level advanced statistical theories and methods with computer practicals are offered at honours level. a common feature of honours programmes amongst south african universities is the statistics project.1 mathematics teaching at school level in south africa lags behind many other countries (reddy, 2006), so that the majority of the first-year students have a poor or insufficient mathematical foundation and cannot cope with the theoretically oriented first-year courses such as statistics (frick, 2008). to alleviate this problem, a number of universities have introduced special tutorials and hot seats (private lessons by postgraduate students) in first-year statistics courses. the use of local instead of foreign textbooks can, to a certain extent, bridge the problem of presenting examples that the students do not relate to and consequently cannot put into context (steffens, 1998). at least one module in a statistics service course is an essential part of almost all programmes at south african universities. one of the objectives of such a course is to equip students with the basic statistical methods for their everyday life, that is, a holistic approach to produce well-rounded graduates. in most cases such an introductory statistics course is offered at first-year level and usually has little or no prerequisites, beyond some mathematics requirement at high school level. the course tends to dwell more on theory and less on applications of statistics, thus fostering an inwardly focused approach where theory plays the dominant role, followed by a few techniques with the hope that the value of the subject will speak for itself. it is argued here that the underlying purpose is implicit rather than explicit, that is, the student graduates from the module with a grasp of some theoretical aspects of statistics but will generally have very little understanding as to how to apply the theory learnt in practice (mclean, 2000). this expectation is far from reality, as many students find statistics courses difficult, with a resulting poor pass rate in such courses (north & zewotir, 2006b). for instance, the average pass rate for third-year engineering students in a statistics service course offered at the university of kwazulu-natal for the period 1997–2005 was 73.8%, whilst the average pass rate for third-year engineering courses over this period was 86% (zewotir & north, 2007). to get a handle on the number of students in the various statistics modules around the country, refer to table 1. due to the sheer shortage of statisticians in the country, bsc and honours graduates in statistics receive lucrative offers from industry, business and government. accordingly, very few south african statistics graduates opt to pursue postgraduate studies in statistics or statistics education (wessels, 2008). currently there are not enough statisticians being produced to meet the demand of personnel with at least an msc degree in statistics. the international review panel report, the review of mathematical sciences research at south african higher education institutions, concludes that ‘the shortage of academic statisticians is so critical that the field is in danger of disappearing through lack of academic capacity’, further noting that ‘the closure of academic departments is a real possibility’ (department of science andtechnology, 2008, p. 3). diab and gevers (2009, p. 78) come to the same conclusion: … there is acute shortage of qualified statisticians with no senior established academic in some of the statistics departments at universities in south africa. there is danger of the imminent collapse of research activity in statistics when the present cohort of senior researchers retire. table 1: number of students in statistics programmes at south african universities, 2006. professional association (back to top) sasa was founded in 1953 and held its first conference in 1958. the association facilitates the advancement of statistical knowledge by creating a forum for collaboration between academic and practicing statisticians in order to nurture and build capacity of statisticians in south africa. accordingly, the association organises an annual conference, has two publications and runs a website. the two publications are the sasa newsletter and the south african statistical journal. these two publications are mailed to members and are placed in libraries of all tertiary institutions in south africa. as can be seen from table 2, the number of sasa members has steadily increased over the last three years. the total number of sasa members was 430 in 2010, an increase of 42% from the membership of 302 in 2007. as a result of the divisions that existed pre-1994, the negligible number of black and asian sasa members in the past has recently started to grow rapidly. in 2007 about 15% of sasa members were black; only three years later, in 2010, this figure had doubled to 30%. interestingly, the number of female black sasa members increased from 4% (13 out of 302) in 2007 to 12% (51 out of 430) in 2010, indicating that the membership of the association is normalising. table 2: number of sasa members by race and gender in 2007 and 2010. conclusion (back to top) statistical thinking, which calls for evidence of critical, investigative, creative, problem-solving and communicative skills, will prove to be difficult in the obe set-up. when students engage with a mathematics course, they are generally accustomed to calculating one definitive correct answer that can be underlined as ‘the correctanswer’, whilst in the statistics class there may be many different answers that could all be considered to be correct, and the emphasis would be on the interpretation of that answer. for example, whether to use the mean, median or mode when determining the measure of central tendency of the data could potentially give different results, but there is not one ‘correct’ answer. one needs to know what the strength of each measure is and which is best to use under which circumstances. this calls for a whole new mindset − one where the habits of questioning, justifying and writing in one’s own words require specific instruction in the introductory statistics course, calling for teachers that have confidence and expertise in this area. teachers thus need to be aware of the need to allow and even reward alternative ways of examining and interpreting data. what is evident from the experiences of other statistics educators around the world is that many teachers do not consider themselves well prepared to teach statistics, nor to address their students’ difficulties (batanero, burrill & chris, 2011). the joint study by the international commission on mathematical instruction and the international association for statistics education aimed to address this omission by promoting research specifically focused on the education and professional development of teachers to teach statistics (batanero, burrill, reading & rossman, 2008). the research of this study suggests that in order to teach statistics effectively, teachers must understand the nature of statistics and its role in all aspects of today’s information-driven society (batanero et al., 2011). preparing teachers to teach statistics is vital to further the improvement of statistics education at all levels, and in all contexts. it is thus in the interest of all south african role-players to work collaboratively in order to promote statistics education at grassroots level. the first steps taken by stats sa and the sasa to jointly improve the statistics-proficiency level of in-service teachers is gaining momentum and making great strides (north & scheiber, 2008). on the other hand, efforts of statistics educators at higher education institutions to collaborate more effectively with experts in other fields to revise and develop statistics service courses is not yet satisfactory. it takes motivation and perseverance to make changes to any curriculum, and is often easier just to retain the status quo, that is, stick with the line of least resistance and carry on doing things the way they have always been done in the past. the result is that statistics service courses have often not evolved at the rate that they should have. another area of concern is that some disciplines offer statistics service courses within their own discipline code, that is statistics is taught by non-specialists. the lesson to be learnt from this is that unless the statistics departments offer service courses with relevance and ease, students will continue to have difficulty in passing, since many students experience anxiety when required to take a statistics course (north & zewotir, 2006a). this is particularly so since many such courses focus on statistical methods (computations) rather than on statistical reasoning (concepts and thinking). to reduce anxiety and provide the most optimal base for future development of statistics education at all levels, it is imperative that all school and introductory statistics modules should be designed in a way such that calculations are delayed until concepts have been mastered, with an emphasis on the use of real or realistic data. preparing teachers to teach statistics meaningfully is thus a vital and important research area that has not received enough attention (wessels, 2008). the outcomes-based reforms that took place in south africa had much in common with curriculum reforms that previously took place in the united kingdom, new zealand, australia, canada and the united states of america (aldridge, fraser & laugksch, 2011). the teaching of statistics reflects, in unexpected ways, the historical, economic and political circumstances of the country, and of course these differences are reflected in the training programmes (loynes, 1987). what is a common thread in many countries, however, is the inclusion of statistics into school syllabi and the desire to improve statistics education at all levels, from elementary (primary) school through to the training of professionals. it is thus argued that no matter how big the challenges to successfully introducing statistics at school level in south africa, this has the potential to improve basic statistical literacy of school leavers. more importantly, it will increase the awareness of statistics as a study area of choice amongst students entering universities and teacher training centres, thus creating an opportunity for statistics education to grow at all levels in south africa. acknowledgements (back to top) we are grateful to the sasa treasurer, prof. p.j. mostert, for providing data on the number of sasa members. competing interests we declare that we have no financial or personal relationship(s) which may have inappropriately influenced us in writing this article. authors’ contributions t.z. performed the analysis of data and drafted the manuscript. d.n. made contributions to collecting relevant data and editing the manuscript. both t.z. and d.n. contributed to the conception and design of the study, and both wrote and edited the final version of the manuscript. references (back to top) aldridge, j.m., fraser, b.r., & laugksch, r. (2011). relationships between the school-level and classroom-level environment in secondary schools in south africa. south african journal of education, 31, 127−144. available from http://www.sajournalofeducation.co.za/index.php/saje/article/view/407/230 appel, m. (2008). mathematical literacy pass rate encouraging for sa. available from http://www.ecdoe.gov.za/news_article/48/mathematical-literacy-pass-rate-encouraging-forsa batanero, c., burrill, g., & chris, r. (2011). teaching statistics in school mathematics: challenges for teaching and teacher education: a joint icmi/iase study: the 18th icmi study. new york, ny: springer. http://dx.doi.org/10.1007/978-94-007-1131-0 batanero, c., burrill, g., reading, g., & rossman, a. (eds.). (2008). joint icmi/iase study: teaching statistics in school mathematics – challenges for teaching and teacher education. proceedings of the icmi study 18 conference and iase 2008 round table conference, 30 june − 04 july 2008. monterrey, mexico: international statistical institute and international commission on mathematical instruction. available from http://www.stat.auckland.ac.nz/~iase/publications.php?show=rt08 botha, r.j. (2002). outcomes-based education and educational reform in south africa. international journal of leadership in education: theory and practice, 5, 361–371. http://dx.doi.org/10.1080/13603120110118831 chance, b.l. (2002). components of statistical thinking and implications for instruction and assessment. journal of statistics education [online], 10 (3). available from www.amstat. org/publications/jse/v10n3/chance.html chisholm, l., volmink, j., ndhlovu, t., potenza, e., mahomed, h., muller, j., et al. (2000). a south african curriculum for the twenty-first century: report of the review committee on curriculum 2005. pretoria: ministry of education. department of education. (1997). senior phase policy document. pretoria: department of education. department of education. (2002). revised national curriculum statement: grades r−9 (schools): mathematics. pretoria: department of education. department of education. (2003). national curriculum statement: grades 10−12 (general): mathematics. pretoria: department of education. department of science and technology. (2008). review of mathematical sciences research in south africa higher education institutions: international review panel report. available from http://www.nrf.ac.za/files/file/report.pdf de wet, j.i. (1998). teaching of statistics to historically disadvantaged students: the south african experience. in proceedings of the fifth international conference on the teaching of statistics, 21−26 june 1998, vol. 2 (pp. 573−577). singapore: nanyang technological university. available from http://www.stat.auckland.ac.nz/~iase/publications/2/topic5e.pdf diab, r., & gevers, w. (eds.) (2009). the state of science in south africa.pretoria: academy of science of south africa. available from http://www.assaf.co.za/epub/assaf/assaftwas.html franklin, c., kader, g., mewborn, d.s., moreno, j., peck, r., perry, m., et al. (2005). guidelines for assessment and instruction in statistics education (gaise) report: a pre-k−12curriculum framework. georgia: american statistical association. available from http://www.amstat.org/education/gaise/gaiseprek-12_full.pdf frick, b.l. (2008). the profile of the stellenbosch university first year students: present and future trends. preliminary research report. draft 4. available from http://stbweb02.stb.sun.ac.za/sol/documents/student%20profile%20report%20draft%204.pdf gal, i. (2002). adult’s statistical literacy. meanings, components, responsibilities. international statistical review, 70(1), 1−25. http://dx.doi.org/10.2307/1403713, http://dx.doi.org/10.1111/j.1751-5823.2002.tb00336.x lehohla, p. (2002). promoting statistical literacy: a south african prespective. in b. phillips (ed.), proceedings of the sixth international conference on teaching of statistics [cd]. voorburg, the netherlands: international statistical institute. available from http://www.stat.auckland.ac.nz/~iase/publications/1/5d1_leho.pdf loynes, r.m. (ed.). (1987). the training of statisticians round the world. voorburg, the netherlands: international statistical institute. available from http://www.stat.auckland.ac.nz/~iase/publications/train1987/ mallows, c. (1998). the zeroth problem. the american statistician, 52, 1−9. http://dx.doi.org/10.2307/2685557 mclean, a. (2000). the predictive approach to statistics. journal of statistics education, 8(3) [online]. available from http://www.amstat.org/publications/jse/secure/v8n3/mclean.cfm north, d., & scheiber, j. 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(1998). statistical education in the african region: private experiences in south africa and namibia. in proceedings of the fifth international conference on the teaching of statistics, 21−26 june 1998, vol. 2 (pp. 572−578). singapore: nanyang technological university. available from http://www.stat.auckland.ac.nz/~iase/publications/2/ topic5d.pdf steffens, f.e., & fletcher, l. (1999). statistics as part of the mathematics curriculum in south africa. in proceedings of the first international conference of the mathematics education into the 21st century project,vol. 1 (pp. 298−305). available from http://math.unipa.it/~grim/esteffensfletcher298-305.pdf sylwester, d. (1993. february). statistical thinking. american statistical association newsletter. p. 2–3. wallman, k.k. (1993). enhancing statistical literacy: enriching our society. journal of the american statistical association, 88, 1–8. http://dx.doi.org/10.2307/2290686 wessels, h.m. (2008). statistics in the south african school curriculum: content, assessment and teacher training. in c. batanero, g. burrill, c. reading, & a. rossman (eds.), proceedings of the 18th icmi study conference and 2008 iase round table conference, 30 june – 04 july 2008. voorburg, the netherlands: international statistical institute and international commission on mathematical instruction. available from http://www.ugr.es/~icmi/iase_study/files/topic1/t1p3_wessels.pdf wild, c., & pfannkuch, m. (1999). statistical thinking in empirical enquiry. international statistics review, 67(3), 223−265. http://dx.doi.org/10.2307/1403699, http://dx.doi. org/10.1111/j.1751-5823.1999.tb00442.x zewotir, t., & north, d. (2007). focus on the statistical education of prospective engineers in south africa. pythagoras, 65, 18−23. available from http://www.pythagoras.org.za/index.php/pythagoras/article/view/87 footnote (back to top) 1.as a typical example, refer to these university school of statistics websites: ukzn: http://statsactsci.ukzn.ac.za/faculty_handbook/statistics_undergraduate aspx wits: http://www.wits.ac.za/academic/science/stats/courses/5726/honours.html uct: http://web.uct.ac.za/depts/stats/undergrad.htm rhodes: http://www.ru.ac.za/statistics/ stellenbosch: http://academic.sun.ac.za/statistics/ a dynamic approach to quadrilateral definitions rajendran govender university of the north email: rajengovender@absamail.co.za michael de villiers university of durban-westville email: profmd@mweb.co.za this study examined 18 prospective secondary mathematics teachers' understanding of the nature of definitions, and their use of the dynamic geometry software sketchpad to improve not only their understanding of definitions but also their ability to define geometric concepts themselves. results indicated that the evaluation of definitions by accurate construction and measurement enabled students to achieve a better understanding of necessary and sufficient conditions, as well as the ability to more readily find counter-examples, and to recognise and improve on uneconomical definitions. introduction research by linchevsky, vinner & karsenty (1992), among others, on definitions in mathematics, has indicated that many student teachers do not understand that definitions in geometry have to be economical (contain no superfluous information) and that they are arbitrary (in the sense that several alternative definitions may exist). it is plausible to conjecture that this is probably due to their past school experiences where definitions were largely supplied to them directly. it would appear that in order to increase future teachers' understanding of geometric definitions, and of the concepts to which they relate, it is essential to engage them at some stage in the process of defining geometric concepts. the research reported here concentrated mainly on student teachers’ (prospective mathematics teachers) understanding of the nature of definitions and the development of their ability to evaluate and formulate definitions in a sketchpad context (see govender, 2002). sketchpad was used to expose the students to the process of defining as a creative activity in which students can be fully involved, rather than perpetuating the view of definition as an imposed body of knowledge immune to any change or development. the following research questions were addressed: • what prior understanding of the nature of definitions do student teachers have before being engaged in a process of formulating some definitions for themselves? • to what extent does the student teachers' understanding of the nature of definitions change while involved in a process of evaluating definitions by means of construction, measurement and dragging within a sketchpad context? • how competent are the student teachers in evaluating other definitions after being engaged in the preceding process? definitions because student teachers often meet mathematics structured only as in condensed formal mathematics textbooks, their learning takes place within this structure. the textbooks used in schools give concisely expressed definitions, and this has an effect both on how our students view definitions and how teachers teach. however, this structured approach can easily lead to a common but false perception that there is only one (correct) definition for each defined object in mathematics. the fact that several different (correct) definitions may exist for a particular concept is seldom addressed in such a structured, pre-packaged approach. 34 pythagoras 58, june, 2004, pp. 34-45 mailto:rajengovender@absamail.co.za mailto:profmd@mweb.co.za rajendran govender, michael de villiers a further misconception that can easily develop from a traditionally axiomatically structured approach is that mathematics always starts with definitions, which can give the false impression that definitions of mathematical objects are given a priori in nature. in such a structured approach, students are not brought to realise that definitions do not exist independently of human experience in some "ideal" platonistic world, so that all we can do is to "discover" them. the fact that definitions are not discoveries, but human "inventions" for the main purpose of accurate mathematical communication, is therefore not addressed. fortunately, the national curriculum statement grades 10-12 (schools) lays a foundation for the use of technology such as dynamic geometry software for exploration and investigation. in particular, the grade 10 learners ought to explore necessary and sufficient conditions for the various quadrilaterals and investigate ways of defining various polygons. furthermore, according to the new fet curriculum, grade 10 learners ought to realise that definitions are not absolute but fixed on the basis of principles that will result in conciseness and efficiency. definitions, axioms and preceding theorems thus serve as starting points for deductive arguments in the expansion of the axiomatic system. for the quadrilaterals, decisions as to which system of definitions to use can depend on either a partition or inclusiveness approach, although the latter is favoured in the interests of efficiency. learners need to be made aware of these factors in determining definitions. definitions are important in mathematics, but they ought to evolve naturally from previous knowledge, models or real experiences that the child can relate to. otherwise, confusion reigns. mathematical definitions are generally very concise, contain mathematical terms, and require an immediate synthesis of the information if understanding is to result. however, although research confirms that most children cannot operate on an abstract or formal level until junior high school, we find many textbooks and teachers presenting formal definitions to children before they reach junior high school. in fact, learners’ spatial thinking needs to be relatively mature (welldeveloped) before they are able to decide what the definition of a rhombus (for example) should be. indeed, according to the van hiele theory, they need to be at least at van hiele level 3 (see burger & shaughnessy, 1986; de villiers, 1997). the van hiele theory the van hiele model has important implications for the teaching of geometry. it can be used firstly to guide students to achieve a more sophisticated level of geometric thinking, and secondly to assess students' abilities. the most obvious characteristic of the theory is the distinction of five discrete thought levels in the development of students' understanding of geometry. the levels, labelled "visualization (recognition)," "analysis," "informal deduction (ordering)," "deduction" and "rigor" describe characteristics of the thinking process. assisted by appropriate instructional experiences, the model asserts that the learner moves sequentially from the initial, or basic level (visualization), where space is simply observed and the properties of figures are not explicitly recognised, through the sequence listed above to the highest level (rigor), which is concerned with formal abstract aspects of deduction. few students are exposed to, or reach, the latter level. according to this model, the learner cannot achieve a specific level of thinking without having passed through previous levels. according to the van hiele theory, the main reason the traditional geometry curriculum fails is that it is presented at a higher level than those at which students are operating; in other words, students cannot understand the teacher nor can the teacher understand why they cannot understand. the general characteristics of the first four levels, the ones commonly displayed by secondary students and most relevant to this study, are given: level 1: recognition: students visually recognise figures by their global appearance. they recognise triangles, squares, parallelograms and so forth by their shape, but they do not explicitly identify the properties of these figures. level 2: analysis: students start analyzing the properties of figures and learn the appropriate technical terminology for describing them, but they do not interrelate figures or properties of figures. level 3: ordering: students logically order the properties of figures by short chains of deductions and understand the relationships between figures (for example, class inclusions). level 4: deduction: students start developing longer sequences of statements and begin to understand the 35 a dynamic approach to quadrilateral definitions significance of deduction and the role of axioms, theorems and proofs. the van hiele theory is a useful framework for interpreting and analysing student teachers’ levels of understanding for the following reasons: • it seeks to explain not only why students have trouble in learning but also what could be done to remove these stumbling blocks. • the differences between the levels can be projected in terms of the objects and structure of thought at each level (see fuys et al., 1986). • each level is characterised by a series of specific actions/performances. for example, by using task-based interviews burger and shaughnessy (1986) identified more fully what students do at the first four levels. due to space limitations, we shall only discuss level 3. level 3 formulate economical, correct definitions for figures. are able to transform incomplete definitions into complete definitions and more spontaneously accept and use definitions for new concepts. accept different equivalent definitions for the same concept. classify figures hierarchically; for example, quadrilaterals. explicitly use the logical form if.... then to formulate and handle conjectures, and implicitly use logical rules such as modus ponens. are uncertain and lack understanding regarding the functions of axioms, definitions and proof. • students attempting definitions of concepts would be influenced by their level of understanding. for example, students who are still at van hiele level 1 tend to give visual definitions, e.g. a rectangle which looks like this (draws or identifies a quadrilateral with all angles 90 degrees and two long and two short sides), while those students at van hiele level 2 tend to give correct, uneconomical definitions (eg. a rectangle is a quadrilateral with opposite sides parallel and equal, all angles 90 degrees, equal diagonals, half–turnsymmetry, two axes of symmetry through opposite sides, two long and two short sides, and so on.). students at van hiele level 3 tend to give correct, economical definitions, e.g. a rectangle is a quadrilateral with two axes of symmetry through opposite sides. (compare de villiers, 1997: 15-17). research design the target population the target population of this study was the 18 mathematics education 3 students at umbumbulu college of education (uce), which is located approximately 15km south of the durban international airport, in the kwa makutha township. uce is a three year teacher college which gives students a professional teacher’s diploma. the college prepares its students for teaching at both the junior and senior secondary level. hence, their mathematical preparation is less than what a regular mathematics student gets at a regular university. on the other hand, there is a lot of emphasis on didactics and pedagogy at the teacher training college. characteristics of the subjects (sample) [backgound of students] quadrilaterals like the parallelogram, square, rectangle, rhombus and kite are taught in all schools in south africa. all the students had done mathematics up to matric level at school. many of them had attempted standard grade mathematics and obtained poor passes. a few had attempted higher grade mathematics and also performed poorly (see table 3.1). at college the students in the mathematics method 1 and 2 courses revisited many of the topics from the junior secondary and senior secondary mathematics school syllabi. their first and second year mathematics education courses included many of the learning theories and other associated topics such as mathematical thinking; cooperative learning; investigative mathematics; obe; and assessment. in the final year of study a great deal of time was allocated to calculus. these students were expected to be conversant with most of the aspects associated with quadrilaterals, in particular the properties of rhombuses, the definition of a rhombus and the associated theorems. all students were black and had only attended schools for blacks in their previous classes. there were 10 males and 8 females in the given sample. table 1 shows: 36 rajendran govender, michael de villiers • the gender distribution in the sample • age of the respective students • the students’ mathematics symbols in the senior certificate examination and the grades on which the subject was written (higher grade or standard grade) • the students’ second year college exam mathematics content results. the overall academic performance for each student in the senior certificate examination is poor. the overall academic performance of most of the students in the second year college exam is clearly below average. the students were preparing themselves to write their final college exit exam. they were all willing participants since they all wanted to become computer literate and also wanted to understand the topic better. the topic is not in the college syllabus and the researcher promised the subjects that he would make them computer literate and issue a certificate of participation to each student. many of the students were computer illiterate. during the afternoons and during the mathematics method period the students were exposed to a basic computer literacy programme. the students had no prior exposure to sketchpad but through organised planning they were brought together in the afternoons in order to familiarise them with the general use and application of the computer software – sketchpad. furthermore, they had not been previously exposed to the process of defining. before engaging them in defining rhombi, the students had to be given some knowledge of the properties of rhombuses and associated theorems. this was well within the capabilities of the students. name age gender mathematics higher grade mathematics standard grade second year final second year supplementary buzani 22 f f 40% 51% ephraim 23 m f 42% 39% innocent 21 m f 43% 44% jwara 20 m f 58% khethu 20 m e 54% letha 21 m e 53% mathunzi 23 m e 59% mthembu 23 f e 61% ntombi 22 f f 56% owen 21 m e 63% petros 23 m f 40% 47% sanele 22 m e 50% siboniso 21 m e 59% sifiso 23 m e 62% siyand 23 m f 43% 50% thandiwe 23 f e 52% xolis;e 23 f e 55% fikile 23 f f 51% table 1: participants in the study research approach the developmental research approach of the freudenthal institute at utrecht university in the netherlands was used (gravemeijer, 1994). basically this approach involves the designing, assessment and consequent redesign of curricula for learning and teaching mathematics. use of activities from de villiers (1999) was made in this study, as well as the associated theoretical framework in relation to different functions of proof in mathematics. 37 a dynamic approach to quadrilateral definitions the van hiele theory of learning geometry also provided a useful conceptual framework for interpreting and analysing the student teachers' levels of understanding (see burger & shaughnessy, 1986). data collection both qualitative and quantitative data were collected between february and may 2000 from 18 students, largely by means of one-to-one task-based interviews. the rhombus was chosen for the task-based activity. the researcher chose this quadrilateral because in his experience as a teacher he had found that learners at schools have a lot of difficulty in understanding this concept, and in meaningfully applying the knowledge gained about a rhombus. they also have difficulty in "connecting" the knowledge gained to other quadrilaterals. the collection of data was of a dual nature, i.e. there were written responses to some questions and verbal responses to others. in addition, there were follow up interviews to some of the written responses during the task-based interview. all interviews were tape recorded and transcribed. bell (1995) is of the view that for the sake of accuracy a tape recorder is a vital device, especially in interviews. the collection of data was divided into three sessions as follows: • ascertaining the student teachers' prior understanding of definitions; • engaging them dynamically in the process of evaluating different given descriptions (definitions) for a rhombus; and • assessing their own ability in evaluating other, new definitions for a rhombus. clarification of some terminology this section is included in this paper because it contains definitions intended to be helpful in the analysis. the arbitrariness of mathematical definitions the arbitrariness of mathematical definitions refers to the existence (or choice) of different, alternative but correct definitions for the same concept. for instance, when defining a trapezoid one can define it as a quadrilateral having at least one pair of opposite sides parallel. on the other hand, one can define it, if one wishes, as a quadrilateral having exactly one pair of opposite sides that are parallel. if you choose the first definition, a parallelogram is also a trapezoid. if you choose the second one, it is not. if the concept that definitions are arbitrary is well understood the above fact will not cause confusion, otherwise it might cause a great deal (see vinner, 1991). necessary and sufficient conditions for a condition in a given description (definition) to be sufficient, it must contain enough information (properties) to ensure that not only do we obtain the elements of the set we want to define, but only those elements (and not any others). however, normally we want to use as little information as possible, i.e. only as much as is really necessary. correct definitions a description (definition) which contains conditions (properties) that are sufficient is said to be correct. in a correct definition, all the conditions may be necessary or some of the conditions may be unnecessary, i.e. it is possible to have unnecessary conditions in correct definitions. incorrect definitions a definition is incorrect if it contains an incorrect property or if it contains insufficient properties. incomplete definitions a definition is incomplete if it contains necessary but insufficient properties. so an incomplete definition is also an incorrect one. economical and uneconomical definitions a correct definition can be either economical or uneconomical. an economical definition has only necessary and sufficient properties. it contains no superfluous information. on the other hand, an uneconomical definition has sufficient, but some unnecessary properties. in other words, it contains more information than necessary (redundant information). rationale for use of the words economical and uneconomical it is generally accepted that definitions in mathematics should be minimal (economical). by this we mean that definitions should not contain parts which can be mathematically inferred from other parts of the definitions. for instance if one decides to define a 38 rajendran govender, michael de villiers rectangle in euclidean geometry by means of its angles it is preferable to define it as a quadrilateral with three right angles and not as a quadrilateral having 4 right angles. this is because in euclidean geometry, if a quadrilateral has 3 right angles one can prove that its fourth angle is also a right angle. so when formally defining the meaning of a term it is customary to give only the minimum required to understand the term. this minimality principle is a crucial structural element of mathematics organised as a deductive system. in fact, it shapes the way in which mathematics progresses when it is presented deductively, namely after the definition, theorems which give you additional information about the concept are formulated and proved. linchevsky et al (1992:54) made the following point in regard to definitions being minimal: “if mathematical definitions were not minimal we would have to prove their consistency. for instance, if you define an equilateral triangle as a triangle whose sides are congruent and all its angles are congruent then you have to proceed by showing that these two properties can "live together." the most appropriate way of doing it in this case is to show that if all sides of a given triangle are congruent then its angles are also congruent. therefore, what is the point of defining an equilateral triangle by both its sides and its angles if you should prove exactly the same theorems you would have to prove when going the minimal way? being minimal is being economical”. however, there are a few cases in geometry where definitions are not minimal. a familiar one, perhaps, is the way in which some textbooks define congruent triangles. for example, “congruent triangles are triangles which have corresponding sides congruent and corresponding angles congruent” (gonin et al, 1974). we know, though, that it is sufficient to require less than that for two triangles to be congruent. this is expressed by each of the four congruence axioms. data analysis and findings the analysis and findings will be presented in the context of each of the critical questions. student teachers' prior understanding of definitions two of the tasks used are given and discussed below. t foll h defi leth xoli . t info rhom nec app h defi mat nto t suff or t t g (n task 1 how would you describe what a rhombus is, over the phone, to someone who is not yet acquainted with it? he students’ responses were classified into the owing categories ere are some examples of incomplete or wrong nitions: a: i would say a rhombus is a four-sided figure. it has both pairs of opposite sides equal. it also has both pairs of opposite sides parallel. sle: a rhombus is a parallelogram with four sides. both opposite sides are equal and both opposite angles are equal and opposite sides parallel hese descriptions clearly contain too little rmation to guarantee the construction of a bus. these student teachers seem to view essary conditions as sufficient conditions and ear to be operating only at van hiele level 1. ere are some examples of typical uneconomical nitions: hunzi : i would tell him/her that it is a four sided, figure with all sides equal. if you join the opposite angles, the angle at the centre where the lines bisect each other will be a right angle and diagonals bisect the angles of the rhombus. mbi: a rhombus is a four-sided figure, with diagonals bisecting each other at right angles and with each pair of opposite sides equal and parallel. hese student teachers provided more than the icient information required to construct a rhombus o deduce the other properties from it, and therefore ask 1 incomplete or wrong definitions uneconomical definitions economical definitions roup =18) 4 13 1 39 a dynamic approach to quadrilateral definitions would appear to be operating at van hiele level 2 (compare burger & shaughnessy, 1986). the following student teacher was the only one to provide a correct, economical definition: sifiso: i would say a rhombus is a closed four sided figure with all its sides equal. their selection of incomplete descriptions (definitions) makes it clear that these student teachers don’t understand that a description (definition) must contain necessary and sufficient properties. whilst many students chose various options to describe a rhombus, none of the students chose the full complement of the correct descriptions, namely c, d, f, g. this therefore suggested that the students did not have a well-developed ability to evaluate definitions. since this definition does not contain any redundant or superfluous information, it would appear that the student teacher is operating at van hiele level 3. co na on ma ch co sel • • • eight students selected: (b) a rhombus is any quadrilateral with perpendicular diagonals. a single student teacher, owen, chose only one description. the others, by choosing more than one, seemed to have at least an intuitive understanding of the arbitrary nature of definitions. summary • the majority of the student teachers appeared to have an intuitive understanding of the arbitrary nature of definitions. • several student teachers understood a definition (description) of a given figure to be a list of properties of that given figure, which in fact is an uneconomical way to describe a figure (thus suggesting van hiele level 2 understanding). • some student teachers incompletely define (describe) figures by viewing necessary conditions as sufficient conditions. constructively evaluating different definitions for a rhombus the student teachers were next given a number of 40 task 2 which of the following descriptions do you think you would be able to use? circle these descriptions. a. a rhombus is any quadrilateral with opposite sides parallel. b. a rhombus is any quadrilateral with perpendicular diagonals. c. a rhombus is any quadrilateral with two perpendicular axes of symmetry (each through a pair of opposite angles). d. a rhombus is any quadrilateral with perpendicular, bisecting diagonals. e. a rhombus is any quadrilateral with two pairs of adjacent sides equal. f. a rhombus is any quadrilateral with all sides equal. g. a rhombus is any quadrilateral with one pair of adjacent sides equal, and opposite sides parallel. none of the student teachers selected the full mplement of correct descriptions (definitions) mely c, d, f and g. however, three students selected ly correct descriptions (definitions). for example, thunzi chose d and f, owen chose g whilst sanele ose d and f. the remaining 15 students chose at least one rrect description (definition), although they also ected incomplete descriptions. for example: twelve students selected: (a) a rhombus is any quadrilateral with opposite sides parallel. eleven students selected: (e) a rhombus is any quadrilateral with two pairs of adjacent sides equal. prerecorded sketchpad scripts from de villiers (1999), and asked the following main questions while playing the scripts: a. does the script construct a correct rhombus? b. which description in the table matches the script? c. drag the figure. does it always remain a rhombus? d. is the given information sufficient for the construction of a rhombus? if yes/no, why? e. is all the given information necessary for the construction of a rhombus? f. is the description correct? if yes/no, why? g. is the description economical or uneconomical? why? rajendran govender, michael de villiers although initially the student teachers required some guidance in constructing the required givens (prerequisite points) for each script, they quickly became independent as they progressed through the 7 scripts. working through each script provided good learning opportunities for the student teachers to check whether the conditions for each script were sufficient to produce a rhombus. due to space limitations, we shall here only discuss the script rhombus 7. an example of an on-screen sketch produced by this script is shown in figure 1 (though appearance may vary depending on the relative positions of the pre-requisite points a and b). when the construction was finished, the researcher firstly questioned the student teachers about the displayed lengths and gradients on the screen. the student teachers showed understanding of the displayed measurements. upon asking the students whether the script constructed a correct rhombus, all responded that it was correct, apparently judging purely from a visual perspective. only upon encouragement did the students check out their claim by measuring the sides to see if they were really all equal. furthermore, all 18 student teachers matched the script rhombus 7 correctly to description g. the student teachers were then requested to drag the figure task 3 one way of testing a description is to construct a figure complying with the description to see if it really gives the desired figure (we will use sketchpad). open the scripts rhombus 1.gss (windows), and go on to check the descriptions a-g. construct the appropriate givens required for each script, and click on the step button repeatedly to make each script construct its figure. when the construction is finished, match each script with a description in the table. drag the figure to see if it always remains a rhombus. in the table below, cross out the script names of any scripts that construct figures that are not always rhombuses. script description (a-g) rhombus 1 rhombus 2 rhombus 3 rhombus 4 rhombus 5 rhombus 6 rhombus 7 on sketchpad and observe whether it always remained a rhombus. after the student teachers had dragged the figure around numerous times, they were confident figure 1: example of an on-screen sketch a b c d m ac = 3.715 cm m ab = 3.715 cm slope ac = 0.442 slope db = 0.442 slope ab = -1.291 slope cd = -1.291 41 a dynamic approach to quadrilateral definitions that this figure always remained a rhombus. the student teachers also indicated that the given information was sufficient as well as necessary. some selected extracts from typical responses are given below: researcher: i now want you to look for a description in the list that fits in with rhombus 7? xolisile: g. researcher: ok, right, g is correct. i want you to focus on the definition or description there and i want you to tell me if the given information is sufficient for the construction of a rhombus? xolisile: yes. researcher: why? xolisile: the script always constructs a rhombus. if you drag it you always get a rhombus. researcher: ok. is the information necessary for the construction of a rhombus? xolisile: yes, it is necessary. all the student teachers identified it is a correct, economical definition because it contains necessary and sufficient conditions, which is indicative of van hiele level 3 thinking. researcher: is it an economical definition? xolisile: yes. researcher: why? xolisile: it is necessary and sufficient. researcher: would you say that the definition is a correct definition of a rhombus? mathunzi: ja, i would say it is a correct definition of a rhombus. researcher: would you say it is a correct economical definition? mathunzi: ja, it is economical. researcher: why do you say economical? mathunzi: aaaah. it is sufficient and necessary. d, des tha ide nam summary the student teachers’ responses to task 4, in comparison to their earlier responses to task 2, clearly suggest the following as a result of being involved with the process of constructive evaluation of definitions in a dynamic geometry environment: • the student teachers appear to have developed a deeper understanding of the arbitrary nature of definitions. • they showed improved ability to select correct alternative definitions for a rhombus. • the scripts and the use of dynamic dragging provided the student teachers with the opportunity to check whether the conditions in the given definitions were sufficient for the accurate construction of a rhombus. • as a consequence, the student teachers exhibited a better understanding of when conditions are: • necessary and sufficient • necessary but not sufficient • sufficient but not necessary (and therefore also some ability to distinguish between economical and uneconomical definitions) student teachers' competency to assess other definitions after the preceding process task 5, consisting of 2 items, was utilised to determine how competent the students were in evaluating other possible definitions for rhombi after the preceding learning experience. it was given to the students immediately after all the interviews were completed. this task comprised only written responses which were done by all the students at one sitting, though all the students had access to sketchpad whilst answering these questions. a discussion of one of the items from task 5 is provided below. 42 task 4 list the descriptions from a-g that you think best describes a rhombus seventeen out of 18 students listed all four (i.e. c, f and g) correct descriptions as the ones that best cribe the rhombus in task 2. only one student, ndiwe, did not choose all four, but managed to ntify three out of the four correct descriptions, ely c, d and f. t desc task 5 consider the following definition: “a rhombus is any quadrilateral with all sides equal, opposite sides parallel, and perpendicular and bisecting diagonals”. a. what are your comments about this definition? b. is it a good definition? c. if not, how would you change it? his item gives a correct but uneconomical ription of a rhombus and therefore was designed rajendran govender, michael de villiers to test the students’ ability to recognise that the description is correct, but uneconomical (i.e. contains more information than necessary). fourteen out of 18 of the respondents recognised that this description was uneconomical (sufficient but has unnecessary properties). this item was also implicitly intended to evaluate the students’ understanding of a “good” definition and it was encouraging to note that all fourteen students indicated that the description was not a good definition. the following are typical examples: jwara's response: a. correct definition. but it is sufficient and unnecessary. b. no. fikile's response: a. this is a correct uneconomical definition because it includes unnecessary properties and can be made economical in a number of ways . b. no, it is not a good definition. sifiso's response: a. this definition is correct but it also contains unnecessary information therefore it is uneconomical. b. no, because it is uneconomical. however, not all fourteen who identified it as an uneconomical definition gave clear responses. indeed some showed either a measure of confusion or difficulty in clear formulation. extracts from these four responses are now presented and discussed individually. mthembu's response: a. the definition is correct but some of the information is not there (uneconomical). b. no. though mthembu says it is uneconomical, he says “some of the information is not there”. this may suggest that he is actually operating at van hiele level 2, and might prefer more properties (rather than less) in the definition, rather than simply not understanding the meaning of the term uneconomical. letha's response: a. the definition is uneconomical for a rhombus. it is insufficient but necessary. b. no. similarly, letha stated that it was uneconomical, but then contradicts himself by claiming that the information is insufficient. it is difficult to ascertain whether the difficulty is conceptual or just a matter of a minor confusion of correct mathematical terminology. siyande’s response: a. uneconomical because the definition is insufficient. b. no. just like letha there is a contradiction in this statement as a definition cannot be (correct) economical, as well as insufficient. it appears that siyande might have been reasoning as follows: if the properties are more than sufficient then it means that it is “not sufficient” which he then equivalently matched with the word “insufficient”. however, without further interviewing it is impossible to determine whether the problem is conceptual, linguistic or a mixture of both. here are some examples of the four responses that did not identify the definition as uneconomical: innocent: a. this is a good definition. in fact it is a correct economical definition as it contains necessary and sufficient conditions. b. yes. ephraim: a. i think this definition is necessary and sufficient because it include all the properties of a rhombus. b. yes. ntuli: a. economical correct, definition contains necessary & insufficient conditions. b. no. (ntuli may have thought it was not a good definition because he personally preferred another correct definition) item 5(c) was intended to assess whether the student teachers could provide a shorter, correct definition of a rhombus. nine of the students came up with shorter, correct economical descriptions by 43 a dynamic approach to quadrilateral definitions leaving out some properties. six of these students chose to define the rhombus in terms of its sides (a fairly standard textbook definition) as follows: a rhombus is any quadrilateral with all sides equal. two students preferred to define a rhombus as a parallelogram with one pair of adjacent sides equal (which is also used in some textbooks). for example: a rhombus is any quadrilateral with one pair of adjacent sides equal, and opposite sides parallel. however, sboniso defined a rhombus just in terms of its diagonals (a seldom used definition in textbooks). for example: a rhombus is any quadrilateral with perpendicular bisecting diagonals. six students came up with different shorter versions, which were either still uneconomical, or insufficient, or contained an incorrect property. for example: uneconomical: mthembu for example gave a definition that was correct, but still uneconomical: a rhombus is any quadrilateral with both pairs of opposite sides equal and parallel and with diagonals bisect each other perpendicularly. note that the definition would have been economical (and a non-standard definition) had mthembu left out the condition that the diagonals bisected each other. ntombi’s response: a rhombus is any quadrilateral with all sides equal, opposite sides parallel and perpendicular, bisecting diagonals, forming two axes of symmetry. ntombi seemed to only list all the properties of a rhombus she knew, making it even more uneconomical. her reasoning appears to be at van hiele level 2. incomplete (insufficient): petros gave a definition that was insufficient. for example: a rhombus is any quadrilateral with opposite sides parallel. thandiwe also gave a definition that was insufficient. for example: a rhombus is any quadrilateral with all pairs of opposite sides parallel and bisecting diagonals. however, it was encouraging to note that half the student teachers presented correct economical definitions in 5(c), which is reflective of van hiele level 3 thinking. summary • the majority of the student teachers were able to identify (realise) the description in task 5 as correct but uneconomical, and half of them were able to change it to a correct economical description of a rhombus in 5(c). • it would appear that this improved ability to evaluate and improve a given definition could in some measure be attributed to the earlier learning activities, i.e. the earlier construction, measurement and dragging activities with sketchpad. concluding remarks the use of construction and measurement to evaluate the correctness of geometric statements (conjectures) before proofs are done is of course common practice among mathematical researchers. as a teaching approach it is also not new. for example, a similar approach was used effectively in the useme teaching experiment during 1977/78 (see human & nel et al, 1989). similarly, smith (1940) reported marked improvement in pupils' understanding of "if-then" statements by letting them first make constructions to evaluate geometric statements. in his research he found that it enabled pupils to learn to clearly distinguish between the "given condition(s)" and the "conclusion(s)", and laid the conceptual groundwork for an improved understanding of the eventual deductive proof. however, this study is markedly different in that it took place within the context of dynamic geometry, where a geometric configuration can be continuously dragged into different shapes to check for invariance. ideally, students should test geometric statements by making their own constructions within sketchpad. however, since this requires a rather high level of technical knowledge of the software, it was decided to provide them with ready-made scripts that they could play through step by step and observe as the figure was gradually constructed. as the scripts are dependent on the arbitrary construction and positioning of the "given points", they sometimes produce crossed quadrilaterals which was a little confusing to some students. accordingly, in the revised version of this activity (see de villiers, in press) use will instead be made of the "hide/show" button facility of sketchpad to produce figures step by step, ensuring that they all initially appear to be a rhombus. only upon further dragging would students 44 rajendran govender, michael de villiers then be able to ascertain whether it always remains a rhombus, and therefore whether the conditions are really sufficient. it should also be noted that since the dynamic geometry software provided conviction to all the student teachers, the role of the eventual deductive proofs (i.e. to prove the sufficiency of the definitions) was conceptualized as that of systematization rather than that of verification. although it was not a main focus of this study, the issue of hierarchical vs. partition definitions for a rhombus arose quite a few times while interviewing (or in discussion with) the student teachers. however, the dynamic nature of the rhombi constructed in sketchpad seemed to make the acceptance of the hierarchical classification of a square as a special rhombus far easier than in a traditional non-dynamic environment, as the student teachers could easily drag the constructed rhombus until it became a square. this is, however, a matter for further research. acknowledgement this research was partially funded by a national research foundation (nrf), pretoria, south africa grant from the spatial orientation & spatial insight (sosi) project, coordinated by prof. dirk wessels (unisa), dr. hercules nieuwoudt (puche) and prof. michael de villiers (udw). the opinions and findings do not necessarily reflect the views of the nrf. references bell, j., 1995, doing your research project. buckingham: open university press burger, w.f. & shaughnessy, m., 1986, “characterizing the van hiele levels of development in geometry” in journal for research in mathematics education, 17(1), pp. 31-48 de villiers, m., 1997, “the future of secondary school geometry” in pythagoras, 44, pp. 37-54 de villiers, m., 1998, “to teach definitions in geometry or teach to define?” in olivier, a & newstead, k. (eds.) proceedings of 22nd pmeconference, university of stellenbosch, 12-17 july 1998, vol. 2, pp. 248-255 de villiers, m., 1999, rethinking proof with geometer's sketchpad, key curriculum press, usa de villiers, m. (in press), rethinking proof with geometer's sketchpad 4, key curriculum press, usa gonin, a.a., archer, i.j.m., slabber, g.p.l., la rey nel, g.d.e., 1974, “congruent triangles” in modern graded mathematics for standard 8 (second edition), elsies river: national book printers limited govender, r., 2002, student teachers' understanding & development of their ability to evaluate & formulate definitions in a sketchpad context, unpublished master's thesis, university of durban-westville gravemeijer, k., 1994, “educational development and educational research” in journal for research in mathematics education, 25 (5): pp. 443-471 human, p. g. & nel, j. h. in co-operation with m. d. de villiers, t. p. dreyer and s. f. g. wessels., (eds.), 1989, useme teaching experiment. appendix a: alternative geometry curriculum material. rumeus curriculum material series no. 11, university of stellenbosch linchevsky, l., vinner, s., karsenty, r., 1992, “to be or not to be minimal? student teachers views about definitions in geometry” in geeslin, w., graham, k. (eds.), proceedings of the sixteenth international conference for the psychology of mathematics education , vol. 2, pp. 48 55. durham, usa department of education, 2002, national curriculum statement grades 10 12 (schools): mathematics (draft), pretoria: department of education smith, r. r., 1940, “three major difficulties in the learning of demonstrative geometry” in the mathematics teacher, 33, pp. 99-134, 150-178 vinner, s., 1991, “the role of definition in the teaching and learning of mathematics” in d. tall (ed.), advanced mathematical thinking, kluwer academic publishers, pp. 65-81 45 a dynamic approach to quadrilateral definitions introduction definitions the van hiele theory level 1: recognition: students visually recognise figures by level 2: analysis: students start analyzing the properties o level 3: ordering: students logically order the properties o level 4: deduction: students start developing longer sequenc level 3 research design the target population research approach data collection clarification of some terminology this section is included in this paper because it contains d the arbitrariness of mathematical definitions necessary and sufficient conditions correct definitions incorrect definitions incomplete definitions economical and uneconomical definitions rationale for use of the words economical and uneconomical data analysis and findings student teachers' prior understanding of definitions task 1 summary constructively evaluating different definitions for a rhombu student teachers' competency to assess other definitions aft uneconomical: mthembu for example gave a definition that was incomplete (insufficient): petros gave a definition that was summary concluding remarks acknowledgement references pyth 37_1 reviewer acknowledgement.indd reviewer acknowledgement open accesshttp://www.pythagoras.org.za page 1 of 2 reviewing is an important part of scholarly work, making a substantial contribution to the field. reviewers’ comments serve two purposes, guided by two inter-dependent objectives: • pythagoras wishes to publish only original papers of the highest possible quality, making a meaningful contribution to the field. reviewers advise the editor on the scholarly merits of the manuscript to help him evaluate the manuscript and to decide whether or not to publish it. reviewers are encouraged to reject a manuscript if it is scientifically flawed, merely sets out observations with no analysis, provides no new insights, or is of insufficient interest to warrant publication. • pythagoras is committed to support authors in the mathematics education community. reviewers help the author to improve the quality of their manuscript. reviewers are encouraged to write their comments in a constructive and supportive manner and to be sufficiently detailed to enable the author to improve the paper and make the changes that may eventually lead to acceptance. the following summary of outcomes of the reviewing process in 2016 shows that our reviewers do well in achieving both objectives: we sincerely thank the following people who have reviewed these manuscripts for pythagoras in 2016. we very much appreciate their time, expertise and support of pythagoras amidst pressures of work. in an effort to facilitate the selection of appropriate peer reviewers for manuscripts for pythagoras, we ask that you take a moment to update your electronic portfolio on http:// www.pythagoras.org.za, allowing us better access to your areas of interest and expertise, in order to match reviewers with submitted manuscripts. if you would like to become a reviewer, please visit the pythagoras website and register as a reviewer. to access your details on the website, follow these steps: 1. log into pythagoras online at http://www. pythagoras.org.za 2. in your ‘user home’ select ‘edit my profile’ under the heading ‘my account’ and insert all relevant details, bio statement and reviewing interest. it is good practice as a reviewer to update your personal details regularly to ensure contact with you throughout your professional term as reviewer to pythagoras please do not hesitate to contact us if you require assistance in performing this task. publisher: publishing@aosis.co.za tel: +27 21 975 2602 fax: +27 21 975 4635 ajayagosh narayanan andile mji andrew maffessanti anita campbell anna posthuma ansie harding anthony essien belinda huntley bruce brown calisto munongi caroline long colin foster craig pournara david andrich dirk wessels duncan mhakure ednei becher erica spangenberg erna lampen eunice moru faaiz gierdien hamsa venkat helen sidiropoulos ingrid mostert johan hugo joseph dhlamini joseph furner kaashief hassan kakoma luneta katy harries the quality of the articles in pythagoras and the credibility and reputation of our journal crucially depend on the expertise and commitment of our peer reviewers. acknowledgement to reviewers no. manuscripts processed in 2016 (outcome complete) 25 accepted without changes 0 (0%) accepted with minor changes (to the satisfaction of the editor)1 6 (24%) accepted after major revisions (re-submit, then re-review)2 1 (4%) rejected after review not acceptable to be published in pythagoras3 5 (20%) rejected without review not acceptable to be published in pythagoras4 13 (52%) no. articles currently in review 17 1. accepted after one round of review, with ‘minor’ changes as specified by reviewers and editor. 2. accepted after two or more rounds of review, with ‘major’ changes specified by reviewers and editor. 3. includes two cases where authors did not resubmit after required to make major changes. 4. all submissions undergo a preliminary review by the editor (and associate editors) to ascertain if it falls within the aims and scope of pythagoras and is of an acceptable standard. includes two cases where authors did not resubmit after extensive feedback prior to reviewing. for ethical reasons, we rejected a flood of articles from a russian agency submitting manuscripts on behalf of researchers. reviewer acknowledgement open accesshttp://www.pythagoras.org.za page 2 of 2 l.m. kaino leila goosen lyn webb lynn bowie marc north margot berger marie joubert mark jacobs mdutshekelwa ndlovu michael de villiers michael mhlolo michael murray nick taylor nyna amin pam lloyd patisizwe mahlabela patrick barmby paul mokilane piera biccard rajendran govender renuka vithal sarah bansilal satsope maoto stanley adendorff temesgen zewotir tim dunne toni beardon tracy craig vanessa scherman vera frith verona leendertz willy mwakapenda if you would like to become a reviewer, please visit the pythagoras website and register as a reviewer. reviewers (continued): proofs that explain from ‘proofs without words’ to ‘proofs that explain’ in secondary mathematics m faaiz gierdien stellenbosch university email: faaiz@sun.ac.za introduction from the mid 1970s onwards in almost every issue of the undergraduate mathematics journals mathematics magazine and college mathematics journal there is at least one ‘proof without words’ (nelsen, 1993). a proof without words can be thought of as a ‘proof’ that makes use of visual representations, that is, pictures or other visual means to show a mathematical idea, equation or theorem (casselman, 2000). it does not contain any words other than literal or numerical symbols and geometrical drawings, for example. there is debate around whether a proof without words really qualifies as a proof. it helps the observer see why a particular mathematical statement may be true, and also to see how one might begin to go about proving it true. it may also have an equation or two, arrows or shading in order to guide the reader in this process. in it there is a clear emphasis on providing visual clues to the reader in order to stimulate thinking with the eventual goal of writing a proof. many proofs without words in the referred journals are directly related to the secondary mathematics curriculum in south african schools although not exclusively so. interpreting a proof without words requires explanations that draw on various mathematical ideas not necessarily evident in the proof without words. when the reader starts to unpack and explain the diagrams or pictures in the proof without words, it can become a ‘proof that explains’ as opposed to a ‘proof that proves.’ more needs to be said about the last two notions. writing explanations for and discussing a suitable proof without words can present opportunities to develop insights about and connections between different mathematical ideas. these are also ways to popularise proof in general in the secondary mathematics curriculum (de villiers, 1990; volmink, 1990). references to proof appear in current south african policy documents on school mathematics reform at the secondary level. for example, “competence descriptions” for learners by the end of grade 12 include “being able to critically analyse and compare mathematical arguments and proofs” and being able to “demonstrate an understanding of proof in local axiomatic systems” (department of education, 2003: 83). the question becomes, what means are available to align learners and teachers with these competence descriptions? elsewhere the document mentions “mathematical process skills” which include generalising, explaining, describing, observing, inferring, specialising, justifying, representing, refuting and predicting (ibid.: 19). as possible visual processes these mathematical skills can stimulate thinking about proofs and also proofs without words. in this regard visualisation is a key construct which will be explored in this paper. a reference to proof also appears in the study of series and sequences (ibid.). the purpose of this paper is to explore an epistemic role for visualisation with respect to proofs without words in secondary mathematics in the current south african education policy context. visualisation as process and product can be a means to examining proofs without words by turning them into proofs that explain. in this way students can develop insights and explanations for the mathematics they encounter in the secondary curriculum. the proofs without words chosen are those that show analytic and visual representations of series and sequences. in the secondary curriculum series and sequences are mainly represented analytically. it will be shown that a thoughtful interpretation and explanation through visualisation of such proofs without words connects different strands in the bureaucratically stated secondary curriculum found in the policy document (department of education, 2003). there is more mathematics embedded and ‘unseen’ in these proofs without words. visualisation as process and product it is difficult to conceptualise a neat division between visualisation as process and product when we interpret a proof without words. visualisation as product can be thought of as the proof without words or the final picture. on the other hand pythagoras 65, june, 2007, pp. 53-62 53 m faaiz gierdien visualisation as process involves employing various techniques to understand and to interpret the proof without words. visualisation has a special attraction in the case of a proof without words because the reader is drawn ‘to fill in the words’ in order to make the theorem or statement in the proof without words true. literature on visualisation sometimes refers to visualising (giaquinto, 1993; 1994), visual reasoning (hershkowitz, arcavi, & bruckheimer, 2001) or simply visualisation (arcavi, 2003). many proofs without words rely on visual means to communicate a mathematical statement. visualisation – as both the product and the process of creation, interpretation and reflection upon pictures and images – is gaining increased popularity in mathematics and mathematics education (arcavi, 2003). what we do upon seeing a proof without words is process a product. depending on the proof without words, the reader can be drawn into “seeing the unseen and perhaps also proving,” according to arcavi (2003). one can think of interpreting, creating and reflecting as examples of visualisation as process, which can also include scribbling notes or diagrams on paper, or making gestures and utterances. interestingly, visualisation as product can include explanatory notes that result from interpretation of and reflection on a proof without words, in addition to the final picture or proof without words. in a proof without words of the infinite geometric series 3 1 ... 2 1 2 1 2 1 642 =+⎟ ⎠ ⎞ ⎜ ⎝ ⎛ +⎟ ⎠ ⎞ ⎜ ⎝ ⎛ +⎟ ⎠ ⎞ ⎜ ⎝ ⎛ (see figure 1), arcavi (2003) argues that a proof without words is (a) neither “without words” nor (b) “a proof.” the reader is most likely to decode the picture through words (a) – either mentally or aloud – and according to hilbert's standard for a proof, it must be “arithmetisable” (b), otherwise it is non-existent (hadamard, 1954, in arcavi, 2003). this explains the cautious use of “visual proof” in the case of the infinite geometric series. what is clear is the controversy around what constitutes a proof. from the former, we infer that what is seen – or visualisation as process or product – might actually be complemented by verbalisation. hence the notion of “without words” in proof without words should not be understood literally. there is continuum between process and product interpretations of visualisation which is illustrated using the proof without words of the same infinite geometric series. the proof without words presented in figure 1 is a product of the proof creator’s visualisation which the reader has to process. it provides us with cues that make our process of visualisation easier. we may not be instantly convinced of the result. also, we potentially see how a proof for the geometric series is done. there are non-trivial bits of numerical manipulations that the reader has to process, for example, interpreting ⎟ ⎠ ⎞ ⎜ ⎝ ⎛×⎟ ⎠ ⎞ ⎜ ⎝ ⎛ 2 1 2 1 and 22 2 1 2 1 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ×⎟ ⎠ ⎞ ⎜ ⎝ ⎛ as areas. the use of areas is an example of arcavi’s (2003) notion of how the reader is attracted to “seeing the unseen” or “filling in the words.” to make the statement true we may be attracted to look at the final picture or product. visualisation as process in the infinite geometric series has its attendant problems, namely, a particularity objection and unintended exclusions (giaquinto, 1993). what is the particularity objection in visualisation as process in the case of this infinite geometric series? we cannot do a visualisation process of the geometric series that goes up to infinity. we can only do a visualisation process up to a particular number of areas. visualisation as process thus cannot include every area in the infinite geometric series but it can specify some areas. also, in the process of visualising the infinite geometric series there are some areas which will be excluded from the content of visualising. this is not to say that a precise number of areas is visualised. there will be numerical vagueness in the visualisation process, but not so much vagueness that no number of areas is excluded. for instance, we cannot visualise 41 specific areas, meaning that there will be “unintended exclusions” (ibid.). from the way figure 1 is shown we are typically unable to carry out a visualisation process that includes exactly 41 figure 1. a proof without words of the infinite geometric series 54 m faaiz gierdien areas. the best we can do is to visualise an arrangement of roughly 6 such areas. there will thus be a problem of unintended exclusions the more we specify the number of areas. this problem does not negate the use of visualisation as process in this infinite geometric series. it does, however, pull us in the direction of the final picture or visualisation as product so that we can hopefully conclude that the sum to infinity equals 1 3 . visualisation as process and product in the case of the infinite geometric series can take us in the direction of analysis. the unseen mathematics in figure 1 is far more than meets the eye. seeing the unseen mathematics depends on the reader’s insights. by visualising the first few steps in the process the reader gets an idea of the common nature of each step: we divide the large unmarked square into quarters, marks the lower left of these quarters and leaves the other three unmarked (to be divided into quarters in the next step). a crucial thought becomes apparent: at each stage there is a shading of one of the four squares. the reader has to come to believe the theorem that the limit of the series 3 1 2 1 2 1 2 1 642 =+⎟ ⎠ ⎞ ⎜ ⎝ ⎛+⎟ ⎠ ⎞ ⎜ ⎝ ⎛+⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ... . also, it is clear that at no particular stage of the division of the squares do the areas of the shaded parts of the figure add up to give 13 . it also seems clear that no area at the top right-hand corner is so small that it will not eventually “fill up” the open space. there will be unintended exclusions. one can think of 13 as the least upper bound of the sequence. the truth of the theorem can be inferred from this, taking it as known that a monotonic increasing function sequence bounded above converges to its least upper bound. the arguments presented here take us into the realm of elementary real analysis, involving the limit of an infinite process (giaquinto, 1994). to get to see why the series has a limit of 13 , the reader’s eye has to digest several pieces of numerical information that are in the picture. this would entail seeing and eventually proving that the limit of the series is 13 through real analysis. on the other hand the reader can simply see or trust that 13 of the area of the outer square is being shaded. according to giaquinto (1994) there is insight garnered from the picture of such an infinite series. looking at the picture or proof without words we understand why the series has the sums it does. this picture is not a proof of what the limit of the infinite series is. implicit in the above are characteristic properties (steiner, 1978, in hanna, 1990), which will be discussed, in more detail later on. this, however, brings us to an important distinction between proofs that demonstrate that a theorem or statement is true and proofs that show why a theorem or statement is true. visualisation as both process and product plays a key role in turning a proof without words into a proof that explains. how do we distinguish a proof that proves from a proof that explains? distinguishing between proofs that prove and proofs that explain one of hanna’s (1983; 1990; 1998a; 1998b) major contributions to literature on the nature of proof in mathematics and mathematics education is a distinction between proofs that prove and proofs that explain. this distinction has a long and interesting history and is stated slightly differently at times. “verifying” is used when proofs demonstrate that a theorem or statement is true and “clarifying” is used when proofs show why a theorem or statement is true (de villiers, 1990). the former has to do with “convincing” or “making certain,” while the latter has to do with “explaining.” this distinction is quite important. for example, the mathematician bolzano (in hanna, 1990) makes a similar distinction “making certain” (gewissmachung) and “building a foundation” (begründung). “making certain” and “building a foundation” are synonymous with a proof that proves or verifies and a proof that explains or clarifies, respectively. hanna (1990) uses “explain” when a proof reveals, and makes use of the mathematical ideas that motivate it and hence refers to an “explanatory proof.” such a proof focuses on “building a foundation” or clarifying, and is consonant with volmink’s (1990) notion of proof as a means of communication. for example, in classrooms, teaching and explaining a proof becomes a form of discourse in which visualisation as process and product can lead to insight and connections among mathematical ideas. on the other hand, a proof that proves does not illuminate the appearance of particular symbols, whether literal or numerical, in a proof. hanna (1998a) cites mathematical induction as the example of a proof that proves or verifies. we have to unpack mathematical induction by firstly examining induction and then mathematical induction. induction is the process of discovering 55 m faaiz gierdien general laws by the observation of and combination of particular instances. it aims at finding regularity and coherence behind observations. how do we insert a mathematical aspect to observations done via induction? according to pólya (1945/1988) there is in mathematics a higher authority than observation and induction: rigorous proof. this is where mathematical induction comes in. it is to the mathematical aspects of mathematical induction that we turn to next. pólya (1954) lists several steps in mathematical induction before its actual technique. these are the inductive phase, the demonstrative phase, examining transitions and, finally, the technique of mathematical induction. during the inductive phase we suspect that a particular mathematical relationship, theorem or statement is true. driven by what we suspect, we formulate a conjecture about the mathematical statement which we test for particular cases to see if it is true. we check to see if the conjecture is true for several cases and we ask how we can test the conjecture more efficiently. during the demonstrative phase we examine whether the conjecture passes a severe test. this is done by taking what is supposedly true to what is incontestably true and finally consequently true. the statement 22 ...321 2 nn n +=++++ which appears in secondary mathematics will be considered. during the inductive phase there is an examination of several numerical values, where we can tabulate the results for n = 1, 2, 3, …. for example, we would end up with a conjecture that 22 ...321 2 nn n +=++++ is probably true. proving this truth would involve testing whether the conjecture is true. in the demonstrative phase we increase our doubts by first, assuming that it is supposedly true. the conjecture is then shown to be incontestably true and then consequently true. examining the transition from n to n + 1, is the last reasoning to conclude that 22 ...321 2 nn n +=++++ is true for all integers. to summarise, see box 1. going through this exercise we end up being certain that the statement is true. it is not difficult to see how all these steps are about convincing and making certain (gewissmachung). a curious student or learner following the steps in a proof via mathematical induction will certainly have questions, such as why is there a 12 in the statement, 22 ...321 2 nn n +=++++ ? this question calls for a ‘proof that explains.’ a visual representation or proof without words of this statement is shown in figure 2. to answer the question about the 12 the student will have to use visualisation processes such as describing and observing a triangle and blocks or square units in order to interpret the proof constructor’s product or proof without words. describing and observing can include ‘filling in the words’ or verbalising and ‘seeing the unseen.’ it becomes clear that processing a visualisation of the arithmetic (giaquinto, 1993) in the statement is supported geometrically. some of the seen and unseen mathematics is the area of a triangle with a height of length n units and a base of length n units. this area turns out to be 2 2n . the missing area to be added is 12 multiplied by n, the number of n square units. these are the shaded half squares, 2 n . it should be noted that the proof without words is about a general theorem in arithmetic: for all positive integers n, the sum of the first n positive integers is a half of n2 + n. the proof without words has a particular number of squares, meaning that in the visualisation processes there will be unintended exclusions. a similar 1 + 2 + 3 + …+ n = 22 2 nn + ∴1 + 2 + 3 + …+ n + (n + 1) = 22 2 nn + + (n + 1) = 2 2 2 2 22 2 +++ nnn = 2 1 2 122 + + ++ nnn = ( ) ( ) 2 1 2 1 2 + + + nn box 1. the transition from supposedly true to incontestably true, to consequently true, for all positive integers 56 m faaiz gierdien point was raised in the case of the infinite geometric series. alternately, we can find the area of a square of side length n, halving this area ( ) and then adding 12 2n of n blocks to yield 22 2 nn + , to find 1 + 2 + 3 + …+ n. there is thus a geometrical justification in terms of the area of a triangle that explains the statement, 22 ...321 2 nn n +=++++ which includes an explanation for the appearance of 12 . there are contrasts between proofs that prove and proofs that explain in the case of the said mathematical statement. in the entire proof that proves via mathematical induction there is no translation back and forth between different representations. only a numerical or analytic representation is used. in contrast, the proof that explains uses far more mathematics with the hope of bringing about understanding. here there is the possibility that the student will develop insights depending on how his or her visualisation as process and product interacts and unfolds. the mathematical statement is about a general arithmetic theorem which is proved via mathematical induction, for all positive integers n. in contrast the proof that explains with its geometric justification makes use of the area of a particular triangle, although the height and base of the triangle is stated as general, namely, ‘n’. more needs to be said about proving and explaining with respect to proofs without words. prove and explain so far it is evident that visualisation as process and product plays an important role in turning suitable proofs without words into explanatory proofs or proofs that explain. central to this is seeking characterising properties in the proofs without words. steiner (1978: 143) and hanna (1990: 10) – who cites steiner – characterise an explanatory proof as follows: n n figure 2. visual representation or proof without words …an explanatory proof makes reference to a characterising property on an entity or structure mentioned in the theorem, such that from the proof it is evident that the result depends on the property. it must be evident, that is, that if we substitute in the proof a different object of the same domain, the theorem collapses; more, we should be able to see as we vary the object, how the theorem changes in response. for example, what characteristic properties are entailed in proofs without words of the following two statements? 6 )2)(1( 2 )1( ...631 ++ = + ++++ nnnnn and 2 3333 2 )1( ...321 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + =++++ nn n ? also, how do we go about finding the characterising properties in proofs without words that motivate, explain and compel the truths of the statements? much of the answer lies with chinese mathematicians for whom a proof consisted of “any explanatory note, which served to convince or to enlighten” (siu, 1993: 346). they practiced “proof as explanation” in ways that were very different from greek mathematicians' axiomatics and deductive proofs, because these had not reached them (hanna, 1998b). explanatory notes as instances of visualisation as process and product thus play an important role in searching for characteristic properties. how can we explain each of the symbols in figure 3, 6 )2)(1( )1( 6 1 )1( 6 1 3 ++=+−+ nnn nn , the mathematical statement for the sum of n triangular numbers? visualisation as product in the form of a proof without words showing the sum of n triangular numbers is shown in figure 3. the triangular numbers – 1, 3, 6, … , 2 )1( +nn – are represented geometrically as the cubes in the layers t1, t2, t3…, tn respectively. the cubes forming the triangular numbers at each stage are arranged in a way where they form three-dimensional objects, which suggests that volume will come into play. in uncovering the characteristic properties of this 57 m faaiz gierdien proof without words, the reader’s eye is guided by the visualisation in the arrangement of the triangular numbers as cubes and the equal signs ending with the generalised pyramid of height (n+1) units and base of area 12 (n + 1) 2 square units. this forms a carefully assembled chain of reasoning and qualifies as a “good mathematical illustration” (casselman, 2000) that entices the reader to visualise the processes that make the mathematical statement true. where does 16 come from? in the third arrangement of the triangular number as cubes, in figure 3, one sixth of the volume of the top small cube is shaded. a small cube on its own will consist of 3 small pyramids having the same height. this is what calculus tells us, namely, the volume of a pyramid having the same height 3 1 =v (area of base × height). the base, however, is halved, meaning that the volume of the shaded part of the cube becomes 6 1 =v (area of base × height). this is indicated by the shaded part in the top, small cube, which forms the first triangular number. in the second triangular number two such slices are shaded. the pattern continues where three such slices are shaded for the third triangular number, and so on. in the second row of the arrangement in figure 3, the shaded pyramids are turned upwards to a generalised pyramid of height (n + 1), with a halved base which is explained by the 16 . the full volume of a cubic arrangement of triangular numbers of side length (n+1) is therefore (n+1)3. in the case of summing the triangular numbers, we are only interested in 16 of the volume. the extra volumes of 16 of 1 cubic unit of which there will be (n+1) have to be subtracted. this illuminates the line )1( 6 1 )1( 6 1 3 +−+ nn . obtaining the right-hand side of the statement 6 )2)(1( ++ nnn is a matter of factoring. as in the previous cases the proof without words contains unintended exclusions because it aims at drawing the reader into seeing a generalised arithmetic theorem. proofs without words for the statement 2 3333 2 )1( ...321 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + =++++ nn n figure 3. proof without words: sum of triangular numbers (from nelsen, 2005) 58 m faaiz gierdien entail characteristic properties that are combinatorial and geometric in nature (see figures 4 and 5). suggestion 1: combinatorial proof without words the combinatorial proof takes its name from the combinations of the sum of positive integers starting with 1 (see figure 4). by examining the combinatorial proof the reader can be encouraged into visualisation processes such as ‘seeing the unseen’ or ‘filling in the words.’ on the left hand side (figure 4), there is the sum of the positive integers in the first row beginning with 1 up to n. in the second row, each of these integers is multiplied by 2. in the third row, each of the integers in the first row is multiplied by 3. this pattern continues. the last two rows on the left-hand side can be factorised and reduced to the following: (1 + 2 + 3 + … + n) × ( which becomes × . on the right hand side of figure 4 the combinations of numbers as indicated are added in the following way: ) ) ∑ = n i i 1 ( )∑ = n i i 1 ( ∑ = n i i 1 1 + (2 + 4 + 2) + (3 + 6 + 9 + 6 + 3) + … . nnnnnnn ++++++++ 23......32 2 non-trivial bits of algebraic manipulation will have to be done to show that any l shape analytically represented as nnnnnnn ++++++++ 23......32 2 sums to . )( 2nn interestingly, these bits are not visualised at all. the l shapes sums are as follows: 1 = 1(1) 2 + 4 + 2 = 2(1 + 2 + 1) 3 + 6 + 9 + 6 + 3 = 3(1 + 2 + 3 + 2 + 1). this generalises as set out in box 2. manipulating ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − + + 2 )1( 2 )1( nnnn n yields )( 2 2 )1()1( 2 22 nn nnnn n nnnn n = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −++ = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −++ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − + + = −+++++++++=++++++++ 2 )1( 2 )1( )1...321()...321(23......32 2 nnnn n nnnnnnnnnnn box 2. generalising for a combinatorial proof 1 2 3 . . . n 1 2 3 . . . n + 2 4 6 . . . 2n + 2 4 6 . . . 2n + 3 6 9 . . . 3n + 3 6 9 . . . 3n + . . . . . . . + . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . + n 2n 3n . . . n2 + n 2n 3n . . . n2 = ∑ + 2 + 3 + … + n = n i i 1 ∑ = n i i 1 ∑ = n i i 1 ∑ = n i i 1 = ( )∑ = n i i 1 2 = 1(12) + 2(2)2 + … + n(n)2 = ∑ = n i i 1 3 = 2 2 )1( ⎟ ⎠ ⎞ ⎜ ⎛ ⎝ +nn figure 4. suggestion 1 – combinatorial proof (adapted from pouryoussefi, 1989) 59 m faaiz gierdien the previous manipulation is a deductive proof that shows that )(23......32 22 nnnnnnnnn =++++++++ the combinatorial proof without words, in fact, uses a result from a previous proof without words about consecutive integers that was discussed earlier, namely: 22 ...321 2 nn n +=++++ by focusing on these combinations of the numbers, we gain a sense of the truth of the original statement, namely: 2 3333 2 )1( ...321 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + =++++ nn n suggestion 2: geometric proof without words in the geometric proof without words, a focus on the area of a square of side length compels the truth in the original statement with some qualification (see figure 5). the reader’s attention can be directed to visualising the area of a square with a side length (1 + 2 + 3 + 4 + 5), or 2 )15(5 + . this is a particular side length meaning that the particularity objection mentioned earlier may be applicable. the area of the square is 2 2 )15(5 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + and not 2 2 )1( ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ +nn . this means that there are unintended exclusions in the geometric proof without words. this does not mean that our visualising experience cannot be in the direction of a general arithmetical or mathematical statement as in this case. the area inferred in the geometric proof is not stated as an arithmetical theorem about all positive integers (giaquinto, 1994) as compared to the combinatorial proof in suggestion 1. the problem of the unintended exclusions does not support a negative view of the utility of visualisation processes which are geometric in this instance. the connection between ‘series and sequences’ and the geometry of the areas of squares are not surprising because we have such a connection for summing the integers 1 + 2 + 3 + …+ n as we saw earlier on. implications for teaching the ideas discussed in this paper have implications for what might happen in teaching. each of the proofs without words became proofs that explain via visualisation as process and product. the latter is therefore a means to align policy statements about learners’ competence descriptions with respect to proof. recall that learners have to be able to critically analyse and compare mathematical arguments and proofs. so what might the teacher do? he or she should encourage learners to do the explaining when poring over a proof without words. they should be encouraged to ‘fill in the words’ and to try to ‘see the unseen’ mathematics through visualisation. they could do so collectively or individually. applicable here are visualisation processes such as generalising, observing, inferring, representing, predicting, describing through writing down what is observed and verbalising collectively and individually. note that that these processes are in concert with the department of education’s “mathematical process skills” according to the south african policy document for secondary mathematics (2003: 19). learners could record their utterances or verbalisation on the sheet containing the proof without words. this would be the product of their visualisation in addition to the proof constructor’s proof without words which they will be interpreting and explaining. the teacher must explicitly tell learners that any proof without words is a proof constructor’s final product that they have to process. the teacher would have the challenging task of orchestrating a discussion that has the goal of linking learners’ visualisation process and product with the proof without words that they are examining. 1 2 3 4 5 2 4 6 8 10 3 6 9 12 15 4 8 12 16 20 5 10 15 20 25 figure 5. suggestion 2 – geometric proof (adapted from pouryoussefi,1989) what is gained by learners explaining what they see in a proof without words? they might see how mathematical ideas in the secondary curriculum are related 60 m faaiz gierdien through different representations. who would have thought that a proof without words of a compact statement such as 22 ...321 2 nn n +=++++ – on series and sequences – can be explained via the area of a triangle, or that an analytic representation such as 6 )2)(1( 2 )1( ...631 ++ = + ++++ nnnnn can be explained using the volume of a pyramid? they could learn from the insights that fellow learners present during explaining. here the teacher plays a critical role because he or she will have to figure out what learners are saying in relation to what they ‘see” and ‘don’t see.’ a broad base of knowledge which is a prerequisite for mathematical insight (hanna, 1983) could be gained by explaining through visualisation as process and product. in a proof that proves, learners would not be able to come up with explanations for the appearance of 12 or a 1 6 as in figure 3. in a proof without words what could be gained by the explanation itself? the explanation can certainly help in terms of bolzano’s “building a foundation” (begründung) (in hanna, 1990). any explanation itself, however, will have to contend with unintended exclusions and particularity objections as shown earlier. in the first proof without words in figure 1, 3 1 ... 2 1 2 1 2 1 642 =+⎟ ⎠ ⎞ ⎜ ⎝ ⎛ +⎟ ⎠ ⎞ ⎜ ⎝ ⎛ +⎟ ⎠ ⎞ ⎜ ⎝ ⎛ it is not possible to exercise visualisation as process that goes to infinity. furthermore, the last three proofs without words (figure 3) are about general arithmetic theorems or series, namely, the sum of consecutive positive integers starting with 1, the sum of consecutive triangular numbers starting with 1 and the sum of consecutive cubes starting with 1. as ‘informal proofs,’ the proofs without words discussed highlight the slippage from dealing with specific numbers to dealing with infinity and general arithmetic theorems. explanations must take this slippage into account. learners might want to know whether there is another method to deal with the problems of unintended exclusions and particularity objections. would this pave the way for proofs that prove? what could done in the case of the learner who cannot ‘see’ the deductive proof for nnnnnnn ++++++++ 23......32 2 = ? )( 2nn this deductive proof does not ‘explain’. if we are to align learners’ competence descriptions with respect to proof then we must in our teaching aim for a level of proof that explains. the deductive mechanisms of mathematical induction and deductive proof do not have the goal of mathematical understanding (hanna, 1983; 1990). concluding remarks this paper has shown that visualisation as both process and product can play an epistemic role in changing selected proofs without words into proofs that explain. it can be a means to help learners to critically analyse and compare mathematical arguments and proofs at the secondary level. what has to be mentioned is the debate around the role of visualisation itself in the learning of mathematics. sfard (1998) cites a prominent member of the mathematics community who states that visualisation is not mathematics. the possibility of the “devaluation of visualisation” (presmeg, 1997) is therefore likely to permeate right to the classroom, curriculum materials and teacher education, according to arcavi (2003). also, there are cognitive difficulties around visualisation. in simplistic terms the issue raised reads as follows: is ‘visual’ easier or more difficult? we saw the cognitive demand was certainly high in turning the combinatorial proof without words of 2 3333 2 )1( ...321 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + =++++ nn n into a proof that explains. in fact, it depends on a previous proof without words. learners would need to attain flexible and competent translation back and forth between visual and analytic representations. learners working on their competences would thus have to be ready for longwinded, non-linear and even tortuous processes (schoenfeld, smith & arcavi, 1993). last but not least, a difficulty arises from the fact that the proofs without words in this paper were taken from mathematics journals associated with tertiary or higher education. in teaching proofs that explain words via visualisation in secondary schools there will be the inevitable “didactical transposition” (chevallard, 1985). there will be a transformation of the knowledge associated with changing proofs without words into proofs that explain. by its very nature this process linearises, compartmentalises and possibly also algorithmetises knowledge, thereby stripping it of any rich interconnections (arcavi, 2003). 61 m faaiz gierdien nelsen, r. b. (1993). proof without words. washington dc: the mathematical association of america. acknowledgement i would like to thank dr helen chick for providing insightful feedback to an earlier draft of this paper. nelsen, r. b. (2005). proof without words: sum of triangular numbers. mathematics magazine, 78(3), 231. references arcavi, a. (2003). the role of visual representations in the learning of mathematics. educational studies in mathematics, 52, 215241. pólya, g. (1954). patterns in plausible reasoning (vol. i & ii). princeton, nj: princeton university press. pólya, g. (1945; 2nd edition, 1988). how to solve it: a new aspect of mathematical method. princeton: princeton university press. casselman, b. (2000). pictures and proofs. notices of the american mathematical monthly, 47(10), 1257-1266. chevallard, y. (1985). la transposition didactique du savoir savant au savoir enseigné. grenoble, france: la penseé sauvage. pouryoussefi, f. (1989). proof without words. mathematics magazine, 62(5), 323. department of education. (2003). national curriculum statement grades 10-12 (general): mathematics. pretoria: department of education. presmeg, n. (1997). generalization using imagery in mathematics. in l. english (ed.), mathematical reasoning. analogies, metaphors and images (pp 299–312). mahwah, nj: erlbaum. de villiers, m. (1990). the role and function of proof in mathematics. pythagoras, 24, 17-24. schoenfeld, a.h., smith, j. & arcavi, a. (1993). learning: the microgenetic analysis of one student’s evolving understanding of a complex subject matter domain. in r. glaser (ed.), advances in instructional psychology, vol. 4 (pp 55–176). hillsdale, nj: erlbaum. giaquinto, m. (1993). visualizing in arithmetic. philosophy and phenomenological research, 53(2), 385-395. giaquinto, m. (1994). epistemology of visual thinking in elementary real analysis. the british journal for the philosophy of science, 45(3), 789-813. sfard, a. (1998). a mathematician’s view of research in mathematics education: an interview with shimshon a. amitsur. in a. sierpinska & j. kilpatrick (eds.), mathematicseducation as a research domain: a search for identity. an icmi study (pp 445–458). dordrecht: kluwer. hanna, g. (1983). rigorous proof in mathematics education. toronto: oise press. hanna, g. (1990). some pedagogical aspects of proof. interchange, 21(1), 6-13. siu, m. k. (1993). proof and pedagogy in ancient china: examples from liu hui's commentary on jiu zhang suan shu. educational studies in mathematics, 24 (4), 345-357. hanna, g. (1998a). proofs that prove and proofs that explain. proceedings of the 13th international conference on the psychology of mathematics education (pp 45-51). paris: pme. steiner, m. (1978). mathematical explanation. philosophical studies, 34, 135-151. hanna, g. (1998b). proof as explanation in geometry. focus on learning problems in mathematics, 20 (2-3), 4-13. hershkowitz, r., arcavi, a., & bruckheimer, m. (2001). reflections on the status and nature of visual reasoning – the case of the matches. international journal of mathematical education in science and technology, 32(2), 255-265. volmink, j.d. (1990). the nature and role of proof in mathematics education. pythagoras, 23, 710. “in order to translate a sentence from english into french two things are necessary. first, we must understand thoroughly the english sentence. second, we must be familiar with the forms of expression peculiar to the french language. the situation is very similar when we attempt to express in mathematical symbols a condition proposed in words. first, we must understand thoroughly the condition. second, we must be familiar with the forms of mathematical expression.” george pólya 62 << /ascii85encodepages false /allowtransparency false /autopositionepsfiles true /autorotatepages /all /binding /left /calgrayprofile (dot gain 20%) /calrgbprofile (srgb iec61966-2.1) /calcmykprofile (u.s. web coated \050swop\051 v2) /srgbprofile (srgb iec61966-2.1) /cannotembedfontpolicy /warning /compatibilitylevel 1.4 /compressobjects /tags /compresspages true /convertimagestoindexed true /passthroughjpegimages true 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<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> >> >> setdistillerparams << /hwresolution [2400 2400] /pagesize [612.000 792.000] >> setpagedevice abstract introduction research methodology findings and discussions summary conclusions learners’ critical sharing of ideas teacher’s intentional intervention acknowledgements references about the author(s) samuel mabotja department of mathematics, science and technology education, university of limpopo, south africa kabelo chuene department of mathematics, science and technology education, university of limpopo, south africa satsope maoto department of mathematics, science and technology education, university of limpopo, south africa israel kibirige department of mathematics, science and technology education, university of limpopo, south africa citation mabotja, s., chuene, k., maoto, s., & kibirige, i. (2018). tracking grade 10 learners’ geometric reasoning through folding back. pythagoras, 39(1), a371. https://doi.org/10.4102/pythagoras.v39i1.371 original research tracking grade 10 learners’ geometric reasoning through folding back samuel mabotja, kabelo chuene, satsope maoto, israel kibirige received: 29 mar. 2017; accepted: 08 sept. 2018; published: 15 nov. 2018 copyright: © 2018. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract this article uses parts of qualitative data from the first author’s study that focused on exploring pirie and kieren’s process of folding back, revisiting previously held understandings of a concept for its extended understanding in order to solve the problem of enhancing grade 10 learners’ reasoning in geometry. in south africa, secondary school leavers and pre-service teachers have difficulties in understanding geometric concepts and little is known about learners’ geometric reasoning. using teaching experiment methodology, we particularly tracked grade 10 learners’ geometric reasoning through folding back. we targeted two groups of learners from a class of 54 in which learners grouped themselves: the first group consisted of two members and the second group consisted of three members. the selection of learners was prompted by the manner in which the learners argued and questioned each other during four weeks of exploratory teaching. data were collected through participant observations, where notes were taken and video recordings made in different sessions. we analysed the field notes and verbatim transcriptions of the video recordings using polkinghorne’s narrative analysis and analysis of narratives. both the teacher’s and the learners’ interventions prompted effective folding back resulting in critical sharing of ideas. we thus argue that learners’ effective folding back is a powerful tool to enhance their geometric reasoning. introduction a few studies have focused on geometric reasoning (battista, 2007; budi, 2011; evans, 2007; national council of teachers of mathematics, 2002) to develop skills of explaining, arguing, verifying (battista, 2007; budi, 2011) and to relate problem-solving to mathematical proof (evans, 2007). thus, geometric reasoning needs to be developed to enable learners to process mathematical (geometric) proof (evans, 2007). proof is defined as a chain of deductions through which the truth of the propositions to be proved is derived from axioms and previously established propositions (fetisov & dubnov, 2012). briefly, proof consists of explicit chains of inference following agreed rules of deduction (hanna & de villiers, 2008). similarly, brown, jones, taylor and hirst (2004) contend that geometric reasoning is important in developing and evaluating deductive arguments about figures and their properties that help learners make sense of geometric situations. emphasis on geometric reasoning can help learners organise their mathematical knowledge in ways that enhance their conceptual development in geometry (national council of teachers of mathematics, 2002). learning procedures and proofs without a good understanding of geometric concepts may leave learners ill-equipped for geometric reasoning (department of basic education, 2011). thus, it could be argued that learners should not only develop conceptual understanding of geometric properties of figures, but also be able to apply such properties to solve geometry problems. pirie and kieran (1994) explain this through their theory of growth in mathematical understanding that comprises eight nested levels. named in their succeeding order from the inner layer these levels are: primitive knowing, image making, image having, property noticing, formalising, observing, structuring and inventising. it is in these levels where pirie and kieran introduce the concept of folding back. folding back is a non-unidirectional process through which mathematical understanding grows through these levels. therefore it is a cognitive shift that happens when previously held understanding, inner layer understanding, acts are informed by outer layer knowing that ultimately leads to further outer layer understanding (pirie & martin, 2000). there are few studies that show that folding back extends learners’ existing understanding of mathematics concepts (delgado, codes, monterrubio, & astudillo, 2014; martin & towers, 2016; wright, 2014). using tall’s (2008) argument, learners’ geometric development builds on experiences that they have met before. previous experiences, referred to as met-befores, form connections in the brain and that would affect how learners make sense of new mathematical situations that are either consistent (martin & towers, 2016) or inconsistent (tall, 2008). nevertheless, there are difficulties learners encounter in geometry (fujita, yutaka, kunimune, & jones, 2014; gunhan, 2014; naidoo, 2013). for instance, fujita et al. (2014) found that, although learners discovered various methods to construct a square, they had difficulties in using properties of shapes to find out why their construction would be correct. their findings further indicated that learners rely on visual perception of the given geometrical shape rather than deduction based on the properties of the geometrical object. similarly, gal and linchevski (2010) confirmed that children prefer to rely on a visual prototype, rather than a verbal definition, when classifying and identifying shapes. the results of gunhan’s (2014) study showed that learners sometimes have insufficient geometrical knowledge and visual perception and, at times, are not able to provide a mathematical argument. he recommended that the school curriculum should place more emphasis on reasoning skills and that learners should be presented with problems in order to apply different reasoning skills. in south africa, secondary school leavers (panaoura & gagatsis, 2009) and pre-service teachers (ndlovu & mji, 2012) were reported to have difficulties in understanding geometric concepts. in view of these learning difficulties, this study aimed at tracking learners’ geometric reasoning through folding back. the findings from this study may contribute to the existing knowledge regarding teacher-learner interactions and the learners’ growth of geometrical understanding. theoretical framework this article is guided by pirie and kieren’s (1994) dynamical theory of the growth of mathematical understanding. this theory deals with the growth of mathematical understanding and comprises eight nested levels, namely primitive knowing, image making, image having, property noticing, formalising, observing, structuring and inventising. ‘by “images” the theory means any ideas the learner may have about the topic, any “mental” representations, not just visual or pictorial ones’ (martin, lacroix, & fownes, 2005, p. 22). levels used in this study are explained as: primitive knowing refers to the knowledge that learners bring to their learning context as prior knowledge. image making is the level where a learner engages in activities intended for developing ideas and images for a concept (martin et al., 2005). image having is a layer where learners can use an understanding of a topic to help them to make mental plans that can be used when working on mathematical tasks (martin et al., 2005). property noticing deals with learners’ examination of an image for relevant properties. formalising is a level where learners consciously notice properties and work with them. the nesting illustrates that the growth in mathematical understanding is neither linear nor unidirectional (pirie & martin, 2000) and that each layer contains the preceding layer and is contained in the layer that follows (pirie & kieren, 1994). for pirie and kieren, this mutual inclusivity of layers symbolises back and forth movement of growth in understanding across these layers, a process they referred to as folding back. although martin (2008) agrees with pirie and kieren’s definition of folding back, he is of the view that the definition ‘remained essentially undeveloped and unelaborated in their work’ (p. 64). according to martin, how and why folding back occurs is not deeply explored. he thus expands the notion of folding back by providing three key elements: source of intervention, form of actions that leads to folding back and outcome of folding back. these elements were adopted to guide this study and are illustrated in figure 1 and the accompanying narrative. figure 1: the framework for describing folding back. each of the three elements explains some practice of folding back that allowed us to describe folding back and trace it to its source. folding back may result from a teacher’s intervention, material intervention, peer intervention by another learner, or self-invoked intervention by a learner who decides to fold back. depending on whether or not the source element is intended to result in folding back, the source of intervention can either be intentional or unintentional. an intentional intervention is said to be explicit or focused if it points directly to the idea, image or existing understanding; otherwise it is said to be unfocused. the second of the three acts of folding back is the form of folding back. it has three categories that express how learners react to interventions by a source. this may take place according to four different categories, namely working at an inner layer using existing understanding, collecting at an inner layer to retrieve previous knowledge for a renewed view that is in line with what is needed for mathematical actions at hand, moving out of topic and working there, and causing discontinuity in the understanding that is developed. working at an inner layer using existing understanding occurs when a learner is aware of limitations of their existing understanding at an outer layer and hence works in a less formal and less sophisticated way. this is done in such a way that less formal understandings are informed and assisted through formal understandings. evidence of working at an inner layer using existing understanding is noticeable when learners either generate new ideas or change how they originally constructed a concept. on the other hand, the act of collecting at an inner layer happens with learners’ sense of ‘knowing that they know what they need to know’, of being aware that they have the necessary understandings, but that they are just not immediately accessible (martin, 2008, p. 77). this knowledge and understandings are used to re-view and re-read concepts to meet the needs of the task at hand. furthermore, folding back may involve stepping out of a topic to allow for development and thickening of another topic so that ultimately it enables working at an outer layer – a process referred to as moving out of topic and working there. while working at an inner layer using existing understanding, collecting at an inner layer, and moving out of topic and working there result in thickening or further and enriched understanding of an existing concept, the act of causing a discontinuity in folding back does not. causing a discontinuity takes place when interventions that cause cognitive shifts, referred to as invocative interventions, do not have an effect on understandings that are developed. this happens when a learner returns to an inner layer but fails to connect understandings that are developed with already existing ones. the three forms of folding back that do not lead to a discontinuity result in two categories of outcomes of folding back. the outcome of folding back is either effective or ineffective. the outcome of folding back is effective if the result of folding back enables further understanding at an outer layer with or without external prompts. otherwise the outcome of folding back is ineffective. research methodology a teaching experiment methodology research design (steffe & thompson, 2000), which researchers use to experience learners’ mathematical learning and reasoning, was used to track whether folding back (martin, 2008; martin & towers, 2016; pirie & kieren, 1994) supported grade 10 learners’ development of geometric reasoning. using data from a master’s study (mabotja, 2017), we targeted data from two groups of learners from a class of 54 in which learners grouped themselves: the first group consisted of two members (john and koena) and the second group consisted of three members (lesiba, lebogang and sipho). these two groups were chosen because of the manner in which learners within the groups argued and questioned each other during the four weeks of exploratory teaching. also, the groups were picked out because their arguments and questioning epitomised instances of folding back and not because folding back was normative among all groups. data were collected through participant observations by the first author through field notes and video recordings of learners’ conversations in different sessions. data collected were conversations between learners. in some instances the teacher was part of the conversations. the conversations were analysed through rewriting them as stories – narrative analysis (polkinghorne, 1995). the stories were not descriptions of these conversations but interpretations and extrapolations of latent meanings so that the conversations could be coherently understood. polkinghorne (1995) and kim (2015) branded this as a smoothing process that allows for filling gaps in the story. this is done so that the story can be plotted into a coherent outcome while being aware that each group and thus each conversation was unique. at the same time, martin’s (2008) theory of folding back provided focus to how data were analysed. the three elements of the theory, namely the source of intervention, the form, and the outcome, were picked out of the conversations as relevant for interpreting the conversations. thus aspects of analysis of narratives (polkinghorne, 1995) were also used in analysing the conversations. quality criteria to ensure quality criteria, transferability, confirmability, credibility and dependability were employed (bitsch, 2005). for transferability, excerpts of conversations among learners provided the researchers with equivalents of descriptions in order to draw more information about data from the same context (bitsch, 2005). confirmability is the degree to which the findings can be confirmed by other people and can be traceable to the original data sources through the excerpts from which findings were drawn. in both instances, since the original data were captured on video, the learning scene could be revisited (martin, 1999; powell, francisco, & maher, 2003). for credibility a prolonged engagement of about 30 days, and being on the research site a month prior to data collection, was used in order to overcome distortion of claims made and to increase subjectivity (guba & lincoln, 1989). for dependability, diachronically reliable evidence was obtained through time triangulation (cohen, manion, & morrison, 2007). the extracts reported on in this article were a week apart from each other. in addition, monitoring of constructs was made by involving two of the authors in using martin’s (2008) framework of folding back to illustrate how the intervention, form and outcome were used in the study. ethical considerations the participants and their parents completed the informed consent forms. informed consent acknowledged the protection of the participants’ rights (creswell, 2012). for video recordings in the classroom and afternoon sessions, permission was requested from the school principal, the parents of the participants as well as the participants themselves. in order to ensure anonymity, pseudonyms were used for the participants and the research site. findings and discussions we organised this section by drawing extracts from the main study’s data and only tracking the two groups of students in order to demonstrate instances of conversation where folding back led to learners’ geometric reasoning or not. according to pirie and martin (2000), brief extracts of dialogue are sufficient to draw attention to the shifts in thinking that took place. the data revealed that learners’ critical sharing of ideas, and the teacher’s intentional intervention, are critical elements for effective folding back. we started by capturing the learning activity that was given and thereafter analysed the learners’ interactions by considering the source of the intervention, form and outcome of folding back (martin, 2008). extract 1: prove that ∆pst ||| ∆pqr the learners were given a learning activity to prove that ∆pst ||| ∆pqr (see figure 2). figure 2: corresponding angles of similar triangles. learners were expected to show that since lines st and qr are parallel, the corresponding angles of the two triangles are equal (mabotja, 2017, p. 73): 1.1 koena: here we can use ps and the other side? 1.2 john: look … okay … you cannot just say ps, so what are the values of ps? but we cannot use it if we are not given values. if we can determine angles of these triangles, then we can prove that these triangles are similar. so ∠p is common, what about other angles? these are angles on a straight line; these are parallel lines. we need a transversal; we use those things of corresponding angles. 1.3 koena: ∠s1 is equal to ∠q, as you see they form a f shape. 1.4 john: f shape this way? [john traces an f on the diagram with a pencil] 1.5 koena: yes. 1.6 john: yes, it is correct. 1.7 koena: if we can put it this way, can’t we take it out? [koena rotates the worksheet as he says that the triangle is rotated 180°] 1.8 john: yes, it is correct, they are this way, and then ∠r is equal to ∠t1. they are corresponding angles. [john traces the f on the diagram with his finger] 1.9 john: then we can say in triangle pst and triangle pqr [talking and writing], ∠t1 is equal to r [talking and writing]. 1.10 koena: corresponding angles are equal. [john and koena simultaneously say angle p is equal to angle r, common] [john talking and writing] 1.11 koena: ∠s1 is equal to ∠q. 1.12 john: they are corresponding angles. 1.13 john: therefore, we can say triangle pst and pqr are similar. 1.14 koena: yes, they are similar. [john and koena simultaneously say angle, angle and angle] [john writes aaa] the two learners in extract 1 were given a task that required them to prove that the two triangles are similar. on the model for growth in mathematical understanding, this task is pitched at the level of formalising understanding in that it required learners to notice and work with the property of similar triangles that corresponding sides have equal ratios. furthermore, for the learners to be able to provide the required proof, they should have the image that when two parallel lines are cut by a transversal, the resulting corresponding angles are congruent. the extract begins with koena’s intentional intervention that they work with side ps and another side. such an utterance could have resulted from knowing that they had to prove that the ratio of ps and its corresponding side is proportional to the other two corresponding sides of the triangles. however, john did not work at this level of the task to grow his understanding. instead, as a form of folding back, he worked at an inner level of property noticing. from the given information on the task, he noticed that since the magnitudes of the sides of the triangles (and probably the proportionality between ratios of corresponding sides) were not known, it could not be argued that the two triangles were similar. he further uttered that ‘if we can determine angles of these triangles, then we can prove that these triangles are similar’; thus suggesting an intentional intervention though not making it explicit. john did not explicitly indicate that the measures of two angles of one triangle should be equal to two corresponding angles of the other triangle for the two triangles to be similar. all the same, from the given information of the task, john noticed that lines st and qr are parallel, and that they ‘need[ed] a transversal; we use those things of corresponding angles’. here it can be claimed that john was aware that when parallel lines are crossed by a transversal, the corresponding angles are equal. at this stage koena also worked at an inner layer using existing understanding as a form of folding back through uttering that ∠s1 and ∠q form an f shape – a mnemonic used in most teaching to identify corresponding angles. this seemed to indicate that koena needed the image of the ‘f shape’ mnemonic to notice that ∠s1 and ∠q are corresponding angles. working at property noticing john traced the ‘f shape’ and koena rotated the diagram to have a clear picture of what they were talking about. the two learners seemed satisfied with their identification of two pairs of corresponding angles that are equal, namely ∠t1 and ∠r (see line 1.8), and ∠s1 and ∠q. after folding back to the property noticing level through using the f shape mnemonic and thickening their understanding through successfully applying the mnemonic on the given problem, the learners returned to the formalising layer without external prompts to conclude that the two triangles were similar. this was an instance where folding back was effective in geometric reasoning. extract 2: finding the magnitudes of a and b in extract 2 the learners must find the magnitudes of a and b in the given diagram (figure 3). figure 3: proportionality in similar triangles. extract 2 shows an instance where moving out of topic and working there resulted in a discontinuity. with critical sharing of ideas on what caused the discontinuity, which emerged after extract 2, folding back was ultimately effective. the extract was generated from the interaction between lesiba and lebogang on the learning activity on finding the magnitude of a and b after showing that triangles ∆trs and ∆tpq are similar. although sipho was a member of the group, he did not contribute actively during this discussion (mabotja, 2017, pp. 76–79): 2.1 lesiba: here we can start by finding the sides that are in proportion. 2.2 lebogang: here we can separate the triangles [observed drawing ∆trs and ∆trs]. 2.3 lesiba: yes, that is correct. then we can say side tr over side tp is equal to ts over tq. 2.4 lebogang: okay, it will be this way . side tr is equal to 15 and tq is equal to 9. 2.5 lesiba: yes, let us continue. 2.6 lebogang: we can say a over 15 is equal to b over 9 . 2.7 lesiba: we can now cross multiply. 2.8 lebogang: it is going to be 9 multiply a is equal to b is equal to 15 multiply b over 4 . here a is equal to eerrhhh … two unknowns. 2.9 lesiba: two unknowns, how? [observed looking into the triangles lebogang has drawn] 2.10 lebogang: yes similar to extract 1, extract 2 began with peer intervention by lesiba. the intervention was intentional and explicit as he suggested that they find the sides of the triangles that were proportional. in order to do that lebogang drew ∆trs and ∆tpq separately. in this case in particular, it can be claimed that lebogang preferred working in a less formal way while working at an inner layer using existing understanding because she worked at an image making level. this was followed by an utterance by lesiba that . lebogang transcribed what lesiba said incorrectly and wrote , instead of . regardless of this error, it can be claimed that they intended to use their knowledge of proportionality between corresponding sides of two similar triangles in order to solve the given problem. also shown in the extract is that the learners did not read the labelling of the diagram well. instead of reading the magnitude of tp and tq as a + 15 and they read them as 15 and 9 respectively as evidenced when lebogang drew the two triangles separately and labelled two sides. but when representing the equation by the magnitude of the sides of the triangle they wrote , which attests to the claim that they attempted to use proportionality of corresponding sides of similar triangles. there was a pause as lesiba observed the two triangles that lebogang drew, and also wanted clarity on why they ended with ‘unknowns’ a and b in (line 2.9). regardless, lebogang simplified the rational equation as (line 2.10), an error that indicated that incorrect algebraic processes were followed. this led to a discontinuity that emanated from algebraic errors committed while working outside the topic of geometry. the extract that follows continues extract 2. it shows how the learners re-started the process of folding back after a discontinuity. it began after another pause in the learners’ conversation and presumably thinking about what they could do next. it began with lesiba’s intentional intervention to show lebogang that they did not read the measures of the sides of triangles correctly. accordingly, this was an explicit intention (lines 2.11, 2.13 and 2.15) (mabotja, 2017, p. 78): 2.11 lesiba: actually can’t we have … let me ask … on tp can’t we have 15 + a? 2.12 lebogang: tp? 2.13 lesiba: 15 + a over … no a over 15 + a is equal to. 2.14 lebogang: a over 15 times a? [confused as to what to write] 2.15 lesiba: 2.16 lesiba: therefore we say a over 15 + a is equal to, like here we say 9 + b over 4 2.17 lebogang: the whole of side of tq is equal to b. the extract shows that, regardless of lebogang’s confusion in lines 2.2 and 2.4 of extract 2, ultimately the students realised that is equivalent to – this was a new way of understanding how the learners read the given information. with this new way of understanding the learners were able to fold back through the form of moving outside the topic of geometry and working on algebra as captured in the next extract (mabotja, 2017, p. 80). 2.18 lesiba: tr divided by tp is equal to ts divided by tq . tr is a divided by 15 plus a is equal to b divided by 4 divided by 9 plus b divided by 4. 2.19 teacher: yes, that equation is correct. 2.20 lebogang: cross multiply. 2.21 lesiba: cross multiply. 2.22 teacher: okay before we cross multiply. 2.23 lesiba: we can say b divided by 4, this way . 2.24 teacher: okay continue. 2.25 lebogang: write b divided by 4, do we divide it? 2.26 lesiba: then we say times, then 4 goes on top … 2.27 lebogang: i don’t understand you … 2.28 lesiba: you don’t understand me, like, you see now is , then we say b divided by 4, times . 2.29 lebogang: yes, write them, but let us first replace with b. 2.30 lesiba: okay i see it since is equal to b 2.31 lebogang: yes this line here [pointing to the line outside the triangle] shows that tq is b. 2.32 lesiba: okay, then we will write it this way . 2.33 lebogang: then we can start substituting. 2.34 lesiba: here is going to be this way , then this one we must change it. 2.35 lebogang: yes, we change it to b over 4 divide by b over 1. 2.36 lesiba: then it will be this way 2.37 lebogang: and then now we can change it by using multiplication. 2.38 lesiba 2.39 lebogang: here b will multiply b. 2.40 lesiba: then this side with 1 over 4 . 2.41 lebogang: then we can cross multiply. the conversation began with lesiba rewriting the equation as its equivalent using the magnitudes of its sides while wrongly reading out what he wrote. the teacher acknowledged the equation written by lesiba, , by uttering that that the equation was right. lebogang intervenes intentionally by suggesting that they should ‘cross multiply’. with this the learners continued working outside the topic of geometry. the teacher interrupted lebogang’s intervention by his utterance in line 2.22. although the utterance was not explicit, it suggested to lesiba that he had to rewrite as , which was not correct. lesiba was supposed to invert the entire compound factor so that was written as ; instead he inverted only to be . the learners did not correct the wrong representation but instead continued to work on it and substituted with b as per lebogang’s suggestion. although the substitution was syntactically wrong, it led to a correct representation of as . in this case, working outside the topic did not lead to a developed understanding of algebraic expressions which martin (2008) would refer to as thickening. nonetheless, following line 2.41 the learners were able to evaluate a and b as required in the given task, resulting in effective folding back. it can thus be concluded that this was an instance where effective folding back happened regardless of a discontinuity as a form of folding back. furthermore, it can be claimed that this effective folding back happened because of critical sharing of ideas and reflection. algebraic errors that were made during folding back through working out of the topic, which did not result in a discontinuity, led to an effective folding back. extract 3: prove that abcd is a parallelogram in this activity learners were expected to prove from the given diagram (figure 4) that quadrilateral abcd is a parallelogram, where bf = af and ce = cd and the angle sizes are as shown. figure 4: proving a quadrilateral to be a parallelogram. since the task required them to notice properties of the given diagram and to work with them to prove that abcd is a parallelogram, the diagram was pitched at a formalising level of understanding. 3.1 lebogang: ∠a is equal to ∠c, opposite angles of a parallelogram are equal. 3.2 teacher: which parallelogram are you looking at? 3.3 lebogang: abcd. 3.4 teacher: and then what are you supposed to do? 3.5 sipho: prove that it is a parallelogram [image having]. 3.6 teacher: so, are you saying ∠a is equal to ∠c? or are you saying if you can find ∠a being equal to ∠c, then you can conclude that abcd is a parallelogram? in this extract (mabotja, 2017, p. 98) a peer, lebogang, was the source of the intervention as she initiated the conversation. she intended for the group to notice that ∠a is equal to ∠c since abcd is a parallelogram. this could only be true if lebogang meant that ∠baf and ∠ecf are equal since this information was given. but since lebogang did not use three letters, with the middle letter being the vertex, to read the angles it was not clear which angles ∠a and ∠c she talked about. this made her intervention unfocused. this could have been the reason why the teacher asked her to specify the parallelogram she was looking at. since lebogang was looking at quadrilateral abcd, she used an invariant property of parallelograms that opposite angels are congruent. this could have happened because lebogang did not intuitively differentiate between what was given and what was to be proved. brown et al. (2004) asserted that this may result from using recently introduced ideas that are not well understood or a requisite to provide proof for something that is obvious. lebogang’s confusion probably resulted in the teacher’s intentional and explicit intervention. the teacher wanted lebogang to specify what the task required her to do (line 3.4). this request could have made her realise that her intervention was unfocused. she did not respond; instead sipho did. when the teacher wanted the learners to clarify if ∠a and ∠c were given as equal or whether the task required them to prove that these angles are equal (line 3.6), none of the learners responded. instead, there was a significant pause during which the learners were probably critically considering the teacher’s question. after this pause lebogang did not pursue her initial intervention (line 3.7 below) thus resulting in a discontinuity at property noticing. it can be asserted that the discontinuity resulted because, using pierie and kieren’s model of growth in understanding, lebogang was at a primitive level, and the knowledge she had, that ∠a is equal to ∠c because they are opposite angles of a parallelogram (line 3.1), was insufficient for her understanding to grow. the vignette that follows shows how folding back proceeded towards completion of the required proof (mabotja, 2017, pp. 99–101). 3.7 lebogang: is ∆abf an isosceles triangle? 3.8 lesiba: yes. 3.9 lebogang: then it means … we can say ∠b is equal to 65° … base angles of iscosceles triangle are equal. 3.10 sipho: where is an isosceles triangle here? 3.11 lebogang: is not this ∆abf an isosceles? 3.12 lesiba: two sides are equal [pointing to the sides]. 3.13 teacher: yes that will be true. 3.14 sipho: let us find all the angles, in that way we can see if it’s a parallelogram. 3.15 teacher: you are thinking of finding all the angles? 3.16 sipho: yes … and in that way we will see whether our opposite sides are equal or not, and if they are equal and then this means it is a parallelogram and if they are parallel. 3.17 lebogang: how are we going to use the angles? [silence] the extract above could have been triggered by the teacher’s intervention when seeking clarification on what lebogang had said. the peer intervention is explicit and intended to show that ∠b equals 65°. lebogang seemed to be at image having level of understanding since her utterance suggested she had mental constructs of isosceles triangles without first developing it. she then folds back to the primitive knowledge level and uses her existing knowledge on properties of isosceles triangles to come to a conclusion on the size of ∠b. this folding back was not effective for sipho, who could not identify the isosceles triangle that lebogang referred to. once lebogang pronounced triangle ∆abf as the one she worked on, and once lesiba showed sipho the sides that were equal (line 3.12), sipho seemed to understand what lebogang was saying, achieving their folding back jointly. sipho then suggested that the group find the sizes of the other angles of the quadrilateral in order to infer that the opposite angles of abcd were equal (line 3.14). sipho showed traits of the property noticing level of understanding and of being aware of a rudimentary fact about parallelograms: that their opposite angles are equal. when the teacher asked if he intended to find the size of all angles (line 3.15), sipho responded in the affirmative. it seems he suggested that they could apply their understanding of isosceles triangles to find the size of the opposite angles of quadrilateral abcd and further show that sides are equal to prove that the quadrilateral is a parallelogram (line 3.16). although sipho’s suggestion was explicit, lebogang did not know how to proceed with it. hence her question: ‘how are we going to use the angles?’ at this stage the learners could have either used the same procedure used for finding the size of ∠b to find the size of ∠d or show that ∠bad is equal to ∠bcd. the conversation that follows shows that, although sipho suggested earlier that the size of ∠cda be found in the same way as the size of ∠abc the suggestion was not considered. instead, in the conversation that follows, the teacher intervened explicitly with a different procedure to be followed (line 3.18). the intervention did not develop sipho’s growth in understanding as none of the people in the group allowed him to continue with his thinking, hence causing a discontinuity (mabotja, 2017, pp. 99–101): 3.18 teacher: okay let us look at all the angles at our disposal. if we add aec and fce? 3.19 lebogang and lesiba: it’s 180. 3.20 teacher: this means that those co-interior angles are supplementary. if we have the co-interior angles that are supplementary, what can we say about the opposite lines? 3.21 lebogang: ohoo, then opposite sides are parallel. 3.22 lesiba: ah! sir, this means we need two pairs of opposite sides that are parallel. 3.23 sipho: and we can find the other pair by [interrupted by lebogang]. 3.24 lebogang: we can find other co-interior angles. 3.25 lesiba: yes, write. 3.26 lebogang: [writes ∠b + ∠c = 180] then co-interior angles are supplementary. 3.27 sipho: then we must also give a pair of opposite sides. 3.28 lesiba: it will be line ab and line dc. 3.29 teacher: okay good, so since we have two pairs of opposite sides that are parallel, what can we generally say? 3.30 lebogang: this means we can say that this is a parallelogram. 3.31 sipho: yes, write there. in this vignette, the source of intervention was the teacher who was a participant-observer in this study. his intervention was intentional because the question he asked drew the learners’ attention to co-interior angles. the intention was explicit because he pointed learners to ∠aec and ∠fce. this was an intentional intervention in which the teacher, abandoning sipho’s thinking of working at an inner layer, directed the learners to fold back and collect at an inner layer. this kind of intervention provided the teacher with an opportunity to explicitly model his thought processes to learners. this helped learners to modify and rebuild their construct of parallel lines while at the image having level of understanding – applying a mental construct about a co-interior angles of parallel lines without first developing it. this use of the sum of co-interior angles that are supplementary in order to show that lines are parallel resulted in an effective form of folding back, namely collecting at an inner layer. in line 3.24 lebogang intentionally intervened and suggested that the same way of thinking be followed to prove that the other pair of opposite sides are parallel. in line 3.26 she writes that ∠b + ∠c = 180. this is another instance where lebogang seemed not to distinguish between what needs to be proved and rudimentary facts about parallelograms that have become intuitive. unfortunately, none of the group members, including the teacher, corrected this because lebogang’s unsubstantiated finding ultimately helped the learners to prove that the quadrilateral abcd is a parallelogram. in this instance, the learners failed to use existing understanding to work at an inner layer. this is more so because sipho’s suggestion that they work at an inner layer (line 3.16) had caused a discontinuity. nonetheless, the outcome of folding back was effective; it resulted in learners’ growth in how to apply the knowledge that if the sum of co-interior angles of two lines is 180° then the lines are parallel. this happened although the group seemed unaware of the need to explore and explain when solving geometric problems – a difficulty that was also highlighted by jones (2002). summary the analysis of the extracts presented in this article show that folding back supported geometric reasoning. this support was also evident in instances where there were discontinuities during the act of folding back. this was the case in extract 2 and extract 3 where the discontinuity was overcome when learners were sources of the intervention through critical sharing of ideas and reflections. in extract 3 the discontinuity was overcome when the teacher was the source of the intervention regardless of whether it was explicit or not for resolving the geometry task. we therefore claimed that teachers’ interventions that are not explicit encourage learners’ self-awareness of the limitations of their current understanding, without prescribing a solution. effective folding back also happened when the form of folding back did not result in developed ideas that learners were working on. for example, the algebraic error committed in extract 2 while learners were working outside the topic of geometry did not affect the outcome of folding back negatively. by the same token, in extract 3 the outcome of folding back in geometric reasoning was positive: learners worked with a rudimentary fact when it was required that they provide proof while working at an inner layer using existing understanding. working with rudimentary facts is not in line with the notion of folding back which provides learners with an opportunity to build a connected understanding of mathematical concepts through returning and thickening their earlier existing images (e.g. martin & towers, 2016) conclusions in this article we pursued the question: how does folding back support learners’ geometric reasoning? we were guided by pirie and kieren’s (1994) dynamic theory for the growth of mathematical understanding and martin’s (2008) elaboration of pirie and kieren’s process of folding back to track two groups of learners from one grade 10 class. the data revealed two critical factors that contributed to effective folding back: the learners’ critical sharing of ideas, and the teacher’s intentional intervention. learners’ critical sharing of ideas learners’ self-questioning and reflections on their thinking in response to questions posed by peers offered them opportunities to revise, modify and extend their initial ideas and ultimately build a connected understanding through folding back (extract 1 and extract 2). using martin and towers’s (2016) words, they managed to return to and thicken their earlier existing images. this was evident when learners justified their geometric actions through using geometric properties relevant to a particular task. as they attempted to justify their thinking they reverted to folding back between the different layers of understanding. it was also a collaborative learning environment that encouraged the learners to be interested not only in their own geometry conceptual development but in their peers as well. teacher’s intentional intervention it was the nature of the teacher’s interventional decisions that led learners to fold back (extract 2 and extract 3). the teacher asked questions that alerted learners to be explicit in their geometric met-befores (tall, 2008) in order to arrive at the solution. from martin and towers’s (2016) argument, the teacher did not ignore problematic met-befores and thus encouraged effective folding back and at the same time allowed thickening to occur. thus, through following learners’ reasoning path the teacher was able to observe learners’ current layers of understanding and how they were used to further develop their understanding. acknowledgements samuel.m. acknowledges a sponsorship grant from the research office of the university of limpopo through the internal research chair in quality teaching and learning. competing interests the authors declare that they have no financial or personal relationships that may have inappropriately influenced them in writing this article. authors’ contributions samuel.m. carried out the research project as a master’s student supervised by k.c. and satsope.m. was the co-supervisor. samuel.m.’s initial draft of the manuscript was reconceptualised by satsope.m. and k.c. and thereafter satsope.m. wrote the introduction, theoretical framework, research methodology, quality criteria, ethical consideration and conclusion sections. k.c. wrote the analysis, aligned and enriched it guided by the theoretical framework. i.k. played a role of a critical reader, improved on the logical flow of ideas and filled in the gaps to thus improve the quality of the manuscript. references battista, m.t. 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(2014). frequencies as proportions: using a teaching model based on pirie and kieren’s model of mathematical understanding. mathematics education research journal, 26(1), 101–128. https://doi.org/10.1007/s13394-014-0118-7 article information author: faaiz gierdien1 affiliation: 1research unit for mathematics education, university of stellenbosch, south africa correspondence to: faaiz gierdien postal address: private bag x1, matieland 7602, south africa dates: received: 25 oct. 2011 accepted: 02 july 2012 published: 07 aug. 2012 how to cite this article: gierdien, f. (2012). pre-service teachers’ views about their mathematics teacher education modules. pythagoras, 33(1), art. #134, 10 pages. http://dx.doi.org/10.4102/ pythagoras.v33i1.134 copyright notice: © 2012. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. pre-service teachers’ views about their mathematics teacher education modules in this original research... open access • abstract • introduction    • literature review       • on (primary) mathematics teacher education modules       • on social practices in the mathematics teacher education modules       • on pre-service teachers’ views in the present study • towards an analytic framework    • on methods       • method of data collection       • validity       • reliability       • method of data analysis • findings    • disagreement, anxiety and apprehension    • anxiety and uncertainty    • verbal awareness of shift from mathematics to mathematics teaching    • from apprehension to excitement    • reality of schools and mathematics as a school subject    • agreement and confidence • discussion • concluding remarks • acknowledgements    • competing interests • references abstract top ↑ this article reports on the views of intermediate and senior phase pre-service teachers (psts) enrolled in mathematics education modules that attempt to teach both content and pedagogy. the psts are students in a four-year bachelor of education (bed) model located in a faculty of education. findings were analysed by means of an analytic framework that takes into account the university–school divide. findings indicate that the psts position themselves in different ways with regard to their preparation for school mathematics teaching. implications are considered, especially the psts’ affective views such as their anxiety and apprehension related to the discursive differences between the content in the university modules and school mathematics. introduction top ↑ in the faculties of education of south african universities there are mathematics teacher educators who teach pre-service mathematics education modules which attempt to include both content and pedagogy. for the purposes of this article these content–pedagogy mathematics education modules will be referred to as mathematics teacher education modules. for instance, we find pre-service teachers (psts) enrolled in a bachelor of education (bed) degree who take mathematics teacher education modules aimed at teaching school mathematics at the intermediate and senior phases (i.e. grades 4−9). this is the result of rapid institutional shifts in the provision of teacher education in south africa such as mergers or the integration of teacher training colleges into universities because of a particular policy implementation. the psts enrolled in these modules hold different views on the value of the modules. for the purposes of this article psts’ views are defined as their comments about different social practices they encounter in the modules. these practices are discussed in more detail below. further details on the context – such as the bed degree programme of the psts concerned, the mathematics teacher education modules they take, and the activities, actions and tasks in the modules – are in order. the psts referred to in the article are students in some of the mathematics education modules that i teach in one of the four year bed programmes of a faculty of education at a university. when they obtain their degrees, the psts will be certified to teach mathematics in the phases where mathematics is a compulsory subject, for example, grades 4−9. i have been teaching these modules consecutively to a cohort of psts during their second, third and fourth (final) years of the bed programme. a module in the sequence runs for one academic year. the content is presented and taught in ways that are informed by different strands of literature on what is called (social) practices found in mathematics teacher education and pre-service or initial teacher education. for example, what are important in the modules are insights that psts derive from connecting (school mathematics) content to pedagogy. in the second year there is a focus on various nuanced interpretations of fractions such as ratio, part-whole, operator, rate, decimal, percentage and measurement. such interpretations span and conceptually connect the components of the school mathematics content. the psts are required to read and write reviews of practitioner-intended mathematics education literature on these different interpretations of fractions. in addition, they have to examine in what way these interpretations are important when they investigate methods for teaching concepts where there is a need to take into account children’s existing knowledge. in the third year the content focuses on ways to foster algebraic reasoning, starting from arithmetic as prior knowledge. in both modules there are attempts to show conceptual connections within and between mathematical ideas that are not made explicit in school mathematics, according to the mathematics education literature. in the fourth year students continue to explore school mathematics from perspectives informed by the use of different information and communications technologies (e.g. excel). the problem statement and research question for this article are as follows. there are differences and similarities between teaching content–pedagogy mathematics education modules in pre-service teacher education (aimed at the intermediate and senior phases) in a university, and teaching school mathematics in those phases. the fact that universities and schools are two different kinds of institutions means that there are different, though at times overlapping, ways of knowing and doing in the two contexts. the overarching differences and similarities can be stated in terms of the university–school divide. taking this situation into account, i pose the following research question: what are psts’ views about the mathematics teacher education modules that i teach? the research question is important because it has the potential to illuminate ways of communicating the role of mathematics teacher education modules with respect to school mathematics teaching. knowing the psts’ views about the mathematics teacher education modules is useful in terms of my own practice as a reflective practitioner (schön, 1983). the psts’ views can also serve as an evaluation of the mathematics teacher education modules per se. in other words, their views should not be interpreted narrowly as being aimed at me on a personal level. the psts’ views can contribute to perspectives on the practices of other mathematics teacher educators at faculties of education where they try to teach both content and pedagogy to psts. there are science teacher educators (e.g. berry, 2004; loughran, 2007) who engage in ‘self-study,’ that is they use their teaching as a site to reflect on their practice. the remainder of the article is organised as follows: firstly there is a literature review starting with mathematics teacher education (modules) in the pre-service preparation of primary school psts. primary school psts were chosen because of the particular population of psts (intermediate and senior phases) used in the study. secondly there is an elaboration of social practice theory (spt) with special reference to practices specific to teaching mathematics teacher education modules in pre-service teacher education. a review of empirical literature on the views of primary (elementary) school psts in terms of practices follows. the different strands of literature will then be integrated to build a framework that can be used to analyse the empirical data, that is psts’ views about the mathematics teacher education modules that i teach. a description of, and justification for, the methodology of the study will be provided. the findings reflecting the views of particular psts’ will be presented. the article ends with a discussion and conclusion based on the findings. literature review on (primary) mathematics teacher education modules the ‘central tasks’ (feiman-nemser, 2001) in pre-service teacher education are activities that aim at preparing psts for school mathematics teaching. these tasks engage the psts in analysing and forming new visions, building an initial repertoire, developing subject matter knowledge for teaching, and developing an understanding of learners and learning. the tasks should not be viewed in isolation, meaning that the one relates to the other in terms of pre-service preparation. for example, if the psts are to form new visions of school mathematics teaching it is necessary to have them read and review mathematics education articles aimed at a practitioner audience. my conjecture is that such actions should enable them to see school mathematics from perspectives that are informed by mathematics education literature. one example is the nuanced interpretations of fractions such as ratio, part-whole, operator, rate, decimal, percentage and measurement mentioned above. at the same time the goal is to have the psts build their initial repertoire in preparation for school mathematics teaching, and lastly, reading these articles should assist the psts in developing subject matter knowledge for teaching. the latter is a point that will be developed below as an instance of what is called ‘mathematical knowledge for teaching’ (mkt). collectively, the central tasks in university modules should be aimed at assisting psts to develop insights into teaching mathematics in a school. evident from the latter is the university–school divide.the case for content–pedagogy mathematics education modules is taken from askew (2008) who argues against the traditional distinction between content and pedagogy in primary school mathematics pst courses. his view coincides with ball, thames and phelps (2008) and hill, ball and schilling (2008) who argue that the knowledge required for teaching mathematics, mathematical knowledge for teaching (mkt), is multifaceted and consists of the following domains: 1. common content knowledge held by all mathematically sophisticated occupations such as accountants and engineers 2. content knowledge specific to the specialised practice of mathematics teaching 3. knowledge of student learning of the mathematical content 4. knowledge of the practices of teaching mathematics 5. knowledge of mathematics-related curricula. each of these domains has some effect on a practicing teacher’s ability to select, organise and sequence tasks at an appropriate developmental level at the appropriate time in the mathematical sequence and student learning. these domains overlap in ways whereby the one supports and informs the other. psts have to encounter these domains in the university, because they potentially afford them with opportunities to see and experience mathematics in ways in which they can enable access to it (morrow, 2007, p. 82) in the school. doing the central tasks implies having the psts do different activities, actions, and assignments in some organised and regular ways. we therefore look for a more comprehensive way to talk about activities, actions and tasks related to the social context of the mathematics teacher education modules. various, related strands of literature on practices turn out to be useful. social practices in the mathematics teacher education modules in the present study the psts encounter social practices which can be explained in terms of social practice theory (spt). according to brodie (2012, p. 3) spt begins with the notion of practices ‘which are constituted in communities’ and not as ‘conceptual structures that are constructed in the mind’. in other words, from the perspective of spt, practices have nothing to do with what are in people’s minds, but much with what people do in communities in order to achieve specific goals. for example, drawing on scribner and cole (1981), brodie and shalem (2011, p. 421) offer what can be called a broad definition of ‘practices as patterned, coordinated regularities of action directed towards particular goals’. brodie and shalem (2011, p. 421) also note that ‘practice is always social practice’ (wenger 1998, p. 47). stated differently, social practices involve people or communities who interact in coordinated and regular ways to get to specific goals.brodie and shalem (2011) provide a definition of social practices that is applicable to the modules where there are attempts to teach both content and pedagogy. for example, they argue that ‘both mathematics and mathematics teaching constitute practices’ (p. 422) which i interpret as meaning social practices. these two social practices ‘intersect through the use of the knowledge and technologies of mathematics, which include symbolising, generalising, solving problems, justifying, explaining, and communicating mathematical ideas and concepts’ (rand mathematics study panel, 2003, p. 422). we find more applications of this definition in the way the rand mathematics study panel (2003, p. 32) defines mathematical practices in terms of activities such as mathematical representation, attentive use of mathematical language and definitions, articulated and reasoned claims, rationally negotiated disagreement, generalising ideas, and recognising patterns. mathematical representation, for example, can include a focus on decompressing mathematical symbols with the aim of focusing on their meanings within school mathematics. also, generalising ideas and recognising patterns would involve pointing out elementarised versions of (algebraic) symbols such as their connections to arithmetic. this panel notes that these practices are not explicitly addressed in schools and strongly recommend that they should be done as preparation for teaching school mathematics. these social practices should be interpreted as ‘process’ dimensions of mathematics and are therefore key features in terms of learning and doing mathematics. to gain a clearer picture of the different actions, tasks and activities that constitute the two social practices (mathematics and mathematics teaching) in the modules, we compare them to julie’s (2002) ‘school-teaching mathematics’ and watson’s (2008) ‘school mathematics’. julie and watson make the point respectively that school mathematics is a ‘variety’ and ‘a special kind of’ mathematics that is subject to the institutional constraints of the school. it has different warrants, authorities, forms of reasoning, core activities and purposes. in turn, these determine many of the ‘content’ features such as a strong focus on single answers (julie, 2002), a high degree of fragmentation rather than structural insights and abstraction (watson, 2008). the presence of these two social practices in the modules implies that there are discursive differences between mathematics content in a school and the ‘content’ features in the modules in the university where there are attempts to teach both content and pedagogy. on pre-service teachers' views in the present study as mentioned earlier, psts’ views are defined as their comments about the different social practices they encounter in the modules. it must also be noted that all views carry affect, some more than others. pre-service teachers’ views can be informed by affective issues such as their own schooling with respect to mathematics and mathematics teaching, differences between the mathematics in the modules and school mathematics in the intermediate and senior phases and teachers’ lack of content knowledge. grootenboer (2005) and peker (2009) found that psts have views on mathematics and mathematics teaching. examples of their views are negative dispositions, fear, anxiety and apprehension. for instance, alridge and bobis (2001), and szydulik, szydulik and benson (2003) found (primary school) psts not to be positively disposed towards mathematics. brady and bowd (2005, p. 43) report on mathematics anxiety, which may contribute to psts’ concerns, especially the ‘apprehension’ they experience when faced with the prospect of teaching the subject during their initial teaching practice. cassel and vincent (2011) describe how (primary school) psts feel overwhelmed and scared about mathematics teaching. another source of uncertainty or anxiety can arise when the psts experience differences between the mathematics in the university modules and the mathematics in the intermediate and senior school phases. in other words, they may not feel confident about teaching school mathematics. science teacher educator berry (2004, 2007) writes about similar challenges that her psts experience where they feel uncertainty as opposed to confidence about teaching school science in the middle grades. stated differently, psts’ lack of confidence or uncertainty can give rise to anxiety and apprehension, which may influence the psts’ views. there is also the possibility that the psts’ views could have been influenced by perspectives such as talk about teachers’ lack of (mathematics) content knowledge. the notion of the lack of content in south african teacher education programmes is thus a view that psts may have encountered before. pre-service teachers’ views about lack of (mathematics) content is an issue that morrow (2007) raises. he criticises some south african teacher education programmes because they construe teaching and learning as generic activities, with scant reference to the content of what is being taught or learned (p. 82). towards an analytic framework top ↑ constant comparison (corbin & strauss, 2008) will be used to develop an analytic framework to analyse psts’ views. constant comparison is suitable for the following two reasons. firstly, it became evident from data excerpts that the psts were making comparisons between and commenting on their experiences of social practices in the modules with what is or could be happening in actual school mathematics teaching. secondly, in terms of an analytic framework i have to keep a ‘distance’, that is, i have to take into account and compare instances where psts’ views are in disagreement as well as in agreement with social practices in the modules. a reason for this has to do with my dual role as the teacher of the modules and the researcher. in addition, the analytic framework has to take into account that the social practices in the modules are framed in ways that aim at teaching both (mathematics) content and pedagogy applicable in the intermediate and senior phases (askew, 2008). a particular consequence of adopting this viewpoint is the differences in the content features of the modules in the university in relation to descriptions of school mathematics teaching given by julie (2002) and watson (2008). the social practices should thus be seen in the light of attempts to bridge the university–school divide. the various strands of literature will now be integrated through relational statements with the aim of providing details on the analytic framework. firstly, a faculty of education in a university and a school are quite different but related institutions, because the former is the site for preparation for school mathematics teaching by means of the modules. social practices in the modules are informed by overlapping strands of literature that come from pre-service preparation, spt, mathematics and science teacher education and psts’ views. figure 1 is a diagrammatic representation that captures the various notions and ideas related to the research question. the analytic framework represented in figure 1 is skeletal. because of constraints regarding space it does not include all the strands of the reviewed literature, but does show the overarching university–school divide vis-à-vis preparation for school mathematics teaching. we cannot lose sight of this divide because of the nature of the bridging, that is the details of the social practices that are present in the psts’ views. the left column denotes the university and specifically a faculty of education where the modules are taught. this column contains skeletal references to mkt and mathematical practices that psts encounter in the modules. as stated earlier, the content of the modules is discursively different from school-teaching mathematics (julie, 2002). the right column denotes the school and specifically a mathematics classroom, with selected references to the literature on school mathematics teaching and its impact on mathematics content. on methods in the present study a distinction has to be made between the method of data collection and the method of analysis. a description and explanation for each of these will be given, because they relate to the way in which the research question is answered. method of data collection during one of my classes with each of the second, third and fourth year modules i gave the psts a questionnaire to complete anonymously and on a voluntary basis. i wanted to know the psts’ views about the mathematics teacher education modules i was teaching. in particular, the psts were asked questions on how they were experiencing the module at that time. what excited them, what inspired them, what frightened them, what did they find particularly difficult, and what challenges had they been able to overcome? they were told that their responses would not affect their marks for the modules. they self-administered the questionnaire, that is completed it in their own time. i left the room whilst they completed the questionnaire. the class representatives brought the completed questionnaires to my office the following day. according to the class representatives 28 psts agreed to answer the questionnaire on their views. not all of them completed the whole questionnaire, however. validity ethical clearance and permission to conduct the study were obtained from the university’s research ethics committee. the psts were informed of the purpose of the study, namely, that i wanted feedback in order to understand their views about the modules better. that, in its turn, would enable me to improve the modules and thus the psts’ experiences of the modules. participation in the study was voluntary. the procedure of volunteering was a way to ensure that the psts participated of their own free will. the psts completed a self-administered questionnaire on their views about preparation they were receiving for teaching school mathematics. it should be noted that the psts’ written responses on their views amount to self-reports. there was no interference on my part as teacher of the modules. comprehensive instructions and details about confidentiality and the purpose of the questionnaire were provided to all the psts in the study. because of the possibly sensitive nature of the study, psts reported anonymously on their views, attitudes and feelings about the modules. as the teacher-researcher i was available to answer questions and address any concerns. i maintained the confidentiality of the psts and kept their completed written responses in a secure place. those psts who wanted feedback provided their contact information and received feedback on the results of the study. all names of psts in the excerpts are pseudonyms. figure 1: a diagrammatic representation of the analytic framework. reliability the claim is that the findings – the psts’ views – are reliable, even though they are in the form of self-reports. if a different group of psts taking the same modules were to be asked their views about their preparation to teach school mathematics, it would be highly likely that there would be consistency in the variety of views ranging from anxiety and apprehension to confidence, with the exception of liezel whose views are isomorphic with the epistemology of practice, although not the actual practice, and the discourse associated with the pre-service mathematics teacher education modules. one plausible explanation for the confident views is that the psts may have figured out what party line to take in terms of my expectations as the teacher of the modules. another plausible explanation for views showing disagreement, anxiety or apprehension is the discursive differences between the variety of mathematics in the modules and the modal, highly fragmented content of school mathematics (in the intermediate and senior phases) as described in the reviewed literature. method of data analysis a few important remarks have to be made about the method of data analysis. firstly, the data analysis has to be seen in relation to bridging the university–school divide. secondly, as a mathematics teacher educator and teacher of the modules i face ‘personal trouble’ around ‘public issues’ (mills, 1959). in south africa a current public issue is practicing teachers’ lack of mathematics content knowledge. a deliberate and personal decision i make in my teaching is to conceptualise the content in the modules in ways which are discursively different from typical school mathematics content (in the intermediate and senior phases) such as a focus on single answers (julie, 2002) and a high degree of fragmentation (watson, 2008). my conviction is that (primary school) psts need opportunities where they can begin to see content that is informed by a multifaceted mkt, that is specific to school mathematics teaching. the fact that there may be disagreement with such a conceptualisation of content is a manifestation of personal trouble that i face as a mathematics teacher educator. thirdly, in a methodological and theoretical sense all attempts to teach content and pedagogy in the modules are driven by literature on mkt and the views of askew (2008). therefore, teaching the modules provides me as a mathematics teacher educator with a space to study teaching (and learning) (ball, 2000) and discover ways in which the university–school divide can be bridged. self-study researchers or science teacher educators such as berry (2004, 2007), loughran and berry (2005) and loughran (2007) also study this divide. conceptually and methodologically there is thus a need for ‘distance’ in terms of data analysis. in other words, the choice of data excerpts (psts’ views) has to show disagreement as well as agreement. the research question requires that the unit of analysis should be psts’ views. in the modules the psts encounter social practices and in the questionnaires they express a variety of views with regards to the modules. as noted earlier, the psts’ views are defined as their comments on the social practices they encounter. in a general sense their views can reflect affect such as disagreement, anxiety, apprehension, uncertainty, confidence or agreement with respect to the modules. in a specific sense their comments can reveal their views on the social practice of mathematics and the social practice of mathematics teaching. the views of eight psts will be presented for analysis; they range from anxiety to confidence with respect to the social practice of mathematics (content) and the social practice of mathematics teaching. the rationale for choosing these eight psts has to do with space as well as the need to bring into view complexities surrounding a conceptualisation of bridging the university–school divide. the two social practices are specific to the university context but are aimed at the school context. pre-service teachers’ views were coded as disagreement, anxiety, apprehension, uncertainty, confidence or agreement with respect to the social practice of mathematics and the social practice of mathematics teaching. findings top ↑ disagreement, anxiety and apprehension anne’s views indicate that she feels anxious and apprehensive and is in fact critical about the modules: this course, with reference specifically to mathematics, has been an ‘ok’ module of the course. to be truly honest, i do not want to teach maths in schools as i do not feel properly trained/educated. these modules have not brought desire into my heart to teach maths. merely trying to understand what is actually being asked is a challenge, and i am a very strong maths student. i would like this course/modules to be revised. i would like to suggest that our course, and specifically key major subjects, be content based. too many teachers are lacking content. now in our 3rd year, we know how to teach and now we need proper content – content that is addressed in schools. (anne, 4th year student) she starts by stating that mathematics’ has been ‘ok’, but quickly expresses her anxiety and apprehension through phrases such as ‘to be truly honest’ and ‘have not brought desire into my heart to teach maths’. she provides specifics by making a comparison between the content of the modules and school mathematics. according to her, the content in the teacher education modules is not the ‘proper content – content that is addressed in schools’. also, she experienced difficulty in following the pedagogy, specifically the questions that were posed in the modules. for instance, she states ‘merely trying to understand what is actually being asked is a challenge, and i am a very strong maths student’. being a ‘very strong maths student’ can refer to her school mathematics or other mathematics experiences outside of the bed programme. there is no further evidence to pin down any specific references when she writes about being a ‘very strong maths student’. what can be said is that she encountered a variety of mathematics in the modules that was quite different from her views of what the content of school mathematics should be like. it seems to be that anne expected a one-to-one correspondence between the content in mathematics teacher education modules and her view of school mathematics content. anne views teaching as a generic activity separated from the mathematics content. for example, she writes ‘now in our 3rd year we know how to teach and now we need proper content – content that is addressed in schools’. she feels that she knows how to teach by the third year and wants the rest of the bed programme years to be devoted to the ‘proper content that is addressed in schools’. her view indicates a fundamental disagreement with attempts in the modules to teach both content and pedagogy. her suggestion is that during the fourth year of the bed programme there should be a teaching of the ‘proper content that is addressed in schools’. teaching the ‘proper content’ of the maths taught in the intermediate and senior phases in the fourth year of the programme is impossible from many perspectives. for example, university conditions are quite different from those found in schools in terms of contact time, holidays and daily routines such as timetables. it is therefore not possible to teach the ‘proper content’ of the mathematics that psts like anne will have to teach learners in the intermediate and senior school phases during the fourth or final year of the bed programme, because a university and a school are simply different environments. secondly, at a conceptual level the modules have goals such as privileging patterned, coordinated regularities of actions related to social practices of mathematics and mathematics teaching. anxiety and uncertainty candice’s views are similar to anne’s in that we also notice evidence of anxiety and a quest for certainty in terms of school mathematics teaching. she writes about the need for ‘practical work’ and ‘lessons that we can use’ when it comes to school mathematics teaching: we as students are never 100% sure of what to study or what we are being taught, for example in maths. i also find that we do not do enough practical work and we do not cover enough about ‘teaching’ and creating lessons and this is what teaching is about. why don’t we get lessons that we can use? (candice, 3rd year student) her anxiety is evident in words like ‘never 100% sure of what to study or what we are being taught in maths’. this anxiety may refer to the overall quality of the modules and the need for more detailed explanations of the activities, actions and tasks in the modules with respect to the social practices of mathematics and mathematics teaching. her anxiety also points to the difficulty of trying to teach content as well as pedagogy in the modules. she provides details about her anxiety when she writes that ‘we do not cover enough about “teaching” and creating lessons and this is what teaching is about’. her view is that time should be spent on ‘creating lessons’ which, according to her, ‘is what teaching is about’. creating lessons about school mathematics ‘teaching’, to use her word, could reduce the anxiety and uncertainty she expressed in her comments. for example, she feels a need for ‘lessons that can we can use’. her comments about creating lessons and not covering enough about teaching in the modules deserve attention. these comments have relevance for the university modules as well as the actual school mathematics classroom setting. let us assume that she wants ‘creating lessons’ to happen as part of or within the modules. designing lessons in the modules are about the social practice of mathematics teaching. they imply designing lessons where the many domains of content knowledge are taken into account, such as knowledge of learners and knowledge of teaching, which constitute the multifaceted mkt. from the perspective of the modules, creating lessons in an epistemic sense entails selecting, organising and sequencing the many nuanced interpretations that a mathematical concept can represent and how it is connected to or networked with other mathematical concepts. in the university modules creating lessons means taking into account what it takes to learn and to teach a particular mathematical concept by, for example, unpacking or decompressing it into its elementarised versions. here creating lessons involves knowing how and when to compress a mathematical concept when teaching it and when to point out connections with other mathematical concepts, that is identifying ‘unifying concepts’. candice’s request for lessons that can be used is therefore understandable, because lessons will help her to reduce her anxiety and gain confidence in school mathematics teaching. one way of complying with her request would be to model lessons that can be used in real classrooms with the explicit understanding that such lessons must be adapted so that children’s responses, for example, can be anticipated and then taken into account as the lesson unfolds during its actual teaching in school mathematics. now and then the psts should be provided with model lessons that are as far as possible situation-specific with respect to the school. verbal awareness of shift from mathematics to mathematics teaching some psts do not feel anxious about the social practices specific to the modules but do notice shifts towards the social practice of mathematics teaching in terms of particular school mathematics content. for example, johan writes that he has been asked to direct his ‘strength’ in ‘solving equations’ to thinking about ‘various ways to teach equations’:i feel that i’m quite strong in mathematics but have not been stimulated to use it but have instead been asked to develop that strength for teaching. this changes the challenges presented to me from solving equations to thinking of various ways to teach equations, etc. i tend to feel that i catch on quickly and feel bored waiting for other students to ‘click’. (johan, 3rd year student) like anne, who sees herself as a very strong mathematics student, johan considers himself as ‘quite strong in mathematics’, referring to mathematics content. in his comments we notice particular (school) mathematics content – solving equations – being considered with an eye on ‘various ways to teach equations’. he feels confident about these shifts or changes in the challenges presented to him. he writes about an instance where the two social practices – mathematics and mathematics teaching – intersect in the modules. his use of ‘challenges’ is an indication that he has somehow become aware of differences between mathematics content and instances of mkt, that is solving equations and thinking of various ways to teach equations. here his views also reflect evidence of one of the central tasks in pre-service preparation, that is developing subject-matter knowledge for teaching. there is more to learn from a comparison of the views of anne, candice and johan. they articulate opposing views about their preparation to teach school mathematics. johan’s description of his challenge in having to shift from equations to teaching equations is different from anne’s suggestion of a need for ‘proper content – content that is addressed in schools’. according to her, school mathematics content is something that exists ad hoc and can be addressed in the final year of the bed programme in terms of preparation. johan seems to have become aware of particular school mathematics content in relation to various ways of teaching equations. however, anne does not see such a need in terms of teaching. teaching or methods of teaching, according to her, are what she has come to know by the third year of the programme and are therefore separate from content. from candice’s perspective, there are not enough activities or tasks related to teaching such as methods of designing lessons that can be used in school teaching. what becomes clear is a complex picture of the psts’ views about school mathematics teaching. from apprehension to excitement in the next section we turn to instances where the psts’ views are in agreement with the social practices of mathematics and mathematics teaching that underpin the modules, but where concerns about school mathematics and its teaching as they know it are raised.tami’s views indicate apprehension and anxiety at the beginning of the modules, which eased but then reappeared as she started thinking about school mathematics teaching in an actual classroom: i found this course extremely challenging at first. now i find it easier. what excites me is the way we approach maths and all the connections we are making. what i find difficult and frightening is i don’t know how i am going to apply everything to the curriculum. i am also concerned about those other aspects of the maths curriculum that we are not going to cover because i don’t know if i will be able to apply the same approach and methods by myself. (tami, 4th year student) affective views are evident in words such as ‘extremely challenging’ and ‘frightening’. these words show the anxiety and apprehension that she experienced in the beginning (‘at first’) of the modules. over the duration of the modules she has come to notice a mathematics content that is different from school mathematics as described by watson (2008). ‘all the connections’ is evidence of the mathematics content in the modules being illustrated through the social practice of mathematics such as mathematical representation where unifying concepts, that is ‘connections’, are pointed out. also, her use of ‘approach’ is evidence of the actions illustrating the social practice of mathematics teaching which occurred in the modules. she wishes for continuity in terms of the ‘approach’ in the modules with respect to the rest of the (school) ‘maths curriculum’ (‘i don’t know if i will be able to app ly the same approach and methods by myself’), that is when it comes to the actual practice of school mathematics teaching. a reality is that the teaching arrangement and organisation within the modules is such that not every ‘aspect of the (school) maths curriculum’ can be ‘covered’ for reasons related to time and the fact that, as the teacher or researcher, i have yet to figure out all the myriad ways that mathematics concepts can unfold in actual teaching, whether in the university or in a school. reality of schools and mathematics as a school subject jana and petro are in agreement with the social practice of mathematics teaching that underpins the modules, but they point out ‘the reality of schools’:the module is very effective and helpful for me especially because i’m learning things that i did not know before and i believe this is the reason why i’m here, to acquire new knowledge and to expand my knowledge. on the other side there is the reality of schools. mathematics gets taught in rote ways and children are forced to memorise formulas etc. (jana, 3rd year student) jana has made a comparison between mathematics content and pedagogy that she had been exposed to in the past and what she claims to have learned in the modules. it is very likely that she is referring to her school mathematics experience, although there is no solid evidence for making this claim. we read further that she refers to the ‘reality of schools’ and ways in which mathematics is taught there. according to jana, there is clearly a difference between the social practices of mathematics and mathematics teaching in school and in the modules. petro’s views are almost similar with regards to the ‘curriculum laid out work that needs to be done’, that is, school mathematics, as compared to the modules: i feel that it will almost be impossible to implement half the things we’ve done. effective as they may be, the time limit in schools and the curriculum laid out work that needs to be done doesn’t allow it. (petro, 3rd year student) she writes how ‘impossible’ it will be ‘to implement half the things’ we have done in the modules (in the university). agreement and confidence asma and liezel are confident about their preparation to teach school mathematics. asma’s confidence is reflected in a particular, coordinated task in the modules which requires the psts to read, discuss and summarise mathematics education journal articles aimed at a practitioner audience. the task is assigned in all modules and serves as a means to get the psts to develop their initial repertoire and their subject matter knowledge for teaching – specifically mkt. reading journal articles connects to the central tasks of pre-service preparation. asma wishes to apply ‘everything’ she has learnt and continues to learn to her teaching in a school:in the future i want to be a maths teacher. i plan to apply everything i have learnt and am learning to my teaching. i also plan to stay current with maths education through the reading of journals. (asma, 4th year student) she intends to ‘stay current’ with ‘maths education’ by reading journals. somehow she seems to have realised the value of reading journal articles as a possible, coordinated, regular action directed towards the practice of (school) mathematics teaching. liezel’s expresses a view where we notice patterned, coordinated regularities of action directed towards the social practices of mathematics and mathematics teaching: the course has enabled me to acquire a new perspective on mathematics and teaching approaches. we often underestimate learners and go with the assumption that we have to tell them what to do all the time. this programme has actually proved the opposite. in mathematics especially, children can be led by means of the correct facilitating strategies and probing questions, to use their own methods by means of inherent experimental processes to formulate and thereby solve the problems posed to them. the module has changed my approach to mathematics. by and large we are taught to follow a product-oriented mathematical approach. in this programme, however, there is emphasis on the opposite (process-oriented) approach as a way to highlight the necessity of the child’s mathematical development. (liezel, 4th year student) [translated from afrikaans] she feels confident about her experience in the ‘course’, which may refer to the bed programme as whole, including the modules i teach, and claims that it has enabled her to acquire ‘a new perspective on mathematics and teaching approaches’. she claims that she now looks at school mathematics with an eye on teaching and on ‘learners’. furthermore, she cautions about ‘underestimating’ learners in terms of what they can do in relation to the mathematics content. it can be argued that her intended, not actual, patterned, coordinated regularities of action reflect the process dimensions of mathematical practices, that is of doing mathematics. this claim is supported by her references to actions such as ‘facilitating strategies’ and ‘probing questions’ where ‘they (learners) use their own methods’, and ‘inherent experimental processes to formulate and thereby solve the problems posed’. here she may be referring to the types of mathematics problems she has in mind, that is ones where the mathematics can emerge through a ‘process-oriented approach’ and possibly having her learners acquire structural insights with respect to school mathematics. she wishes to teach in ways where the ‘the child’s mathematical development’ is taken into account, that is she has knowledge of student learning of the mathematical content. this is an instance of the multifaceted mkt such as knowledge of content and learners, as well as knowledge of content and teaching. what liezel espouses is different from the mathematics found in school, according to watson (2008) and according to petro (‘curriculum laid out work’). also, her views are different from anne’s, that is the ‘proper content – content that is addressed in schools’. liezel articulates the social practices of mathematics and mathematics teaching, such as solving problems and communicating mathematical ideas and concepts, according to the rand mathematics study panel (2003). she positions herself in ways that differ from those of the other psts. it is important to note that her positioning is not relative to actual school mathematics teaching but rather to what she has probably encountered in the university. discussion top ↑ the discussion focuses on the university–school divide because the psts’ views about the modules are about social practices related to bridging this divide. firstly, there will be comments on views that reveal anxiety and apprehension, followed by those that reveal instances of confidence related to social practices in the modules. the discussion ends by mentioning implications for the university–school divide. in an overall sense the discussion aims at highlighting personal troubles that i have as mathematics teacher educator around public issues with regard to the preparation of primary school psts. mathematics teacher educators who teach modules aimed at mathematics teaching at the intermediate and senior phase must be aware of, and take into account, psts’ feelings of anxiety and apprehension. empirically we see the examples of anne and tami. these examples are confirmed by the literature, which also states that psts feel scared and are overwhelmed by mathematics and mathematics teaching. tami had considered the modules ‘extremely challenging’ in the beginning, whilst anne had feelings of anxiety almost right up to the end of the modules. on the one hand there is liezel, whose views can be described as confident with regard to the intended and not the actual social practices related to (school) mathematics and mathematics teaching. more activities and tasks are needed to provide situation-specific instances of mathematics teaching, such as creating lessons within the university modules. ideally the university environment, that is the modules, provides an intellectual space for creating lessons. here a school timetable is absent and there are possibilities for mathematical practices such as the thoughtful use of representation to show how, why and where school mathematics ideas (in the intermediate and senior phases) can be connected through teaching. certainly what is proposed can be criticised as coming from an ivory tower. on the other hand, reflection on such teaching and the availability of sufficient time will probably not be possible in a school situation, where there are limited time slots (watson, 2008), which can influence school mathematics teaching. at a structural level the different views of the different psts on their preparation to teach school mathematics speak to the university–school divide. the university is a place that valorises theoretical insights and questions such as ‘what could or should school mathematics teaching be like?’ for a pst like anne, the kind of mathematics that focuses on connections and considerations for teaching (eloquently articulated by liezel) and counts in the university, is not the ‘proper content’ that is ‘addressed in school’ or that counts in the school. if the school mathematics content that anne refers to is interpreted as school-teaching mathematics (julie, 2002) or simply as school mathematics (watson, 2008), she does not take into account mkt, for example. examples of mathematics teaching in the modules are situation-specific to the university, that is they count within the confines of the university. jana and petro see value in such examples of mathematics teaching but express reservations when it comes to the school situation. on the other hand, anne has experienced a rupture in terms of her prior mathematics experiences. candice, however, wishes to bridge the university–school divide; hence she is interested in creating lessons that can be used in school (the mathematics classroom). in practical terms psts such as anne and candice should be helped to notice the larger picture of mkt. they and other psts also need knowledge that is situation-specific and related to a context such as a school classroom in which they meet a problem. in practical terms, liezel’s formulations of the social practices of mathematics and mathematics teaching and asma’s views about wanting to remain up to date with mathematics education are necessary in terms of transitioning, albeit verbally, from their earlier mathematics experiences. when they enter the school, they will have to wrestle with the overlay of a school classroom on the mathematics content. they will have to contend with textbook writers, test developers, education bureaucrats and parents. it makes sense to ask what support they will receive in their transitioning from the university to the school. it can also be said that liezel and asma may have figured out what the idealised pst should say or how they should behave. concluding remarks top ↑ this article reports on the views of a selected number of intermediate and senior phase psts about mathematics teacher education modules that aim at teaching both content and pedagogy. it is evident from the findings that the psts position themselves differently in relation to the social practices that are privileged in the modules. their positioning should be interpreted in relation to the social practices of mathematics and mathematics teaching exemplified in a university context and not in a school context. making the psts’ views public is a means to show what it means to be at the coalface of offering the psts a particular vision of school mathematics teaching as i come to understand it based on my on-going reading and understanding of mathematics teaching enacted in and confined to the university. the psts’ views can only tell us the extent to which the psts are able to verbalize the social practices of mathematics and mathematics teaching that are privileged in the modules. their views reported here cannot tell us what they will or might do in a school context.the social practices of mathematics and mathematics teaching in the modules are ultimately about conceptualising the nature of bridging of the university–school divide. in many ways these practices are informed by strands of literature specific to pre-service teacher education and mathematics teacher education. for the purposes of learning, the university–school divide cannot be collapsed. the university context can never become the school context. also, the vision of (school) mathematics teaching in the university context that is offered to the psts is far from complete and perfect. such a vision is necessary and so is the university as a place for the education of the pst. there is an important lesson to be learned, at least for me, in the complaint of one of the psts that ‘we do not cover enough about “teaching” and creating lessons’. it is in ‘creating lessons’ that we can all stand to learn. acknowledgements top ↑ competing interests i declare that i have no financial or personal relationship(s) that might have inappropriately influenced me in writing this article. references top ↑ aldridge, s., 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(2008), school mathematics as a special kind of mathematics. for the learning of mathematics, 38(3), 3−7. available from http://www.jstor.org/stable/40248612 microsoft word 65-75 mudaly.docx pythagoras, 71, 65-75 (july 2010) 65 thinking with diagrams  whilst writing with words    vimolan mudaly  university of kwazulu‐natal  mudalyv@ukzn.ac.za    this paper describes how the visual aspects of mathematics influence the understanding  of mathematical concepts. it attempts to contribute to the discussion that surrounds the  ideas  of  visualisation,  more  specifically  in  mathematics  classrooms.  two  teaching  interventions  are  described  that  may  indicate  how  visual  strategies  can  contribute  to  teaching in the classroom. an important finding of the research reported here is the ease  with which dynamic geometry software can be used as a visual tool to develop analogical  reasoning skills.  mathematics teaching is generally conducted in a verbal way, where the teacher orally engages his or her learners with new or old concepts. these words are often abstract cues which the learners have to decipher, and with the added language problems that learners inherently have, these cues often confuse learners. mathematics, being the king amongst abstract languages, conjures up nightmares in the minds of those that are mathematically challenged. the previous debates surrounding the abstract versus visual intrigued us, but the evidence provided in real life indicates that learning is aided by visual or pictorial images (ball & ball, 2007; hadamard, 1973; kadunz & sträßer, 2004; naidoo, 2007; shear, 1985; singh, 2007; steen, 1990; waisel, wallace, & willemain, 1997). it is common knowledge that a large part of our communication in daily life is dependent on visual imagery. lowe (2000) states that the use of pictures to represent technical subject matter is not a new idea: “ancient pictures from many different countries show that visual information has long been an important means of communicating ideas about our world and how it works” (para. 2). moreover these “mental images arise from perceptual representations that are created from stored information  not from information currently being registered by the senses” (borst & kosslyn, 2008, p. 849). in other words, the existing a priori knowledge determines how these mental images are interpreted and communicated in daily life. from birth, children identify with objects based on the meanings they attach to what they see. they know who mum and dad is, and any other male or female would not be accepted with similar enthusiasm. bells make beautiful sounds and should be vigorously shaken in order to produce music, hence the rattles that babies play with. teenagers know that a “tick” on an item of clothing is highly sought after and makes a huge fashion statement. adults identify road signs and make appropriate adjustments in their driving as they commute from point a to point b. a sudden appearance of a “road works ahead” sign (depicted by a man with a shovel), necessitates a particular driving strategy. often, a visual survey of the environment immediately following such a sign will indicate some form of labour taking place on the road. a similar sign, placed incorrectly on a road will evoke a similar change in behaviour from a driver of a motor vehicle. these visual images create in the mind of the viewer a particular reaction, not necessarily similar to the reaction created by the verbal utterance of a word. viewed pictures often create clearer images in our minds because of the symbols attached to what we see, accompanied by other sensory perceptions. it is this act of creating images in the mind that gives credence to the words “seeing with the mind’s eye” or “hearing with the mind’s ear” (kosslyn, ganis & thompson, 2006, p. 195). more importantly, though is the idea that whilst “seeing with the mind’s eye”, the mind is capable of creating new information or the transformation of old knowledge (mudaly & rampersad, 2010). anecdotal evidence also shows that our mind reacts to mental or physical images in particular ways. for example, in the well-known kanizsa thinking wit 66 illusion (see example, th association drawn mind through ver essential in just ‘seeing thinking an about visua experience. without a m quite differe student is to ideas arise revolutionar essential fo negotiate ne of mathema ideas and ap that learning ideas, then i outset. eng representati the followi children, bu consider th water is in fact a rate unti this is the t analogical p a rectan a cross s in the po th diagrams e figure 1), he mental c is formed. d-maps, table rbal means. recording of g’ an object a d analysis th al literacy, w aristotle (a mental image ent from the o build up, in out of old ry idea, it r or the conne ew ideas. en atics educati pply this und g is an active it follows tha glish further ions must com ing example ut it is expect he question: s pumped into s d increases il the pool is f target proble problem can ngular swimm section of th ool varies wit whilst writing we see a w cue directs o in other wor es and diagr whilst ph f the informa and its benefi hat follows th which is mu as cited in z e”. the emp e word act. n his or her m ones (davi reassures us ection of old nglish (1997) ion is that ch derstanding t e constructio at children m argues that mprise the st provides an ted that there o a pool at a s for a certai filled. sketch em. in order be given to t ming pool is e pool is sho th time, from g with words hiter-than-w our mind to rds, we are s ams aid in c hysical and m ation acquire figure 1 fits is certainl he viewing o uch more th askis, dubin phasis should davis and m mind, a colle is & maher that the pr d ideas to n ) also reinfor hildren see t o the solutio on process ba must develop if children a tructural rela idea of the u e is a reasona a constant ra in period of t h the cross s to encourag them. being filled own in figur m the moment figu white triangle o some old seeing what connecting fa mental imag d through vi 1: kanizsa ill ly not what t of an image, han simply m nsky, & daut d actually be maher (1997 ection of mat r, 1997, p. rovision of c new, foreign rces this idea the connecti on of new pro ased on reco meaningful are to make ations betwee use of analog able resembl te. the depth time, the rate section of the ge learners to using a hose re 2. descri t that the emp ure 2: the po e, when no s information we know. b acts that mig ges are nece sual means. lusion this paper is , either phys making a vi termann, 199 e placed on 7, p. 89) stat thematical id 89). whils correct men ones. the a when she s ions and rela oblems” (p. 1 gnising simi and cohesiv the appropr en ideas, not gical reasonin lance to wha h of the wate e decreases a e pool. o provide a r epipe which be fully, in w pty pool beg ool uch triangle n that we al but i hasten ght not neces essary, word all about. th ically or me sual connec 96, p. 3) said the word thi te that “the j deas”. they f st this does tal or physi se visual sti tates that “on ationships be 191). she arg larities betw e mental rep riate links to the superfic ng. this has at can be exp er (d) increas and then d in easonable so delivers wat words, how t gins to fill. d e really exist lready know to add that ssarily be un ds and sente this paper is entally. this ction with so d that “no so inks. clearl job of a mat further state not appear ical visual s imuli act as ne of the ma etween math gues that if w ween new and presentations o the new id cial surface d s not been te pected in a cl ses but not u ncreases at a olution, the f ter at a cons the depth (d) ts. in this w and an carefully nderstood ences are about the s paper is ome past oul thinks ly, this is thematics that new to be a stimuli is tools to ajor goals hematical we accept d existing from the eas, such details. sted with lassroom. niformly. a uniform following tant rate. of water vimolan mudaly 67 with some guidance, it is expected that the learners would be able to provide the following solution: d increases uniformly or steadily for the deep-end, and thereafter d increases uniformly, but at a reduced rate until it is full. it can now be anticipated that children may provide one of the two diagrams in figure 3 as a solution to the target problem: figure 3: possible solutions to the pool problem in arguing for a connection between different ideas in order to establish meaningful learning, it must be considered within the context of dreyfus’s (1991) plea to give increased importance to visual reasoning. wheatley (1997, p. 281) further argues that “visual reasoning plays a far more important role in the work of today’s mathematicians than is generally acknowledged”. this is similar to lakoff’s (1987, p. 278) argument that “understanding abstract structure is understanding structure in terms of schemas”. stokes (2002, p. 10) claims that research reported in educational literature suggests that using visuals in teaching results in a greater degree of learning. the basic premise of this body of research is the concept of visual literacy, defined as the ability to interpret images as well as to generate images for communicating ideas and concepts. developing diagrams and mental images allows the individual to free up more mental space for new imaging and to construct new relationships (wheatley, 1997, p. 285) this paper also demonstrates the idea that children develop analogical reasoning skills (gholson, smither, buhrman, & duncan, 1997, p. 151) if presented with a ‘training problem’ prior to their actual attempt at solving the real problem. gholson et al. (1997, p. 151) state that the representation that results from this type of acquisition process called the base. following acquisition, they receive a transfer problem, called the target. if the target is similar to the problem presented in acquisition, children usually perform reasonably well in transfer relative to appropriate controls. this paper further alludes to the fact that children are quite capable of solving difficult mathematical problems if their analogical reasoning skills are developed in a dynamic geometry environment. it is perhaps at this point necessary to draw a distinction between that which is visualised and visual literacy. visualised would refer to that which can be seen physically or mentally, like the mental picture that comes to mind at the mention of the word parabola. visual literacy would refer to many pictures of parabolas, each one conjuring images that show parabolas having minima and maxima, the different shapes of the parabola, the effect of changing the values of c and the coefficients of x2 and x. an entire series of possibilities flashes through the mind. of course, this depends on the extent to which one has completed the section on parabolas. the distinction may be subtle and may invoke some debate but zaskis, dubinsky and dautermann (1996) critically reflect on this dichotomy. however, i take the stance that visual literacy is visualisation combined with logical thought. zaskis et al. (1996) in a sense allude to this distinction when they describe their visualisation/analysis model (see figure 4). they describe a process which begins with the visualisation of a picture or diagram, just some external stimulus that results in the mind seeing a picture. the next step they describe is that of analysis which involves a series of mental processes (similar to piaget’s reflective abstraction). the moment the learner moves from step 1 to step 2 they begin the process of visual literacy. the spiral nature of the model implies that the solution to a problem should begin with visualisation and then followed by an analysis process. d d thinking wit 68 the visualis makes new therefore th and so the meaning ou consider as seeing this represent th begins at th this would g(x) = 2sin construed to g(x) is an following p this is perh with visuali natural tend diagram. vi external pic the process processes (r image (sigh priori know versa. it see subsumed b of meaning fig th diagrams sation/analys meaning of he learner ma process con ut of similar p figure 4 s an example formula of he v1 in fig he coordinate then represe (x + 45o) + o be a simpl exercise in attern: f(x) = sin x haps what pr isation. this dency for th isual literacy cture or diagr s of rational refer to figur ht) or when wledge takes ems that the by a process construction gure 5: proce whilst writing sis model as f v1. hence akes greater tinues. the pictures and 4: visualisat e the functio the function gure 4. with e (0;0), pass ent a1 in fig 1. this requi le visualisati visual litera x  p(x) = 2 resmeg (1992 notion conv he mind to y would refe ram or a men and logical re 5) of mean thinking abo s place. men new knowle of visual-an n. esses involve proc cons visual/physica image (sight) mental image g with words described b in re-assessi sense of wh conic shape diagrams. tion/analysis n f(x) = sin n would be h a little thou ing through gure 4. now ires mental m on of the fu acy. it is po 2sin x  h( 2) means wh veys the idea engage in lo er to the inte ntal image. it thought is in ning constru out an imag ntal images edge influenc nalytical thin ed in visualis i k iteratio and ext cess of meanin struction al or (insight) by zaskis et ing the pictu hat s/he sees. implies tha s model. sour n x in the dom to simply v ught, one ma the coordina w attempt to manipulation unction. the ossible that (x) = 2sin (x hen she talks that if visua ogical and r ernal process t is concerne nfluenced by uction (muda ge (insight), influence th ces the next nking. these sation. sourc influence on ne knowledge n between inte ternalisation p ng al. (1996) in ure or diagram . this picture at one canno rce: zaskis e main [0; 360 visualise the ay be able to ates (90; 1), mentally vi ns that are no ability to se the visualis + 45)  g(x about logica al literacy is s rational thou ses that the m d with a prod y the way th aly & rampe a process of he learner’s conjecture an images play ce: mudaly & ew ernalisation rocesses visual-analyt thinking forma transfo knowle ntimates that m, new infor e/diagram th t continue fo et al., 1996, p 0]. an imme shape of th o mentally p (180; 0), (2 sualise the fu ot straight fo ee the transfo sation schem x) = 2sin (x + al rationality successfully ught at the s mind engage duct of some he learner en ersad, 2010). f reflection u physical, dr nd the old in y an importan & rampersad tical ation or ormation of edge t after a1, th rmation is a hen can be ca forever to de p. 447 ediate conseq he graph. th picture a func 270; -1) and function with orward and c ormation fro ma might fo + 45) + 1 y which is int taught then sight of a p es in after vi e visual stimu ngages in the on seeing a using the le rawn image ncorrect ones nt role in the d, 2010, p. 39 he learner dded and alled v2, erive new quence of his would ction that (360; 0). h formula cannot be m f(x) to ollow the tertwined there is a picture or ewing an ulus. e iterative a physical earner’s a and vice s become e process 9 vimolan mudaly 69 two experimental tasks task 1 the following problem (de villiers, 2003, p. 23; mudaly, 1999, p. 62; mudaly, 2007, p. 68) was posed to grade 9 learners in an intervention in a school in kwazulu-natal. sarah, a shipwreck survivor manages to swim to a desert island. as it happens, the island closely approximates the shape of an equilateral triangle. she soon discovers that the surfing is outstanding on all three of the island’s coasts and crafts a surfboard from a fallen tree and surfs every day. where should sarah build her house so that the total sum of the distances from the house to all three beaches is a minimum. (she visits them with equal frequency.) the previous discussions of this intervention in mudaly (1999) and mudaly (2007) examined learners’ responses from a mathematical modelling perspective. this paper extends the analysis to a visualisation perspective. the question “where do you think that sarah should build her house?” elicited a common response among most learners, namely that sarah should build her house at the centre of the triangle. the learners were asked why they felt that the house should be built at the centre. kumarasen, for example, responded by saying that: “…if you build anything in the centre then there is always a short distance around it”. manivasan, whose reason was “ …because everything will be equal”. rowan believed that it should be at the centre because “it will be close … it will be the same distance to all the beaches” and therefore the sum will be a minimum. karishma felt that the sum would be a minimum if the house was at the centre of the triangle because “it will be closer to all three beaches”. without a diagram or a picture to guide them the learners attempted to picture in their minds a possible island and a probable solution. a possible explanation here relates to the fact that these learners used their a priori knowledge of circles to envision a point that was equidistant to the three sides of the triangle. equality of length was instantly equated to the idea that a radius was equal in a circle. the next task required the learners to investigate the problem with a ready-made sketch on sketchpad (see figure 6) where the distances to the sides and their sum, was already provided. learners were asked to move (“drag”) point h (the position of the ‘house’) around in the triangle and to observe whether any changes were taking place. providing an already constructed diagram, instead of the learners first constructing it, was prudent because it saved much time. figure 6: learners’ attempt to find the optimum position of the house on seeing the picture the learners were further convinced that the house should actually be built at the centre of the triangle. they were then prompted to drag the point h around and observe the measures on the left. after the learners had moved the point around, they had to make a conjecture regarding their hf + hd + he = 9.8 cm he = 2.0 cm hd = 3.3 cm hf = 4.5 cm e f d b a c h thinking with diagrams whilst writing with words 70 observations. all the learners found that when moving the point around, the distances to the three sides of the triangle changed, but their sum did not. all of them seemed quite surprised at the result. kumarasen stated that “at first you think it should be at the centre and the sum will be small. but now it can be anywhere”. this could probably be attributed to the result so clearly contradicting their initial expectation. it was also noticed that the majority of them began to smile after they came up with their conjecture, which indicated that the discovery was not an unpleasant, but a pleasant surprise. the dynamic nature of sketchpad allowed the learners to ‘see’ and analyse several different diagrams and its resultant sum in a matter of a few minutes. although, similar results could have been obtained using pencil and paper constructions, sketchpad simply facilitated this process far quicker than would have been expected. but the idea that it conveys is that without seeing the diagrams and analysing the results, as it appeared on the screen may not have achieved a high level of conviction in the minds of the learners. more importantly though, it was the visual cues that the learners saw on the computer screen that convinced them of the result. there was iteration between the physical and mental images and, although this process occurred very quickly, it enabled the learners to analytically create new information. task 2 in a second investigation grade 10 learners were asked the following question (de villiers, 2003, p. 27; mudaly, 1999, p. 114; mudaly, 2007, p. 70): in a developing country like south africa, there are many remote villages where people do not have access to safe, clean water and are dependent on nearby streams or rivers for their water supply. with the recent outbreak of cholera in these areas, untreated water from these streams and rivers has become dangerous for human consumption. suppose you were asked to determine the site for a water reservoir and purification plant so that it would be the same distance away from four remote villages. where would you recommend the building of this plant? as was the case in the previous task, these student responses are theorised using a visualisation framework of analysis as compared to the discussion in mudaly (1999) and mudaly (2007). the intention of the task was to teach learners about cyclic quadrilaterals (which they would normally encounter in their grade 11 year). therefore all diagrams had to be provided, otherwise the learners would have had to already have known the properties of the cyclic quadrilateral and this would have contradicted the purpose of the investigation. the diagram they saw on the screen with the easy-to-use buttons is shown in figure 7. figure 7: learners trying to find the centre of the quadrilateral the initial question attempted to determine whether the learners had any ideas about the solution. the researcher was aware that the learners had insufficient information to construct a reasonable solution. so when they were asked to predict a site for the reservoir, all learners “guessed” a solution somewhere in the “middle” of the quadrilateral (similar to task 1), but none could find a precise solution. more importantly, few of them even thought of or tried to test their solution. the inaccuracy of their visual construct point within quadrilateral v3 v4 v1 v2 vimolan mudaly 71 perception created surprise when the learners later discovered that their conjectures were not correct. it is not strange that in both instances the learners felt that the object should be constructed in the ‘middle’ or ‘centre’. it is highly probable that learners think in terms of a circle and perhaps one should be asking why only the centre implies equidistant to all points. is it perhaps, that learners are only exposed to symmetrical geometric shapes in classrooms? nonetheless, their previous knowledge systems indicated to them that the point was in the middle. in attempting to analyse the contextual problem the learners visualised a quadrilateral where the point “somewhere in the middle” would be ideal for the building of the reservoir. it can be conjectured that previous images of working with circles contributed to their responses. they were then asked to move v3 so that they could find the point at which the distances to the vertices were the same. this proved to be a tricky task. it was evident that the learners had no idea as to how they could proceed further with finding an accurate solution to the problem. it was fine that they knew that the point could be dragged around to find an approximate solution, but determining points in this way is time consuming as well, and if one looks at finding solutions correct to a few decimal places then finding accurate solutions is essential. moreover, the act of dragging and viewing the images on the screen had the effect of showing them that their initial conjecture was incorrect. this must have resulted in a mental dilemma, where they were fairly convinced that the point should be “somewhere in the middle” and then discovering that this is not the case. new mental images had to be created but with no definitive position for the point in question. instead of considering four villages, the learners were asked to work with two. the simplified version of the problem was easy to solve for the learners. they could easily visualise the solution – which they knew to be the midpoint of the segment between the two villages. but by using the idea that the midpoint may not be suitable as a real-life solution, the learners were asked if they could determine another point which was also equidistant from the two villages. the purpose was to investigate whether learners could find other points and realise that there were infinitely many and the points all lay in a line, that is, the perpendicular bisector. it should be pointed out to the reader that traditionally in south africa, the perpendicular bisector is usually introduced in grade 8 or 9 (or sometimes earlier), but only as a construction. in other words simply as a line that passes perpendicularly through the midpoint of a line segment, but its equidistance properties are traditionally never investigated (and perhaps only alluded to in passing). the learners were quite clear about the fact that if one point was not suitable then another could be used another point that satisfied the condition of being equidistant from two points. the researcher was initially unsure whether the learners would be able to make the deduction that all points that are equidistant from two points will lie on the perpendicular bisector. but the learners made this connection to the perpendicular bisector quite easily. notice the analysis that had taken place. they considered the midpoint to be the solution and then discovered that there were in fact many solutions. the constant iteration between visualisation and analysis lead them to eventually conclude that the any point on a perpendicular bisector would be equidistant to the two end points of the segment (refer to figure 8). figure 8: using the perpendicular bisector at this point the researcher was interested to see whether the learners would now immediately use the newly acquired concepts of equidistance and perpendicular bisector they arrived at for the problem with the four villages. all learners confidently stated that we should use the perpendicular bisectors of each je = 6.18 cm ie = 6.18 cm jd = 4.86 cm id = 4.86 cm jq = 4.63 cm iq = 4.63 cm hide midpoint q i j d e thinking with diagrams whilst writing with words 72 side of the quadrilateral. as has been expected, the learners had used their knowledge of the perpendicular line between two points to conjecture a result for the four points. the learners showed no particular emotion when they discovered that the four perpendicular bisectors met at a common point. it was as if they had expected it to happen and it was taken for granted that these lines would meet. none of them stated that this result really surprised them; in fact, most of them conveyed the idea that this was expected. this is perhaps also indicative of the social context within which learners construct meaning for problems given to them. learners intuitively sense that the problem would not have been asked if there was no precise solution already “pre-existing”, and this kind of (unrealistic) anticipation is likely to become a determinant force in their behaviour, unless they are given sufficient experiences to counter it. furthermore, it implied that their ‘new discovery’ was in line with what they already ‘knew’, so there was no reason to doubt or question their finding. the physical image on the screen simply corroborated their existing knowledge. next the researcher informally introduced the learners to the circumcircle, but no formal definitions of circumcentre and circumcircle were given. a circle was constructed from the point of concurrency (the circumcentre) as center with the circumference passing through one of the vertices of the cyclic quadrilateral. the learners showed surprise in their facial expressions and the way in which they responded when they saw that the circle now passed through all the vertices of the quadrilateral. the diagram below (figure 9) is an example of what the learners saw on the computer screen as they worked with the circumcircle. figure 9: circle now passes through all the vertices of the quadrilateral the intention of the next aspect was mainly to see if learners would realise that the perpendicular bisectors of a quadrilateral were not necessarily always concurrent, that is, that for a quadrilateral it was not always possible to find an equidistant point. the dynamic facility of sketchpad not only facilitated this finding quite easily, but assisted some learners to realise that an equidistant point would only exist when the circle passed through the four vertices (that is, the quadrilateral is cyclic). with sketchpad it was easy to change the size and shape of the given quadrilateral, so that the learners could observe whether the perpendicular bisectors were concurrent and whether the circumference of the circle still passed through the vertices of the quadrilateral. whilst moving a vertex of the quadrilateral around the learners discovered that the result they expected was not always true. it can be conjectured that this created another mental dilemma for them. by moving the vertex to various positions, a new mental image was formed. they now discovered that the result was only true when the vertices of the quadrilateral lay on the circumference of a circle. it must be noted that this constant iteration between the different images was creating new knowledge that may have been difficult to achieve if pencil and paper methods were used. v1 v4 v3 v2 vimolan mudaly 73 the learners were next presented the same problem, but now within the context of a triangle. the questions that they were asked were not different from the previous ones and the responses they made were, as expected, in accordance with what they saw in the previous case. the learners confidently responded that they would again use perpendicular bisectors to locate the required point. in fact, the level of their conviction was very high and this could be determined from the way they responded and the language they used. they were convinced that the perpendicular bisectors would be concurrent if the vertices were concyclic. this was just an extrapolation of the previous result. this was schofield’s response: schofield: it must be the same for the four-sided figures. researcher: what do you mean? schofield: for the quadrilateral it only worked for some…when the circle touched the four corners…it might be the same for the triangle. faeeza had made similar responses. but faeeza’s response was quite interesting. she initially stated that by constructing the perpendicular bisectors we could find the most suitable position. but she immediately changed her mind to include something that she had just learned for quadrilaterals – a qualification she felt was necessary. she felt that the perpendicular bisectors would only meet if a circle passed through the vertices of the triangle. one can imagine the pictures that were being processed in their minds. their new conjectures were becoming influenced by the new knowledge they had acquired in the previous task. interestingly enough, their mental images were changing as they viewed the sketchpad diagrams on the screen. these images started with an image that represented their conjecture and changed as they viewed the results on the screen. it is this analysis that ensured that meaning-making was occurring. of course, when they constructed the perpendicular bisectors they could see that the perpendicular bisectors were concurrent and the vertices were concyclic for all triangles that they created by dragging a vertex. although this may not represent every possible triangle but it was a sufficiently large number to convince them. they initially doubted their observations, perhaps because it contradicted something they just learned. schofield was initially very sure. his conviction arose from his experience with the quadrilateral. he was surprised to see that the lines were always concurrent for a triangle, but was still eager to try it for more triangles: “no…i think move it a bit more”. he was still unconvinced that the result was true for all triangles. after working through more examples, his response to the question “are you satisfied that the perpendicular bisectors are always concurrent?”, he replied “yes…but i never would have guessed that!”. this process also took only a few minutes but he was very convinced because he saw the changes on the computer screen (“no doubt…i saw it myself!”). the implication of this may be that he would not have accepted the truth of the result had he been just told, but the fact that he could dynamically see the result convinced him that the result was true. a more pertinent implication is the way these learners experienced the acquisition of new knowledge. the need for proof the researcher found that the majority of learners expressed a desire for an explanation in the first task. in fact 94 % of the learners said that they would want an explanation and only one learner took a while before saying that she would also like an explanation. it seemed that despite being convinced, they also wanted to understand why the result is true. it seemed that wanted something more than just being able to observe and accept the validity of the statement. the researcher acknowledges that by asking the question “do you want to know why this is true?” the learner may have been led to answering in the affirmative. however their body language and response seemed to indicate that they truly wanted to know why. although all the learners were absolutely sure of the statement, they nevertheless seemed to express a need for further understanding. these are some of the learner responses to the question: “why do you want an explanation for this?”. rodney: to satisfy my curiosity. manivasan : so i can understand it. [his emphasis] karishma: because i’m curious and i’d like to know what’s going on. debashnee : because i’m a curious person and i would like to find a solution for things. i would like to do the same for this. thinking with diagrams whilst writing with words 74 again it would seem that there was a deeper urge to find an explanation rather than to check whether the result was really true. it further seems clear that the learners’ desire for further explanation or a deeper understanding had not been satisfied by the empirical exploration on computer. this exploration only seemed to convince them, but did not appear to have satisfied some deeper need for explanation and understanding. in fact, it is probable that the visual evidence only spurred them on to desire a more formal deductive proof. conclusion this paper does not investigate the way learners mentally structure the mathematics that they learn. but, it did provide insight into the idea that given the opportunity to segment a difficult mathematical question into little, more solvable ones made the learning and content more meaningful to them. the visual strategies employed in attempting to elicit responses that directed their learning seemed to work well as they constructed hypotheses and tested them. new knowledge in the form of visual images (as they worked on sketchpad) was constantly replacing the old untested hypotheses. sketchpad provided the visual justification for them to make specific conjectures, and having high levels of conviction did not necessarily remove the need for proof. understanding generally was attained after ‘seeing’ the solution and the visual impact of the dynamic diagrams allowed the learners to quickly draw adequate conclusions. whilst i did not get students to attempt these problems using pen and paper, i would conjecture that they would have taken far more time to complete the tasks. clearly, “effective mathematical learning requires active student participation in meaningful experiences” (english, 1997, p. 3). more importantly, the research showed that ‘seeing’ the pictures, albeit through the use of sketchpad, enhanced the learning tremendously by creating meaningful opportunities for the learners to grasp the concepts presented. the second task demonstrated that analogical reasoning skills can easily be developed using sketchpad. the use of simpler problems to eventually answer more complex questions was attained through the power of the visuals and the effective use of the dynamic geometry software. references arcavi, a. 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can be opened with acrobat and adobe reader 5.0 and later.) >> /namespace [ (adobe) (common) (1.0) ] /othernamespaces [ << /asreaderspreads false /cropimagestoframes true /errorcontrol /warnandcontinue /flattenerignorespreadoverrides false /includeguidesgrids false /includenonprinting false /includeslug false /namespace [ (adobe) (indesign) (4.0) ] /omitplacedbitmaps false /omitplacedeps false /omitplacedpdf false /simulateoverprint /legacy >> << /addbleedmarks false /addcolorbars false /addcropmarks false /addpageinfo false /addregmarks false /convertcolors /converttocmyk /destinationprofilename () /destinationprofileselector /documentcmyk /downsample16bitimages true /flattenerpreset << /presetselector /mediumresolution >> /formelements false /generatestructure false /includebookmarks false /includehyperlinks false /includeinteractive false /includelayers false /includeprofiles false /multimediahandling /useobjectsettings /namespace [ (adobe) (creativesuite) (2.0) ] /pdfxoutputintentprofileselector /documentcmyk /preserveediting true /untaggedcmykhandling /leaveuntagged /untaggedrgbhandling /usedocumentprofile /usedocumentbleed false >> ] >> setdistillerparams << /hwresolution [2400 2400] /pagesize [612.000 792.000] >> setpagedevice article information author: dorit patkin1 affiliation: 1department of mathematics, kibbutzim college of education, israel correspondence to: dorit patkin email: patkin@netvision.net.il postal address: 149 namir road, tel-aviv 62507, israel dates: received: 08 july 2011 accepted: 12 oct. 2011 published: 16 nov. 2011 how to cite this article: patkin, d. (2011). the interplay of language and mathematics. pythagoras, 32(2), art. #15, 7 pages. http://dx.doi.org/10.4102/pythagoras.v32i2.15 copyright notice: © 2011. the authors. licensee: aosis openjournals. this work is licensed under the creative commons attribution license. issn: 1012-2346 (print) issn: 2223-7895 (online) the interplay of language and mathematics in thisoriginal research... open access • abstract • introduction    • theoretical background       • daily language and mathematics language       • difficulties in mathematics       • literacy activities in teaching and their integration in mathematics lessons • research method    • methodology       • research population       • research procedure       • ethical considerations • findings    • additional activities and materials    • findings obtained from diary analysis       • pleasure       • comprehension       • positive attitude towards the continuation of teaching in such a way       • sense of self-confidence • discussion and conclusions • acknowledgements    • competing interests • references abstract (back to top) this article deals with an intervention unit which focused on the positive effect of integrating poems, stories and fables (the literary genre) for improving mathematical language, fostering the teaching of mathematics language, increasing satisfaction with the process and strengthening the relationship between use of daily language and mathematics language. the study was conducted amongst pre-service mathematics teachers, specialising in literacy activities designed to foster mathematics language. one of the study outcomes was a ‘bank’ of words with double meanings which can result in misguided perceptions and common errors. another outcome was a pool of ideas for literacy activities in mathematics which can develop wide, integrative activities. exposure to and awareness of such a bank of words may help mathematics education practitioners to cope with difficulties in mathematics teaching and learning. introduction (back to top) current literature supports the existence of four language components in learning mathematics: reading, writing, speaking and listening (timor & patkin, 2010). in the teaching and learning of mathematics one should consider that mathematics has its own unique language, and that learners find some of the mathematical terms ambiguous. there are words which are used in everyday life as well as in mathematics language, but their meaning in each is different or ‘double’. in many cases natural language − the spoken everyday language − is blended into mathematics language. consequently, students ‘drag’ the meaning of words in everyday language into the mathematics language, which often results in mistakes and misconceptions. for many years only reading and writing of mathematical symbols were used during mathematics lessons, without any language skills required. this has changed in the last 20 years. the idea of communication in mathematics is expressed clearly in principles and standards for school mathematics (national council of teachers of mathematics (nctm), 2000, p. 60): ‘they [learners] communicate to learn mathematics and they learn to communicate mathematically.’ if we re-examine the definition of language (webster’s universal college dictionary, 1997), it seems that mathematics language is a formalised set of conventions that are used for the specific purpose of problem-solving. this study attempted to examine the interplay between mathematics language and natural language (in this study hebrew, the formal language of studies at the college of education concerned). the research involved a group of pre-service mathematics teachers who were exposed to the new attitude in mathematical pedagogy, emphasising mathematical discourse and communication as well as views of mathematics as a language. the pre-service teachers were asked to develop additional materials to help them whilst teaching mathematics in their classes. theoretical background daily language and mathematics language this article focuses on the interplay between two languages: everyday, natural language and mathematics language. this interplay is ‘performed’ by using the natural language in mathematics, integrating literacy activities for the purpose of fostering mathematics language and enhancing the relationship between these two languages through stories and poems. the poem look, we are two numbers (free translation and emphasis by d.p.) by israeli poet yehuda amichai is an example of a literary composition which includes mathematical terms: look, we are two numbers, standing together and being added or subtracted, because finally the sign changes from time to time. it was so difficult until we succeeded to stand together, and we also knew multiplications of happiness, fractions too, as usually happens to numbers. even now, under us, the world is a fraction line – don’t be alarmed, look how beyond that line has now bloomed for us – the common denominator. (amichai, 1975, p. 36) this is a love poem describing being a couple. amichai made surprising use of the mathematics language and created original images for defining the relationship of a couple with its fluctuations ‘from time to time’. in describing the relationship he included metaphors, all taken from the world of the basic four operations of mathematics. the mathematical expressions mentioned in the poem embody opposites, reflecting contrasts in human and emotional relationships. he employed mathematical terms as ‘scaffolds’ for the everyday language. according to him, these terms are not merely mathematical, they also have a meaning in daily life. applying words by moving and passing back and forth from the everyday to the mathematics language might enhance mathematics language and consequently students’ achievements in this field. mathematics language should be fostered in various ways for that purpose. webster’s universal college dictionary (1997) provides a number of definitions of the term ‘language’, two of which can be related to the definition of mathematics as a language. the first deals with communication using a system of arbitrary vocal sounds, written symbols, signs or gestures in conventional ways with conventional meanings. the second is any set or system of formalised symbols, signs, sounds or gestures used or conceived as a means of communicating. these definitions address the concept of language as a formalised or arbitrary means of communication comprising specific terminology such as signs, graphic representations, numbers and icons. one should bear in mind that every language is based on unique symbols and syntax rules (webster’s universal college dictionary, 1997) which define the use of its symbols, which can serve to represent meanings in a specific area. if we relate to mathematics as a language built of symbols and syntax rules used for describing objects, actions and relations between sizes, it can be utilised both in writing and orally. this is done both by numerical representations and graphic or iconic representations for abstraction of reality, for use in axioms and in a linguistic-verbal representation. hence, it becomes necessary to understand the mathematical word lexicon and terms in the teaching and learning process. mathematics encompasses unique linguistic forms. there is extensive use of key words which imply the four operations of mathematics (addition, subtraction, multiplication and division), and since the language of mathematics comprises symbols and signs, failing to distinguish between them on visual grounds may impede learning. for example, there can be confusion between 2 and 5, 6 and 9, the signs > and <, or between + and -. moreover, there is a mix of various meanings of words used in everyday life on the one hand, and in mathematics lessons on the other (as in the poem of amichai). all of this is difficult and confusing for learners (cobb, yackel & mcclain, 2000). difficulties in mathematics throughout the school years many students have difficulties in learning mathematics. in the last two decades the prevailing attitude amongst mathematics educators has been that mathematics is a language and therefore all language components must be taken into consideration in mathematics teaching (nctm, 2000; skemp, 1972). difficulties stem not only from failure to acquire mathematical knowledge but from the use of language in general and the abstract nature of mathematics language in particular (patkin & gazit, 2011). current literature supports the existence of four language components in mathematics learning: reading, writing, speaking and listening (kazemi, 1998; nctm, 2000; siegel, borasi, sanridge & smith, 1996; usiskin, 1996). despite the fact that mathematics is not perceived as a spoken language, the new standards for teaching mathematics indicate the need for verbalisation (‘mathematical discourse’) (huinker & laughlin, 1996). skemp (1972) argues that many difficulties and problems in mathematics education are due to the fact that many words have different meanings in different languages, and some words have different connotations in different countries. there are also some words which have double or even triple meaning in the same language. thus, confusion and obstacles are to be expected. skemp employed the french term faux amis to describe this, which means ‘false friends’. he gives the example of the word ‘biscuit’, which in the united states of america refers to a different item of pastry than in britain. in the usa one should say ‘cookie’ in order to obtain the same item, ‘biscuit’, as in britain. hence, during their lives learners acquire a variety of words, which they apply in order to express thoughts and ideas. sometimes they use words which represent mathematical terms, whose ‘daily’ meaning is not in line with their mathematical meaning. all over the world, everyday words in english (such as field, group or set) are included in mathematics language − and this turns them into ‘false friends’. in hebrew, for example, the word ‘cube’ has a different meaning in mathematics language and in daily life. when you tell a child in kindergarten to bring a cube from the cubes corner, this refers to one of the differently shaped game blocks with which they can build towers, bridges, et cetera. in mathematics language, however, this word refers unequivocally to a cube: a solid figure with six identical square faces, all right-angled. according to usiskin (1996) the reasons for experiencing problems in mathematics learning are related to aspects of its language. the fact that students are not exposed to the ‘language of mathematics’ at home or in their close environment makes teaching harder. it is important to start teaching mathematics at an early stage, so that it can become a mother tongue or second language for the learners. moreover, mathematics is often taught out of context. this makes learning meaningless and inapplicable to the learners’ lives. as a result, the learners study a ‘dead’, useless language. in addition, teaching mathematics comprises the use of abstract concepts which are not always clear and meaningful to the learner. usiskin argues that mathematics has much in common with other languages because of the following: 1. mathematics does not only describe ideas, but also fosters the organisation of these ideas within the learner. 2. the number of symbols and signs in mathematics (e.g. ⊥, = , ≅) is similar to the number of letters in other languages 3. mathematics has its own syntactic rules, with expressions such as ‘3+4’, and verbs (e.g. subtract). 4. mathematics has a ‘private property’ of vocabulary like any other language, as well as its own unique features. 5. mathematics lends and borrows words, like any other language. it makes use of the latin alphabet in algebra, the greek alphabet in geometry (e.g. ellipse, parabola), the word ‘radius’ from latin, et cetera. it also lends words; for example, the word ‘triangle’ in mathematics refers to a two-dimensional shape whereas in everyday english it is also used to describe a romantic loop with three people involved. the linguistic approach, which illustrates the difficulties in learning mathematics and the perception of mathematics as a language with all its aspects, has a strong relation to literacy activities (patkin, millet & ezer, 2001). literacy activities in teaching and their integration in mathematics lessons the concept that literacy can be enhanced in all learning subjects and that it promotes thinking in the studied discipline was developed in the 1990s (fulwiler & young, 1990; kazemi, 1998; norwood & carter, 1994). according to this concept, there is a tendency to integrate literacy activities such as writing, speaking and reading in every subject, in view of expanding relations between disciplines and extracurricular systems. this approach emphasises that comprehension of a complex subject is accomplished by its presentation within multidisciplinary systems. the professional literature underscores that literacy activities can also be integrated into disciplines not related to the field of humanities. at the beginning of the 1980s evans (1984) discussed the integration of writing activities in primary school mathematics lessons. she demonstrated that the attainments of students who participated in such a project were higher than those who did not. moreover, rose (1989) described a design to integrate literacy in mathematics at high schools in california, illustrating that writing in mathematics lessons allowed students to advance at their own pace whilst using their everyday language and personal experiences. in recent years mathematics educators have encouraged practicing teachers and their students to develop, in addition to writing, a ‘mathematical discourse’ during mathematics lessons (kazemi, 1998). they also advocate use of a reflective written record in order to promote mathematical thinking and learners’ ways of coping with mathematical material (hart, schultz, najee-ullah & nash, 1992; norwood & carter, 1992, 1994; schiebelhut, 1994). a study amongst different populations of mathematics teachers illustrates that integration of literacy in teaching may contribute to understanding of this subject and reduce fear of it (ezer, patkin & millet, 1999). the question in a study which focused on integration of the fable the lion and the hunter by lafontaine was: ‘what is the relation between lafontaine’s quarter and the rational number quarter? or what is the difference between mathematical justice and the jungle justice?’ in lafontaine’s fable, four animals went hunting and agreed to divide the prey into four equal parts. according to mathematical justice, each animal was supposed to receive 1/4 of the prey. the lion, however, took everything because he made a ‘just’ division, according to the jungle laws. thus, although the animals agreed about equal division, all of the parts went to the lion. findings indicated that the use of an intriguing fable that stimulates mathematical thinking led to an educational lesson which promoted a positive approach to mathematics teaching and use of mathematics language (millet, patkin & ezer, 2002). the present study explored a bank of words with multiple meanings and literacy activities by means of an intervention unit. the research questions addressed in this article are: 1. which literacy activities can pre-service teachers create or use in order to be more aware of the problems with language and mathematics? 2. how do the reflections of the pre-service teachers provide insights into the importance of integrating literacy activities in the teaching of mathematics? research method (back to top) this study used qualitative methods; diaries and outcomes activities were qualitatively analysed, allowing thorough and valid research. the use of several data sources enabled triangulation and validation of the data (payne, 1999). methodology research population the study was conducted at a college of education in israel. it involved 22 pre-service teachers, 10 of them specialising in primary school and the others in junior high school mathematics. research procedure the intervention unit and its stages: the intervention unit was conducted as an annual course dealing with ‘mathematics teaching and assessment’, with 30 weekly sessions each 2 hours long, 15 sessions in each semester. the course focused on observations of the pre-service teachers’s recorded mathematics lessons as well as the investigation and analysis thereof. preparation of the theoretical basis of the course was grounded on two parts: reading about uses of everyday language in mathematics lessons, and reading about a selected topic in mathematics. during the course the pre-service teachers were asked to write a diary describing their feelings, difficulties they encountered throughout the year, dilemmas, ideas and reflections. the aim was to monitor their feelings stemming from their experience with the course and its products. the diaries were qualitatively analysed.the first semester: the first session began with the poem of amichai. the students were divided into small working groups of three or four people. in every subgroup one student had to read the poem aloud (in hebrew). the second step in this working group was to write the poem in their own words, and then to mark all words in the poem which represented mathematics concepts and to explain the meaning of these words in the poem. in the second session the students had to check the meaning of these words in a regular dictionary and in the dictionary of mathematics, and compare them to their explanations and definitions. in the following sessions each pre-service teacher had to choose a mathematical topic from the curriculum, and they were asked to read papers discussing the uses of everyday language in general and in mathematics in particular, for example communications in the language of mathematics (kazemi, 1998; rose, 1989; schiebelhut, 1994; usiskin, 1996). they were also requested to read studies dealing with the teaching of the chosen topic, its historical background, and related misconceptions and common mistakes. the activity during the first semester consisted of consolidating the pre-service teachers in the two topics and exposing them to use and integration of literacy activities in teaching. they were taught to operate a programme which introduced literacy activities into mathematics lessons, namely leading mathematical conversations (verbalisation of mathematical relations), verbalisation of thinking processes in order to identify misconceptions and barriers, writing and reading students’ reflective written records, writing procedure assessment, performing inquiry assignments, developing strategies for understanding mathematical texts and interviewing students. at the end of each session all the pre-service teachers had to indicate all words with a different or double meaning in a list. semester break: the pre-service teachers were required to look for a short story, poem, folk tale or fable which included words in hebrew with a double or triple meaning or one meaning in everyday life and another as a mathematics concept, and to write down the mathematics concepts in the chosen text. then they had to choose two concepts and look for the theoretical background, including the component of the history of these concepts, the place of these concepts in the curriculum, and student mistakes and misconceptions relating to these concepts. they also had to design an intervention which comprised four to five lessons on the chosen topic using and integrating literacy activities. second semester: the pre-service teachers had to practice implementation of the programme in their own classes with children. the weekly sessions consisted of discussions, reflection and analysis of findings, that is executing the programme from theory to practice. ethical considerations permission to conduct the study was granted by the college of education. the pre-service teachers gave their consent to participate in the study. the aims and objectives of the study were discussed with them. the names of the pre-service teachers were not divulged. the diaries were stored away safely by the author. findings (back to top) additional activities and materials one of the course objectives was to expose pre-service teachers to the linguistic approach to mathematics teaching (timor & patkin, 2010). this was achieved by using literacy activities designed to foster mathematics language, enhancing the pre-service teachers’ awareness of words with one meaning in everyday language and another in mathematics lessons. one of the main products was a bank of words in hebrew with one meaning in everyday language and another in mathematics language, together with their definitions.throughout the course the pre-service teachers gathered an extensive collection of words which have one meaning in daily life and another in mathematics language. table 1 shows examples from this ‘double-meaning words bank’ of the pre-service teachers, representing words with double meaning in each branch of mathematics: arithmetic, algebra, geometry, solid geometry, statistics, differential and integral calculus, et cetera. their exact definitions were taken from chambers dictionary (magnusson, 1993) for everyday usage and the penguin dictionary of mathematics (nelson, 1998) for the mathematical definitions. it is important to mention that amongst these words are some which have a double meaning in mathematics and a double or triple meaning in ordinary language (e.g. ‘base’). table 1: examples from the ‘double-meaning words bank’ of the pre-service teachers.† using the double-meaning words from the ‘bank’, discussions were conducted about the ways these words increase the need for accurate definitions of terms as well as the ability to distinguish between the different meanings and definitions which these terms sometimes have. the pre-service teachers and students in their classes reached precise definitions of concepts. for example, one of the pre-service teachers developed an intervention unit with the concept ‘height’, which has a different meaning in mathematics language (the height of a triangle, etc.) and in daily life. for example, in hebrew when somebody asks what the height of the building is, he means its length. conversation about this word led to inquiry and the conclusion that height exists in triangles (there are three heights), but not in every polygon. all of this can be used as a tool for reducing mistakes in mathematicsanother example which led to a very interesting activity was the word ‘division’ (e.g. when eight candy bars have to be divided between four children). in daily life the division must not necessarily be into four equal parts; there might even be an extreme situation where one child does not get any candy whereas another receives all eight bars. conversely, in mathematics, if we divide eight candy bars between four children the tacit assumption is that the division is into four equal parts. activity of this type is similar to that presented in the study of millet, patkin and ezer (2002), which focused on a literacy activity in mathematics on the basis of lafontaine’s fable, described earlier. another outcome was a bank of fables, poems and other types of literature which can be used in intervention units or sessions for the interplay of everyday language and mathematics language, such as fables by aesop and lafontaine, stories from the bible, et cetera. findings obtained from diary analysis the diaries were completed in free writing during the intervention unit (two semesters, including semester break). the pre-service teachers willingly wrote in the diaries and were pleased to share their feelings. twenty of the 22 participating pre-service teachers actively wrote in the diaries, indicating mathematical terms which they found difficult, experiences during the intervention unit, and words with various meanings which were new to them. consequently they concluded that perhaps the different meanings of the words of which they had not been aware before constituted a reason for misunderstanding and misconceptions of these terms. analysis of the diaries revealed four key indices, namely pleasure, comprehension of the importance of knowledge about double meanings, a positive attitude towards the continuation of teaching in such a way, and a sense of self-confidence. the following excerpts from the diaries provide examples of each index: pleasure i am very satisfied. they (the students in my own class) have never believed that they will have to tell and explain words. they were accustomed to solving problems and that was all. talking about reflective writing in mathematics was non-existent because it might have wasted precious time in the lesson ... but now i have realised that with some imagination one can plan many lessons filled with surprises which are growing from the students’ natural language, helping them to express their ideas. one of my students told me that: ‘i knew, but now i also understand’. [pre-service teacher r] comprehension by means of the activity which transcended mathematics, i reached additional areas with the students. the studied mathematical subject was height of triangles. one of the students asked what the height of the eiffel tower is and a second student answered that in order to measure this we must know which measurement units to use and must make an estimate. on the other hand, when we dealt with the concept of height of a triangle from the mathematical point of view, we examined the options of the place of height in all triangle types. the activity enhanced the difference in the interpretation of the word ’height’. [pre-service teacher r] positive attitude towards the continuation of teaching in such a way i felt that my teaching level had changed extremely, both from the point of view of using the means concerning the students and from my personal feeling, because i helped the students to acquire tools for coping with general knowledge and not merely with specific knowledge. [pre-service teacher l]i have undergone a change and i think that i am beginning to understand what is going on. the discussions and talks conducted during the entire academic year constituted part of my professionalisation process ... until several days ago it was difficult for me to internalise and understand what was transpiring in my class. however, now when i am writing i actually think that one of the problems in my class stemmed from the fact that the students were not exposed to words with different meaning in different languages like the term ‘to reduce’. in hebrew as in english, to reduce means making it smaller, to reduce weight etc. … in mathematics, to reduce means to write equal fractions in other way. [pre-service teacher m] sense of self-confidence … the activity in which we engaged during the course gave me a lot of power. power to understand that we can make a change ... acquaintance with mathematics as a language is a gift for life … we have learnt to try everything, as new as it might be, and particularly to continue and persevere with the change, not to be afraid of it. thank you for a real change in the way of teaching objects in mathematics. it has displayed and will continue displaying its fruits in the field. [pre-service teacher d]i indicated that all the class students as well myself sensed a change following the talks about words with double meaning and the activities. [pre-service teacher e] … asking the students to bring examples of words with different meanings and explain them as part of the integration of literacy activities in lessons throughout the course, made me sense that the concept which was taught became clear and comprehensible. [pre-service teacher y] to summarise, the findings indicated progress in terms of comprehension and ability to define words and terms correctly. moreover, the pre-service teachers indicated a sense of pleasure resulting from progress achieved in the course as well as enhanced motivation to teach in this way. discussion and conclusions (back to top) this study describes the interplay of language and mathematics amongst pre-service teachers in elementary and junior high school. an examination of the literature supports the existence of four language components in mathematics learning (usiskin, 1996; uso-juan & martinez-flor, 2006). this study involved a group of pre-service mathematics teachers who were exposed to the new attitude in mathematical pedagogy, which emphasises mathematical discourse and communication and views mathematics as a language requiring use of oral and written components (timor & patkin, 2010). the intervention unit exposed the pre-service teachers to numerous words with double or triple meanings which were ‘false friends’. when teaching a mathematical subject comprising words which have different meanings in everyday use and in mathematics, they discussed these words with their students. this process made them more attentive and sensitive to the possibility that double or triple meanings caused problems in the comprehension of their students. throughout the entire year whilst teaching in their classes they searched for words with different or confusing meanings, adding words from their own personal knowledge. the search for this word lexicon was performed by means of all the literacy activities described above. the students also directly addressed the students, asking them to search by themselves for short stories, poems, et cetera, to present them orally or in writing and to emphasise words with one or more meaning in everyday life and another in mathematics language. integrating literacy activities into mathematics by raising awareness of the various meanings that words can have can, as reported in the diaries, eliminate mistakes (nctm, 2000; patkin & gazit, 2011). speaking plays an important part in learning. it requires students to organise their thoughts and to focus them. mathematical speech, which is a combination of everyday language and mathematics language, is manifested by reflective mathematical talks (usiskin, 1996). as one of the pre-service teachers reported in his diary: the talks about the different meanings helped me to be more accurate when making definitions and it influenced my way of teaching. it helped my students construct their mathematical comprehension and apply in practice their theoretical studies … [pre-service teacher l] discussing questions like ‘is it correct to refer to the wooden blocks from which kindergarten children build towers as “cubes”? ’ can demonstrate to the learners the risks embodied in failing to be meticulous in the application of words. there is no problem calling this section of the kindergarten the ‘cubes corner’ or the ‘solid figures corner’. one might say that we should not teach young children (up to age three years) the mathematical vocabulary because it is too difficult for them. however, just as young children learn to differentiate between different colours and the names of many animals, it is important to emphasise accuracy in mathematics terminology, otherwise they develop misconceptions and mistakes with these concepts. we should take advantage of the opportunity to actively construct mathematical knowledge through mathematical talk, by means of which we identify difficulties, compare solutions, ask leading questions and present assumptions for discussion and thought (siegler, 2006). encouraging students to make use of verbalisation, writing reflective records, writing procedures and building stories which include mathematical problems creates conditions for the learning of mathematics whilst participating in social class processes. during the course the pre-service teachers attested that the literacy activities in mathematics lessons not only enriched their personal knowledge but also extended their mathematical thinking. using everyday language as scaffolds assisted them to teach the ‘second’ language – mathematics language. the course outcomes reinforced and enhanced the need for teaching of this kind. for example, the poem of amichai (1975) ‘crossed’ the border and made use of mathematical terms, assuming that readers are familiar with these. the poem describes addition, subtraction and multiplication; it discusses fractions, two numbers, fraction line and common denominator. in the poet’s mind these terms are not merely mathematical terms but rather words with a double meaning. to summarise, it is possible and recommended that learners be encouraged to move back and forth from one language − natural, everyday language − to a second language, mathematics language. thus we can promote mathematical thinking and comprehension and, as a result, improve students’ attainments in this discipline. it is recommended that research of wider scope (including quantitative data) be conducted to validate the findings of this study. acknowledgements (back to top) competing interests the author declares that she has no financial or personal relationships which may have inappropriately influenced her in writing this article. references (back to top) amichai, y. 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(1996). talk your way into writing. in p.c. elliott, & m.j. kenney (eds.), communication in mathematics: k−12 and beyond (pp. 81–88). reston, va: national council of teachers of mathematics. kazemi, e. (1998). discourse that promotes conceptual understanding. teaching children mathematics, 4(7), 410−414. magnusson, m. (ed.). (1993). chambers biographical dictionary. cambridge: the university press. millet, s., patkin, d., & ezer, h. (2002). what is what is between lafontaine’s quarter and the rational number quarter? or what is the difference between math justice and the jungle justice? in a.d. cockburn, & e. nardi (eds.), proceedings of the 26th conference of the international group for the psychology of mathematics education, vol. 1 (p. 362). norwich: school of education and professional development university of east anglia. national council of teachers of mathematics. (2000). principles and standards for school mathematics. reston, va: national council of teachers of mathematics. nelson, d. (ed.). (1998). penguin dictionary of mathematics. london: penguin books. norwood, k.s., & carter, g. (1994). journal writing: an insight into students’ understanding. teaching children mathematics, 1(3), 146−148. patkin, d., & gazit, a. (2011). effect of difference in word formulation and mathematical characteristics of story problems on mathematics pre-service teachers and practicing teachers. international journal of mathematical education in science and technology, 42(1),75−87. http://dx.doi.org/10.1080/0020739x.2010.519790 patkin, d., millet, s., & ezer, h. (2001, april). literacy-in-mathematics as perceived by mathematics educators. paper presented at the 79th annual meeting of the national council of teachers of mathematics, orlando, florida, usa. payne, s. 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(pp. 464−510). new york, ny: john wiley & sons. skemp, r.r. (1972). the psychology of learning mathematics. london: penguin books. timor, t., & patkin, d. (2010). pre-service teachers’ perceptions of mathematics as a language. journal of the korea society of mathematical education series d: research in mathematical education, 14(3), 233−247. usiskin, z. (1996). mathematics as a language. in p.c. elliot, & m.j. kenney (eds.), communication in mathematics: k−12 and beyond (pp. 231–243). reston, va: national council of teachers of mathematics. uso-juan, e., & martinez-flor, a. (2006). current trends in the development and teaching of the four language skills. the hague: mouton de gruyter. http://dx.doi.org/10.1515/9783110197778 webster’s universal college dictionary. (1997). new york, ny: gramercy books. abstract introduction context orientation concept development and learning professional development professional networks threads of development concluding remarks acknowledgements references about the author(s) kenneth ngcoza department of education, rhodes university, grahamstown, south africa sue southwood centre for higher education research, teaching and learning, rhodes university, grahamstown, south africa citation ngcoza, k., & southwood, s. (2019). webs of development: professional networks as spaces for learning. pythagoras, 40(1), a409. https://doi.org/10.4102/pythagoras.v40i1.409 original research webs of development: professional networks as spaces for learning kenneth ngcoza, sue southwood received: 12 dec. 2017; accepted: 31 may 2019; published: 27 june 2019 copyright: © 2019. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract this article is the result of a professional collaboration between two educationists (working originally in the fields of mathematics and science education), who share a passion for exploring collaborative approaches to the professional development of educators. it extends ideas explored in earlier work by focusing on the concept of professional networks as ‘webs of development’, and identifying fundamental ‘threads’ holding a range of different professionals working together in complex spaces of development. the article offers a framework juxtaposing aspects emerging from different research projects, which attempts to hold the complexity around engagement in spaces of development. the framework acknowledges and attempts to capture a sense of the joint responsibility of different professionals involved in the arena of education while conceptual threads of engagement are regarded as weaving through: connectivity, collaboration, negotiation, dialogue and appreciation. such ideas are presented as a stellar framework for potential research in the future, offering an holistic, learning-focused approach to the concept of professional networks as ‘webs of development’. we offer this theoretical article as an invitation for collegial engagement and dialogue – a potential space for learning. keywords: development; learning; professional networks; space; community. introduction all … are caught in an inescapable network of mutuality, tied in a single garment of destiny. …. this is the inter-related structure of reality. (martin luther king jr, 1967, p. 1–2) the concept of the inter-related structure of social reality, made famous over 50 years ago by martin luther king (1967), and scientifically articulated by the likes of capra (1996), focuses on the complexity of life, the underlying connectedness, the systemic nature of our existence. conventional linear thought and mechanistic reductionism necessarily yield to ideas of complexity, viewing the world as a systemic organism. rather than studying parts to understand the whole, understanding of the whole is attempted through analysis of the relationships and connections making up the whole. yet, the way in which we approach life is so often to deny this complexity. for instance, there is a tendency to split life into compartments or boxes, give them labels, and even give those who work in them labels, and then proceed to operate within those boxes, often ignoring and thereby negating the relationships and connections between them (katz & earl, 2010). the discipline-fragmented curriculum in most educational institutions is evidence of this. as breen points out in his article on dilemmas of change, we ‘zoom’ in, ‘fixing’ one part, negating the ‘complexity of the phenomenon’. we deal with ‘the complicated rather than the complex and so only a part and never the whole’ (breen, 2005). in this article, we focus on research conducted in the fields of mathematics and science education (brodie, 2016; chauraya, 2016; ngcoza, 2007; southwood, 2000) around the complex notion of collaborative teacher professional development. supporting our own research, chauraya (2016) emphasises the importance of providing teachers with opportunities for collaborative participation and collectivity, offering opportunities for teachers to grow together (brodie, 2016). a meta-analysis of such research led to the identification of essential aspects or dimensions, fundamental conceptual ‘threads’ regarded as ‘holding’ the complexity of what are referred to here as ‘webs of development’. such ‘professional networks’, are recognised not only as promoting dynamic spaces for learning, but as spaces for building capacity not only within educators, but also between educators and ultimately beyond. context the original research endeavours (ngcoza, 2007; southwood, 2000) on which the ideas in this article are based took place in contexts of the professional development of mathematics and science educators both at primary and senior levels of schooling. the projects share a qualitative methodology, the orientation described as essentially interpretive (cohen, manion, & morrison, 2018), employing narrative inquiry (clandinin & connelly, 1998, 2004) as the fundamental research method. narratives – the ‘medium through which we order and make sense of experiences and events’ – were explored to help develop insights into and understandings of ‘relational spaces of connectivity’ (baerenholdt & simonsen, 2004, p. 10). it is in such spaces that teachers grow through the collective development of knowledge (brodie, 2016). the research conducted by the authors, on which this article is based, involved a wide range of professionals within the mathematics and science education arena. while most of this work focused on the experience of classroom educators, other members of the educational community involved included teacher professional developers, educational managers, for example school principals, and education advisors involved in forming policy as well as practice. experiences of these different members of educational networks, who had been working together for an extended period of time, were explored through narrative inquiry. the narratives of experience were captured, analysed and fed back in generative cycles of inquiry and meaning-making, identifying emerging dimensions of engagement; that is, interpretation was generated through a ‘dynamic ongoing construction of meaning – a process of negotiation’ (southwood, 2000, p. 33). orientation the ideas presented in this article are located within a discourse of social practice informed by the ideology of actor network theory (ant) (callon & latour, 1981; edwards, 2002; latour, 1999; law, 1999; singh-pillay & alant, 2015). ant reflects the interrelated complexity of life and learning, involving a shift of focus from individuals to the collective (chauraya & brodie, 2017). such an approach suggests symbiotic connections among participants. this is critical since educators, their practices and their contexts are regarded here as ‘interrelated’ – inextricably linked – and learning spaces are understood as sociocultural contexts to which participants bring and share diverse cultural and social knowledge (mavuru & ramnarain, 2017). in vygotskian (1978) thinking, the notions of knowledge and learning are regarded as products of sociocultural activities rather than of isolated minds. interactions between participants are regarded as the building blocks of networks. this ideology is based on the notion of spatiality, a concept that combines conditions and practices of individual and social life. the concept of space is regarded here as: a dimension of social relations and imaginations: it is not an objective structure but rather a social experience … a conception conducted by way of people’s social practices in their involvement with the world. it is a social construct … constituted by, as well as constitutive of, social relations and social practices. (baerenholdt & simonsen, 2004, pp. 1–2) it is a notion of space that is not absolute but rather relational, focusing on the idea of connectivity (brodie, 2016). important threads of this re-imagination include the notion of spatial flows and an intellectual context where space is frequently being imagined as a product of networks and relations (jones, 2005). we are concerned here with a three-dimensional approach to space – space within, space between and space beyond. the space ‘within’ is about the space ‘inside’ participants. the space ‘between’ is about the space of interaction between participants and the space ‘beyond’ is that space which is neither within nor between, but space that is yet to be defined. of course, these spaces are inextricably interwoven, but contrary to denying the complexity, this model is designed as a framework to hold the complexity, attempting to capture the experience in layers of depth and richness. concept according to imenda (2014, p. 188), concepts reflect ‘theoretical concerns and ideological conflicts’. hence, this research focused on the concept of professional networks as spaces for development in education. in their seminal work, chauraya and brodie (2018) refer to such professional networks as professional learning communities central to which are collegial conversations to foster learning. in attempting not only to recognise but to celebrate complexity, we use the imagery of webs, to illustrate the ideas presented. we begin by articulating ideas pertaining to development and learning with specific reference to the context of professional networks. then we describe our ideological orientation by identifying the main structural spaces and threads of the web, acknowledging that these do not stand in isolation but are intricately enmeshed with each other. we offer the image of the web as a model for conceptualising spaces of educator development (nel & luneta, 2017), and a framework highlighting conceptual threads that might be more consciously woven in to support future research. development and learning development is not simply change but a rather a complex, dynamic and dialectical process of qualitative change (verosov, 2014; vygotsky, 1978, p. 189). verosov (2014, p. 132) highlights that the term ‘complexity’ entails ‘the qualitative re-organisation of a certain system which includes several essential aspects’. inspired by work on the evolution of professional learning communities (brodie, 2016; chauraya, 2016; chauraya & brodie, 2018), the concept of development is defined here in a broad and holistic way as ‘a process of interacting, understanding, growing together; of rebuilding and reconceptualising hope and trust; and of coping with, experimenting and contributing to meaningful change’ (jain, 1997, p. 5). in order to develop we need to learn. learning is vital, an essential process in life without which we stagnate or regress. if we are not learning, we are not moving, we are not growing, and we are not developing. the concept of learning is regarded here as a complex process, a process of overlapping, interlocking, juxtaposed layers of understanding, that deepens, widens and enriches our understanding of the world, leading to qualitative change (verosov, 2014). learning is viewed as ‘an ongoing, interactive and mutually enhancing process of questioning, discovering, reflecting, sharing and inventing what it means to be human, both individually and collectively’ (jain, 1997, p. 5). it is non-linear, non-homogenous, neither spatialised nor distributed. it is about growth, not deficiency. it is about celebration, not demoralisation. it is a process of engagement, not transmission (kennedy, 2005). the concept of learning drawn on here moves beyond the constructivist focus on individual cognition, encompassing vygotskian (1978) ideas around socioculturism, locating it in the spaces both within and between people. while we are concerned with the concept of learning as a life-long process, it is not viewed here as a linear, continuous process, a movement from one point to another along a continuum of development. rather, we are exploring a way of looking that reflects the complexity of development, seeing it as an iterative process. in so doing, we are keen to work with a concept used in the wider field of development – the idea of ‘contiguum’ – implying ‘complementarity within a context’ rather than ‘separation in time’ (pirotte, husson, & grunewald, 1999, p. 45). thus, in this article the idea of learning as a contiguous process underpins the notion of collaborative professional development. professional development the concept of development is situated here in terms of educators (in this case mathematics and science) as professionals: ‘a … process of professional growth and fulfilment, resulting in an improved quality of educational understanding and practice’ (southwood, 2000, p. 19). building on ideas of learning organisations (senge, 1990), communities of practice (lave & wenger, 1991) and professional learning communities (brodie, 2013, 2016; chauraya & brodie, 2018; tam, 2015), the process of professional development (loucks-horsley, love, stiles, mundry, & hewson, 2003) is regarded as a cultural practice engaged in by participants of various communities, a collaborative praxis in which participants are involved in negotiating meaning and developing improved understandings in professional communities of learning. implicit in this approach is the active involvement of educators in an ongoing process of professional learning and growth, as reiterated by brodie (2016) and chauraya (2016) in their studies on professional learning communities of mathematics teachers. it is a concept of development that is about educators, for educators, by educators, with educators. educators are regarded here as change agents whose role is to implement the curriculum in the classrooms (hameed, 2013). hence, there is a need to optimise teacher learning, perceived here as a long-term collaborative process (chauraya & brodie, 2018; ono & ferriera, 2010). the process of professional development through professional networks advocated here is a collaborative one, where educators ‘negotiate’ their own development as well as that of others (southwood, 2000). professional networks the basic premise of this article rests on the potential strength that can be gained from building, and building on, constructive relationships within, between and beyond educators. professional networks, or networked learning communities (katz & earl, 2010), are regarded here as webs of interaction, appropriate environments for enabling and enacting processes of collaborative professional development. these comprise individuals who come together from different environments to engage in development activity informed by their own knowledge base and experience, co-constructing new knowledge together, learning with, from and for others. such environments aim to foster co-engagement and interaction in sociocultural negotiation of meanings in constructive and reciprocal ways (ngcoza, 2007) and have, we believe, the potential to not only be spaces of interaction, but webs of development. webs of development building on the insights and understandings gained from our own and others’ research (e.g. hameed, 2013; mukedzi, 2013; ngcoza & southwood, 2015; singh-pillay & alant, 2015), and incorporating thinking from the wider development arena (e.g. lewis, 2001; pirotte et al. 1999), we have developed a conceptual model (imenda, 2014) for researching the concept of professional networks of learning. to imenda (2014, p. 189), a conceptual model or framework entails combining a number of related concepts to explain an event or phenomenon. in the context of our article, the intention was to identify dimensions and processes of engagement within ‘webs of development’. while the concept of ‘webs’ has the unfortunate potential to conjure up imaginings of ‘entrapment’ (illich, 1971), it is intended to evoke and encompass the complex, contiguous and mutually supportive nature of the relationships involved without the connotations of technicality suggested by the notion of ‘network’. similarly to katz and earl’s (2010) networked learning communities, ‘webs of development’ necessarily imply the involvement of a learning community, in which different degrees and areas of experience and expertise are valued, shared and built on. a space where all are learners, all have something to teach and contribute, and where all have a common interest that forms the foundation of their work – learning. they share a basic desire to learn and apply their learning in the development of education praxis in their own contexts. they learn together. it is not just about learning from but about learning with, supporting not only one’s own development but that of colleagues too. by recognising needs, identifying successes and building on these, it is possible to move forward in developing more effective spaces for learning, that is, communities of learning and for learning. in such spaces, the heterogeneity of participants is recognised and responsibility for both their own and other educators’ professional development is realised. further, emphasis is placed on the importance of developing contexts of mutual support that foster co-engagement, co-learning and co-ownership (ngcoza, 2007). threads of development a meta-analysis of the earlier research (ngcoza, 2007; southwood, 2000) led to the identification of five fundamental threads of engagement, holding the spaces and fuelling the pathways in ‘webs’ of development. engagement in this context is defined as active participation (katz & earl, 2010; sedlacek & sedova, 2017), interaction and collaboration with members of the community (brodie, 2016; chauraya & brodie, 2018). essentially, it is the nature of engagement within professional learning networks that is of interest to us. the threads that emerged through the analysis included: connectivity, collaboration, dialogue, negotiation and appreciation. these threads are represented in figure 1. each of these is touched on here but will be focused on more strongly in future research. figure 1: web of conceptual threads. the construct of connectivity is identified as a strong conceptual thread, pertaining to the ‘stuff that holds the network together’. within a discourse of sociocultural learning (mavuru & ramnarain, 2017; vygotsky, 1978), and as reiterated by brodie (2016) and her colleagues, there can be no learning and no development without connectivity. the concept of connectivity is thus employed here in the sense of both intellectual and emotional connection – the sharing of interests and concerns that motivate the coming together and engagement of professionals with a view to learning. within the ant orientation, it is recognised that such ‘networks’ are essentially transient, existing in a process of constant making and re-making (callon, 1986). it is also assumed that such relational networks are not necessarily coherent and may exhibit conflict. it is, we believe, through the exploration of relations between participants that we can gain greater understanding of the notion of webs of professional development. the notion of collaboration is recognised as being fundamental to mutually supportive spaces of professional development as suggested here. while much work has been done in the area of collaboration and teacher development (christiansen, goulet, kretz, & maers, 1997; fullan, 1985; musanti & pence, 2010; stoll, 1992) there has been much debate around its meaning (hargreaves, 1994). while it is often used synonymously with the term cooperation, in this case the latter is regarded as necessary to, but is not regarded as equating to, collaboration. while cooperation is necessary for collaboration to occur, not all cooperation is necessarily of a collaborative nature. collaboration is understood here as a dynamic engagement in mutually desired activity. it is the notion of mutuality that fundamentally distinguishes collaboration from cooperation. ‘cooperation alone may serve to perpetuate the status quo, while collaboration, by its inherent dynamic, is more likely to challenge it’ (southwood, 2000, p. 30). while the concept of collaboration is regarded as important, it is not enough. building on the fundamentally interrelated nature of the work of educators, the notion of negotiation is proposed rather as the basis for professional co-engagement: ‘negotiation necessarily implies a dialogical relationship: the intersubjectivity of person/s and context, action and meaning, knowledge and experience’ (southwood, 2000, p. 83). negotiation is recognised as a fundamental thread of the web, composite threads including negotiation of meaning and the negotiation of people and context, linked inextricably to negotiation of power (southwood, 2000). negotiation can be explored in terms of how actors involve themselves and are involved in the network, and how understandings and relationships are developed. negotiation is recognised as a fundamental process within a complex and contiguously evolving dynamic: negotiation takes place in a space of dialogue. dialogue is imagined here as encompassing the interaction of mind and ideas as well as words … transcending the boundaries of time, space and culture. … [it] entails imagination, empathy and making of connections. (alexander, 2005:2) informed by frierian ideas of ‘dialogic education’ (arnett, 1992; freire, 1970) dialogue is recognised as offering the potential for promoting free and critical learning, and encouraging epistemological curiosity. by opening up spaces of dialogue, ways of thinking may be disrupted and re-negotiated, ideas can be deliberated and understandings can be developed. through dialogue, prejudices may be faced and engaged with, meaning sought, criticality developed, feelings and thoughts identified, positions recognised and a language of critique and possibility can be developed (southwood, 2012). within such a dialogical approach, traditional monological relationships between the ‘knowing’ expert imparting knowledge and wisdom to the ‘unknowing’ learner are challenged by participatory notions of teaching and learning – ‘a partnership in which all parties take active responsibility’ – ‘partners in the pursuit of higher learning. … it is through dialogue that understandings and practices are re-imagined and re-negotiated’ (southwood, 2012, pp. 91–92). the developmental orientation underpinning the work outlined here reflects an essentially positive approach to development, appreciation, an approach that focuses on growth not deficiency, celebration not demoralisation. it builds on strengths rather than weaknesses, opportunities rather than gaps, and challenges rather than problems. it looks at where we are now and where we want to get to, and promotes development of the space between. an appreciative mode of engagement is adopted in the belief that such a positive and constructive approach to development is powerful in ‘locating the energy for change’ (elliott, 1999). building on the field of appreciative inquiry (cooperrider & whitney, 1999), the development processes imagined here build on what is effective towards that which is desired. such processes are recognised as being iterative and contiguous rather than linear and continuous. by developing further insights around the ways in which the different threads play out in these contexts, we can look at how future webs of development can potentially be fuelled. we need to ask such questions as what constitutes constructive engagement in professional networks – what does it look like and what motivates it? meta-analysis of the earlier research projects has resulted in the development of a simple conceptual model which may be useful in framing future research. the framework outlined here offers a conceptual space for further exploration. the visual model offered in figure 2 is developed from that offered by ngcoza (2007), and is presented here as a possible way of conceptualising the notion of professional networks – and, alluded to earlier, emphasising connections within, between and beyond communities of practice (lave & wenger, 1991). the stellar network offers a symbolic model articulating the potential interconnection of different possible professional learning (sub-) communities – communities of practice. the triangles can be seen to hold the space within each educational (sub-) community; the pentagon can be viewed as holding the space between the (sub-) communities, while the circle can be seen to hold the spaces beyond. figure 2: web of potential interaction. while the model may appear rigid and somewhat deterministic, it needs to be regarded symbolically. it could be viewed more topologically, so as to offer a greater feeling of the movement and potential relational flow of engagement. while the triangles at the points of the star hold the spaces of the sub-communities, the central pentagon represents the overlap between all the different components of the web. the outlines of the circle, the star and the pentagon are perforated to indicate potential flows of engagement within, between and beyond communities. the illustration (figure 2) represents conceptual threads identified as holding the space together. it is important to note here that these are not considered to be necessarily the only threads holding together such spaces but are those that emerged out of the research drawn on here. the two webs (figure 1 and figure 2) represent overlapping, interrelating and interweaving ideas. this web may be juxtaposed with the first, its orientation irrelevant. together the two stellar frames depict the notion of ‘webs of development’, the ‘threads’ regarded as a conceptual mesh underpinning the potential for development. concluding remarks this article has attempted to capture a conceptual space, a space of thinking, in words and images. the task is a formidable one and the concern is that by attempting to countenance all the complexity, we may just get lost in it! in line with the holistic, relational orientation of the approach and the web of ideas offered, we have drawn on a range of traditional ‘disciplines’ including mathematics, physics, geography, economics and politics. the fundamental threads identified here cut across all of these. the challenge we have attempted to face here is that of confronting the complexity without dismantling it, and to identify and work with the fundamental threads running through it, the mesh that holds it together. in attempting not to lose sight of ourselves, we have attempted to capture the essence; that is, we have construed learning and development as fundamental to what we as educators are about. we have acknowledged, celebrated and worked with the complexity of professional networks as webs of development. we have explored the development of spaces for learning, spaces where educators are active participants in the co-construction of knowledge (ngcoza & southwood, 2015), connecting and developing through collaborative processes of learning, characterised by dialogue, not by imposition but through negotiation. through processes characterised by appreciation, success is acknowledged and built on, provoking development from where we are to where we want to go – ‘the space of the possible’ (cohen & stewart, 1994). acknowledgements we would like to thank the many professionals with whom we have worked and would like to celebrate their passion, energy and desire to improve spaces of teaching and learning. competing interests the authors declare that they have no financial or personal relationships that may have inappropriately influenced them in writing this article. authors’ contributions since this article is based on the authors’ doctoral studies in mathematics and science education, which were conducted at the same institution, both authors (k.n. and s.s.) have made a significant contribution in terms of the research drawn on as well as in conceptualising and writing the manuscript. ethical considerations this article followed all ethical standards for a research without direct contact with human or animal subjects. funding this article is based on some of the research supported in part by a bursary from the national research foundation. data availability statement data sharing is not applicable to this article as no new data were created or analysed in this study. disclaimer the views and opinions expressed in this article are those of the authors and do not necessarily reflect the official policy or position of any affiliated agency of the authors. references alexander, r.j. 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(1978). mind in society. cambridge, ma: harvard university press. 66 p52-63 austin & webb final 52 pythagoras 66, december, 2007, pp. 52-63 developing inquiry-based teaching and learning in family maths programme facilitators pam austin and paul webb faculty of education, nelson mandela metropolitan university email: pamela.austin@nmmu.ac.za and paul.webb@nmmu.ac.za the inquiry-based family maths professional development programme, offered by the nelson mandela metropolitan university, attempts not only to support the transformative education practices targeted by the south african national department of education, but also to extend them beyond the school walls to the community at large. this study investigates the extent to which this programme develops facilitators’ ability to implement inquiry-based learning. the research undertaken uses both qualitative and quantitative methods in an empirical study of 39 facilitators. the facilitators’ inquiry beliefs and ability to implement inquiry learning was measured by means of questionnaires, observation schedules and interviews. data generated by the study reveal that both the facilitators’ understanding and practice of inquiry improved as they progressed through the novice, intermediate and veteran categories of the family maths professional development programme. introduction the family maths programme was conceptualised and designed at the university of california, berkeley during the late 1970s (kreinberg, 1989) and has been adopted by various universities, nongovernmental and governmental organisations in a number of countries around the world. examples of these countries include canada, australia, new zealand, sweden, costa rica, and puerto rico (thompson, 2005). in south africa, the family maths programme has been offered by the cooperative organisation for the upgrading of numeracy training (a johannesburg-based ngo), the university of the free state and the nelson mandela metropolitan university. the overall aims of the intervention programme are to redress inequalities in the schooling system, to dispel negativity towards mathematics, to make school mathematics relevant to learners in their everyday lives, and to promote an inquiry-based approach to teaching and learning (damerow, dunkley, nevres & werry, 1984; thompson & mayfield-ingram, 1998). since the new south african dispensation of 1994, recently revised national curriculum statements have aimed at transforming heavily entrenched, traditional approaches and replacing them with a new vision for education based on the introduction of outcomes-based education. these practices are to a large extent underpinned by constructivist and inquiry-based philosophies of teaching and learning (moll, 1994). however, despite the fact that the facilitation of inquiry learning is a core methodology promoted in south african revised national curriculum statements (department of education, 2002), research suggests that traditional teacher-centred practices and rote memorisation of algorithms remain common practice in many mathematics classrooms (taylor & vinjevold, 1999). the family maths programme, as offered by the nelson mandela metropolitan university in south africa, not only attempts to support the transformative education practices targeted by the department of education, but also to extend them beyond the school walls to the community at large. this is done by offering a creative education practice that provides opportunities for teachers, parents, learners and community members to solve problems through discussion and the use of handson, minds-on, process oriented, inquiry-based activities in a relaxed, non-threatening environment. this empirical study was undertaken in an attempt to determine the extent to which this programme is able to develop inquiry learning practices and skills in family maths facilitators. background the international family maths programme attempts to eliminate much of the pressure, anxiety and fear of failure experienced by both parents and their children, to secure parental and community involvement in learners’ education, and dispel the misperception that school mathematics is unrelated to a child’s everyday experience (thompson & mayfield-ingram, 1998). the programme adopts an inquiry learning approach which supports interactions amongst learners that focus on pam austin and paul webb 53 problem-solving (llewellyn, 2005), and which encourages learners from diverse backgrounds to participate fully in the learning process (national science education standards, 1996). the programme in general also aims to develop the language necessary for meaningful communication in mathematics, develop problem-solving skills and increase confidence and enjoyment of the subject (kreinberg, 1989). although all family maths programmes throughout the world adhere to the basic principles described above, adaptations are made from time to time to accommodate local contexts, both in emphasis and execution. in the case of the family maths programme at the nelson mandela metropolitan university, there are two contextual demands. firstly, it must respond to the issue of second-language teachers, learners and parents, and secondly, the university’s ability to respond to requests from teachers, principals, departmental official and donors to offer the programme more widely is limited by staff capacity. the response to the issue of limited staff capacity has been to ‘train’ family maths facilitators in both rural and urban sites in the eastern cape (east london) and western cape (george and beaufort west) in south africa. the professional development course gives facilitators the opportunity to engage in inquiry-based experiences and to develop higher order thinking skills as they ask questions, conduct problemsolving activities, and interpret and discover solutions while constructing mathematical understanding. this study investigates the effect of the family maths approach on the ability of facilitators (teachers and teacher educators) to facilitate inquiry-based teaching and learning as they progress through the two-year professional development programme. in terms of the second-language issue, a ‘homelanguage’ approach for group discussions is promoted wherever possible in order to create opportunities for meaningful learner-centred discourse. this approach is considered important, not only because the generic family maths programme aims at developing the language necessary for meaningful communication in mathematics, but also because the promotion of classroom discussion has been shown to have a profound effect on children’s cognitive development (wegrif, mercer & dawes, 1999; webb & treagust, 2006). as such, the promotion of social discourse is central to the modelling process adopted by the family maths professional development programme for facilitators at the nelson mandela metropolitan university. the research takes place against the backdrop of national concerns regarding poor achievement and negative attitudes towards mathematics (asmal, 2000; reddy, 2006). it takes into account the fact that a large number of under-qualified primary and secondary school teachers do not have the knowledge and skills to teach the subject competently (asmal, 2000; taylor & vinjevold, 1999), which is exacerbated by teaching and learning that takes place in a second language, in under-resourced classrooms (taylor & vinjevold, 1999). it also takes cognisance of research findings that teachers in south africa appear not to communicate attitudes of curiosity, respect for evidence, or critical reflection – qualities that are necessary for the development of higher-order cognitive skills (enslin, 1990; webb & treagust, 2006). there has also been extensive research in recent years on the relationships between the beliefs of mathematics teachers and their actual practice in the classroom (brodie, 2001; ensor, 1998; ernest, 1989, 1991; hoyles, 1992; lerman, 1986, 2002; pehkonen & törner, 2004; thompson, 1992; skott, 2001a, 2001b, 2004; speer, 2005; wilson & cooney, 2002). most of these studies focus on the correlation or disparity that researchers have identified between what teachers believe they should be doing in their classrooms (their ‘espoused beliefs’ or ‘professed beliefs’) and what researchers infer, based on observational and other data (teachers’ practice or ‘attributed beliefs’). we also investigate the espoused and attributed beliefs of the participants in terms of inquiry-based teaching and learning. in the light of the above, the aim of this study is to make a contribution to the many debates regarding the promotion of inquiry-based mathematics approaches in the south african context, and to contribute to the thinking of those who are grappling with, and attempting to rectify, the concerns noted above. design the research undertaken was an empirical study of 39 facilitators participating in the family maths professional development programme. they were assigned to three categories: novice (less than one year’s participation), intermediate (one to two years’ participation) and veteran (more than two years’ participation). the participants were predominantly in-service teachers and teacher educators in the departments of education in the developing inquiry-based teaching and learning in family maths programme facilitators 54 western and eastern cape. the findings were triangulated by comparing the data generated by both qualitative (interview data) and quantitative (questionnaire and observation) instruments. firstly, a facilitators’ ‘inquiry learning belief system questionnaire’ was used to measure the participants’ inquiry beliefs and understandings of aspects of the inquiry process. the questionnaire focused on the first three steps of the inquiry process, namely, engaging the participants, allowing them to explore the concept, and encouraging them to explain mathematical concepts and terms. their ability to implement inquiry learning was measured by using observation schedules to observe and record their inquiry skills while they conducted family maths workshops for teachers, learners and parents. one research instrument is based on the ‘workshop interaction coding system observation instrument’ which measures the extent to which the facilitators use inquiry verbal feedback techniques during interactions with participants (brophy & good, 1970). the ‘workshop observation instrument’ was used to measure the facilitators’ ability to capture and focus participants’ attention on critical parts of the problem-solving process. attempts were made to minimise the participants’ perceptions of what was being measured, because knowledge of what was ‘expected’ might change their behaviour. the contents of the observation instruments were therefore not revealed, in order to create as authentic a setting as possible. semi-structured interviews with facilitators, using standardised, open-ended questions, provided opportunities to use probing questions to obtain clarity and additional information from the interviewees’ responses. this yielded deeper insights into their beliefs and perspectives on inquiry-based learning, their understanding of mathematical problems, and their perceptions of their own inquiry-learning facilitation skills. quantitative statistical data were generated from the facilitators’ inquiry learning belief system questionnaire (n=88) and the workshop observation instrument (n=39). these data were analysed and subjected to analysis of variance (anova) techniques to provide descriptive and inferential statistics. comparisons were made between statistically significant mean facilitator scores across the three categories of facilitators in each of the three stages of the inquiry process. all three measures of statistical significance – where p≤.0.01 (highly significant), p≤.0.0 (significant) and p≤.0.1 (weakly significant) – were used for clarity. however, all differences at p≤.0.1 are considered as ‘statistically significant in this report. quantitative data were generated from the workshop interaction coding system instrument (n=39) and qualitative data were generated through the semi-structured interview schedules (n=39). these data were analysed and classified according to broad categories to provide descriptive and inferential statistics. the data generated by the three groups (novice, intermediate and veteran) were compared and subjected to statistical analyses wherever possible. results data analysis of the belief questionnaire, workshop observations and semi-structured interviews with facilitators indicates that their inquiry beliefs and practices improved over time as they progressed from the novice to veteran category. there is also clear evidence that the majority of facilitators who first embark on the family maths programme appear to have very little knowledge or experience criteria group mean scores novice intermediate veteran encourages initiative (1b) 3.45 3.70 3.88 encourages questioning (2c) 3.18 3.58 3.50 poses contradictions (2e) 3.05 3.63 3.45 allows time after questions (2f) 3.30 3.60 3.50 seeks elaboration (3a) 3.14 3.35 3.63 participants refine explanations (3d) 3.12 3.65 3.63 table 1. comparison of respondents’ mean scores with regard to questions on the belief system questionnaire. pam austin and paul webb 55 in terms of inquiry learning as a teaching and learning strategy, despite this being a strategy recommended for all teachers by the south african government (department of education, 2002). it is also clear that in all categories of comparison (veteran, intermediate or novice), the facilitators’ inquiry workshop implementation strategies did not always support their stated belief system regarding inquiry learning. belief as measured by questionnaires analysis of the mean scores of the data generated by the four-point scale beliefs questionnaire reveals a statistically significant difference between the three groups of facilitators in terms of engaging participants in problem situations, allowing participants to explore the concept, and encouraging participants to explain the concepts and define mathematical terms. the statistically significant mean scores of the novice, intermediate and veteran groups are indicated in table 1. the statistical significance (p values from anova) of the differences between the groups is shown in table 2 it is evident that there is a statistically significant difference in the mean value in each case between the novice group of facilitators and one or both of the other two groups of facilitators who have been participating on the programme for a longer period of time. these data suggest that the facilitators’ understandings (beliefs) of inquiry learning improved – in that they became a better match of what is expected – as they progressed from the novice through intermediate to veteran categories of the family maths programme. in other words, the novice group of facilitators held an inquiry belief system which was the least developed in terms of the criteria which are indicators of teachers who will probably promote inquiry learning the most effectively. in all cases the veteran and intermediate groups of facilitators criteria probability values novice intermediate veteran encourages initiative (1b) novice 0.0595* 0.0286** intermediate 0.0595* 0. 4126 encourages questioning (2c) novice 0.0198** 0.1871 intermediate 0.0198** 0.7678 poses contradictions (2e) novice 0.0207** 0.0215** intermediate 0.0207** 0.5182 allows time after questions (2f) novice 0.0295** 0.3141 intermediate 0.0295** 0.6498 seeks elaboration (3a) novice 0.0938* 0.0090*** intermediate 0.0938* 0.1767 participants refine explanations (3d) novice 0.0002*** 0.0122** intermediate 0.0002*** 0.9091 * = statistically significant at the 90% level of confidence ** = statistically significant at the 95% level of confidence *** = statistically significant at the 99% level of confidence table 2. comparison of probability values indicating statistically significant differences between facilitator mean group scores for criteria on the belief system questionnaire for inquiry learning. developing inquiry-based teaching and learning in family maths programme facilitators 56 were more advanced in their understanding of what best promoted inquiry learning. teacher practice as measured by observations the data generated by means of the workshop observation schedule were analysed statistically to provide descriptive and inferential statistics. the mean scores for each criterion were calculated and comparison of the mean scores of the three facilitator categories suggested a progression in both understanding and implementation of inquiry learning strategies as they proceeded through the two-year family maths facilitator professional development programme. brooks and brooks (1993) outline a five stage instructional model for assessing inquiry teaching according to the goals of the national science education standards. each stage comprises characteristics of teaching according to inquirybased education. the workshop observation instrument used in this study was adapted from the first three stages of this model to the learning cycle approach to instruction which is outlined in science for life and living by layman (1996). as noted above, the workshop observation instrument assessed the first three consecutive steps of the five-stage model in relation to the inquiry process, namely, the extent to which the facilitator promoted inquiry learning by engaging the participants (step 1), allowing participants to explore the concept (step 2), and encouraging participants to explain the concept and define the terms (step 3). the facilitators’ competence in each of these steps was rated on a scale of one to four. the following coding representations were used, namely, (1) poor or no implementation of inquiry criteria; (2) limited implementation of inquiry criteria; (3) satisfactory implementation of inquiry criteria; and (4) implementation of inquiry criteria that exceeds expectations. the mean scores for each criterion in each of the three inquiry learning steps were calculated and are shown in table 3 (step 1: engages the participants), table 4 (step 2: allows participants to explore) and table 5 (step 3: encourages participants to explain the concept). mean scores and levels of statistically significant differences are indicated in tables 6 and 7 respectively. no statistically significant differences were recorded for a number of criteria, indicating that levels of competence observed during the workshops were similar for all facilitators. however, comparison of the mean scores still suggests that progression is made as facilitators proceed from novice through to veteran category. criteria which showed no statistically significant differences included: ‘creates a relaxed, nonthreatening environment’ (1a); ‘uses manipulative, interactive and physical materials’ (1b); ‘allows participant responses to drive lessons, shift instructional strategies and alter content’ (2a); ‘encourages participant inquiry by posing thoughtful, open-ended questions’ (2b); ‘encourages participants to question each other’ (2c); ‘engages participants in experiences that pose contradictions to their initial hypotheses’ (2d); ‘allows time after posing questions’ (2e); ‘seeks elaboration of participants initial responses’ (3a); ‘encourages use of cognitive terminology such as classify, analyse, predict’ (3b); and ‘asks probing questions to elicit meaningful explanations’ (3c). the probability levels of confidence for criteria that are statistically significantly different are shown in table 7 and suggest a progression in understanding and implementation of inquiry learning strategies as facilitators proceed through criteria for step 1 group mean scores novice intermediate veteran creates a relaxed environment 2.97 3.00 3.21 encourages student autonomy 2.60 3.10 3.14 uses materials (manipulatives) when teaching 3.30 3.22 3.14 familiarises self with particular understanding 2.40 2.70 2.86 encourages participants’ discussion 2.47 2.50 2.93 nurtures participants’ natural curiosity 2.40 2.50 2.86 table 3. comparison of mean facilitator scores, across facilitator categories, on their ability to engage participants in problem-solving activities. pam austin and paul webb 57 the sequence of stages of novice, intermediate and veteran categories. step 1: engages the participants during this stage the facilitators were expected to introduce activities that engaged learners and parents with a problem or phenomenon. these types of activities are expected to provide participants with an open-ended opportunity to interact with the materials and each other and are also aimed at capturing participants’ interest and enabling them to make connections with what they know and can do. as such, the ‘step 1’ section of the workshop observation instrument attempts to identify the degree to which the facilitator engaged the participants in the problem-solving activity according to specific criteria, as shown in table 3. step 2: allows participants to explore the concept step two of the national science education standards (1996) model of inquiry instruction identifies the degree to which the facilitator allows participants to explore the concept of the problemsolving activity according to specific criteria. mean scores for each group of facilitators for the second step of inquiry learning were calculated from the data generated by the workshop observation schedule (table 4). step 3: encourages participants to explain the concept and define the terms step three of the national science education standards (1996) model identifies the degree to which the facilitator encourages participants to explain the concept and define the terms related to the problem-solving activity, according to specific criteria. mean scores for each category of facilitators for the third step of inquiry learning were calculated from the data generated by the workshop observations schedule (table 5). overall mean scores for practices observed the mean scores for the workshop observations in general, which includes all three steps of the criteria for step 3 group mean scores novice intermediate veteran seeks elaboration of particular responses 2.47 2.60 2.79 encourages use of cognitive terminology 2.13 2.00 2.29 asks probing questions 2.27 2.60 2.64 gives opportunity to refine explanations 2.20 2.40 2.64 table 5. comparison of facilitator mean scores, across facilitator categories, on their ability to encourage participants to explain the concept and define the terms. criteria for step 2 group mean scores novice intermediate veteran allows particular responses to drive lessons 2.47 2.70 2.64 poses thoughtful, open-ended questions 2.40 2.50 2.79 encourages participants to question 2.13 2.30 2.43 poses contradictions in initial hypotheses 2.13 2.40 2.38 allows time after posing questions 3.20 3.00 3.07 focuses and supports inquiries 2.71 3.00 3.21 sum of mean scores 15.04 15.90 16.52 table 4. comparison of facilitator mean scores, across facilitator categories, on their ability to allow participants to explore concepts. developing inquiry-based teaching and learning in family maths programme facilitators 58 national science education standards inquiry model are indicated in figure 1. findings suggest a steady progression in the implementation skills of the facilitators on the family maths professional development programme as they advance through the sequential stages from novice, to intermediate, to veteran. analysis of variance statistical analyses (anova) of the data generated by workshop observations reveal that there were no statistically significant differences between the veteran, intermediate and novice groups for all criteria (see tables 3, 4 and 5). however, there were statistically significant differences between the three categories in terms of ‘encouraging and accepting student autonomy and student initiative’ (1b), ‘familiarising themselves with the participants understandings of concepts’ (1d), ‘encouraging participants to engage in discussion with the facilitator and one another’ (1e), ‘nurturing participants natural curiosity’ (1f), ‘focusing and supporting inquiry while interacting with the participants’ (2f), and ‘giving participants opportunities to refine their explanations and definitions’ (3d). in each case the veteran group criteria group mean scores novice intermediate veteran encouraging autonomy (1b) 2.60 3.10 3.14 participants understanding (1d) 2.40 2.70 2.86 engaging discussion (1e) 2.47 2.50 2.93 nurturing curiosity (1f) 2.40 2.50 2.86 focussing/supporting inquiry (2f) 2.71 3.00 3.21 refining explanations (3d) 2.20 2.40 2.64 table 6. comparison of statistically significant different mean scores for criteria observed during facilitation of family maths workshops. m e a n s c o r e 38 39 40 41 42 43 44 45 46 novice intermediate veteran facilitator categories figure 1. overall mean scores of novice, intermediate and veteran groups in terms of inquiry-based practice. pam austin and paul webb 59 scored the highest and the novice group the lowest. these data are reflected in table 6. the mean scores recorded show a sequential increase, and the statistical significance between these scores is portrayed in table 7. table 7 reveals probability values at the 90%, 99% and 95% levels of confidence between the novice and intermediate groups and between the novice and veteran groups respectively. these indicate significantly different levels in the ability of facilitators to encourage and accept student autonomy and initiative as an important inquiry strategy as they progress through the two year family maths professional development programme. there is also a significant difference at the 95% level of confidence between the veterans’ and novices’ ability to familiarise themselves with the participants’ understandings of concepts. the veteran group of facilitators show a much higher level of competence in familiarising themselves with workshop participants’ understanding of concepts than the novice group of facilitators. in terms of encouraging participants to engage in discussion with the facilitator and one another, the statistically significant difference between the veterans and novices is at the 95% level of confidence. the data for the category ‘nurturing participants’ natural curiosity’ shows a statistically significant difference between the veterans and the novices. there is a similar level of confidence between the scores of the veteran and novice group in terms of ‘focusing and supporting inquiry while interacting with the participants’. the difference between the veterans and the novices in ‘giving participants opportunities to refine their explanations and definitions’ is also significant at the 99% level of confidence. criteria probability values novice intermediate veteran encourages autonomy (1b) novice 0.0056*** 0.0012*** intermediate 0.0056*** 0. 8047 participants’ understanding (1d) novice 0.2002 0.0355** intermediate 0.2002 05046 engages discussion (1e) novice 0.8842 0.0391** intermediate 0.8842 0.0712 nurtures curiosity (1f) novice 0.6715 0.0386** intermediate 0.6715 0.1409 supports inquiries (2f) novice 0.0825* 0.0016*** intermediate 0.0825* 0.1886 participants refine explanations (3d) novice 0.3061 0.0160** intermediate 0.3061 0.2218 * = statistically significant at the 90% level of confidence ** = statistically significant at the 95% level of confidence *** = statistically significant at the 99% level of confidence table 7. probability values which indicate statistically significant differences between mean scores for criteria observed during facilitation of family maths workshops. developing inquiry-based teaching and learning in family maths programme facilitators 60 comparison of mean scores between facilitators’ inquiry learning belief system and implementation of inquiry learning of the three categories of facilitators, the novices show the least amount of understanding regarding inquiry learning (belief) and have the lowest rating with regard to implementation of inquiry learning strategies (practice). the intermediate group of facilitators generally show a greater understanding than the novice group and also implement inquiry learning more effectively in the workshop situation. the veteran group of facilitators show the greatest understanding of inquiry learning and also show the greatest skill in the implementation of inquiry learning strategies. table 8 represents the mean scores of each of the categories of facilitators with regard to their inquiry belief system and their implementation of inquiry learning. in order to compare what facilitators say they believe regarding inquiry learning with their implementation skills of inquiry learning in the workshop situation, mean scores were given for each of the categories in terms of their ability to encourage participants to engage in the problem-solving situation, explore the concepts and explain the concepts and terms. comparison of beliefs and practices the novices’ belief questionnaire responses suggested that they had the least understanding of inquiry learning, and observation of novices practice revealed that they also have the lowest rating with regard to implementation of inquiry learning strategies. the veteran group of facilitators showed the greatest understanding (belief) of inquiry learning, and also showed the greatest skill in the implementation of inquiry learning strategies (practice). these data are represented graphically in figure 2. in all categories of comparison in figure 2 the probability value is less than 0.05 (p ≤ 0.05) and there is, therefore, a 95% level of confidence that the statistical differences between the mean scores of the belief system questionnaire and the workshop observation instrument scores are not due to chance. it can therefore be concluded that in all categories – novice, intermediate and veteran – facilitators’ stated inquiry belief systems are novice intermediate veteran belief practice belief practice belief practice engage 3.39 2.58 3.62 2.83 3.70 3.02 explore 3.15 2.42 3.34 2.65 3.48 2.76 explain 3.16 2.38 3.45 2.53 3.56 2.69 total 9.70 7.38 10.41 8.01 10.74 8.47 table 8. comparison of facilitator mean scores with regard to inquiry learning belief versus practice. i n q u i r y l e a r n i n g 0 2 4 6 8 10 12 novice inter veteran practice belief facilitator categories figure 2. comparison of participating facilitators’ beliefs and practice in terms of inquiry learning. pam austin and paul webb 61 expressed to a lesser degree in their workshop implementation. facilitators’ verbal responses measured by workshop interaction coding system instrument the facilitators’ verbal feedback techniques were measured using the brophy and good ‘dyadic interaction coding system’. this instrument was used to classify answers, questions, clues and rephrasing of questions. it provides a coding sheet against which the facilitators’ verbal feedback techniques are recorded. analysis of these data revealed that across all categories of facilitators, higher order responses of ‘giving clues’ and ‘rephrasing questions’ were more frequent amongst facilitators than merely ‘repeating questions’. these findings are encouraging, as the family maths programme strongly discourages giving answers to participants. interviews the data generated via interviews were classified into broad categories and analysed within the framework of reviewed literature. the responses from facilitators of the intermediate and veteran groups suggest that the majority felt confident in terms of implementing inquiry learning because they believed that the family maths programme had been effective in developing their questioning skills and their ability to give meaningful clues. however, a large number of novice group facilitators felt that they had not mastered the skills of ‘questioning’ or ‘giving clues’ and would benefit from further training in this regard. most of the facilitators from the intermediate and veteran categories acknowledged that inquirybased learning had changed their way of teaching both in the workshop situation and in the classroom. one novice facilitator gave the following response: “i still find it hard to move away from the teacher-centred approach”. intermediate and veteran facilitators gave responses such as: “inquiry learning has changed my way of teaching” and “inquiry-based learning has made me think differently about my own teaching strategies”. discussion the apparent disjuncture between the facilitators’ beliefs and practices may be of concern to some. however, while some researchers have suggested that beliefs are a major force in affecting teaching practice (schoenveld, 1992; thompson, 1992), others assert that they are not (hoyles, 1992). skott (2001a) maintains that mathematics teachers can simultaneously hold multiple, and possibly conflicting, beliefs about their practice in the course of classroom interaction. this view implies that understanding of the micro aspects of the classroom, such as classroom atmosphere and interactions between the teacher and specific groups of learners are essential to the understanding of differences between their beliefs and practices. as such, it would be interesting and profitable to investigate and interrogate the reasons that the facilitators in this study give for the apparent disjuncture between their beliefs and practices, and use these explanations as a framework for developing a more grounded professional development programme. more definitively, analysis of the data generated by the questionnaire, observation schedules and interviews suggest that facilitators’ understanding of inquiry improved as they progressed through the novice, intermediate and veteran categories of the family maths programme. however, on occasions, the intermediate category of facilitators achieved slightly higher mean scores than the veteran facilitators (see tables 1, 3, and 4). one possible explanation is that while intermediate facilitators participate in regular training workshops and benefit from regular support from family maths trainers, many of the veteran facilitators – on successful completion of the two professional-year development programme – conduct family maths workshops independently, without ongoing support from the trainers. there is, therefore, a possible tendency for veteran facilitators to regress to the more traditional beliefs which they may have held over a period of many years. regression after exposure to new teaching and learning strategies is fairly common in educational research literature (webb, 1992). this notion of regression suggests that the need for ongoing support for facilitators after completion of the family maths professional development programme is a factor that should not be underestimated. encouragingly, data analysis of the verbal response techniques of facilitators indicates that only one percent of both the novice and intermediate groups of facilitators succumbed to giving participants solutions to problems before participants had been given the opportunity to solve the problem within the group situation. none of the veteran facilitators gave solutions to the participants; they persevered by repeating the question, giving clues and rephrasing the question. this is particularly significant in light of the fact developing inquiry-based teaching and learning in family maths programme facilitators 62 most of the participants were themselves not educated in inquiry-based methodology, nor trained to teach in these ways; many found that making the paradigm shift was not as overwhelming as they initially thought. most participants learnt to combine inquiry-based teaching and learning strategies with the creation of a relaxed, non-threatening environment. it appears that confidence and self esteem increased as facilitators and participants actively challenged and engaged with one another, thereby honing their interpersonal and verbal skills during the learning experience. findings of this nature are echoed by the reports of mercer, wegerif and dawes (1999), abrami, chambers, poulsen, de simone, d’apollonia and howden (1995), and a number of other researchers focussing on discussion and exploratory talk. for these reasons it seems that the active promotion of classroom discussion techniques should be highlighted as an important aspect of continuous professional development programmes in order to break the cycle of teachercentred practices and rote learning. interview, questionnaire and observation data all indicate that facilitators’ perceptions, attitudes and abilities with regard to inquiry learning developed as they proceeded through the facilitator categories of the family maths programme. the emphasis of the programme on inquiry techniques and active engagement is particularly relevant to the implementation of the new south african outcomes based education curricula, which are underpinned by constructivist and inquiry-based approaches to meeting curriculum outcomes. conclusion the findings of this study suggest that the family maths professional development programme at the nelson mandela metropolitan university can promote the ability of teachers and teacher educators to engage in multiple aspects of inquirybased teaching and learning over the two-year course. this in turn implies that appropriately selected aspects of the approach have the potential to assist in the design of other teacher development programmes aimed at dispelling negativity towards mathematics and making school mathematics more relevant to learners. references abrami, p.c., chambers, b., 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(pp 127-148). dordrecht: kluwer academic publishers. “those who are accustomed to judge by feeling do not understand the process of reasoning, because they want to comprehend at a glance and are not used to seeking for first principles. those, on the other hand, who are accustomed to reason from first principles do not understand matters of feeling at all, because they look for first principles and are unable to comprehend at a glance.” blaise pascal microsoft word 57-67 du toit.doc pythagoras, 70, 57-67 (december 2009) 57 metacognitive strategies  in the teaching and learning of mathematics    stephan du toit;  gary kotze  faculty of education, university of the free state  dutoitds@ufs.ac.za;  kotzeg@ufs.ac.za      the broad aim of  this study was  to  investigate  the use of metacognitive strategies by  grade  11  mathematics  learners  and  their  teachers.  two  objectives  were  stated:  to  investigate  which  metacognitive  strategies  grade  11  mathematics  learners  and  mathematics  teachers  can  employ  to  enhance  metacognition  among  learners,  and  to  investigate  the  extent  to  which  grade  11  mathematics  learners  and  teachers  use  metacognitive strategies. questionnaires were used to obtain quantitative data about the  use  of  metacognitive  strategies  by  learners  and  teachers.  the  findings  indicate  that  planning strategy and evaluating the way of thinking and acting were used most by both  teachers and  learners. journal‐keeping and  thinking aloud were used  least by  teachers  and learners.        the purpose of teaching mathematics is to empower learners to “make sense of society” (departement of education (doe), 2003, p. 9). various stakeholders in society, for example parents, employers and tertiary institutions, exert pressure on mathematics education because mathematical competence “contributes to personal, social, scientific and economic development” (doe, 2003, p. 9). south african learners do not perform very well in mathematics. the aim of the department of education was for 50 000 learners to pass mathematics with more than 50% in the 2008 national senior certificate (doe, 2008, p. 12; naude, 2007, p. 17). this aim was achieved, a total of 63 038 learners scored above 50% in the 2008 ncs mathematics examination. when the total number of learners that wrote the grade 12 mathematics is considered, a more distressing picture emerges. a total of 270 097 learners wrote mathematics in 2008, therefore only 23,34% of those learners achieved more than 50% in the examination (doe, 2008, pp. 10, 12). on international level, an even worse scenario emerges. south africa’s grade 8 learners scored the lowest of 46 countries with a score of 264 in the 2003 trends in mathematics and science study (timss), 11 points lower than in 1999 (gonzales et al., 2004, p. 7). south africa did not participate in the 2007 timss. how could learners’ mathematical competence and performance be improved? campione (1987, p. 136) observes that knowledge about a domain, specific procedures for operating in that domain, and general task-independent regulatory processes are three prerequisites for effective performance within some domain. de corte adds affective components as another prerequisite (1996, pp. 34-36) by stating that expert performance in a given domain necessitates the integrated acquirement of the following four categories of aptitude, namely a structured, accessible domain-specific knowledge base; heuristic methods; affective components; and metacognition. metacognitive strategies in the teaching and learning of mathematics 58 defining metacognition papaleontiou-louca (2003, p. 9) states that, in the field of cognitive developmental research, metacognition has become a foremost topic since 1973. in this regard, schoenfeld (1992, p. 9) describes “metacognition” as a term that was coined in the 1970s and only occasionally appearing in the literature of the early 1980s, but appearing with growing frequency through the decade, becoming (with problem solving) probably the most clichéd and least understood buzz words of the 1980s. definitions of metacognition vary. schoenfeld (1992) asserts that metacognition has multiple and almost disjoint meanings (for example, knowledge about one’s thought processes, self-regulation during problem solving) which make it difficult to use as a concept. (pp. 2, 38, 39) hacker (1998, p. 11) states that there is general agreement that the definition of metacognition should at least include the following aspects: knowledge of one’s knowledge; the conscious monitoring and regulating of one’s knowledge; and cognitive and affective states. metacognition is the knowledge and beliefs about cognition, in addition to the skills and strategies enabling the self-regulation of cognitive processes (de corte, 1996, pp. 35, 36), while papaleontiou-louca (2003) defines metacognition as …all processes about cognition, such as sensing something about one’s own thinking, thinking about one’s thinking and responding to one’s own thinking by monitoring and regulating it. (p. 12) these various definitions of metacognition have in common the emphasis on the knowledge of cognition and the monitoring and regulation of cognitive processes. the summaries of the different facets of metacognition by hacker (1998, p. 11) and schoenfeld (1992, pp. 38, 39) contain an additional reference to the awareness and regulating of one’s affective state. metacognition and academic performance cognitive monitoring enhances learning (paris & winograd; 1990, p. 15). butler and winne (1995, p. 245) assert that there is agreement among theoreticians that the most effective learners are self-regulating. self-regulation is viewed as synonymous to metacognitive strategies (boekaerts & simons, 1995, p. 85). in support schraw (1998, p. 114) states that academic performance is improved by metacognitive regulation as learners utilise resources and existing strategies better. a study conducted by camahalan (2006, p. 194) found that students’ academic achievement is more likely to improve when they are given the chance to self-regulate and are explicitly taught metacognitive learning strategies. metacognitive strategies metacognitive strategies refer to the conscious monitoring of one’s cognitive strategies to achieve specific goals, for example when learners ask themselves questions about the work and then observe how well they answer these questions (flavell, 1981, p. 273). boekaerts and simons (1995, p. 91) view metacognitive strategies as the decisions learners make before, during and after the process of learning. there are various metacognitive strategies aimed at developing learners’ metacognition (costa, 1984, pp. 59-61; blakey & spence, 1990, pp. 2, 3; brown, as quoted in boekaerts & simons, 1995, p. 91). planning strategy at the start of a learning activity, teachers should make learners aware of strategies, rules and steps in problem solving. time restrictions, goals and ground rules connected to the learning activity should be made explicit and internalised by the learners. consequently, learners will keep them in mind during the learning activity and assess their performance against them. during the learning activity, teachers can encourage learners to share their progress, their cognitive procedures and their views of their conduct. as a result, learners will become more aware of their own behaviour and teachers will be able to identify problem areas in the learners’ thinking (costa, 1984, p. 59). when learning is planned by someone else, it is difficult for learners to become self-directed (blakey & spence, 1990, p. 3). stephan du toit & gary kotze 59 generating questions blakey and spence (1990, p. 2) state that learners should ask themselves what they know and what they do not know at the beginning of a research activity. as the research activity progresses, their initial statements about their knowledge of the research activity will be verified, clarified and expanded. ratner (1991, p. 32) views the questioning of given information and assumptions as a vital aspect of intelligence: learners should pose questions for themselves before and during the reading of learning material and pause regularly to determine whether they understand the concept; if they can link it with prior knowledge; if other examples can be given; and if they can relate the main concept to other concepts. here muijs and reynolds (2005, p. 63) argue that the connection of prior knowledge and new concepts should take place during the lesson and not only when a new concept is introduced. this integration of prior knowledge and new concepts enables the learner to understand the unified and interconnected nature of knowledge, while also facilitating profound understanding of subject matter (ornstein & hunkins, 1998, p. 240). integration adheres to the second of the principles for quality mathematics education (nctmp), stated by the national council of teachers of mathematics (nctm), (nctm, 2000, p. 2), namely a coherent curriculum in which students’ mathematical concepts are linked and built on one another. choosing consciously teachers should guide learners to explore the results of their choices before and during the decision process. therefore, learners will be able to recognise underlying relationships between their decisions, their actions and the results of their decisions. non-judgmental feedback to learners about the consequences of their actions and choices promotes self-awareness (costa, 1984, p. 60), and it enables the learners to learn from their mistakes, thereby supporting the fourth principle of the nctmp of “learning… understanding, actively building new knowledge from experience…” (nctm, 2000, p. 2). setting and pursuing goals artzt and armour-thomas (1998, p. 9) define goals as “expectations about the intellectual, social and emotional outcomes for students as a consequence of their classroom experiences”. these goals support the first principle of the nctmp of high expectations and support for learners. learners who are selfregulating strive to attain a self-formulated goal while self-regulated behaviour can be adapted with changing circumstances (diaz, neal, & amaya-williams, 1990, p. 130). evaluating the way of thinking and acting metacognition can be enhanced if teachers guide learners to evaluate the learning activity according to at least two sets of criteria (costa, 1984, p. 60). initially, evaluative criteria could be jointly developed with the learners to support them in assessing their own thinking. as an example, learners could be asked to assess the learning activity by stating helpful and hindering aspects and their likes and dislikes of the learning activity. accordingly, learners keep the criteria in mind when classifying their opinions about the learning activity and they motivate the reasons for those opinions (costa, 1984, p. 60). guided selfevaluation can be introduced by checklists focusing on thinking processes and self-evaluation will increasingly be applied more independently (blakey & spence, 1990, p. 3). identifying the difficulty costa (1984, p. 60) advises teachers to discourage the use of phrases like “i can’t”; “i am too slow to…”; or “i don’t know how to…”. rather, learners should identify the resources, skills and information required to attain the learning outcome. as a result, learners are assisted to distinguish between their current knowledge and the knowledge they need. they also have more conviction in seeking the right strategy for solving the problem. paraphrasing, elaborating and reflecting learners’ ideas teachers should support learners to restate, translate, compare and paraphrase other learners’ ideas. consequently, learners will be better listeners to other learners’ thinking and also to their own thinking (costa, 1984, p. 61). the teacher can, for example, respond: “what you are explaining to us is…”; “i understand that you are suggesting the following…”. metacognitive strategies in the teaching and learning of mathematics 60 carpenter and lehrer (1999, p. 22) assert that the ability to articulate one’s ideas requires profound understanding of significant aspects and concepts. they view the ability to reflect as a prerequisite for articulation and that articulation requires the identification of the essence and critical elements of an activity. clarifying learners’ terminology learners regularly use vague terminology when making value judgments, for example “the question is not fair” or “the question is too difficult”. teachers should elucidate these value judgments, for example “why is the question not fair?” or “why is the question too difficult?” (costa, 1984, p. 61). problem-solving activities in problem solving, existing knowledge is applied to an unfamiliar situation to gain new knowledge (killen, 2000, p. 129). problem-solving activities are ideal opportunities to enhance metacognitive strategies, as good problem solvers are generally self-aware thinkers. learners with superior metacognitive abilities are better problem solvers. the ability to analyse their problem-solving strategies and reflect on their thinking reveals the learners’ metacognitive skills (blakey & spence, 1990, p. 2; panaoura, philippou, & christou, 2003, p. 3). after the problem-solving process, teachers should encourage learners to clarify their course of action, instead of merely correcting the learner (costa, 1984, p. 61). goos & galbraith (1996, p. 231) state that non-cognitive aspects, like learners debilitating beliefs about the nature of mathematics and about themselves, could have a positive or negative effect on cognitive and metacognitive processes involved in problem solving. when the whole class works on a problem, the teacher, instead of steering the learners to the answer, helps the learners to take full advantage of those aspects that they have produced. during this process of guiding the learners, the teacher will ask questions like: “are you all convinced that you understand the problem?”; and “which of the suggestions to solve the problem should we attempt first, and why?”. after the class has worked on the problem for about five minutes, the teacher could ask them whether the process is going well, and if not, to reassess the strategy. if the class decides to reject that strategy, the teacher could ask whether anything helpful could be recovered from their effort. when a solution is reached, the teacher reviews the whole problem-solving process and indicates where the class went wrong initially. teachers also lead the class in finding alternative solutions to the problem (schoenfeld, 1987, p. 202). in this regard, muijs and reynolds (2005, p. 64) list reflection as one of the elements of constructivist teaching strategies. they describe reflection, a key learning moment, as the comparing of solutions between learners. they also regard reflection as the process learners engage in when they think about problem-solving strategies and their effectiveness. schoenfeld (1987, p. 202) considers whole class problem solving as promoting self-regulation, because the teacher’s role as a moderator compels learners to focus on control decisions made by themselves, and not by the teacher. another aspect of whole class problem solving that schoenfeld (1987, p. 202) discusses is the opportunity it affords to pose problems that evoke beliefs about mathematics. an example is mentioned of the belief that problems can be solved relatively quickly if the subject matter is well understood. to challenge this belief, a problem is assigned that would probably take the class a few days, or even weeks, to solve. schoenfeld’s (1987, p. 206) aim with small group problem solving is to provide learners with a range of problem-solving strategies (heuristics), and then to train them to use those strategies effectively. when learners are only taught about heuristics and then have to work on problems at home, the teacher is not present in the midst of problem solving when his/her input could have promoted the use of self-regulation skills, for example, the teacher informs the learners that they are going to be asked the following three questions whenever they work on a problem: “what exactly are you doing?”; “why are you doing it?”; and “how does it help you?”. gradually, it becomes a matter of practice for the learners to start asking the questions themselves, thereby improving their problem-solving skills and operation on a metacognitive level. stephan du toit & gary kotze 61 thinking aloud teachers should promote the habit of thinking aloud when learners solve problems (costa, 1984, p. 61). talking about their thinking will help learners to identify their thinking skills (blakey & spence, 1990, p. 2). muijs and reynolds (2005, p. 64) use the term “articulation” to describe learners’ expression of their own thoughts and ideas. they recommend that learners should discuss complex tasks and present their ideas to fellow learners. they furthermore suggest that group work could be very effective in promoting articulation. in this regard, blakey and spence (1990, p. 2) mention paired problem solving, where one learner describes his/her thinking processes while his/her partner helps him/her to clarify his thinking by listening and asking questions. a main aspect of vygotsky’s developmental theory is that children start using language not only to communicate, but also to regulate their activities by guiding, planning and monitoring (diaz et al., 1990, p. 135). three consequences for self-regulation through the use of language can be identified. firstly, children organise and restructure their perceptions in terms of their goals. secondly, children’s actions are less impulsive as they allow them to act reflectively according to their goals. finally, language not only enables children to regulate their way of perceiving stimuli, but also to regulate their behaviour (diaz et al. 1990, pp. 135, 136). camp, blom, hebert and van doornick, (1977, p. 160) developed a program called think aloud to improve self-control. children are taught to use the following four questions when solving problems: “what is my problem?”; “how can i do it?”; “am i using my plan?”; and “how did i do”? journal-keeping keeping a personal diary throughout a learning experience facilitates the creation and expression of thoughts and actions. learners make notes of ambiguities, inconsistencies, mistakes, insights, and ways to correct their mistakes. preliminary insights can be compared with changes in those insights as more information is gathered or obtained through feedback from assessment, thereby supporting the fifth principle of the nctmp, namely, that assessment should support the learning of mathematics (costa, 1984, p. 61; blakey & spence, 1990, p. 3; nctm, 2000, p. 2). cooperative learning cooperative learning creates the opportunity for learners to work together in small groups to enhance learning. it entails more than group work, as group work is considered as a modification of whole-class discussion. in cooperative learning, the teacher gives indirect guidance as the group works together to achieve specific learning outcomes (killen, 2000, p. 73). cooperative learning may promote awareness of learners’ personal thinking and of others’ thinking. when learners act as “tutors”, the process of planning what they are going to teach, lead to independent learning and clarifying the learning material (blakey & spence, 1990, p. 2). modelling the nctm lists effective teaching as a third principle of the nctmp (nctm, p. 2000, p. 2). modelling occurs when teachers demonstrate the processes involved in performing a difficult task, or when teachers tell learners about their thinking and the motivation for selecting certain strategies when solving problems (muijs & reynolds, 2005, p. 63). modelling and discussion enhance learners’ thinking and talking about their own thinking (blakey & spence, 1990, p. 2). schoenfeld (1987, p. 200) refers to the importance for teachers of not always presenting the finished, neat presentation of the answers on the board, but to sometimes model the problems and working through the problem step by step. consequently, the processes yielding the correct answer (for example false starts, recoveries from false starts and interesting insights) are exposed and the chief purpose of the modelling approach is achieved, namely the centering of learners’ awareness on metacognitive behaviours. costa (1984, p. 61) suggests that modelling could be the most effective strategy used to enhance metacognition among learners because they learn best by imitating adults. teachers will, by thinking aloud throughout planning and problem-solving activities, demonstrate their thinking processes. teachers, therefore, have a great responsibility because “a fair proportion of the learning problems in mathematics are actually taught to the children…” (moodley, 1992, p. 8). van der walt and maree (2007, metacognitive strategies in the teaching and learning of mathematics 62 p. 235) found that mathematics teachers employed question-posing strategies and think-aloud models, but that they did not sufficiently promote the implementation and practice of these strategies among learners. aspects that denote teachers’ modelling behaviour include explaining their planning, goals and objectives to the learners and motivating their actions; acknowledging their temporary inability to answer a question, but developing pathways for finding the answer; making human mistakes, but demonstrating how to correct those mistakes; requesting comments and assessment of their actions; acting in accordance with an explicitly stated value system; the ability to explain what their strengths and weaknesses are; and expressing an understanding and valuing of learners’ ideas and feelings (costa, 1984, p. 61). regarding the expression of understanding and the valuing of learners’ ideas and feelings, muijs and reynolds (2005, p. 65) state that flexibility, an element of the constructivist teaching strategies, is the process whereby learners partly guide the progress of the lesson as teachers interact with learners. vygotsky’s developmental theory proposes that the development of self-regulation originates and is enhanced by the teacher-learner social interactions (diaz et al., 1990, p. 128). diaz et al. (1990, p. 139) identify three characteristics of teacher-learner interactions that promote self-regulation, namely the use of reasoning and supplying reasons for commands; the gradual withdrawal of teacher control; and the combination of the previous two aspects in an atmosphere of emotional warmth and affective nurturance. de abreu, bishop and pompeu (1997, p. 235) also stress the importance of affect in arguing that, although learners experience mathematics cognitively and affectively, they only have the opportunity to express the cognitive aspect. aims of the study the broader aim of the study was to investigate the use of metacognitive strategies by grade 11 mathematics teachers and their learners in the teaching and learning of mathematics in the motheo district. the following three research questions were formulated: o which metacognitive strategies can grade 11 mathematics learners and their mathematics teachers employ to enhance learners’ metacognition? o to what extent does grade 11 mathematics teachers implement and encourage learners to use the identified metacognitive strategies? o to what extent do grade 11 mathematics learners implement the identified metacognitive strategies? research design form of inquiry information gathered from a literature study provided an answer to the first research question, and survey research as a form of inquiry was used to collect the quantitative data required to answer the second and third research questions. a learner and a teacher questionnaire, based on the literature study, were constructed to determine the extent of the use of the metacognitive strategies in the teaching and learning of mathematics. questionnaire the learner questionnaire and the teacher questionnaire comprised 37 and 47 questions respectively that were based on the use of the metacognitive strategies. the learner questionnaire determined the extent to which learners use the metacognitive strategies, except modelling, in the learning of mathematics. the teacher questionnaire investigated the extent to which teachers use the metacognitive strategies in the teaching of mathematics, and encourage the use of the metacognitive strategies in the learning of mathematics. in both questionnaires, respondents could choose any of the following options on a 4-point likert-scale: almost never, sometimes, usually, almost always. table 1 reflects the correspondence between questionnaire items and the metacognitive strategies. stephan du toit & gary kotze 63 table 1: correspondence between questionnaire item number(s) and metacognitive strategies metacognitive strategy learner questionnaire item(s) teacher questionnaire item(s) planning strategy 1, 2 1, 2 generating questions 3, 4, 5 3, 4, 5 choosing consciously 6, 18 6, 18 setting and pursuing goals 7, 27 7, 27 evaluating the way of thinking and acting 8, 9, 10, 11, 28, 29, 30 8, 9, 10, 11, 28, 29, 30 identifying the difficulty 12, 13, 14, 15, 16 12, 13, 14, 15, 16 paraphrasing, elaborating and reflecting learners’ ideas 17 17 clarifying terminology 15, 16 15, 16 problem-solving activities 19, 31, 32, 33, 34, 35, 36, 37 19, 41, 42, 43, 44, 45, 46, 47 thinking aloud 20 20 journal-keeping 21, 22, 23 21, 22, 23 cooperative learning 24, 25, 26 24, 25, 26, modelling 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39,40 piloting of the questionnaire four grade 11 learners were asked to complete the pilot learner questionnaire and to note any ambiguous or vague questions. two current grade 11 teachers and five experienced mathematics teachers completed the learner and the teacher pilot questionnaires. sampling this study focused on grade 11 mathematics teachers and grade 11 mathematics learners in the motheo district. of the five districts in the free state, this district was the leading district regarding the pass percentage in the senior certificate examination of 2006 (free state department of education, 2007a, p. 4). in the mathematics higher grade (hg) senior certificate examination of 2006 in this district, averages of the top 10 schools for the examination were between 69,75% and 60,13% (free state department of education, 2007b). only those schools with more than 20 learners who had written the senior certificate examination in mathematics hg (five schools) were selected for the study. table 2 contains information about the position of the five selected schools according to the 2006 senior certificate examination mathematics hg results; the number of learners who wrote the examination; and each school’s average. table 2: achievement in the 2006 senior certificate examination in mathematics hg school position in district according to mathematics hg results number of learners who wrote mathematics hg average percentage obtained in mathematics hg 1 2 25 68,96 2 4 80 67,46 3 6 97 65,46 4 7 26 63,24 5 8 77 62,81 metacognitive strategies in the teaching and learning of mathematics 64 in total, 394 learners and their teachers from 16 classes in the five selected schools participated in the study. thirteen teachers participated in this study; three teachers had two classes each. the respondents numbered 83% of the total number of learners (respondents and non-respondents). limitations of the study since it was not one of the aims of the study to generalise the findings to the whole population, the findings have limited value. the following aspects are considered as limitations of this study: the use of two metacognitive strategies was determined by only one item each in the teacher and learner questionnaire. therefore, the reliability of those subscales could not be determined (see table 1); and two items on the teacher and learner questionnaire were used to obtain information about the use of more than one metacognitive strategy (see table 1). reseach findings reliability of the questionnaire the cronbach alpha procedure is regarded as the most suitable type of reliability for survey research where items are not scored right or wrong and where each item could have different answers (mcmillan & schumacher, 2001, p. 246, 247). the reliability scores of the learner questionnaire and teacher questionnaire were 0,88 and 0,95 respectively, indicating a high reliability on both questionnaires. the use of metacognitive strategies one of the research questions was to determine the extent to which each metacognitive strategy is used by teachers and learners. the precise means (x) and standard deviations (sd) of the extent to which teachers and learners use metacognitive strategies (according to the questionnaires constructed for this study) are indicated in table 3. table 3: the extent to which metacognitive strategies are implemented by teachers and learners teachers learners strategy x sd x sd planning strategy 4,00 0,00 3,07 0,64 generating questions 3,47 0,47 2,52 0,70 choosing consciously 3,38 0,53 2,83 0,66 setting and pursuing goals 3,41 0,56 2,44 0,69 evaluating the way of thinking and acting 3,55 0,35 3,15 0,48 identifying the difficulty 3,28 0,63 2,83 0,50 paraphrasing, elaborating and reflecting learners’ ideas 3,16 0,83 2,53 0,97 clarifying terminology 2,91 1,01 2,61 0,72 problem-solving activities 3,24 0,52 2,68 0,53 thinking aloud 2,73 1,15 2,28 1,02 journal-keeping 2,78 0,97 2,38 0,79 cooperative learning 3,23 0,70 2,24 0,61 modelling 3,22 0,48 mean score 3,28 0,43 2,72 0,38 stephan du toit & gary kotze 65 the metacognitive strategies that were implemented most by the teachers were planning strategies (4,00); evaluating the way of thinking and acting (3,55); and setting and pursuing goals (3,41). the learners employed evaluating the way of thinking and acting (3,15); planning strategies (3,07); choosing consciously (2,83); and identifying the difficulty (2,83) most. planning strategies and evaluating the way of thinking and acting were used to the greatest extent by both teachers and learners. this could indicate that teachers and learners were well-organised and aware of their strengths and weaknesses in mathematics. the metacognitive strategies that were implemented least by the teachers were thinking aloud (2,73); encouraging journal-keeping (2,78); and clarifying terminology (2,91). the learners employed cooperative learning (2,24); thinking aloud (2,28); and journal-keeping (2,38) to the least extent. thinking aloud and journal-keeping were used to the least extent by both teachers and learners. this could imply that the keeping of a reflective journal is not encouraged among learners and that learners are not keeping a written record of mistakes they tend to make and insights they gain. when considering that learners use evaluating the way of thinking and acting to the greatest extent, it seems that learners can identify their strengths, weaknesses, mistakes and successes in mathematics, but they do not keep a written record of this self-knowledge. the fact that the learners used cooperative learning, which requires the articulation of one’s ideas, to the least extent, could explain why thinking aloud was used to the second least extent, as learners would be more inclined to verbally express their thoughts in a group setting than individually. each mean score for the extent to which a specific metacognitive strategy was implemented, was higher among the teachers than the learners. the teachers’ mean score (3,28) for the extent to which all the metacognitive strategies were used falls in the category “usually” to “almost always” on the 4-point likert-scale, whereas the learners’ mean score (2,72) falls in the category “sometimes” to “usually”. teachers used metacognitive strategies to a greater extent than the learners, as the teachers’ metacognitive mean of 3,28 as compared to the learners’ metacognitive mean of 2,72 indicates. conclusion and recommendations from our experience of teaching, many mathematics learners do not like mathematics because they regard it as too difficult, and they cannot see the relevance of mathematics for their everyday or future lives. learners also regularly enquire about effective study methods in mathematics. the use of metacognitive strategies could address these concerns as teachers, by valuing learners’ ideas and feelings (modelling), could assist in improving learners’ attitudes towards mathematics. learner self-regulation could also be improved by the keeping of a reflective journal. by assigning real-life problems (problem solving activities), teachers have the opportunity to show the relevance of mathematics in learners’ everyday and future lives. the metacognitive strategies identified in this study could serve as a guide in ensuring effective teaching and assisting learners to study and learn mathematics effectively. it is recommended that teachers and learners are assisted with the implementation of all the identified metacognitive strategies in the teaching and learning of mathematics, especially those that were used to the least extent by teachers and learners. modelling and problem solving activities in a cooperative learning context are also regarded as focus areas. further research could investigate the following aspects: the factors that play a role in the extent to which specific metacognitive strategies are used by teachers and learners; the reasons why certain metacognitive strategies are used to a greater or lesser extent by both the teachers and the learners; and the influence of the teacher-learner ratio, teaching experience, teaching qualifications, and allocated time of teaching on the use of the metacognitive strategies by the teachers. the relation between learner age, home language, and language of instruction, gender, and race on the use of the metacognitive strategies by learners could be further researched. in a speech delivered by naledi pandor, the minister of education (pandor, 2008), she stated that the government intends to launch an intensive teacher support programme for the improvement of teaching and learning. as the link between better academic performance and the use of the metacognitive strategies has been established by previous research, we believe that teacher support programmes must include training in the use of the metacognitive strategies to ensure better teaching and learning of mathematics. metacognitive strategies in the teaching and learning of mathematics 66 references artzt, a. f., & armour-thomas, e. 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can be opened with acrobat and adobe reader 5.0 and later.) >> /namespace [ (adobe) (common) (1.0) ] /othernamespaces [ << /asreaderspreads false /cropimagestoframes true /errorcontrol /warnandcontinue /flattenerignorespreadoverrides false /includeguidesgrids false /includenonprinting false /includeslug false /namespace [ (adobe) (indesign) (4.0) ] /omitplacedbitmaps false /omitplacedeps false /omitplacedpdf false /simulateoverprint /legacy >> << /addbleedmarks false /addcolorbars false /addcropmarks false /addpageinfo false /addregmarks false /convertcolors /converttocmyk /destinationprofilename () /destinationprofileselector /documentcmyk /downsample16bitimages true /flattenerpreset << /presetselector /mediumresolution >> /formelements false /generatestructure false /includebookmarks false /includehyperlinks false /includeinteractive false /includelayers false /includeprofiles false /multimediahandling /useobjectsettings /namespace [ (adobe) (creativesuite) (2.0) ] /pdfxoutputintentprofileselector /documentcmyk /preserveediting true /untaggedcmykhandling /leaveuntagged /untaggedrgbhandling /usedocumentprofile /usedocumentbleed false >> ] >> setdistillerparams << /hwresolution [2400 2400] /pagesize [612.000 792.000] >> setpagedevice abstract introduction the nature of quantitative literacy and proportional reasoning research on proportional reasoning method results and discussion conclusion acknowledgement references about the author(s) vera frith centre for higher education development, university of cape town, south africa pam lloyd centre for higher education development, university of cape town, south africa citation frith, v. & lloyd, p. (2016). proportional reasoning ability of school leavers aspiring to higher education in south africa. pythagoras, 37(1), a317. http://dx.doi.org/10.4102/pythagoras.v37i1.317 original research proportional reasoning ability of school leavers aspiring to higher education in south africa vera frith, pam lloyd received: 19 nov. 2015; accepted: 05 aug. 2016; published: 03 dec. 2016 copyright: © 2016. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract the ability to reason about numbers in relative terms is essential for quantitative literacy, which is necessary for studying academic disciplines and for critical citizenship. however, the ability to reason with proportions is known to be difficult to learn and to take a long time to develop. to determine how well higher education applicants can reason with proportions, questions requiring proportional reasoning were included in one version of the national benchmark test as unscored items. this version of the national benchmark test was taken in june 2014 by 5 444 learners countrywide who were intending to apply to higher education institutions. the multiple choice questions varied in terms of the structure of the problem, the context in which they were situated and complexity of the numbers, but all involved only positive whole numbers. the percentage of candidates who answered any particular question correctly varied from 25% to 82%. problem context and structure affected the performance, as expected. in addition, problems in which the answer was presented as a mathematical expression, or as a sentence in which the reasoning about the relative sizes of fractions was explained, were generally found to be the most difficult. the performance on those questions in which the answer was a number or a category (chosen as a result of reasoning about the relative sizes of fractions) was better. these results indicate that in learning about ratio and proportion there should be a focus on reasoning in various contexts and not only on calculating answers algorithmically. introduction one valued attribute of a university graduate is a degree of quantitative literacy appropriate to their discipline. this means that graduates should be able to engage confidently with data in an informed and critical way. however, many programmes of study expect students to have this ability as undergraduates. for example, as seen in social sciences and health sciences contexts, progress made in addressing challenges facing society can be examined by means of measuring change in statistical indicators such as poverty rates, government spending on social grants, infection rates and lifestyle risks. meaningful comparison of the values of indicators at different times, or between different indicators, usually entails consideration of both absolute and relative quantities, that is, reasoning with proportions is necessary. reasoning with proportions is clearly also an essential ability in scientific disciplines. thus it is important for students to be able to reason logically and confidently with relative quantities and furthermore to be able to explain this reasoning. from our experience of teaching quantitative literacy to university students, we have observed that proportional reasoning is a troublesome concept; it is difficult to learn and takes a long time to learn. consequently, we regard proportional reasoning as a ‘threshold concept’ (meyer & land, 2003) for all academic quantitative literacy (frith & lloyd, 2014; lloyd & frith, 2013). this threshold concept, once mastered, opens a gateway to thinking differently about quantities. although solving problems involving ratio and proportion is part of the curriculum for both mathematics (department of basic education, 2011a) and mathematical literacy (department of basic education, 2011b) in primary and high school, it is useful to know to what extent students are able to reason with proportions when they enter higher education. this information is also useful for school teachers whose goal it is to ensure that learners are adequately prepared for tertiary education as well as for critical citizenship. in this article we discuss the results of an assessment of the proportional reasoning ability of learners who aspire to enter higher education institutions and who wrote the academic and quantitative literacy component of the national benchmarks tests (nbt). this is mainly intended to inform both school teachers and university lecturers about the general ability of these learners to reason with proportions. it also highlights some of the factors that affect the difficulty of this kind of reasoning. the nature of quantitative literacy and proportional reasoning in the literature there are many different definitions of quantitative literacy (or numeracy), which is the same competency that is embodied in the school subject mathematical literacy (department of basic education, 2011b). different definitions emphasise different aspects of this complex concept, but the idea that it is mainly concerned with mathematics and statistics used in context is essential to all of them (e.g. chapman & lee, 1990; jablonka, 2003; johnston, 2007; steen, 2004). in this study we use the following definition: quantitative literacy (numeracy) is the ability to manage situations or solve problems in practice, and involves responding to quantitative (mathematical and statistical) information that may be presented verbally, graphically, in tabular or symbolic form; it requires the activation of a range of enabling knowledge, behaviours and processes and it can be observed when it is expressed in the form of a communication, in written, oral or visual mode. (frith & prince, 2006, p. 30) the formulation of this definition is strongly influenced by the definition of numerate behaviour underlying the assessment of numeracy in the adult literacy and lifeskills survey (gal, van groenestijn, manly, schmitt & tout, 2005) and the view of academic literacy and numeracy as social practice. in this definition, the ‘range of enabling knowledge, behaviours and processes’ refers to the competences necessary for quantitative literacy practice, which include number sense, some basic mathematical abilities and quantitative reasoning in context. one of the most crucial competences, in our view, is that of proportional reasoning, that is, reasoning with numbers in relative, not absolute terms. according to lamon (2007), in her review of research on proportional reasoning and rational numbers, the term ‘proportional reasoning’ has become an ill-defined term ‘referring to anything and everything related to ratio and proportion’ (p. 637). however, she pointed out that the research on ratio and proportion has implicitly defined proportional reasoning in terms of two problem types, ‘comparison problems’ and ‘missing value problems’. in a comparison problem two ratios and are given and it must be determined which is larger or whether they are equal. missing value problems are ones where three of the four values in a proportion are given and then the fourth value must be found. a broader definition for proportional reasoning was proposed by lamon (2007, p. 638): supplying reasons in support of claims made about the structural relationships among four quantities, (say a, b, c, d) in a context simultaneously involving covariance of quantities and invariance of ratios or products; this would consist of the ability to discern a multiplicative relationship between two quantities as well as the ability to extend the same relationship to other pairs of quantities. she stressed the need to supply reasons because many students can provide a correct numerical answer to a proportion problem using an algorithmic procedure, but this does not necessarily mean that they actually employed proportional reasoning. research on proportional reasoning in the middle of the last century piaget’s theory established proportional reasoning as a defining characteristic of the formal operations stage of development of thinking (inhelder & piaget, 1958). as a consequence, research into the development of proportional reasoning of children and adolescents has been carried out and has continued since then. tourniaire and pulos (1985), in their review of the literature of the previous 25 years, claimed that the existing research was disjointed and difficult to apply to mathematics education. the rational number project in the late 1980s and early 1990s produced numerous articles on proportional reasoning (e.g. cramer, post & currier, 1993; harel, behr, post & lesh, 1991; lesh, post & behr, 1988). when lamon (2007) reviewed the research on the topic she noted that there was only a small number of researchers engaged in long-term research agendas in the field at that time. however, the research mentioned above all points to the fact that fractions, ratios and proportions are the most protracted in terms of development, the most difficult to teach, the most mathematically complex, the most cognitively challenging, the most essential to success in higher mathematics and science. (lamon, 2007, p. 629) tourniaire and pulos (1985) summarised the research done on the difficulties that children and adults have in reasoning with fractions, proportions and ratios and concluded: despite its importance in everyday situations, in the sciences and in the educational system, the concept of proportions is difficult. it is acquired late. … moreover, many adults do not exhibit mastery of the concept. (p. 181) they also claimed that we could expect the majority of learners to be able to successfully solve proportion problems only in late adolescence. lamon (2007, p. 637), even more dramatically, said that her ‘own estimate is that more than 90% of adults do not reason proportionally’. she went on to say that ‘many adults, including middle school teachers … and preservice teachers … struggle with the same concepts and hold the same primitive ideas and misconceptions as students do’ (lamon, 2007, p. 633). courtney-clarke and wessels (2014) found that only 25% of preservice teachers in a study in namibia could recognise the relative size of two common fractions in a comparison problem. one might assume, however, that the fraction of people who can reason proportionally would be greater within prospective higher education students than in the general population, but ‘proportional reasoning remains problematic for many college students’ (lawton, 1993, p. 460). in a study at us colleges, thornton and fuller (1981) found that only about 75% of science students had a sound grasp of the ratio concept and lawton (1993) found that only about half of the psychology students she studied could solve simple proportion problems. a recent south african study (harries & botha, 2013) of third-year medical students’ ability to perform dosage calculations involving proportions found that only 23% were able to do these correctly. clearly it is therefore necessary to make more sustained efforts to teach proportional reasoning at both school and higher education levels. lamon (2007) however pointed out that there are no quick fixes for students who have not developed a proportional reasoning ability appropriate to their stage of development. short-term teaching interventions ‘have been largely ineffective’ and ‘indicate that building fraction, ratio and proportion knowledge will involve a long-term learning process’ (lamon, 2007, p. 645). another very important observation made by lamon (2007) is that reasoning intuitively about proportions can be made more difficult for students by their having learned algorithmic methods in mathematics classes. she reported that young children have been shown to have quite powerful intuitive reasoning strategies, but that a few years of mathematics instruction weakens this ability and replaces it with rules and algorithms that are frequently applied incorrectly. this results in challenges in teaching for understanding. factors affecting the difficulty of proportional reasoning problems tourniaire and pulos (1985) and lamon (2007) reviewed the numerous studies of factors that influence the difficulty of proportional reasoning problems. an obvious factor affecting difficulty of problems is the size and type of numbers to be worked with. however, recognising this difficulty, researchers have often used easy numbers (less than 30) and integral ratios in their studies in order to avoid the interaction effect of difficult numbers (heller, post & behr, 1985). among the many other factors that have been studied are those of context and structure of the problems to be solved. the context of a task includes the event in which the task is situated and the language used to describe both task and event (van den heuvel-panhuizen, 2005). included in this notion of context are issues such as whether the measures used in the problem are discrete or continuous, whether the context of the problem involves mixtures (such as dilutions) and whether the event is familiar. tourniaire and pulos (1985) suggested that problem-solving strategies used by students in solving mixture and non-mixture problems are different and that discrete quantities in mixture problems are more easily visualised. lawton (1993) concluded that college students are more likely to use proportional reasoning when the measures used in a mixture problem are discrete. bell, fischbein and greer (1984) found that familiarity with the context is important, as seen in the competent handling of price stories by 12and 13-year-olds. heller et al. (1985) concluded that grade 7 students found the less familiar context of petrol consumption more difficult than that of speed. tourniaire and pulos (1985) agreed that familiarity with the context makes problems easier to solve, but only if the student is also familiar with using ratios in that context. bell et al.’s (1984) work on problem structure confirmed the known misconceptions about multiplication and division rate problems, such as that multiplication always makes bigger and division always makes smaller. they reported that division of a smaller number by a bigger one often leads to reversals of operations by students. in rate problems requiring division, the easiest to solve are those that are of the partition type (e.g. , where the distance can conceivably be partitioned into as many subsets as there are time units). problems in which the divisor is itself a rate (e.g. ), which are of the quotition type, are more difficult. in a small study of preservice elementary teachers, conner, harel and behr (1988) looked at the effect of two structural variables on the level of difficulty of missing value problems, namely the position of the missing value and the coordination of the measure spaces as stated in the question. the measure spaces contain the variables being considered in the question and they are coordinated when the variables are mentioned in the same sequence within the two statements making up the question. for example, the measure spaces (apples and cost) are coordinated in the question ‘if 5 apples cost r6, 8 apples cost how much?’ whereas in the question ‘if 5 apples cost r6, what will be the cost of 8 apples?’ they are not coordinated. the two statements making up the question can also be rearranged so that the open statement (containing only one value) comes before the closed statement (containing two values), for example ‘what will be the cost of 8 apples if 5 apples cost r6?’ the results of the study indicated that the coordination of measure spaces did not pose as much of a problem as did the position of the missing value within the open statement, as they hypothesised that students are able to transform the structure of the problem by changing the order of the quantities in the open statement so that the measure spaces are coordinated. however, in follow-up research harel and behr (1989) suggested that there may be differences in the strategies used in such problem-structure transformations by high performers and low performers. the results reported in this article highlight the effect of two factors affecting proportional problem difficulty that have been described above, namely context and problem structure. method ethical clearance for this project was obtained from our faculty research ethics committee in a process that included the approval of the consent form that is signed by all candidates of the nbt, allowing the use of their results for research purposes. before we describe the questions used in this research to investigate prospective students’ proportional reasoning, we need to clarify our terminology. there is some debate about the meanings of the terms ‘ratio’, ‘rate’, ‘fraction’ and ‘proportion’ (lamon, 2007), but in describing the structure of the questions, we will use only the terms ‘rate’ or ‘fraction’ to refer to any number in the form , where a and b can be any numbers or measurements (with b ≠ 0 ). a rate or fraction may be represented as a decimal fraction, a percentage or in some other conventional way. some examples are speed, cost per item, birth rate (births per 1 000 of the population) and crime rate (crimes per 100 000 of the population). in order to study learners’ proportional reasoning, we mainly used questions that did not require calculation, but reasoning based on an understanding of the way that changes in the values of numerator and denominator will affect the overall value of a fraction (or rate). we were not focusing narrowly on learners’ ability to work algorithmically with the concept of proportion, but on their reasoning; thus, only two of the nine questions required an answer to be calculated. in order to avoid the numerical difficulty effect, the numbers used in the questions were positive integers and mostly small in size; any large numbers used were simple multiples of powers of 10. these questions are examples of problems of both missing value and comparison types, but in some cases the latter are more complex than determining only the order of two fractions, requiring the values of denominators to be compared. these are examples of what harel et al. (1991, p. 127) described as ‘advanced multiplicative reasoning in which ratios and products are compared in terms of changes and compensations’. the questions were administered as part of the nbt academic literacy test, which includes assessment of quantitative literacy. the questions were multiple choice items with four alternative answers. the version of the test that included these items was written by 5 444 candidates in june 2014 and the alternative chosen by each candidate for each question was recorded. the questions were interspersed among trial items placed at the end of the test and did not contribute to candidates’ scores. the first three questions used in this study were missing value questions and had the same mathematical structure, being of the form: ‘if a is equivalent to b, then c is equivalent to what?’, where quantities a, b and c are given and the context is clearly one of direct proportion. the simplicity of the numbers and the familiarity of the contexts differed between questions. question 1 and question 2 were simple calculations in a familiar and less familiar context respectively, while question 3 required candidates to recognise the answer in the form of a mathematical expression, rather than to calculate it. in this case the context was also less familiar and the numbers did not divide easily. question 4, question 5 and question 6 were comparison type questions in which the numerator and denominator of four rates were given and candidates had to identify the rate with the biggest value. in question 6 they had to identify the largest of four common fractions. question 4 and question 5 were presented in a similar way, using the context of speed and tb infection rates (cases per 1 000 of the population) respectively. question 4 is reproduced below to illustrate how these questions were presented: the table summarises the distance travelled and the time taken by four different cars. which car drove at the fastest speed? car a car b car c car d distance (km) 110 110 100 100 time (minutes) 40 60 60 40 (a) car a            (b) car b            (c) car c            (d) car d question 7 and question 8 were presented in the same way as question 4 and question 5 and used the same or very similar contexts. however, for these questions the problem was not to compare the values of the rates, but to compare the values of the denominators (that is, time and population respectively). table 1 provides a summary of the relevant characteristics of these questions. table 1: summary of the characteristics of the proportional reasoning questions. the performance of the candidates on each of the above questions was described by determining the proportions who selected each alternative answer. these proportions were calculated for all candidates and then separately for the candidates in each of four performance bands. the performance bands were determined using the quartiles of the total scores for the 50 scored items in the version of the nbt quantitative literacy test (in which the research items were included as unscored items). so, for example, a candidate was classified as being in the highest quarter if their overall score for those 50 items was equal to or above the third quartile of all the overall scores and a candidate was in the lowest quarter if their overall result was less than or equal to the first quartile. results and discussion the percentages of all candidates selecting the different alternative answers for each question are listed in table 2. for each question, the percentage selecting the correct answer is given in bold. table 2: percentages of candidates (n = 5 444) who selected the different alternative answers to questions. correct answers in bold type. missing value problems question 1 was a straightforward, simply stated missing value problem (with uncoordinated measure spaces) using easily divisible small whole numbers and the familiar context of the cost of a number of items. if 4 items cost r5, how much will it cost to buy 20 items? (a) r1        (b) r16        (c) r20        (d) r25 we expected that all candidates would be able to solve this, probably intuitively, without consciously applying a learned procedure. however, nearly 20% of them did not solve it correctly and 13% chose the answer r20. in figure 1 it can be seen that, in fact, more than 20% of candidates in the lowest quarter chose this particular incorrect answer (alternative c). in total, nearly 30% of the candidates in the lowest quarter did not solve the problem correctly, which is a far higher proportion than the 6% of candidates in the highest quarter who answered incorrectly. this perhaps indicates a greater ability of the higher performers to transform the problem structure of uncoordinated measure spaces in order to facilitate reasoning, as suggested by harel and behr (1989). however, it appears that all candidates who answered incorrectly did not examine their answer critically in the context to see if it made sense. figure 1: results for missing value, by performance band. question 2 had the same mathematical structure as question 1, but used the less familiar context of the south african birth rate (expressed as number of births per 1 000 people) with the missing value being the population size. using this context also meant that the language would be less familiar and that larger numbers were used. however, these were still easily divisible whole numbers. also, due to the less familiar context the problem statement and presentation of data required three short sentences, thus increasing the language demands of the question. however, unlike the situation in question 1, the measure spaces were coordinated. from table 1 we see that only 61% answered correctly and that nearly 20% chose option d, which was an absurdly large number (80 000 000) for the population of a typical south african city, once again revealing that these candidates did not examine their answer critically in the light of the context. thus, the more complex and less familiar context affected candidates’ performance negatively, even though the problem was structurally the same and the measure spaces in this case were coordinated so that no transformation of the problem structure was required. in figure 1 it is clear that the negative effect of the change of context (and the associated complexity of the language and size of the numbers) on the performance of the candidates in the lower performance bands is greater than for those in the upper bands. question 3 was the same as question 1 and question 2 in terms of mathematical structure and, as for question 1, the measure spaces were uncoordinated: you need 50 g of a chemical to do 4 repetitions of an experiment. which of the following is the calculation for finding how many repetitions you can do if you have 125 g of the chemical available? (a)       (b)       (c)       (d) this question used a less familiar context and numbers that were simple whole numbers, but not divisible by each other. the most significant difference between this question and the previous two was that the alternative answers were in the form of fractional expressions representing the calculation of the answer, not the numerical answer itself. compared to question 1, less than half as many candidates could answer this correctly (only 38% of them). though many could do the calculation in question 1, the majority could not express the multiplicative procedure for a very similar calculation in the form of an expression (although some of the difference in performance can presumably be ascribed to the less familiar context and more complicated language of the question). this inability to recognise the symbolic representation of the relationships between the quantities involved is consistent with observations made by bell et al. (1984) in their study of young adolescents. nearly one-quarter of all candidates chose alternative a, which represents the product of the first two numbers given in the question, divided by the third. figure 1 shows that in the lower performance bands the percentage who chose option a was greater than the percentage who chose the correct answer. it appears that many of these writers may have been misapplying some learned method, without understanding the meaning of the problem. comparison problems (comparing rates) question 4, shown earlier in the ‘method’ section, was a simple problem requiring the comparison of rates in the context of speed, distance and time. four combinations of distance and time were given and the problem was to identify which represented the greatest speed. the numbers involved were simple, but not divisible by each other, making it necessary to solve the problem by reasoning that the fastest speed resulted from the greatest distance covered in the least time. from table 1 it can be seen that just under 80% of the candidates could do this correctly, but it is notable that over 10% chose an answer (alternative d) that represented the smallest distance covered in the shortest time. the relatively good performance on this question probably results from the familiar context that most candidates would have experienced in their daily lives. this question provides an example of a partition rate problem, which is considered to be easier than a quotition rate problem (bell et al., 1984). question 5 was identical in terms of mathematical structure to question 4, but situated in the less familiar context of number of tb cases per 1 000 of the population and requiring four short sentences to state the problem, which was presented as follows: the infection rate for tb is expressed in terms of ‘cases per 1 000 of the population’. this means that to find the rate the number of cases is divided by the population. the below information summarises the number of cases and the size of the population (number of people) for four communities. which community had the biggest infection rate? community a community b community c community d number of tb infections 10 11 10 11 number of people in the community 1000 1000 900 900 (a) community a    (b) community b    (c) community c    (d) community d once again the change to a perhaps less familiar and more complex and reading-intensive context that does not easily allow for intuitive reasoning has resulted in a decrease in the proportion answering correctly (64.8%). part of the complexity of the context in question 5 is that, in calculating the rate, a smaller number is divided by a bigger one and this could lead to a reversal of operations, as described by bell et al. (1984). as was the case for the missing value problems (question 1 and question 2), the change of context has adversely affected the candidates in the lower performance bands and hardly affected those in the highest quarter (compare the charts for question 4 and question 5 in figure 2). figure 2: results for comparison problems requiring the comparison of the rates, by performance band. question 6 was context-free and required candidates to identify the largest of four fractions as follows: which of the following is the biggest number? (do not do any calculations.) (a)           (b)           (c)           (d) this question has the same mathematical structure as question 4 and question 5, but the answers are given as a symbolic mathematical representation. as was the case for question 3, where the calculation for a missing value question was given in the form of an expression, this fractional representation appears to have made the problem more difficult, though not for the candidates in the highest quarter (see figure 2). only 58% of all candidates could identify the largest fraction. this means that 4 in every 10 candidates aspiring to higher education do not appreciate that a smaller denominator and a larger numerator will produce a larger fraction. almost a quarter of candidates chose alternative a, which had the smallest numerator and the smallest denominator, and 10% chose the fraction with the largest numerator and denominator. the performance was considerably worse in this context-free question, which required applying an understanding of the mathematical concept of fractions, than in both question 4 and question 5, where the problem was presented in a context. even though a more complex context is seen to make a problem more difficult, it still appears that the presence of a context facilitates the reasoning. for both missing value problems (question 1, question 2 and question 3) and comparison problems (question 4, question 5 and question 6), there is a similar pattern of decreasing performance overall (see table 1) and similar differences between performance bands (see figure 1 and the first three charts in figure 2). so, as the contexts of the problems become increasingly complex, and when alternative answers are given as mathematical expressions, there is a decrease in performance. question 7 was also a comparison problem, and similar to question 6, but somewhat different in that the data was given in the form of a bar chart with stacked bars of different heights, each stack showing two subsets of a total set. in addition, the alternative answers required the candidates to choose the correct explanation of the reasoning involved in the identification of the stack in which a specified subset was the smallest fraction. the question required writers to identify the subset that was ‘the smallest proportion of the total’. only 34% of the candidates answered this item correctly (see table 1). more than 50% of all candidates (and nearly 70% in the lowest quarter and 31% in the top quarter) chose answers that reflected reasoning with absolute quantities rather than relative quantities. these candidates ignored the word ‘proportion’ and effectively selected the smallest numerator (the shortest bar in the chart). in their literature review, tourniaire and pulos (1985, p. 185) noted that this strategy of simply comparing the numerators (and ignoring the denominators) is ‘a developmentally primitive strategy’, but that ‘it is still used with some frequency in adolescence’. we have also seen that this strategy is used by about half of the students in a university quantitative literacy course for law students (lloyd & frith, 2013). the results for question 7 reveal that even in the top quarter, under two-thirds of the candidates could successfully compare fractions by reasoning alone (admittedly in a fairly complex, reading-intensive question). in comparing the performance in question 6 and question 7 (58% and 34% of candidates answering correctly, respectively), it once again appears that the graphical presentation of the data and the alternatives being in a less familiar format than the numerical representation have affected performance adversely. comparison problems (comparing denominators) while questions 4 to 7 were more traditional comparison problems requiring the comparison of the rates (the size of the fractions), question 8 and question 9 required the sizes of the denominators to be compared. they were presented in the same way as question 4 and question 5 (set in the context of speed-distance-time and crime rate respectively), but, this time, with the numerators and rates given. thus in question 8 four pairs of distance and speed values were given and the problem was to identify the longest time, while in question 9 four pairs of numbers of crimes and crime rates were given and the problem was to identify the biggest population. in both of these problems the performance was worse than in the equivalent problems requiring comparison of the rates (see table 1). only 51% of all candidates answered question 8 correctly (compared to 78% for question 4) and only 25% answered question 9 correctly (compared to 65% for question 5). it is clear that this change to the structure of the problems has made them much more difficult. in addition, the (perhaps) less familiar context of crime rate made question 9 more difficult than question 8 which used the context of speed-distance-time. we do, however, recognise that question 9 was the second last item in a very long test and the number of candidates who left it out was greater than for other items. in question 8 the alternative that represented the largest speed and largest distance (c) was the second most popular choice after the correct answer. the equivalent (incorrect) alternative was even more frequently chosen in question 9, where the one representing the largest rate and largest numerator (b) was chosen by 41% (compared to 25% choosing the correct alternative). for this question it was only in the highest performance band where more candidates (45%) chose the correct answer than alternative b (33%), and even in this case it was not the majority (see figure 3). the other alternative that was more popular than the correct answer with the lower performance bands was d, which represented fewer crimes and a bigger crime rate. figure 3: results for comparison problems requiring the comparison of the denominators, by performance band. it is clear from these results that the structure of these comparison problems affects difficulty. although we have not found evidence of research on this type of problem, research on missing value problems has shown that the position in the fraction (numerator or denominator) of the number to be found affects the difficulty (conner et al., 1988). in a similar way, we see that when the focus is on the denominators the comparison problem becomes more difficult than when the focus is on the fraction as a whole. we are confident that the large sample lends credibility to our observations, but we acknowledge that there are shortcomings in both the format of the questions and the form of the data used for analysis. the format of multiple choice questions could impact the performance of candidates with little experience in this form of assessment. also, the data consists only of the quantitative results of students’ performance on a limited number of multiple choice questions written at the end of a long test. clearly, the data would benefit from being supplemented by qualitative information, but this was beyond the scope of this study. conclusion it is widely accepted across different levels of the education sector that quantitative literacy is important for critical citizenship, a feature of which is the ability to engage confidently with social and other data. appropriate application of proportional reasoning is often essential for gaining a full understanding of data in society and is also an important requirement for many academic disciplines. this study has shown that many students aspiring to higher education perform poorly on proportional reasoning questions. this is not wholly unexpected in light of the literature reporting, on the proportional reasoning abilities of mainly young children and early adolescents. however some of the difficulties experienced were greater than we anticipated. only approximately 80% of candidates could identify the correct answer to the simplest question involving calculating a missing value in a highly familiar context using easy numbers. in some questions requiring more complex proportional reasoning, less than half could answer correctly. two factors that we have seen to affect the difficulty of proportional reasoning problems are the context and structure of the question. embedded in the notion of context are issues of familiarity with the event in which the task is situated and the associated language required to describe the event and the task, as well as the types of numbers used. our study indicated that contexts that are likely to be less familiar to candidates, or are more complex in terms of number type or language demands, make problems more difficult. the structure of a problem includes the type of problem in which quantities are to be reasoned about (missing value or comparison), whether there is coordination of the measure spaces and the location of the missing value. the performance on missing value and comparison types in comparable contexts is similar, but, in the case of comparison problems the difficulty is influenced by the location of the quantity to be reasoned about: whether it is the denominator or the fraction itself. in both missing value and comparison problems the representation of the answers, as mathematical expressions or as sentences that explain the reasoning involved, has the most adverse effect on performance. this appears to indicate that many candidates lacked a conceptual understanding of the mathematics involved. the vast body of literature indicates that the development of proportional reasoning requires repeated exposure to problems in a variety of contexts over a long period of time. although the curricula for mathematics and mathematical literacy specify dealing with proportions, the explicit teaching of this does not go beyond grade 9 (in the case of mathematics) or grade 10 (in the case of mathematical literacy). the emphasis appears to be on solving missing value problems where calculations are done, presumably, using a calculator and by means of a learned algorithm. the results of this study support the existing research that shows that the intentional development of proportional reasoning should begin in the intermediate phase and continue into late adolescence and beyond, in the case of higher education. a greater focus needs to be on reasoning about proportions (rather than application of algorithms), in different contexts, with exposure to a variety of ways of presenting problems, in terms of both the language used and the representation of the data provided. opportunities for the development and practice of this kind of reasoning should be exploited at all levels across the curriculum. acknowledgement we thank the nbt project team at the centre for educational testing for access and placement at the university of cape town who provided the opportunity to conduct this research, with the goal of contributing to the nbt project’s purpose of assessing the relationship between entry level proficiencies and school-level exit outcomes. authors’ contributions both authors contributed equally to the conceptualisation and realisation of the research project and the writing of the article. competing interests the authors declare that they have no financial or personal relationship(s) that may have inappropriately influenced them in writing this article. references bell, a., fischbein, e., & greer, b. 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(1985). proportional reasoning: a review of the literature. educational studies in mathematics, 16, 181–204. van den heuvel-panhuizen, m. (2005). the role of contexts in assessment problems in mathematics. for the learning of mathematics, 25(2), 2–9, 23. available from http://www.jstor.org/stable/40248489 article information authors: eunice k. moru1 makomosela qhobela2 poka wetsi1 john nchejane1 affiliations: 1department of mathematics and computer science, national university of lesotho, lesotho 2department of science education, national university of lesotho, lesotho correspondence to: eunice moru. email: ekmoru@yahoo.com postal address: department of mathematics and computer science, national university of lesotho, maseru, lesotho dates: received: 26 mar. 2014 accepted: 17 nov. 2014 published: 12 dec. 2014 how to cite this article: moru, e.k., qhobela, m., poka, w., & nchejane, j. (2014). teacher knowledge of error analysis in differential calculus. pythagoras, 35(2), art. #263, 10 pages. http://dx.doi.org/10.4102/pythagoras.v35i2.263 copyright notice: © 2014. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. teacher knowledge of error analysis in differential calculus in this original research... open access • abstract • introduction    • statement of the problem       • research questions • theoretical considerations and literature review    • error types in mathematics    • students’ errors in differential calculus    • teacher knowledge    • teacher knowledge for error analysis • methodology    • the sample    • questionnaires    • interviews    • data analysis       • ethical considerations       • reliability and validity • findings    • teachers identified the errors correctly    • the teachers’ generalised error identification resulted in opaque analysis    • some teachers’ interpretation lacked multiple perspectives    • the teachers’ evaluation of errors was either local or global    • when remedying some errors the teachers emphasised accuracy and efficiency more than conceptual understanding • discussion • conclusion • acknowledgements    • competing interests    • authors’ contributions • references abstract top ↑ the study investigated teacher knowledge of error analysis in differential calculus. two teachers were the sample of the study: one a subject specialist and the other a mathematics education specialist. questionnaires and interviews were used for data collection. the findings of the study reflect that the teachers’ knowledge of error analysis was characterised by the following assertions, which are backed up with some evidence: (1) teachers identified the errors correctly, (2) the generalised error identification resulted in opaque analysis, (3) some of the identified errors were not interpreted from multiple perspectives, (4) teachers’ evaluation of errors was either local or global and (5) in remedying errors accuracy and efficiency were emphasised more than conceptual understanding. the implications of the findings of the study for teaching include engaging in error analysis continuously as this is one way of improving knowledge for teaching. introduction top ↑ aside from the challenges of a variety and the complexities of students’ errors, analysing such errors is a fundamental aspect of teaching for mathematics teachers (ball, thames & phelps, 2008; brodie, 2014; legutko, 2008; makonye, 2012; olivier, 1989; peng & luo, 2009; shalem, sapire & sorto, 2014). students’ errors are complex because they are a function of many variables. they may originate from the students themselves, the teachers, the curriculum, the environment or the interaction amongst these variables (radatz, 1979). so it is difficult to impute an error to any one variable. the way students build up their knowledge is also complex. the constructivist views of learning assert that students learn by transforming and refining their prior knowledge into more sophisticated concepts (smith, disessa & roschelle, 1993). so the understanding that prior knowledge is the main resource of knowledge construction is necessary for the teacher. the teacher also needs to understand that this prior knowledge has conceptions that may conflict with the new ideas that the teacher hopes students will learn (davis & vinner, 1986). effective teaching therefore requires a teacher with the understanding that some conceptions though flawed in one context may be productive in another (brousseau, 1997; smith et al., 1993). the task of identifying such contexts is important to the teacher so that students are not encouraged to make errors that are a result of an overgeneralisation of certain conceptions. statement of the problem ‘calculus is an important subject area within mathematics, and this underlies the argument for introducing it to non-specialists’ (orton, 1983, p. 235). however, students continue to encounter problems in learning some of its basic concepts (makonye, 2012; orton, 1983; thompson, 1994). this has called to the attention of both the mathematics educators and researchers the need to find ways in which mathematics teaching may be improved (ball et al., 2008; brodie, 2014; legutko, 2008; olivier, 1989; shalem et al., 2014). as in other parts of the world students at the national university of lesotho encounter problems in their learning of calculus and in most cases some of the reasons that are given for bad performance in the departmental meetings include: ‘[w]e have poor quality of students’, ‘students are lazy to learn’ and ‘students are not serious about their studies’. whilst to some extent these may be true we cannot totally place the blame of poor performance on students alone; in one way or another teachers may also contribute. over the years we have observed that when teachers mark students’ work the focus is more on the correctness or incorrectness of an answer: where the student has committed an error that stage will be marked with a cross and where the correct answer is given a tick is put instead. teachers seem to be less concerned about knowing the origin of errors. the question to ask is: how effective can teaching be in remedying an error that has not been diagnosed? as highlighted earlier students’ errors are a function of many variables such as the student, the teacher, the curriculum, the environment and their interaction, but as a matter of focus we concentrated on just one variable, the teacher, by engaging in a study that has the potential to improve teachers’ knowledge of teaching. since error analysis is one way to contribute to effective teaching of mathematics (brodie, 2014; moru & qhobela, 2013; olivier, 1989; peng & luo, 2009; shalem et al., 2014), we thought that engaging in a study that investigated teacher knowledge of error analysis would be of help in alleviating the problem. this is because engaging in such a study would force teachers to think deeply about the errors students commit not only in calculus but in other courses as well. sharing ideas during the development of the article would also force teachers to start talking about students’ errors and may thus cause them to start paying attention to them. the hope is that once a teacher is introduced to an effective way of teaching, they would start looking for more ways of effective teaching. hence, these may help in alleviating the problem of unsatisfactory performance by students in mathematics. research questions the main research question for the reported study is: what teacher knowledge of error analysis do the teachers possess? in particular: how do the teachers identify students’ errors? how do the teachers interpret students’ errors? how do the teachers evaluate students’ errors? how do the teachers remedy the students’ errors? the four sub-questions above are in line with peng and luo's (2009) four error phrases for error analysis, which are used in the study. these are: identify, interpret, evaluate and remediate. identification means knowing of the existence of the error. interpreting the error means knowing the underlying rationality or the possible causes of an error; evaluation involves assessing students’ levels of performance according to the error and remediation demands presentation of teaching strategy to eliminate the error. theoretical considerations and literature review top ↑ this section discusses the key ideas needed in the interpretation of the results of the study, namely error types in mathematics, students’ errors in differential calculus, teacher knowledge and teacher knowledge for error analysis. error types in mathematics olivier (1989) differentiates errors from slips and misconceptions. he defines an error as wrong answers due to planning that are systematic in that they are applied regularly in the same circumstances. slips on the other hand are described as wrong answers due to processing. unlike errors they are not systematic but are sporadically and carelessly made by both experts and novices. they are easily detected and corrected. misconceptions are referred to as underlying conceptual structures that give rise to errors. thus it could be argued that errors are indicators of the existence of misconceptions. this is the sense in which errors will be perceived in this article. errors have been classified differently by various researchers and mathematics educators. legutko (2008) classifies them as mathematical and didactical: a mathematical error is made by a person (student, teacher) who in a given moment considers as true an untrue mathematical sentence or considers an untrue sentence as mathematically true. didactic errors refer to a situation when teachers’ behavior is contradictory to the didactic, methodological and common sense guidelines. (p. 149) mathematical errors include giving an incorrect definition of a mathematical concept and a wrong application of the definition, making a generalisation after observing a few particular cases and incorrect use of mathematical terms. didactical errors include unsuitable selection of examples used in the formation of a concept, incoherent structure of teaching such as teaching concepts of a higher order before concepts of lower order. orton (1983) classified mathematical errors as structural, executive and arbitrary: structural errors were described as those which arose from some failure to appreciate the relationships involved in the problem or to grasp some principle essential to solution. executive errors were those which involved failure to carry out manipulations, though the principles involved may have been understood. arbitrary errors were said to be those in which the subject behaved arbitrarily and failed to take account of the constraints laid down in what was given. (p. 4) mathematical errors may also be procedural or conceptual (eisenhart et al., 1993; long, 2005; shalem et al., 2014). procedural errors are associated with procedural knowledge and conceptual errors are associated with conceptual knowledge. procedural knowledge refers to mastery of computational skills and knowledge of procedures for identifying mathematical components, algorithms, and definitions. … conceptual knowledge refers to knowledge of the underlying structure of mathematics – the relationships and interconnections of ideas that explain and give meaning to mathematical procedures. (eisenhart et al., 1993, p. 9) to these we can add that procedural knowledge has two components: (1) knowledge of the format and syntax of the symbol representation system and (2) knowledge of rules and algorithms, some of which are symbolic, that can be used to complete mathematical tasks. it could therefore be argued that ‘the fluent execution of algorithm represents an aspect of procedural fluency’ (long, 2005, p. 63). however, algorithms in themselves cannot be said to be devoid of mathematical concepts and if conceptual features inherent in the algorithms are not fully grasped, errors are likely to occur (long, 2005). this suggests that it is not easy to categorise an error as either conceptual or procedural. the categorisation given to an error therefore has to be supported by a justification, which may clarify how the classification has come about. students’ errors in differential calculus in a study by orton (1983) students were asked to find the stationary points of the curve y=x3 − 3x2 + 4, that is, the points on the graph where the gradient is zero. students in this study managed to find the gradient function (the derivative) f′(x) = 3x2−6x, but in solving for x values of stationary points in 3x2 − 6x − 0, 24 students out of 110 solved the equation incorrectly. half (12) of the 24 students lost one solution x = 0 by dividing throughout by x to get 3x − 6 = 0 whilst six incorrectly factored 3x2 − 6x = 0 into 3x(x − 6) = 0 students who lost one solution were not aware that dividing both sides of an equation by an expression involving a variable may not produce equivalent equations. the equation 3x2 − 6x = 0 has two solutions, x = 0 and x = 2, from 3x(x − 2) = 0 whilst 3x − 6 = 0 has only one solution, x = 2 and is therefore not equivalent to 3x2 − 6x = 0 thus, the errors were classified as both structural and executive. an executive error was further committed when finding the gradient of the tangent to the curve at x = 3: although the 3 was substituted correctly numerical manipulations were incorrect. in a differentiation task of the same study when expanding 3(a + h)2 students lost the middle term 6ah, this was possibly due to the fact that in most cases students’ expansion of (a + h)2 is given as a2 + h2. students distribute the power over the brackets as would be the case in (ah)2 = a2h2. students should have thought in reverse by realising that in real numbers one cannot find the factors of the sum of two squares. but in complex numbers such roots would exist because (a + hi) (a − hi) = a2 + h2 since by definition i2 = −1. in a study by thompson (1994) students interpreted the difference quotient an average rate of change or the gradient over an interval, as the derivative instead of the derivative being defined as an instantaneous rate of change or the gradient at a point. in makonye (2012) students were asked to find the derivative of f(x) = −3x2 from the first principles (i.e. using the definition of derivative); the students failed to make proper substitution for f(x + h). for example, instead of substituting −3(x + h)2 for f(x + h), they substituted −x(x + h − 3) students had a problem with evaluating function values. in the same study when finding the gradient of the tangent line (the derivative) to the curve at point, the gradient of the chord (a portion of the secant line), which is the average rate of change, was computed instead. this matches with the results of thompson's study where the difference quotient used to find the gradient of the chord was interpreted as the derivative, which is the limit of the difference quotient as teacher knowledge the two kinds of teacher knowledge that have received much attention in the work of shulman (1986, 1987) are subject matter content knowledge (smk) and pedagogical content knowledge (pck). in differentiating between the two kinds of knowledge, shulman defines smk as the amount and organisation of knowledge in the mind of the teacher. this type of knowledge he suggests may be equal to that of a colleague who is a subject matter major whereas pck goes beyond the subject matter to the dimension of subject matter knowledge for teaching. he defines pck as follows: it represents the blending of content and pedagogy into an understanding of how particular topics, problems, or issues are organized, represented, and adapted to the diverse interests and abilities of learners, and presented for instruction. (shulman, 1987, p. 8) this seems to suggest that smk is necessary in teaching but it is not a sufficient condition. for smk to operate in teaching it must be blended with pedagogy. in teaching, for example when a teacher uses a representation in explaining a concept, it means that one can now talk about the knowledge of teaching as the activity is now beyond a mere possession of subject matter knowledge. ball et al. (2008) developed a practice-based theory of content knowledge for teaching from the work of shulman. their research findings indicate that pck has at least two sub-domains, knowledge of content and students and knowledge of content and teaching, whilst smk could be divided into common content knowledge and specialised content knowledge. they define common content knowledge as the mathematical knowledge and skill used in settings other than teaching, which involves knowing the material to be taught and recognising when students give wrong answers or when the textbook gives an inaccurate definition. being able to size up the nature or cause of an error, in particular an unfamiliar error, belongs to specialised content knowledge. familiarity with common students’ errors and deciding which of the several errors students are likely to commit are examples of knowledge of content and students. it also involves knowledge of common students’ conceptions and misconceptions about mathematical content. someone with knowledge of content and teaching can sequence topics, choose examples to use in deepening students’ understanding and evaluate the advantages and disadvantages of representations used to teach a specific area and identify what different methods and procedures are appropriate for instruction. according to moru and qhobela (2013) teacher knowledge of error analysis should also include multiple interpretations of students’ errors. this is because students have diverse experiences and ways of thinking; hence, it is not always possible for the same error to originate from the same source. multiple interpretations will accommodate a variety of remedial strategies. they add that such knowledge should also involve asking students to explain their answers because correct answers may even originate from incorrect steps or procedures. however, the opportunity for students to explain their answers is not always possible, especially with large class sizes or when someone is marking an assignment in the absence of students; hence, a teacher should also be able to contemplate the sources of such errors. shalem et al.'s (2014) elements for error analysis include awareness of an error, diagnostic reasoning of learners’ thinking in relation to an error, use of everyday links in explanations of errors and multiple explanations of errors. these elements overlap with peng and luo's (2009) identification and interpretation and moru and qhobela's (2013) interpretation of an error from multiple perspectives. the use of everyday links in explanations of error is very important because new knowledge is built on the already existing conceptual structure of which everyday experiences are part. teacher knowledge for error analysis in investigating teacher knowledge for error analysis, peng and luo (2009) used the four error phrases identify, interpret, evaluate and remediate. in their study, they found out that teachers managed to identify a student's error, ‘but interpreted it with wrong mathematical knowledge, which led to meaningless evaluation of the student’ performance and unspecific presentation of teaching strategy’ (p. 24). in the same study, the task given to students was: if and calculate from the given information the student constructed the equations and , then solved simultaneously. the teachers were aware that according to the rules of limits the constructed equations were incorrect. they however objected to their use with improper reasoning. in a study by moru and qhobela (2013) teachers were given an opportunity to analyse some common student errors in sets. in this study teachers managed to identify the errors of the content they were familiar with (e.g. empty set, union of sets, intersection of sets, etc.); they however failed to identify errors for the content that they were not familiar with (e.g. treating infinity as a number and cardinality of infinite sets). depending on the nature of students’ tasks teachers’ strategies and explanations of dealing with the errors were inclined towards calling on procedural knowledge. only a few cases of conceptual knowledge were noted. in a study by turnuklu and yesildere (2007) teachers had sound subject matter knowledge of primary mathematics (directed numbers and fractions), but did not have adequate knowledge for the teaching of mathematics, which implies that having a deep understanding of mathematical knowledge is necessary but not sufficient to teach mathematics. methodology top ↑ the sample the sample consisted of two lecturers who are also the co-authors of this article. these lecturers were involved in the teaching of calculus in the department of mathematics and computer science at a university in lesotho. both lecturers (abbreviated t1 and t2) had also conducted tutorials for social science students. their teaching experience ranged from two to three years. t1 is a subject matter specialist whilst t2 is a mathematics education specialist. because lecturing is a form of teaching, the lecturers are sometimes referred to as teachers in this article. questionnaires the first draft of the questionnaire was constructed by the first author. the second author critiqued the questionnaire and continuous discussions occurred between them until the final draft was arrived at. the questionnaire was administered by the first author. each research participant (t1 and t2) had one month to complete the questionnaire as it was very demanding in terms of time and thinking. the questionnaire consisted of errors committed by the majority of second-year social science students taught by the first author. in total, 103 students sat the course examination. the examination was taken in may 2013. in the teaching the students had covered amongst others the following content: concept of limits, definition of derivative and finding the derivatives of functions either by the use of the definition or by differentiation rules, geometrical and algebraic interpretation of derivatives as gradient functions and finding equations of tangent lines to the curve. the questions in which the students committed most errors in the examination are: question 1: use the definition of derivative to find f′(x) for f(x) = 3 question 2: find all the points on the curve where the tangent is horizontal almost half of the students (49%) committed errors when responding to question 1 and 61 students (59%) committed errors when responding to question 2. the sample student responses were chosen by the first and the second authors who further selected the scripts in such a way that there was some variation in the committed errors. for teachers’ analysis of students’ errors the teachers were given a questionnaire consisting of a question and the corresponding student response. they were asked to study the question and the corresponding student response and then asked to perform the following tasks: (1) identify the error, (2) write the possible causes of the identified error(s), (3) show how the identified error can impact on the mathematical performance of the student either in doing or learning mathematics and (4) suggest the strategies or explanation that one would provide in remedying the error. interviews the follow-up structured interviews were constructed and conducted by the first author. the interviews emerged from the teachers’ responses to the questionnaire. these were conducted with the intention of seeking clarification on some questionnaire responses. the expectation was that these interviews would also start a conversation amongst colleagues about the importance of paying attention to students’ errors. the interviews for teacher knowledge of error analysis for each question were held separately with the individual lecturers (t1 and t2). on average the first stage took one and a half hours and the second just one hour. this was with the intention of achieving in-depth data with regard to the teachers’ knowledge of analysis of students’ error. data analysis the analysis of the teachers’ knowledge of students’ errors was done by the first and the second authors. they were investigated in their ability to: (1) identify the error, (2) suggest the possible causes of the error, (3) judge how the error may impact on the performance of the student in doing or learning mathematics and (4) offer remedial strategies for the errors. after studying the teacher knowledge for error analysis, assertions for the displayed knowledge were constructed. ethical considerations permission to use students’ examination scripts was sought from the head of the department of mathematics and computer science. this is because students’ examination scripts are the property of the university. the agreement made was to conceal the students’ identities. with regard to the involvement of t1 and t2 their informed consent was sought by the first author. reliability and validity data was collected by the first author for both the questionnaires and interviews. this is because the first author had already established a rapport with the two lecturers who are not only colleagues in the department but also former students of the first author. to ensure reliability of the research instruments the second author assessed the clarity of the questions to see if they would be interpreted the same by different people for consistency of results. the suggested comments were discussed until an agreement was reached. t1's and t2's responses to the questionnaire and interviews show that the study did produce consistent results. this is because the two lecturers interpreted the questions and instructions in the questionnaire and interviews in the intended way. in order to verify the validity of the questionnaire, it was given to the second author together with the research questions and the four error phrases by peng and luo (2009) to assess if the questionnaire had the potential to provide data that could answer the research questions and also to see if the potential data could be explained in terms of the four error phrases. author 3 and author 4 (t1 and t2) were not involved at this stage as they were the sample of the study. questionnaires and interviews were used to allow their data to complement each other. after the first stage of data analysis, which involved the first and the second author, each teacher was given the first draft of the article. the teachers were asked to check if the interpretation given represented their views (respondent validity). they were asked to point at areas where they felt the analysis was not a true reflection of their work. the process was repeated twice. during the last stage each member of the research team was given the document to judge coherence and inconsistencies, if any. low inference descriptors were also used (seale, 1999). these include verbatim accounts of what people say, for example, rather than the researchers’ reconstructions of the general sense of what a person said, which would allow researchers’ personal perspectives to influence the reporting. (seale, 1999, p. 148) findings top ↑ data analysis of teacher knowledge of error analysis yielded the following five assertions: (1) teachers identified the errors correctly, (2) the generalised error identification resulted in opaque analysis, (3) some of the identified errors were not interpreted from multiple perspectives, (4) teachers’ evaluation of students’ errors was either local or global and (5) in remedying errors accuracy and efficiency were emphasised more than conceptual understanding. the presentation follows the order in which the assertions have been listed. teachers identified the errors correctly in all students’ work, the teachers managed to identify almost all the errors. a few that were left during the completion of the questionnaire were recovered during the interviews. this was a sign that failure to identify the errors in the first stage of data collection was not a sign of lack of knowledge but just a slip on the part of the teachers. the first example (see figure 1) shows how one of the errors left out was recovered by t2 when discussing his analysis of s74's work. this will be followed by an example from s61's work showing how t1 recovered the errors he had not identified during the first stage of data collection. figure 1: error analysis of s74's response. the question asked is now followed by a response: researcher (r): can we look at the minus f (x) (i.e. the numerator of the difference quotient). t2: oh! and the minus f (x), even though i did not realise, is another problem because the minus has been distributed on the first term and not on the second term. it is a recurring problem. which means that the student throughout used the bracket to distribute on the first term only. this shows that an error that was left out was just an omission and not related to lack of subject matter knowledge. the same thing happened to t1 when discussing his analysis of s61's work (see figure 2). figure 2: error analysis of s61's response. thus, the discussion during t1's interview was directed towards wanting to see if he would recognise that he had left some errors that appeared before line 3. the discussion with him went as follows: r: you have shown that the errors committed start from line 3 downwards, so how do you think the expression in line 3 is related to line 2 and maybe also to line 1? t1: i was thinking it is related to line 2 but to write f (x) = 3 followed by is wrong. but even if we were to relate it to line 1 to say that is also wrong. it should be everything went wrong from the beginning. the limit operator , is missing at every stage. the results show that the errors were omitted not because of lack of knowledge but because of human error or lack of concentration. other errors that were identified but have not been presented include: (1) incorrect substitution, (2) improper division by zero and (3) losing one solution by dividing by a variable. incorrect substitution is the error that was committed by students in the studies by thompson (1994) and makonye (2012). losing one solution by dividing by a variable was an error committed by students in the study by orton (1983). the teachers’ generalised error identification resulted in opaque analysis in figure 3, t1 says that every step in the question is wrong. this however seems problematic because the particular error has not been identified but only the general error. the problem also arises with the other parts of the framework, evaluation and remedy. he says that the impact is that the student will fail to understand mathematics in general and in remedying the error the whole concept of tangent has to be revisited: r: here you say that every step is wrong. is it everything or some parts? t1: i mean everything is wrong in the sense that in step 1 the function is equated to zero. so here the student is not answering the question and therefore everything is wrong. the student is finding the roots of the function and not the point where the gradient of the tangent is zero. figure 3: error identification in s57's response. t1 bases his answer on the fact that the student writes the wrong thing in step 1. so any working that results from an error also becomes an error, although not of the same kind as the first. so the problem here was that there is no step that t1 could identify as correct according to the demands of the question. a closer look at this analysis reflects that step 1 (or line 1) is the main error as everything that follows emanates from it. the discussion continues: r: in remedying the error you say that the whole concept of tangent has to be revisited. when you say the whole concept of tangent what do you mean by this? t1: the whole means to start a little bit behind. i talk about everything that will lead us here. r: that will lead us where? t1: that will lead us to finding where the gradient is zero. this student has the problem with the functions so the student has to know that a function is different from a tangent. though t1 has now identified the main error as that of finding the zeros of the function instead of the points where the gradient of the tangent line is zero, what t1 believes needs to be done is still not clear to the researcher. to say the whole concept of tangent means ‘to start a little bit behind’ does not explicitly give us information about where to start. thus, the analysis has been characterised as opaque because one cannot see through the intentions of the teacher. some teachers’ interpretation lacked multiple perspectives after identifying the errors, the teachers interpreted the errors from a single perspective instead of from multiple perspectives. this resulted in also offering remedial strategies that were confined to a single interpretation. interpretation from multiple perspectives is important because it is unlikely that the same error could be committed by students who had the same type of thinking. this is because students are individuals whose thinking also varies. the teachers’ analysis of s25's work (figure 4) demonstrates the stated assertion. figure 4: error identification in s25's response. t2 shows the error in lines 3 and 4 to be that the student does not understand what it means to say that the ‘tangent is horizontal’. this means the point on the tangent line where the gradient is 0 and not where x = 0. this is similar to t1's interpretation that the student is not able to differentiate the tangent with the x-coordinate. in an interview he said: t2: i think the student missed that part of what becomes zero for the tangent to be horizontal. what has to be zero is the derivative and not the value of x. t2 suggests that the student knew that an equation involved a zero but did not know which part exactly was a zero. this is a sign that the student could not connect the procedure to the concepts surrounding the procedure. an alternative perspective to this is that since each term of the slope function had an x as a factor, it is possible that the student might have determined the solution by inspection and then substituted 0 as a matter of completeness. the danger of this method is the other solution is lost and in terms of presentation it is not very clear how the solution was arrived at because the writing may not always explicitly reflect the thought process. since both teachers’ interpretations did not observe alternative sources of the errors, an opportunity of introducing teaching strategies that could address the errors from a different perspective was missed. the teachers’ evaluation of errors was either local or global in some cases, the teachers’ evaluation of students’ errors was local whilst global in others. local evaluation in this case means looking at an impact of an error by making reference to the mathematical content that is directly related to it. global means judging the impact of an error by not being topic specific but by making reference to an aspect of mathematics that cuts across topics (see table 1). the examples in table 1 are taken from the work of s25, s37 and s57. table 1: identified errors with corresponding teachers’ evaluation. table 1 shows that t1 is local about the impact that the error committed by s25 will have in learning or doing mathematics and is global about the one committed by s57. he shows that s25 will not manage to solve problems related to tangents even when the skills and tools are available. in an interview he said that by skills and tools here he meant that the student knew that they had to differentiate and also how to differentiate but the problem here is that of not knowing what it means to say the tangent is horizontal. with regard to s57 he is not specific about the content that will give problems; rather, he considers that whole danger to be in the mathematics that the student is exposed to. t2 goes global with s37 and local with s57. his evaluation of s37's error is that it will put s37 in danger of carrying out wrong calculations in mathematics in general. however, with s57 he becomes more local by relating the problem to specific concepts the student has not mastered: intercepts and slope of the tangent line. the two types of evaluation have a place in the teaching of mathematics. the local view leads to the remedying of the problem by using immediate related concepts within a particular topic and the global view cuts across topics. some knowledge and skills in mathematics have to be acquired independently of a particular topic because they are a necessity in doing mathematics in general. however, they end up being pictured globally because they first started being applied locally. thus, the two perspectives complement each other in this regard and are justified in the context of mathematics. when remedying some errors the teachers emphasised accuracy and efficiency more than conceptual understanding accuracy is an important aspect of mathematics teaching and learning. however, it can only be mastered meaningfully if other methods that do not produce accurate results are appreciated and used to develop an understanding of why they are preferred in certain situations but not others. in the analysis shown in figure 5 a student uses a graph that is not drawn to scale to solve a problem. the solution points are thus not accurate. the teachers suggest that the student should be shown more accurate and faster methods without convincing the student why their method is not appreciated in this particular case. figure 5: error identification in s37's response. t1 suggests that the student should be shown simpler equivalent methods that are more accurate as the graphical methods make mathematics appear more difficult than it actually is. in an interview he said: t1: we are learning calculus so that we can have shorter ways. … in calculus we want to get the solutions quicker and faster. the emphasis here is not on where the student is in terms of understanding or thinking. the efficiency of the method is the focus, rather than the understanding of the student. when asked if the graphical method has to be discouraged completely he said: t1: i am not against it (graphical method). it is just that it has lots of steps and once you have lots of steps. there is high probability that you may get lost before you get to the answer. although t1 says that he is not against the graphical method, he does not seem to be convinced about its usefulness. hence, the tone set is to discourage students from using it. geometrical representation is a very powerful representation of mathematical concepts and thus cannot just be dismissed. what both teachers fail to do in this case is to address the problem from the student's perspective. the student is not shown how to draw an accurate graph and from there move to more accurate methods after showing that no matter how accurately one tries to draw the graph there will always be other sources of error such as the sharpness of the pencil used in drawing, the scale chosen and the parallax error when taking readings. we believe that the student could appreciate different representations of mathematical concepts better if they are taught first-hand the advantages of using one. here the focus is on what the question requires and what the teacher values instead of where the student is in terms of understanding. discussion top ↑ the study has shown that teachers had elements of both smk and pck. the teachers managed to identify students’ errors without difficulty. thus, the teachers possessed a component of smk called common content knowledge. in sizing up the causes of the identified errors the teachers interpreted some errors from a single perspective. this was a reflection that their specialised content knowledge, an element of smk, was not fully developed. during the interviews it was evident that the teachers were familiar with some of the students’ common errors and this type of knowledge is knowledge of content and students, a component of pck. this we believe was because the teachers were involved in the teaching of calculus at this level. in their discussions the teachers evaluated the instructional advantages and disadvantages of the representations used. however, in some parts the evaluation was at variance with the demands of the mathematical community. for example, the analytic method was emphasised more than the geometrical method instead of allowing the representations to reinforce each other. the teachers’ knowledge of content and teaching, a component of pck, was therefore not fully fledged in this regard. unlike in the study of peng and luo (2009) where teachers identified the errors and interpreted them with wrong mathematical knowledge, in this study teachers identified the errors and interpreted them with the appropriate mathematical knowledge. in the study of moru and qhobela (2013) teachers managed to identify errors from the content they were familiar with. their remedial strategies were confined to promoting procedural knowledge. in the reported study the teachers seemed to be familiar with all the content. they remedied the errors by promoting either procedural knowledge or conceptual knowledge depending on the nature of the error. the similarity the current study shares with that of turnuklu and yesildere (2007) is that teachers had a deep understanding of mathematical knowledge but that was not sufficient for them to perform satisfactorily all the stages of error analysis. thus, engaging in error analysis continuously would be very useful as this requires the type of knowledge that is necessary for teaching. conclusion top ↑ the study has shown that error analysis, although a necessity, is a complex process. it is complex because errors are symptoms of misconceptions (olivier, 1989) and misconceptions that have strong experiential foundation are said to be strongly held and resistant to change (smith et al., 1993). thus, the findings of the reported study have some implications for teaching. these include engaging in error analysis continuously in order to enrich knowledge of teaching, familiarisation with student errors and error analysis from the literature, as some errors in mathematics are shared across contexts, and making an effort to gain an understanding about learning theories, in particular the constructivist views as they are concerned with how knowledge is constructed by the learner. acknowledgements top ↑ we would like to thank the head of the department of mathematics of computer science at the national university of lesotho for allowing us to use the students’ examinations scripts in the study as they are the property of the university. competing interests the authors declare that they have no financial or personal relationship(s) that may have inappropriately influenced them in writing this article. authors’ contributions e.k.m. 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(2007). the pedagogical content knowledge in mathematics: preservice primary mathematics teachers’ perspectives in turkey. iumpst: the journal, 1. available from www.k-12prep.math.ttu.edu abstract introduction teacher knowledge of error analysis the project the sample data for this article – lessons and learners’ interviews operationalising the analytical criteria findings: engaging with errors in classroom teaching and learner interviews discussion conclusion acknowledgements references appendix 1 footnotes about the author(s) ingrid sapire school of education, university of the witwatersrand, south africa yael shalem school of education, university of the witwatersrand, south africa bronwen wilson-thompson school of education, university of the witwatersrand, south africa ronél paulsen department of mathematics education, university of south africa, south africa citation sapire, i., shalem, y., wilson-thompson, b., & paulsen, r. (2016). engaging with learners’ errors when teaching mathematics. pythagoras, 37(1), a331. http://dx.doi.org/10.4102/pythagoras.v37i1.331 original research engaging with learners’ errors when teaching mathematics ingrid sapire, yael shalem, bronwen wilson-thompson, ronél paulsen received: 08 feb. 2016; accepted: 04 sept. 2016; published: 31 oct. 2016 copyright: © 2016. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract teachers come across errors not only in tests but also in their mathematics classrooms virtually every day. when they respond to learners’ errors in their classrooms, during or after teaching, teachers are actively carrying out formative assessment. in south africa the annual national assessment, a written test under the auspices of the department of basic education, requires that teachers use learner data diagnostically. this places a new and complex cognitive demand on teachers’ pedagogical content knowledge. we argue that teachers’ involvement in, and application of, error analysis is an integral aspect of teacher knowledge. the data informed practice improvement project was one of the first attempts in south africa to include teachers in a systematic process of interpretation of learners’ performance data. in this article we analyse video data of teachers’ engagement with errors during interactions with learners in their classrooms and in one-on-one interviews with learners (17 lessons and 13 interviews). the schema of teachers’ knowledge of error analysis and the complexity of its application are discussed in relation to ball’s domains of knowledge and hugo’s explanation of the relation between cognitive and pedagogical loads. the analysis suggests that diagnostic assessment requires teachers to focus their attention on the germane load of the task and this in turn requires awareness of error and the use of specific probing questions in relation to learners’ diagnostic reasoning. quantitative and qualitative data findings show the difficulty of this activity. for the 62 teachers who took part in this project, the demands made by diagnostic assessment exceeded their capacity, resulting in many instances (mainly in the classroom) where teachers ignored learners’ errors or dealt with them partially. introduction teachers come across errors in the mathematics classroom virtually every day. when they respond to learners’ errors in their classrooms, during or after teaching, teachers are actively carrying out formative assessment (black & wiliam, 2006). responding to learners’ errors is a specialised activity of formative assessment, which relies on teachers’ deep knowledge of content, and requires teacher’s professional judgement on how to respond to learners’ needs when teaching that content. working with learners’ errors diagnostically in context implies that the ‘cognitive architecture’ (hugo, 2015, p. 81) of teachers’ mathematics knowledge is strong and that their knowledge is stored in a form of ‘networked schemas’ in their long-term memory, ready to be selected economically, for example in a form of principles, representations and other symbolic forms. yet in south africa there is empirical evidence showing that teachers’ mathematical and pedagogical content knowledge is weak (taylor, van der berg & mabogoane, 2013). teachers have always had to assess learners’ work and recognise the errors present in this work, but prior to the introduction of the annual national assessment (ana), a written test under the auspices of the department of basic education, there were no specific demands on teachers to use learners’ errors as building blocks for teaching and learning. with the introduction of ana, teachers are now required ‘to interpret their own learners’ performance in national (and other) assessments’ (departments of basic education & higher education and training, 2011, p. 2) and develop better lessons on the basis of these interpretations. this requirement implies that teachers are expected to use learner data diagnostically, which places a new and complex cognitive demand on teachers’ pedagogical content knowledge (for example, deciding on what to focus, and how, and what to leave out or postpone). in a previous article (shalem, sapire & sorto, 2014), we developed analytical criteria for teachers’ explanations of learners’ errors in standardised mathematics assessments following the framework of mathematics knowledge for teaching, (ball, hill & bass, 2005; ball, thames & phelps, 2008; hill, ball & schilling, 2008). in that article we operationalised the criteria for an analysis of teachers’ engagement with errors during interactions with learners in their classrooms and in one-on-one interviews with learners. the empirical data for this article were collected during the first two phases of the data informed practice improvement project, a three-year teacher development project which was a collaboration between the school of education of a johannesburg university and the gauteng department of education. the specific problem we want to bring to light, in relation to the diagnostic work required for this specialised activity as part of formative assessment, is that when teachers respond to learners’ errors in context, their ‘pedagogical load’ is increased in complexity because of the ‘cognitive load’ placed on the teachers (hugo, 2015, p. 83). when working with learners’ errors, teachers need to increase learners’ ‘germane load’, which involves reflecting on and making meaning of the patterns underlying errors. by creating connections between concepts and the related errors, teachers increase the learners’ germane load. this pedagogic activity is essential to enable learners to generalise learning from errors. in hugo’s (2015) terms, generalising learning means shifting information ‘from the limited world of working memory into knowledge networked within the infinite world of long-term memory’ (p. 85). at the same time as increasing germane load, teachers need to reduce the ‘extraneous load’ on learners by limiting factors that could increase this load. extraneous load is increased by things such as incorrect mathematical explanations, misleading statements or examples, all of which can lead to incorrect generalisations. the playoff between increasing germane load and reducing extraneous load places cognitive load on the teachers, which they need to manage at the same time as they manage their pedagogical load. the research questions addressed in this article are: how did teachers engage with learners’ errors in mathematics classes and in one-on-one interviews with learners? what does this reveal about the relationship between the pedagogical and cognitive loads involved in using errors for teaching? our first step is to build a conceptual framework which shows what constitutes the schema of teachers’ knowledge of error analysis. teacher knowledge of error analysis studies on teacher knowledge, in the field of mathematics education, agree that there is a professional knowledge of mathematics for teaching (adler, 2011; anderson & clark, 2012; ball et al., 2005; bertram, 2011; grossman, 1990; rowland & turner, 2008; shalem, 2013; shulman, 1986). this knowledge is ‘tailored to the work teachers do with curriculum materials, instruction and students’ (ball et al., 2005, p. 16). these studies maintain that teachers need specialised knowledge of what they teach, a broad sense of diverse methods of teaching and, most importantly, ways of explaining and representing the content they teach, with the view to imparting it to learners of a specific age and cognitive level of development. shulman (1986) was the first to introduce this idea, when he introduced the term ‘pedagogical content knowledge’ (pck) to describe the unique specialisation involved in teaching a given subject. pck, he says, is ‘that special amalgam [blend] of content and pedagogy that is uniquely the province of teachers, their own special form of professional understanding’ (p. 8). many theorists have followed shulman’s innovative idea and developed different categorisations of teachers’ knowledge of mathematics for teaching. elaborating on and extending shulman’s work, ball et al. (2008), explain that teachers’ knowledge consists of six core domains. domains one and two elaborate the specialisation of subject matter knowledge (common content knowledge and specialised content knowledge). this refers to knowing subject matter in ways that are specific to teaching (e.g. using mathematical language precisely but also age appropriately, justifying the use of specific representations), which general mathematicians do not necessarily need to focus on. the knowledge of what counts as a correct solution, taking into account the age and cognitive development of learners, is included in these two domains. the next four domains elaborate on the specialisation of pck from the perspective of learners, curriculum and pedagogy. two of the four refer to teaching subject matter knowledge from the perspective of curriculum demands (knowledge of content and curriculum and horizontal content knowledge). the other two refer to mediating content in the light of what learners of a specific age are likely to know about the concept being taught as well as of misconceptions arising during learning (knowledge of content and students and knowledge of content and teaching). teachers’ awareness of errors and their diagnostic activities about learners’ reasoning in relation to errors are included in these domains, albeit in a particular sequence. ball et al. (2008) emphasise that teachers’ awareness of errors and their diagnostic activities about learners’ reasoning build on the first two specialised domains: recognizing a wrong answer is common content knowledge (cck), whereas sizing up the nature of an error, especially an unfamiliar error, typically requires nimbleness in thinking about numbers, attention to patterns, and flexible thinking about meaning in ways that are distinctive of specialised content knowledge (sck). in contrast, familiarity with common errors and deciding which of several errors students are most likely to make are examples of knowledge of content and students (kcs). (p. 401) the inter-dependence between these domains has serious implications for the expectation that teachers should work diagnostically with learners’ errors. studies on teaching dealing with learners’ errors show that teachers’ interpretive stance is essential for the process of remediation of error, without which teachers simply reteach without engaging with the mathematical source of the error or with its metacognitive structure (brodie, 2014; gagatsis & kyriakides, 2000; peng, 2010; prediger, 2010). according to ball et al. (2008), teachers need to judge if there is a pattern in student errors. they also need to size up ‘whether a nonstandard approach would work in general’ (p. 400). when teachers size up a learner’s error or interpret the source of its production, they are working diagnostically with the subject matter being taught, for which they need to recruit different ‘networked schemas of knowledge’ of specific aspects of error analysis corresponding to the concept or the procedure they teach. recruiting different aspects of error analysis places simultaneous cognitive and pedagogical demands on teachers and thus makes for a challenging form of pck. building on the work of cognitive load theorists such as sweller, kirschner and clark (2007), hugo (2015) shows that because the capacity of working memory is limited, structuring one’s knowledge along ‘networked schemas’ and not by ‘tiny elements [of information] at a time’ is essential. conceptual network schemas (connections between concepts) are developed through systematic formal learning and are stored in long-term memory. they include tiny and contingent elements but because these elements are ordered in a schema, over time, the conceptual frame stored in long-term memory becomes the organising tool for processing new contingent elements. this is the nub of the challenge in diagnostic work of formative assessment: many aspects of a lesson can be planned and extensive preparation can be done for every lesson, but learners’ errors can raise unanticipated questions, for which teachers cannot necessarily prepare. in these situations, teachers need to make quick decisions as to how to conduct their general pedagogy in the course of a lesson and how to attend to learners’ errors specifically. the cognitive load of these situations, which can only be inferred, consists of the work of synthesising and making decisions about aspects drawn from curriculum knowledge (what of the actual content knowledge to focus on), curriculum coverage (how to deal with pressures such as lack of knowledge or time constraints) and pedagogical knowledge (what to anticipate in learner’s response, how to listen to learners and how and when to respond to learners). the pedagogical load of focusing the response to a learner’s error on the germane load and controlling the extraneous load and the cognitive load attended to it are managed with more ease by teachers whose mathematics knowledge is strong, since their knowledge of mathematical errors and misconceptions is structured in networked schemas. teachers with weak mathematics knowledge, however, experience a high extraneous cognitive load in recognising and interpreting errors, thinking about them and responding to them in the context of the engagement. these teachers may be unable to fully grasp the learner’s position; they may be hesitant or even unable to adapt their own knowledge in order to respond appropriately and are more likely to avoid dealing with errors. in the sections that follow we describe the project and the methodology we followed in order to study how a sample of teachers engaged with learners’ errors during lessons and in one-on-one interviews with learners. our analysis of the findings examines what they reveal about the relationship between the pedagogical and cognitive loads involved in using errors for teaching. the project working with teachers on interpretation of learner standardised assessment data was the central goal of the project, which provided a context for professional conversations where 62 grade 3–9 mathematics teachers from a variety of johannesburg schools discussed mathematical assessment data. in this pioneer community of practice research project, teachers were organised into groups by grade level, forming eight grade 3−6 groups and six grade 7–9 groups. the groups worked together in weekly meetings, mapping mathematics test items onto the curriculum (shalem, sapire, & huntley, 2013), analysing learners’ errors, designing lessons, teaching and reflecting on their instructional practices, preparing and conducting interviews with learners and constructing test items. the teaching and interview activities were intended to give teachers an opportunity to apply error analysis when interacting with learners, and hence to develop their understanding of the role of errors and misconceptions in the learning of mathematics. the sample data for this article – lessons and learners’ interviews the teachers all took part in planning the lessons that were taught by a group representative in each round. over the course of the project 39 teachers were filmed teaching planned lessons on behalf of their groups. we selected 17 lessons across all of the groups as our sample for analysis. interviews were also planned collectively but carried out individually. each teacher selected one, two or three learners to interview about an interesting error that they identified in a test they had drawn up collectively, administered individually in their own classes and then marked and analysed together in their project groups. for the purpose of analysis of teachers conducting learner interviews the sample was made up of 13 interviews conducted by teachers, corresponding to the classroom lessons sample. consent was obtained from all participants in the lessons and interviews (teachers, learners and parents of learners), prior to the videoing of the activities. the lessons and interviews were not all equal in length. the number of minutes, in total, that formed the sample for the quantitative analysis is summarised in table 1. a lesson required that time was spent teaching and engaging with learners on more than just errors. it is not always appropriate to address all errors in the context of a full lesson. for the purpose of analysis of teachers engaging with errors during teaching only the ‘error episodes’ were coded minute by minute. the error episodes were identified as intervals of time starting with an error expressed by a learner in the context of a lesson and ending when engagement with the error terminated. a total of 129 minutes were coded in the classroom teaching sample as ‘error episodes’ (out of 906 minutes teaching time). for the purpose of analysis of teachers engaging with learners’ errors during interviews, the full length of each interview was coded, minute by minute, since the aim of the interview was to engage with learners’ errors. the total time of the coded interviews was 223 minutes. table 1: overall time coded (minutes). operationalising the analytical criteria to analyse how the teachers’ engaged with learners’ errors during teaching and in one-on-one interviews we used four criteria: teachers’ procedural explanations in relation to the error (proc), teachers’ conceptual explanations in relation to the error (con)1, teachers’ awareness of the error (awa) and teachers’ diagnostic reasoning when engaging with the learner in relation to the error (diag). to capture variability in the quality of the teachers’ explanations of the errors, each of the criteria was divided into four categories: full, partial, inaccurate and not present. the classroom and interview activities had independent coding sheets, which contextualised the criteria and categories to the teaching and interview contexts (see appendix 1 for the category descriptors for the interview activity). the criteria were applied to each minute and were methodologically differentiated in the coding process. nonetheless, it is the relationship between the teaching activities, as they develop across the lesson or the interview, that leads to a successful error analysis engagement. the four coding criteria (full, partial and inaccurate exemplars) are demonstrated below using examples relating to classroom teaching and learner interviews. procedural understanding of the error teachers need to recognise and be able to explain the steps needed to get to the correct answer, the sequence of the steps and the appropriate conceptual links between the steps. because this knowledge underlies recognition of error, we include it under the first of ball et al.’s (2008) domains, common content knowledge. procedural activity explanations of errors need to be given with sufficient clarity and accuracy if the learners are to grasp correct procedures and become competent in performing them. a proc code was assigned to the aspects in the teachers’ utterances that demonstrate an attempt by the teacher to unpack a mathematical procedure while probing the learner’s error. in this excerpt from a grade 6 lesson the teacher probes to expose the error in the learner’s expression of a decimal number. the teacher noticed the flaw in the learner’s use of terminology relating place value: she was confusing tens and tenths and hundreds and hundredths. by giving simple prompts such as ‘in the place values of the decimals what do i have? what words?’ the teacher worked with the learner procedurally until the learner used the correct expression (‘one tenth’ and ‘two hundredths’). a partial proc code was assigned to this excerpt. the teacher’s prompts did result in technical corrections, hence the inference made is that the teacher managed the pedagogical load (she got the learner to use the correct mathematical expressions). however she did not address the number concept related to this terminology she only made procedural corrections to the learner’s expression. the inference made here is that the teacher did not manage the cognitive load posed by the episode. in this example the germane load of the learner is not increased to the extent to which it could be. a further complication in this episode is that we see the teacher using incorrect language herself (she says ‘zero comma twelve’ – she does so in several other instances in the lesson) which may increase the extraneous load of the learners. teacher:now what do i have there? yes, thembi? learner:ma’am, it’s zero comma twelve. teacher:zero comma twelve. now explain to me why zero comma twelve? [learner writes 0,12 on board.] i’m waiting. learner:ma’am, because there are no units and there are tens and hundreds. teacher:there’s no units, good. learner:and there are tens and hundreds. teacher:and then the other part after the fraction is a decimal … i mean, after the comma is a decimal, right? and in the place values of the decimals what do i have? what words? learner:one. teacher:one what? learners:one tenth. teacher:one tenth and …? learners:two hundredths. teacher:and two hundredths. conceptual understanding of the error whereas teachers’ knowledge of what counts as the explanation of the correct answer enables them to recognise the error, looking for explanations that will enable them to interpret learners’ solutions and evaluate their plausibility points to a teachers’ conceptual knowledge of errors. in ball et al.’s (2008) words, the key aspect here is that teachers are looking for patterns in student errors and ‘sizing up whether a nonstandard approach would work in general’ (p. 400). conceptual aspects related to the recognition of whether a learner’s answer is correct or not could thus span two domains, the first common content knowledge and the second specialised content knowledge. a code con was assigned to the aspects in teachers’ utterances that demonstrate an attempt by the teacher to unpack a concept, or a conceptual feature of a procedure, while probing the learner’s error. in a grade 7 lesson on balancing equations the teacher asked the class probing questions to ascertain learners’ conceptual understanding of the meaning of the equal sign. this is taken as an error episode because while the teacher’s pedagogical load consists of asking for the meaning of the equal sign, the explanations given by learners were limited to the operational (find the answer) rather than the relational (balance the equation) meaning of the equal sign. at the end of this interaction in which learners expressed their thoughts about the meaning of the equal sign, the teacher moved on, even though the explanations had not touched on the relational meaning of the equal sign. the inference we make in this episode is that the teacher managed the cognitive load though not optimally. a partial con code was assigned to this excerpt since the teacher engaged conceptually in discussion with learners about the meaning of the equal sign but did not fully complete this discussion. in this example there was no distraction created in the form of extraneous load but the germane load of the learners is not increased to the extent to which it could have been. the teacher did not clarify the different understandings of the equal sign nor did she explain that the core conceptual understanding is one of balance and equality on both sides of an equation as is evident in the following dialogue: teacher:what we’re going to be doing today is discussing the equal sign. you’ve often used it in maths since you were tiny so now we’re going to try and find out what this sign actually means to you. okay who can tell me – what does this sign actually mean? [points to equal sign on board.] put up your hands if you know the answer. right isabella? learner:it means after the equals sign you put your answer. [teacher repeats her answer – writes something on board.] teacher:right, what else does it mean? to somebody else, what does it mean? okay so you say that after the equal sign you put your answer. what else does it mean? aandile? learner:it means that two numbers are the same, are equal. teacher:ok two numbers can be equal. [writes on board.] alright what else does it mean to somebody? learner:it means that you’ve got your answer. teacher:so it means that once you’ve got your answer sign you’ve finally got your answer. perfect, so you’ve got your answer. [teacher writes ‘got answer’ on board.] right anything else, does it mean anything else to somebody? mandla? learner:the equal sign means that it’s the end of that sum. teacher:the end of the sum? learner:ja. teacher:ok. so it’s the end of that sum. [teacher writes ‘end of sum’ on board.] jeff? learner:it’s also the end of the equation. teacher:the end of the equation. ok perfect. anything else? no? ok right, let’s carry on. awareness of error sizing up the source of the error, in particular recognising common misinterpretations of specific topics (olivier, 1996) or learners’ levels of development in representing a mathematical construct, is an aspect of pck related to teachers’ knowledge of errors. a conversation in which the nature of the error is not made explicit or elaborated has very little educational value. from the point of view of error analysis, this knowledge domain involves teachers explaining specific mathematical content primarily from the perspective of how learners typically learn the topic or ‘the mistakes or misconceptions that commonly arise during the process of learning the topic’ (hill et al. 2008, p. 375). knowledge of content and student enables teachers to explain and provide a rationale for the way the learners were reasoning when they produced the error. because contexts of learning (such as age and social background) affect understanding and because in some topics learning develops through initial misconceptions, teachers need to develop a repertoire of explanations, with a view to addressing differences in the classroom. a code awa was assigned to the aspects in teachers’ utterances that demonstrate an attempt by the teacher to identify the error around which the conversation is focused. the emphasis of this code is on teachers’ discussion with the learner of what the error is about, in response to what is verbalised by the learner in the course of discussion. in a grade 9 interview the teacher engaged with the way in which a learner had plotted points using a scale, in particular the way in which the learner had chosen to deal with numbers that did not fit into the scale he had chosen for his axes. the teacher’s pedagogical load consists of giving the learner a possible explanation for how he had adjusted the values using a mathematical method (‘multiplied by ten’), even though doing this was not appropriate in the context. the learner rejected the explanation, asserting that he had ‘ignored the zero’ and ‘removed the comma’. neither of these responses is appropriate but the teacher did not engage with them, she echoed what he said and moved on. an inaccurate awa code was assigned to this excerpt since in her discussion the teacher did not refer to ‘scale’ at all and does not unpack the erroneous ways in which the learner adjusted his coordinate values in order to fit them into his chosen scale. this is an example where the learners’ extraneous load is increased. the error relates to the use of a scale and so in not touching on this topic, the teacher does not establish a conversation around the error in focus, creating further difficulty for the learner. the inference made here is that the teacher did not manage the cognitive load posed by the episode. learner:as in for twenty-five [points to number on horizontal axis], i did a two point five, to, two hundred and fifty [points to number on vertical axis], because there was no twenty-five. so i used these variables [points to numbers on horizontal axis] just to say this is five, even though it’s zero comma five, i said this is five, and like that… teacher:ok, so you ignored the zero [points to zero] and… learner:yes. teacher:actually you multiplied by what? multiplied by ten or the…? learner:no, i didn’t multiply it, i just ignored the [crosses out a number on horizontal axis]… teacher:you removed the zero. learner:yes. teacher:ok. learner:just like the two point five, i removed the comma [draws a comma]. yes. i did it like that. diagnostic reasoning when engaging with the learner in relation to the error the pedagogical work of probing learners’ thinking, by taking them through the error and supporting them with examples, representations and, when appropriate, bringing an example from every day to enable them to understand the concept, forms another aspect of pck related to teacher knowledge of errors. in terms of teacher knowledge of error analysis, the key idea that this criterion puts forward is that teachers go beyond stating the actual error by using probing questions to try and follow (with the learner) the way the learner is reasoning about the error. this criterion could also span two of ball et al.’s (2008) knowledge domains, knowledge of content and students and knowledge of content teaching. this kind of knowledge would enable a teacher seeking to find out the learner’s mathematical reasoning behind the error. in response to the error the teacher probes further and asks the learner to explain the steps of their reasoning. a code diag was assigned to the aspects in teachers’ utterances that demonstrate an attempt by the teacher to probe the learner’s reasoning behind the error. the emphasis of this code is on the questions the teacher uses to probe the learner to explain the steps in their reasoning. a grade 8 teacher set up an interview to probe the learner’s thinking in relation to an error that arose in an activity where learners had to generate equations using a given set of symbols (see figure 1). the learner had not been able to make correct equations using the symbols given in the task. figure 1: grade 8 test item with learner’s working. the teacher managed the pedagogical load presented by this error in the following way. she asked several probing questions to follow the thinking of the learner. she asked both broad (‘just explain to me your answer’) and specific (‘so that’s what you meant when you wrote here that the envelope and the heart must have a bigger value’) questions. a full diag code was assigned to this excerpt since the teacher used systematic probes to identify the learner’s error. based on the learner’s responses and in order to follow through the diagnosis the teacher then chose to introduce a supportive example in which the learner had to use the numbers (rather than symbols) to make equations. this allowed for alternative explanations because it offered a different representation, hence increasing the germane load of the learner by setting up the stage to follow up on the error. the inference made here is that the teacher managed fully the cognitive load posed by the episode. teacher:ok, alright, so that’s what you meant when you wrote here that the envelope and the heart must have a bigger value? learner:yes. teacher:ok i see. alright. and can you just explain to me your answer here, or your working out for me? learner:well i thought that if the envelope would be a bigger value, i’d have to add a smaller value with it so that’s why i added just the smiley face to think that maybe that would equal that answer. teacher:ok. learner:and with the heart, i thought that you need two of these symbols [points at the smiley face and envelope] in order to equal a bigger value with another bigger value in order to get the answer. teacher:ok, i see, i see, alright. now what i’m going to do is i’m going to give you a sum with numbers. learner:ok. teacher:and we’re going to work with that sum of numbers a little bit and then we’re going to see if we can link it in somewhere to this question [points to question being discussed]. ok so i’m just going to put that aside for the moment [pushes away first paper and brings forward another paper]. findings: engaging with errors in classroom teaching and learner interviews in answering the research questions, the evidence presented below suggests, firstly, that the teachers shied away from engaging with learners’ errors during teaching and, secondly, that when they did engage with learners’ errors, both during the teaching and interviews, they were not always successful in coming to grips with the nature of the error nor did they enable learners to clarify their own thinking and develop a deeper understanding of the mathematical concepts underpinning the error. in other words they did not increase the germane load of the learners. we argue that this is evidence of teachers not coping with the cognitive load they face when errors arise in discussions, which impacted on the way they managed their pedagogical load. this can be seen both in the time spent engaging with errors and in the quality of activity when engaging with errors. figure 2 shows the time identified as error episodes (time actively spent engaging with a learner error over the course of a normal lesson) in relation to the total amount of teaching time in a lesson during the three rounds of teaching. despite the noted increases between rounds the percentages of time during which teachers engaged with errors remained low. across the three rounds the grade 7–9 groups spent more time engaging with learners’ errors than the lower grade groups and the time they spent increased in successive rounds. the interviews (only one round) were focused on errors (as planned) and so teachers were actively engaged with errors throughout the interviews.2 figure 2: time spent engaging with errors in classroom lessons. the cognitive difficulty (which we use to infer cognitive load) of the activity for the teachers can be shown by a comparison between the amount of time teachers spent engaging with errors using procedural and conceptual explanations. the graphs (figure 3 and figure 4) show the differences between the occurrence and the quality of the procedural and conceptual activities, when teachers interact with learners on an error, in the classroom and during learner interviews. figure 3: procedural explanations in relation to the error (averages across all rounds). figure 4: conceptual explanations in relation to the error (averages across all rounds). both in the classroom and in the interviews, the teachers’ engagement with errors is predominantly procedural: there are more not present and inaccurate codes and fewer partial and full codes in both contexts. the presence of not present codes in the lessons (36% proc, 65% con) compared to 0% (proc and con) in the interviews suggests that many of the teachers glossed over errors in the course of a lesson, which may be based on a pedagogical choice or an inability to engage meaningfully with wrong answers. examples where wrong answers are not probed or explained can be found in the majority of lessons. it is further notable that during lessons, when engaging with learners’ errors, the teachers gave more inaccurate procedural explanations (36% in lessons and 13% in interviews) than conceptual explanations of the learners’ errors; in the interviews they gave more inaccurate conceptual explanations (24% in lessons and 33% in interviews) than procedural explanations of the learners’ errors. notwithstanding, the conceptual explanations are more consistent in quality in the interviews and more evident than in the classroom. this suggests that the teachers felt more confident to address conceptual issues in the interviews (for which the interviews had been planned) than in the course of a lesson. qualitative analysis also sheds light on the cognitive difficulty of the activity for the teachers. importantly, it shows how the teachers coped with the pedagogical and cognitive loads presented by the error episodes. the four examples discussed earlier in this article gave insight into how teachers managed the loads in different ways, some more successfully than others. teachers found it very difficult to home in on the error underlying the learners’ statements made during conversations, both in classroom lessons and during interviews. this is evidence of teachers not coping with the cognitive load they face when errors arise in discussions. in figure 5, which shows the differences between the occurrence and the quality of the awa activity, we see very few partial and full codes, indicative of poor awareness of the nature of the mathematical errors in both the classrooms and interviews, although interviews do show slightly better quality in this criterion. the code not present is highly frequent in the lessons and during interviews (55% and 54% respectively), showing that teachers found it difficult to express the nature of the error arising in the conversation mathematically. figure 5: awareness of the error (averages across all rounds). teachers often used incorrect or sloppy mathematical language resulting in inaccurate explanations (more so in the classroom), further evidence of the inability to correctly express an awareness of the mathematical error. when teachers use inaccurate mathematical language when attempting to address an error, their ability to identify the error around which the conversation is focused is compromised. for example, in a grade 9 lesson, a learner answered ‘eighteen and twenty-one’ (referring to a coordinate point). in this lesson the teacher and the learners both use this incorrect/vague language. in a grade 6 lesson a teacher says ‘one comma twenty-three’ for 1,23. this poor expression undermines the concept of decimal place value that he is trying to teach. saying ‘one comma twenty-three’ confuses the relationship between places before and after the decimal comma. precision in mathematical expression (both verbal and written) is vital since it supports a deeper understanding of the concepts under discussion; unclear or imprecise language provides further evidence of teachers not coping with cognitive load. figure 6 shows the differences between the occurrence and the quality of the diag activity. here the teacher’s skill of probing, listening to the learner and developing the conversation is under scrutiny. in both the lessons and interviews, the data shows that the teachers do not engage with the learners’ language of expression, evidence that teachers struggle to cope with both the cognitive and pedagogical loads presented by errors voiced by learners. the category inaccurate for this criterion (most highly evidenced in both classrooms and interviews) captures times when teachers probed the error broadly but did not seek the learner’s mathematical reasoning behind the error. this is different from inaccuracy in the other criteria (proc, con and awa) where mathematical content may be compromised. teachers often asked broad questions such as ‘why did you say that?’ or ‘tell me about what you did here’. on many occasions they followed up the broad probe with a question, ‘so is it correct/right/wrong/incorrect?’ in so doing, teachers were merely asking learners to make a judgement call. in this way broad probes did not lead to exploration of, or elaboration on, issues related to errors. for example, in one grade 9 lesson the teacher used the word ‘correct/correctly’ 23 times, usually as part of a probe. when a teacher asks ‘is that correct?’ the learner generally realises it is not and answers ‘no’ but often seemingly without understanding. diagnostic conversations tend to terminate or continue at the same level, without focusing on or moving closer to the misconception underlying the error. this is evidenced in the low level of partial and full diagnostic explanations both in classrooms and interviews, although the interviews showed slightly better quality in this criterion. figure 6: diagnostic reasoning in relation to the error (averages across all rounds). in the classroom teachers’ probing frequently involves calling on a learner to re-do work that has been done or then calling on another learner to correct work that had been done. this is often guided by the teacher using leading questions. when learners respond they often repeat the same error. sometimes the ‘correcting’ learner makes another mistake which may or may not be identified. in their haste to move on, teachers often do not probe further, they either reteach using the same or another similar example, without engaging with the content of the learners’ response. no explanations are given; work is simply corrected and the conversation moves on. the error, or limitation, in this type of interaction is noted and sometimes even identified, but is not worked through. diagnosis is thus rare and not often adequately acted upon; in all of these cases teachers have not coped with the cognitive and pedagogic load created by the need to engage with learners’ mathematical errors. discussion the above findings point to the cognitive difficulty involved in applying error analysis, when teachers engage with learners, more so in teaching than in interviews. we use this to infer that cognitive load impacts on pedagogical load. to cope with the pedagogical load, teachers may choose to ignore errors. at times they ignore errors completely, sometimes accepting incorrect work, or they continue teaching or asking questions on a planned route, not acknowledging the error but trying to teach or reteach the ‘right stuff’. in other instances, teachers do acknowledge errors. acknowledgement of errors might involve indicating recognition of an error vocally (for example, saying ‘no’) without addressing or questioning the error. in limited attempts at engaging with errors, teachers may indicate recognition of an error vocally and request a correction from another learner or offer the correct answer themselves. finally, teachers may engage diagnostically with errors, ask probing questions, which range from broad to more error-focused questions or use open-ended exploratory questions. classroom and interview responses might follow a different set of ‘rules’ according to their contextual nature. table 2 summarises the key differences between the two contexts. table 2: contextual differences for engagement with errors. the comparison shown in table 2 might imply that application of error analysis in an interview is more straightforward, but, as we have shown, teachers found the activity difficult in both contexts. although there were examples of meaningful interaction on the part of teachers with their learners’ errors in the lesson and interview activities, such evidence was sparsely scattered in the data set. we expected to find stronger and more consistent evidence of diagnostic reasoning in the context of a learner interview, yet the interview activity, undertaken during the last six months of the project, highlighted teachers’ difficulty in dealing with learners’ mathematical errors in conversation. teachers conducted these interviews after more than two years in the project, working with colleagues and district officials under the guidance of mathematics education experts (university staff members or postgraduate students). these could be considered ideal circumstances for optimal performance, and yet the teachers struggled to engage meaningfully with their learners about errors they had made. this might lead one to ask whether teaching through errors is too much to expect of teachers. when teachers engage with their learners’ errors they use their diagnostic reasoning in the activity of formative assessment. this kind of pedagogic content knowledge (shulman, 1986) is a higher and more difficult level of teacher knowledge, which largely depends on teachers’ ability to unpack a mathematical procedure, a concept or a conceptual aspect of a procedure while probing the learner’s error. if this argument is accepted then hugo’s (2015) analysis, which shows that cognitive load economises or increases pedagogic load depending on the strength of the teacher’s ‘cognitive architecture’ (p. 81) of their mathematics knowledge for teaching, explains why teachers find working with learners’ errors during teaching a complex task to achieve. conclusion the schema of teachers’ knowledge of error analysis and the complexity of its application was discussed in relation to ball et al.’s (2008) domains of knowledge and hugo’s (2015) explanation of the relation between cognitive and pedagogical loads. evidence of the difficulty of this activity has been shown quantitatively as well as through qualitative examples from the classrooms and interviews. the evidence highlights that the cognitive and pedagogic loads of applying error analysis in context exceeded the capacity of the teachers. some teachers did engage in discussions that could increase the learners’ germane load, but in most instances teachers actually increased the extraneous load when they engaged with errors, making it more difficult for learners to absorb what was being taught. the difficulty the teachers experienced in responding meaningfully to errors in context could be related to their mathematical knowledge gaps, linguistic ability or lack of experience in focusing on what the learner says and responding directly to what has been said. teachers’ involvement in activities such as analysing learners’ errors on standardised tests, engaging with learners’ errors when planning and teaching a lesson, discussing them with colleagues and through interviews with learners’ to probe their reasoning has some merit. the data presented from the three rounds of teaching indicate that teachers can learn to engage with learners’ errors over time but that such learning is very slow. poor overall demonstration of awareness of error and poor use of probing questions (in diagnostic reasoning) suggest the need for caution in advocating for developing teacher competence in addressing learners’ errors through informal group-guided discussion. further research into the relationship between teachers’ knowledge of mathematical content and their ability to engage diagnostically with learners’ errors and demonstrate an awareness of mathematical errors needs to be done. nevertheless, the findings suggest that teachers need formal and structured opportunities to improve their mathematical content knowledge, to inform the diagnostic work required by this kind of formative assessment. acknowledgements we acknowledge the funding received from the gauteng department of education and in particular would like to thank reena rampersad and prem govender for their support of the project. the views expressed in this article are those of the authors. competing interests the authors declare that they have no financial or personal relationship(s) that may have inappropriately influenced them in writing this article. authors’ contributions i.s., the lead author, was also involved in the coding and analysis of the data. y.s. was involved in the conceptualisation, analysis of the data and the writing of the article. b.w.-t. was involved in the coding and analysis of the data and contributed to the writing of the article. r.p. was involved in the coding and analysis of the data and contributed to the writing of the article. references adler, j. 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(2013). what makes schools effective? report of the national schools effectiveness study. cape town: pearson. appendix 1: appendix 1: interview coding criteria. footnotes 1. we see procedural and conceptual explanations as activities that can be characterised distinctly, while acknowledging that they essentially occur simultaneously and cannot be split into a false dichotomy (kieran, 2013). 2. there was only one set of learner interviews and so no progression over time can be analysed. mudaly 36 pythagoras 60, december, 2004, pp. 36-43 modelling of real-world problems is often the starting point for proof vimolan mudaly department of mathematics education, university of kwazulu-natal, south africa email: vims@mweb.co.za in this paper i claim that modelling should be seen as the first stage of the proving process. i discuss an experiment conducted with grade 10 (15 year old) learners in a small suburb in south africa. there is little emphasis placed on modelling in our schools and it is just beginning to make an appearance in our new outcomes based curriculum. the research shows that as a result of the modelling process learners felt the need to know why the result was true. there is ample evidence that a lot of work on a similar topic has been done elsewhere in the world, but not much has been done in south africa. the research was conducted using sketchpad as a mediating tool. this in itself was a difficult task because our learners have not really been exposed to dynamic geometry environments. introduction proof is often only seen as a means of simply verifying the truth of mathematical statements. although proof should be seen as serving many functions, it would seem that establishing certainty in a statement has been its main function. according to davis and hersh (1981: 249) this can be traced back to the greek mathematicians who saw the proof process as that of validation and certification. a survey conducted in 1984 by de villiers (1999: 18) revealed that more than 50 % of higher education diploma students in mathematics education agreed that the only function of proof was that of “making sure”, that is, the verification of the truth of the results. this is a perception that is commonly propagated in mathematics classrooms, where teachers focus on the verifying of mathematical truths that are being investigated. seldom is there a link established between mathematical modelling of real-world problems and proving. this is also the finding of hodgson and riley (2001: 724), who state that that "proof and real-world problemsolving are typically considered to be separate and distinct endeavours". it has always been difficult to gauge the relationship between real-world problem-solving and proof, yet the clear value of real-world problem-solving in the process of proving cannot be underestimated. hodgson and riley (2001: 724) further state that: our experience has been that real-world problems supply an important ingredient that seems to be missing from typical classroom instruction on proof. as such, real-world problems may actually be one of the most effective contexts for introducing and eliciting proof. realworld problems are commonly used as vehicles to introduce or deepen students' understanding of mathematical concepts and relationships. hodgson and riley's argument that real-world problems could be the basis for mathematical proof stems from one step in the modelling process, namely, the testing of the solution. they believe that it is essential for students to ask "why is the statement true?" after they have arrived at a solution. in their experiment the students found that their solution was incorrect and this initiated the desire for an explanation. it is my contention that had they not gone through the process of modelling, it is unlikely that they would have wanted an explanation. similarly, klaoudatos and papastavridis (2001) discuss a teaching experiment based on context oriented teaching (cot). according to klaoudatos and papastavridis, cot is “a model based on a problem solving framework and on the selection of the appropriate task context” (p. 1). they observe that cot provides the student who has little understanding of the mathematics involved in solving a particular problem, with a starting point and a sense of direction. essentially, they conclude that starting with a context oriented question (which is an adapted real-world question), the learners use context oriented heuristics to develop context oriented concepts. context oriented conjectures are formulated, which leads to context oriented proofs. despite framing their arguments within the idea of vimolan mudaly 37 contexts, they still show that the modelling activity serves as a starting point for this proof. a further significant argument which shows this direct link between modelling and proof is made by blum (1998) when he states that applications in mathematics (solving of real-world problems) provide contexts for what he refers to as reality-related proofs. he clearly points out that: formal proofs are mostly the final stage in a genetic development – historically as well as epistemologically as well as psychologically. (p. 4) however, klaoudatos and papastavridis (2001) as well as blum (1998) discuss proving in relation to modelling in teaching situations which have been explicitly designed so as to connect the two. the question still remains whether the modelling activity will still serve as a starting point for proof if the specific modelling activity is not constructed with the intention of arriving at a proof. furthermore, it is relevant to ask whether the experiences gained from the european contexts will be similar in south african classrooms, with different traditions, and teacher-learner relationships and numbers. i address both these questions below. real-world problems the direct connection between classroom mathematics and real-world mathematics is a tenuous one, because it is often difficult to relate classroom mathematics to what happens in the real world. if the word “real” in this instance is not only interpreted as a connection to the real world, but as a reference to the problem situations which appear to be real in the learner’s mind, then the relationship between real-world and classroom mathematics becomes a bearable one. furthermore, if mathematics is to be related to reality, not only is reference being made to real-world problems but also to the fact that the mathematics must make sense to children. it must remain as close as possible to the concepts that children already have and know. the work they do must appeal to them within the frames of reference that they understand. selden and selden (1999) state that: from the perspective of realistic mathematics education, students learn mathematics by mathematizing the subject matter through examining 'realistic' situations, i.e., experientially real contexts for students that draw on their current mathematical understandings. (p. 9) clearly, the problem presented precedes the abstract mathematics that is to be learned. according to van den heuvel-panhuizen (1998): the reason … why the dutch reform of mathematics education was called ‘realistic’ is not just the connection with the real-world, but is related to the emphasis that rme [realistic mathematics education] puts on offering the students problem situations which they can imagine. (p. 1) the realistic mathematics educators place immense emphasis on the idea of making a mathematical idea real in the mind of the learner. so when working with real-world problems the learner should first be exposed to a problem situation that they are familiar with and it must appear to be real. this then allows them to use their previous experiences to interact and engage with the problem at hand. it is this interaction with a familiar situation that leads to the development of a predetermined mathematical concept. through further interrogation of the problem situation and its results the learner develops a better understanding of the concept (zulkardi, 2003: 6). although it may be difficult to replicate realworld conditions in a mathematics classroom, it is essential to expose learners to different real-world scenarios so that they learn mathematics in contexts that are familiar to them. bonotto (2004) states that: progressive mathematization should lead to algorithms, concepts and notations that are rooted in a learning history which starts with students’ informal experientially real knowledge. in our approach everyday-life experience and formal mathematics, despite their specific differences, are not seen as two disjunctive and independent entities. instead, a process of gradual growth is aimed for, in which formal mathematics comes to the fore as a natural extension of the students’ experiential reality. (p. 42) besides the ideal of showing learners how mathematics is related to the real world it also serves to increase interest in the subject matter. bowman (1997: 8) states that after allowing his students to work with real-world problems in his class the level of student interest increased to the extent that “they were especially excited about being able to solve a mathematics problem that even the so-called ‘math geniuses’ in calculus could not solve”. modelling of real-world problems is often the starting point for proof 38 the process of modelling and its relationship to proving in order to understand the relationship between proof and modelling, one needs to have some knowledge of the processes involved in modelling. modelling is not an easy task. it often involves a process of creating a miniature problem, which is analogous to the larger problem, but enables the modeller to draw exact conclusions, which can be extrapolated to the original real-life problem. although a model attempts to simulate the original problem it cannot truly replicate all the constraints that might be imposed by the problem itself. modelling usually begins with a real-life situation which may be relatively controlled (for example, determining the profit of a manufacturing company), or sometimes in environments in which the modeller cannot control all the conditions (for example determining the population increase of fish in a river). in all cases the modeller is hoping to predict future behaviour of the system under prevailing conditions. this research was based on the following diagrammatic explanation of the modelling process of michael olnick (1978:4). refer to figure 1. the model depicted in figure 1 was adapted to include other technologies in the modelling process. for the purposes of this research the technology used was the computer software package sketchpad. refer to figure 2. de villiers (1993: 3) describes three different categories of model application namely, direct application (‘immediate recognition of a model to be used’), analogical application (‘development of a model that is similar to an existing model’), and creative application (‘a completely new model is created using new techniques and concepts’). the experiment described below is entrenched in the creative application. in evaluating a model and its results the modeller begins to ask why does the result hold true? (or even why does the result not hold true?). it is this question that clearly defines the relationship between modelling and proving. asking why indicates a desire to seek some sort of explanation. it is also clear that the question is not whether the result is true because the modeller has already convinced him/herself that the result is true during the interpretative stage. once convinced the modeller develops a certain curiosity as to why the result is true and possible under such conditions. in attempting to answer this question the modeller begins to develop an explanation for the observed result and hence establishes a proof valuable in increasing the understanding of the problem. the purpose of this study and the theoretical framework the purpose of this study was to determine whether sketchpad could be useful as a mathematical tool when teaching children to model. furthermore, this study tested curriculum material that was developed (de villiers, 1999) and refined as a result of previous empirical and theoretical research. the material in the form of worksheets allowed the learner to work through the problem by guiding the child through stages that are easy and practical. as the child progressed through the worksheets, the child was allowed to record his/her conclusions and conjectures and this led to an explanation (proof). the theoretical and empirical part of this research focused on the following major research questions: 1. what is the role and function of mathematical modelling in mathematical sciences, and its potential role in mathematics education? real world mathematical system real-world conclusions mathematical conclusions abstraction experiments logical argument interpretation figure 1. diagrammatic representation of olnick’s modelling process vimolan mudaly 39 2. can learners acquire knowledge about geometric concepts and shapes such as equidistance, perpendicular bisectors and concurrency via creative modelling? 3. are secondary school learners able to create and use mathematical models to solve geometric problems in the real world without the use of sketchpad? if so, what strategies do they use? 4. are learners able to use the provided sketchpad sketches effectively to arrive at reasonable solutions? 5. do learners display greater understanding of the real-world problems under question when using sketchpad? my one-to-one task-based interviews were analysed using qualitative methods. this method made it easier to document the high level of information that individual children display when working through a specific problem. furthermore, this method allowed the researcher greater control to observe and take note of how each learner answered questions based on the computer manipulation they experienced. the tasks used in the interviews had been conceptualised within an action research paradigm. these tasks were conceptualised mainly as a means of teaching children the different functions of proof developed by de villiers (1999), and also included aspects of modelling. this research attempted to determine how well they coped with the tasks provided, whether they construct meanings as conceptualised, and whether they would be able to mathematically model a solution. based on initial trials the tasks had to be redesigned in order to achieve the predetermined goals of the learning activity. in other words, action research acted as a guiding methodology. the task that the learners had to work through was based on a relevant real-world topic. all learners were exposed to the different media in south africa and from prior questioning it was established that all learners were aware of the seriousness of the water born illness called cholera. learners were also aware that cholera was mainly concentrated in rural areas, particularly in areas where no fresh water was available. hence, identification with the problem was not new and difficult. all the sketchpad sketches were presented as ready-made models to the learners. the task of constructing such models is also an essential modelling skill and would be an interesting task to ask of learners. for example, it would require them choosing and implementing a reasonable scale and utilising sketchpad’s tools to accurately construct a dynamic scale drawing. the decision to present the sketches directly to them was based on the following reasons: • the construction of accurate, dynamic sketchpad sketches requires a fairly high level of technical expertise. • even if learners had had a sufficiently high level of expertise, it would have been very time real world mathematical system real-world conclusions mathematical conclusions abstraction experiments logical argument interpretation figure 2. sketchpad modelling process computer software sketchpadcomputer model manipulation simulation simulation conclusions modelling of real-world problems is often the starting point for proof 40 consuming to construct the sketches. • the research was aimed at ascertaining whether learners could use these given sketchpad sketches effectively to solve real-world problems, and not on their ability to construct such sketches themselves. • moreover, the research was aimed at investigating whether learners could, through using these given sketches, acquire understanding of important concepts such as bisector, circumcentre, circum-circle, cyclic quadrilateral, and some properties of these. all of the activities were entrenched in the constructivist belief that "the learners have their own ideas, that these persist despite teaching and that they develop in a way characteristic of the person and the way they experience things, leads inevitably to the idea that, in learning, people construct their own meaning" (brookes, 1994: 12). there is little doubt that children enter mathematics classes with views and ideas of certain mathematical occurrences that they experience in real life. they, for example, may be aware that water reservoirs are generally constructed at the top of the highest hill in a village because the pressure required to feed the water to large distances must be great. they may be aware that in order to establish the height of a mountain, it is basically impossible to start from the top of the mountain with a tape measure and work downwards towards the foot of the mountain. they may not know how the height is calculated, but they could have a sense that mathematics would play a role. often, mathematics educators make the flawed assumption that learners are empty vessels into which knowledge must be poured. the constructivist view is completely opposite and is based on the theory that learning is an active process and that learners construct their own meaning. this therefore implies that learners themselves are responsible for their own learning. closely linked to this socio-constructivist theory is the problem-centred learning (pcl) approach, developed in south africa in the mid 1980s by researchers at the university of stellenbosch. the pcl approach is based on a socio-constructivist theory of the nature of knowledge and learning and hinges on the following (olivier, murray and human, 1992: 33): • the learner is active in the process of acquiring knowledge. • in acquiring this knowledge, the learner makes use of past experiences and existing knowledge. • learning is a social process and the learner acquires new knowledge through interaction with other learners and educators. research conducted by the research unit for mathematics education at the university of stellenbosch (olivier, murray and human, 1992) showed that the: majority of children invent powerful nonstandard algorithms alongside schooltaught standard algorithms; that they prefer to use their own algorithms when allowed to do so and that their success rate when using their own algorithms is significantly higher than the success rate of children who use the standard algorithms or when they themselves use standard algorithms. (p. 33) this research clearly contributed to the general constructivist theory of children creating their own knowledge from their own experiences and not from the experiences of the educator or textbook author. the research itself in a modelling experiment conducted with grade 10 learners to teach concepts such as perpendicular bisectors and concurrency, it was found that these learners displayed a definite desire for a proof. i initially envisaged that it would not be necessary to develop and pursue the actual proof of their discovery, because the aim of the investigation simply was to determine whether sketchpad could be used as a modelling tool in developing new concepts (such as concurrency and perpendicular bisectors). later, i felt that it was useful and necessary to investigate whether learners could actually be guided to a simple proof based on their modelling activity. the proof was based on materials developed by de villiers (1999: 32). a real-world problem was given to the students, contextualised within the south african rural background. the question was: in a developing country like south africa, there are many remote villages where people do not have access to safe, clean water and are dependent on nearby streams or rivers for their water supply. with the recent outbreak of cholera in these areas, untreated water from these streams and rivers has become dangerous for human consumption. suppose you were asked to determine the site for a water reservoir and purification plant so that it would be the same distance away from four remote villages. vimolan mudaly 41 where would you recommend the building of this plant? the students were presented with already constructed diagrams using sketchpad as a mediating tool. the diagrams were constructed in a way that allowed the students to eventually model a solution to the above question. after the students had gone through the process of working on a dynamic quadrilateral and discovering that only the perpendicular bisectors of a cyclic quadrilateral were concurrent, they were asked a similar question related to three villages. they readily conjectured that perpendicular bisectors would be concurrent for only those triangles that were cyclic. they were enormously surprised when they used the drag function of sketchpad to discover that the perpendicular bisectors of all triangles were concurrent. when asked if they wanted to know why this is true all students gave the impression that they wanted an explanation in order to extract some understanding from it. the proof was not required because they doubted the result but only because they felt that it would satisfy some innate curiosity around the reason for the result. an example of a learner’s proving process and strength of conviction only one example of the proving process is cited to convey the gist of what transpired after the learners felt the need for an explanation. teacher look at this triangle on the screen again. construct the perpendicular bisector of any side. desigan can i do it for ab? teacher yes. (after the construction) desigan, what can you tell me about all the points on this perpendicular bisector? desigan it is equidistant from a and b. teacher what is equidistant? (trying to ascertain for sure what the 'it' was) desigan all the points on this line (pointing to the perpendicular bisector). teacher what does that really mean to you? desigan if you measure the distance from any point on this line to this a and b, the distance will be the same. in this segment i was simply attempting to get the learner to recall the concepts of perpendicular bisector and equidistant. in a way, it was also a means of determining whether the learner actually understood and remembered what he had done earlier in the interview. teacher now construct any other perpendicular bisector. desigan (constructing) teacher what can you tell about the points on this line now? desigan all the points are the same distance away from b and c. teacher now look at this point of intersection. what can you say about this point in particular? desigan eh … eh… teacher think carefully about the point. desigan that point there is the same distance away from a and b and, b and c. teacher a and b and, b and c? desigan yes, it is the same distance away from a, b and c. teacher are you sure? desigan it lies on this line so it must be equidistant from a and b and it lies on that line so it must be equidistant from a and c (note the error). teacher if it lies on that line would it be equidistant from a and c? desigan no b and c. teacher so are you sure that this point of intersection is the same distance away from a, b and c? desigan yes. initially, it seemed that this learner realised that the point of intersection was equidistant in a fragmented way. in other words, he could see that the point of intersection was equidistant from a and b and b and c separately. he did not instantaneously see the connection between all of the vertices to the point of intersection. teacher you have to think about this very carefully. what can you say about the perpendicular bisector of ac? desigan all the points will be equidistant from a and c. teacher yes, that is correct. but look at the other perpendicular bisectors. desigan (silence for a while) ….oh yes, it must pass through the point where these two lines meet (pointing to the perpendicular bisectors). teacher really? desigan yes because if all the points on this perpendicular bisector of ac are the modelling of real-world problems is often the starting point for proof 42 same distances away…then … then this point of intersection is also the same distance away .. then… teacher yes? desigan then the line must pass through the point of intersection. teacher are you absolutely sure that this would happen? desigan yes, i'm positive. teacher do you want to see whether that is true? desigan yes. teacher construct the perpendicular bisector of ac then. desigan (after constructing) this is so easy. teacher was it really that easy? desigan i didn’t take so long to get it right! eventually, when he realised that there was a connection between the three vertices and the point of intersection, the rest of the explanation became simple. it was clear that because these learners had initially worked in the modelling process, with the concept of equidistance, the actual proof became easy to understand. this is supported by the fact that the learner felt that this explanation was quite easy, and, furthermore, he felt that he alone had got it correct. this must be attributed to the high level of conviction that could be achieved using sketchpad as a mediating tool. it must also be pointed out that during the modelling process itself the learner was encouraged to find the link between the real-world situation and the modelling activity itself. when the learners were asked what this result meant in terms of the three villages, some of the responses were as follows: desigan you can join the three villages and then find the perpendicular bisectors. where they meet is the important point for us to use. teacher do you think that it is easy to just join these villages and find the perpendicular bisectors? desigan i don't think that it’s easy … i'm sure they can draw it on a page first and then do an exact drawing … or even use this programme to get the exact position. teacher do you think that this would be easy to do in real life? desigan i don't know… we must consider a lot of factors … like we discussed in the beginning. vischalan if the villages are situated like this triangle then all you have to do is join the villages, find the midpoints between them and construct the perpendicular bisectors. the point of concurrency will be the most suitable point. teacher do you think that it is easy to just join these villages and find the perpendicular bisectors? vischalan yes … you can use a map of the area. teacher do you think that this would be easy to do in real life? vischalan my uncle told me that you can use gps (global positioning system) to find any point you want. i think the government has pictures of every part of the country. it was evident from some of their responses that these learners were quite capable of transposing real-world problems into mathematical systems and returning to the real world as they see it. conclusion within the context of teaching a real-world problem, the evidence presented by the learners involved in this study, indicates that there is some reason to believe that using modelling situations in a classroom may lead to mathematical proving opportunities. although it cannot be conclusively stated that all modelling activities will lead to the proving process, this experiment does reveal that given certain modelling opportunities, learners may, as a result of the high levels of conviction established, want an explanation for the results they observe. the researcher also concedes that in the pseudo-real-world problem that the learners had to solve, the inductive process was made easier by the use of sketchpad, and indeed the deductive process was also catalysed by what the learners could see whilst working with sketchpad. but it is the contention of the researcher that it is exactly this combination which facilitated a level of understanding not easily achieved by ordinary pencil and paper methods. as a result of this high level of understanding, which began with the modelling process, the learners felt a need for an explanation (proof). vimolan mudaly 43 references blum, w., 1998, “on the role of ‘grundvorstellungen’ for reality-related proofs – examples and reflections”, in gailbraith p. et al., mathematical modelling – teaching and assessment in a technology –rich world. retrieved october 13, 2003 from url http://www.lettredelapreuve.it/cerme3papers/ tg4-blum.pdf bonotto, c., 2004, “how to replace the word problems with activities of realistic mathematical modelling”, in henn, w. & blum, w., eds, proceedings of the icmi study 14: applications and modelling in mathematics education, university of dortmund bowman, j., 1997, making math relate to the real world. a math teacher challenges his students with interesting story problems. retrieved september 12, 2004 from url: http://www.4teachers.org/testimony/bowman/ brookes, d. w., 1994, the nature of conceptual learning and thinking. unpublished article, university of durban-westville davis, p. j. & hersh, r., 1981, the mathematical experience, boston: the harvester press de villiers, m., 1993, “modelling as a teaching strategy”, pythagoras 31, pp 3-4 de villiers, m., 1999, rethinking proof, emeryville, usa: key curriculum press hodgson, t., & riley, k. j., 2001, “realworld problems as contexts for proof”, mathematics teacher 94(9), pp 724 – 728 klaoudatos, n., & papastavridis, s., 2001, “context orientated teaching”, in matos, j.f., blum, w., huston, s.k., carreira, s.p., eds, modelling and mathematics education, chichester: horwood mudaly, v., 2004, the role and function of sketchpad as a modelling tool in secondary schools. unpublished doctoral thesis, university of kwazulu-natal olinick, m., 1978, an introduction to mathematical models in the social and life sciences (1st edition), addison-wesley. olivier, a., murray, h. & human, p., 1992, “problem-centred learning: the case of division”, pythagoras 28, pp 33-36 selden, a. & selden, j., 1999, tertiary mathematics education research and its future. a technical report number 1999-6 from the tennessee technological university. retrieved october 15, 2003 from url: www.math.tntech.edu/techreports/reports.html van den heuvel-panhuizen, m., 1998, realistic mathematics education – work in progress. retrieved august 19, 2004 from url: www.fi.uu.nl/en/rme zulkardi, 2003, how to design mathematics lessons based on the realistic approach? retrieved october 17, 2003 from url: www.geocities.com/ratuilma/rme.html “the silence of infinite space terrifies me.” pascal 63 p30-37 de villiers layout final 30 pythagoras 63, june, 2006, pp. 30-37 a generalisation of the spieker circle and nagel line michael de villiers school of science, mathematics, & technology education, university of kwazulu-natal profmd@mweb.co.za introduction many a famous mathematician and scientist have described how their first encounter with euclidean geometry was the defining moment in their future careers. some of the most well known are probably isac newton and albert einstein. often these encounters in early adolescence have been poetically described as passionate love affairs. for example, the mathematician howard eves describes his personal experience as follows: “...euclid's elements ... i leafed through the book, and found that, from a small handful of assumptions ... all the rest apparently followed by pure reasoning ... the experience had all the aspects of a romance. it was love at first sight. i soon realized i had in my hands perhaps the most seductive book ever written. i fell head over heels in love with the goddess mathesis ... as the years have gone by i have aged, but mathesis has remained as young and beautiful as ever” (in anthony, 1994: xvi-xvii) perhaps noteworthy is that very few famous mathematicians and scientists have ever mentioned arithmetic or school algebra as having been as influential as geometry in attracting them to mathematics. one of the reasons may be the algorithmic nature of high school algebra as pointed out by howard eves as follows: “ ... i still think that geometry is the high school student's gateway to mathematics. it's not algebra, because high school algebra is just a collection of rules and procedures to be memorized” (in anthony, 1994: xvii). moreover, the fundamental mathematical idea of proof, and that of a deductive structure and of logical reasoning, is usually introduced and developed largely within high school geometry, and hardly at all in algebra. though elementary number theory and algebra can provide exciting opportunities for some conjecturing and proof, it is unfortunately not common practise in high school. at present, it is mainly geometry that provides a challenging, non-routine context for creative proof that requires learners to explore and discover the logical links between premises and conclusions. the current reduction of euclidean geometry from the new south african school curriculum at the general education and training (get) and further education and training (fet) levels has been largely motivated by the need to introduce some more contemporary topics. some of these are cartesian and transformation geometry, as well as a little non-euclidean geometry such as spherical geometry, taxi-cab geometry and fractal geometry. however, it would seem disastrous for the future development of mathematicians and scientists in our country to argue, as some do, for the complete removal of euclidean geometry from the curriculum. often the argument seems a purely political one: learners find geometry difficult compared to algebra; we have to improve the pass rate; so let's get rid of geometry! of course, the problem of geometry education is a very complex one, and is not one that i will attempt to address in this article, though some of my mathematics education research and thoughts in this regard appear in de villiers (1997). it is also not a problem limited to our country, but is fairly international. suffice to say that ignoring the problem will not solve it, but that it has to be faced head on, and will require the concerted, combined efforts of mathematicians, mathematics educators, teachers and researchers. this article instead modestly aims to acquaint the reader with some results from 17th and 19th century geometry, and to combat the perspective that geometry is dead by showing that new discoveries can and are still being made. specifically it will discuss a possibly new generalisation of the spieker circle and the associated nagel line, which is parallel to that of the generalisation of the nine-point circle and euler line discussed in de villiers (2005). not only should these results be accessible to a fair number of undergraduate students, prospective and practising high school teachers, but also to the more mathematically talented high school learner. unlike cutting edge research in other areas of mathematics, the results are relatively easy to understand and appreciate, even without proof, because of their visual nature. apart from the remarkable concurrencies of the medians, altitudes and perpendicular bisectors of a michael de villiers 31 triangle mentioned in de villiers (2005), there is a fourth concurrency theorem mentioned in a few south african textbooks, namely: the angle bisectors of the angles of a triangle are concurrent at the incentre, which is the centre of the inscribed circle of the triangle (see figure 1). figure 1: incentre nagel point many mathematics teachers are not aware that there are many famous special centres associated with the triangle, and not only the four, i.e. the centroid, orthocentre, circumcentre and incentre, normally mentioned in textbooks. in fact, clark kimberling's two websites are worth a visit, where over 1000 special centres are associated with the triangle (see kimberling)! antonio gutierrez's site also provides some beautiful, draggable dynamic geometry sketches of some of the more famous triangle centres (see gutierrez). one such notable point is the nagel point, which is the point of concurrency of the lines from the vertices of a triangle to the points on the opposite sides where they are touched by the escribed circles (see figure 2). this interesting point is named after its discoverer, the german mathematician christian heinrich von nagel (1803-1882) and some biographical information about him can be obtained from: http://faculty.evansville.edu/ck6/bstud/nagel.html pascal's theorem the french philosopher and mathematician blaise pascal (1623-1662) discovered and proved the following remarkable theorem at the age of sixteen: all six vertices of a hexagon lie on a conic, if and only if, the intersections of the three pairs of opposite sides are collinear (lie on a straight line) – see figure 3 on following page. this is one of the first entirely projective theorems discovered and proved, and does not involve any measurement of sides or angles. note that when the opposite sides of the inscribed hexagon are parallel they are assumed to meet at infinity, and all points at infinity are assumed to lie on the line at infinity. pascal's proof has unfortunately not survived, but he probably used classical euclidean geometry, and not modern projective methods. y y x x i figure 2: nagel point n a generalisation of the spieker circle and nagel line 32 spieker circle and nagel line the discovery of the nine-point circle and the associated euler-line has often been described as one of the crowning glories of post-greek synthetic geometry (see de villiers, 2005 for more details). however, less well known seems to be an interesting analogue or parallel result involving the spieker circle and the nagel line. the spieker circle is named after theodor spieker whose 1890 geometry book lehrbuch der ebenen geometrie was one of the books that greatly inspired the young einstein (see pyenson, 1985). the rather remarkable parallelism between the nine-point circle and euler line on the one hand, and that of the spieker circle and nagel line on the other hand, is contrasted in the table below, and illustrated in figure 4. (the reader is reminded that the median triangle is the one formed by the midpoints of the sides of a triangle.) the spieker circlethe nine-point circle n p g o g s h i a b c a b c figure 4: nine-point & spieker circles figure 3: pascal’s theorem michael de villiers 33 the nine-point circle is the circumcircle of abc's median triangle and has radius half that of circumcircle of abc. the spieker circle is the incircle of abc's median triangle and has radius half that of incircle of abc. the circumcentre (o), centroid (g) & orthocentre (h) of any triangle abc are collinear (euler line), gh = 2go and the midpoint of oh is the centre of the ninepoint circle (p) so that hp = 3 pg. the incentre (i), centroid (g) & nagel point (n) of any triangle are collinear (nagel line), gn = 2gi and the midpoint of in is the centre of the spieker circle (s) so that ns = 3 sg. the nine-point circle cuts the sides of abc where the extensions of the altitudes through the orthocentre meet the sides of abc. the spieker circle touches the sides of the median triangle where they meet the lines from the nagel point to the vertices of abc. the nine-point circle passes through the midpoints of the segments from the orthocentre to the vertices of the triangle. the spieker circle touches the sides of the triangle whose vertices are the midpoints of the segments from the nagel point to the vertices of abc. the above observations are not new, and appear together with proofs in coolidge (1971) and honsberger (1995). more generally, this is an example of a limited, but an interesting kind of duality not only between the incircles (and escribed circles) and circumcircles of triangles and other polygons, but also between the concepts of side and angle, equal and perpendicular diagonals (e.g. for quadrilaterals), etc. this limited duality or analogy is explored fairly extensively in de villiers (1996) and has been useful in formulating and discovering several new results (see for example de villiers, 2000). having recently rediscovered a generalisation of the nine-point circle to a nine-point conic and an associated generalisation of the euler line (de villiers, 2005), the author wondered how one could generalise the spieker circle (and the nagel line) in a similar way. the following is the result of that investigation. let us first prove the following useful lemma that will come in handy a little later. the first algebraic proof is my own original one while the second geometric one was kindly sent to me by michael fox from leamington spa, warwickshire, uk. figure 5: lemma a generalisation of the spieker circle and nagel line 34 lemma given a', b', c' as the images of any three points a, b, c after a half-turn about o, then the six points a', b', c', a, b, and c lie on a conic. algebraic proof place o at the origin so that the general conic equation we need to determine reduces to ax 2 + 2hxy + by2 + c = 0 . divide through by c, so that only three unknowns now need to be determined. due to the symmetry of the conic equation it follows that if ( x1, y1) satisfies the equation so does its image (− x1, − y1 ) under a halfturn. therefore, only three points are needed to find the conic, one from each symmetric pair, e.g. a or a', b or b', c or c'. geometric proof consider the hexagon ab'cc'ba' shown in figure 5. the image of ab' is a'b, therefore ab' // a'b; similarly bc' // b'c. thus bl'b'l is a parallelogram, and its diagonals bisect each other. but the midpoint of bb' is o, hence lol' is a straight line. these are the intersections of the opposite sides of the hexagon, so by the converse of pascal's theorem, the vertices a, b, c, a', b', c' lie on a conic. spieker conic given a'b'c' as the median triangle of a triangle abc, and a'd, b'e and c'f are three lines concurrent at n. let l, j and k be the respective midpoints of a'n, b'n, and c'n, and x, y and z be the midpoints of the sides of a'b'c' as shown in figure 6. for purposes of clarity, an enlargement of the median triangle and only the relevant points are shown in the bottom part of figure 6 (see following page.) since both xk and lz are parallel and equal to half b'n, it follows that xklz is a parallelogram. similarly jxyl and zjky are parallelograms. let s be the common midpoint of the respective diagonals xl, yj, and zk of these parallelograms. further let the intersections of an, bn and cn with the sides of the median triangle be p, q, and r, and their respective reflections through s be p', q', and r'. if a conic is now drawn through any five of p, q, r, p', q', and r', then the conic cuts through the sixth point, and is inscribed in the median triangle (as well as the triangle obtained from the median triangle through a halfturn around s). proof since p, q, r, p', q', and r' are symmetrically placed around s by construction, it immediately follows from the preceding lemma that all six points lie on the same conic. furthermore, it is obvious that projecting the lines a'd, b'e and c'f onto the altitudes of the median triangle, reduces the conic to the spieker circle. since the spieker circle is inscribed in the median triangle (as well as its half-turn around s), and since any conic and tangents remain a conic and tangents under projection, it therefore follows that the general spieker conic is also inscribed in both triangles. nagel line generalisation given the above configuration for any triangle abc, then the centre of the spieker conic (s), the centroid of abc (g) and n are collinear, and ns = 3 sg. proof the projection of the spieker conic onto the spieker circle, also projects s onto the the centre of the spieker circle, and the point n onto the nagel point, and since collinearity is preserved under projection, s, g and n are collinear. however, since ratios of segments are not necessarily invariant under projection, this is not sufficient to prove ns = 3 sg. however, this follows directly from the ninepoint conic result and associated euler generalisation discussed in de villiers (2005). in figure 6, the nine-point conic result implies that x, k, y, e, l, z, f, j, and d also lie on a conic, and that it has the same centre s as the spieker conic. hence, the euler line corollary of this inscribed nine-point spieker conic, directly proves the nagel generalisation above, so that the centre s of this nine-point conic, the centroid g of abc and the point of concurrency n, are collinear, and ns = 3 sg. concluding comments it is hoped that this article has to some extent expelled the myth that the ancient greeks already discovered and proved everything there is to find and prove in geometry. apart from these results being easily accessible to undergraduate students, they are probably also within reach of talented high school students, particularly those at the level of the third round of the harmony sa mathematics olympiad. michael de villiers 35 figure 6: spieker conic a generalisation of the spieker circle and nagel line 36 moreover, this article has hopefully also demonstrated that possible new geometric discoveries such as the nine-point and spieker conics discussed here can still be made. in fact, it is quite likely that using dynamic geometry software in teaching geometry at high school or tertiary level may enable learners and students to more easily make their own discoveries, as the author has found on several occasions when working with prospective and in-service mathematics teachers. in particular, dynamic geometry software encourages an experimental approach that enables students to make and test geometric conjectures very efficiently. in recent years there has been a general increase in geometry research on many fronts. we’ve seen the development and expansion of knot theory and its increased application to biology, the use of projective geometry in the design of virtual reality programs, the application of coding theory to the design of cd players, an investigation of the geometry involved in robotics, use of search theory in locating oil or mineral deposits, the application of geometry to voting systems, the application of string theory to the origin, nature and shape of the cosmos, etc. even soap bubble geometry is receiving new attention as illustrated by the special session given to it at the burlington mathsfest in 1995. even euclidean geometry is experiencing an exciting revival, in no small part due to the recent development of dynamic geometry software such as cabri, sketchpad and cinderella. indeed, philip davies (1995) already ten years ago predicted a possibly rosy, new future for research in triangle geometry. just a brief perusal of some recent issues of mathematical journals like the mathematical intelligencer, american mathematical monthly, the mathematical gazette, mathematics magazine, mathematics & informatics quarterly, forum geometricorum, etc. easily testify to the greatly increased activity and interest in traditional euclidean geometry involving triangles, quadrilaterals and circles. of note too is a specific yahoo discussion group which is specifically dedicated to current research in triangle geometry and traditional euclidean geometry. readers are invited to visit: http://groups.yahoo.com/group/hyacinthos/ it is therefore unfortunate that, with the exception of a handful of south african universities, hardly any courses are offered in advanced euclidean, affine, projective or other geometries. in this respect, we seem to be lagging behind some leading overseas universities where there is a resurgence of interest in geometry not only at the undergraduate, but also at the postgraduate and research level. not only does this tendency in south africa narrow the potential field of research for a young mathematical researcher, but especially impacts negatively on the training of future mathematics teachers, who then return to teach matric geometry, having studied no further than matric geometry themselves. in contrast to the present south african tertiary scene, the mathematics department at cornell university, for example, is currently running more geometry courses at the graduate level (our postgraduate level) than any other courses (according to a personal communication to the author from david henderson about four or five years ago). moreover, peter hilton, one of the leading algebraic topologists (now retired from binghamton university), is well known for frequently publicly stating that geometry is a marvelous and indispensable source of challenging problems, though algebra is often needed to solve them. it is also significant that the recent proof of fermat's last theorem by wiles relied heavily on many diverse fields in mathematics, including fundamental geometric ideas (see singh, 1997). note: a dynamic geometry (sketchpad 4) sketch in zipped format (winzip) of the results discussed here can be downloaded directly from: http://mysite.mweb.co.za/residents/profmd/spieker.zip (this sketch can also be viewed with a free demo version of sketchpad 4 that can be downloaded from: http://www.keypress.com/sketchpad/sketchdemo.html) references anthony, j.m. (1994). in eve's circles. notes of the mathematical association of america, 34. washington: maa. coolidge, j.l. (1971). a treatise on the circle and the sphere (pp 53-57). bronx, ny: chelsea publishing company (original 1916). honsberger, r. (1995). episodes in nineteenth & twentieth century euclidean geometry (pp 713). the mathematical association of america. washington: maa. davies, p.j. (1995). the rise, fall, and possible transfiguration of triangle geometry. american mathematical monthly, 102(3), 204-214. de villiers, m. (1996). some adventures in euclidean geometry, university of durbanwestville (now university of kwazulu-natal). michael de villiers 37 de villiers, m. (1997). the future of secondary school geometry. pythagoras, 44, 37-54. (a pdf copy can be downloaded from: http://mysite.mweb.co.za/residents/profmd/futur e.pdf ) de villiers, m. (2000). generalizing van aubel using duality. mathematics magazine, 73(4), 303-306. (a pdf copy can be downloaded from: http://mysite.mweb.co.za/residents/profmd/aube l.pdf ) de villiers, m. (2005). a generalisation of the nine-point circle and euler line. pythagoras, 62, 31-35. gutierrez, a. triangle centers at: http://agutie.homestead.com/files/trianglecenter_b .htm kimberling, c. triangle centers at: http://faculty.evansville.edu/ck6/tcenters/index.htm l kimberling, c. encyclopedia of triangle centers at: http://faculty.evansville.edu/ck6/encyclopedia/ pyenson, l. (1985). the young einstein: the advent of relativity. boston: adam hilger. singh, s. (1997). fermat's last theorem. london: fourth estate publishers. at the age of 12, i experienced a second wonder of a totally different nature: in a little book dealing with euclidean plane geometry, which came into my hands at the beginning of the school year. here were assertions, as for example the intersection of the three altitudes of a triangle in one point, which -though by no means evident -could nevertheless be proved with such certainty that any doubt appeared to be out of the question. this lucidity and certainty made an indescribable impression on me. – albert einstein (autobiographical notes) abstract introduction conceptual framework and literature overview research methodology analysis of results discussion of findings conclusion acknowledgements references about the author(s) gabrielle de freitas department of science and technology education, faculty of education, university of johannesburg, johannesburg, south africa erica d. spangenberg department of science and technology education, faculty of education, university of johannesburg, johannesburg, south africa citation de freitas, g., & spangenberg, e.d. (2019). mathematics teachers’ levels of technological pedagogical content knowledge and information and communication technology integration barriers. pythagoras, 40(1), a431. https://doi.org/10.4102/pythagoras.v40i1.431 original research mathematics teachers’ levels of technological pedagogical content knowledge and information and communication technology integration barriers gabrielle de freitas, erica d. spangenberg received: 18 may 2018; accepted: 10 nov. 2019; published: 11 dec. 2019 copyright: © 2019. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract many mathematics teachers struggle to effectively integrate information and communication technology (ict) in their teaching and need continuous professional development programmes to improve their technological pedagogical content knowledge (tpack). this article aims to identify mathematics teachers’ levels of tpack and barriers to integrating ict as a means to inform their continuous professional development needs. the tpack framework of mishra and koehler was used as a lens for this the study. both quantitative and qualitative research methods were utilised. ninety-three mathematics teachers, who completed a quantitative questionnaire, reported higher levels of content, pedagogical, and pedagogical content knowledge, with comparatively lower levels of technology, technological pedagogical, and technological content knowledge. ten of these participants also participated in semi-structured interviews and revealed six primary barriers to integrating ict in the classroom, namely curriculum-related time constraints, technological infrastructure, impact of ict use on the learning process, ineffective professional development, teachers’ pedagogical beliefs and poor leadership. continuous professional development programmes addressing specific ict-integration barriers can effect significant changes in teachers’ tpack, which may promote better teaching and learning of mathematics. keywords: mathematics; technological pedagogical content knowledge; teachers; information and communication technology; continuous professional development. introduction technological advances in south africa over the past two decades have led to information and communication technology (ict) becoming a significant role player in the educational landscape (guerrero, 2010). icts are more readily available and form part of the general resources in many mathematics classrooms. the effective use of icts for teaching and learning adds value to the mathematics curriculum and is associated with improved learner understanding (nkula & krauss, 2014). the incorporation of icts in the mathematics classroom may also have important implications for mathematics performance in south africa, which is viewed as under-performing and below international standards (mccarthy & oliphant, 2013). despite the benefits of icts for mathematics teaching, niess et al. (2009) argue that strategies for the effective integration of icts in the teaching of mathematics are lacking. bester and lautenbach (2014, p. 2175) refer to the ‘fumbling use of technology by practicing teachers who did not grow up with technology’. thus, increased access to and reliance on technology has precipitated an international call for continuous professional development (cpd) to support mathematics teachers’ effective use of icts in the classroom (lundall & howell, 2000; stoilescu, 2011). in particular, sub-saharan african governments underscore the value of professional development for teachers focusing on the use of ict to improve teaching and learning (hennessy, harrison & wamakote, 2010). given the extent of the master’s thesis on which this article draws (de freitas, 2018), we only report on mathematics teachers’ technological pedagogical content knowledge (tpack) and barriers to integrating ict. mathematics teachers should prepare their learners to become members of a global, technological society by fostering 21st century learning goals in their classrooms (mccarthy & oliphant, 2013; spaull & kotze, 2015). however, many south african teachers’ lack of mathematics tpack could entrench poor achievement in mathematics. in south africa, learners are often exposed to traditional delivery of content by teachers and are denied teaching strategies that promote collaboration, communication and the sharing of ideas through icts due to insufficient and inappropriate professional development opportunities for teachers to enhance their tpack. nevertheless, much international research focuses on the development of in-service teachers’ tpack (doering, veletsianos, scharber, & miller, 2009; stoilescu, 2011). in the united states, doering et al. (2009) explored how in-service social studies teachers’ metacognitive awareness of their tpack changed through participation in a professional development programme using online learning environments in the classroom. they found that when teachers are encouraged to think explicitly about tpack and develop metacognitive awareness of their professional knowledge, this leads to positive changes in their teaching practice. however, the study did not focus on the advancement of in-service mathematics teachers’ tpack. stoilescu (2011) explored flexible ways of using the tpack framework for in-service secondary mathematics teachers in toronto, canada. in the study, ict approaches were used to assess learners’ work and to provide them with feedback. the findings show that teachers have difficulty persuading learners to use icts meaningfully in the mathematics class. they recommend that in-service teachers should receive opportunities to further their tpack knowledge and skills. evidence from these studies on tpack highlights the need for professional development and illustrates the pivotal role it plays in bringing about change in teaching and learning. locally, cassim (2010) conducted a secondary analysis of the data from sites 2006, with an emphasis on exploring the pedagogical use of icts for teaching and learning among grade 8 mathematics teachers. the results showed that although south african teachers indicated they were enthusiastic to learn new ways to make teaching and learning interesting, they encountered four barriers that hindered their pedagogical use of icts, namely confidence, time, access to resources, and professional development. in addition, leendertz, blignaut, nieuwoudt, els, and ellis (2013) found that mastery of tpack by mathematics teachers contributed to more effective mathematics teaching in south african schools. however, these studies did not relate mathematics teachers’ current levels of tpack and their barriers to integrating ict to cpd programmes. therefore, this article aims to identify mathematics teachers’ levels of tpack and barriers to integrating ict in order to inform the design of cpd programmes. the authors argue that mathematics teaching in south africa could improve by providing well-designed cpd programmes that address mathematics teachers’ current levels of tpack and specific ict-integration barriers. thus, the research questions are: how do the mathematics teachers’ knowledge domains of the tpack framework correlate with each other? what barriers do mathematics teachers’ face when integrating ict in their teaching? this article adds to current research studies on ict integration in the mathematics classroom. the suggested implications for the design of cpd programmes to meet in-service mathematics teachers’ levels of tpack and to overcome the ict-integration barriers they face may improve mathematics teaching. improved teaching may translate to enhanced mathematics competence in learners. learners leaving school with good mathematical knowledge and skills may add to a better-skilled workforce, which could eventually contribute to the south african knowledge-based economy. mathematical problem solving, critical thinking and innovation are crucial skills for economic growth and development, as well as for south africa’s global competitiveness. subsequently, an outline of the tpack framework is provided, followed by a discussion on barriers affecting cpd of mathematics teachers. thereafter, the research design, and methodology are delineated and an interpretation of the findings is provided. conceptual framework and literature overview the tpack framework considering international interest in how to teach effectively, mishra and koehler (2006) extended the work of shulman (1986) and developed the tpack framework, which underpins this article. the tpack framework describes how teachers can strategically integrate icts in their teaching to create meaningful learning experiences (landry, 2010). tpack supports teachers in the design and integration of context-specific ict-based teaching strategies. a teacher who demonstrates the ability to negotiate the dynamic interaction of mathematics content, pedagogy and technology possesses a unique form of expertise, namely tpack. a teacher with tpack expertise is superior to a mathematician (content specialist), an experienced teacher (pedagogical expert) or a computer scientist (authority on technology) (guerrero, 2010), in the sense that they can integrate all three knowledge domains and employ them in their teaching of mathematics. the tpack framework consists of three primary knowledge domains (koehler, mishra, kereluik, shin, & graham, 2014). the first domain, technology knowledge (tk), associates teaching and learning with the incorporation of knowledge of traditional analogue and digital technologies. thus, a teacher with tk understands (1) how ict integration can improve teaching methods and enhance learners’ content knowledge, (2) how ict tools fit into teaching and learning, but also in learners’ daily life, and (3) how to use current icts. secondly, the pedagogical knowledge (pk) domain relates to knowledge of teaching strategies to support learner understanding. lastly, the content knowledge (ck) domain describes the disciplinary knowledge a teacher conveys to learners. the tpack framework has four additional domains describing the dynamic interaction between the three primary knowledge domains. the first is pedagogical content knowledge (pck). shulman (1986) describes pck as knowledge of teaching strategies, learners’ prior knowledge, interests and abilities and aspects of teaching specific content. secondly, technological pedagogical knowledge (tpk) describes the reciprocal relationship between technology and teaching practices and outlines how technology promotes or constrains specific pedagogical processes. thirdly, technological content knowledge (tck) considers representational and functional capacities of icts to promote or constrain mathematics content. the last and most comprehensive intersecting domain, tpack, refers to the relationship between technology, pedagogy and content. tpack enables teachers to design and integrate relevant, context-specific mathematics activities for learners (koehler et al., 2013). for example, by using dynamic software programmes, such as geometer’s sketchpad, teachers allow learners to explore and visualise interactive mathematics graph functions and to make informal conjectures about the characteristics of functions, which promotes inquiry-based learning. barriers affecting the continuous professional development of mathematics teachers structured cpd to support mathematics teachers’ tpack has been linked with the successful integration of icts in schools (lundall & howell, 2000) and has a significant impact on learner achievement (anthony & walshaw, 2009). the effective use of icts in the mathematics classroom enhances learner productivity by saving time on calculations, reinforcing the relationship between curriculum and reality, scaffolding learners’ exploration and experimentation, and providing immediate feedback (leendertz et al., 2013). leendertz et al. (2013) found that if mathematics teachers master tpack, their teaching of the subject is more effective. mathematics teachers thus have a responsibility to engage in ongoing cpd in how to use icts to offer learners opportunities to succeed in mathematics (beswick, 2007). in-service teachers who did not grow up with technology often feel threatened when they lack opportunities to develop professionally in the use of icts (hennessy et al., 2010). therefore, effective cpd for in-service teachers may contribute towards ameliorating teachers’ negative attitudes towards the use of icts in teaching mathematics (crompton, 2011). numerous barriers exist that impact on the quality and success of cpd programmes. these barriers need to be considered in conjunction with teachers’ levels of tpack to ensure that what teachers learn during cpd interventions translates to their practice. firstly, teachers engaging in cpd possess different ict skills, goals, attitudes about their abilities and notions of themselves as ict integrators (morsink et al., 2011). as such, cpd cannot assume a one-size-fits-all approach (guerrero, 2010). secondly, morsink et al. (2011) cite inconsistent models of cpd as a barrier towards teacher development and training in terms of effective ict integration. this may be exacerbated by the fact that teachers often claim lack of time and access to icts as barriers to their participation in both formal and informal cpd (bennison & goos, 2010) and is evident in teachers’ failure to develop their identities as fluent users of icts, even after engagement in cpd. often cpd efforts are disconnected from classroom practice and the role of reflection is disregarded. polly and hannafin (2011) advocate that reflection plays an important role in cpd, which should create opportunities for teachers to examine their own teaching praxis and should be integral to classroom activity and situated in teachers’ work. thus, sharing of ideas, peer coaching and collaborative problem solving are requisite to teachers’ efforts to integrate icts into teaching (galanouli, murphy, & gardner, 2004). furthermore, developing expertise in ict integration is a time-consuming, long-term process that requires commitment and ongoing effort from teachers (morsink et al., 2011). cpd programmes should focus on the appropriate use of icts by allowing time for teachers to review, evaluate and explore the affordances of different technologies and mathematical software. teachers need to develop an understanding of when to use ict as a part of instruction (crompton, 2011). polly and hannafin (2011) advocate that teachers should select the content and activities they want to focus on during cpd. when teachers perceive ownership, they are more likely to adopt and integrate the cpd pedagogies in their own teaching. cassim (2010) suggests that teachers should design ict-based lessons in collaboration with their colleagues by forming a community of practice. knowledgeable teachers should also host informal ict-mediated workshops to support less knowledgeable teachers. mathematics teachers also experience internal barriers to their development of tpack. ling koh, chai and tay (2014) argue that teachers’ prevailing knowledge bases serve as epistemic resources for their development of tpack, meaning mathematics teachers’ beliefs about how to teach mathematics align with how they were taught mathematics at school. many mathematics teachers who did not grow up with technology limit their knowledge of teaching strategies and interpretation of transferring mathematics content using ict to merely demonstrations, verification, memorisation and practice. however, the incorporation of icts as a learning tool benefits active engagement of learners in a conducive atmosphere by creating opportunities for authentic, cooperative and inquiry-based learning (martin & herrera, 2007). teachers could implement their tpack in their teaching by using icts around four areas as proposed by niess et al. (2009): (1) designing of authentic learning environments and experiences that incorporate appropriate icts to augment learning and innovation in mathematics, (2) planning of lessons that include applications of suitable icts to enhance learning in mathematics, (3) expanding assessment methods and techniques by means of icts and (4) developing professionally by utilising icts. therefore, the authors advocate cpd programmes aimed at improving teachers’ tpack, which are grounded in the context in which ict integration is applied (ford & botha, 2010) by taking cognisance of existing barriers to ict integration, rather than isolated, once-off isolated professional development programmes. research methodology research design both quantitative and qualitative research methods were employed (feilzer, 2009) in this study. the use of quantitative methods enabled the researcher to measure and assess the level of tpack of senior phase mathematics teachers, while the qualitative methods allowed the researcher to explore teachers’ views about ict-integration barriers affecting their teaching. this study was contextualised in gauteng, the economic hub of south africa. although it is the smallest province, gauteng receives yearly the second largest budget after kwazulu-natal for education (unicef south africa, 2017). due to this large financial investment, it was expected that gauteng should include many schools that have access to resources which include ict infrastructure. teachers in gauteng schools might also receive more training opportunities in the use of ict than teachers in other provinces. purposive sampling was used to identify 93 senior phase mathematics teachers from 41 schools to participate in the quantitative survey. they were selected according to the following criteria: firstly, all teachers had to teach mathematics at senior phase level (grade 7, 8 or 9); secondly, mathematics teachers were from public and private schools; thirdly, schools were located in gauteng; lastly, participation was voluntarily. for the sample of 93 participants, 73 (78.4%) participants were female. three ethnic groups were included, namely 10 black african teachers (10.8%), 81 white teachers (87.1%) and 2 asian teachers (2.1%). the mean age of the sample was 40.5 years (standard deviation, sd = 12.1) with a range between 23 and 71 years. participants’ years of experience using icts to teach mathematics are presented in table 1. table 1: frequencies for number of years using information and communication technologies to teach mathematics. thereafter, a sub-sample of 10 participants from the original sample was again purposively selected to engage in one-on-one semi-structured interviews. the researcher wanted to compare teachers who were frequent users of ict in their teaching practice with those who reported having no or limited ict integration in their teaching practices. therefore, participants were selected who had presented different points in their responses on the tpack questionnaire. the researcher was also interested in selecting teacher participants who taught at different types of schools (for example, private versus public, primary versus high, and english versus afrikaans medium schools). by examining the differing opinions, experiences and skills levels of participants, the researcher sought insight into the varying levels of teachers’ tpack. quantitative data collection: questionnaire a tpack questionnaire was used to collect quantitative data, which consisted of two sections. the first section, including 12 questions, surveyed participants’ biographical information. the second section included 26 likert-type scale items with a four-point scale (1 = strongly disagree, 2 = disagree, 3 = agree, 4 = strongly agree) to assess participants’ levels of tpack according to the seven primary knowledge domains, namely tk (items 1–6), ck (items 7–8), pk (items 9–14), pck (items 15–16), tpk items 17–19), tck (items 19–21) and tpack (items 22–26) (de freitas, 2018, pp. 238–239). a fifth response option, namely i cannot respond, was not weighted and excluded from the data analysis. although the original surveys offer a mid-point response, neither agree nor disagree, it was substituted with the response, i cannot respond. the omission of the middle response forced participants with relatively weak opinions to a specific direction. according to sturgis, roberts, and smith (2014), providing a mid-point response may favour (1) participants who are indecisive, (2) those who have an opinion but try to avoid thinking constructively about a directive response, or (3) those who wish to camouflage their ignorance. they also noted that four-point scales compared to five-point scales yield similar reliability, and substituting a neutral response with the i cannot respond response meets the key objective of any survey, namely valid inference. figure 1 illustrates an example of a test item in the questionnaire. figure 1: example of test item 15. the questionnaire was adapted for in-service mathematics teachers within a south african context from two standardised tpack instruments, namely the survey of preservice teachers’ knowledge of teaching and technology (schmidt et al., 2009) and a tpack survey developed by chai, koh and tsai (2011). schmidt et al. (2009) created a self-report survey instrument for measuring preservice teachers’ tpack across four subjects (mathematics, social studies, science and literacy). the survey included 47 tpack items based on a five-point likert-type scale and was designed for repeated use by preservice teachers as they progress through their teacher education programmes. chai et al. (2011) further developed schmidt et al.’s (2009) survey instrument for use with singaporean preservice teachers, who were trained to teach at least two subjects. the ck items in the questionnaire focused on teachers ‘first teaching subject’ and ‘second teaching subject’. information about the specific teaching subjects of each teacher was collected in the demographic data. the survey specifically contextualised tpack items according to a constructivist-orientated use of ict for self-directed and collaborative learning. the newly developed survey consisted of 34 items. the quantitative data were collected over a period of six months, from june to november. participants had an option to complete either an online version of the questionnaire, using google forms, or a hardcopy version in their own time. while 342 schools were invited to participate, only 41 schools agreed. the researchers sent emails to the senior phase mathematics teachers at these schools. the researchers personally collected 46 completed hardcopy questionnaires from participants, while 47 online versions were automatically sent to the researchers via google forms. qualitative data collection: semi-structured interviews semi-structured interviews were conducted during march and april the following year aiming to corroborate the quantitative findings pertaining to participants’ levels of tpack and to determine participants’ views on barriers regarding the integration of icts in their teaching practice. reflecting on the literature review, initial interview questions were developed. after the analysis of the quantitative data, the interview questions were refined and were as follows: how would you describe your level of expertise as a mathematics teacher in terms of your content knowledge? what teaching strategies do you use in your mathematics lessons? how do you apply them? how do you use ict in your mathematics classroom? what do you perceive as barriers for integrating icts into the mathematics classroom? what strategies can you use to overcome these barriers? all interviews took place at locations convenient to the participants, during school holidays, or after school hours. interviews were approximately 45 minutes long and were audio-recorded with the consent of the participants and later transcribed. data analyses procedures the raw quantitative data collected from the questionnaire were captured on a microsoft excel spreadsheet. thereafter, data were cleaned by omitting incomplete, incorrect or inaccurate data from the data set. the quantitative data were analysed using the software package statistical package for the social sciences (spss, version 23). descriptive statistics were employed for the scale variables for the seven tpack domains. the researchers also conducted correlational analyses, namely pearson correlation coefficient (r), to determine the relationship between the seven tpack domains. the qualitative data were analysed by first transcribing the audio-recordings of the interviews and conducting pre-coding. the researchers read through the transcripts and underlined significant text to gain an overall impression of the text data. thereafter, relevant responses were separated from irrelevant responses. saldaña’s (2013) method of inductive coding was applied. codes sharing the same characteristics were grouped together under the same categories and classified according to the theme. the findings from the qualitative analysis were re-examined in relation to the literature review. quality measures validity an overview of literature focusing on the tpack framework and cpd of mathematics teachers contributed to the theoretical validity of the study. the questionnaire was based on two standardised tpack surveys, which had been previously validated by the developers. however, due to changes in the number of items used and the wording of some items, an exploratory factor analysis (efa) was conducted to assess the internal structure of the questionnaire, thus validating the extent to which the test items sufficiently match and exemplify the construct (watson, 2017). the kaiser-guttman rule was used to identify a number of factors (schmidt et al., 2009). to address face validity, the supervisors of this study reviewed the questionnaire to ensure that the constructs were clearly conceptualised. the questionnaire was amended with regard to language and wording of items to make it more suitable for a south african context. thereafter, the questionnaire was piloted with four mathematics teachers with regard to clarity, readability and terminology before being administered to the participants in the sample. the piloting process contributed to the coherence and consistency of the questions. reliability the internal consistency of the tpack domains was determined with the cronbach’s alpha (a) coefficient. the questionnaire used in this study was adapted from the questionnaires developed by schmidt et al. (2009) and chai et al. (2011), which both demonstrated acceptable reliability measures. the cronbach’s a coefficient for each domain in the present study was calculated as indicated in table 2. the total cronbach’s a coefficient for the instrument was 0.93, thus above 0.7, and considered as reliable (gliem & gliem, 2003). table 2: cronbach’s alpha coefficients for the seven knowledge domains of the tpack questionnaire. trustworthiness the rigour and trustworthiness of qualitative research are evaluated through the lenses of credibility, dependability, transferability and confirmability (baumgartner, 2016). two data sources, namely questionnaires and interviews, were used to confirm the emerging findings. these findings were compared with trends identified in the literature review. furthermore, to improve the credibility related to the interviews, the researchers read the transcriptions several times before coding the material. this allowed the researchers to understand and gain insight from the qualitative data to confirm that their interpretations were correct, and that they accounted for the context and spirit of meaning the participants had intended. the researchers documented the procedures and steps during the study so that others could replicate the processes and confirm the findings, thus contributing to the transferability of the study (creswell, 2014). furthermore, by comparing the findings of the interviews against the quantitative results from the questionnaire, the researchers could confirm that trends in the findings had been accurately identified. lastly, by comprehensively narrating qualitative findings from the interviews, and by using thick descriptions, the researchers further enhanced the trustworthiness of their findings. ethical consideration the ethical committee of the faculty of education of the university granted ethical clearance for this study, as well as the gauteng department of education. the reference number for ethical clearance is 2015-080. the researchers complied with all prescribed ethical measures. data confidentiality by using anonymous reporting in this study also contributed to the trustworthiness of the findings. analysis of results quantitative data analysis exploratory factor analysis is suitable for analysing the underlying concepts of a theoretical construct (landry, 2010) and was employed to examine the interrelationships (pallant, 2011) between the seven tpack knowledge domains. efa was specifically used to determine whether participant responses revealed each of the seven knowledge domains of the tpack framework to contain a single factor. establishing the knowledge domains of the tpack that teachers foregrounded in their responses to the questionnaire assisted in achieving the objective of the study, namely to identify their levels of tpack. following the process described by schmidt et al. (2009), the kaiser-guttman rule was used to identify a number of factors and their composition. the kaiser-guttman rule posits that factors with eigenvalues greater than 1 should be accepted. because of high loadings among clustered items, and using the kaiser-guttman rule, a five-factor solution was produced during the factor analysis. table 3 shows a summary of the efa results. table 3: exploratory factor analysis results. this five-factor solution differs from the two questionnaires on which the current tpack survey was based. in the original tpack survey by schmidt et al. (2009), 10 factors were identified. in the survey developed by chai et al. (2011), eight factors were identified. chai et al. (2011) comment that although the tpack construct is conceptualised as having seven constructs, many researchers have successfully validated only the constructs of tk and ck and find it difficult to differentiate pck, tpk, tck, and tpack through factor analysis. similarly, voogt, fisser, pareja roblin, tondeur, and van braak (2012) report that it is difficult to reproduce the seven knowledge domains of the tpack framework using efa. the number of items per knowledge domain in the tpack questionnaire for this study were unequal. furthermore, the low participant-to-item ratio may have influenced the inconsistencies in the factor analysis. furthermore, the mean and standard deviation pertaining to each of the seven knowledge domains were calculated and are presented in table 4. table 4: mean and standard deviation for each of the seven knowledge domains included in the tpack framework. statistical analyses were undertaken to measure mathematics teachers’ levels of tpack according to the seven knowledge domains included in the tpack framework. specifically, correlational analyses utilising pearson correlation coefficient (r) were employed to investigate the strength and direction of the relationships between the tpack domains as indicated in table 5. a significance level of 0.05 was assumed throughout. table 5: correlations table of the seven technological pedagogical content knowledge domains. qualitative data analysis structural coding, which categorises content-based or conceptual phrases to segments of data (saldaña, 2013), was employed during the first-cycle coding process to provide an overview of the data and the broad topic barriers to integrating ict was identified. coded segments were then summarised together for further analysis. in the second cycle, pattern coding was used to organise coded data identified during first-cycle coding by developing a category label that attributed meaning to the organisation of the codes (saldaña, 2013). in-depth analysis of the barriers to integrating icts led to categories related to curriculum-related factors, technological infrastructure, impact on learning process, professional development, teachers’ pedagogical beliefs, and leadership. the themes, categories and codes that transpired from the transcripts of the one-on-one interviews are presented in table 6. table 6: themes, categories and codes for qualitative data analysis. theme: barriers to integrating ict. discussion of findings the three primary knowledge domains: technology, content, pedagogy technology knowledge (tk) participants’ tk is positively correlated with their tpk (r(93) = 0.53, p < 0.01). this relationship is reflected in the findings of schmidt et al. (2009), who reported a positive correlation (r(124) = 0.46, p < 0.01) between american preservice teachers’ tk and tpk. sahin, celik, akturk and aydin (2013) also found a positive correlation between the tk and tpk knowledge domains for preservice teachers in turkey (r(163) = 0.40, p < 0.01). these results suggest that advancing teachers’ knowledge of various icts and improving teachers’ technical proficiency in using icts (tk) may simultaneously lead to better understanding of how icts can be used to change the way teaching and learning occurs (tpk). these knowledge structures might therefore not be independent from one another but could be addressed in an integrated manner. participants’ tk also correlated positively with their tck (r(93) = 0.74, p < 0.01). this finding corroborates the findings of schmidt et al. (2009) and sahin et al. (2013), who revealed a positive correlation between preservice teachers’ tk and tck. the strength of the relationship was weaker in those two studies than in the present one (r(124) = 0.54, p < 0.01 and r(163) = 0.53, p < 0.01). participants’ tk also correlates positively with their tpack (r(93) = 0.71, p < 0.01). this finding is similar to that of schmidt et al. (2009) and sahin et al. (2013), who also reported positive correlations between tk and tpack for preservice teachers in america (r(124) = 0.42, p < 0.01) and turkey (r(163) = 0.41, p < 0.01). the correlations between the domains indicate that teachers’ tk is associated with their tck and tpack, and to a lesser extent with their tpk. these results confirm the findings of finger, jamieson-proctor and albion (2010), who reported that poor tk is linked with limited tpk, tck and tpack. improved tk enables teachers to understand ict, apply it effectively, identify relevant icts for teaching, and adapt to changes and advances in ict. the statistical correlations between tk and tpk, tck and tpack may indicate that a possible avenue to improve teachers’ tpack is to focus first on advancing their tk. content knowledge (ck) participants’ ck correlates moderately and positively with participants’ pk (r(93) = 0.41, p < 0.01). this finding is consistent with those of sahin et al. (2013), who reported a positive correlation between turkish preservice teachers’ ck and pk (r(163) = 0.61, p < 0.01). the strength of the relationship in this study is weaker than the correlation reported by sahin et al. (2013). this result suggests that teachers’ improved understanding of mathematics content (ck) is associated with improved understanding of how learners construct mathematical knowledge and develop mathematical skills (pk). pedagogical knowledge (pk) a strong positive correlation is noted between participants’ pk and pck (r(93) = 0.64, p < 0.01). this result corroborates with previous research findings indicating a positive correlation between preservice teachers’ pk and pck. sahin et al. (2013) report a strong correlation of r(163) = 0.8 (p < 0.01) and schmidt et al. (2009) report a moderate correlation of r(124) = 0.56 (p < 0.01). a moderate positive correlation is found between participants’ pk and tpk (r(93) = 0.37, p < 0.01). this finding aligns with that of schmidt et al. (2009), who reported a positive correlation between preservice teachers’ pk and tpk (r(124) = 0.56, p < 0.01). improved knowledge about educational goals, learner management, planning and implementing teaching strategies and assessing of learners’ understanding (pk) by teachers may equip them to understand how icts can be used to achieve pk goals. the four additional domains: pck, tpk, tck and tpack the tpack framework based on the results for the ck, pk and pck domains, participants’ responses illustrated in table 4 indicate that the participants possess adequate knowledge in each of these domains. there is a strong positive correlation between teachers’ tpk and tpack (r(93) = 0.70, p < 0.01). this relationship corroborates the findings of schmidt et al. (2009), who reported a similar correlation coefficient between preservice teachers’ tpack and tpk (r(124) = 0.71, p < 0.01). sahin et al. (2013) also reported a strong positive correlation between these domains for turkish preservice teachers (r(163) = 0.72, p < 0.01). these findings suggest that cpd to foster teachers’ understanding of the relationship between technology and instructional practices, and of how technology promotes or constrains specific pedagogical processes (tpk), may improve the overall tpack of teachers. there is also a strong positive correlation between participants’ tck and tpack (r(93) = 0.81, p < 0.01). this result is in line with findings by sahin et al. (2013), who reported a strong positive correlation between preservice teachers’ tck and tpack (r(163) = 0.79, p < 0.01). therefore, to advance teachers’ tpack, cpd activities may need to focus on improving teachers’ understanding of representational and functional capacities of icts to better explain mathematics content (tck). the analysis reveals that participants reported higher levels of ck, pk and pck, with comparatively lower levels of tk, tpk, tck and tpack. therefore, teachers’ cpd needs may relate to an increased focus on knowledge and skills related to how to use ict effectively in the teaching of mathematics, by developing their knowledge in tpk, tck and tpack. the correlational analyses as discussed above reveal statistically significant, positive relationships between most of the knowledge domains, with only one negative relationship between ck and tck. these results suggest that tpack should thus be viewed from an integrative perspective (doering et al., 2009) as each of the knowledge domains influences the others. an integrative view proposes that tpack is not a distinct form of knowledge, but is rather integrated in other forms of knowledge during teaching. according to the integrative view, gains in the primary knowledge domains (technology, pedagogy or content) or the intersecting domains translate to a shift in tpack. thus, if a teacher lacks tk, then it is impossible for the teacher to approach teaching through a tpack framework. to advance mathematics teachers’ tpack, intervention strategies should focus on improving each of the underlying knowledge domains. discussion of findings of the qualitative data qualitative findings from one-on-one interviews with mathematics teachers reveal six primary barriers to integrate ict effectively in the mathematics classroom. these barriers include curriculum-related time constraints, technological infrastructure, impact on the learning process, professional development, teachers’ pedagogical beliefs and leadership. these barriers are significant as they inform mathematics teachers’ cpd needs. firstly, curriculum-related time constraints serve as a persistent barrier to teachers’ effective use of icts in the classroom. participant d (female, 11 april) describes that time is her ‘enemy’ and prevents her from researching, learning about and experimenting with new icts, while participant a (female, 30 march) reports ‘i have known that there’s better ways or easier ways to do things but it’s time … you’re on the treadmill and you’re just repeating things and you haven’t had time’. participant e (female, 12 april) laments that ‘there is a huge pressure to get through the syllabus’, which prevents her from experimenting with new, creative learning resources, including icts. literature confirms that teachers’ lack of time to learn about and experiment with new technologies contributes to them feeling under-prepared to integrate icts (cassim, 2010; tondeur et al., 2012). thus, mathematics teachers require cpd that is sensitive to the time demands of their busy schedules and which allows for learning that is sustainable within the context of their teaching. secondly, lack of access to reliable, stable technological infrastructure also impacts on mathematics teachers’ use of icts. this is succinctly summarised by participant f, who suggests that: ‘my barrier to technology is that it’s unreliable. like you can prep[are] a lesson and then you haven’t charged your ipad and then it’s gone or your projector isn’t connecting. … there’s all those other aspects, especially in our country, that’s also going to be a massive barrier.’ (participant f, female, 23 april) insufficient access to technology contributes to teachers feeling ill-prepared to incorporate icts in their teaching (tondeur et al., 2012). borko, whitcomb and liston (2009) argue that the affordances and constraints of icts in education are inherent in the technologies themselves. therefore, although teachers may have access to technology, they cannot utilise it effectively in their teaching due to inadequate or unreliable technology, which is not adopted uniformly in the school setting. literature indicates that teachers’ interpretation of contextual demands and opportunities impact on how they draw upon and integrate their existing knowledge sources (ling koh et al., 2014). therefore, if teachers perceive contextual factors to be a hindrance to their effective use of icts, this may further entrench their reluctance to adapt their pedagogical approaches to be more inclusive of icts. thirdly, mathematics teachers may be reluctant to use icts in the classroom due to the perceived negative impact of technology on the learning process. participant e (female, 12 april) expresses frustration that the incorporation of tablets in the mathematics classroom requires additional discipline and micromanaging on the part of the teacher to ensure that the technology is being used in the correct way and for the correct purposes, arguing ‘i’m forever checking what they’re [learners are] doing’. participant g (male, 23 april) corroborates that the inclusion of icts in the classroom requires the teacher to constantly be ‘supervising’ learners. furthermore, participant c (female, 05 april) reports that in some cases, icts act as a hindrance to learners’ mathematical understanding. she goes on to say: ‘i think it’s [ict is] also a huge distraction to the learners who are going to be distracted anyway’. similarly, literature reveals that the incorporation of ict devices in the classroom can interfere with learners’ abilities to pay attention and to understand content (goundar, 2014). therefore, cpd should equip teachers with knowledge and skills to harness the potential of icts for enhancing learner understanding while limiting the potential distractions that icts may introduce in the classroom. fourthly, participant b (female, 03 april) cites ‘lack of professional development’ as a significant barrier to her use of icts. ford and botha (2010) argue that insufficient training and lack of effective cpd opportunities for teachers have contributed to the failure of e-education projects in south africa, and the fact that icts are yet to transform teaching and learning in schools (polly & hannafin, 2011). hennessy et al. (2010) report that teachers feel threatened when they lack opportunities to develop professionally in the use of modern icts. furthermore, while some participants cite lack of cpd as a barrier, others describe cpd that is techno-centric in nature and that does not meet their needs. participant b states that although she has received training in icts, the training is ineffective as it focuses solely on how to operate technology, rather than to equip her to integrate and use icts for the purposes of teaching. she states that ‘there’s no input as to how am i going to use this in my class, how do i integrate this in my class’ (female, 03 april). therefore, cpd for mathematics teachers should be frequent and occur on a continuous basis. furthermore, teachers require exposure to cpd activities that integrate their knowledge of content, pedagogy and technology, rather than training them in the use of technology in isolation from their daily practice. cpd that relates to mathematics teachers’ daily practice may be more effective in the long term. in addition, mathematics teachers’ pedagogical beliefs and attitudes towards icts may create a persistent internal barrier, which could impact on their effective use of icts. participant d (female, 11 april) concedes that her pedagogical choices are largely influenced and shaped by the teaching strategies she was exposed to during her own schooling. she states, ‘i’m still doing the maths lessons like it was 30 years ago. … i used much of my knowledge of how it was explained to me’. furthermore, participant a (female, 30 march) adds ‘i do know that there’s a big part of all of us that is subconsciously teaching the way we were taught’. holmes (2009) also argues that teachers are unwilling to engage with icts, which require them to change their pedagogical practices. ling koh et al. (2014) suggest that teachers’ prevailing knowledge bases serve as epistemic resources for their development of tpack. participant a (female, 30 march) explains that she has a ‘natural aversion’ and ‘resistance’ to ict and that technology does not ‘even interest me’. she readily admits that teaching with icts feels ‘laborious’ and goes on to say that ‘it shows my own lack of interest in that [ict-based] kind of learning’. hennessy et al. (2010) refer to technophobia in teachers as a prominent factor that hinders teachers’ readiness and confidence to use icts in their teaching. moreover, naidoo and govender (2014, p. 2) state that ‘the use of technology-based tools depends on the teacher’s attitude towards these tools’. effective cpd for in-service mathematics teachers should therefore aim to ameliorate teachers’ negative attitudes towards the use of icts in teaching mathematics (crompton, 2011). mathematics teachers require cpd that not only equips them with knowledge and skills of how to use icts, but also transforms their beliefs about the value of ict and role of ict in teaching. if teachers possess positive attitudes towards icts, the long-term efficacy of cpd may be improved as teachers may be more willing to experiment with and integrate icts in their teaching. lastly, poor school leadership may negatively influence mathematics teachers’ effective use of icts. in the absence of an effective leader, teachers need to become their own driving force. participant c (female, 05 april) comments that at her school, there is ineffective or insufficient driving force from heads of department and the school principal regarding ict integration in the classroom. she suggests ‘you’ve got to have a visionary leader. you’ve got to have somebody who says change is necessary and who makes it happen and who sustains that change’. those in leadership positions at schools play a role in policy decisions regarding ict use, and may be responsible for fundraising and acquiring the ict infrastructure and resources available to teachers in schools. leaders in schools also have a responsibility to support and encourage teachers in their ict use and ensure that policy requirements are implemented. modisaotsile (2012) argues that those in positions of leadership at schools have a responsibility to ensure that decision-making processes, policy determination, problem-solving processes and general governance of schools are participatory in nature. encouraging the involvement of different stakeholders, including teachers, parents and learners, may contribute to more effective policies and implementation thereof. implications for cpd programmes for mathematics teachers the quantitative and the qualitative data analyses distill six implications for cpd programmes for mathematics teachers. firstly, cpd programmes need to be teacher-owned rather than expert-driven. thus, teachers should have input into what they want to learn about. for example, during one-on-one interviews, participant e (female, 12 april) describes that she is a trained accounting teacher, who was asked by the principal at her school to teach grade 8 mathematics. she states ‘maths teachers come up with tricks to help them [learners] remember, ways to explain it. i don’t have any of that background’. cpd programmes to advance teachers’ tpack should focus specifically on improving their understanding of how to explain particular mathematics concepts to learners and should be driven by teachers’ unique, individual needs. secondly, in order to meet mathematics teachers’ unique needs, cpd programmes should relate to teachers’ daily practice. research demonstrates that stand-alone, isolated cpd opportunities are ineffective in bringing about desired changes (o’sullivan & deglau, 2006). in contrast, cpd may be more effective when it pays attention to the teaching-learning context and is situated in the spheres of political, social, curricular and school-level systems. furthermore, to relate to teachers’ practice, cpd programmes should adapt to teachers’ time constraints and employ short, frequent and continuous episodes throughout the year, rather than isolated, sporadic and longer interventions which may be ineffective. thirdly, teachers require cpd programmes that strengthen the professional community of educators. the efficacy of cpd programmes is improved when they are participatory in nature. each teacher who participates in cpd programmes should be provided with a voice through sharing, discussing, reflecting on, critiquing and debating personal teaching experiences and related challenges and successes. peer lesson observation is integral to the formation of professional communities. wahlstrom and louis (2008) suggest that increased visibility of classroom practice through teacher peer observation translates to improved instruction, enhanced teacher self-efficacy and better teacher attitudes towards cpd. teachers may be more courageous and willing to try new things when they can share and learn from the experiences of their colleagues. cpd programmes should provide teachers with the opportunity to observe lessons presented by other teachers, including teachers from other subjects. by examining case studies of their own and others’ teaching practice, teachers also extend their knowledge and understanding of what ‘good practice’ looks like, which may facilitate a broader understanding of how mathematics content can be explained, using different pedagogical approaches and ict resources. the fourth implication is that cpd should align with 21st century learning goals such as being able to collaborate and communicate with others, think critically and solve problems, use icts in innovative ways, take initiative, and bring together various perspectives when learning (law, lee, chan, & yuen, 2008). cpd programmes should promote learning and skills for teachers that mirror the learning and skills required by their learners. in this sense, teachers should be treated as active, participatory learners who construct their own meaning and understanding (o’sullivan & deglau, 2006). drawing on the work of the koehler et al. (2014), the fifth implication is premised on the notion that in order to advance teachers’ tpack, cpd programmes should build on the foundation of in-service teachers’ existing pck knowledge and move towards tpack. when building on teachers’ pck and moving towards tpack, technology should be introduced as a means to scaffold and enrich existing teaching and learning strategies and should build on teachers’ years of experience in a natural way. cpd programmes should therefore start with the fundamentals of pck, in line with shulman’s (1986) model. early stages of cpd, centred around discussion among teachers, should focus on (1) the most useful forms of knowledge representation to make mathematics content comprehensible to learners, (2) the value of different analogies, illustrations, examples, explanations and demonstrations that are most powerful in producing learner understanding, and (3) what makes specific topics easy or difficult for learners to understand, by exploring learners’ thinking and understanding. in addition, initial phases of cpd discussions should explore and examine the concepts and preconceptions that learners of different backgrounds (different schools) and ages bring with them to the classroom. cpd discussions should focus on establishing curriculum requirements. these initial discussions may enable teachers to make sense of and prioritise multiple factors that impact on learners’ understanding and subsequently support their choices of different instructional strategies. this type of approach mirrors that of harris and hofer (2009), who advocate an activity-types approach to cpd, where ict selections are only made after the curriculum learning goals are finalised. building on teachers’ pck, cpd programmes should then focus on improving teachers’ tk by improving their ict literacy in general. based on the quantitative results, improved tk may lead to enhanced tpk, tck and also tpack. therefore, combining teachers’ areas of strength in ck and pk, with an improved understanding of tk may contribute to extending teachers’ tpack. lastly, qualitative findings show that cpd programmes should prepare and support teachers to serve in leadership roles if they are motivated to do so (o’sullivan & deglau, 2006). lack of visible, effective leadership at schools may contribute to teachers’ negative attitudes towards icts. to overcome this threat, teachers should be empowered to serve as leaders themselves. therefore, by creating teacher-owned, contextually based professional communities, mathematics teachers will be better equipped to advance their own tpack and lead others towards developing their tpack. cpd should not only contribute to the development of teachers’ skills but also empower teachers to lead their colleagues to integrate icts effectively. teachers should therefore play a central role in designing and implementing initiatives for their own learning and should be encouraged to view their own classrooms as sites of inquiry. conclusion despite international trends towards effective use of icts in education, south african mathematics teachers often struggle to employ icts as a transformative learning tool to support learners’ mathematical understanding. the reason may be that cpd programmes training teachers in the use of icts do not adequately meet mathematics teachers’ needs in terms of their levels of tpack and barriers to ict integration. therefore, this article aimed to identify mathematics teachers’ levels of tpack and barriers to integrating ict in order to inform the design of cpd programmes. the findings revealed that in-service mathematics teachers possess adequate ck, pk and pck knowledge, while they reported comparatively lower levels of tk knowledge and related tpk, tck and tpack. furthermore, correlational data indicate that tpack should be viewed from an integrative perspective. qualitative data revealed that mathematics teachers face six primary barriers in terms of their effective use of icts, including curriculum-related factors, insufficient technological infrastructure, perceived negative impact of ict on the learning process, lack of or insufficient cpd, incorrect or detrimental teacher pedagogical beliefs and ineffective school leadership. six implications for the design of cpd programmes are suggested. in order to meet in-service mathematics teachers’ levels of tpack and to overcome the ict-integration barriers they face, cpd programmes should be teacher-owned, relate to teachers’ daily practice, strengthen the professional community of educators, align with 21st century learning goals, build on teachers’ pck while moving towards advancing their tpack and, lastly, prepare and support teachers to serve in leadership roles. this article adds to current research studies on the integration of ict in mathematics teaching. suggested implications for the design of cpd programmes for mathematics teachers may translate to improved cpd interventions and teaching practices in mathematics, better alignment with international ict educational trends and eventually improved learner achievement. research on cpd programmes for mathematics teachers may also promote more passionate, knowledgeable, professional and skilled mathematics teachers facilitating mathematics in schools. unfortunately, considering the localised nature of this study by focusing on a single province in south africa, with an emphasis on a single demographic group in terms of financial resources, the conclusions are restricted to this sample. also, as only 41 out of 342 schools agreed to participate in the study, the results cannot be generalised to other contexts. thus, before generalisations of the results can be achieved, the findings should be confirmed through similar studies in other provinces, with teachers from more rural schools. such studies could determine whether teachers from rural areas have different levels of tpack knowledge and face different barriers regarding ict integration from those in the sample of this study. further research could examine changes in teachers’ tpack prior to and after engagement in cpd programmes designed according to the suggested implications in this study. these implications may hold the key to effective cpd, which could promote longitudinal changes in teachers’ tpack and teaching practices. in order to keep abreast of international changes in the educational use of icts for teaching mathematics, there is a dire need to train in-service south african mathematics teachers effectively in how to best use icts. cpd programmes should aim to increase the number of teachers with sufficient and well-established mathematical content knowledge. in addition, cpd programmes should develop mathematics teachers’ knowledge of how to convey content to learners in an understandable way (pedagogical knowledge). furthermore, cpd programmes should advance mathematics teachers’ understanding and skills related to how and when to use icts to support their instruction, thus technology knowledge. teacher educators, school leadership and mathematics heads of departments should consider mathematics teachers’ unique cpd needs in combination with the knowledge domains included in the tpack framework to prepare and further train in-service mathematics teachers as part of being lifelong learners. change in pedagogical choices that reflect the successful use of icts is needed. this change should be based on a renewed emphasis on reflecting on best practice while promoting the standard of education in south africa. acknowledgements competing interests the authors declare that no competing interests exist. authors’ contributions since this article is based on g.d.f.’s master’s study and e.d.s. was the supervisor of the study, g.d.f. collected the data, while e.d.s. provided academic inputs and technical editing to the manuscript. both authors contributed in terms of the conceptualising and writing the manuscript. funding information 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(2017). establishing evidence for internal structure using exploratory factor analysis. measurement and evaluation in counseling and development, 50(4), 232–238. https://doi.org/10.1080/07481756.2017.1336931 performance of grade 6 learners an investigation into the mathematics performance of grade 6 learners in south africa gary s kotzé and japie p strauss university of the free state email: kotzeg.hum@mail.uovs.ac.za and strausjp.hum@mail.uovs.ac.za the aim of this article is to investigate mathematics performance of intermediate phase learners. the quality of learners’ scholastic achievement in mathematics are analysed based on the empirical evidence obtained from an international survey. the method of inquiry is based on an analysis of existing statistical data concerning present-oriented problems. the effects of particular variables on mathematics learning are explored, such as male and female learners and those from less advantaged social backgrounds. overall performance and competence levels are analysed. although there are no outstanding contributing factors that influence mathematics achievement of grade 6 learners in south africa conclusions are drawn that may influence school system organisation and also the quality of mathematics teaching. introduction this article is a companion paper to a preceding article, contextual factors of the mathematics learning environment of grade 6 learners in south africa (kotzé & strauss, (2006), that analysed and related data based on the economic, social and cultural contexts of grade 6 learners and their families to learners’ mathematics learning environment. the focus in this study is the mathematics achievement of grade 6 learners in south africa. data are based on an international research project undertaken by the southern africa consortium for monitoring educational quality, better known as sacmeq ii (sacmeq ii research report, n.d.). since mathematical literacy implies the need for a range of competencies at several levels (pisa, 2003) separate skills cannot be singled out in different mathematical contexts. the sacmeq ii assessment approach is based on the premise that different levels distinguish between different competencies. defined achievement levels reflect explicit targets and expectations of national curricula and international standards. the lowest band of test scores represents low achievement (andrich, 2002a; andrich, 2002b). levels range from simple computations based on definitions, to making connections to solve straightforward problems, to mathematical thinking requiring learners to engage in different reasoning patterns. aim and problem statement the sacmeq ii structure for evaluating mathematics proficiency is based on competency and achievement levels. scaling procedures are implemented to describe learners’ mathematical numeracy levels according to increasing levels of competence and corresponding achievement levels. levels are identified based on item response theory (irt), where the proportion of individuals passing a test item varies as a function of the individual’s ability. thus the proportion of individuals getting a valid item correct (that is the p-value) is correlated with ability. an example would be to administer the same test item to two different groups. suppose the p-value for a group of twelve-year olds is 0.35. thirty-five percent of the twelve-year-olds answer the item correctly while sixty-five percent do not. yet the same item may have a p-value of 0.55 for thirteen-year-olds. in other words, the p-value of an item is not a function of the item but a function used with a specific sample. therefore the test item can be regarded as the independent variable, and the respondents of a specific sample as the dependent variable. in the case where it is assumed that all items have the same discrimination power, this relationship between the probability of a correct response and ability is called the rasch model (andrich, 2002b: 112). this model is regarded as ideal for assessing performance with item structures that do not adhere to normal distribution of classical test theory (ctt) (salvia & ysseldyke, 2004: 3). the value of this approach is that test items are ordered according to levels of difficulty, that is, the greater the total score, the greater the ability implied. in addition, the difficulty structure reflects an implied continuum of achievement. each successive task along the continuum should include elements in earlier tasks, as well as more challenging material. 24 pythagoras 65, june, 2007, pp. 24-31 gary s kotzé and japie p strauss in this study the level and quality of learners’ scholastic achievement in mathematics are analysed based on the empirical evidence obtained from the sacmeq ii survey. data are presented to answer major questions about the mathematics achievement of grade 6 learners in south africa. the following questions are formulated: • what are the mathematics achievement levels of grade 6 learners in south africa? • what is the relationship in achievement levels between important subgroups? research design as the inquiry is concerned with the description and explanation of empirical facts, the investigation is grounded in a positivistic paradigm. the research method is based on an analysis of existing statistical data concerning present-oriented problems (bishop et al., 2003; romberg & collins, 2000). the effects of particular variables on mathematics learning are investigated. mathematics content was based on domains of broad mathematical concepts underlying mathematical thinking (figure 1). the main steps used for the construction of the mathematics tests are outlined in figure 2. the procedure for the construction of the tests commenced with a comparison of common curriculum elements. test construction attempted to ensure that the structure was congruent with the content (domains) and behaviour (skills) by means of analysing curricula, syllabi, examinations and textbooks used in the sacmeq ii countries. after restrictions were eliminated, rasch scaling procedures were implemented to analyse competency levels and establish relationships between competency and probability. a test blueprint was developed followed by writing trial test items. high levels of face validity and construct validity were ensured by achieving congruence between the test blueprint (that is, a framework for test construction) and the descriptions of increasing levels of competence. the framework was developed based on a rasch analysis of item difficulty levels in combination with a skills audit of test items. the initial five levels for the test blue print were extended to eight levels and will be dealt with in more detail below. representative test items after the skills audit new levels were identified and derived. the extra levels that were added tended to reflect deeper levels of understanding of specific mathematics competencies. in order to obtain a better background of different levels, a brief analysis of item levels is provided. representative items from the eight levels followed by a brief motivation are provided below. level 1: pre-numeracy example a b c d 73 + 27 = …?… 46 90 100 110 motivation this item requires learners to perform simple addition provided that only one application of the arithmetic operation is required. level 2: emergent numeracy example a 3,531 b 3,631 c 3,742 subtract … 6,000 – 2,369 ? d 4,369 motivation learners at this level can subtract with carrying and successfully perform arithmetic tasks that require fixed processes. number operations and number line, square roots, rounding and place value, significant figures, fractions, percentages, and ratios measurement measurements related to distance, length, area, capacity, money, and time. space-data geometric shapes, charts (bar, pie, and line), and tables of data. figure 1. domains for the sacmeq ii mathematics test (sacmeq ii research report, n.d.) 25 an investigation into the mathematics performance of grade 6 learners in south africa 26 level 3: basic numeracy example what shape are these dice? a – spheres, b – cubes, c – cylinders, or d – pyramids. motivation at this level learners know the names of threedimensional regular shaped objects and are able to link them to everyday objects. level 4: beginning numeracy example how many 80 g small packets can you fill from a 2 kg packet? a b c d 20 25 40 160 motivation at this level learners show the ability to combine operations. in this example the task involves a conversion of measurement units followed by division. these steps require knowledge of units and the relationship between the units. level 5: competent numeracy example use the price list to find the total cost of 5 litres of milk, 3 kg of tomatoes and 2 chickens. price list item unit cost milk 1 litre 0,60 tomatoes 1 kg 0,50 chicken each 6,70 a b c d r7.80 r16.90 r17.90 r18.90 describe domains (content) construct test blueprint analyse official curricula school syllabi textbooks and examinations describe skills (behaviours) write trial test items implement trial testing select test items implement main data collection conduct rasch item calibration and test equating develop summary descriptions for each competency level complete skills audit to identify the specific skills required for success on each item cluster items with similar difficulties and requiring similar skills into competency levels cross-check the descriptions of the competency levels with the test blueprint to ensure face validity and construct validity figure 2. key steps involved in test construction for the sacmeq ii project gary s kotzé and japie p strauss motivation at this level learners are able to combine operations such as required above. the sequence and the ability to control the information are important. level 6: mathematically skilled example calculate the area of the shaded part in the figure below. a b c d 13 m2 28 m2 48 m2 50 m2 motivation learners at this level are able to apply specific knowledge of mathematics in a range of areas such as number, measurement and data analysis. the item requires learners to bring to the task some external knowledge of mathematics but it involves conventional mathematics skills that are generally taught in the curriculum. in this example the learner needs to know properties of geometry figures and the notion of conservation of area and combine this with knowledge of areas. if learners possess this kind of knowledge the solution is relatively straightforward. level 7: problem-solving example the table represents a relation between x and y. what is the missing number in the table? x 1 2 4 5 y 3 9 11 a b c d 4 5 6 7 motivation this level requires learners to be able to extract and convert information (for example, with respect to measurement units) from tables, charts, visual and symbolic presentations in order to identify and then solve multi-step problems. level 8: abstract problem-solving 2 m 2 m 5 m 10 m example a figure consists of 6 equal squares. the area of the whole figure is 384 cm2. find the perimeter of the figure. a b c d 64 cm 81 cm 112 cm 192 cm motivation this level requires learners to identify the nature of an unstated mathematical problem embedded within verbal or graphic information. the learner has to translate the information given into a mathematical approach and then identify the correct mathematical strategies to obtain a solution. at this level the mathematical knowledge and skills required to solve the problem is not immediately obvious. it is argued that the emergence of extra levels was understandable because the test blueprint had been targeted at learners, whereas the skills audit covered test items given to both teachers and learners. unfortunately, in south africa, teachers were not afforded the opportunity to write the test as in other sacmeq ii countries, which is a great impediment in addressing mathematical literacy in south africa. analysis of test test scores were analysed with a software interpretation of the rasch scaling procedures that automatically adjusted the scores to a scale with an arbitrary zero point and a standard deviation of one. the result was that many learners were assigned negative scores. as this is not an ideal situation from an educational perspective, a linear transformation assisted in assigning 500 and 100 to the mean and standard deviation of learners’ scores respectively. therefore the basic feature of the mathematics achievement rasch model is that the programme performs a linear transformation of the test scores into a scale on which the predetermined mean is 500 and the standard deviation is 100. by means of the rasch analysis the ability of the learners is matched to the difficulty of the test items, which allows learners and items to be mapped on to the same scale. learners could therefore be grouped in the same ‘ability’ or ‘difficulty’ range as the items that had similar difficulty. sixty-three items were used to test the mathematics performance of grade 6 learners. 27 an investigation into the mathematics performance of grade 6 learners in south africa presentation of data and discussion the data analysis firstly involved a comparison of learners’ overall scores in ‘all’ the test items. this is only a relative comparison that can be used to assess the distribution of learners’ achievement, for instance among schools within provinces or among provinces. it cannot, however, be implemented to make judgments about the quality of performance in any of these instances. the overall mathematics performance for grade 6 learners in south africa is presented in table 1. for south africa the overall mean rasch score obtained by grade 6 learners for mathematics was 486.2. this is below the pre-determined rasch score of 500. three provinces exceeded the mean. the standard error of sampling (se) of the means as reflected in the table above were fairly large which implies that results should be interpreted with due caution. every result should be interpreted in terms of the standard error of sampling (se). in the case for south africa in total, the performances of 95% of the population from which the sample was taken will fall in the performance range 486.2 ± 2(7.18). for the second dimension of comparison, all the test items were organised into clusters according to levels of difficulty (discussed earlier) and the learner competencies required in answering them correctly. the percentages of grade 6 learners who reached various competence levels in mathematics are presented in table 2. for mathematics the modal competence reached by 44.4% of the learners was level 2 (emergent numeracy). in addition, 7.8% of the learners achieved only level 1 (beginning numeracy). a total of only 24% of the grade 6 learners achieved competence above level 3 in mathematics and a mere 1.3% were competent at level 8. in light of the fact that grade 6 is a key transition stage between the intermediate and the senior phases of the south african school system, these figures and tendencies are extremely worrying. in a normal distribution it would have been expected that a considerably lower percentage of the e as te rn c ap e f re e s ta te g au te ng k w az ul u n at al m pu m al an ga n or th er n c ap e li m po po n or th w es t w es te rn c ap e s ou th a fr ic a mean 449.5 447.7 552.3 510.3 433.6 461.1 446.2 419.8 591 486.2 se 10.72 5.99 25.97 17.45 10.8 8.23 18.77 10.58 23.89 7.18 table 1. overall mathematics performance (sacmeq ii research report, n.d.) percentage of learners reaching the mathematics competence level 1 2 3 4 5 6 7 8 % se % se % se % se % se % se % se % se ec 11.6 2.25 52.1 4.42 25.5 2.92 7.2 2.26 1.7 1.56 1.6 1.10 0.0 0.00 0.3 0.26 fs 6.8 1.40 56.0 3.75 33.1 3.50 2.4 1.15 1.8 1.00 0.0 0.00 0.0 0.00 0.0 0.00 gau 2.4 0.95 24.0 8.67 23.7 6.01 13.1 3.61 17.1 7.95 15.1 4.46 3.5 1.72 1.2 0.67 kzn 3.4 0.88 39.1 6.16 23.6 2.79 13.0 2.72 8.5 2.89 8.4 3.36 2.8 1.33 1.3 0.70 mpu 10.6 2.37 64.6 5.52 20.4 4.03 2.7 2.07 1.2 1.24 0.4 0.41 0.0 0.00 0.0 0.00 nc 6.4 1.51 50.4 5.08 31.7 3.38 9.1 2.17 2.1 1.14 0.4 0.36 0.0 0.00 0.0 0.00 lp 13.5 2.69 57.3 4.12 21.1 3.00 3.3 1.35 0.6 0.41 0.2 0.24 1.4 1.44 2.6 2.64 nw 18.3 4.75 61.8 2.52 17.4 3.88 2.3 1.03 0.2 0.20 0.0 0.00 0.0 0.00 0.0 0.00 wc 0.5 0.54 13.1 4.39 25.7 4.82 19.4 2.21 12.2 2.13 16.1 3.45 8.6 2.63 4.4 2.29 sa 7.8 0.77 44.4 2.32 23.8 1.37 8.8 0.96 6.1 1.47 5.8 1.09 2.1 0.46 1.3 0.48 table 2. mathematics competence levels of learners 28 gary s kotzé and japie p strauss learners should attain level 3 and that more learners would demonstrate higher levels of competence. hence, comparisons between regions, between sub-groups and between particular characteristics prevalent in different regions are investigated. three important sub-groups are identified to compare them with the mathematics achievement. table 3 reflects the data on these sub-groups. gender and learner achievement reveal that the modal achievement for mathematics is level 2 (emergent numeracy). the modal achievement for boys was 44.7%, which is slightly higher than the 44.2% for girls. however, the percentages of both boys and girls achieving the highest level for mathematics are remarkably comparable. more boys than girls achieved at the lower end of the scale while more girls tend to perform better on the higher end. the modal achievement of school location and learner achievement reveal the following: learners from both isolated/rural and small town schools are level 2 (emergent numeracy) achieved by 59.6% and 46.9% of the learners respectively. learners from the large city schools (21.1%) achieved level 3 (basic numeracy). for isolated/rural and small town schools the percentages of learners who achieved various levels of achievement declined towards the higher end of the scale and ended with no learners from isolated/rural school and only 0.4% from small town schools achieving at level 8. there seem to be factors other than school location that influence achievement because, regardless of the location of school, the majority of learners achieved around the same level (level 2). socio-economic differences and learner achievement were determined and expressed in terms of the levels of home possessions, as discussed in kotzé & strauss (2006). two subgroups were identified: learners in the low economic status sub-group (low ses) were those who came from homes whose possessions were half or less than the suggested list of items. learners in the high economic status sub-group (high ses) were those who came from homes where the possessions were more than half the suggested list of items. the presentation in table 3 indicates that for both the low and the high socioeconomic status learners the modal achievement in mathematics is level 2 (emergent numeracy), that is 54.9% for the low socio-economic status subgroup and 34.2% for the high socio-economic subgroup. a significant number of learners from low socio-economic backgrounds achieved levels from one to three whereas learners from high socioeconomic levels were fairly evenly spread across higher levels of achievement. findings and conclusion data from grade 6 learners’ scores for mathematics provide the following answers to the problem statement: • the overall learner achievement was noticeably on the lower end of the acceptable limits on the sacmeq ii benchmark. percentage of learners reaching the mathematics achievement level 1 2 3 4 5 6 7 8 % se % se % se % se % se % se % se % se gender boys 8.2 0.96 44.7 2.22 25.0 1.69 8.5 0.97 4.9 1.24 5.6 1.21 2.0 0.48 1.1 0.43 girls 7.6 0.97 44.2 2.87 22.7 1.64 9.0 1.31 7.1 1.78 5.9 1.20 2.1 0.55 1.4 0.60 socio-economic level low ses 10.7 1.14 54.9 2.04 25.5 1.54 6.0 1.09 1.7 0.63 1.1 0.63 0.1 0.09 0.0 0.04 high ses 5.0 0.73 34.2 3.22 22.1 2.19 11.5 1.40 10.3 2.34 10.3 1.66 4.0 0.88 2.5 0.92 school location isolated/rural 11.8 1.48 59.6 2.08 22.7 1.70 4.6 1.09 1.0 0.49 0.3 0.15 0.1 0.10 0.0 0.00 small town 7.7 1.35 46.9 3.87 27.5 2.17 9.8 1.98 4.0 1.62 2.3 0.90 1.4 0.65 0.4 0.30 large city 2.4 0.75 20.0 4.21 21.1 3.22 14.2 2.01 15.5 4.04 17.1 2.71 5.7 1.55 4.0 1.60 south africa 7.9 0.77 44.4 2.34 23.5 1.37 8.8 0.97 6.1 1.49 5.8 1.10 2.1 0.47 1.3 0.48 table 3. mastery levels for sub-groups (sacmeq ii research report, n.d.) 29 an investigation into the mathematics performance of grade 6 learners in south africa • the relationship between difficulty levels and learner competencies reveals a significant low level of general numeracy, understanding and skills. test items focusing on higher order meta-cognitive skills were poorly answered. variations (inequities) in learner achievement were significantly wide both within the provinces and among the provinces. • data addressing the achievement levels of important subgroups revealed that although overall achievement in mathematics was generally low, the girls in this study demonstrated higher numeracy competencies than the boys when higher levels are compared. it seems as if the relationship between the learners’ socio-economic status and their level of achievement is fairly strong. it also appears as if there are apparently deeper problems regarding the inadequate performances of learners in general. the general findings that emerged are now discussed. there are no outstanding contributing factors that influence mathematics achievement of grade 6 learners. poor results are probably due to a range of factors that have financial and logistic implications. it appears that a major emphasis was given in classrooms to the mastery of basic skills. deeper analysis of the data may also assist in identifying some of the most important factors contributing to the levels of achievement. achievement comparisons can also provide indications about what can be accomplished, for example, it was possible for learners from some provinces to attain higher levels of achievement. this should serve as an incentive for improvement in other provinces. the diversity of learners must be recognised. the effect of age and gender, and the role of parental involvement and family background could be identified as possible influences on achievement. however, learner characteristics such as aptitude, perseverance, prior knowledge, affective barriers as well as attitudes and values towards mathematics could not be determined in this investigation. learners’ economic and social contexts influence their schooling. the correlation between the index of home possessions (indicating the home economic status) and achievement showed a considerably high correlation. the school plays a bigger role than the home in determining mathematics achievement in a developing country like south africa. mathematics achievement often lags behind in disadvantaged schools. the question may be posed whether mathematics teaching is a bigger problem in disadvantaged schools than ordinary schools, or whether other contributing reasons exist. on the other hand, there may also be special factors prevalent in such schools because some of the disadvantaged schools show much higher achievement than others, in spite of difficult circumstances. school location plays a role in achievement. among the provinces with lower economic levels lower scores were associated with rural schools and higher scores were associated with urban schools. one of the limitations to this investigation is the fact that the role of the educator was not taken into consideration with regard to academic content knowledge of mathematics as a discipline. educator subject knowledge is crucial and has been shown to be a good predictor of learner achievement. the lack of such information may jeopardise academic advancement of mathematics in south africa. a further limitation is that the rasch theory as applied in this study suggests that achievement in mathematics is a one-dimensional trait, even though the items covered topics from a variety of strands in the curriculum, including the three indicated domains. in addition, measurement of the traits is based on 63 achievement items – a small number on which to base an assessment of mathematical proficiency. lastly, information was collected through the administration of test items presented in objective multiple-choice format. it is a cost-effective and efficient assessment technique to collecting great volumes of data from many respondents. in this case validity and reliability of data are also guaranteed to a much greater extent than with other data-gathering techniques. nevertheless, its limitations are obvious. information about how learners approach the solution of a given problem is more important than whether or not they are able 30 gary s kotzé and japie p strauss to recognise the correct solution. valuable conclusions regarding problem-solving abilities, creativity, values and attitudes have been lost. the data discussed may serve as valuable sources of information against which stakeholders can compare and contrast quality education. the following observations that resulted from this study may serve as recommendations for stakeholders in assisting them in future planning: • those who are involved in the teaching of mathematics need to find their own solutions to improving quality. by focusing on targeted interventions, specific weaknesses may be addressed. • a sound knowledge base can make a major difference to the quality of mathematics education. initiatives often require research and/or knowledge that are specific to context and local circumstances. those who work in the locality or region can generate such initiatives. • investment in sources, networks and structures designed to develop and share educational knowledge can yield significant returns by enabling schools to make much better use of limited resources. however, much can be achieved by making better use of existing resources. the investigation provided a more concrete analysis of what learners can actually do. a search for any general theory of successful education reform is difficult to accomplish and unlikely to succeed. in conclusion, it seems that incremental improvements are important: we should not be afraid of growing slowly, we should only be afraid of standing still. references andrich, d. (2002a). a framework relating outcomes based education and the taxonomy of educational objectives. studies in educational evaluation, 28: 35-59. andrich, d. (2002b). implications and applications of modern test theory in the context of outcomes based education. studies in educational evaluation, 28: 103-121. bishop, a.j., clements, m.a., keitel, c., kilpatrick, j. & leung, f.k.s. (2003). second international handbook of mathematics education. dordrecht: kluwer. kotzé, g.s. & strauss, j.p. (2006). contextual factors of the mathematics learning environment of grade 6 learners in south africa. pythagoras, 63: 38-45. pisa (2003). programme for international student assessment. retrieved december 11, 2005, from http://www.pisa.gc.ca/math romberg, t.a. & collins, a. (2000). the impact of standards-based reform on methods of research in schools. in a.e. kelly & r.a. lesh (eds.), handbook of research design in mathematics and science education. new jersey: lawrence erlbaum associates. sacmeq ii research report (no date). southern africa consortium for monitoring educational quality research report. paris: unesco. salvia, j. & ysseldyke, j.e. (2004). assessment in special and inclusive education. retrieved june 16, 2003, from http://suen.ed.psu.edu “you may never know what results come from your actions, but if you do nothing, there will be no results.” mahatma gandhi 31 << /ascii85encodepages false /allowtransparency false /autopositionepsfiles true /autorotatepages /all /binding /left /calgrayprofile (dot gain 20%) /calrgbprofile (srgb iec61966-2.1) /calcmykprofile (u.s. web coated \050swop\051 v2) /srgbprofile (srgb iec61966-2.1) /cannotembedfontpolicy /warning /compatibilitylevel 1.4 /compressobjects /tags /compresspages true /convertimagestoindexed true /passthroughjpegimages true /createjdffile false /createjobticket false /defaultrenderingintent /default /detectblends true /colorconversionstrategy /leavecolorunchanged /dothumbnails false /embedallfonts true /embedjoboptions true /dscreportinglevel 0 /syntheticboldness 1.00 /emitdscwarnings false /endpage -1 /imagememory 1048576 /lockdistillerparams false 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<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> >> >> setdistillerparams << /hwresolution [2400 2400] /pagesize [612.000 792.000] >> setpagedevice abstract introduction literature review research design and methodology findings conclusion acknowledgements references about the author(s) odette umugiraneza department of mathematics and computer science education, university of kwazulu-natal, south africa sarah bansilal department of mathematics and computer science education, university of kwazulu-natal, south africa delia north school of statistics and actuarial science, university of kwazulu-natal, south africa citation umugiraneza, o., bansilal, s. & north, d. (2018). exploring teachers’ use of technology in teaching and learning mathematics in kwazulu-natal schools. pythagoras, 39(1), a342. https://doi.org/10.4102/pythagoras.v39i1.342 original research exploring teachers’ use of technology in teaching and learning mathematics in kwazulu-natal schools odette umugiraneza, sarah bansilal, delia north received: 03 june 2016; accepted: 10 sept. 2018; published: 14 nov. 2018 copyright: © 2018. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract it is often claimed that technology can be used as a tool that can facilitate teaching and learning and contribute to learners’ achievement. this article reports on a study about how kwazulu-natal mathematics teachers use, access and integrate technology in the teaching and learning of mathematics. a questionnaire containing closed and likert scale questions regarding the use of technology, was distributed to 75 kwazulu-natal mathematics teachers. the findings reveal that the technology used most commonly by the group for teaching mathematics is calculators. almost all the teachers reported that they never use computers in their teaching of mathematics. although the teachers reported that they do not use computers in teaching and learning, about 80% of the participants conveyed a positive view that using technology improves learners’ understanding of mathematics. the findings further indicate that the teachers’ propensity to use technology in instructional practice is associated with demographic factors related to teaching experience, gender, level of study and participation in professional learning activities. the study also showed that teachers who have access to internet instructional resources have higher levels of confidence in teaching mathematics and hold broader beliefs about the nature of mathematics and the aims of teaching mathematics than the teachers who do not use the internet for instructional purposes. introduction the rapidly growing influence of technology in the 21st century has led to calls for teaching and learning to be transformed to prepare learners to compete within the global knowledge economy. learning in the 21st century requires the collaboration of well-trained teachers, working in well-equipped classrooms and using technology innovatively to support a constructive learning atmosphere (molnár, 2008). technology allows learners to move beyond focusing on basic information to more global issues by providing them with access to innovative applications and tools (van melle & tomalty, 2000). the teaching environment can thus be transformed by teachers if they integrate technology effectively in preparing lessons, designing learning activities and conducting assessments. the potential of technology to transform the classroom is recognised by the south african department of education (doe) which supports the idea of introducing information and communication technology (ict) in south african schools (department of basic education [dbe], 2016; doe, 2007). teachers are urged to develop learners with ‘relevant modern skills that match the needs of our changing world’ (dbe, 2016, p. 3). learners should be able to ‘access, analyse, evaluate, integrate, present and communicate information; create knowledge and new information by adapting, applying, designing, inventing and authoring; and function effectively in a knowledge society by using appropriate ict … skills’ (doe, 2007, p. 3). the education department states that ict can recreate a classroom atmosphere while also advancing higher-order thinking skills in learners (dbe, 2010). for example, it enables teachers and learners to increase the level of comprehension, reasoning, problem-solving, thinking and employability (doe, 2004, 2007). the doe further highlights five targets of the use of ict which involve ‘entry (basic ict skills), adoption and adaptation (integration of ict in teaching and learning), and appropriations and innovation (specialisation and innovation in ict education)’ (doe, 2007, p. 9). thus, teachers are encouraged to develop their capability and innovation to make the best use of the potential of digital devices in augmenting learner performance (ndlovu & lawrence, 2012). it has become incumbent upon teachers to attain relevant and appropriate ict knowledge and skills to be able to integrate it appropriately in teaching, learning and administration (doe, 2007). however, the digital divide, which is the disparity in the level of development of and access to ict between different sectors, presents a challenge to educational innovations. insufficient basic ict infrastructure in rural schools poses a challenge for teachers, which is not necessarily the case in urban schools (dzansi & amedzo, 2014). ndlovu and lawrence (2012) emphasise that ict policy has been poorly implemented across south african schools, more specifically for those schools that serve disadvantaged areas, thus adding to the digital divide. many disadvantaged schools cannot keep up with the well-resourced schools in terms of integrating ict into their teaching and learning approaches. the limited use of ict is not simply caused by the shortage of resources, but it is dependent on the ways in which the teachers utilise the available educational tools in their teaching (ndlovu & lawrence, 2012). research highlights particular teacher factors such as age, experience, confidence, beliefs, as well as gender which seem to influence the extent to which teachers take up technology in their teaching practices (ali, 2015; beswick, 2007; brändström, 2011; cavas, cavas, karaoglan, & kisla, 2009; choi, 1992). this article addresses the use of technology in teaching mathematics and statistics. recent advances in technology have unlocked entirely new directions for education research. in this study, we try to make a contribution towards finding out more about the use of technology in kwazulu-natal schools. the study also explores the relationship between teachers’ use of technology and their confidence and beliefs about the ways in which mathematics should be taught. to our knowledge, no previous study has focused on these issues. furthermore, the study looks at some factors that may have a relationship with the use of technology. it is hoped that the knowledge contributed by this study will help the education department in their planning and provision for teacher support in the use of technology. we also hope that this study will help other researchers identify areas in the field of mathematics teachers’ use of technology which need more attention. literature review the integration of technology in teaching and learning is not intended to replace traditional methods, but to support schools to improve teaching and learning (tishkovskaya & lancaster, 2012). some technology tools include ‘power points, web-based games, the internet, projectors, smart boards, elmos, calculators, videos, dvds and music’ (moore, 2012). the gaise college report (gaise college report asa revision committee, 2016) includes graphical calculators, statistical software packages, educational software, applets, spreadsheets, classroom response systems, web-based statistics related resources, data repositories, online texts, and data analysis routines in their list of recommended technology tools. icts, especially computers and internet technologies, support new ways of teaching and learning rather than simply allowing teachers and students to do what they have done before in a better way (noor-ul-amin, 2013). however, for teaching and learning to improve, technologies must be used as cognitive tools for learning and not simply as an alternative delivery platform (herrington, reeves, & oliver, 2010). moore (2012, p. 14) reports that integrating technology in a mathematics classroom can promote the development of computational skills while also developing higher order mathematical skills. the view of forster (2006) is that using technological tools can improve the learning of mathematics by allowing learners to pay attention to underlying properties and relationships instead of focusing on tedious complicated calculations that may sometimes detract from the intended outcomes. ict provides opportunities for learning by helping learners to access, spread, renovate and share ideas and information, which is transmitted in integrated communication styles and designs. technological tools can also open up access to a wider variety of problem-solving strategies than those limited to paper and pencil strategies (bansilal, 2015). tools such as online videos allow the students to vary the pace at which they can learn new material in mathematics (bansilal, 2015). by providing access to different representations that help visualisation of mathematical objects, certain mathematics software can contribute to a deeper understanding of the concepts. technology also opens up possibilities for developing statistical concepts by enabling the visualisation of the concepts (sorto & lesser, 2009); it can make the demonstration of complex abstract ideas easier while also providing multiple examples (chance, ben-zvi, garfield, & medina, 2007). in teaching statistics, technology can aid students in learning to think statistically by facilitating access to real (and often large) data sets and fostering active learning. thus it can allow a learner to explore concepts and analyse data, manage and visualise data, perform inference, and check conditions that underlie inference procedures (gaise college report asa revision committee, 2016). purcell, heaps, buchanan and friedrich (2013) describe the importance of internet and digital tools in teachers’ work of teaching. they state that ‘the greatest impact of the internet and other digital tools on their role as teachers has been access to more content and material for use in the classroom and a greater ability to keep up with developments in their field’ (p. 51). noor-ul-amin (2013) argues that networked computers with internet connectivity can increase learner motivation as it combines the media richness and interactivity of other icts with the opportunity to connect with real people and to participate in real world events. kramarski and feldman (2000) report that instruction that integrates the use of the internet in classrooms improves learners’ motivation in learning and has positive effects on learners’ reading comprehension. brändström (2011) examined the influence of the use of the internet on planning and instruction by interviewing five upper secondary school teachers. the findings revealed that the teachers consider the internet as a valuable source of information and an important additional teaching tool. it also reduces teachers’ work while facilitating quick exchanges (higgins, 2003). some studies have reported that the use of technology also increases teachers’ confidence in the content (brändström, 2011; buabeng-andoh, 2012; cassim, 2010; cox, preston, & cox, 1999; leendertz, blignaut, nieuwoudt, els, & ellis, 2013; mumtaz, 2000; o’dwyer, russell, & bebell, 2003; remesh, 2013; sabzian & gilakjani, 2013; yang, 2013). for instance, in cox et al.’s (1999) study, teachers reported that using ict increased their confidence. o’dwyer et al. (2003) further found that higher teacher confidence is associated with the largest increased use for delivering instruction and, in particular, increased use for class preparation. further findings showed a significant relationship between teachers’ confidence and ict applications (albion, jamieson-proctor, & finger, 2011; tasir, abour, halim, & harun, 2012). research conducted in south africa reports that the use of computers tends to feature fairly extensively in the learning areas of language and mathematics, natural sciences and technology, and less in humanities and arts (lundall & howell, 2000). on the one hand, they found that in grades 1 to 7 computers tend to be used mainly for drill and practice and problem-solving exercises; on the other hand, from grade 8 upwards computers tend to be used for a greater variety of purposes in the teaching and learning process. they also mention that drill and practice exercises, although less prominent, continue to be used in grades 8 to 12. leendertz et al. (2013) investigated the level of technological pedagogical content knowledge (tpack) of mathematics teachers and how tpack contributes towards more effective grade 8 mathematics teaching in south african schools. their findings indicate that, with the improvement of tpack of mathematics teachers, their confidence increases in their ability to apply technology for teaching mathematics in south african schools. teachers acknowledged that ict promotes conversations with colleagues and peers regarding teaching and learning practices and gives a platform to express their teaching and learning accomplishments. ict also enabled them to conduct their administrative work more efficiently, allowed them to facilitate interactive lessons, and promoted confidence in using a variety of teaching and learning strategies designed for teaching (leendertz et al., 2013). sometimes the failure by teachers to integrate technology in their classrooms is because of problems that are beyond their control (marwan, 2008; mumtaz, 2000). some challenges experienced by teachers when trying to implement ict include insufficient ability of ict specialist teachers to teach students computer skills, lack of computer accessibility, lack of time as well as lack of financial support (mumtaz, 2000). similarly, buabeng-andoh (2012) identified poor ict skills, low teacher confidence, insufficient pedagogical teacher training, absence of suitable educational software, limited access to ict, inflexible structure of traditional education systems as well as limiting curricula design as some of the reasons that inhibited take-up of technology by teachers. the application of technology in teaching can lead to complexity because of the demands of learning newer technologies (koehler & mishra, 2009). cavanagh, reynolds and romanoski (2004) examined how the ict learning culture reconciles student learning and curriculum implementation in the classroom. in their study, they found that students expressed high confidence in their capacity to use ict in their learning, but teachers were uncertain about the extent to which the learning was sustained by the learners. teachers’ beliefs about teaching and learning play a major role in their decisions about how to teach the content. hollingsworth (1989) articulated that the way teachers implement new methods or programmes in their classrooms relates to whether teachers’ beliefs correspond with the suggested new methods. ernest (1989) emphasises the important role of teachers’ beliefs, particularly in mathematics education, where these beliefs depend on individual teachers. ernest argues that teachers have particular beliefs about the nature of mathematics and how it is best taught. for instance, beliefs that mathematics is computation stems from ideas about the nature of mathematics whereas beliefs that teaching mathematics should be shaped by alternative ideas stem from beliefs about teaching mathematics. beswick, callingham and watson (2012) found that while some teachers agreed that mathematics is the same as computations and that telling learners the answer is an efficient way of facilitating their mathematics learning, other teachers of mathematics believe they should be involved with learners’ thinking. beswick et al. are of the view that teacher’s beliefs about general principles related to the nature of mathematics, and the learning and teaching of mathematics (rather than the use of specific approaches), are what matter to student learning. several studies have focused on teachers’ beliefs about technology (cavas et al., 2009; choi, 1992; mueller, wood, willoughby, ross, & specht, 2008; o’dwyer et al., 2005) as a factor that motivates teachers’ use of technology. some studies found a significant relationship between teachers’ beliefs towards technology and their instructional technology practices (ali, 2015; mumtaz, 2000; palak & walls, 2009). further factors that have been explored with respect to teachers’ use of ict are gender and age. the findings of choi (1992) revealed that females and young teachers hold a slightly higher computer literacy level than male teachers and older teachers. however, the older age group tended to have more positive attitudes toward the instructional use of microcomputers in comparison with the younger age groups. however, the results of the study indicated no relationship between the teachers’ attitudes and their knowledge of microcomputers. on the other hand, almekhlafi and almeqdadi (2010) found that male teachers were more likely to use technology than female teachers. gender and age were also discussed by cavas et al. (2009) who found that turkish science teachers’ attitudes towards ict did not differ regarding gender, but differed regarding age, computer ownership at home and computer experience. these authors state that factors influencing the use of technology include availability of computers in the classroom, sharing of resources, a supportive administration, strong support staff, environmental, personal, social and curricular issues. similar findings indicated that school factors, personal factors as well as beliefs towards technology influence teachers’ use of technology (cubukcuoglu, 2013; mumtaz, 2000). mumtaz (2000) identified an important technical sustenance of 20 hours per week that was necessary for teachers and found that a positive attitude of the principal contributed to teachers’ use of technology. these authors agree it is important to support teachers in using technology in teaching and learning. sabzian and gilakjani (2013) identified two contributing factors to teachers’ low self-confidence in using technology. the authors found that limited computer instruction could lead to teachers’ low confidence level when they initiate computer activities and result in high anxiety about using computers. the second was poor motivation which could result in insufficient knowledge in using instructional technology even if computers are provided in the classroom for teaching and learning. ali (2015) points out that teachers’ poor knowledge in using technology may be due to a lack of professional training with computers and lack of teacher-centred experiences in education and the lack of technological devices. these studies emphasise the need for programmes that can provide effective computer instruction to teachers while also helping them gain experience in the use of the technological tools. using factor analysis, leendertz, blignaut, ellis and nieuwoudt (2015) validated a questionnaire for ict development of mathematics teachers. they found that the first factor was related to ‘teachers’expectation’ (reliability of 0.92), which means that mathematics teachers expect the doe, provincial departments and schools to work together to improve an ict strategic plan in order to increase technology use. based on their study, they emphasise that professional development courses are urgently needed to support teachers in integrating ict into teaching and learning. the site of the training does not have to be confined to the school as lundall and howell (2000) point out that many schools indicated that some teachers have access to technology-related professional training opportunities that take place outside the school. this article addresses the use of technology in teaching mathematic in kwazulu-natal schools. as illustrated by the literature, recent advances in technology have unlocked entirely new directions for education research and we briefly surveyed some of the more pertinent studies in this area. we first looked at the ways in which digital classrooms support students’ learning, before moving to the use of particular tools for instruction such as the internet which is a focus of this study. we then reviewed studies that investigated the association between using technology and particular demographic factors. the literature review also included studies about challenges faced by teachers in trying to increase the use of ict in their classrooms. this review serves as a useful foundation to look at the use of technology by a group of kwazulu-natal mathematics teachers, and to identify the factors that are associated with it. research design and methodology this quantitative study is a part of a larger doctoral study by the first author, developed to investigate teachers’ knowledge of, beliefs about and confidence in teaching mathematics (umugiraneza, bansilal, & north, 2016, 2017, 2018a, 2018b). knowledge and skills in appropriate technological tools explored in this study are recognised as a part of the knowledge required to teach these concepts. the participants in this quantitative study were 75 mathematics teachers from grades 4 to 12 from kwazulu-natal, who were part of a group of teachers who attended an in-service course. the course was designed to help develop statistical knowledge and skills of mathematics teachers particularly from schools with a poor overall performance in mathematics (north, gal, & zewotir, 2014; north & scheiber, 2008). a questionnaire based on an existing instrument developed by beswick et al. (2012) was used to probe various aspects of teachers’ knowledge, beliefs and confidence related to the teaching and learning of mathematics and statistics. the original questionnaire (beswick et al., 2012) focused on mathematics which we extended to the teaching of statistics and the use of technology in classrooms. the questionnaire included several parts such as questions related to teachers’ confidence and beliefs, teaching practices (lesson planning, teaching methods and assessments, etc.), predicting learners’ responses and the use of technology. in this article, we focus on the teachers’ responses to items regarding the implementation of technology in their instructional practice. teachers were required to respond to statements on a four-point likert item scale with categories 1 (‘never’), 2 (‘rarely’), 3 (‘sometimes’) and 4 (‘often’) to indicate how often they integrated technology in teaching mathematics. they were also asked about their access to calculators, computers and the internet and the extent to which these were used for teaching mathematics in their classrooms. teachers were also asked to rate their level of confidence in teaching mathematics topics using a three-point likert item scale with categories 1 (‘low’), 2 (‘moderate’) and 3 (‘high’). their beliefs about teaching and learning mathematics were rated using a three-point likert item scale with categories 1 (‘disagree’), 2 (‘neutral’) and 3 (‘agree’). this research was underpinned by the following research questions: (1) to what extent do mathematics teachers incorporate technology into their teaching practices? (2) to what extent are teachers positive about using technology in the teaching of mathematics? (3) is there any relationship between demographic factors and the use of technology in instructional practices? participants table 1 presents a description of the participants in terms of various demographic factors. table 1: participants by demographic factors. table 1 shows that the study involved an almost equal number of female and male teachers, with the majority being 40 years or younger. more teachers (60%) were teaching grades 10–12 (fet) than those who taught grades 4–9 (get), while 60.0% completed a bachelor’s degree and 40.0% completed postgraduate studies. only 21.3% were from quintile 4 or 5 schools. table 1 further indicates that 68.0% of the participants have met with a local group of teachers to study and discuss mathematics and statistics teaching on a regular basis as part of their professional learning, and 45.3% said that they integrate the national curriculum statement grade r–12 in their teaching process. data analysis the data were analysed using ibm spss statistics 23 version (george & mallery, 2016). the package was used to evaluate the connection between the use of technology for educational instruction purposes and teachers’ confidence and beliefs. moreover, it was used to identify the important factors that may influence teachers’ ability to use technology. we used cross tabulation with chi-square test of independence at significance level alpha equal to 0.05, to explore some relationships. the chi-square test is known as a general test designed to evaluate when the difference between observed frequencies and the expected frequencies under a set of theoretical assumptions is statistically significant (michael, 2001). this test is a standard statistical procedure to test whether there is evidence of a statistically significant relationship between two categorical variables, as opposed to the two categorical variables operating independently. it is assumed that if the p-value is less than 0.05, we conclude that a significant difference does exist. this test was accordingly used to determine whether there is a statistical significant relationship between teachers’ use of technology (using technology in class or consulting the internet for educational instructions) and teachers’ confidence (including beliefs). effectively then, we were exploring whether using internet or technology in the classroom for educational instructions was a reliable relationship with the level of confidence in their ability to teach a variety of mathematics and statistics topics and positive beliefs about teaching in teaching mathematics and statistics. we further used a comparison of means (a standard test used to compare differences between means of two or more groups) to explore whether there appears to be a statistically significant relationship between teachers’ demographics and their use of technology in the different instructional practices. this test was used to examine the magnitude of the difference between two groups in terms of using technology. effect size reported in the output of the comparison of means is a name given to a family of indices that measure the magnitude of a treatment. it can help to see how much of a practical significance any result has (becker, 2000; cohen, 1988; kotrlik & williams, 2003). hence, it was used to examine the magnitude of the difference between two groups in terms of using technology. most of the effect sizes are less than 0.3; this indicates that the difference between groups of demographic factors in terms of using technology in teaching practice is small. differences observed will thus be deemed to be significant if the p-value is less than 0.05 and the effect size is bigger than 0.3. mean plots are used to see if the mean varies between different groups of the data. they were further used to explore the factors that may influence teachers to integrate technology in their teaching practice. ethical considerations all ethical considerations stipulated by the university of kwazulu-natal were adhered to. out of the group of 136 teachers who were approached to take part in the study, only 75 opted to participate. the participants were guaranteed anonymity and were also given the choice to withdraw from the research if they wanted to. permission to carry out the research was granted by university of kwazulu-natal with the protocol number hss/1529/015d. findings we start by exploring the extent to which the teachers have access to calculators, computers and the internet in teaching mathematics, followed by details about the instructional purposes for which the technology is used. thereafter we report in more detail on the differences in confidence and beliefs of teachers who use the internet for instructional purposes, and those who do not. this section is organised according to the research questions of the study. research question 1: to what extent do mathematics teachers use technology in their teaching practices? access to technology table 2 displays the results regarding the use of calculators and computers. of the 75 teachers who were surveyed, only 49 (65%) teachers reported that in the schools where they were teaching calculators were used to teach and learn mathematics and statistics, even though calculators were commonly available. when asked about access to computers, there were even fewer teachers who enjoyed this privilege. there were only 33 (44%) teachers who reported that computers were available in the schools where they teach; only 21 (28%) said that computers were used to teach mathematics and statistics at the schools. twenty (26.7%) had access to the internet and 19 (25.3%) said that internet was used for educational instruction. table 2: access to technology. reports about the availability of computers at schools suggest similar figures to those reported by the teachers in this study. in 2015, it was found that 33.2% of schools had computers (south africa institute of race relations, 2015). even though, in the current study, the availability of computers in schools was reported at approximatively 44%, only 28.5% of the teachers reported that these were used for teaching mathematics and statistics, which represents a limited use of technology. the use of computers and calculators in teaching mathematics and statistics was disaggregated by the grade in which teachers were teaching. table 3 indicates that 84.4 % of the teachers who were teaching grades 10–12 mostly used calculators to teach mathematics, compared to 36.7% of teachers for grades 4–9. on the other hand, only 40.0% of teachers who were teaching in grades 10–12 reported that they used computers in mathematics and statistics teaching and learning, whereas only 10.0% of teachers in grades 4–9 reported that they used computers in the classroom. table 3: the use of calculators and computers by grade. it is evident that in the schools represented in the study, the use of computers in the classroom is still at very low levels and much effort is needed to sensitise teachers to using computers for improved teaching of mathematics and statistics. this finding shows that the doe (2007) recommendation that the use of ict in the classroom should aim to develop a range of skills ranging from basic ict skills to developing specialisation and innovation in ict education is unlikely to be met under these conditions. it is clear that teachers would need much assistance and continuous professional teacher development on the implementation of information technology pedagogical knowledge in relation to integrating ict in the teaching of mathematics (cassim, 2010). given that more than half the teachers do not have computers available at their schools, it is unrealistic to expect that these teachers would be able to take on the vision of the doe in using ict to improve the learning outcomes in the education system (doe, 2007). the instructional purposes for which the technology is used mishra and koehler (2006) agree that the connection between technology and teaching can transform the conceptualisation and the practice of teacher education, teacher training and teachers’ professional development. teachers can use technology in different ways, such as in simple drill and practice tasks. drill and practice mathematics software offers teachers a relatively simple way to use technology in the classroom (kuiper & de pater-sneep, 2014). teachers could also use technology in more complex tasks such as using simulations in investigating real-life data. table 4 indicates how often technology (computers) is implemented in different teaching practices. it can be noted from table 3 that most teachers reported that they never used technology for any of the instructional activities mentioned. it is clear that most of the teachers in the study group were not using technology at all, not even in the most rudimentary way. activities such as collecting and retrieving data from computers are associated with exploring data in real-life applications. the use of statistics in understanding and making informed decisions in real life is an important outcome of the subject, and these findings show that teachers need more help in this regard. table 4: exploration of the use of technology in teachers’ practice. the use of the internet and teachers’ confidence and beliefs the data allowed us to look in more detail at the specific use of the internet for instructional purposes and to test whether this use was linked to certain factors. ndlovu and lawrence’s (2012) view is that access to ict enables quality use for educational purposes. it is expected that a teacher who makes use of the internet as an additional teaching tool will most likely earn their students’ respect and regard, which in turn may motivate teachers to develop more innovative indeas about teaching. some studies contend that teachers with more access to the web for instructional purposes had higher levels of self-determination and that teachers with better computer access had lower computer nervousness and more computer self-efficacy (liu & kleinsasser, 2015). thus, access to technology may be a factor that builds up teachers’ knowledge. we now investigate the links between the use of the internet and teachers’ confidence and beliefs. teachers’ confidence in teaching mathematics recent studies articulate that there exists a connection between teachers’ confidence and the use of technology (brändström, 2011; o’dwyer et al., 2003; sabzian & gilakjani, 2013). sabzian and gilakjani (2013) argue that the lack of computer instruction often accounts for teachers’ low confidence levels when they initiate computer activities. in this study, we explored whether teachers who use the internet for educational instruction purposes are confident in their ability to teach mathematics. we considered topics such as percentage, fraction, decimal, inference and prediction, measurement, pattern and algebra, mental computation, pie graphs and histograms, range and variations, ideas of sampling and data collection, and so on. the results showed a statistical significant relationship between using the internet for educational instructional purposes and teachers’ confidence in teaching mathematics or statistics topics. it can be noted from table 5 that teachers who use the internet for instructional purposes expressed a high confidence in teaching percentages (χ2 = 6.082(2), effect size = 0.285, p-value = 0.048), ratios and proportions (χ2 = 9.835 (2), effect size = 0.362, p-value = 0.007), pie charts and histograms (χ2 = 12.231 (2), effect size = 0.320, p-value = 0.048), pattern and algebra (χ2 = 13.747 (2), effect size = 0.428, p-value = 0.001), measurement (χ2 = 6.399 (2), effect size = 0.292, p-value = 0.041) and mental computation (χ2 = 8.573 (2), effect size = 0.338, p-value = 0.014). table 5: using internet for instructional purposes and teachers’ confidence. the values of effect sizes in table 5, table 6 and table 7 ranged between 0.2 and under 0.4 which indicate moderate practical significance (becker, 2000; cohen, 1988; kotrlik & williams, 2003). table 6: using internet for education instructional purpose and teachers’ beliefs about teaching mathematics. table 7: teachers’ beliefs about using technology in teaching and learning. teachers’ beliefs about the goals of teaching mathematics we further examined whether there is a significant relationship between using the internet for educational instructional purposes and teachers’ beliefs about the nature of mathematics. it can be noted from table 6 that teachers who reported that they use the internet were more likely to agree about some broad goals of teaching mathematics (as identified by beswick et al., 2012) than those who did not. teachers who reported that they use the internet agreed that mathematics teaching should assist learners to develop an attitude of inquiry (asking questions, being curious about solutions) (χ2 = 6.362 (2), effect size = 0.291, p-value = 0.042), statistics teaching should assist learners to develop a positive attitude to problem-solving (χ2 = 6.050 (2), effect size = 0.284, p-value = 0.049), and statistical literacy, thinking and reasoning are the main goals in statistical teaching and learning (χ2 = 7.458 (2), effect size = 0.315, p-value = 0.024). the findings from table 5 further show that the use of the internet for educational instructional purposes is associated with a stronger belief in the value of linking teaching to other key areas (χ2 = 11.797 (2), effect size = 0.404, p-value = 0.003) as well as the need for applying statistics in real-life settings outside of the classroom situation (χ2 = 8.701 (2), effect size = 0.397, p-value = 0.013). research question 2: to what extent are teachers positive about using technology in the teaching of mathematics? table 7 indicates that 60 (80%) of the 75 teachers in the study had a positive view regarding the use of technology to facilitate teaching and learning mathematics and statistics topics and 49 (65.3%) expressed a positive belief that it improves learners’ understanding. we further used a comparison of means to identify factors that may be associated with teachers’ positive beliefs towards technology. it can be noted from table 8 that teachers younger than 40 were more confident about the potential of technology to influence learning and understanding of statistics positively than was the case for teachers who were older than 40 (f = 4.912, p-value = 0.030, effect size = 0.251). they further believed that using technology helps to increase learners’ learning and understanding of statistics (f = 8.886, p-value = 0.004, effect size = 0.329). younger teachers were thus more positive about the use of technology to enhance understanding of statistics than older teachers. these young teachers are the same group that have 10 or fewer years of teaching experience and it will be shown that they are the group who are more likely to make use of technology in the classroom. table 8: teachers’ beliefs about technology and effect of demographic factors. these results support the findings of cavas et al. (2009), who reflected on science teachers’ attitudes towards the use of technology in education. they found that the attitudes of young science teachers in their study (age group 20–35) were more positive about using technology in the classroom, which was significantly different from teachers in other age groups (36–49 and 50+). however, in another study, choi (1992) found that older teachers displayed more positive attitudes towards computer use in education than was the case for the younger teachers in that study. we further note that teachers who use the national curriculum statement in their teaching have positive beliefs that technology influences learning and understanding of statistics 40 (f = 7.164, p-value = 0.009, effect size = 0.299) and that using technology helps to increase learners’ learning and understanding of statistics (f = 4.995, p-value = 0.028, effect size = 0.253). this indicates the importance of consulting the curriculum as the factor that encourages teachers to use technology in their teaching process. teachers largely agreed that the use of technology helps learners to develop their understanding of mathematics and statistics topics. forty-nine out of 75 (65.3%) teachers said they believed that they would integrate technology into teaching and learning mathematics and statistics in the classroom. furthermore, the findings indicate that teachers who reported that they meet with a local group of teachers and discuss mathematics and statistics teaching on a regular basis as a part of their professional learning expressed positive beliefs that technology enhances learners’ understanding (f = 10.541, p-value = 0.002, effect size = 0.355) and that using technology helps to increase learners’ learning and understanding of statistics (f = 4.328, p-value = 0.041, effect size = 0.237). this finding indicates that in professional learning, teachers continue to acquire new skills while collaborating with other teachers and can share the best practice and integrate the innovations in the classroom. the doe (2007) supports this idea that teachers’ desires and benefits should be the driving force for their professional growth. research question 3: is there any relationship between demographic factors and the use of technology in instructional practices? technology knowledge, as with other aspects of teacher knowledge, is not constant. it develops over time according to teachers’ professional development or training, teaching experience as well as teachers’ attainment of a higher level of education, and so on. the comparison of means (a standard test used to compare differences between means of two or more groups) was used to identify factors associated with teachers’ tendency to integrate technology into their teaching practice as reported in table 4. the teachers’ demographic factors that were tested included school quintile, gender, age, teaching experiences, education level, workshop attendance, grades taught, level of education and instruction practices. the analysis reported in table 9 was made by comparing the means at a significance level (alpha) equal to 0.05, between the variables that were explained in table 4 and the demographic factors reported in table 1. table 9: factors associated with teachers’ use of technology. the findings reveal that the difference between the means is statistically significant for the factors of gender, level of study, teaching experience, attending workshops and school quintile and their ability to integrate technology in different instructional practices at alpha equal to 0.05. table 9 reports only significant effect p-values less than 0.05. it can also be noted from table 9 that the effect sizes between 0.2 and 0.4 (in bold) indicate that the difference between groups in terms of using technology has moderate practical significance. on the other hand, it can be noted from table 8 that the effect sizes less than 0.2 indicate that the difference between groups in terms of using technology has moderate practical significance (kotrlik & williams, 2003). we also discuss the statistical relationship between some factors and the use of technology reported in table 9 by examining which demographic group may be more likely to use technology in instructional practice than other groups. regression analysis was made using mean plots to compare the magnitude of each group in terms of using technology; however, only those that reflected a moderate difference are reported. we found that teachers who took postgraduate courses may be more likely to use technology than teachers who have a bachelor’s degree or below. it can be noted from figure 1 that means scores for teachers who attended postgraduate courses are greater in terms of taking a test or quiz, retrieving and exchanging data and demonstrating statistical principles than those with bachelor’s degrees in terms of using technology (e.g. 2.230 versus 1.340 and 2.350 versus 1.510). this result was similar to findings in a previous study that education level contributes to teachers’ use of technology in instructional practices (mathews & guarino, 2000). figure 1: using technology for instructional practice by level of education. the education department introduced a funding policy by using a system of categorising schools into five quintiles in order to inform decisions around financial allocations. quintile 1 schools are those serving the poorest children while quintile 5 schools cater for children who come from well-resourced backgrounds. looking at table 9, there also appears to be a statistically significant difference between teachers’ school quintile and their ability to integrate technology in different instructional practices. we observe from figure 2 that teachers who teach in quintile 4 or 5 schools are more likely to use technology in instructional practices than teachers from the quintile 1–3 schools. a general trend in the use of technology as the quintile ranking of the school increased can be noted from figure 3: as the quintile ranking of the school increases, the use of technology in the various instructional activities at that school increases. figure 2: using technology for instructional practice by school quintile. thus, mean scores for teachers who teach in quintile 4 and 5 are greater in terms of drilling and practice and demonstrating statistical principles than for those who teach in quintile 1, 2 and 3 schools in terms of using technology (e.g. 2.630 versus 1.690 or 1.570, 3.060 versus 1.750 or 1.570). it is evident, therefore, that teachers who teach in the poorest schools are using technology to a lesser extent than those in the more well-resourced schools, which illustrates the digital divide between the poorest and the richest schools. however, it is important to note that teachers need more than access to use technology; they also need support in using the technology to teach more effectively. ndlovu and lawrence (2012) point out that it is not simply the availability of technology that brings about improvements in learning, but the ways in which this technology is used. many studies have also reported that poorly resourced schools have less access to ict facilities than well-resourced schools (ndlovu & lawrence, 2012), and the results of the current study also support such findings. the findings further showed that male teachers are more likely to integrate technology into their educational practice than female teachers, given that the mean scores of male teachers were higher than those of female teachers (e.g. 2.500 versus 1.405, 0.921 versus 0.165, etc.). this finding supports results from a study in africa (buabeng-andoh, 2012, p. 39), which explored factors that influence ‘teachers’ adoption and integration of information and communication technology’. his finding also showed that there was a significant difference between ghanaian male and female teachers in technical ict capabilities, where he found evidence that male teachers’ scores were higher than those of female teachers in relation to the use of ict in the classroom for instructional purposes. furthermore, it was noted that teachers whose teaching experience is 10 years or fewer were more likely to use technology than the teachers with teaching experience more than 10 years (e.g. 2.180 versus 1.250, 2.330 versus 1.400, etc.). this finding was also reported in another study (almekhlafi & almeqdadi, 2010), that is, that novice teachers are more likely to use technology and the internet in several teaching practices, which may be because they grew up in the technological era. a statistically significant difference was also apparent between using technology in instructional practice and professional learning. this means that teachers who attended mathematical workshops may be more likely to use technology than those who did not (e.g. 2.148 versus 1.357, 1.967 versus 1.143, etc.). however, the effect sizes (table 9) are small for all instructional practice. this means that the difference in terms of using technology between those who attended workshops and those who did not is small in practice. however, it appeared that the effect sizes (table 9) are small for all instructional practice. this means that the difference in terms of using technology between those who attended workshops and those who did not is small in practice. mueller et al. (2008) found that attending professional development workshops influences teachers’ use of technology. perhaps workshops that focus on the use and application of technology in the teaching of mathematics specifically may prove to have a bigger influence on whether teachers opt to use technology or not. mueller et al. (2008) noted that ‘professional development’ and the ‘continuing support of good practice’ play a valuable role in sustaining the use of ict in the classroom. conclusion digital classrooms to support students’ learning have been the focus of research recently and this article reveals some of the challenges that schools in poorer communities in south africa experience in this regard. results from this study indicate that approximately a quarter of teachers have access to ict for teaching mathematics. the use of ict is even lower in the earlier grades (4–9), where only 10% of the teachers said they used ict for teaching mathematics. furthermore, the data showed that teachers are generally more comfortable with integrating calculators when teaching mathematics and statistics, as compared to using computers. this indicates that teachers may need training in the integration of computers into the teaching of mathematics and statistics in the classroom. even though the practice of integrating technology into teaching instruction was not well developed among these teachers, they exhibited a positive view with respect to teaching using technology. of interest is the finding that teachers who reported that they use the internet for instructional purposes held more positive views about the broad goals of mathematics and were also more confident about teaching mathematics than those teachers who did not. beswick et al. (2012) assert that it is teachers’ beliefs about general principles about the learning and teaching of mathematics that make a difference to student learning. this study suggests that teachers who have access to internet resources have progressive views about what the goals of mathematics and statistics should be. they also have stronger beliefs about the role of real-life applications in learning statistics and the need for connections across various subjects. the study also found that teachers who use the internet have higher levels of confidence in teaching mathematics. this may be because teachers who have access to a wider set of resources have a greater chance of learning more about the broad goals and applicability of mathematics beyond the confines of the classroom. knowing more about the connections between mathematics and the real world helps people to better understand the role of fundamental concepts such as percentages, and this may in turn improve their confidence about teaching these concepts. a problem that has been exposed is that although some schools are reported to have computers, these computers are not used in instructional practice, but are used for administrative purposes. it is not clear whether this is because teachers do not have the necessary skills or are reluctant to use the computers, or whether it is because school management is restricting the teachers’ access to the technology. if computers are available but are not being used, the possible reasons for this state of affairs need to be urgently probed. interventions that seek to increase access to technology will not be successful if the roll-out of computers does not result in a concomitant increase in the teachers’ use of the technology. this study has provided evidence that teachers who attend workshops are more likely to use technology in their instructional practices than those who do not; hence, interventions that aim to increase the use of ict in schools must be accompanied by continuous support. it is the support through workshops that will enable teachers to develop confidence in using technology and this may lead to more progressive attitudes by school management regarding the use of computers in classrooms. an important finding of the study is that teachers display different levels of technological readiness and enthusiasm according to their age, experience, gender and how well resourced their school is. older teachers appear to need more support to help them become more confident to take on the technology. younger teachers are more confident and will not need as much support as their older counterparts. in addition, the study has also provided further evidence of the digital divide between schools with different quintile rankings. the digital divide presents a barrier to achieving equity in the provision of quality education to all learners. the removal of the digital divide requires more than just resources because it is the way in which the resources are used that makes the difference in the quality of the learning experience that is offered. the study shows that teachers from quintile 1 schools need much more sustained attention and support, different in form and substance from those from quintile 4 and above. successful integration of technology can have a transformative effect on schools and the education system as a whole. the study shows that teachers who have made a start at using the internet for their teaching have also developed broader understandings about the value and aim of teaching mathematics. hence, helping teachers to take on technological resources is likely to assist them to develop new pedagogies that can help learners engage productively with the content of the subject. continuous professional development will be required to help teachers integrate the newly acquired technological knowledge into their pedagogical knowledge so that they can develop in all the components specified in mishra and koehler’s (2006) tpack framework. in order for the doe to realise their vision of helping their learners to function effectively in a knowledge society by using appropriate ict in their schools (doe, 2007), teachers need sustained support and assistance to develop the necessary ict capabilities. any intervention that involves provision of technological resources such as internet access, mobile tablets or laptops will need to be accompanied by the relevant teacher professional development training courses, as well as training and sustained support for using and maintaining the infrastructure. acknowledgements competing interests the authors declare that they have no financial or personal relationships that may have inappropriately influenced them in writing this article. authors’ contributions o.u. performed the analysis of data and made a first draft of the manuscript. s.b. refined the draft. d.n. was responsible for checking the accuracy and suitability of the statistical analysis. o.u., s.b. and d.n. contributed to the conception of the study. references albion, p.r., jamieson-proctor, r., & finger, g. 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(2013). transforming k-12 classrooms with digital technology. hershey, pa: information science. article information authors: mario sánchez aguilar1 juan gabriel molina zavaleta1 affiliations: 1mathematics education department, national polytechnic institute, mexico correspondence to: mario sánchez aguilar postal address: national polytechnic institute, legaria 694, colonia irrigación, c.p. 11500, d.f., mexico dates: received: 15 mar. 2012 accepted: 04 nov. 2012 published: 03 dec. 2012 how to cite this article: aguilar, m.s., & zavaleta, j.g.m. (2012). on the links between mathematics education and democracy: a literature review. pythagoras, 33(2), art. #164, 15 pages. http://dx.doi.org/10.4102/ pythagoras.v33i2.164 copyright notice: © 2012. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. on the links between mathematics education and democracy: a literature review in this review article... open access • abstract • introduction • aim and structure of the article • method    • where should we look?       • layer 1 – journals       • layer 2 – conference proceedings       • layer 3 – books    • what should we look for?    • what was excluded from the review? • definitions of democracy used in these texts    • the political dimension of democracy    • the juridical dimension of democracy    • the economic dimension of democracy    • the socio-cultural dimension of democracy • links between mathematics education and democracy    • link 1: mathematics education as a provider of critical mathematical skills    • link 2: mathematics education as a source of values and attitudes    • link 3: mathematics education as a social gatekeeper • strategies to foster a democratic competence in students    • fostering critical mathematical skills    • fostering democratic values and attitudes • tensions and difficulties inherent in mathematical education for democracy    • a tension between open and empowering materials    • a tension between democracy and authority    • the paradox of empowerment vs disempowerment    • the paradox of relevance vs indifference • the fundamental role of the teacher    • skills needed    • attitudes needed    • rights and obligations • selected criticisms    • more empirical research is needed    • possible discriminatory outcomes    • an imposition of emancipation    • the applicability and relevance of a critical ideology • discussion    • what are the main ideas that we found in the literature?    • some topics that need to be researched       • longitudinal studies       • teacher education       • textbook development       • the concept of power vs the concept of democracy       • how values and attitudes are constituted       • curriculum studies • a final comment • acknowledgements    • authors’ contributions • references • appendix 1 • bibliographic details of the references consulted for the development of the literature review    • layer 1 – journals       • africa       • australasia       • europe       • international       • latin america       • north america    • layer 2 – conference proceedings       • icme 11       • icme 10       • cerme 7       • cerme 6       • mes 1       • mes 2       • mes 3       • mes 5       • mes 6    • layer 3 – books abstract top ↑ this article reports the results of a literature review focused on identifying the links between mathematics education and democracy. the review is based on the analysis of a collection of manuscripts produced in different regions of the world. the analysis of these articles focuses on six aspects, namely, (1) definitions of democracy used in these texts, (2) identified links between mathematics education and democracy, (3) suggested strategies to foster a democratic competence in mathematics students (4) tensions and difficulties inherent in mathematical education for democracy, (5) the fundamental role of the teacher in the implementation of democratic education and (6) selected criticisms of mathematical education for democracy. the main contributions of this article are to provide the reader with an overview of the literature related to mathematics education and democracy, and to highlight some of the theoretical and empirical topics that are necessary to further development within this research area. introduction top ↑ the connection between democracy and mathematics education is not a new topic in the field of mathematics education research. these connections have been discussed in our field for over 20 years (see e.g. d’ambrosio, 1990; skovsmose, 1990). so why is it important to do a literature review on mathematics education and democracy now? we think that there are at least two reasons for this.firstly, although some authors, such as gutiérrez (2010), claim that our field is currently experiencing a ‘sociopolitical turn’ (that is, a growing awareness of the need to consider the political, power and identity issues associated with mathematics education), our field is still dominated by views that consider the relationship between the teacher, the students and the mathematical content as the main space of inquiry (pais & valero, 2012; valero, 2007). similarly, amongst practitioners there exists a limited view of the role that mathematics education can play in forming critical citizens able to live in and sustain democratic societies. thus we believe that a literature review addressing the links between mathematics education and democracy may help to challenge and contest these perspectives, and increase awareness about the need to widen the area of inquiry that has been favoured traditionally. secondly, although discussion of the connections between mathematics education and democracy is not new, it has branched out into related topics such as equity and social justice, indicating that this discourse is far from complete. in fact, there are several research topics related to mathematics education and democracy that need further exploration. one contribution of this literature review is to highlight some of the theoretical and empirical research that is necessary to further develop this field. aim and structure of the article top ↑ the literature related to mathematics education and democracy is extensive. this article consists of a review of that specialised literature, presenting it in a summarised and organised way. this review is aimed at readers who are unfamiliar with this research area, and would like an introductory overview of it. the review is based on the analysis of a collection of texts produced in different regions of the world. the analysis of these articles is focused on six aspects:1. the definitions of democracy that are used. 2. identified links between mathematics education and democracy. 3. suggested strategies to foster democratic competence in mathematics students. 4. tensions and difficulties inherent in mathematical education for democracy. 5. the fundamental role of the teacher in the implementation of democratic education. 6. selected criticisms of mathematical education for democracy. the review concludes with a discussion of the literature analysed, where we indicate some of the theoretical and empirical research that is required to further develop this area of research. method top ↑ this method used to select the texts for this review was guided and defined by two key questions, (1) where should we look? and (2) what should we look for? next we describe how these two questions were operationalised. where should we look? from the beginning we wanted to develop a ‘democratic review’ including different voices and views on the relationship between mathematics education and democracy. we tried to do this by performing a literature search that included not only international journals, but also regional journals with influence and authority in different regions of the world. in our review we also included proceedings of some international conferences as well as selected books on mathematics education research. these literature sources were organised into three layers, described in more detail below. layer 1 – journals in this layer we included international and regional journals. in each case we selected the journals that we consider to be the most influential and representative. the regional journals were divided into the regions of latin america, north america, australasia, europe and africa. it is important to clarify that, although these journals are produced in particular regions of the world, it is likely that at least some of them may not represent the thinking of the researchers within the geographical areas where they are produced, simply because they may contain articles written by authors from outside that region. this is particularly true for some of the journals included from europe and north america.this review included five journals in the ‘international’ category: educational studies in mathematics, zdm: the international journal on mathematics education, for the learning of mathematics, international journal of science and mathematics education and international journal of mathematical education in science and technology. the latin american journals included in this review were: educación matemática, revista latinoamericana de investigación en matemática educativa and bolema: boletim de educação matemática. we also included two north american journals, namely journal for research in mathematics education and the journal of mathematical behavior. the australasian region was represented in this review by the journal mathematics education research journal. for the european region we selected the journals research in mathematics education and philosophy of mathematics education journal. finally, we included the journal pythagoras to represent the african region. layer 2 – conference proceedings in this layer we included proceedings of international conferences that were freely available on the internet. we examined the most recent proceedings of the international congress on mathematical education (icme) and of the congress of european research in mathematics education (cerme). we also included the proceedings of the international conference mathematics education and society (mes). in particular, we reviewed the proceedings of the icme 10 (retrieved from http://www.http://dg.icme11.org/ and the papers associated with the discussion groups and topic study groups of the icme 11 (retrieved from http://dg.icme11.org/ and http://tsg.icme11.org/). we also reviewed the proceedings of the cerme 7 (retrieved from http://www.erme.unito.it/doc/cerme7/cerme7.pdf) and the cerme 6 (retrieved from http://ife.ens-lyon.fr/editions/editions-electroniques/cerme6/). in the case of the proceedings of the mes conference, we reviewed the proceedings of the conference mes 1 (retrieved form http://www.nottingham.ac.uk/csme/meas/measproc.html), mes 2 (retrieved from http://nonio.fc.ul.pt/mes2/), mes 3 http://mes3.learning.aau.dk (retrieved from http://mes3.learning.aau.dk/), mes 5 (retrieved from http://pure.ltu.se/portal/files/2376308/proceedings_mes5.pdf) and mes 6 (retrieved from http://www.ewi-psy.fu-berlin.de/en/v/mes6/proceedings/index.html). the proceedings of the conference mes 4 were not included in the review because they were not freely available on the internet.due to its importance and influence, we would have liked to include in our review the proceedings of the conferences organised by the international group for the psychology of mathematics education (pme). however, these proceedings are not available online and they were not accessible through the library of our university; hence, they could not be included in this review. this can be considered as one of our review’s limitations. layer 3 – books in this layer we included books on mathematics education research. these books were identified though a study of the bibliographies of the selected articles contained in the first layer. as we went through the lists of references used in these articles, we noticed that certain books were cited frequently; after examining them directly, we decided which ones to include in this third layer of this review. what should we look for? the layers just described illustrate where we looked when searching the materials used in this review. what we want to do now is to clarify how we selected the sources used in this review.the documents selected for the review complied with the following condition: in the title, abstract, keywords or the body of the paper they used the key term ‘democracy’, or related terms such as ‘democratic’ and ‘democratisation’. we set this condition to try to ensure that the selected texts would address the relationship between mathematics education and democracy. however, we are aware that this condition may have excluded some sources that discuss political issues related to democracy, such as equity and social justice, but fail to explicitly use the term ‘democracy’. this should be considered as another limitation of our review. we used search engines to find key terms within the documents. for example, for the articles contained in layer 1 we used the web-based search engines included in the web pages of the journals. these tools allowed us to quickly locate relevant articles contained in large collections of documents. for the documents contained in layer 2 and layer 3, we located the key terms by using the ‘search’ function of the pdf reader program. for some documents in layer 3 it was necessary to locate the key terms manually. what was excluded from the review? not all the materials identified in layer 1, layer 2 and layer 3 were included in our literature review. next we explain the various reasons for exclusion.it is important to note that not all the documents containing a key term such as ‘democracy’ or ‘democratic’ were relevant to the review. consider, for example, an article by hanna and sidoli (2002) where the phrase ‘mathematical education and democracy’ appears in the body of the article. within the article, however, the phrase is used to refer to the title of an article written by skovsmose (1990). in fact, hanna and sidoli (2002) do not address the relationship between democracy and mathematics education at all, but rather focus on providing a statistical profile of articles published in the journal educational studies in mathematics. another category of articles that was not considered in this review was those that mentioned a possible relationship between mathematics education and democracy, but only superficially. for example, there were texts claiming that mathematics is important for the education of citizens, but the reasons why mathematics is important were not clarified. we considered only materials written either in spanish, portuguese or english; as a result, articles written in other languages were excluded, as was the case for three articles from the special issues on mathematics education and democracy in the journal zdm: the international journal on mathematics education, issues 30(6) and 31(1), which were written in german. table 1: overview of the references consulted for the development of the literature review: layer 1 journals. table 2: overview of the references consulted for the development of the literature review: layer 2 conference proceedings. table 3: w of the references consulted for the development of the literature review: layer 3 books. the philosophy of mathematics education journal was not reviewed in its entirety. we only surveyed the two special issues on social justice, june and september 2007, and the special issue on critical mathematics education, october 2010. since our literature review was essentially a web-based review, there are other journals that were not fully scrutinised because not all their issues are available online. this is the case with the journals pythagoras, educación matemática, bolema: boletim de educação matemática, zdm: the international journal on mathematics education, journal for research in mathematics education and the journal of mathematical behavior.table 1, table 2 and table 3 show an overview of the references consulted for the development of the review. readers interested in the details of the consulted references (title, journal, volume, etc.) can refer to appendix 1, located at the end of this review. after selecting the texts, we posed two questions to guide our detailed analysis: 1. what are the links between mathematics education and democracy that are identified in these texts? 2. what proposals or strategies do they recommend to strengthen these linkages? whilst reading the materials, we also realised that it was necessary to focus on other relevant aspects of the mathematics education–democracy relationship addressed in some of the texts. for example, what tensions or difficulties are encountered in implementing mathematical education for democracy, or the centrality of the role of teachers in the implementation of democratic education. thus, we decided to widen our focus to include the six aspects listed in the explanation of the structure of this article. in the remainder of the article we will report the results obtained by focusing our review on these six aspects. we will end this article discussing some of the topics that need further research in order to develop this area of research and acknowledge the limitations of this review. definitions of democracy used in these texts top ↑ different definitions of democracy were found in the texts analysed. skovsmose and valero (2001) affirm that this richness of definitions suggests that the open nature of democracy is such that a precise definition of the concept is not possible. however, we think is important to try to identify the interpretation of democracy that each author adopts since the ideas, concepts and proposals that they present are usually related to their own interpretation of democracy.one of the most elaborate definitions of democracy is that of murillo and valero (1996), cited in valero (1999), in which democracy is interpreted as an ideal form of social organisation with four dimensions: democracy can be defined as an ideal way social organization establishes a series of political, juridical, economic and cultural values, norms and behaviors aiming at providing a better living for the whole population of a given state. this definition highlights a conception of democracy not as an actual reality, but as a goal to reach. … this definition also considers four different dimensions of democracy. the political dimension includes the series of procedures to form governments by means of regular, free elections as the corner stone of representative democracy. the juridical dimension sets and protects the different basic legal human rights and duties. the economic dimension deals with the material conditions of living and the organization of the economy by the state. and the socio-cultural dimension which considers the space where democratic values are embedded and embodied in people’s interactions. (p. 20, [author’s own emphasis]) in this article we use this definition of democracy as a framework that allows us to present, in an orderly manner, other explicit definitions of democracy located in the articles included in the review. we decided to use this definition because it offers an overarching characterisation of democracy covering the different dimensions of democracy that are identified in other definitions. the political dimension of democracy authors such as woodrow (1997) and almeida (2010) refer to the political dimension of democracy; that is, the type of democracy where citizens elect their representatives to participate in discussions about public affairs and make decisions related to those public affairs. representatives are elected through free elections in which citizens exercise their right to vote. this interpretation of democracy assumes that citizens do not directly participate in the discussion of public affairs, but they do so through the representatives of their choice. skovsmose (1994) criticises this interpretation of democracy because it puts the election of the government at the centre of the discussion, and makes other conditions or dimensions of democracy irrelevant. skovsmose (1998) proposes an alternative interpretation of democracy, inspired by the concept of direct democracy. here democracy is conceptualised as a form of political democracy in which citizens participate directly in the discussion of public affairs. this position may seem impractical if we think of a state, but skovsmose conceives the application of this type of democracy in all types of institutions, such as workplaces, schools and classrooms. it is in these kinds of institutions that skovsmose’s position seems clearly applicable and feasible. furthermore, this conception of democracy puts the type of skills that a citizen must possess in order to fully participate in the public discourse at the centre of the discussion. this point will be addressed later in the review, when we discuss the links between mathematics education and democracy. the juridical dimension of democracy respect for the rights and freedoms of individuals is another element included in some definitions of democracy. for example, harris (1998) mentions that a characteristic of democracies is that they are social formations where people have largely equal rights as citizens. woodrow (1997) and almeida (2010) refer to democracy as concerned with the protection of the individual’s human rights and freedoms within society. such freedoms include freedom of speech, freedom to work, freedom from hunger, freedom from oppression and freedom to worship. the economic dimension of democracy skovsmose (1998) points out that democracy is subject to the fulfilment of certain conditions. one of these conditions is the fair distribution of goods; in other words, democracy is not possible in a context where material goods are unevenly distributed. although d’ambrosio (2003) does not mention an explicit definition of democracy, he also emphasises the importance of sharing the cultural and natural goods amongst all human beings in order to ensure the survival and intellectual enhancement in a democratic society. the socio-cultural dimension of democracy democracy not only refers to the fair distribution of goods, equal rights for citizens and the free election of representatives. it also refers to a type of social organisation that can accommodate different views and ways of thinking. hannaford (1998) pays special attention to this aspect of democracy, referring to two types of democracy. the first is a single-minded democracy, where people’s values and ways of thinking are homogeneous. in this kind of democracy there is no room for people with alternative ways of thinking. the second kind of democracy is that in which there are almost as many ways of thinking as there are people. hannaford claims that the latter type of democracy is slower and seems less efficient; however, history has shown that in the long term it is more efficient than depending on only one idea.as the above discussion shows, the concept of democracy is multidimensional; that is, it is a concept that refers to freedoms, rights, obligations, the distribution of material and cultural goods, and respect for diversity of ideas and ways of thinking. the question now is: what are the links between mathematics education and democracy? in the following section of the article we will present the links that we have identified through the literature review. links between mathematics education and democracy top ↑ as noted by de mattos and batarce (2010) and valero (1999), the term ‘mathematics education’ has at least two meanings. the first one refers to mathematics education as a field of research, whilst the second one refers to mathematics education as a set of practices associated with the teaching and learning of mathematics. such practices are not confined to the classroom. as noted by valero (1999), they include external educational practices that affect the learning and teaching of mathematics, such as curricular policymaking, mathematics textbook writing and pre-service and in-service education. in this section, where we address the links between mathematics education and democracy, we have adopted the second connotation. we have identified three links between mathematics education and democracy in our literature review. firstly, mathematics education can provide students with mathematical skills to critically analyse their social environment, and also to identify and evaluate the uses and misuses of mathematics in society. the second link relates to the fact that the mathematical education that students receive in a classroom can promote or inhibit values and attitudes that are essential to build and sustain democratic societies. the third link is the acknowledgment that mathematics education can function as a sort of social filter that restricts the opportunities for development and civic participation of some students. link 1: mathematics education as a provider of critical mathematical skills one idea that skovsmose (1990, 1998) particularly highlights, but which is also mentioned by d’ambrosio (1990, 2003) and orrill (2001), is the pervasive role that mathematics plays in modern societies. as stated by d’ambrosio (1990, p. 21), ‘our life is regulated by mathematical indices.’mathematics is applied in economics, politics, marketing, administration, education, et cetera. mathematics is an integral component of society. in fact, society is largely shaped by mathematics. thus many decisions that are socially relevant may be strongly influenced by mathematical models and applications, for example which municipalities in a country are considered poor enough to receive additional financial aid from the state, how much an employee should produce in order to maintain their position within a company, or what level of pollution levels in a city should lead to a recommendation that the inhabitants avoid exercising outdoors. the important point here is that it would be difficult for citizens to assess whether these decisions are fair or appropriate if they have not received a proper mathematical education. in sum, a mathematical education helps citizens to identify how mathematics is being applied to support such decisions, and to reflect on the consequences, positive and negative, that this application can produce. in order to maintain a democratic society, it is important that citizens are capable of critically analysing such questions and their answers because if they are to understand the economic and juridical dimensions of democracy, for example how the economic resources are distributed in a country or the defence of labour and environmental rights, it is vital that they understand the mathematics underlying those decisions. when we refer to the particular case of the application of mathematics in politics, we are addressing the connection between mathematics education and the political dimension of democracy. for example, almeida (2010) remarks: one of the ways that the government or elected representatives convince the citizens that their policies are the correct ones is by producing reports which include a mass of numerical and statistical data. there are many instances where this data is misleadingly summarised. (p. 13) along the same lines, wagner and davis (2010) assert: as politicians and bureaucrats use numbers to claim objectivity, to mask their biases, and to legitimize their decisions, it could be said that the citizens, who have been enculturated in schools to put their trust in number, are being duped by number, not empowered to make informed decisions, and, of course, claims of objectivity are made by more people than politicians and bureaucrats. children and adults need their number sense to be part of their critical sense. (p. 49) these quotes illustrate the importance of having mathematically literate citizens, able to critically analyse the reports and statements issued by the politicians who govern them. a democracy without this kind of citizenry is a fragile democracy. link 2: mathematics education as a source of values and attitudes in the previous section we emphasised the importance of having mathematically educated citizens, able to critically analyse how mathematics is applied in their societies. however, an adequate mathematical education is not sufficient to produce critical citizens. a critical citizenry also requires the promulgation of democratic values and attitudes. values like tolerance and respect for diversity, and attitudes about truth that demand the critical analysis of information. the second link between mathematics education and democracy identified in our review is the claim made by several authors that the mathematics classroom can be any place where, alongside mathematics learning, it is possible to transmit and acquire (perhaps subconsciously) both democratic and undemocratic values and attitudes. this link is closely related to the socio-cultural dimension of democracy, which refers to the social space where democratic values are produced. bishop (2002) refers to the transmission of values in the mathematics classroom as follows:underpinning any discussion about social justice and democratisation in mathematics education lies the issue of ‘values’. this is problematic at the present time because we neither know what currently happens with values teaching in mathematics classrooms, or why, nor do we have any idea how potentially controllable such values teaching is by teachers. in addition, many mathematics teachers are not even aware that they are teaching any values when they teach mathematics. changing that perception may prove to be one of the biggest hurdles to be overcome if we are to move to a more just mathematics education. (p. 1) a key concept for understanding the process of transmission of values and attitudes is that of classroom absolutism. skovsmose (1998) uses this concept and explains it as follows: the phenomenon that communication between students and teacher is structured by the assumptions that mathematics (school mathematics) can be organised around exercises and questions which have one and only one correct answer, and that, ultimately, it is the teacher’s job to make sure that mistakes are eliminated from the classroom. (p. 200) authors such as valero (1999) and almeida (2010) affirm that this type of classroom interaction creates authoritarian relations between teachers and students: relationships in which students learn to uncritically accept the claims and dictates of the authority. skovsmose (1990) argues that the nature of mathematics classroom interactions can teach the students to follow explicitly stated prescriptions. this process takes place through instructions such as ‘solve the equation …’, ‘find the length of …’, ‘calculate the value of …’ skovsmose argues that these kinds of instructions have little to do with the actual processes of investigations, being more closely related to the instructions that characterise the routine work processes. hence, skovsmose suggests that this sort of mathematical education, more than producing critical citizens, prepares students to perform routine work and become part of the workforce. in turn, hannaford (1998) suggests that the teaching of mathematics in which students are taught that there is no room for mistakes and there is only one correct answer does not promote plurality and respect for the diversity of ideas. link 3: mathematics education as a social gatekeeper the third link that we have identified relates to the fact that mathematics education can function as a kind of social filter. it is a social filter in the sense that it not only restricts students’ opportunities for development, but may even limit their civic participation. several researchers acknowledge this situation (e.g. amit & fried, 2002; anderson & tate, 2008; christiansen, 2006; knijnik, 2002; malloy, 2008; skovsmose & valero, 2008). for instance, thomas (2010), referring to the australian situation, states: australia and some other nations risk becoming societies divided by access to mathematical knowledge. a minority will have access to high levels of mathematics and will be the highly paid professionals and leaders. the majority will have ‘benchmark’ levels of mathematics and will be poorly paid, often unemployed or underemployed, and in ‘service’ industries. this is not the basis for either a clever country or a democracy but it is the basis for a divided society. (p. 137) indeed, by preventing students’ access to higher education, lack of a mathematics education limits students’ opportunities for a professional career or finding a decent and well-paid job. in short, it decreases their chances of economic and social success. skovsmose (1998) has even suggested that lack of a mathematics education may contribute to the growth in modern societies of a new lower class. the lack of mathematics education can even limit people’s participation in civic society. for example, when mathematics is used in political discussions of social problems, only those who understand the mathematics being used can criticise its use and participate in the discussion, effectively leaving citizens who lack such knowledge out of any deliberations. as noted by christiansen (2008), ‘the use of mathematics [in political discussions] may exclude someone from (feeling confident) taking part in the discussion’ (p. 72). johansen (2002) points out that even some politicians acknowledge that the lack of mathematical skills can be an impediment for citizens’ participation in public debates and democratic processes. orrill (2001) and skovsmose (1994, 1998) go so far as to argue that the lack of such knowledge is a threat to democracy because people who are not mathematically literate cannot fully participate in civic life. unless a population has such mathematical knowledge the potential criticism that may exercise social controls over society’s leaders is threatened. in the words of skovsmose (1994): democracy may be destroyed by a dictatorship which obstructs formal democratic procedures. … democracy can be undermined in ways other than by just neglecting rules of election. democracy refers not only to formal, but also to material and ethical conditions and to possibilities for participation and reaction. in particular, democracy can be destroyed if a critical citizenship cannot brought into life. (p. 38) strategies to foster a democratic competence in students top ↑ the links between mathematics education and democracy are not only a matter of promulgating specific mathematical skills, but also a matter of promoting democratic values and attitudes needed to create the democratic competence (skovsmose, 1990) necessary to apply and critically analyse the use of mathematics in society. how to promote such democratic competence in mathematics students, however, is the subject of much debate. fostering critical mathematical skills the term critical mathematical skills refers to the mathematical knowledge that allows students to use mathematics to analyse social problems or to address issues relevant in their personal lives. such critical mathematical skills enable students to identify and judge how mathematics is applied to address socially relevant issues, as well as to reflect on the consequences of their application.one suggestion made by almeida (2010), christiansen (2008) and moreira (2000) is that mathematics teaching should include activities that will encourage students to use mathematics as a ‘thinking tool’. these activities can be used both to assist students in developing an understanding of mathematical tools and ideas, and to analyse social problems. for example, christiansen (2008) asked pre-service teachers to represent in different ways the share of land that black people and white people in south africa owned in 1981, and then to reflect on the impression given by each representation. besides promoting a reflection on the different ways quantitative information may be represented, this activity also made these future teachers aware of the racial problems in south africa. moreira (2000) used similar methods to introduce students to mathematical applications that allow them to analyse various economic, political and social problems, such as trends in the number of people with aids, the impact of fishing policies on endangered species, and the advantages and disadvantages of adopting nuclear power as a source of energy. malloy (2008) in turn suggests that students should be confronted with moral issues that surround the uses mathematics: we must present them with problems that not only tackle issues that affect their communities, but also reveal the motivations and the hidden agenda (curriculum) in their world. when students use and apply mathematical knowledge in such situations, they are learning to think critically about world issues and their environment through mathematics. through this process students will have an understanding of inequities in society, and will be able to critique the mathematical foundations of social situations. (p. 28) another suggestion, offered by skovsmose (1990), is to promote critical mathematical skills through teaching mathematical modelling and applications in order to prepare students to identify and evaluate the applications of mathematics in society. however, as skovsmose also points out: it is not possible to develop a critical attitude towards the application of mathematics solely by improving the modelling capability of students. … to develop a more critical attitude towards this model building we have not only to understand the mathematical construction of the model; we have also to know about its assumptions. we must be able to point out which economical ideas are hiding behind the curtain of mathematical formulas. (p. 112) thus skovsmose suggests the use of empowering teaching-learning materials (that is, didactical activities) as a means for students to develop critical mathematical skills through the use of open teaching-learning materials whose main characteristics are that (1) the material has to do with a topic of subjective relevance for the students, (2) the material initiates a variety of activities, not pre-structured and fully fixed, and (3) several decisions have to be taken when involved in the teaching-learning process, which normally necessitate a discussion between teacher and students (skovsmose, 1990, pp. 118–119). orrill (2001) goes even further and argues that we should avoid the compartmentalisation of the mathematical knowledge in the school curriculum. in other words, he argues that the teaching of mathematics should be spread across the curriculum. the logic behind this idea is that in real life mathematics is everywhere; it should not be isolated into a single subject. skovsmose and valero (2001) express the same idea this way: there is also a need to consider that mathematical competencies do not operate in isolation outside school but as part of integrated units assembled in schooling. this implies interdisciplinarity among the school subjects as an important research issue. competencies in one discipline interact − or counteract − with competencies developed in other disciplines. even more, competencies development in a school setting interact − or counteract − with competencies formed and used outside the school. (p. 49) fostering democratic values and attitudes explicitly or implicitly, proposals to use mathematics to promote democratic values and attitudes require challenging traditional absolutism in the mathematics classroom. such proposals aim both at modifying the kinds of interactions that occur between the teacher and the students, and at changing the mathematical activities that mediate such interactions.a basic idea behind the promotion of democratic values and attitudes in students is the one proposed by vithal (1999): that within the mathematics classroom it is possible for students to experience democratic life. in the mathematics classroom students can learn, amongst other things, to listen to others’ ideas, to argue, to take decisions and to critically analyse arguments made by authorities (the mathematics teacher for example). ernest (2002) also makes this point: teaching approaches should include discussions, permit conflict of opinions and views but with justifications offered, the challenging of the teacher as an ultimate source of knowledge (not in their role as classroom authority), the questioning of content and the negotiation of shared goals. (p. 6) skovsmose (1990) claims that it is essential to change the fixed and pre-structured mathematical activities within the classroom that characterise traditional mathematics teaching. he argues that the use of open materials (see description above) is compatible with the kind of investigative activities known as project work. he further posits that these kinds of mathematical activities give students more power to make decisions about what to study and how to study it. it also covers the mathematics lessons with uncertainty because it is difficult to predict how students’ projects will evolve. in other words, this approach fundamentally changes the roles and the power relations between teachers and students. hannaford (1998) makes another important proposal to promote dialogue and negotiation in the mathematics classroom. he argues that students should be taught to listen, to think, to argue effectively, and to respect others because democracy depends on those values. in a similar vein, almeida (2010) recommends the use of (informal) mathematical proof as a means to introduce students to a culture of interrogating explanations. students should be invited to consider the explanations provided to them, and to question their level of plausibility. it is especially important for them to learn to look critically at the information and explanations provided by the teacher. according to almeida (2010), if students uncritically accept the information that teachers provide simply because they are authority figures, then it is likely that as citizens they will tend to accept uncritically the information and proposals politicians provide. vithal (1999) similarly envisages the mathematics classroom as a democratic microsociety where the students can learn to both live together with and talk back to authority figures. tensions and difficulties inherent in mathematical education for democracy top ↑ attempts to implement a mathematical education for democracy curriculum to promote democratic competence in mathematics students are not without obstacles, tensions, difficulties and contradictions. a tension between open and empowering materials it is difficult to design activities that are both open and empowering at the same time. for example, what do you do when a student is truly interested in an activity but it does not address any socially relevant problem? similarly, because teachers want students to understand the functions and assumptions behind a real mathematical model, it is difficult for them to avoid proposing activities that are too structured and guided. skovsmose (1990) sums up this problem: open material could result in open and democratic educational situations – but no empowerment is guaranteed; and empowering material could result in critical understanding – but no openness is guaranteed. (p. 120) a tension between democracy and authority harris (1998) and woodrow (1997) also note the ever-present dilemma of democracy and authority: for there to be democracy, some kind of authority is necessary; these elements are complementary. for example, to protect human rights, an authority must exist to defend and guarantee them. vithal (1999) illustrates very clearly how this tension between democracy and authority may occur at different levels within the mathematics classroom: at the classroom level, within working groups, and even in the teacher–student, teacher–researcher domain. this tension brings two important points into focus: firstly, in order to promote democracy sometimes it is necessary to engage in non-democratic practices; and secondly, it is important to recognise that democracy requires some kind of authority, but it is also important to be aware that authority can turn into authoritarianism. the paradox of empowerment vs disempowerment attempts to introduce mathematical activities into school curricula aimed at promoting democratic competencies in the students often face obstacles and resistance. for example, almeida (2010) notes how such activities may cause delays in delivering the traditional curriculum. such situations raise a paradox, as noted by christiansen (2008), that the intention to empower students can be transformed into their actual disempowerment. for instance, we may propose activities that use mathematics to help the students to analyse social problems in their own communities. behind this type of instruction is an assumption that these activities will empower students. however, whilst students may develop mathematical skills through such empowering activities, there is no guarantee that these skills will be those assessed on their exams. this could cause students to obtain poor marks or even to fail, ultimately disempowering them. the paradox of relevance vs indifference d’ambrosio (2003) observes that it is unfortunate that many mathematics educators are not familiar with unesco’s world declaration on education for all (unesco, 1990), which enshrined the right to education for all human beings. due to the fact that human rights are an intimate part of democracy’s legal dimensions, he notes that it is regrettable that there is not a widespread interest in the community of mathematics educators to know and try to implement the resolutions and mechanisms established in that document. the paradox is that, despite the vital importance of the unesco declaration, many mathematics educators are indifferent to this right, and to other dimensions of democracy. this paradox is similar to the issue raised by ernest (2007) concerning the status of social justice within the mathematics education community:why do some individuals believe in social justice? there is great divergence in interest and commitment to social justice among mathematics educators. some view it as central to their professional concerns, whereas others take no personal or professional interest in pursuing social justice issues. why this divergence? (p. 3) the fundamental role of the teacher top ↑ two elements key to implementing mathematical education for democracy initiatives are the attitudes and skills of the teachers responsible. consequently, the competencies that these teachers require to implement this educational approach in their classroom are the subject of much debate. skills needed the specific skills that teachers need to possess can be divided into two groups: mathematical skills and pedagogical skills. with respect to the former, christiansen (2008) holds that such teachers should be mathematically creative. this statement makes sense if they must select and even design empowering activities for their students. christiansen also states that this requires that teachers understand the mathematical potential that such activities entail. they must understand why algorithms are applicable, must know the different ways a problem may be solved and must have a good understanding of how concepts and mathematical structures are interconnected. d’ambrosio (2003) also stresses the importance of being conscious of the dual role that mathematics plays in society. mathematics can be used either as a tool to improve the welfare of humanity, or applied to increase inequality and injustice. awareness of its dual role is especially important when teaching students to identify the uses and abuses of mathematical applications in society. a mathematics classroom that aims to promote democratic values and attitudes should model deliberative interaction, argumentation, critical analysis of the information, and respect for the ideas of others. these features require that teachers possess the pedagogical skills to manage and promote such dynamics in the classroom. as almeida (2010) points out, this requires that teachers use effective questioning techniques and appropriately manage class discussions. christiansen (2008) argues that teachers should be aware of the pedagogical potential of activities both for the individual student and for the class as a whole. this requires, amongst other things, specific pedagogical knowledge about how students learn mathematics, and about concept development. attitudes needed besides having specific mathematical and pedagogical knowledge, teachers need to possess particular attitudes in order to promote democratic competencies in students. for instance, as harris (1998) states, one of the important qualities that teachers must possess is commitment, particularly to social renewal along rational democratic lines. harris, like d’ambrosio (2003), identifies one of the major challenges of mathematics educators: to propose directions to counteract ingrained practices that exclude marginalised individuals and cultures on the periphery and deny them access to knowledge.christiansen (2008) stresses the importance of having critical teachers who are willing to speak up when they detect that a potentially empowering curriculum is being blocked or mathematical creativity is being hindered by limited assessment criteria representing traditional values or by recipe-like instructions about how to teach. another necessary attitude that almeida (2010) highlights is the egalitarian treatment of students in the classroom. he claims that it is the responsibility of teachers to treat students as equal partners in the teaching–learning process. for example, students’ misconceptions and mistakes should not be considered as failures of an intellectual inferior, but rather should be analysed to try to understand the students’ reasoning processes and conceptions that led them to these results. along similar lines, vithal (1999) suggests that teachers can be useful models of authority for students not only to learn about their individual limits, but also to learn that it is possible to raise their voice against authorities. but for this to happen, teachers must be willing to give up part of the authority they traditionally have enjoyed in the classroom and understand that challenges to their authority are part of what constitutes a democratic education for their students. finally, another of the necessary attitudes that some texts address is a willingness of teachers to work in academic environments that are full of uncertainty. this is because, as skovsmose (1990) observes, the evolution of a lesson based on open activities may be unpredictable, so the control that teachers usually have over the mathematical knowledge that is assigned and discussed in the classroom can be diluted and replaced with uncertainty. rights and obligations obligations, as conceptualised by christiansen (2008), are those qualities that teachers responsible for implementing a mathematical education for democracy should possess. framed in a juridical sort of rights and obligations discourse that is one dimension of democracy, christiansen also refers to the rights of teachers:in extension of these obligations, it must also be a democratic right for teachers to have a say in how curricula, guidelines and recommended teaching materials are put together; a right to have the many years of experience from the teaching profession being put to use. a right to be taken serious if they choose to criticise curricula and required teaching methods for being too idealistic and too demanding to realise in practice. do we secure these rights? (pp. 81–82) we think that this conceptualisation is very important as it highlights not only the qualities that the teachers should possess, but also the rights that they should enjoy. selected criticisms top ↑ more empirical research is needed mathematics education for democracy has drawn its share of criticisms. some of these criticisms are aimed at applications of this approach, whilst others refer to unwanted results that it could produce. we reviewed enough works for it to be evident to us that most of this literature consists of programmatic theoretical and rhetorical statements rather than careful empirical research. there is a clear need for empirical studies to test and expand these theoretical ideas. vithal (1999) explicitly addresses this issue:there is now a considerable literature exploring the connections between mathematics education and democratic society, much of it theoretical about what could or should occur. the question is what happens when an attempt is made to deliberately realise such a link in a mathematics classroom. (p. 27) possible discriminatory outcomes one of the paradoxes of empowerment vs disempowerment is that mathematics education for democracy may lead to results that are contrary to expectations. woodrow (1997) points to some studies that show that this approach may produce discriminatory results. for example, he cites a study where some pre-service teachers were empowered to create new curricula based on exploratory and investigative work. however, when faced with the school reality and having found out that school mathematics can be quite ritualistic and confirmatory, one of these empowered teachers became so disillusioned that he left teaching. an imposition of emancipation mathematics education for democracy consists of a series of mathematical activities and modes of interaction that are considered to be beneficial and empowering. they are based on the assumption that teachers with the appropriate training can tell what kinds of education will further the civic development of students. yet entailed in this proposition is the assumption that their superior position gives them the right to modify the curriculum, and to decide what is beneficial for their students. christiansen (2008) questions this ‘right’: who has, when it comes down to it, the right to influence the purpose and content of education? does our insistence on these ‘critical examples’ end up being ‘imposition of emancipation’? how would the historically advantaged feel if the educational system really came to function on the premises of the historically disadvantaged? if our cultural capital … was depreciated overnight? would we not object to the purpose and content forced upon us − even if claimed to be emancipatory? (p. 76) christiansen’s criticism is similar to ernest’s (2010) critique of critical theory, namely that its judgements also require an epistemologically and ethically privileged standpoint. the applicability and relevance of a critical ideology valero (1999) makes a criticism of the critical ideology that underlies the link between mathematics education and democracy that seems particularly relevant to us as latin americans. we refer to the theoretical position that holds that mathematics is ubiquitous in modern societies, and that mathematical models and applications influence many of the decisions that affect and shape modern societies. valero (1999) analyses this ideology from a latin american perspective: critical ideology overemphasizes the role of mathematics in society. in latin america, the power structure has lead to a clientelist political system where decisions are made based on personal loyalty of clients to patrons, political convenience, power of conviction through the use of language or violent and physical imposition. in this ‘rationality’, mathematics does not necessarily constitute a formatting power that greatly influences decision-making. (p. 22) harris (1998) makes a similar criticism that is also relevant in the latin american context: there is a peculiar pointlessness in advocating schooling for democracy and authority within broader social contexts where schooling itself undemocratically favours some individuals and groups and disadvantages others, or where the potential for individual autonomous development is otherwise fundamentally stifled. under those circumstances ‘schooling for democracy’ and ‘education for autonomy’ are either slogans, fashionable ideals or hypocritical rhetoric. (p. 176) discussion top ↑ this literature review was motivated and driven by our own curiosity and personal interests. we live in mexico, a country with a developing but still fragile democracy. when we discovered that there was a research area focused on studying the relationship between mathematics education and democracy, we were very interested in studying it. we wanted to know what the links between mathematics education and democracy are, but we also wanted to know what knowledge and strategies our field has produced that may be used to promote democracy. what are the main ideas that we found in the literature? the first key idea that we found in the literature is that indeed there are connections between mathematics education and democracy, connections that we have already presented. however, these connections are not always positive. mathematics education can promote democratic competences and values, but it can also inhibit them, and create social inequalities. the second key idea we encountered is that there is a generalised interest in the research community to promote equality, democratic values and democratic competencies in mathematics students and teachers. this is a task that is far from trivial. the relationship between mathematics education and democracy is a relationship fraught with tension, and theoretical and practical difficulties. the third idea that became clear to us is that the majority of research on mathematics education and democracy has been developed at a theoretical level. more empirical research is necessary in order to test and expand these theoretical ideas. as noted by vithal (2000, p. 1), ‘hard evidence to support (or refute) theoretical propositions about empowerment, emancipation, democracy, social justice, equity and so on through mathematics education are still rather thin.’ some topics that need to be researched there are a number of research topics, theoretical and empirical, pertinent to the further development of mathematics education for democracy that are absent, are emerging, or have been little explored in the literature. the three links between mathematics education and democracy that we have identified may help to put these theoretical and practical research topics into perspective, within the limitations of this literature review. longitudinal studies as we have seen, several didactical proposals to promote critical mathematical skills in students have been made. some of these proposals that call for the use of real mathematical models in the teaching of mathematics have been tested in actual mathematics classrooms. however, our review found no examples of longitudinal studies that show the long-term effects of this type of instruction. there is some evidence that the students who experience this type of instruction discuss and reflect on the implications and consequences of the application of such mathematical models. but how long do such reflections and attitudes last for these students? is it true that those students who receive this type of mathematical instruction will develop into critical citizens? unfortunately, this type of empirical research is absent in the literature reviewed. teacher education in order to implement a mathematical education for democracy, special teachers are needed. such teachers should be capable of providing their students with opportunities to develop critical mathematical skills and democratic attitudes. however, very few studies address how to structure and implement a teacher education to produce teachers of this kind. although we are aware of the existence of empirical and theoretical studies in the field of mathematics teacher education that seek to promote a more socially just mathematical education (see e.g. the double special issue on social justice published in the journal of mathematics teacher education, volume 12, numbers 3 and 6, 2009), we think that because of the fundamental role played by the teacher, this is an area that still needs to be developed further. textbook development one issue that is closely related to the implementation of a mathematical education for democracy is the production of textbooks. textbooks are important tools that help to organise and structure mathematics lessons. it is necessary to produce textbooks that will foster critical mathematical skills in students. such books may contain, amongst other things, regional examples of applications of mathematics in the analysis and solution of social problems, or even examples of the use of mathematics amongst political actors and governance systems. the concept of power vs the concept of democracy during the construction process of this literature review, we found some articles addressing the concept of power and its connections with the concept of democracy. in particular we refer to the articles by ernest (2002) and valero (2007). there we discovered two theoretical issues that we think deserve to be discussed in more depth in our community.firstly we learned that power is a concept underlying the discussion about mathematics education and democracy, especially when we refer to link 1 mentioned in this review, where it is assumed that students are empowered through the acquisition of certain mathematical skills. valero (2007) discusses and explains how this conception of empowerment, which presupposes power transference, is problematic and leads to contradictions. on the other hand, when we read the concept of epistemological empowerment as presented in ernest (2002) we found some similarities with the concept of democratic competence. this is because to achieve epistemological empowerment it is necessary not only to gain mastery over some mathematical knowledge, but also to possess certain values and attitudes such as personal engagement with mathematics and confidence. as ernest (2002) put it: only when all of these powers are developed will they [the students] feel they are entitled to be confident in applying mathematical reasoning, judging the correctness of such applications themselves, and critically appreciating (including rejecting, in some cases) the applications and uses of mathematics by others, across all types of contexts, in school and society. (p. 11) our point here is that it is necessary to further discuss if and how the theoretical concepts related to power complement or contradict the concepts that have been used in the discussion of mathematics education and democracy. perhaps concepts such as empowerment and democratic competence need to be revisited and even reformulated. how values and attitudes are constituted bishop (2002) affirms that values are an issue that underlie any discussion of social justice and democratisation. through our literature review we noted that the empirical research investigating how values and attitudes are constituted is virtually non-existent. there are questions related to the values and attitudes required for the constitution of democratic competence that need to be addressed. for example, where do values and attitudes come from and how are they constituted? do they come from the textbooks? are they generated through interpersonal relationships in the classroom? does the mathematics teacher transmit them? this is certainly a research topic that needs to be explored. however, the study of this topic may be theoretically complex since there are different conceptualisations of values and attitudes in mathematics education. curriculum studies there are authors like bopape (1998) who mention that in the past the curriculum has perpetuated race, class gender and ethnic division. however, during our review we found only one empirical study showing how a mathematics curriculum can serve as a tool for sorting citizens. we refer to the work of rogers (1998), which is a study that shows how different types of mathematics were taught to different groups of people in england during the industrial revolution. we believe it is necessary to carry out more contemporary studies, such as the one developed by ho (2012), that demonstrate how the mathematics curriculum can serve as a stratification tool in modern societies. a final comment top ↑ we would like to finish this article by pointing out the need to disseminate more widely, and mainly in latin american countries, the research related to mathematics education and democracy. the latin american region is a geographical area constituted by several countries with developing democracies. this is why we consider it relevant that the ideas we have learned through this literature review be disseminated more broadly in these countries. we have the impression that, with few exceptions, this perspective of mathematics education is not widely known in the region. we hope that in future we can help to disseminate these ideas. acknowledgements top ↑ we would like to thank james b. greenberg for proofreading the final version of the manuscript. authors’ contributions m.s.a. 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(1998). linking mathematics education and democracy: citizenship, mathematical archaeology, mathemacy and deliberative interaction. zdm: the international journal on mathematics education, 30(6), 195−203. http://dx.doi.org/10.1007/s11858-998-0010-6  skovsmose, o., & valero, p. (2001). breaking political neutrality: the critical engagement of mathematics education with democracy. in b. atweh, h. forgasz, & b. nebres (eds.), sociocultural research on mathematics education: an international perspective (pp. 37–55). mahwah, nj: lawrence erlbaum. skovsmose, o., & valero, p. (2008). democratic access to powerful mathematical ideas. in l.d. english (ed.), handbook of international research in mathematics education (2nd edn., pp. 415–438). new york, ny: routledge. thomas, j. (2010). policy issues in the teaching and learning of the mathematical sciences at university level. international journal of mathematical education in science and technology, 31(1), 133−142. http://dx.doi.org/10.1080/002073900287453 unesco. (1990). the world declaration on education for all and framework for action to meet basic learning needs. new york, ny: unesco. available from http://www.unesco.org/education/pdf/jomtie_e.pdf valero, p. (1999). deliberative mathematics education for social democratization in latin america. zdm: the international journal on mathematics education, 31(1), 20−26. http://dx.doi.org/10.1007/s11858-999-0004-z valero, p. (2007). what has power got to do with mathematics education? philosophy of mathematics education journal, 21. available from http://people.exeter.ac.uk/pernest/pome21/valero what has power got to do with math ed.doc vithal, r. (1999). democracy and authority: a complementarity in mathematics education? zdm: the international journal on mathematics education, 31(1), 27−36. http://dx.doi.org/10.1007/s11858-999-0005-y vithal, r. (2000). re-searching mathematics education from a critical perspective. in j.p. matos, & m. santos (eds.), proceedings of the second mathematics education and society conference (pp. 1−22). lisbon: universidade de lisboa. available from http://nonio.fc.ul.pt/mes2/renuka.pdf wagner, d., & davis, b. (2010). feeling number: grounding number sense in a sense of quantity. educational studies in mathematics, 74(1), 39−51. http://dx.doi.org/10.1007/s10649-009-9226-9 woodrow, d. (1997). democratic education: does it exist – especially for mathematics education? for the learning of mathematics, 17(3), 11−16. available from http://www.jstor.org/stable/40248247 appendix 1 top ↑ bibliographic details of the references consulted for the development of the literature review top ↑ layer 1 – journals africa christiansen, i.m. (2006). mathematical literacy as a school subject: failing the progressive vision? pythagoras, 64, 6−13. http://dx.doi.org/10.4102/pythagoras.v0i64.94 australasia goos, m., dole, s., & geiger, v. (2011). improving numeracy education in rural schools: a professional development approach. mathematics education research journal, 23(2), 129−148. http://dx.doi.org/10.1007/s13394-011-0008-1 walshaw, m. (2007). research as a catalyst for the promotion of equity. mathematics education research journal, 19(3), 1−2. http://dx.doi.org/10.1007/bf03217458 europe almeida, d. (2010). are there viable connections between mathematics, mathematical proof and democracy? philosophy of mathematics education journal, 25. available from http://people.exeter.ac.uk/pernest/pome25/d.%20f.%20almeida%20%20are%20there%20viable%20connections.docx boylan, m. (2007). teacher questioning in communities of political practice. philosophy of mathematics education journal, 20. available from http://people.exeter.ac.uk/pernest/pome20/boylan teacher questioning in communities.doc dowling, p. (2007). organising the social. philosophy of mathematics education journal, 21. available from http://people.exeter.ac.uk/pernest/pome21/dowling organising the social.doc frankenstein, m. (2010). critical mathematics education: an application of paulo freire’s epistemology. philosophy of mathematics education journal, 25. available from http://people.exeter.ac.uk/pernest/pome25/marilyn frankenstein critical mathematics education.doc nikolakaki, m. (2010). investigating critical routes: the politics of mathematics education and citizenship in capitalism. philosophy of mathematics education journal, 25. available from http://people.exeter.ac.uk/pernest/pome25/maria%20nikolakaki%20%20investigating%20critical%20routes.doc valero, p. (2007). what has power got to do with mathematics education? philosophy of mathematics education journal, 21. available from http://people.exeter.ac.uk/pernest/pome21/valerowhathaspowergottodowithmathed.doc walls, f. (2007). using rights-based frameworks for rethinking teacher-directed pedagogies of mathematics. philosophy of mathematics education journal, 20. available from http://people.exeter.ac.uk/pernest/pome20/wallsusing right-based frameworks.doc international bishop, a. j. (1999). mathematics teaching and values education − an intersection in need of research. zdm: the international journal on mathematics education, 31(1), 1−4. http://dx.doi.org/10.1007/s11858-999-0001-2 d’ambrosio, u. (1990). the role of mathematics education in building a democratic and just society. for the learning of mathematics, 10(3), 20−23. available from http://www.jstor.org/stable/40247989 de mattos, a.c., & batarce, m.s. (2010). mathematics education and democracy. zdm: the international journal on mathematics education, 42(3/4), 281−289. http://dx.doi.org/10.1007/s11858-009-0232-2 greer, b. (2009). estimating iraqi deaths: a case study with implications for mathematics education. zdm: the international journal on mathematics education, 41(1/2), 105−116. http://dx.doi.org/10.1007/s11858-008-0147-3 hannaford, c. (1998). mathematics teaching is democratic education. zdm: the international journal on mathematics education, 30(6), 181−187. http://dx.doi.org/10.1007/s11858-998-0008-0 harris, k. (1998). mathematics teachers as democratic agents. zdm: the international journal on mathematics education, 30(6), 174−180. http://dx.doi.org/10.1007/s11858-998-0007-1 lengnik, k. (2005). reflecting mathematics: an approach to achieve mathematical literacy. zdm: the international journal on mathematics education, 37(3), 246−249. http://dx.doi.org/10.1007/s11858-005-0016-2 o’donoghue, j. (1995). numeracy and further education: beyond the millennium. international journal of mathematical education in science and technology, 26(3), 389−405. http://dx.doi.org/10.1080/0020739950260308 skovsmose, o. (1985). mathematical education versus critical education. educational studies in mathematics, 16(4), 337−354. http://dx.doi.org/10.1007/bf00417191 skovsmose, o. (1988). mathematics as part of technology. educational studies in mathematics, 19(1), 23−41. http://dx.doi.org/10.1007/bf00428383 skovsmose, o. (1990). mathematical education and democracy. educational studies in mathematics, 21(2), 109−128. http://dx.doi.org/10.1007/bf00304897 skovsmose, o. (1992). democratic competence and reflective knowing in mathematics. for the learning of mathematics, 12(2), 2−11. available from http://www.jstor.org/stable/40248044 skovsmose, o. (1994). towards a critical mathematics education. educational studies in mathematics, 27(1), 35−57. http://dx.doi.org/10.1007/bf01284527 skovsmose, o. (1998). linking mathematics education and democracy: citizenship, mathematical archaeology, mathemacy and deliberative interaction. zdm: the international journal on mathematics education, 30(6), 195−203. http://dx.doi.org/10.1007/s11858-998-0010-6 skovsmose, o. (2001). landscapes of investigation. zdm: the international journal on mathematics education, 33(4), 123−132. http://dx.doi.org/10.1007/bf02652747 skovsmose, o. (2007). doubtful rationality. zdm: the international journal on mathematics education, 39(3), 215−224. http://dx.doi.org/10.1007/s11858-007-0024-5 spielman, l.j. (2008). equity in mathematics education: unions and intersections of feminist and social justice literature. zdm: the international journal on mathematics education, 40(4), 647–657. http://dx.doi.org/10.1007/s11858-008-0113-0 thomas, j. (2000). policy issues in the teaching and learning of the mathematical sciences at university level. international journal of mathematical education in science and technology, 31(1), 133−142. http://dx.doi.org/10.1080/002073900287453 tutak, f.a., bondy, e., & adams, t.l. (2011). critical pedagogy for critical mathematics education. international journal of mathematical education in science and technology, 42(1), 65−74. http://dx.doi.org/10.1080/0020739x.2010.510221 valero, p. (1999). deliberative mathematics education for social democratization in latin america. zdm: the international journal on mathematics education, 31(1), 20−26. http://dx.doi.org/10.1007/s11858-999-0004-z valero, p. (2007). a socio-political look at equity in the school organization of mathematics education. zdm: the international journal on mathematics education, 39(3), 225−233. http://dx.doi.org/10.1007/s11858-007-0027-2 vithal, r. (1999). democracy and authority: a complementarity in mathematics education? zdm: the international journal on mathematics education, 31(1), 27−36. http://dx.doi.org/10.1007/s11858-999-0005-y vithal, r., & skovsmose, o. (1997). the end of innocence: a critique of ‘ethnomathematics’. educational studies in mathematics, 34(2), 131−157. http://dx.doi.org/10.1023/a:1002971922833 wagner, d., & davis, b. (2010). feeling number: grounding number sense in a sense of quantity. educational studies in mathematics, 74(1), 39−51. http://dx.doi.org/10.1007/s10649-009-9226-9 woodrow, d. (1997). democratic education: does it exist – especially for mathematics education? for the learning of mathematics, 17(3), 11−16. available from http://www.jstor.org/stable/40248247 latin america bernardi, l.s., & caldeira, a.d. (2012). educação matemática na escola indígena sob uma abordagem crítica. [mathematics education in an indigenous schools undera critical approach]. bolema: boletim de educação matemática, 26(42b), 409−431. available from http://www.periodicos.rc.biblioteca.unesp.br/index.php/bolema/article/view/5765campos, c.r., jacobini, o.r., wodewotzki, m.l.l., & ferreira, d.h.l. (2011). educação estatística no contexto da educação crítica. [statistics education in the context of critical education]. bolema: boletim de educação matemática, 24(39), 473−494. available from http://www.periodicos.rc.biblioteca.unesp.br/index.php/bolema/article/view/5104 jacobini, o.r., & wodewotzki, m.l. (2006). uma reflexão sobre a modelagem matemática no contexto da educação matemática crítica. [a reflection on mathematical modeling in the context of critical mathematics education]. bolema: boletim de educação matemática, 19(25), 1−16. available from http://www.periodicos.rc.biblioteca.unesp.br/index.php/bolema/article/view/1876 pinheiro, n.m., & bazzo, w.a. (2009). caso simulado no ensino-aprendizagem de matemática: ensinar sob uma abordagem crítica. [simulated case in the teaching-learning of mathematics: teaching under a critical approach]. boletim de educação matemática, 22(32), 101−122. available from http://www.periodicos.rc.biblioteca.unesp.br/index.php/bolema/article/view/2074 north america amit, m., & fried, m.n. (2002). high-stakes assessment as a tool for promoting mathematical literacy and the democratization of mathematics education. the journal of mathematical behavior, 21(4), 499−514. http://dx.doi.org/10.1016/s0732-3123(02)00147-5 gutiérrez, r. (2010). the sociopolitical turn in mathematics education. journal for research in mathematics education, 41(0), 4−32. a vailable from http://www.nctm.org/uploadedfiles/journals_and_books/jrme/articles/jrme_special_equity_issue/jrme2010-08-5a.pdf parks, a.n., & schmeichel, m. (2012). obstacles to addressing race and ethnicity in the mathematics education literature. journal for research in mathematics education, 43(3), 238−252. available from http://www.jstor.org/stable/10.5951/jresematheduc.43.3.0238 skovsmose, o. (2006). research, practice, uncertainty and responsibility. the journal of mathematical behavior, 25(4), 267−284. http://dx.doi.org/10.1016/j.jmathb. 2006.11.002 layer 2 – conference proceedings icme 11 gellert, u., & jablonka, e. (2008, july). the demathematising effect of technology: calling for critical competence. discussion group 3: mathematics education: for what and why? paper presented at the 11th international congress on mathematical education, monterrey, mexico. available from http://dg.icme11.org/document/get/63 icme 10 skovsmose, o. (2004). critical mathematics education for the future. in m. niss (ed.), proceedings of the tenth international congress on mathematical education (pp. 1−19). roskilde, denmark: imfufa, roskilde university. available from http://vbn.aau.dk/files/17034122/criticalmathematicseducation cerme 7 andersson, a., & norén, e. (2011). agency in mathematics education. in m. pytlak, t. rowland, & e. swoboda (eds.), proceedings of the seventh congress of the european society for research in mathematics education (pp. 1389−1398). rzeszów, poland: university of rzeszów. available from http://www.cerme7.univ.rzeszow.pl/wg/10/cerme7_wg10_anderssonnoren.pdf cerme 6 valero, p. (2009). mathematics education as a network of social practices. in v. durand-guerrier, s. soury-lavergne, & f. arzarello (eds.), proceedings of the sixth congress of the european society for research in mathematics education (pp. lv−lxxx). lyon, france: institut national de recherche pédagogique. available from http://ife.ens-lyon.fr/publications/edition-electronique/cerme6/plenary2-valero.pdf mes 1 bopape, m. (1998). the south african new mathematics curriculum: people’s mathematics for people’s power? in p. gates, & t. cotton (eds.), proceedings of the first mathematics education and society conference. nottingham: the centre for the study of mathematics education, nottingham university. available from http://www.nottingham.ac.uk/csme/meas/papers/bopape.html moreira, l., & carreira, s. (1998). no excuses to command, no excuses to obey, no excuses to ignore. some data to reflect upon. in p. gates, & t. cotton (eds.), proceedings of the first mathematics education and society conference. nottingham: the centre for the study of mathematics education, nottingham university. available from http://www.nottingham.ac.uk/csme/meas/papers/moreiracarreira.html skovsmose, o. (1998). aporism, and the problem of democracy in mathematics education. in p. gates, & t. cotton (eds.), proceedings of the first mathematics education and society conference. nottingham: the centre for the study of mathematics education, nottingham university. available from http://www.nottingham.ac.uk/csme/meas/plenaries/skovsmose.html mes 2 moreira, l. (2000). some reflections on democracy with regard to curricula and vice versa. in j.p. matos, & m. santos (eds.), proceedings of the second mathematics education and society conference. lisbon: universidade de lisboa. available from http://nonio.fc.ul.pt/mes2/moreira.docvithal, r. (2000). re-searching mathematics education from a critical perspective. in j.p. matos, & m. santos (eds.), proceedings of the second mathematics education and society conference. lisbon: universidade de lisboa. available from http://nonio.fc.ul.pt/mes2/renuka.pdf mes 3 bishop, a.j. (2002). research, policy and practice: the case of values. in p. valero, & o. skovsmose (eds.), proceedings of the third mathematics education and society conference. copenhagen: centre for research in learning mathematics. available from http://mes3.learning.aau.dk/papers/bishop.pdf carvalho, v., mendonça, m.c.d., santos, s.a., & santos-wagner, v.m. (2002). in p. valero, & o. skovsmose (eds.), proceedings of the third mathematics education and society conference. copenhagen: centre for research in learning mathematics. available from http://mes3.learning.aau.dk/papers/carvalho.pdf ernest, p. (2002). what is empowerment in mathematics education? in p. valero, & o. skovsmose (eds.), proceedings of the third mathematics education and society conference. copenhagen: centre for research in learning mathematics. available from http://mes3.learning.aau.dk/papers/ernest.pdf johansen, l.ø. (2002). why teach math to the “excluded”? in p. valero, & o. skovsmose (eds.), proceedings of the third mathematics education and society conference. copenhagen: centre for research in learning mathematics. available from http://mes3.learning.aau.dk/papers/johansen.pdf knijnik, g. (2002). two political facets of mathematics education in the production of social exclusion. in p. valero, & o. skovsmose (eds.), proceedings of the third mathematics education and society conference. copenhagen: centre for research in learning mathematics. available from http://mes3.learning.aau.dk/papers/knijnik.pdf popkewitz, t. (2002). whose heaven and whose redemption? the alchemy of the mathematics curriculum to save (please check one or all of the following) (a) the economy, (b) democracy, (c) the nation, (d) human rights, (d) the welfare state, (e) the individual. in p. valero, & o. skovsmose (eds.), proceedings of the third mathematics education and society conference, addendum. copenhagen: centre for research in learning mathematics. available from http://mes3.learning.aau.dk/plenaries/popkewitz.pdf mes 5 chartres, m. (2008). are my students engaged with critical mathematics education? in j.f. matos, p. valero, & k. yasukawa (eds.), proceedings of the fifth mathematics education and society conference (pp. 186−196). lisbon: centro de investigação em educação, universidade de lisboa – department of education, learning and philosophy, aalborg university. available from http://pure.ltu.se/portal/files/2376308/proceedings_mes5.pdf mes 6 martin, d.b. (2010). not-so-strange bedfellows: racial projects and the mathematics education enterprise. in u. gellert, e. jablonka, & c. morgan (eds.), proceedings of the sixth mathematics education and society conference, vol. 1 (pp. 57−79) berlin: freie universität berlin. available from http://www.ewi-psy.fu-berlin.de/en/v/mes6/documents/proceedings/band_1_finale.pdf sinclair, n., & pimm, d. (2010). mathematics, democracy and the aesthetic. in u. gellert, e. jablonka, & c. morgan (eds.), proceedings of the sixth mathematics education and society conference, vol. 2 (pp. 57−79). berlin: freie universität berlin. available from http://www.ewi-psy.fu-berlin.de/en/v/mes6/documents/proceedings/15072010.pdf layer 3 – books anderson, c.r., & tate, w.f. (2008). still separate, still unequal: democratic access to mathematics in u.s. schools. in l.d. english (ed.), handbook of international research in mathematics education (2nd edn., pp. 299–318). new york, ny: taylor & francis. christiansen, i.m. (2008). some tensions in mathematics education for democracy. in b. sriraman (ed.), international perspectives on social justice in mathematics education (pp. 69−86). missoula, mt: information age publishing. d’ambrosio, u. (2003). the role of mathematics in building a democratic society. in b.l. madison, & l.a. steen (eds.), quantitative literacy. why numeracy matters for schools and colleges (pp. 235−238). princeton, nj: the national council on education and the disciplines. available from http://www.maa.org/ql/qltoc.html gellert, u., jablonka, e., & keitel, c. (2001). mathematical literacy and common sense in mathematics education. in b. atweh., h. forgasz, & b. nebres (eds.), sociocultural research on mathematics education: an international perspective (pp. 57−73). mahwah, nj: lawrence erlbaum. malloy, c.e. (2008). looking throughout the world for democratic access to mathematics. in l.d. english (ed.), handbook of international research in mathematics education (2nd edn., pp. 20–31). new york, ny: routledge. orrill, r. (2001). mathematics, numeracy, and democracy. in l.a. steen (ed.), mathematics and democracy. the case for quantitative literacy (pp. 8−20). princeton, nj: the national council on education and the disciplines. available from http://www.maa.org/ql/mathanddemocracy.html skovsmose, o. (1994). towards a philosophy of critical mathematics education. dordrecht: kluwer academic publishers. skovsmose, o., & valero, p. (2001). breaking political neutrality: the critical engagement of mathematics education with democracy. in b. atweh, h. forgasz, & b. nebres (eds.), sociocultural research on mathematics education: an international perspective (pp. 37−55). mahwah, nj: lawrence erlbaum. skovsmose, o., & valero, p. (2008). democratic access to powerful mathematical ideas. in l.d. english (ed.), handbook of international research in mathematics education (2nd edn., pp. 415–438). new york, ny: routledge. article information author: andreas j. stylianides1 affiliation: 1faculty of education, university of cambridge, united kingdom correspondence to: andreas stylianides email: as899@cam.ac.uk postal address: university of cambridge, faculty of education, 184 hills road, cambridge cb2 8pq, united kingdom dates: received: 13 june 2011 accepted: 03 july 2011 published: 03 aug. 2011 how to cite this article: stylianides, a.j. (2011). towards a comprehensive knowledge package for teaching proof: a focus on the misconception that empirical arguments are proofs. pythagoras, 32(1), art. #14, 10 pages. doi:10.4102/pythagoras.v32i1.14 note: this article is a substantially revised version of the author’s plenary lecture at the 15th annual congress of the association for mathematics education of south africa (amesa), which appeared in the proceedings of the congress (stylianides, 2009b). copyright notice: © 2011. the authors. licensee: aosis openjournals. this work is licensed under the creative commons attribution license. issn: 1012-2346 (print) issn: 2223-7895 (online) towards a comprehensive knowledge package for teaching proof: a focus on the misconception that empirical arguments are proofs in this original research... open access • abstract • introduction • knowledge for teaching proof    • mathematical knowledge about proof    • knowledge about students’ conceptions of proof    • pedagogical knowledge for teaching proof • an instructional intervention    • background    • overview of activities in the intervention       • activity 1: the squares problem       • activity 2: the circle and spots problem       • activity 3: the monstrous counter-example illustration       • ethical considerations    • what knowledge was important for the teacher to have to successfully implement the instructional intervention in her classroom?       • element 1: understanding how the three activities fitted together to form a coherent whole and to support students’ progression along the intended learning trajectory       • element 2: understanding the nuances of implementing the activities – the notion of ‘conceptual awareness pillars’ • conclusion • acknowledgements    • competing interests • references • footnotes abstract (back to top) the concept of proof is central to meaningful learning of mathematics, but is hard for students to learn. a serious misconception dominant amongst students at all levels of schooling is that empirical arguments are proofs. an important question, then, is the following: what knowledge might enable teachers to help students overcome this misconception? earlier research led to construction of a significant but rather incomplete ‘knowledge package’ for teaching in this area. major elements of this knowledge package are summarised and its further development is contributed to by discussing a research-based instructional intervention found to be effective in helping secondary students begin to overcome the misconception that empirical arguments are proofs. implications for mathematics teacher education are considered. introduction (back to top) a large body of research shows that students of all levels of schooling (including high-attaining secondary students) ‘prove’ mathematical generalisations by using empirical arguments (e.g. chazan, 1993; coe & ruthven, 1994; healy & hoyles, 2000; senk, 1985; sowder & harel, 1998), whereby ‘empirical arguments’ are those that purport to establish the truth of a generalisation by validating it only in a proper subset of all possible cases in its domain. yet, the fact that a generalisation is found to be true in some cases does not guarantee − and thus does not prove − that it is true for all possible cases. this is a fundamental difference between an empirical argument and the notion of proof in mathematics (stylianides, 2007b). however, many students have difficulty understanding this distinction. in addition to establishing conclusively and convincingly the truth of a generalisation, a proof can also help someone understand why the generalisation is true. a proof’s potential to promote conviction (justification) and understanding (explanation) in part accounts for it being considered central to meaningful learning in mathematics (ball & bass, 2000, 2003; hanna, 2000; harel & sowder, 2007; stylianides, 2008; stylianides & stylianides, 2008). according to harel and sowder: [m]athematics as sense-making means that one should not only ascertain oneself that the particular topic/procedure makes sense, but also that one should be able to convince others through explanation and justification of her or his conclusions. (harel & sowder, 2007, pp. 808–809) unless students realise the limitations of empirical arguments as methods for validating generalisations, they are unlikely to appreciate the importance of proof in mathematics (stylianides & stylianides, 2009). in order to achieve this learning objective, however, teachers must have good knowledge in the area of proof, for the quality of learning opportunities that students receive in classrooms depends on the quality of their teachers’ knowledge (ball, thames & phelps, 2008; goulding, rowland & barber, 2002; ma, 1999). what knowledge, then, might enable teachers to help their students begin to overcome the misconception that empirical arguments are proofs? knowledge for teaching proof (back to top) following shulman’s (1986, 1987) influential work on the nature of teachers’ knowledge for teaching, a significant body of mathematics education research has begun to unravel the kinds of knowledge that teachers need for effective mathematics teaching. shulman’s work has stimulated at least two major and related research strands in the mathematics education literature on teacher knowledge. the first research strand has examined the mathematical demands that mathematics teaching places on teachers’ knowledge (e.g. ball & bass, 2000, 2003; ball et al., 2008; davis & simmt, 2006; ma, 1999), and has helped identify important mathematical ideas essential for teachers to know in order to support student learning of particular mathematical topics or concepts, such as proof (for a study that focused on proof, see stylianides & ball, 2008). this strand informs the field’s understanding of teachers’ mathematical (subject matter) knowledge about proof, one of three broad kinds of teacher knowledge for teaching proof considered here. the second research strand stimulated by shulman’s work has examined students’ common ways of thinking about particular mathematical ideas, helping identify important student conceptions (including misconceptions) that are essential for teachers to know in order to help their students develop their current conceptions. the extended mathematics education research on students’ conceptions of proof (e.g. chazan, 1993; coe & ruthven, 1994; healy & hoyles, 2000; knuth, choppin, slaughter & sutherland, 2002; senk, 1985; sowder & harel, 1998; stylianides & al-murani, 2010) informs understanding of teachers’ knowledge about students’ conceptions of proof, the second broad kind of teachers’ knowledge for teaching proof considered here. this kind of knowledge can be regarded as an example of an aspect of shulman’s notion of pedagogical content knowledge, namely, knowledge ‘of what makes the learning of specific topics [in this case proof] easy or difficult’ for students (shulman, 1986, p. 9). the third broad category of teacher knowledge considered in this article, pedagogical knowledge for teaching proof, has received significantly less research attention than the other two, even though also related to shulman’s work. prior research has offered few insights into pedagogical practices that are important for teachers to know and implement in their classrooms to help students develop conceptions of proof that better approximate conventional understanding. such fine-grained, domain-specific pedagogical knowledge in the area of proof can be regarded as an example of another aspect of shulman’s notion of pedagogical content knowledge, namely knowledge of ‘the ways of representing and formulating the subject matter that make it comprehensible to others’ (shulman, 1986, p. 9). effective teaching of proof requires at least these three broad kinds of knowledge, which are interrelated. for example, good knowledge about students’ conceptions of proof is not possible without a robust understanding of the mathematical ideas that underpin these conceptions. similarly, a teacher’s ability to implement pedagogical practices in the classroom to help students develop their conceptions of proof requires a robust understanding of what the students’ conceptions are and other forms (more advanced) that these can take. what follows is a summary of what the field of mathematics education currently considers important for teachers to know in relation to the three broad kinds of knowledge, focusing on the common student misconception that empirical arguments are proofs. this is used to discuss how the field might develop a more comprehensive knowledge package for teaching in this area. i use the term knowledge package to describe a cluster of related kinds of knowledge (about mathematics, students and pedagogy) that are important for teachers to have in order to teach effectively a particular idea in their classrooms.1 mathematical knowledge about proof in our review of prior literature on teachers’ mathematical knowledge about proof for teaching (stylianides & ball, 2008), we identified understanding the distinction between empirical arguments and proofs to be an important element of this knowledge. unless teachers at all levels of schooling develop a good understanding of this distinction, it is unlikely that large numbers of students will overcome their misconception that empirical arguments are proofs. according to martin and harel: [i]f [elementary] teachers lead their students to believe that a few well-chosen examples constitute a proof, it is natural to expect that the idea of proof in high school geometry and other courses will be difficult for the students. (martin & harel, 1989, pp. 41−42) also, elementary teaching practices that promote or tolerate a conception of proof as an empirical argument instill mental habits in students that significantly deviate from conventional mathematical understanding in the field. dewey (1903, p. 217) cautioned educators against such practices when he said that whatever the preliminary approach to learning, it should not inculcate ‘mental habits and preconceptions which have later on to be bodily displaced or rooted up in order to secure proper comprehension of the subject’. knowledge about students’ conceptions of proof a significant body of research has investigated students’ conceptions of proof and developed various taxonomies of these (e.g. balacheff, 1988; sowder & harel, 1998); these in turn inform our understanding of what is important for teachers to know. good knowledge about students’ conceptions of proof can help teachers evaluate their students’ current understandings about proof, and thus prepare teachers to design instruction that aims to help students develop conceptions of proof that better approximate conventional understanding. the following is a taxonomy (hierarchy) of students’ conceptions of proof, presented in increasing level of mathematical sophistication (for elaboration, see stylianides & stylianides, 2009): • naïve empirical conception: validating a mathematical generalisation (pattern, conjecture, etc.) by checking a proper subset of all possible cases in its domain, selected primarily on the basis of convenience. • crucial experiment conception: validating a mathematical generalisation by checking a proper subset of all the possible cases in its domain; these cases are not selected on the basis of convenience but rather based on some kind of rationale, notably the search of possible counter-examples to the generalisation. • nonempirical conception: recognising empirical arguments as insecure methods for validating mathematical generalisations; being able to understand and write proofs. naïve empiricism and crucial experiment are two kinds of empirical arguments (balacheff, 1988), the former implying no awareness of the possibility that the examined generalisation could be false. accordingly, although both represent an empirical approach to validating mathematical generalisations, the first is less advanced than the second. the nonempirical conception includes several stages, the most advanced being students’ ability to understand and write proofs. this article focuses on what might be considered to be the first stage of a nonempirical conception – recognising the limitations of empirical arguments as insecure methods of validation, and seeing the need to learn about more secure methods (i.e. proofs). pedagogical knowledge for teaching proof prior research has developed a useful basis about general pedagogical practices for engaging students in mathematical reasoning, argumentation, and proof (e.g. ball & bass, 2000, 2003; stylianides, 2007a, 2007b; stylianides & ball, 2008; yackel & cobb, 1996; zack, 1997). however, this body of research needs to be developed further before it can inform design of effective instruction to help students develop more accurate conceptions of proof, including the nonempirical conception. for example, in our earlier work (stylianides & ball, 2008), we elaborated on the importance of teachers using a variety of proving tasks that can offer students learning opportunities to develop an understanding of different proving strategies and reasoning skills. although implementation of particular proving tasks in the classroom can support generation of different kinds of arguments, thus offering an opportunity to teachers and students to discuss the differences between empirical arguments and proofs, it is unclear how this discussion can be organised to help students overcome their deeply rooted misconception that empirical arguments are proofs. indeed, prior research and practice have shown that addressing this misconception is a stubborn problem in mathematics education, even at university level (goulding & suggate, 2001; martin & harel, 1989). this problem cannot be addressed without a carefully designed instructional intervention. by instructional intervention is meant a purposeful and cohesive collection of activities (and respective implementation strategies) for achieving particular learning outcomes. ability to successfully implement an instructional intervention in the classroom to help students begin to overcome the misconception that empirical arguments are proofs would be an essential element of teachers’ pedagogical knowledge about proof and, by implication, an important complement to the existing knowledge package for teaching proof. what could such an instructional intervention look like, and what demands would successful implementation thereof place on teachers’ knowledge? a research-based instructional intervention found to be effective in helping students begin to overcome the misconception that empirical arguments are proofs is discussed in the next section. this will exemplify also the point that successful implementation of the instructional intervention requires good knowledge about mathematics and students – emphasising the inextricable relationships amongst the different kinds of knowledge comprising the knowledge package for teaching proof. an instructional intervention (back to top) background in a four-year design experiment in an undergraduate mathematics course in the united states of america (for discussion on design experiment methodology, see e.g. schoenfeld, 2006), gabriel stylianides and i developed an instructional intervention. we showed this to be effective in helping undergraduate students begin to understand the limitations of empirical arguments and to see an ‘intellectual need’ (harel, 1998) for learning about more secure methods of validation (i.e. proofs) (stylianides & stylianides, 2009). presented at a later stage in this article is how a secondary mathematics teacher implemented a modified version of the original instructional intervention in a year 10 class in england (14–15-year-old students) to achieve the same learning goals. implementation of the intervention by the secondary teacher was part of a school-based project organised as a design experiment, and following up on the aforementioned university-based design experiment. i conducted the school-based project in collaboration with two secondary mathematics teachers in two classes in a state school in england over a period of two years (when the students were in years 10 and 11). the project aimed to generate theoretical and practical knowledge about how secondary teachers can better support students’ learning of proof, and focused on high-attaining students. given the findings of prior research that even high-attaining secondary students in england possessed weak understanding of proof (coe & ruthven, 1994; healy & hoyles, 2000; küchemann & hoyles, 2001−2003), and limited research knowledge on how to improve students’ understanding of proof, i considered it strategic to focus on a student population that would offer better chances for success (high-attaining students), as a first step in a long-term research programme. the theoretical framework underpinning the design and implementation of the focal instructional intervention is elaborated on in stylianides and stylianides (2009, pp. 316−324). the implementation of the intervention by the secondary teacher lasted approximately 60 minutes, and extended over two consecutive lessons. the teacher followed detailed lesson plans that i prepared and discussed with her prior to the lessons; our discussion led to minor modifications of the lesson plans, primarily in order to accommodate her timetable constraints (stylianides, 2009a).2 whilst reading the description of implementation of the intervention, i invite the readers to consider this question: what knowledge was it important for the teacher to have in order to successfully implement the intervention in her class? i will return to this question later in the article. overview of activities in the intervention the intervention included three activities: the squares problem, the circle and spots problem, and the monstrous counter-example illustration (refer to figures appearing respectively throughout the article). the teacher used these three activities to facilitate students’ progression along a ‘learning trajectory’ (simon, 1995; see also clements & sarama, 2004) from a naïve empirical conception, to a crucial experiment conception, to a nonempirical conception. each of these three stages in the learning trajectory corresponds to a conception in the taxonomy discussed earlier. activity 1: the squares problem the hardest part of the squares problem (figure 1) was part three, which asked students to find and explain their answer for the number of different 3-by-3 squares in a case that was difficult for them to check practically (n = 60). the teacher made sure that the students understood what the problem said, and then asked them to work on it in small groups. figure 1: the squares problem. in small groups the students identified the pattern that the number of different 3-by-3 squares in an n-by-n square was given by the formula (n – 2)2. they verified the pattern for n = 4 and n = 5, and some of them for n = 6 as well. based on this confirming evidence, they concluded that the pattern would hold for all values of n, including n = 60. thus the students validated the pattern empirically on the basis of naïve empiricism. the whole-group discussion that followed further illustrated dominance of the naïve empirical conception in the class, with all small groups answering the three parts of the problem using the formula (n – 2)2. after some discussion about the meaning of the formula, the teacher (following the lesson plan) asked the students each to write down their thoughts about whether and why they could be sure that applying this formula would give the correct answer for part three of the problem. some illustrative responses from students in one of the small groups were as follows: because we have found a formula and tried it against smaller squares so we can make sure that the formula is right [bob]. i am sure that this solution works because it worked for every one we did [calvin]. i am sure that the answer is correct because it has been proved for a number of smaller grids [dan]. (stylianides, 2010a, pp. 10–11) these comments were representative of those of the rest of the class. the students were convinced of the truth of the pattern on the basis of naïve empiricism; the pattern worked for the first few cases and so they felt it would also work for n = 60. after the students’ individual reflections, the teacher proceeded with the next item on the lesson plan, which was to summarise the students’ written responses. the teacher’s summary was based on a quick inspection of students’ written responses as she circulated during their individual reflection time. accuracy of the teacher’s summary was confirmed by a more careful analysis of students’ responses by both her and me at the end of the lesson: ‘i get a feeling that most of you have said, “well, i think we have sort of answered this question that 582 is the right answer: we have found a pattern by checking smaller grid sizes and then we have used that pattern, assuming that it would continue all the way up to 60-by-60.” that’s the stage where we are right now: we’ve seen a pattern working, somebody said they tried the 6-by-6 and it worked for that too, and so we continued our pattern up to the 582.’ (stylianides, 2010a, p. 11) the state of student conceptions of proof in the class, as described in the aforementioned summary (where naïve empiricism predominated), had been anticipated in the lesson plan and was consistent with findings of prior research on the issue. after the teacher’s summary, the class moved on to the next activity. according to the lesson plan, the issue about the correctness of the pattern in the squares problem would remain tentatively unresolved. the class would revisit and resolve the issue after the students had been assisted to realise the limitations of empirical arguments (both naïve empiricism and crucial experiment). our intention was for the students to realise the limitations of empirical arguments on their own, after experiencing a series of ‘cognitive conflicts’ (see stylianides & stylianides, 2009) and reflecting on situations in the next two activities, where empirical methods of validation were inadequate (for the readers’ information, i note that the [n – 2]2 pattern in the squares problem was actually correct). activity 2: the circle and spots problem the teacher introduced the circle and spots problem (figure 2), and helped the students understand what it said. she discussed with them the meaning of the terms maximum and non-overlapping regions, and clarified for them that spots meant points and that the phrase around the circle referred to the circle’s circumference. she also mentioned that the points on the circumference did not have to be equidistant. then the teacher asked the students to work on the problem in their small groups. figure 2: the circles and spots problem. in our planning it had been anticipated that, similar to what students had done in the squares problem, they would check simpler cases, identify a pattern, trust the pattern based on naïve empiricism, and apply it to offer a definite answer for n = 15 (where n stands for the number of points). however, there is a key difference between the two problems: unlike the emerging pattern in the squares problem, which is true, the emerging pattern in the circle and spots problem fails for n = 6. our plan was for the teacher to use the anticipated cognitive conflict that students would experience with the unexpected discovery of the counter-example in order to help them move from the naïve empirical to the crucial experiment conception. after about 10 minutes of small group work the teacher brought the class together and asked the students whether they thought they had an answer for n = 15. mac said his group thought the formula included powers of 2. the teacher asked the class to state the maximum number of non-overlapping regions they found for different numbers of spots, and drew a table on the board with the following numbers: 4, 8, and 16, for n = 3, 4, and 5 respectively. then she pointed out to the class that as mac had mentioned, the values were all powers of 2; in each case, the power was one less than the number of spots: 22 for n = 3, 23 for n = 4, and 24 for n = 5. the teacher asked: ‘so what will it be for 15 spots, then?’ several students offered to answer the teacher’s question. based on what i observed during students’ small group work, i presumed they would propose application of the 2n – 1 formula for n = 15. however: ken: [said loudly] ‘can i just say that is wrong, because on 6 [spots] there are only 30 [regions].’ teacher: ‘we were about to say that the answer would be 2 to the power of 14. however, you are telling me that for 6 spots it doesn’t work out to be ... with this pattern for 6 six spots it would be 2 to the power of 5, that would be 32, but did anyone manage to find this number of spots?’ [some students said they found 31.] ‘when we were back to the squares problem, we said that because the pattern worked for some of the different grids, the 5-by-5, 6-by-6 squares, and so on, we were willing to trust it. but this time we have shown that it works for 3, it works for 4, it works for 5, but actually, ken, you are right: if we had 6 spots on a circle and we joined them all up, the number of non-overlapping regions that we get is not what we expect to get, it’s not 32. it’s actually 31.’ (stylianides, 2010a, pp. 12–13) as she talked, the teacher used a powerpoint slide to illustrate a case in which the maximum number of non-overlapping regions for n = 6 was 31, noting also that this number of regions was the maximum one could get with 6 spots.3 she also noted that if one drew the spots in a regular hexagon, the maximum number of regions would be 30, which is again smaller than 32. then, following the lesson plan, she asked the students to write down their thoughts on what the circle and spots problem had taught them. the same three students as quoted before wrote as follows: you can’t always trust a formula until you have tested it many times over for lots of different examples [bob]. this test has taught us that if you see a pattern [it] doesn’t make it correct [calvin]. the circle and spots tells us that we can’t always trust a formula that works on the first few [dan]. (stylianides, 2010a, p. 13) as suggested by these reflections, which were again representative of the whole class, the students had started to move away from naïve empiricism. for example, dan started feeling uneasy trusting a pattern based on checks of the first few cases. also, bob’s comment approximated the crucial experiment conception, since he seemed to raise a concern about the number and nature of cases one had to check before trusting a pattern. indeed, an important issue for many students at this stage of the lesson was how many cases would be enough for them to check before trusting a pattern. this issue had been anticipated in our planning, and we prepared a powerpoint slide with the following fictitious student comment, that the teacher used to orchestrate discussion: the circle and spots problem teaches me that checking 5 cases is not enough to trust a pattern in a problem. next time i work with a pattern problem, i’ll check more cases to be sure. (stylianides, 2010b, slide 1) the teacher invited reactions to this comment from the class. as examples of responses, dan suggested trying spread cases, such as for 1 spot, 75 spots and 100 spots, and larry said: ‘you should test it [the formula] as many times as you have time to do.’ the teacher asked larry: ‘so when you have tested it as many times as you have time to do, can you then trust it?’ larry revised his response: ‘no … not 100 per cent!’ then pauline said: ‘try it out with smaller numbers and bigger numbers.’ the teacher observed that pauline’s comment was similar to dan’s earlier one. indeed, the two comments were similar and illustrative of the crucial experiment conception, which predominated in the class at this stage of the intervention. activity 3: the monstrous counter-example illustration the teacher introduced the monstrous counter-example illustration4 (figure 3) on a powerpoint slide, which she presented in segments to give to the students a chance to process the information. once the students were comfortable with the meaning of the statement, the teacher showed the counter-example. the students were amazed and experienced a new cognitive conflict, for they had not anticipated that a pattern that held for so many cases (of the order of septillions) could ultimately fail. figure 3: the monstrous counter-example illustration. the teacher then directed the students’ attention to their previous discussion about the fictitious student comment: teacher: ‘we said in the circle and spots problem that, okay, it’s not enough to just check a few cases, you need to try different ones. well, this expression, what does this tell us?’ emily: ‘if you kept trying, you might have to go that high until you find one [a counter–example].’ teacher: ‘but i can imagine that it took the computer quite a long time to check all of those cases. and when do you stop checking?’ larry: ‘when you’ve found one!’ [several students laughed.] teacher: ‘and when do you trust a pattern then?’ adam: ‘when you cannot find one, until you are dead!’ (stylianides, 2010a, p. 15) the previous discussion illustrates that the students began to develop distrust in empirical arguments of any kind, including crucial experiment, and proposed checking the pattern indefinitely. yet whilst students began to realise the limitations of empirical arguments, they lacked knowledge of more secure methods of validation. this caused frustration in some, as illustrated in adam’s comment to the effect that one could die whilst checking cases before being in a position to trust a pattern. thus we may say that the students reached a point where they saw an intellectual need to learn about more secure validation methods, that is, they started to reason within the realm of the nonempirical conception, which was the intended endpoint of the intervention. it is beyond the scope of this article to give details about what happened next in the class. ethical considerations the data for this article were collected when i held an academic fellowship at the university of oxford. the data collection and broader study were scrutinised according to the university’s procedures for ethical approval, which met the british educational research association and british psychological society standards. school, teacher, and student participation in the study was voluntary and was solicited through consent forms to the school’s head teacher, the school’s head of mathematics, the teachers, their students, and the students’ parents or guardians. all student names mentioned in the description of the instructional intervention were pseudonyms. what knowledge was important for the teacher to have to successfully implement the instructional intervention in her classroom? i return now to the question i invited the readers to consider whilst reading the description of implementation of the instructional intervention. the elements of mathematical knowledge about proof and those of knowledge about students’ conceptions of proof discussed previously were necessary for successful implementation of the instructional intervention. specifically, if the teacher had difficulty in understanding the idea that empirical arguments are not proofs (cf. mathematical knowledge about proof), she would most likely have accepted students’ empirical arguments as proofs in the squares problem, and would have seen no purpose in proceeding with the rest of the instructional intervention. similarly, if the teacher did not understand the taxonomy of student conceptions of proof and how its different components compare in terms of mathematical sophistication (cf. knowledge about students’ conceptions of proof), she would have had difficulty evaluating her students’ thinking at different stages during the intervention. for example, without this knowledge she might not have recognised the crucial experiment approach to validation that surfaced in students’ work on the circle and spots problem as a significant progression in students’ learning, compared to the naïve empirical approach that dominated their earlier work on the squares problem. despite the importance of the teacher’s mathematical knowledge about proof and about students’ conceptions of proof, as explained earlier, these two kinds of knowledge are by themselves inadequate to capture the breadth of knowledge that she used (consciously or not) during implementation of the intervention. the teacher also drew on a strong pedagogical knowledge, some elements of which were generic and characteristic of her broader teaching practice: for example, her good questioning strategies and ability to create and maintain a positive classroom environment in which the students felt comfortable contributing and challenging ideas, making mistakes, and so on. in addition to these generic elements, however, the pedagogical knowledge which she exhibited included at least two key elements that were specific to the intervention. these two key elements, which presumably would be required by other teachers who might successfully implement the intervention in their classrooms, are discussed next. element 1: understanding how the three activities fitted together to form a coherent whole and to support students’ progression along the intended learning trajectory the three activities formed a coherent whole that supported students’ progression along the intended learning trajectory. research on prior iterations of the instructional intervention showed that possible changes in organisation of the activities in the intervention are unlikely to have such a powerful and positive effect on students’ learning (see stylianides & stylianides, 2009, pp. 331–333). for example, this research suggested that if the teacher omitted the circle and spots problem and went directly from the squares problem to the monstrous counter-example illustration, many students who at the time held the naïve empirical conception would treat the monstrous counter-example in the illustration as an exception, that is, as an extreme case outside of their experience. accordingly, the counter-example in the illustration would fail to become ‘pivotal’ for students (see zazkis & chernoff, 2008); that is, it would fail to generate a cognitive conflict for them, thereby having no substantial effect on their naïve empirical conceptions. more generally, a teacher would need to understand the over-arching idea which permeated the choice and sequencing of activities in the intervention, namely to confront students with a series of cognitive conflicts, each linked to a pivotal counter-example and intended to facilitate students’ progression to the next conception in the intended learning trajectory. the counter-example in the circle and spots problem corresponded to a case that was immediately outside of the convenient set of cases that (based on past experience) students with the naïve empirical conception would check in this problem (n ≤ 5). as a result, the discovery of the counter-example (for n = 6) had good potential to challenge (as it did) students’ naïve empirical conception, thereby facilitating their progression to the crucial experiment conception. at that stage students considered, as we saw in the previous section, that more strategically selected checks would help discover possible counter-examples to pattern problems. in other words, they felt that crucial experiment would offer a secure method of validation. students’ crucial experiment conception was then challenged by the counter-example in the monstrous counter-example illustration. this counter-example corresponded to a remarkably large case that one would not normally expect to discover even if one strategically selected which cases to check. the new counter-example was pivotal for the students and facilitated their further progression to the nonempirical conception. to conclude, a teacher would have to understand that possible changes in the organisation of activities in the intervention would probably disrupt the intended series of cognitive conflicts, thereby having a negative effect on the potential of the intervention to support the intended learning trajectory. the choice and sequencing of activities in the intervention were purposeful and based on an empirically tested theoretical framework for how instruction can generate cognitive conflicts for students in the area of proof by means of pivotal counter-examples (see stylianides & stylianides, 2009, pp. 319–323). element 2: understanding the nuances of implementing the activities – the notion of ‘conceptual awareness pillars’ understanding the organisation of the activities in the sequence is important, but is nevertheless inadequate to support successful implementation of the intervention by itself. successful implementation also requires that teachers understand the nuances of implementing the activities and, in particular, the following important premise that underpinned their design. students are more likely to experience a cognitive conflict in the area of proof when there is a sharp contrast between their existing conceptions of proof and a situation that contradicts these conceptions: the more aware students are of their existing conceptions, the more likely it is that they will experience a cognitive conflict when they encounter a counter-example intended to challenge these conceptions (stylianides & stylianides, 2009). specifically, in order to help the students become more aware of their conceptions of proof at different stages during the intervention, the teacher used (following the lesson plans) four ‘conceptual awareness pillars’, that is, ‘instructional activities that aim[ed] to direct students’ attention to their conceptions about a particular mathematical topic’ (stylianides & stylianides, 2009, p. 322), in this case proof. the teacher used conceptual awareness pillars both before and after the two counter-examples, which were intended to generate cognitive conflicts for students. the conceptual awareness pillars that came before the counter-examples aimed to prepare students for the upcoming cognitive conflicts, whilst those that followed them aimed to focus students’ attention on the validation issues raised by the counter-examples and on thinking about possible ways to resolve the cognitive conflicts. the first conceptual awareness pillar was at the end of the squares problem, when the teacher asked the students to respond to the prompt about whether and why they could be sure that applying the formula (n – 2)2 for n = 60 would indeed give the correct answer to part three of the squares problem. by each student writing down their reasons for their certainty about the correctness of the formula, the students became more aware of their validation methods at the time (predominantly naïve empirical). the second conceptual awareness pillar was again at the end of the squares problem, when the teacher reported back to the students the dominant (most common) validation method in the class (according to her analysis of their individual responses to the prompt in the first conceptual awareness pillar). with this report, the teacher helped the students become even more aware of the validation method that predominated in the class at the time, thereby preparing them to experience a cognitive conflict when they encountered the counter-example in the circle and spots problem. the third conceptual awareness pillar followed the counter-example in the circle and spots problem, and took the form of a class discussion around the fictitious student comment on how many cases would suffice to be checked before one could trust a pattern in mathematics. by orchestrating a discussion around this fictitious comment, the teacher helped direct students’ attention to the implications of the counter-example in the circle and spots problem for their conceptions about validating patterns in mathematics. as shown in the earlier description of implementation of the intervention, this discussion resulted in several students resolving the prior cognitive conflict by progressing to the crucial experiment conception. by articulating their ideas of what it meant for them to validate a pattern in mathematics, the students became more aware of their new conceptions of proof, thereby reaching a state of mind amenable to experiencing (as intended) a new cognitive conflict in the context of the monstrous counter-example illustration. the fourth (last) conceptual awareness pillar followed the presentation of the monstrous counter-example illustration, and took the form of a new class discussion around the same fictitious student comment as before. by orchestrating this discussion, the teacher directed her students’ attention to issues of validation as opposed to other possible issues they could consider that would not be directly relevant to the goals of the lessons (e.g. the issue of using computers in mathematics). in addition, the teacher used the discussion to focus her students’ thinking on the implications of the monstrous counter-example illustration for the veracity of the conceptions of proof they articulated in their earlier discussion of the fictitious student comment (predominantly in the form of crucial experiment). consideration of these implications and the ways in which they could be resolved resulted in the class progressing to the nonempirical conception of proof. to conclude, a teacher would have to understand the rationale for including all four conceptual awareness pillars in the intervention, so that she or he could implement them in the classroom without jeopardising their effect on students’ learning experience. research on prior iterations of the instructional intervention showed that possible omission of some or all of these conceptual awareness pillars is likely to weaken the potential the two counter-examples in the intervention have to be pivotal for students and to create the intended cognitive conflicts (stylianides & stylianides, 2009). conclusion (back to top) this article focused on the common and deeply rooted student misconception that empirical arguments are proofs, and aimed to contribute to development of a more comprehensive knowledge package for teaching proof. specifically, the importance of expanding teachers’ knowledge to include, in addition to knowledge about mathematics and students, fine-grained, domain-specific pedagogical knowledge that can allow them to help students overcome this misconception was highlighted. the implementation of a research-based instructional intervention found to be effective in helping secondary students begin to overcome this misconception was discussed, and two key elements of pedagogical knowledge for teaching proof that are important for teachers to have when they implement the intervention in their classrooms were identified. although the two elements of pedagogical knowledge identified are specific to the focal intervention, the notions of pivotal counter-examples and conceptual awareness pillars that underpin these elements are generic and could very well find application in other instructional interventions, not only in the area of proof but also in other areas of the school curriculum where students have persistent misconceptions. it is also important to recognise that these two elements of pedagogical knowledge are based (implicitly or explicitly) on certain premises about how students learn mathematics best, and how teaching can support that learning. for example, a premise that underpinned both elements (as well as the design of the instructional intervention as a whole) was that deeply rooted student misconceptions cannot be changed simply by ‘telling’ from the teacher, but rather by purposeful didactical engineering and careful design of activities that can drive students through a series of cognitive conflicts. thus, a different perspective on teaching and learning could lead to the design of a different instructional intervention, which in turn could require different elements of pedagogical knowledge from the teacher. yet, the limited progress that teaching practice and research have made thus far to address the pervasiveness amongst students of the misconception that empirical arguments are proofs elevates the merit of the specific instructional intervention, which has opened a window of optimism by showing how it might be possible to attack this stubborn problem in students’ mathematical education. students tend to have several other deeply rooted misconceptions in the area of proof, and this creates many challenges for educators in designing instructional interventions to successfully address them. indeed, it took us five research cycles of design, implementation, analysis and refinement over a four-year period (stylianides & stylianides, 2009) before we managed to theorise and develop the instructional intervention, a modified version of which was presented in this article. it would therefore be unrealistic to expect teachers to develop such interventions on their own, and also the knowledge about how they could successfully implement the interventions in their classrooms. how can the field of mathematics teacher education support teachers in promoting student learning of proof? one way would be to make ‘educative curriculum materials’ (davis & krajcik, 2005) available to teachers, which can incorporate existing research knowledge about instructional interventions for promoting student learning of proof, like that discussed here. by definition, educative curriculum materials would not only present the activities in the instructional interventions, but also help teachers develop the kinds of knowledge (about mathematics, students and pedagogy) that could allow them to implement effectively these activities in the classroom. however, despite the important role that educative curriculum materials can play in promoting teacher knowledge for teaching proof, there is still a long way to go in the development of such materials. for example, research has shown that a popular reform-oriented textbook series in the united states offered limited support to teachers about how to implement the proof tasks included in the series in their classrooms (stylianides, 2007c).5 finally, another way in which the field could support teachers to acquire important knowledge for teaching proof would be to integrate into teacher preparation programmes a coherent set of learning experiences for prospective teachers, that: (1) would address all elements of a comprehensive knowledge package for teaching proof; and (2) would also consider prospective teachers’ affective worlds, notably their beliefs about teaching proof. with regard to the former, we saw, for example, that various elements of the specific knowledge package discussed here are interconnected, and therefore that possible consideration of only some of these elements is likely to be insufficient to prepare teachers for successful implementation of the respective instructional intervention. with regard to the latter, it is important to help prospective teachers develop beliefs about proof that are consistent with desirable school-based instructional practices and objectives for teaching proof to students (e.g. national council of teachers of mathematics, 2000). incompatible teacher beliefs are likely to hinder development of these practices and promotion of these objectives (thompson, 1984). acknowledgements (back to top) the author would like to thank the uk’s economic and social research council for their support. competing interests the work reported in this article received support from the uk’s economic and social research council (grant number res-000-22-2536). the opinions expressed here are those of the author and do not necessarily reflect the position, policies or endorsement of the council. references (back to top) balacheff, n. 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(2010a). dataset produced by the esrc-funded project (res000222536a): enhancing students’ proof competencies in secondary mathematics classrooms (main dataset file). ukda-store. available from http://store.data-archive.ac.uk/store/ stylianides, a.j. (2010b). dataset produced by the esrc-funded project (res000222536a): enhancing students’ proof competencies in secondary mathematics classrooms (appendix 1c). ukda-store. available from http://store.data-archive.ac.uk/store/ stylianides, a.j., & al-murani, t. (2010). can a proof and a counterexample coexist? students’ conceptions about the relationship between proof and refutation. research in mathematics education, 12(1), 21−36. doi:10.1080/14794800903569774 stylianides, a.j., & ball, d.l. (2008). understanding and describing mathematical knowledge for teaching: knowledge about proof for engaging students in the activity of proving. journal of mathematics teacher education, 11, 307−332. doi:10.1007/s10857-008-9077-9 stylianides, g.j. (2007c). investigating the guidance offered to teachers in curriculum materials: the case of proof in mathematics. international journal of science and mathematics education, 6, 191−215. doi:10.1007/s10763-007-9074-y stylianides, g.j. (2008). an analytic framework of reasoning-and-proving. for the learning of mathematics, 28(1), 9−16. available from http://www.jstor.org/stable/40248592 stylianides, g.j., & stylianides, a.j. (2008). proof in school mathematics: insights from psychological research into students’ ability for deductive reasoning. mathematical thinking and learning, 10, 103−133. doi:10.1080/10986060701854425 stylianides, g.j., & stylianides, a.j. (2009). facilitating the transition from empirical arguments to proof. journal for research in mathematics education, 40, 314−352. thompson, a. (1984). the relationship of teachers‘ conceptions of mathematics and mathematics teaching to instructional practices. educational studies in mathematics, 15, 105−127. doi:10.1007/bf00305892 yackel, e., & cobb, p. (1996). sociomathematical norms, argumentation, and autonomy in mathematics. journal for research in mathematics education, 27, 458−477. doi:10.2307/749877 zack, v. (1997). ‘you have to prove us wrong’: proof at the elementary school level. in e. pehkonen (ed.), proceedings of the 21st conference of the international group for the psychology of mathematics education, vol. 4 (pp. 291–298). lahti, finland: university of helsinki. zazkis, r., & chernoff, e.j. (2008). what makes a counterexample exemplary? educational studies in mathematics, 68, 195–208. doi:10.1007/s10649-007-9110-4 footnotes (back to top) 1.ma (1999) used the term knowledge package to describe networks of relationships amongst different concepts that a teacher must understand in order to make proper decisions about which concepts are required for the learning of other concepts, which concepts can be learned simultaneously, etcetera. thus, ma used knowledge packages to describe concept maps or organising structures within teachers’ mathematical knowledge for teaching. my use of the term is broader than that: it can include, but is not limited to, concept maps that show the relationships between proof and other closely connected notions such as pattern, conjecture, and argument. 2.the description of the implementation is a shortened and slightly adapted version of the description that was previously published in a teachers’ journal (stylianides, 2009a), and is used here with permission from both editors. its use here is different and serves a new purpose, namely to explore issues about teachers’ knowledge for teaching proof. readers can also find the powerpoint slides and lesson plans that the secondary teacher used in implementing the intervention in stylianides (2009a). 3.the question in the circle and spots problem asked whether there is an easy way to tell for sure what the maximum number of non-overlapping regions into which the circle can be divided is for n = 15 (see figure 2). although the students’ inability to generate 32 regions for n = 6 did not guarantee that it was impossible to generate this number of regions with 6 spots, it did suggest that the emerging pattern offered an insecure way to find the maximum number of regions for n = 15. in this sense, the teacher’s explanation to the class that the maximum number of regions for n = 6 was 31 (which is true) should be interpreted as a confirmation of students’ emerging view that the pattern they identified for n ≤ 5 offered an untrustworthy means to answer the question for n = 15, rather than as an authoritative act to impose conviction of a certain truth in her class. after all, the class did not have the means to prove the complex formula that describes the correct pattern. 4.the name ‘monstrous counter-example’ was not mentioned in the class; this name is used in the article for ease of reference to the illustration. 5.this textbook series has recently undergone revision, and it is possible that the issues identified by stylianides (2007c), whose examination focused on the earlier version, have now been addressed. parker 2 pythagoras 63, june 2006, pp.2-17 official pedagogic identities from south african policy – some implications for mathematics teacher education practice diane parker university of kwazulu-natal parkerdc@ukzn.ac.za in south africa the national curriculum statement for grades 10 – 12 (general): mathematics (doe, 2003) together with the norms and standards for educators (doe, 2000a) are key policy documents that provide the official basis for mathematics education reform and for the construction of new pedagogic identities. in this paper i use a framework based on the work of bernstein (1996, 2000) to theorise the construction of pedagogic identities. i use this to build on graven’s (2002) description of the new official pedagogic identity of the south african mathematics teacher, and on adler et al. (2002) and others to raise questions related to teacher knowledge and the challenges of developing specialist mathematics teacher identities through initial teacher education programmes. introduction the past decade has been characterised by major transformations in south african society. there has been a concerted effort by the state to radically transform the apartheid educational terrain through new policies and practices. a major political project has been to radically transform the pedagogic identities of teachers working within the system and to produce new teachers who meet these transformation ideals. a major concern of education reform is to change “the bias and focus of official knowledge” and to construct new pedagogic identities in teachers and learners. the new pedagogic identity emerges as reflections of differing discursive bids “to construct in teachers and students a particular moral disposition, motivation and aspiration, embedded in particular performances and practices” (bernstein, 2000: 65). new policy statements overtly give details of the kind of teacher and learner envisaged by the new curriculum: … (t)eachers and other educators are key contributors to the transformation of education in south africa. the national curriculum statement grades 10-12 ... visualise teachers who are qualified, competent, dedicated and caring. they will be able to fulfil the various roles outlined in the norms and standards for educators. and the kind of learner ... is one who will be imbued with the values and act in the interests of a society based on respect for democracy, equality, human dignity and social justice as promoted in the constitution. … (l)earners emerging from the further education and training band must … have access to, and succeed in, lifelong education and training of good quality; demonstrate an ability to think logically and analytically, as well as holistically and laterally; and be able to transfer skills from familiar to unfamiliar situations. (doe, 2003: 5) these quotations, from the introduction to the national curriculum statement for grades 10 – 12 (general): mathematics (ncsm), the curriculum for the schooling sector of further education and training (fet)1 in south africa, give a symbolic picture of ‘ideal’ teachers and learners. they point to the vision of the kind of moral disposition, motivation and aspiration desired in teachers and learners by the south african state and more generally by south african society. the role of teachers as agents of transformation for a new democratic order is clearly articulated. the ncsm goes on to describe some of the particular performances and practices in which these should be embedded, and indicates both the nature of mathematical knowledge to be acquired and how it should be acquired and assessed. other policy, the norms and standards for educators (nse) (doe, 2000a), describes what it means to be a “competent professional educator” 1 south africa schooling is divided into ‘bands’. early childhood education (ece) – pre-school; general education and training (get) – grades 1 to 9; and further education and training (fet) – grades 10 to 12. diane parker 3 in south africa. it provides a vision of a professional teacher who is able to integrate a complex set of seven teacher roles with social, economic and moral responsibility. the roles include being: mediators of learning; interpreters and designers of learning programmes and materials; leaders, administrators and managers; scholars, researchers and lifelong learners; community members, citizens and pastors; assessors; and subject specialists. the nse describes in generic terms the “applied and integrated competences” that constitute the roles. these are: foundational competence (knowing that/what); practical competence (knowing how); reflexive competence (knowing why), integrated so that teachers know what to do, why it should be done, when to do it, and how to do it in the moment of practice. the criteria for recognition and evaluation of qualifications (criteria) (doe, 2000b), is a further policy, which compliments the nse. the nse has a largely symbolic function presenting a holistic picture of an ideal teacher towards which teacher education curricula should aim. the criteria plays a largely regulative function making it mandatory for higher education institutions involved in teacher education to design curricula in line with the nse. from the perspective of the department of education (doe), these norms, standards and criteria indicate to all providers (public and private) the kinds of teacher qualifications and learning programmes that the doe will consider for employment. and for the public providers, the kinds of programmes and qualifications the doe will consider for funding (parker, 2003). the nse together with the ncsm projects a symbolic image of what is expected of mathematics teachers in the new reformed system. this is an official image of a desired pedagogic identity, a policy image, rather than a constructed reality based in practice. the competent professional mathematics teacher in post-apartheid south africa is characterised through these images and is expected to be produced though curriculum reform in teacher education, as regulated through the criteria. teacher education is thus charged with a major challenge: to produce new teachers in this new image through newly designed pre-service and in-service teacher qualifications, and so, to institutionalise the ‘bias and focus’ of official knowledge. how do mathematics south african teacher education providers respond to this transforming context and to the challenges presented by these new policies? what positions do they take in response to the policy, and how do they design and organise their mathematics teacher education curricula so as to produce new specialist mathematics teachers for this new social and political context? these are some questions that frame the major research project from which the current paper emerges. in order to answer these and other related questions, is was necessary to break the wider research project into various phases. the first phase, reported in parker and adler (2005), focused on an investigation of the institutional and policy changes that occurred in relation to teacher education during the first ten years of the post-apartheid order, and theorised the pedagogic space produced for mathematics teacher education within this transforming context. the issue of specialising the consciousness of specialist mathematics teachers2 through such teacher education programmes led to the theorisation of pedagogic identities and knowledge discourses in the design of teacher education curricula aimed at producing teachers in and for this new context. an analysis of key policy documents including the nse, the criteria and the ncsm was necessary as a basis for analysing the official pedagogic identity projected from south african policy. in particular a detailed analysis of the ncsm (parker, forthcoming) provides a resource for producing a description of the mathematical identities projected from south african policy described in the current paper. the second phase of the project, reported in parker (2006), involved a comprehensive survey of all specialist mathematics teacher education programmes offered by higher education institutions in south africa, so as to investigate how various institutional providers of mathematics education responded to the changes. in particular what knowledge resources and practices they have selected, how they have organised these in their curricula and what pedagogic identities they have attempted to institutionalise through their initial mathematics teacher education programmes. a third phase of the project, still to be reported, focuses on case studies selected on the basis of the survey. this paper is located within the first phase of the wider project. it outlines the context of teacher education reform in south africa, briefly theorises the notion of ‘pedagogic identity’ and provides an analysis of the official pedagogic identity of 2 when i refer to specialist mathematics teachers in this paper i am referring to teachers for grades 10 -12 of fet. teachers for the various phases of the get are more often identified as generalists, and would possibly require a different type of mathematical education from that which is suggested here. official pedagogic identities from south african policy – some implications for mathematics teacher education practice 4 specialist fet mathematics teachers in south africa, projected by policy. the official pedagogic identity is discussed in the light of debates in the field around knowledge for specialist mathematics teaching and teacher learning. tensions between different demands produce challenges for mathematics teacher educators in relation to the way in which they could construct their curricula. how they select and privilege knowledge and practices for teacher learning will have consequences for the construction of a specialised identity of ‘mathematics teacher’ in and for south africa within this new context. the context of teacher education in south africa teacher education has undergone rapid transformation that has included a delocation and relocation of pedagogic practices from colleges of education regulated and controlled by the state, to relatively autonomous universities and technikons located in the higher education sector. this movement has created a space for mathematics teacher educators/researchers and mathematicians to play a major reform role by designing new curricula (criteria) for the development of new mathematics teacher identities (parker and adler, 2005). in the terms of the nse the ‘specialist role’ is marked out as the “the overarching role into which the other roles are integrated, and in which competence is ultimately assessed” (doe, 2000a: 12). in terms of initial qualifications for fet mathematics teachers, there is no prescription of what ought to be taught, how it ought to be taught, or what “the disciplinary basis of content knowledge, methodology and relevant pedagogic theory” (ibid.: 28) is in substantive terms. it is left up to the teacher educational professionals to produce the criteria for the specialisation. the policy sees fet teaching as a specialist domain and specifies the possibility of providing single subject (discipline-based) qualifications. this produces the possibility of focused qualifications designed to integrate highly specialised knowledge for developing mathematics teachers. there are two ways to qualify as a fet mathematics teacher in south africa: a three-year general formative degree with at least two years study in mathematics, followed by a professional certificate in education (pgce), or a new undergraduate bachelor of education (b.ed) which integrates the academic and professional components into a four-year degree. i am interested in the possibilities inherent within the field for the development of initial mathematics teacher identities through a specialist b.ed programme, particularly in the potential for different forms of specialised curricula to produce different forms of “specialist consciousness” (bernstein, 1996, 2000) in mathematics teachers. in south africa, there are multiple dimensions to this teacher education task. as adler (2004: 6) points out, we work in a “socio-cultural and political context deeply scarred by apartheid education”. in the field of mathematics the unequal distribution of knowledge and ‘ability’ is starker than in most areas of the school curriculum, and is a product of unequal opportunities under apartheid. the national strategy for mathematics and science (doe, 2001: 12) highlights the dismal performance of black african candidates in mathematics. in the interests of transformation it is necessary to create access routes into mathematics teaching for students who would not normally ‘make the grade’ for entry into university mathematics courses. this is a major challenge for teacher educators: it is not only necessary to develop an identity as ‘mathematics teacher’, it will also be important to develop an identity as ‘able mathematics learner’. theorising pedagogic identities: official and local theoretically, pedagogic identities are ‘forms of consciousness’, and any particular educational reform represents an approach to regulating and managing moral, cultural and economic change, which are expected to become the lived experiences of teachers and students, through the shaping of consciousness (bernstein, 2000). for bernstein, the power (classification) and control (framing)3 relations of any pedagogic practice regulate the acquisition of pedagogic identity. the selections of knowledge(s), performances and practices and their evaluation rules (criteria for recognition and realisation)4 relay a particular social order and way (mode) of knowing and being, whether explicitly or tacitly. the acquisition of the specialised consciousness produces particular orientations to meaning – ways of recognising and realising what is constituted as the ‘legitimate text’. this comes “to have the force 3 classification and framing are key concepts for bernstein (1990, 1996, 2000). classification “provides us with our voice and the means of its recognition” and framing is “the means of acquiring the legitimate message”. classification is a product of power and framing of control. 4 according to bernstein (1990: 15), “( r)ecognition rules create the means of distinguishing between and so recognising the speciality that constitutes a context, and realisation rules … regulate the creation and production of specialised relationships internal to that context.” diane parker 5 of the natural order and the identities that it constructs are taken as real, as authentic, as integral, as the source of integrity” (bernstein, 1996: 21). educational reforms require changes in the recognition and realisation rules of the pedagogic practice and therefore can be seen as “the outcome of the struggle to produce and institutionalise particular identities” (bernstein, 2000: 66). for bernstein (2000) local identities are social identities, constructed through social location. these vary with age, gender, social class, occupational field and economic and symbolic control. they are not necessarily stable positions and shifts can be expected depending on maintaining the discursive/economic base of the identity. this fits with castells’s (1997) concept of identity as a source of individual meaning and experience that should be distinguished from social ‘roles’. roles are defined by norms structured by institutions and organisations of society, whereas identities are sources of meaning for the actor, constructed through a process of individualisation. identities organise meaning and roles organise functions. meaning is the symbolic identification by social actors of the purpose of their actions. this helps point to the difference between an official pedagogic identity and a local pedagogic identity of a teacher. the official pedagogic identity is constructed through descriptions of what ‘ought to be’ based on particular projections by institutions of the roles, knowledge codes and social modes individuals ought to take up (official knowledge). local pedagogic identity is constructed sociologically in local educational and historical contexts. thus while official teacher identities can be designed on the basis of ‘teacher roles’, local teacher identities cannot – teacher identities emerge, enabled or constrained, within the pedagogic context (graven, 2002). in this understanding local pedagogic identities are not individual (cognitive) attributes, neither are they simply constructed politically or as a result of a curriculum prescription, they are constructed through an interplay of the ‘voice-message’ system (bernstein, 1996), an interplay between official and local knowledge and practices within an educational community. thus the ‘legitimate’ text (e.g. what is accepted as ‘good mathematics teaching practice’) is constructed through a relay between specialists in the field of teacher education, novice teachers, and experienced teachers within the social contexts of educational practice. teacher identity is therefore embedded in the social practices of an education community within ‘a particular social order’ and develops in this context through relationships “of reciprocal recognition, support, mutual legitimisation and finally through negotiated collective purpose” (bernstein, 1996: 73). according to bernstein (1996, 2000) individual pedagogic identities are constructed both inwardly and outwardly. the introjected identity faces inwardly and is most often related to the acquisition of stable inner loyalties related to esoteric forms of thinking and doing, for example, working in principled ways with disciplinary knowledge, or developing a therapeutic identity related to notions of child development and internal, or sacred, religious and cultural values. the projected identity faces outwardly and is most often related to external demands from the state and the market for producing particular kinds of citizens, and for regulating and controlling them. the challenge for teacher educators is to design programmes that enable the construction of introjected identities leading to ‘good mathematics’ and ‘good mathematics teaching’. this needs to happen within the economic constraints and competitive environment of the higher education sector, and should be balanced with projected identities that meet some of the transformational ideals of the state: particularly the need to provide access to powerful mathematics to a wider range of south african students. what is considered ‘good mathematics’ and ‘good mathematics teaching’ practice within these contexts becomes a major issue: who defines what this means, on what basis is that decision made, and how is access to the criteria (recognition and realisation rules) for these new notions of mathematics and mathematics teaching made possible? any notion of ‘good practice’ that a particular institution attempts to institute will have an ideological basis, and the particular selections of knowledge contents and practices together with how these are made available to students, can be analysed to identify it. whether this is an ideology that is based on and driven by political and social concerns, academic and intellectual concerns, or practical and professional concerns, or some combination of these, is of interest and will have consequences for the kind of specialisation of consciousness that may be made possible within the educational context. in a context of the poverty of mathematics education alluded to earlier, this becomes a crucial concern. improving access to meaningful relationships with powerful forms of mathematics within the schooling system will to a large extent be dependent on producing teachers official pedagogic identities from south african policy – some implications for mathematics teacher education practice 6 who have acquired this identity, as interested and able mathematics learners themselves. the experiences student teachers have, in the mathematics classroom/lecture theatre, the teacher education lecture theatre and out in the school in practice will influence their specialisation of consciousness. whether their understanding of the nature of mathematics, their relationship with the subject matter, and what they consider and construct as ‘good’ mathematics teaching practices, is substantially changed from prior, and probably internalised, notions forged during their 12 years of schooling and determined by the apartheid educational order or not, becomes a central question underpinning the research project, and will become a focus of the case studies for the third phase of the project mentioned earlier in this paper. in order to investigate how teacher educators in the various institutions have responded to policy and what ideology lies behind the image of ‘good practice’ they project from their institutions (as embedded within the organisation of their curricula), it was necessary to first analyse the official identities projected from the mathematics curriculum policy. this is the focus of the next section of the paper. the official pedagogic identity of specialist mathematics teachers projected from sa policy policy documents can be analysed to identify the particular ‘bias and focus’ of official knowledge and to examine the official pedagogic identities they project, and therefore to unpack what it might mean to produce the kind of teacher expected. this could be critically reflected on in terms of research in the mathematics education field to produce a local resource for the construction of curricula for specialist mathematics teachers. a clear picture of the projected official pedagogic identity requires a detailed document analysis. i have insufficient space here to provide details of this document analysis,5 and thus simply sketch some of the characteristics of the policy image based on my analysis of the ncsm (parker, forthcoming). the analysis required working through all four chapters of the ncsm, sentence by sentence, categorising these using a framework based on bernstein’s concepts discussed earlier, and building on work done by graven (2002). graven’s (2002) analysis of the official pedagogic identity projected from the south african policy base, focuses on senior phase general education teachers (grades 7–9), and 5 see parker (forthcoming) for a discussion of the detailed analysis of the ncsm. effectively illuminates some of the main differences in the ‘outgoing’ roles of teachers and their future ‘incoming’ roles as designed within the new education system. she shows that there is a movement in thinking about teaching and learning within south african education from a performance-based to a competence-based pedagogy,6 and from a collection to an integrated knowledge code.7 she uses this together with an analysis of specific curriculum statements for the grade 7 to 9 mathematics ‘learning area’ to identify four different orientations to mathematics, and from this four mathematical roles teachers are expected to fulfil, each with its own mathematical demands. these orientations to mathematics are summarised as: mathematics for critical democratic citizenship; mathematics as relevant and applicable to aspects of everyday life and local contexts; mathematics for its beauty and intrinsic value; mathematics as a way of communicating in, thinking about and viewing the world; and, mathematics as conventions and skills to master in order to gain access to further studies. my analysis of the new ncsm (parker, forthcoming) shows that while there are some differences much of graven’s (2002) analysis still holds for the ncsm. the logic of competence (bernstein, 1996) is clearly visible, particularly in the first chapter of the statement. a shift in approach to mathematics teaching is visible – a socio-constructivist, learner-centred, discussionbased approach is advocated. this is clearly articulated through the reference to the roles of a teacher described in the nse and the kind of teacher and learner advocated by the curriculum (as mentioned in the introduction to this paper). these role descriptions demand significant changes from teachers in relation to their orientation to knowledge and learning, and in their conception of what it means to teach. in particular, 6 graven draws on bernstein’s (1996) distinction between two pedagogic models underpinning a curriculum: competence based and performance based. in general competence models are directed at what the learner knows and can do at the end of the learning process, whereas performance models focus on specific learning contents and texts. see bernstein (1996: 58-63) for a useful comparison in relation to: time, space and discourse; orientation to evaluation; pedagogic control; pedagogic text; pedagogic autonomy; pedagogic economy. 7 see bernstein, 1977, “on the curriculum”. according to bernstein there are two broad types of curriculum: collection and integrated, although these can be thought of as a continuum rather than a straight dichotomy. in a collection type the contents stand in a closed relation to each other (bounded and insulated from one another) – here the learner has to ‘collect’ a group of favoured contents in order to satisfy some criteria of evaluation and classification of knowledge contents is strong. in an integrated type the contents stand in an open relation to one another (blurred boundaries and hybrid) – here the learner follows a course structured around some overarching ‘big idea’, and classification is weakened. diane parker 7 the teacher is projected as a learning mediator: she no longer teaches given content knowledge, she facilitates learning. she is responsible for interpreting and designing learning programmes to meet the needs of her learners – the teacher is expected to interpret the broad outcome descriptions and assessment standards in the new curriculum statements and select contents and learning activities to provide learners with appropriate experiences to achieve the outcomes. the new roles thus place high demands on teachers. teachers do not teach: they mediate learning through the skilful development and use of learning materials. the control of the pedagogic space is displaced from the teacher towards the text (activity/learning material) and the learner is required to take responsibility for his/her own learning (individually and in groups). this represents a move from directly teaching given texts towards the management of knowledge, learning and learning spaces. thus there is a shift in the locus of classroom control and a visible flattening of hierarchical relations in the classroom. in other words, a movement towards what bernstein (1996) describes as invisible pedagogy which he associates with a competence based curriculum. this is in contrast to the markedly different practices still existing in schools under the old curriculum, where teachers follow a content laden syllabus prescribed by the department of education and the curriculum is strongly externally controlled (framed) through a high stakes matriculation examination which focuses on an orientation to received knowledge.8 the locus of control is with the teacher and the classroom relations are more hierarchical and authoritarian – in bernstein’s (1996) terms, a visible pedagogy is in place which can be associated with a performance based curriculum. the ncsm document indicates a commitment to integration in general terms as one of the underlying principles of the curriculum: integration is achieved within and across subjects and fields of learning. the integration of knowledge and skills across subjects and terrains is crucial for achieving applied competence [… and …] seeks to promote an integrated learning of theory, practice and reflection. (doe, 2003: 3) 8 see boaler (2002) for a useful discussion on connected and received knowledge and the relationships with mathematics that are implied by each. however, a close look at the assessment standards and contents of the ncsm shows that the real emphasis on integration is within mathematics rather than across fields of learning. for example, the idea of ‘function’ is a key integrating principle. this marks out a significant change in the organisation of the contents of the new fet school mathematics curriculum from the get curriculum discussed by graven (2002) or the existing (outgoing) secondary school mathematics curriculum. mathematics remains fairly strongly classified in relation to contents outside of the field of mathematical sciences, but there is a weakening of classification values within the field itself. instead of ‘topics’, such as algebra, trigonometry, geometry and calculus, that were well insulated from one another in the old curriculum and organised vertically, the contents of the ncsm are organised in terms of four learning outcomes – number and number relationships; functions and algebra; space, shape and measurement; and data handling and probability – and are connected horizontally through mathematical processes such as “making conjectures, proving assertions and modelling situations” (ibid.: 10). conceptual progression in the disciplines underpinning the subject mathematics as defined in the ncsm is emphasised and the more overtly political and controversial radical integration aspects of the original get version discussed by graven (2002) are de-emphasised. there is a focus on application but in contexts where it is appropriate to the core disciplines that form the ‘subject’. this curriculum broadens the focus of school mathematics learning from entry into a single discipline (pure mathematical topics) into a region (the mathematical sciences: mathematics, applied mathematics and mathematical statistics). there is a focus on access to the discourse of abstract mathematical knowledge, its structure and processes for entry into further studies in the mathematical sciences. each of the components of the mathematical sciences is relatively strongly insulated within the ncms, i.e. there is a principle of internal classification which enables clear distinctions to be made, for example, between statistics and mathematics, and between mathematics and applied mathematics. statistics is most strongly insulated appearing in the document under a single outcome: data handling and probability, which is an entirely new area in the fet curriculum. other previously insulated topics in mathematics are spread across the other three learning outcomes and integrated horizontally in official pedagogic identities from south african policy – some implications for mathematics teacher education practice 8 terms of mathematical structures, conventions and processes. thus in the ncsm there are significant shifts in the specialised contents and processes to be taught and in the underlying philosophy of the mathematics projected. mathematics is seen as a fallibilistic discipline (ernest, 1991), and mathematics learning is seen as relational and meaningful in its own right, and useful and meaningful to life. the ncsm provides a definition of mathematics that projects an image of mathematics as practice, a “human activity practised by all cultures” that enables creative and logical reasoning. it sees mathematical knowledge as constructed by “observing patterns, with rigorous logical thinking, […] lead(ing) to theories of abstract relations” (doe, 2003: 9). it is thus a systematic way of seeing the world and thinking about the world using structured abstract principles. further it is “developed and contested over time through both language and symbols and by social interaction and is thus open to change” (ibid.: 9). mathematical problem solving is seen as a key element which “enables us to understand the world and make use of that understanding in our daily lives” (ibid.: 9). the idea of empowerment as a purpose of mathematics learning is visible: access to mathematical knowledge empowers learners “to make sense of society” by enabling learners to “respond responsibly and sensibly to personal and broader societal concerns” and to engage “responsibly with quantitative arguments relating to local, national and global issues” (ibid.: 10). this is a broad conception in which mathematics is characterised as a “discipline in its own right and pursues the establishment of knowledge without necessarily requiring applications in real life” (ibid.: 9). at the same time, it is also specifically emphasised that mathematics is more than a cannon of specialised knowledge contents, “competence in mathematical process skills such as investigating, generalising, and proving is more important than the acquisition of content for its own sake” (ibid.: 9). while there is a focus on application of mathematics, the idea of unproblematic transferability of everyday knowledge into mathematics so prominent in the first version of the get curriculum (graven, 2002), is absent – the focus is on the “establishment of proper connections between mathematics as a discipline and the application of mathematics in the real world” (doe 2003: 10, emphasis added). mathematical modelling is seen as the means to analysing and describing the world mathematically. other proper connections are in relation to the use of mathematical tools for problem solving in other subject areas, such as physical, social and management sciences. thus there is a focus on mathematics as a discipline, a practice and a tool – it is a specialised knowledge form with its own unique conventions, symbolism and structure; it is a specialised practice involving specialist processes of thinking, reasoning, proving; and it is a powerful tool for problem solving in a variety of contexts including mathematical (for example, abstract problem solving) and nonmathematical (for example, as applied in issues of public health, finance, or other subject areas such as the physical sciences). furthermore, mathematics has a history – it is viewed as socially constructed and therefore as a fallible discipline. in terms of the pedagogic discourse to be realised at the classroom level the ncsm implies new relationships between teachers and learners and between these actors and the subject matter to be taught – changes in both the instructional and the regulative discourse (the what and how) – both in general terms and in very specific terms in relation to what is seen as legitimate mathematical knowledge (concepts) and ways of knowing it (habits of mind and the regulatory order for its learning). whereas the earlier curriculum was very much product oriented working on the basis of ‘received’ knowledge (as discussed by boaler, 2002) – a hierarchy of concepts, facts and skills expressed as definitions, products and methods to be learnt and practiced – this curriculum is not. it is more practice oriented and focused on producing “connected knowledge”. it focuses on the practices of mathematics (e.g. investigating, making conjectures, justifying, generalising, etc.) as well as the skills (e.g. factorising) and the products (e.g. ‘laws of exponents’); and on making meaning though problem solving contexts. the implication of this curriculum is that teachers’ mathematical identities should be constructed as “connected”, they should have “productive dispositions” (kilpatrick, et al., 2001) towards mathematics and be able to engage in a “dance of agency” (pickering as used by boaler, 2002). this does not seem to be a reform curriculum that is based on ‘generic’ knowledge and a ‘watering down’ of mathematics, rather it seems it is a curriculum that is very concerned with mathematics and mathematical ways of being and seeing – but these are not images that are necessarily common in the south african context. the new fet mathematics teacher needs to be diane parker 9 competent in these extended curriculum areas – she needs to develop a number of specialised pedagogic identities, each related to a specialist knowledge discourse: an identity as mathematician; applied mathematician, statistician and mathematical historian. access to the grammar of mathematics, applied mathematics and statistics as distinct knowledge discourses, knowledge about their historical development and ways of coming into being, and the ability to apply these meaningfully to problem solving situations, are a key. these mathematical identities are related to the novice teacher’s access to practice in the field of mathematical sciences in and for itself (and not necessarily for the purpose of teaching). they have to do with the novice teacher’s growth as an ‘able mathematics learner’ and thus her development of subject loyalty in relation to the disciplines themselves. it is this loyalty that may be a key to her interest in, involvement in and passion for the mathematical sciences that could, given the appropriate opportunities, become the basis for the development of a different set of identities related to mathematics teaching. initial teacher education, through the four-year degree programme, is thus faced with a complex task – a need to provide curricula to create paths for the acquisition of mathematical science discourses for teachers who in their own schooling have probably experienced an impoverished mathematical education. however, the development of these consciousnesses is insufficient for a south african teacher hoping to institute the new curriculum, in particular the requirement that they are able to carry out the teacher roles mentioned at the beginning of the paper. thus, teachers also need to develop practices for teaching these mathematical discourses as distinct from learning them. that is, in addition to acquiring the criteria (recognition and realisation rules) for these specialised forms of consciousness in the mathematical sciences (in and for themselves), the new teacher needs to develop a specialised pedagogy in relation to each “for the complex task of transforming this knowledge into appropriate opportunities for learning in school” (adler et al., 2002: 151). and this is related to the mathematical work of teaching in practice and the development of mathematical knowledge for teaching (ball et al., 2004; adler and davis, 2006), a knowledge discourse and its practice, that is different from, and possibly works in an opposite direction to, the discourses and practices of the mathematical sciences.9 the changes in the mathematics curriculum represent major shifts for most prospective mathematics teachers whose mathematical identities were constructed under an ‘old’ (outgoing but still existing) education system (graven 2002). teachers are required to implement these new ideals in their classroom practice. this means that they are required to develop new images of ‘good practice’ for mathematics teaching (recognition rules), and new pedagogic identities (forms of consciousness) that enable them to carry out these practices (realisation rules). teacher educators will need to construct curricula for producing these outcomes. in terms of the theoretical ideas introduced earlier, while the curriculum statements can project images of ideal mathematics teachers, these intended identities will not necessarily be acquired. what happens in practice will depend on what occurs in real educational contexts and how the student teachers respond to these. the design of teacher education curricula can only work at the level of officially projected identities. however, these can influence the emergence of new teacher identities through the relations they set up with the particular knowledge discourses and practices they make available. what resources are used as a basis for the specialisation of the consciousness and how these are made available to the student teachers will be a crucial issue. acquisition of the recognition and realisation rules for a specific practice (say learning mathematics or teaching mathematics) will depend on the evaluation rules of the pedagogic discourse – the criteria of what is seen to be the ‘legitimate text’. so a different specialised consciousness could be acquired depending on the selection and organisation of knowledge contents and how they are made available to teachers: i.e. what is recognised as legitimate knowledge and practice, and the pedagogic modes of its transmission. in terms of the various paths to becoming a teacher in south africa mentioned earlier, it is in the new four-year b.ed programme that such a (re)education in the mathematical sciences and in mathematics teaching becomes a possibility – that is, teachers coming to know and work within the 9 see ball and bass (2000) for a discussion on the idea that mathematicians work at compressing knowledge, while mathematics teachers need to decompress it; ball, bass and hill (2004) for a discussion on the need for teachers to learn to ‘unpack’ familiar mathematical ideas; and, adler and davis’s (2006) extension of this idea in their understanding that teachers are required to unpack mathematical knowledge for the purposes of teaching. official pedagogic identities from south african policy – some implications for mathematics teacher education practice 10 mathematical sciences in and for themselves, and, teachers working with transformed school mathematical knowledge within a classroom and knowing and practising mathematics for teaching. gaining access to these forms of knowledge provides a possibility for breaking the cycle of poverty in mathematics education that is a feature of the south african educational context. key areas of curriculum contestation in relation to these teacher education tasks are linked to questions of what knowledge should be selected?; how should it be organised in the teacher education curriculum?; and, who should be involved in teaching this selection to teachers. for example, should teachers be taught mathematics relevant to the school curriculum by mathematics educators modelled in a way that they ought to teach it? on the other hand, should they be taught mathematical sciences by academics within the disciplinary departments of the university at a level above school mathematics and possibly divorced from school mathematics? or would some combination of these be best? in terms of teachers learning to select and transform mathematical knowledge for teaching, similar questions can be asked about mathematics teacher education academics and experienced mathematics teachers. i will not enter into this debate here, but rather signal it as a consideration for further research. specialising the consciousness of a mathematics teacher: resources, discourses and criteria for recognition and realisation in my wider research project mentioned earlier in the paper, an empirical focus is on identifying the knowledge resources and discourses that teacher educators do select for their specialist mathematics teacher education programmes and the way these are organised, co-ordinated and made available to new teachers within their educational contexts. the major focus is on the production of the criteria for the recognition and realisation of ‘good mathematics’ and ‘good mathematics teaching’ practices within the various teacher education programmes across the field in south africa. in the context of designing initial four-year teacher education programmes the preceding discussion becomes important. the development of the teacher as an ‘able mathematics learner’, learning the mathematical sciences in and for themselves and thus developing disciplinary identities, must be part of the initial education programme, particularly in the light of the generally low level of personal mathematical competences developed in our prospective teachers through their prior schooling experiences (parker, 2004), and the high demands of the new curriculum (parker, forthcoming). in the wider research field of mathematics teacher education, learning mathematics (becoming an able mathematics learner and developing loyalty to the subject) is often conflated with learning to teach mathematics (becoming a teacher of mathematics and developing knowledge of teaching and learning mathematics) and practising as a mathematics teacher (becoming a mathematics teacher and using mathematical knowledge for teaching in practice). for example, see ball and bass’s (2000) criticism of the ‘fragmented curriculum’ of teacher education programmes in terms of the difference between working as a mathematician (compressing knowledge), which they seem to want teacher education programmes to discard, and working as a teacher (decompressing knowledge) which they want to privilege. another example is ensor’s (2000) work which is concerned with teachers’ mathematics education (and teacher education) practices and not teachers learning mathematics in and for itself. in both types of teacher education mentioned, prior mathematical competence is taken as given. to reiterate, much of the reported work relates to in-service mathematics teacher education, or initial teaching where the teacher has previously developed an identity as ‘able learner of mathematics’ and a certain level of subject loyalty and hence an identity as ‘mathematician’. in the light of this i suggest that teacher educators in south africa should be careful not to conflate the process of developing the specialist fet mathematics teacher as a learner of mathematics with her development as a learner of mathematics for teaching. in this paper i do not have space to elaborate on any findings from my wider research project; however i do propose some tentative conclusions for a model based on my initial analysis of research in the field. i suggest that practising mathematics teaching (learning a professional practice) and practising mathematics (learning mathematics) are two distinctly different types of activity related to distinct knowledge discourses (bernstein, 2000). i propose that initial mathematics teachers require both, particularly in times of reform where new mathematical learning identities and teaching identities need to be formed. although these are connected discourses, i would suggest they should not be learnt at the same time and in the same space, since they work diane parker 11 in opposite directions (as ball and bass, 2000 so clearly show with their discussion on compressing and decompressing mathematical knowledge). i also identify a third distinct discourse, created in the growing research domain of mathematics education, which focuses on developing knowledge about teaching and learning mathematics (learning mathematics education). thus there are at least three different mathematically related pedagogic identities that a novice specialist mathematics teacher should develop through any teacher education programme. an identity as a student of mathematical sciences (becoming an able mathematics, applied mathematics and mathematical statistics learner, thinker and actor); an identity as a student of mathematics education (becoming someone interested in learning from research in the field of mathematics teaching and learning); and an identity as a mathematics teacher (becoming someone who can utilise their knowledge to help learners develop productive mathematical identities and be motivated to learn the discipline at higher levels). each of these identities is a product of access to different knowledge discourses, and in each case recognition and realisation rules for what comes to be seen as the ‘legitimate’ discourse and its practices need to be developed. knowledge resources and practices need to be selected and organised in the curriculum for these purposes. a key debate and issue of contention in the empirical field is centred on the extent to which these should be integrated or not in teacher education practice, and related to this who should take responsibility for developing them in teachers (mathematicians/mathematics education specialists/teachers). the discussion above, leads me to suggest that there are at least three specialist (mathematically related) knowledge discourses that initial teachers need to acquire – each with its own ways of thinking and doing, and different organisational structures (vertical and horizontal) and grammar (strong/weak) (bernstein, 2000). i suggest that these should be co-ordinated in the teacher education curriculum to bring a ‘notion of best teaching and learning practice into practice’ (an adaptation of ensor’s (2000) language). each discourse requires a different kind of specialisation, probably best developed at different times and in different spaces, and finally coordinated in the practices of the classroom alongside a competent teacher. in this way, distinctions can be made, boundaries between the different discourses can be set up and transgressed and they can be used as knowledge resources to be recruited in practice. i suggest that the curriculum designed for the construction of each of these identities should be based on knowledge produced in the growing domain of mathematics education research, and not simply on the basis of interpretations of what is ‘good’ from policy or local teaching experiences and resources. i do not have space here to elaborate on the possible modalities for the acquisition of each of these identities, to discuss the different types of discourses, nor to theorise what type of specialised consciousness different modalities might produce. that is part of my wider research project, and is left for later dissemination. however, what is clear to me is that each one requires specialised mathematical work and not generic practices, and each one needs to be designed, with careful consideration given to the criteria for the selection of the privileged reservoir for recognising the practice and repertoire for realising the practice (ensor, 2000). conclusion what does it mean to know mathematics, to teach mathematics and to develop mathematical and other forms of knowledge and practice for teaching? this is a key question for mathematics teacher educators to ask and extremely difficult to answer in any straightforward manner. however, the answers we give to this question will be crucial for designing curricula for our student teachers to acquire the criteria for the realisations of the specialisation – effective specialist fet mathematics teacher in the context of curriculum reform, teacher educators, education academics and academics in the mathematical sciences, who have an interest in producing specialist mathematics teachers for grades 10 – 12, have a responsibility to contest for space and time in the four-year curriculum – to argue for the specialised focus, to compete for resources to project their particular ‘bias and focus’ into the official pedagogic identity projected from their institutions. a responsibility to research and produce criteria for novice teachers to navigate the acquisition of the recognition and realisation rules for specialist mathematical pedagogic identities. this requires the development of criteria for what constitutes ‘best practice’ in mathematics and mathematics teaching: a clear notion of what kind of knowledge(s) and practice(s) mathematics teachers should acquire to be in a position to put this ‘best practice into practice’, and how these should be acquired and co-ordinated in the teacher official pedagogic identities from south african policy – some implications for mathematics teacher education practice 12 education programme, and who should be involved in their transmission. the modalities of practice and knowledge discourses selected and co-ordinated in the four year degree curriculum do matter, and may have profound effects on the construction of new specialist mathematics teacher identities for and in south africa, and hence on the possibility of breaking the cycle of poverty in mathematics education more broadly. references adler, j. (2004). research and maths teacher education in ten years of saarmste: trends and challenges. in a. buffler & r. laugksch. (eds.), proceedings of the 12th annual conference of the southern african association for research in mathematics, science and technology education (pp 6-15). durban: saarmste. adler, j., and davis, z. (2006). opening another black box: researching mathematics for teaching in mathematics education. journal for research in mathematics education, 37 (4), 270-296. adler, j., slonimsky, l., & reed, y. (2002). subject-focused inset and teachers’ conceptual development. in j. adler, & y. reed (eds.), challenges of teacher development: an investigation of take-up in south africa (pp 135-152). pretoria: van schaik publishers. bernstein, b. (1977). class codes and control: volume 3 towards a theory of educational transmission. chapter 4: on the curriculum. (pp 79-84). london: routledge & kegan paul. bernstein, b. 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(2005) constraint or catalyst? the regulation of teacher education in south africa. journal of education, 36, 5978. acknowledgements this paper is based on work supported by the national research foundation (sa) under the thuthuka (researchers in training) programme, grant number 2052952 – mathematics teacher education. any opinion, findings and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the national research foundation. a version of the paper is to be published in g. anthony & b. grevholm (eds.). (forthcoming) research on teacher education, recruitment, competencies and teachers' identity. (icme-10 thematic afternoon conference proceedings.) with equal passion i have sought knowledge. i have wished to understand the hearts of men. i have wished to know why the stars shine. and i have tried to apprehend the pythagorean power by which number holds sway about the flux. a little of this, but not much, i have achieved. bertrand russell ( autobiography) microsoft word 64 front cover final.doc 2 pythagoras 64, december, 2006, pp. 2-5 mathematical literacy: a new literacy or a new mathematics? renuka vithal and alan j. bishop university of kwazulu-natal, south africa, and monash university, australia vithalr@ukzn.ac.za and alan.bishop@education.monash.edu.au why a special issue? mathematical literacy is a ‘hot’ topic at present in most countries, whether it is referred to by that name, or in some cases as numeracy, or quantitative literacy, or matheracy, or as some part of ethnomathematics, or related to mathematics in society. questions continue to be asked about what is meant by mathematics in any concept of mathematical literacy and the use of the very word ‘literacy’ in its association with mathematics has been challenged. its importance, however, lies in changing our perspective on mathematics teaching, away from the elitism so often associated with much mathematics education, and towards a more equitable, accessible and genuinely educational ideal. as south africa begins to implement a new mathematical literacy curriculum this year for all learners from grades 10-12 who do not take the subject ‘mathematics’, there is much discussion about what exactly is mathematical literacy and issues about how it can be realised have emerged. the editors of this special issue sought interesting and provocative papers on different aspects of mathematical literacy, in order to create an issue which will reflect some current thinking and practices, as well as promising innovative approaches. at this time there is a great need to share the variety of experiences, concerns and aspirations related to this topic, and we welcomed the opportunity to demonstrate that variety, and to contribute to the current debates. any development in mathematics education raises questions about the curriculum, for the teachers and their teaching, and for the learners and their learning. mathematical literacy is no exception. for example, there are new curricular issues about the range of contents and contexts to be addressed, the necessary changes in assessment practices, the extent of curricular control by any textbooks or materials used, how it is being featured in teacher education and training programmes, and the ways in which mathematical literacy links with other subjects in the school curriculum. teachers need to see practical examples of relevant classroom activities, with discussions of how these new practices differ from the old, and how they can avoid losing what was good and effective in what they previously did. what new materials are, or would be, helpful for teachers teaching mathematical literacy? what are the teaching implications of striving for “a more equitable, accessible and genuinely educational ideal” in the classroom? what new language issues will arise? what pre-service and in-service need does this new ideal demand, and how should these be met? also what are the implications for the learners? how can they be empowered to bring their unique perspectives from home and from their community and society into any mathematical classroom activity? we learn from other countries’ experiences that this particular development not only makes greater demands on the teachers, it also demands more from the students. by engaging them more in their own learning, they need to accept a greater share of responsibility for what they learn and for how they learn. there are a number of different forces that have led to a strong concern with the quality of the mathematical knowledge, skills, values and attitudes of learners in the last few decades. that is, not only a concern with those expected to continue in further studies and the professions related to mathematics, science, technology and economics but the mathematical competence of the general school population. these questions often revolve around how mathematically literate learners are and its importance for how they will enact their citizenry in a rapidly advancing scientific and technological world once they leave schooling. the imperatives driving this concern could be grouped into three broad categories. the first may be referred to as contextual forces, for instance, how this concern has emerged and is taken up in different countries, by the state in renuka vithal and alan j. bishop 3 mathematics curriculum policy, or by mathematics teacher organisations, and in research agendas. the second category may be derived from the theoretical shifts and emergence of new perspectives such as the developments in issues of gender in mathematics education, ethomathematics, realistic mathematics or critical mathematics education which include social, cultural, political, historical and economic analyses for how and why children learn mathematics the way they do. a third category may be characterised as the broad range of innovative practices and pedagogies being developed and enacted to address the claimed inadequate teaching and learning of mathematics – from small scale interventions to radical changes in practices. these analytical categories are, however, interlinked. mathematical literacy and the south african context the focus on mathematical literacy in the south african context needs to be understood both historically in how access to mathematics was denied to the vast majority of black people, as well as with reference to the current imperatives for general scientific, technological and economic development (vithal & volmink, 2005). the huge backlog of deliberate underdevelopment that post apartheid south africa inherited in 1994, in both human (qualified mathematics teachers) and physical resources (classrooms, materials, etc.) over several decades is proving very difficult to redress in education in the current context of a globalising world in a country with high levels of poverty and unemployment. although some strides have been made, still one fifth of secondary schools in the country do not offer mathematics beyond grade 10. of the approximately half a million learners who write the high status and high stakes grade 12 end of schooling national matric examination that determines entry into further education and work opportunities, some 40% of learners do not take mathematics (cde, 2004). not surprisingly learners have not been performing very well in these national exams and also in various national and international mathematical tests at lower levels of the schooling system. the focus and significance given to mathematics achievement nationally and in the media has been driven by south africa’s repeated extremely poor performance on a number of international studies such as in the third international mathematics and science study (howie 1997, 2001) and other national evaluations and assessments (see cde, 2004). heads of educational institutions, government officials, and politicians often refer to these in public. since 1994 south africa has had three waves of curriculum reforms for the grades 0 to 9 curriculum and is in the process of implementing a new curriculum for grades 10-12 as of 2006. in both these curricula, mathematical literacy features as a competence to be acquired by all learners. an important change in the new curriculum for the grade 10-12 band is that all learners will have to take mathematics or mathematical literacy as a subject from grades 10 to 12. that is, for the first time all students who leave the schooling system from the end of 2008 will have taken some mathematics up to the end of schooling in grade 12. this has meant a major and substantial intervention to reskill and upgrade teachers to be able to deliver the new mathematical literacy curriculum. from the outset the new curriculum has had to engage challenges about what exactly is mathematical literacy and how does it differ from mathematics. the notion of mathematical literacy has differed in name and conception in different countries in curriculum policies and in research. for example the term numeracy is used in the uk (brown, 2003) while qualitative literacy appears in the usa (steen, 2001). in the south african curriculum mathematical literacy is defined as follows: mathematical literacy provides learners with an awareness and understanding of the role that mathematics plays in the modern world. mathematical literacy is a subject driven by life-related applications of mathematics. it enables learners to develop the ability and confidence to think numerically and spatially in order to interpret and critically analyse everyday situations and to solve problems. (doe, 2003: 9) this definition needs to be read and understood against the broader principles underpinning the south african national curriculum statements which include: “social transformation, outcomesbased education, high knowledge and high skills, integration and applied competence, progression, articulation and portability; human rights, inclusivity, environmental and social justice; valuing indigenous knowledge systems; and credibility, quality and efficiency” (doe, 2003: 1). the national curriculum explicitly takes as its point of departure the constitution of south africa, the values of which are infused also in the mathematical literacy: a new literacy or a new mathematics? 4 mathematics curricula. this can be observed in the critical and developmental outcomes which are interpreted specifically for mathematical literacy as intending “to enable learners to: • use mathematical process skills to identify, pose and solve problems creatively and critically • work collaboratively in teams or groups to enhance mathematical understanding • organise, interpret and manage authentic activities in substantial mathematical ways that demonstrate responsibility and sensitivity to personal and broader societal concerns • collect, analyse and organise quantitative data to evaluate and critique conclusions • communicate appropriately by using descriptions in words, graphs, symbols, tables and diagrams • use mathematical literacy in a critical and effective manner to ensure that science and technology are applied responsibly to the environment and to the health of others • demonstrate that a knowledge of mathematics assists in understanding the interrelatedness of systems and how they affect each other • be prepared to use a variety of individual and co-operative strategies in learning mathematics • engage responsibly with quantitative arguments relating to local, national and global issues • be sensitive to the aesthetic value of mathematics • explore the importance of mathematical literacy for career opportunities • realise that mathematical literacy contributes to entrepreneurial success” (doe, 2003: 10) these critical and developmental outcomes are expected to be achieved through four learning outcomes: number and operations in context – the ability to use knowledge of numbers and their relationships to investigate a range of different contexts which include financial aspects of personal, business and national issues; functional relationships and the ability to solve problems in real and simulated contexts; space, shape and measurement including handling instruments, estimating and calculating physical quantities and working with two and three dimensional shapes and objects; data handling and the ability to apply knowledge of statistics and probability to communicate, justify, predict and critically interrogate findings and draw conclusions. many of the intentions in the critical, developmental and learning outcomes may be found in the broad emerging international literature that explores multiple dimensions of mathematics education both as a field of practice and as a field of study. in exploring not only the mathematical aspects of mathematical literacy but also its social, cultural, political, psychological, economic, historical and societal dimensions, has led to different interpretations being made of mathematical literacy and produced a range of pedagogies. the papers in this special issue capture some of this variety of concerns and foci. no doubt much research and many more analyses will follow in the wake of the implementation of this new mathematical literacy curriculum in south africa. iben maj christiansen sets the scene with a critical analysis of the whole issue of mathematical literacy as a school subject and questions the possibilities for achieving the educational ideals often promised in such curricula. in hamsa venkatakrishnan and mellony graven’s paper in which mathematical literacy in two different contexts of south africa and england are presented, we read of interesting and contrasting approaches to the same problem. in particular we gain a deeper understanding of the policy-practice imperatives that these curricula are having to negotiate where such concerns are being debated at a national level. drawing on their experience of materials development, lynn bowie and vera frith focus attention on the importance of distinguishing between mathematics and mathematical literacy and address the issue of the role of technology in mathematical literacy as well as the need for a proper understanding of the contexts used to teach mathematical literacy. the issue of context which is central to any conception of mathematical literacy is taken further by renuka vithal in exploring a specific pedagogy, that of project work for developing mathematical literacy. she discusses a range of practice and conceptual tools to show how a mathematical literacy from a critical perspective may be realised by teachers in a south african classroom but also some of the tensions this produces. bruce brown and marc schäfer retain the focus on teacher education by raising issues arising renuka vithal and alan j. bishop 5 in the training of mathematical literacy teachers, using an approach based on mathematical modelling. they point to the challenges of preparing teachers who do not having the requisite mathematical skills and knowledge choosing to become mathematical literacy teachers. the paper of vera frith and robert prince reports on the use of a research task in a mathematical literacy teacher education course. a curriculum component on data handling was structured around this research task which required the teachers on the course to put mathematical literacy into a practice context. in cyril julie’s paper he explores various myths, inclusions and exclusions, in relation to mathematical literacy. in concluding this special issue, his paper does not seek closure, rather he opens up further issues which will need to be investigated before the teaching of mathematical literacy can be said to have achieved its goals. references brown, m. (2003). research and national policies in primary numeracy. in b. putsoa, m. dlamini, b. dlamini, & v. kelly (eds.), proceedings of the 11th annual conference of the southern african association for research in mathematics, science and technology education research, 11-15 jan, waterford kamhlaba, swaziland. centre for development and enterprise (cde). (2004). from laggard to world class. reforming maths and science education in south african schools. cde research policy in the making. research report no 13. department of education (doe). (2003). national curriculum statement grades 10-12 (general) mathematical literacy. pretoria: department of education. howie, s. (1997). mathematics and science performance in the middle school years in south africa: a summary report on the performance of south african students in the third international mathematics and science study. pretoria: human sciences research council howie, s. (2001). mathematics and science performance in grade 8 in south africa 1998/9 timss-r 1999. pretoria: human sciences research council. steen, l.a. (2001). (ed.) mathematics and democracy: case for quantitative literacy. national council on education and the disciplines, the woodrow wilson national fellowship foundation. vithal, r., & volmink, j. (2005). mathematics curriculum roots, reforms, reconciliation and relevance. in r. vithal, j. adler, c. keitel (eds.), mathematics education research in south africa: challenges and possibilities pretoria: human sciences research council. mathematics is a logical method ... mathematical propositions express no thoughts. in life it is never a mathematical proposition which we need, but we use mathematical propositions only in order to infer from propositions which do not belong to mathematics to others which equally do not belong to mathematics. – ludwig wittgenstein abstract introduction method and research questions which features relate to sense-making in mathematics classrooms? which features of modelling support learner choice and contribute to sense-making? under which conditions can sense-making be enhanced in mathematics classrooms? a possible framework for conceptualising sense-making through learner choice in mathematics classrooms conclusion acknowledgements references about the author(s) piera biccard department of curriculum and instructional studies, university of south africa, south africa citation biccard, p. (2018). mathematical sense-making through learner choice. pythagoras, 39(1), a424. https://doi.org/10.4102/pythagoras.v39i1.424 original research mathematical sense-making through learner choice piera biccard received: 09 apr. 2018; accepted: 04 sept. 2018; published: 24 oct. 2018 copyright: © 2018. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract this article explores a conceptual relationship between learner choice and mathematical sense-making. it argues that when learners can exercise choice in their mathematical activities, mathematical sense-making can be enhanced. the literature around mathematical modelling suggests a link between sense-making and learner choice. a three-tiered conceptual analysis allowed ‘purposiveness to thinking’ from the author through engagement with selected literature. research questions related to a three-tiered analysis: generic, context-specific, and conditional accounts of sense-making in mathematics classrooms were formulated. the analysis resulted in a framework showing how sense-making may be constrained or enhanced in mathematics classrooms through learner choice. this article may add to our holistic understanding of sense-making in mathematics classrooms. it may contribute to mathematics teacher education by proposing that teachers are resourced to facilitate learners’ conceptual and procedural choice in primary or secondary mathematics classrooms. introduction learners often view learning mathematics as non-sense-making (dienes, 1971; schoenfeld, 1991). non-sense-making is distinct from nonsense (no meaning is possible) and is closer to the term senseless (having no meaning). schoenfeld (1991, p. 316, 320) coined the phrase ‘suspension of sensemaking’ or ‘significant nonreason in students’ school mathematics’ to describe learners’ disengagement with mathematics. the senselessness experienced by learners when trying to engage with mathematics may stem from a disconnection between the learners’ procedural and conceptual understanding. teachers also mistake procedural competency for conceptual understanding where they see the latter as a natural consequence of the former. often the senselessness of mathematics comes from this assumption, especially when the problem changes from ‘basics’ (manipulation) to ‘application’ (word problems). curricula are also often set up to mask procedural ability for conceptual understanding. reusser (2000) explains that non-sense-making takes a number of forms such as learners finding answers to unsolvable problems, for example ‘there are 125 sheep and 5 dogs in a flock. how old is the shepherd?’ (p. 23), or students using keyword strategies inflexibly, for example always adding if the word ‘more’ is in the problem. another area where sense-making is suspended is in realistic contexts. in reality-based problems the numerical answer needs to be interpreted against the real context (e.g. calculating that ‘5 remainder 2’ buses are necessary to transport a group of people. very few primary school learners reinterpret the ‘remainder 2’ as the need for a sixth bus). for schoenfeld (1991) many hours of completing worksheets with similar format have a significant effect on non-sense-making. if teachers want to assist learners’ sense-making they need to help them experience acts of sense-making (sierpinska, 1994). blind application of procedures (schoenfeld, 1991) makes it very difficult for learners to be involved in authentic sense-making. mathematics teachers would like to see learners engage in mathematical sense-making. the term ‘sense-making’ is often interwoven with ideas of deepening understanding and application of mathematical concepts (van velzen, 2016). this article uses a conceptual approach to explore learner choice as a property of sense-making. it proposes that choice may elicit enhanced understanding and application of mathematics. the author, therefore, tries to draft a ‘programmatic description’ (scheffler in soltis 1978, p. 9) of what should take place in mathematics classrooms. this may contribute to the field by simplifying the necessary conditions for promoting sense-making in mathematics classrooms. this article is not a full description of sense-making but rather proposes a certain way of looking at sense-making in mathematics classrooms through the role of learner choice. furthermore, this article proposes that learner choice may be a feature of sense-making in mathematics classrooms. for weick, sutcliffe and obstfeld (2005) sense-making is explicit and individuals create understanding through a retrospective reflection on decisions and actions. in contrast, klein, moon and hoffman (2006) argue that sense-making can precede actions since ‘sensemaking is a motivated, continuous effort to understand connections (which can be among people, places and events) in order to anticipate their trajectories and act effectively’ (p. 71). the stance of the author is that both ideas may be useful. learner decisions and actions may affect sense-making in mathematics classrooms both prospectively and retrospectively. this article asks the question ‘can learner choice enhance sense-making in mathematics classrooms?’ perhaps learner choice and sense-making are circular and not linear ideas, that is, they inform each other. the quote ‘any mathematical experience in which students make choices about how to use mathematics to create representations of a real-world process is a form of mathematical modeling’ (gann, avineri, graves, hernandez, & teague, 2016, p. 97) led to the author’s interest in exploring learner choice in mathematics classrooms. since mathematical modelling led to the ideas in this article, a brief definition is necessary at this point. the definition of lesh and doerr (2003) is that models are: conceptual systems that consist of elements, relations, operations and rules governing interactions that are expressed using external notations system and that are used to construct, describe, or explain the behaviours of other systems – perhaps so that the other system can be manipulated or predicted intelligently. a mathematical model focuses on structural characteristics (rather than for example physical or musical characteristics) of relevant systems. (p. 10) model-eliciting problems are reality-based problems where the product that learners are required to create or design will elicit a model of/for the problem. modelling is known to increase sense-making (lesh & doerr, 2003; lesh, yoon, & zawojewski, 2007). this article argues that one of the reasons why modelling supports sense-making is because model-eliciting problems provide avenues for learner choice when using or creating mathematics. when purely procedural methods are drilled, learners are passive in exercising agency and choice. what does it mean to have a choice? generally, it means both the act of choosing (to select) and the power of choosing (having an option) (merriam-webster online dictionary). the idea of decision-making, freedom or choice is also considered by other scholars in mathematics education. polya’s (in kilpatrick, 1985) earlier seminal work on problem-solving classification ranked problems according to the degree of choice the learners have in solving the problem. freudenthal (1991, p. 117) conceptualised the term ‘spontaneous differentiation’ where students choose for themselves at which level they will work, rather than ‘imposed differentiation’, where a teacher decides in advance at which level or with which methods the students will work. other realistic mathematics education scholars (gravemeijer, 1994a; treffers, 1987) echo the importance of differentiation based on learners using their own methods (which implies learner choice). terms such as ‘own methods’ are considered to be consistent with learners being allowed to make decisions or have choice when solving problems. hiebert et al. (2003) cite that learners benefit from choice in solving problems and documented this aspect in the timss video analysis study. hiebert et al. (1997, p. 24) also point out that as a result of students using their own methods ‘they develop general approaches for inventing specific procedures or adapting ones they already know to fit new problems’. fosnot, dolk, zolkower, hersch, and seignoret (2006) consider teacher facilitation of learners’ own mathematical constructions as an advanced form of pedagogy as do cognitively guided instruction scholars (franke, carpenter, levi, & fennema, 2001). stein, engle, smith, and hughes (2008) set out how important it is for teachers to use learners’ own methods to build connections between important concepts. what does learner choice look like in mathematics classrooms? learner choice entails learners being in the driving seat of the methods, procedures, representations and explanations in the mathematics class. learners will have the option of entering a problem from their knowledge base, tackling the problem using their own ‘mathematical toolbox’ (jensen, 2007, p. 144). the teacher will then facilitate vertical mathematisation (connecting and developing more abstract mathematical ideas; see treffers, 1987) through focusing on connections between the various approaches used by learners. this means that the learners build a ‘floating capacity’ (webb, boswinkel, & dekker, 2008) of the abstract concept. floating capacity from an iceberg metaphor refers to the many informal representations of an abstract concept while the tip of the iceberg is the concept in abstracted form. the floating capacity develops and supports the understanding of the tip. this article is structured in the following way. the section on method and research questions provides some discussion of the method followed for the analysis of the concepts under scrutiny. it also sets out the questions that were formulated in undertaking the study. the section that follows the method focuses on the occurrence of sense-making in mathematical classrooms generally. it is followed by a section looking at a specific mathematical activity, that is, modelling, focusing specifically and descriptively on what features of modelling activities support learner choice to enhance learner sense-making. following this, some basic tenets for sense-making in mathematics classrooms are proposed. the final section concludes with a possible framework provided by the author for understanding sense-making through learner choice in mathematics classrooms. method and research questions bousso, poles and da cruz (2014) explain that it is important to engage with concepts because they are used to develop theory, they can be operationalised and they can enhance practice. wilson (1963) explains that understanding concepts is not related to facts, values, definitions or meanings of the words but rather about the actual and possible use of the words and their ‘logical mystery’ (p. 13). soltis (1978) set out three features for the analysis of education concepts. he proposed a generic analysis (what features must x have to be called x?), a differentiation analysis (what are the different basic meanings of x?) and a conditional analysis (what context conditions govern the use of x?). in this article, soltis’s scheme was adapted and abridged and provided some ideas for method and logical structure of the article in three related parts. for this article, the three-part analysis comprised a generic analysis, a context-specific analysis and a conditional analysis. the three-part analysis allowed the researcher to explore a conditional relationship between learner choice in mathematics classrooms and their sense-making. the analysis is interpretive and non-technical and begins by the researcher setting out a premise (kahn & zeidler, 2017) (in this case: learner choice is a property of sense-making). a conceptual analysis allows one to unpack central constructs in an area under investigation (kahn & zeidler, 2017). to unpack the central constructs, the following three questions were formulated. in this article, a generic analysis seeks to answer the question: what are some of the features related to sense-making in mathematics classrooms? according to soltis (1978), the first step is to draw from general knowledge some general features. various scholars have described sense-making using a range of terminology and examples. the current study undertakes to distill some common features of sense-making through the generic analysis. a context analysis will then answer the question: what features of mathematical modelling support learner choice and contribute to sense-making? in this case a context-specific analysis only is done. this article will focus on sense-making in one particular type of task – mathematical modelling. in this section, mathematical modelling as a specific type of mathematics activity is described and analysed for features of enhanced sense-making. a conditional analysis will answer the question: under which conditions can it be said that sense-making may be enhanced in mathematics classrooms? soltis (1978) explains that the point of departure of a conditional analysis is to puzzle over the ‘context appropriate for the use of the concept’ (p. 104). in this study, some necessary conditions for enhancing sense-making in mathematics classrooms were considered based on the generic and context-specific analysis. these questions were answered through engagement with selected literature on mathematical modelling and sense-making. this engagement allowed ‘purposiveness to thinking’ (wilson, 1963, p. ix). the centrality of learner choice in mathematical sense-making is an idea from gann et al. (2016) who linked choice to mathematical modelling. the author of this article undertook to determine if learner choice is a general element of sense-making in mathematics. in order to understand the role of learner choice in sense-making, the terms sense-making, choices and decision-making were located in mathematics education literature. this assisted in ‘the process of becoming conscious’ of the meanings of words (wilson, 1963, p. 15). however, the literature review led to exploring words beyond ‘choice’ or ‘decision-making’ to words such as ‘modelling’, ‘problem-centered learning’, ‘informal methods’ and ‘flexible thinking’. which features relate to sense-making in mathematics classrooms? teaching and learning as explicated by brousseau (1997) in the theory of didactical situations forms the basic theoretical tenet of this article. brousseau elaborates on a ‘didactical contract’ that exists in mathematics classrooms. in many cases, teachers take the responsibility of showing or explaining mathematics to learners (and where the learners are expected to copy, memorise and repeat what they are shown). at the same time learners expect the teacher to show or explain. this, however, affects the level of knowledge, skill and understanding that the teacher wants the learner to achieve since the teacher is doing the mathematical work in the classroom. schoenfeld (1992) states that when learners experience mathematics as having to follow a single procedure the teacher has in mind (traditional didactical contract), they will experience mathematics as a discipline that does not always make sense. brousseau suggests that problems should be devolved through an adidactical situation. the teacher creates a new milieu in which problems are handed over to learners. the learners have to accept the responsibility for solving the problems knowing that the teacher has set the problem with an aim of learning something specific. this is similar to freudenthal’s (1991) conception of learners having to ‘re-invent’ mathematics for themselves through teacher guidance. in general terms, to devolve a situation or a problem means to ‘pass on (something, such as responsibility, rights, or powers) from one person or entity to another’ (merriam-webster online dictionary). this section looks at different conceptions of sense-making in mathematics classrooms. the analysis looks at various definitions and conceptions of what sense-making could look like in mathematics classrooms. schoenfeld (2014) surmised that sense-making is about perceiving structure. the national council of teachers of mathematics (nctm) defined sense-making more generally as ‘developing understanding of a situation, context, or concept by connecting it with existing knowledge’ (nctm, 2016, p. 1). sense-making is more about flexible application of mathematical knowledge (van velzen, 2016) than it is about performing procedures. in mathematics classrooms sense-making can be compromised by learners’ inflexible concept cores (trzcienieka-schneider, 1993). according to trzcieniecka-schneider (1993), ‘people with more plastic concept cores have higher consciousness of cognitive control of events’ (p. 258). increased concept flexibility could be developed by encouraging learner choice and explaining their choices. trzcienieka-schneider, citing earlier research where 98 learners solved ill-formulated problems, acknowledged that when pupils are discouraged from looking for their own (possibly unconventional) methods and examples, they develop stiff concept cores. tabachneck, koedinger and nathan (1994) explain that the abstraction process of formal mathematics is often nonsensical and that following rote procedures can lead to errors. informal strategies (such as those students invent on their own) have been suggested for sense-making in mathematics classrooms (gravemeijer, 1994a; schoenfeld, 1991; tabachneck et al., 1994; treffers, 1987). informal strategies, in particular, are more flexible (tabachneck et al. 1994) and suggest a degree of learner choice. dewey (1910, pp. 64–65) set out the concept of ‘freedom’ which could be akin to the concept of choice. he considered freedom as being capable of independent thought without the ‘leading strings of others’ (not following teacher presented methods). he furthermore set out that freedom is the ‘ability to turn things over’. if learners are encouraged to explore concepts from their own and different vantage points, it may allow them to ‘turn things over’. boaler (2016, p. 189) also refers to intellectual freedom as learners using intuition to develop new perspectives on mathematical and ‘transform their relationship with learning’. schoenfeld (1991) sets out how teacher ‘assistance’ and ‘structuring’ (leading strings) result in a procedural and nonsensical routinising of mathematics. teacher assistance comes in the form of limiting the vocabulary used or teaching a ‘key word method’ (p. 323). sometimes the rules teachers use do not make sense, but learners know that if they use them, they will get the ‘right’ answer. this perpetuates the idea that mathematics is about passively following rules that do not need to make sense. boaler (2015) stated that learners who are taught passively ‘do not engage in sense making, reasoning, or thought … and they do not view themselves as active problem solvers’ (p. 40). nieuwoudt (2015) found that when grade 4 learners (who initially displayed negative attitudes to solving word sums) were encouraged to solve word problems collaboratively, they were able to come up with different methods and were able to make sense of general problem-solving models. schoenfeld summarises that advanced sense-making is evident (irrespective of the age level or content) where the teacher does not provide answers in advance, various methods (elicited by the learners) are compared and mathematical understanding is arrived at by social consensus. the teaching for robust understanding framework (schoenfeld, 2016) encompasses what is proposed as a sense-making framework that can be used in any mathematics classroom. although it is beyond the scope of this article to discuss the framework in detail, a brief synopsis is presented. the framework was developed after schoenfeld and colleagues considered many aspects of what should be present in mathematics lessons that promote ‘robust understanding’. they analysed existing frameworks and distilled certain foundational ideas. their framework comprises five dimensions (schoenfeld, 2016). a mathematics dimension that considers how connections are forged between procedures, concepts and contexts. a cognitive demand dimension, that relates to the opportunities students have to make sense of mathematics through being challenged appropriately. the equitable access to content dimension focuses on who takes part in mathematics classroom discussions and how active engagement of all students is supported while the agency, ownership and identity dimension considers the opportunities that learners will have to see themselves as mathematical thinkers. the final dimension of formative assessment focuses on how students’ current thinking is used to develop mathematical ideas. learner choice may be relevant to each dimension. the mathematics content and the cognitive demand should allow learners to engage with concepts that are rich in connections. this means learners should be able to enter the mathematical discussion from their own perspective. furthermore, learners should be introduced to the ideas of other learners. this promotes equitable access to content. learners who are allowed to make choices of method, procedures and representations and have the opportunity to explain their reasoning develop identities of themselves as mathematical thinkers. eliciting learner thinking through challenging tasks that are rich in mathematical connections allows teachers to use and develop learner ideas. in summary, learners should be involved in ‘doing the mathematics’ in mathematics lessons. tasks that promote active mathematical thinking by encouraging learners to use their own informal methods before memorising procedures are necessary for sense-making. which features of modelling support learner choice and contribute to sense-making? some features of mathematical modelling as they relate to learner choice and sense-making are presented in this section. the first feature of modelling is that problems are devolved. model-eliciting problems typically include a messy real-world situation where students make assumptions and limit the information they use based on these assumptions. handing over problems and the responsibility for solving them implies that these decisions and choices are also handed over to students. blomhoj and jensen (2007, p. 49) hold that the dilemma of ‘teacher directed autonomy’ needs to be overcome and the students need to be responsible for most of the decisions. the second feature of modelling that relates to learner decision-making is that the starting and ending point for problems is reality. cirillo, pelesko, felton-koestler and rubel (2016) differentiate mathematical modelling from modelling mathematics. this is an extension of lesh et al.’s (2007) deweyian conceptions of making mathematics practical versus making practice mathematical. many learners experience modelling mathematics in their classrooms but not mathematical modelling. for cirillo et al. modelling mathematics takes place when teachers find something in the world (contexts, representations etc.) on which to model the mathematics they want to teach – so the process starts in mathematics (e.g. using blocks or a number line to teach subtraction). on the other hand, mathematical modelling entails starting in the real world (with a problem) and then finding, choosing or creating mathematics (a model) that allows us to understand some aspects related to the structure of the problem. the following example of a modelling problem may assist the reader who is unfamiliar with them. big foot (lesh & doerr, 2003) is a model-eliciting problem where learners (working in groups) are given a shoeprint measuring about 50 cm in length. they are given the task of explaining to the police what this person’s height could be. learners need to create a model from which the police can determine anyone’s height from their shoe print. this problem can be used across many grades and age groups since the choice of mathematics is open to learners. the learners will use the mathematics that they are capable of applying. furthermore, learners have the choice of which aspects of reality to include (do they consider only adults or only tall people in their measurements? what do they do with outliers?), which methods to use and what forms of representation to create (tables, graphs etc.). for bliss and libertine (2016, p. 12) ‘a modeling problem must also provide room for students to interpret the problem and have choices in the solution process’. one of the strengths of mathematical modelling tasks is that they reveal what situations learners can mathematise and what previously taught mathematics they can apply. the third feature of modelling where learner choice is embedded is that learners produce a model and do not only apply known models. this means that learners create models of other systems. with model-eliciting activities, teachers provide learners with a system (usually real) and through the question about that system ask learners to create a model in order to solve a problem about that system. reiley (2017, p. 446) concludes that ‘students should be able to make decisions about what they are presenting as the product of their task’. in producing a model, learners do not (necessarily) follow known procedures; they do not always apply a known method to a problem and find an answer in two steps. modelling is a structuring process to deal with a problem where a genuine search for meaning is necessary. the problems are open and pathways are many. learners need to discover relationships first, before finding the variables to describe them (treilibs, burkhardt, & low, 1980). gann et al. (2016, p. 105) explain that the focus of mathematical modelling is not to learn (or make sense of) a specific mathematical topic, but it is ‘learning to make decisions and assumptions and on using mathematics to understand a real-world-scenario’. treilibs et al. (1980) found that poor modellers avoid decisions of any sort entirely. this may be because they are not exposed to decision-making in mathematics classrooms that are teacher-centred and teacher-directed. when they are exposed to open problems that need structuring they are unsure how to proceed. this decision-making feature of modelling may be central to learner sense-making. a fourth feature of model-eliciting problems that encourages decision-making and supports sense-making is that learners work in groups. competencies of the group are likely to be greater than those of individuals (hatano, 1996; zawojewski, lesh, & english, 2003) since working with others encourages spontaneous verbalisation (artz & armour-thomas, 1992). mathematical sense-making can be seen as being distributed (pea, 1993) across learners, representations, explanations, materials and tools in the group activity. an important feature of any group work is the concept of ‘negotiation’ (lave & wenger, 1991). weick et al. (2005) state that sense-making is ‘an issue of language, talk, and communication’ (p. 409) where a ‘share[d] understanding … lift[s] equivocal knowledge out of the tacit, private, complex, random and past to make it explicit, public, simpler, ordered and relevant to the situation at hand’ (obstfeld, 2004, in weick et al., 2005, p. 413). in model-eliciting problems, the space is complex and necessitates discussion and decision-making collaboration with others which may contribute to enhanced sense-making. a final feature of modelling is that learners make use of their own ideas. authors such as english and watters (2004) and hamilton (2007) propose that even young learners and learners who do not have the necessary ready-made mathematical tools can be involved in mathematical modelling. hamilton contends that learners will invent their own version of the tools because they have decided that these tools are necessary. learners will decide which tools to use or invent. in modelling mathematics, the teacher makes the choices regarding relevant concepts and procedures for the learners, while in mathematical modelling, the learners structure and model the messy complex problem. when learners create their own tools and representations to solve problems, this process may enhance sense-making. modelling requires that learners make choices about the problem context as well as the mathematics that can be used to solve it. the features of modelling that contribute to learner choice and enhance sense-making can be summarised as: problems in their entire complexity are devolved (handed over) to learners, problems are set in authentic realistic contexts (where learners make their own assumptions), learners produce a model and learners use, create or decide on the tools they need. modelling involves more competencies than only being able to follow set methods and procedures. furthermore, learners work collaboratively in groups in what is largely a decision-making process. under which conditions can sense-making be enhanced in mathematics classrooms? a conditional analysis for the context conditions for which sense-making may be enhanced is now considered. in mathematics classrooms, the context conditions discussed in this section are the type of problems learners solve, as well as the classroom environment. mazur (2013) makes a clear distinction between contextual problems that are authentic or non-authentic and the impact of these on sense-making in classrooms. his concepts are similar to modelling mathematics and mathematical modelling (cirillo et al. 2016). according to mazur, a non-authentic contextual problem has the form as indicated in figure 1 where a known procedure is used to solve the problem. figure 1: non-authentic problems. the given problem has an unknown answer that can be found using a known procedure. learners simply recall the correct formula and apply it to the problem once they have removed the words in the problem. it may not necessarily be ‘the answer’ per se that is unknown, but also not knowing that the answer is justified or suitable. this is evident when learners give senseless answers to problem contexts (e.g. giving an answer of 7½ people). when mazur defined authentic problem solving, he explained that the problem has a known outcome but the solution path is unknown (see figure 2). figure 2: authentic problems. for kramarski, mevarech and arami (2002, p. 226) authentic tasks ‘can be approached in different ways … and often ask solvers to use different representations in their solutions’. these differences infer learner choice in their approaches and representations. mazur (2013) also explains that it is deceptively easy for a teacher to move authentic problems to non-authentic problems through limiting the choices and assumptions learners need to make. the process is subtle and often teachers are not aware of the consequences of adding some structure to a problem. structuring comes in the form of providing assumptions, suggesting which parts of the problem are significant or even reminding students of a formula. for example, if big foot is an authentic problem, a teacher could assist learners to get started by suggesting that they measure their own feet and heights and to calculate how many of their foot-lengths fit into their height instead of learners deciding that this is an option. the teacher can also remind the learners that a ratio can be calculated by dividing height by foot-length. if learners struggle with this, the teacher eventually ‘assists’ by providing a table of different heights and foot-lengths and learners simply have to divide them. this is how a didactic transposition (chevallard, 1989) takes place on a micro level. when teachers impose structure on the problem, it constrains the decision-making and choices learners have. this may tie in with brousseau’s formulation of the didactical contract. mathematics teachers often feel compelled to maintain their show/explain/assist role in the classroom. with limited learner choice, sense-making is relegated to remembering and applying at most and boaler reminds us that learners who experience mathematics as remembering ‘are the lowest achieving in the world’ (boaler, 2015, p. 41). according to mazur (2013), opportunities for learners to practise sense-making and decision-making are lowered every time teachers ‘help’ and every time teachers remove the higher-order levels of the problem because it is too messy for learners. this entails narrowing down learner choice. decisions around structuring of problems and the trade-off of these decisions may affect the level of interpretation and sense-making. in terms of a classroom environment that supports sense-making, schoenfeld (1991) asserted that the development of ‘meaningfulness and understanding comes from interaction and negotiation and that that process is inherently social’ (p. 339). boaler (2015) explains that when learners have to work silently in mathematics classrooms, they do not provide their own ideas or perspectives. boaler clarifies that mathematical discussions enhance sense-making because ‘reconstruction deepens understanding’ (p. 50) while leatham, peterson, stockero and van zoest (2015, p. 5) emphasise that ‘using student thinking to further mathematical understanding typically involves verbal interactions.’ treffers uses the term ‘interactivity’ (1987, p. 249) which means that learners work with or alongside other learners. he maintains that the productions and constructions of other learners could stimulate learners to either shorten their own path (vertical mathematisation) or to become aware of positive or negative aspects of their own ideas. it may also make learners aware of other options they had in solving the problem or in solving similar problems in the future. in describing interactivity, gravemeijer (1994b) says: explicit negotiation, intervention, discussion, cooperation, and evaluation are essential elements in a constructive learning process in which the student’s informal methods are used as a lever to attain the formal ones. (p. 451) the constructive learning process may come about through group collaboration, since it promotes ‘students’ mathematical understanding by creating opportunities for students to reexamine the validity of their reasoning’ (francisco, 2013, p. 434). working within a group situation or sharing ideas of other learners further provides learners with avenues to connect mathematical ideas by reflecting on the choices made by other learners. boaler (2015) summarised the problems in mathematics classrooms as: learning without thought, learning without talking and learning without reality. this article proposes that learning without choice may underpin all three. in summary, problems where the solution procedure is not always known and explicitly followed may enhance sense-making in mathematics classrooms. problems that are so tightly structured that only one possible method can be used may also limit sense-making. the role of group processes may encourage sense-making since learner choice needs to be negotiated and validated by members of the group. a possible framework for conceptualising sense-making through learner choice in mathematics classrooms bliss and libertini (2016) conceptualised how mathematics problems can be developed into modelling problems through a process of adding labels, meaning and interpretation as in figure 3. figure 3: one way of transforming a mathematics problem into a modelling problem. in this additive process, the problem is being opened up to allow for learning, interaction and sense-making through adding interpretations, meanings and labels. an example of this process may be that the mathematics problem is to calculate rate, for example ‘simplify 10 ℓ:100 km’. adding labels would lead to a word problem, for example ‘a car uses 10 ℓ of fuel for every 100 km; at what rate is fuel consumed?’ if some meaning were added: ‘dan is choosing between two cars and wants to buy the car with better fuel consumption. car a uses 11 ℓ per 120 km and car b uses 12 ℓ per 130 km. which car offers better fuel consumption?’ a modelling problem may involve asking if crossing a border to the next country 20 km away for cheaper fuel is worth the effort. a possible explanatory framework that extends their diagram and that can help us think about sense-making through learner choice is presented in figure 4. figure 4: supporting sense-making through learner choice. figure 4 shows how a mathematics problem can be opened up to allow for learner choice. with the added meaning and interpretation comes an increased potential for learner choice thereby possibly increasing learner sense-making. however, mazur (2013) showed that it is possible to reverse the process and limit the assumptions, choices and sense-making of a modelling problem by restricting learners’ access to a messy real-world problem. this reversal is shown as removing assumptions from the problem, adding a teacher method or model and removing the context of the problem. in the above fuel consumption example, if the modelling problem is structured by the teacher, the assumptions learners need to make are provided by the teacher, for example the size of the fuel tank, the price of fuel, and so on. the teacher may further suggest a method (e.g. first calculate how much it will cost to travel 20 km to the next country, etc.) and so the move from a modelling problem to a word problem may take place. practically, teachers and researchers may be able to use the framework to gauge to what extent lessons or lesson activities may be open enough to allow for learner choice and enhanced sense-making. the framework may also assist teachers or researchers to develop existing tasks to enhance the sense-making potential of the task. the framework may also be used to reflect on classroom observations. further work in developing rubrics or observation guides based on the framework may also be useful. questions around the level of interpretation need to be asked while also exploring who is making assumptions or who provides the models in the classroom. conclusion the nctm (2016, pp. 3–4) sets out that sense-making in mathematics classrooms depends on ‘worthwhile tasks that engage and develop students’ mathematical understanding, skills and reasoning, a classroom environment in which serious engagement in mathematical thinking’ takes place and where ‘purposeful discourse’ aimed at encouraging sense-making is evident. sense-making takes place through active learner engagement with concepts and not procedural fluency only. increasing learner choice and assisting teachers in their supporting role may create the need for different types of problems. boaler (2014, p. 2) refers to mathematics problems that give learners ‘space to learn’. this article sought to look at some aspects of opening this ‘space’. the element of choice appears to be evident in creating a sense-making space. although this idea emanated from a models and modelling perspective, it may be possible to transfer it to define significant pedagogical spaces in other areas of mathematical teaching and learning. the article set out to conceptually describe sense-making in mathematics classrooms and to understand some of its features. the description or highlighted features are not an all-encompassing account. wilson (1963) reminds us that conceptual analysis does not always result in right or wrong or complete answers but only ‘a number of logical sketches of greater or less merit’ which may contribute to something worthwhile (p. 48). by exploring the research questions, this article may add to our understanding of sense-making but does not produce a ‘one size fits all’ definition’ (kahn & zeidler, 2017, p. 542). from the field of mathematical modelling it appears that sense-making is enhanced because complex problems are devolved (brousseau, 1997) to learners and the model construction process is largely a decision-making one. learners are engaged with choice in order to produce models. furthermore, sense-making in mathematics classrooms is enhanced through less teacher structuring and learners using their own informal methods (which also develops cognitive flexibility). certain types of problems may enhance sense-making through providing opportunities for learners to interpret contexts. in interpreting contexts, learners are challenged to make assumptions and engage in more significant mathematical thinking. finally, a possible framework for understanding learner sense-making is suggested but will need further interrogating by means of conceptual and empirical research. inductive reasoning in conceptual analysis usually results in conclusions that are likely but not certainly true (kahn & zeidler, 2017). encouraging learner choice and freedom to get involved in mathematical discussions with peers may allow for greater levels of sense-making. learner choice may come about in methods or representation or simply in engaging in alternative procedures or finding connections between ideas and procedures. featuring more learner choice in lessons may involve re-negotiating the didactical contract that exists in classrooms. this necessitates that a teacher ‘lets go’ of doing most of the mathematical work in the classroom. acknowledgements competing interests the author declares that she has no financial or personal relationships that may have inappropriately influenced her in writing this article. references artz, a.f., & armour-thomas, e. 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(2003). a models and modeling perspective on the role of small group learning activities. in r. lesh & h.m. doerr (eds.), beyond constructivism: models and modeling perspectives on mathematics, problem solving, learning, and teaching (pp. 337–358). mahwah, nj: lawrence erlbaum. meaningful contexts or dead mock reality: which form will the everyday take? godfrey sethole tshwane university of technology (ga-rankuwa) email: sethole@hotmail.com this paper describes the experiences of two teachers, bulelwa and kevin, who attempt to take on board the notion of incorporating the everyday into the mathematics, as the new south african curriculum requires. both of them, though in different ways, attempt to accommodate the two discourses, namely mathematics and the everyday. because of their own different experiences and the different values they place on the everyday in mathematics, they handle the incorporation of the everyday quite differently. the paper argues that the practical experiences of these teachers provide insight into some of the practical challenges teachers have to negotiate in order to provide legitimacy for the everyday in mathematics. the definition of mathematics in the new south african curriculum, curriculum 20051 (c2005), entails both an epistemological (what mathematics is) and pedagogical (how mathematics should be taught) position. it states (2001:16): mathematics has its own specialised language that uses symbols and notations for describing numerical, geometric and graphical relations. mathematical ideas and concepts build on one another to create a coherent whole. while sound mathematical development is very important, this learning area statement recognises that access to mathematics is a human right in itself, and it is not value or culture-free. in teaching mathematics, try to incorporate contexts that can build awareness of human rights, and social, economic and environmental issues relevant and appropriate to the learners’ realities. the epistemological position, in the first paragraph of this definition, foregrounds mathematics as a unique and distinct entity. in bernstein’s (1996) terms, this definition puts mathematics in a “closed” relationship with other subjects. the pedagogical position, in the second paragraph, clearly addresses the 1 the south african government’s plan was to phase in the new curriculum gradually. it was hoped that by the year 2005 every grade at school level would be implementing the new curriculum, hence the name 2005. teaching approach. it prioritises an approach to mathematics teaching which is compatible with a realistic approach. thus, whilst mathematics remains a uniquely distinct entity, the non-mathematical everyday realities may be used as platforms to access it. there is an acknowledgement, however, that incorporating the everyday in mathematics may derail mathematical goals and therefore should be treated “judicially” (taylor and vinjevold, 1999:116). this argument is central to skovsmose’s notion of “mathematical archaeology” (1994) and adler’s (2000) elaboration of the concept of transparency. skovsmose (1994:94) admits that “mathematics may be integrated to such a degree that it disappears for both the children and the teachers. then it becomes important that time is spent on getting hold of the imbedded competence. mathematics has to be recognised and named and, that is the task of mathematical archaeology.” in analysing lave and wenger’s notion of transparency, adler (2000:214) argues that “resources in schools need to be seen to be used (visible) and seen through to illuminate mathematics.” if context is considered a resource in a mathematics classroom, then adler’s suggestion is that context is beneficial, provided it does not distract learners from mathematics. this paper cites and reflects on the experiences of two grade 8 mathematics teachers, bulelwa and kevin2. kevin and bulelwa teach in two different schools, in different contexts and settings, and handle 2 bulelwa and kevin are not teachers’ real names 18 pythagoras 59, june, 2004, pp. 18-25 godfrey sethole the expectation of using context to teach mathematics quite differently. in particular, the paper aims to: 1. highlight the different ways in which they handle the expectation of using contexts to teach mathematics. 2. reflect on the reasons they handle the contexts (in the teaching of mathematics) in they way they do. in conclusion, the paper will argue that the experiences of bulelwa and kevin provide us with insights into the practical challenges of incorporating these two different discourses, namely the everyday and mathematics. bulelwa and kevin were participants in a bigger international project, the learners’ perspectives study (lps). an outline of this study provides the broader context for this paper. the learners’ perspective study the learners’ perspective study is an international study which drew participation from ten countries, including south africa. as the name suggests, the main aim/focus of the study was to document the learners’ experiences of well-taught lessons. each country was to document learners’ experiences in three different schools at grade 8 level (clarke, 2001:05). because it was the experience of well-taught classrooms which was of interest, the major task was to identify schools with “good” mathematics teachers at grade 8 level. good teachers are those who are considered reputable and efficient by the school community (the principal, colleagues and students) as well as other mathematics teachers in the area. in south africa, data collection took place from march 2001 to may 2002 in three grade 8 classes located around durban, in the province of kwazulunatal. the demographic profiles of the three schools differed significantly: one school was predominantly indian, another predominantly african and the third school predominantly white. the decision to focus on three good teachers of different racial backgrounds was a deliberate one. it was motivated by the desire to observe the way in which teachers working in different cultural settings take up the challenge of implementing c2005. each data collection session entailed videotaping at least ten consecutive grade 8 mathematics lessons for a period of three weeks. for each school, we (as researchers) focused on one grade 8 classroom and therefore one teacher. each lesson was followed by a post-lesson interview with a focus group (normally a group of 4 –6 learners) for that particular day. after each data collection session (i.e. at the end of a three week period) each of the teachers was interviewed. it is within this context that the teachers’ views were accessed. this paper draws on data collected at two of the three schools, a predominantly african and predominantly white school. bulelwa, who teaches in a predominantly african, school holds a bachelor of science degree with maths and statistics as majors and a postgraduate diploma in education. the school is electrified, has a photocopying machine and a tidy reception area. for a township school, this school is fairly well-resourced and stands in contrast with the shacks in the school’s neighbourhood. kevin holds a four-year teaching diploma with mathematics as major. his first language is english and he teaches in a predominantly white school located in a fairly quiet suburb. the school is electrified and has a photocopying room and a fully equipped computer laboratory with about thirty monitors. access into the school is controlled; the school clerk has a clear view of the gate and can open it from her office. the next section focuses on some of the local and international debates in relation to the incorporation of the everyday into mathematics as well as the theoretical framework i used to make sense of the data. summoning the non-mathematics into and for mathematics to illustrate the seriousness of tensions within the mathematics community relating to the merits and demerits of the realistic approach, cooper and dunne (2000:07) admit: “we know already, as a result of responses to our work at conferences, that there will be a few individuals within the world of mathematics education who will wish that we had not produced this account of the difficulties children experience with ‘realistic’ items … at worst, they may have a commitment to the merits of ‘relevant’ and ‘realistic’ school mathematics, … more acceptably, they may fear that those demanding a return to a regime emphasising rote learning … may find support in the work reported here.” a large body of literature, highlighting both the benefits and the limitations of using the everyday in mathematics, has amassed in recent years. the following section cites and reflects on some of the literature. 19 meaningful contexts or dead mock reality: which form will the everyday take? benefits and limitations of the realistic approach skovsmose is amongst a number of mathematics educators who sees merit in incorporating the everyday into mathematics. in developing a critical mathematics philosophy, skovsmose (1994) regards mathemacy as an integrated competence. to this end, he argues that the mathematics curriculum cannot be constituted by making reference to mathematics only. freudenthal (1973), whose main thrust is in the teaching of mathematics, asserts that mathematics is too important to frame its instruction to suit the needs of the mathematicians. freudenthal maintains that “meaningful contexts that are not exercised daily are as easily forgotten as mathematics or even faster.” (1979:73). recently mukhupadhay (1998) highlighted a classroom experience in which a mathematical task led to a reflection on an everyday aspect. adult learners were required to sketch an equivalent of a popular doll, barbie, according to real-life scale. on completing the task, all groups suggested that real life barbie was unreal and unnatural, with a pelvic area way too small to bear a child. mukhupadyai argues that this exercise validates the usefulness of school mathematics beyond the usual abstract and contextempty existence. though the studies and debates above are argued from different perspectives, they all tend to support the inclusion of the everyday into mathematics. in south africa, a number of scholars have registered their discomfort with a teaching approach which favours the inclusion of the everyday in school mathematics (see, for example, taylor & vinjevold, 1999:113; muller & taylor, 1985:17; c2005 review report, 2000:37). others are fairly critical but sympathetic (for example, volmink, 1993; vithal, 1997; nyabanyaba, 1999) arguments against the inclusion of the everyday in school mathematics, or which suggests that such an action is insignificant in relation to the way learners engage with mathematics, have also been cited. floden et al (1987), for example, are quite unambiguous in their criticism of incorporating the everyday into mathematics. they suggest that the everyday restricts the students’ scope of vision and exaggerates reliability of close-to-home experience in the learning of mathematics which makes it difficult for learners to understand the academic disciplines. saljo and wynhamn (1993) conclude from their study that the influence of the everyday in mathematics is insignificant. they challenged 332 swedish students to determine the cost of posting a letter on the basis of the official table of postage rates from the swedish post office. the results of their study suggest that the task is interpreted as a mathematics task in a mathematics classroom and as a non-mathematical task in a social studies classroom (1993:332), rendering the incorporation of “relevant” data insignificant. this selective and situation-laden way of interpreting data is detectable in cooper and dunne’s study as well. they note that one middle-class child treats realistic tasks presented to her as “…merely differently presented exemplars of standard arithmetic problems.” the inclusion of the everyday for this child does not evoke an “everyday” response. in spite of its everyday nature, the task is still seen as a mathematical task. the common thread amongst these studies is a position which suggests that the boundary between mathematics and the everyday is significant and cannot be traversed easily in the learning of mathematics. bulelwa and kevin are two teachers who attempt to manage an expectation of incorporating the everyday into mathematics. the nature of the tasks they use suggests their views on the role and value of the everyday in mathematics. paul dowling’s (1998) notion of mythology provides a language and lens through which to engage with the mathematics – everyday relationship. he asserts that the particular ways in which mathematics and other practices relate produces a set of myths. to this end, dowling (1998:02) maintains that mythology is a result of “particular forms of realisation between the mathematical and other practices”. two of the six myths he cites are particularly useful for my current discussion, namely the myth of reference and the myth of participation. the myth of reference and the myth of participation the myth of reference entails a belief that mathematics is a place-holder for or “refers” to the non-mathematical. it recognises mathematics as a different activity to non-mathematical, everyday practices. in incorporating non-mathematics into mathematics, the non-mathematics is pruned off its other details, noise or “substantial residue of the nonmathematical” (dowling, 1998:08). as dowling suggests, “in its pure, ideal-type form, the myth of reference consumes the non-mathematical setting within a mathematical play leaving only a trace to remind us that there is something outside of mathematics” (1998:16). thus, the myth of reference 20 godfrey sethole is characterised by mathematical tasks which are conveniently covered by a veneer of the everyday. thus mathematics is presented “disguised” as if it is about a particular practice and not itself. the myth of participation entails a belief that mathematics is a tool needed to engage in the everyday. in order for an individual to participate fully in a practice or a culture, mathematics is a necessity. in this respect, dowling (1998:16) maintains: “this is precisely the nature of the myth: the claim to unity of culture which is, thereby, incomplete without the mathematics.” in brief, mathematics is a ticket to participation. this myth is characterised by school activities which foreground the non-mathematical and limits the explicit references to mathematics. thus, these activities appear as genuine and meaningful practices, with a “high modality of the non-mathematical.” as dowling suggests, the myth of participation constructs mathematics as a “reservoir of use-values.” in drawing a distinction between the two myths, dowling observes that the myth of participation, like a tool, is for something else; unlike the myth of reference, in which mathematics is about something else (my emphasis). bulelwa: foregrounding the social concerns over the maths skills even though my colleagues and i observed and videotaped thirteen lessons in bulelwa’s class, our main focus for the interview drew mainly from a lesson in which she had used aids as a context for teaching number patterns. in teaching this section, bulelwa set the scene by introducing a general discussion about aids. during this discussion, she advised the learners to practise safe sex because aids is real. the main aim of her lesson, as she explained to the learners, was to highlight the relevance and applicability of mathematics in real life situations. below is part of the worksheet that bulelwa had given to the learners. her seven-item worksheet, which she had given to the learners the previous week, consisted of tasks that required them to engage, describe and communicate mathematical observations. for example, item (a) read as follows: describe the pattern of population increase every forty years as shown in the first table. there were also questions that required learners to draw on their everyday experiences, for example item (g), which read: what can we do as a society to break the pattern of the increasing number of aids sufferers (i.e. decrease the number of aids sufferers)? regarding this worksheet, the following two observations need to be highlighted: aids as a context: kwazulu natal, the province in which this school is located, was once reported to have the highest infection rate of the hiv-virus. even closer to the school, an aids activist in a neighbouring township, gugu dlamini, was beaten to death following her public declaration of her hivpositive status. at a national level, president thabo mbeki has also faced a lot of criticism and negative publicity following his “unconventional” views about the cause and the prevention of aids. the details pertaining to these debates are well beyond the scope of this paper. the main point to tease out, though, is that aids, even as a context to advance mathematics mathematicians have studied number patterns for many years. it was discovered that there are links between mathematics and our natural environment and sometimes events occurring in our societies. for this reason an understanding of algebra is central to using mathematics is setting up models of real life situations. study the tables given and answer the questions that follow. year 1960 2000 2040 2080 2120 world population growth 3 000 million 6 000 million 12 000 million year 1997 1998 1999 2000 2001 world increase in the number of aids sufferers 16,7 33,4 66,8 million million million 21 meaningful contexts or dead mock reality: which form will the everyday take? is not a playful reality or benign issue. it is a sensitive and emotional socio-political issue. the mode of expression: the mode of expression or the tone used to communicate the text to the reader does not bracket out the non-mathematicians. the letters and expressions which characterise mathematics are absent. the task makes no explicit reference to specialised mathematical knowledge. instead, it is presented as a genuine and meaningful activity in which mathematics will be a necessary tool. in this case, the incorporation of the everyday into mathematics yields a myth of participation (dowling, 1988). in other words, mathematics is a prerequisite for predicting the number of aids sufferers in years to come. in contrast, bulelwa felt that she was prioritising the mathematics, and the context of aids, however useful, was meant to be a vehicle. for example, when our colleague renuka asked why she chose aids as a context, she responded: bulelwa: well, actually it was still number patterns. i wanted to choose something connected to real life. it’s not that we learn mathematics in isolation. just like when we started, we had an outbreak of cholera. i brought some statistics from the department; you know…the actual statistics from the department. so i taught them at the time how to get a table, a statistical table and analyse information. so it was learning mathematics, but with something that was happening at the time. renuka: how did you feel about the whole issue of mathematics and context? did you feel that there was one which you were prioritising? bulelwa: i felt i was prioritising mathematics because most of the questions i asked were of a mathematics nature except the last two questions… “how it was transmitted” and “ what can we do?” because obviously if doing a lesson in class and the obe context it need not just end up in a classroom situation. if you are dealing with the situation like this you need also to go out into the communities. so what i found out is that they (the learners) had more knowledge on aids…that they could handle most of the questions. that’s why it was difficult for them to handle a question that was long. it is clear that bulelwa is conscious of the mathematical purpose of the lesson. yet, her choice of aids as a context was not arbitrary; she was equally conscious of her role as an “advisor” to the learners. her personal experience, in which a cultural practice and lack of knowledge about the transmission of hiv virus led to an unfortunate consequence, was related during the interview. bulelwa: even more so when it comes to our tradition as africans where we use traditional healing methods. because that’s where most of our communities have been hurt. because you’ll find yourself….that it is a true story where you find that the whole family has been wiped out. you know, mother, father, and grown up kids. you know i am not talking about small kids …(looks for words) renuka: (finds the words) oh yes…who were born with aids. bulelwa: yes, i am talking about kids in their 20’s and 30’s. now you ask yourself, “how can the whole family be wiped out?”. and i know of two families. but when you follow the story you find that one family member had hiv, but now because it was our custom and tradition that from time to time we call a traditional healer, …so maybe when things are not going well and you use traditional methods of healing – so part of it is to use a razor and cut…. renuka: (impulsive comment) yes, it is the first time i learnt about it and i found it quite interesting. bulelwa: yes and it did not come from me, it came from the kids themselves. they know… renuka: so the main problem was that if every member of the family was sharing the same blade then it was transmitted like that. it is notable that bulelwa is enacting two different and perhaps competing identities within the same lesson. she has a “shifting identity” (setati, 2002:48). on the one hand, she is a mathematics teacher whose aim is to encourage a particular mathematical agenda: number patterns. on the other hand, she is a responsible citizen who wishes to alert the learners to aids and its effects, as the new curriculum advises. could it be, though, that she is trading the mathematical skills for the social concerns? 22 godfrey sethole kevin: foregrounding maths skills over social concerns? during our three week visit at his school, kevin treated two sections, word problems and geometry. in setting the scene for the lesson on word problems, kevin highlighted to the learners that the lesson was aimed at using mathematics to solve real-life problems. he actually formulated his own examples using the names of some of the learners in class. like bulelwa, kevin had compiled the tasks for the learners on a worksheet, which each learner was in possession of. the type of tasks kevin had compiled are what freudenthal (1973) refer to as “dead mock reality.” these are tasks whose context is obviously fictional. as stated before, the new policy favours the use of meaningful, realistic examples; however, it does not explicitly discourage “mock reality” context. the following is one of the tasks in kevin’s worksheet: r p r m m v c 3 i c i t s a e m a v s c g provide? for example, if they were to say 3 cd’s cost r9.00 3 kevin: (repeats) if we said 3 cd’s cost r9.00 godfrey: the procedure: yes, suggesting that 1 cd costs r3.00. would you be happy with that? kevin: ehh…i’d probably want it to be a little more accurate but then you would have…i think quite often i use examples that would not be quite what you would expect. i want them to be able to pick up that three cd’s costs this much and the way you get the price of one is that you divide by three. to me that’s the important thing. renuka: the procedure kevin: (in agreement) the procedure in his suggestion that he would want the answer to be a ‘little more accurate’ kevin clearly sees the value of using the everyday context for the teaching of mathematics. he regards the acquisition of mathematical skills as more important, though. in fact, as is evident from the interview, that the context is tshepo is n years old. mpho is twice as old as tshepo will be in three years time. how old is mpho, in terms of n? the names used in this task are real and the task ecruits the age comparison setting. however, the resence of the letter alphabet n and lack of details in elation to setting suggest that the task is more about athematics than the age comparison setting. the athematical operation (n + 3) × 2 is an “exchange alue” (dowling, 1998:06) for an age comparison ontext. in this case, the mathematical operation (n + ) × 2 is presented as referring to something other than tself, the myth of reference (dowling, 1998:6). in presenting the solution of the task in the lassroom, kevin provided explicit rules on the way n which these tasks were to be engaged. he indicated o the learners that they would have to use letter ymbols like n in terms of which the ages of mpho nd tshepo could be determined. this process makes xplicit what the teacher foregrounds, namely the athematical procedures. even though the context of ge is recruited, it is clear that this only serves as a eneer for mathematical procedures. as shown in this napshot interview with me (godfrey); kevin is onscious of the pseudo-realistic nature of the task: odfrey: in designing these word problems, does it matter what kind of answer the learners trivial does not seem to strike him as significant. he prioritizes the induction of his learners to the art and language of a mathematical discourse and is less keen on raising issues of social concern. could it be that he trades off these social issues for mathematical skills? having reflected on the different ways in which bulelwa and kevin manage the curriculum expectations with regard to the use of the everyday, the next section focuses on the implications that these two different approaches have with regard to their learners’ mathematical competences. from pedagogy to mathematical knowledge it was indicated at the beginning that the epistemological position of the definition portrays mathematics as a unique subject with its own explicit grammar. kevin’s task and views are leaned towards supporting his learners gain access to this mathematics. the age comparison setting task is motivated more by mathematical interests than the everyday non-mathematical ones. providing a rationale for his backgrounding of the everyday, kevin highlighted, during the interview, that …we are still linked to this matric syllabus…matric exam. if that changes, then i’ll feel happy with a lot of stuff they are doing but 3 r9.00 is an unrealistic price for one cd let alone three cds. 23 meaningful contexts or dead mock reality: which form will the everyday take? they haven’t changed that and what they are asking people to do now, that you will still be able to write the same matric exam. and i want someone to turn around and say, “... well, in the matric exam in 2005 this is what is going to be examined.” kevin’s approach is mainly influenced or shaped by the matric or grade 12 mathematics syllabus. he sees himself as exposing learners to what is prioritized at grade 12 level. in south africa, the matric results receive considerable media publicity. there is thus an extent to which, albeit unintentionally, matric results serve as a marketing tool for most high schools. that kevin seeks to align his teaching with matric maths requirements could be understood from this context. thus, in spite of the expressed desire to recruit aspects of the everyday relevant to learners’ experiences (when he introduced the lesson), kevin finds himself having to accommodate and even foreground the mathematics. bulelwa’s intentions to teach mathematical skills competes with (or is obscured by) her intentions to raise issues of social importance as suggested by the new curriculum. her use of aids as a context, as well as the question on the prevention of aids implies that there is “a substantial residue of the nonmathematical setting which remains after the mathematical routines have run their course: (dowling, 2000:9). the use of a meaningful context, it should be highlighted, is compatible with the pedagogic position of the definition as espoused in the revised national curriculum statement. bulelwa does not take this expectation lightly. with reference to aids as a context, bulelwa emphasised during the interview: i was actually concerned but the subject itself is a concern for debate. so even if there is somebody who has aids, i felt that it would enlighten them more, it would make them feel that you don’t have to…(looks for the right phrase) it’s a subject that we need to open up for debate. we need to discuss how it gets transmitted, for those who don’t know about it yet. because…in our community we still find people who are illiterate, who feel that there is no aids. in spite (or because) of her passion and desire to implement what is advocated in the new curriculum, her students may be denied access to mathematics content knowledge. who needs a realistic approach? the point the new south african curriculum makes is that mathematics needs to be seen as a body of knowledge which is not devoid of human experiences. however, mathematics and the everyday are two different discourses and putting them in an “open relationship” may result in the dominance of one discourse over the other. this paper has highlighted, at a practical level, the experiences and challenges of two teachers in this respect. in no way can these experiences be regarded as general. however, the experiences of the two teachers provide some insights into the practical challenge of incorporating the everyday into mathematics. summoning a range of their own resources as experienced teachers, bulelwa and kevin take up the challenge of incorporating the everyday into mathematics in two different ways. firstly, foregrounding the social aspects of the everyday, as was the case in bulelwa’s class, seemed to render the mathematics invisible or inaccessible. in other words, the meaningfulness of the context in relation to the lives and experiences of the teacher and learners may render the context so visible that the mathematics may not be accessed. secondly, foregrounding the mathematical goals, as was the case in kevin’s class, may motivate teachers to recruit what may be considered dead-mock reality. in other words, the everyday may solely be treated as a sugar-coat for or vehicle towards mathematics. in real life, for example, no david or mary can be of age n or n+1. these experiences help us appreciate the practical challenge of incorporating the everyday into mathematics. instead of regarding bulelwa and kevin as having “succeeded” or “not succeeded,” we should appreciate that the task of making mathematics relevant is a challenging one. the expectation that the incorporation of the everyday into mathematics will occur unproblematically seems simplistic. references adler, j., 2000, “conceptualising resources as a theme for mathematics teacher education” in journal of mathematics teacher education, 4 (1), may bernstein, b., 1996, pedagogy, symbolic control and identity: theory, research critique london: taylor and francis. 24 godfrey sethole clarke, d.j., 2001, application to conduct research summary: learners’ perspective study (unpublished report) cooper, b. & dunne, m., 2000, assessing children’s mathematical knowledge: social class, sex and problem-solving, open university press: buckingham department of education, 2001, revised national curriculum statement for mathematics, department of education: pretoria dowling, p., 1998, the sociology of mathematics education: mathematical myths/pedagogic texts, falmer press: london floden, r.e., buchman, m.& shwille, r.j., 1987, breaking with everyday experience, summer 88(4) freudenthal, h., 1973, mathematics as an educational task, d reidel publishing company: holland mukhopadhay, s., 1998, “when barbie goes to classrooms: mathematics in creating a social discourse” in c. keitel (ed.) social justice and mathematics education: gender, class, ethnicity and politics of schooling pp. 150 161 muller, j. and taylor, n., 1985, “schooling and everyday life: knowledges sacred and profane” in social epistemology 9 (3), pp. 257-75 nyabanyaba, t., 1999, “how ‘real’ is ‘relevance?’ tensions emerging in teachers’ talk within and about relevance,” in pythagoras, 48, pp. 17 – 25 saljo, r. & wyndahmn, f., 1993, “the school as a context for problem solving” in s. chanklin and j. lave understanding practice, new york: cambridge university press, pp. 327 – 343 setati, m., 2002, unpublished phd dissertation, university of the witwatersrand skovsmose, o., 1994, towards a philosophy of critical education, kluwer academic publishers: dordrecht taylor, n., 1999, “curriculum 2005: finding a balance between the everyday and school knowledges” in n. taylor and p. vinjevold. (eds.), getting learning right. joint education trust: johannesburg volmink, j., 1994, “mathematics by all” in s. lerman (ed.) cultural perspectives on the mathematics classroom, kluwer academic publishers: dordrecht vithal, r. et al, 1997, “student teachers doing project work in primary mathematics classrooms” in p. kelsall & m de villiers (eds.) proceedings of the third national congress of amesa, pp. 261 – 276 25 the learners’ perspective study summoning the non-mathematics into and for mathematics bulelwa: foregrounding the social concerns over the maths sk kevin: foregrounding maths skills over social concerns? from pedagogy to mathematical knowledge who needs a realistic approach? references microsoft word 36-45 mogari et al.doc 36 pythagoras, 69, 36-45 (july 2009) investigating the status of supplementary tuition  in the teaching and learning of mathematics    david mogari  institute for science and technology education, university of south africa    mogarld@unisa.ac.za  hanlie coetzee  dept. of mathematics, walter sisulu university   hcoetzee@wsu.ac.za   riette maritz  dept. of mathematics, university of south africa  maritr@unisa.ac.za    the study seeks  to  investigate  the status of supplementary  tuition  in  the  teaching and  learning  of  mathematics  and  mathematical  literacy.  the  study  followed  a  descriptive  survey  design  involving  the  use  of  learner  and  teacher  questionnaires.  a  convenient  sample  of  mathematics  and  mathematical  literacy  teachers  together  with  a  stratified  sample  of  their  grade  11  learners  were  drawn  from  a  purposive  sample  of  high‐ performing high schools in the east london district of the eastern cape, south africa. the  results show that supplementary tuition  is popular, especially among girls, and  it  is  in  three forms (i.e. private tuition, vacation school and problem‐solving classes): problem‐ solving  classes  dominated  by  working  on  past/model  examination  papers  is  the  most  preferred;  in  some  instances  supplementary  tuition  is  offered  for  a  fee;  it  is  not  only  confined  to  poor  performing  learners;  and  participation  in  supplementary  tuition  is  influenced by a variety of factors.      a number of asian countries have showed remarkable mathematics performance in recent international comparison studies, e.g., the third international mathematics and science study-repeat (timss-r) (howie, 1999), programme for international student assessment (pisa) (oecd, 2001) and the southern african consortium for monitoring educational quality project ii (sacmeq) (moloi, 2000). the good performance by the japanese learners, for example, has impressed countries such as the united kingdom to such an extent that they made some efforts to emulate the japanese school system. howie (1999) reports that the japanese learners tend to perform well in the international comparison studies mainly because of the extensive use of supplementary tuition, which is defined in the current study as extra tuition given to learners outside the normal school time which can be in the form of private tuition, vacation school and problem-solving classes. during the day learners are exposed to the mathematics content through the teacher-dominated lessons and the afternoon lessons focus on conceptual understanding through problem solving-oriented lessons. in south korea, in addition to 10 hours of normal teaching, a typical day starts with an hour of morning studies and end up with 4 to 6 hours of evening classes (lee, 2002). the use of supplementary tuition has also been recorded in countries such as hong kong, sri lanka, taiwan, india, cambodia and malaysia (bray, 2003). the study by baker, akiba, letendre and wiseman (2001) showed that more than 50% of grade 7 learners in japan, hong kong, latvia, russia, czech republic, slovakia and slovenia receive supplementary tuition. bray and kwok (2003) indicate that more than 70% of grade 7 learners in japan, hong kong, latvia, russia, czech republic, slovakia and slovenia have received supplementary tuition at some point in their schooling david mogari; hanlie coetzee & riette maritz 37 career. due to the importance of mathematics in various careers such as engineering, economics, architecture, computer science, commerce and so on, a higher premium has been placed on it. it is for this reason that in germany, for example, where the prevalence of supplementary tuition is relatively low, extra tuition in mathematics seems to be the most sought after (mischo & haag, 2002, p. 264). to this end, the use of supplementary tuition has not only improved the learner performance in the international comparison tests, but it also benefited learners in their school work. jacob and lefgren (2002), in the united states of america, found evidence of substantial positive effects of supplementary tuition (in the form of extra lessons offered during summer holidays) on mathematics which remained considerable even two years after completing the programme. studies conducted by posner and vandell (1999) and macbeath, kirwan and myers (2001) found structured after-school programmes in great britain to be particularly beneficial to learners from disadvantaged backgrounds. in israel, a supplementary high school intervention programme targeting 10th to 12th graders effected an increase in the mean matriculation rate by 3,3% and improved the learners’ probability of earning a matriculation certificate by 12%, which amounted to the overall improvement of 22% (lavy & schlosser, 2004). supplementary tuition has also been shown to have a positive influence on attitudes and self-concept, even at low participating levels (camp, 1990). furthermore, it is noticeable that countries where supplementary tuition is prevalent have shown considerable technological advancement accompanied by economic growth. typically, countries such as japan, singapore and south korea have not only shown their learners performing well in international comparison tests, but are also rapidly progressing technologically. there have been some negatives associated with supplementary tuition in some quarters. bray (1999, p. 9) mentions that supplementary tuition, due to the significant impact it has had on learning (camp, 1990; jacob & lefgren, 2002; lavy & schlosser, 2004; macbeath et al., 2001; posner & vandell, 1999), has generally developed into a huge business enterprise. parents are prepared to pay considerable amounts of money on supplementary tuition with a view to getting their children to have a competitive edge (bray, 1999, p. 39). in fact it has been established that there is a strong relationship between learners’ opportunity to participate in supplementary tuition and their socio-economic background (ireson, 2004; paviot, heinsohn & korkman, 2007). according to smyth (2008) this has led to participation in supplementary tuition being disproportionally concentrated among learners from middle-class families. it also stands to reason that learned parents are likely to get better paying jobs, thus they are in a better financial position to have money to pay for supplementary tuition. bray (1999, p. 65) notes that south korean parents spent about 25% of their income to ensure that their children get good education involving supplementary tuition. petterson (1993) reports that in 1988 japanese families paid an average of us$240 per month per child for supplementary tuition. in certain instances the benefits of being a supplementary tuition provider have caused teachers to conduct themselves in an undesirable manner. for example, foondun (2002) has found that in countries such as russia, egypt and nigeria teachers somehow manipulate the mainstream system to their advantage by slacking at their teaching and this creates more charged tutoring opportunities for themselves. similar practices have also been found in cambodia (lee, 2002). supplementary tuition has also contributed to the exodus of teachers with rare skills from the mainstream education. upon realising the financial benefits of being a supplementary tuition provider, teachers in costa rica and lithuania resigned their positions in schools and opted to be a provider of supplementary tuition (bray, 1999). supplementary tuition in south africa there has been a deluge of supplementary tuition in mathematics in south africa. one possible reason for the proliferation of supplementary tuition is that learners may be receiving extra tuition because of the shortage of qualified mathematics teachers. the issue of limited qualified mathematics teachers was alluded to by the minister of education in her address at the agrey klaaste mathematics, science and technology teacher of the year award. she indicated that of the 27 000 mathematics teachers in service in 2005, 18 000 (66,7%) had proper qualifications (pandor, 2005). a similar observation has been made by blaine in business day newspaper of november 24, 2007, where she asserts that too few mathematics teachers in south africa know enough about mathematics or have been trained properly. in this regard the current study seeks to establish why learners take supplementary tuition. the status of supplementary tuition in mathematics 38 another possible reason for the high demand of supplementary tuition in south africa may be that the syllabi are too full. moloi (2000) noted that it takes time to learn mathematical skills and especially how to transfer these skills from familiar to unfamiliar situations. moloi, therefore, enquired whether it would benefit anyone if time spent on mathematics during school time, is increased. the need to obtain examination results that meet admission requirements in most tertiary studies seems to have prompted the use of supplementary tuition. in general, the majority of learners taking supplementary tuition are those who are doing well or want to do well, particularly in their further studies (reddy, berkowitz, & mji, 2005). the current study intends to verify this. as in other countries, supplementary tuition in south africa is normally offered at a fee. bray (1999, p. 39) notes that supplementary tuition in some societies is offered by university students to raise money for their studies, retired teachers to supplement income from their pension savings and the unemployed graduates. reddy et al. point out that the department of science and technology (dst) commissioned the human sciences research council (hsrc) in 2003 to conduct a study on supplementary tuition. among others, the study sought to assess the extent, nature and cost of supplementary tuition in mathematics, physical science and computer studies for secondary school learners. the study reported that supplementary tuition was mostly conducted after school hours and focused a great deal on the curriculum and past examination papers. in terms of the number of learners participating in supplementary tuition, there were no definitive statistics available. reddy et al. (2003, p. 9) alluded to the difficulty of determining the exact numbers of learners involved in supplementary tuition mainly because of the different modes of teaching used to offer supplementary tuition. it is for this reason the present study is being pursued to determine the dominant mode of teaching used in supplementary tuition. in sum, the study will attempt to investigate the status of supplementary tuition in the teaching and learning of mathematics (mathematical literacy is also implied) by determining why learners do take or not take supplementary tuition, the nature of supplementary tuition offered and which learners take supplementary tuition. methodology research design given the nature of the study, a descriptive survey design was used to attempt to address the aims of the study. sample a purposive sample of nine former model c suburban schools was identified on the basis of their performance in the east london school district of the eastern cape province in south africa. these schools had obtained an average grade 12 mathematics pass rate of above 80% in previous years. from these schools, a stratified sample of 430 grade 11 learners was proportionally selected. this was the first grade 11 class to study the new curriculum, namely the national curriculum statement (ncs). in terms of ncs’s policy, learners who do not study mathematics are compelled to register for mathematical literacy. when sampling, it was ensured that there was a comparable split of the number of mathematics and mathematical literacy learners (the idea to include mathematical literacy mainly emanated from the fact that its introduction in the school curriculum compounded the perennial problem of shortage of properly qualified mathematics teachers. the education authorities then considered training teachers with limited mathematics background to teach mathematical literacy as a way to deal with the problem. furthermore, mathematical literacy was made compulsory to learners as an alternative to mathematics. given the learning difficulties most learners generally have with a figure-oriented subject such as mathematical literacy, one would obviously expect such learners to seek extra help to overcome the cognitive challenges posed by the subject). for teachers, a convenience sample of those teaching mathematics and mathematical literacy in grades 10-12 teachers took part in the study. the teachers were derived from these classes since it is a common practice in most schools to have a grade 12 teacher also responsible for either grade 11 or 10 as well. furthermore, teachers responsible for mathematics, normally also teach mathematical literacy. david mogari; hanlie coetzee & riette maritz 39 instruments two questionnaires (for teachers and learners, respectively) were adapted from a study improving the teaching of science and technology in the new south africa (rochford, sokopo, & kleinsmith, 1997). the modifications on the questionnaires were made to satisfy the aims of the current research and according to the experience and outcomes of the pilot study which was conducted a year earlier in a school with features similar to those of the main study. the learner questionnaire was in two parts where the first had three sections that elicited the biographical data of learners, details on supplementary tuition exposed to and reasons for not taking supplementary tuition, respectively. the second part of the questionnaire sought to determine reasons why learners were not satisfied with their performance. the teacher questionnaire also had two parts where the first consisted of two sections that derived the biographical data of the teachers and teachers’ views about the mathematics syllabus. part two of the questionnaire drew the teachers’ opinion on possible reasons for unsatisfactory performance in mathematics. the appropriateness of the questionnaires for the purpose of the study was determined by giving them to other mathematics teachers in the district to administer to their learners and also comment on. the reliability was determined by comparing the results yielded in both the pilot and main study to establish any consistency. data collection teachers completed their questionnaire individually while the learners completed the questionnaires at their respective schools under the supervision of their mathematics teachers. all the completed questionnaires were then packed for collection by the researcher. data analysis the data were analysed according to the assumptions made and linked to the aim(s) and objectives of the study. three assumptions were made for “why learners take or do not take supplementary tuition?” firstly, it was assumed that most of the learners who take supplementary tuition are dissatisfied with their performance in mathematics. the presumed dissatisfaction could have many causes, e.g. ineffective or sub-standard teaching, big classes, learners who are over-committed in other areas, long (select a better word) syllabi, lack of resources, etc. these avenues were explored. secondly, it was also assumed that learners have high expectations of their performance in mathematics, possibly because of the enrichment strategy. lastly, it was assumed that some learners and/or teachers may view the formal allocation of school time for mathematics as inadequate. for the question “what is the nature of the supplementary tuition offered?” the investigation focused on the form of supplementary tuition most preferred and why it was popular, as well as the amount of time spent on it. the focus of problem solving classes was also sought. the last question, namely, “who takes supplementary tuition?” mainly determined the gender and ratio of mathematics learners to those who take mathematical literacy. for the latter, the distinction between the mathematics and mathematical literacy learners was based on the fact that learners opting for mathematics in grades 10-12 are normally those who have shown prowess in the subject since mathematics is optional after grade 9 in south africa. findings the data show that of the learners who took part in the study, about 53% studied mathematics and the rest had enrolled for mathematical literacy. in terms of gender, mathematics was studied by more females than males. that is, 44% were male and the rest were females. for mathematical literacy, there was an equal split. the study also determined the satisfaction levels of learners with their performance. the data revealed that 48% of learners were not satisfied with their performance and about 42% of them felt that teachers were not giving them individual attention. figure 1 provides a graphical summary of the satisfaction level of the learners. the status of supplementary tuition in mathematics 40 figure 1: satisfaction levels figure 1 shows that more mathematics learners than mathematical literacy learners are not satisfied with their performance. about the same number of learners in the two subjects is satisfied with their performance and there were more mathematical literacy learners who were unsure about their views than those of mathematics. regarding the perceptions of learners about the importance of mathematics or mathematical literacy in their future careers, the data indicate that 90% regard a good pass in mathematics/mathematics literacy as important for their future careers. figure 2 presents an analysis of data on how the learners of the respective subjects responded. figure 2: the importance of a good mathematics/mathematical literacy pass figure 2 shows that an overwhelming number of learners considers a good pass in mathematics or mathematical literacy important for their future careers. given the importance learners attach to their future careers, it was necessary to establish whether they were content with the time allocated to mathematics/mathematical literacy in the school time table. figure 3 presents the results. in total, a substantial number of mathematics/mathematical literacy learners (42%) were not happy with the amount of school time allocated as compared to 20% who seemed happy with the time spent on mathematics. between mathematics and mathematical literacy, about 49% of mathematics learners wanted more school time for the subject as compared to slightly above 30% of mathematical literacy learners. according to the policy document of the national curriculum statement (2005), mathematics and mathematical literacy were each allocated 5 hours per week with 33 to 35 school weeks in a school year. satisfaction levels for mathematics & mathematical literacy 0% 10% 20% 30% 40% 50% 60% mathematics mathematical literacy total satisfied not satisfied unsure importance of a good pass in mathematics/mathematical literacy 0% 20% 40% 60% 80% mathematics mathematical literacy total important not important not sure david mogari; hanlie coetzee & riette maritz 41 figure 3: views of learners on school time for mathematics/mathematical literacy it emerged that teachers struggle with the volume of work in the new mathematics syllabus in grades 10 and 11 (the ncs syllabus had not been introduced in grade 12 at the time of study). about 92% of those teaching grade 10 mathematics mentioned that they struggle to complete the syllabus while 83% indicated that they were having problems with the grade 11 syllabus. fewer grade 12 teachers (29%) experienced problems with the old mathematics syllabus. problems were minimal with the completion of the mathematical literacy syllabus. only 6% of the grade 10 and 11 teachers stated that they experienced moderate problems completing the syllabus and none has serious problems. a substantial portion of teachers (43%) felt compelled to work fast despite learners finding it difficult to cope. about 47% of teachers used school time allocated for other activities, e.g. assembly time, library and music periods, and so on to catch up with the mathematics syllabus. a majority of teachers (91%) gave learners extra help after school hours where 40% of them mentioned that it was on regular basis and the rest said extra help was only given when needed. there were teachers (about 24%) who said they charged to give extra help to learners. the general view among learners (about 47%) was the new curriculum was much more demanding and as such the syllabus might not be completed. learners (about 7%) also complained that their teacher was often away on other school commitments. about 11% of learners thought that their teacher did not explain the contents of work clearly and there was no individual attention given by the teacher. it is for these reasons that learners felt that teachers should devise means to catch up by arranging for extra lessons. there were mathematics learners (about 49%) who wanted more school time devoted to the subject. teachers, on the other hand, indicated that they were frustrated by the misconceptions learners carried from the lower classes and also expressed some concern about the learners’ lack of commitment to their school work in mathematics. what also emerged from the data is that 91% of the teachers are adequately qualified to teach mathematics in grades 10-12 because they had post matriculation mathematics and about 18% of these teachers had a post-graduate qualification in mathematics. the data further show that 44% of the teachers had been teaching for longer than 20 years and only 21% had fewer than 9 years teaching experience. with regards to time spent on supplementary tuition by learners, the data show an average 1,67 hours per week. the mean number of hours spent on supplementary tuition from the beginning of the year up to mid-september, when the survey was done, was 18,23 hours. supplementary tuition offered in the participating schools is generally in three forms which are private tuition, vacation school and problem-solving classes. it turns out that problem solving classes, which were mainly dominated by the revision of past/model examination papers, was most preferred (about 83%) followed closely by private tuition at 81% and lastly vacation school at 38%. given that this was the first grade 11 group to be taught the new syllabus, the education authorities supplied schools with model examination papers so that teachers could adequately prepare learners for the end-of-year examination. the idea of vacation school was not popular because learners felt that they needed a break during the holidays. in fact, few vacation schools were offered in the area and about 38% of learners were not sure whether vacation school can help them improve their performance. in general, the proportion of mathematics learners who wanted supplementary tuition was significantly (t = 5,27, df = 288, p < 0,0005) more school time for mathematics/mathematical literacy 0% 10% 20% 30% 40% 50% 60% mathematics mathematical literacy total yes no unsure the status of supplementary tuition in mathematics 42 more than that of mathematics literacy. nevertheless, regardless of the popularity of supplementary tuition there was a range of reasons given by learners who did not take supplementary private tuition. the common ones were: cannot afford supplementary tuition; too much homework in other subjects; being too busy; transport problems; and having an excellent and experienced teacher at school. the data also showed that supplementary tuition was much more popular amongst mathematics learners compared to mathematical literacy learners. discussion the current study sought to investigate the status of supplementary tuition in the teaching and learning of mathematics/mathematical literacy. learners in the current study were taught by properly qualified and experienced teachers and yet they still felt it was important to participate in supplementary tuition. most probably, it is the reinforcement value associated with supplementary tuition that influenced learners to consider it positively. in this regard, it is asserted that the need for supplementary tuition by learners is not dependent on the teachers’ qualification and their years of teaching. the learners and teachers came from high performing schools that obtained an average of 80% plus in the matriculation results over the years. notably, the schools are located in a suburban area and are former model c schools. from a historical point of view, these schools were never disadvantaged in terms of resources and they generally drew learners from families that were not financially deprived. this is consonant with the findings by reddy et al. (2005, p. 2) that supplementary tuition generally targets high performing learners in urban metropolitan schools who can afford it. the findings from the current study, therefore, dispel the perception that generally associates supplementary tuition with low performing learners (ireson, 2004; paviot et al., 2007; smyth, 2008). what is also apparent from the current study is that there seems to be a range of factors that encourage learners to participate in supplementary tuition. these are: o the urge to do well in mathematics/mathematical literacy, o the significant role learners reckon mathematics/mathematical literacy play in their future careers, o time devoted to mathematics/mathematical literacy on a school time table is considered inadequate, o learners not being able to grasp content because teachers rush through it as a result of limited time available, o some teachers offer supplementary tuition to supplement their income, o seemingly some teachers provide supplementary tuition to help learners catch up, and o to provide extra help particularly to learners whom they consider having learning deficiencies. from this range of factors, there are those that are consistent with the view by baker et al. (2001, p. 9) that remedial strategy is the dominant reason for taking supplementary tuition and also the importance that learners attach to mathematics to gain access to certain careers. on the other hand, none of the factors is consonant with a finding by ireson (2004) that social desirability of tutoring tends to urge learners to participate in supplementary tuition and also with an argument by smyth (2008) that participation in supplementary tuition must be seen in the context of the competitive examination system in operation. furthermore, considering that in south africa mathematical literacy is compulsory even to learners with a deficient capacity and phobia for figures, one would expect far more mathematical literacy learners to participate in supplementary tuition than mathematics learners. the current study presents a different picture altogether. that is, preference for supplementary tuition is strong among mathematics learners. it may be due to the cognitive challenges mathematics poses, inadequate time available on school time table for mathematics, and an urge by learners to perform well, so that they can pursue their dream careers. there may still exist then/also quite some apathy, confusion and lack of quality/commitment among both learners and teachers in this regard, as part of the more perennial problem of an ailing education system – and this will be much bigger in non-former model c schools. it would also seem that participation in supplementary tuition is gendered. more girls than boys prefer supplementary tuition. bray (1999) came up with similar findings in the study that was carried out in egypt, malaysia, malta, taiwan and japan. a possible explanation for this is derived from fennema and hart (1994) and volman and van eck (2001), who report that girls are generally prejudiced in mathematics classes. on one hand, largely in classes taught by males, girls tend to be snubbed or ignored david mogari; hanlie coetzee & riette maritz 43 in the lesson’s activities. volman and van eck indicate that such a practice tend to instil among girls a sense that they lack capacity for mathematics or teachers simply do not have confidence in the girls’ ability to cope with mathematics. gender-stereotypical behaviour against girls, on the other hand, tends to be displayed by their male counterparts. for example, mogari (2002) found such behaviour manifesting in the cooperative learning groups during mathematics lessons. girls were denied full participation in the lesson’s activities by boys and subsequently boys outperformed girls. the overwhelming recognition by learners of the importance of a good pass in mathematics may well have prompted girls to seek supplementary tuition to circumvent the teaching and learning difficulties they experience in their classes. of the three forms of supplementary tuition that are found in the current study, revising past examination papers is the most popular and is closely followed by private tuition, with vacation tuition being the least popular. it stands to reason that revision of examination papers is popular because of its easy availability and the low cost involved. in fact, past examination papers with worked solutions are on sale at a reasonable price in most stationery stores in south africa. furthermore, the department of education has also made available past/model examination papers through the internet and newspapers, and this mode of delivery is cost effective. there are schools that are supplied daily with free editions of newspapers containing supplements of various learning materials including past/model examination papers. those schools that are not supplied with free newspapers buy them. probably the relative ease with which past/model examination papers is made available to learners has somehow influenced the learners’ preference for this form of supplementary tuition. what is also notable is a fairly considerable proportion of learners (about 38%) being negative about vacation school. some learners feel they need a break from their busy learning schedule while others are apprehensive about its potential to help them better their achievements. with regards to the former, learners feel they would rather utilise a form of supplementary tuition that does not encroach on their private time. a possible explanation, for the latter, is that in most cases the lesson structure of vacation school is similar to the ordinary, normal school lessons. perhaps learners think they might still encounter learning difficulties they usually experience during the normal school lessons. for private tuition, it is probably highly preferred because it provides an opportunity for a learner to have special individual attention that may tend to be very helpful. this observation is consistent with the view by ireson (2004) that learners value the one-on-one attention that they get in supplementary tuition and see it as an opportunity to enhance understanding. but it is speculated that the financial expenses associated with private tuition, made it not to be the most preferred form of supplementary tuition. another possible reason for the popularity of working on model/past examination papers is its flexible nature, where learners can work individually or in a group whenever it is convenient for them. however, there are factors that tend to impede learners from participating in supplementary tuition regardless of their willingness to do so. the factors, among others, are social, economic and educational in nature. the fact that some learners could not participate in supplementary tuition due to financial problems corroborates allusions by ireson (2004) and paviot et al. (2007) that there is a strong relationship between learners’ ability to participate in supplementary tuition and their socio-economic background and by bray (1999) that supplementary tuition seems to create and perpetuate social inequalities. for the latter, it should be noted that learners with financial difficulties cannot benefit from supplementary tuition regardless of its proven effectiveness (camp, 1990; jacob & lefgren, 2002; lavy & schlosser, 2004; macbeath et al., 2001; posner & vandell, 1999). learners who are unable to participate in supplementary tuition have to rely solely on the normal school tuition which, as evident in the current study, seems never to be devoid of problems. it is suggested that further investigations identify all these problems and possibly suggest ways and means to find sustainable solutions to them. what has also emerged serendipitously from the study are some of the challenges posed by the new curriculum. firstly, given its much more context and activity based nature, the new curriculum requires different timetabling arrangements. it would appear the time set aside for mathematics lessons has to be increased so that the entire prescribed syllabus could be covered. secondly, teachers need to adapt their teaching knowledge and skills to the instructional requirements of the new curriculum. this will enable them to plan the lessons accordingly and be able to appropriately time the different steps of the lessons. through these, each offered lesson will properly fit the allocated slot on the time table, thus enabling the the status of supplementary tuition in mathematics 44 teachers to teach effectively and complete the syllabus. lastly, teachers need to be given the necessary training so that they could properly implement the new curriculum and achieve its desired goals. in conclusion, the study has revealed some rather significant findings about supplementary tuition. that is, it is not only confined to poor performing learners; parents are anxious about ensuring their children acquire plausible education; learners’ participation in supplementary tuition is driven by an urge to acquire good results and enhance opportunities to study what may be referred to as financially rewarding science-related careers (e.g. engineering; actuarial science; medicine and so on); some teachers participate in supplementary tuition for financial benefits; girls use supplementary tuition to make up for the teaching and learning challenges they experience in their classes; and there are still problems associated with ncs implementation in the schools. references baker, d. p., akiba, m., letendre, g. k., & wiseman, a. w. (2001). worldwide shadow education: outsideschool learning, institutional quality of schooling, and cross-national mathematics achievement. educational evaluation and policy analysis, 23(1), 1-17. bray, m. (1999). the shadow education system: private tutoring and its implications for planners. paris: unesco, international institute for educational planning. bray, m. (2003). adverse effects of private supplementary tutoring: dimensions, implications and government responses. comparative education research centre, the university of hong kong. bray, m., & kwok, p. (2003). demand for private supplementary tutoring: conceptual considerations and socio-economic patterns in hong kong. economics of education review, 22(6), 611-620. camp, w. (1990). participation in student activities and achievement: a covariance structural analysis. journal of educational research, 83(2), 272-278. department of education. 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(2001). knowledge and skills for life: first results from the oecd programme for international student assessment (pisa) 2000. paris: oecd publications. pandor, n. (2005). address by the minister of education, (ms naledi pandor, mp) at the agrey klaaste mathematics, science and technology educator of the year award. department of education. retrieved from http://education.pwv.gov.za. paviot, l., heinsohn, n., & korkman, j. (2007). extra tuition in southern and eastern africa: coverage, growth, and linkages with pupil achievement. international journal of educational development, 28(2), 149-160. david mogari; hanlie coetzee & riette maritz 45 petterson, l. (1993). japan’s ‘cram schools’. educational leadership, february, 56-58. posner, j., & vandell, d. l. (1999). low-income children’s after-school care; are there beneficial effects on after-school programs? developmental psychology, 35, 868-879. reddy, v., lebani, l., & davidson, c. (2003). schools out …or is it? out of school interventions for mathematics, science and computer studies for secondary school learners. pretoria: human science research council. reddy, v., berkowitz, r., & mji, a. (2005) supplementary tuition in mathematics and science: an evaluation of the usefulness of different types of supplementary tuition programmes. pretoria: human sciences research council. rochford, k., sokopo, z., & kleinsmith, c. (1997) improving the teaching of science and technology in the new south africa: concurrence between the policy preferences of lecturers, teachers and students. global journal of engineering education, 2(1), 103-118. smyth, e. (2008). the more, the better? intensity of involvement in private tuition and examination performance. educational research and evaluation, 14(5), 465-476. volman, m., & van eck, e. (2001). gender equity and information technology in education: the second decade. review of educational research, 71(4), 613-634. abstract introduction theoretical framework methodology data analysis ethical consideration discussions and conclusions implications and recommendations acknowledgements references appendix 1 about the author(s) judah p. makonye marang centre for science and mathematics education, university of the witwatersrand, south africa citation makonye, j.p. (2017). pre-service mathematics student teachers’ conceptions of nominal and effective interest rates. pythagoras, 38(1), a307. https://doi.org/10.4102/pythagoras.v38i1.307 original research pre-service mathematics student teachers’ conceptions of nominal and effective interest rates judah p. makonye received: 30 may 2016; accepted: 02 mar. 2017; published: 26 apr. 2017 copyright: © 2017. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract the general public consumes financial products such as loans that are administered in the realm of nominal and effective interest rates. it is debatable if most consumers really understand how these rates function. this article explores the conceptions that student teachers have about nominal and effective interest rates. the apos theory illuminates analysis of students’ levels of conception. seventy second-year mathematics students’ responses to grade 12 tasks on effective and nominal interest rates were analysed, after which 12 students were interviewed about their mathematical thinking in solving the tasks. the findings varied. while some students could not do the tasks due to erratic use of formulae (algebra), i ascertained that some students obtained correct answers through scrupulous adherence to the external prompt of formulae. most of those students remained stuck at the action and process stages and could not view their processes as mathematical objects. a few students had reached the object and schema stages, showing mature understanding of the relationship between nominal and effective interest rates. as most students remained at the operational stages rather than the structural, the findings accentuate that when teaching this topic, teachers ought to take their time to build learners’ schema for these notions. they need to guide their learners through the necessary action-process-object loop and refrain from introducing students to formulae too soon as this stalls their advancement to the object and schema stages which are useful in making them smart consumers of financial products. introduction this article is about mathematics major student teachers’ conceptions in the area of financial mathematics with particular reference to effective and nominal interest rates. second-year mathematics major students’ responses to grade 12 financial mathematics tasks are analysed using the process of object theories (dubinsky, 1991; gray, pitta & tall, 2000; gray & tall, 1994; piaget, 1985; sfard, 1991; tall, 2007). to help understand how students think about mathematics processes and objects, the author of this article, who is a lecturer to these students, uses this methodology to inform good preparation for teaching of student teachers. financial mathematics is not only an enriching mathematics topic in its own right, but it is also a topic of mathematics with practical applications in daily life for everyone. yet many mathematics teachers do not understand the basics of this topic let alone how to teach it (pournara, 2013). in this article i argue that exploring student teachers’ conceptions of financial mathematics can help researchers and teachers to deal with key epistemological factors which could inform stakeholders to better handle mathematics topics in general and the financial mathematics topics in particular. the nominal interest rate is the annual interest rate without any reference to compounding. the effective interest rate is derived from the nominal interest rate and yields the actual return on investment over a compounding period, which is often more than once per year; it could be half-yearly, quarterly, monthly or even daily. it is unusual to call a rate of interest a nominal rate unless it is compounded more (or less) frequently than once per annum. exploring students’ epistemological difficulties through script analysis and interviews is an essential component of quality teaching (makonye, 2012; moru, qhobela, poka, & nchejane, 2014; nesher, 1987). it is important because it informs researchers and practitioners about the difficulties experienced in specific mathematical processes and objects (dubinsky, 1991; gray et al., 2000; gray & tall, 1994; sfard, 1991) that mathematics learners encounter. learners’ conceptions provide their teachers with insight into their thinking, which can be used to inform teaching (borasi, 1994; gallagher, 2004). their conceptions can reveal what they think about certain mathematical work. pedagogical content knowledge (shulman, 1986) constitutes the teacher’s knowledge of the likely mistakes that learners are prone to when they encounter particular mathematical concepts as individuals or in groups. pedagogical content knowledge is the interface between subject and pedagogical knowledge and is referred to as specialised content knowledge (shulman, 1986), which refers to having the pedagogical knowledge to teach particular content to the learners. shulman (1986) concludes that educators would not be effective if they were not knowledgeable about how learners form amateur concepts around particular scientific concepts (vygotsky, 1986). adler and ball (2009) propose that mathematics knowledge for teaching is multidimensional and topic specific. teachers with mathematical pedagogical content knowledge are more aware of learners’ likely thinking patterns, which empowers them to introduce teaching interventions that help learners to learn mathematics more successfully. so teachers ought to establish learners’ thinking patterns that produce errors. teachers with pedagogical content knowledge can formulate situations that probe learners to elicit their erratic thinking on particular mathematical objects. this confrontation of learners with their misconceptions helps to induce cognitive conflict (drews, dudgeon, hansen, lawton & surtees, 2005). once such misconceptions are manifested, teachers can devise pedagogical approaches to help learners understand that their thinking is in fact incomplete. for these reasons, it is important for mathematics teachers and their students to negotiate wrong conceptions as they inhibit mathematics learning. over the last three decades, many articles have been written on the misconceptions that learners show when learning mathematics (e.g. cockburn & littler, 2008; davis, 1984; erlwanger, 1973; green, piel & flowers, 2008; olivier, 1989; shahrill, 2013). these articles report that the errors exhibited, whether shared or idiosyncratic, follow carefully reasoned patterns and are quite predictable, if one understands them. in particular, olivier (1989), in his seminal paper, posits that the misconceptions are in the main due to ‘patchwork’ (p. 5), ‘generalising over numbers’ (pp. 6–7), ‘generalising over operations’ (p. 8), ‘meanings’ (p. 9) and ‘interference’ (p. 11). this implies that most misconceptions are formed as learners attempt to assimilate or accommodate new mathematical objects into their existing schema. despite this research, there have been very few articles that explore student teachers’ and teachers’ misconceptions in school mathematics and how they form them. the conceptions that mathematics teachers’ have are inadvertently passed on to learners even if incomplete or wrong. further, teachers avoid teaching mathematics concepts about which they do not have enough knowledge. objectives in relation to process-object theories of constructing mathematical concepts (dubinsky, 1991; gray & tall, 1994; sfard, 1991), the research aims to explore the conceptions that teacher students majoring in mathematics have about the grade 12 financial mathematics concepts of effective and nominal interest rates. research question what are mathematics teacher students’ conceptions of nominal and effective interest rates in relation to apos and process-object theories? significance of the research since the onset of democracy, the average performance of south african learners on periodic international comparative mathematics tests has been consistently under expectation (howie, 2001; reddy et al., 2012). one would suspect that such attainment on the part of learners might not be wholly their fault. it might be that some of the teachers who teach these students are not necessarily more knowledgeable others in vygotskian terms. indeed, some mathematics teachers were recently requested to write their learners’ grade 12 mathematics examinations (bansilal, mkhwanazi & brijlal, 2014). they scored an average mark of 57% and a quarter of the teachers attained below 39%. similarly, many teacher students do not fully understand the school mathematics they teach. it is important to study how teacher students formulate the conceptions or misconceptions of mathematics that they have. the process-object theories help to explain how learners assimilate and accommodate new mathematical knowledge. theoretical framework in this study, the process-object theories of forming mathematical structures inform the exploration of teacher students’ conceptions about effective and nominal interest rates. according to gilmore and inglis (2008), these influential theories could be divided under three main views: the apos theory (arnon et al., 2014; dubinsky, 1991); process-object theory (sfard, 1991) and procept theory (gray & tall, 1994). dubinsky (1991) and arnon et al. (2014) propose a four-stage process of mathematics concept development (apos) due to ‘individuals’ tendency to deal with perceived mathematical problems by constructing mental actions, processes, objects and organising these into schemas to make sense of the situations and solve problems’ (dubinsky, 1991, pp. 101–102). action an action is a physical or mental transformation of a mathematical entity in response to outside stimuli. actions may require initiation or mediation by a teacher or peer to direct steps that are explicitly taken towards a goal. it is the beginning stage of learning and making sense of a mathematical situation. thus, an action conception leads to an operation external to a learner’s mind. process when an individual reflectively repeats an action they may interiorise it into a process. interiorisation occurs when someone can carry out an action mentally. when someone reflects upon an action without actually engaging with it, they are said to have interiorised that action into a mental process (aineamani, 2015; cottril et al., 1996). when a mathematical entity is viewed as a process, it requires a computation to be done on that entity (sfard, 1991). a process is operational and invokes the metaphor of a verb. a process conception leads to an operation internal to a learners’ mind. object when a mathematical entity is ‘seen as an object’ it is seen as if it were ‘a real thing that exists in space and time’ (sfard, 1991, p. 4). as aineamani (2015) explicates, ‘a learner can be shown expressions such as 2x + 4 and 7x to show him/her the features that make an algebraic expression an object’ (p. 60). a mathematical object is structural and invokes the metaphor of a noun. in real life, loans and loan purchases such as car loans and mortgages, bank savings, bond investments, inflation, depreciation and others governed by regular compounding of interest rates are mathematical objects in the sense of apos theory. according to cottril et al. (1996), ‘an object is constructed through the encapsulation of a process. … [it] is achieved when the individual becomes aware of the totality of the process, realizes that transformations can act on it’ (p. 4). thus when a process becomes an object, it becomes a thing, it achieves permanence, it becomes an entity in its own right; it has become a noun (davis, 1984). dubinsky (1991) refers to this transformation as the encapsulation of a process into an object. as aineamani (2015) argues, ‘reification (or encapsulation) enables the learner to see a familiar mathematical expression in a totally new light’ (p. 61). schema when actions, processes and objects are revisited and a learner has a bird’s eye view of them, they form a schema of the mathematical entity. this schema is organically linked to other schemas. analysis and reflection on a schema can generate yet another cycle of actions and processes so that new, more advanced mathematical objects and schemas can be formed. if individuals can carefully compare financial products such as mashonisa loans, ponzi get-rich-quick schemes, bank loans, car loans and so on and make an informed decision, they have the schema on objects that are governed by nominal and effective interest rates. individuals without this schema are in danger of losing out on their life’s savings to schemers. the schemers do their mathematical calculations using nominal and effective interest rates very carefully to hide the disastrous financial effects for their unwary clients. dubinsky (1991) explains that when someone is developing the understanding of a mathematical idea, it does not necessarily happen in a linear process; rather, it is dialectic. similarly, sfard (1991) argues that at first processes or operations are performed on a familiar object which is the interiorisation stage. at the second stage, referred to as condensation, the learner is able to reproduce the operation in their mind without actually doing it. at the third stage, if the learner suddenly sees the familiar process in new light; the process becomes a static structural object that can become the subject of even more advanced processes. sfard refers to this as the reification of a process into an object. gray and tall (1994) agree with the process-object theories of mathematics concept construction, but suggest that the movement from process to object and vice versa is enabled by a procept. they define an elementary procept as the ‘amalgam of three components: a process which produces a mathematical object, and a symbol which is used to represent either process or an “object”’ (p. 224). the procept is a mathematical symbol which often is a barrier to success in learning mathematics if students do not understand it (gilmore & inglis, 2008). a major procept in this study is the nominal effective interest rate formula (see figure 1). figure 1: the nominal-effective interest rate formula. but what really is the difference between mathematical processes and objects? sfard (1991) argues that processes focus on operations and procedures. in this research, finding the effective interest rate or the nominal interest rate using the formula shown in figure 1 is a manipulation and therefore a process. in this formula (see figure 1), i is the effective interest rate, i(m) the nominal interest rate and m the number of times interest is paid per annum. piaget (1985) argues that there are primarily two types of objects: (1) a physical one such as a triangle, which is visible, known as figurative or perceived objects, and (2) operative which cannot be seen, but can be only be thought of, such as the number 5. piaget refers to these as conceived or operative objects. they exist because we can mentally do something with them. constructivists argue that learners are not explicitly taught the misconceptions they have, but make them by themselves (confrey & kazak, 2006; davis, 1984). since learning does not occur in a vacuum, students’ conceptions, however mistaken, are the result of their reigning knowledge with which they connect and interpret new knowledge. if new knowledge is connected to current knowledge that is incorrect, another error will occur. in addition, even if reigning knowledge is correct, problems can result while connecting it to new knowledge. new knowledge has to be integrated with something a learner already knows, however tenuous. behaviourist learning theories (mcleod, 2007; todes, 2002) view students’ conceptions that are errors as pernicious in the learning process. their stance is that once detected, errors must be weeded out and corrected to make them extinct. on the other end of the spectrum, constructivists view errors as useful resources in teaching and learning mathematics (borasi, 1994; nesher, 1987). they regard errors as learners’ attempts to construct meaning from a learning context. to constructivists, once a learner shows an error, the teacher must be strategic about it. they must refrain from immediately supplying a correct answer as this would be missing a learning opportunity for the learner. rather, the teacher needs to help the learner to reconsider their positions by requesting the learner to compare their answers with those of their peers. that way, the learners encounter peer-induced cognitive conflict which can help them to reconcile their thinking. according to davis (1984), a student’s thinking that results in errors is not random; on the contrary, errors turn out to be very regular and systematic. they have specificity and determinism and it is often possible to predict exactly which wrong answer is most likely to be given by a particular student. so systematic wrong answers given by a student often provide clues as to how the student is thinking about a class of mathematical processes and objects. according to davis, students use frames they have to interpret and process new knowledge. assimilating new knowledge in old frames is often problematic as the frames might be overstretched to generalise to new platforms on which they are not appropriate. methodology the research used a qualitative research design. eisner (1991) proposes that a good qualitative study assists to ‘understand a situation that would otherwise be enigmatic or confusing’ (p. 58). at first the teacher students wrote tasks on nominal and effective interest rates which required them to do calculations as well as offer written explanations regarding the differences between those interest rates (see appendix 1). then, some were interviewed about their notions of nominal and effective interest rates to support their answers. seventy second-year mathematics major students of both genders were given previous years’ grade 12 financial mathematics examination tasks on effective and nominal interest rates. their scripts were collected and responses analysed. students’ responses to tasks were first analysed under the categories of correct, partially correct and incorrect, as well as not attempted. after this, 12 student teachers were interviewed in pairs to elicit the thinking behind their responses, whether the responses were correct or wrong. the students chosen for interview constituted a stratified sample by performance and gender. the interviews were analysed so that the stages where students faltered in concept formation could emerge. reliability and validity to ensure reliability of the research, data were collected through both written tasks and interviews. this allowed for probing of students in the interviews to see if they stuck to their written answers and to determine their thinking on nominal and effective interest rates. there was also internal consistency reliability to assess the degree to which different tasks involving the same concept produced comparable results (see appendix 1). construct validity helps to guarantee that the measure essentially measures the intended construct, in this case students’ conceptions on nominal and effective interest rates. this was the most important form of validity in this research. i selected nominal and effective interest rate tasks from 2012 and 2013 mathematics matric examinations as well as from textbooks approved by the department of basic education. this increased the face validity of the research. to be faithful to the theoretical framework, students’ responses in scripts and interviews were analysed focusing on whether the conceptions found were at the action, process, object or schema stages of concept formation or in transition between one stage and another. thus, teacher students’ conceptions were analysed through the lens of how they constructed knowledge with the process to object constructions. data analysis data were analysed both deductively and inductively. deductive analysis was informed by the apos framework for building mathematics knowledge. inductive analysis occurred through grounded theory. grounded theory is a continuous process of ‘constant comparison’ (glaser & strauss, 1973, p. 36) to come up with categories and themes underpinning the data. this happens through de-contextualising and re-contextualising data to come up with meanings that help to answer the research questions. when students were given written tasks on effective and nominal interest rates, the work was marked and their performance is shown in graphs (see figure 2 and figure 3). figure 2: percentage performance of teacher students on nominal and effective rate tasks. figure 3: boxer and whisker plot for teacher students’ performance. figure 2, figure 3 and table 1 show that the performance for the students on the written tasks was wanting. in all cases the mean mark per item was lower than half the possible mark. this is the reason why it was important to probe and elicit students’ conceptions of the constructs under question in light of the apos theory. table 1: some statistics on students’ performance on the tasks. i now briefly analyse some written work. one of the items was: question 2: mrs ndlovu invested r10 000 in the bank with interest compounding monthly. after one year, she had r10 750 in the bank. calculate 2.2 the effective interest rate and 2.3 the nominal interest rate. for calculating effective interest rate one student wrote: this emanates from the following formula: p is the present value, r is the nominal interest rate, y is the number of years invested, n is the number of compounding periods per year and fv is the amount the present value accumulates to in y years. the student used formulaic reasoning. given that r10 000 grew to r10 750 in one year, there was no need to go the formula way. they should have used the object stage thinking that r750 interest was earned on a principal of r10 000, so the effective interest rate must have been 7.5%. thus the student was operating at the action level where they used the formula as a way to process external stimuli. the substitution was quite correct if they wanted to find the nominal interest rate, but then the student divided on both sides by ‘log’ as if ‘log’ was an algebraic variable representing a number so that it can be ‘cancelled’: this was a conception of equation balancing, prefaced by the rule ‘you do the same thing to both sides of an equation and they are still equal’ (pimm, 1987, p. 20). i regard this conception as at action level. students who do not fully understand the limits of formulae are at the action stage, not having interiorised it into a process. in trying to answer the same item, some students mixed up the periods, for example , in this case quarterly interest rates (denoted by the number 4) and monthly payments (denoted by the number 12). this is a failure to operate at action level. improper action level performances are made. students do not understand the use of the formula, how to choose its input variables, the time and interest rates per period, so that they can process properly. some students seemed to have proper action and process conceptions as evidenced by correct answers found through use of the formula. however, i wished to be convinced whether they had formed in their minds permanent objects or schemas of nominal and effective interest rates. such doubts could only be laid to rest through interviews. i sat in a classroom and interviewed students (see box 1). box 1: participant n and participant p. participant n is confused as she thinks that if the interest is stated once per year then it must be nominal. she is at the action level of conception as she clearly states that she sticks to the formula. her interpretation that the nominal interest rate is 7.25% is purely directed by her faith in the formula. to her, the formula seems to be the ‘object’ – the end in itself – which really is not the case. participant n clearly shows that her mathematical thinking on these concepts is governed by formulae – ends in themselves – but she is clearly unhappy because she complains that any mistake in the substitutions leads to disaster (see box 2). box 2: participant s. participant s thinks the interest rates are different because they have different formulae. that means that she is at the action conception level. she has not yet interiorised the actions into processes. to her they are different ‘objects’ because they look different not because they essentially would earn different amounts of interest at the end of the year (see box 3). box 3: participant q and participant r. while participant q’s responses are not much different from the other students, participant r said the ‘effective interest rate is better because it is more reliable’. this he said even though the effective interest rate yielded the same interest amount as the nominal interest rate. clearly his view is a belief. at the end of the interview with this pair, he said, ‘most banks give nominal, others give effective interest rate. the reason is to protect them from liability.’ this was an important remark, but it was clear that he could not say why the effective interest rate protects these banks from liability. he also said, ‘effective tells you how much you get each month (error). nominal you get it once a year.’ this pair was very cooperative but though they had correct numerical answers they still held an action conception of these concepts. i did not see the ‘aha!’ i was anticipating by doing this interview with these good students (see box 4). box 4: participant t. clearly participant t had achieved the object and schema conception level i was looking for (see box 5). box 5: participant y and participant k. participant y and participant k did not rely on formulae; they used common sense as we were speaking to each other. they never, unlike participant q and participant r, clutched for their bags to look for formulae from books or calculators when asked a question. to me they had reached schematic conception of nominal and effective interest rates. this is because they had interiorised the actions to processes and encapsulated the processes to objects. they were willing to revisit their conceptions and did not regard them as fixed; this is exemplified by participant y saying ‘i have no idea but you get the same….’ her conceptions came out as appropriate even when she said she was not too sure. ethical consideration ethical approval of the research was obtained from the ethics committee in education of the faculty of humanities acting on behalf of the university senate. after obtaining approval each participant gave their written informed consent to participate in the research. they also gave informed consent for the publication of the research in a research journal. in particular the students were keen to see the research published so that it would help them in preparing to teach the topic of nominal and effective interest rates. discussions and conclusions statistical data on performance on nominal and effective interest rate tasks (see figure 2 and figure 3) show that some teacher students had not made adequate advances on the apos stages (arnon et al., 2014; dubinsky, 1991) in constructing appropriate conceptions. script analysis and interview data show that students’ stages for understanding effective and nominal interest rates sometimes shifts; it moves back and forth often between the action and process conceptions as proposed by dubinsky (1991). while some teacher students reached the object and schema stages of concept formation on nominal and effective interest rates many others were at action level, and a few at the process level. in their productions, students were mainly aided by the procept (gray & tall, 1994) of the formula that connects actions, processes and objects. this manifested in the interviews where the same students responded with object-based thinking as well as process-based and action-based thinking on the same task. for example, participant r (and participant q) said ‘i do not know the difference [between nominal and effective interest rates] … i just work it out’, which is action-based thinking. at the same time they said ‘we just work it out…we just work it out. … 12% effective is better because it’s more reliable’, which seems both object-based and action-based thinking. trying to interpret this further participant q had not really reached object level conception as he could not justify it. on investigating and assessing students’ written and interview responses to effective and nominal interest rates tasks, four categories of students’ conceptions emerge. the first category consisted of students who failed to get correct answers because they could not scrupulously use the formula in order to obtain correct answers. thus these students failed to operate at the processes stage; they had not interiorised the actions (dubinsky, weller, mcdonald & brown, 2005; sfard, 1991) so they were at the lowest action stage. the second group of students was quite scrupulous and meticulous in using their formulae, which resulted in them obtaining correct answers. however, on interviewing some of these students i realised that they were fixated at the process stage in that when asked to explain or demonstrate their understanding of effective and nominal interest rates they showed clearly that they had not reached the object level of conception and were comfortably poised at the process stage. they thus have interiorised the action but not yet condensed and reified the process into an object, a standalone mathematical entity (sfard, 1991). a third, but tiny, group of students operated at the object level of understanding in that when asked to explain the difference between the effective and nominal interest rates they never considered using their calculators, but just reasoned and verbalised their conceptions. this was in contrast with the second group who were quick to grab their calculators and textbooks to get the formula in order to the answer the same questions. the rest of the discussion expands these categories. there was also a fourth category with very few students. category 1: activity conceptions in this category, blind and erratic substitution into financial mathematics formulae was common. participant n and participant s fall in this category. for example, participant n said ‘sometimes you forget the + sign in a formula … minus should be there … sometimes it should be plus’ and added ‘the most difficult part of financial maths is defining the difference between nominal and effective interest rates.’ one would assume that students like participant n would have difficulty in using the formula and as a result get wrong answers. these students are still at the activity level (dubinsky et al., 2005). other practitioners strongly believe that if a learner cannot use a formula to correctly get an answer, they have not reached the activity level. i think some learners who cannot manipulate the formula may actually be at the object level, reasoning that more frequent interest payments result in the effective rate paying out more interest than the nominal interest rate. such students may have difficulties in aligning the formula in terms of time and periodic interest rates to substitute in the formula. that is why one cannot always be absolutely sure that a student is operating at a particular apos stage. category 2: process conceptions these are the students who got correct answers for wrong reasons. these students were satisfying the ritual of getting the correct answers to get the approval of their teachers and getting good grades but without understanding the gist of what they were doing. many of these students got correct answers through using formulae they did not understand. examples of these students are participant r and participant q (see box 3). participant s was at the process stage, when referring to the different interest rates, she said ‘they are different because there are two different formulas for that’. category 3: object conceptions these were the few students who actually had achieved the object of learning conception. they had already encapsulated the processes into objects ready to incorporate them in their schemas. for example, on item 2: researcher: what is the difference between effective and nominal interest rates? participant t: effective is the actual interest you will be given. nominal is the stated. if nominal is 12% stated, the effective could be a little more than 12% so participant t operated at object level. he completely understood and gave reasons for his stance. participant y said, ‘the effective is better, i am thinking of the equation. the effective gives more interest because its compounded more.’ participant y also had arrived at this level. participant p also understands as he said, ‘in nominal … it’s fixed in a period of time; will remain the same. effective interest rate accumulates over time directed by the nominal. the accumulated interest is the effective rate.’ category 4: schema conceptions students like participant t and participant y had clearly arrived at the schema stage of conception. this was shown in their brightly lit eyes, and their exclamations of ‘hoo’. they knew what was happening and the traps that lay in the questions. the research question was: what are mathematics teacher students’ conceptions of nominal and effective interest rates in relation to apos and process-object theories? i report that most teacher students are at the action conception on these concepts and seem unaware that their conceptions are unsatisfactory and incomplete. they need to advance their conceptions to the desirable object and schema stages. the fact that formulae help them to get correct answers seems to stall their efforts to learn more. thus, most teacher students in this research showed that their understanding is operational (at action or process stages) rather than structural (at object or schema stages) (sfard, 1991). the operational conception is indicated by the increased amount of time they spend on solving tasks with the use of calculators and formulae compared to those students who use object-based strategies. much fewer teacher students in this research have object and schema conceptions on nominal and effective interest rates. gilmore and inglis (2008) have shown that object-based thinking is more economical and more powerful than process-based thinking. this is also shown in this research as object-based teacher students used discourse to come up with balanced reasons for their positions. implications and recommendations some students said to me: ‘at school we were just given formulae’. this suggests that some of the students’ incomplete conceptions result from teachers who want learners to obtain answers quickly, and are not concerned about developing the lasting and more powerful object and schema conceptions. the implication for teaching mathematics of this is that teachers must not rush to introduce calculators and formulae to learners when they are teaching nominal and effective interest rates. this hinders learners from constructing the object and schema conceptions which are the ultimate goals for teaching these topics, in order for learners to be financially literate. rather, teachers must encourage their learners to learn about these concepts inductively, through engaging in numerical investigations and exercises that help to make the apos constructions. practical real-life street investigations on loans and loan products (e.g. mashonisa loan sharks versus bank loans, ponzi schemes, fixed deposit savings and others) go a long way in developing mature conceptions related to nominal and effective interest rates in mathematics teacher students. the recommendations are that more studies be done on how the apos theory may be used for research in financial mathematics education. acknowledgements my special thanks go to my colleagues marie weitz and bharti pharshotam for encouraging me to do the 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(2007). developing a theory of mathematical growth. zdm: the international journal on mathematics education, 39, 145–154. todes, d.p. (2002). pavlov’s physiology factory. baltimore, md: johns hopkins university press. vygotsky, l. (1986). thought and language. cambridge, ma: mit press. appendix 1 written tasks james invested r50 000 in the bank for 1 year. the nominal interest rate was 8,5%. find the effective rate. (3) mrs ndlovu invested r10 000 in the bank with interest compounding monthly. after one year, she had r10 750 in the bank. 2.1 what can you say about the effective and nominal interest rates here? calculate: 2.2 the effective interest rate (2) 2.3 the nominal interest rate (3) interview questions what is nominal interest rate? what is effective interest rate? why do we need to distinguish between the two? what is the difference? someone invests r1000 for a duration of one year and is awarded an interest of r120. what is the annual nominal interest rate? what is the annual effective interest rate? in the above case, the r1000 was deposited at r% compounded monthly. which would be a better investment, the one with the annual nominal interest rate or the own with the interest rate that was compounded monthly? abstract introduction purpose of study theoretical framework research methodology ethical considerations quality criteria results and discussions conclusion acknowledgements references about the author(s) satsope maoto department of mathematics, science and technology education, school of education, university of limpopo, south africa kwena masha centre for academic excellence, university of limpopo, south africa kgaladi maphutha department of mathematics, science and technology education, school of education, university of limpopo, south africa citation maoto, s., masha, k., & maphutha, k. (2016). where is the bigger picture in the teaching and learning of mathematics? pythagoras, 37(1), a338. http://dx.doi.org/10.4102/pythagoras.v37i1.338 original research where is the bigger picture in the teaching and learning of mathematics? satsope maoto, kwena masha, kgaladi maphutha received: 22 may 2016; accepted: 20 sept. 2016; published: 16 nov. 2016 copyright: © 2016. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract this article presents an interpretive analysis of three different mathematics teaching cases to establish where the bigger picture should lie in the teaching and learning of mathematics. we use pre-existing data collected through pre-observation and post-observation interviews and passive classroom observation undertaken by the third author in two different grade 11 classes taught by two different teachers at one high school. another set of data was collected through participant observation of the second author’s year 2 university class. we analyse the presence or absence of the bigger picture, especially, in the teachers’ questioning strategies and their approach to content, guided by tall’s framework of three worlds of mathematics, namely the ‘conceptual-embodied’ world, the ‘proceptual-symbolic’ world and the ‘axiomatic-formal’ world. within this broad framework we acknowledge pirie and kieren’s notion of folding back towards the attainment of an axiomatic-formal world. we argue that the teaching and learning of mathematics should remain anchored in the bigger picture and, in that way, mathematics is meaningful, accessible, expandable and transferable. introduction our major concern in this article lies with pursuing the question: where is the bigger picture in the teaching and learning of mathematics? it is prompted by the kinds of tasks, questions, classroom interactions and targeted content that ground mathematics teaching and learning within and across the different educational levels. in most cases, these teaching and learning activities seem to lack coherence, lack focus on important mathematics and lack appropriate articulation. one of the contributing factors that always affects students’ mathematics performance is the fact that a variety of teaching styles are to be found in operation in mathematics lessons, each depending on the teacher’s knowledge (skills and attitudes) of mathematics and the teacher’s knowledge about teaching mathematics. hiebert and grouws (2007) claim that the nature of classroom mathematics teaching significantly affects the nature and level of students’ learning. this claim seems obvious, especially as students have different learning styles and for a teacher to produce quality learning requires working well in all learning style modes. for tall (2006), success comes from focusing on the most generative ideas, not from covering detail again and again. thus, if attaining success comes in that manner, it suggests that in teaching or developing any mathematics concept we need to seek out the generative ideas or big ideas (charles, 2005) that are at the root of more powerful learning. a big idea, as proposed by charles (2005, p. 10), is ‘a statement of an idea that is central to the learning of mathematics, one that links numerous mathematical understandings into a coherent whole’. to some extent charles’s definition of a big idea contributes to our thinking of what and where a bigger picture in the teaching and learning of mathematics is. big ideas make connections. good teaching, we argue, should make these connections explicit, meaningful, accessible, expandable and transferable. for each big idea there are mathematical understandings. mathematical understandings of a number of different big ideas are important ideas that students need to learn because they contribute to an understanding of the bigger picture of mathematics. this is explained well by charles (2005), who says that some mathematical understandings of big ideas can be identified through careful content analysis, but many must be identified by hermeneutic listening to students (davis, 1997), recognising common areas of misconceptions, alternative conceptions or confusion, and analysing issues that underlie those misconceptions, alternative conceptions or confusion. according to tall (2011), part of a bigger picture is also seeing a student becoming capable of looking at a recurring pattern and seeing many different ways of interpreting and representing it, for example, in words, numerically, algebraically or graphically. purpose of study the purpose of the study was to pursue the question: where is the bigger picture in the teaching and learning of mathematics? theoretical framework over the years, researchers have generally defined good mathematics teaching implicitly, with focus on various processes, such as reasoning and problem-solving (wilson, cooney & stinson, 2005), students’ and teachers’ explorations, constructing and mathematising (tarlow, 2014). rather than a discipline or body of knowledge (concepts, skills) to be transmitted, mathematics is defined from the perspective of ‘mathematising’: the activity of interpreting, organising and constructing meaning of situations with mathematical modelling or by presenting a notation (tarlow, 2014). based on these views, this study is located within a broader framework of what tall (2008) refers to as three worlds of mathematics: the ‘conceptual-embodied’ world based on perception, action and thought experiment, the ‘proceptual-symbolic’ world of calculation and algebraic manipulation and the ‘axiomatic-formal’ world of set-theoretic concept definitions and mathematical proof. each ‘world’ has its own sequence of development and its own forms of proof that may be blended together to give a rich variety of ways of thinking mathematically (tall, 2008). tall’s (2008) three worlds of mathematics offers an overview of the process inherent in mathematics and its teaching and learning. however, in this article we take a different line from that of tall as we interrogate the influence of the bigger picture at the teacher-learner interface. at a higher level from which tall frames his argument, the impression he gives is that there is sequential progression from the ‘conceptual-embodied’ world to the ‘proceptual-symbolic’ world and ultimately to the ‘axiomatic-formal’ world. while he acknowledges that at the formal-axiomatic world, structure theorems can lead back to worlds of embodiment and symbolism, it is not immediately clear how the process links to teaching activities at classroom level. our view is underpinned by the earlier position of gray, pitta, pinto and tall (1999), in which they represented the increasing compression from procedure through multi-procedure, process and procept. that representation clearly shows the sequential developments of mathematical thinking from pre-procedures to procepts. it is here that we found pirie and kieren’s (1994, p. 69) notion of folding back more appropriate. their theory of growth in mathematical understanding comprises eight levels, namely primitive knowing, image making, image having, property noticing, formalising, observing, structuring and inventising. folding back is the process through which mathematical understanding grows through these levels. this is a non-unidirectional process of coming to understand the bigger picture of teaching and learning mathematics. that is ‘when faced with a problem or question at any level, which is not immediately solvable, one needs to fold back to an inner level in order to extend one’s current, inadequate understanding’ (pirie & kieren, 1994, p. 69). pirie and kieren’s (1994) folding back and tall’s (2008) three worlds of mathematics manifest themselves in the mathematics content structure and the questioning approaches used by teachers. in terms of the latter, the issue is how questions that teachers raise can be pitched at a level that encourages the development or formation of the bigger picture of the concept under focus. research methodology within the interpretive paradigm, this study pursued the question: where is the bigger picture in the teaching and learning of mathematics? we use pre-existing data collected through pre-observation and post-observation interviews and passive classroom observation by the third author in two different grade 11 classes taught by two different teachers, thabiso and lerato (pseudonyms), at one high school. pre-observation interviews were conducted to gather information on the teachers’ beliefs about the nature of mathematics and how students learn and should be taught, given the current demand for curriculum reforms. classroom observations, which commenced a week after a pre-observation interview, exposed the teachers’ teaching approaches and classroom interactions, which offered an opportunity to establish consistency between the interview responses and the classroom practices. observations also offered an opportunity to search for alignment between the teachers’ teaching philosophy and classroom practice. a post-observation interview sought to get some clarity on issues that emerged during classroom observation. another set of data was collected through participant observation in the second author’s year 2 university class. in this article, we focus particularly on the teachers’ questioning strategies and the approach to content. we analyse these three different mathematics’ teaching cases, guided by tall’s (2008) framework of three worlds of mathematics: the ‘conceptual-embodied’ world, the ‘proceptual-symbolic’ world and the ‘axiomatic-formal’ world. within this broad framework we acknowledge pirie and kieren’s (1994, p. 69) notion of folding back towards the attainment of axiomatic-formal world. we trace whether mathematical connections and understandings were encouraged in both content structure and the questioning strategies of the teacher and, lastly, whether those connections were explicit, meaningful, accessible, expandable and transferable. that is, we looked for the bigger picture in both the structure of the content and the questioning strategies used. ethical considerations permission was granted by the education department, the school and the two teachers who participated. the teachers were made aware of the fact that they were free to voice their opinions, give advice and withdraw, if they felt so inclined. because the research was not directly focused on the learners, we requested the teacher-participants to explain the research to the learners and their parents, via the school principal. participation was voluntary and pseudonyms have been used to identify the two school teachers (thabiso and lerato), ensuring anonymity and confidentiality. approval for data collection was obtained from the department in which the study was located, the students and the relevant university structures beyond the department. in both locations, the nature and purpose of the study were declared, inclusive of potential audiences and substantive foci. erickson (1998, p. 1161) writes: ‘consent that is genuinely informed and without coercion reduces the risk of social harm because it affirms the dignity and respects the agency of those who will be involved in the study’. at the university, the students themselves were beneficiaries of the results of this study. they were inducted into the dynamics of what it means to learn mathematics with a view to teaching young learners. quality criteria it was possible for the three different mathematics’ teaching cases to involve prolonged engagement, persistent observation, peer debriefing and member checks with the teachers because the third author was teaching at the same school. with respect to the data collected at the university, the second author was teaching the class and the first author was the internal moderator for the module. we thus had enough opportunity to hear the teachers’ voices, which contributed to establishing credibility of this study (bitsch, 2005; guba & lincoln, 1989). sufficient descriptive data added to both confirmability and transferability of this study (guba & lincoln, 1989). results and discussions we organise our results and discussions of the three cases (two cases of high school mathematics teaching and one case from a year 2 university mathematics education class) below, commencing in each case with a brief biography, followed by two excerpts from the teachers’ mathematics lessons, our analysis of the lessons using the identified theoretical framework and a reflection on where the bigger picture is in relation to the teaching and learning of mathematics concepts being addressed. case 1: thabiso thabiso, a male teacher aged 43 years, held a bachelor in science honours degree, majored in mathematics. he had 18 years of teaching experience and had taught mathematics in the further education and training band (fet, that is, grade 10, 11 and 12 classes). he attended dinaledi project workshops as part of his professional development. figure 1 captures two excerpts from his teaching in one of the grade 11 classes. figure 1: thabiso’s two teaching excerpts. in both excerpts, learners were expected to operate in the symbolic world, beginning with processes or actions that are symbolised and coordinated for calculation and manipulation (tall, 2008). the teacher’s questioning strategy encouraged learners to practise the same routine procedures that they would use when dealing with similar problems that do not deviate from the norm, as evidenced in the exercises that followed. the teacher concentrated on a single procedure that could lead to a spectrum of learning outcomes in his class, a proceptual divide (tall, 2008). such an approach encourages procedural learning without meaning. it does not provide learners with an opportunity to choose their own strategies, which would necessitate folding back (pirie & kieren, 1994) for them to build deeper understanding. in excerpt 1 it appears that the teacher was confident that the learners understood what quadratic equations are and how they are derived. for him, it was not necessary to explain to the learners when to use a quadratic formula to solve the quadratic equation. the teacher assumed that learners could work with symbolic representations thus introduced to use the quadratic formula to solve the equation x2 – 3x + 2 = 0. the learners were expected to use these met-befores – ‘current mental facility based on specific prior experiences of the individual’ (tall, 2008, p. 6) – to identify the value of a as 1, the value of b as –3 and the value of c as 2. even if that was the case, the teacher seemed not to trust the learners’ met-befores. this became evident when he announced that learners had to listen attentively and should take notes that they could use for revision. after the teacher’s explanation on the steps to follow, learners were given an exercise that required them to practise the routine procedure learned from the given example. question 1c) deviated a little from the pattern as it required learners to first realise that they needed to simplify the given expression to its standard form before they could follow the routine procedure. it is doubtful whether the learners comprehended why ± should be separated and what the meaning of the equal sign in the context of quadratics was. it would also be interesting to find out whether learners would know how the values of x relate to the original question and whether they could check solutions meaningfully. in excerpt 2 the teacher expected the learners to build on their experiences with working with triangles. the learners were expected to recall that they could only use the sine rule if two angles and a side are given or two sides and the non-included angle are given, that is, their met-befores. they had to connect that experience with a concept of the sine rule to find the unknowns. it could have been more empowering if learners were exposed to the given triangle to solve for the unknowns without being reminded of which formula to use. meaningful learning occurs when learners succeed in choosing effective mathematical strategies to solve given problems. where is the bigger picture? reflecting on excerpt 1, we wonder whether learners would know, before even attempting any given quadratic equation to solve, that the equation might have two solutions, or one solution, or perhaps no solution. the bigger picture with regard to quadratic equations lies in their origins. egyptian, babylonian and chinese mathematicians dealt with areas of quadrilaterals and were interested in finding the length and breadth of a rectangle with known area (gandz, 1940; mcmillan, 1984; yong, 1970). this is generally represented as ax2 + bx = c or x(ax + b) = c where x is the breadth, ax = b is the length and c is the area. that is, we are looking for the value of x for which ax2 + bx – c = 0. the challenge that remains is to express the equation as a product so that the null factor law can apply (the null factor law states: if the product of two factors is equal to zero, one or the other, or both, of the factors must be equal to zero. symbolically it could be written as: if pq = 0, then p = 0 or q = 0 or both p and q are zero). this is the reason why factorisation is the preferred approach to solving quadratic equations. the general formula is ideally used when it becomes difficult to factorise the expression. in fact, the formula itself is a result of using ‘completing a square’ as a strategy for factorisation. thabiso’s approach did not make that explicit and, as a result, could hinder meaningful learning for most learners and opportunities for the learners to transfer their conceptual understandings. the technical approach was not placed in the appropriate context. a similar observation is made with regard to excerpt 2 regarding the solution of triangles. the sine rule is one of the many strategies that is used to solve the triangles. once the rule is derived, it is important that it is adequately analysed to establish the conditions under which it applies and the opportunities it gives us in solving triangles. in other words, in the bigger picture of solution of triangles, when is it appropriate or more ideal to use the sine rule? determining whether the sine rule is appropriate is more important than its actual application. thabiso’s approach was to ask the learners to simply plug in the values to calculate the missing values. once again, the technical approach was not placed in the appropriate context within which the sine rule would have been seen as a particular strategy more suitable for a particular situation. when approached in thabiso’s way, mathematics is viewed as a collection of rules or formulae that learners must memorise, often out of context. its role as a way of observing and interpreting our daily experiences is stripped away. in his defence, one might argue that at this stage thabiso was simply helping learners to develop the skill of using the formulae or the rules and that the context would be brought in at a later stage when all different skills had been acquired. this is where we differ from tall (2006, 2008). our argument is that the bigger picture should always provide the backdrop against which mathematics activities are designed and implemented. the meaningfulness of mathematics cannot be deferred. case 2: lerato lerato, a female teacher aged 51 years, held a primary teachers diploma (ptd), advanced certificate (ace) in mathematics education (majored in fet mathematics teaching). she had 26 years teaching experience and taught mathematics in the fet band (that is, grade 10, 11 and 12 classes). she attended in-service training workshops at the mathematics, science and technology college (mastec) in 2008 as part of her professional development. figure 2 captures two excerpts from her teaching in one grade 11 class. figure 2: lerato’s two teaching excerpts. in both excerpts, learners operated in the symbolic world but were subjected to a different questioning strategy as compared to thabiso’s case. lerato, like thabiso, led the learners to perform a procedure to find the correct answer. it is, however, doubtful whether with her questioning, learners would have managed to build up the symbolic mental imagery that is the basis of true understanding (tarlow, 2014). in other words, although the questions allowed flexibility in terms of searching for solutions, it appeared that, for some learners, encapsulation from process to procept was still problematic (tall, 2008). in excerpt 1, the expected learners’ met-befores were to know the standard form of quadratic equations and to factorise. the questions that required learners to share their challenges in factorising, and in making x the subject of the formula, provided opportunities for learners to fold back to their previous experiences (pirie & kieren, 1994). in addition, they offered the teacher some space to give direction by explaining to the learners where they experienced difficulties. this marked lerato’s attempt to employ instructional scaffolding (vygotsky, 1962), a process that assists growth with regard to understanding of concepts. another expected met-before was making x the subject of the formula by completing the square. the method of completing the square requires learners to follow specific steps. thus, those who could respond to question 6 would be those who drew on met-before knowledge and had mastered the routine procedure for this method of completing the square. although it might have appeared to lerato that the solution should be easy, for some learners this caused profound difficulties. once learners realised that they were repeating the same steps for each given quadratic equation, the value of x was then introduced as quadratic formula, which they were expected to use to solve the given exercise. in excerpt 2, lerato reminded the learners about the expected met-befores: to use the horizontal reduction and special angles, to locate the given angle and to check the sign of the ratio. learners were also expected to know that angles are conventionally measured counter-clockwise from the right hand horizontal axis and that angles measured in a clockwise direction are considered negative. although an opportunity was provided to fold back to previous experiences, the tasks chosen still did not support learners’ mathematising (tarlow, 2014). where is the bigger picture? the bigger picture with regard to quadratic equations was outlined in thabiso’s case. in this section the focus is on the questioning strategy that lerato used to facilitate learning. the questions that lerato raised were generic and required learners to solve for x in the general quadratic equation. the process led to the derivation of the quadratic formula. the approach had the potential to incorporate the bigger picture inherent in the solution of quadratic equations. this way of solving quadratic equations is just one of many. it is in the context of multiple approaches that the bigger picture becomes clear. if lerato’s questioning strategy focused on a multiplicity of solutions, then the learners would have been in a position to interrogate those solutions with a view to identifying strengths and weaknesses for each. that analysis would have offered them multiple opportunities for folding back (pirie & kieren, 1994). the question “when can we use the quadratic formula?” captures another aspect of the bigger picture. this prompts the question that, if the general formula method works so well, why would we ever use factoring? if pursued, it would reveal that the bigger picture involves a realisation on the part of learners that general formulas exist only for polynomials with degree less than 5, as proved by the french mathematician galois (moore, 1978). what is still not clear is whether the learners would realise that, by finding the values of x for which ax2 + bx − c = 0, they are, in fact, finding the x-intercepts of the graph. the learning environment created should be such that it accommodates situations where the quadratic formula also gives learners a zero or a negative inside the square root, thus affording an opportunity to introduce them to use of complex numbers. operating within a context that allows relating such knowledge simultaneously (even graphing on the xy plane) would provide deep learning, which reverses the sequence of construction of meaning back and forth between the three worlds of mathematics, as suggested by tall (2008). the reduction formulae are meant to take advantage of the circularity of the angles, something that is rarely acknowledged. lerato’s questioning in excerpt 2 prevented the learners from seeing fundamental relationships between lengths, angles and areas of triangles in a broader sense. the use of ‘degree’ as the unit for measuring angles (the babylonian astronomers’ unit, emerson, 2005) and use of the unit ‘radian’, commonly used when graphing trigonometric functions, would remain a mystery to most mathematics learners. it is between the relational understandings of these two units, degree and radian, where the bigger picture in dealing with the reduction formulae resides. if teaching does not expose learners to folding back experiences (pirie & kieren, 1994), the two concepts of plane angle, one treating plane angle as a base quantity and the other defining it as a dimensionless ratio of two lengths, will remain incompatible (emerson, 2005). an understanding of the fact that trigonometric functions are periodic, that is, repeating over and over again as θ gets larger or smaller by 2π, is also necessary to comprehend the bigger picture. case 3: kwena the teacher, a male aged 48 years, held a doctoral degree in mathematics education. he majored in mathematics and applied mathematics for his bachelor of science (bsc) degree, did a bsc honours degree in applied mathematics and a higher education diploma as a teaching qualification. figure 3 captures two excerpts from his teaching in a year 2 class for a bachelor of education in senior phase and fet teaching programme. figure 3: kwena’s two teaching excerpts. while the learning in the two excerpts started from different worlds, the proceptual-symbolic world in excerpt 1 and the axiomatic-formal world in excerpt 2 (tall, 2008), it was the questioning strategy employed that supported the learners’ mathematical development through a process of mathematising (tarlow, 2014) and construction of mathematical ideas. the teacher’s questions provided a richer learning environment that encouraged taking a multi-procedural approach, which can lead to procedural efficiency (tall, 2008). learners were offered some flexibility to investigate possibilities from different entry points. it is doubtful whether all the learners managed to transcend beyond the process into procept (tall, 2008). a possibility remains that some learners stayed stuck in either the symbolic or the formal world, with emphasis on the logic and with a less conceptual insight. in excerpt 1, just like in thabiso’s case, the learning was approached from the point of view of orientating learners on one procedure to solve a system of linear equations represented by ax = b, where a and b are matrices and x is a column matrix of unknowns. it was an entry into the investigation which allowed more than one possible strategy, with no predictable procedure to find a solution. in excerpt 2, in order to respond to the first two questions, confirming limits of functions using the ε and δ methods and stating the mean value theorem and providing its detailed proof, required that connections be made within the formal world. the subsequent questions in both excerpts added a different learning flavour, making provision for learners’ self-questioning. as learners searched for efficient solutions and interpreted and constructed the meaning of the concepts at hand, they were encouraged to formulate the critical questions that they would ask their teacher and those that they would pose to the whole class. unlike lerato’s approach, the instructional scaffolding (vygotsky, 1962) was approached from a different angle. the approach involved learner interactions in different forms that did not necessarily progress sequentially, including student-material interaction, student-student interaction, student-teacher interaction and whole-class interaction. evident was the dialogical and interactional nature of scaffolded instruction that encouraged a nonthreatening participation in a shared community of practice (van lier, 2004). where is the bigger picture? matrices emanated from the study of solutions of a system of linear equations. the idea was to find different strategies that could be used to solve the unknowns efficiently and reliably. however, as mathematical objects, their properties are also amenable to the four mathematical operations addition, subtraction, division and multiplication. at the core of all these operations, is the solution of the matrix equation ax = b, where a is the coefficient matrix and x and b are column matrices. ideally, we should always ask how each skill that we acquire, or each strategy that we develop, assists us in the solution of the matrix equation above. like with all other areas of mathematics, each skill or strategy has its own strengths and weaknesses. hence, the end is not the mastering of the strategy itself, but whether it presents itself as the most efficient solution at that particular time. that is, an analysis of the strategy should always be integral to mathematics learning activities. this way, each learning opportunity enhances pirie and kieren’s (1994) folding back and collecting, while also enriching tall’s (2008) proceptual stage of mathematising. the same scenario applies to the limits of functions. given the function f(x) we are to establish whether exists. a variety of skills and strategies are required to determine the limits of functions. strategies used include numerical approaches, graphical approaches and symbolic approaches. the use of tools, such as theorems, constitutes a strategy that uses a theoretical approach. the learning or development of such tools away from their intended purposes works against the development of the bigger picture. but even more challenging is the learning of skills and strategies without interrogating their relative efficiencies in addressing the problem at hand. the exclusion of this aspect makes it a challenge to transcend from multiple strategies to the proceptual stage. both excerpts require students to analyse the strategies used in resolving the problems at hand. while the first excerpt is wide open in terms of the strategies around the bigger picture, the second excerpt is anchored around a particular strategy: the mean value theorem. this makes the first excerpt relatively stronger in addressing the bigger picture. on the other side, the application is closed in the first excerpt, as a particular scenario is given, whereas it is open in the second excerpt, as students are expected to come up with their own problems. this makes the second excerpt relatively stronger with regard to the bigger picture. the insistence on reflections on own experiences and the generation of reflective questions in both excerpts makes the realisation of the bigger picture in both cases more likely. the approach in both cases, if well developed, has the potential to contribute to an integrated development of the bigger picture with regard to the concepts at hand. conclusion in this article we pursued the question: where is the bigger picture in the teaching and learning of mathematics? we used three teaching cases to analyse the presence or absence of the bigger picture, especially, in the teachers’ questioning strategies and their approach to content. in both tall’s (2008) three worlds of mathematics and pirie and kieren’s (1994) growth in mathematics understanding, the bigger picture of mathematics is captured in the process of growth of a mathematical idea or concept. almost all mathematics ideas evolve from a primitive or concrete stage through to their axiomatic formal stage. however, it was pirie and kieren’s folding back that was found to be more practical at the teacher-learner interface. in that context, our argument is that the bigger picture of the idea at hand should influence how the lesson unfolds. the content structure and the nature of the questions the teacher raises should reflect the way the bigger picture is anchored in the teaching and learning of mathematics. in that way mathematics is meaningful, accessible, expandable and transferable. the bigger picture should always provide the backdrop against which mathematics activities are designed and implemented. in that way the meaningfulness of mathematics cannot be deferred. in the three teaching cases that we have used in the study, we found that all the topics covered had potential for the incorporation of a bigger picture of mathematics. however, the analysis of the content and questioning strategies in those lessons revealed that either the influence of a bigger picture was non-existent or it had minimal influence. in the case of thabiso’s approach, the influence of the bigger picture was non-existent on both the content structure and the questioning strategy with regard to quadratic equations and the solution of triangles. in both lerato’s and kwena’s cases, engagement with the content was framed by their questioning strategies. in lerato’s case, the questioning strategy had glimpses of influences of the bigger picture of quadratic equations and that of the solution of trigonometric equations. in kwena’s case, the questioning strategy revealed a concerted effort to incorporate the bigger picture of matrices and that of limits in the lessons. however, a lot is still needed to improve on that practice. generally, all three teachers can benefit from exposure to the influence of the bigger picture in the teaching and learning of mathematics. both tall’s (2008 & 2011) and pirie and kieren’s (1994) ideas can be used to interrogate different topics in mathematics in order to show specific ways in which the bigger picture can be used in classroom lessons. hashemi, abu, kashefi, mokhtar and rahimi (2015) used tall’s ideas to improve the teaching of derivatives and integrals. furthermore, the non-incorporation of the bigger picture cannot solely be blamed on the teachers. the assessment regime in the system of education, the learning support materials and the training of teachers need to incorporate the bigger mathematics picture in their orientations. acknowledgements competing interests the authors declare that they have no financial or personal relationships that may have inappropriately influenced them in writing this article. authors’ contributions s.m. was the project leader and moderator of k. masha’s assessment instruments. k. masha facilitated lessons in the year 2 university class and captured data from that activity. k. maphutha collected data from the two teachers’ mathematics classes and assisted in peer debriefing and member-checks with the teachers involved. s.m. and k. masha conceptualised the article and thereafter s.m. produced the first draft. the two then proceeded to work on the final draft. references bitsch, v. 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(2014). assessing teacher change in facilitating mathematizing in urban middle schools: results of an effective professional development program. current issues in middle level education, 19(2), 9–12. van lier, l. (2004). the ecology and semiotics of language learning. dordrecht: kluwer academic. vygotsky, l.s. (1962). thought and language. cambridge, ma: mit press. wilson, p.s., cooney, t.j., & stinson, d.w. (2005). what constitutes good mathematics teaching and how it develops? nine high school teachers’ perspectives. journal of mathematics teacher education, 8, 83–111. yong, l.l. (1970). the geometrical basis of the ancient chinese square-root method. isis, 61(1), 92–102. available from http://www.jstor.org/stable/229151 article information author: anita campbell1 affiliation: 1academic support programme for engineering in cape town (aspect), university of cape town, south africa correspondence to: anita campbell email: anita.campbell@uct.ac.za postal address: private bag x3, rondebosch 7701, south africa dates: received: 15 apr. 2015 accepted: 03 sept. 2015 published: 24 nov. 2015 how to cite this article: campbell, a. (2015). exploring boot camps for ‘gatekeeper’ service courses in mathematics. pythagoras, 36(2), art. #298, 9 pages. http://dx.doi.org/10.4102/pythagoras.v36i2.298 copyright notice: © 2015. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. exploring boot camps for ‘gatekeeper’ service courses in mathematics in this original research... open access • abstract • introduction • development of the mathematics tutored reassessment programme • considerations for the design of the tutored reassessment programme • impact theory • five theory-based design principles    • principle 1: transition into university as an issue of identity    • principle 2: consider students’ human needs    • principle 3: students have different levels of actual and potential development    • principle 4: use both acquisitionist and participationist approaches    • principle 5: quality assurance of success • methodology    • ethical considerations • findings and discussions    • questionnaire and interviews data    • pass rates from the july 2014 mathematics trp    • pass rates from the january 2015 mathematics trp    • financial implications • conclusions • acknowledgements    • competing interests • references • appendix 1    • tutored reassessment programme (trp) interview schedule – january 2015    • sample questions    • students’ attitudes to learning    • how students cope with their studies    • students’ views on becoming an engineer abstract top ↑ pressure to increase the throughput of university students in ethical ways has been a catalyst for innovations to improve learning and student success. student dropout occurs mostly in the first year of study and poor performance is a major contributor to dropout even if the underlying reason for the poor performance is not academic under-preparedness. this article discusses the design and implementation of a mathematics tutored reassessment programme (trp or ‘boot camp’) to improve the pass rate of students writing supplementary examinations for first year engineering mathematics. interviews with students and tutors suggest that the trp cultivated positive affective changes in students. a notable result from this case study was that students who qualified for a reassessment with marks in the range 40%–44% (and who would not normally have been granted a supplementary examination) outperformed students qualifying with marks of 45%–49%, for whom attendance at the trp was optional. theoretical motivations for five principles guiding the design of the trp are discussed. introduction top ↑ the knowledge economy strategy used in many countries has put pressure on universities to produce more graduates as a way to stimulate economic growth (deiaco, hughes & mckelvey, 2012). low graduation rates of students in the south african higher education institutions (council on higher education, 2013) suggest that improving throughput rates is necessary, rather than just increasing the number of students accessing higher education. strategies to help students pass their service courses timeously are an important part of the goal to improve student throughput at university. a service course is a compulsory or optional course offered by a department in which students will not take their major courses. the first year mathematics courses for engineering students considered in this case study are examples of service courses. in these first year mathematics courses, students are taught fundamental concepts in calculus and other topics that are directly or indirectly relevant to many courses in their engineering degrees. compulsory service courses perform a ‘gatekeeper’ role: students cannot graduate without passing the service courses. although these courses are not the focus of the degree, students who cannot meet the demands of the service course are assumed to be less likely to meet the demands of further courses. this assumption reflects research findings (e.g. van eeden, de beer & coetzee, 2001) that past academic performance is the strongest indicator of future academic success. however, the placement of service courses in the early stages of a degree programme means that student achievement in these courses is likely to be impacted by factors such as how well and quickly students ‘acculturalise’ to a higher education institution (race, 2014) and not only their capacity to succeed in higher education. numerous studies (e.g. kuh, kinzie, buckley, bridges & hayek, 2006; tinto, 2012) point to practices that can ease the transition from school to university for students, such as integrating support services and engaging students through active learning techniques. however, as noted by pym and paxton (2013), the problem of adjusting to university is exacerbated for first-generation students, particularly if they are experiencing an english-only learning environment for the first time. catching up after falling behind in studies is especially difficult in courses that have a final examination after one semester. the aim of this case study is to show how a tutored reassessment programme (trp or ‘boot camp’) can be implemented to improve student success in ‘gatekeeper’ service courses. the article begins with the history of the development of the mathematics trp for service courses in engineering mathematics and a description of the design used at a university in the western cape, south africa, for the july 2014 and january 2015 programmes. a discussion of some of the tensions regarding the notion of student success is followed by the impact theory of the programme and five theory-based principles that underpin the current design. students’ results following the supplementary examinations in july 2014 and january 2015 are presented and discussed. development of the mathematics tutored reassessment programme top ↑ the department of chemical engineering at the university has a long-established academic development focus, with two full-time academic development lecturers (case et al., 2015; heydenrych & case, in press). one of this department's successful initiatives to increase student throughput was boot camps for courses with low pass rates (case, von blottnitz, fraser, heydenrych & petersen, 2013). during vacations, students would have two to three weeks of intense review of a course, led by a senior tutor. typically the senior tutor would be a postgraduate student with recent tutoring experience in the course. the senior tutor would prepare the review material in consultation with the lecturer. the lecturer would be responsible for setting and marking the examination, which would be unseen by the senior tutor to prevent ‘teaching to the test’. the lecturer's involvement would be minimal to preserve their research time. final pass marks would be recorded as 50% on students’ academic transcripts to prevent the boot camp being strategically used as a means to increase their overall mark average, since students who passed the course did not have an opportunity to improve their marks. the success of the boot camps in chemical engineering led to this model being adapted for use in the service courses considered in this case study, engineering mathematics 1a and 1b, which are semesterised courses for engineering students with final examinations in june and november and supplementary examinations in july and january, respectively. pilot programmes for the engineering mathematics courses were run in july 2013 and january 2014 and an external review of the january 2014 programme (chapman, 2014) influenced the design of the mathematics trp used in july 2014 and january 2015. the trp consisted of five days of intense revision in the week prior to the supplementary examinations. the schedule was the same for each of the five days of contact session: in the morning a review lecture by an experienced lecturer was followed by whiteboard tutorials, where 24 to 28 students worked in self-formed groups of two to five students around large wheel-mounted whiteboards spaced around a large room with movable furniture. while students wrote their answers to worksheet questions on the whiteboards, two or three tutors (senior or postgraduate students with recent tutoring experience in the course) walked between groups, offering advice, questioning students and providing help when students were stuck. interaction between students was encouraged by having answers displayed openly on the whiteboards. compared to tutorials where students record answers on paper while sitting at desks, the use of whiteboards stimulated more questioning between students and also between tutors and students. after a lunch break, the afternoon session started with a shorter review lecture followed by an individually written test, immediately peer-marked by students and followed up with explanations of common problems at a board by tutors. tutors were also available for one-to-one help at the end of the day. figure 1 summarises the structure of the mathematics trp. figure 1: daily structure of the mathematics tutored reassessment programme. students who obtained final marks of 45%–49% after the first examination qualified for a supplementary examination under normal rules and attendance at the camp was optional for these students. students with final marks of 40%–44% were allowed to write the supplementary examination provided that they attended the entire trp. accommodation at university residences was provided for students who needed accommodation. funding grants allowed any student to attend without paying fees for tuition or accommodation. considerations for the design of the tutored reassessment programme top ↑ the trp was designed with a rather different perspective on student success, which will be described below. in addition, because this study uses the same impact theory used by outside evaluators in evaluating the july 2014 trp, this impact theory will be described. finally, i describe the five theory-based design principles that emerged from reflecting on the weaknesses and successes of the two pilot studies in june 2013 and january 2014. this study has viewed student success as something broader than simply passing or graduating from a particular course, which according to harper (2012) is the dominant view in higher education literature. there are many definitions of student success, reflecting different theoretical perspectives on education and giving rise to different types of questions. the view of student success adopted by this project is closer to that of allie et al. (2009) who take a strong participatory position. they define success as ‘the demonstration of the ability to use the relevant discourse to be able to participate in a workplace community’. in the context of first year mathematics for engineering students, we could consider a ‘workplace community’ to consist of other courses students take in which they need to use the discourse of mathematics. this definition of success would favour the inclusion of activities that give students practice in their ability to use the discourse of first year mathematics and their ability to participate in a community where there are shared goals. in these first year mathematics courses, there are essential skills and practices that students are expected to ‘pick up’ by the time they pass the course, such as communicating in a style that is recognised as mathematics, having a confidence to tackle problems and ask useful questions to help to make progress when solving a problem and reflecting on an answer to see if it is feasible. these are unwritten outcomes. however, given the ease with which students can look up information or make complex calculations using technology, the unwritten outcomes are arguably more important than the course outline as defined by the faculty handbook entry, which simply lists topics, for example ‘curve sketching. applications of the mean value theorem. rates of change and optimization involving functions of a single variable’ (faculty of engineering and the built environment, 2015, pp. 149–150). furthermore, rather than focus on proficiency in mathematics alone, the design of this intervention aimed to address all three areas of under-preparedness identified by matoti (2010) as impediments to student success. these are: academic under-preparedness (proficiency in english and basic mathematics, effective study skills), cultural under-preparedness (fitting into the dominant culture of the university) and emotional under-preparedness (self-efficacy and self-regulation). the programme addressed these issues by selecting a diverse group of tutors, by having the majority of the students living together in residence and by providing opportunities for students to improve their self-efficacy by experiencing success in answering questions posed by tutors and peers in the whiteboard sessions and in the individual tests. additional worksheets were made available on a website so that students could practice in the evenings and in the days between the end of the camp and the examination. impact theory top ↑ the impact theory of the intervention is the same as for a similar intervention in january 2014 that was externally evaluated and is depicted in figure 2 (chapman, 2014). the intervention is intended to increase students’ subject knowledge and application skills, improve their attitude towards learning and increase their confidence in their ability to persist in engineering. as a result, a higher pass rate for the compulsory service course should be attained and, ultimately, higher numbers of students should graduate. figure 2: impact theory of the tutored reassessment programme model. five theory-based design principles top ↑ with the above considerations and theories in mind, the design team reflected on the weaknesses and success of two pilot studies in june 2013 and january 2014 and five theory-based principles for the trp emerged from this reflection. a pluralistic theoretical framework (sfard, 1998) acknowledges that education theories are not mutually exclusive and a learning programme designer can simultaneously use multiple educational theories. the principles are outlined below. principle 1: transition into university as an issue of identity the adjustment to university has been likened to an unavoidable ‘rite of passage’ (clark & lovric, 2008; palmer, o’ kane & owens, 2009) that is inherently challenging. however, hernandez-martinez et al. (2011) found that this time is also viewed by students as an opportunity for growth and to test out new identities. the students attending the programme were high achievers in high school mathematics but had failed university mathematics. the idea that students were seeking opportunities to grow inspired the belief that they would be willing to sacrifice part of their vacation time to attend an intense revision programme and participate actively in it. principle 2: consider students’ human needs in his humanistic theory of learning, the psychologist abraham maslow described a hierarchy of human needs leading to the need for ‘self-actualisation’, a state of becoming what you have the potential to be (maslow, 1970). the trp design ensured that the basic human needs for food and shelter were met by offering university accommodation with meals to all students. three activities provided students with opportunities to have some control over their learning and to feel a sense of belonging. in the whiteboard tutorials, working in groups was encouraged but students could choose to work independently. after test reviews, there was opportunity for one-to-one consultation with tutors. when working alone outside of contact sessions, students could engage with a tutor or peers in online spaces where resources were shared. the need for self-esteem was addressed by a positive and respectful approach by tutors and by providing the opportunity for students to develop peer relations in the interactive whiteboard sessions. principle 3: students have different levels of actual and potential development lev vygotsky's social constructivism theory claims that students need the guidance of a teacher or collaboration with peers to reach new levels of actual development, where they can solve problems independently (vygotsky, 1978). the whiteboard tutorials provided the guidance and collaboration opportunities to allow for this development. in order to maximise the zone of potential development for students during the whiteboard tutorials, the review lectures served as refreshers for students and gave the group a shared experience on which to draw as they worked through the whiteboard tutorials. the safe space of the lectures kept students from being too far from their comfort zone that they were unable to function in the whiteboard tutorials, which may have occurred if students attempted the whiteboard tutorials without the review lecture. principle 4: use both acquisitionist and participationist approaches an acquisitionist approach considers learning as a process of gaining knowledge whereas a participationist approach takes the view of learning as a process of becoming able to do something that could be recognised by a community (sfard, 1998). this principle recognises that students are transitioning from school where an acquisitionist approach to learning is dominant. while the examination structure for these mathematics courses is clearly based on testing whether students have acquired the knowledge outlined in the course syllabus, the trp also aimed to develop mathematical behaviour, such as laying out work in an orderly way, using questioning to stimulate ideas when stuck, and having tenacity when feeling challenged. the interaction with tutors in all the activities was the primary way in which mathematical behaviour was developed; hence, it was important to select mathematically strong tutors who understood, modelled and encouraged good mathematical behaviour. feedback from students in pilot studies suggested that input in the form of a review lecture boosted engagement with the interactive whiteboard tutorials. it was also stressed that students would also have to spend time independently reviewing the coursework from the semester as the review camp would not be able to cover all sub-topics. students unable to make progress independently would have access to peers and online resources (such as review lectures) from which knowledge could be acquired if they were staying in residence. formative assessment with many opportunities for practice and feedback was based on the acquisitionist idea of gaining knowledge. despite reservations regarding the limitations of examinations, the design incorporated practice time for students to experience examination-like conditions. principle 5: quality assurance of success this principle followed from the particular perspective on success adopted by the trp designers. although the main goal of students attending the trp would be to pass, it would be of little benefit if students achieved a pass but then failed the subsequent mathematics course. the selection and delivery of lecture and tutorial material aimed to give students adequate preparation to enhance chances of success in the subsequent course. keeping the examination paper unseen by all the tutors and worksheet developers eliminated any possibility of teaching to the test that may lead to an inflated pass rate. the external examination process used in the normal course examination was followed and the final course mark was determined by replacing the original examination result with the newly obtained examination result. a class mark still contributed to the final course mark. methodology top ↑ the methodology of a case study matches a ‘situational perspective’ (case & light, 2011) rather than hypothesis-driven research looking for cause and effect relationships. the purpose is to gain insight into the mathematics tutored reassessment programme for first year undergraduate university engineering students rather than making statistical generalisations. data were obtained through three data sources: students’ responses to an online questionnaire, individual interviews with students and tutors and an analysis of the pass rates of students who did and did not attend the programme. the three data sources triangulated the emerging data. the online questionnaire was developed in collaboration with the experienced lecturer who gave the review lectures and observed some whiteboard and test marking sessions. students were asked to rate the different components of the programme and the resources and to answer ‘yes’, ‘no’ or ‘unsure’ to the following questions: ‘do you feel the trp helped you gain confidence? did you make new friends at the trp? did you consider changing out of engineering when you got your results? do you think that explaining work to others helps you understand? would you have attended if you had to pay r500 for tuition?’ students could also add free response comments to these questions, as well as to the question ‘what would you change about your trp experience?’ interviews were semi-structured and followed an interview schedule of potential questions to ensure that each interview spanned the same topics (see appendix 1). the questions were peer-reviewed by two educational researchers. audio recordings of the interviews were used to expand research notes taken during the interviews and to transcribe selected quotes. the interview questions asked students about their attitudes to learning, how they coped with their studies and their views on becoming an engineer. the online questionnaire results and interview summaries were discussed with the experienced lecturer who gave the review lectures and consensus was reached on the interpretation of the results as reported here. linking to the impact theory in figure 1, the questionnaire and interviews probed changes in students’ attitudes towards learning, their coping abilities and their beliefs that they could become engineers, while the pass rates provided an indication of students’ subject knowledge and application skills. ethical considerations ethical clearance was obtained from the faculty-based research ethics committee at the participating university. participants were informed via consent forms that the purpose of this research was to evaluate the ways in which tutored supplementary examinations can help improve student success, that their identities would be kept anonymous, that their participation was voluntary and neither participation nor non-participation would affect any of their course results and that they could withdraw form from the study at any time. the researcher in this study (the author) was not involved in any assessment of participating students that contributed towards course results. findings and discussions top ↑ questionnaire and interviews data interviews with students who attended the trp indicated that attending the camp increased the students’ commitment and hours of preparation for the examination, particularly for students who stayed in residence and were able to continue conversations through meal times: student t: i definitely spent more hours studying [at residence] than [i would have] at home. we were talking maths in the dining room, at night… student l: i wouldn’t have worked so much if i was staying at home. providing an environment conducive to learning links to principle 2 (consider students’ human needs) and appears to have enabled greater time on task for those staying in residence. working in the intense way described by these students may help students to establish (or re-establish) an identity as a university student who can handle a challenge, a quality linked to principle 1 (transition into university as an issue of identity). the daily programme of the trp was unanimously praised by the students and tutors who were interviewed, for example: student k: the structure was perfect. i wouldn’t change a thing. many responses to the online question ‘what would you change about your trp experience?’ showed high levels of satisfaction with the programme. for example: [change] nothing. [i]t was well coordinated. perhaps an extra day for consolidation would have been helpful, and maybe test questions could be a little tougher. … other than that it was genuinely a wonderful experience. the two-hour morning lectures were felt by a tutor to be too long but students did not comment negatively about this: tutor m: many students said the lectures were too long. tutor n: the lectures went overtime, sometimes by quite a bit. the students didn’t seem to mind. student t: we had time in lectures to try out problems. [the lectures were] very valuable. the review lecture helped students to feel less daunted in the whiteboard tutorials and to make effective use of the whiteboard tutorial time: student l: kenny's lectures were brilliant. when we started on the tutorial questions, we knew what was going on. tutor n: the students really worked well when they came in after the lecture, surprisingly well, really. i expected that they would want a break, but they didn’t. if there were unresolved problems by the end of the whiteboard tutorials, students valued the time to address misconceptions or knowledge gaps during the review lecture prior to the test: student k: the chance for clarification in the afternoon is especially useful if the section is tough or tricky. student t: in the afternoon lecture… things clicked. the variation in satisfaction with the timing of the lecture relates to principle 3 (students have different levels of actual and potential development). catering for a diverse range of needs is a challenge in lectures. online quizzes were suggested by a tutor as a way to keep more advanced students busy during sections of lectures on topics they were confident in. students liked the test as a way to rate their performance: student k: the structure was perfect. i wouldn’t change a thing. when asked to rate the different activities in the programme, 22 out of 29 (76%) students rated the lectures as ‘excellent’, 4 as ‘okay’ and 3 said they did not attend the lectures. none rated the lectures as unhelpful. principle 4 (use both acquisitionist and participationist approaches) acknowledges the acquisitionist perspective that is dominant in school and in the way the course is assessed. students’ high levels of satisfaction with lectures may reflect the dominance of an acquisitionist perspective in most students; however, as the following quote shows, the influence of the participationist approach in the design appears to have helped at least one student: during the semester i would try homework problems at home but often i couldn’t do them. … i’d feel overwhelmed and think ‘everyone else seems to know what's going on. i’m so lost, i can’t do this’. but now i can! i was working alone one night in the boot camp, trying [a problem] and not getting [it] but then i thought, ‘no! i can do this,’ and i stuck with the problem and thought, ‘what am i trying to do? what is the next step?’ and … i solved it! (student k) the behaviour of the tutors during the tutorials – questioning to find a way forward when students are stuck, correcting notation use – appears to have provided a model for how to make progress with problems, showing students how to behave mathematically. the trp was very successful in building students’ confidence and self-efficacy regarding their ability to complete challenging problems, with 35 out of 39 (90%) students in the january 2015 online survey and all 8 out of the 8 (100%) students in the july 2014 survey answering yes to the question ‘do you feel the trp helped you gain confidence?’. pass rates from the july 2014 mathematics trp a total of 27 out of 41 students with qualifying marks of 40%–49% passed, giving an overall pass rate of 66%. students who attended the full camp had a pass rate of 68%; students who did not attend had a pass rate of 65%. the results for students who attended can be further broken down into those with lower qualifying marks of 40%–44%, for whom attendance was compulsory, and those with qualifying marks of 45%–49%, for whom attendance was optional. surprisingly, the highest pass rate of 71% came from the 17 students with the lowest qualifying marks of 40%–44%. the 12 students in this group who passed would not have been allowed to write the supplementary examination if they did not fully attend the trp. of the 24 students with qualifying marks of 45%–49%, 15 (63%) passed (3 attended, 1 partly attended, 11 did not attend) and 9 failed (2 attended, 1 partly attended, 6 did not attend). these results are summarised in table 1. table 1: final results after the july 2014 mathematics tutored reassessment programme. pass rates from the january 2015 mathematics trp compared to july 2014, the january 2015 mathematics trp had a higher number of students qualifying (59 vs 41) as well as a greater percentage of students attending (47% vs 39%). the longer time between the release of marks and the supplementary examination in january compared to july may account for the increase in the percentage of students attending, as students living outside of cape town had more time to make travel plans. a total of 35 out of 59 students with qualifying marks of 40%–49% passed, giving an overall pass rate of 59%. unlike the july camp, students who did not attend the full camp had a pass rate substantially lower than those who did attend (47% in january 2015 and 65% in july 2014). the best pass rate was for students with the higher qualifying marks who did attend (67%), closely followed by students with qualifying marks of 40%–44%, for whom attendance was compulsory. however, the combined pass rate for students in the 45%–49% qualifying range was only 55%, compared with the 64% pass rate for students in the 40%–44% qualifying range. students who did not attend the trp but still wrote the examination only had a 47% pass rate. the difficulty of studying in isolation during a time when families are on holiday may have been a significant factor impacting on these results. the results in table 2 suggest that there is an advantage for students who attended the trp. table 2: final results after the january 2015 mathematics tutored reassessment programme. financial implications for this case study, the cost of tuition and residence was covered by a grant to improve graduation rates of engineering students. requiring students to pay for this intervention could impact students both positively (by increasing the motivation to pass) and negatively (by adding pressure and expense to vulnerable students). ways of incorporating the cost of residence during vacation periods into annual residence fees are worth researching further, particularly as the goal of increased participation in higher education is likely to increase the need for financial support for students. in the online survey for the january 2015 students, 56% (22 students) said they would still have attended if they had to pay r500 for tuition, while 13% (5 students) said they would not and 31% (12 students) were unsure. for the july 2014 students, the responses were similarly split with 3, 2 and 3 students answering yes, no and unsure, respectively. conclusions top ↑ from a perspective of improving throughput rates in engineering degrees, the tutored reassessment programme (trp) for first year mathematics was successful because it increased the number of students obtaining credits for first year mathematics courses without spending an additional semester and more course fees on repeating the course. however, 16 students in the first semester and 33 students in the second semester failed with marks below the qualifying range of 40%–49% and were ineligible to write the supplementary examination. given that if students fail, it is most likely to occur in their first year of study and that the adjustment to university is especially difficult in a student's first semester, it is possible that students with final marks lower than 40% might be able to improve to the required achievement level with the intense revision afforded by the review camp. in this case study, lower qualification marks did not correspond with lower chances of passing after participating in the trp. this raises the question of what to choose as the qualification level. for example, if the qualification level was 35%, 11 more students would have qualified for the july 2014 trp and 20 more would have qualified for the january 2015 trp. before expanding the intake range, however, it would be useful to know the success rate of trp students in future courses. there is limited value to helping students to pass one course only for them to fail subsequent courses. future studies could explore the effects of giving a computer-marked entrance test to the trp to students who do not automatically qualify for a supplementary examination. an entrance test would allow students to demonstrate that they had learnt from their mistakes and completed some self-study after the examination, as well as to filter out students who were taking a chance on passing the examination without doing proper preparation. a limitation of the trp is that it is not possible to cover all topics of a 12–13 week course in five days. the selection of examples and topics to include in the review camp was based on judgments made by the trp staff and senior tutors, none of whom saw the examination prior to students writing. naturally, the questions used in the trp lectures, tutorials and tests would not have covered the scope of all the work in the course and students were advised to review their course tutorials, notes and test papers in addition to the trp questions. this case study showed that students’ chances of passing are enhanced if they are given necessary academic support. a concern that future studies should investigate is whether students who are successful in the trp manage to succeed in future courses. it would be doing students a disservice if helping them to pass first year mathematics though a trp only prolonged their stay at university before facing exclusion in later years due to not coping with further mathematics courses. the urgency of the need to graduate more engineers and the research showing that adjusting to university is most difficult in a student's first year, especially for first-generation students (pym & paxton, 2013), suggest that an intervention such as the trp is worth pursuing. acknowledgements top ↑ i would like to thank sue clegg and moragh paxton for mentoring me while developing this article and the centre for higher education development at the university of cape town for supporting the mentorship and providing funding to attend a conference where helpful feedback was obtained on a related presentation. i am also grateful to kenny rafel and the tutors and students who participated in this research. this publication is based on research that has been supported in part by the university of cape town's research committee (urc). competing interests the author declares that she has no financial or personal relationships that may have inappropriately influenced her in writing this article. references top ↑ allie, s., armien, m.n., burgoyne, n., case, j.m., collier-reed, b., craig, t.s., et al. 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(1978). mind in society: the development of higher mental processes. cambridge, ma: harvard university press. appendix 1 top ↑ tutored reassessment programme (trp) interview schedule – january 2015 welcome student, hand out and explain consent forms, ask if the session may be audio recorded. questions seek to explore the potential impact of the trp on three areas: students’ attitudes to learning. how they cope with their studies. their views on becoming an engineer. sample questions what did you think of the structure of the trp, i.e. the lecture followed by whiteboard tutorial, review lecture, test and test review? students’ attitudes to learning 2. do you use a study group? did you use one in the first semester, during the trp and in this semester? 3. how much time did you spend alone preparing for the supplementary exam? 4. have you changed your approach to learning this semester compared with last semester? if there is a difference, to what extent do you think the trp influenced the change? how students cope with their studies 5. what does it mean to you to be successful in a course? 6. how stressful (on a scale of 0–5) was the first semester, the first semester exam, the trp, the supplementary exam and this semester so far? 7. what do you find helps you to cope with your studies? what is unhelpful or works against you coping with your studies? students’ views on becoming an engineer 8. how sure were you at the start of the year that you would qualify with an engineering degree? 9. did that change after hearing your first semester results? 10. did that change (again) after the trp? 11. what do you think it takes to graduate with an engineering degree (besides passing all courses)? 12. would you complete your engineering degree even if you didn’t want to work as an engineer? teaching geometry in schools teaching geometry in schools: an investigative rather than instructive process rasheed sanni university of the witwatersrand email: riosan1@yahoo.co.uk research has documented the prevalence of lessons characterised by homework check, followed by teacher lecture and demonstration, followed in turn, by learner practice sequence of classroom instructional activities in our classrooms. this sequence of classroom activities does not allow for the development of sound mathematics practices and mathematical proficiency. meanwhile, curriculum reforms in south africa as well as in other parts of the world recommend classroom activities where teachers create opportunities for, listen to and extend learners. this paper presents a sequence of activities to be used in the teaching of geometry and surface areas of solid shapes in a grade 8 classroom. the sequence portrays the teaching of these concepts as an investigative rather than instructive process. introduction outcomes-based education (obe) is the foundation for curriculum 2005 (c2005) in south africa, and guides curriculum development and learning outcomes based on critical and developmental outcomes. according to the department of education policy documents, critical outcomes require learners to be able to (1) identify and solve problems and make decisions using critical and creative thinking; (2) work effectively with others as members of a team, group, organisation and community; (3) organise and manage themselves and their activities responsibly and effectively; (4) collect, analyse, organise and critically evaluate information; (5) communicate effectively using visual, symbolic and/or language skills in various modes; (6) use science and technology effectively and critically showing responsibility towards the environment and the health of others; and (7) demonstrate an understanding of the world as a set of related systems by recognising that problem-solving contexts do not exist in isolation (department of education, 2002a). to achieve these, the revised national curriculum statement emphasises a learnercentred, activity-based approach to the teaching of mathematics. this is clearly the tone in the set learning outcomes and assessment standards at various phases of the different grade levels of the educational system. it identifies the following learning areas, which include interrelated knowledge and skills, on the basis of which learning outcomes and subsequent assessment standards are set. the knowledge areas include, (1) numbers, operations and relationships; (2) patterns, functions and algebra; (3) space and shape (geometry); (4) measurement; and (5) data handling. skills include, (1) representation and interpretation; (2) estimation and calculation; (3) reasoning and communication; (4) problem-solving and investigation; and (5) describing and analysing (department of education, 2002b). a critical look at these will reveal some positive relationships between the curriculum and either of the rand mathematics study panel’s (2002) mathematics practices and kilpatrick, swafford and findel’s (2001) strands of mathematical proficiency. it then suggests that the provisions of the curriculum are strong enough to evolve the doers of mathematics that would exhibit mathematics practices and proficiency in mathematics, within and outside of the classroom. however, the sequence of mathematics classroom practices which research has documented to be prevalent in our classrooms entails checking homework, followed by teacher lecture and demonstration, followed in turn by learner practice in a sequence of classroom instructional activities. this is unfortunately neither that which the curriculum proposes, nor that which allows for the attainment of the laudable outcomes. this explains why learners would tell almost spontaneously that the area of a rectangle 20,000 metres by 20 metres is 400,000 square metres, but could not find out how much land has to be taken from a cocoa plantation in order to build a 20-km-long 2-lane highway within the plantation; or a learner can successfully find the surface area of a cuboids pythagoras 65, june, 2007, pp. 39-44 39 teaching geometry in schools: an investigative rather than instructive process 10 × 8 × 10 cm in class, but cannot help her mother estimate the number of one-metre square tiles required to cover the walls and floor of her (mother’s) shop which is 10 m long, 8 m wide and 10 m high. this issue of discontinuity between school learning and cognition out of school is observed by resnick (1987, in engestrom, 1996) thus: the process of schooling seems to encourage the idea that the “game of school” is to learn symbolic rules of various kinds, that there is not supposed to be much continuity between what one knows outside school and what one learns in school. there is growing evidence then that not only may school not contribute in a direct and obvious way to performance outside school, but also that knowledge acquired outside school is not always used to support inschool learning. schooling is coming to look increasingly isolated from the rest of what we do. (p 151) as a means of overcoming this, curriculum reforms, especially in the new south africa, create more challenging responsibilities for the teachers and generally call for change. for instance, it has been widely recognised that traditional mathematics teaching emphasises and encourages memorisation and application of algorithm and procedures. procedural fluency is only one of five strands of mathematical proficiency. others, such as conceptual understanding, strategic competence, adaptive reasoning and productive disposition (kilpatrick et al. 2001) are not emphasised, nor are rand mathematics study panel’s (2002) repressentation, justification and generalisation skills focused on in traditional teaching. provisions and demands of the new south african curriculum suggest a drastic change from traditional modes of teaching to reform (learner-centred) teaching, which focuses on the nurturing of proficiency and development of mathematical practices. it is therefore imperative that ways are found not only to suggest changes in teachers’ practices, but also to provide necessary support and assistance for such desired changes to manifest in the classroom. one such mode of support is through the presentation of lessons that support the nurturing of proficiency and development of mathematical practices in learners, which is the focus of this paper. investigative teaching of geometry research findings have shown that learning is enhanced by their engagement with relevant materials, even if learners have certain learning difficulties (bransford, brown & cocking, 1999; mastropieri & sruggs, 1992). hence, the situated theorist’s position is that everybody will learn if given an enabling environment to participate in a community of practice. this is similar to the calls by brown, collins and duguid (1989), hanks (1991) and lave (1996), that learning is enhanced through participation of individual learners – a process referred to as enculturation and apprentice-ship. all these point to the importance of learners’ active participation in instructional process. abraham’s (1997) learning cycle is an inquirybased teaching approach, developed for science teaching, which i have found useful in investigative teaching of concepts in geometry and indeed in other areas of mathematics. the approach, which derives from the cognitivist and constructivist position on teaching and knowledge-seeking, divides instruction into four progressive stages. step one is engage, where the teacher creates an enabling environment to engage learners in activities that generate curiosity and interest in the planned topic of the day. usually, an inquiry question is presented to the learners at this stage. the second stage is explore, where learners explore the question(s) raised at the engage stage and generate answers. at this stage, the learners are placed in groups, and the teacher acts as a facilitator and usually asks further questions to guide learners’ explorations, and provides hints about how to proceed, without showing learners “exactly how to go about solving the problem” (stein, smith, henningsen & silver 2000). usually, this stage is characterised by a series of questions and introductory activities that are similar to the topic presented in their worksheets (see appendix 1 and 2). then comes the third stage, explain, where opportunities are provided for learner groups to present solutions or answers to the inquiry question(s), giving justifications and explanations for their claims. and then comes the last stage, extend, where learners extend their concepts and skills to other situations by applying what they have learned in the explain stage. usually, and particularly so in mathematics, further tasks in which these skills can be exhibited are provided for the learners. it could be in groups, pairs, or individually. at each of these stages, and indeed at the end, evaluation of the process would go on simultaneously. it is important to note that activities at all of these stages are interlinked, and will bring about learners’ active participation. this emphasis represents an important merger between mathematics as an investigation and mathematics as a body of knowledge, where learners acquire “knowledge through 40 rasheed sanni investigation and experimentation in order to facilitate verbalising, understanding and applying principles in the real world” (luera, killu & o’hagan, 2003: 195). according to freudenthal (1991), mathematics must be connected to reality. it is through this approach that the learners can develop and apply mathematics to problem that makes sense to them (van den heuvel-panhuizen, 2003; wigley, 1994; freudenthal, 1991). below, i provide an example of a learner-centred, inquiry-based investigative lesson in a unit of mathematics in the space and shape (geometry) learning area of c2005. this unit is chosen because of its applicability to all learners, especially from grade level 7 through to 12, and because elements of it are discernable from the assessment standards at these levels. more importantly, it has been used because learners find it difficult to understand description of surface areas, even though this is one of the most physical ways of describing an object. lesson description skills required • length measurement. • addition and multiplication of numbers. • unit analysis. lesson objectives by sustained investigation, inquiry-learning activities, the learners will be able to: • describe the concept of surface area. • explain the features of a shape that influence its surface area. • measure different edges of a prism. • decouple regular prisms and identify the different planes in it. • find the area of the composite surfaces. • by addition, compute the surface area of solids. • generate and use mathematical equations to compute surface areas of plane shapes and other possible variables in them materials needed • three cardboard boxes of 3 different sizes – a: 5 × 4 × 6 cm; b: 2 × 3 × 5 cm; and c: 4 × 4 × 7 cm – and their nets, for different learner groups. • rulers. • 1×1 cm square rubber tiles or stickers. • 1×1 cm square dotted papers. • scissors. • gum. lesson presentation engage this stage requires the teacher’s initiative in presenting a relevant scenario that would arouse learners’ curiosity and stimulate their interest in getting involved. i consider the one below for the lesson: christmas is approaching and mr white and his family have decided to make their living room wear a new look. john, a member of the family, came up with an idea: “tiling the walls and floor would be good.” “that’s nice, sky blue walls and light brown glittering floor,” says his sister flo. “how many tiles will we need?” asked mr white. the teacher then poses the following question to the class: how many wall tiles, floor tiles would be needed to make the living room, 8½ metres long, 7 metres wide and 3 metres high, wear a new look? assume that they are using tiles that are 0.5 by 0.5 metres, and packaged in packs of 10 tiles. if a pack of floor tiles sells for r135 and a pack of wall tiles sells for r110, estimate the cost of the required tiles. the inquiry question here is to determine the number of tiles required and then the cost. the learners are supposed to work in groups of 4 and the teacher circulates and provides guidance but never tells them the way to go about it. explore 1 at this stage, learners in their various groups discuss ideas that emerge within the groups as they attempt to work through worksheet 1 (see appendix 1). they engage in different forms of activities using the materials provided. these activities are guided by the teacher and especially through work-sheet 1. once learners have worked through the worksheet successfully, the class is ready to move onto the next stage. explain 1 this is the stage where learners discuss their solution to the problem in worksheet 1. this should be done by a presenter from each group, and detailed explanation and justification should be demanded. the teacher should ensure this by pressing other learners to ask questions and specifically demand explanations. before the end of this plenary session, the teacher should ensure that an interaction evolves in which learners realise the importance of nets of shapes in mathematical explorations. this leads to the main problem in explore 2. 41 teaching geometry in schools: an investigative rather than instructive process explore 2 exploration continues in this stage. as in explore 1, activities are guided by the teacher and especially with worksheet 2 (appendix 2). the inquiry question in this exploration is to determine the number of tiles that the whites will need, and the cost. an important question regarding the door and windows in the living room will possibly erupt as learners work through this worksheet. depending on the learners’ level of competence, the teacher may find it necessary to advise them to ignore this detail in working on the number of tiles required for the wall. learners should be given free hand to approach this exercise by counting squares on the square paper or by fixing squares stickers on the net. more competent learners may even recognise the pattern and choose to compute the areas. they might work in their original groups or in pairs. explain 2 this is the stage where learners discuss their solution to the original problem in a plenary session. presentation should be done by a different presenter from each group, and detailed explanation should be demanded. as in explain 1, the teacher needs to ensure that logical explanations or justifications are given for the position taken by different groups. pressing them and specifically demanding explanation will do this. one or two groups might notice the linkages to a pattern relating to the area of rectangles. during the learners’ presentations, the teacher should guide the learners in understanding that the number of 1-by-1 squares that a plane surface can take is referred to as, the ‘surface area of the plane surface’. furthermore, where two or more plane surfaces combine to form a shape, then the surface area of that shape is obtained by summing the surface areas of all the plane surfaces involved. the latter is what is referred to as, ‘total surface area’. extend here, an extension of concepts and generalisations of ideas is necessary. learners infer from the group presentations that the surface area of a rectangular prism is a function of its length, breadth and height, or its base area and height. using the two activities, the teacher should now show how the work relates to (i) length, breadth and height, and (ii) base area and height. this can then be used to generalise an equation for the surface area of a rectangular prism, and perhaps other prisms (triangular base, cylinder, etc), depending on the learners’ level of competence. learners might now be asked to find the number of tiles that will be required: (i) to cover the walls of a company’s warehouse that is 25 metres long, 15 metres wide and 12 metres high. (ii) both inside and outside of a regular hexagonal reservoir with an outer measurement of 10 metres long and 20 metres high, assuming the thickness of the wall is one metre. this could be done in pairs. to further extend knowledge in this area, learners could do a project using buildings within the school, or beyond, where the teacher has identified shapes that are not rectangular but some other geometrical shape like a trapezium, or has other fixtures like fireplaces or alcoves. explain 3 this phase allows the learners to approach surface area problems without having to engage in practical counting of squares. it should be noted that some of the pairs might still approach task (i) by disjointedly computing the areas of the walls and summing up. this is fine, but the teacher needs to help the learners link the two tasks in a logical manner, especially if some of the class have already approached the solution holistically. an attempt at task (ii) above, and the project that follows, should actually be commended and appreciated. in fact, this could generate lengthy engage explain 3 extend explain 2 explore 2 explain 1 explore 1 evaluate figure 1. investigative learning (adapted from luera et al.’s (2003) inquirybased learning cycle 42 rasheed sanni discussions, based on shapes and fixtures that learners find and bring to class. such discussions could provide the inquiry question for the next lesson (figure 1). evaluation although, there ought to have been some form of evaluation at each of the different levels above, it is imperative that learners be asked questions or given tasks that allow a further demonstration of knowledge, like tasks (i) and (ii) and the suggested project on the previous page. this could be in another worksheet or textbook, or tasks earlier identified and selected by the teacher for that purpose. it is important that such tasks cover a variety of possible scenarios that learners might face in real life situations, within or outside of the school, and even in different units of measurement. these tasks should be handed in, graded, returned and discussed in class because, according to piaget (1964) and other cognitivists, the discussion that learners’ responses generates is even more important than the questions asked by the teacher or tasks set up and implemented in the classroom. conclusion the foregoing scenario is an illustration of application of abraham’s (1997) learning cycle in the planning and implementation of mathematics lessons. although it is widely used in science teaching, it is equally useful in teaching mathematics. the approach explicates the teaching of geometry as an investigative rather than instructive process. an investigative approach to teaching ensures adequate teacher preparation for the lesson, offers opportunities for learners to recognise previous knowledge, and accommodates learners’ alternative conceptions. hence, it provides learning experiences that help them to revise alternative notions and develop new concepts, and ensures adequate involvement in the lesson. also, according to lorsbach (2002), it naturally leads to other investigations that promote further exploration of other mathematical concepts. it therefore appears that one way for mathematics teachers to exemplify the current reform in mathematics teaching is for them to teach the subject via the learning cycle. this emphasises an investigative rather than instructive mode of instruction that will enhance the nurturing of all five strands of mathematical proficiency and develop mathematical practices. acknowledgements i wish to acknowledge, with sincere gratitude, the quality of supervision and mentorship i enjoy from my supervisor, professor karin brodie. i also acknowledge the financial support that i receive from the mellon foundation and national research foundation towards my doctoral research, out of which this study and paper stemmed. finally, the professional support we get from other members of staff in the marang centre for mathematics and science education at wits is acknowledged and appreciated. references abraham, m. (1997). the learning cycle approach to science education. research matters – to the science teacher. retrieved march 10, 2005, from http://www. narst.org/research/cycle.htm bransford, j., brown, a. & cocking, r. (eds.) (1999). how people learn: brain, mind, experience and school. washington, dc: national research council. boaler, j. (1997). experiencing school mathematics: teaching styles, sex and setting. buckingham: open university press. brown, s.j., collins, a. & duguid, p. (1989). situated cognition and the culture of learning. educational researcher, 18(1), 32-41. department of education. (2002a). introducing the mathematics learning area. in revised national curriculum statement grades r-9 (schools): mathematics (pp 4-6). pretoria: department of education. department of education. (2002b). the senior phase. in revised national curriculum statement grades 10-12 (schools): mathematics (pp 61-91). pretoria: department of education. engestrom, y. (1996). toward overcoming the encapsulation of school learning. in h. daniels (ed.), an introduction to vygotsky. chapter 6 (pp 143-150). london: routledge. freudenthal, h. (1991). revisiting mathematics education. dordrecht: kluwer academy. hanks, f. (1991). foreword by william f. hanks. in j. lave & e. wenger, situated learning: legitimate peripheral participation (pp 13-24). cambridge: cambridge university press. kilpatrick, j., swafford, j. & findel, b. (eds.) (2001). adding it up: helping children to learn mathematics. washington dc: national academy press. lave, j. (1996). teaching, as learning in practice. mind, culture and activity, 3(3), 149-163. 43 teaching geometry in schools: an investigative rather than instructive process lorsbach, a. (2002). the learning cycle as a tool for planning science instruction. retrieved march 10, 2005, from: http://www.coe.ilstu.edu/scienceed/ lorsbach/2571rcy.htm appendix 1: activities in worksheet 1 a) you are to select one of the boxes (a, b, c) you have in your group. luera, g. r., killu, k. & o’hagan, j. (2003). linking math, science and inquiry-based learning: an example from a mini-unit on volume. school science and mathematics, 103(4), 194-202. cut the box, to have one of the shapes a', b’ & c' – these are called the nets of the boxes. measure the different edges on a', b', or c' that you have chosen. mastropieri, m.a. & sruggs, t. e. (1992). science and students with disabilities. review of educational research, 62, 377-411. write these measurements on the edges. b) carefully paste the 1-by-1 cm squares stickers neatly on the chosen net. count the number of these stickers that can be fitted. you might find it useful counting these in the different parts of the shape and then add. national council of teachers of mathematics (2000). principles and standards for school mathematics. reston: va: national council of teachers of mathematics. piaget, j. (1964). development and learning. in r. e. ripple & v. n. rockcastle (eds.), piaget rediscovered. ithaca: cornell university press. rand mathematics study panel (2002). mathematical proficiency for all students: towards a strategic research and development program in mathematics education (pp 24-35). arlington, va: rand education, science and technology policy institute. c) compare the number of tiles in each part with the dimension of the part, and note your observation. appendix 2: activities in worksheet 2 a) the figure on the activity sheet is a sketch of the whites’ living room. draw the net. stein, m., smith, m., henningsen, m. & silver, e. (2000). implementing standard based mathematics instruction: a casebook for professional development. new york: teachers college press. b) identify the walls on your net. c) estimate the number of 30-by-30 cm tiles that will cover the walls. d) identify the floor on your net. van den heuvel-panhuizen, m. (2003). the didactical use of models in realistic mathematics education: an example from a longitudinal trajectory on percentage. educational studies in mathematics, 54, 9-35. e) estimate the number of 30-by-30 cm tiles that will cover the floor, if the tiles are packaged in packs of 11 tiles. you might find it useful to estimate the number of tiles in different parts of the shape and then add them together. f) if a pack of floor tiles sells for r135 and a pack of wall tiles sells for r110, estimate the cost of the required tiles. wigley, a. (1994). models for mathematics teaching. in a. bloomfield & t. harries (eds.), teaching and learning mathematics (pp 22-25). association of teachers of mathematics. be ready to explain to the class how you arrived at your answer. “the best of science doesn’t consist of mathematical models and experiments, as textbooks make it seem. those come later. it springs fresh from a more primitive mode of thought, wherein the hunter’s mind weaves ideas from old facts and fresh metaphors and the scrambled crazy images of things recently seen. to move forward is to concoct new patterns of thought, which in turn dictate the design of the models and experiments. easy to say, difficult to achieve.” edward o. wilson 44 << /ascii85encodepages false /allowtransparency false /autopositionepsfiles true /autorotatepages /all /binding /left /calgrayprofile (dot gain 20%) /calrgbprofile (srgb iec61966-2.1) /calcmykprofile (u.s. web coated \050swop\051 v2) /srgbprofile (srgb iec61966-2.1) 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<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> >> >> setdistillerparams << /hwresolution [2400 2400] /pagesize [612.000 792.000] >> setpagedevice article information authors: cyril julie1 lorna holtman1 monde mbekwa1 affiliations: 1school of science and mathematics education, university of the western cape, south africa correspondence to: cyril julie email: cjulie@uwc.ac.za postal address: private bag x17, bellville 7535, south africa dates: received: 23 june 2011 accepted: 09 july 2011 published: 14 sept. 2011 how to cite this article: julie, c., holtman, l., & mbekwa, m. (2011). rasch modelling of mathematics and science teachers’ preferences of real-life situations to be used in mathematical literacy. pythagoras, 32(1), art. #13, 8 pages. http://dx.doi. org/10.4102/pythagoras. v32i1.13 copyright notice: © 2011. the authors. licensee: aosis openjournals. this work is licensed under the creative commons attribution license. issn: 1012-2346 (print) issn: 2223-7895 (online) rasch modelling of mathematics and science teachers’ preferences of real-life situations to be used in mathematical literacy in this original research... open access • abstract • introduction    • background • research question    • measurement with rating scales • research method    • instrumentation, sample and procedure       • ethical considerations • results    • variance and unidimensionality    • differential item functioning    • rank ordering of the items • discussion and conclusion • acknowledgements    • competing interests    • authors’ contributions • references • footnote abstract (back to top) in order to ascertain the real-life situations that teachers, as stakeholders, would find suitable and appropriate to deal with in mathematical literacy (a compulsory subject for students who are not doing mathematics at the further education and training level of the south african education system), we embarked on a study known as the relevance of school mathematics education (rosme). the principle underpinning this article is that there are times when it is necessary to assess the functionality and quality of questionnaires used to ascertain affective domain issues. the study provides an analysis technique which is not affected by the sample of individuals completing a questionnaire, provided that the instrument meets particular requirements. it thus improves the rigour of measurement. various statistics obtained in this study showed that the instrument used to determine the real-life situations which teachers prefer for mathematical literacy reasonably identifies this variable. however, it is cautioned that much more care needs to be exercised in construction of such instruments. the results also indicated the real-life situations which teachers most and least preferred to be included in mathematical literacy, providing useful information for policy-makers and textbook authors on contextual situations to be included in learning materials. introduction (back to top) it is widely recognised that affective domain issues such as beliefs, attitudes, interest, motivation and perceptions are important determinants for effective teaching and learning. although other means, such as interviews and observations, are used to ascertain affective domain issues in educational research, the dominant research approach is the use of surveys with likert scale-type questionnaires. this approach is followed due to the advantages offered by survey research. despite criticisms offered against questionnaires, pring (2005, p. 39) argues for the value of survey research, particularly as it pertains to the interpretation of questionnaire items, stating that ‘the meanings which the respondents attribute to the questions are not something private and subjective, but the meanings which anyone conversant with the language would attribute to them’. in terms of affective domain issues, the survey instruments used are normally concerned with a latent trait or variable which is operationalised through the questionnaire items. thus effectively a questionnaire comprises a set of items which are realisations of the latent trait being investigated, and respondents are requested to rate each item on the scale. this implies that questionnaires dealing with affective domain issues are effectively rating scales. this is, for example, the case with the questionnaire used by swanepoel and booyse (2006, p. 190) to ascertain the ‘views of south african secondary-school principals regarding the involvement of their teachers in processes of school change’. similarly, the rating scale underpinning of a questionnaire comes to the fore in the study conducted by lessing and de witt (2007, p. 57) on ‘the perceptions of teachers on the value’ of a workshop presented to them. this also is the case in vermeulen’s (2007) study on the mathematical anxiety of grade 10 learners, for which an adapted ‘attitudes toward mathematics’ inventory was used. in much research, as in the case of the aforementioned three studies, the salient aspects of the operation of such scales and their quality are not discussed and reported. this is understandable at least in terms of the space limitations of research articles. what is normally reported, however, is the derivation of the questionnaire from theoretical bases and practical observations, reliability coefficients and various measures undertaken to ensure construct validity. in the study reported here, the view is that there is at all times a need to assess the operation and quality of questionnaires used to ascertain affective domain issues. one of the major reasons for this is that in order to make meaningful, confident and believable recommendations for policies and practices in education, a crucial requirement is that the measurement instrument employed be beyond suspicion with regard to its functioning or, as expressed in legal parlance, ‘it should be beyond reasonable doubt’. this is especially important given the current scepticism of the value of educational research rendering low returns by not providing convincing guidance on issues of importance to governments, policy-makers and those involved in professional practices in education (pring, 2005). this article reports in-depth on the functioning of a survey instrument designed to ascertain the hierarchical order that science and mathematics teachers assign to real-life situations to be used in school mathematics. background it is well known that mathematical literacy was introduced as a compulsory subject for those learners not enrolled for mathematics in the further education and training phase (grades 10−12). this curriculum focuses heavily on the use of mathematics in real-life situations or contexts. a burning question regarding these contexts is who decides on the contexts to be used in mathematical literacy? it is obvious that there are a number of important stakeholders in education that should decide on the real-life situations that should be used in mathematical literacy. the voices of teachers, however, have been silent in this regard, despite the fact that they can provide valuable insights about desirable real-life situations which could possibly be used in mathematics. a study by zevenbergen, sullivan and mousley (2002) brought this to the fore in their report on a group of indigenous australian teachers who did not find the context of a police identification parade suitable and appropriate as inspiration for an open-ended mathematical activity that dealt with averages. using learning resources with which teachers subjectively identify has the potential to improve their engagement with the subject matter and intended objectives of the curriculum for mathematical literacy. as julie asserts: what engages teachers and what [does] not is a complex issue. immediacy in terms of what i can use in my situation as it is currently is emerging as a facet of teacher behaviour regarding relevance. (julie, 2002, p. 7) notwithstanding the isolated australian example, the issue of contexts that teachers would find desirable to be dealt with in mathematical literacy is an under-researched area in the research literature. knowledge about such contexts could benefit experts in developing learning resources for mathematical literacy. in order to ascertain contexts that teachers, as stakeholders, would find suitable and appropriate to deal with in mathematical literacy, we embarked on a study known as the relevance of school mathematics education (rosme). in this article we discuss the functioning of the instrument used to ascertain the teachers’ preferred contexts to be used in mathematical literacy. research question (back to top) the interrelated research questions flowing from the aforementioned narrative and pursued in this research were the following: 1. how well does an instrument used to ascertain the real-life situations preferred for teaching mathematical literacy function? 2. how are the items arranged hierarchically in terms of level of agreement? from the measurement and survey construction literature (e.g. deselle, 2005; fowler, 1995), the functioning of an instrument is defined as whether (1) the instrument represents a single trait or construct; (2) the items form a hierarchy; (3) there are persons and items that do not contribute towards the construct being operationalised; and (4) the items comprising the questionnaire are unique (there are no redundant items). measurement with rating scales as is clear from the introduction, rating scales measure latent traits or constructs. the constructs being measured are not directly observable. for example, when teachers have attended a course or workshop, the appropriateness of such an experience for their practice cannot be ascertained through observation. the construct of importance in this case is ‘the appropriateness of course x for teachers’ practice’. in order to ascertain this ‘appropriateness’, some instrument is developed to operationalise the abstract construct. such an instrument consists of a set of items and the respondents are requested to express their level of agreement with each item on some hierarchically ordered response scale. the response scale contains more than one category, which allows for the making of judgements on the level of endorsement that each respondent and the entire cohort give to the item or the scale. for a scale to represent the construct being measured, it should fulfil certain requirements. wright and master (1982, p. 3) list these requirements as: 1. the reduction of experience to a one dimensional abstraction 2. more or less comparisons amongst persons and items 3. the idea of linear magnitude inherent in positioning objects along a line 4. a unit determined by a process which can be repeated without modification over the range of the variable. the first requirement implies that there must be some certainty that the scale does not measure more than one construct. this assurance starts with the researchers carefully selecting items for the scale, based on the range of literature deemed to provide information on the construct of interest, their own experience and knowledge and those of other experts informing the domain of interest of the construct. decisions concerning the defining items occur through a process of constructive and competitive argumentation. the second requirement necessitates that the scale should be able to distinguish between respondents who have various levels of endorsement for the construct. a further demand relating to the scale is that the items should form some sort of hierarchy. this is also initially decided upon at a theoretical level by the scale constructors. the linearity requirement is to ensure that the words such as ‘highly disagree’, ‘disagree’, etcetera which are being used as response categories comply with the equality of distances when translated into numbers. about the fourth requirement, wright and masters (1982, p. 3) assert that what is required is ‘a theory or model for how persons and items must interact to produce useful measures’. in essence, these requirements are the conditions which social scientists place on a measurement scale as a response to the conundrum of whether what is in other people’s heads can be measured or not. in addition to the aforementioned conceptual approach in developing useful measurement scales, recently there has also emerged the bolstering of such development with quantitative assessment of the derived instruments. rasch methods are one such set of procedures used to evaluate whether an instrument is indeed useful for measuring a latent trait. according to linacre (2008, p. 12): ‘rasch analysis is a method for obtaining fundamental linear measures (qualified by standards errors and quality-control fit statistics) from stochastic observation of ordered category responses’. the rasch model is a probabilistic model and explains how a person’s level of endorsement of an item on a scale dealing with a latent trait is predicted by the person’s endorsement of the scale. for polytomous rating scale data, such as under discussion in this article, the model is: where pnij is the probability that person n encountering item i is observed in category j, bn is the “ability” measure of person n, di is the “difficulty” measure of item i, the point where the highest and lowest categories of the item are equally probable. fj is the “calibration” measure of category j relative to category j-1, the point where categories j-1 and j are equally probable relative to the measure of the item. (linacre, 2008, p. 99) this model is taken as the ideal which the data must fulfil for the rating scale to be viable. however, collected data invariably deviate from this ideal. with rasch analysis the quest is to determine how closely the data fit the model; thus a model is not developed to fit the obtained data. as stated by de roos and allen-meares, the rasch model is: … a normative model for constructing interval-level unidimensionality of data on a group of subjects for a set of items they have completed. the of items on a line that indicates greater and lesser amounts of the variable being measured constitutes operationalization of the variable. the rasch model defines the ideal delineation of items that would constitute an objective, interval-level measuring instrument. (de roos & allen-meares, 1998, pp. 95−96) the rasch procedures report their outcomes in logits derived from the conversion of a raw score according to the aforementioned mathematical formula. in addition to application of rasch modelling in a variety of academic domains, substantial work has also been done on rasch modelling itself, as indicated by andrich (1988) and bond and fox (2001). given this description of the rasch model, data obtained through implementation of a conceptually developed scale can be tested to provide an indication of whether it satisfies the four requirements of a useful scale listed earlier. rasch analysis has been used to analyse the robustness of instruments in a variety of settings. it has also been used in various educational projects, such as the large-scale assessment project trends in mathematics and science study (timss), teacher professional development studies, learner performance on school tests, as well as in medical education and other health-related studies. boone and scantlebury (2006, p. 267) used rasch analysis to interrogate the functioning of an achievement test in science, finding items that ‘functioned differently for african american students compared with their white peers’. they further recommend the use of the rasch model, since by ‘using the rasch model, science educators can improve the quality of quantitative measurement at the individual and the systemic level’ (p. 267). mpofu et al. (2006, p. 329) used rasch modelling to investigate health risk behaviour of a south african cohort of teenagers, reporting that ‘the results from the analysis … suggest that an underlying or latent variable defines health risk in south african teenagers’. these studies indicate that rasch modelling is used to assess and improve the functioning of measurement instruments for researchers to use, to make more informed decisions about issues of import and thereby improve the measures of intervention. in 2006 the mathematics education research journal devoted an entire edition (vol. 18, no. 2) to the use of rasch procedures in research in mathematics education. regarding the issue under purview in this article, callingham and bond (2006, p. 1) argue that ‘the rasch rating scale model allows likert scale attitude data to be thought about in developmental rather than merely descriptive ways’. rasch analysis of the data obtained for this study was performed using the winstep suite of computer programs (linacre, 2008). research method (back to top) instrumentation, sample and procedure the instrument under discussion is a 20-item questionnaire developed by a group of mathematics educators and postgraduate students in mathematics education (julie & mbekwa, 2005). the major criterion used to identify items was that issues inherent in the cluster should be amenable to mathematical treatment. two of the items were strictly intra-mathematical items (‘mathematics that will help learners to do mathematics at universities and technikons’ and ‘to do their mathematics with calculators and computers’), and one dealt with mathematical practice (‘the kind of work mathematicians do’). in order to focus the instrument on real-life situations these items were removed, so the instrument under scrutiny in this article therefore consisted of 17 items. for each of these items teachers had to indicate their preference, by selecting a response 1, 2, 3 or 4 with ‘1’ indicating ‘strongly disagree’ and ‘4’ indicating ‘strongly agree’. data were collected by the researchers and their colleagues from teachers attending university in-service programmes and by postgraduate students in their own and neighbouring schools. the postgraduate students taught in urban and peri-urban areas throughout the western cape province of south africa. given that teachers who attended this specific university teach primarily in low socio-economic areas, and the questionnaire specifically requested that they should indicate the real-life situations they would prefer learners in grades 10–12 to engage with in mathematical literacy, it can be assumed that their responses were targeted at learners from these environments. the sample was thus a convenience sample. science teachers were included in the sample, based on the plausible assumption that such teachers have an interest in the real-life situations that are dealt with in mathematical literacy. it is also a common occurrence that science teachers are assigned to teach mathematical literacy based on workload considerations in schools. in order to ascertain the usefulness of the sample, an assessment was done on whether there were misfitting persons; these are ‘persons who have not used these items in the way it was intended’ (wright & masters, 1982, p. vi) and their responses are deemed as being idiosyncratic. this might, for example, happen when respondents give the same response for all the items or respond to the first few items and leave the rest blank, and so forth. for the instrument under discussion, 67 questionnaires were returned. rasch analysis was done for misfitting persons, and 18 misfitting persons were found through an iterative process of analysis and removal of such persons. further analyses were done with the remaining 49 respondents. the demographic information relating to the 49 teachers of mathematics, science and biology is presented in table 1. respondents were not asked whether or not they were mathematics or mathematical literacy teachers or both. table 1: demographic data. one can glean from table 1 that the majority of teachers were teaching in the further education and training phase (grades 10–12) and had at least 10 years of teaching experience. ethical considerations at the time of data collection respondents were assured of confidentiality and anonymity, and that their participation was voluntary and they could withdraw at any time. they were further informed that there would be no penalties related to their results for assessment of courses they were following. results (back to top) variance and unidimensionality a procedure to determine the unidimensionality of an instrument measuring a latent variable is the principal component analysis of standardised residuals. this procedure is not ‘usual factor analysis’ but ‘shows contrasts between opposing factors, not loadings on one factor’ (linacre, 2008, p. 250). this procedure points to possible items which may distort the unidimensional aspect of an instrument. the decision criteria for the results emanating from this procedure were as follows: variance explained by measures > 60% is good. unexplained variance explained by 1st contrast (size) < 3.0 is good. unexplained variance explained by 1st contrast < 5% is good. (linacre, 2008, p. 335) data obtained from implementation of the instrument rendered the variance explained by measures forthcoming from the empirical data as 27.2%. the expected variance (for the data to fit the rasch model) to be explained by measures was 28.0%. this difference was not deemed significant. the unexplained variance emanating from the data was 72.8% and the rasch model’s expectation is 72.0%; this can also be deemed as not significant. as is clear from the decision criteria, the cut-off point of 60% was not met. however, these percentages of the variances were to be expected, since the respondents were fairly homogeneous with regard to their teaching context and the issue under discussion. the reported standardised residual variance for the first contrast of 3.1 is above the recommended cut-off point. analysis of graphs of the spread of items indicated that two items, one dealing with ‘youth dances’ and the other with ‘youth fashion’, stood out as operating as a group independent of the rest of the items. these two items can be considered as conceptually linked around the notion of the behaviour of young people. the respondents were mature adults, and it can reasonably be assumed that they viewed the two activities as related. further analysis was done to ascertain the absence of which of these two items produced a better unidimensional instrument. this analysis rendered that removal of the item dealing with ‘youth dances’, with standardised residual variance for the first contrast of 2.8, enhanced the unidimensionality of the instrument. further analysis proceeded using the instrument now reduced to 16 items through removal of that dealing with ‘youth dances’. a variety of other indicators can be calculated by the rasch procedures, and these were interpreted to give an indication of the functioning of an attitudinal instrument such as that under consideration here. the results emanating from these procedures are discussed in the next section. differential item functioning another important criterion for a measuring scale is that the items should not function differentially for different categories of participants comprising the sample. given that for the instrument under scrutiny here the participants were teachers of different genders, the items should not function differentially for females and males. analysis of differential item functioning along gender lines was conducted for the cohort of teachers. this analysis rendered that two items (‘pension and retirement’ and ‘health’) might be easier for female than for male teachers to endorse, and that a further two items (‘agriculture’ and ‘emergency services’) might be easier for male than for female teachers to endorse. although differential item functioning (dif) is noticeable for these items, ‘for statistically significance dif on an item, prob. < .05’ (linacre, 2008, p. 266). none of the reported probabilities for these items were less than 0.05 and hence dif between female and male teachers was not statistically significant for all the items of the scale. dif was not performed for the other demographic dimensions since the sample was fairly homogeneous in respect of their teaching environments. rank ordering of the items as pointed out earlier, in a useful scale the items operationalising the abstract construct under discussion should form a hierarchy, so that it is possible to conclude which of the items respondents would find easy and which they would find difficult to endorse. with rasch modelling three values can be determined to ascertain the hierarchical property of a scale: the measure of an item, and the infit mean square and the outfit mean square values respectively. the measure of an item is the location on the scale. for a rating scale it indicates the level of difficulty for endorsing the item. the difficulty of endorsement ‘of an item is defined to be the point on the latent variable at which it’s high and low categories are equally probable’ (linacre, 2008, p. 221). reeve and fayers (2005) give the criterion for the spread of items to be deemed acceptable. these authors point out that the measures should be in the range -2 to +2 logits. with the range for the instrument in this study being -0.94 to 1.46, as given in table 2, this criterion was fulfilled. table 2: measure and fit statistics. in rasch analysis mean infit and outfit squares (see table 2) are calculated to indicate ‘items which do not contribute to the definition of a coherent and useful variable’ (wright & masters, 1982, p. vi). for items to have a good fit to the rasch model, the decision criteria are: values greater than 2.0 degrades measurement; values greater than 1.5 neither constructs nor degrades measurement; values from 0.5 to 1.5 are productive of measurement and those less than 0.5 misleads us into thinking we are measuring better than we really are. (linacre, 2008, pp. 221−222) it is observable from table 2 that the fit statistics for all the items were within the ‘productive of measurement’ range. in fact, both the infit and outfit mean square values for all the items fell within this acceptable range. it is thus concluded that the reconstructed scale to measure the construct ‘teachers’ preference for real-life situations to be used in mathematical literacy’ forms a continuum. the rasch model can be used simultaneously to estimate a person’s ability (ability to endorse an item) and an item’s difficulty (or endorsability of the item). the winsteps software (linacre, 2008) analysis presents these two estimates in a ‘person-item map’ which provides an indication of the informativeness of the measures. figure 1 gives the person map of items for teachers’ preferences for real-life situations to be used in mathematical literacy. the 49 teachers appear on the left-hand side, with teachers with a high level of endorsement of the scale at the top and those with a low level of endorsement at the bottom. figure 1: person-item map. the items, with those ‘hard to endorse’ at the top and ‘easy to endorse’ at the bottom, appear on the right-hand side. noticeable from this figure is that the mean for the persons (m = 0.66, sd = 0.72) is higher than the mean for the items (m = 0.00, sd = 0.59), which suggests that the respondents hierarchically endorsed the same items. a further observation is that items in four sets of items (t11 and t5; t14 and t9; t17 and t3; t12, t13, t4 and t6) share the same location. essentially this may imply redundancy of items, and that the reliability of the instrument will not be influenced if only one of the shared items is used. however, for an instrument dealing with affective issues, care should be taken with replacement, and conceptual considerations in addition to computational ones should drive decisions about replacement of items. for example, t5 (youth fashion) and t11 (lottery and gambling) share the same location and are somewhat remotely conceptually linked, but are different in terms of the mathematics related to them. they are thus not candidates for replacement. on the other hand, t9 (pension and retirement) and t14 (inflation) can be considered as conceptually linked, because of the relationship they share in construction of pension and retirement scheme mathematical models. at school level, however, they point to different mathematical topics and thus removing any one would not be sensible. in figure 1 gaps are apparent at five places (between t18 and t11; between t16 and t1; between t14 and t17; between t17 and t12 and between t2 and t7.)1 these gaps indicate that the items in these regions are not evenly spread. this might be a result of the homogeneity of the respondents, the small sample and strong preferences, both negative and positive expression. for example, for t10 (‘health’), the item found to be the easiest to endorse, 96% of the respondents selected the categories ‘agree’ and ‘strongly agree’. t18 (‘youth fashion’) was the hardest to agree with, with only 8% selecting ‘strongly agree’. the rasch model reports a ‘person reliability’ measure which ‘is equivalent to the traditional “test” reliability’ (linacre, 2008, p. 393). the person reliability for the teacher context preferences for real-life situations to be used in mathematical literacy was 0.65. ‘low values indicate a narrow range of person measures’ and person reliability of 0.5 ‘divides a sample into 1 or 2 groups’ (linacre, 2008, p. 393). the homogeneity of the sample accounts for the person reliability being low and points in the direction of a need for a more diverse sample for further development of the instrument. ‘”item reliability” has no traditional equivalent and low values indicate a narrow range of item measures, or a small sample’ (linacre, 2008, p. 393). the item reliability obtained was 0.84 and gives a high level of support that the hierarchical ordering of the items in table 2 will be reproduced with a different sample of teachers working in a similar context as the respondents in this study. the results obtained on the functioning of the instrument and the adjustments effected indicate that this was reasonable. the revised instrument resulted from removal of misfitting persons and an item contributing to violation of the unidimensional character of the initial instrument. this has helped identify a unidimensional trait representing mathematics and science teachers’ preferred contextual situations to be used in mathematical literacy. discussion and conclusion (back to top) the overall objective of the rosme project is to ascertain and trace over time the real-life situations which learners, teachers and parents would prefer to be used in mathematical literacy. the teacher instrument is specifically aimed at ascertaining the contexts that teachers prefer. expression by teachers of preferred contextual situations is a subjective issue. however, for a variety of reasons – of which economic factors and expediency are the most important – it is desirable to have some robust, easily implementable measurement instrument. this is because such an instrument will enable the assessment of real-life contexts that teachers prefer to be used in mathematical literacy. also, the instrument would allow for the tracking of teachers’ interests in contexts over time, in the same way that the timss and programme for international student assessment instruments track the performance of learners in school mathematics. tracking is important for informing decision-makers and learning resources developers of relevant real-life situations to include in such materials. as boone and scantlebury (2006, p. 253) assert, ‘statistical confidence in [such] measures’ validity and reliability is essential.’ the results of the infit and outfit mean squares and standardised residual variance are indicative of the rosme instrument’s ability to ascertain the contextual situations that mathematics and science teachers prefer, bolstering this ‘statistical confidence’. the fit statistics show that the instrument used to determine the contexts that teachers prefer for mathematical literacy reasonably identifies this variable. given that attitudinal instrument development is an iterative process, this finding points in the direction of further development with a more heterogeneous group of teachers in terms of the socio-economic context within which they teach, to ascertain the universality of the instrument. in pursuing this path we will heed the advice of wright and masters (1982, p. 102), that: ‘when items do not fit, that signifies … not the occasion for a looser model, but the need for better items’. low endorsement of items points to areas in need of continuous professional development. so, for example, low endorsement was accorded to the item ‘mathematics involved in a lottery and gambling’. a plausible reason for this low endorsement is the attachment of negative consequences of this activity. a teacher motivated the low endorsement status as follows: ‘if you want to instil positive value these [lottery and gambling] might be the opposite effect’. in this instance teachers might not, as yet, have a sense of the mathematics involved in lottery and gambling and how this can be used productively to inculcate the positive values they desire. niss (2007, p. 1306), in his assessment of the state of research related to problematiques in mathematics education, concludes that there is a ‘need for investing a considerable amount of effort’ into researching issues related to the affective domain in mathematics education. in research on issues related to affective issues pertaining to school mathematics, instruments are normally used without reporting the viability of these in measuring the trait under consideration. our analysis of one such instrument shows that much more care needs to be exercised in the construction of these. they should, at a minimum, reasonably identify the latent traits they purport to measure in order to provide useful information on attitudinal issues related to school mathematics. acknowledgements (back to top) this research is supported by the national research foundation under grant number fa2006042600032. any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the national research foundation. competing interests we declare that we have no financial or personal relationships which may have inappropriately influenced us in writing this article. authors’ contributions c.j. was the project leader and was responsible for experimental and project design. c.j. performed the analysis. l.h. and m.m. made conceptual contributions. c.j., l.h. and m.m. wrote the manuscript. references (back to top) andrich, d. (1988). rasch models for measurement. thousand oaks, ca: sage publications. bond, t.g., & fox, c.m. (2001). applying the rasch model: fundamental measurement in the human sciences. mahwah, nj: lawrence erlbaum. boone, w., & scantlebury, k. (2006). the role of rasch analysis when conducting science education research utilizing multiple-choice tests. science education, 90(2), 253−269. http://dx.doi.org/10.1002/sce.20106 callingham, r., & bond, t. (2006). research in mathematics education and rasch measurement. mathematics education research journal, 18(2), 1−10. http://dx.doi.org/10.1007/bf03217432 de roos, y., & allen-meares, p. (1998). application of rasch analysis; exploring differences in depression between african-american and white children. journal of social service research, 23, 93−107. deselle, s.p. (2005). construction, implementation, and analysis of summated rating attitude scales. american journal of pharmaceutical education, 69(5), 1–11. http://dx.doi.org/10.5688/aj690597 fowler, f.j. (1995). improving survey questions: design and evaluation. thousand oaks, ca: sage publications. julie, c. (2002). making relevance relevant in mathematics teacher education. in i. vakalis, d. hughes-hallett, c. kourouniotis, d. quinney, & c. tzanakis (eds.), proceedings of the 2nd international conference on the teaching of mathematics (at the undergraduate level), 01−06 july (pp. 1−8). hersonissos, crete, greece: university of crete, [cd]. wiley publishers. julie, c., & mbekwa, m. (2005). what would grade 8 to 10 learners prefer as context for mathematical literacy? the case of masilakele secondary school. perspectives in education, 23(3), 31−43. available from http://journals.sabinet.co.za/webz/images/ejour/persed/persed_v23_n3_a6.pdf?sessionid=01-38432-1480009827&format=f lessing, a., & de witt, m. (2007). the value of continuous professional development: teachers’ perceptions. south african journal of education, 27(1), 53−67. available from http://journals.sabinet.co.za/webz/images/ejour/educat/educat_v27_n1_a4.pdf?sessionid=01-38432-106046365&format=f linacre, j.m. (2008). winsteps® rasch measurement computer program user’s guide. beaverton, or: winsteps.com. available from http://www.winsteps.com/a/winsteps-manual.pdf mpofu, e., caldwell, l., smith, e., flisher, a., mathews, c., wegner, l., et al. (2006). rasch modeling of the structure of health risk behavior in south african adolescents. journal of applied measurement, 7(3), 323−334. niss, m. (2007). reflections on the state and trends in research on mathematics teaching and learning: from here to utopia. in f.k. lester (ed.), second handbook of research on mathematics teaching and learning (pp. 1293−1312). charlotte, nc: national council of teachers of mathematics. pring, r. (2005). philosophy of educational research. london: continuum. reeve, b., & fayers, p. (2005). applying item response theory modeling for evaluating questionnaire item and scale properties. in p. fayers & r. hays (eds.), assessing quality of life in clinical trials: methods and practice (pp. 55−76). new york, ny: oxford university press. swanepoel, c., & booyse, j. (2006). the involvement of teachers in school change: a comparison between the views of school principals in south africa and nine other countries. south african journal of education, 26(2), 189−198. vermeulen, n. (2007). mathematical literacy: terminator or perpetuator of mathematical anxiety? in m. setati, n. chitera, & a. essien (eds.), proceedings of the 13th annual national congress of the association for mathematics education of south africa, vol. 1 (pp. 368−380). johannesburg: amesa. wright, b.d., & masters, g.n. (1982). rating scale analysis. chicago, il: mesa press. zevenbergen, r., sullivan, p., & mousley, j. (2002). contexts in mathematics education: help? hindrance? for whom? in p. valero & o. skovsmose (eds.), proceedings of the 3rd international mathematics education and society conference (pp. 1−9). copenhagen: centre for research in learning mathematics. available from http://www.mes3.learning.aau.dk/papers/zevenbergen_et_al.pdf footnote (back to top) 1.only first items in the row are mentioned. article information authors: marc north1,2 iben m. christiansen1 affiliations: 1school of science, mathematics and technology education, university of kwazulu-natal, south africa 2centre for research in mathematics education, university of nottingham, united kingdom correspondence to: marc north email: marc.north@nottingham.ac.uk postal address: room b44, dearing building, jubilee campus, university of nottingham, ng8 1bb, united kingdom dates: received: 26 nov. 2014 accepted: 25 apr. 2015 published: 29 june 2015 how to cite this article: north, m., & christiansen, i.m. (2015). problematising current forms of legitimised participation in the examination papers for mathematical literacy. pythagoras, 36(1), art. #285, 11 pages. http://dx.doi.org/10.4102/pythagoras.v36i1.285 copyright notice: © 2015. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. problematising current forms of legitimised participation in the examination papers for mathematical literacy in this original research... open access • abstract • introduction • the relevance of dowling’s theoretical language to the empirical terrain of mathematical literacy • domains of practice, positions and apprenticeship in mathematics    • domains of mathematical practice    • positions    • apprenticeship in mathematics • domains of practice and positions embodied within the grade 12 mathematical literacy exemplar examinations3    • why a focus on examinations?    • background information on the structure of the examinations    • demonstration of the method used for categorising questions    • categorisation of the examination paper questions according to the domains of mathematical practice    • discussion    • ml in the secondary schooling framework: the promotion of educational disadvantage? • conclusion: towards an alternative vision for mathematical literacy • post script • acknowledgements    • competing interests    • authors’ contributions • references • footnotes abstract top ↑ in this article we argue that in south africa the current format of legitimised participation and practice in the examination papers for mathematical literacy restricts successful apprenticeship in the discipline of scientific mathematics and limits empowered preparation for real-world functioning. the currency of the subject, then, is brought into question. we further argue that the positioning of the subject as a compulsory alternative to mathematics and the differential distribution of these two subjects to differing groups of learners facilitates the (re)production and sustainment of educational disadvantage. we draw on dowling’s theoretical constructs of differing domains of mathematical practice and positions and focus analysis on a collection of nationally set exemplar grade 12 examination papers to identify legitimised forms of participation in the subject. we conclude by arguing for a reconceptualised structure of knowledge and participation in mathematical literacy and make preliminary recommendations in this regard. introduction top ↑ the secondary school subject mathematical literacy1 (ml) has faced increasing criticism in recent years, with some positing the subject as a second-rate qualification to mathematics – ‘mathematical literacy (which is little more than arithmetic) is to mathematics what spelling is to writing’ (oberholzer, 2012, slide 14) – and others calling for the removal of the subject from the secondary school curriculum framework (for e.g., see jansen, 2012). we contend that current criticism is grounded in three main concerns relating to the structure, status and practices of the subject. firstly, the ml examinations are perceived to be considerably less demanding than those in mathematics – and the high pass rate of 87.1% in the subject compared to 59.1% in mathematics in the 2013 academic year provides some validation for this concern. this state of affairs is seen to contribute to the exodus of increasing numbers of learners from mathematics to ml (enrolment in mathematics has decreased significantly since the introduction of ml – from 60.1% in 2006 to 42.7% in 2013) (department of basic education [dbe], 2014a, pp. 125, 159; department of education [doe], 2008, p. 27). secondly, there is an accompanying concern that participation in the subject does not afford access to the same and equally varied and comprehensive avenues of study or career choice as mathematics. in this regard, ml is perceived as a limiting qualification; this is why increasing enrolment figures in the subject at the expense of enrolment in mathematics is of such concern. it is in response to concerns such as these that the current minister of education instituted a ministerial panel to investigate, amongst other things, ‘the currency of mathematics and mathematical literacy and whether this is the best option for the south african schooling system in terms of preparing learners for the workplace and for higher education studies’ (dbe, 2013, p. 4). the third concern draws directly from the two previous concerns and is the topic of particular and primary relevance in this article. this concern relates specifically to the mismatch between the stated curricular intention of the subject for preparing and empowering participants for more effective and empowered functioning in complex contextual sense-making practices encountered in everyday life, the workplace and in a democratic society (dbe, 2011, p. 8) – what venkat (2010, p. 55) refers to as a life-preparedness orientation – and current legitimised forms of participation in the examinations for the subject that prioritise engagement with elementary mathematical structures in largely contrived reconstructions of real-world practices. this mismatch is explicitly recognised and highlighted in the findings and recommendations of the above mentioned ministerial panel who argue for heightened emphasis on the curriculum policy intentions, particularly in relation to areas of teaching, assessment and examinations (dbe, 2014g, pp. 12, 54–58). a consequence of this mismatch is that participation in the subject ml is reserved primarily for ‘weaker’ learners who are perceived to be unable to cope with the demands of scientific mathematics contents. in the south african context, such learners are, predominantly, learners who are located in poorly resourced schools situated in lower socio-economic environments. given the positioning of ml as a qualification that involves engagement with only limited forms of mathematical participation, these (increasing numbers of) ‘weaker’ learners who engage in the subject are denied access not only to an educational experience that would better prepare them for life and the world of work, but also to a vast array of study and career opportunities which would facilitate upward social and economic mobility. we contend that the existing structure of mathematically legitimised participation in the subject contributes to the (re)production and sustainment of a degree of educational and social disadvantage. our intention in this article is to employ a theoretically informed analysis of practices in the subject – as evidenced in the structure of participation legitimised in nationally set grade 12 exemplar examinations for the subject – to validate this claim. at a general level this article builds on the work of others such as christiansen (2006), julie (2006), frith and prince (2006) and venkatakrishnan, graven, lampen and nalube (2009), all of whom have problematised components of policy and practice associated with the subject mathematical literacy. specifically, however, this work is an extension and elaboration of the work of christiansen (2007), but with important differences. firstly, christiansen performed an analysis of the curriculum for the subject prior to the implementation of the curriculum at classroom level and, as such, was only able to make predictions regarding possible formats of classroom practice. this article, by contrast, has been written eight years after the initial implementation of the subject and after five cohorts of learners have passed through the subject structure. furthermore, this article is focused on analysis of empirical practices contained in national examinations for the subject – which, in turn, are seen to directly influence and determine the structure of legitimised forms of participation with the contents of the subject at classroom level. in alternative terms, where christiansen’s analysis focused on the field of official recontextualisation, the analysis in this article is focused on the field of pedagogic recontextualisation (cf. bernstein, 1996). secondly, the original ncs curriculum that formed the primary focus of christiansen’s analysis has been replaced by a restructured curriculum contained in the curriculum and assessment policy statement (caps) (dbe, 2011). this caps curriculum is characterised by a modified statement of intention and philosophy for the subject and by a reformulated statement of intended focus for pedagogic and assessment practices. the analysis in this article, then, is more directly relevant to and reflective of current empirical practices and forms of legitimised participation in the subject. the following structure applies in the article. in the next section we provide an overview of aspects of dowling’s (1998) theoretical language relating to domains of mathematical practice, positions and apprenticeship in mathematics. in the subsequent section we employ these theoretical constructs in analysis of an amended or reworked 2014 exemplar national grade 12 ml examinations. here we argue that the examinations prioritise a form of participation in the subject characterised by engagement with, primarily, public and descriptive domain of mathematics type practices. as a consequence, learners are relegated to positions of dependency and objectification in pedagogic processes and are denied the opportunity for successful apprenticeship in the discipline of mathematics. we argue further that it is this characteristic of the structure of endorsed participation in the examinations that contributes to the devaluing of the qualification and facilitates a degree of educational disadvantage. in the final section of the article we suggest an alternative structure of endorsed participation for the subject characterised by the promotion of a life-preparedness orientation (venkat, 2010) and argue tentatively that this alternative structure may prompt a more empowering educational experience for participants in the subject. the relevance of dowling’s theoretical language to the empirical terrain of mathematical literacy top ↑ a key issue we seek to highlight in this article is that despite curriculum intentions for the prioritisation of contextual sense-making practices, participation in ml is endorsed and evaluated primarily according to mathematical structures and mathematically legitimised forms of practice. the work of dowling (1998) provides a useful means for identifying and describing the characteristics of the dominant form of legitimised participation and practice in the subject – as elaborated below. importantly for the contents of this article – focused as it is on the empirical terrain of the subject ml – a key aspect of dowling’s work involves analysis of the relationship between mathematical and extra-mathematical knowledge, contents, discourse and practices. a central argument in this regard is that academic (generally) and mathematical (specifically) activities are incommensurate with everyday activities and that academic mathematical knowledge cannot be used as a theory for facilitating adequate or appropriate understanding of everyday practices. for dowling, exclusive or predominant participation in particular forms of contextualised mathematics practices inhibits mathematical understanding and affords only a limited degree of life-preparation (1995a, p. 9, 1995b, p. 209). dowling argues further that the consequence of this is particularly experienced in the schooling system where mathematics focusing on relevance is commonly made available to learners who are deemed to have lower mathematical ability (many of whom are located in predominantly working-class environments) whilst abstract mathematics is made available to supposedly higher ability learners (many of whom are located in better resourced schools situated in middle-class environments). the ‘weaker’ learners from poorer socio-economic environments are, thus, exposed to a form of mathematics that is limiting, both in terms of mathematical and real-world understanding and also in terms of future study and career opportunity. it is in relation to this situation that emphasis on relevance in mathematics is deemed to facilitate the production and sustainment of a degree of educational difference and disadvantage (dowling, 1994, p. 138, 1998, pp. 236–241, 2010a, slide 2; hoadley, 2007, p. 684). given the high degree of correlation between these identified areas of focus in dowling’s theoretical language and the structure of participation in the subject ml that is promoted in the examinations – namely, as a subject reserved for learners of supposedly weaker mathematical ability and, yet, characterised by the prioritisation of mathematical structures in encounters with heavily mathematised reconstructions of real-world problem-solving scenarios – it is appropriate to employ elements of this theory in analysis of empirical examination-related practices in the subject. domains of practice, positions and apprenticeship in mathematics top ↑ the discussion in this section of the article is specifically concerned with what dowling (1998) refers to as the structural level of his theoretical language – namely, the dimension of the language that facilitates identification and description of the positions filled by participants in an activity and the practices that those participants engage in within the activity.2 domains of mathematical practice dowling (1998) identifies four domains of mathematical practice (see figure 1), each of which is characterised by differing strengths of institutionalisation (i.e. the degree of specialisation) of mathematical contents and mode of expression employed in the messages through which the practices of the activity are transmitted. importantly for the discussion on the empirical terrain of the subject ml, contents refers not only to specific knowledge or skills (i.e. the topics of mathematics), but also to the nature of the context – mathematical or extra-mathematical – from which the content is drawn (sethole, goba, adler & vithal, 2006, p. 119). mode of expression, on the other hand, refers to the means through which these contents are transmitted, together with the words, method and language used in the transmission of a message. drawing on the above, strong institutionalisation (i+) of the mode of expression and content in a statement or problem is characterised by explicit reference to, engagement with and use of specialised mathematical entities or explicitly intra-mathematical contexts. weak institutionalisation (i–), by contrast, is characterised by restricted reference to specialised entities (dowling, 2008a, p. 15) and/or by engagement with extra-mathematical contexts and contents (sethole et al., 2006, p. 119). figure 1: domains of mathematical practice. the esoteric domain of mathematical practice is characterised by a high degree of institutionalised practice (i+) and comprises explicitly specialised, generalisable and abstracted mathematical contents, contexts, principles, symbols and statements: ‘the esoteric domain comprises the specialised forms of expression and content which are unambiguously mathematics’ (dowling, 1994, p. 130). for dowling (2008b, p. 4), this is the domain that contains the non-negotiable part of school mathematics and what is considered to be legitimate mathematical practice. given the high degree of specialisation of both content and mode of expression, the mathematical principles that regulate the practices of the activity and the principles against which the practices of the activity are evaluated are explicit in this domain (dowling, 1994, p. 129) and learners who engage with mathematical knowledge and practices in this domain have full access to these principles. furthermore, dowling (1998) argues that it is only in this domain that full access to these principles is possible: because ambiguity is minimised in the esoteric domain, specialised denotations and connotations are always prioritised. it is, therefore, only within this domain that the principles which regulate the practices of the activity can attain their full attention. the esoteric domain may be regarded as the regulating domain of an activity in relation to its practices. (p. 135) however, school mathematics contains more than just this highly specialised non-negotiable domain of practice. rather, pedagogic practice facilitates the casting of a gaze beyond the esoteric domain to establish links between this domain and the extra-mathematical world: ‘the practice [mathematics] must also constitute a more weakly institutionalised region in order to permit entry into it; this is the public domain’ (dowling, 2010a, slide 2). the result of this mathematical gaze, as extra-mathematical settings are appropriated in the mathematics classroom and colonised according to mathematical principles and structures, is the development of the public domain of school mathematics as a collection of recontextualised and reformulated or mathematised problems (dowling, 2008b, p. 4). problems posed in this domain are weakly institutionalised in terms of both content and mode of expression, with the consequence that the practices of this domain appear to be and are experienced as being about something other than mathematics: ‘mathematics parading as something other than itself’ (dowling, 2001, p. 20). crucially, the public domain is not to be equated with reality or with the real–world. rather, it is the space where the everyday represents a recontextualised, virtual and distinctly mathematised and, hence, mythologised representation of reality. the expressive domain of practice is also constituted through the imposition of a mathematical gaze from the esoteric domain on the terrain of the extra-mathematical and represents an alternative form of recontextualisation than in the public domain. in this domain, non-mathematical modes of expression (i−) are appropriated for use within explicitly intra-mathematical contexts and are employed to give expression to specialised mathematical contents (i+) (dowling, 1998, pp. 135–136). elsewhere, dowling (2010a, slide 2) refers to this domain as the domain of pedagogic metaphors, where fractions are equated to pieces of cake and where words such as sharing are employed to facilitate understanding of abstract concepts such as division. as with the public and expressive domains of practice, the descriptive domain is a further form of esoteric domain gaze recontextualisation. in this domain, specialised mathematical modes of expression (i+) are employed to model non-specialised contents and/or extra-mathematical contexts (i−) (dowling, 1998, p. 136). this is the domain that reflects modelling practices – where mathematics is employed in the generation of descriptions of extra-mathematical contents and contexts. importantly, the generative, regulative and evaluative esoteric domain principles that define the recontextualisation process and, consequently, the structure of legitimate participation in these domains, cannot be fully realised in practices that remain exclusively in these domains and which do not make a deliberate move into the esoteric domain and towards a degree of abstraction and generalisation: the esoteric domain must signify differently because of the recruitment of a non-mathematical setting, so that, once again, the principles of the esoteric domain cannot be made fully explicit within [these] domain[s]. (dowling, 1998, p. 137) furthermore, the identified domains of practice are not mutually exclusive in the sense that engagement with public domain contents precludes engagement with esoteric domain contents. rather, and as is discussed in more detail below, for dowling the development of mathematical knowledge and, particularly, successful apprenticeship in the discipline of mathematics, are facilitated through traversal of the entire terrain. positions the activity of mathematics also constitutes positions in relation to how knowledge and available practices are distributed to participants in the activity (dowling, 1998, p. 131). dowling identifies four possible positions – subject, apprentice, dependent and object. the subject of an activity has mastered the practices and regulating principles of the activity. this position is the most dominant position and every other position is to a greater or lesser extent subordinated to or objectified by the subject (dowling, 1998, p. 140). by contrast, participants in the apprenticeship position engage in the practices of the activity with the intention, at some point in the future, of becoming potential subjects of the activity: successful apprenticeship to an activity is achieved (metaphorically) upon the completion of a one-hundred-and-eighty-degree rotation of the apprentice who thereby ‘moves’ from ‘outside’ to ‘inside’ the activity and becomes its subject. (dowling, 1998, p. 123) the dependent position is a subordinated position to the apprentice in respect to the subject. this position is occupied by participants who are unable to access (or are denied access to) the regulating principles of an activity, commonly through the interference of extra-mathematical elements that obscure these principles. in such instances, participants are dependent on the subject to make visible and explicit the regulating principles according to which any mathematisation processes of non-mathematical elements have been conducted. this position is particularly characteristic of practices that remain primarily within the expressive or descriptive domains of mathematical practice, where the inclusion of non-mathematical expression and contents can serve to inhibit access to the regulating esoteric mathematical principles that structure a problem. participants in the dependent position are not construed as potential future subjects (as with the apprentice position). consequently, the final career outcome of such participants is less certain: the apprentice will become the subject, but the only certainty for the dependent is their reliance on the subject to mediate the practices of the activity (dowling, 1998, p. 141). importantly, participants in the dependent position may be fully aware that they are operating outside of the public domain and that encountered problems are mathematical in nature, but are thereby reliant on a subject of the activity to make visible and accessible the underlying generative and regulating esoteric domain knowledge, principles and practices. not so with the objectified position. this position occurs primarily in relation to public domain practices characterised by the recontextualisation of real-world practices according to the principles of the esoteric domain – via an imposed mathematical gaze by either the subject or another party on an extra-mathematical context. when practices are recontextualised in this way, participants are invited to recognise themselves in the problems, as though the problems are their own and relate to and have relevance to their lives: learners are invited to become objects in the problems (dowling, 1996, p. 402) – for example, as a shopper who needs to solve a problem involving a cost comparison. this is in contrast to the apprentice position where the individuality and identity of the participant remain exterior and irrelevant to the context and problem (dowling, 1996, p. 402). participants who are objectified in a practice believe (or are led to believe) that they are operating inside of the domain of the practice (for example, as shoppers in a supermarket) rather than in the domain of mathematics. for this reason they are construed as neither subjects nor dependents of mathematical practices since they believe that they are engaged in non-mathematical activities. the consequence is that such participants operate with no or only restricted independent and/or unaided awareness of and access to the esoteric domain principles that regulate the practice. furthermore, since the recontextualised practice represents a mythologised version of the real-world practice, and because the mathematics is hidden in the practice, within this position the learners learn sufficiently neither about mathematics nor about real-world practices (dowling, 1998, p. 141). apprenticeship in mathematics in establishing an explicit connection between the differing domains of practice and positions in these domains, consideration must be given to the conditions under which successful apprenticeship in mathematics is to be achieved. ensor and galant (2005, p. 297) argue that apprenticeship in mathematics is achieved ‘when learners … have grasped the “generative principles” (dowling, 1998) of whatever discourse they have been inducted into, and are able to produce appropriate learning performances’. since it is only in the esoteric domain of mathematical practice that learners are exposed to the generative, regulative and evaluative principles that define and structure legitimate knowledge and participation in the discipline of mathematics, successful apprenticeship in school mathematics is only possible if participants engage in and with esoteric domain practices (dowling, 1998, p. 140). the reverse is true of practices that remain in the public domain and, to a lesser extent, the expressive and descriptive domains. participants exposed exclusively to public domain practices do not gain direct access to the esoteric mathematical principles underpinning the practices since these are obscured and overshadowed by the interference of weakly institutionalised contents and modes of expression. such participants are more likely to be positioned as dependents or objects: the mythologising of the public domain practices as valid representations of reality and the objectification of participants in the problem-solving process render the participants dependent on the subject of the activity to make explicit the underlying (mathematical) regulating principles and the criteria according to which mathematisation processes have been conducted (by either the subject or by another party) in the generation of public domain contents (dowling, 1998, p. 141). a similar situation applies for practices embedded exclusively or primarily within the expressive and descriptive domains (and which do not make an explicit reach to esoteric domain contents), where the inclusion of non-mathematical elements serves to inhibit access to and understanding of the mathematically regulated principles of encountered problems. however, in these domains participants are fully aware that the problems are regulated according to mathematical principles and so are not objectified through the problem-solving process. nonetheless, participants remain dependent on a subject to make more explicit and to facilitate access to the mathematical principles and structures that regulate participation in the practice. however, this does not mean that the teaching of mathematics should confine itself only to the esoteric domain. rather, potential subjects for an activity are attracted to an activity through the public domain: ‘the public domain is, in this sense, the principal arena in which an activity selects its apprentices’ (dowling, 1998, p. 149). as such, if no projection is made from the esoteric domain to the public domain, then no new apprentices will be ‘hailed’ into the activity (dowling, 1998, p. 141). importantly, the move from the public domain to the esoteric domain is not a direct process. rather, the expressive domain of practice provides a bridge for the transition between these domains by facilitating engagement with more explicit mathematical contents through reference to familiar non-mathematical expressions. equally, the descriptive domain provides a bridge from the esoteric to the public domain: once esoteric domain contents have been mastered, a mathematical gaze is able to be cast over the practices of world to facilitate the description of these practices according to mathematical structures. as summarised by dowling (2008a): there is no natural route into the esoteric domain of mathematics … nor, of course, can mathematics education begin and remain exclusively in the esoteric domain; there has to be a way in and this will always be via the public domain. pedagogic action must then construct trajectories that lead into the esoteric domain via the expressive and that lead to the public domain from the esoteric via the descriptive. … in general, in respect of any specialist region of mathematics, the whole of the map should be traversed in one way or another. (p. 27) as such and in summary, apprenticeship of students into mathematics, in dowling’s terms, involves the successful move from public to esoteric domain. interruption of this trajectory inhibits students’ ability to master mathematics. (ensor & galant, 2005, p. 297) domains of practice and positions embodied within the grade 12 mathematical literacy exemplar examinations3 top ↑ why a focus on examinations? according to dowling (1998, p. 120), ‘activities are produced by and reproduced in human subjects – who move, routinely, between activities – and by texts’. focus on the national examination papers thus provides a particular site of identification and analysis of the dominant practices and positions legitimised and prioritised for the subject ml. the examinations provide a useful site of analysis for two further reasons. firstly, the examinations reflect current official state opinion on the structure of legitimised and endorsed participation with the contents of the subject. secondly, the structure of endorsed participation espoused in the national examinations has a ‘backwash effect’ (allais, 2007) on pedagogic practice by informing the dominant orientation and forms of participation legitimised by teachers as they prepare learners for the examinations. that said, the limitations of the conducted analysis are acknowledged, particularly with respect to any conclusions made regarding pedagogic practice within classroom settings. background information on the structure of the examinations examinations in ml are characterised by two examination papers that are differentiated according to cognitive demand. paper 1, classified as a basic skills paper, is focused on the assessment of proficiency of basic skills and knowledge of both mathematical and contextual contents; it comprises questions posed primarily at the two lowest levels of the four-level assessment taxonomy. paper 2, by contrast, characterised as an applications paper, is focused on assessment of the ability to engage with both mathematical and non-mathematical techniques and considerations in contextual problem-solving processes. this paper comprises questions posed primarily at the two highest levels of the assessment taxonomy (dbe, 2014f, pp. 5, 7). crucially, the caps curriculum document prioritises as a primary goal in the subject, engagement with authentic contexts and resources that bear a high degree of resemblance to real-world practices (as opposed to contrived, mathematised or fictitious contexts) and a focus on the development of an enhanced understanding of these contexts (as opposed to a dominant emphasis on the development of mathematical knowledge) (dbe, 2011, pp. 8–11). there is, thus, every expectation that this impetus is reflected in the exemplar examinations. demonstration of the method used for categorising questions consider the question extract shown in figure 2. importantly, the department of basic education did not respond to our request for permission to use an extract from the exemplar examination papers. as such, figure 2 shows a re-modelled version of question 1.2 in the exemplar paper 1 examination (dbe, 2014b, p. 4). although the contextual scenario and question phrasing are different in this remodelled version (the exemplar examination makes reference to the context of landline telephone tariffs), we have made every effort to ensure that the core concepts and domain of practice prioritised in each question bear close resemblance to the original examination question. that said, we acknowledge the potential challenge to the validity of the analysis process of the exemplar examinations based on the method employed and demonstrated through analysis of a reformulated task. figure 2: remodelled version of a paper 1 examination question. the first thing to notice is that the questions are pre-empted by and based on a contextual scenario; this is a common strategy and occurrence in the examinations as well as in pedagogic practices in the subject. although our analysis is focused primarily on the type of practice prioritised in engagement with these scenarios, it is worth noting that many of the contextual scenarios employed draw on deliberately constructed fictitious situations and resources. for example, although the scenario of electricity costs is realistic in south africa, there are no such systems as cheep-cheep and bright-sparks (and no such telephone system as the scamtho 250 cited in the examination paper) and the tariff structures associated with electricity or telephone contracts are seldom as simple (or as simply presented) as portrayed here. in the context of the examinations, employed contextual scenarios are largely deliberately constructed to facilitate evaluation of particular mathematical and calculation-based processes. with respect to the questions developed for engagement with this contextual scenario, question 1.2.1 is characterised by the usage of, primarily, non-specialised references to everyday forms of expression, with no explicit signification given on the structure of the institutionalised mathematical content required for answering the question (and it is only through the inclusion of the vocabulary signifier calculate that an indication is given of a requirement for a form of mathematical engagement with the scenario). in this question, then, participants are led to believe that this is an actual real-world scenario and that they are engaging with the scenario in a way that reflects real-world practice. as a result, this question is categorised as reflecting a form of mythologised practice associated with the public domain. question 1.2.2 (a), by contrast, makes reference to a resource involving largely non-specialised contents (namely, electricity costs), but employs a specialised mode of expression through reference to missing variables that have been imposed on the unspecialised context. as such, this question and the resource required for the successful completion of the question are categorised as reflecting a form of practice associated with the descriptive domain. a similar classification applies to question 1.2.2 (b), where a specialised mathematical mode of expression (i.e. a graph) is referenced for use in relation to an extra-mathematical context and unspecialised contents. categorisation of the examination paper questions according to the domains of mathematical practice the information in table 1 shows the count and percentage of the questions in the examination papers categorised according to the identified domains of mathematical practice. table 1: categorisation of the exemplar examination questions according to dowling’s (1998) domains of practice schematic. discussion the complete absence of both esoteric and expressive domain contents in the examinations is immediately noticeable from the information shown in table 1: participants in this subject are clearly not being apprenticed in the discipline of scientific mathematics. this finding is unsurprising given that the subject is directed at learners who are unable to cope with the demands of the scientific mathematics course and is focused, instead, on engagement with elementary mathematical principles in contextualised problem-solving scenarios. however, herein lies a contradiction: even though apprenticeship in esoteric domain mathematics practices is entirely absent, the fact that all of the examination questions are able to be categorised in the domains of mathematical practice schema signifies that legitimate participation and practice in the examinations are regulated by mathematical structures and principles. learners who supposedly are unable to engage with esoteric mathematical contents continue to be regulated and evaluated according to mathematically defined and imposed knowledge, structures and principles. this observation gives rise to a further question of what, precisely, participants in the subject are being apprenticed into – clearly not mathematics, so what then? perhaps, as suggested by the curriculum statement for the subject, the answer lies in more empowered and effective participation in real-world practices: ‘the subject mathematical literacy should enable the learner to become a self-managing person, a contributing worker and a participating citizen in a developing democracy’ (dbe, 2011, p. 8). however, the spread of all of the questions within the domains of mathematical practice schema undermines and negates this intention. by way of explanation, consider that there are 21 instances in the combined examinations that reflect contextual scenarios (and associated resources) comprising characteristics of constructed or fictitious situations, deliberately developed to facilitate evaluation of engagement with particular mathematical processes. only four instances (paper 1: 3.1 and 5.1.5; paper 2: 2.1 and 2.1.3) reflect an expectation for engagement with real-world practices through exposure to unaltered and authentic contextual resources. 4 this is significant in that the examinations posit engagement with mathematised, virtual and mythologised representations of reality as the dominant terrain of contextual engagement in the subject, which again reinforces the dominance of esoteric mathematical principles as the basis of legitimate participation and negates the potential for engagement in the subject to serve as a means for empowered functioning in real-world practices. whether or not dowling’s framework is employed as a lens for analysis, it remains obvious that virtually every question in the examinations is driven almost exclusively by mathematical goals – either the assessment of a specific mathematical technique or the recollection of a form of mathematical knowledge. in this sense, the majority of the supposedly real-world contexts employed are superfluous – mere window dressing, since the dominant orientation involves the assessment of mathematical techniques and knowledge and not authentic and enhanced contextual sense-making practices. enhanced understanding of contextual environments and contextually legitimate forms of participation is thus negated in the examinations. as predicted by christiansen (2007, p. 91), participants in the subject ml, then, are seemingly stranded in ‘no-mans-land between mathematics and life-related content’, both denied apprenticeship in mathematics and restricted in preparation for real-world functioning. this situation is exacerbated by the fact that the esoteric mathematical generative and evaluative principles that regulate the criteria for successful and legitimate participation in the subject are rendered hidden and inaccessible to the participants through a dominant focus on questions in the examinations that require engagement with public and descriptive domain practices. in other words, despite the complete exclusion of esoteric domain practices, the authors of the examination papers deliberately prioritise engagement with forms of practice that reflect varying degrees of esoteric domain recontextualisation. in this regard, the number of questions associated with public domain practices dominate throughout both examination papers (although more marks are allocated to descriptive domain practices in the paper 1 examination) and instances of objectification are commonplace: ‘1.4 write down another reason, excluding the profit, why the committee decided to use venue abc.’; ‘3.1.4 (c) justify megan’s claim that the price of a 9-year-old pre-owned smart car could be worth r50 000.’ (dbe, 2014c, pp. 4, 7). this emphasis on public domain practices signifies the prioritisation on the part of the examiners of the ‘myth of participation’ (dowling, 1998) as a key component of the structure of legitimate pedagogic action in the subject. the prioritisation of descriptive domain practices is also significant, reflecting a heightened expectation for participants to employ distinctly mathematical modes of expression in engagement with unspecialised contents in a variety of contextual settings, albeit with awareness of the mathematically legitimated basis of any generated descriptions. it is particularly significant that this emphasis on descriptive domain practices occurs in conjunction with the complete exclusion of esoteric domain contents: although learners are expected to employ specialised, mathematically legitimated forms of expression in solving problems, at no point are they afforded formal and explicit access and exposure to the esoteric mathematical principles that regulate and structure these expected forms of expression. this emphasis on descriptive domain practices is somewhat in contradiction to the equally dominant emphasis on public domain practices (at the complete expense of esoteric and expressive domain practices): by emphasising public domain practices, the authors are encouraging participants to identify with problem scenarios, to draw on their own experiences in solving those problems and to envision how they might engage with the problems if encountered in their own daily life settings; however, by then placing almost equal emphasis on descriptive domain problems the examiners are also encouraging the dominance of mathematical modes of expression and mathematically orientated descriptions in problem-solving processes. this could lead to confusion on the part of some learners about the domain of practice and associated criteria according to which the structure of legitimate participation in different questions in the examinations will be evaluated by the examiners.5 the prioritisation of public and descriptive domain practices to the complete exclusion of esoteric domain practices ensures that learners in the subject are positioned as dependents in the learning process, commonly objectified by the problem-solving scenarios, but seldom (if ever) given direct and explicit access to the specialised esoteric principles that define and regulate the structure of legitimate participation in these scenarios. learners in the subject are continuously required to engage mathematical principles in contextual problem-solving situations, but seldom (if ever) exposed to processes involving generalisation and abstraction of these principles. learners in the subject are consistently required to engage in mathematised problem-solving scenarios, but seldom (if ever) empowered to engage in the mathematisation processes. instead, learners are reliant on their teachers to uncover and make explicit the mathematically structured principles (decided on by others) according to which legitimate and endorsed participation in the problem-solving processes are defined and evaluated. time and time again, learners are exposed to mathematised forms of contextual situations that bear only limited resemblance to reality and, yet, are presented as opportunities for empowered real-world functioning. all of the above indexes not only a limited and limiting form of participation in the discipline of mathematics, but also stunted preparation for enhanced real-world functionality. however, as noted in the introduction, the pass rate in ml is significantly higher than that in mathematics. this begs the question that if ml is not affording access to mathematics nor to life preparation, then what is the high pass rate indicative of? we contend that it is the distinction between the two examination papers and, specifically, the presence of the basic skills paper (which only assesses questions posed at the two lowest levels of the taxonomy of cognitive demand) that is contributing to the significantly high pass rate in the subject. in this regard, the high pass rate is indicative only of the ability of the learners to engage in simplistic and low-level numeracy-type calculations – hence the criticism that ml is an easier qualification than mathematics and offers less opportunity for career recruitment. as a final observation, it is worth mentioning that the dominance of public and descriptive domain practices in the examinations highlights a degree of inconsistency with the intention of the caps curriculum and with the statement of intention and philosophy for the subject espoused in that curriculum. the caps curriculum, both through stated intention (see dbe, 2011, pp. 8–14) and also through curriculum and assessment structure (see dbe, 2011, pp. 12–14, 96–97, 104–109), prioritises a dominant intention for contextual sense-making practices over the development of mathematical knowledge: mathematics is posited as one of several elements required for effective engagement in sense-making practices of authentic real-world contexts. this intention suggests engagement with contextual environments, practices, considerations and forms of participation that do not fit within dowling’s domains of mathematical practice schema – namely, with practices in which the generative principle of the structure of legitimate participation does not rest exclusively in the domain of esoteric mathematics. this curricular intention is clearly not achieved in the examination papers. ml in the secondary schooling framework: the promotion of educational disadvantage? since it is the contents and the traits of practices associated with the esoteric domain contents that are privileged for ‘recruiting to careers’ (dowling, 1996, p. 393), mathematics in south africa is posited as a gateway subject to participation in numerous fields of tertiary study and future career choice. ml, by contrast – due to the limited degree of mathematical apprenticeship and restricted empowerment for real-world functioning afforded through participation in the subject – is perceived as a limiting qualification with a significantly lower study and employment currency than mathematics. now consider that within the secondary schooling curriculum framework, participation in the subject ml is reserved primarily for learners of supposedly weaker mathematical ability, many of whom are located in poorly resourced schools situated in poorer socio-economic environments. these learners are encouraged to participate in a qualification that provides not only a limited learning experience in the classroom, but also limited opportunity for future study and career choice and, hence, for social and economic advancement. further, given that the number of learners enrolled in the subject exceeds those in mathematics and is still increasing, it is of little wonder that there is heightened concern from certain quarters of the mathematics education community regarding the suitability and viability of the subject as a compulsory alternative to a scientific mathematics course. thus, despite curricular intentions for ‘social transformation’, ‘social justice’ and ‘inclusivity’ (dbe, 2011, pp. 4–5), the inclusion of ml in the south african schooling system, and particularly the orientation of the subject towards public and descriptive domain mathematical practices, facilitates rather than alleviates an element of educational disadvantage. conclusion: towards an alternative vision for mathematical literacy top ↑ given the identified problematic structure of existing mathematically legitimised forms of participation in the subject, we suggest that an alternative conception of the structure of legitimate knowledge and participation is necessary. although it is still in development, we posit tentatively at this stage that this revised knowledge structure is dominated by a life-preparedness orientation (venkat, 2010) which, in turn, we characterise as comprising a dominant agenda for contextual sense-making practices and a dominant intention for the critical evaluation of both mathematical and contextual structures encountered in the real-world problem-solving process. in this revised conception, any and all mathematics appropriated in the midst of contextual sense-making practices is done so in service to the goal of the development of an enhanced and broader understanding of appropriate and legitimate forms of participation in those contexts. mathematics is, thus, replaced by the ability to identify, engage with and model contextually appropriate and endorsable forms of participation in real-world practices as the structuring principle of legitimate participation in the knowledge domain. it is our contention that this revised structure of legitimate participation has the potential to negate the mythologising associated with current mathematically-legitimised forms of participation in the subject and to offer a more empowered form of participation and more effective preparation for enhanced real-world functioning in the world beyond the walls of the classroom. further details of our preliminary thinking on the components of this envisioned knowledge structure can be found in north (2015). post script top ↑ at the time of final edits to this article the official end-of-year 2014 grade 12 mathematical literacy examinations (dbe, 2014d, 2014e) have been written. public and descriptive domain practices continue to dominate in these examinations (to the exclusion of esoteric and expressive domain practices), albeit with a higher count of questions (55.1%) and higher mark allocation (59.4%) focused on public domain practices. we interpret this change as indicative of an attempt by the examiners to place heightened emphasis on problem-solving encounters that appear to bear closer resemblance to contextual practices (through a reduction in the degree of institutionalisation of expression and content in the problems). of particular interest is the inclusion of a question (question 4.4) that we contend is not able to be classified according to domains of practice schema due to the contextually prevalent form of practice and participation required and legitimised in the question. this signifies potential, albeit minor, consideration of an orientation towards life-preparedness. acknowledgements top ↑ this article was written whilst m.n. was a fellow in the sasol inzalo fellowship programme and would not have been possible without the generous support, financial and other, of the sasol inzalo foundation. competing interests the authors declare that we have no financial or personal relationships that might have inappropriately influenced us in writing this article. authors’ contributions m.n. (university of nottingham) authored the article based on research conducted for his doctoral dissertation and was responsible for all alterations made during the review process. i.m.c. (university of kwazulu-natal) was the doctoral supervisor and provided substantial feedback on the component of the doctoral dissertation on which this article is based and also on the contents of this article. references top ↑ allais, m.s. (2007). making educational judgements: reflections on judging standards of intended and examined curricula. pretoria: umalusi. bernstein, b. (1996). pedagogy, symbolic control and identity: theory, research and critique. london, uk: taylor and francis. christiansen, i.m. (2006). mathematical literacy as a school subject: failing the progressive vision? pythagoras, 64, 6–13. http://dx.doi.org/10.4102/pythagoras.v0i64.94 christiansen, i.m. (2007). mathematical literacy as a school subject: mathematical gaze or livelihood gaze? african journal of research in mathematics, science and technology education, 11(1), 91–105. retrieved from http://hdl.handle.net/10520/ejc92659 cooper, b., & dunne, m. (2000). assessing children’s mathematical knowledge: social class, sex and problem-solving. buckingham: open university press. department of basic education (dbe). (2011). curriculum and assessment policy statement: mathematical literacy. pretoria: author. available from http://www.education.gov.za/linkclick.aspx?fileticket=q8+skgy43rw=&tabid=570&mid=1558 dbe. (2013). government gazette (vol. 577, issue 36677). pretoria: government printers. dbe. (2014a). 2013 national senior certificate examination: national diagnostic report. pretoria: author. available from http://www.education.gov.za/linkclick.aspx?fileticket=tnib+pqqxu4=&tabid=358&mid=1325 dbe. 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(2014g). the ministerial task team report on the national senior certificate (nsc). pretoria: author. available from http://www.education.gov.za/linkclick.aspx?fileticket=vifw9jginno=&tabid=358&mid=1301 department of education. (2008). education statistics in south africa 2006. pretoria: doe. available from http://www.dhet.gov.za/dhetstatisticspublication/doestatsataglance2006.pdf dowling, p. (1994). discursive saturation and school mathematics texts: a strand from a language of description. in p. ernest (ed.), mathematics, education and philosophy: an international perspective (pp. 124–142). london: the falmer press. dowling, p. (1995a, january). against utility in school mathematics and in education research: a voice from the twilight zone. paper presented at the annual conference of the southern african association for research in mathematics and science education, cape town. dowling, p. (1995b). discipline and mathematise: the myth of relevance in education. perspectives in education, 16(2), 209–226. dowling, p. (1996). a sociological analysis of school mathematics texts. educational studies in mathematics, 31(4), 389–415. http://dx.doi.org/10.1007/bf00369156 dowling, p. (1998). the sociology of mathematics education: mathematical myths/pedagogic texts. london: routledge falmer. dowling, p. (2001). mathematics education in late modernity: beyond myths and fragmentation. in b. atweh, h. forgasz, & b. nebres (eds.), sociocultural research on mathematics education: an international perspective (pp. 19–36). new jersey, nj: lawrence erlbaum associates. dowling, p. (2008a, may). what is mathematics? the problem of recontextualisation. paper presented at the university of thessaly. available from http://www.pauldowling.me/publications/dowling2008b.pdf dowling, p. (2008b, november). mathematics, myth and method: the problem of alliteration. paper presented at the kings college, london. available from http://www.pauldowling.me/publications/dowling2008a.pdf dowling, p. (2010a, october). the problem of recontextualisation. powerpoint presentation to the japan alumni association. available from http://www.pauldowling.me/presentations/dowling2010.ppt ensor, p., & galant, j. (2005). knowledge and pedagogy: sociological research in mathematics education in south africa. in r. vithal, j. adler, & c. keitel (eds.), researching mathematics education in south africa: perspectives, practices and possibilities (pp. 281–306). cape town: hsdrc press. frith, v., & prince, r. (2006). reflections on the role of a research task for teacher education in data handling in a mathematical literacy education course. pythagoras, 64, 52–61. http://dx.doi.org/10.4102/pythagoras.v0i64.99 hoadley, u. (2007). the reproduction of social class inequalities through mathematics pedagogies in south african primary schools. journal of curriculum studies, 39(6), 679–706. http://dx.doi.org/10.1080/00220270701261169 jansen, j. (2012, 08 november). we can do the maths. times live. available from http://www.timeslive.co.za/opinion/columnists/2012/11/08/we-can-do-the-maths julie, c. (2006). mathematical literacy: myths, further inclusions and exclusions. pythagoras, 64, 62–69. http://dx.doi.org/10.4102/pythagoras.v0i64.100 north, m. (2015). the basis of legitimisation of mathematical literacy in south africa. unpublished doctoral dissertation. university of kwazulu-natal, pietermaritzburg, south africa. oberholzer, a. (2012, may). the dilemma of mathematics and mathematical literacy. paper presented at the umalusi maintaining standards conference, johannesburg. sethole, g., goba, b., adler, j., & vithal, r. (2006). fine-tuning a language of description for mathematics items which incorporate the everyday. in d. clarke, c. keitel, & y. shimizu (eds.), mathematics classrooms in twelve countries: the insider’s perspective (pp. 118–130). rotterdam: sense publishers. venkat, h. (2010). exploring the nature and coherence of mathematical work in south african mathematical literacy classrooms. research in mathematics education, 12(1), 53–68. http://dx.doi.org/10.1080/14794800903569865 venkatakrishnan, h., graven, m., lampen, e., & nalube, p. (2009). critiquing the mathematical literacy assessment taxonomy: where is the reasoning and the problem solving? pythagoras, 70, 43–56. footnotes top ↑ 1. in south africa, participation in the subject mathematics (comprising scientific mathematics contents) is compulsory up to the end of grade 9. at the beginning of grade 10, learners are required to choose between participation in either mathematics or mathematical literacy. mathematics is characterised by the study of scientific, abstract or esoteric mathematical contents; mathematical literacy, on the other hand, is characterised by engagement with everyday problem-solving situations and the utility of mathematics in those situations. a detailed discussion of the intention and philosophy of the subject, together with analysis of the existing body of literature on the subject, is provided in (north, 2015, chapter 8). a similar discussion and analysis of international conceptions of mathematical literacy, quantitative literacy and numeracy (and the distinctions between these) is found in chapters 5–7 of that same work. 2. dowling also constitutes textual and a resources levels for his theoretical language. see dowling (1998, pp. 131–132, 150–154) for elaborated discussions of these levels. 3. the decision to focus analysis on a set of exemplar examinations and not on the final end-of-year nationally set examinations is deliberate and is driven by logistical constraints with respect to the availability of examinations based on the recently implemented caps curriculum process. at the time of writing, the first official final grade 12 national examinations for the caps curriculum have not yet been written, and the recently distributed exemplar grade 12 examinations that are the focus of the analysis process in this article are the only currently available examination papers that directly reflect the structure and form of the forthcoming final grade 12 caps-related examinations. it is our belief that since these exemplar examinations serve as an example of and precursor to the final examinations and are set by the same national examining panel, the structure of endorsed participation legitimised in or through engagement with these examinations can be taken to be reflective of the structure of legitimate participation to be prioritised in the final examination papers. 4. question 3.1 references a table of seemingly authentic statistical data (together with a source reference for the data); question 5.1.5 references a tax bracket table; question 2.1 includes information and a photograph of the salt telescope in sutherland; question 2.1.3 shows a cross-sectional representation of the telescope (which an internet search revealed is an authentic resource sourced from the website for the south african astronomical observatory (see http://www.saao.ac.za; interestingly, this source reference for the diagram is not given in the examination paper!). 5. see (cooper & dunne, 2000) for a detailed discussion of the ways in which learners of varying mathematical competence and different social class backgrounds might respond differently to assessment tasks that include realistic elements. abstract purpose of the article background literature review methodology results conclusion acknowledgements references footnotes about the author(s) carol a. bohlmann centre for educational testing for access and placement, university of cape town, south africa robert n. prince centre for educational testing for access and placement, university of cape town, south africa andrew deacon centre for innovation in learning and teaching, university of cape town, south africa citation bohlmann, c.a., prince, r.n. & deacon, a. (2017). mathematical errors made by high performing candidates writing the national benchmark tests. pythagoras, 38(1), a292. https://doi.org/10.4102/pythagoras.v38i1.292 original research mathematical errors made by high performing candidates writing the national benchmark tests carol a. bohlmann, robert n. prince, andrew deacon received: 16 mar. 2015; accepted: 27 feb. 2017; published: 25 apr. 2017 copyright: © 2017. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract when the national benchmark tests (nbts) were first considered, it was suggested that the results would assess entry-level students’ academic and quantitative literacy, and mathematical competence, assess the relationships between higher education entry-level requirements and school-level exit outcomes, provide a service to higher education institutions with regard to selection and placement, and assist with curriculum development, particularly in relation to foundation and augmented courses. we recognise there is a need for better communication of the findings arising from analysis of test data, in order to inform teaching and learning and thus attempt to narrow the gap between basic education outcomes and higher education requirements. specifically, we focus on identification of mathematical errors made by those who have performed in the upper third of the cohort of test candidates. this information may help practitioners in basic and higher education. the nbts became operational in 2009. data have been systematically accumulated and analysed. here, we provide some background to the data, discuss some of the issues relevant to mathematics, present some of the common errors and problems in conceptual understanding identified from data collected from mathematics (mat) tests in 2012 and 2013, and suggest how this could be used to inform mathematics teaching and learning. while teachers may anticipate some of these issues, it is important to note that the identified problems are exhibited by the top third of those who wrote the mathematics nbts. this group will constitute a large proportion of first-year students in mathematically demanding programmes. our aim here is to raise awareness in higher education and at school level of the extent of the common errors and problems in conceptual understanding of mathematics. we cannot analyse all possible interventions that could be put in place to remediate the identified mathematical problems, but we do provide information that can inform choices when planning such interventions. purpose of the article the national benchmark tests (nbts), as with all high-stakes testing, are at times viewed with anxiety and scepticism. parents, test candidates and teachers appear to want more information, even though a large amount of information is readily available on the national benchmark tests project (nbtp) website (www.nbt.ac.za). particularly in relation to mathematics, the website attempts to provide information to ensure that all stakeholders have the information they need to allay anxiety and dispel some myths (see specifically the section ‘preparing your learners for the mathematics (mat) test’). there are also requests for test exemplars and information regarding special preparation classes. we do not support ‘extra lessons’ in preparation for writing the nbts, since comprehension does not necessarily increase with coaching. once school leavers are admitted to university they will not have access to coaching. if an analysis of the results of the nbts can raise awareness of problems that learners face, and if these can then be remediated, the result will be an improvement in the standard of mathematics in general and, in particular, in performance in mathematically demanding first-year courses. if this also translates to improved nbt results, it would indicate that the school sector has been responsive to the message the nbtp has sought to give. higher education needs to be more responsive to the needs of the students that they admit by providing ‘responsive curricula’. examples of such interventions include extended programmes, bridging courses, tutorials provided by lecturers or specially appointed tutors and referral to external sources of additional support such as online mathematics programmes providing particular assistance in areas not well covered at school, for example trigonometry. we provide background to the nbts together with an analysis of the results of the mathematics tests written in 2012 and 2013. the results are considered at a national level, and we cannot analyse specific interventions that could be put in place. this is more properly the field of teachers and the academics who are involved in teacher education who can hopefully be more fully informed by the results presented here. background the nbtp was commissioned in 2005 by higher education south africa (hesa), now called universities south africa, with the following objectives (griesel, 2006, p. 4): to assess entry-level academic and quantitative literacy and mathematics proficiency of students. to assess the relationship between higher education entry-level requirements and school-level exit outcomes. to provide a service to higher education institutions requiring additional information to assist in admission (selection and placement) of students. to assist with curriculum development, particularly in relation to foundation and augmented courses. at the end of grade 12 all school leavers write the national senior certificate (nsc); those wishing to enter higher education also write the nbts if required to do so by the institutions to which they intend applying. all nbt candidates must write the academic and quantitative literacy (aql) test; those who intend to study in an area requiring mathematics need to write the mathematics (mat) test as well. the norm-referenced nsc mathematics exam necessarily attempts to reflect the entire school mathematics curriculum. while the criterion-referenced nbt mat tests do not test anything outside the school curriculum, they are not constrained to include all nsc mathematics topics, and thus focus on those aspects of the school curriculum that have a greater bearing on performance in first-year mathematics courses. clearly, the nsc mathematics exams and the mat tests should be regarded as complementary forms of assessment. the two assessment regimes are complementary in the sense that the nsc attempts to answer the question ‘to what extent do nsc candidates meet the curriculum statement expectations as expressed in the subject assessment guidelines?’ while the nbtp attempts to answer the question ‘to what extent do students aiming to enter higher education meet the core academic literacy, quantitative literacy and mathematics competencies required by school leavers on entry to higher education study?’ whereas the aql tests are intended as tests of generic skills in the domains of academic and quantitative literacy, the mat tests are explicitly designed to measure the mathematical preparedness of candidates for mathematically demanding curricula in higher education. the curriculum and assessment policy statement (caps), as was also the case with curriculum 2005, emphasises the ability of mathematics to provide the necessary conceptual tools for analysing, making and justifying decisions (department of basic education (dbe), 2008a, 2011a), important competencies in higher education. the mat tests assess the degree to which learners have achieved the ability to manipulate numbers, synthesise a number of different mathematical concepts and draw strictly logical conclusions in abstract symbolic contexts. lecturers agree that these higher-order skills underlie success in higher education mathematics. the nbts have an important role to play in the south african educational landscape (in both basic and higher education) by providing useful information additional to that provided by the nsc. initially there was scepticism as expressed in 2009 by the then department of basic education director-general duncan hindle: we would need to be convinced about the need for additional testing. we need to be shown where the nsc is not adequate, and we need to be convinced that the nbt is a credible test. in the end we run the nsc at huge expense. … is it really justifiable to introduce something else? (paton, 2009) the then deputy director-general penny vinjevold questioned their purpose: ‘what will they be used for?’ she asked (paton, 2009). an important use has been in the provision of additional diagnostic information with which lecturers in higher education and teachers in schools can engage. spaull and taylor (2015) highlight the growing evidence of exceedingly low levels of learning in many developing countries, including india, indonesia, malaysia, mexico, pakistan, thailand, turkey and south africa. they also remark that not only are the levels of learning typically low, but the actual learning associated with a year of schooling differs widely across countries. comparing nsc results across schools and across provinces, it appears that within south africa there are wide differences in the learning demonstrated by grade 12 learners. the nbts are tools that enable us to measure levels of learning relevant to higher education. a council on higher education (che) study (scott, yeld & hendry, 2007) analysed the performance patterns of the 2000 intake into higher education. the results are further substantiated by professor adler. under the auspices of marang (the wits centre for maths and science education), on 5 may 2011 prof. adler commented in wits maths connect on ‘the implications for universities of the new caps particularly with respect to admissions requirements’. she attached to this document additional information emanating from a 2010 academy of science of south africa (assaf) forum, called ‘mind the gap’, which noted the high attrition rate in first-year life and physical sciences degrees followed by low graduation rates, with only a small group completing in regulation time (adler, 2011). these results, which are confirmed in the analyses of the 2006 intake cohort in the che (2013) publication, indicate that most students at south african universities take more than the specified three or four years to complete their studies. in the period after the introduction of the nsc in 2008 up to 2012, less than a quarter of grade 12 learners writing mathematics achieved more than 50%. in the same period the proportion of learners achieving between 70% and 100% fell from 8.3% in 2008 to 5.9% in 2011 and increased to 7.0% in 2012. in 2012, only 15 800 learners achieved between 70% and 100% for mathematics compared with 24 900 in 2008 (snyman, 2013, p. 510). with challenging school conditions and changing school curricula (curriculum 2005, examined in grade 12 from 2009 until 2013, and then the caps, examined for the first time in 2014), teachers find it hard to meet the challenges of changed content, changed emphases and different forms of assessment. in many cases aspects of the mathematical curriculum are not taught, or poorly taught, leaving learners less well prepared for higher education study. over many years, experience in first-year courses reflects a lack of alignment between school outcomes and higher education expectations. any information that can provide insight into this lack of alignment should be given consideration. table 1 (compiled from table 10.1 in chapter 10 of the 2013 nsc diagnostic report, and table 10.1 of the 2014 nsc diagnostic report) gives the mathematics achievement rates over a five-year period (dbe, 2013a, 2014a). the table shows that from 2010 to 2014 the percentage of those who passed at 40% or more increased by just over 4%. table 1: national senior certificate mathematics achievement rates 2010–2014. while the increase in the proportion of students achieving 40% or above is encouraging, it is sobering to consider the low proportion of candidates achieving results that enable them to be admitted to degree programmes requiring mathematics. even though they are admitted they may ultimately take longer than expected to complete their degrees, or drop out altogether. the dbe is concerned about the problem (dbe, 2014b). it notes a lack of algebraic skills and the fact that increased attention should be given to higher-order thinking skills. it is stated that many errors have their origins in poor understanding of the basics and foundational competencies taught in the earlier grades. interventions to improve learners’ performance should thus also focus on knowledge, concepts and skills learnt in earlier grades and not only on the final year of the further education and training (fet) phase. teachers are also encouraged to ensure that mathematical terminology is well taught. understanding terminology is linked to the understanding of the language of mathematics, which is linked to competence in the language of instruction. performance in the nbt academic literacy tests suggests that there are problems with the comprehension of even simple english – see the reference to proficiency in academic literacy in the 2013 nbtp national report (nbtp 2012-2013). broad references to algebraic skill, the language of mathematics, knowledge of basic competencies and foundational competence provide insufficient information. additional information on specific skills and competencies is identified below that may require particular attention. analysis of mat test results identifies problem areas and their extent among otherwise high performing learners. teachers, and mathematicians involved in teacher education programmes, are best able to design strategies that will remediate aspects of mathematics that are barriers to success in mathematically demanding programmes in higher education. the findings presented here could also assist higher education in providing appropriate support, since this is clearly necessary. given that the highest achieving school leavers are participating in higher education, representing 16% of the age cohort, for the majority of students the curriculum structures are clearly not working (adler, 2011). literature review mathematical assessment assessment is an important tool in informing teaching and learning. the dbe requires teachers and district officials to monitor learner performance and report progress. several regional and international evaluations that include mathematical performance have taken place, such as the trends in international mathematics and science study (timss, 1995, 1999, 2003), and the annual national assessments1 (anas). analysis of the 2013 ana results has been referenced against the goals set in the action plan to 2014 (dbe, 2015a). there has been some criticism of the purpose and quality of the anas, for example by the south african democratic teachers’ union (sadtu, 2014), and also of the interpretation and use of the results (spaull, 2014). there is also the danger of ‘collateral damage’ (long, 2015), that is, the unintended consequences such as undermining teachers and distorting the focus of teaching. the anas are systemic tests, unlike the nbts which are criterion-referenced tests. however, it is noteworthy that the overall mathematical performance of the sampled learners was at the ‘not achieved’ level (less than 15% in both 2012 and 2013) (dbe, 2014b). the reason the anas are mentioned in this context is the fact that in grade 9, in 2012 and in 2013, only 2% of the candidates achieved 50% or more. of these candidates, not all would plan to enrol in higher education. however, some would do so, and it is then less surprising that the proportion of candidates in the ‘proficient’ band for the mat test is around 10%. (a proficient score indicates that candidates who are admitted to higher education should be able to cope with regular programmes of study.) educational goals and assessment goals are linked: learning involves the acquisition of skills and knowledge; assessment identifies the level of knowledge or skill acquired. if assessment is to be meaningful, it needs to advance learning and not simply record its status. if teachers can engage with clear examples of the problems their learners exhibit in assessment tasks, they will be better able to communicate to the learners the underlying mathematics that would facilitate better comprehension. for example, if a teacher knows the theory of addition (to give a very trivial example) they could help their learners understand what it means to add ‘like’ terms,: that denominator relates to the name (i.e. identifies ‘like’ terms), numerator relates to ‘how many’, and so on, then fewer learners would think that the importance of foundational knowledge mathematics should be a gateway, and not a gatekeeper, to success in higher education. students entering science, technology, engineering and mathematics fields need to be proficient in the requisite mathematics. at university the prior domain knowledge and previous learning experiences that students bring to their studies are acknowledged as significant factors influencing student success (crawford, gordon, nicholas & prosser, 1998). teachers who are able to equip their learners with appropriate domain knowledge will help them move through the gateway, rather than run up against the barriers of poor comprehension and competence. analyses of mat test results indicate areas where prior knowledge is in fact lacking. teachers who are pressed for time, and teachers who need or want to ensure specific test results, tend to teach to the test. unfortunately, teaching (and possibly learning) may be driven by the extent and type of assessment involved (jennings & bearak, 2014; kahn, 2002). aspects of the specified nsc curriculum that are not examined may be neglected. aspects of mathematics that are tested in the nbt mat tests may be paid more attention as a result of increased awareness of what is deemed to be important. raising awareness of specific conceptual gaps will hopefully result in greater attention to these areas at school. increased understanding should lead to a greater chance of success in mathematically demanding first year courses. errors and misconceptions the advent of the anas has given rise to additional expectations from teachers. see for example the following dbe statement (2011b): ana is intended to provide regular, well-timed, valid and credible data on pupil achievement in the education system. assessment of pupils’ performance in the get band (grades 1–9) has previously been done at school level. unlike examinations that are designed to inform decisions on learner promotion and progression, ana data is meant to be used for both diagnostic purposes at individual learner level and decision-making purposes at systemic level. at the individual learner level, the ana results will provide teachers with empirical evidence on what the learner can and/or cannot do at a particular stage or grade and do so at the beginning of the school year. schools will inform parents of their child’s ana performance in march. the above statement suggests that data from external assessments are intended to be used diagnostically. shalem and sapire (2012) suggest that the idea of informing local knowledge using a systemic set of evidence, diagnostically, is not without problems in terms of application. teachers have always needed to recognise learners’ errors, a skill without which they would not have been able to assess learners’ work. the difference now is that teachers are required ‘to interpret their own learners’ performance in national (and other) assessments’ (dbe & dhet, 2011, p. 2) and develop better lessons on the basis of these interpretations. diagnosing errors is a necessary first step, but it is not always clear to teachers how they should use the available information. shalem and sapire (p. 12) pose the following questions: ‘how does teachers’ tacit knowledge about learners’ errors (which they have acquired from years of marking homework and tests, encountering learners’ errors in teaching, or through hearing about errors from their colleagues) inform or misinform their reasoning about evaluation data? in what ways should teachers work with proficiency information when they plan lessons, when they teach or when they design assessment tasks?’ smith, disessa and roschelle (1993) note that assessment research reflects the importance of student misconceptions. the analyses of national assessment of educational progress (naep) mathematics results and the open-ended california assessment program tests both showed that misconceptions have a strong influence on how student learning is currently evaluated. whereas researchers may previously simply have separated correct responses and errors, it is now more common, even in large-scale assessments, to actively search for misconceptions to explain frequent student errors (smith et al., 1993). the mat diagnostic information can highlight some errors and misconceptions and at the same time help teachers use the information. since methods of assessment should enable learners to demonstrate what they know rather than what they do not know, not all mat test items present candidates with possible misconceptions as options, that is, not all mat items have options that include identifiable error types or misconceptions. doing so would ‘trap’ many candidates into selecting the apparently obvious option; if it is not provided they are more likely to try to solve the problem and find an answer. to give a trivial example, candidates could be asked to choose the correct answer for the following item: the volume of the cylinder in the diagram, in cm3, is (a) 200 (b) 201 (c) 202 (d) 203 if the correct answer is 202, then none of the others would reflect any misconception such as using diameter instead of radius, or calculating area instead of volume. the decision to limit the number of questions probing misconceptions minimises to some extent possible diagnostic opportunities, but yields more accurate results in terms of a candidate’s mathematical proficiency. methodology obtaining data nbt data are obtained from the nbtp at the university of cape town, while the nsc data are obtained from dbe reports. the data are analysed by year, for candidates writing in each dbe province as well as under the independent examinations board (ieb). the total number of candidates who wrote both the nscs and nbts from 2009 to 2013 has increased by 16 453 candidates over this period to a total of 182 156 candidates. there were 41 314 and 45 245 candidates in 2012 and 2013 respectively. the mat tests take three hours and comprise 60 multiple choice questions. items are scored dichotomously: a correct response to an item is given a score of ‘1’ and an incorrect response to an item is given a score of ‘0.’ the total raw score is obtained by summing the scored item responses. test items2 are analysed using item response theory (irt) and classical test theory. a three-parameter (a, b, c) irt model is used to analyse and score item responses, where a = discrimination, b = difficulty and c = guessing/pseudo-chance (yen & fitzpatrick, 2006, p. 114). many different tests are written in each testing cycle, all adhering to the same specifications. test equivalence is assured through an equating process. a psychometric report for each test provides measures of test reliability, item behaviour and test behaviour. the psychometric data show, for each item, the ranking of candidates in three groups: the lower, middle and upper thirds. more information on the tests themselves can be obtained from the 2013 nbtp national report (nbtp 2012–2013). to determine whether learners are able to make the transition between mathematics at secondary and tertiary level, the competencies that are required, but not necessarily made explicit, by higher education need to be assessed. the choice of competencies was until 2013 influenced by the four learning outcomes (lo1, lo2, lo3 and lo4) that appeared in the learning programme guidelines of the national curriculum statement for mathematics for grades 10–12 (dbe, 2008a). in 2014, grade 12 learners were assessed in terms of the caps (dbe, 2011a) and the mat tests for 2014 were adapted accordingly. the mat tests are embedded in the nsc curriculum, but cut across the different learning areas. this means that whereas the nsc grade 12 exam assesses separately different learning areas such as algebra, trigonometry and euclidean geometry and measurement, a mat test may include a geometry question that will be solved using trigonometry and algebra. the mat test specification spreads questions into six clusters: algebra, functions, transformations, trigonometry, spatial reasoning and data processing. the nsc subject assessment guidelines previously specified a taxonomy of categories of mathematical demand, which indicated that learners needed to perform at the levels of knowing (recall or basic factual knowledge), performing routine procedures, performing complex procedures and problem-solving (dbe, 2008b), carrying weights of approximately 25%, 30%, 30% and 15%, respectively. in the caps (dbe, 2011a) the following taxonomy is proposed: knowledge 20%, performing routine procedures 35%, performing complex procedures 30% and problem-solving 15%. the mat tests are also cognitively differentiated, starting with lower-order questions to facilitate an easy introduction into the test and then progressing to questions with greater cognitive demand. the highest level (counting for about 8%) comprises items that involve greater insight; about 45% of the items comprise knowledge, recall and application of straightforward procedures. test results place candidates into three benchmark categories: basic, intermediate and proficient (determined during standard setting, which took place in 2009, 2012 and again in 2015) (for further details see nbtp, 2012-2013). for convenience and additional clarity the intermediate category is further divided into upper intermediate and lower intermediate bands. when the benchmarks were first set in 2009 it became clear that a significant proportion of applicants to higher education would be in need of support. less than 10% of all candidates in the cohort analysed attained the proficient benchmark level in mat tests (i.e. obtained scores of 62% or more using the 2009 benchmarks, which were in place for test candidates in 2012). furthermore, table 2 shows that less than 15% of all candidates were in the upper intermediate group (i.e. obtained scores of between 48% and 61%) and 36% of all candidates were in the lower intermediate group (i.e. obtained scores of between 34% and 47%). table 2: distribution across benchmark levels for 2012 national benchmark test mathematics candidates. benchmarks were reset in 2012, resulting in the following distribution for candidates in 2013 (see table 3). the percentage falling in the basic category was higher in 2013 than in 2012. table 3: distribution across benchmark levels for 2013 national benchmark test mathematics candidates. figure 1 compares performance in the mat tests for the 2012 and 2013 cohorts. the percentage in the basic category increased from 41.4% to 49.0%; however, there was also a small increase (2.1%) in the proficient category. figure 1: proportion of learners within nbt mathematics performance levels for nbt 2012 and 2013 intake cycles. figure 1 shows that in 2012 and 2013, respectively, the mat candidates falling into the intermediate upper and proficient benchmark bands (those students for whom universities may expect to have to provide minimal or no additional support) constituted 22.5% and 24.0% of the total writer cohorts, clearly, institutions will need to accept candidates from the remaining bands and their support needs will be substantial. this suggests that the schooling system does not adequately prepare students for the mathematical rigours of higher education. while higher education needs to provide support for incoming students, possibly by providing differentiated curricula, it is important that basic education attempts to address the problem at the level at which it occurs so that students can be better prepared for the demands of university. diagnostic information for this article, we considered the results of prospective students, nationally, who wrote the mat tests between may and november of 2012 (38 730 candidates) and 2013 (48 318 candidates) and who achieved scores in the top third band of that particular cohort (this is possible since the psychometric test data report option choices of candidates according to lower third, middle third and upper third bands). when 20% or more of the top third of all the candidates selected a particular incorrect option, we considered possible causes for their choice: whether the choice could be attributed to a specific misconception or to an identifiable flaw in reasoning. all data relate to candidates who applied to register for courses requiring mathematics, not necessarily to those who were actually admitted. all these candidates are representatives of learners aspiring to study courses in which mathematics is required. if mathematical topics that have been identified appear to create problems for even the ‘good’ test candidates, they are most likely problematic for all others as well. teachers need to become aware of the errors and misconceptions in order to assist learners. the difficulties or misconceptions experienced by these nbt candidates indicate the need for various forms of intervention that could be undertaken to address the problems outlined here. item responses from different mathematical topics that reflect similar misconceptions or errors in reasoning have been grouped together. suggestions for possible interventions are noted, but in-depth analysis of such interventions is beyond the scope of this article. teachers, and academics involved in teacher training, would be best placed to consider at what grade (even at primary school) and in how much depth various interventions could be focused. they will be aware for example of the work of parker & leinhardt (1995) on understanding percentages, of davis (2013) on understanding what cancelling means and many more. it may be common knowledge that for example, as noted in the 2014 nsc diagnostic report (dbe, 2014a), there is limited understanding of basic exponential laws, but it is sobering to note that in the nbt mat tests as well, errors such as a3 + a5 = a8 are evident in the very group of candidates aspiring to enter university. while these common errors and misconceptions may be familiar to teachers, quantifying and characterising these for the cohort of students entering university is less well understood. results interpreting the data the purpose of tables 4 to 8 is to show how common errors and misconceptions in one conceptual area (such as an algebraic operation) also occur in many other clusters of items in the mat tests. in the tables, items are listed according to their unique identity numbers (for example a5 refers to a specific item in one of the algebra subsections). in each row of each of the following tables, we give an indication of the proportion of candidates who demonstrate incomplete or incorrect understanding in relation to a specific item. an item may appear in several tests, and the table indicates the number of tests in which a particular error was identified. for convenience, items in which the same type of error has been made have been reported together. for the different error categories there are brief suggestions regarding intervention approaches that might be considered in response to the problems identified. in-depth discussion of possible strategies is beyond the scope of this article. table 4: algebraic processing. implications of mistakes made in algebraic processing algebraic processing skills are fundamental to all aspects of mathematics in higher education and table 4 shows that even the better writers demonstrate problems in understanding algebra. if a student has for example understood differentiation, but is unable to correctly apply the necessary algebraic processing procedures, further application and problem-solving is undermined. analysis of the mat tests results show that when solving algebraic equations, test candidates have forgotten the difference between an expression and an equation, and the need to apply identical procedures to both sides of an equation in order to find its solution. they have difficulty dealing with signs when subtraction is involved. it is necessary to revise expansion of brackets preceded by a minus, so that learners understand that, for example, −(2 − 3) = (−1)(2 − 3). it may be necessary to revise all operations involving negative integers. a number of other algebraic concepts are also problematic. ‘cancelling’ is poorly understood (it appears to be the same as ‘crossing out variables or numbers that are the same’). this also relates to solving equations, where cross-multiplication is applicable (because the same procedure is in effect applied to both sides) whereas it is not applicable when simplifying a mathematical expression. factorisation, fractions and equivalent fractions must be clarified, along with what ‘cancelling’ actually means. ratio and proportion are poorly understood. learners do not remember (from earlier grades) or know that a proportional statement is an equation involving two ratios. all learners know that (a + b)2 = a2 + 2ab + b2 but do not necessarily understand that 2ab means twice the product of the first and last term. when either of these terms is slightly different from a single number or algebraic term they do not know what to do. binomial expansion needs to be taught using many different types of terms in the binomial. exponential laws are also often memorised and not understood. it is necessary to ensure that learners know the terminology of power, base and exponent, and understand how the exponential laws have been derived. revision of all operations in which powers are involved is needed. implications of mistakes made with functions the concept of a function is fundamental to all first-year mathematics courses, whether they are pure mathematics courses or mathematics courses in other disciplines such as commerce or statistics. table 5 shows that many learners have problems understanding the function concept. table 5: functions and the equations that define them. the definition of a function, different representations of functions and the terminology associated with functions need to be revised. ideally, generic graphs should be used to clarify function terminology such as domain, range, function graph (where the graph lies above, on or below the x-axis), function value (which is then positive, zero or negative), turning points, asymptotes, intercepts (of a graph) and roots (when the equation representing the graph is equal to zero), period and amplitude. because the range of the function defined by y = sin x is [−1, 1], the assumption is easily made that any sine function has the same range, unless the concept of range is properly understood. because the maximum value of a quadratic function occurs at the turning point of the graph, the assumption is made that any maximum must be related to the turning point. similarly, because the y-intercept of a linear graph is the constant (in the equation y = mx + c) there is a tendency to assume that the y-intercept is also the constant in any other graph, such as y = ax + k. various combinations of graphs should be shown with the distances between them at different points, to demonstrate that the maximum distance can occur at a point other than the turning point. different types of graph should demonstrate different y-intercepts. generic graphs can also be used to illustrate transformation principles such as horizontal shrink or stretch, vertical shift (upwards or downwards) and reflection in the x-axis, y-axis, or other lines such as y = x. the effect of the parameters involved in different algebraic representations of functions needs to be demonstrated graphically, so that learners understand how the numerical value and the sign of the parameters affect the shape of the graph, rather than just depend on calculators to draw specific graphs. if learners understand the graphical meaning of parameter change and the link between algebraic and graphical representation of transformed graphs they may be less likely to memorise (incorrectly) various transformation rules. implications of mistakes made with trigonometric functions trigonometric ratios are initially taught in terms of right triangles. this understanding is then poorly expanded later to trigonometric functions in the cartesian plane, as can be seen from table 6. learners have remembered the ratios, but cannot move beyond these to a context where the function value is not necessarily positive. table 6: trigonometric functions. implications of mistakes made with geometric concepts table 7 shows that by the end of high school, many geometric concepts are not yet well understood or remembered. table 7: basic geometric concepts. it is important to reinforce the link between circles, pi, radius and diameter. it should not be assumed that learners have remembered or understood basic geometric concepts, such as perimeter, area, surface area and volume, especially in relation to objects that are shown from a perspective different from the standard perspective, or where composite shapes are involved. the terminology of geometry is possibly problematic: are learners familiar with the meaning of geometrical terms such as face, vertex, rhombus and so on? this needs to be addressed in the earlier grades. implications of mistakes made in number sense calculator dependence has resulted in a limited understanding of the number system. it is important to teach the structure of the number system, especially numbers (natural, integers, rational, etc.) in relation to one another. table 8 indicates some of the problems that have arisen when the number system, and several other number concepts such as percentage, are not well understood. there is a lack of awareness of how big or small a negative number is. surds and fractions are similarly misunderstood. where a number is represented algebraically, test candidates appear not to use a test case to determine its possible magnitude. table 8: number sense. multiplication and division of numbers are not always well taught in primary school, and the results are still evident much later. learners need to understand what the process of division actually means, in order to understand why division by zero is impossible. multiplication and division by negative numbers need to be revised, to enable learners to see that the relative positions of numbers on the number line change, which would clarify their understanding of < and >. it should not be assumed that percentage is a well understood concept. for many candidates, ‘percentage’ appears to exist in isolation and there is no attempt to associate it with the quantity to which the percentage is applied. the meaning of percentage, and how percentage can be applied, needs to be revised. this concept is taught in earlier grades but has apparently been forgotten. conclusion this article presents some common errors made by high performing candidates in a large-scale study and indicates problems in the conceptual understanding and mathematical skills of these candidates. while teachers would anticipate some of these, it is important to note that the problems are exhibited by the top third of prospective applicants to higher education who wrote the nbt for mathematics. this group constitutes a large proportion of first-year students in mathematically demanding programmes. the purpose of this article is to raise awareness in both higher and basic education about the type and extent of the problem. it is not the intention to engage in an analysis of all possible interventions that could be put in place. the diagnostic information provided identifies problems mat candidates demonstrate regarding some of the essential mathematical concepts and procedures deemed necessary by higher education mathematicians. the interventions suggested in response to the diagnostic information from mat tests can be of use to the school sector in foregrounding areas where mathematical comprehension is weak. these topics, as well as the related terminology and language, need to be given greater attention in the classroom. it is not the intention of this article to be prescriptive with respect to the suggested interventions; teachers should themselves determine how best to use the diagnostic information from the mat tests to create learning environments which could be more responsive to the needs of higher education. policymakers rather than teachers need to consider that some topics may possibly need to be excluded from the school curriculum, without necessarily detracting from its value, in order to achieve greater understanding of key concepts relevant to higher education. acknowledgements competing interests the authors declare that we have no financial or personal relationships that might have inappropriately influenced us in writing this article. authors’ contributions r.p. and a.d. were responsible for compilation and analysis of all data from the nbts referred to in the document. c.b. was responsible for the purpose, background, literature review, methodology and analysis of the mat tests to provide 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(2006). item response theory. in r.l. brennan (ed.), educational measurement (4th edn., pp. 111–153). westport, ct: greenwood/praeger. footnotes 1. the anas are standardised national assessments for languages and mathematics in the senior phase (grades 7–9), intermediate phase (grades 4–6) and in literacy and numeracy for the foundation phase (grades 1–3). the question papers and marking memoranda (exemplars) are supplied by the national dbe and the schools manage the conduct of the tests as well as the marking and internal moderation (dbe, 2015b). 2. in the nbts, questions are referred to as ‘items’. ethnomathematics 2 pythagoras 65, june, 2007, pp. 2-9 does ‘african mathematics’ facilitate access to mathematics? towards an ongoing critical analysis of ethnomathematics in a south african context1 kai horsthemke and marc schäfer university of the witwatersrand and rhodes university email: kai.horsthemke@wits.ac.za and m.schafer@ru.ac.za mosibudi mangena, the minister of science and technology, said in an address to the annual congress of the south african mathematical society at the university of the potchefstroom, november 2, 2004: “there is one thing we need to address before anything else. we need to increase the number of young people, particularly blacks and women, who are able to successfully complete the first course in mathematics at our universities.” how is this to be achieved? a popular trend involves a call for the introduction and incorporation of so-called ethnomathematics, and more particularly ‘african mathematics’, into secondary and tertiary curricula. although acknowledging the obvious benefits of socalled ethnomathematics, this paper critically analyses three aspects of ethnomathematics that have been neglected in past critiques. our focus is not on the relationship as such between ethnomathematics and mathematics education. our critique involves (1) epistemological and logical misgivings, (2) a new look at practices and skills, (3) concerns about embracing ‘african mathematics’ as valid and valuable – just because it is african. the first concern is about problems relating to the relativism and appeals to cultural specificity that characterise ethnomathematics, regarding mathematical knowledge and truth. the second set of considerations concern the idea that not all mathematical practices and skills are necessarily culturally or socially embedded. with regard to the validity and viability of ‘african mathematics’, our misgivings not only concern the superficial sense of ‘belonging’ embodied in the idea of a uniquely and distinctly african mathematics, and the threat of further or continuing marginalisation and derogation, but the implicitly (self-)demeaning nature of this approach. this paper serves as a reminder that a critical position in the deliberations of ethnomathematics needs to be sustained. it warns against the bandwagon syndrome in a society where political correctness has become a prominent imperative. this paper is framed by many unanswered questions in an attempt to inspire and sustain a critical discourse in the ethnomathematics movement. 1 an abridged version of this paper was presented at the third international conference on ethnomathematics, auckland, new zealand in february 2006. background the vision underlying the policy thrusts of the south african education system is captured in the preamble to the constitution which aims to “heal the divisions of the past and establish a society based on democratic values, social justice and fundamental human rights”, to “lay the foundations for a democratic and open society” and to “improve the quality of life of all citizens and free the potential of each person”, in order to “build a united and democratic south africa able to take its rightful place as a sovereign state in the family of nations” (republic of south africa, 1996). “education and the curriculum”, it is stated in the introduction to the revised national curriculum statement grades r-9 (schools): mathematics, have an important role to play in realising these aims. the curriculum aims to develop the full potential of each learner as a citizen of a democratic south africa. (department of education, 2002: 1) yet, the “review of educational progress”, after ten years of democracy in south africa, revealed that, counter to the vision of policy, inequalities had increased, exacerbated by the deepening of poverty and its impact on education … there remained serious questions with regard to access to quality education. … an educational debate on access via school fees [arose in public in 2004], raising old questions about class, access and race within a new frame of emerging elites and ongoing inequalities. … [the] managerial discourse around ‘quality’ … is characterised by inflexibility and a desire kai horsthemke and marc schäfer for control. it narrows the view of quality and affects democratic processes surrounding discussions and debates about quality. in retrospect, it seems that the attempt to improve the quality of education, to bring about equality and support democracy in a globalising, market-oriented society, has met with mixed results. policies seem to have been idealised and are remote from contextual realities. democracy in education appears to exist in name only and falls short in its actualisation. (kenton conference, 2005) our paper attempts to explore these tensions and some of the proposed solutions, with special attention to mathematics education in south africa. in particular, it seeks to (re)visit some of the more critical issues surrounding the incorporation of so-called ‘african mathematics’ into the mathematics education curriculum of south africa. this paper arises out of our concern about the apparent intellectual complacency with regard to the theoretical and philosophical underpinnings of ethnomathematics – there is (certainly in south africa) a view that a critical interrogation of ethnomathematics is passé, indeed undesirable. this, in our view, is a problematic attitude. we argue that if what inspires ethnomathematics is to strengthen its position, this requires ongoing critical reflection on its very assumptions and underpinnings. in essence, we are committed to the notions of transcultural truth and knowledge, in mathematics as elsewhere. we are concerned about the fragmentation of knowledge, and we reject the idea of ownership of knowledge, along ethnic and indigenous divisions. like the idea of ‘indigenous knowledge’, ethnomathematics has, in many circles in south africa, unfortunately become a bandwagon-type of concept that has been uncritically co-opted by politicians and policy-makers to further their own agendas, without any reference to its philosophical and theoretical underpinnings. it tends to be employed as a buzzword in simplistically justifying the polarisation of (mathematical) knowledge into ethnic/indigenous, and ‘world’/ mainstream. although we do not dispute that mathematical learning arises and takes place in diverse socio-cultural contexts of meaning-making, we – like many others – find the uncritical ‘ethnification’ of mathematical knowledge problematic. the south african context according to the revised national curriculum statement grades r-9 (schools): mathematics, after providing a definition of mathematics and an account of the mathematics learning area, being mathematically literate enables persons to contribute to and participate with confidence in society. access to mathematics is, therefore, a human right in itself. (department of education, 2002) mosibudi mangena, the minister of science and technology, said in an address to the annual congress of the south african mathematical society at the university of the potchefstroom, 2 november 2004: there is one thing we need to address before anything else. we need to increase the number of young people, particularly blacks and women, who are able to successfully complete the first course in mathematics at our universities. (mangena, 2004) how is this to be achieved? a popular trend involves a call for the (re)introduction and (re)incorporation of a ‘culturally informed mathematics’, so-called ‘ethnomathematics’, and more particularly ‘african mathematics’, into secondary and tertiary curricula (see emeagwali, 2003; zaslavsky, 1979 & 1994). after providing an account of the african origins of mathematics (seepe, 2000: 125-128; see also van sertima, 1999: 314-316; both writers seem to vacillate between an account of the “african origins of mathematics” and that of the “origins of african mathematics”), sipho seepe defends the desirability for a “culturally informed mathematics”, an ethnomathematics approach as part of the ‘democratisation’ of curricula (seepe, 2000: 131-133). he also reports on the dst/ csir’s2 collaborative national audit of indigenous (south african) technologies (seepe, 2000: 133134) that “was followed immediately by the launch of the indigenous knowledge systems (iks) programme”, a “programme [that] is seen as a critical component in the restructuring and democratisation of the south african science and technology system, which has hitherto remained eurocentric” (seepe, 2000: 133, 134). seepe concludes: the challenge facing (south) african scholars is to build on this initiative and 2 these acronyms stand for department of science and technology and council for science and industrial research, respectively. 3 does ‘african’ mathematics facilitate access to mathematics? towards an ongoing critical analysis of ethnomathematics in a south african context engage themselves in unravelling the mathematical and scientific basis of these technologies. in other words, the challenge is to locate and identify the scientific skills, knowledge and process embedded in the cultural practices of the african majority. once these are identified, they can be used to restructure, redesign and reformulate the present curricula. a restructured curriculum should assist in the affirmation of the african child. since it is in culture and language that learners find an intellectual home, the utilisation of indigenous technology and african knowledge systems might be the key to unlocking the door that has prevented the masses from accessing mathematics, science and engineering. (seepe, 2000: 134; see also department of education, 2001) however laudable his intentions, seepe, like many others, seems to have fallen into the trap of polarising the notion of mathematical knowledge into ‘mainstream’ (in his case “eurocentric”) and ‘ethnic’ (in his case “african”). the notion of eurocentrism (with particular reference to mathematics) has unfortunately become a buzzword that is bandied about without much thought or insight. • what is a ‘eurocentric’ science and technology system or, for that matter, system of mathematics? • what makes it a eurocentric system? • what makes it eurocentric? furthermore, how exactly is “the utilisation of indigenous technology and african knowledge systems” supposed to unlock “the door that has prevented the masses from accessing mathematics, science and engineering”? seepe leaves these, and other, questions unaddressed and provides no further detail. finally, the question of the african origins of mathematics is moot and not very helpful, and we do not intend to engage with it. rather, our concern here resides with the issue of the origins of african mathematics, in the sense that we question the existence of a uniquely and distinctly african mathematics (just as much as we question a uniquely and distinctly european mathematics). we argue that the interpretations and applications of mathematical concepts may be distinctly african (or european), but that to claim a uniquely ethnic or cultural ownership of mathematics is misleading and merely reinforces fragmentation and marginalisation. ‘ethnomathematics’ what follows is a brief discussion of a recent illustration of ethnomathematics. according to paulus gerdes, in his stieg mellin-olsen memorial lecture presented in bergen, norway, on 31 august 2005, ethnomathematics embodies forms of valuing that include, or are contained in, understanding, recognition, sources of inspiration, as well as (african) renaissance. explaining its central concerns, he quoted stieg mellin-olsen: “to this day it has not been questioned at all whose culture, or which intellectual material, should be the basis for mathematics education” (stieg mellin-olsen, proceedings of the conference on mathematics and culture, bergen, september 1995; quoted in gerdes, 2005). using a concrete example of the value and distinctness of ethnomathematics, gerdes explained that the ‘right angle’ (‘epopera’) in mozambican basket weaving is not 90° but 60° – that is, the only angle that permits continuous folding, weaving and stability within the woven structure. the ‘classic’ form design of the soccer ball (hexagons and pentagons) first introduced in the 1970 world cup in mexico may have already been in use in thailand and cambodia for some 2000 years, as gerdes claimed, but this indicates the transcultural value and validity of design involving mathematical insights, rather than unique and distinct knowledge. gerdes’ reference to ‘knowledge’, throughout, concerns ‘practical’ knowledge or skills – so, invoking “complicated”, pre-practice “calculations” (say, by indigenous basket weavers) appears to be no more than another way of describing a process of learning from trial and error. what about gerdes’ “examples of exclusively oral transmission of pottery design patterns” (by practitioners who had not engaged in pottery for years, as a result of the displacement forces of modernisation)? do these indicate complex, extra-practice ‘calculations’? hardly: they might, rather, be said to indicate postpractice recollection. moving on to sona/cokwe sand drawings, gerdes explained that the chief values in these drawings are considered to be symmetry and monolinearity, exceptions to which, however, do exist: asymmetry and bilinearity. again, this indicates skills and creativity, certainly. yet, the ability of so-called ‘innumerate’ (or, to use gerdes’ term, “unmatherate”) people to count and to work with numbers in a broadly abstract fashion does not amount to ‘indigenous’ mathematics – any 4 kai horsthemke and marc schäfer more than knitting inventiveness and skills attest to mathematical prowess. on one level, then, ‘ethno-’ and related mathematics are descriptions of mathematical practices or skills through a cultural lens. on another level (after all, seepe and others refer to “skills, knowledge and process”), they embody an epistemological relativism and/or invoke a notion of the cultural specificity of truth. the pertinent concerns might be grouped under the headings that follow. pedagogical/pragmatic there is a view, for example, that mathematics may (already) be imbued with ethnomathematics, and that this is now an empirical matter, an open empirical question (adam, alangui & barton, 2003). this view is frequently accompanied by (reference to) accounts of how ethnomathematics permeates conventional mathematics and school curricula (for example, adam, 2004). there is, however, scant evidence that ethnomathematics as a general concept actually works. research suggests that ethnomathematics may be meaningful only in a very narrow and localised context. for example, current research into the incorporation of xhosa beadwork into a mathematics learning programme showed, inter alia, that only a very limited number of learners were able to identify with this practice and hence incorporate and assimilate it meaningfully into their learning experience. to many learners the practice of beadwork was foreign and old-fashioned, a practice that only their grandparents indulged in. this begs the question whether ethnomathematics is, indeed, a more appropriate way of doing mathematics. there appears to be little empirical evidence for giving an affirmative answer to this question. as a means for providing contextual, cultural and historical meaning to mathematics, ethnomathematics may be very useful, but to claim more than this is questionable. at the recent 1st african regional congress of icmi held in johannesburg in june 2005, a number of ethnomathematics researchers from botswana presented their work. in the discussion that ensued after the presentations, it was interesting to note the observation that learners themselves frequently rejected the incorporation of ‘african mathematics’ (not to be confused with applied or practical mathematics), viewing it as irrelevant, exotic, backward, and culturally alienating. the assumption that the label ‘ethno-’ (or ‘indigenous’) will automatically be embraced by learners is clearly a dangerous one. in the debate of product versus process, we warn against an over-emphasis on product (see, for example, seepe, 2000: 134). ethnic artefacts, baskets, pottery, sand drawings, and the like, arguably have a meaningful function as teaching and learning aids. yet, it is unclear whether, as products, they occupy an equally significant function in a competitive, global ‘knowledge economy’ as, say, the products of general data handling and analysis. political this concerns mainly issues of (lack of) access, exclusion and inclusion, and failure in learning mathematics (mellin-olsen 1987, chapter 5). according to stieg mellin-olsen, the failure in learning mathematics [is] a result of the pupil’s lack or appreciation of the thinking-tools of the curriculum. … [s]uch failure [is] political: some pupils are prevented from an important field of knowledge because of the design of the curriculum or the mechanisms of the examination system. (1987: 191) in south africa, unlike mellin-olsen’s norway, the situation has an additional dimension, that of past and present inequality, poor quality education and lack of democratic process “in a globalising, market-oriented society”, where educational policies “seem to have been idealised and are remote from contextual realities” (kenton conference, 2005). after all, “[m]athematics is a product of investigation by different cultures – a purposeful activity in the context of social, political and economic goals and constraints”, according to the revised national curriculum statement grades r-9 (schools): mathematics (department of education, 2002: 4). we need to recognise, however, that a familiar context in one class will be alien in another. the question is whether context-sensitivity requires, or at least renders desirable, a focus on ‘indigenous mathematics’? the politicisation of ethnomathematics may be interrogated on the basis of the following considerations: • the bandwagon syndrome: indigenous knowledge systems, and ethnomathematics in particular, may be embraced for reasons of ‘political correctness’; • the confusion of categories: mathematics (and the “interrelated knowledge and skills” that constitute it; see department of education, 2002: 4) is 5 does ‘african’ mathematics facilitate access to mathematics? towards an ongoing critical analysis of ethnomathematics in a south african context a science, and its laws, principles, functions and axioms have little, if anything, to do with issues of social justice. philosophical/epistemological this concerns the philosophical assumptions behind the invocation of ethno-, indigenous or a uniquely and distinctly african mathematics. does the idea of ethnophilosophy (and, in particular, african philosophy of education) make any sense? there is at least a strong presumption against the plausibility of any such idea that, characteristically, tends to equate ‘philosophy’ with ‘world view’ rather than with ‘critical [thinking] activity’. the problem with a purportedly uniquely and distinctly african philosophy is that either what is so presented is not obviously ‘philosophy’ in any profound or informative sense (since everyone would be a ‘philosopher’ on this understanding of the term) or it is not distinctly and uniquely african (see horsthemke & enslin, 2005). where is ethnomathematics/‘african mathemattics’ vis-à-vis mainstream mathematics? how is it rationalised? does the concept in question refer to practices and skills, or to bodies of theoretical/ factual/propositional knowledge? what about the possibility and desirability of a movement called ‘gynomathematics’ or, indeed – and ideally even joining forces with the former – ‘afrogynomathematics’? (what about the mathematics of insomniac egyptian pyramid builders? this tongue-in-cheek question simply illustrates a concern about the fragmentation of mathematics.) are all mathematical practices equally valuable and/or valid? how, then, would one distinguish between good and bad mathematical reasoning? are mathematical skills essentially/characteristically embedded in culture and society? a possible response is that some skills – like basic numeracy skills – are universal, that is, translocal or transcultural. additionally, there are obvious problems with relativism relating to knowledge and truth. are there different, alternative bodies of mathematical knowledge in the theoretical/factual/ propositional sense, different deductive logics? (ramagupta’s mathematics was not ‘indian mathematics’, but mathematics per se.) some doubts about the notion of ‘african’ mathematics, in particular realistically, when ethnomathematicians and indigenous knowledge apologists speak of the cultural specificity of mathematics, they are actually referring to traditions relating either to practices or to beliefs. we argue that ethnomathematics is, at best, a rhetorical tool for establishing relevance and promoting sensitivity to cultural differences – but neither a viable pedagogical nor epistemological construct. we contribute to past and extant critiques of ethnomathematics (like vithal & skovsmose, 1997; rowlands & carson, 2002; rowlands & carson, 2004) in that our critical focus is not on the relationship as such between ethnomathematics and mathematics education. our critique of, and concern about, ethnomathematics emanates 1. from an epistemological and logical perspective, 2. on the basis of a new look at practices and skills, 3. from concerns about embracing ‘african mathematics’ as valid and valuable – just because (or on the mere grounds that) it is african. the first concern is about problems relating to the relativism and appeals to cultural specificity that arguably characterise ethnomathematics, regarding mathematical knowledge and truth. the second set of considerations concern the idea that not all mathematical practices and skills are culturally or socially embedded. with regard to the validity and viability of ‘african mathematics’, our misgivings not only concern the superficial sense of ‘belonging’ embodied in the idea of a uniquely and distinctly african mathematics, and the threat of further or continuing marginalisation and derogation (vide ‘african time’), but the implicitly (self-)demeaning nature of this approach. a counterargument, however, might be that by taking apparent ‘ownership of math’, the west (europe) has achieved precisely what africa is trying to do now. it could be argued, for example, that our academic discourse is distinctly western, if not eurocentric: consider the dominance of the english language – the medium of our discourse. because mathematics has become part of this discourse it has arguably become distinctly western. this western/eurocentric ownership, the argument concludes, has not led to any selfmarginalisation of any kind. the response to this contention would be that, if the idea of mathematics is to have any sense at all, reference to ‘western’ – let alone ‘eurocentric’ – mathematics, perhaps by way of contrast with ‘african’ and other so-called ‘indigenous’ mathematics, is misguided. if something constitutes mathematics (or mathematical procedure), that is, the science of 6 kai horsthemke and marc schäfer number, quantity and space either studied in its own right or applied to other disciplines like physics or engineering, the questions around ownership or dominance are simply inappropriate. they have to do with (legitimate) concerns about historical and political processes and events rather than with mathematics as such. in addition, like the reference to a ‘eurocentric’ and ‘african’ mathematics, the sheer sweeping generalisation involved in attributing certain characteristics to ‘europe’ is as disconcerting as it is questionable. what is the referent here? ‘europe’ is an invented idea, not an object. are pacifist traditions or the green movement in some way un-european? needless to remark, there exist both petty and profound enmities between members of different european nations. given a similar diversity and stark cultural, economic and political differences within africa, even ‘afrocentrism’ seems unviable, unless both were interpreted in some sense as inwardlooking to the exclusion of what is outside/ different (see horsthemke & enslin, 2005: 69). regarding the first concern referred to above, what is wrong with relativism? logically speaking, epistemological relativism and relativism about truth are coherently expressible only as a relative knowledge-claim or truth – which undermines the relativist enterprise as much as expressing it as a universally applicable knowledge-claim or truth. in the former instance, the problem would be one of self-marginalisation. in other words, nonrelativists would not be, and could not be expected to be, impressed by relativists’ claims or pronouncements. in the latter case, the problem would be that of self-contradiction. in other words, relativists would thereby be committing themselves to at least one nonrelative knowledge-claim or truth. empirically, too, embracing relativism has undesirable consequences. these become obvious when, for the sake of argument (and bearing in mind that this cannot coherently be done in any non-relative fashion), we assume that relativism is true. what would be some of these consequences? first, we could not judge that the beliefs and practices of other societies are epistemically and veritistically inferior to our own, in terms of their knowledge and truth functionality. we could not say that something is a false belief or a superstition, or that something is a laborious, time-wasting practice. (consider judging the beliefs and practices of a flat-earth society.) second, we could decide whether beliefs are true or false and practices are the correct or incorrect ones simply by consulting the standards of our society or epistemic community. third, the idea of progress (mathematical, scientific and other) is called into doubt, as is the idea of ‘reform’. we would not be able to say that a new paradigm constitutes an improvement on the older paradigm it has replaced. in view of these consequences, not even considering the paradoxicality of denying the objectivity and universality of knowledge and truth, it appears to make more sense to assert that there is considerably less disagreement than it seems and that social and ethnic groups share a considerable body of mathematical knowledge and practices. some thoughts about truth and ‘the social’ first, ‘consensus’ versus ‘truth’ in mathematics: mathematical concepts like that of area (space within a bounded surface) are universal. descriptions of that bounded surface (such as number of square units etc.), on the other hand, may well be cultural products and, as such, a matter of cultural or social consensus. similarly, the pythagorean theorem is a human construct; yet, the relationship described exists independently of human in(ter)vention. ‘knowledge’ is clearly not the same as ‘consensus’. the latter may well be the product of a dialogical relationship, and this is what marcelo borba seems to be suggesting: “the teacher/researcher has a particular ability and responsibility to help the students find the intersections between their realms of meaning and the teacher’s” (borba, 1990/1997: 269). however, knowledge clearly goes beyond consensus: there may be consensus about what is false, untrue, not the case. second, the debate about the nature and status of mathematical truth waged between those who support a ‘discovery’ approach and those who endorse an ‘invention’ model. obviously, the concepts employed in mathematics are ‘human’ in origin, yet to what they refer and are applied goes beyond human presence, agency or ‘invention’. that is, while the terms and symbols denoting mathematical phenomena are, in an important sense, not discovered, the events and complex relations to which they refer are, again in an important sense, not invented or socially constructed. they are objectively accessible, translocal or transcultural phenomena. to pre-empt any misunderstandings: we do not wish to play down the effects of colonialism or of its modern heir, globalisation. nor do we intend to denigrate africa’s contribution to ‘world mathematics’ (seepe, 2000: 127). we agree with many of the basic concerns that underpin ethnomathematics and an ‘indigenous knowledge 7 does ‘african’ mathematics facilitate access to mathematics? towards an ongoing critical analysis of ethnomathematics in a south african context systems’ approach, like concerns about the arrogant and patronising attitudes of ‘mainstream’ (‘malestream’?) mathematicians, scholars and researchers, as well as the demand for relevance, that mathematics education should be sensitive to cultural differences, and so on. we differ, however, in our contention that while it is correct that many mathematical practices and skills (african or other) are “culturally and socially embedded”, some skills – like basic numeracy skills – are not. there is ample empirical evidence that numeracy is not developed in any manner that places a premium on cultural or social influences and context. in short, we contend that there is no such thing as a body of knowledge called ‘ethnomathematics’. consequently, we argue that the focus on ‘african mathematics’, whether for reasons of ‘political correctness’ or social justice, that is, as a means of redressing “inequality, poor quality education and a loss of democratic process” (kenton conference, 2005), is seriously and significantly misguided. just as mathematical beliefs and ideas may differ among or across cultures, the manifestation of mathematical practices and skills may so differ. however, the former amount to knowledge only if they are true and if they are adequately justified. similarly, while they may differ in their manifestation, mathematical activities and practices like “representation and interpretation; estimation and calculation; reasoning and communication; problem-posing; problem-solving and investigation; and describing and analysing” (department of education, 2002: 4) are transcultural, in that “they appear to be carried out by every cultural group ever studied” (bishop, 1988, in borba, 1990/1997: 266). it follows that the term ‘indigenous’ has, at best, limited applicability. a similar point could be made about the prefix ‘ethno’. if ethnomathematics constitutes knowledge in the propositional or factual sense, then it is unclear what purpose the prefix is meant to serve – other than artificially severing ethnomathematics from mathematics as such. if it constitutes activities or practices, then – while their actual manifestations may differ among or across cultural or ethnic groups – the fact that these are carried out by all cultural or ethnic groups renders them universal. it follows that the term ethnomathematics encompasses, at the very most, the different ways in which mathematical activities and practices manifest themselves. these activities and practices need not be treated as anthropological curiosities but can enrich the teaching and learning of mathematics as such, as well as mathematical research. concluding comments there appears to be a perception amongst proponents of ethnomathematics and of indigenous knowledge systems that discourse within the movement is now largely a matter of descriptive, empirical investigation (see, for example, adam, alangui & barton, 2003 and adam, 2004). we have argued, against this perception, that critical and rigorous self-reflection and analysis of all its assumptions is crucial if ethnomathematics is to contribute meaningfully to curriculum development and implementation, in (south) africa as elsewhere. after all, despite disagreement, there appears to be a shared, implicit assumption that even the most difficult and theoretical mathematical problems are amenable to discussion and argument. moreover, there seems to be basic agreement on some transcultural standard of correct and incorrect reasoning in mathematics, as in other areas of intellectual life. references adam, s. (2004). ethnomathematical ideas in the curriculum. mathematics education research journal, 16(2), 49-62. adam, s., alangui, w. & barton, b. (2003). a comment on: rowlands and carson ‘where would formal, academic mathematics stand in a curriculum informed by ethnomathematics? a critical review’. educational studies in mathematics, 52, 327-335. borba, m.c. (1990). ethnomathematics and education. for the learning of mathematics 10(1), 39-43. reprinted in a.b. powell & m. frankenstein (eds.) (1997). ethnomathematics: challenging eurocentrism in mathematics education (pp 261-272). new york, suny press. department of education. (2001). national strategy for mathematics, science and technology education. pretoria: government printer. department of education. (2002). revised national curriculum statement grades r-9 (schools): mathematics. retrieved april 8, 2005, from http://www.education.gov.za/ content/documents/559.pdf emeagwali, g.t. (2003). african indigenous knowledge systems (aik): implications for the curriculum. in t. falola (ed.). ghana in africa and the world: essays in honor of adu 8 kai horsthemke and marc schäfer boahen. trenton, new jersey: africa world press. retrieved april 8, 2005, from http://www.africahistory.net/aik.htm gerdes, p. (2005). forms of valuing embedded knowledge and creativity of so-called ‘illiterate’, ‘unmatherate’ people like ‘indigenous’ artisans. stieg mellin-olsen memorial lecture, scandic hotel, bergen, norway, august 31, 2005. rowlands, s. & carson, r. (2002). where would formal, academic mathematics stand in a curriculum informed by ethnomathematics? a critical review. educational studies in mathematics, 50, 79-102. rowlands, s. & carson, r. (2004). a comment on adam, alangui, and barton’s “a comment on: rowlands and carson ‘where would formal, academic mathematics stand in a curriculum informed by ethnomathematics? a critical review’”. educational studies in mathematics, 56(2-3), 329-342. horsthemke, k. & enslin, p. (2005). is there a distinctly and uniquely african philosophy of education? in y. waghid, b. van wyk, f. adams & i. november (eds.). african(a) philosophy of education: reconstructions and deconstructions (pp 54-75). stellenbosch: stellenbosch university department of education policy studies. kenton conference. (2005). (in)equality, democracy, and quality. retrieved april 8, 2005, from http://www2.ru.ac.za/academic/depart ments/education_resources/conference/about/i ndex.php seepe, s. (2000). africanization of knowledge: exploring mathematical and scientific knowledge embedded in african cultural practices. in p. higgs, n.c.g. vakalisa, t.v. mda & n.t. assie-lumumba (eds.). african voices in education (pp 118-138). lansdowne: juta. van sertima, i. (1999). the lost science of africa: an overview. in m.w. makgoba (ed.). african renaissance – the new struggle. cape town: mafube publishing. mangena, m. (2004). address to the annual congress of the south african mathematical society at the university of potchefstroom. retrieved april 8, 2005, from http://www.dst. gov.za/news/speeches/minister/mathematical_s ociety.htm vithal, r. & skovsmose, o. (1997). the end of innocence: a critique of ‘ethnomathematics’. educational studies in mathematics, 34, 131157. zaslavsky, c. (1979). africa counts: number and pattern in african culture. westport, conn.: lawrence hill & company. mellin-olsen, s. (1987). the politics of mathematics education. dordrecht: d. reidel. republic of south africa. (1996). the constitution. pretoria: government printer. zaslavsky, c. (1994). ‘africa counts’ and ethnomathematics. for the learning of mathematics, 14(2), 3-8. “it is truly difficult to make a democracy... it is not what i say that says i am a democrat, that i am not racist or machista, but what i do.” paulo freire 9 << /ascii85encodepages false /allowtransparency false /autopositionepsfiles true /autorotatepages /all /binding /left /calgrayprofile (dot gain 20%) /calrgbprofile (srgb iec61966-2.1) /calcmykprofile (u.s. web coated \050swop\051 v2) /srgbprofile (srgb iec61966-2.1) /cannotembedfontpolicy /warning /compatibilitylevel 1.4 /compressobjects /tags /compresspages true /convertimagestoindexed true /passthroughjpegimages true /createjdffile false /createjobticket false /defaultrenderingintent /default /detectblends true /colorconversionstrategy /leavecolorunchanged /dothumbnails false /embedallfonts true /embedjoboptions true /dscreportinglevel 0 /syntheticboldness 1.00 /emitdscwarnings false /endpage -1 /imagememory 1048576 /lockdistillerparams 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<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> >> >> setdistillerparams << /hwresolution [2400 2400] /pagesize [612.000 792.000] >> setpagedevice abstract background reporting mathematics results curriculum and assessment policy statement reporting framework conceptual model for using assessment data challenges to the use of assessment results framework for standards-based reporting implementation of a standard setting process development of standards-based reports use of reports by teachers for enhancing learning and teaching areas for further research conclusion acknowledgements references appendix 1 about the author(s) meshack moloi department of primary education, tshwane university of technology, south africa anil kanjee department of primary education, tshwane university of technology, south africa citation moloi, m., & kanjee, a. (2018). beyond test scores: a framework for reporting mathematics assessment results to enhance teaching and learning. pythagoras, 39(1), a393. https://doi.org/10.4102/pythagoras.v39i1.393 original research beyond test scores: a framework for reporting mathematics assessment results to enhance teaching and learning meshack moloi, anil kanjee received: 25 aug. 2017; accepted: 29 apr. 2018; published: 25 july 2018 copyright: © 2018. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract in this article we propose a framework for reporting mathematics results from national assessment surveys (nas) such that effective use of the resulting reports can enhance teaching and learning. we explored literature on factors that may contribute to non-utilisation of assessment data as a basis for decision-making. in the context of south africa, we identified the form and formats in which results of nas are reported as a possible limiting factor to the effective use of summative assessment results for formative purposes. as an alternative, we propose a standards-based reporting framework that will ensure accurate measurement of, and meaningful feedback on, what learners know and can do. we illustrate how, within a properly designed reporting framework, the results of a nas in mathematics can be used for formative purposes to enhance teaching and learning and, possibly, improve learner performance. background national assessment surveys (nas) have been implemented in south africa since the abolishment of the apartheid education system in 1996, and have evolved over time, changing in name, purpose, design, scope and frequency (department of education [doe], 2005; kanjee, 2007). national assessments are defined as ‘regular and systematic measurement exercises designed to determine what students have learned as a result of their educational experiences’ (unesco, 2000, p. 14). they are different to public examinations in that their goal is to inform policy for the education system as a whole, rather than to certify individual learners. these assessments may be administered to an entire cohort (census testing) or to a statistically chosen group (sample testing) and may also include background questionnaires administered to learners, teachers or education officials to obtain additional information for use in interpreting learner scores. braun and kanjee (2007) note that the utility of the data generated from these assessments depends on the quality and relevance of the assessment, the thoroughness of the associated fieldwork, as well as the expertise of those charged with the analysis, interpretation, reporting and dissemination of results. between 1996 and 2015, the form, format and frequency of nas in south africa have changed significantly – from national sample-based surveys administered in selected grades to assess mathematics and language performance every 3 to 4 years to annual national census-based assessments (department of basic education [dbe], 2013). in addition to nas, individual provinces such as the western cape and north west also administer provincial assessments and common tests, respectively, which focus on different subject areas and grades (hoadley & muller, 2016). while there have been marked improvements in the administrative and logistical processes of the assessments, a challenge that remains unresolved pertains to the meaningful reporting and effective use of the results from these assessments for enhancing teaching and learning. the phenomenon of non-utilisation or under-utilisation of national assessment data in influencing decision-making in south africa has been noted as a matter of concern (kanjee & moloi, 2014; kanjee & sayed, 2013). yet there has been a growing body of research which indicates that, when the results of nas are reported, disseminated and utilised properly, there are observable improvements in learner performance (klinger, deluca & miller, 2008; ravela, 2005; schiefelbein & schiefelbein, 2003). it would appear, therefore, that one challenge facing teachers in south africa is the inadequacy in meaningful reporting and effective utilisation of evidence from assessment. meaningful reporting includes finding effective ways of converting raw data into information that could inform decision-making. at classroom level, ‘meaningful information’ refers to information that the teacher could use for determining what learners at a particular grade level know or do not know, and can or cannot do, and to develop relevant interventions to address specific learning needs of learners. in this article, we propose a framework for reporting results from nas for use at the school level, and demonstrate how this framework can be applied to identify specific learning gaps of learners and provide guidelines to address identified learning gaps. although the reporting framework is exemplified in mathematics, its applicability extends to any school subject. first, we contextualise the proposed framework by providing a brief overview of reporting of assessment results as regulated in the south african curriculum and assessment policy statement (caps). next, we provide a conceptual framework for reporting and using assessment data, highlighting the challenges impeding effective use of data. this is followed by a description of the proposed reporting framework, its underlying philosophy and an exemplar school report, in which we highlight its practical application and implications for enhancing teaching and learning. we conclude the article by listing areas for further research to optimally use summative assessment results for formative purposes. reporting mathematics results the view taken in this article is that mathematics as a subject embodied in most school curricula is often characterised as a hierarchical cumulative body of knowledge. as such, the foundations of relevant mathematics content at a particular grade level are developed in the previous grade and the acquisition of complex capabilities builds on relatively basic concepts. for instance, young children progressively develop a ‘number concept’ often demonstrated by first being able to organise concrete objects before they can manipulate abstract concepts. given this unique nature of the subject, assessment and use of assessment results in mathematics seem to present specific challenges to mathematics teachers (webb, 1997). in order to enhance learning of mathematics knowledge and skills, as well as to identify and address specific learning gaps revealed by assessment results, teachers must have full mastery of the mathematics content area as well as a thorough understanding of the hierarchical nature of the subject. similarly, for assessment data to be useful for teachers to enhance learning in mathematics, it becomes more critical that the data be organised and reported in a manner that reflects the nature of mathematical knowledge and how learning in mathematics takes place. in practice, this implies that learner performance results reported with the intention of enhancing teaching and learning must, at a very minimum, provide information on what learners at a particular point know and can do and, at the same time, what they are potentially ready to learn (vygotsky, 1962). one limitation in reporting the results of nas is the tendency to adopt a norm-referenced approach in which schools, and even learners, are ranked and compared with one another according to their performance in the tests (green, 2002). the ‘league tables’ that often emanate from norm-referenced reporting are notorious for attracting resistance to assessment and evocation of negative feelings among teachers. this undesirable phenomenon was reported in the united kingdom (goldstein, 2001) and was also observed in south africa when teacher unions boycotted the administration of the annual national assessment (ana) because they perceived the assessment as ‘an onslaught on teachers with no intention to improve the [education] system’ (nkosi, 2015). in this article we argue that the vital element that links nas results to enhancing teaching and learning is a reporting framework that provides accurate measurement and meaningful feedback on what learners know and can do (griffin, 2009). importantly, the reporting framework must reflect the structure of mathematical knowledge as well as the process of learning in mathematics. it must embrace what griffin (2009) defines as ‘criterion-referenced interpretation’ and involve measurement coupled with ‘skills audits’ in which responses to clusters of items in a test are interrogated to identify an underlying construct. for example, a grade 6 learner who only responds correctly to test items that involve counting forward with whole numbers is demonstrating mathematical understanding that is at a lower level than a learner who, in addition, also responds correctly to items that involve doing calculations using fractions. we are aware of the critical distinctions that some make between nas and school-based assessments in terms of how the assessments are impacted upon differently by the socio-economic contexts within which learning takes place (nichols & berliner, 2007). disparities among school-based testing procedures (webb, 1997), possible variations in curriculum coverage across schools and other differences may lead to questioning the fairness of the nas. within the limits set by these caveats, we take the view advanced by dunne, long, craig and venter (2012) that a good balance between nas and school-based assessment is possible with proper test design and effective reporting of results. proper test design encapsulates considerations of the extent to which the test adequately elicits meaningful information on what learners know, can or cannot do in the subject area of interest. effective reporting involves ‘packaging’ and presenting the results in ways that enable the target users to initiate appropriate interventions for improvement. in particular, the south african ana model, where all learners in a grade, and not just typical representative samples, participate in the nas, enhances both the feasibility and the practicability of the balance that dunne et al. recommend. a standards-based reporting framework (srf) that allows criterion-referenced interpretation of test results in these conditions stands to benefit policymakers, teachers, parents and even learners. it is important to recognise that the value of the results of an assessment is optimal when they are used within the confines of the purpose for which the assessment is designed. on the one hand, school-based assessments include formative assessments whose purpose is to inform teaching and learning while a lesson is in progress and are, therefore, developmental in design. on the other hand, schools also conduct summative assessments which basically measure the extent to which learning has taken place after several lessons were delivered. testing that characterises nas falls under the latter category of assessments. our argument is that, within a properly designed reporting framework, the results of summative assessments can be used for formative purposes to enhance the quality of teaching and learning. curriculum and assessment policy statement reporting framework assessment results in basic education in south africa, both school-based as well as results of common examinations and nas, are recorded and reported according to a framework that is prescribed in the caps document. the ‘framework’ has three key elements designed in seven levels, namely rating codes, descriptions of competence and percentages (table 1). we examined the caps framework against our proposed conceptual model and noted some disparities which we consider to be of material importance. table 1: curriculum and assessment policy statement framework for reporting assessment results. the caps framework prescribes that assessment data will be organised into fixed percentage bands with the lowest band ranging from 0% to 29% and the highest from 80% to 90%. within this framework, a learner obtaining a minimum score of 50% is deemed to be functioning at the ‘adequate achievement’ level (dbe, 2011). we argue that by organising and summarising results using percentages, the caps framework does not provide any information on the specific knowledge and skills that learners have or have not mastered. for example, a score of 56% provides no information on what should be done for enhancing teaching and learning. we extracted a table that summarises nas results in a typical ana report compiled according to the caps framework to point out some of the conceptual challenges that compromise caps-based reports (table 2). table 2: percentage of grade 6 learners by achievement level in mathematics. in table 2, which contains information that was put in the public arena, the raw score bands have been summarised using the seven codes and the corresponding descriptions of competencies. no substantive qualitative analysis has been presented that provides detailed information on what learners at each score band in table 2 know and can do. in a survey to assess the extent to which south african teachers used the ana results, kanjee and moloi (2014) reported that up to 26% of the teachers in their study were of the view that the ana reports did not provide any new information that they did not already know. an inference that could be made from these perceptions was that these teachers were, logically, not likely to utilise these results. our view is that perceptions of inadequacy in the content of the nas reports could contribute to non-utilisation of the results for enhancing teaching and learning, which in turn could lead to perpetuation of underperformance in the system. the fixed percentage bands as exemplified in table 2 do not accommodate variations in the difficulty of tests. for instance, learners who score in the range of 0% – 29% are categorised as functioning at the ‘not achieved’ (l1) level and those who score in the range of 80% – 100% as functioning at the ‘meritorious’ (l7) level, regardless of the difficulty of the specific test. we should be aware that on an easy test percentage correct raw scores tend to be higher than in a difficult test; however, in the same test learners of higher ability are expected to score higher than their counterparts of lower ability (bond & fox, 2007). so, a meritorious achievement in an easy test may not necessarily be the same in a difficult test. it is also not possible to set two different tests that have exactly the same level of difficulty, even if the exact same test specifications are followed. it is for this reason that test equating measures have been introduced to adjust for differences in test difficulty (kolen & brennan, 1995). the net effect of these inconsistencies is that the users of caps-based reports may have either superficial or distorted knowledge about the performance of learners. moreover, the use of this reporting framework implies that a higher conceptual workload is placed on teachers and school leaders by expecting them to be able to record, report, categorise and address learner needs across seven levels of performance. it could prove unrealistic to expect a teacher to keep track of and provide differentiated support across seven categories of learners in a class. to mitigate the observed shortcomings of the caps reporting framework and ensure that assessment results are reported in ways that provide quality information to support users and enhance the teaching and learning process, in this study we propose an alternative model that is underpinned by a theory of data use proposed by breiter and light (2006). conceptual model for using assessment data breiter and light (2006) developed a conceptual model for using data to inform decision-making in the management of education districts. central to their model is a definition of decision-making as a ‘highly complex, individual cognitive process that can be influenced by various environmental factors’ (breiter & light, 2006, p. 208). they discourage notions of decision-making that require innumerable disparate pieces of data and suggest rather that decision-making involves intelligibly reducing (collecting and organising) large amounts of data, converting the data (summarising and analysing) into information and transforming the information into context-related knowledge to inform action (prioritising and synthesising). their model comprises four key elements, namely data, information, knowledge and decision-making. while not necessarily focusing specifically on assessment data, the model also accounts for the multiplicity of data and data sources that decision-makers in education must deal with. we adapted the model by breiter and light and developed a conceptual model to report assessment results so that the information can be used to enhance the quality of teaching and learning in schools (see figure 1). figure 1: conceptual model for using assessment data. the basic element of the model by breiter and light (2006) is data. this includes, but may not be confined to, raw statistical data like test scores, for instance. once teachers, school leaders or other decision-makers become aware of a situation of educational importance that needs to be addressed such as the persistent underperformance in mathematics and related issues, for example language ability or home background, appropriate data, often presented in numerical formats, need to be collected and analysed to gain detailed insight into the nature of the phenomenon. we agree with breiter and light that once data is collected it must be organised in ways that will make it meaningful to the users. but data do not speak for themselves; hence the continued reporting of mathematics assessment results in raw scores in our schools seems to have influenced neither teaching nor learning. data must be reported in ways that allow key users, such as district officials, school leaders and teachers, to decode the data (coburn, honig & stein, 2009). information, in the model, refers to data that has been appropriately analysed and summarised so that it sheds light on the nature and extent of the identified problem. thus, any report must communicate relevant information that will either add to what is known or will illuminate a new area of interest or further investigation. for example, a mathematics school report should provide information on what individual or groups of learners know or do not know and can or cannot do in mathematics, which domains of mathematics pose particular challenges to learners and whether different groups of learners (e.g. boys vs girls or rural vs urban) display comparable levels of proficiency. later in this article we show how reporting assessment data using meaningful performance standards provides information that empowers key users to make relevant decisions about the challenges of teaching and learning in schools. knowledge builds on available information by synthesising what is new with what is already known or available to change the undesirable situation and weighing what the priorities are. for example, a teacher who interprets assessment results and identifies relevant teaching strategies to address revealed learning gaps and explores possible interventions to rectify the situation has knowledge. we contend that there is a relationship between the depth and quality of knowledge about the education system and the quality of available information. assessment information that is either incomplete or inaccurate will lead to partial or distorted knowledge about the education system and is likely to result in ineffectual interventions for improvement. decision-making is the deployment of acquired knowledge to impact the situation as desired and, in the case of knowledge that comes from assessment, to improve learning outcomes. breiter and light (2006) argue that decision-making does not begin with data but with knowledge of needs, for instance needs of learners, teachers or even district officials. it is knowledge that directs the decision-maker to the types of data to collect, the time of collecting it and the methods of transforming the data into actionable decisions. it is important to note that, because of the dynamic nature of the education enterprise, there is a dialectical relationship between knowledge and the context in which teaching and learning take place. on the one hand, there is knowledge of what the assessment results reveal and what needs to be done to turn things around. on the other hand, there is knowledge of new phenomena that may require the collection of new data to understand their nature and thus begin a new cycle of data collection, generation of information and development of necessary knowledge to make relevant interventions. decision-making involves leveraging on existing knowledge and prioritising what needs to be done to achieve the desired goals. for instance, when it is known that learner performance in mathematics in a school or district is particularly and continually unacceptable and the factors that contribute to the situation are also known, policymakers and practitioners are confronted with deciding on the best action to take to remedy the situation and count on existing evidence to justify their interventions. development and implementation of relevant intervention for any decision to have an impact on practice, relevant interventions that address the key challenges identified must be developed and implemented. in practice, the nature, extent and duration of the intervention may vary depending on what decisions are taken across different contexts. for example, interventions to improve mathematics performance could focus on a specific phase or grade, for example foundation phase or grade 3; these interventions could address specific content areas, for example geometry, or groups of students, for example second language speakers, or the interventions could be conducted as additional lessons before new concepts are introduced or as additional exercises during lessons. the key point is that the intervention developed must be based on addressing challenges identified from the information collected, provided the information is clear, meaningful, easy to read and relevant. in addition, it is critical that some form of evaluation be conducted to monitor progress in implementing interventions. improved learning is the ultimate goal within classrooms and schools. within a learner-centred paradigm, improvement in learning and realisation of observable learner performance hinge largely on the quality of feedback that is given to learners (saddler, 2010). while breiter and light (2006) were specifically referring to feedback in formative assessment in classrooms, we argue that the principle applies to test results as well. when feedback, in the form of information-rich assessment reports, is clear and specific in terms of where the learners are and what the expectations are such that learners are enabled to take control of their learning, it can serve to move learners to the next step. feedback that provides evidence of what knowledge and skills learners have mastered and which they have not guides teachers to support learners meaningfully and relevantly according to their identified needs (sloane & kelly, 2003). it creates a conducive environment wherein teachers and learners work together to realise their shared instructional goals. in this environment learner performance is highly likely to improve. the implications of adopting the proposed conceptual model for use of assessment data to enhance learning in mathematics are twofold. firstly, an assessment framework that is based on this conceptual model must have a facility that makes it possible to transform assessment data to information and add value to information to convert it to knowledge. secondly, because our focus is on mathematics, the framework must be sensitive to the nature of mathematics as a body of knowledge and a school subject and to how learning in mathematics takes place. we argue that these requirements can be met by a standards-based reporting framework. challenges to the use of assessment results some of the reasons identified for the nonor under-utilisation of information from nas include poor or non-dissemination of the findings, lack of confidence in the validity of such information among those who have to act upon it, and lack of capacity and absence of appropriate tools to help teachers use the data (kellaghan, greaney & murray, 2009). other researchers (hambleton & pitoniak, 2006; hambleton & slater, 1997; underwood, zapata-rivera & van winkle, 2010) also blame reports from national assessments for being complex, often couched in statistical jargon that users cannot decipher, difficult to read, and even more difficult to interpret. in south africa, kanjee and moloi (2016) reported that, although the results of nas had been considered in some policy-related decisions, there had been limited focus on using the results to support improvements in teaching and learning. more pertinent to the objectives of this article was the finding from kanjee and moloi (2016) that a significant percentage of the teachers endorsed ‘agreed’ and ‘strongly agreed’ when presented with the statement: ‘teachers do not know how to use ana results to assist learners’. what was more concerning was that approximately 60% of the teachers in this category were teaching in affluent schools that were reputed for high performance. the corresponding percentage of teachers in poorer and often under-performing schools went up to 85%. an inference that could be made from these perceptions was that these teachers were not likely to utilise the results of these national assessments. the situation could be exacerbated by the finding that in the same study, up to 65% of the teachers strongly negated a statement that district officials provided guidance and training on the use of ana results. effectively, it would appear that teachers are left to their own devices when it comes to interpreting and using the ana results. in his study on how provinces, districts and circuits utilise data from ana, govender (2016) reported wide variations in the two provinces and districts that he sampled. although the education officials were aware of the utility value of the data, govender (2016) notes that the majority reported that they lacked technical and practical capacity to analyse and interpret the data in meaningful ways. again, like in the case of kanjee and moloi (2014), this finding implies that district officials are unable to provide relevant guidance and support to schools and teachers to enhance their use of assessment results for improving teaching and learning. in another study kanjee and mthembu (2015) explored the extent to which foundation phase teachers in one district in south africa demonstrated understanding of concepts and practices related to both formative and summative assessment. their study sample included teachers from schools serving communities that ranged from low to high socio-economic status. kanjee and mthembu (2015) reported that the teachers demonstrated very low levels of assessment literacy, more so in formative than in summative forms of assessment. although the sample was quite small and not representative, it is to be noted that these findings were in agreement with the observations that govender (2016) made. both district officials and teachers in south africa appear to lack adequate capacity to utilise assessment data in ways that will enhance the quality of teaching and learning. overall, research suggests that, invariably, the interpretation and utilisation of assessment results to enhance teaching and learning in schools are often limited by the competencies of key users, including teachers, school leaders and education department officials (griffin, 2009; timperley, 2009). the implication is that reports presenting assessment results must not be dependent on assumed competencies of key users. thus, these reports should be easy to read, easy to understand, and provide some indication of possible ‘next steps’ that users can follow to identify and address specific learning gaps or support learners in improving on their current levels of performance. however, limited information currently exists on how such reports need to be developed, nor on what type of analysis is required or how information is best presented to increase the utility value of these reports for teachers. exploring the use of standards-based reports to address the limitations of reporting as discussed in the previous sections we propose a standards-based reporting framework. green (2002) notes that a standards-based report presents assessment results according to demonstrable mastery of knowledge and skills displayed by learners as evidence of achieving expected learning outcomes. a standards-based report does not ‘average’ learner scores in a test but identifies what learners know and can do in relation to what the expected standard specifies. implicit in a standards-based report is a priori statement of what is expected of a learner at a particular level or grade. drawing from the analysis of observed learner scores, the report provides easy-to-read, easy-to-understand and clear guidelines or clues on next steps for teachers (ravela, 2005). the ‘knowledge and skills’ expected from learners are generally referred to as ‘standards’ (goodman & hambleton, 2004, p. 148). in educational circles, a distinction is made between ‘content standards’ and ‘performance standards’ (cizek, 1996; rodriguez et al., 2011). rodriguez et al. (2011, p. 18) define ‘content standards’ as ‘what students need to learn’. in the context of south africa, ‘content standards’ are spelt out in the caps by grade and by subject (dbe, 2011). ‘content standards’ specify the nature and scope of content knowledge, including skills, that a learner must acquire in a given grade. hambleton (2000) defines performance standards as: well-defined domains of content and skills and performance categories for test score interpretation [that] are fundamental concepts in educational assessment systems aimed at describing what examinees know and can do. the primary purpose [of the affected assessments] is not to determine the rank ordering of examinees, as is the case with norm-referenced tests, but rather to determine the placement of examinees into a set of ordered performance categories. (p. 2) thus, while content standards answer the ‘what’ question, performance standards answer the question about ‘how much’. an apt description of the purpose of performance standards proffered by hambleton (2000) is that they are qualitative and descriptive statements of how much learning has taken place and how much of it is ‘good enough’. our interpretation of hambleton is that performance standards provide a framework of evidence to be used for placing learners at particular points on a continuum of knowledge and skills according to what they are able to demonstrate when given opportunity to do so, like in a test. important features of performance standards are performance levels (pls) and performance level descriptors (plds). zieky and perie (2006) describe pls as: general policy statements that indicate the official position on the desirable number and labels of categories to be used in classifying learners according to their knowledge and skills in a particular subject and grade. (p. 3) because the knowledge and skills are mapped on a continuum that stretches from low to high, carefully selected scores, known as cutscores, are determined to mark and distinguish two consecutive levels of competence or pls on the continuum (kaftandjieva, 2010). plds are defined as detailed descriptions of ‘the knowledge, skills, and abilities to be demonstrated by students who have achieved a particular pl within a particular subject area’ (zieky & perie, 2006, p. 4). morgan and perie (2005) affirm that plds are ‘working definitions of each of the performance levels [that] … define the rigor associated with the performance levels’ (p. 5). standard setting the link between ‘standards’ and effective reporting of learner performance is provided by the process of standard setting. cizek and bunch (2007) define standard setting as: a process of establishing one or more cutscores on a test for purposes of categorising test-takers according to the degree to which they demonstrate the expected knowledge and/or skills that are being tested. (p. 13) a cutscore is defined as a point on a score scale which distinguishes two consecutive levels of competence (kaftandjieva, 2010). from this definition learners who obtain scores lower than the cutscore will typically be less competent in the affected subject than those with scores above the cutscore. in practice, the standard setting process involves both quantitative and qualitative inputs. it involves technical analysis of learner responses in raw data form as well as content expert inputs from teams of professionals from relevant stakeholder groups who serve to validate the technical results (tiratira, 2009). teams of content experts, preferably teachers of mathematics in this case, develop concise descriptions of typical knowledge and skills that characterise a learner who functions at a particular level. in addition, the teams suggest implications for progression and intervention for learners who, from their test scores, are categorised to be functioning at a particular level. the standard-setting exercise serves to transform raw data into meaningful information as envisaged in our conceptual model in figure 2. details on the technical processes of standard setting are available in moloi (2016). figure 2: standards-based reporting framework for national assessment surveys results. framework for standards-based reporting having pointed out the shortcomings in the raw score reporting system as currently used in south africa (dbe, 2013), and highlighted the value of using performance standards as an alternative, we now propose a framework to implement a standards-based reporting system. the primary purpose of this srf is to present user-friendly reports that promote the formative use of summative nas results to enhance teaching and learning. for example, end-of-year annual national assessment results that are reported using performance standards should provide teachers with detailed information on specific learner strengths and weaknesses. this information can then be used by teachers to plan and prepare lessons that address identified gaps or reinforce specific knowledge of learners. the srf comprises five key sections: (1) rationale for the srf and how it should be applied in practice, (2) guidelines on the quality, form and format of the raw data obtained from nas, (3) process to be followed for conducting standard setting exercises, (4) content and key elements required to compile standards-based reports and (5) practical proposals on how teachers could use the reports to enhance learning and teaching. rationale for, and application of, the standards-based reporting framework the purpose of the srf is to propose key specifications and practical guidelines for developing information-rich reports for users to provide relevant feedback to enhance teaching and learning in all schools. the srf addresses the limitations of current guidelines and reporting practices specified in the national curriculum documents for data from nas. the framework establishes a coordinated system of processing results from nas to compile relevant reports of high utility value for use at the different levels of the education system. depending on the purpose and focus of the nas, the reports can be used by officials at the national, provincial and district levels as well as by school leaders and teachers. the practical application of the srf requires: (1) obtaining valid and reliable data from nas, (2) conducting appropriate standard setting exercises, (3) compiling relevant standards-based reports for the targeted audience and (4) using reports to develop and implement relevant options to enhance teaching and learning. a diagrammatic representation of the key elements and the flow of the srf is shown in figure 2. data obtained from national assessment surveys a primary requirement for the application of the srf is the availability of relevant and valid data from nas. the raw data must be in the form of item-level learner results and could be in a scored or raw data format. more importantly, the data must be linked to specific teachers, schools, districts or provinces, depending on the level for which reporting will be conducted. complete data will include scores for all the schools and learners who participated in the nas. the data must be valid in the sense that, in the case of mathematics, the learners’ scores are inclusive of all the domains of mathematics that the assessment covered, have high reliability coefficients, and be free of errors. the quality of the reports that are based on the srf depends on the quality of the data used. implementation of a standard setting process the standard setting process involves selected panels of subject content experts, for example teachers of mathematics or mathematics curriculum specialists, to establish both quantitative and qualitative indicators of expected performance standards for the subject. the process provides information to identify those learners who meet the standards, those who fall below the expected standard and those who exceed the standard. it is important to ensure that the panels are representative across the varying contexts within which teaching and learning takes place in the education system. for instance, in the case of south africa proportionate representation of urban, semi-urban, rural and farm schools, or quintile categories, in the panels is necessary. the panels receive training on how to develop pls and plds and how to determine valid cutscores. the panels develop generic pls that categorise learners according to thoroughly discussed generic competencies with clear indications of what the implications are for intervention and progression at each level. in addition, the panels develop subject-specific pls and plds. they must determine and validate cutscores that mark transitions from one pl to the next in terms of subject knowledge and skills. using the pls, plds and cutscores determined by the panels, relevant standards-based reports can be developed for national, provincial, district or school level use. development of standards-based reports developing standards-based reports and reporting accordingly constitute the realisation of the aim of the srf. standards-based reporting presents assessment results in report formats that are easy to read, easy to understand and easy to use in decision-making. compilation of standards-based reports is a process of converting the information from the assessment into useful knowledge that is synthesised and prioritised in ways that enable users to make evidence-based decisions as envisaged in the conceptual model presented in figure 2. a standards-based report comprises the following sections: (1) particulars of the institution, (2) how to use the report, (3) performance level definitions and implications, (4) subject-specific performance levels and descriptors and (5) presentation of results by performance levels. each of these sections is described below, its purpose clarified and, where appropriate, its potential in contributing to enhancing teaching and learning is specified. in order to demonstrate its practical application, an exemplar of a standards-based report is provided in appendix 1 focusing on reporting at the level of a school. similar reports can also be compiled at the level of provinces, districts and classrooms. particulars of the institution for ease of identification, basic particulars such as the name of the school, the district and province under which the school falls, the grade and the subject must appear on the first page of the report. as noted in the exemplar in appendix 1, this report has been compiled for ‘city primary school’ which is located in southern province using grade 6 mathematics data from the 2015 national assessment study. how to use the report a note on how the standards-based report should be used is included to specify the steps that teachers should follow to enhance teaching and learning in their classrooms. performance level definitions and implications the use of performance levels, their definitions and implications in standards-based reports is a seismic shift from traditional raw score reporting. a standards-based school report will clarify up front what these features mean and how they help the teacher address learning needs in a differentiated approach as opposed to traditional one-size-fits-all approaches. subject-specific performance levels and descriptors in a standards-based school report all the results from a particular assessment are presented according to performance levels and, preferably, in iconic formats such as pie charts, bar and linear graphs for visual impact. performance levels do not only enable teachers to adopt differentiated approaches to interventions, but, in the case of mathematics, they mirror the hierarchical nature of the subject. requisite knowledge and skills at a particular performance level lay the foundations for the next higher level. for more nuanced analysis, school results in a standards-based report are usually disaggregated by sub-groupings such as gender, school poverty quintile category, subject domains, cognitive levels, urban-rural sub-divisions and others. an illustrative example of a standards-based school report with some of the features discussed in this article is shown in appendix 1. presentation of results the results in the standard-based report are aggregated by pl and may be presented by specific sub-groups (e.g. boys and girls) as well as by subject-specific sub-domains and cognitive levels. moreover, a summary of the results, for example by school or class, should also be provided to provide an overview of performance, while additional comparisons by district or province, where available, should also be reported in order to provide schools with a context within which to interpret results. use of reports by teachers for enhancing learning and teaching non-utilisation of information from nas is one criticism that led to the development of the srf and dissemination of standards-based reports. standards-based reports are designed with the needs of the end-user in mind. greater value from the report will be derived if teachers operate within professional learning communities than if they work as individuals. it is recommended that the report be reviewed and discussed by all school staff responsible for mathematics, including the head of department and school management team members. for instance, in the exemplar school report in appendix 1 (a1), key information is contained and presented in a user-friendly format in the generic pls and the subject-specific plds. by referring to the pls and the plds, the teacher can easily detect whether learners in the school or class are performing at the requisite grade level, identify learners with specific learning needs and plan targeted interventions. from the overall performance the report drills down to performance disaggregated by relevant sub-categories such as gender, subject domains, content cognitive demand levels, and others as necessity dictates. the report places at the disposal of a teacher powerful and specific information that they can use to decide on what to prioritise and how to differentiate interventions. as reported in figure 1-a1 of the school report, 25% of grade 6 learners in city primary school are functioning at the partly achieved level. according to table 1-a1, ‘partly achieved’ means that these learners demonstrate partial understanding of the knowledge and skills required to function at the grade 6 level for mathematics (table -a1), and are ‘unlikely to succeed in the next grade without support’. moreover, these learners require specific intervention to address their identified knowledge gaps, and additional support to progress to the required grade, that is, the ‘achieved’ level. table 3-a1 indicates that the performance of these learners is similar to other learners in the district and province, while table 4-a1 indicates that there are more boys than girls in this pl category. the findings reported in table 5-a1 indicate that these learners have performed relatively well for ‘number and operations’ but need more assistance with ‘probability’, ‘data handling’, and ‘measurement’. in addition, results in table 6-a1 also indicate that these learners struggled the most with questions that focused on ‘application’ and ‘reasoning’. from the reported findings on learner knowledge and skills, teachers can draw specific learner performance trends in their school and then proactively plan and prepare their lessons, and assessments to address identified learning deficiencies or improve on learner’s strengths. in practice, three options exist for using the results of summative assessments in a formative manner to support learners to address their learning gaps or improve on their strengths. planned interventions by teachers can be implemented: (1) at the beginning of a school year or of a school term, depending on whether results are from previous year or term, (2) just before a teacher introduces a new topic for which findings from the reports have shown pose particular challenges to learners or (3) using both options (1) and (2). the discourse in this article highlights important seminal work on how standards-based reporting can influence effective utilisation of summative assessment results to enhance teaching and learning. while there is educationally sound motivation from research literature for the potential efficacy of standards-based reports from nas, especially within the context of developed nations, empirical research on the aspects of the framework and its application in a setting like south africa would provide necessary evidence to serve as a basis for taking the framework forward. in this regard, some of the specific areas for further research into the efficacy of the framework and how the use of standards-based reports could be sharpened need to be highlighted. areas for further research for ease of implementation of the srf and to ensure that assessment data obtained from summative assessments can be optimally used for formative purposes, we suggest four areas for further research. first, the srf must be implemented in practice to: (1) determine its utility value across the different school types that characterise the education system in south africa and (2) identify specific challenges and successes in its application by different role players, that is, teachers, school leaders, and education department officials at the district, provincial and national level. second, exploration is needed of the use of srf to focus greater attention on, and implement specific interventions for, addressing the challenge of equity in classrooms, schools and districts. the use of standards-based reports can provide teachers, school leaders and education department officials with more useful and valid indicators that move beyond accountability measures that highlight specific disparities between learners at the different pls. in this regard, additional research is required to explore the use of standards-based reports as indicators for determining the support that is needed to reduce the percentage of learners functioning at the lowest pls, that is, ‘not achieved’ and ‘partially achieved’, and to increase the percentage of learners functioning at the ‘achieved’ and ‘advanced’ pls. thus, instead of monitoring change using mean percentage scores, which obscure which learners are progressing, the use of srf focuses on those learners who need the most support, that is, the poor and marginalised. third, we recommend the development of a formatted excel spreadsheet that teachers can populate with test results, and the software compiles a typical standards-based report that allows teachers to: (1) easily identify the specific questions on which learners perform poorly, (2) identify which learners need more assistance and (3) decide on possible next steps to follow for using summative data in a formative manner. for example, a report generated in the form of figure 3 provides a shaded item map that shows whether learners got an item correct (unshaded) or incorrect (shaded). from figure 3, teachers can immediately see which learners performed ‘well’ and in which test questions learners had the ‘most difficulty’. in this case, learners had the most difficulty in questions 4, 5, 6, 9, 12, 13 and 15. figure 3: example of excel data entry sheet showing an item correct map. similarly, the software should also generate a table that provides teachers with ideas for next steps, tailored to learners for each performance level. for example, the information generated in table 3 provides intervention ideas for learners functioning at the ‘partly achieved’ level. the information presented in table 3 indicates the question number that most learners got incorrect, the specific competency or skill assessed in the question, and the pages in the dbe workbook or commercial textbook which teachers can use as revision exercises for addressing specific learning gaps. table 3: example of an excel sheet indicating next steps for learners functioning at the ‘partly achieved’ level. fourth, we are of the view that the introduction and use of performance standards in reporting assessment data paves the way for individualised testing. the question raised by kingsbury, freeman and nesterak (2014) is appropriate in this regard: ‘if we believe that education should meet each student’s academic needs, why wouldn’t we use assessments that adjust to their individual achievement levels?’ (p. 1). with the assistance of enabling ict in general and appropriate item response theory techniques in particular, the prospects of strategies such as computerised adaptive testing (cat) are growing and need to be exploited optimally (weiss & betz, 1973; weiss & kingsbury, 1984). in cat an individual learner does not have to respond to all items in a test. instead, the learner responds to an item that is considered to be either easy or of medium difficulty from a pool of items of a wide range of levels of difficulty. if the learner answers the item correctly, a more difficult item is administered until the probability of answering a more difficult item is shown to be at its lowest and testing is discontinued. the learner is then assumed to be functioning at the level of the most difficult item that they answered correctly. several features seem to make cat a preferable option in terms of enhancing the utilisation of assessment data. firstly, cat has short turnaround times as the candidate does not have to answer a fixed number of test items and, therefore, allows time for immediate intervention and remediation. secondly, cat is a relatively more cost-effective testing option than traditional testing as it uses a limited number of test items. thirdly, computer applications like cat allows for a wider range of items to be used, items that are more likely to approximate practice and real-life contexts, ensuring the validity of the testing process. we recommend that every report must be accompanied by both the generic pls and the subject-specific pls and plds. while information in the report may be presented in suitable graphical and tabulated formats, reading and interpreting the results against the pls and plds enhances the significance and implications of what the results show. the ultimate goal is to ensure that teachers will find the detailed information in the plds more meaningful and thus be able to utilise this information for providing relevant support and feedback to improve chances of effective learning for all learners in their classrooms. conclusion we reviewed literature that shows that, while the phenomenon of national assessments has been on the increase, there has not been convincing evidence that the results of these summative assessments optimally influence what happens in the classrooms in terms of teaching and learning. in the context of south africa, we explored how prescriptions for recording and reporting the results of these assessments tend to fall short of the main purpose of assessment, which is to provide evidence-based feedback that will enable appropriate interventions to enhance teaching and learning. we pointed out how the reporting framework that is prescribed in the national curriculum may limit the extent to which nas results could be used meaningfully as evidence to inform decision-making for planning and delivering appropriate interventions to enhance teaching and learning in the classrooms. in particular, we highlighted that the reporting approach that averages assessment results in raw scores such as ‘percentage correct responses’ is deficient in information. it lacks necessary qualitative information on what learners know and can do as evidenced from the assessment. consequently, users of the reports, particularly teachers, are not empowered to intervene in ways that will enhance teaching and learning and, potentially, improve performance. we propose an alternative reporting framework that is intended to add value to reporting of results from nas, and how this information can be used to provide quality feedback to inform evidence-based decision-making at different levels. the key design features of the proposed srf are a clear rationale for providing relevant feedback to all users, the requirement for standard-setting exercises to enrich quantitative data with qualitative expert-provided information and easy-to-read guidelines on how standards-based reports should be compiled and used. using a mathematics school report as an exemplar, we demonstrated in a fair amount of detail how information from summative national assessments – presented and disaggregated by subject domains and various learner categories as guided by the srf – can be used effectively for differentiated formative purposes to address identified learner needs at strategic stages during the school year. this kind and level of detail in reporting, derived from a carefully designed standards-based framework, goes beyond sheer traditional raw test scores, provides information-rich feedback and has potential to enhance teaching and learning in all schools. through the use of carefully defined hierarchical subject-specific performance levels and descriptors, the srf leads to generating reports that adequately reflect the hierarchical nature of mathematics where knowledge of basic concepts lays foundations for understanding complex concepts. in the same vein, the hierarchy suggested in the framework reflects how learning in mathematics should be facilitated and carefully planned to provide ‘scaffolding’ that helps learners continually move to the next cognitive level. we recognise that the alternative srf needs to be piloted to obtain adequate empirical feedback about its efficacy. this is a limitation that we plan to address in a large-scale pilot of the srf. the known challenges of capacity among teachers cannot be ignored; hence, we propose that the use of the srf must be coupled with professional support, monitoring of the implementation progress, especially as it pertains to the needs of teachers and learners in low-resourced schools, and, where necessary, provision of appropriate icts to reduce workloads so that teachers can spend most of their time on effective utilisation of assessment results to enhance learning. acknowledgements acknowledgement is due to the department of primary education, tshwane university of technology, for providing an enabling professional learning community where staff had opportunity to critique one another’s manuscripts. this article benefitted immensely from those interactions. competing interests we declare that we have no financial or personal relationships that might have inappropriately influenced our writing of 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(2007). improving learner achievement in schools: applications of national assessments in south africa. in s. buhlungu, j. daniel, r. southall, & j. lutchman1 (eds.), state of the nation: south africa 2007 (pp. 470–499). pretoria: hsrc press. retrieved from http://www.hsrcpress.ac.za/product.php?productid=2183 kanjee, a., & moloi, q. (2014). south african teachers’ use of national assessment data. south african journal of childhood education, 4(2), 90–113. https://doi.org/10.4102/sajce.v4i2.206 kanjee, a., & moloi, q. (2016). a standards-based approach for reporting assessment results in south africa. perspectives in education, 34(4), 29–51. https://doi.org/10.18820/2519593x/pie.v34i4.3 kanjee, a., & mthembu, a. (2015). assessment literacy of foundation phase teachers: an exploratory study. south african journal of childhood education, 5(1), a346. https://doi.org/10.4102/sajce.v5i1.354 kanjee, a., & sayed, y. (2013). assessment policy in post-apartheid south africa: challenges for improving education quality and learning. assessment in education: principles, policy & practice, 20(4), 442–469. https://doi.org/10.1080/0969594x.2013.838541 kellaghan, t., greany v., & murray, t.s. (2009). using the results of a national assessment of educational achievement. washington, dc: the world bank. kingsbury, g.g., freeman, e.h., & nesterak, m. (2014). the potential of adaptive assessment. educational leadership, 71(6). retrieved from http://www.ascd.org/publications/educational-leadership/mar14/vol71/num06/the-potential-of-adaptive-assessment.aspx klinger, d.a., deluca, c., & miller, t. (2008). the evolving culture of large-scale assessments in canadian education. canadian journal of educational administration and policy, 76, 1–34. retrieved from https://journalhosting.ucalgary.ca/index.php/cjeap/article/view/42757 kolen, m.j., & brennan, r.l. (1995). testing equating methods and practices. new york, ny: springer. moloi, m.q. (2016). a national framework for reporting the results of large-scale surveys in south africa. unpublished doctoral dissertation, tshwane university of technology, pretoria, south africa. retrieved from http://tutvital.tut.ac.za:8080/vital/access/services/download/tut:2377/source1 morgan, d.l., & perie, m. (2005). setting cut scores for college placement. new york: college board. retrieved from https://research.collegeboard.org/publications/content/2012/05/setting-cut-scores-college-placement nichols, s.l., & berliner, d.c. (2007). collateral damage: how high-stakes testing corrupts america’s schools. cambridge, ma: harvard education press. nkosi, b. (2015). sadtu: boycott national assessments in schools. mail&guardian. retrieved from https://mg.co.za/article/2015-09-02-sadtu-calls-for-boycott-of-national-assessments ravela, p. (2005). a formative approach to national assessments: the case of uruguay. prospects, 35(1), 21–43. https://doi.org/10.1007/s11125-005-6816-x rodriguez, m., rubio, f., landsdale, j., vukmirovic, z., meckes, l., & gysling, j. (2011, april). standard setting issues and practice in latin america. paper presented at the symposium on standard setting in an international context: issues and practice, at the annual meeting of the ncme, new orleans, la. retrieved from http://edmeasurement.net/research/rodriguez%20rubio%20landsdale%20meckes%202011.pdf saddler, d.r. (2010). beyond feedback: developing student ability in complex appraisal. assessment & evaluation in higher education, 35(5), 535–550. https://doi.org/10.1080/02602930903541015 schiefelbein, e., & schiefelbein, p. (2003). from screening to improving quality: the case of latin america. assessment in education, 10(2), 141–154. https://doi.org/10.1080/0969594032000121252 sloane, f., & kelly, a. (2003). issues in high-stakes testing programs. theory into practice, 42, 12–17. https://doi.org/10.1207/s15430421tip4201_3 timperley, h. (2009). using assessment data for improving teaching. in australian council for educational research conference proceedings (pp. 21–25). perth: acer. retrieved from http://research.acer.edu.au/cgi/viewcontent.cgi?article=1036&context=research_conference tiratira, n.l. (2009). cutoff scores: the basic angoff method and the item response theory method. the international journal of educational and psychological assessment, 1(1), 27–35. retrieved from https://docs.google.com/open?id=0byxug44ovrlpwutirlzns2furms underwood, j.s., zapata-rivera, d., & van winkle, w. (2010). an evidence-centred approach to using assessment data for policymakers. princeton, nj: educational testing service. unesco. (2000). assessing learning achievement. education for all: status and trends 2000. paris: unesco. vygotsky, l.s. (1962). thought and language. cambridge, ma: mit press. webb, n.i. (1997). criteria for alignment of expectations and assessments in mathematics and science education. washington, dc: council of chief state school officers. weiss, d.j., & betz, n.e. (1973). ability measurement: conventional or adaptive? research report 73–1. minneapolis, mn: department of psychology, university of minnesota. retrieved from http://www.iacat.org/sites/default/files/biblio/we73-01.pdf weiss, d.j., & kingsbury, g.g. (1984). applications of computerized adaptive testing to educational problems. journal of educational measurement, 31, 361–375. https://doi.org/10.1111/j.1745-3984.1984.tb01040.x zieky, m., & perie, m. (2006). a primer on setting cut scores on tests of educational achievement. educational testing service. retrieved from https://www.ets.org/media/research/pdf/cut_scores_primer.pdf appendix 1 example school report source: adapted from moloi, m.q. (2016). a national framework for reporting the results of large-scale surveys in south africa. unpublished doctoral dissertation, tshwane university of technology, pretoria, south africa. retrieved from http://tutvital.tut.ac.za:8080/vital/access/services/download/tut:2377/source1      2018 national assessment: grade 6 mathematics results city primary school     southern province the evidence presented in this report is intended to support school leaders and teachers identify learner strengths and weaknesses, and plan appropriate interventions for improvement. how to use this report it is recommended that the report be reviewed and discussed by all school staff responsible for mathematics, including the head of department and school management team members. the information provided in table 1-a1 to table 6-a1 and in figure 1-a1 below should be carefully reviewed to: determine whether learners in the school are performing at the requisite grade level. identify specific learning needs of learners who at risk and those who are on track. plan targeted interventions for supporting all learners improve learning, based on their learning needs identified, especially those at the lower performance levels. performance level definitions and implications table 1-a1 below provides information on the four levels used to report the mathematics performance of grade 6 learners, and it implications for progression and interventions for improving learning. table 1-a1: performance levels for grade 6. table 2-a1 lists the specific mathematics knowledge and skills that learners functioning at each performance level are expected to demonstrate. table 2-a1: mathematics knowledge and skills at each performance level. results for school figure 1-a1 below presents the overall percentage of grade 6 learners functioning at each of the performance levels in mathematics for your school. it shows learners who are at risk (i.e. at the not achieved and partly achieved levels) and those who are on track (achieved and advanced levels). figure 1-a1: distribution of learners across mathematics performance levels. table 3-a1 compares the overall percentage of grade 6 learners in this school functioning at each of the performance levels in mathematics against the district and province results. table 3-a1: school performance (%) in mathematics by district and province. table 4-a1 provides information on the percentage of boys and girls functioning at the different performance level within the school, the district and the province. table 4-a1: mean score (%) by district, province and gender. table 5-a1 lists the mean scores of learners functioning at each of the performance levels in the five mathematics content domains. this information indicates the knowledge levels of learners in each performance level for the different content domain areas. table 5-a1: mean score (%) by content domain and performance level. table 6-a1 presents mean scores of learners functioning at various cognitive levels. this information indicates the extent to which learners are demonstrating complex cognitive capabilities in mathematics. table 6-a1: mean score (%) by cognitive level and performance level. http://www.pythagoras.org.za open access page 1 of 1 reviewer acknowledgement no. of manuscripts processed in 2019 (outcome complete) 49 accepted without changes 0 (0%) accepted with minor changes (to the satisfaction of the editor)1 7 (14.3%) accepted after major revisions (re-submit, then re-review)2 4 (8.2%) rejected after review – not acceptable to be published in pythagoras3 8 (16.3%) rejected without review – not acceptable to be published in pythagoras4 30 (61.2%) no. of manuscripts currently in review 7 1.accepted after one round of review, with ‘minor’ changes as specified by reviewers and editor. 2.accepted after two or more rounds of review, with ‘major’ changes specified by reviewers and editor. 3.includes two cases where the authors did not resubmit after required to make major changes. 4.all submissions undergo a preliminary review by the editorial team to ascertain if it falls within the aims and scope of pythagoras, is of sufficient interest to our readers, offers substantially new knowledge, and is of sufficient quality to be sent for review, and is not in any way scientifically flawed. we rejected 11 manuscripts because they were mainly about mathematics and not mathematics education, nine manuscripts because they were scientifically flawed (mostly using inappropriate statistical methods), and six manuscripts, mostly from eastern countries, because the authors’ english language constructs simply could not carry the mathematics education concepts they were trying to communicate and fixing them seemed impossible, despite our commitment to support authors and despite our professional language editing services. acknowledgement to reviewers in an effort to facilitate the selection of appropriate peer reviewers for pythagoras, we ask that you take a moment to update your electronic portfolio on https:// pythagoras.org.za for our files, allowing us better access to your areas of interest and expertise, in order to match reviewers with submitted manuscripts. if you would like to become a reviewer, please visit the journal website and register as a reviewer. to access your details on the website, you will need to follow these steps: 1. log into the online journal at https:// pythagoras.org.za 2. in your ‘user home’ [https://pythagoras.org. za/index.php/pythagoras/ user] select ‘edit my profile’ under the heading ‘my account’ and insert all relevant details, bio statement and reviewing interest(s). 3. it is good practice as a reviewer to update your personal details regularly to ensure contact with you throughout your professional term as reviewer to pythagoras. please do not hesitate to contact us if you require assistance in performing this task. publisher: publishing@aosis.co.za tel: +27 21 975 2602 tel: 086 1000 381 the quality of the articles in pythagoras and the credibility and reputation of our journal crucially depend on the expertise and commitment of our peer reviewers. reviewing is an important part of scholarly work, making a substantial contribution to the field. reviewers’ comments serve two purposes, guided by two inter-dependent objectives: • pythagoras wishes to publish only original manuscripts of the highest possible quality, making a meaningful contribution to the field. reviewers advise the editor on the scholarly merits of the manuscript to help evaluate the manuscript and to decide whether or not to publish it. reviewers are encouraged to reject a manuscript if it is scientifically flawed, merely sets out observations with no analysis, provides no new insights, or is of insufficient interest to warrant publication. • pythagoras is committed to support authors in the mathematics education community. reviewers help the authors to improve the quality of their manuscript. reviewers are encouraged to write their comments in a constructive and supportive manner and to be sufficiently detailed to enable the authors to improve the manuscript and make the changes that may eventually lead to acceptance. the following summary of outcomes of the reviewing process in 2019 shows that our reviewers do well in achieving both objectives: we sincerely thank the following people who have reviewed these manuscripts for pythagoras in 2019. we very much appreciate their time, expertise and support of pythagoras amidst pressures of work. anita campbell belinda huntley benita nel bhekumusa khuzwayo bruce brown caroline long cerenus pfeiffer david andrich david mogari deonarain brijlall dirk wessels divan jagals duan van der westhuizen erlina ronda eunice moru faaiz gierdien hamsa venkatakrishnan hannah barnes helen sidiropoulos helena miranda iben christiansen ifunanya ubah ingrid mostert jacques du plessis jayaluxmi naidoo jill adler jogy alex joseph baidoo judah makonye karin brodie kathryn mellor kenneth ngcoza lynn bowie marc north marie joubert marthie van der walt mdutshekelwa ndlovu mellony graven michael de villiers michael mhlolo michael murray million chauraya neil eddy odette umugiraneza patrick barmby paul mokilane piet human rajen govender rina durandt sarah bansilal satsope maoto shaheeda jaffer sibawu siyepu suela kacerja tracy craig vanessa scherman vera frith vimolan mudaly willy mwakapenda yip-cheung chan http://www.pythagoras.org.za� https://pythagoras.org.za� https://pythagoras.org.za� https://pythagoras.org.za� https://pythagoras.org.za� https://pythagoras.org.za/index.php/pythagoras/user https://pythagoras.org.za/index.php/pythagoras/user https://pythagoras.org.za/index.php/pythagoras/user mailto:publishing@aosis.co.za pyth 39(1)_table of contents.indd open accesshttp://www.pythagoras.org.za index i issn: 1012-2346 (print) issn: 2223-7895 (online) editor-in-chief alwyn olivier stellenbosch university (retired), south africa associate editors anthony essien university of the witwatersrand, south africa faaiz gierdien stellenbosch university, south africa rajendran govender university of the western cape, south africa dirk wessels stellenbosch university, south africa editorial board jill adler university of the witwatersrand, south africa bill barton university of auckland, new zealand marcelo borba são paulo state university, brazil doug clarke australian catholic university, australia jeremy kilpatrick university of georgia, united states of america gilah leder la trobe university, australia stephen lerman southbank university, united kingdom frederick leung university of hong kong, sar, china liora linchevski hebrew university of jerusalem, israel john malone curtin university, australia andile mji tshwane university of technology, south africa willy mwakapenda tshwane university of technology, south africa john olive university of georgia, united states of america david reid university of bremen, germany paola valero stockholm university, sweden renuka vithal university of kwazulu-natal, south africa anne watson oxford university, united kingdom volume 39 number 1 december 2018 ‘islamic geometric patterns’. retrieved from: https://www. kisscc0.com/clipart/islamicgeometric-patterns-islamic-artislamic-arc-szgqj7/ page i of ii information for authors and readers review article acquiring mathematics as a second language: a theoretical model to illustrate similarities in the acquisition of english as a second language and mathematics maureen ledibane, kotie kaiser, marthie van der walt original research the effects of examination-driven teaching on mathematics achievement in grade 10 school-based high-stakes examinations onyumbe okitowamba, cyril julie, monde mbekwa metacognitive awareness and visualisation in the imagination: the case of the invisible circles divan jagals, martha van der walt affordances for learning linear functions: a comparative study of two textbooks from south africa and germany kathryn mellor, robyn clark, anthony a. essien students’ understanding of geometry terminology through the lens of van hiele theory jogymol alex, kuttikkattu j. mammen teachers’ learning and assessing of mathematical processes with emphasis on representations, reasoning and proof satsope maoto, kwena masha, lekwa mokwana tracking grade 10 learners’ geometric reasoning through folding back samuel mabotja, kabelo chuene, satsope maoto, israel kibirige ‘eighteen hands high’: a narrative reading of animal farm from a mathematical perspective liveness mwale, willy mwakapenda mathematical sense-making through learner choice piera biccard conversations in a professional learning community: an analysis of teacher learning opportunities in mathematics million chauraya, karin brodie beyond test scores: a framework for reporting mathematics assessment results to enhance teaching and learning meshack moloi, anil kanjee exploring teachers’ use of technology in teaching and learning mathematics in kwazulu-natal schools odette umugiraneza, sarah bansilal, delia north correction corrigendum: metacognitive awareness and visualisation in the imagination: the case of the invisible circles divan jagals, marthie van der walt reviewer acknowledgement guidelines for authors ii 1 13 23 33 45 53 63 73 83 92 101 115 128 129 130 abstract introduction the teaching and learning of functions textbooks and tasks in the teaching and learning of mathematics theoretical and analytical framework research design and methodology analysis and discussion of findings conclusion and implications acknowledgements references about the author(s) kathryn mellor school of education, university of the witwatersrand, south africa robyn clark school of education, university of the witwatersrand, south africa anthony a. essien school of education, university of the witwatersrand, south africa citation mellor, k., clark, r., & essien, a.a. (2018). affordances for learning linear functions: a comparative study of two textbooks from south africa and germany. pythagoras, 39(1), a378. https://doi.org/10.4102/pythagoras.v39i1.378 original research affordances for learning linear functions: a comparative study of two textbooks from south africa and germany kathryn mellor, robyn clark, anthony a. essien received: 18 may 2017; accepted: 31 july 2018; published: 26 sept. 2018 copyright: © 2018. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract textbook content has the ability to influence mathematical learning. this study compares how linear functions are presented in two textbooks, one of south african and the other of german origin. these two textbooks are used in different language-based streams in a school in gauteng, south africa. a qualitative content analysis on how the topic of linear functions is presented in these two textbooks was done. the interplay between procedural and conceptual knowledge, the integration of the multiple representations of functions, and the links created to other mathematical content areas and the real world were considered. it was found that the german textbook included a higher percentage of content that promoted the development of conceptual knowledge. this was especially due to the level of cognitive demand of tasks included in the analysed textbook chapters. also, while the south african textbook presented a wider range of opportunities to interact with the different representations of functions, the german textbook, on the other hand, included more links to the real world. both textbooks linked ‘functions’ to other mathematical content areas, although the german textbook included a wider range of linked topics. it was concluded that learners from the two streams are thus exposed to different affordances to learn mathematics by their textbooks. introduction mathematics textbooks have been found to have a strong influence on classroom practice (fan, 2013; howson, 1995; stylianides, 2014; valverde, bianchi, wolfe, schmidt, & houang, 2002) and the tasks that they include can stimulate different levels of mathematical thinking and learning (watson, & ohtani, 2015; stein, smith, henningsen, & silver, 2000; sullivan, clarke, clarke, & o’shea, 2010). at a south african school (the context of this study), learners in two language-based streams are taught mathematics using different textbooks up until the end of grade 9. in grade 10, however, learners who choose to write the national senior certificate examinations merge from both streams and are thereafter taught together. although both groups of learners have encountered similar mathematical content en route to grade 10, the textbooks utilised respectively in the two language-based streams do not necessarily approach this content in the same way. at the time the study was undertaken, one stream, which was taught in german according to the german baden-würtemberg curriculum, used the elemente der mathematik textbook series (griesel, postel, & suhr, 2005). the other, which was taught according to the south african caps curriculum, used classroom mathematics grade 9 (rhodes-houghton et al., 2013). what sparked our interest in this study was the fact that the first author of this article was teaching grade 9 mathematics in the english stream at this school, with the knowledge that she would be moved to teach a grade 10 class the subsequent year. it was thus imperative to understand the mathematics that learners were exposed to in the separate streams in preparation for teaching a merged grade 10 class. thus, in a larger study (mellor, 2015), a comparative textbook analysis was undertaken as a means of attempting to interrogate the affordances for learning experienced by the two groups before progressing to grade 10, with the ultimate aim of helping her, and other grade 10 teachers at the school, have a deeper understanding of their learners’ mathematical backgrounds, thus avoiding potential ‘mathematical mismatches’ between the teachers and their learners (skemp, 1976). the topic of functions was selected as the focus for the textbook analysis because it is generally agreed that functions form one of the most important unifying ideas in mathematics education (knuth, 2000), and are ubiquitous in high school mathematics in general (even, 1990). emphasis is placed on the topic in both the south african and german curricula. furthermore, various national assessments (such as the annual national assessment results for grade 9 and the national senior certificate results for grade 12) have indicated that the topic of functions is a recurring area of weakness for learners in south africa (department of basic education, 2012, 2013a, 2013b, 2014a, 2014b). as linear functions, specifically, is the first type of function studied in depth in both the south african and german mathematics curricula, it seemed an appropriate place to begin the investigation. the overall purpose of this study was therefore to analyse and compare the manner in which linear functions are presented in the south african and german mathematics textbooks used at the observed school by learners en route to grade 10. due to variations between the south african and the german mathematics curricula adhered to in the two language-based streams, similar mathematical topics are not necessarily taught at the same grade level across the two streams. the topic of linear functions is introduced and taught as an area of focus in grade 7 in the german stream while this occurs in grade 9 in the english stream at the school. in order to compare textbook content on linear functions pitched at a similar level for purposes of this comparison, the english stream grade 9 textbook, classroom mathematics grade 9 (cm) was therefore paired with the german stream grade 7 textbook, elemente der mathematik 3 (edm). to this end, the study was informed by the following questions: how are linear functions presented in classroom mathematics grade 9 and elemente der mathematik 3? what affordances for learning about linear functions are made available in classroom mathematics grade 9 and elemente der mathematik 3? it must be noted that this investigation did not aim to endorse either textbook’s approach as superior. furthermore, this study was not aimed at making any value judgements with regard to the academic strategy of using different language-based streams in the research school. rather, the study intended to provide a thorough comparison of both textbooks’ content originating in two different education systems, with the aim of understanding the affordances for learning made available to the two groups before progressing to grade 10. the teaching and learning of functions it has been argued that the concept of functions is one of the most important in all mathematics (dubinsky & harel, 1992). the topic lays a foundation for further study within the field of mathematics itself, while it also contributes as a tool in other subjects due to its capability to model phenomena in the real world (ayalon, lerman, & watson, 2013). functions and graphs form one of the initial points in mathematics where learners encounter using one symbolic system to express and understand another (leinhardt, zaslavsky, & stein, 1990). linear functions, in particular, form a crucial aspect in the learning of elementary algebra (pierce, stacey, & bardini, 2010). the study of functions is, however, often the point at which a learner may conclude that ‘mathematics is meaningless and difficult’ (pierce, 2005). schoenfeld (as cited in leinhardt et al., 1990) argues that to avoid difficulties and misconceptions in the study of functions, a deep understanding of the correlation between the simplest form – a graphical line and its algebraic equation – is required. when selecting criteria by which to compare the textbook content on linear functions and the affordances for learning made available to learners in the two textbooks, this study therefore investigated whether this fundamental topic is presented in a manner promoting procedural or conceptual knowledge, as well as how these types of knowledge are sequenced. against this background, the integration of the multiple representations of functions, and the presence of links established between linear functions and the real world, as well as to other mathematical content areas, were also considered. the debate as to whether procedural or conceptual knowledge should be predominantly promoted in mathematics instruction has been well documented over the past century (hiebert & lefevre, 1986; kilpatrick, swafford, & findell, 2001; piaget, 1978; skemp, 1976). according to hiebert and lefevre (1986), the distinction between the two types of knowledge has generally juxtaposed formulaic skill against understanding. they describe procedural knowledge as knowledge that can be learnt by rote alone, while conceptual knowledge depends on the development of links and relationships. these ideas mirror skemp’s (1976) work on instrumental and relational understanding, which describes instrumental understanding as being made up of ‘rules without reasons’, while relational understanding is knowledge of ‘what to do and why’ (p. 23). these extremes lie at two poles of a continuum. skemp considers these different types of knowledge as instructional goals that a teacher can direct their learners toward. kilpatrick et al. (2001), however, argue that procedural fluency is vital to support conceptual understanding and to attain mathematical proficiency; thus, the two cannot be taught independently. research on how learners develop procedural and conceptual knowledge has historically suggested four causal relationships (rittle-johnson & schneider, 2014): the ‘concepts-first’ view (which argues that children first learn conceptually and then derive procedural knowledge by repetition), the ‘procedures-first’ view (which suggests that children first learn procedure and thereafter learn concept in a process of abstraction), the ‘inactivation’ view (which posits that procedural and conceptual knowledge develop independently), and the ‘iterative’ view (which puts forward that the causal relationship is bi-directional). of these, the ‘iterative’ view is the most commonly accepted (rittle-johnson & schneider, 2014). this study therefore attempted to investigate whether the linear functions content in the two textbooks predominantly promoted procedural or conceptual knowledge of linear functions, and how this (linear functions content) is sequenced in the two books under consideration. a distinctive aspect of the study of functions is its integration of different types of representations. in mathematics, representations include verbal, numerical, graphical and symbolic descriptions of a concept, which allow for it to be interpreted, communicated and discussed (tripathi, 2008). van dyke and craine (1997) contend that in the study of functions, it is critical that learners comprehend the underlying equivalence between the different representations: the same function can be represented in tabular, algebraic and graphical form. this is reiterated by leinhardt et al. (1990), whose research indicates that school learners have difficulty in linking these various representations, resulting in compartmentalised knowledge that is not integrated into a coherent conceptual understanding of ‘function’. rider (2007) and knuth (2000) argue that developing conceptual understanding of functions requires learners to understand their various representations, to understand the representations’ respective strengths and weaknesses, and to be able to translate between them fluently. the desire that learners connect the various representations of a function is reminiscent of hiebert and lefevre’s (1986) and kilpatrick et al.’s (2001) definitions of conceptual knowledge which emphasise the importance of creating links between different segments of information. utilising the different representations is like ‘examining the concept through a variety of lenses, with each lens providing a different perspective that makes the picture (concept) richer and deeper’ (tripathi, 2008, p. 439). in this study, as previously indicated, the extent to which the various representations of functions are integrated in the two textbooks was therefore also interrogated. a further criterion that contributes to the mathematical learning process is the use of contexts that link mathematics to the real world. the realistic mathematics education movement is a prime example of how using tasks that relate to the real world can be utilised as ‘a source of learning mathematics’ (van den heuvel-panhuizen, 2000). by using instructional material that is connected to reality, mathematics is hoped to be ‘imaginable’, so that it can remain close to learners’ experiences and be perceived as relevant to society. particularly in terms of teaching functions, leinhardt et al. (1990) suggest that using real-world applications serves the dual purpose of deepening understanding of the topic and ‘perhaps increas[ing] students’ motivation by giving familiar meanings to the problems they encounter’ (p. 20). durant and garofalo (1994) furthermore describe that functions have traditionally been taught abstractly. this does not allow learners to perceive the context they originate from, nor to understand their usefulness. as such, the literature indicates that using learning material that relates to the real world in the teaching of functions can be viewed as a means of promoting learning. it can also contribute to developing ‘productive disposition’ (kilpatrick et al., 2001) – having the ability to see the value of mathematics. in light of this, the two textbooks in this study were also examined in terms of the extent to which they created links between linear functions and the real world. textbooks and tasks in the teaching and learning of mathematics as indicated previously, textbooks have been shown to have a large influence on classroom practice (stylianides, 2014; valverde et al., 2002) as they are designed to help teachers structure their teaching and suggest a pathway for learners to follow when exploring a topic (johansson, 2005). their influence has led to a wealth of literature on textbook analysis being developed over past decades (e.g. fan & zhu, 2007; shield & dole, 2013). stylianides (2014) contends that textbooks can be analysed from various perspectives, among others the learner’s perspective, the teacher’s perspective or a mathematical perspective (in terms of examining a textbook’s potential to aid learning). fan (2013) argues that examining how a topic is presented in a textbook is only the first step – thereafter, causal issues need to be examined. for example, one could investigate why a particular textbook’s way of treating a topic in a specific context may be better than another. mathematical tasks, in many ways the building blocks of mathematics textbooks, have also been a focus of much international research in past years. watson and ohtani (2015), for instance, posit that mathematical tasks can influence learners’ learning as well as their perception of the nature of mathematics, and are in fact the ‘bedrock of classroom life’ (p. 3). various frameworks exist that aid the classification of tasks. for example, shimizu, kaur, huang and clark (2010) differentiate tasks with categories such as ‘complex’, ‘rich’ or ‘authentic’. furthermore, the task types and mathematics learning research project in australia classified tasks according to four types, namely exemplifying tasks, contextualised tasks, open-ended tasks and interdisciplinary tasks (sullivan et al., 2010). in this study, however, stein et al.’s (2000) task analysis guide was selected to classify tasks due to its appropriateness to investigate our research questions. theoretical and analytical framework this study was framed theoretically and analytically by an adapted version of stein et al.’s (2000) task analysis guide. this guide provides a framework by which to evaluate learning material in terms of the level of cognitive demand of tasks – the type of thinking required by learners to successfully solve problems. it is the cumulative effect of learners’ engagement with instructional tasks that allows them to learn mathematics and forms their conception of the nature of mathematics and their perception of whether they can make personal sense of the subject (stein et al., 2000). as shown in figure 1, the task analysis guide’s four categories (‘memorisation’, ‘procedures without connections’, ‘procedures with connections’, and ‘doing mathematics’) are split into two sections, namely ‘lower-level demands’ and ‘higher-level demands’. ‘lower-level demand’ tasks include those that can be solved by memorisation or by using procedures that can be applied independently of understanding the underlying concept (procedures without connections). ‘higher-level demand’ tasks require the ability to use procedure in a manner that relies on an understanding of the underlying concept (procedures with connections) as well as tasks that are non-algorithmic and require learners to explore underlying concepts (doing mathematics). figure 1: a summary of task analysis guide. this task analysis guide was for us the springboard from which to evaluate the opportunities for learning in this study’s analysed textbook content. in order to engage with this study’s research questions specifically, the guide was adapted using the literature on procedural and conceptual mathematical knowledge, while also considering literature on the multiple representations of functions. firstly, the task analysis guide was used to investigate the interplay of procedural and conceptual knowledge in the two textbooks in the following manner: textbook elements that resonate with the description of ‘lower-level demands’ were described as promoting procedural knowledge, while those that resonate with the description of ‘higher-level demands’ were described as promoting conceptual knowledge. this use of the task analysis guide was based on the language used to describe procedural and conceptual knowledge by key theorists on the topic, particularly hiebert and lefevre (1986) and kilpatrick et al. (2001). hiebert and lefevre describe procedural knowledge as knowledge that can be gained by rote learning and that can exist without being connected to any existing schemas, echoing the descriptions in stein et al.’s (2000) ‘lower-level demand’ tasks. furthermore, hiebert and lefevre describe conceptual knowledge as being rich in links, with connections to pre-existing knowledge. such knowledge is gained by meaningful learning that promotes knowledge integration into existing schemas. kilpatrick et al.’s description of conceptual knowledge, and skemp’s (1976) comparable description of ‘relational understanding’, also highlight conceptual knowledge’s reliance on connections. this view of conceptual knowledge resonates well with the ‘higher-level demand’ categories in stein et al.’s task analysis guide. importantly, the task analysis guide category of ‘procedures with connections’ highlights that making links between multiple representations of mathematical ideas helps to develop mathematical meaning. thus, with these adaptations, the task analysis guide allowed the linear functions textbook content to be analysed and then compared. in addition, the trends in international mathematics and science study (timss) textbook investigation (valverde et al., 2002) guided this study’s research methodology. the global timss in the mid-90s investigated and compared how textbooks contribute to implementing education policy and curricular goals in 48 education systems. as the timss research examined textbooks in terms of their educational opportunity with the goal to compare textbooks from different countries, their methodology resonates well with this study’s aim of examining cm and edm’s linear functions content in terms of their affordances for learning. in the timss, each analysed textbook chapter was partitioned into smaller units of analysis, called ‘blocks’, by separating content according to its purpose. ten categories were used to differentiate the purpose of textbook content: narrative, related narrative, unrelated instructional narrative, related graphic, unrelated graphic, exercise set, unrelated exercise set, activity, worked example, and other. for example, a single textbook page in their study could have been made up of content with four different purposes, thus resulting in four separate blocks. each block was then analysed as an individual component. the timss methodology lent itself to be used in conjunction with the adapted version of stein et al.’s (2000) task analysis guide for the purposes of this article’s study, as it provided a means of delineating the selected textbook content into individual units that could then be classified and analysed. research design and methodology the approach used in the larger study (mellor, 2015) was as follows: once the relevant chapters on linear functions were identified in cm and edm, their contents were examined and then compared by means of a qualitative content analysis. the first step in this process was to partition the selected chapters into much smaller units of analysis. to do so, the content of each selected page was segmented into various ‘blocks’ of information, separated by their purpose. the five purpose categories in table 1 were used. table 1: the different purposes of textbook content used to partition the selected chapters. after the chapters had been partitioned into ‘blocks’, each individual block was then coded so that the content could be accurately described, allowing for analysis to take place thereafter. making use of the adapted task analysis guide, as described above, each block was first coded as promoting conceptual or procedural knowledge of linear functions. during this coding process, we realised that stein et al.’s (2000) four categories were not sufficient to code all textbook content. although this method was effective to code exercises and worked examples, it did not cater for sections of instructional narrative or graphics. in light of this, a fifth category was added (a posteriori) to the original categories of the task analysis guide. it was titled ‘reading for understanding’. when coding content that fell into this category, we drew on both the ideas in the task analysis guide and the literature on procedural and conceptual knowledge to determine if it promoted procedural or conceptual knowledge. after coding for procedural or conceptual knowledge, each block was then coded according to the representations of functions included in the block. the following representations of functions were considered: flow diagram, table, equation, graph, ordered pairs, and words. finally, each block was also coded in terms of whether it made links to other mathematical content areas (for example, geometry, area, etc.) or to the real world (mellor, 2015). in order to promote validity and reliability of the coding process, two inter-raters provided feedback on the coding, and discrepancies were resolved by discussion. table 2 illustrates a selection of content ‘blocks’ similar to those found in the analysed chapters, as well as how these were coded. once all relevant textbook content had undergone this coding procedure, analysis of the data took place. table 2: examples of the types of content ‘blocks’ found in classroom mathematics grade 9 and elemente der mathematik 3, and how these were coded. analysis and discussion of findings valverde et al. (2002) differentiate between a textbook’s macrostructure – broad features of the book as a whole – and its microstructures – the features of individual chapters. although not the focus of this study, a brief summary of the two textbooks’ macrostructures is included to provide context for the detailed analysis of the elected chapters’ microstructures thereafter. the analysis of the microstructures focuses on the affordances for learning presented in the textbooks regarding the interplay between procedural and conceptual knowledge, the integration of the multiple representations of functions and links established to other mathematical content areas and the real world. individual textbook contexts: the macrostructures the two textbooks differ markedly in terms of their macrostructures. classroom mathematics grade 9 has 27 chapters overall, three of which relate to linear functions. these three chapters are located at different points in the book. in comparison, elemente der mathematik 3 is made up of only six chapters overall with one chapter on linear functions. although a similar total number of pages are dedicated to linear functions in both books, the pages are set out in a contrasting manner: when segmenting the chapters for analysis, cm had 78 blocks across the 38 analysed pages, while edm had 166 blocks in its 39-page chapter. the content of edm is thus far denser, with edm having about double the number of content blocks of cm within a similar amount of space. a further analysis of these content blocks follows in the next section. microstructure: interplay of procedural and conceptual knowledge of cm’s 78 content blocks, just over half were coded as promoting procedural knowledge (54%) while only about a third of edm’s 166 blocks were coded as procedural (37%). as is visible in figure 2, both textbooks appear to adopt hiebert and lefevre’s (1986) view of a mathematically competent learner in the sense that they promote a combination of both conceptual and procedural knowledge. additionally, kilpatrick et al.’s (2001) view on mathematical proficiency is helpful in making sense of the different foci of the respective chapters: when it comes to linear functions, cm does not promote conceptual understanding as much as edm does. figure 2: comparison of blocks that promote procedural and conceptual knowledge in classroom mathematics grade 9 and elemente der mathematik 3. the number of blocks coded in each category per book is also given. with the aim of gaining a more holistic understanding of how the respective textbooks utilise conceptual and procedural knowledge, the linear sequencing of the content was also analysed. as indicated earlier, the chapters on linear functions are arranged differently in the two textbooks: cm has three chapters inserted at different locations of the textbook while edm has only one chapter with six subsections. examining the linear sequencing of the coded content blocks in the separate sections highlighted the interplay between the individual blocks (mellor, 2015). figure 3 depicts these relationships. for example, chapter 7 of cm (functions and relationships 1) begins with three content blocks that were coded as promoting conceptual knowledge, and ends with three content blocks that were coded as promoting procedural knowledge. figure 3: analysis of coded content blocks, sequenced as in the respective chapters, in terms of procedural and conceptual knowledge. analysis of the three chapters of cm did not indicate a consistent method of arranging content that promotes procedural and conceptual knowledge. the chapters ‘functions and relationships 1’ and ‘functions and relationships 2’ can be viewed as a unit as they cover the same content. mellor (2015) explains further as follows: it therefore appears that in these two chapters the content is arranged predominantly according to the ‘concepts-first’ view. this view posits that learners first acquire knowledge of concepts and use this as a foundation to develop procedural knowledge (rittle-johnson & schneider, 2014). contrastingly, the chapter on ‘graphs’ appears to subscribe to the ‘iterative’ view of the development of conceptual and procedural knowledge. this suggests that the causal relationship between conceptual and procedural knowledge is bi-directional, where an increase in conceptual knowledge influences procedural knowledge and vice versa (rittle-johnson & schneider, 2014). this view is particularly in line with kilpatrick et al.’s (2001) view of mathematical proficiency that suggests that conceptual and procedural understanding (as well as their other three strands) are interdependent and develop in conjunction with each other. (p. 28) in edm, there appears to be a more consistent pattern of content blocks supporting concept or procedure. like the ‘graphs’ chapter in cm, the subsections of the linear functions chapter also indicate a predominantly ‘iterative’ view, with procedural and conceptual knowledge being more consistently integrated. there is, however, also a trend of the subsections beginning with conceptual content. as a whole, the content of edm appears to combine conceptual and procedural content more consistently than cm across the analysed chapters (mellor, 2015). further points of interest regarding the pattern of edm’s content in figure 3 occur at the noticeable clusters of conceptual or procedural blocks. these occur at the end of subsections 6.1, 6.2, 6.3 and 6.5, and in the first half of subsection 6.6. a closer examination reveals that in the clusters for subsections 6.2 and 6.3, this is solely due to the nature of the ‘regular’ exercises at that point. in 6.1, 6.5, and 6.6, however, the conceptual clusters are due in part to longer, special investigative tasks that invite learners to explore the topic of linear functions. these small sections, entitled ‘im blickpunkt’ [in focus] present content to extend learners such as using graphical calculators to work with linear functions, learning about the line of best fit, or using functions to investigate how to save electricity. classroom mathematics grade 9 does not include this type of enrichment content, which is a contributing factor to the two textbooks’ contrasting proportional breakdowns of content that promotes conceptual and procedural knowledge (see figure 2). analysis of individual content blocks provided further explanation of the different approaches to conceptual and procedural knowledge found in the two textbooks. figure 4 provides a summary of the coded content blocks from both chapters in terms of their cognitive demand. classroom mathematics grade 9 has noticeably more content that promotes ‘procedures without connections’, while edm has more ‘procedures with connection’ and ‘doing mathematics’ content. no content blocks in either book were categorised as ‘memorisation’. ‘reading for understanding’ content was categorised almost exclusively as promoting conceptual understanding in both cm and edm, and made up a similar percentage of content. the remaining content blocks in the compared chapters are made up predominantly of exercises and, to a lesser extent, worked examples. thus, it can be argued that it is the nature of the tasks that these textbook chapters include (in terms of exercises and worked examples) that influences the type of mathematical knowledge that the two books promote. figure 4: comparison of each category used in the coding process for procedural and conceptual knowledge. comparing specifically selected tasks from the two textbooks indicated that although the two textbooks include similar tasks with similar desired outcomes, the level of cognitive demand of the tasks is generally higher in edm than in cm. consider the following comparison between figures 5 and 6. figure 5: translating between function representations in classroom mathematics grade 9. figure 6: task from elemente der mathematik 3 on translating between function representations, (b) translation of the task. the tasks in figures 5 and 6 test similar content knowledge – that of being able to represent functions in four different ways (albeit with cm investigating only one function and edm investigating four different functions). both questions appear toward the end of their respective chapter or section. however, figure 5 promotes ‘procedures without connections’ (thus contributing to content that promotes procedural knowledge), while figure 6 promotes ‘procedures with connections’ (thus contributing to content that promotes conceptual knowledge). schoenfeld, smith and arcavi (1993) (as cited in schwarz & dreyfus, 1995) state that if tasks focus on translating between representations of functions in a manner that becomes highly procedural, this skill no longer contributes considerably to conceptual understanding. this is in line with the classification description in the task analysis guide which states that tasks should be classified as ‘procedures without connections’ if the required procedure is evident ‘based on … placement of the task’ (stein et al., 2000, p. 16). in the set of exercises that lead up to and include the task in figure 5, the act of translating between representations does become very procedural. furthermore, the task only requires one functional relationship to be manipulated, and uses only a table presenting consecutive integers as its starting point, reducing opportunity for conceptual links to be developed. in comparison, the task in figure 6 requires learners to engage simultaneously with four ‘starting points’ when manipulating representations in a layout that highlights the interplay between them. additionally, the provided table form of the function goes up in twos, and the respective functions need to be linked to a real-world context. both these aspects contribute to the task’s potential to strengthen conceptual understanding. as argued in mellor (2015), the comparison of these two tasks thus illustrates how similar content can be presented in a manner that creates different affordances to strengthen procedural or conceptual understanding thereof. the use of graded questions also differs in the two textbooks’ linear functions content. both books include a highly similar task on isosceles triangles that requires learners to describe the functional relationship between the base angles and the third angle. these two tasks are provided in figures 7 and 8. figure 7: task from classroom mathematics grade 9 on isosceles triangles. figure 8: task from elemente der mathematik 3 on isosceles triangles, (b) translation of the task. the comparison between the exercises in figures 7 and 8 again highlights a notable difference between the manner in which the two textbooks pose their tasks: in cm, the tasks often include sub-questions that guide the learner while in edm, this occurs less frequently. in this particular case, the task in figure 7 is a ‘procedures with connections’ task as although it does develop deeper levels of understanding, it clearly suggests pathways for learners to follow that remove it from the category of ‘doing mathematics’. in comparison, the exercise in figure 8 gives little guidance to the learner, thus promoting non-algorithmic thinking and metacognition. here, learners could perhaps come to realise on their own how useful the tabular form of a function can be. this would arguably have more impact on their conceptual (and procedural) understanding than being explicitly told to create one as in iii) of figure 7. this comparison thus exemplifies how the use of sub-questions can lead to a decline in the cognitive demand of a task. the comparisons between figures 5 and 6 and figures 7 and 8 provide a potential explanation of the discrepancies between the two textbooks highlighted by the coding process depicted in figure 3: although the contents tested are in many respects similar, the manner in which the tasks are designed contributes to their ability to promote either conceptual or procedural knowledge and thus influences the affordances for learning made available in the books. we acknowledge that these examples were hand-picked and that bias affected the choices. however, after having engaged deeply with the content of the two textbooks, we concluded that these are representative examples of how the two books set their tasks. classroom mathematics grade 9 does, therefore, appear to set tasks of a lower cognitive demand than edm although the task content outcomes are largely similar. this could therefore be a strong contributing factor to the proportional breakdown of procedural and conceptual knowledge in the chapters being compared. microstructure: integration of the multiple representations of functions both the task analysis guide (stein et al., 2000) and kilpatrick et al. (2001) stress that in mathematics using multiple representations is an indication of conceptual understanding. this is especially the case in terms of the study of functions, where an understanding of their various forms is an integral aspect of the topic (knuth, 2000; leinhardt et al., 1990; van dyke & craine, 1997). in light of this, we examined how the two textbooks integrate various representations of functions (flow diagram, table, equation, graph and words) into each content block of the analysed chapters. the manner in which this is done creates different affordances for learning linear functions. the results of this investigation are presented in figure 9. figure 9: percentage of content blocks that utilise one or more function representations (flow diagram, table, equation, graph, written words). the two textbooks have different foci in terms of multiple representations. about half of cm’s content blocks make use of only one representation of functions. of this, the notable contributors are ‘graph only’ questions (28% of total blocks) while ‘equation only’ questions make up about 15% of all content blocks. the next most frequent category is shared between blocks that integrate two and three representations, each making up about a fifth of total blocks. finally, just over 10% of blocks incorporate either four or five representations. this indicates that although there is a clear focus on content that involves either graphs or equations, the book does place importance on integrating other representations into its content. in comparison, only about a fifth of edm’s content blocks utilise just one representation. over half of its content blocks integrate two representations of functions. this modal category is composed overwhelmingly of content blocks that combine graphs and equations (45%). blocks that utilise three representations make up its second highest category (just over 20% of the data) with only 2 out of the 166 coded blocks using four representations. thus, although cm focuses on individual skills related to linear functions more than edm does, the book also emphasises translating between three or more representations to a higher degree than the german book: a third of cm’s coded content integrates three or more representations while only a quarter of edm’s content encourages this. as indicated previously, the ability to translate between multiple representations in mathematics is seen to contribute toward conceptual understanding (kilpatrick et al., 2001). it was therefore surprising that although cm includes many tasks that integrate multiple representations, analysis of the chapter as a whole indicated a predominance of procedural knowledge. translating between multiple representations can, however, become highly procedural if focused on repeatedly in the same fashion. tasks such as in figure 5 occur frequently in cm, in both the exercises and worked examples, which contributed to these content blocks being coded as promoting procedure. this can, however, hold value as kilpatrick et al. (2001) describe: without procedural fluency, students have trouble deepening their understanding of mathematical ideas or solving mathematical problems. the attention they devote to working out results they should … compute easily prevents them from seeing important relationships. (p. 122) translating between the multiple representations of functions is an acknowledged weakness in high school mathematics (leinhardt et al., 1990; rider, 2007) and it thus appears that in cm, this skill is being promoted to the point that it becomes procedural. if learnt with understanding, this would hopefully lead to flexible mathematical knowledge where learners can use the various representations to their advantage. thus, if the conceptual links behind the translations are stressed in the classroom rather than purely the procedural steps, the content of cm has the potential to allow learners to translate between the representations of functions in a manner where this becomes second nature. although the content of edm does not promote the translation between representations to the point that cm does, the book focuses in depth on the relationship between linear functions and solving linear equations. consider, for example, the task in figure 10. figure 10: (a) task from elemente der mathematik 3 linking linear graphs to linear equations, (b) translation of the task. such a task has the potential to develop strong connections and deep understanding of the relationship between linear equations and linear graphs. although edm may not develop the procedural fluency that cm does in translating between representations of functions, tasks such as this present the opportunity for procedural steps to be understood conceptually in a manner that was not evident in cm. thus, overall, edm can be seen as promoting deep, interconnected knowledge of linear functions in equation, graphical and tabular form, while cm promotes procedural fluency in a broader range of representations. microstructure: links from linear functions to other mathematical topics and the real world hiebert and lefevre (1986) argue that conceptual understanding of a mathematical topic comes about when ‘the holder recognises its relationship to other pieces of information’ (p. 4). apart from ‘internal’ links between multiple representations, the tasks discussed in figures 7 and 8 illustrated how both books have used linear functions to add meaning to the study of isosceles triangles (and vice versa). the two textbooks are quite similar in this regard, with both linking individual tasks to topics such as rate, area or volume, number systems and patterns in similar ways. as discussed, edm also particularly emphasises the connection between linear functions and solving linear equations to deepen conceptual understanding of both topics. the textbook also touches on the concept of linear regression toward the end of the chapter, suggesting the use of linear functions in statistical analysis and prediction. in terms of linking functions to other mathematical topics to deepen conceptual understanding, both books include opportunities to do this, although edm does so to a stronger degree than cm. however, although these types of links have the potential to develop learners’ conceptual understanding of mathematical content, they do not necessarily contribute to the learners’ understanding of the value of linear functions, nor how functional relationships manifest themselves in the real world. while perhaps not originally meant in this sense, hiebert and lefevre’s (1986) description of conceptual understanding, cited above, could go beyond the realm of content knowledge to include a concept’s relationship to and value in the real world. as kilpatrick et al. (2001) argue in their discussion of mathematical expertise, conceptual and procedural knowledge are just two strands of being mathematically proficient– understanding why the learned knowledge is worthwhile is another (they call this aspect of mathematical proficiency ‘productive disposition’). incorporating real-world contexts can contribute to learners perceiving mathematics as useful and relevant (durant & garofalo, 1994; leinhardtet al., 1990; van den heuvel-panhuizen, 2000). in terms of including content that demonstrates the use of linear functions in the real world, edm and cm use different approaches. edm includes more blocks linked to the real world than cm – 32% of content blocks in edm compared to 24% in cm. furthermore, their placement of these blocks is not the same, particularly in terms of how the chapters are introduced. elemente der mathematik 3 uses real-world situations in the majority of its worked examples that form the introduction to the chapter’s individual subsections. in comparison, cm does not use real-world contexts in its introductions (or worked examples). these contrasting introductory styles begin the chapters with different tones. in edm, the mathematical content is used to make sense of realistic contexts, while in cm, the mathematics is positioned as being quite abstract and part of a pre-existing system to be learned. elemente der mathematik 3 therefore appears to illustrate the real-world value of linear functions to a higher degree than cm. overall, the analysis indicated that edm promotes the interlinking of topics to a higher degree than cm, both in terms of links between mathematical topics and links between mathematics and the real world. conclusion and implications this study has argued that cm and edm create different affordances for learning about linear functions. both textbooks provide opportunity to learn about linear functions procedurally and conceptually, but with different emphases. elemente der mathematik 3 presents this content in a more conceptual manner whereas cm presents the work more procedurally. analysis indicated that this difference was evidenced by the cognitive demand of the included exercises. cm’s use of guiding questions and the (at times) repetitive nature of the tasks led to promoting knowledge of a predominantly procedural nature. contrastingly, edm included more tasks that provided opportunity for linear functions to be engaged with conceptually. the textbooks’ approaches to sequencing content in terms of procedural and conceptual knowledge also differs, with cm making use of both ‘concepts first’ and an ‘iterative’ arrangement of content while edm appears to consistently use an ‘iterative’ approach, although with a trend of beginning individual subtopics more conceptually. this study also found that cm includes a broader range of function representations than edm. however, cm also includes a higher percentage of tasks that only deal with one type of representation. in comparison, over half of edm’s coded content links two types of representations, predominantly linear equations to graphical representations of functions. in general, the findings indicated that although the german book covers fewer representations, those included are studied more in depth and conceptually than is apparent in cm. lastly, edm creates more links between linear functions and other mathematical content areas as well as to the real world than cm does. according to hiebert and lefevre (1986) and skemp (1976), conceptual knowledge is developed when links and connections are created to existing mathematical knowledge. this suggests that learners using edm are more likely to have greater affordances to integrate the topic of linear functions conceptually with their existing mathematical knowledge than learners using cm (mellor, 2015). also, edm’s many links to the real world, particularly in introductory content, may provide additional affordances to learn, as working on realistic problems is considered to aid learners’ development of mathematical tools and understanding (van den heuvel-panhuizen, 2000). overall, this study has exhibited that two different textbooks can present the topic of linear functions in different ways, and thus create different affordances for this content to be learnt. at the school where these textbooks were used in two language-based streams, learners entering a combined class in grade 10 were thus exposed to different methods of making sense of mathematics. what do these findings mean for a teacher who is tasked with teaching grade 10 learners in such a school? we argue that the key to successfully teaching learners in a combined class is the awareness and acknowledgement of the different affordances learners have been exposed to in terms of their mathematics content prior to grade 10. this awareness may mean tailoring lesson plans by using the types of examples from the german book that would benefit the english stream learners, and vice versa. acknowledgements k.m. thanks the university of the witwatersrand for the postgraduate merit award that provided financial support during the course of this research. the article is based on the b.sc. honours research report of k.m. which was jointly supervised by r.c. and a.a.e. competing interests the authors declare that they have no financial or personal relationships that may have inappropriately influenced them in writing this article. authors’ contributions k.m. wrote the majority of the article. all three authors contributed to the conceptualisation of the methodological approach used in the study. data analysis was carried out by k.m., r.c. and a.a.e. based on the jointly developed analytical framework. references ayalon, m., lerman, s., & watson, a. 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(2015). themes and issues in mathematics education concerning task design. in a. watson & m. ohtani (eds.), task design in mathematics education: an icmi study 22 (pp. 3–15). new york: springer. https://doi.org/10.1007/978-3-319-09629-2 abstract introduction theoretical framework methodology ethical considerations quality criteria findings and discussions conclusion and implications acknowledgements references about the author(s) satsope maoto department of mathematics, science and technology education, university of limpopo, south africa kwena masha centre for academic excellence, university of limpopo, south africa lekwa mokwana department of mathematics, science and technology education, university of limpopo, south africa citation maoto, s., masha, k., & mokwana, l. (2018). teachers’ learning and assessing of mathematical processes with emphasis on representations, reasoning and proof. pythagoras, 39(1), a373. https://doi.org/10.4102/pythagoras.v39i1.373 original research teachers’ learning and assessing of mathematical processes with emphasis on representations, reasoning and proof satsope maoto, kwena masha, lekwa mokwana received: 05 apr. 2017; accepted: 26 dec. 2017; published: 19 mar. 2018 copyright: © 2018. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract this article focuses mainly on two key mathematical processes (representation, and reasoning and proof). firstly, we observed how teachers learn these processes and subsequently identify what and how to assess learners on the same processes. secondly, we reviewed one teacher’s attempt to facilitate the learning of the processes in his classroom. two interrelated questions were pursued: ‘what are the teachers’ challenges in learning mathematical processes?’ and ‘in what ways are teachers’ approaches to learning mathematical processes influencing how they assess their learners on the same processes?’ a case study was undertaken involving 10 high school mathematics teachers who enrolled for an assessment module towards a bachelor in education honours degree in mathematics education. we present an interpretive analysis of two sets of data. the first set consisted of the teachers’ written responses to a pattern searching activity. the second set consisted of a mathematical discourse on matchstick patterns in a grade 9 class. the overall finding was that teachers rush through forms of representation and focus more on manipulation of numerical representations with a view to deriving symbolic representation. subsequently, this unidirectional approach limits the scope of assessment of mathematical processes. interventions with regard to the enhancement of these complex processes should involve teachers’ actual engagements in and reflections on similar learning. introduction this article is based on a study aimed at the improvement of the quality of teaching and learning mathematics using authentic real-life mathematics explorations. this involves paying closer attention to relationships between the manipulation of concrete material and the arising numerical values. we particularly focus on the early stages of the study and use two mathematical processes: representation, and reasoning and proof (national council of teachers of mathematics [nctm], 2000), incorporating adaptive reasoning (kilpatrick, swafford & findell, 2001). the study addresses some of the expectations with regard to the same processes within the context of the south african mathematics curriculum and assessment policy statement (department of basic education, 2011). our specific concern was how teachers’ own learning of mathematical processes influence assessments of their learners. we limited the scope of this article by focusing on assessment of patterns arising from 2d and 3d shapes. it has been a while since curricula developers and implementers strove for emphasis of both subject content and mathematical processes in mathematics teaching, learning and assessment (kilpatrick et al., 2001; nctm, 2000; orton & frobisher, 1996). there are many processes to be experienced and developed. orton and frobisher (1996) classified general processes into four categories: communication, operational, recording, and reasoning, which they claimed could contribute to some mathematical processes. they argued that these mathematical processes are not unique to mathematics, but play an important role in the establishment of new ideas and structures within mathematics. the nctm (2000) described problem-solving, reasoning and proof, communication, connections, and representations as process standards. kilpatrick et al. (2001) proposed five intertwining strands of mathematical proficiency: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition. the development of mathematical proficiency takes time and it would be interesting to analyse teachers’ assessments for process engagements. it is recognised that, in addition to assessing content knowledge, assessments should provide insight into students’ ability to engage with mathematical processes (davis, smith, roy & bilgic, 2014; gulkilik & arikan, 2012; hunsader et al., 2014). gulkilik and arikan (2012) determined pre-service secondary mathematics teachers’ views about using multiple representations in mathematics lessons. they found that, although the participants had concerns about the usage of multiple representations, they believed that using them was necessary for mathematics instruction. davis et al. (2014) examined students’ opportunities to engage in reasoning and proving within exposition and task components of two us reform-oriented secondary algebra textbooks. hunsader et al. (2014) analysed the extent to which students have opportunities to engage in the processes within the tests accompanying published textbooks. they intended to support teachers’ ability to make their own decisions about the efficacy of their assessment. studies that investigate classroom instruction, or the actual assessments administered to students, continue to be important in order to support the development of specific pedagogical content knowledge. because of this view, we were prompted to analyse 10 high school mathematics teachers’ learning of mathematical processes and how to assess them. in particular, we focused on the design of appropriate assessment rubrics and the facilitation of learning that takes into account those expectations. it is important that teachers’ values about assessment and their associated practices are consistent with each other and are both pedagogically sound. theoretical framework this article is framed by the process standards as set out by the nctm (2000) and kilpatrick et al.’s (2001) theory of mathematics proficiency. the nctm’s process standards describe ways in which all students learn the algebra content by engaging in problem-solving, reasoning and proof, communication, making connections, and multiple representations. according to kilpatrick et al.’s (2001) theory, mathematics proficiency comprises five intertwined strands: conceptual understanding, procedural fluency, strategic competency, adaptive reasoning, and productive disposition. while the bigger study utilises all these mathematical processes and strands, we focus on representation, reasoning and proof incorporating adaptive reasoning to analyse how teachers learn these processes and, subsequently, learn how to assess learners on the same processes. we analyse the teachers’ learning from their written responses to a pattern searching activity. the nctm (2000) points out that representation refers both to process and to product. the emphasis is that students should have opportunities to view and to create multiple representations of mathematics graphically, numerically, algebraically, and verbally. reasoning and proof should be a natural activity, an ongoing part of classroom discussions, no matter what mathematics topic is being studied. particularly, ‘students should expect to explain and justify their conclusions; … to clarify their thinking, … and to develop standards for high-quality mathematical reasoning’ (nctm, 2000, p. 342). meanwhile, kilpatrick et al. (2001, p. 5) say that adaptive reasoning comprises the ‘capacity for logical thought, reflection, explanation and justification’. methodology we used case study research design to analyse 10 high school mathematics teachers’ learning of representations, reasoning and proof as mathematical processes and how to assess them within the context of their university studies towards a bachelor in education honours degree in mathematics education. data constructed in this manner, where we could actually talk directly to participants and see them behave and act within their context and ultimately reflect on their own learning, are a major characteristic of qualitative research (creswell, 2007). we used two sets of data. the first set of data comprised the teachers’ written assessment task (figure 1) submitted as part of the assessment requirements for a module that was delivered on a part-time basis. with the first six questions we took the teachers through a practical exercise that allowed them to learn, with understanding, the assessment of mathematical processes as expected in the other two questions. figure 1: pyramid activity. the second set of data arose from the grade 9 lesson activity of one of the teachers (figure 2). figure 2: matchsticks pattern activity. the class had 14 learners, 9 girls and 5 boys. they formed three groups: two of five members each and one with four members. this teacher’s class was selected to provide a window through which to see how the participants’ newly acquired knowledge and skills unfolded in an actual classroom environment. our analysis of the two sets of data was guided by the nctm’s (2000) process standards of representation, reasoning and proof, and kilpatrick et al.’s (2001) notion of adaptive reasoning. for the first data set we drew our analysis from the teachers’ responses to item 5 and item 8 (see figure 1). for item 8 we analysed the teachers’ designed assessment criteria, targeting descriptions of the cells with the highest score to find out what and how they would assess. for the second data set, our analysis was guided by drawings that the learners produced in response to item 4 and their associated conversations to explain and justify their conclusions. ethical considerations this qualitative study complied with the university’s ethical requirements. approval was obtained from the relevant university structures and the 10 participants. the nature and purpose of the study were declared, inclusive of potential audiences and substantive foci. erickson (1998) writes: consent that is genuinely informed and without coercion reduces the risk of social harm because it affirms the dignity and respects the agency of those who will be involved in the study. (p. 1161) we agreed on anonymity of the participants, hence no individual identities were divulged and pseudonyms were used. the teachers were beneficiaries of the study. they were inducted into the dynamics of what it means to learn and to assess mathematical processes with a view to teaching and assessing their own learners. quality criteria prolonged engagement (semester), persistent observation, ongoing probing during a number of whole-class discussions, peer debriefing and member checks provided sufficient opportunity to hear the teachers’ voices, which contributed to establishing the credibility of this study (bitsch, 2005; guba & lincoln, 1989). recursive discussions with the teachers and sufficient descriptive data added to the confirmability and transferability of this study (guba & lincoln, 1989). findings and discussions this section is organised according to the nctm’s (2000) process standards of representation, reasoning and proof. the latter incorporates kilpatrick et al.’s (2001) notion of adaptive reasoning. each of the two process standards is further organised in relation to the teachers’ own learning of process skills and into assessment of their learners in this. representation representation as a process standard is attained when a learner can demonstrate the ability to: create and use representations to organise, record, and communicate mathematical ideas. select, apply, and translate among mathematical representations to solve problems. use representations to model and interpret physical, social, and mathematical phenomena. the activities that the participants engaged with in both data sets provided opportunities for exposure to all aspects of representation. the given pattern that the participants engaged with in the first data set is unique in the sense that the cubes were organised in a specific way to make the pyramid. the number of cubes that are required to construct any other pyramid can be found using a variety of methods. the pyramids can be dismantled and the individual cubes counted, cubes in a layer could be counted and subtotals added, the structure could be analysed and the counting done using the outcomes of that structural analysis, etcetera. the structural analysis of the pattern has the potential to lead to techniques that preserve mathematical ideas among the various representations. only two methods became conspicuous from the teachers’ responses: counting cubes per layer of the pyramid, and counting cubes per layer and adding the subtotals. in each of these approaches we analyse the teachers’ own learning of the mathematics process of representation and their envisaged assessment of the process. cubes per layer seven respondents limited their calculations of the number of cubes in a particular layer of the pyramid. while the activity clearly asks for the number of cubes needed to build any pyramid similar to the one given, the participants interpreted that to require a number of cubes in any layer of a pyramid. responses within this method also showed minor differences in terms of the additional techniques used. the teachers’ own learning in using only layers: these are how three of the seven teachers explained their work: mkk: a general formula should be formulated that satisfies any number of cubes in each layer. i need a strategy to enable me to find the correct formula. mak: i extended the pyramid by extra layer. write down the number of cubes in each layer of the pyramid. find the relationship between the number of cubes. mln: the first pyramid has 1 cube the second 1 + 4 = 5 the third 5 + 8 = 13 the fourth 13 + 12 = 25 the number of cubes in the previous pyramid plus the pyramid number times 4 gives the number of cubes in the next pyramid, i.e. for pyramid 3, the total cubes will be t3 = 2(3)2 + 2(3) + 1  = 25 the approaches fell short of addressing the total number of cubes required to build any pyramid. the numerical patterns generated from the approaches were a reflection of the pictorial patterns of the layers in a pyramid and not pictorial patterns of the pyramids themselves. once the numerical patterns were generated, the respondents applied three different techniques to arrive at the generalisation. the first technique (1 response) could be regarded as trial and error, as the generalisation was first presented and a few items were tried out. there is no clear evidence as to how such a generalisation was arrived at. for example, mln had tn = 2n2 + 2n + 1 as the general formula. mln used n – 1 to compute the value for n. the value that is calculated as t3 is for the fourth pyramid. however, it is presented as the value for pyramid three. that is clearly a misrepresentation that defeats the purpose of harnessing this process skill. with the second technique (2 responses), generalisation was arrived at through the use of ‘met before’ techniques of finding the differences between consecutive terms in the numerical patterns until a constant difference is found. it was mak and mkk who used the technique, arriving at the conclusion that tn = 2n2 – 2n + 1. while this generalisation is a true generalisation of the numerical pattern of the number of cubes in a layer, this representation has no structural relevance to the pictorial representation of the layer. the manipulation of the numerical patterns did not mimic the concrete manipulation of the pictorial representation of the layers. using the third technique (2 responses), generalisation was also given as tn = 2n2 – 2n + 1. this outcome emanates from a different approach in analysing the numerical pattern. each term in the sequence is seen as a sum of two consecutive squares – the square of n and that of (n – 1). stt and mmw provide examples of how that was arrived at. stt: make a sequence of squaring the natural numbers: 1; 4; 9; 16; 25 etc. add the square numbers starting from the second term (e.g. 4 + 1; 9 + 4; …). this means i squared the level number and add level plus one squared (e.g. 12 + (1 – 1)2 = 1; 22 + (2 – 1)2 = 5). mmw: independent 1 2 3 4 … 10 n dependent 1 5 13 25 … ? [here mmw shows how entries in the table were calculated.] 12 22 32 42 52 1 4 9 16 25 + 1 + 4 + 9 + 16 l2 + (l − 1)2 = l2 + l2 − 2l + 1 = 2l2 − 2l + 1 the technique deployed by both stt and mmw can be regarded as sums of two consecutive squares. expressing the generalisation in this way allows for a closer review of the pictorial representations of the layers of the pyramid with a view of determining what that translates into. the model allows for the interpretation of physical mathematical phenomena, a core aspect of representation. the last two participants of the seven (mst and kas) in this category had insufficient responses for what the activity had asked for. however, the opportunity to learn about the processes was not adversely affected. the issue is the depth of that learning and how that was translated into the designed assessment criteria in the context of representation. the teachers’ approach to assessment of learning: the focus in this section falls on five of the seven participants whose responses were regarded as sufficient. while mln had a completely irrelevant response (no connection between the response and item 5), the other four teachers had similar responses. two issues were targeted for assessment and those were the ability to recognise the pattern and the ability to generalise the pattern. furthermore, the expectation of the four teachers in terms of the two criteria closely resembles how they dealt with the problem in item 5. here are how the four teachers expressed performance at the highest level in relation to the two criteria. mkk: able to recognise observable, hidden and underneath cubes to extend the pattern show outstanding understanding of relating number of cubes and layers stt: ability to recognise cubes observable and hidden show a complete understanding mak: all occurrences mentioned and logically done create a pattern using the observable, behind and underneath cubes and able to generalise the pattern mmw: ability to: recognise cubes, hidden and those underneath and could extend the pattern derive a formula the responses were, to a limited extent, in line with one of the three aspects of representation: ‘create and use representations to organise, record, and communicate mathematical ideas’. the limitation of the responses is in relation to the amount of time spent on the manipulation of the way the layers are structured. in responding to item 5, these teachers were content with the numbers without checking whether those numbers adequately communicated the structure of the layers. the drawing of the structures of the layers, analysing the structures, and subsequently recording the observations are the actions that make representation a process standard. however, there is no clear evidence that an in-depth attempt was made in that regard. mak’s expectation that all occurrences should be mentioned and logically done can be tricky if considered out of context. at the lower levels he expects one, two, or three occurrences, while remaining unclear about what these occurrences are. however, if one reflects back on mak’s own engagement with the activity, these occurrences could be the different layers of the pyramid. the responses in relation to generalisation also show the participants’ obsession with symbolic representation that has no direct relation with the structural aspects of what is represented. it is not immediately clear what mkk and stt refer to by understanding, but that could be in line with what mmw regards as a formula that will help find the number of cubes per layer. approached in this way, the opportunity for translation among mathematical representations is very limited. again, the appreciation of the generalisation as a model for the pyramids is lost and therefore the opportunity to fully develop representation as a process standard is limited. cubes per layer and the totals for the pyramid three participants fell in this category. they first calculated the number of cubes per layer and then added the subtotals to find the required number of cubes for any pyramid. the teachers’ own learning in using layers and pyramid totals: all three participants arrived at the generalisation of . the conclusion was reached using similar approaches, with mll hinting at an additional strategy. ckg: consider the number of layers per pyramid and number of cubes per layer. number of cubes per pyramid is: 1; (1 + 5 = 6); (1 + 5 + 13 = 19); (1 + 5 + 13 + 25 = 44); (1 + 5 + 13 + 25 + 41 = 85) 1; 6; 19; 44; 85; … first difference: 5; 13; 25; 41 – not constant second difference: 8; 12; 16; 20 – not constant third difference: 12 – 8 = 4; 16 – 12 = 4; 20 – 16 = 4 is constant, then the pattern is cubic hmj: write the pattern. find the differences … first differences not common means – not a linear pattern, second differences not common – is not a quadratic pattern, the third difference is common therefore is a cubic pattern. first difference = 5; 13; 25; 41; 61; 85; 113; 145; 181 second difference = 8; 12; 16; 20; 24; 28; 32; 36 third difference = 4 mll: i used the conjecture i formulated in question 3 as basis for generalisation. number of cubes per layer form a quadratic pattern, thus the rule is of the second degree. squared the term number and multiplied it by two, subtracted different numbers for different layers. different numbers i subtracted formed a linear pattern which enabled me to generalise the number of cubes per layer. alternatively: total number of cubes needed to build pyramids with layers 1 to 5 will be 1; 6; 19; 44; 85. differences between consecutive terms: from prior knowledge, a pattern with a constant difference yields a linear pattern, and with second constant difference yields a quadratic pattern. the one with a third constant difference yields a cubic pattern: tn = an3 + bn2 + cn + d, the challenge is to find the values of a, b, c, and d. the three participants clearly worked towards establishing the number of cubes in a pyramid. counting the cubes in each layer was considered as a step in that process. the need for justification for manipulation of the numerical patterns is evident in the responses. the implication of those manipulations on the structure of the layers is not, however, reflected in the way the subtotals were found, especially in ckg’s and hmj’s responses. mll hinted at an attempt to analyse the structure of the layers without concluding that aspect. meanwhile, all the three responses show that once the numerical sequence was established, the general strategy of determining the differences among consecutive terms was used to arrive at a conclusion that the general pattern is a cubic one. that strategy was then correctly used to arrive at the generalisation of . the teachers’ approach to assessment of learning: the three participants, while they generally highlighted both the ability to generate a pattern and generalise it, placed different emphasis on the two aspects. the expected performance at the highest levels in relation to the given criteria was as follows: ckg: correctly count front, behind, hidden cubes and correctly draw next pyramid correct manipulation of numbers, variables and operations and make a relevant conclusion hmj: use front, back and underneath to extend pattern able to generalise correctly, logically and appropriately mll: total cubes per layer and per pyramid are correct. correct description of the patterns in words or using conjectures or any mathematical calculation, for both cubes per layer and per pyramid correct generalisation for both the number of cubes per layer and per pyramid it was clear in all the responses that there was more than generalisation that was expected. hmj wanted to see the logic, ckg wanted to see the manipulations of numbers and variables and the relevance of that in the conclusion, while mll wanted descriptions or conjectures for both the layers and the totals. if strictly and consistently applied, these assessment rubrics could encourage the realisation of the core aspects of representation. however, in their current format, it is not clear that navigation between multiple representations is being encouraged. it is only when we take a look at how mll facilitated learning in his classroom that this aspect became clearer. mll’s facilitation of the matchsticks activity in engaging with the assigned activity to construct a series of chained squares, their associated number patterns and the generalisation thereof, three groups of learners in mll’s classrooms used different approaches and representations. initially, two patterns were generated. however, at the end there were three as a result of the amendment of one of them. three distinct representations from two groups of learners were observed from data set 2. the associated conversations also articulate the aspects of representation. the two groups presented these drawings on the board. this was the drawing for group 1: this was the drawing for group 2 and group 3: jahi: [group 1] each square required four matchsticks and for every additional square four matchsticks were required. 16 squares: 16 × 4 = 64 matchsticks 40 squares: 40 × 4 = 160 matchsticks 72 squares: 72 × 4 = 288 matchsticks any number of squares x × 4 jahi: [saw how other learners frowned at the presentation, turned to one of them.] bongi, what’s wrong? bongi: are your drawings forming a chain? jahi: [looked at the drawings, turned to bongi and then at her group silently asking for assistance.] kabo: [from group 1 walking towards the board.] oh, a chain, they must be joined together. but it will not make a difference on our answer. [redrew this amended drawing.] group 2 and group 3 had the same series of matchstick squares, with same pattern and conjectures. however, group 3 could not generalise the pattern and group 2’s generalisation did not show any relationship with their conjecture. thalan presented group 2’s responses. thalan: the first square required four matchsticks and for every additional square, three match-sticks were required. 16 squares: 4 + 3 × 15 = 49 match-sticks 40 squares: 4 + 3 × 39 = 121 match-sticks 72 squares: 4 + 3 × 71 = 217 match-sticks any number of squares: x × 3 + 1 [he was questioned why the 4 disappeared in the generalisation and where the 1 came from. this had an effect on the group as thalan changed the presentation and wrote for 16 squares: 16 × 3 + 1 = 49] rof: [from group 3] i have a different general rule. for 16 squares, they needed 4 + 3 × 15 = 49 matchsticks and 15 which is multiplied by 3 is one less than 16. so the general rule is 4 + 3(x – 1) [wrote it on the board]. therefore, for any number x there will be 4 + 3(x – 1) matchsticks. the learners agreed with rof that his general rule was working and was observable in the conjecture. but the question raised was why there were two different generalisations that are both working. it was at this stage that the teacher, mll, asked the learners to simplify the expression and see what would happen. there was much excitement when the expression reduced to the other group’s general rule of x × 3 + 1. in mll’s classroom activity, the learners’ descriptions, drawings, their general rules and numerical representations were all being interrogated with a view to establish direct connections among them. the 4 in the general formula was expected to be in the numerical representation (conjecture) and in the actual drawings. the learners were not only interested in the final general rule but clear translation among the different representations. the elements of the omnidirectional value in engaging with representations as a mathematics process were becoming evident. these kinds of learning interactions that we observed in mll’s class take time to inculcate. the classroom learning environment allowed for demonstration of knowledge and skills commensurate with mll’s assessment criteria (recognition, description in words or using conjectures or any mathematical calculation, and generalisation of the number pattern). reflective thoughts on representation at this stage of the study, we observed that, when faced with an authentic mathematical problem that requires different facets of representation, teachers tend to rush into generation of numerical representations. this stage is then followed by manipulations with a view to deriving symbolic representations. the process is almost exclusively unidirectional and, thereby, limits the opportunity for the richness of each representation’s contribution in developing insights into other representations. reasoning and proof reasoning and proof as a mathematics process standard requires of learners to: recognise reasoning and proof as fundamental aspects of mathematics make and investigate mathematical conjectures develop and evaluate mathematical arguments and proofs select and use various types of reasoning and methods of proof. during reasoning and proof, viewed from adaptive reasoning as a strand of mathematical proficiency, learners should develop capacity for logical thought, explanation, and justification. all of these traits were observable in almost all the responses in both data sets, although to varying degrees. the activity itself was designed to facilitate the learning of this process strand. for the data set that involved teachers (see figure 1), item 4 required that they explain how item 3 was resolved while item 6 required critical reflection on how item 5 was resolved. the responses to the latter item were used earlier. for the current process standard, we use responses to item 3 and item 4. one of the 10 participants (mst) gave a very brief response to the two activity items indicating that he used number series to arrive at an answer of 186 (total number of cubes). while 186 is incorrect, the lack of richer explanation made it difficult to establish his thought process. the responses of the other nine participants fell into three categories, viz. numerical approaches, numerical and symbolic approaches, and numerical, symbolic and pictorial approaches. numerical approaches the numerical perspective of mathematics is very powerful as it constitutes the elementary nature of the discipline. our interest at this point is how the participants’ own learning, which is dominated by numerical approaches, translates into the assessment of learners. teachers’ own learning in using numerical approaches: there were five participants who used a numerical approach. they used the existence of the numerical pattern as a justification for their response to item 3. the strategy involved the establishment of the total number of cubes per layer, and subsequently adding those numbers to find the total for the pyramid. mkk’s and stt’s responses are used to demonstrate the strategy. mkk 3. 670 cubes  identify the number of observable and hidden cubes in each layer. no. of layers 1 2 3 4 … 9 10 cubes/layer 1 5 13 25 … 145 181 total cubes per pyramid 1 6 19 44 … 489 670 alternatively: 12 = 1 22 = 4 + 12 = 5 32 = 9 + 22 = 13 42 = 16 + 32 = 25, etc. stt 3. the number of cubes needed to build a pyramid that is 10 cubes high is 670. level cubes/level total cubes/pyramid 1 1 1 2 5 6 3 13 19 ⋮ ⋮ ⋮ 10 181 670 for the total cubes per pyramid (e.g. 1 + 5 = 6, 1 + 5 + 13 = 19) i calculated the differences between successive terms and the common difference was found in the second difference. i used the common difference to generate the sequence for the number of cubes per level. i cumulated the generated pattern to get the total cubes for pyramid that is 10 levels high. both mkk and stt went into detail to demonstrate how they arrived at their answer to item 3. that, on its own, demonstrated that the participants do value reasoning and proof as important aspects of mathematics. it is evident that more work was also done elsewhere before the results were captured in their tables. the first step of their response was to generate the number of cubes per layer and later add those to get the number for the pyramid. that is, the participants established and reasoned for their response by actually calculating all the previous sums until the 10th. the power of numerical pattern was preferred. the only structural property of the pictorial pattern that was used is the layered nature of the pyramid. the structure of the layers themselves did not feature. we also observed that mkk did pursue an alternative approach to establish the answer; however, that was also numerical. while both participants achieved their goal of determining the actual number of cubes required to build the required pattern, their limited focus on numerical approach constrains the opportunity of encountering the varied ways through which the answer can be established. the critical aspect of mathematics is not its ability to resolve a problem but its broadness in facilitating such a resolution. justifying and proving the solution of the given problem using a single algebraic strategy of numerical pattern is less convincing as it falls short of room for a selection and use of various types of reasoning and methods of proof. the teachers’ approach to assessment of learning: indicators on what the participants valued in relation to assessment of learning are derived from their responses to item 7 and item 8 of the given activity. we again use mkk’s and stt’s responses. they both identified similar skills to be assessed with one difference. ability to: generate a number pattern. sequentially/logically record data. find relationship of layer and total cubes per layer. give a convincing explanation in a narrative/mathematical form. the only difference was that mkk included ‘interpretation of word problems’ while stt had ‘ability to generalise’. as was observed with their own learning, the two participants value the generation of number pattern and relating that to layers and total number of cubes. the reasoning and proof is limited to ‘giving convincing explanation’. while this acknowledges the significance of reasoning and proof, it falls short of encouraging the varied nature of how that could be undertaken. numerical and symbolic approaches by numerical and symbolic approaches, we refer to a scenario in which the strategy is considered credible when a symbolic generalisation is established. a numerical pattern is generated and analysed with a view to generalise it and only then is it applied to resolve prevailing problems. teachers’ own learning in using numerical and symbolic approaches: there were three participants in this category. in addition to what the participants in the previous category did, this group went further to establish the generalisation for number of cubes per level and in other cases also for the pyramid. the generalisability of the number of cubes was considered reason enough to accept the response to item 3. we use kas’s and mll’s responses to demonstrate the strategy. kas no. of layers 1 2 3 4 5 6 7 8 9 10 no. of cubes per layer 1 5 13 25 41 61 85 113 145 181 total cubes per pyramid 1 6 19 44 85 146 231 344 489 670 t1 = 12;  t2 = 22 + 1;  t3 = 32 + 4;  t4 = 42 + 9;  t5 = 52 + 16;  t10 = 102 + 81 = 100 + 81 = 181 square the number of layer and add the square of the previous number of layer. mll number of layer cubes/layer total 1 1 = 2(1)2 – 1  = 2(1)2 – 2(1) + 1 2 5 = 2(2)2 – 3  = 2(2)2 – 2(2) + 1 3 13 = 2(3)2 – 5   = 2(3)2 – 2(3) + 1 4 25 = 2(4)2 – 7   = 2(4)2 – 2(4) + 1 10 2(10)2 – 2(10) + 1 = 181 i created a conjecture that relates the number of layers, the numbers of cubes per layer and per pyramid. the conjecture holds for the first 5 terms of the pattern, hence i applied it to a pyramid 10 layers high. in kas’s case, the generalisation was with regard to the number of cubes per layer. while the origin of the idea is not clear, kas used the sum of squares to arrive at the number of cubes per layer. the number of cubes for a pyramid is not problematised as it is the sum of cubes per layer. mll worked with a conjecture for both the layers and the pyramid. once it was established that the conjecture worked for a few items in the pattern, the idea was extended to establish the answer for item 3. the three participants in this group demonstrated their appreciation of reason and proof as valuable aspects of mathematics. the need to establish and test the conjectures was attempted which made their responses qualitatively different from those in the first category. however, more still needs to be done to address other aspects of reasoning and proof as the process standard requires. the multiplicity of methods of proof still need to be demonstrated. it is in the category that follows where the emergence of the idea is observable. the teachers’ approach to assessment of learning: for the sake of continuity, kas’s and mll’s responses are used. kas: skills to assess: sequential/logical recording of data ability to relate numbers interpretation generalisation, and substitution mll: could assess ability to: count the number of cubes per layer and per pyramid. recognise a number pattern from the number of cubes per layer and per pyramid. extend the number pattern by both drawing and working with the pattern recognised. describe the pattern recognised on cubes per layer and per pyramid. generalise the described pattern for any pyramid. verify the generalised rule for the patterns. evident in the two responses is that generalisation is not seen as an end in itself. there is clear expectation that the learning should include the application of the generalisation. furthermore, mll expects the ability to carry out the extensions not only numerically but also pictorially. the latter requires exploration and understanding of the structural aspects of the pictorial patterns. what remains unclear, however, is the ability of learners to relate structural properties of the pyramids with the numerical or symbolic patterns. this shortfall limits the potential for meaningful interpretation of the various strategies that can be used to address the problem. numerical, symbolic, and pictorial approach approaching learning from a variety of perspectives has a potential for triangulation, deeper and meaningful learning. in many respects, this is closer to the ideals of reasoning and proof as a mathematical process. teacher’s own learning with respect to numerical, symbolic and pictorial approach: only one participant, mln, unpacked and analysed the pictorial pattern and thus attempted to relate the structural properties of the layers and their numerical patterns. this is an important aspect in the development of reasoning and proof as a mathematics process standard. analysis of pictorial patterns allows for meaning making and has the potential for multiplicity of reasoning and methods of proof. 3. 1st top row = 1 cube 2nd top row = 5 cubes 3rd top row = 13 cubes ⋮ 10th top row = 181 cubes number of cubes needed to build a pyramid that is 10 cubes high is: 1 + 5 + 13 + 25 + 41 + 61 + 85 + 113 + 145 + 181 = 670 5. the number of cubes in the given pyramid was obtained by first drawing the view of each row of the pyramid and then add them. … number of cubes for a pyramid 10 cubes high was obtained by counting the number of cubes per row and then find the sum. in this response it is clear how the numbers of cubes per layer were obtained. the relationship between the numerical and the pictorial patterns is clearer as compared to the previous cases. to an extent, the explanation is more convincing as the reorientation of the pictorial pattern reveals the hidden cubes. as an attempt to justify the response to the activity item 3, mln’s presentation had fewer uncertainties as both the pictorial and numerical patterns complemented each other. this double method of representation advanced the opportunity to acquire reasoning and proof as a mathematics process standard. teacher’s approach to assessment of learning: mln’s approach to assessment was distinct as compared to other participants. instead of the usual expectation of generating a pattern and ability to generalise, she emphasised geometric and problem-solving skills. this is how she responded to the task list of issues that should be considered for assessment. mln: skills to assess drawing skills: understanding how to draw the figure and the actual drawing skill arrangements: area models and set models counting: use of patterns in counting problem-solving: understanding of the problem; taking risks; and justifying results there is an obvious attempt to integrate geometric and algebraic skills in the assessment. the inclusion of problem-solving skills also provides for further opportunities in which individual learners could bring additional skills into the problem. if properly applied, then this approach has the potential to encompass all the attributes of ‘reasoning and proof’. mln’s invested energies in engaging with the given activity congruently translated into his expectations of what the learners should be able to do. reflective thoughts on reasoning and proof the development of attributes for reasoning and proof require a culture of multiplicity in the resolution of problems. both teachers and learners need to appreciate that it is not necessarily the final answer that is essential but the variety of ways of accomplishing the task matters. we have observed that while the participants demonstrated an appreciation for reasoning and proof, the preoccupation with the final answer limited explorations of other forms of proofs. the practice translates into limited scope of assessment. conclusion and implications mathematics process standards, as covered in this study, are broad and long term in nature. in engaging with them we acknowledge the need to take into account the bigger picture (maoto, masha, & maphutha, 2016) that underpins each of those standards. the learning activities presented as part of this article were not meant to address these mathematical processes fully, but to set in motion a concerted effort to expose the participants to such mathematics processes. in closing, we revert back to the questions raised in the study. teachers’ challenges in learning mathematical processes the presentation of the pictorial pattern that was used to engage the teachers offered multiple scenarios of engagement in the process of deriving symbolic representations. in addition to the various forms of experiences and interpretations that individual participants brought to the activity, the outcomes of the engagements were plenty. however, what became evident is the challenge of rushing the process towards numerical and symbolic representations. once the numbers were derived from the pictorial representation, different levels of manipulation of the numbers became the main focus away from the structural aspects of the original figural pattern. for instance, while the activity item clearly asks for the number of cubes needed to build any pyramid similar to the one given, we observed 7 of the 10 participants interpreting and limiting themselves to calculating the number of cubes in a particular layer of the pyramid. this bottleneck in the generated pattern might be due to the perception of mathematics as being about numbers. the unidirectional approach that they used made it impossible for them to realise the discrepancies in their responses. meanwhile, the other three participants remained focused and continued to find the totals for the pyramid as required; they missed the bigger picture in resolving the problem due to their confinement to numerical and symbolic patterns. justifying and proving a solution through a single strategy lacks rigour that is required for one to appreciate reasoning and proof as a mathematics process. in our case the function type determined by the non-linear nature of the task challenged the participants’ reasoning capabilities. while it was easier for them to generalise a linear pattern (generated from counting the cubes per layer), they had difficulty in generalising what emerged as a non-linear pattern. this confirms jurdak and mouhayar’s (2014) view that the development of reasoning in pattern generalisation is mainly due to experience rather than maturation. the slight moderation of mln’s attempt to integrate the pictorial and numerical approaches better presented a double method of representation that complemented each other and thus advanced the opportunity to acquire reasoning and proof as a mathematics process standard (see figure 3). figure 3: pyramid layers with cubes coloured differently for emphasis. mln’s unpacking and analysis of the pictorial pattern provided her an opportunity to develop capacity for logical thought, explanation and justification. investing more time in manipulating the different forms of representations, especially the pictorial or concrete representations, does contribute to richer insights into the ideas and concepts at play. justification of the strategy is in its consistency of success in resolving a problem. while this provides some comfort, it is still not a replacement for a corroboration that is arrived at through varied forms of reasoning and proof. authentic learning of mathematical processes requires that all representations be equally explored as that would lead to a smooth omnidirectional approach to learning. that kind of learning could be facilitated by teachers who have excellent translation abilities among multiple representations so that they could assist their learners to access and to comprehend abstract mathematical ideas. teachers’ learning and assessing of mathematical processes to some extent, we observed that when complex knowledge and skills are involved, teachers’ own approaches to learning do reflect depth of their expectations of what would be learned. it was mll and mln who presented relatively better quality engagements in their learning and that led to qualitatively better expectations from their learners. however, what we observed from mll’s actual implementation of his newly acquired skills in a classroom offers some hope. learners’ ways of engaging with the activity offered more and better opportunities to acquire representation as a process skill. links among different representations were being interrogated much more than mll did in his own learning. the manipulatives (matchsticks) enhanced learners’ learning as they interrogated and established connections among their multiple representations. this aligns to uribe-flórez and wilkins’s (2017) findings that the use of manipulatives encourages multiple representations and has a long-term effect on students’ learning as opposed to immediate achievement. we observed the value of an omnidirectional approach to the development of mathematics processes. more importantly, it emerged that meaningful assessment is learned in the context of the concepts that should be assessed. deeper understanding of the concept opens possibilities for varied and meaningful assessment. interventions with regard to the enhancement of these complex processes should involve teachers’ actual engagements in and reflections on similar learning. acknowledgements competing interests the authors declare that they have no financial or personal relationships that may have inappropriately influenced them in writing this article. authors’ contributions s.m. was the project leader and facilitated lessons for the assessment module towards a bachelor in education honours degree in mathematics education at the university. l.m. collected and analysed data for the second data set. s.m. conceptualised the article and wrote the draft manuscript. k.m. played a role of a critical reader, improved on the logical flow of ideas and filled in the gaps that existed, thus improving the quality of the article. s.m. and k.m. then engaged in reconceptualising and critically analysed the two data sets to produce the final draft of the paper. references bitsch, v. 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(2017). manipulative use and elementary school students’ mathematics learning. international journal of science and mathematics education, 15(8), 1541–1557. https://doi.org/10.1007/s10763-016-9757-3 abstract introduction literature review context methodology findings and discussion conclusion recommendations acknowledgements references about the author(s) benita nel mathematics education department, university of south africa, south africa school of education, university of the witwatersrand, south africa kakoma luneta faculty of education, university of johannesburg, south africa citation nel, b., & luneta, k. (2017). mentoring as professional development intervention for mathematics teachers: a south african perspective. pythagoras, 38(1), a343. https://doi.org/10.4102/pythagoras.v38i1.343 original research mentoring as professional development intervention for mathematics teachers: a south african perspective benita nel, kakoma luneta received: 09 june 2016; accepted: 19 june 2017; published: 22 sept. 2017 copyright: © 2017. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract only a small percentage of mathematics teacher professional development activities in south africa include pedagogy and instruction approaches required to teach the content. in the light of the poor mathematics results, it is of pertinent importance that teachers need to be developed in terms of the mathematics they teach and the pedagogical component of it. a teachers’ professional development programme that used mentoring as one of the interventions was investigated where the mentor conducted classroom observations and had post–observation conferences with the teachers. the data from the observations, post observation conferences and interviews of both the mentors and the mathematics teachers was used to inform the off-site workshops on mathematics content and the instructional skills required to effectively teach the content. mentoring supported the teachers in terms of mathematics content and instruction, team teaching and lesson preparation. the teachers’ individual content and instructional needs were also assessed and supported. the study revealed that mentoring that takes cognisance of teachers’ content and instructional needs enhanced their lesson preparation and understanding of mathematics as well as the skills required to teach it effectively. the study recommends mentoring as an effective intervention in professional development programmes especially when it is informed by the teachers’ instructional and content needs. introduction in the south african context, mentoring in professional development programmes has the potential to equip teachers in mediating teaching and learning (guskey, 2000). in this article, we argue that one key factor in ensuring effective professional development, specifically in terms of pedagogy and instruction, is to invite mentors into the classroom to support teachers to rethink the teaching and learning taking place in order to devise appropriate interventions customised to the needs of teachers. fundamentally, this means authentic and accurate needs analysis which can direct appropriate interventions, either one-on-one, or in out-of-class workshops, linked to the specific needs of the teachers (luneta, 2012). only a small percentage of teacher professional development initiatives in south africa include mathematics pedagogy and instruction (reddy, 2006). it can therefore be concluded that most of the professional development (pd) revolves around content, curriculum and assessment. a concern can then be expressed that most pd initiatives are not focusing on mathematical content and instructional challenges that individual teachers experience in classrooms and it is this shortfall that probably perpetuates the poor content delivery and performance of south african learners in mathematics internationally. the introduction of mentoring, which included on-site classroom support to practising teachers (sibanda & jawahar, 2012), allowed for constructive feedback on how to improve their teaching (edwards, 1998; luneta, 2012). this constructive feedback from the mentor should also be supported by a conducive learning environment where effective communication takes place and the mentee is treated as a colleague (ambrosetti & dekkers, 2010). mentoring as pd with on-site intervention is more effective than interventions detached from the classroom (de clercq & phiri, 2013). nel (2015) asserts that interventions that are not linked to the classroom and the needs of individual teachers have limited impact on pd that is aimed at teachers’ mathematics content knowledge and the instructional skills required to teach the subject effectively. this article is based on the larger study of nel (2015) which evaluated a mathematics pd programme initiative which included mentoring as an intervention that was implemented in a rural mining district in south africa. the programme sought to address gaps relating to the teachers’ instructional skills as well as some of the mathematics content knowledge that was required. the research question was: how can mentoring as a pd intervention for mathematics teachers in south africa support teachers in their development? literature review south africa is faced with a challenge of redressing inequalities of the past (graven, 2014) where a significant number of teachers were trained in underdeveloped teacher training colleges of education created by the apartheid government (gordon, 2009). although the poor performance of learners in mathematics has been high on south africa’s agenda for some time now (centre for development and enterprise, 2013; kriek & grayson, 2009), no significant improvement has been noted in learners’ mathematical knowledge outputs. the quality of teachers’ instruction has a direct impact on their learners’ achievements (mckinsey & company, 2007; venkat & spaull, 2015), so in teacher pd programmes the emphasis should be on improving the instruction of the individual teacher. however, this cannot be done with a one-size-fits-all approach, as in off-site workshops detached from the classroom. luneta (2012) points out that in-service teacher education programmes in south africa predominantly make use of off-site approaches, especially continuous professional development of mathematics teachers. according to de clercq and phiri (2013, p. 80), however, often off-site teacher development is ‘rarely able to make teachers change paradigm and improve their classroom practice’. furthermore, the inability of the subject advisors to supervise teachers’ performance (de kadt, 2010) adds to the improbable ability of most pd programme to change teachers’ practices. there are also claims that many pd programmes are not geared to ‘address the context in which teachers are operating in the classrooms’ (islam, 2012, p. 20). interventions that are directly linked to the classroom situation as well as the specific needs of the individual should be implemented to ensure effective teacher development and a possible improvement in learner performance, especially in mathematics. santamaria (2009) asserts that teachers should make instructional decisions that suit their learners’ needs. the new curricula introduced in south africa also required a move away from teacher-centredness to a more learner-centred approach where the teacher acts more as a facilitator in the classroom. the quality of learning can also be enhanced if we improve the interaction between teachers and learners in the classroom (mckinsey & company, 2007). those teachers, referred to earlier, who were trained in the teacher training colleges, were exposed to the use of rigid rote-learning methods of teaching, as well as an educational philosophy and pedagogy that did not encourage learners in the class to critically analyse or raise questions pertaining to the content or the instructions they received (gardiner, 2008). that style of teaching did not favour spontaneous interaction and relevant communication between teachers and learners in mathematics classrooms. the current move worldwide is to have mathematics classrooms that are informed by teachers that are grounded in the mathematics they teach as well as the instructional skills required to do so (schoenfeld & floden, 2014; sullivan, 2011). teachers with such skills would exhibit kilpatrick, swafford and findell’s (2001) five desirable strands of good mathematics teachers and learners’, schoenfeld and floden’s (2014) framework of teaching for robust understanding in mathematics (tru math) and stein, engle, smith and hughes’s (2008) five key pedagogical principles model for mathematics effective classrooms. all of the provinces in south africa have schools situated in rural areas with their unique conditions and challenges, where teachers work under solitary conditions and without much human support (gardiner, 2008). the pd programme under investigation was also implemented in a rural setting where a shortage of qualified teachers (department of education, 2005; islam, 2012) was rampant. the situation was particularly severe with regard to qualified mathematics, science and english teachers (masinire, maringe & nkambule, 2014). these three subject areas are interlinked such that when a learner has a challenge with reading and understanding english as language, it often has a negative effect on understanding mathematics and science. a challenge particularly in rural high schools is the lack of teachers’ subject content knowledge (human sciences research council, 2005) and limited resources also contribute to teachers’ low morale (gordon, 2009). although workshops are offered by training institutions to teachers in rural areas, the transfer of new ideas into teachers’ classroom practice has often been questionable (gardiner, 2008). in addition, other programmes like mentoring could ensure that what was learnt in pd is transferred and modified to suit classroom practice. mentoring refers to a professional relationship in which an experienced person, the mentor, assists another one (the mentee) in developing specific skills and knowledge that will enhance the less experienced person’s professional and personal growth (aspfors & fransson, 2015). in this study, the mentor was responsible for a number of mathematics teachers in rural schools which were geographically scattered. we have argued in this research that mentoring is an intervention that can be located in the classroom and be customised to the specific needs of the individual teacher. it entails a relationship between a more experienced person (mentor) and a less experienced individual (mentee) (shulman & sato, 2006) with the intent of growing and developing the mentee (allen, finkelstein & poteet, 2009). this type of teacher development has the potential of being on site, needs driven and context specific. such an undertaking could mean that the incorporation of mentoring in a pd programme can be a solution to the generic training of the past. guskey (2000) agrees that the inclusion of mentoring, pedagogy and instruction in a pd programme increases the potential of yielding better equipped teachers. on-site classroom support yields more significant success than externally imposed expectations (onchwari & keengwe, 2008). the mentor can give constructive feedback to the mentee and incorporate team-teaching or co-teaching, and in that way demonstrate other ways of instruction developed through reflective practices after a lesson, as well as have a better approach to interacting with learners (badiali & titus, 2010). mentoring thus lends itself to the mentee receiving individual attention and support, and the sharing of knowledge as well as experiences (murray, 2010). good mentoring provides space for self-reflection within the school itself, a strategy that can boost the teachers’ commitments towards their own development (islam, 2012). epistemologically mentoring can counteract workshop and seminars that are detached from classrooms and that do not address unique instructional and content needs of individual mathematics teachers (luneta, 2006). possible challenges with mentoring have always been ineffective implementation, a mismatch between the mentor and the mentee, the lack of commitment to the process, and time and financial constraints as well as inappropriate needs analysis (rhodes, stokes & hampton, 2004). mentoring is also quite labour-intensive and not easy to roll out on a large scale. it is, however, important to contextualise the engagement. according to sullivan (2011), current debates on mentorship and professional development in mathematics education are pointing to the recognition of the critical complementarity of the two constructs and that mentorship should and ought to be addressed as professional development. context this article reports on part of a broader intervention involving mathematics teachers. the mathematics pd programme ran for three years in a particular rural mining district in a province in south africa. the programme used different interventions, namely workshops and mentoring, coupled with classroom visits and cluster meetings, in the quest to develop the teachers’ mathematical competence and performance that addressed gaps within their learners’ mathematical knowledge and skills (khosa, 2010). the topics dealt with in the workshops emanated from the analysis of pre-tests that teachers wrote before the start of the programme, but also through requests from participants as well as classroom observations. the topics included financial mathematics, circle geometry, probability, applications in calculus, exponents, algebraic inequalities, trigonometry and contextualised quadratic number patterns at the further education and training level. mentoring is an intervention that supported the workshops by having scheduled one-on-one sessions between the mentors and the teachers after classroom visits and lesson observations. there were two mentors involved in the programme. the first mentor left at the end of the second year and the second mentor then took over for the remaining year. the participants got along well with the first mentor but not with the second one for reasons unknown to the researcher. an interview was done only with the first mentor as the second mentor passed away soon after the programme, hence the mention of one mentor at times and two at other times. in the classroom visits the mentor determined areas where the individual teacher needed support and these were addressed in the workshops or on a one-on-one basis. during classroom visits the mentor at times also co-taught with the teacher. the classroom visits gave the mentor first-hand insight into the classroom conduct and instructional abilities of the teacher and provided an opportunity for a needs assessment per teacher. workshop topics were then determined from the needs analysis. the cluster meeting did not form part of this study. the different teachers were given pseudonyms to protect their anonymity. they were allocated the following codes to distinguish between the different participants: mk, nk, rb, sb and sr, and the mentor was allocated ak. mk, sb and sr were heads of the mathematics departments of their schools and the other two teachers were post level one teachers. all of them had very full workloads and were teaching two or even more subjects at the time. all the schools are rural schools and very far apart in a mining-dominated region in south africa. methodology this study followed a qualitative approach where events and people involved in the pd programme were scientifically discussed and where thick and rich descriptive data were generated without the use of numerical data (best & kahn, 1998). the specific pd programme under investigation is seen as a case study where in-depth understanding of the programme was achieved through intensive data collection. yin’s (1984, p. 23) definition of a case study was used: ‘a research method that uses empirical inquiry to investigate a contemporary phenomenon within its real-life context’. in this study, the ‘phenomenon’ was mentorship and teachers and mentees’ real-life contexts. the target population was all the teachers in the further education and training band involved in a mathematics pd programme. the five schools involved in the programme were all situated in a rural setting with mostly one teacher teaching both mathematics and science in the further education and training band at their respective schools. one of the schools’ mathematics teachers was not involved in the research as he left the district for another post. at the end, there were five teachers involved in the research as one school had two teachers who took part in the programme. due to the small number of mathematics participants in the programme, all the further education and training mathematics teachers in the four remaining schools took part in the research. primary and secondary data sources were used in the study. the researcher was not the mentor and did lesson observations separate from that of the mentor. the primary data were generated from interviews with the participants and with the mentor teachers using different semi-structured interviews for the teachers and the mentor respectively. reports written by the mentor after every school visit, lesson observation reports done by the mentor, workshop reports, the project proposal and the programme executive manager’s reports (secondary data sources) were among the many data sources. the lesson observation schedule that the researcher compiled and used during class visits (primary data source) was also used as a data source. the different components of the observation schedule included aspects like whether learners were involved in active learning, the way in which the teacher facilitated learning, the type of support given to learners, the correctness of teacher’s content knowledge, whether lesson plans were done and whether the teacher was on schedule in terms of the year programme. the semi-structured, open-ended, face-to-face interviews were used with both the teachers as well as the mentor. all the teachers were asked the same questions on the same topics to make comparisons between the interviews possible (best & kahn, 1998). in order to elicit the teachers’ different perspectives, the open-ended interviews assisted in making these interviews flexible to probe into comments made (hatch, 2002). the interview questions included questions on whether and how they felt they developed as teachers in terms of preparing and conducting lessons, their assessment skills, timeous completion of the syllabus, the role of the mentor and recommendation on how to improve pd programmes. in the compiling of the interview schedules as well as during the data analysis phase, the researcher used a thematic approach to then draft the questions as well as structure the data analysis. the thematic approach, a qualitative data reduction process (grbich, 2006), was used to analyse data. here related data segments were grouped or consolidated around a central concept or category, which can also be referred to as codes. these predetermined categories were derived from the literature review pertaining to effective pd programmes as well as using the programme under investigation’s aims and objectives. the following predetermined categories were used to analyse the data: support through lesson observations, completion of syllabus, teachers’ written lesson preparations, how teachers supported learners in class and learner participation, and homework given to learners. analysis of the mentor reports was also done by using the inductive approach using the same categories as stated earlier. these different data sources provided rich data and aided in data triangulation (krefting, 1991) and added to the trustworthiness of the study (babbie & mouton, 2001; lietz, langer & furman, 2006). ethical considerations ethical clearance was obtained from the provincial department of education as well as the university through which the qualification was done (reference: 2013/iste/30). thereafter, consent was sought from the schools, participating teachers as well as the learners involved. the researcher explained the aims and objectives of the study to the participants and assured them that their participation is voluntary. before signing the consent forms, the participants were also informed that they can withdraw from the study at any time without being penalised in any way and that their identities will be kept anonymous by using pseudonyms in the writing up of the study. they were also informed that confidentiality pertaining to the collected data will be upheld. findings and discussion the findings are presented according to each classification of the data. we used descriptive analysis (ed. maree, 2011) which is ‘the transformation of raw data into a form that will make them easy to understand and interpret; rearranging, ordering, and manipulating data to generate descriptive information’ (zikmund, 2003, p. 1). the raw data from the lesson preparation and observations, and the completion of content, the supporting of learning in the classroom, learner participation, lesson preparations and the validity of homework given to the learners were analysed so as to provide thick descriptions of the events (gall, gall & walter, 1999). support through lesson observations the mentor assisted the participants of the programme in their development as teachers by observing their lessons followed by feedback and reflection sessions. these sessions were integrated with workshops where developmental issues were interrogated. the lesson observations by the mentor revealed areas where teachers needed support and the mentor assisted them in those areas through one-to-one conversations after the observations, co-teaching and workshops on lesson planning. the participants gave insight in the interviews into how they experienced the sessions where the mentor observed their lessons. here for instance is an excerpt of the interview with nk: they came to school, observed the lesson; they even teach. … it was very nice because there were some of the aspects that i was having a problem with. so … but after their visit, i was just fine. even if we go together to the class, i give them [mentors] chance to say something or do something on the board. it was very nice and maybe even with these learners. if they see a different person talking with them, they become better. (nel, 2015, p. 196) the interviewee had a positive experience with the mentor observing his lesson and mentioned that co-teaching also took place, an effective method of supporting teachers in their development. through co-teaching the teachers had the opportunity to see how the mentor taught certain sections and reflected on what methods and strategies could be incorporated in their own teaching. co-teaching allows the mentor to model good practices to the mentee in their own classroom setting (mclaughlin & talbert, 2006). these good practices include different instructional practices which are customised to the mentee within their own classroom setting and which can also reveal to the mentee that these practices are implementable in their rural settings. co-teaching, unlike ideas given to teachers attending off-site one-size-fits-all workshops detached from the teachers’ classroom (de clercq & phiri, 2013), can inspire teachers to try out instructional strategies that they were not used to in the past. one can also gather that the teachers welcomed the input of the mentor in the lesson and gave the mentor opportunities to ‘say something’ or ‘do something on the board’. in this instance, the teacher acknowledged that there were areas where he experienced difficulty in his classroom teaching or practice and the mentor’s intervention added value to his development. this interview excerpt above also alludes to a possible improvement in the learners’ understanding of concepts through the co-teaching. the mentor became part of these classroom interactions and was later not perceived as a stranger by either the teacher or the learners, as stated in an interview with rb: what i’ve learnt about mentoring is that for every lesson that you do, there is a correction for it; there is an improvement – every lesson. there is no 100% lesson. there is an improvement for every lesson because someone who is sitting and watching can improve on your lesson. (nel, 2015, p. 196–197) this quote reveals that mentoring allows for learning through reflective conversations with the mentor (badiali & titus, 2010), and that the teacher realised that there were areas in the lessons where he could improve. edwards (1998) highlights the giving of feedback to the mentee as one of the dominant roles of the mentor. in reaction to the feedback, the teacher could reflect on their teaching and rethink possible changes to it. the teacher also indicated that even if one improves one’s lesson presentation, there are still aspects in a lesson that can be improved. this quote further confirms kilpatrick et al.’s (2001) argument that teachers’ understanding and the rethinking of their own teaching is a continuous process and improvement is never-ending. both previous excerpts also reveal that the mentor’s input was welcomed by the teachers as the assistance was perceived as helpful. this implies a good match between the teachers and the mentor. another participant, sr, mentioned the importance of the feedback after the lesson observation: she was going with us to the classroom to observe the lessons … what i can say was it was effective because when we come from the classroom, we sit down and we discuss the shortcomings of the lesson and other things. (nel, 2015, p. 198) feedback is important. this customised feedback has the potential to sustain teacher development as it is linked to the individual needs of teachers (hattingh, 2009). these regular lesson observations and supportive discussions are some of the pertinent benefits of mentoring (harrison, 2005). constructive feedback also assists the teacher to reflect on their practice in class. the teacher’s choice of the word ‘effective’ also indicates that the teacher experienced the classroom observation sessions as beneficial to their growth as a professional. it needs to be remembered that these are all teachers with big workloads (having more than 30 periods per week) and through this pd what is added is a mentor taking up their limited free periods or their lunch breaks to discuss observed lessons with them. perceiving the input of the mentor in an ‘effective’, that is positive way, implies that the pd initiative added value to these teachers in addition to the efforts from their side. however, in the interview with the mentor, on the question as to what could be done to improve the mentoring, ak revealed that she did not always find sufficient time for the feedback session after the lesson observations: we didn’t have a lot of time to sit and talk to them – the teacher after [the lesson observation]. … i think there should have been more time even if they (the teachers) had to miss a lesson. the right time to talk about a lesson was straight after it. and you need a full period not quickly before they go to the next period. you know it doesn’t work. and just more time because you know they’re quite far apart – the schools you know. (nel, 2015, p. 211) the mentor teachers also observed and recognised the insufficient time for the post observation meetings. the question of time is acknowledged in other studies that assert that the effectiveness of mentoring is often enhanced by the amount of time spent between the mentor and the mentee; however, time constraints are a frequent challenge for mentors and ‘insufficient time can often have an adverse impact on achieving their learning objectives’ (myall, levett-jones & lathlean, 2008, p. 1840). the mentor also highlights the importance of reflection sessions immediately after the lesson observation while the detail of the lesson is still fresh in the minds of both the teacher and the mentor. it is also evident from the interviews that enough time should be set aside for the reflection sessions immediately after the lesson so that it can allow for thorough reflection. this ensures that lesson observations as a tool to develop teachers can be used optimally. however, it is also possible that the extensive distance between the rural schools contributed to the lack of time available after the lesson as the mentor had to rush to the next school for other lesson observations. training of mentor teachers in each school would be the most appropriate solution to the problem. completion of syllabus it is very important for teachers to complete each grade’s syllabus as they are structured and sequenced to support the progression of concepts. it is, however, alarming to find that in south africa there are instances where the mathematics syllabus is not completed in a given year (makgato & mji, 2006), leaving some topics not covered in certain grades. the mentor reports did not indicate much in this category as teachers in general did not update their files on which sections of the syllabus they had already covered. it should also be noted that schools in rural areas are not necessarily monitored regularly by the district officials due to the remote location of these schools (adedeji & alaniyan, 2011). however, the interviews shed some light on this category. teacher sr was initially unable to finish the syllabus within the stipulated year, but gained knowledge about new approaches of how to cover the work. this enabled him to complete the syllabus timeously: it is a problem to finish the work plan because i am having a lot of classes here. … ja, after [the programme] chipped in, then i managed to finish everything … because they just showed us how to teach this topic, the skill of teaching this topic because before we were dwelling much, before [the programme] we were dwelling much on the same topic. this took up a lot of time. but now they have just given us the direction that we just do this, concentrate on this and that and that. so that is how we can cover the syllabus. you see those skills are very, very important from those two people [the two mentors]. they just give you the overall plan, how to attack it. (nel, 2015, p. 184) what became evident through this interview was that the teacher initially did not manage to complete the syllabus for a particular year. this was due to him spending too much time on a particular topic and eventually running out of time later on in the year. the mentoring then assisted him in pertinent areas within particular topics which he needed to emphasise in class to enable learners to grasp concepts. the mentor assisted him in managing his time in class which led to him completing the syllabus within the required time frames. the mentor who came into his classroom could pick up this challenge of the teacher and could support him in such a way that his classroom practice changed. workload also turned out to be an issue as this teacher, besides being the head of department at the school, was also teaching grades 10, 11 and 12 mathematics in addition to physical science for two other grades. this was alluded to when he mentioned in the excerpt that he had ‘a lot of classes’. this teacher had a big teaching load due to the shortage of qualified mathematics teachers at this rural school. so, although this teacher was assisted by the mentoring to complete the syllabus within the required timeframes, i have my reservations if the timeous completion of the syllabus can be sustained if he continues to have this big teaching load. teachers’ lesson preparation written lesson plans describe what is planned to be covered in the classroom (mckay, 2010). individual lessons cannot be successful if they are not planned thoroughly as well as integrated into the medium-term and long-term plans (croft, coggshall, dolan, powers & killion, 2010). therefore, lesson preparation is an important aspect of the teacher’s daily practice. at the start of the programme it was evident that the teachers did not develop formal lesson preparations, though reports indicated that some form of preparation was done from the way lessons were conducted. the lessons were written on rough pieces of paper without clearly specified lesson aims and objectives, instructional approaches, learners’ activities and lesson evaluation strategies. this was also evident in the excerpt of mk on the question of whether the teachers prepared written lesson plans: you know the way we used to do some lesson plans … it’s just only things that maybe we think is right to do. but when the project now came in, we [were taught]: for a lesson preparation you must do one, two, three. that is how a lesson plan must look like. (nel, 2015, p. 175) this teacher reveals that he did not do lesson preparation according to a prescribed template with detailed sections, but merely according to what he thought was correct. through the pd programme he was taught what to include in the lesson preparation. the mentor then followed this up during school visits to guide the teachers on this aspect. sr mentioned: before the project i did not have lesson preparation, but now during the project, they showed us how to prepare mathematics. … so it’s more advisable for any teacher that whenever you go to the class, you must have a lesson, you must have a plan for the particular lesson. (nel, 2015, p. 181) this teacher acknowledged that he did not do lesson preparation before the programme was implemented. this was a teacher who was in the profession for quite a number of years. the challenge, however, is the centrality of lesson preparation to teaching and learning where what will be done in the classroom is outlined on a daily basis (mckay, 2010). in daily lesson preparations teachers should outline the methods and strategies used to achieve their learning outcomes (adedeji & alaniyan, 2011). through the programme he realised that he needed to prepare a lesson plan for every lesson he want to conduct. the training of the teachers in the programme was done after the mentor realised during school visits that teachers did not do formal written out lesson preparations. so the gap was identified by the mentor and support was put into place to address the matter. unfortunately this aspect of not doing written lesson preparation was only addressed in the last year of the three-year programme, leaving very little time for monitoring and refining this practice even more. how teachers should support learners in class and learner participation in class teachers’ key role is the facilitation of learning in the classroom. they therefore need to constructively support learners and ensure active participation in class. development in these areas was ignited when the following was observed by the mentor during lesson observation of mk, as documented in a mentor report: on exponents the learner wrote 3,25 instead of 3.25. (nel, 2015, p. 123) the teacher did not notice this, and the mentor alerted him to this. it can be assumed that the teacher did not check what the learners wrote in their books to which the mentor alerted the teacher. in later reports the same teacher’s conduct was described as giving learners ‘immediate’ feedback on classwork activities. this indicates that the teacher had observed the importance of providing feedback soon after the learners have worked through the assessment tasks rather than wait for some days or the following day. this highlights a shift to supporting learners more in class than before through the intervention of the mentor. mk used group work as a teaching strategy well, although the mentor noticed that it was time consuming. the teacher was advised by the mentor as to how to alleviate the time issue. teachers were advised by the mentor to work out the solutions to classwork exercises beforehand to ensure that they did not get into a situation where they struggled to find solutions to the problems in front of the learners (progress report on sb): [we were taught to have] detailed worked out solutions for problems given in class and … [for] homework. (nel, 2015, p. 122) in other mentor reports on mk there were indications that few learners participated in the lessons. however, in later reports it was noted that co-facilitation took place where more learners were involved in the lesson (mentor report on nk). this alludes to the mentor demonstrating how learners can be encouraged to participate within the confinement of the teacher’s own classroom. the linking of pd initiatives with the teacher’s own classroom is important for effective pd (de clercq & phiri, 2013). this approach lends itself to possible customised ways of bettering teaching in the teacher’s own classroom, unlike recommending best practices to a group of teachers who are detached from their schools’ context. homework given to learners mathematics is an application subject where learners need to practise their knowledge of and skills in the different topics to ensure complete mastery. therefore, classwork should be supplemented with homework exercises to ensure that learners progressively develop their knowledge and skills. this is why investigation was undertaken to establish whether the participants gave learners homework on a regular basis. in the mentor reports of three of the participants, mention was made that they gave homework regularly, but that the other two participants gave homework sporadically. however, there was a challenge with regard to the time taken to go through or correct the homework in the following lessons. one teacher took an entire period to mark the homework, which resulted in this teacher not making sufficient progress with the syllabus. the feedback from the mentor addressed that the teacher should not spend so much time on correcting learners’ work and should devise other methods of correcting learners’ work such as teaching learners to mark their own work or exchange books. the mentor ak revealed that the classroom visit and mentoring observations and post observation discussions enabled teachers to some extent to cover the syllabus: i just tried to point out to the teachers … how much time you spend the next day on going over homework because a lot of teachers spend the whole lesson going over the previous day’s homework. so they never finish the syllabus. (nel, 2015, p. 208) conclusion the mentor did lesson observations which were received by the teachers in a positive way, where they experienced the mentor support and guidance as helpful and supportive. through the lesson observations individual needs of teachers were picked up by the mentor and immediate support was provided during reflection sessions. the lesson observations also allowed for co-teaching between the teacher and the mentor which led to the development of both the teacher and the learners in the classroom. the exercise also enabled the mentor to identify the areas in mathematics where both the teachers and the learners needed more support. the teachers in the interviews acknowledged that through the process of co-teaching they gained insights into other ways of teaching certain topics as well as the appropriate instructional approaches that supported learners’ knowledge acquisition. another advantage of the mentor observing the teachers in the classroom was constructive feedback and reflection sessions that followed the lesson observation, which were quite informative and educational. the teachers’ instructional and content needs identified during the lesson observation and co-teaching was also dealt with in the pd programmes that followed. such activities allowed for need-based workshops customised to the specific needs of the participants on the programme. due to the presence of the mentor in the classroom, the mentor also established possible reasons why some of the teachers struggled to complete the syllabus. remedial actions were implemented to address that challenge, both in class and in the workshops that followed. it can be safely concluded that both the teachers and the mentor benefited from the programme. it enabled them to identify the instructional and content gaps. the moving away from the teacher-centred approach to one where learners actively participate in class was a development process that was started through mentoring. some of the teachers did develop by getting learners more actively involved in the classroom, but in general more can still be done to promote learner participation. more development is thus needed in this learner-centred instructional approach, keeping in mind that some teachers were exposed to earlier teacher training college methods which promoted passive learning. the provision of homework on a regular basis so as to ensure that concepts are scaffolded and errors are corrected timeously was another area where mentoring assisted and the teachers felt developed. recommendations it can be recommended that pd programmes in south africa include mentoring in their interventions to increase the effectiveness thereof. they must, however, take into account limited time for feedback after lesson observations as well as immediate feedback where possible straight after the lesson observation. another recommendation is to try and ensure a match between the teachers and the mentor. the mentor should be able to form a good relationship with the teachers. this might possibly ensure a conducive environment for openness to trust and guidance. it can also be recommended that heads of departments be trained to act as mentors for their teams as mentoring is costly and time consuming. the heads of departments would be the most appropriate and effective mentor teachers as long as they are trained in the skills of mentorship and have the required content and instructional skills for the subject. such an initiative will cut down on the cost of hiring and transporting a mentor teacher from another school. the head of department as a mentor teacher would also be able to offer school-based pd programmes that are contextualised to the teachers as well as the learners. acknowledgements the first author thanks university of south africa and the university of johannesburg for the financial support for her phd studies. competing interests the authors declare that we have no financial or personal relationships that might have inappropriately influenced us in writing this article. authors’ contributions this was part of b.n.’s doctoral study where k.l. was the supervisor. b.n. did the initial conceptualisation of the manuscript, data analysis, interpretation, conclusion and writing. b.n. did further revisions to the conceptualisation of the manuscript, critical revision and interpretation of the data and final editing. references adedeji, s.o., & alaniyan, o. 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(2003). health economics research method 2003/2. available from http://pioneer.netserv.chula.ac.th/~ppongsa/2900600/lmrm02.pdf article information author: ole skovsmose1,2 affiliations: 1department of learning and philosophy, aalborg university, denmark2department of mathematics, state university of sao paulo (unesp), brazil correspondence to: ole skovsmose postal address: rua 4, no. 555, apto 44, centro, 13500-030, rio claro, sp, brazil dates: received: 14 mar. 2012 accepted: 20 june 2012 published: 14 aug. 2012 how to cite this article: skovsmose, o. (2012). students’ foregrounds: hope, despair, uncertainty. pythagoras, 33(2), art. #162, 8 pages. http://dx.doi.org/10.4102/ pythagoras.v33i2.162 copyright notice: © 2012. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. students’ foregrounds: hope, despair, uncertainty in this original research... open access • abstract • introduction • foreground • multiple and collective foregrounds • foregrounds and meaning • dreams in cages • a ruined foreground? • globalisation and exclusion • acknowledgements    • competing interests • references • footnotes abstract top ↑ a foreground is formed through the possibilities, tendencies, propensities, obstructions, barriers, hindrances, et cetera, which his or her context provides for a person. simultaneously, a foreground is formed through the person’s interpretations of these possibilities, tendencies, propensities, obstructions, barriers, hindrances. a foreground is a fragmented, partial, and inconsistent constellation of bits and pieces of aspirations, hopes, and frustrations. it might be both promising and frightening; it is always being rebuilt and restructured. foregrounds are multiple as one person might see very different possibilities; at the same time they are collective and established through processes of communication. in this article educational meaning is discussed in terms of relationships between the students’ foregrounds and activities in the classroom. i illustrate how students’ dreams might be kept in cages, and how this has implications for how they engage or do not engage in learning processes. i investigate how a foreground might be ruined, and in what sense a ruined foreground might turn into a learning obstacle. finally, i discuss processes of inclusion and exclusion with reference to the notion of foreground. introduction top ↑ long ago, in denmark, i was visiting a classroom where the students, about seven years old, were doing mathematics. the teacher was very positive and encouraging and often smiling. the teacher had a rather large stomach. ‘have you swallowed a football?’ one of the children once asked him. there was a relaxed atmosphere in the classroom. the teacher followed a traditional pedagogical pattern. one day at the blackboard he explained how to add numbers: ‘twenty-eight plus seventy-four ... the eight and the four add up to twelve. down there you write two, and up there you put the one.’ after he had carefully explained the proceedings a couple of times and answered questions from the students, he announced: ‘and now you do the exercises on page thirty-five.’the students had the textbooks in front of them, opened on page 35, and they started doing the exercises. i was sitting next to john, and i, trying to operate as a supportive observer, asked him: ‘so what are you going to do?’ john did not even look at me. he was already deeply concentrating on doing the exercises on page thirty-five. i asked a second time, but he completely ignored me. he was progressing rapidly through the exercises. his handwriting was not very clear but as far as i could see, correct answers were produced with efficiency. whilst he was writing, he half-covered what he wrote with his left hand so that the boy sitting on the other side of him and i had difficulties seeing and following his calculations. i looked around in the classroom: several other students seemed as focussed as john. when the first students had finished the exercises, they immediately got up and went to stand in a small row next to the teacher’s desk. whilst the students were doing the calculations, the teacher had walked around in the classroom giving supportive comments. he smiled at the students and they smiled back. but when the first student had finished the exercises, the teacher had already taken his seat at his desk. the first student showed what she had done. it was not john; he was number three in the row. the teacher looked at the solutions and nodded: ‘very well done!’ and he made a huge tick in the student’s notebook. the second in the row got at huge tick as well. everybody in the row received a huge tick and positive comments. this procedure turned out to be the same in the following lessons, and so did the students’ competition for becoming the first in the row. john was one of the candidates, but there were also several others. during my visits to that classroom i paid particular attention to a small group of girls who were sitting in the back of the classroom. they were not really participating in the competition to be among the first in the row. they worked on the designated exercises, but at their own rhythm. they wrote things in their notebooks. sometimes it was wrong, and they had to erase it. but that was not to be done immediately. they had erasers of different sizes and colours and smells. maybe the eraser at hand had the smell of a banana. but it could be that the mistake had to be erased with an eraser with the smell of strawberry. so one had to ask around for an eraser with the appropriate smell. certainly such an eraser was available, as the group of girls were well equipped with erasers. they also had different pencils and different pencil sharpeners. it might well happen that a number like ‘2’ needed to be written with a newly sharpened pencil. so if an exercise seemed to result in such a number, better do the sharpening of the pencil in advance. there were in fact so many things to take care of! the group of girls had created a small pleasant world of their own, not disturbed too much by the general rush hour sweeping through the classroom. they had shifted their intentions away from participating in the performance game. but the consequence was not that they did nothing. they had created their own priorities, and they were engaged in a productive form of underground construction of meaning. they had established their own vision about what was important to do. my observation of this group of girls made me pay particular attention to students’ intentions for learning, and the possible redirection of such intentions. it made me formulate the notion of foreground.1 foreground top ↑ one can see the foreground of a person as a complex combination of two sets of features. on the one hand, a foreground is formed through the possibilities, tendencies, propensities, obstructions, barriers, hindrances, et cetera, which his or her context provides for a person. one can therefore see the foreground as structured through social, economic, political and cultural parameters. being born into a certain context makes available a configuration of life opportunities that are defined through a set of statistical parameters, which signify expectations about length of life, length of schooling, affluence or poverty, et cetera. such parameters form part of the structuring of the foreground of a person. on the other hand, one should not consider the foreground of a person as a simple objective affair. the foreground is formed through the person’s experiences and interpretations of possibilities, tendencies, propensities, obstructions, barriers, hindrances. in this sense the foreground becomes a complex mixture of subjective and external factors.foregrounds are not panoramic and coherent pictures of possibilities. they are fragmented, partial, inconsistent, dizzy; they are criss-crossed by ruptures. foregrounds may be frightening, as they contain uncertainty, nightmares, danger, threats. foregrounds may include dreams which, from the beginning, might be recognised as being completely out of reach. a foreground is a perplexing constellation of aspirations and frustrations. a foreground is an ongoing rebuilding of such constellations. there is no stability with respect to foregrounds, and maybe it is most adequate to see a foreground as a process. this observation makes it relevant to talk about ‘foregrounding’ and to see ‘to foreground’ as a verb.2 in its introduction the world bank’s world development report 2006 refers to two south african children born on the same day in 2000:3 nthabiseng is black, born to a poor family in a rural area in the eastern cape province, about 700 kilometers from cape town. her mother had no formal schooling. pieter is white, born to a wealthy family in cape town. his mother completed a college education at the nearby prestigious stellenbosch university. (world bank, 2006, p. 1) the report went on to point out that: … nthabiseng has 7.2 percent chance of dying in the first year of her life, more than twice pieter’s 3 percent. pieter can look forward to 68 years of life, nthabiseng to 50. pieter can expect to complete 12 years of formal schooling, nthabiseng less that 1 year. […] nthabiseng is likely to be considerable poorer than pieter throughout her life. […] growing up, she is less likely to have access to clean water and sanitation, or to good schools. (world bank, 2006, p. 1) nthabiseng and pieter’s foregrounds are structured through different values of the parameters designating propensities in life. naturally, they can interpret their situations and possibilities in different ways. their futures are not determined by statistics (not in any strict sense of determining), but the statistical framing of their aspirations and hopes are radically different. the difficulties and obstructions they are going to encounter in life will be radically different. as their foregrounds are different, their ways of acting might be very different, in particular with respect to schooling. if one wants to understand their priorities and their ways of acting in school, one has to understand how they might experience going to school. what kind of meaning would they associate to schooling? what kind of possibilities does schooling provide for them? what new elements might schooling bring to their foregrounds? foregrounds and backgrounds are related. naturally, one can claim that the foreground of a person is shaped by the background of the person. this is clearly illustrated by the case of nthabiseng and pieter. principal statistical parameters seem to become extended from the past and into the future. however, we have to be careful if we are going to stipulate any causal transaction from background to foreground. pieter is, for instance, expected to live much longer than nthabiseng, but in what sense can we see this as caused by their personal backgrounds? the difference of the foreground-parameters with respect to nthabiseng and pieter is produced through an extensive exploitation of one group of people with respect to another. the foreground-parameters have to be interpreted in terms of complex socio-political relationships. they can be formed through exploitation and domination. in case one tries to simplify the explanation by relating the background and foreground of, say, nthabiseng, one could be trapped by the assumption that the explanation of nthabiseng’s difficulties, for instance with respect to education, have to be searched for in her personal background. this would invite a deficit-interpretation of her achievements in school. instead i suggest that foreground-parameters should be analysed as expressions of complex relational patterns of domination, exploitation and exclusion. this, in turn, would show the inadequacy of a deficit-interpretation. multiple and collective foregrounds top ↑ as part of the project learning from diversity, organised by helle alrø, paola valero and myself, many foreground investigations were conducted.4 the project was undertaken in different communities, in different situations and through different forms of collaboration. in brazil we made foreground investigations with respect to indian students. there are many different types of indian communities in the country: some are very remote and, for instance, operate without any money system; others are situated close to cities and in what seem to be favelas (‘favela’ is the portuguese word for township). there are indian communities where the young people go to the disco in the nearby city, dance like other young people dance, and find their parents to be hopeless and old-fashioned like other parents of teenagers. it was in one such indian community that we completed some foreground investigations (see skovsmose, alrø & valero in collaboration with silvério & scandiuzz, 2007). these investigations revealed many things, but here i will emphasise first of all what can be called the multiplicity of a foreground. in fact, it is an open question whether we should use the singular or plural when we refer to the foreground(s) of a particular person. a foreground refers to opportunities which the social, political, cultural and economic contexts make available for the person, and to the way in which they are interpreted and experienced by the person. however, it is possible to change perspectives, to see new possibilities, to create new possibilities; it is possible to fall into a gloomy mood that annihilates any form of hope. the foreground is not any well-defined territory of perceived opportunities. a person might, simultaneously, envisage different foregrounds that might contradict one another. it is important to acknowledge the multiplicity of foregrounds, also when we have to do with a specific person at a specific moment. foregrounds are packed with contingencies, they represent uncertainties, they contain risks, they are unpredictable. they represent hope and aspirations as well as defeat and misery. they seem ready to split up and multiply. in particular, this multiplicity can be dominant when we consider a person in a borderland position5. such a position is experienced by a person who can envisage radically different opportunities, as for instance these young indian students to whom i have just referred. they are familiar with the life of young people in the city, and they know what it means to prepare for further studies. they know what the brazilian society may offer by way of living conditions. they know the indian community and the indian language. they know about working in the fields. they can easily envisage very different scenarios. they can see themselves leaving the indian community and moving to the city, getting a job and getting married. but they can also remain within the indian community. in this sense they are positioned in a borderland which opens to radically different foregrounds. one of the indian students expressed clearly that remaining in the indian community was a high priority. he liked to work in the fields and to be part of the solidarity associated with working for everybody. another student expressed his aspirations to study medicine. the reason he gave was that health was one of the principal problems in indian communities in brazil. he wanted to return as a qualified doctor. foreground can be expressed, it can be changed, it can be reworked. there is not much continuity in this process of foregrounding. the student who preferred to work in the fields might come to see other possibilities. the student who wanted to study medicine might become interested in working in the city or becoming a researcher at the university. foregrounds are multiple, not least for students in a borderland position. sikunder ali baber, who is from pakistan, completed his phd in denmark. he studied the conditions of immigrant students in denmark, particularly students from pakistan (baber, 2007). he paid special attention to the students’ foregrounds, and to how they saw their possibilities. one of the principal points of his study was that immigrant students’ foregrounds were somehow polarised. they felt that they, as immigrant students, had to perform much better in school than the average in order to have any chance in denmark. if your school performances were average or below, you had no possibilities, except helping in your father’s small shop. you were a lost case. for a ‘regular’ danish child, however, being average provided a comfortable position. the point of baber’s study was not to document whether or not the immigrant students’ interpretation was correct or not, statistically speaking. the point was to clarify features of their foregrounds; and these features had implications for the meaning they might associate to going to school, for their experience of being excluded, for their choice of stepping out of the race in school or not. foregrounds contain strong collective features. above i talked about the foreground of a person; we said that the foreground of nthabiseng was radically different from the foreground of pieter. one could, however, also talk about the foreground of a group of persons. thus we could pay attention to the statistical parameters which form part of the constitution of nthabiseng’s foreground. these values outlining expectations of length of life, income, and schooling she shares with many other people from her neighbourhood. in this sense, nthabiseng shares foreground with many others, and it seems possible to talk about the foreground of a group of people. furthermore, experiences can be shared. interpretations also have strong collective features. thus, nthabiseng’s aspirations with respect to education are emerging as a result of a collective process. foregrounds are constructed through shared life-experiences, shared visions and aspirations, shared interpretations of threats and possibilities, and shared frustrations. they are constructed through communication. naturally, foregrounds can also be imposed on a group of people, through exploitation and stereotyping. foregrounds are elaborated though complex social processes. this means that one can talk about the student’s foreground (for instance, when one wants to emphasise the individual features of a foreground); or about the students’ foreground (when one wants to emphasise the collective aspects of a foreground). one can talk about the student’s foregrounds, when one wants to address the multiplicity of foregrounds. lastly, and in general, one can talk about students’ foregrounds. whatever formulation one chooses, there are always some important features that tend to be left out. foregrounds and meaning top ↑ once, i was giving a lecture in brazil about the conception of foreground and about the importance of paying attention to the students’ foregrounds in order to understand their construction of meaning and how they might act or not act in the classroom. one person from the audience raised his voice and said, ‘i have proof that what you are saying is right.’ i was not sure that i had understood him correctly: ‘you have proof that what i’m saying is right?’ ‘yes’, he answered. i immediately gave him the floor.he said that he was a mathematics teacher and that in his school there was a boy who had given up. he showed no interest in doing anything. he was well known in the school, and his doing nothing was clearly demonstrated to everybody in the classroom independent of which teacher happened to give the lesson. the boy was recognised amongst his friends as a trendsetter. he did not spoil the class with loud comments, but the way he put down his head on his folded arms when the teacher was about to say the first words was a manifest public announcement of his lack of interest. the teachers had tried everything to get him interested, but his head remained down on his folded arms. one day the principal of the school asked the boy to come to his office. here he asked the boy, ‘what do you want? what do you want to become?’ the boy smiled and said, ‘i want to get into the military.’ the principal nodded and said that there was a military school at a military camp not so far away. he asked the boy to get into his car: ‘let us go there.’ and off they went to see the military camp, and to get an idea of what the school demanded and what it meant to be trained for the military. the boy and the principal walked around, saw something, talked with somebody. it was not a long excursion; they were back in school a few hours later. from that day on, the boy did not rest his head on his arms any longer. after the excursion he changed completely and became a perfectly attentive student. it appeared that just showing an interest in his foreground established a new relationship between the student and what was taking place in the school. the point of the story is not that the boy had experienced some particular and well-defined relevance of the topics dealt with in the school whilst he visited the military school, for instance that reading a map is crucial in any military education. it appears that the very fact that the principal showed an interest in the student’s aspirations turned the situation around. meaning is constructed, and this also applies to the meaning that students may construct with respect to activities in the classroom. i interpret meaningfulness in terms of relationships, and in particular in terms of relationships between, on the one hand, what is taking place in the classroom and the activities the students are invited to participate in, and, on the other hand, the foreground of the students. however, such relationships need not be elaborated in complete detail in order for students to experience meaningfulness. some approaches in mathematics education have been searching for meaning by relating the activities in the classroom to a particular aspect of the background of the students. i do not find this to be irrelevant, but i find that meaning construction with respect to learning has a more intimate relationship to the foreground of the person. what can be experienced as meaningful education by the students is an open question. however, showing an interest in the student’s foreground means showing interest in what is crucial for establishing meaningfulness. as part of the project learning from diversity we also interviewed young people from a brazilian favela.6 we asked the young people what they wanted to do in the future, and how they saw mathematics with respect to this. for instance, one point of this foreground investigation was to understand how the students might see the relationship between the mathematics they experienced in school and their aspirations for the future. it was generally recognised that mathematics was relevant for many different kinds of further studies and that mathematical techniques were essential for engineers, doctors, dentists, et cetera. what was completely obscure, however, was the particular nature of this relevance. it was not possible for them to point out any particular relationships between issues in the curriculum and out-of-school practises assumed to be applying mathematics. the relevance of mathematics could only be expressed in general terms; it could not be exemplified. or, as pointed out by one student, when we consider the solutions of quadratic equations of the form ax2 + bx + c = 0, and we are asked to calculated the discriminant, delta = b2 – 4ac, we are lost. it is not possible to imagine any real-life situation – in engineering, medicine, economy, computing, whatever – where one needs to calculate delta. one can make different interpretation of this delta syndrome. one can, for instance, claim that the syndrome emerges from a general misunderstanding, namely that the relevance of mathematics can be broken down into the relevance of particular elements of mathematics. however, this can bring about misleading questions such as: what is the practical relevance of calculating delta? instead one may claim that the relevance of mathematics has to be acknowledged through the relevance of the overall features of mathematics, and not via its particular elements. this might sound reasonable, but still there remains the question of how the particular activities in the classroom then might be related to the overall features of mathematics. and this, again, makes us return to the point that meaningfulness, as experienced by the students, emerges from relationships between what is taking place in the classroom and their foregrounds. the delta syndrome reminds us that this relationship is of a most complex nature and often obscure. students in different situations have different options for dealing with the delta syndrome. naturally, students can operate within the mathematics classroom as if one has to do with a competition, as was symbolised by the competition in getting first in the row next to the teacher’s desk. thus, students can ascribe meaning to activities through their instrumental value. thus instrumentalism can provide much energy to some students’ activities.7 in general instrumentalism with respect to learning mathematics refers to an activity, not motivated by aspirations of understanding mathematics, but by aspirations of obtaining something such as passing a test, entering further education, or being the winner of the page 35 race. through an instrumental approach one might become able, in a determined way, to concentrate on some classroom activities which otherwise might appear meaningless. some students will get strong support from home in dealing with the delta syndrome in an instrumental way. parents could help with homework, whether it makes sense or not, always making clear the importance of good performance in school for future career opportunities. but for students with less support from home, the delta syndrome might be devastating. thus pieter and nthabiseng have very different conditions for experiencing meaning in mathematics education, as well as for coping with the delta syndrome. dreams in cages top ↑ once i visited a school in a poor part of barcelona. in catalonia the official language is catalan, and it is forbidden to speak spanish at school. catalan represents a middle class culture, whilst spanish is the language of immigrants from other parts of spain or from spanish speaking countries in south or middle america. immigrant groups populate the poor parts of barcelona, and the school i visited was located in such a neighbourhood. the mathematics teacher spoke spanish in the classroom, and this clandestine form of communication was very much appreciated by the students. it was a powerful sign of solidarity. there was a very good atmosphere in the classroom.the students were fifteen or sixteen years old, and i asked them to do a little experimentation (my english was translated into spanish by a research colleague). i wanted them to pay attention to their imagination, to their hopes and dreams with respect to the future. in other words, i wanted to conduct a foreground investigation, although on a very small scale. i asked the students: ‘just close your eyes, and start to imagine. imagine yourself in the future. imagine yourself in ten years. imagine that you can become exactly what you dream of becoming.’ there was a little laughing, but everybody was sitting with closed eyes. i continued: ‘now, imagine something you would really like to be and do in ten years …. we have good time, just close your eyes and do some dreaming. and imagine that you succeed in doing what you dream of doing.’ i could see their mouths moving a little bit. some were smiling, some looked grave and focused. many changed expression now and then. ‘no, no, don’t look’, i said, ‘just continue imagining.’ so they were sitting there and imagining, with their eyes closed. after a while i asked them to open their eyes. they looked around with shy smiles. i asked if somebody wanted to tell about their imaginings. my idea was the following: firstly, i wanted to listen to what they would dream of when asked to dream freely. secondly, i wanted to ask them to dream with open eyes, and imagine more realistically how they saw themselves in ten years’ time. in this way, i would get an impression of the more realistic features of their foreground. thirdly, i wanted to ask them how they saw the relationship between what they were doing in mathematics and what they had imagined, whatever it might be – either wild or realistic dreaming. after a while a girl said, ‘i was dreaming about becoming a hairdresser.’ and one of the boys said he was dreaming about working as an electrician. he would like to put up wires in houses. i was rather surprised. i had expected to listen to something about becoming famous, becoming a singer, becoming a professional football player. had the students misunderstood? i had after all asked them to do some ‘wild dreaming’. i was about to say something, and to explain that later we would come back to a more realistic perspective of the future. but then i understood: i had, in fact, been listening to their wild dreaming. ghettoising means tying people to the ground. this can be done in a concrete way, as in case of palestinians who cannot leave the ghettoes as they do not have human rights or passports. people can also be tied to the ground by not having resources for moving anywhere. ghettoising makes dreaming hurtful. in a ghetto, dreams are kept in cages and this brings us to consider what a ruined foreground can mean. however, let me just add one more thing with respect to the barcelona example: after i had listened to the students’ wild dreaming, i asked if they saw relationships between what they were doing in the mathematics classroom and what they wanted to do in the future. they said that they could see many relationships. i asked them to exemplify. the boy who wanted to become an electrician explained how an electrician could figure out if there was enough wire in the roll he had to take with him to complete a certain task. the students explained that one need not unroll the whole wire and measure its length. the wire could remain in the roll. one could, instead, measure the diameter of the roll, count the number of rounds of wire in the role, multiply these two numbers and multiply the result with 3. then one knew how much wire one had available in the roll. this was in fact the common practice among electricians, and this practice had been explained by the mathematics teacher. maybe the boy wanted to become an electrician only after the teacher had explained how to measure the length of a wire without unrolling the whole wire. meanings are constructed, and foregrounds are constructed. a ruined foreground? top ↑ let us consider again the girls sitting at the back of the classroom, erasing the numbers they might have written incorrectly. they had withdrawn from the general classroom activity. they were not taking part in the race to become number one in the row. they might have imagined that the same kind of race was going to be repeated year after year after year during the mathematics lessons. they might also have realised that the first in the race would be john or lisa or birgitte (or maybe a few others) year after year after year. they might have realised that they would never be able to compete for the number one position. how were they to cope with such a recognition? a desperate option might be to try to improve one’s racing capabilities. the desperate option might, however, have as implication that one would be defeated during all the school years to come. the desperate option might be far from reasonable. one should not try to fight for something which is a pure illusion. so better defend oneself and redirect one’s ambitions. one could simply change focus, and this was what the erasing girls did. they withdrew from complying with the official classroom practice, and this disengagement can be seen as healthy self-protection. furthermore, their disengagement did not disturb the classroom practice: the girls established their own neat underground practice. it was a practice for silent girls, and some years later the group was in fact recognised by the teachers as ‘the group of silent girls’.i find that one key to understanding students’ achievements at school is their foregrounds, including their interpretation of possibilities. in particular it is difficult to fight for something that appears unattainable. this applies to students who find it impossible to complete a first-in-the-row race, and it applies to students in a ghetto. in this sense a ruined foreground may form a most profound learning obstacle. a ruined foreground can be the most direct cause of failure in school. let us look at some examples. in his doctoral thesis herbert khuzwayo (2000) studied the history of mathematics education in south africa during the apartheid period, from 1948 to 1994. in particular he investigated what could be called ‘white research on black education’. one element in this ‘research’ was to find out why, according to some statistics, black students could not figure out mathematics. naturally such statistics had been both produced and interpreted within a racist framework and formulated through a deficit-model, and the conclusion was that the cause of black students’ failure was associated with the black students themselves. one can, however, get a different reading of achievement figures if one considers the students’ foregrounds. what did black students’ foregrounds look like during the apartheid era? could they consider studying engineering? or any other kind of technical studies? many possibilities were simply eliminated from their foregrounds – maybe due to the explicit apartheid rules; maybe due to engraved social and economic suppression. black students’ foregrounds were ruined. what sense could it make to black students to struggle with mathematical issues in order to qualify for further studies that were in any event inaccessible? for them there were no possibility to pursue any form of further technical studies requiring mathematical skills. one reasonable approach would be to do as the girls did, keep quiet and stay out of the school race, which could bring only defeat; another would be to become an activist. ruined foregrounds of black students during the apartheid era established the most brutal form of learning obstacle. looking around the world today, however, one finds many examples of ruined foregrounds. let us consider a different situation. if we go back to, say, the 1930s in denmark, we find that women, it seemed, could not do well in mathematics. how are we to explain this phenomenon? one could try to relate this to the background of the girls. their failure in mathematics could have been related to their upbringing. however, we could also try to consider their foregrounds. how did girls during the 1930s in denmark see their future opportunities? they could easily observe that mathematics was for men, as all further studies that drew heavily on mathematics, like engineering for instance, were extremely male-dominated. it might have been very difficult for girls at that time to envision themselves as dealing with technical issues. mathematics did not play any part in any practices that could be associated with their foregrounds. a most reasonable thing, then, was not to struggle with this subject. better simply to pull out of the race. however, denmark did not remain the same. new opportunities arose, and girls’ performances with respect to mathematics changed. once, mathuma bopape invited me to visit different townships schools in the pietersburg area in south africa.8 one school looked more broken-down than many others. we were standing in a classroom that looked as if it had suffered a light bombing, and up there, right above our heads, there was a hole in the ceiling. it may be that some houses in the neighbourhood had needed the tiles a bit more than the school building. when it was raining the chairs in the classroom had to be moved away from this part of the classroom. if we should try to explain the observation that black children have lower achievement rates in school than white children, i think it would be wise to remember the hole in the ceiling. for me it is the most obvious learning obstacle one could think of. this hole, however, has been overlooked my much educational research, in particular by ‘white research on black education’. by providing different patterns of explanation this research has been engaged in explaining away the obvious: that black children’s foregrounds have been ruined. but not only this: their actual learning conditions have been ruined as well. globalisation and exclusion top ↑ global networking might include some people in the flow of information and goods. but globalisation might also, in the most brutal way, exclude many others, who apparently do not have any role to play in a capitalist supply-demand dialectics. one feature of globalisation is ghettoising. zygmunt bauman makes the following observation when talking about the ‘problem of capitalism’. previously this problem had taken the form of exploitation; however, bauman (2004) emphasise that the ’most blatant and potentially explosive malfunction of capitalist economy, is shifting in its presents planetary stage from exploitation to exclusion’ (p. 41).he continues: it is exclusion, rather than the exploitation suggested a century and half ago by marx, that today underlies the most conspicuous cases of social polarization, of deepening inequality, and of rising volumes of human poverty, misery and humiliation. (bauman, 2004, p. 41) this is a strong reminder of the fact that the exclusion of economically ‘irrelevant’ groups is part of the global order today. bauman describes the drama of inclusion–exclusion in terms of extreme conditions for constructing identities in the following way: at one pole of the emergent global hierarchy are those who can compose and decompose their identities more or less at will, drawing from the uncommonly large, planet-wide pool of offers. at the other pole are crowded those whose access to identity choice has been barred, people who are given no say in the deciding their preferences and who in the end are burdened with identities enforced and imposed by others; identities which they themselves resent but are not allowed to shed and cannot manage to get rid of. stereotyping, humiliating, dehumanizing, stigmatization identities … (bauman, 2004, p. 38) conditions for constructing identities are polarised. globalisation, representing the capitalist order of today, establishes some groups of people as being in a position of making an apparently free build-up of identity (although certainly subjected to the capitalist logic of consumption), whilst others have to cope with imposed and stigmatised identities. this constitutes that part of the new global order where exploitation has turned into exclusion. let us now repeat bauman’s formulation, elaborating them a little further and talking about students’ foregrounds instead of their identity.9 in this way i want to emphasise that the discussion of inclusion–exclusion is closely related to the discussion of foregrounds. at one pole of the global hierarchy that is emerging through the processes of globalisation we find those groups of students who have the opportunity of composing and decomposing their foregrounds more or less at will – although we cannot ignore the fact that these compositions are taking place within the capitalist layout of globalisation. there are plenty of resources from which they can bring together a variety of possibilities in life. they can draw on the uncommonly large, planet-wide pool of offers to which they, maybe due to the affluence of their families, have direct access. they can get the support needed to make the best of their time in school. they can go to private schools if this turns out to be most beneficial. they can choose the education that prepares them for the opportunities in life that to them seem most attractive. at the other pole are crowded those many students whose access to the free formation of foregrounds has been barred. here are students who have no (or very little) say in establishing their future possibilities in life, and who in the end are troubled with foregrounds imposed, as well as ruined, by others. these are the students who – tied to the ground in a ghetto or located in a borderland position – find their dreams to be put in cages. these are the students whose foregrounds might represent real learning obstacles. there is no lack of stereotyping that accompanies ghettoising and exclusion. students from poor conditions might easily be classified as being troublemakers and obstructive, at best they might be classified as suffering a range of deficiencies due to their personal backgrounds. if we consider, for instance, immigrant students in denmark, then stereotyping, humiliation, dehumanisation, and stigmatisation form part of the public formation of their foregrounds. dramatic processes of inclusion–exclusion are established though schooling. and if we follow bauman in claiming that the malfunction of capitalist economy is now operating through exclusion, it becomes crucial to consider the role of schooling with respect to this malfunction. processes of inclusion and exclusion operate to a great degree through schooling. these processes can be experienced by every student: john, the silent girls, pieter, nthabiseng, anybody. it becomes important to consider how foregrounds might be ruined; how they might be reconstructed; how it might be possible to add new elements to them; and how schooling might provide students with new possibilities. it becomes important to consider how foregrounds might represent hopes, despairs, and uncertainties. acknowledgements top ↑ this article is based on my lecture at the symposium mathematics education, democracy and development: challenges for the 21st century on 04 april 2008 at the university of kwazulu-natal, durban. i want to thank renuka vithal for inviting me to participate in the symposium and to nyna amin for responding to my lecture. i want to thank denival biotto filho, renato marcone, raquel milani, and miriam godoy penteado for making suggestions and critical comments on the manuscript. competing interests i declare that i have no financial or personal relationship(s) which may have inappropriately influenced me in writing this article. references top ↑ alrø, h., skovsmose, o., & valero, p. (2009). inter-viewing foregrounds: students’ motives for learning in a multicultural setting. in m. césar, & k. kumpulainen (eds.), social interactions in multicultural settings (pp. 13−37). rotterdam: sense publishers.baber, s.a. (2007). interplay of citizenship, education and mathematics: formation of foregrounds of pakistani immigrants in denmark. unpublished doctoral dissertation. aalborg university, denmark. bopape, m. (2002). mathematics school based in-service training (sbinset): a study of factors contributing towards success or failure of sbinset in the south african school context. unpublished doctoral dissertation. aalborg university, denmark. bauman, z. (2004). identity: conversations with benedetto vecchi. cambridge: polity press. khuzwayo, h. (2000). selected views and critical perspectives: an account of mathematics education in south africa from 1948 to 1994. unpublished doctoral dissertation. aalborg university, denmark. laursen, i.l. (2008). at fremrette – et studie af etniske minoritetskvinders subjektivt fortolkede livsverden og arbejdslivsforventninger i komplekse samfund [forward-directing: a study of ethnic minority women’s subjectively interpreted life-world and work expectations in a complex society]. unpublished master’s thesis. department of learning, education and philosophy, aalborg university, denmark. mellin-olsen, s. (1981). instrumentalism as an educational concept. educational studies in mathematics, 12, 351−367. http://dx.doi.org/10.1007/bf00311065 penteado, m.g., & skovsmose, o. (2009). how to draw with a worn-out mouse? searching for social justice through collaboration. journal of mathematics teacher education, 12(3), 217−230. http://dx.doi.org/10.1007/s10857-009-9103-6 skovsmose, o. (1994). towards a philosophy of critical mathematics education. dordrecht: kluwer academic publishers. skovsmose, o. (2005a). travelling through education: uncertainty, mathematics, responsibility. rotterdam: sense publishers. skovsmose, o. (2005b). foregrounds and politics of learning obstacles. for the learning of mathematics, 25(1), 4−10. available from http://www.jstor.org/stable/40248476. reprinted in u. gellert, & e. jablonka (eds.), mathematisation – demathematisation: social, philosophical, sociological and educational ramifications (pp. 81−94). rotterdam: sense publishers. skovsmose, o. (2011). an invitation to critical mathematics education. rotterdam: sense publishers. skovsmose, o., alrø, h., & valero, p. in collaboration with silvério, a.p., & scandiuzzi, p.p. (2007). “before you divide you have to add”: inter-viewing indian students’ foregrounds. in b. sriraman (ed.), international perspectives on social justice in mathematics education. the montana mathematics enthusiast, monograph 1, 151−167. available from http://www.math.umt.edu/tmme/monograph1/skovsmose_etal_final_pp151_168.pdf skovsmose, o., & greer, b. (eds.) (2012). opening the cage: critique and politics of mathematics education. rotterdam: sense publishers. skovsmose, o., & penteado, m.g. (2011). ghettoes in the classroom and the construction of possibilities. in b. atweh, m. graven, w. secada, & p. valero. (eds.), mapping equity and quality in mathematics education (pp. 77−90). new york, ny: springer. skovsmose, o., scandiuzzi, p.p., valero, p., & alrø, h. (2008). learning mathematics in a borderland position: students’ foregrounds and intentionality in a brazilian favela. journal of urban mathematics education, 1(1), 35−59. available from http://ed osprey.gsu.edu/ojs/index.php/jume/article/view/4/4 stentoft, d. (2009). challenging research conceptions in (mathematics) education: telling stories of multiplicity and complexity. unpublished master’s thesis. department of learning, education and philosophy, aalborg university, denmark. world bank (2006). world development report 2006: equity and development. washington and new york: a copublication of the world bank and oxford university press. available from http://wwwwds.worldbank.org/external/default/wdscontentserver/iw3p/ib/2005/09/20/000112742_0050920110826/rendered/pdf/322040world0development0report02006.pdf footnotes top ↑ 1.a first development of the notion of ‘foreground’ is found in skovsmose (1994). see also skovsmose (2005a, 2005b, 2011). in the presentation i give here, i draw on these resources. the story about the silent girls has already been referred to in skovsmose (2005b). 2.in her master’s thesis, iben lindgaard laursen studied the foreground of immigrant women in denmark. she emphasised the importance of seeing ‘to foreground’ as a verb. she used the danish word fremrette, which means ‘forward-directing’ when directly translated into english. in the english abstract of her thesis, she translates fremrette as ‘envisage’ (laursen, 2008). 3.renuka vithal referred to these paragraphs in the world bank report in her lecture at the symposium mathematics education, democracy and development: challenges for the 21st century, faculty of education, university of kwazulu-natal, durban, on 04 april 2008. 4.see, for instance, alrø, skovsmose and valero (2009). important contributions to the project ‘learning from diversity’ were also provided through the doctoral studies by sikunder ali baber (2007) and diana stentoft (2009). 5.for an introduction of the notion of borderland position see skovsmose, scandiuzzi valero and alrø (2008). see also penteado and skovsmose (2009), where the notion is further explored. 6.see skovsmose, scandiuzzi, valero and alrø (2008). 7.instrumentalism has been carefully addressed in mellin-olsen (1981). 8.i refer to this event in skovsmose (2005a). see also bopape (2002). 9.for a careful discussion of the notion of identity and related notions, including foreground, see stentoft (2009). article information author: sarah bansilal1 affiliation: 1department of mathematics education, university of kwazulu-natal, south africa correspondence to: sarah bansilal postal address: 8 zeeman place, malvern 4093, south africa dates: received: 22 oct. 2011 accepted: 04 june 2012 published: 17 aug. 2012 how to cite this article: bansilal, s. (2012). using conversions and treatments to understand students’ engagement with problems based on the normal distribution curve. pythagoras, 33(1), art. #132, 13 pages. http://dx.doi.org/10.4102/ pythagoras.v33i1.132 copyright notice: © 2012. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. using conversions and treatments to understand students’ engagement with problems based on the normal distribution curve in this original research... open access • abstract • introduction    • some literature on the normal distribution curve    • the analytic framework • methodology    • ethical considerations and recruitment procedures    • reliability and validity    • the test items    • remarks about, and solutions to, test items       • question 1: we need to find p(x < 400) = ?       • question 2: here we need p(450 < x < 600) = ?       • question 3: we need to find an x-score so that p(x > ?) = 0.1 • findings and discussion    • findings for question 1       • × blank or unrelated algorithm       • ■ partial treatments (pt)       • ■ complete or full treatments (ft)       • ■ partial conversions (pc)       • ■ complete or full conversions (fc)    • findings for question 2    • findings for question 3    • performance on the three questions    • success rates in conversion transformations and treatment transformations    • direction of conversions • summary    • implications of the findings • acknowledgements    • competing interests • references abstract top ↑ including probability and statistics in the core curriculum of mathematics in south african schools has made it necessary to train teachers to teach statistics at high school level. this study concentrates on practising mathematics teachers who were students in an in-service programme. the purpose of the study was to investigate students’ success rates on different questions of a multi-part task based on the normal distribution curve. the theory that i used to understand the students’ difficulties is duval’s theory about movement within and between semiotic representation systems, called treatment transformations and conversion transformations respectively. the first two parts of the problem were unknown percentage problems and involved a treatment followed by a conversion. the third was an unknown value problem and required a conversion before the students could undertake a treatment transformation. the findings reveal that the success rates the students achieved in treatment transformations were higher than those they achieved in conversion transformations. the study also revealed that the direction of the conversions played a role in success rates. recognising the different challenges the two types of transformations pose requires that teachers pay particular attention to actions that involve movement between different representation systems. introduction top ↑ including probability and statistics in the core curriculum of mathematics in south african schools has made it necessary to train teachers to teach statistics at high school level. although the normal distribution curve is not part of the school curriculum, it is part of a basic course in statistics that aims to equip teachers to teach probability and statistics up to grade 12 level.this exploratory study was conducted with 290 in-service secondary school mathematics teachers who had enrolled in an in-service mathematics programme. it focuses on one multi-part problem, which was part of the course summative assessment and includes ‘unknown percentage’ and ‘unknown value’ problems (watkins, scheaffer & cobb, 2004). in unknown percentage problems, students first transform a given value into an associated z-score using the standardisation process. thereafter, students identify the probability associated with the z-score and interpret the value in terms of the graph. this involves working simultaneously with properties of the standard normal distribution and the properties of particular z-table values. in unknown value problems, students have the percentage and they have first to identify the z-score from a table of z-values that corresponds to the given percentage by working with the properties of the standard normal distribution. thereafter, they calculate the x-score by ‘unstandardising’ it, or reversing the standardising process. in this article, i refer to these in-service teachers as students because they were participants in the programme. in analysing the students’ performance, i drew on duval’s (2006) framework for transforming semiotic representations, where he distinguishes between transformations that occur within the same system of representations (treatments) and those that involve a change of register (conversions). the purpose of this study was, firstly, to investigate whether there were differences in students’ success rates on the two types of transformations (conversions and treatments) that are inherent in one multi-part problem and, secondly, to investigate whether the direction of the conversion transformations influenced success rates. students experience unknown value and percentage problems as challenging for many reasons, including because they involve applying and not just recalling the properties of the normal distribution curve. in this article i report one particular aspect of the challenges. i am looking at students’ proficiency in carrying out treatment and conversion transformations and investigating whether the differential engagement with these two types of transformations could account for the differences in success rates. in doing so, i do not suggest that this is the only factor that accounts for the challenges associated with these types of problems. some literature on the normal distribution curve reading and canada (2011) think that distribution of data is a fundamental concept in its own right, but that it is complex despite its relatively straightforward definition. one can see probability distributions as even more complex and understanding the differences between data distributions and probability distributions is a key step in statistical reasoning (cohen & chechile, 1997). the authors’ comment that, despite the emphasis on hands-on data analysis and alternative methods of inference, the concept of probability distributions should be part of all introductory statistics courses. unlike data distributions, probability distributions are formal theoretical models statisticians use to describe the likelihood of a variable taking on a value or a range of values. it is this theoretical nature that brings out contrasts between probability and data, thereby helping students develop ideas about stochasm (cohen & chechile, 1997, p. 2). wilensky (1997) regards probability distributions as a key concept in probability and statistics because of their importance in understanding statistical models in scientific research and because they stand ‘at the interface between the traditional study of probability and the traditional study of statistics’ (p. 175), and therefore provides an opportunity to make strong connections between the two fields. concern about the lack of research into students’ understanding of the normal distribution led to pfannkuch and reading (2006) publishing a special issue of the statistics education research journal. it focused on reasoning about distributions and provides suggested research questions that could address various aspects of reasoning about distributions, including one about the ‘difficulties that students encounter when working with analysing and interpreting distributions’ (p. 5). bakker and gravemeijer (2004) regard a distribution as a conceptual entity for thinking about variability in data. pfannkuch and reading (2006) warn that any discussion about the nature of distributions needs to include a conceptual perspective (which clarifies the notions that underpin distributions and why they are important) and an operational perspective (which explains how distributions capture, display and manipulate specific sets of data). reading and reid (2006) included both perspectives in their development of a two-cycle hierarchy of reasoning about distributions, based on the application of the structure of observed learning outcomes (solo) taxonomy. the first cycle involved understanding key elements whilst the second, more cognitively sophisticated levels, involved using those elements. pfaff and weinberg (2009) believed that actively generating data before analysing them would increase understanding of the statistical concepts. one may see this as indicative of the operational perspective that pfannkuch and reading (2006) described. however, their study (pfaff & weinberg, 2009) found that, despite the fact that their students actively generated data, their students’ performance in their post-activity assessments was no better than it was in their pre-activity performance. carlson and windquist (2011), in their comment about these unexpected results, argued that pfaff and weinberg were correct in concluding that ‘the physical act of generating data was not sufficient to produce learning’ (p. 3). however, they disagreed with the conclusion that the authors (pfaff & weinberg, 2009) drew that ‘active learning approaches in general are ineffective’ (carlson & windquist, 2011, p. 3). north and zewotir (2006) move beyond considering only the approach to teaching statistics. they question the content that introductory statistics courses should cover. they call for a re-think of the statistics courses for social scientists and argue for courses that focus on how to use descriptive statistics instead of focusing on calculations like those based on grouped data. they advise that the courses should devote more time to understanding principles and developing statistical reasoning by using rich contexts. (north & zewotir, 2006). however, the situation of social scientists, who are learning how to interpret and use statistics when studying socioeconomic phenomena, is different to that of teachers who are learning how to teach statistics to school children – the context of the current study. reading and canada (2011) describe two studies about the statistical reasoning of elementary teachers. both studies ‘firmly cast the teacher in the role of the learner’ (p. 229) in the current study, the teachers were also the learners in a basic course in statistics that aimed to equip them to teach probability and statistics up to grade 12 level. the module covered aspects of statistics like central tendencies, grouped data, distributions, bivariate data, regression, probability concepts and probability distributions. a concept like the normal distribution curve is not part of the school curriculum. however, one can see it as an example of what ball, thames and phelps (2008) call horizon knowledge. this is an ‘awareness of how mathematical topics are related over the span of mathematics included in the curriculum’ (p. 403) and is one of the six domains that comprise their model of mathematical knowledge for teaching. having knowledge of the horizon can help teachers make decisions about how to teach concepts like variation, distributions and other statistical topics. the analytic framework a set of elementary signs, a set of rules for producing and transforming signs as well as an underlying meaning structure that derives from the relationship between the signs within the system characterise a semiotic system (ernest, 2006). radford (2001) has argued that using signs and tools modify our cognitive functions. on the other hand, ernest (2006) says that a focus on signs and sign use is the characterising feature of a semiotic perspective of mathematical activity that provides a way of conceptualising the teaching and learning of mathematics. each semiotic system has its own specific way of working. duval (2006) points out that the role semiotic systems of representation play is not only to designate mathematical objects or to communicate but also to work on, and with, mathematical objects. duval asserts that two different types of transformations of semiotic representations can occur during any mathematical activity. the first type, called treatments, involves transformations from one semiotic representation to another within the same system or register (duval, 2006, p. 110). duval (2002) argues that the treatments that one can perform depend on the register one uses and: the procedures for carrying out a numerical operation depend just as much on the system of representation used for the numbers as on the mathematical properties of the of the operation. (p. 111) he illustrates his argument with the fact that the algorithm for adding fractions is different for a decimal notation and a fractional notation of the same numbers (0.2 + 0.25 as opposed to ). furthermore, when dealing with treatments, the semiotic system eases the connection of different representations because the rules of the semiotic system link different representations of the same object. the second type, called conversions, involves changing the system but retaining the reference to the same objects (duval, 2006, p. 112). in order to illustrate the differences between treatments and conversions further, i will use an example from transformation geometry. consider a point a (2; 3) on the cartesian plane with the required transformation on a being a clockwise rotation of 90° around the origin. a person can perform the transformation on a by applying the algebraic rule (x, y) → (-y, x) to get the result a (-3; 2). this transformation is an example of a treatment because it does not require a change in the system of representation because, after applying the formula, the object is being described by the same representation. a study by bansilal and naidoo (2012), on learners’ engagement with transformation geometry, identified a learner who considered the representation of a in a different register by identifying the location of the point a(2; 3) on the cartesian plane before performing the rotation transformation. this movement (from the two-coordinate description of a to the location of the point a in the cartesian plane) is an example of a conversion transformation because the register has changed but not the object (point a). thereafter the learner worked out the resulting location of the point when he rotated it 90° through the origin by interpreting the motion within the new register. he then identified the location of the rotated point and thereafter assigned the coordinates based on its new position (bansilal & naidoo, 2012). this example illustrates how it is possible for one to perform a transformation using the same representation system (a treatment) and how one could perform it using a representation from a different register. however, the second case needed a conversion transformation to move to the different register of representation before one could perform a treatment using the second system of representation. duval gives conversions a more central role in understanding mathematics than he does to treatments and regards conversions as a cognitive threshold that is the main cause of learning difficulties in mathematics. he argues that one cannot reduce a conversion of a representation (change of register) to a treatment. therefore, conversions account for one of the sources of incomprehension in mathematics. he believes that ‘we cannot deeply analyse and understand the problem of mathematics comprehension for most learners if we do not start by separating the two types of representation transformation’ (duval, 2006, p. 127). duval’s contention is that treatments command more attention in mathematics whilst conversions cause the greatest difficulties in mathematics. he argues that conversions only become relevant because we need to choose ‘the register in which the necessary treatments can be carried out most economically or most powerfully’. another reason he suggests for using conversions is that they provide ‘a second register to serve as a support or guide for the treatments being carried out in another register’ (p. 127). the visualiser/analyser (va) model of zazkis, dautermann and dubinsky (1996), which specifies two elements (visualisation and analysis) as two interacting modes of thought, may help us develop an insight into the effort students require to understand conversion transformations. the model describes a series of movements between visual and analytic representations, each of which is mutually dependent in problem solving rather than unrelated opposites. in their model, the thinking begins with an act of visualisation, v1 (see figure 1). it could consist of looking at some ‘picture’ and constructing mental processes or objects. the next step is an act of analysis, a1, which consists of some kind of coordination of the objects and processes constructed in step v1. this analysis can lead to new constructions. in a subsequent act of visualisation, v2, learners return to the same ‘picture’ they used in v1. however, because of the analysis in a1, the picture has changed. as learners repeat the movement between the v and a, they use each act of analysis, based on the previous act of visualisation, to produce new and richer visualisations that they then subject to more sophisticated analyses. this creates a spiral effect. in this model, the acts of analysis deepen the acts of visualisation and vice versa. it is also important to note that, according to this model, as learners repeat the horizontal motion in the model, the acts of visualisation and analysis become successively closer. at first, the passage from one to the other may represent a major mental effort. however, the two kinds of thought become gradually more interrelated and the movement between them becomes less of a concern. the va model suggests that the repetition of these successive visual and analytic acts move closer together over time. the implication of this is that this fusion occurs when learners are able to see the properties of the object emerging from the various representations as a whole and can appreciate that the different representations of the same object emphasise different properties of the object. however, it is still one object, like seeing the object from different perspectives. at the stage when learners can see past the differences in representations and understand the connections between the properties revealed by the different registers, then conversion transformations are less likely to present barriers. therefore, the va theory suggests, that it is at this stage when the two kinds of perspectives merge, that the ease of conversion transformations may be facilitated. on the other hand, when learners view representations from two registers as being separate and unconnected, conversion transformations would be more laborious because the learners do not appreciate the links between the properties that each representation conveys. figure 1: visualisation/analysis model. methodology top ↑ the study utilised an interpretive approach because the main goal of the study was to understand the students’ interpretations of reality (cohen, manion & morrison, 2000) when it comes to solving problems based on the normal distribution curve.the participants were 290 practising teachers who had enrolled in an in-service programme designed to upgrade and retrain mathematics teachers in the further education and training (fet) band. the programme was for an advanced certificate in education (ace) with a mathematics fet specialisation. the programme consisted of eight modules, four of which were specific to mathematics, two of which were generic education modules and two were mathematics education modules. this article focuses on one of the four mathematics modules devoted to a study of introductory probability and statistics suitable for teachers of fet mathematics. the test items was selected in the module specifically for assessment and research purposes and presented the three-part task as part of a summative classroom assessment, which included questions from other sections of the module. one can regard the analysis of the students’ responses as content analysis to throw ‘additional light on the source of communication, its author, and on its intended recipients, those to whom the message is directed’ (cohen et al., 2000, p. 165). in this case, the students’ responses are the source of the communication intended to convey their engagement with the concept. the research questions that focused on one multi-part problem based on the normal distribution are: • are students more likely to succeed in completing the treatment or conversion transformations the problem requires? • what role does the direction of the conversion transformations play in the students’ success rates? the data analysis process involved studying the responses of the 290 students in order to understand the ‘what’, the ‘why’ and the ‘how’ that underlies the data (henning, 2004). dey (1993, p. 30) describes data analysis as ‘a process of resolving data into its constituent components to reveal its characteristic elements and structure’. the students’ responses were broken down into constituent parts that reflected phases of treatments and conversions. i did this to classify and make connections between the data elements (henning, 2004, p. 128). this means presenting ‘the operations by which data are broken down, conceptualised, and put together in new ways’ (strauss & corbin, 1998, p. 120) in order to assess their responses in terms of movement within the same system or between different systems. the students’ responses were then categorised into various categories according to their written explanations. the findings (see below) explain the specific coding, with examples. ethical considerations and recruitment procedures the participants in this study were the teachers who had enrolled in the particular ace programme. all students signed informed consent forms and agreed that their responses could be used on condition that no real names or personal details would be revealed. no student refused permission. reliability and validity the test items were carefully selected after discussing them with a colleague from the united states of america (usa). i ensured that the questions were ones that the students would have encountered in their learning during the course. the language was sufficiently basic to ensure that most students would understand it. i coded the responses myself. however, discussions with an experienced statistics education researcher constituted peer debriefing to improve the credibility of the analysis. peer debriefing occurs when researchers describe the research to peers who ask the ‘why’ and ‘so what’ questions and may suggest alternative frameworks. the test items the tasks used an application of the properties of the standard normal distribution as its basis. when the distribution of a variable in a set of data is approximately normal, one can use the properties of the standard normal distribution curve to make inferences about the variable under discussion. the standard normal distribution has a mean of 0 and a standard deviation of 1. one refers to the scores as z-scores in the standard normal distribution. converting to standard units, or standardising, is the two-step process of re-centring and re-scaling that turns any normal distribution into the standard normal. firstly, one re-centres all the values in the normal distribution by subtracting the mean from each. this results in a distribution with a mean of 0. thereafter, one divides all the values by the standard deviation (re-scaling). this results in a distribution with a standard deviation of 1. this process of re-centring and re-scaling allows one to solve problems like the unknown percentage problem (question 1 and question 2) and unknown value problem (question 3) (watkins et al., 2004). students encountered both types of questions during class discussions and assessments. in unknown percentage problems, students first transform a given value into an associated z-score by re-centring and re-scaling. the next step, in which students identify the probability associated with the z-score and interpret its value in terms of the graph, involves working simultaneously with properties of the standard normal distribution and the properties of particular z-table values. unknown value problems require students first to identify the z-score from a table of z-values that corresponds to a given percentage. thereafter, they calculate the x-score by ‘unstandardising’ it, or reversing the standardising process. the questions under scrutiny in this study are: a university entrance examination scores are scaled so that they are approximately normal. the mean is about 505 and the standard deviation is about 111. 1. find the probability that a randomly selected student has a score below 400. 2. find the probability that a randomly selected student has a score between 450 and 600. 3. the school will offer scholarships to students scoring in the top 10%. what score should be used to decide who should be offered scholarships? remarks about, and solutions to, test items note that these types of questions were familiar to the students because part of the course was devoted to solving such problems using applications of the normal distribution curve. defining the random variable x is important for computing the probabilities associated with the random variable.in this case, the random variable is the entrance examination scores, which have a normal distribution. in order to solve this problem, students received a formula sheet that contained the standardisation formula . the students could use scientific calculators. different statistics textbooks use different tabulation values of a standard normal curve area for a given positive value z0, like p(0 < z < z0) or p(z < z0) or p(z > z0), where these are associated with the area of the corresponding sectors. in the lectures and the assessments, the z-table the students had used was p(0 < z < z0) for positive z0. in order to answer these questions, it is necessary for students to use properties that apply to the standard normal distribution, like having a mean of 0, a standard deviation of 1 and an area (under the curve) of 1. the area under the curve is the probability. the symmetry of the curve means that the area to the left of 0 is equal to the area to the right of 0. because of symmetry at 0, p(-z0 < z < 0) = p(0 < z < z0) and p(z < -z0) = p(z > z0), where z0 is positive and -z0 is negative. question 1: we need to find p(x < 400) = ? the unknown percentage problem requires students to calculate the corresponding z-score from the given x-score using the process of ‘standardising’: . students then identify the percentage that corresponds to the z-score from the z-table and interpret it. figure 2 shows the categorisation of the steps as treatments and conversions. table 1 presents a summary of the solution with explanatory comments and diagrams. question 2: here we need p(450 < x < 600) = ? figure 3 is a diagram that explains the decomposition of the problem into treatments and conversions.table 2 presents a summary of the solution with explanatory comments and diagrams. question 3: we need to find an x-score so that p(x > ?) = 0.1 the unknown value problem requires students first to identify the z-score from a table of z-scores that corresponds to the given percentage. the students can then calculate the x-score by ‘unstandardising’, or reversing the standardising process. figure 4 shows the categorisation of the steps as treatments and conversions. the diagram in figure 4 has the arrows reversed from question 1 to show that the direction of the solution is opposite to that of question 1. table 3 presents a summary of the solution with explanatory comments and diagrams. findings and discussion top ↑ one can regard the standardisation procedure as a treatment transformation because it is within the same register. the x-score is the input and the z-score is the output of the procedure. as duval predicted, most students did not experience problems at this point. for question 1 and question 2, students could complete the standardisation procedure in one register with the visualisation serving only as ‘a second register to serve as a support or guide for the treatments being carried out in another register’ (duval, 2006, p. 127).for question 3, the situation is a bit different because the conversions are necessary because we need to choose ‘the register in which the necessary treatments can be carried out most economically or most powerfully’ (duval, 2006, p. 127) which permitted the process of unstandardising of the z-score. one could not access the z-score without doing a c onversion operation which would allow movement from the percentage value to the z-score. figure 2: question 1 broken down in terms of conversions and treatments. figure 3: question 2 broken down in terms of conversions and treatments. figure 4: question 3 broken down in terms of conversions and treatments. table 1: solution to question 1 with explanations. table 2: solution to question 2 with explanations. table 3: solutions to question 3 with explanations. it is necessary to distinguish between direct and inverse problems (groetsch, 1999) in this study. a direct problem is one that asks for an output when students have the input and the process. in an inverse problem, students have the output and the problem could ask for the input or the process that led to the output. one can regard question 1 as a direct problem and question 2 as a two–step direct problem. one can regard question 3 as an inverse problem because it consists of a conversion that takes a p-value and converts it to a z-score. the z-table is organised according to the z-scores. for a given z-value, students can read off a corresponding p-value. in question 3, the students had a probability value and had to scan the tables until they identified a suitable z-score that corresponded to the given probability. secondly, the formula in the formula sheet was the standardisation formula . in question 1 and question 2, the students used the formula in the form presented. the value x was the input and the output was z. however, for question 3, the students had output z and they had to calculate the input. therefore, one can regard question 3 as a combination of two inverse problems and as an inverse problem in the way that groetsch (1999) described.in order to present the analysis, the students’ responses are labelled to serve as references, for example, s17 which means the response was that of student 17. students’ responses are labelled from s1 to s290. the students’ responses are verbatim, although the layout has been changed because of limited space. findings for question 1 × blank or unrelated algorithm here responses were coded blank if students made no attempt. a response was coded as unrelated algorithms if students wrote a formula where the algorithms did not relate to the standardisation procedure. two examples follow: ■ partial treatments (pt) here responses were coded as partial treatments (pt) if students wrote the appropriate standardisation formula but did not substitute the correct values or substituted the correct values but did not compute the result correctly, for example: ■ complete or full treatments (ft) here responses were coded as complete or full treatments (ft) if students completed the standardisation and arrived at the correct figure of -0.945 or if they wrote the value 0.945 as the value they would read off from the z-table. if they went on to other steps that were incorrect, then the responses were coded as ft. for example, some students (12) did not read off a p-value from the z-table and interpreted the z-score as a probability. an example follows: some students continued and used the resulting ‘probability value’ (obtained as for s1) to determine a z-score in the z-table. an example follows: here the student used the z-score (-0.946) as a probability value, found the z-score that corresponded to the ‘probability value’ and presented the z-score (1.83) as a probability (even though it was greater than 1). ■ partial conversions (pc) responses were coded as partial conversions (pc) if students determined a p-value from the z-table that corresponded to the z-score even if the value was not accurate as long as there was a reading of a p-value from a related z-score. an example follows: where 0.95 corresponds to a z-score of 33.65% ■ complete or full conversions (fc) here responses were coded as complete or full conversions (fc) if students interpreted the p-values of the z-table in terms of the area under the curve to provide correct (or nearly correct) answers.each step depends on the previous step. therefore, a student who completed an fc, would have done the pc, ft and pt steps. table 4 shows that, of the 290 students, 223 (77%) were able to recognise the correct standardisation formula. only 199 (69%) were able to complete the standardisation procedure correctly. fifty-five (19%) performed partial conversions and 79 (27%) completed the conversions and the question. table 4: results for question 1. in order to get a clearer idea of how the students progressed from the treatment steps to the conversion steps, we can consider the cumulative totals: • the number of students who managed partial treatments will include those who completed the treatments • those who completed the treatments will include those who managed partial conversions • those who managed partial conversions will include those who completed full conversions. the bar graph in figure 5 gives these numbers. of the 290 students, 223 (77%) students began the appropriate standardisation procedure. of these 223 students, 199 (89%) completed the standardisation treatments and of these, 134 (67%) were able to complete the first part of the conversions. seventy-nine (59%) of the last group were able to complete the conversions correctly. findings for question 2 the following codes were used for question 2. it is not necessary to give examples of responses in all categories because they are similar to those for question 1 except that there are two sets of treatments and conversions. × blank or unrelated algorithm ○ partial treatments (pt), where students chose the appropriate standardisation formula (in one or in both cases) but did not complete both. ● full or complete treatments (ft), where students completed the standardisation procedure in one or in both cases but completed no further correct steps. □ partial conversions (pc), where students read off a p-value from the z-table in one or in both cases, but did not combine the two p-values correctly, for example: ■ full or complete conversions (fc), where students interpreted the p-values of the z-table in terms of the area under curve to provide correct (or nearly correct) answers.table 5 shows that, of the 290 students, 174 (60%) started one or both standardisation procedures, whilst only 156 (54%) were able to complete one or both standardisation procedures correctly. only 40 students (14%) completed the questions correctly (two of whom had a final answer that differed slightly from the expected one). table 5: results for question 2. in order to get a clearer idea of how the students progressed from the treatment steps to the conversion steps, i considered the cumulative totals from right to left:• the number of students who managed partial treatments will include those who completed treatments • those who completed treatments will include those who managed partial conversions • those who managed partial conversions will include those who completed full conversions. the bar graph in figure 5 gives these numbers. of the 290 students, 174 (60%) were able to recognise the correct standardisation formula, whilst only 156 (90%) of these student were able to complete it correctly once or twice. of these 156 students, 96 (62%) completed only the first part of the conversions once or twice (they read off the p-value for the corresponding z-score). only 40 (42%) of these were able to complete the conversions and arrive at the correct result. findings for question 3 the following codes were used for question 3:× blank or unrelated algorithm □ partial conversions (pc), where students interpreted the percentage value given as a p-value, which was the correct one (p = 0.4), but did not carry out any further correct steps or could have interpreted the percentage as an incorrect p-value. ■ full or complete conversions (fc), where students read off p-values in a z-table to generate a z-score which was correct or incorrect; students who completed full conversions all continued.○ partial treatments (pt), where students chose the appropriate formula for unstandardising a z-score. ● full or complete treatments (ft), where students completed the procedure for unstandardisation correctly or nearly correctly. table 6: results for question 3. the response of s133’s was coded almost correct compared to that of s135, where the final answer was not close to the expected one.table 6 shows that, of the 290 students, 108 students did not respond and 34 used an irrelevant algorithm. therefore, 142 (49%) did not even begin partial conversions. seventy-eight (27%) tried but did not generate the correct p-value whilst 20 (7%) students completed partial conversions by correctly extracting the p-value from the information the students had. three (1%) students completed the conversions and started the unstandardising treatments, whilst 47 (15%) students managed complete treatments and obtained a correct or almost correct solution (the final answer that 26 students reached differed slightly from the expected answer). in order to get a clearer idea of how the students progressed from the conversion steps to the treatment steps, i considered the cumulative totals from right to left: • the number of students who completed partial conversions will include those who completed full conversions • those who completed full conversions will include those who completed partial treatments • those who completed partial treatments will include those who completed full treatments. the bar graph in figure 6 gives these figures. there were 148 (51%) students who started the conversions (obtained p-values). of these 148 students, 50 were able to complete the conversions by reading off p-values and chose the correct formula for unstandardising. that is, 34% completed the conversions (read off the p-values for the corresponding z-score) and started treatments whilst 47 (94%) of the 50 students were able to complete the treatments and solve the problem (the final answers of 26 students differed slightly from the expected one). performance on the three questions students clearly found that question 2 was more challenging than question 1 was. only 40 students got question 2 correct whilst 79 students managed to complete question 1 correctly – almost twice as many. furthermore, there were 67 blank or incorrect algorithms for question 1 compared to 116 for question 2. this showed that more students did not attempt to solve question 2 than those who failed to attempt question 1. it is clear that question 2 was more complex than question 1 because it involves regions bounded by two given x-scores. therefore, there were two sets of treatments as well as two sets of partial conversions and completing the conversions meant that students had to take a global view of the two areas and decide how they would use them to generate the required percentages. consequently, solving question 2 would have been more demanding than just carrying out treatments followed by conversions, as question 1 required. question 3 was challenging for the 142 (49%) students who did not start correctly. forty-seven completed the whole question correctly or almost correctly. this was more than the 40 who completed question 2 correctly or almost correctly but fewer than the 69 who completed question 1 correctly or almost correctly. if one compares performance on question 3 with that on question 1, 67 students did not start question 1 correctly. on the other hand, there were more than twice as many (142) students who did not begin question 3 correctly. there are two possible reasons for this. firstly, the inverse nature of the question meant that the steps to the solution were reversed, which made it more complex (bansilal, mkhwanazi & mahlabela, in press; groetsch, 1999; nathan & koedinger, 2000). secondly, students had to complete the conversions before the treatments. this created a bigger first barrier than the situation where the first barrier was not as great as the second was. duval’s (2006) theory maintains that conversion transformations are more difficult than treatment transformations are because they require crossing into another register of representation. conversions are more complex because they involve movement in each of the two registers and movement across them, whilst treatments require movement in one register only. success rates in conversion transformations and treatment transformations the bar graph in figure 5 provides a visual representation of the progress of students through the stages for question 1 and question 2. it shows the number of students who did a pt, ft, ft pc and ft fc respectively and excludes the students who made no response or used a wrong formula. note that, in this graph, the first set includes the second, which includes the third, which includes the fourth and derives from the figures tables 4 and table 5 provide. figure 5: number of students progressing at each stage to the final solution for question 1 and question 2. figure 6: number of students progressing at each stage in question 3 to the final solution. the cumulative picture for question 3 (see figure 6) shows the number of students who completed a pc, fc, fc pt and fc ft respectively. the first set includes the second, which includes the third, which includes the fourth. these figures derive from the information table 6 provides.there are clear trends in performance on question 1 and question 2. of the 290 students, 223 (77%) performed a pt on question 1. of these, 199 (89%) completed the treatments. of this group, 134 (67%) went on to complete a pc and 79 (59%) of this group were successful. for question 2, the numbers from table 2 are 290 (original), 174 (pt), 156 (ft), 96 (pc) and 40 (fc). the flow diagrams below show these figures: question 1: 100% → (pt) 77% → (ft) 89% → (pc) 67% → (fc) 59% question 2: 100% → (pt) 60% → (ft) 90% → (pc) 62% → (fc) 42% the attrition rate at each stage of question 2 was higher than that for question 1, except for the progression from partial treatments to full treatments, where 90%of students who managed partial treatments for question 2 completed the treatments. the corresponding percentage for question 1 was 89%. however, for all other stages, the progression rate from one to the next was higher for question 1 than it was for question 2. on both questions, the highest attrition rate was in the progress from pc to fc. it showed that only 59% of students who started conversions for question 1 completed them, whilst for question 2 only 42% of students who started the conversions were able to complete them. when one considers the performance on question 3, the numbers from table 6 are 290, 148 (pc), 50 (fc), 50 (pt), 47 (ft). the flow diagram below shows the figures: question 3: 100% → (pc) 51% → (fc) 34% → (pt) 100% → (ft) 94% here, as for question 1 and question 2, the highest attrition rate was in the movement from pc to fc. only 34% of the group who started conversions were able to complete them and all of these students went on to start treatments. thereafter, there were few challenges for this group and only three students did not complete the procedure. the treatment procedure for question 3 was not a problem for those students who completed their conversions. forty-seven of the 50 students (94%) who completed conversions were able to complete treatments. the conversions were problems in question 1 and question 2. they were insurmountable for many, because only 79 of the 199 (39%) and 40 of the 156 (25%) of the students who completed treatments were successful with conversions. a comparison between trends in responses across the questions supports duval’s assertion that conversion transformations can be more complex than treatments. for question 1 and question 2, the percentage of students who proceeded from full treatments to full conversions was 39% and 25% respectively, whilst for question 3 the percentage of students who proceeded from full conversions to full treatments was 94%. it is clear that, for the group as a whole, the students’ success rates in conversion transformations were lower than in treatment transformations. however, not all the students would have experienced conversions as more difficult than treatments. the movement between the two registers was not a problem for some students. direction of conversions the direction of conversions is another factor that duval contends affects the complexity of mathematical activities. duval maintains that a ‘conversion in one direction can be without any cognitive link with this in the reverse direction’ (duval, 2008, p. 47), suggesting that the direction of the conversions is important. duval has shown that, when the original and destination registers of conversions change, students’ performances vary considerably. in one case of linear algebra, 83% of students were able to move successfully between a two-dimensional table representation of a vector to a two-dimensional graphical representation, whereas only 34% of students were able to move in the opposite direction.the direction of the conversions seems to have been a factor that influenced the students’ success rates. sixty-nine students completed question 1 correctly, whilst only 40 students did so on question 3. of the students who started conversions for question 1, 59% were able to complete them, whilst only 34% of the students who started conversions for question 3 were able to do so. the reason for the lower completion rate for the conversions for question 3 could lie in the fact that the conversion transformation of question 1 involved moving from the z-scores to the probability value (or area) that travelled in the opposite direction to the conversion in question 3 (moving from the probability value to the z-score). in addition, 89% of the students who completed conversions for question 3 went on to complete the treatments. therefore, the conversions were bigger hurdles. the percentage of students who proceeded from full treatments to full conversions in question 1 was 39%. one of the factors that made question 3 more challenging was the direction of the conversions, which was different in the two cases. duval’s own observations about linear algebra (2008) support this. however, we need further research to help us understand why conversions in one direction were more challenging to complete than were conversions in another. summary top ↑ this article presented an analysis of 290 students’ responses to a three-part task using applications of the normal distribution curve. duval’s framework was used to explain the students’ difficulties with solving the task.question 1 and question 2 of the task are ‘unknown percentage problems’ and question 3 is an example of an ‘unknown value problem’ (watkins et al., 2004) and one can regard it as an inverse problem (groetsch, 1999). different parts of the solutions to the questions were categorised into conversions and treatments, depending on whether the operation required students to move across a register or stay within the same register. the students’ responses were coded according to whether they performed partial treatments, complete treatments, partial conversions or complete conversions. the findings show that question 2 was more difficult than question 1: twice as many students completed question 1 correctly compared to question 2. it was argued that question 2 was more challenging because students had to complete two sets of conversions and two sets of treatments. the results of these transformations had to be synthesised together to produce an answer. it was also found that question 3 was more challenging than question 1 was. seventy-nine students obtained correct answers for question 1 and only 47 obtained correct, or close to correct, answers for question 3. it was argued that one factor could be the inverse nature of question 3, whilst question 1 was a direct problem. the other factor could be that students needed to complete the conversion transformations for question 3 before the treatment transformations. furthermore, because the conversions were bigger hurdles, more students could not progress further. the students encountered the treatment transformations first in question 1. more students succeeded with this hurdle than with the first hurdle in question 3, allowing them to progress. duval’s theory that conversions are more challenging than treatments is supported by the findings in this study. when the attrition rate is examined at each stage in each of the three questions, there were clear patterns in the performance of the students. on question 1 and question 2, 59% and 42%, respectively, of the group that started conversions were able to complete them. this compares to approximately 90% of the group who started treatments who were able to complete at least one treatment. in addition, only 34% of the group who started conversions for question 3 were able to complete them, whereas 94% of the group who started treatments were able to complete them. this shows that completing the conversions was harder than completing the treatments in all three of the questions. furthermore, this study supports duval’s (2006) examples in linear algebra that show that the direction of conversions also plays a role in the difficulty level of questions. he writes that ‘when the roles of source register and target register are inverted within a semiotic representation, the problem is radically changed for students’ and that ‘performances vary according to the pairs (source register, target register)’ (p. 122, brackets added). this was true for question 1 and question 3. in question 1, if one considers the group of 134 who completed the treatments, then 79 of these (or 58%) succeeded in completing the conversions when the movement was from z0 to p(z < z0). in question 3, when the movement was from p(z > z0) to z0, the success rate was 34% (50 of the 148 had identified some sort of p-value). this shows that the students found the second conversion more difficult. if one considers the percentages for the whole group of 290, then 79 of the 290 (or 27%) were able to complete conversions for question 1 whilst only 50 of the 290 (or 17%) were able to complete conversions for question 3. implications of the findings duval (2006) differentiated between treatments and conversions and commented that ‘we cannot deeply analyse and understand the problem of mathematics comprehension for most learners if we do not start by separating the two types of representation transformation’ (p. 127). this study has also shown that conversions and treatments in this problem offer different levels of challenges to students. therefore, educators should note the additional challenge of moving between systems of representations. the findings suggest that educators may need to support conversion transformations more than treatment transformations to help learners to overcome the challenges. one aspect that deserves notice is that this group of students did not receive any computer-aided instruction, nor could they work through computer simulations of normal curves, as normally happens in probability and statistics modules nowadays. if they had had some exposure, they might have had a better idea of the visual aspects of the normal distribution curve and may have been able to switch between representations more easily. applets or other computer simulation activities could allow students to engage with the properties the different representations reveal. they could also help students to explore situations that show links between the changes in the z-scores with the changes in the area values in the different modes of representation. drawing on zazkis et al.’s (1996) va model, perhaps such opportunities will help students move more effortlessly between the different registers, thus reducing the barriers related to carrying out conversion transformations. the solutions to these questions involved coordinating two different registers, which were initially separate. however, zazkis et al. (1996) suggest, in their va model, that even though movement between two modes may start as distinct and separate, they eventually merge. zazkis et al. confine their discussion to the movement between the acts of visualisation and analysis. however, we can apply it to the two registers that we have identified here to suggest that, at some point, the students will regard the combination of these two registers as one that enriches their ‘cognitive architecture’ (duval, 2006), and which will enable them to move on to further layers of movement between more complicated registers. finally, this article delved into students’ engagements with the treatment and conversion transformations associated with one particular problem. readers may want to consider whether one could look at other areas in similar ways and whether they could help to explain the students’ difficulties in those areas. it is hoped that this study will encourage other researchers to look for evidence to support or contradict these findings in other areas. additionally, it is hoped that such further research would help to illuminate further the challenges that learners experience when they work with problems that involve moving across different registers of representation. acknowledgements top ↑ i acknowledge a grant from the united states agency for international development (usaid), administered through the non-governmental organisation higher education for development for research on the different modules in the ace certification programme. there was no specific grant for this article.i also acknowledges the contribution from thomas schroeder (university at buffalo, state university of new york [suny], usa), who assisted with a preliminary report on this project, sketched the normal distribution curves in the article and acted as peer debriefer during the analysis process. competing interests i declare that i have no financial or personal relationship(s) that may have inappropriately influenced me when i wrote this article. references top ↑ bansilal, s., & naidoo, j. 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(1996). using visual and analytic strategies: a study of students’ understanding of permutation and symmetry groups. journal for research in mathematics education, 27(4), 435–457. http://dx.doi.org/10.2307/749876 microsoft word 22-29 willy mwakapenda.doc 22 pythagoras, 71, 22-29 (july 2010) integrating mathematics and other learning  areas: emerging tensions from a study involving  four classroom teachers    willy mwakapenda & joseph dhlamini    tshwane university of technology    mwakapendawwj@tut.ac.za & jjthemba@yahoo.com    this paper presents findings from a pilot study that investigated the extent to which teachers make  connections between mathematical concepts and concepts from other disciplines. data from concept  maps and interviews were collected. the analysis revealed that the kinds of connections teachers  made are closely tied to teachers’ disciplines of specialisation. the findings suggest that for some  teachers,  though  desirable,  it  may  not  be  feasible  to  require  them  to  make  connections  with  disciplines  that are not within  their areas of specialisation. this presents  tensions  for  learners  learning  mathematics  in  classrooms  where  opportunities  for  making  connections  between  mathematics and other learning areas are available but are neither taken up nor appropriately used  by teachers.  many school curriculum reforms have included the notion of integration (presmeg, 2006). in south africa, there are calls for schooling to shift from following structured curricula marked by a separation of bodies of knowledge (snyder, 2000), to interdisciplinary curricula. according to davison, miller and metheny (1995), the concept of integration is interpreted in many different ways. the absence of uniform commonalities in interpretations of this concept has led to the unavailability of a common definition for integration. however, with specific reference to mathematics, arguments by adler, pournara and graven (2000) have identified three levels of integration: “integration of the various components of mathematics; between mathematics and everyday real world knowledge; and where appropriate, across learning areas” (p. 3). they have argued that while integration is desirable, the extent of the demands placed upon teachers makes integration less feasible. in order for teachers to integrate what they are teaching with other learning areas, teachers need not only to have a sufficient knowledge of their own learning areas, but they also need to have and be aware of a broad range of knowledges within and outside the curriculum. however, there are claims within the new curriculum that integration across learning areas should be more feasible at the lower grades than at the higher because of the difficulty of finding sufficiently generative contexts at the higher levels. according to the department of education (2006), contexts are “situations or conditions in which content is taught, learnt and assessed”. these are derived from “different sources” such as: the nature of the learning area being taught, the socio-economic environment of learners, national and other events, interests, nature and needs of learners, and the integration of appropriate assessment standards from other learning outcomes and other learning areas (p. 27). it is generally understood that contexts are a useful way in which to “integrate” learning areas, and that ways of proceeding with integration determine what kinds of integration are possible. the preceding remarks indicate that the introduction of the new curriculum has placed demands on teachers to adopt integrated teaching, to organise their teaching so that it promotes integration of one learning area with another. however, there have been concerns that teachers are not adequately trained to handle new curriculum demands, given that they are qualified in specific disciplines. adler et al. (2000) have noted that willy mwakapenda & joseph dhlamini 23 the teacher is expected to posses a broad general knowledge of matters unrelated to his or her subject and possibly also to be an expert in other subject areas. this is clearly seldom possible and might leave the teacher feeling powerless to cope with the new demands. (p. 6) in this paper, we focus on how teachers integrate mathematics and the following subjects (learning areas) in the new south african school curriculum: arts and culture (ac), economic and management science (ems) and science. from the given background, it is clear that there is a challenge for teachers to incorporate new pedagogical approaches that emphasise the need to integrate mathematics and these learning areas. there is a need to understand how teachers think about integration across subjects. how they think about integration may shape how they integrate across disciplines in actual classroom practice. this paper reports on a pilot study involving four teachers: one grade 9 ac teacher, one grade 7 ems teacher, one grade 7 mathematics teacher, and one grade 11 science teacher. it is important to establish how teachers deal with situations in which they are called upon to reflect on their mathematical knowledge during lesson planning and teaching. the paper reports on aspects of the following key research questions in relation to the pilot study: o within the context of a new curriculum in south africa, what connections do teachers make between mathematical concepts and concepts in their disciplines of specialisation? o what do these connections mean within the context of mathematics classroom practice in the new south african curriculum? theoretical framework the study from which this paper emerged was guided by a theory of situated learning. situated theories are founded on the premise that knowledge is situated, and is a product of the activity, context, and culture in which it is developed and used (brown, collins, & duguid, 1989). within this perspective, one learns differently in different situations, so learning is situated within a context. adler et al. (2000) have argued that transferring knowledge from one setting (context) to another is always problematic: “knowledge and skills cannot be neatly lifted out of one setting and imported ready-to-use into a new setting” (p. 11). the use of situated theory highlights the fact that implementing the notion of integration is potentially problematic, as mathematics and arts and culture, for example, present two different learning contexts. how one links concepts from one discipline to another depends on one’s understanding of the possibilities of connections that are available. such connections are highly dependent on the contexts involved (dhlamini, 2009; mwakapenda, 2008). integration is seen to be a key driving principle of the new south african curriculum (adler et al., 2000). after the new curriculum was introduced in south africa, it became clear that teachers had to adopt new pedagogical approaches to teaching and learning. teachers are expected to implement integrated teaching in their lessons. however, as already pointed out, there is a concern that teachers do not have the requisite knowledge and are not well oriented to manage the demands of integrating across disciplines (adler et al., 2000; czerniak, weber, sandmann, & ahern, 1999; huntley, 1999). this raises pedagogical concerns, particularly given that teacher’s content knowledge is a prerequisite in order to facilitate connections between disciplines or subjects (huntley, 1999). shulman (1986) presents key ideas related to the pedagogical content knowledge that teachers need in order to teach a discipline. shulman argues that a teacher needs to be familiar with the curriculum and concepts that her students are learning in other subjects at the same time that they are studying her subject. within the new south african context, little is known about how teachers work with the new pedagogical demands to integrate across subjects. what connections exist between mathematics and other disciplines such as arts and culture, ems, and science? according to beckmann, michelsen and sriraman (2005), there is a “here is a “historical lineage of connections between mathematics, arts and science” (p. 2). arts and culture provide possibilities to visualise mathematical thinking and expressing mathematical thoughts which might be difficult to comprehend theoretically. mathematics, on the other hand, can contribute to the solution of significant unresolved cultural and social problems, for instance, global birth control and epidemic control (beckmann et al., 2005). on the other hand, ems as a learning area is fairly new at the current grade 7 integrating mathematics and other learning areas 24 level in south africa. it is also already an integrated learning area consisting of different subjects: economics, leadership and management, entrepreneurship and financial knowledge. in bernstein’s (1982) terms, ems is a weakly classified subject, that is, it has the potential to allow other learning areas to integrate with it. it is widely acknowledged that mathematics and science are inextricably connected. for example, there are many school mathematics topics such as “vectors” that appear in mathematics as well as in science textbooks. conventional science topics such as “energy” frequently incorporate mathematical concepts such as “equation”. in order to develop science knowledge, scientists frequently invoke the use of mathematics. such strong connections have even prompted mathematicians such as gauss to refer to mathematics as “the queen of the sciences” (bell, 1951, p. 2). bell (1951) has provided illustrative examples of connections between mathematics and science disciplines. as can be seen from the above, there are inherent connections between mathematics and arts and culture, ems, and science. however, to what extent are these and other connections known to classroom teachers? what kinds of connections do teachers see between these disciplines? study design and data collection a qualitative descriptive research methodology was used, and because qualitative research is inherently multi-method in focus (naidoo & parker, 2005), data was collected through concept mapping tasks and interviews with teachers. taking the perspective of malone and dekkers (1984), this study considered that concept maps are “windows to the minds” of the teachers we work with for reflecting on teachers’ perceptions and meanings. it is viewed that maps “facilitate a sharing of meaning unhampered by any lack of verbal skills” (p. 231). the concept mapping tasks which were administered to teachers were as follows: the arts and culture (ac) teacher was asked to draw a concept map involving the following concepts (selected from a topic in a grade 9 arts and culture textbook): angle; area; colour; dance; design; dimension; melody; parallel; pattern; percentage; positive. the ems teacher and the mathematics teacher were asked to draw concept maps showing links between the following concepts taken from the ems curriculum: graph; ratio; percentage; salary; product; expand. the science teacher was asked to draw, in her own time, a concept map linking the concepts: work, equation, energy, power, solutions and function. these concept mapping tasks took place in different schools at different times. teachers’ responses to the concept mapping tasks were followed up with interviews. research participants the research involved a sample of teachers, purposively selected, from participating schools in gauteng. the ac teacher was teaching arts and culture on a fulltime basis and had teaching experience of approximately 15 years. the teacher indicated that he was quite familiar with both the old and the new curricula. he acknowledged having attended new curriculum training workshops. the ems teacher had teaching experience in economics and business economics at higher levels. the ems teacher was also an ems masters student at a tertiary institution and had over 5 years experience teaching ems in grade 7. the mathematics teacher had over 10 years’ experience in teaching mathematics in grade 7. the mathematics teacher was part of the transition from the old traditional curriculum to the new integrated curriculum and was an honours student at a tertiary institution. the science teacher was qualified as a mathematics and physical science teacher, and had been teaching for nearly 20 years and was the head of department for science and natural science at the time of the study. results and observations the arts and culture teacher figure 1 shows the concept map drawn by the arts and culture teacher. as can be seen from figure 1, there are three components in the map: it shows concepts that seem to belong to the top part “pure maths”; secondly, those that are “pure arts and culture” concepts, and those that “integrate between mathematics and arts and culture”. the teacher felt that the concepts colour, melody, dance and design were pure arts and culture concepts, and so could not be linked to mathematics. this is interesting, willy mwakapenda & joseph dhlamini 25 particularly in light of graumann’s (2005) comments below in relation to the connection between mathematics and music: rhythm and notation is a relatively simple mathematical field, which provides a good opportunity for application-oriented practice of fractions. the determination of pitches and scales respectively tunes is a big chapter in which the development in the theory music from pythagoras to twelve-tone music can be opened up by mathematics. in this context, the ancient theory of music can serve as a field of application for fractions as well. (p. 3) figure 1: the arts and culture teacher’s concept map the absence of connections between these concepts and mathematics was further confirmed in the interview with the arts and culture teacher when she repeatedly stressed that the concepts colour and music could not be linked to mathematics. however she was passionate about integration and acknowledged that “teachers should be trained on integration” because some of her colleagues were “still struggling with integration”. she noted that they sometimes relied on other teachers as well as learners for assistance with integration. according to her, “the learners will definitely help you, you can give them a problem, they will give you answers that you did not expect, they will integrate”. she identified ‘area’ and ‘percentage’ as the only concepts that can be linked to mathematics. during the interview, she insisted that ‘percentage’ is a mathematics concept. she stated: “in arts i don’t talk about percentage”. the teacher noted that integration is encouraged in curriculum documents through the statement of learning outcomes (lo) acknowledged that some of the los encouraged the integration of mathematics with arts and culture. she particularly quoted assessment standard 9 in the grade 9 arts and culture curriculum in which the concepts “positive” and “negative” are mentioned, namely that there are “positive and negative effects of television, radio, documentaries or films on our lives” (department of education, 2003). she acknowledged that such topics are likely to stimulate mathematical discussions, thus opening up opportunities for connections between the two subjects. she noted that learners can “learn two things at the same time” when such topics emerge during lessons. the ems teacher and the mathematics teacher figures 2 and 3 show the concept maps drawn by the ems teacher and the mathematics teacher. as indicated earlier, the ems teacher and the mathematics teacher were given the same concept mapping task, that is, to draw a concept indicating how the concepts graph; ratio; percentage; salary; product; expand are connected. an examination of the two concept maps shows that the connections made by these two teachers are more tied to everyday situations than to mathematics. integrating mathematics and other learning areas 26 figure 2: the ems teacher’s concept map figure 3: the mathematics teacher’s concept map the mathematics teacher used the everyday context of a business: “gotha’s safety doors”. he then linked the given concepts to this context. although this is an interesting way of making connections, it needs to be pointed out that the nature of some of the connections made may be problematic, in a mathematical sense. for example, what does the statement involving the concept expand: “expand from 10 workers to 15 workers” (see figure 3) mean? increasing the workforce from “10 to 15 workers” might be considered as “expanding a business”, in an everyday context. however, the concept expand has a rather different meaning in mathematics. it is interesting that the mathematics teacher did not seem to have incorporated more mathematical connections in relation to this concept. the way the mathematics teacher thinks about the concept product (see figure 3) is also interesting. the links made by the ems teacher in relation to the concept product (see figure 2) also has connections that concern more of the everyday world than that of mathematics. figure 2 displays an understanding of product that has an everyday and realistic association: “tangible”, “end product” and products that have some “durability”. while the concept maps in figures 2 and 3 show that teachers are able to make links between concepts in ems and mathematics, the nature of the links appears to be highly contextualised. the understanding of the connections between concepts in this way (i.e. highly contextualised, located in a business and real world context) may be limited given that it appears to be strongly tied to one knowledge domain. an understanding that is tied to one discipline may make it difficult for teachers to integrate successfully and meaningfully as the curriculum requires. the science teacher figure 4 shows the science teacher’s concept map involving the concepts: work, equation, energy, power, solutions and function. a look at the map shown in figure 4 shows that it has detailed connections between concepts that are more linked to science (i.e. work, energy, power). however, the concepts solutions, function and equation are not explicitly indicated on the map. when asked to explain the absence of the concept solutions on the map, the teacher said the following during the interview: solutions? is calculations isn’t? i do not know what you want me to do. can i give an example? … you see the problems that matriculants do are quite complicated hence i did not want to put them on the map. … you see when we give the questions, and it is long problems hence when i give a problem i don’t say find a solution. i give activities where they will read and analyse, and extract information. if you want me to i can bring a textbook and we will look at an example. as can be seen from the above interview excerpt, the teacher pointed out that she did not seem to immediately see the connections between the concept solutions and work, energy and power. however, the willy mwakapenda & joseph dhlamini 27 science teacher recognised the need to use mathematics in teaching the concepts work, energy and power. the question that remains is: why didn’t she include the “mathematics” concepts on her map? when asked why she did not include the concepts, she said, “i didn’t know how to put them for you in a nice way”. some of her responses, such as: “solutions? is calculations, isn’t?” indicate that the teacher was possibly uncertain on whether and how the mathematics concepts may be linked to science. this is surprising, given that the concept solution is frequently used in mathematics classrooms. this teacher had been a mathematics teacher for several years. figure 4: the science teacher’s concept map however, the science teacher made attempts to relate her science knowledge to mathematics. this can be seen in figure 5, given by her during the interview. figure 5: the science teacher’s use of mathematics clearly mathematics concepts are seen to be useful for solving problems in science. this is evident from the equation form in which the formula is written, the substitutions that have been made, and the changing of “grams” into “kilograms” and the manipulations (e.g. multiplication and division) required to obtain an answer. however, as can be seen below, the teacher pointed out that changing the subject of the formula may not be essential in science activity: we do not mark the maths, we mark science, we want to see if learners understand that you are to use a particular equation, for example ep = mgh, and if you are able to substitute your ep, your m and g, when you are supposed to calculate h. if i can see that they understand this concept i am happy. how you deal with it by making h the subject is the maths teachers’ problem. say for example you are given that ek = 20 j, m = 100 g, and you are to find v. the first problem would be to substitute ek correctly into the correct equation, the second problem would be to establish that grams are not the correct si units. thirdly they might try to change the subject of the formula which might be a challenge. if they do correct substitution into the correct equation, they will get 4 out of 5 marks if the final answer is incorrect... but if they try to change the subject of the formula and mess it up they get 1 out of 5 for initially using the correct formula. this is the maths teachers’ problem: to ensure that the subject of the formula is changed properly, not mine. my concern is science, not mathematics. i just do not want to see them messing up my science. integrating mathematics and other learning areas 28 the above remarks suggest that according to the teacher, some aspects of mathematics may not be particularly important to science. specifically, they may not be required for successful performance in science as long as those aspects “do not mess up the science”. it appears that the science teacher may allow learners to “get away” with what might be mathematical flaws provided these do not interfere with the science activity in question. this contradicts with the intentions of the south african new curriculum which encourages teacher to look inside as well as outside their learning areas. discussion and implications the above analysis has demonstrated that the arts and culture teacher did not consider concepts such as melody and design as being linked to mathematics in spite of available evidence that these two concepts have important mathematical connections. although the ems teacher and the mathematics teacher were both able to link ems concepts (e.g. product, expand) to mathematics, the nature of the links made were more related to the everyday context than to deeper mathematical meanings associated with the given concepts. while the science teacher was able to make detailed connections between concepts that were more about science (work, energy, power), the more mathematical concepts (solutions, equation, function) were not explicitly included in her concept map. in a follow-up interview, the science teacher made it clear that making attempts to ensure that mathematical aspects of a science activity are appropriately considered may be at the expense of “messing up the science”. the findings from these four brief cases have implications for classroom practice. these cases reveal some of the complexities related to translating curriculum expectations into practice particularly in relation to the requirement that teaching needs to integrate across learning areas (naidoo & parker, 2005). we need to acknowledge that working in integrated ways in the school curriculum opens up new opportunities and realities for schooling. integration appears to be a complex activity for teachers who are likely to meet with challenges both within and beyond their specific curriculum disciplines. there is a need to understand the identities of teachers and learners (their beliefs about mathematics and learning) in order to facilitate the implementation of the notion of integration. there is a critical challenge here that concerns the mathematics, and in general, the knowledge that is needed for teaching and learning. for example, at present, we do not know much about what happens when the mathematics needed in order to enable learners to understand a science concept that requires mathematics is not well understood, both by learners and teachers themselves. the analysis in the paper demonstrates that when teaching science (or indeed any non-mathematical subject), there is a need to unpack the mathematical aspects that might be important to effect the learning that is being sought. we need to document and analyse the mathematical aspects that are inherent in a discipline or unit being taught, and to make available this knowledge to teachers. however, this analysis shows integration may not succeed in cases where the teacher is socialised to believe that their teaching roles are restricted to those subjects in which they have been trained or are “responsible” for. the analysis of the case of the science teacher reveals some of the tensions that learners may face when studying mathematics in classrooms where opportunities for making connections between mathematics and other learning areas are available but are either neglected or inappropriately used by teachers. acknowledgement this paper emerges from the research project exploring understanding of school mathematics and the use of concept mapping, supported by a grant from the national research foundation (nrf). we acknowledge contributions from prevein marnewicke and dumazi mkansi, who also participated in the concept mapping research. willy mwakapenda & joseph dhlamini 29 references adler, j., pournara, c., & graven, m. 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/hrv (za stvaranje adobe pdf dokumenata najpogodnijih za visokokvalitetni ispis prije tiskanja koristite ove postavke. stvoreni pdf dokumenti mogu se otvoriti acrobat i adobe reader 5.0 i kasnijim verzijama.) /hun 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can be opened with acrobat and adobe reader 5.0 and later.) >> /namespace [ (adobe) (common) (1.0) ] /othernamespaces [ << /asreaderspreads false /cropimagestoframes true /errorcontrol /warnandcontinue /flattenerignorespreadoverrides false /includeguidesgrids false /includenonprinting false /includeslug false /namespace [ (adobe) (indesign) (4.0) ] /omitplacedbitmaps false /omitplacedeps false /omitplacedpdf false /simulateoverprint /legacy >> << /addbleedmarks false /addcolorbars false /addcropmarks false /addpageinfo false /addregmarks false /convertcolors /converttocmyk /destinationprofilename () /destinationprofileselector /documentcmyk /downsample16bitimages true /flattenerpreset << /presetselector /mediumresolution >> /formelements false /generatestructure false /includebookmarks false /includehyperlinks false /includeinteractive false /includelayers false /includeprofiles false /multimediahandling /useobjectsettings /namespace [ (adobe) (creativesuite) (2.0) ] /pdfxoutputintentprofileselector /documentcmyk /preserveediting true /untaggedcmykhandling /leaveuntagged /untaggedrgbhandling /usedocumentprofile /usedocumentbleed false >> ] >> setdistillerparams << /hwresolution [2400 2400] /pagesize [612.000 792.000] >> setpagedevice article information author: bruce brown1 affiliation: 1education department, rhodes university, south africa correspondence to: bruce brown email: b.brown@ru.ac.za postal address: po box 94, grahamstown 6140, south africa dates: received: 17 july 2014 accepted: 06 apr. 2015 published: 19 june 2015 how to cite this article: brown, b., 2015, ‘the relational nature of rational numbers’, pythagoras, 36(1), art. #273, 8 pages. http://dx.doi.org/10.4102/pythagoras.v36i1.273 copyright notice: © 2015. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. the relational nature of rational numbers in this original research... open access • abstract • introduction • transitions from whole to rational number understanding • the research projects • rational numbers as relations    • relational comparisons in the common introduction to rational numbers    • prior learning and rational comparisons    • changes in the concept of ‘quantity’ in rational number learning    • teachers and the relational nature of rational numbers    • the relational nature of rational numbers in repeated subdivisions • the number line: magnitude as a relational concept    • identity and embedding    • the relational nature of rational numbers and the number line • possible implications for teaching • conclusion • acknowledgements    • competing interests • references abstract top ↑ it is commonly accepted that the knowledge and learning of rational numbers is more complex than that of the whole number field. this complexity includes the broader range of application of rational numbers, the increased level of technical complexity in the mathematical structure and symbol systems of this field and the more complex nature of many conceptual properties of the rational number field. research on rational number learning is divided as to whether children's difficulties in learning rational numbers arise only from the increased complexity or also include elements of conceptual change. this article argues for a fundamental conceptual difference between whole and rational numbers. it develops the position that rational numbers are fundamentally relational in nature and that the move from absolute counts to relative comparisons leads to a further level of abstraction in our understanding of number and quantity. the argument is based on a number of qualitative, in-depth research projects with children and adults. these research projects indicated the importance of such a relational understanding in both the learning and teaching of rational numbers, as well as in adult representations of rational numbers on the number line. acknowledgement of such a conceptual change could have important consequences for the teaching and learning of rational numbers. introduction top ↑ the 21st century has seen a resurgence of research, in both mathematics education and cognitive psychology, into the teaching, learning and understanding of rational numbers. a number of different approaches are taken in this research. some emphasise the unity between whole and rational numbers, based on the observation that both represent magnitudes: quantities that can be represented by a point on a number line (siegler, thompson & schneider, 2011). they advocate ways of teaching that build on this unifying property. others, such as privileged domain theories (gelman & williams, 1998) and conceptual change theories (ni & zhou, 2005), emphasise the discontinuity between rational number concepts and whole number concepts and investigate the conflicts these introduce into rational number learning. in particular, vanvakoussi and vosniadou (2004) and mcmullen, laakkonen, hannula-sormunen and lehtinen (2015) have identified a ‘whole number bias’ in peoples’ responses, when comparing the magnitudes of two rational numbers. with regard to prior knowledge, it is generally accepted that rational number learning builds on a foundation of whole number understanding. additional prior knowledge that is important for rational number learning but is not directly related to whole number understanding has been identified by mcmullen, hannula-sormunen and lehtinen (2013, 2014). this involves a comparison relational scheme, which provides a relational foundation for rational number understanding. this article seeks to identify the fundamental nature of the discontinuity between whole and rational number concepts and relate this to the unifying conception of magnitude on a number line. it advances the argument that the relational property, as well is being important for the early teaching and learning of rational numbers, is also important for rational number understanding of senior phase and further education and training learners and of adults. furthermore, the relational property introduces a fundamental conceptual difference between whole number and rational number quantities. this difference involves a change from a quantity as an absolute count, to a quantity as a relative comparison. in addition, there is a corresponding increase in the level of abstraction of the concept of quantity. transitions from whole to rational number understanding top ↑ in the literature of both cognitive psychology and mathematics education, it is acknowledged that the development of rational number understanding involves a number of changes in the child's experience related to numbers. these may be viewed in different ways and this section will start with a brief review of these different approaches in the literature. the changes in children's experience result in the development of more complex operational schemata for numbers. wright (2014) includes operations such as unitising (including equal subdivision of wholes into part units and consolidation of collections of wholes into grouped units), iteration, rational stretching and shrinking, partitioning and allocation. these operations allow a number of functionally different conceptual structures relating to rational numbers, identified by kieren (1976, 1988) and behr, lesh, post and silver (1983). the structures are characterised as the part-whole, measurement, quotient, operator and ratio sub-constructs. in addition, the child develops more complex numerical representations, such as fraction and decimal notation, and a more complex relation between symbol and number, seen in the multiple fraction representations of the same rational number. the conceptual change orientation to rational number learning (mcmullen et al. 2015; ni & zhou, 2005; vanvakoussi & vosniadou, 2004) identifies and investigates a number of differences in the properties of whole and rational numbers. many of these are properties of whole numbers, that are not properties of rational numbers. examples (siegler, fazio, bailey & zhou, 2013) are: each whole number may be separated from the others by an open interval, the product of two whole numbers is generally greater than each and the whole numbers can be counted in their natural order. two differences that have become an important focus of conceptual change research are the density of the rational numbers in the real numbers and the complexity of the process of ordering and comparing two rational numbers. according to vanvakoussi and vosniadou (2004) and mcmullen et al. (2015), responses when comparing the magnitudes of two rational numbers appear to be related to the manner in which these numbers are represented symbolically as fractions or decimals, as well as to the distance between their magnitudes on the number line. it appears as if the use of two whole numbers to represent fractions may have an important influence on comparisons of their magnitudes, a property termed the ‘whole number bias’. the child's earlier development of whole number knowledge thus has an important influence on their developing rational number concept and this influence extends into the developed understanding of adults. the privileged prior nature of whole number learning is thus held to interfere with later rational number learning, a view that is shared by privileged domain theories (gelman & williams, 1998). whole number knowledge does not appear to be the only prior knowledge of importance for rational number learning. the work of mcmullen et al. (2013, 2014) indicates that a cognitive comparison relational scheme, which is not directly related to whole number knowledge, contributes an additional foundation for rational number learning. the power of using such relational understanding for early formal rational number learning has been demonstrated by cortina, visnovska and zuniga (2014). they showed that the capacity to view the measurement unit and the quantity to be measured as separate entities contributed greatly to learning the inverse ordering relationship for unit fractions. the integrated theory of siegler et al. (2011) challenges the view that whole number learning interferes with fraction learning. siegler et al. maintain that the property of magnitude is the ‘only property that unites real numbers’ (siegler et al., 2013) and emphasise the importance of basing rational number teaching on the concept of magnitude. in this approach, the number line is used as a representational model for the conception of rational numbers as magnitudes. this model emphasises the one-dimensional, ordered nature of the number system. a teaching approach based on magnitude rather than the standard part-whole model has been researched by moss and case (1999) and has shown positive improvements in learning. each of these views on rational number learning offers important insights for the teaching and learning process. but taken together they may be read as conflicting and contradictory. this article advances the view that relational understanding provides a unifying perspective on these different views. it argues that conceptual change is indeed important in the learning transition from whole number to rational number thinking and that this discontinuity stems from the nature of the rational number concept. the interference of whole number properties in rational number learning then derives from the fundamental change in the nature of the concept of quantity, from absolute count to relative comparison. this change is brought on by incorporating the relational nature of rational numbers into the child's thinking about numbers. finally, this article supports the opinion that the property of magnitude unites whole and rational numbers, but it will argue that the concept of magnitude, which allows this unity, relies on a relational interpretation of number. the achievement of this unity will thus occur with the successful negotiation of this fundamental transition in rational number learning. the research projects top ↑ this article draws from four exploratory research projects investigating rational number teaching and learning at different levels: in grade r, grade 3, grades 4–7 and with teachers and student teachers. the overarching focus of these research projects was to investigate the manner in which teachers and children made meaning of rational numbers and the ways in which teaching influenced the meaning that children made of rational numbers. each project focused on a different question relating to this issue. in the grade r project, ways of developing grade r children's foundational knowledge for rational number learning were explored. the grade 3 research focused on learners’ everyday knowledge relating to rational number learning. in grades 4–7, ways of teaching and learning rational numbers were investigated in a school. the fourth project focused on student teachers’ and teachers’ knowledge of rational numbers. the primary focus of this article, the fundamental nature of the discontinuity between whole and rational number concepts and the corresponding transition in teaching and learning, emerged as an important concern during the course of these projects. these were all small-scale, qualitative research projects. the first three involved english-medium primary or pre-schools in the eastern cape. the schools were all historically advantaged, although the first had an even mix of children from historically advantaged and disadvantaged backgrounds. this article will discuss the whole number-rational number discontinuity, drawing on episodes from these projects to illustrate the emergence of this concern. rational numbers as relations top ↑ relational comparisons in the common introduction to rational numbers it is still common practice (kilpatrick, swafford & swindell, 2001; lamon, 2007) for children to be introduced to rational numbers by working with parts of wholes. fractions are then used to quantify the ‘size’ of the part in relation to the ‘size’ of the whole. in this way, the child experiences particular instances of rational numbers describing parts of objects, such as half of an apple, a third of a cake and even three-quarters of a slice of bread. by extensive repetition, it is expected that the child will abstract this descriptive relationship to form a conception of a rational number, in much the same fashion that earlier they abstracted the concept of a counting (whole) number from multiple instances of counts of objects. concrete or drawn objects are soon supplanted by partially shaded diagrams that the child learns to describe using part-whole ‘double counts’. prior learning and rational comparisons the research of mcmullen et al. (2013, 2014) indicates the importance of the scheme of relational comparison as prior knowledge for the early learning of rational numbers. such a scheme was evident in both the grade r and the grade 3 research. in the grade 3 project, in-depth clinical interviews with two teachers and two learners were carried out to explore children's everyday knowledge for rational numbers. in one task, children were asked to allocate dough to containers of different sizes for cooking. they both competently carried out this task, relating the relative sizes of the containers to the relative sizes of the dough. but neither child was able to describe what they had done using quantities or relational terms; the most that they could offer was that one was bigger than the other. the grade r investigation reported in this article involved a number of clinical interviews with four children, each of whom could count confidently from 0‒50. a number of tasks were formulated that may contribute to the development of the children's relational capacity. children were asked to carry out the tasks and their engagement with the rational relation was investigated. the first task involved collections of easily identifiable objects, such as models of people and drawings of bicycles and tricycles. for each collection, children were asked to count the objects in the collection and also to count the number of specific components of the objects in the collection (arms for people, wheels for bicycles and tricycles). toy models were presented for the collections of people and pictures of bicycles and tricycles were shown. the tasks for each object were presented in sequence, starting with a single object, then presenting collections with more and more objects. finally a number of objects was specified (without a physical or diagrammatical presentation) and the child was asked to work out how many components would be in a collection with that number of objects. firstly, no help was given for the final task, but if little progress was made, the child was encouraged to draw. for each sequence, the children confidently counted wholes and components. also, they stated how many components were included in each whole. for example, two arms for each person. when asked to compare the counts of wholes and components, they would only state that there were more components than wholes, and if more detail was requested, they reiterated the relationship for each whole. in the final task of the sequence, they were not able to determine the number of components without a drawing, but they were able to make an appropriate drawing and correctly count the components from this. in a further task sequence, the children were asked to count handles and wheels of a number of tricycles. in this case, they were not able to formulate the relationship of 2 handles to 3 wheels (even for a single tricycle) and also they found great difficulty in drawing a collection with a given number of handles. it appeared as if the children were working with a constitutive object-component relationship for each object (how many arms were needed to form a person), rather than a numerical relationship between quantities. for when asked to describe the relationship, they either gave the number of components for each whole or they stated that there were more components than wholes. and without prompting, they did not give the total number of objects and components for comparison. in the final sequence, the only response was that there were more wheels than handles, showing some difficulty in working with component-component relationships. in subsequent tasks, drawings of scoops of ice cream in different containers were used. firstly, the children were shown a drawing of a single container and asked to count the number of scoops in this container. then they were shown pictures of different numbers of containers and asked how many scoops of ice cream would be needed for the containers. children worked with three sequences, first with cones (2 scoops per cone), then cups (3 scoops per cup) and lastly bowls (6 scoops per bowl). each child was able to identify and describe the relation for each container (each cone has 2, each cup has 3 and each bowl has 6). for each sequence, they were then shown a single empty container and then two empty containers. they all needed some help with these first two counts (in the form of the researcher verbally reiterating the number of scoops per container shown by the first picture) as they became familiar with the task. they generally worked by indicating empty containers with their fingers and counting finger taps for each ice cream scoop in the container. having successfully completing the two-container task, they were all able to confidently and quickly count how many scoops were needed in different numbers of cones or cups, showing an effective use of the 1–2 and 1–3 relation. for the bowl, they found it difficult to count finger taps, tending to lose count and so count either more or less than six for some bowls, but they could confidently draw and then count the final result. the children did not use fractions to describe what they were doing. they also did not describe the relationships between their counts of objects and components (or containers and ice cream scoops) in multiplicative terms. but they were able to confidently use and describe the number of components for each object and the number of ice cream scoops per container. note that it is the invariant numerical relationship between object and component, or container and scoop, that is quantified by a rational number. from the children's responses, it appeared as if this numerical relationship may be founded on their initial awareness of the constitutive relationship between object and components (or container and contents), the necessary relationship for properly forming the object or filling the container. this prior knowledge could form an experiential foundation for the child's developing concept of rational number quantities. changes in the concept of ‘quantity’ in rational number learning it is commonly accepted (see verschaffel, greer & de corte, 2007, for a review) that a young child's learning of whole numbers builds on the conception of a quantity as a completed count of discrete objects. the process by which this conceptualisation is developed has been extensively researched. when children use whole numbers to identify the numerator and denominator of a fraction, they will thus be working conceptually with two completed counts of discrete objects. in the classical introduction to fractions, these are: the completed count of discrete pieces in the whole and the completed count of discrete pieces making up the identified part. but the rational number is neither of these quantities. rather, it quantifies the relationship between these two whole number quantities. what is more, because of the density of the rational numbers, it is not possible to reduce this relationship to a single count. no matter what counting unit is chosen, infinitely many rational relationships would require a subdivision of this unit to be accurately quantified. quantifying these relationships would then require two counts (one of them a count for the subdivision). for this reason, quantifying a rational relationship is intrinsically more complex than quantifying a count. to properly conceptualise rational number quantities, we thus need to change our view of ‘quantity’ from something relating only to single whole number counts, to something more suitable to rational numbers. the grade 4–7 research project involved regular weekly meetings, for the duration of the second and third terms, with eight teachers who taught grade 4‒7 mathematics in the school. teaching materials and samples of learners’ work were collected for analysis. in the third term, teachers were individually interviewed and weekly grade 4 classes were attended, in which the researcher was free to interact with the children. these teachers considered an important element of rational number learning to be the use of fractions to describe the size of the identified portion in a part-whole subdivision. to contribute to this learning, children were asked to complete worksheets containing tasks such as that shown in figure 1. it was expected that the comparison relation in part (b) would be determined by a simple visual comparison of the shaded regions (valid because the wholes were the same size). figure 1: example fraction task: what is being quantified? in answering this question, most children confidently counted the total number of subdivisions in each block, as well as the number of shaded subdivisions, to arrive at fractions (a: 5/8 and b: 4/6) describing each part-whole configuration. but in part (b) many children responded a > b, a relationship that would fit a comparison of counts of the total number of blocks and of the numbers of shaded (or unshaded) blocks. in the following question, where the comparison was between two unit parts, many of these children had responded 1/8 > 1/6. when it was mentioned to these children that part (b) was asking them to use the fraction numbers to describe what they saw (which looked bigger) they immediately and confidently changed the relation to a < b. without any further prompting, many also changed their following response to 1/8 < 1/6. it appeared from this interaction that the children's initial response was because they did not understand that part (b) was asking for a simple visual comparison. that is, that they were not clear that the purpose of part (b) was to show one way of making sense of the fraction quantities constructed in part (a). rather, they seemed to see this task as asking them to use their understanding of whole numbers as counts, and the relational signs as describing order between whole numbers, to compare these composite fraction symbols. once it became clear to them that the comparison was not of the counts, but of a different (visual) comparison of size, they confidently and correctly completed the worksheet. once the orientation of the task had been clarified, the comparison task helped the children realise that fraction quantities were not the same as simple counts. most of the class readily came to this realisation. but the comparison did depend on the fact that in each question, the wholes were the same size, a constraint that was not clearly specified in the task. thus, the importance of the whole was downplayed in this task. in the same research project, a different teaching sequence was observed that brought in the size of the whole for consideration. it was interesting that the children did not appear to engage readily with this aspect of the task. this teaching sequence involved concrete experiences of sharing, as related to fractions. in the first two lessons, the teacher grouped the children into twos, threes or fours and asked them to share some given food items (such as 3 hot dogs or 1 cookie) fairly amongst the group. the children took great care to ensure that the portions were all equal. they were also able to confidently use fractions to describe the shares, showing familiarity and competence with fraction notation. in the third lesson, the class was grouped into pairs and each pair was given an apple to share. in preparing the lesson, the teacher had deliberately chosen apples of different sizes for sharing. the children divided their apples quite accurately into halves and reported that each had received a half of their apple. then the teacher asked if all the children had the same amount of apple. the initial response of the children was that they did: they all had half an apple. when asked to physically compare their halves they noticed the difference in size, but they found this difficult to reconcile with the fact that they had shared fairly in their pairs to each get half an apple. the teacher explained that the original apples were of different sizes and so the halves would be different, but this explanation did not appear to dispel the confusion. the fair sharing in pairs, together with the fact that each had the same fraction of an apple appeared to outweigh the explanation given after the sharing, that the whole apples were initially different sizes. we were left with the question of whether the effect of the whole may have been more readily acknowledged if the children had been explicitly asked to compare the sizes of the wholes before the sharing and if the aim of the activity had been presented as ‘looking to see what would happen if different size wholes were shared’. later work with individual children based on a supplementary task developed to explore the relational theme suggested that children may naturally identify the size of the objects being halved as a reason for halves being different sizes. the task given was: james has a chocolate. he gives half to peter. he gives half of what is left to bob. are these two halves the same?. here the immediate response was that these halves were not the same. the reason given was that, for bob, what was divided was smaller. these interactions suggest that, with suitable activities, grade 4 children may come to see rational number quantities as related to the amount in a part, rather than a simple count of pieces. also that they may come to understand that the size of these amounts varies with both the rational number describing the amount and the size of what is considered as the whole. three important components may be identified in this relationship: the part amount, the whole amount and the comparative rational number. activities suitable for the grade 4 child may allow the investigation of the effect of changing any single element on another, whilst holding the third fixed. it is questionable whether learners of this age would be able to effectively engage with the effects of covariation of two elements; this would require considerably more sophistication in thinking than would be expected of a grade 4 child. teachers and the relational nature of rational numbers a relational understanding is also useful for teachers, allowing more flexible and appropriate interpretations of children's responses. an example of this is provided in the teacher's interpretation of answers (shown in figure 2) to the following question in a grade 5 class test: use drawings to show how you would share 5 slices of bread among 2 people. what fraction would each person get? figure 2: responses to sharing five slices of bread between two people. the first two responses were unsurprising, but the third was different. nonetheless, the teacher suspected that the child may have appropriately thought through the question. after the test, she asked the child to clarify the response. the child explained that each slice was cut into two equal parts and five of these ten pieces were then given to each person. so each person got 5/10 of the bread. in this response, the fraction describing each person's share was obtained by comparing the share with the total amount of bread. this is in contrast to the other two, which compared the share to one slice of bread. here the rational number describing the share changes, depending on the choice of reference unit. it describes the relationship between the share and the chosen unit. that is, the rational number is a relative quantity and the chosen reference unit needs to be known in order for the absolute magnitude of the share to be known. understanding this relationship allowed the teacher to respond appropriately to the child's answer, resulting in a positive learning experience. the relational nature of rational numbers in repeated subdivisions the importance of relative comparisons in the early learning of rational numbers raised the question of whether such comparisons may be important in the developed adult concept. to further explore this question, a further research project was undertaken, in which a number of simple rational number tasks were administered first to five teachers and later to 22 student teachers. the participants were observed as they worked on these tasks. their responses are discussed in this section. in tasks relating to repeated subdivision, adults did appear to relate fraction quantities to a reference whole, although this whole was not always appropriately chosen. they were appropriate in adults’ responses to the simple question described before: james has a chocolate. he gives half to peter. he gives half of what is left to bob. are these two halves the same? here the immediate response was ‘no’: peter received more because he received half of the whole chocolate, whilst bob received half of a half. many adults immediately added that bob received a quarter of the chocolate. converting the representation of bob's amount to a fraction of a whole allowed a direct comparison of the quantities as parts of the same whole. others did not, identifying instead the different sizes of what was subdivided. with prompting, these adults made the conversion and quantified the comparison. however, when asked to work with repeated subdivisions in a more abstract task, many adults did not choose appropriate reference units. adults were asked to draw each of the fraction quantities in the repeated subdivision: 2/3 of 3/4 of 1/2 and then to draw the final amount. the response of all the adults (both teachers and students) was to separately draw each of these fractions as parts of a whole of the same size and shape. to draw the final amount, some of the adults (mainly grade 9–12 teachers) drew a single diagram showing the successive subdivision of a single whole, successfully identifying the composite quantity. to enable the other adults to draw the final amount, it was generally necessary to draw their attention to the repeated nature of the subdivision as indicated by the term ‘of’. a further task was the following puzzle: you make three inventions to improve the fuel efficiency of a machine. one saves 30% of fuel, a second saves 45% and a third saves 25%. how much fuel can you save if you use all three inventions at once? in their first approach to this puzzle, all the adults first added the percentages together to come to a combined saving of 100%. they then noted that a 100% savings was not practically possible and many looked for alternative approaches. some, however, saw this as demonstrating the difference between mathematics and the ‘real world’ and were happy to say that 100% gave a mathematical solution that was unrealistic. those who searched for an alternative approach generally resorted to some sort of averaging procedure. one group of adults was given this question immediately after the chocolate subdivision question and they were told that these problems could be usefully related. but this had very little effect on their solutions. this group were then prompted to see the inclusion of the inventions as a process in which one was included after the other, and to calculate the savings accordingly. in response, half of the group appropriately calculated composite proportions of fuel use (70%, 55% of 70% and 75% of 55% of 70%). the remainder needed a second prompt: asking what the initial fuel use was before the second invention was included. once the process of finding cumulative proportions became clear, the adults responded convincingly to the puzzle (with varying degrees of calculational efficiency). these responses indicated that, for both abstract and situated repeated subdivision tasks, adults naturally interpreted the rational quantity in relation to a reference unit. but, unless they were clearly and explicitly stated as being different, the reference unit was taken as the same for each rational quantity in the task. this made abstract repeated subdivisions rather confusing. these results suggest that a relational comparison may indeed be part of adults’ conception of rational numbers, but that the reference unit may generally be standardised, resulting in a lack of flexibility in the relational use of this concept. the number line: magnitude as a relational concept top ↑ identity and embedding the observation that the property of magnitude unites whole and rational numbers is fundamental for the integrated theory of siegler et al. (2011). but deducing from this that there is no fundamental conceptual change in the child's transition from whole number thinking to rational number thinking depends on a conflation between the ideas of identity and of embedding. for magnitude is indeed a common property of these numbers, when viewing both whole numbers and rational numbers as embedded in the real number field. but the child, who has no conception of real numbers, does not view these numbers as real numbers and so this may not be a common property when viewed in terms of conceptual development. to be precise, this is a common property of the images of whole and rational numbers under the respective embeddings of the whole and rational number structures into the real number field. this configuration of embeddings is represented diagrammatically in figure 3. figure 3: embedding whole and rational numbers in real numbers. an identification of a mathematical object with its image under an embedding is common in axiomatic mathematical practice. in this case, the embedding allows the domain to inherit the additional structure of the range space. but, when investigating conceptual development, this inherited structure originates in the additional conceptual structure of the range, not in the original conceptual structure of the domain. the property of magnitude that unifies whole and rational numbers is the capacity of these quantities to be represented by a point on the real number line. this is a property of the images of these numbers when embedded in the real numbers. but the question remains whether this is a property of these number concepts before they are subsumed in the development of the later and richer real number concept. this question may be answered by the investigation of the relational nature of the number line as a model of quantity. the relational nature of rational numbers and the number line to investigate adults’ use of the number line as a conceptual model for rational numbers, students and teachers were presented with a blank number line marked only with a zero point and asked: where can you place 15/4 on the given number line? in each case, the respondent's first action was to mark ‘1’ as a unit on the line. the immediacy with which this was carried out suggested that this action was an automatic first response to the number line task. once the unit had been marked, adults determined the approximate position of 15/4, based on the distance between 0 and 1. when asked if 15/4 could have been placed anywhere else, they replied ‘no’. the observation that in each case, the unit (1) was placed first, and that this placement was immediate, suggests that a reference interval, such as [0.1], may be an implicit component of the number line that forms a conceptual model for rational numbers. the placement of the unit identifies an interval that determines the scale of the line and this uniquely determines the position of any other number on the line. it may be noted that, even though the research participants each placed a ‘1’ as the unit, the scale would have been determined by the placement of any non-zero number on the line. in a second query, i called their attention to the fact that they had first placed the 1 and asked where they could place the 15/4 before the 1 was placed. generally, the response to this was ‘anywhere’. these adults were thus aware that a number line with a zero but no reference interval (or scale) did not fully define absolute magnitude: the 15/4 could then be placed anywhere. the placement of a magnitude as a point on the line only becomes absolute once a single reference quantity has been placed, through the need to maintain the correct rational relationship with the reference quantity. the need for a reference unit to fully specify the point for any number on the line (even a whole number) suggests that this conceptual model of magnitude is a relational (relative) model, rather than an absolute model. that is, working with whole or rational numbers as magnitudes uniquely placed on the number line will require a conceptual transition from numbers as absolute counts, to numbers as relational, in this case relative to the chosen reference unit. this is the same transition that this article argues is fundamental for the child's learning of rational numbers. taking magnitude to be fundamental to the abstract concept of rational numbers thus implies that the relational property is fundamental to the concept of a rational number, even in the abstract. as a further observation, the immediate placement of ‘1’ as a unit by the adult participants suggests that the reference unit is an implicit aspect of the concept that is also generally standardised. that is, flexibility in the choice of the reference unit may not be a well-developed aspect of the rational number concept. this is supported by the observation that adults first responded as if the position of the number 15/4 was fully determined and also by the observation that common diagrams of fractions in repeated subdivisions, such as 2/3 of 3/4 of 1/2, showed each fraction as a part of a whole of the same size. in both cases, when the possibility of changing the reference unit was brought to their attention, adult participants were able to usefully engage with the flexible, relational nature of the rational number. but this flexible relational engagement generally needed to be prompted. possible implications for teaching top ↑ this view of the conceptual change between whole and rational numbers has a number of implications for teaching. an important change at the outset would be to strengthen and build on the informal relational comparison scheme. then, as the child's whole number competence grows, to develop ways to quantify these comparisons by relating counts, in this way deliberately developing rational numbers as a level of abstraction above that of whole numbers. the use of activities involving the measurement sub-construct and unit fractions as a means to quantify subdivision would also contribute to this development. the grade r research project included the investigation of a number of such activities at the grade r level. this will be reported in a subsequent article. also, as is evident in the discussion of adults’ use of the number line, using number lines and teaching approaches building on magnitude will also contribute to developing this relational understanding. another explicit focus that may prove valuable for the rational number transition would be the move to relative, rather than absolute quantities: looking at fractions as ‘how much of the whole’ (relative) rather than ‘how much in total’ (absolute). in the work with children in these projects, and particularly in the work at grade 4 level, it appeared as if acknowledging the fundamental difference between whole numbers as counts and rational numbers as relational comparisons may be a powerful lever for helping the child to achieve the transition to rational number thinking. in a number of instances, discussing whole numbers and rational numbers as different but related concepts seemed to make it easier for the child to consider whole and rational numbers alongside each other and so explore their consonances and dissonances to learn the interrelations between these two concepts. from a motivational perspective, acknowledging this difference when learning rational numbers allowed the child to attribute differences in properties of whole and rational numbers to differences between the types of numbers being considered, rather than to their own misunderstanding of number. as a result, children appeared to have more control and be more confident as they navigated this conflicting conceptual process. further research related to this issue may prove valuable for the practice of teaching. conclusion top ↑ the rational number conceptual field shows more complexity than the whole number field, both in the scope of application of rational numbers (relating in different ways to a broader range of phenomena) and in the level of technical proficiency required to master the symbol systems of this field. this article argues for a third increase in complexity: that rational numbers are fundamentally relational in nature and that the move from absolute counts to relative comparisons requires a further level of abstraction in our understanding of number and quantity. that is, that the conceptual development from whole numbers to rational numbers does not simply involve the ‘filling in’ of the set of possible values for quantities (to form a dense, although not yet complete, set). rather it involves a change in understanding of what a quantity is (from an absolute count to a relative measure) and a corresponding change in understanding of how a quantity may be measured. this argument is based on an in-depth analysis of data from four small-scale qualitative research projects. such an approach allows the development of deep insight into the particulars of the learning situation, but the conclusions reached are grounded only in the data of the particular situation. to increase the empirical base of these conclusions, further research would be necessary. in particular, further investigation of the relational nature of the number line for indicating magnitude will be important. also, teaching experiments designed to investigate teaching that explicitly acknowledges the difference between the nature of counting and rational numbers would be fruitful. acknowledgements top ↑ competing interests the author declares that he has no financial or personal relationships that may have inappropriately influenced him in writing this article. references top ↑ behr, m., lesh, r., post, t., & silver, e.a. 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(2014). towards a hypothetical learning trajectory for rational number. mathematics education research journal, 26(3), 635–657. http://dx.doi.org/10.1007/s13394-014-0117-8 66 p64-75 mudaly final 64 pythagoras 66, december, 2007, pp. 64-75 proof and proving in secondary school vimolan mudaly university of kwazulu-natal email: mudalyv@ukzn.ac.za this paper argues that mathematics should be a human activity in which the process of guided invention takes the learner through the various stages and steps of the discovery of mathematical ideas and concepts. the central premise is that proof functions as an explanation, where experimental mathematics is used to empirically convince the learner that the conjecture holds true. inductive reasoning used while working with the computer software sketchpad assists learners in arriving at and testing their conjectures. introduction there are various definitions and conceptions of the word ‘proof’. to most mathematicians, proof plays the role of verification. but does this conception of proof satisfy a similar need for the masses of people not inclined to become mathematicians? in his book, i believe, rampa (1977) makes the following comment: oh humans always want proof of everything, they even want proof that they are humans, but how can you prove a thing…if a thing is true it needs no proof because it is self evident that the thing is there, but if a thing is not true and if it is not there then no amount of 'proof' will prove that it is there so there is no point in trying to prove anything. (1977: 01) is this a reflection of the way proof is viewed by most people? this paper argues otherwise. proof-making in geometry is a difficult task in mathematics classrooms, and this surely must have contributed to the ‘math-o-phobia’ that has plagued schools. the evidence available shows that proving in mathematics should be left only to mathematicians. a study by suydam (1985: 483) showed that about 50% of learners saw no need to prove what they considered obvious. senk (1985: 454) found that only 30% of learners attained 70% mastery of six geometry problems involving proofs. usiskin (1982) also determined that although 50% of secondary school graduates completed a year of geometry fewer than 15% mastered proof-writing. bell (1976: 23) carried out an investigation of 160 grammar school girls and discovered that only 10% of them attained van hiele stage 3 – the stage at which learners could give an acceptably complete, deductive argument (proof). reynolds (in bell, 1979: 370) studied the “proof concepts of grammar school learners” and concluded that, in general, formal axiomatic proof was not understood even by 17-year old learners specialising in mathematical and scientific subjects. williams (driscoll, 1988: 156) surveyed eleventh grade learners and found that fewer than 30% exhibited any understanding of the meaning of proof, and that almost 60% were unwilling to argue, for the sake of argument, from any hypothesis they considered false. these statistics and the experiences of mathematics educators in general have created a sense of urgency in attempting save proof within euclidean geometry. more importantly, an attempt should be made to resurrect the beauty of – and the need for – proof in euclidean geometry. this paper does not engage in discussion regarding the need for proof, as much has already been written on the topic. what is it about the way in which euclidean geometry is traditionally taught that creates the impression that there is a need for drastic change? many authors have been drawn to this topic and the general consensus seems to lie in the way proof is taught. much of our teaching of proof centres on a content-driven curriculum. the emphasis is placed on the factual aspects of proof and has therefore precipitated the belief that learning of proof is simply the transmission of knowledge from the source (the educator or textbook) to the recipient (the learner). the teacher undoubtedly becomes the authoritative source of all knowledge, along with the textbook. a quick perusal of a few mathematics educational journals, will reveal that there are many methods of teaching proof which deviate from the traditional methods. this paper does not intend to imply that the statistics listed in the opening paragraph are entirely the result of poor traditional teaching. vimolan mudaly 65 there are many reasons for learners’ poor performances in constructing proof in examinations and class work: • insufficient emphasis on proof heuristics. teachers prefer the direct teaching method, namely, presenting proof directly. • negative attitude as a result of not understanding the role/meaning of proof when the concept was first introduced to learners. • learners may be working at the incorrect van hiele level when attempting proof. • the negative attitude of learners towards mathematics in general. • the negative attitude of some teachers towards their learners and the subject. however, traditional teaching of proof has been dominated by an authoritarian method. as davis and hersh (1983: 282) state: “then there is a desire on the part of some teachers to appear brilliant. (what i’m telling you is pretty easy and obvious to me, and if you’re not getting it, you really must be pretty stupid)”. strangely enough, the view that proof is necessary only for verification has been even more dominant. the traditional role of proof has been seen mainly in terms of verification of the correctness of mathematical statements. in other words, proof serves the explicit function of convincing sceptics about the truth of a statement. coe and ruthven (1984: 42) summarise this in their claim that “the most salient function of proof is that it provides grounds for belief”. in fact, a survey in 1984 by de villiers (1990: 18) revealed that more than 50% of higher education diploma students in mathematics education agreed that the only function of proof was that of “making sure”, that is, the verification of the truth of the results. despite the dominance of this view, several authors have cautioned against stereotyped thinking. bell (1976) states that: conviction is normally reached by quite other means than following a logical proof; proof is essentially a public activity of validation which follows the reaching of conviction, though it may be conducted internally. (1976: 24) similarly, hersh (1993: 390) observes that, “more than whether a conjecture is correct, mathematicians must know why it is correct”. reid (1996: 185) echoes this sentiment: “i would like to question the common assumption that the role of deductive reasoning or proving in mathematics is the verification of conjectures”. the view that proof is necessary mainly for verification often ignores a simple fact: the learner is ultimately responsible for his/her own learning and therefore his/her participation is essential. it is important to bear in mind that despite the learner’s prior knowledge, s/he cannot easily make meaning of new concepts. this paper presents an argument for proof using reasoning, and make the following proposition: if evidence can be presented to a learner that would support the knowledge we want the learner to learn, in a visually-active way, then learning is made easier. this hypothesis is based on the adage: ‘i hear and i forget, i see and i remember, i do and i understand’. the approach is based on real-world and problem-centred approaches in mathematics, and is underpinned by constructivist theory. it thus accepts that “the learners have their own ideas, that these persist despite teaching and that they develop in a way characteristic of the person and the way they experience things, leads inevitably to the idea that, in learning, people construct their own meaning” (brookes, 1994: 12). learners can easily determine a correspondence between what they know and the new knowledge they ‘see’ unfolding as they work through a realworld exercise. often there may be a conflict between the old knowledge and the new knowledge they are discovering. cognitive restructuring of knowledge takes place, where the new knowledge is assimilated using existing schemas that were already established. this is closely linked to the problem-centred learning (pcl) approach developed in south africa in the mid 1980’s by researchers at the university of stellenbosch. the pcl approach is based on a socio-constructivist theory of the nature of knowledge and learning and hinges on the following aspects (olivier, murray & human, 1992): • the learner is actively engaged in the process of acquiring knowledge. • the learner draws on past experiences and existing knowledge. • learning is a social process in which new knowledge is acquired through interaction with other learners and educators. (1992: 33) the hans freudenthal institute has, since 1971, been developing the theory of realistic mathematics education. this is strongly influenced by hans freudenthal's concept of mathematics as a human activity, and takes into consideration what mathematics really is, how it should be taught and how learners should actually learn. selden and selden (1999) state that: from the perspective of realistic mathematics education, students learn proof and proving in secondary school 66 mathematics by mathematising the subject matter through examining 'realistic' situations, i.e., experientially real contexts for students that draw on their current mathematical understandings. (1999: 9) this paper draws attention to an important consideration, that mathematics must be a human activity, which is presented in a way that resembles the manner in which it was discovered. in other words, mathematics teaching must be organised in such a way that the process of guided invention takes the learner through the various stages and steps of the discovery of mathematical ideas and concepts. proof as a means of explanation the paper argues that proof serves as a means of explanation. tiles (1991) provides a definition that encapsulates the function of proof: by proof is meant a deductively valid, rationally compelling argument which shows why this must be so... (1991: 7) this function of proof helps the individual make sense of a mathematical result and satisfies the individual’s curiosity as to why it may be true. this aspect has been neglected because proof has been seen as performing only the function of verification. coe and ruthven (1984: 42) claim that less emphasis has been placed on explanation because much writing about proof “has been from a philosophical rather than a pedagogical perspective”. however, hanna (1996: 16) states that “with today’s stress on ‘meaningful’ mathematics, teachers are being encouraged to focus on the explanation of mathematical concepts”. gale (1990) states that “the main goal of all science is to first observe and then to explain. in mathematics the explanation is the proof” [emphasis by author]. schoenfeld (1985) sums up this important function of proof succinctly: ‘prove it to me’ comes to mean ‘explain to me why it is true’, and argumentation (proof) becomes a form of explanation, a means of conveying understanding. (1985: 172) although it is possible to achieve a high level of conviction that a conjecture holds true by using experimentation, this does not provide a deeper understanding as to why the conjecture may be true (de villiers, 1990: 19). experimentation, especially if it is computer-driven, may provide a large degree of certainty but it does not necessarily provide the insight or understanding of how the result may be true as a consequence of other already established results. hersh (1993: 396) states that “what proof should do for the student is provide insight into why the theorem is true” and at the high-school level, “the primary role of proof is explanation” (ibid: 398). anderson (1996) provides an appropriate summary of the explanatory role of proof in establishing a deeper understanding of why certain results always hold true: proof should be seen as being about explaining, albeit carefully and precisely. it is where instrumental understanding gives way to relational understanding. it should be seen as the essence of mathematics and all learners who study mathematics should meet it at some time, at some level. (1996: 32) slomson (1996: 12) expressed the idea that “good proofs not only convince us of the truth of mathematical statements, but also helps us to understand what is going on”. a number of authors emphasise the important role of proof as a means of explanation in mathematics: the mathematician’s reaction shows quite clearly that a proof which does nothing but prove in the sense of mere verification must be unsatisfactory. a proof is also expected to generalise, to enrich our intuition, to conquer new objects, on which our mind may subsist. (otte, 1994: 310) the functions of proof are to generate knew knowledge and to advance mathematical understanding [emphasis by author]. (kitcher, 1984: 189) the best proof, even in the eyes of practicing mathematicians, is one that not only establishes the truth of a theorem but also helps understand it. such a proof is also more persuasive and more likely to be accepted. (hanna, 1996: 135) this paper thus focuses attention on the role of proof as a form of explanation. teaching experiments this paper reports on two different teaching experiments conducted with learners at a secondary school. the experiments build on research conducted by de villiers (1990, 1991), in the context of dynamic geometry. the purpose of the first experiment was to determine whether learners have any need for conviction and explanation within the context of dynamic geometry (mudaly, 1999). the study also tested vimolan mudaly 67 curriculum material that was developed as a result of previous empirical and theoretical research. the material was designed with the aim of helping learners discover solutions to problems through a process of guided discovery in stages that are easy and practical. as the learners progressed through the worksheets, they were encouraged to record their conclusions and conjectures and so develop an explanation (proof). seventeen learners aged about 14 years (grade 9) were interviewed in february 1997. they were selected randomly from a group of 153 learners by their computer studies class teacher, who picked every ninth learner appearing in the attendance register. the purpose of the second study was to determine whether sketchpad could be useful as a mathematical tool when teaching learners to model (mudaly, 2004). although this study tested curriculum material that had already been developed (see de villiers, 1999), it was refined as a result of previous empirical and theoretical research. ten learners between the ages of 15 and 16 years (grade 10) were interviewed in march 2000. they were selected randomly from a group of 60 learners by their computer studies class teacher. study one research methods learners were given a computer-based task to work through, which was based on an equilateral triangle. a sketch of the equilateral triangle was presented ready-made to the learners, although the task of constructing it for themselves might have been an interesting one. all measurements were clearly visible on the screen of the computer, so that learners could easily view any changes that might have taken place. furthermore, the learner was actively involved in finding a solution which implied that s/he was ultimately responsible for his/her own learning. this method gives credence to experimental mathematics, which involves a well-planned, sequentially structured scheme in which the learner is guided towards “discovering” a solution for him/herself. this is in keeping with scientific experiments conducted in school laboratories; the experiment is generally performed many times, and a conclusion can be drawn if the result seems to be the same every time. while traditional teaching has been teacher-centred, didactic, directive, corrective and mostly concerned with the transmission of knowledge in well-defined areas, experimental teaching in mathematics is grounded in teaching skills where the teacher acts as a facilitator. the method employed in this study also drew on the inquisitive nature – a fundamental component – of human beings. humans have a fundamental need to find explanations, and children in particular are inclined to ask questions such as ‘why’. most parents have been irked by endless questions such as: why shouldn’t i sit up late? why is the sky so blue? why must i eat my peas? why is that man so fat? why do i have to go to school? why do we have to do geometry? why is mathematics so difficult? furthermore, even if a child is inductively convinced about the truth of a statement, it does not constitute a reasonable proof. in other words, the fact that the sun rises every morning does not explain why it rises. there is no doubt that even if there are thick clouds in the sky we know that the sun is there behind the clouds, but it does not explain why it rises. thus, the step that follows would require an explanation, albeit a guided one, as to why the result is true. it could also be interesting to give learners a problem in which the result only holds true for certain cases. in the problem conjecture testing establish level of conviction use of computer use of computer use of computer unsure guided explanation figure 1. flow diagram illustrating the process of experimental mathematics using computer software. proof and proving in secondary school 68 process, the learner would undergo an experience which is bound to increase learning. a flow diagram illustrates the steps involved when using computer software in experimental mathematics (figure 1). the emphasis on this study is on the level of conviction attained by the learner. the experiences that are provided are owned by the learner. the learning process is thus structured around the way in which the learner experiences the problem. critical questions that must be considered are: can a conjecture (intuitive) be drawn? has the learner tested the conjecture sufficiently? is the learner convinced with the result? to what extent is the learner convinced? can the learner explain the result? in this experiment, the learners were given the following problem to work through: sarah, a shipwrecked survivor, manages to swim to a desert island. as it happens, the island closely approximates the shape of an equilateral triangle. she soon discovers that the surfing is outstanding on all three of the island's coasts and crafts a surfboard from a fallen tree and surfs everyday. where should sarah build her house so that the total sum from the house to all three beaches is a minimum? (she visits them with equal frequency.) study one results at first, all the learners intuitively guessed that sarah should build her house at the centre. the learners were asked why they felt that the house should be built at the centre. kumarasen, for example, responded by saying that “… if you build anything in the centre then there is always a short distance around it”. kumarasen seemed quite convinced of his conjecture and so was manivasan, whose reason was “… because everything will be equal”. rowan believed that it should be at the centre because “it will be close … it will be the same distance to all the beaches” and therefore the sum will be a minimum. karishma felt that the sum would be a minimum if the point p was at the centre because “it will be closer to all three beaches”. ansuya’s reason was similar when she said “because it seems the easiest way to get to any of the three beaches”. an equilateral triangle representing the island was then drawn on the computer using the software programme, sketchpad. point p was placed within the triangle. perpendicular line segments were drawn from p to each side of the triangle. these line segments were measured and their sum determined. the learner was then allowed to move point p around and careful observation was encouraged. the learner could see the measurements of the perpendicular segments change as the point p was moved about. further, they could see that the sum did not change. the surprise at discovering this result was clearly visible. the following extracts were some of the comments made by the learners. kerushnee: (emphatically) yes, i find the result very surprising. ansuya: (confidently) yes, i thought it would change. kumarasen: yes, because at first you think it should be at the centre and the sum will be small. but now it can be anywhere. floyd: (emphatically) i didn’t expect it. it is surprising! it might be true that this result which the learners observed encouraged them to want to know why it was the case. the majority of learners expressed a desire for an explanation. in fact, 16 of the learners (94 percent of the total) said that they would want an explanation, and only one learner (6 percent) cogitated a while before saying that she would also like an explanation. the extracts from the interviews illustrates this desire that the learners had for an explanation. researcher: do you think then, now that you are a 100% convinced, that there is a need for an explanation? manivasan: yes. researcher: would you want an explanation? manivasan: yes. researcher: why? manivasan: so i can understand it [learner’s emphasis] researcher: do you think, now that you are very convinced … is it necessary to know why this is the case? rodney: yes. researcher: why do you want an explanation for this? rodney: to satisfy my curiosity. researcher: why do you think there is a need for an explanation? karishma: because i’m curious and i’d like to know what’s going on. vimolan mudaly 69 researcher: why do you think there is a need for an explanation? debashnee: because i’m a curious person and i would like to find a solution for things. i would like to do the same for this. researcher: do you desire an explanation for what is going on? ryham: yes. researcher: you really would want to know why? rhyam: yes. researcher: why? rhyam: i like to find out why things are taking place. an interesting part of this experiment focussed on whether learners were able to formulate an explanation, albeit a guided one, on their own. researcher: okay, i can see that you have done that (referring to the writing down of expressions for the areas of the three small triangles). the next step asks you to add all three up. do you know what to do? nicholas: yes. researcher: (after a while) you’ve got a1, a2 and a3 and you’ve got expressions for them. now add these expressions... (after a while) have you done that nicholas? nicholas: yes. researcher: now simplify it... have you done that? nicholas: yes. researcher: i’ve noticed that you removed half ‘a’ as a common factor. nicholas: yes. researcher: describe what you have done. nicholas: i’ve removed half ‘a’ as a common factor and i’ve got half ‘a’ into h1 + h2 + h3. researcher: nicholas can you tell me how these three triangles relate to the area of the large triangle? nicholas: the area of the three triangles when you add it up, will give you the area of the big triangle. researcher: if that is the case and we found the sum of the areas of the three triangles, then what can we conclude? nicholas: (silence) researcher: that the areas of these triangles equal the ... ? nicholas: ... area of the big triangle. researcher: now look at e4. i want you to write down this expression. nicholas: (after a while) i noticed that the big triangle also had half ‘a’ in it. so i cancelled off the half ‘a’ from the big triangle and half ‘a’ from the three small triangles. researcher: and what have we arrived at? nicholas: the height of the three triangles… when you add it up it gives you the height of the big triangle. researcher: what does this mean to you? nicholas: no matter what the heights of the three smaller triangles are, it will always equal the height of the big triangle. researcher: so what does it mean in terms of sarah’s house? nicholas: it means that no matter where she puts her house the total distances will always be constant. the ability of the learners to formulate conjectures and subject them to critical tests was admittedly lacking. however, it is more difficult to relate the formal proof, as it is taught in the ostensive way, to learners’ past experiences. ultimately, proof is necessary to see why the conjecture always holds true, rather than whether or not it will hold true. this suggests that proof in schools should reflect the idea of an explanation instead of the idea of verification. when a teacher states directly that a theorem is correct, the learner sees no need to verify it. in contrast, where a proof is presented as a means of explanation and further understanding – as is being advocated by many mathematicians and mathematics educators – it may be possible to address the current crisis in school geometry. in the context of this study, it becomes evident that learners inherently attempt to find explanations, however simplistic they might be. their desire to know why is authentic, and not just a hypothetical assumption of some theorist, as was demonstrated in this teaching experiment. when the learners were asked to attempt an explanation they were always willing to try. this may be because their level of conviction was always very high (the learners, after seeing the results on the computer generally indicated that they were 90% proof and proving in secondary school 70 to 100% convinced.) conviction alone is insufficient when teaching proof. in this experiment the learners were given a worksheet that gave appropriate directions for developing an explanation. guided through each step, the learners had a good understanding of the explanation. in fact, many of them indicated that this was a good explanation, which is encouraging considering their antipathy towards geometry classes. this experiment is still in process, and it should be noted that those learners that have gone through this process are keen to get back into the computer laboratory for further interviews. study two research methods in the second study, the learners investigated perpendicular bisectors of quadrilaterals and triangles. the question they were given was: in a developing country like south africa, there are many remote villages where people do not have access to safe, clean water and are dependent on nearby streams or rivers for their water supply. with the recent outbreak of cholera in these areas, untreated water from these streams and rivers has become dangerous for human consumption. suppose you were asked to determine the site for a water reservoir and purification plant so that it would be the same distance away from four remote villages. where would you recommend the building of this plant? they were then given a modelling exercise in which they were required to work with two villages, four villages and then three villages. the reason for this particular order is not discussed in this paper. study two results at first, the learners attempted to solve the problem directly, before moving on to the modelling exercise. unlike quadrilaterals, the perpendicular bisectors of every triangle they constructed, using sketchpad, showed that the perpendicular bisectors were concurrent. this raised the curiosity of the learners and i then asked them whether they would like to know why the perpendicular bisectors of the triangle were always concurrent. all learners indicated a desire for an explanation, and shared a similar reason: they were surprised at what they had experienced. judging by the tone of their voices, it could be said that most of them were quite enthusiastic about working through an explanation. one learner initially seemed uncertain, but eventually admitted that he was surprised at the results and would be interested to find out why. several learners felt that it would be interesting to know “why the result was always true” and that “it would be useful to know why the result is true”. christina’s need for an explanation was significant: researcher: would you want to know why this result is always true? christina: yes…i can see it is true but maybe if there is a proof for it i'll understand it better. she showed a very high level of conviction when she stated, “i can see it is true” and yet she felt that her understanding would be increased if she worked through a proof. this indicates that the level of conviction obtained from working with dynamic geometry software may stimulate further curiosity, which can be used as a starting point for proof. roxanne also felt that an explanation would show her why the result obtained for all triangles was different from that obtained for quadrilaterals. researcher: would you want to know why this is always true? roxanne: yes… maybe it will explain why it was different. the fact that she saw that the results were distinctly different for the cases for two and four villages kindled in her the desire to want to know why this was the case. this also clearly indicates that different individuals show different needs when working with proof in geometry. whilst some felt that it would be useful just to know why, others felt that an explanation will give them greater understanding. researcher: would you like to know why the perpendicular bisectors are always concurrent? pravanie: i guess that it would be useful to know. researcher: would you like to know why this is always true? faeeza: it might be interesting to know…i can’t believe it (showing surprise). researcher: would you like to know why this is always true? nigel: definitely…maybe i could trick my friends too. vimolan mudaly 71 researcher: would you like to know why this is always the case? schofield: what do you mean sir? researcher: do you want to know why the perpendicular bisectors are always concurrent? schofield: i don’t know... mmm... yes, maybe it will be interesting. researcher: you think that this would be interesting? schofield: yes…i was surprised at the results. it should perhaps be noted again that ausubel’s learning theory suggests that meaningful learning occurs as a result of stimulating the learners’ curiosity during a discovery process (ausubel, novak & hanesian, 1978). it seems as if the difference in their findings for the quadrilateral and triangle stimulated their curiosity, and created a desire for some form of explanation. the explanation for the concurrency of perpendicular bisectors of all triangles was based on materials developed by de villiers (1999: 32). below are extracts of interviews with the learners. researcher: construct the perpendicular bisector of any side. desigan: can i do it for ab? researcher: yes. (after the construction) desigan, what can you tell me about all the points on this perpendicular bisector? desigan: it is equidistant from a and b. researcher: what is equidistant? desigan: all the points on this line (pointing to the perpendicular bisector). researcher: what does that really mean to you? desigan: if you measure the distance from any point on this line to this a and b, the distance will be the same. in this segment, attempts were being made to stimulate the learners to recall the concepts of perpendicular bisector and equidistance. in a way, it was also a means of determining whether the learners actually understood and remembered what they had done earlier in the interview. vischalan displayed a similar understanding of the concept of equidistance. researcher: look at this triangle on the screen. construct the perpendicular bisector of side ac. (after the construction) what can you tell me about all the points on this perpendicular bisector? vischalan: they are the same distance away from a and b. researcher: what is the term used to describe same distance away? vischalan: equidistance. researcher: so what are you saying about all points on this line? vischalan: all the points on this line (pointing to perpendicular bisector) are equidistance from a and c. researcher: equidistant – not equidistance – from a and c. what does that really mean to you? vischalan: if you calculate the distance from any point to a and then to c the distance will be exactly the same. it was clear that the learners had developed a good grasp of this concept (equidistance) and that the researcher could therefore continue with the rest of the explanation. the next part of the explanation was similar in that it required the learners to construct another perpendicular bisector to relate the point of intersections of the two perpendicular bisectors to the three vertices. this relationship between the intersection and the three vertices did not take long to achieve, although in desigan's case it was obvious that he made a mistake at one point in the interview but he did correct himself. researcher: now construct any other perpendicular bisector. desigan: (constructing) researcher: what can you tell about the points on this line now? desigan: all the points are the same distance away from b and c. researcher: now look at this point of intersection. what can you say about this point in particular? desigan: eh … eh… researcher: think carefully about the point. desigan: that point there is the same distance away from a and b, and b and c. researcher: a and b, and b and c? desigan: yes, it is the same distance away from a, b and c. researcher: are you sure? proof and proving in secondary school 72 desigan: it lies on this line so it must be equidistant from a and b and it lies on that line so it must be equidistant from a and c. researcher: if it lies on that line would it be equidistant from a and c? desigan: no, b and c. researcher: now construct perpendicular bisector of ab. vischalan: (constructing) researcher: what can you tell about the points on this line now? vischalan: all the points are equidistant from b and a. researcher: now look at this point of intersection. what can you say about this point in particular? vischalan: that is the point of concurrency of these two perpendicular bisectors. researcher: yes, that is true, but think carefully about the point. what is special about it? vischalan: it is equidistant from a, b and c. researcher: really? why? vischalan: it is equidistant from a and c, and then it is equidistant from a and b, then it must be equidistant from a, b and c. it was also quite interesting to note the level of reasoning that these learners were able to achieve and their ability to employ the basic transitive property. for example, if ba ⊗ and cb ⊗ , then ca ⊗ (where ⊗ represents a general binary relationship). it implies that these learners had reached the stage of van hiele level 3 (usiskin, 1982). they could see the deductive logic in the explanation as they were being guided through it. being able to ascertain that if the point was equidistant from a and b and then from b and c, therefore the point must be equidistant from a, b and c is characteristic of van hiele level 3. furthermore, they seemed convinced that their reasoning was correct. the researcher attempted to get them to measure the distance just to check, but the learners felt that this was not necessary. researcher: so are you sure that this point of intersection is the same distance away from a, b and c? vischalan: yes. researcher: don't you want to measure and check? vischalan: no…it's not necessary. the next aspect was particularly important because it would be the real test as to whether the learners understood this concept of equidistance. researcher: this you have to think very carefully about. what can you say about the perpendicular bisector of ac? desigan: all the points will be equidistant from a and c. researcher: yes, that is correct. but look at the other perpendicular bisectors. desigan: oh yes, it must pass through the point where these two lines meet (pointing to the perpendicular bisectors). researcher: what can you say about the perpendicular bisector of bc? vischalan: (silence) researcher: think about it… what can you say about the perpendicular bisector of bc? vischalan: i think … it will pass through this point of intersection here. the researcher was aware that the learners may have just guessed the response because they already knew that the perpendicular bisectors of the triangle were concurrent. therefore the response that followed was essential in determining whether they were making a response with understanding or not. researcher: really? do you really think so? vischalan: yes, i’m quite sure. researcher: why? vischalan; well if i construct the perpendicular bisectors, all the points on that line must be equidistant from b and c. researcher: yes, go on. vischalan: what do you mean? researcher: you just said that all the points on that line must be equidistant from b and c. so what does that mean? vischalan: that point of intersection has to pass through the point of intersection … it has to because that point is also equidistant from b and c. vimolan mudaly 73 researcher: really? why? desigan: yes, because if all the points on this perpendicular bisector of ac are the same distances away... then... then this point of intersection is also the same distance away... then… researcher: yes? desigan: then the line must pass through the point of intersection. it was clear that these learners had actually grasped the concept of equidistant points. nevertheless, it was not surprising that the learners wanted to see whether their conjecture was true. this indicated that they where still sceptical. they knew that they were correct, but they wanted to see it nonetheless. furthermore, it was interesting to note that the learners were actually taking ownership of the explanation. researcher: do you want to see whether that is true? desigan: yes. researcher: construct the perpendicular bisector of ac then. desigan: (after constructing) this is so easy. researcher: was it really that easy? desigan: i didn’t take so long to get it right! researcher: do you want to see whether that is true? vischalan: yes. researcher: construct the perpendicular bisector of bc then. vischalan: (after constructing) i was right again. it was encouraging to note that the actual explanation became much easier because of the way the different problems were modelled. the learners made use of their existing knowledge to deductively construct an explanation, even though they were guided through it. their high levels of understanding (confirmed by the way in which they responded) helped them to arrive at an explanation with ease. it should be noted that this is not the normal textbook proof conducted in south african schools (which is based on congruency). however, it is a valid proof that appears to have increased learners’ understanding. conclusion this research has highlighted some valuable insights regarding teaching and learning geometry theorems and problems. given the fundamental importance of proof within mathematics as a discipline, proof should remain an essential part of the secondary school curriculum. moreover, the teaching (and learning) approach used in this empirical research seems to have provided learners with an improved, and more meaningful, understanding of the role of proof. this study has focussed mainly on the introduction of proof to learners as a means of explanation, rather than as verification. the research demonstrates how learners have a need for an explanation (deeper understanding) which is independent of their need for conviction. it would appear that the learners exhibited an intrinsic desire for an explanation, even though they had a high level of conviction with respect to their conjecture. such conviction often reduces a problem to the realm of the obvious, in other words, ‘i can see that it is true so why do i need an explanation for it?’ if the learners were so sure of the result then it should have made no difference to them whether there was some logical explanation for it or not. yet they expressed a strong desire for an explanation. it seemed that they had recognised the fact that they had merely observed the result through experimentation. the learners were aware of the difference that existed between observation, through experimentation, and knowing why it was really true. they undoubtedly wanted to know why the result was true and not whether the result was true. from the learners’ responses it seemed that the explanation provided insight into the reason why it was true. more significantly, this research has found that with appropriate guidance, learners can construct reasonable explanations for their conjectures. the learners involved in these teaching interventions, showed that, with guidance, they could construct a proof. in a sense, the act of moving points on a screen and seeing the results displayed on the screen is a type of proof in itself. constructing a logical argument thereafter became much easier, because seeing the images on the screen allowed the learners to see the generalisation in the particular diagrams they were constructing. although the two studies worked with very small groups of learners, it is possible to extrapolate these results to most learners. these groups were drawn from ‘below average’ classes, which permits a certain level of generalisation in the south african context. the fact that the proof and proving in secondary school 74 learners in this study were guided through the explanation (proof) should not detract from the findings, as this approach is in keeping with vygotsky’s zone of proximal development (morris, 2007). although the one-on-one interactions do not resemble typical classroom teaching, it is safe to hypothesise that even more productive interactions would take place in the dynamic classroom context. if learners are able to hypothesise and attempt explanations individually, then they should be able to solve problems even more effectively in a collective situation. this would require a classroom environment in which interaction with peers is encouraged. investigation of this aspect in future research may provide further useful insights. references anderson, j. 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(1982). van hiele levels and achievement in secondary school geometry. unpublished report, department of education, university of chicago, chicago, il. retrieved november, 2006, from http://socialsciences. uchicago.edu/ucsmp/van_hiele_levels.pdf “even fairly good students, when they have obtained the solution of the problem and written down neatly the argument, shut their books and look for something else. doing so, they miss an important and instructive phase of the work. ... a good teacher should understand and impress on his[/her] students the view that no problem whatever is completely exhausted. one of the first and foremost duties of the teacher is not to give his[/her] students the impression that mathematical problems have little connection with each other, and no connection at all with anything else. we have a natural opportunity to investigate the connections of a problem when looking back at its solution.” george pólya article information authors: deonarain brijlall1 sarah bansilal1 deborah moore-russo2 affiliations: 1department of mathematics education, university of kwazulu-natal, south africa2graduate school of education, university at buffalo, the state university of new york, united states correspondence to: sarah bansilal postal address: 8 zeeman place, malvern 4093, south africa dates: received: 16 mar. 2012 accepted: 15 oct. 2012 published: 04 dec. 2012 how to cite this article: brijlall, d., bansilal, s., & moore-russo, d. (2012). exploring teachers’ conceptions of representations in mathematics through the lens of positive deliberative interaction. pythagoras, 33(2), art. #165, 8 pages. http://dx.doi.org/10.4102/ pythagoras.v33i2.165 copyright notice: © 2012. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. exploring teachers’ conceptions of representations in mathematics through the lens of positive deliberative interaction in this original research... open access • abstract • introduction • rationale and research questions    • classroom communication • methodology    • the questionnaire       • issues of ethics and trustworthiness • findings    • analysis and discussion of data • conclusion • acknowledgements    • competing interests    • authors’ contribution • references abstract top ↑ this article reports on an exploration of teachers’ views on the meaning of mathematical representations in a democratic south africa. we explored teachers’ conceptions of ‘mathematical representations’ as a means to promote dialogue and negotiation. these conceptions helped us to gauge how these teachers viewed representations in mathematics. semi-structured questionnaires were administered to 76 high school mathematics teachers who were registered for an upgrading mathematics education qualification at a south african university. common themes in teacher conceptions of representations were investigated as part of an inductive analysis of the written responses, which were considered in terms of practices that support dialogue and negotiation. findings suggest that these conceptions are in line with progressive notions of classroom interactions such as the inquiry cooperation model. furthermore, the findings suggest that teachers can support the development of classroom environments that promote democratic values. introduction top ↑ what are the specific elements of a mathematics classroom that allow it to be characterised as democratic or as a classroom that seeks to prepare all children for life in a democratic society? are there connections between democracy and mathematics classrooms? skovsmose (1998) asserted that mathematics education could be related to a discussion of democracy in terms of citizenship, mathematical archaeology, mathemacy and deliberative interaction. he illustrated how these four aspects, which concern classroom practice in mathematics education, also concern democracy. in this article we delve into one of the above aspects, namely deliberative interaction, which skovsmose views as possible when ‘an interaction in the classroom which supports dialogue and negotiation’ is developed (p. 200). figure 1 illustrates the components inherent in skovsmose’s notion of deliberative interaction. figure 1: components of deilberative interaction. deliberative interaction excludes a view of mathematics as an unchanging body of knowledge, which a teacher transmits to learners. such a view presupposes that mathematical tasks have only one correct answer and often only one correct, or one preferred, method to arrive at that answer. this view sets the classroom as an autocracy in which the teacher serves as the sole authority. alrø and skovsmose (1996) used the phrase classroom absolutism to refer to the type of communication between the teachers and learners that is structured by assumptions that (1) school mathematics can be organised around mathematics activities with unique answers, and (2) the teacher’s task is to ensure that mathematical errors are removed from the classroom. in trying to identify the role of representations when teaching and learning mathematics within such a paradigm, we extend the notion of classroom absolutism to include two further assumptions, (1) that mathematical learning can be organised around classroom activity with one possible representation for a mathematical notion or task, and (2) it is the duty of the teacher to ensure that other representations are eradicated from mathematical learning.in the democratic micro-society of a mathematics classroom it is imperative for the teacher to move away from such classroom absolutism because learners should be afforded different ways to express themselves. we are not implying that teachers accept any and all responses to mathematical tasks as final answers. our message is this: the teacher should (1) be aware of the different mathematical representations that can be used to achieve mathematically acceptable arguments, and (2) be willing to work with learners’ developing mathematical ideas and personal mathematical representations to facilitate a clearer understanding of mathematics and the way it is conventionally represented. educational environments that discourage classroom absolutism often have a prevailing view of mathematics as a process rather than a product. mathematics is much more than the production of answers; it is the process of determining how to quantify, model, et cetera a situation. rather than producing an equation or a table or a graph, educational environments that discourage classroom absolutism should emphasise that different representations of the same mathematical concept are possible and that doing mathematics is often the process of determining what is asked or needed and the affordances and limitations of any mathematical representation that could be used in the answer. such classrooms exemplify the inquiry cooperation model (skovsmose, 1998, p. 200), which refers to a ‘pattern of communication where the student and teacher meet in a shared process of coming to understand each other’ whilst learning about mathematics. mathematical representations, as vehicles of communication, play a central role in such classrooms. in the mathematics education community, the concept of mathematical representation has been based on different theoretical perspectives (english, 1997; goldin, 1998; presmeg, 1997). we adopt the widely used definition that a representation is a configuration that can represent something else (goldin, 2002). the representations used to communicate ideas, including those involving mathematical concepts, are socially embedded and culturally created (greeno, 1997). therefore, the manner in and extent to which representations mediate mathematical understanding depend as much on the individuals engaged in the task as they do on the task itself. the use of multiple mathematical representations and the fostering of an environment that facilitates and values various representations provide a space where learners can engage with substantial mathematics and develop the tools to become citizens who are productive and active, two qualities of democratic mathematics education (ellis & malloy, 2007). rationale and research questions top ↑ the most recent reform in curriculum and assessment policy in south africa aims at producing learners who are able to communicate effectively using visual, symbolic and/or language skills in various modes (department of basic education, 2011, p. 2). this is expected to occur throughout their learning where opportunity for representation arises. teachers need to engage in meaningful discourse with their learners so as to better recognise and appreciate the learners’ use and understanding of specific representations. such shared exchanges result in a process in which two groups come to understand the other’s viewpoints as well as the discursive resources and mathematical representations employed to communicate those viewpoints. in this way, deliberative interaction is exemplified. however, to build on and from the representations of learners, teachers must have both a deep understanding of the different representations (including the affordances and drawbacks of each) and the flexibility to use the representation that is most appropriate for the mathematical situation and the learners. the issue under investigation in this study is whether teachers have deep, flexible understandings of mathematical representation that enable them to create democratic environments in their mathematics classrooms. this concern will inform mathematics teacher educators and the relevant educational department authorities whether these teachers are prepared for changing educational policies (department of basic education, 2011). accordingly, this study was designed to explore teachers’ understanding of mathematical representations. for this study, we formulated the following research questions:1. what do teachers in a democratic south africa believe is meant by the expressions ‘mathematical representations’ and ‘representation in mathematics’? 2. how do teachers view representations in mathematics? we ask these questions because we believe that the use of a variety of mathematical representations for differing purposes is a powerful tool for teachers to foster deliberative interaction in the micro-society of the classroom. the model presented in figure 2 is an attempt to show the role of mathematical representations in creating democratic classroom environments. figure 2: various mathematical representations supporting deliberative interaction in the context of democracy. teachers need to be able to make flexible use of representations before they are able to create an environment that allows learners the freedom to use developing representations. thus, teachers’ representational fluency impacts on their ability to foster deliberative interaction. in this study, ‘representational fluency’ means that an individual has an abundance of mathematical representations at their disposal for use when reasoning and communicating in the mathematics classroom. in classrooms where deliberative interaction is embraced, learners’ communication and representational fluency are encouraged and developed through shared negotiation between and amongst learners and the teacher. development of representational fluency in learners will better prepare them to interpret mathematical tasks, share their mathematical ideas, and interpret the mathematical communication of others. hence, representational fluency can contribute positively to developing active citizens, by giving them a sense of freedom of expression, which is a concern of democracy, within the mathematics classroom. with these qualities we hope that the pupils become active citizens, taking ownership of their learning and thus becoming responsible citizens. responsibility is a prerequisite for upholding democracy. these connections are indicated in the model in figure 2, which shows the links between the use of mathematical representations and the development of a democratic society. classroom communication mathematical representations can facilitate dialogue between teachers and their learners, if teachers choose not to conform to classroom absolutism. of course, classroom communication may be constrained by the pre-described assigned roles of the teacher and the learner (skovsmose, 1998). however, if the teacher accepts a shared, negotiated dispensation with the learners then the mathematical representations used become ideal entities to promote interactive dialogue. vithal (1999) argued that the learners in her study who used drawings and graphs to represent their expenses and interviews as part of their project, demonstrated that the classroom ‘could serve as the arena for acting out a democratic life’ (p. 29).it is reasonable to suppose that the development of learners’ understanding of mathematical ideas and their capacity to use representations to communicate and reason about ideas are influenced by the nature of their teachers’ conceptions of mathematical representations. teachers need to believe that representations can be used as tools to understand mathematical concepts and solve problems but also as modes of communicating about these problems and concepts (roth & mcginn, 1998). in the sciences, ochs, jacoby and gonsales (1994) studied the work of a group of physicists to display how professionals use representations to create a shared world of understanding. in mathematics education, moore-russo and viglietti (2012) investigated teachers in collaborative problem-solving situations and found that even when presented with the same task, individuals within the groups use various resources to communicate and reason mathematically, often adopting and adapting the representations used by their group members. for teachers to value such situations, they need to foster a democratic classroom environment that departs from classroom absolutism. stenhagen (2011) and allen (2011) have suggested that teachers, teacher educators and curriculum designers place emphasis on teacher beliefs and philosophy in classroom instruction. this article explores teacher beliefs about mathematical representations, with the aim to discover the beliefs teachers have about the practice of teaching in general and the use of representations in particular. the findings should provide insight into the deliberative interactions in mathematics classrooms. teachers using representations in a way that creates deliberative interactions build possibilities for the classroom to serve as an opportunity for learners to become members of a democratic micro-society and, in doing so, preparing to be active citizens in a democratic society. methodology top ↑ this study was qualitative in nature. it has been argued that interpretive researchers use mainly qualitative research methods in order to gain a more in-depth understanding of the participants’ perceptions of the phenomenon (cohen, manion & morrison, 2007; henning, 2004). this ties in with our method of inquiry since we intended to find out what teachers in a democratic south africa believe is meant by the expressions ‘mathematical representations’ and ‘representation in mathematics’. the research instrument used was an open-ended questionnaire. by allowing for free responses, the instrument allowed the research team to elicit the opinions of the teachers without influencing them to provide the answers they felt might please us. a non-probability sampling strategy was used. this is in line with the study because qualitative researchers do not count generalisation as their primary aim but instead seek to represent a particular group (cohen et al., 2007; maree & pieterse, 2007).the study participants were 76 teachers from historically disadvantaged schools, pursuing an advanced certificate in education, specialising in high school mathematics teaching in grades 10−12, at a south african university. all had successfully completed the first semester course on differential calculus. the questionnaire the semi-structured questionnaire, with twelve items, was administered to the 76 participants in the second semester of their study. for this article we consider only the teachers’ responses to the first two items of the questionnaire, namely:• item 1: what does the phrase ‘mathematical representation’ mean to you? • item 2: what comes to mind when someone talks about ‘representation in mathematics’? the teachers’ responses to the two items were analysed for emerging themes through a general inductive analysis. using theoretical memoing (glaser, 1998), the research team members individually classified the teachers’ responses, then collaboratively developed categories based on their memos. these initial categories established the themes described in table 1. table 1: themes for mathematical representations. the research team used the 12 categories to individually revisit the data to ensure that their constant comparison method constituted a saturation of categories. using a teacher’s response to an item as the unit of analysis, two of the team members independently coded all teachers’ responses. working independently, the two coded each response as providing evidence, or not, for each of the 12 themes shown in table 2. the 76 teachers’ responses to the two items provided 152 units of analysis. the overall inter-coder agreement for the teachers’ responses was 0.95; the related cohen’s kappa value was 0.80, above the 0.60 that is accepted to represent good agreement (altman, 1991; landis & koch, 1977). after inter-coder agreement was determined, all disparities in assigned codes initially given to the responses were treated in the following manner: each disparity was identified and then two members of the research team discussed coding until a consensus was reached for each response. the consensus codings were used for all subsequent data analysis. table 2: the presence of themes in teachers’ responses (n = 76). once the data set was completely coded, the research team discussed what they saw emerging from the data and collapsed the initial categories into broader themes. the team members then individually revisited the data once more to verify that the themes made sense of the data (thomas, 2006). finally, the whole team finalised the descriptions of the 12 themes that were used for data coding. issues of ethics and trustworthiness ethical clearance was obtained from the university research office for the collection of the data. to comply with the terms of the university research policy, consent to participate in the study was provided by all the participants. in qualitative research, reliability and validity are conceptualised as trustworthiness criteria (golafshani, 2003). to eliminate bias and increase researcher truthfulness, triangulation in this study was achieved via independent coding and with agreement being reached by consensus. in addition, the researchers sought convergence of different responses to form common themes from the categories. findings top ↑ after themes were identified and the data set was coded, the research team generated descriptive statistics to complete the analysis of the data. the first consideration was which categories were most frequently evidenced in the teachers’ responses. information regarding the 12 identified themes that were evidenced in the teachers’ responses is summarised in table 2 in order from most to least common themes. note that the columns in table 2 provide information for each item as well as cumulative information on both items. in order to read table 2, consider the first row: 32 teachers’ responses to item 1, 52 teachers’ responses to item 2, and 59 teachers’ responses to only one of item 1 or item 2 were coded as evidencing the examples theme. the data for the examples theme is illustrated in figure 3. figure 3: venn diagram showing the coding evidencing the examples theme. during the coding process, it was apparent that many teachers’ responses provided evidence that the teachers’ beliefs regarding representations addressed many of the 12 identified themes. for this reason, details regarding the number of themes noted in each teacher’s responses to the two items are provided in table 3. table 3: number of themes present in teachers’ responses (n = 76). in order to read table 3, consider the first row: 2 teachers’ responses to item 1, 3 teachers’ responses to item 2, and 0 teachers’ responses to only one of item 1 or item 2 were coded as evidencing 0 themes. the final column shows how teachers responded to both items. analysis and discussion of data in this discussion, note that we use the exact responses of the teachers, without editing for language or clarity. the notation t1 is used to denote the first teacher in the list, and t76 denotes the last teacher. we will primarily emphasise the themes that are most pertinent to promoting positive interactions and a democratic environment in the classroom. the theme examples was most commonly noted: almost 80% of the teachers gave examples of representations in their responses. in considering the examples theme, it is noteworthy that 84 responses (32 to item 1 and 52 to item 2) provided examples of representations although only 59 teachers used examples on one item only. this means that a significant proportion (84 − 59 = 25) of the teachers used examples in their responses to both items. some teachers mentioned only examples of a single representation, for example, t6 wrote ‘writing mathematics in graphical form’ for item 1 and ‘graphs’ for item 2. on the other hand, many other teachers listed a variety of examples of representations, such as t31: here mathematical knowledge is represented using verbal, pictures, symbols and manipulatives. the representation theme was second most common, addressed by 54% of the teachers. in this theme, teachers described what representations are and how they are used, especially in response to item 1. a typical example is t14: all representations are important based on the concept in which you are dealing with. the teachers in this sample displayed knowledge of numerous types of representations (as seen in the responses coded under the examples theme) as well as a belief that mathematical ideas can be represented in different ways. this variety theme was the third most common, with 40% of the teachers showing evidence of it in their responses. the variety theme was applied to teachers’ responses that explained that there are many different ways to represent mathematical concepts, ideas or relationships. such responses show that the teachers strongly believe that there are different ways to represent mathematical concepts. this is evidence that these teachers do not subscribe to an absolutist view of classroom communication. one example of the variety theme was displayed by teacher t17: using a variety of ways to capture concepts and relationships. being able to develop, share and preserve thoughts in mathematics. what was encouraging, in terms of promoting democracy, was that this teacher perceived mathematical representation as ‘using a variety of ways’. this abundance of alternatives is crucial to create a democratic mini-society (classroom) since these ‘variety of ways’ establish a ‘sharing’ of mathematical thoughts, which is a positive contribution to a democratic classroom. in this response, teacher t17 does not specify if the teacher or learner initiates this variety of ways of sharing. this could imply that either the teacher or the learner could employ a variety of ways; sharing is thus regarded as a two-way phenomenon, with importance placed on both key players in the classroom. in this case, the teacher would have no dominance, but be regarded as an equal to the learner in the classroom. the communication theme was noted by 38% of the teachers. whilst mathematical concepts are important, representations are the vehicles through which these concepts are shared with others. as evidenced by responses that fall under the communication theme, 21 teachers saw representations as things that convey or express mathematical information. an example of a response in this theme is t31’s response to item 1: learners can use the representations themselves to communicate their understanding of the (mathematical) concepts to the rest of the class or in smaller groups. here the perception of t31 is that mathematical representation is learner driven. this is in keeping with the principles of the south african school curriculum (department of education, 2003). t26 put it another way: the way you conveying the knowledge of maths to one another. we interpreted this as being related to communication since the knowledge is being conveyed to others. this comment also reveals that the teacher does not see the communication as being one-sided; rather it is communication with ‘one another’. this view of communication of mathematics ideas as being both from and to the teacher is aligned to the inquiry cooperation model (skovsmose, 1998) because representations are being used as a form of communication where the learner and teacher meet in a shared process of coming to understand each other, whilst learning about mathematics. twenty-six per cent of the teachers had responses that were coded as evidence of the aid for understanding theme. this reveals that the teachers see representations as facilitating the understanding of mathematical concepts or relationships. t3 expressed the view that mathematical representation means: [a] way of delivering and presenting the concepts such that the concept is very understandable to learn, encouraging learners to participate willingly and stay in every learner’s mind to his life-long period. it is notable that t3 included the need for mathematical concepts to be made understandable to learners. this provides evidence that this individual is a caring and accountable teacher who wants to make learning accessible for all. this trait is valuable in acknowledging the purpose and function of effective schooling as desired by any democratic society. t3 specifically uses the words ‘to participate willingly’ and mentions ‘every’ learner. two aspects emanate implicitly from t3’s response, (1) participation by all learners and (2) freedom of expression. the first aspect evokes the concept of participatory democracy, which requires that all individuals be afforded the opportunity to take part in the decisions that affect their lives (devenish, 2005). the second aspect alludes to free will, as evidence by t3’s use of the word ‘willingly’. freedom of expression is entrenched in the bill of rights within the south african constitution and is fundamental to liberal democracy (devenish, 2005). freedom of expression is indispensable in establishing mathematical truth in proof or problem solving, and it is a means of fulfilment of human personality since mathematics is a human activity (department of basic education, 2011, p. 8). with the recent emphasis worldwide on the need for links between mathematics education and real-life situations, it is no surprise that 24% of teachers identified the role of representations in portraying real life situations. the linking the learning of mathematics to real life is of paramount importance. this is highlighted in the curriculum and assessment policy statements (department of basic education, 2011), which state in the first specific aim that real-life situations should be incorporated into all sections whenever appropriate. such linking will prevent the classroom from being a micro-society in which only mathematical abstractions prevail. t51 places emphasis on real life in his response to item 1: depending on real life situation. one problem might require a graph to solve (a mathematics task), another may require a table, while some may require a flow chart. t51 indicated that the type of mathematical representation employed is dependent on the real-life situation to which it applies. this shows that this teacher places greater emphasis on the need for contextualisation than on the particular mathematical representation. this could mean that the teacher places the context first in making the choice of which mathematical representation to use to foster the learning of a particular mathematical concept. in some of the most common responses, teachers mentioned that representations are used for problem solving (17%) and discussed the tools (12%) that they use to create mathematical representations. teacher t61 was one who associated mathematical representations with problem solving: simplify problems by interpreting, analysis using sketches or mind maps. she also perceived mathematical representation as a means to simplify the problem situation. her aim to make mathematics problem solving more understandable and, hence, more accessible to her learners indicates her respect for them. across the two items, nine teachers associated representations with the tools (equipment or resources) used to create them. for example, t21 wrote: … being able to use different approach in sketching, use of computer, … whilst t46 wrote: mathematical representations refer to visual images which are ordinarily associated with pictures in books and drawings on a overhead projector. these responses suggest that these teachers see classroom resources and tools as an advantage in trying to present various forms of mathematical representations of mathematical concepts and ideas. they seem to want everybody to have access to tools and resources, a privilege that most, if not all, of the teachers from historically disadvantaged backgrounds in this study were denied. the remaining themes were flexibility, visualisation, differentiation & selection, and interrelation. from the results, it is clear that many of these high school teachers have a rich idea of the roles played by representation in mathematics. table 3 presents the 12 themes that were discerned in the teachers’ responses. from these data we know that (28 + 17 + 5 + 1 =) 51 of the 76 teachers thought of representations in multiple ways since their responses evidenced three or more themes. these results demonstrate that the majority of the teachers had a rich, broad understanding of representations, as opposed to a narrow or limited understanding, and their roles as would be associated with an absolutist view of mathematics. conclusion top ↑ despite the teachers being previously disadvantaged, with access to few resources and a varying quality of initial teacher preparation, their views on mathematical representation provide evidence of their willingness to embrace a democratic approach to teaching mathematics. the responses have revealed that many of the teachers see representations as being interrelated and the need to move between representations showed a fluid, dynamic and flexible understanding of mathematics, once more aligned to a democratic classroom. this abundance of alternatives offered by mathematical representations is crucial to creating a democratic classroom environment since this ‘variety of ways’ establishes a ‘sharing’ of mathematical thoughts thus allowing for contributions by both learners and the teacher. the choice of mathematical representation available for classroom activity encourages free will in expression of the relevant mathematical idea. this aspect alludes to an individual’s freedom of expression regarding mathematical concepts using mathematical representations. the use of mathematical representations caters for greater learner involvement and participation during classroom activities, which enhances participatory democracy. the responses of these teachers displayed that mathematical representations are potentially a means of encouraging a form of classroom interaction that promotes dialogue and negotiation in a democratic south africa. we are encouraged by the teachers’ flexible and open-minded approach to the use of representations in the mathematics classroom. we believe that with the display of mathematics teachers’ knowledge of various kinds of representations, and the various ways in which representations can be used in their classrooms, will enhance their teaching practices. their responses suggest that they see the learning of mathematics as a shared process and not a one-way transmission of a product from the teacher to the learner. the findings from this study also suggest that the teachers want to engage in the inquiry cooperation model (skovsmose, 1998), rather than following the absolutist tradition, and are keen to use a variety of representations to facilitate understanding of mathematics processes. the findings also showed that the teachers believed that learners and teachers could use representations as a tool for communication and were positive about freedom of expression in their classroom. all of the abovementioned findings augur well for the creation of deliberative interactions by these teachers in their classrooms, which we believe will support the creation of a democratic environment by enhancing the development of active citizens. more specifically, the data suggest that the teachers believe that mathematical representations can (1) be used to reason and preserve thought in mathematics classrooms, and (2) be used as a tool for sharing thoughts and communicating ideas related to mathematical tasks. moreover, the study suggests that teachers believe that representational fluency (1) creates opportunities for willing participation by both learners and teachers during mathematics classroom interactions, and (2) aids individuals as they express mathematical ideas freely. these views and beliefs on mathematical representations all facilitate communication, freedom of expression, negotiation and shared meaning, and understanding, which are vital attributes of deliberative interaction, as displayed in figure 2. these observed attributes are envisaged to prepare active citizens in the mathematics classrooms, thus addressing some of the concerns of democracy. we are mindful however that the study is based on the teachers’ reports of their views of mathematical representations and not on their actual classroom practice, which may not be aligned with these positive reports. further study should continue in this line of research to determine whether teachers’ apparently democratic leanings towards mathematical representations and their uses translate into democratic classroom practices and the facilitation of democratic learning environments. acknowledgements top ↑ we acknowledge that the collaboration on this research was aided by funding from a grant entitled enhancing secondary mathematics teacher education from the united states agency for international development, administered through the non-governmental organisation higher education for development. competing interests we declare that we have no financial or personal relationship(s) that may have inappropriately influenced us in writing this article. authors’ contribution the idea to work in this field of representations in mathematics education was encouraged by d.m-r. (state university of new york) and d.b. (university of kwazulu-natal) promoted the conceptual framework of democracy and mathematics education for the research design. the creation and implementation of the research instruments were done collaboratively by d.b., s.b. (university of kwazulu-natal) and d.m-r. data collection was carried out by d.b. and s.b. the analysis of data was led by d.m-r. and worked on collaboratively with d.b. and s.b. d.b. wrote the manuscript and it was refined by d.m-r. and s.b. references top ↑ allen, k. (2011). mathematics as thinking − a response to “democracy and school math”. democracy & education, 19(2), 1–7. available from http://democracyeducationjournal.org/home/vol19/iss2/10/alrø, h., & skovsmose, o. (1996). on the right track. for the learning of mathematics, 16(1), 2–29. available from http://www.jstor.org/stable/40248191 altman, d.g. (1991). practical statistics for medical research. london: chapman and hall. cohen, l., manion, l., & morrison, k. 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(1999). democracy and authority: a complementarity in mathematics education. zdm: the international journal on mathematics education, 31(1), 27−36. http://dx.doi.org/10.1007/s11858-999-0005-y 63 p38-45 kotze layout final 38 pythagoras 63, june 2006, pp. 38-45 contextual factors of the mathematics learning environment of grade 6 learners in south africa g.s. kotze and j.p. strauss school of education, faculty of humanities, university of the free state, and research institute for educational planning, university of the free state kotzeg.hum@mail.uovs.ac.za and strausjp.hum@mail.uovs.ac.za the aim of this article is to provide insight into the personal characteristics and family backgrounds that south african grade 6 learners bring to the learning process as well as how they interact with the resources and practices that characterise the schools they attend. the research method is historiographic, offering insights into current conditions. an analysis of content is done through examining evidence gathered from an international project. gender, socio-economic groupings and other major aspects associated with differences in mathematics achievement among learners and schools are addressed. an overall picture of contextual factors that may impact on mathematics achievement is presented. although the essential conclusions of the investigation are largely straightforward, valuable contextual information emerged that may have implications for mathematics education within the context of south africa as a developing country. introduction this investigation is based on data collected on personal characteristics and academic attributes of gade 6 learners in south africa. this article is a companion paper to an investigation into the mathematics performance of grade 6 learners in south africa. in this series of two articles, the mathematics achievements of learners in south africa are analysed. the research is based on an international project, the southern and eastern african consortium for monitoring educational quality (sacmeq), a network of 15 countries from southern and eastern africa aiming to monitor and evaluate the quality of their education systems. to be mathematically literate implies that learners are able to identify, understand and engage in mathematics and to make well-founded judgements about the role that mathematics plays. mathematical literacy also relates to an individual’s capacity to use and engage with mathematics in ways that meet the needs of that individual’s life as a constructive and reflective citizen (pisa, 2003: 7,15). these contexts for the use of mathematics address both external contexts (‘situations’ and ‘the world’) and internal contexts (‘an individual’s life’ and ‘everyday activity/ practice’). the former refers to the natural, social and cultural setting in which an individual lives in order to organise the phenomena of the physical, social and mental world (pisa, 2003: 25). the ‘individual’s life’ may include personal experience in his or her private life, occupational life, social life with peers, relatives and others as well as life as a citizen of a community. hence internal and external factors (such as personal, situational and cultural) that may contribute to the learning environment against which learners measure themselves are investigated. aim and problem statement the aim of this article is to provide insight into the individual characteristics and family backgrounds that mathematics learners bring to the learning process and how they interact with the resources and practices that characterise the schools they attend. in order to achieve the aim the following general questions are formulated: • what aspects of grade 6 learners and their backgrounds – that is, personal and home-related characteristics of learners, including their family backgrounds such as the economic, social and cultural capital of learners and their families – might impact upon these learners? • what aspects of grade 6 learners’ lives at school, including their habits and life inside school might have implications for learners – in other words, what aspects of the school context might impact upon these learners? g.s. kotze and j.p. strauss 39 subsequently the following questions are posed regarding grade 6 mathematics learners in south africa: • what are the personal characteristics of the learners? • what is the socio-economic context of the learners? • what are the school context characteristics of the learners? information was gathered from learners concerning their attitudes, perceptions, backgrounds and in-and-out-of-school activities. this information was analysed and interpretations between regions and between low and high socioeconomic groups were made. particular characteristics prevalent in different regions and differences between the approaches to learning of various groups, including male and female learners and those from less advantaged social backgrounds, were identified. research design the research strategies implemented are partly historiographic (an effort is made to cast light on current conditions and problems through a deeper and fuller understanding of what has occurred in the past) and partly based on an analysis of content (sources of evidence are examined). the specific procedures of the conducting of the sacmeq ii survey comprised three parts, namely the development of the main phases, sample design procedures, and the construction of the tests (sacmeq ii research report, s.a.). conceptualisation the social, political and economic contexts in which schooling takes place have a considerable impact on learning. clarke (1996: 327) states that schools represent society’s greatest collaborative enterprise because everyone has a stake. knowledge, skills and values are influenced to a large extent by the kinds of interactions that take place between learners’ personal characteristics and their social environment (troutman & lichtenberg, 2003: 23). learners in south african schools come from extremely diverse social, political, economic and cultural backgrounds, and this can be linked to the concern about the mathematics education of south african learners learner diversity may be observed with regard to race, religion, ethnicity as well as other potential disadvantages and disabilities, and exceptional ability. swart and pettipher’s classification (2005: 8) of diversity includes learners with exceptional intelligence; learners who are mentally impaired; gifted and talented learners; learners with specific learning disabilities; learners with physical challenges and chronic health problems; learners with communicative disorders; and learners with emotional and behavioural disorders. this learner diversity implies that provision should be made for unique needs in regular education classrooms. socio-economic characteristics span factors such as parent educational level, material possessions and the quality of the home environment. socio-economic status relates to parental education, familial income, education, employment, books in the home and other similar indicators that are used to infer learners’ social class that, in turn, has to do with diverse ethnic, racial and linguistic backgrounds (troutman & lichtenberg, 2003: 26, 561). personal characteristics include aspects such as gender and age. research studies reveal that there is considerable concern amongst the mathematics education research community regarding gender issues (van de walle, 2001: 459). the role of the family in shaping learners’ beliefs and attitudes toward schooling in general and mathematics in particular is significant. this is strongly advocated by mcleod (1992: 587) who argues that mathematics education can benefit from careful analyses of cross-cultural comparisons on issues related to the social context of instruction. eccles and jacobs (in leder, 1992: 613), in an examination of social determinants of mathematics attitude and performance, identify parents as a critical force and state that parents exert a more powerful and more direct effect than educators on learners’ attitudes towards mathematics. school context characteristics focus on learner engagement at school, especially on the practices that characterise the schools they attend. an important research conclusion (efa, 2004) is that there are opportunities to significantly improve the ways human and material resources are managed and used in schools, recognising that the school is a complex institution that operates within a wider socio-cultural and political context. contemporary changes in education, such as outcomes-based education, have resulted in a greater variation within the context of the mathematics classroom with respect to what is taught, as well as how and to whom it is taught. naudé (2004: 128, 140) states that an effective learning environment provides opportunities for successful learning experiences, facilitates strategic planning, providing immediate contextual factors of the mathematics learning environment of grade 6 learners in south africa 40 feedback and reinforcement that eventually makes learners aware of the results of their efforts. presentation and discussion of data data were gathered from the nine provinces in the south african school system (sacmeq ii research report, s.a.). where necessary codes were allocated to interpret certain variables in the tables, for example, recoded variables describe parents’ level of education, the materials used in the construction of learners’ homes, the number of books and reading materials, and the number of possessions in learners’ homes, among others. the desired target population definition for the sacmeq ii project was all grade 6 learners in 2000 (at the first week of the eighth month of the school year) who were attending registered mainstream primary schools (sacmeq ii research report, s.a.). every result should be interpreted in terms of the standard error (se). at national level, the average age of the grade 6 learner was 156,9 months (about 13 years and one month), in other words, the age of 95% of the population from which the sample was taken fell in the age mean 156,9 ± 2(0,69) months. in terms of the age requirements for admission to an ordinary public school (turning seven in the year of admission to grade 1) it is expected that a learner in grade 6 who has not repeated a grade should be 12 years old (144 months) towards the completion of the grade 6 year. thus, the grade 6 learner on average was one year older than the official norm (table 1). gender age (months) boys girls province mean se % % se eastern cape 164,2 2,03 50,9 49,1 2,60 free state 160,7 1,88 49,1 50,9 1,63 gauteng 151,4 1,54 47,3 52,7 3,72 kwazulu natal 155,1 1,50 46,6 53,4 2,26 mpumalanga 160,6 1,85 51 49,0 1,69 northern cape 159,7 2,76 54,2 45,8 3,66 limpopo 161,3 1,61 43,6 56,4 3,33 north west 149,0 2,40 48 52,0 2,15 western cape 149,5 1,14 43,8 56,2 2,47 south africa 156,9 0,69 47,5 52,5 1,00 table 1. personal characteristics: age and gender the data concerning the age of learners also provide insight into initial enrolment at the officially prescribed age, regular attendance and the progression of most learners from one grade to another at the appropriate time so that everyone is able to complete the curriculum. repetition may also account for over-aged learners. in table 1 the ratio in percentages of boys and girls who participated in the study is provided for the provinces, followed by the figure for south africa overall. the overall south african average for girls constituted 52,5% of the grade 6 learners. a relatively lower overall presentation of male grade 6 learners is observed. further research into gender distribution in the education system may shed light on phenomena like these. in this paper the reasons for this grade 6 gender phenomenon will be merely speculative. socio-economic characteristics involve, among others, parents’ level of education, quality of the home environment, and the quantity and quality of support given to their children. table 2 reflects the level of education of learners’ parents. the key to the codes was developed as follows: the index for the level of education was calculated by assigning a score of 1 for a parent who had no schooling or adult education. the score 2 was assigned for some primary education, 3 for completed primary education, 4 for some secondary education, 5 for completed secondary education and 6 for completed post-secondary or university education. to compare the education of a two-parent household with the education of a single parent situation, the average of the education of a home with two parents was calculated. the mean levels of education of the parents are presented in table 2 on the following page. the national mean level of parent education is 3,8 on a scale of 1 to 6. this implies that the average grade 6 learner’s parents did not complete secondary education. it may be concluded that the average grade 6 learner comes from a home where the parents lack basic secondary education. the generally low levels of education for parents may be ascribed to the education policy of the government before 1994. quality and adequate education are envisaged for all south african learners with the implementation of outcomesbased education. the data reveal a dire need for adult basic education and training (abet) programmes that provide education for young people beyond school-going age as well as adults. in order to examine the impact that learners’ home and social environments may have on academic achievement, some of the research instrument questions addressed the issue of educational resources in the home. books play an important role in the development and education of g.s. kotze and j.p. strauss 41 parent education (index) books at home (number of books) possessions at home (index) general quality of learners’ homes (index) province mean se mean se mean se mean se eastern cape 3,5 0,14 24,1 4,33 5,4 0,44 10,6 0,37 free state 3,3 0,31 27,4 6,80 6,6 0,39 11,4 0,96 gauteng 4,6 0,20 45,2 7,51 8,5 0,46 13,8 0,50 kwazulu natal 3,9 0,16 29,2 6,88 6,5 0,60 12,1 0,49 mpumalanga 3,5 0,16 30,3 6,61 5,3 0,55 11,4 0,43 northern cape 3,6 0,18 12,0 2,92 6,7 0,35 11,0 0,52 limpopo 3,5 0,14 37,8 6,45 5,2 0,51 10,6 0,36 north west 3,5 0,11 22,1 4,78 5,9 0,39 11,8 0,31 western cape 4,8 0,15 49,2 7,54 9,6 0,39 14,0 0,25 south africa 3,8 0,07 32,9 2,36 6,6 0,20 11,9 0,19 a child. the number of books to which learners have access can influence their abilities. the average number of books that were reported to be available in learners’ homes is reflected in table 2. to obtain an index of other possessions in the home, learners were asked to tick a given list of items. the list included: other reading materials and electronic media such as weekly or monthly magazines, daily newspapers, radio, television set, video cassette recorder, cassette player, telephone, refrigerator or freezer, car, motorcycle, bicycle, piped water, electricity, and a table to write on. items were calculated ranging from a minimum of zero for no item ticked to a maximum of 13 if all the items were ticked. the mean index is presented in table 2. the national average for books in the home of the average grade 6 learner is more or less 33 books. the mean index for possessions in the home of an average grade 6 learner is 6,6 at national level. these results are in line with previous results linking educational levels with socio-economic status. the availability of books and the number of possessions at home correlate with the level of education of the parents or guardians and the economic ability to purchase books and possessions. provinces that displayed fewer books and possessions in homes corresponded with those provinces that had lower levels of parent education. the final aspect relating to socio-economic characteristics is the general quality of the home environment. this information is presented in table 2. the parents’ socio-economic status, as reflected by general housing conditions, was investigated. general housing conditions included lighting, the condition of floors, walls and roofs in learners’ homes or where they stayed during the school week. for each of the aspects the minimum value of the index was 1 for absolutely basic or poor conditions, and the maximum was 4 if the conditions were perfect. therefore, the minimum value of the index for general quality was 4 if all the aspects were absolutely basic or poor, and the maximum was 16 if all the aspects were perfect. the national mean of the index was 11,9 on the scale from 4 to 16. the extremes for the provinces were western cape (14,0), gauteng (13,8) and kwazulu natal (12,1) on the one end of the general quality scale. according to the interpretation of the allocated scores, this means that on average grade 6 learners in these provinces stayed in homes where the majority used a paraffin, oil or gas lamp; the majority of the floors were made of wood or cement but few had carpets; the majority of the walls were made of stone, metal or wood; while the majority of the roofs were grass thatch, mud, metal or asbestos, but few were tiles. on the lower end of the general quality scale were northern cape (11,0), limpopo (10,6) and eastern cape (10,6). the scores indicate that on average grade 6 learners stayed in homes where the majority either had no light or used fire or candles. learners in this category lived in homes where the majority of the floors were made of earth, clay or canvas; the majority of the walls were made of cardboard, plastic, canvas, reeds, sticks or grass, while only a few walls were of stones, mud bricks or wood. the majority of the roofs were cardboard, plastic, canvas, grass thatch or mud but few were cement or concrete. table 2. socio-economic characteristics: parent education contextual factors of the mathematics learning environment of grade 6 learners in south africa 42 place where learners stay during the school week parent/guardian relatives/family hostel/board self/children province % se % se % se % se eastern cape 73,9 4,30 15,1 2,61 3,4 1,52 7,5 1,95 free state 73,0 6,64 14,2 3,70 2,2 1,61 10,6 3,98 gauteng 87,3 3,21 7,3 1,78 0,6 0,36 4,8 1,88 kwazulu natal 82,4 4,33 9,6 1,95 3,7 1,80 4,4 1,40 mpumalanga 62,7 7,48 16,9 2,55 4,8 1,57 15,6 5,63 northern cape 86,2 3,28 9,3 2,61 0,4 0,44 4,0 1,17 limpopo 70,8 6,29 9,8 2,35 8,8 3,33 10,6 2,63 north west 79,2 5,31 14,9 3,47 2,4 1,71 3,6 2,12 western cape 87,6 2,50 8,8 2,69 1,9 1,20 1,7 0,90 south africa 78,1 1,82 11,4 0,87 3,6 0,73 6,9 0,86 from the findings, it is evident that, to varying degrees, a significantly high percentage of the learners have to do homework in conditions that are not conducive to effective learning. many learners live in conditions that expose them to the elements (cold and heat) and social hazards. from the analysis, it would appear that a considerable number of learners stayed in temporary informal settlements that have become a typical phenomenon around towns and large cities. another aspect in this category is the place where grade 6 learners stay during the school week. learners had to respond to the following options: living in their parents’/legal guardians’ home; staying with other relatives or another family member; living in hostel/boarding school accommodation or somewhere by themselves or with other children. the findings are presented in table 3. the modal place of residence was a home with parents or legal guardians. a further 11,4% of the learners stayed with relatives or another family member, while 3,9% stayed in a hostel or in boarding school accommodation. however, 6,9% of grade 6 learners either stayed on their own or with other children. fairly high percentages of learners from mpumalanga (15,6%), free state (10,6%) and limpopo (10,6%) indicated that they stayed on their own. the latter learners are deprived of parental care, support and assistance with homework, which might influence their learning and attendance of school. the next aspect addresses learners’ absenteeism. learners were asked to write down the number of days that they were absent during the month preceding the survey. subsequently they had to provide a reason for their absence, choosing from the following options. these responses are reflected in table 4 together with the data concerning the repetition of grades. days absent illness family reasons school fees work province mean se % se % se % se % se eastern cape 1,4 0,21 21,2 2,6 10,6 2,78 4,8 1,53 6,2 1,13 free state 1,4 0,21 25,4 5,96 10,6 3,02 1,3 0,65 4,3 1,59 gauteng 1,0 0,27 16,9 2,58 4,9 1,6 1,5 0,77 1,1 0,59 kwazulu natal 2,2 0,40 37,7 3,98 10,6 2,08 1,7 0,78 3,7 1,47 mpumalanga 1,9 0,33 25,5 5,42 8,5 2,05 5,2 2,24 9,2 2,52 northern cape 1,5 0,25 33,8 3,82 5,5 2,36 3 1,29 3,8 1,39 limpopo 2,2 0,39 20,1 3,58 8,8 1,64 6,2 1,87 9,9 2,19 north west 0,6 0,11 15,8 3,16 4,8 1,43 1,7 0,76 3,1 1,18 western cape 1,6 0,45 26,1 4,64 13,5 3,98 0 0 1,5 0,96 south africa 1,6 0,13 24,6 1,4 9 0,86 3 0,47 4,9 0,57 table 3. school context characteristics: after-school care table 4: absenteeism and reasons 43 the overall south african mean for days absent was 1,6 days per month. with regard to absenteeism at the national level the learners had been absent for an average of two days, which translated into an average of 10% of school days. the main reason for absence was ‘illness’, followed by ‘family reasons’ and then ‘work they had to do at home’. the payment of school fees did not play an important role in learner absenteeism. information concerning mathematics issues (the frequency with which learners received mathematics to be done and practised at home) is addressed next. table 5 summarises the relevant data. home assistance ‘most of the time’ with school work receiving mathematics homework do mathematical calculations questions on mathematics own mathematics textbook province % se % se % se % se eastern cape 48,7 4,82 30,5 3,98 28,9 2,40 42,3 4,85 free state 54,8 10,94 26,1 7,66 29,6 7,59 49,1 12,36 gauteng 58,5 8,67 24,6 3,99 27,5 4,58 51,1 13,55 kwazulu natal 46,2 6,31 29,3 5,40 34,9 4,86 39,9 7,69 mpumalanga 50,2 6,36 38,9 3,70 37,7 4,52 34,6 6,63 northern cape 43,4 5,90 27,4 4,84 26,5 4,00 28,4 7,25 limpopo 62,8 5,48 25,4 2,40 33,6 3,09 43,1 7,52 north west 43,3 6,40 33,8 4,55 32,3 4,00 24,7 6,49 western cape 70,6 5,16 30,9 6,74 27,6 6,35 36,9 9,37 south africa 53,8 2,41 29,2 1,71 31,4 1,63 41,1 3,00 the overall south african percentage for home involvement in mathematics homework is 53,8%. this reflects the extent to which homework is valued and the extent to which understanding of what was taught during class is consolidated. however, doing homework also avails learners of an important opportunity to interact with family members over their schoolwork and, through these interactions, learners receive a wider spectrum of perspectives. grade 6 learners’ responses to whether they received mathematical assistance at home ‘most of the time’ are indicated in table 5. overall, for south africa, the levels of home assistance with mathematics were noticeably low. the national averages for assistance at home with mathematical calculations was 29,2% while a similar pattern is reflected concerning questions on school mathematics work to be done at home (31,4%). the generally low levels of assistance that learners received from family members correlate with the low levels of parents’ educational backgrounds, as pointed out earlier. parents seem to have attained low levels of schooling and, therefore, may not feel competent enough to render the necessary assistance. the low levels of education may also influence the interest that parents take in their children’s schoolwork. finally, information was gathered to determine whether learners had sufficient access to classroom materials to participate meaningfully in their lessons. the question concerning the possession of mathematics textbooks is addressed in table 5. for south africa overall, 41,1% of grade 6 learners had their own mathematics textbooks. the distribution varied among the provinces with the highest average in the free state and the lowest in northern cape. generally, the percentage of books owned or available to individual learners was very low. this phenomenon had a direct influence on practicing mathematics at home as not all learners could refer to their textbooks. mathematics to be done at home was normally written down on the chalkboard and copied by the learners. this is not the ideal way of spending teaching time in a mathematics classroom. findings and conclusion answers were sought to three major questions regarding the impact that personal characteristics, socio-economic aspects and school context characteristics might have upon grade 6 mathematics learners. from the data gathered, certain insights emerge. table 5. learners’ involvement with mathematics homework contextual factors of the mathematics learning environment of grade 6 learners in south africa 44 a considerable number of learners were overaged (about one year older than was expected for a grade 6 learner if they started school at the official age of seven years and had not repeated a grade). this may be ascribed to possible delayed enrolment and/or grade repetition. this may imply that major changes in school system organisation and improvements in teaching quality is needed. although girls (52,5%) outnumbered boys (47,5%), with the exception of a few provinces it seems as if gender imbalances are not a major issue. gender disparity at this level of schooling is not prevalent. however, disparities between the sexes are usually more prevalent in secondary and higher education than in primary education (efa, 2004: 103). it may be argued that mathematics acts as a gatekeeper in depriving women and minority students of equal opportunities in further education and employment. contemporarily, gender issues focus on ensuring girls equal access to education resulting in gender parity and equality in education. the majority of the learners (78%) stayed with parents whose average education was mainly primary and barely secondary. it is generally accepted that higher education levels of parents foster a greater appreciation of the value of education and an ability to help children learn. the average grade 6 learner lived in a home where there were little more than 30 books. this low availability of books in homes can be related to the level of education of the parents or guardians and the economic ability to purchase books. these results also correlate with the number of possessions at home where the mean for possessions at the home of an average grade 6 learner was just under 7 on a scale of 13 at the national level with a noticeable drop in the number of possessions in rural settings in comparison with urban environments. data gathered with regard to the general quality of grade 6 learners’ homes reveal that a considerable number of learners stay in informal settlements. conditions in such migrant settlements are not conducive to effective learning and may impact upon learners’ schoolwork. a sizeable number of learners (about 7%) indicated that they stayed either on their own or with friends during the school week. the socio-economic backgrounds of learners underline other educational research findings (van de walle, 2001: 456). khuzwayo’s research (2005: 313) indicates that learners from higher socioeconomic backgrounds tend to perform better on tests of educational achievement than those from poorer backgrounds – mainly because children from wealthier homes have greater access to a range of human and material resources that enhance, facilitate and reward school learning. also, in more affluent areas where parents can afford to pay extra, data reveal positive results such as the acquisition of social and analytic skills by learners. research about the nature of differences (social class, race, ethnicity, language background, gender and other demographic characteristics) is needed to help schools better educate a portion of the population that is growing. keitel (2005: 330) advocates alternative forms of mathematics education research in which greater focus is placed on socio-cultural aspects of learning. the provinces with the highest absenteeism rates were limpopo, kwazulu natal and mpumalanga. in view of the contemporary impact of the hiv/aids pandemic in africa, this tendency of illness needs to be monitored so that appropriate measures can be taken. general support of and interest in learners’ academic activities were moderate. the average percentages were as follows: for ensuring that homework was done, 18,4%; assistance with homework, 34,5%; and interest in completion of schoolwork at home, 39%. homework provides an opportunity to enhance academic development and more time devoted to homework usually results in higher scores. less than half of the grade 6 learners (41,1%) owned a mathematics textbook. the problem was most acute in rural provinces. in conclusion, there were major differences between the nine provinces; between low and high socio-economic groups and between rural and urban schools. particular characteristics prevalent in certain provinces as well as differences between various groups emerged from the data. in the second article in this series, correlations between certain variables and achievement will be addressed. the essential conclusions of this investigation are: • to understand the diverse needs of learners especially multiple disadvantaged learners in a developing country. • to focus on the place where teaching and learning actually takes place – the classroom. • to point out that the cognitive value added by schooling (as opposed to home background) should not be underestimated. g.s. kotze and j.p. strauss 45 although these conclusions are largely straightforward, they have implications for mathematics performance that will be addressed in the second article. compared to the vision of quality education, access and equity of education for all (world declaration on education for all, 2004) the results of the investigation indicate that much more should be done to ensure that children learn to a sufficient level to perform well as future citizens. references clarke, d. (1996). assessment. in a.j. bishop, k. clements, c. keitel, j. kilpatrick, and c. laborde (eds.), international handbook of mathematics education (pp 327-370). london: kluwer 1. education for all (efa). (2004). global monitoring report. paris:unesco. keitel, c. (2005). reflections on mathematics education research in south africa. in renuka vithall, jill adler & christine keitel (eds.), researching mathematics education in south africa (pp 329-344). cape town: hsrc press. khuzwayo b a history of mathematics educarion research in south africa: the apartheid years. in renuka vithall, jill adler & christine keitel (eds.), researching mathematics education in south africa (pp 307-327). cape town: hsrc press. leder, g.c. (1992). mathematics and gender: changing perspectives in douglas a grouws (ed.), handbook of research on mathematics teaching and learning (pp 597-622). new york: macmillan. mcleod, d.b. (1992). research on affect in mathematics education: a reconceptualization. in douglas a grouws (ed.), handbook of research on mathematics teaching and learning (pp 575-596). new york: macmillan. naudé, d. (2004). acquisition of mathematics literacy. in irma eloff & liesel ebersohn (eds). keys to educational psychology (pp 119-144). cape town: uct press. programme for international student assessment (pisa). (2003). retrieved november 11, 2004, from http://www.pisa.gc/math sacmeq ii research report s.a. (forthcoming). paris: unesco. swart, e., & pettipher, r. (2005). a framework for inclusion. in e. landsberg, e. kruger & n. nel (eds.), addressing barriers to learning (pp 323). pretoria: van schaik. troutman, a.p., & lichtenberg, b.k. (2003). mathematics – a good beginning. belmont: thompson-wadsworth. van de walle. j.a. (2001). elementary and middle school mathematics. new york: longman. for the things of this world cannot be made known without a knowledge of mathematics. roger bacon abstract introduction design of the testing programme missing data methodology analysis results discussion and recommendations acknowledgements references about the author(s) caroline long department of childhood education, university of johannesburg, south africa johann engelbrecht department of mathematics, science and technology education, university of pretoria, south africa vanessa scherman department of psychology of education, university of south africa, south africa tim dunne department of statistical sciences, university of cape town, south africa citation long, c., engelbrecht, j., scherman, v. & dunne, t. (2016). investigating the treatment of missing data in an olympiad-type test – the case of the selection validity in the south african mathematics olympiad. pythagoras, 37(1), a333. http://dx.doi.org/10.4102/pythagoras.v37i1.333 note: our respected and loved colleague, tim dunne, sadly passed away in a car accident after this article had been submitted for publication. we acknowledge his valuable contribution not only to the article but also to the field of rasch measurement theory. original research investigating the treatment of missing data in an olympiad-type test – the case of the selection validity in the south african mathematics olympiad caroline long, johann engelbrecht, vanessa scherman, tim dunne received: 02 mar. 2016; accepted: 15 aug. 2016; published: 31 oct. 2016 copyright: © 2016. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract the purpose of the south african mathematics olympiad is to generate interest in mathematics and to identify the most talented mathematical minds. our focus is on how the handling of missing data affects the selection of the ‘best’ contestants. two approaches handling missing data, applying the rasch model, are described. the issue of guessing is investigated through a tailored analysis. we present two microanalyses to illustate how missing data may impact selection; the first investigates groups of contestants that may miss selection under particular conditions; the second focuses on two contestants each of whom answer 14 items correctly. this comparison raises questions about the proportion of correct to incorrect answers. recommendations are made for future scoring of the test, which include reconsideration of negative marking and weighting as well as considering the inclusion of 150 or 200 contestants as opposed to 100 contestants for participation in the final round. introduction mathematics competitions globally have grown into an immense vibrant network that engage millions of students and teachers, contributing significantly to the development and maintenance of mathematical knowledge and the educational process. however, performance in mathematics competitions does not always correlate with classroom performance (ridge & renzulli, 1981). in fact, kenderov (2006) sees competitions as providing a tool to identify and develop students with higher abilities and talent who do not experience any challenge in the standard curriculum. the outcome of this curtailed curriculum experience in many a classroom is that the mathematical abilities and talent of students with great potential then remain undiscovered and undeveloped. while they play an important role, competitions are not unconditionally promoted. one critique of competitions is that they provide unnecessary pressure, stress and feelings of failure from excessive competitiveness (davis, rimm & siegle, 2011). kenderov (2006) supports this view and argues that although students who perform well in competitions often become good mathematics researchers, many highly creative students do not function well under time pressure. he further states: what matters in science is rarely the speed of solving difficult problems posed by other people. more often, what matters is the ability to formulate questions and pose problems, to generate, evaluate, and reject conjectures, to come up with new and nonstandard ideas. (kenderov, 2006, p. 1592) the south african mathematics olympiad (samo), organised by the south african mathematics foundation (samf), is the premier mathematics olympiad in the country and an important event in the school calendar. participation has grown from just over 5000 contestants in the first event in 1966 to about 82 000 contestants who participated in the samo 2015. objectives of the samo are to generate enthusiasm and interest in the subject, to enrich the study of mathematics, to promote mathematical problem-solving proficiency, to equip contestants for university level mathematical thinking and, in addition, to identify and inform selection of the finest young mathematical minds for international competition. the reported benefits and critique of competitions apply in part to the samo. however, ongoing evaluations have over the years introduced changes to ensure the furthering of the main aims of the samo competition. this research study forms part of the evaluation in the interest of furthering the central objectives stated previously. this study focuses on the second round results of the samo 2012 for the junior division (grade 8 and grade 9). we investigate specific aspects of the second round test for the purpose of providing information on the fitness for purpose of the test and for the future refining of the processes of contestant selection for the third and final round of the competition. the primary question guiding this study is: how do the testing procedures and processes, including administration, marking and analysis processes, support the selection of the most deserving 100 contestants for participation in the third round; in other words: are there contestants excluded from the third round who should reasonably be included? subsidiary questions, supporting the primary question, include: to what extent are the psychometric properties robust enough to claim measurement of mathematical excellence? how does the treatment of missing data affect the selection outcomes? how do various test design features, administration, scoring and analysis procedures affect the outcomes? design of the testing programme at the design stage the construction and the selection of items is the task of the olympiad committee comprising master teachers of mathematics, past olympiad winners and mathematicians from tertiary institutions. items in the samo test are ranked from 1 to 20 according to the level of difficulty as judged by the item writing team. items 1 to 5 are considered to be easy or accessible to most contestants; items 6 to 15 are considered to be moderately difficult and items 16 to 20 are expected to be most difficult. each band of items is scored differently. the standard scoring procedure in samo is depicted in table 1. table 1: weighting and penalty procedure and random guess score. the question here is whether the weighting of items (as in table 1) skews the selection outcome to some extent, whether the a priori judgement of weighting is valid and whether this weighting makes a difference to the ranking of contestants. these questions are explored in the results section and suggestions are offered in the discussion. one of the features of the test programme is ‘negative’ marking. in the administration phase of the samo 2012 test, the instructions to contestants state that incorrect answers will be penalised. the rationale for negative marking is as follows. for each multiple-choice item with five options, the probability of answering correctly through random guessing is 20%. taking this logic further over all the items in the test means that any contestants simply guessing all the way through the test will obtain scores close to an average score 20%. the intended outcome of the negative marking is to ensure that the random guessers score on average zero on each item. the intended behavioural effect is to eliminate random guessing. by knowing there is a penalty for wrong answers the contestant, when confronted with an item for which they find no apparent correct answer, is induced to omit the answer rather than risk the negative penalty, as opposed to a minimal chance for a prospective positive score. in the case where a contestant is not completely unsure of the answer but thinks they may select a correct option, they may take the risk. the question arises here whether, when analysing the data, the induced missing value should be allocated a zero score or should be taken as missing and what effects these two different treatments have on the rankings of contestants. the time allocated for the first round paper of the samo is one hour for 20 multiple-choice questions. for the second round the time is increased to two hours to complete the 20 multiple-choice questions. the test designers and administrators are of the opinion that the time is adequate for the test to serve its purposes. one reported caveat is that some good mathematical thinkers may require extra time per item simply because they are excessively thorough. this comment is in line with kenderov’s (2006) call for acknowledging and encouraging original mathematical thinking. the third round paper is four hours for six extended format questions. this format shows a radical departure from the first two rounds. missing data missing data is common in research and alludes to planned and desired information that is not available for examination and analysis (tsikriktsis, 2005). thus, explanations for the missing data may be difficult to deduce (mallinckrodt et al., 2003). it is due to the very nature of the phenomenon that it cannot be adequately described (mcknight, mcknight, sidani & figueredo, 2007). there are several reasons why data could be missing: these conditions may relate to the participants, the study design and the interaction between the participants and the study design. the contestant could have missed an item, saved the item for later and run out of time or felt reluctant to answer the question (sijtsma & van der ark, 2003). missing data may be described in terms of three mechanisms of missingness: missing completely at random (mcar), missing at random (mar) and missing not at random (mnar). missing completely at random (mcar) asserts a completely unsystematic pattern. the probability of a missing data element at any observation on any variable is unrelated to any of the data values intended in the data set, missing or observed. we may write an equation to state the marginal probability and the joint conditional probabilities: prob (xk is missing) = prob (xk is missing | xk and xobs) an example of the mcar phenomenon occurs when a contestant accidentally skips an item. the accidental skipping of an item has a probability that is not related to proficiency in the test construct, for example, and can therefore be classified as mcar. data values are said to be mar when the propensity for an element in a data case to be missing may be related to the observed values for that case, but the propensity is not related to the values of its own unobserved (missing) variables. here the word random may appear to have an unfamiliar connotation. equivalently, if any two data cases share identical values on their commonly observed elements, the pair will presumably have the same statistical behaviour on the other observations, whether observed or not. the probability of a missing value may depend upon the value of the observed elements. for example, the missing data mechanism may be related to general language proficiency or may be related to speed and time available, rather than the difficulty of the item or proficiency of the contestant. the relationship between proficiency, speed and difficulty impacts on the performance in any timed test. in both instances, mcar and mar, the response mechanism is termed ignorable. thus the researcher can make a reasoned argument to ignore the unknown factors leading to the missing data and thus permit a simpler approach to the available data (pigott, 2001). when unobserved data are neither mcar nor mar, the data are termed to be missing not at random (mnar). mnar means that the data is missing for a specific reason (i.e. the unknown value of a variable that may become missing in a data case may affect the probability of the value becoming missing). this situation arises in the context of this olympiad study, when the missing status for an unobserved (missing) data value is directly attributable to contestant proficiency or item difficulty. in the case of negative marking being a deterrent to guessing, mnar may be the case. however, there is an argument to be made that overcautious contestants may forego the reasonable probability of answering correctly. missing data, of any kind, influences the interpretation of the results. in the case of this study, we posit that the missing data impacts somewhat on the item difficulty estimates and on the ranking of the contestants. it therefore has consequences for validity and reliability claims (mcknight et al., 2007). it remains one of the major challenges for analysis, due to the fact that information has been lost. potential solutions to address missingness include regarding omitted items as not administered, or by contrast as incorrect, and allocating a zero score (ludlow & o’leary, 1999). in this samo 2012 junior second round test, 92.8% of the total item-person data points (20 × 4141 = 82 820) were recorded, with some 7.2% of the data missing, which may be perceived as relatively inconsequential. nevertheless, the approaches to handling of missing data account for slight differences in the estimated locations of items and greater corresponding shifts in estimated person locations, which in turn account for variation in the selections of the top 100 contestants (at the highest person locations). in this article each of these approaches is explored with reference to specific items and the consequences or the differing effects on the selection of contestants are discussed. methodology our data sources consisted of 4141 junior contestants (grade 8 and grade 9) who participated in the samo 2012 second round test comprising 20 multiple-choice items with five options, one of which was correct. in view of the primary research question, that is whether or not the testing procedures and processes support the selection of the most deserving 100 learners, with particular reference to the procedure for handling missing data, two parallel analyses were conducted on two versions of the dataset. in one analysis, zeroes were assigned for all missing data (zm) and in the second all missing data were handled through the standard procedures of the rumm 2030 software (mm). for both sets of data, a rasch analysis was conducted and the set of statistical methods applied to provide information on the test as a whole and on the individual items. parallel results are reported throughout the analysis. rasch measurement theory the application of the rasch measurement theory (rmt) to item level data collected on the second round tests enabled us to answer questions concerning the robustness of the data, in terms of the psychometric properties required for measurement-like outcomes. the model is clearly explained elsewhere (andrich, 1988, 2004; dunne, long, craig & venter, 2012; rasch, 1960/1980; wilson, 2005); some pertinent aspects are discussed here. the dichotomous model, scoring correct responses as one and incorrect answers as zero, is operationalised in the rumm2030 software features and applied in the early stages of the analysis to verify the functioning of the test as a whole and the coherent functioning of individual items. the model assumes that the probability of a contestant answering any dichotomous item correctly is ‘a logistic function of the relative distance between the item location and the [contestant] location on a common linear scale’ (tennant & conaghan, 2007, p. 1359). the rasch analysis aligns both item difficulty and person proficiency on the same scale, by assigning estimated locations whose difference governs probabilities of zero or one item scores. for person (v) and item (i), the probability of a correct response is governed by: the left-hand side of the equation is read as ‘the probability of [person] v being correct [scoring 1] on item i given the [person’s] ability, βv , and the item’s difficulty, δi‘. the right-hand side involves only an exponential transformation of the difference between person ability, βv , and the item difficulty, δi. the function of the denominator in equation 1 is to constrain the sum of the (two) probabilities for any dichotomous item to 1 (andrich, 2006, p. 63). one consequence of the model (equation 1) being applied independently to all the data of each candidate and all items is an expectation that the parameters for candidate ability and item difficulty are all on a common interval scale, called the logit scale. if the data collectively fit the complete model derived from equation 1, this common scale permits very specific stochastic interpretations of the observed data that are highly desirable, including measurement-like interpretations of the functioning of the test and its constituent items. analysis in this analysis there are two analytic processes that we regard as somewhat distinct but related in that they both impact on the outcomes of the contest and influence which second round contestants will be selected for the third round. the first process is the analysis of the theoretical coherence of the test and the test items: the second is the method of scoring. we report on the model and the software selected for the analysis, the summary statistics (means, standard deviations, chi-square statistics and associated exceedance probabilities and person separation index) and then the fit of items and persons to the model. the second process involves the handling of missing data and the possibility of randomly guessed correct items within the observed data. its purpose is to manage the fact that despite the attempts to dissuade guessing, there would inevitably be one or more chance correct multiple-choice question responses guessed by a contestant for whom the conditional probability of a correct response on any difficult item seems very low. we expect that a contestant will have a low probability of success on difficult items precisely because the contestant appears to perform at a low ability level on the test as a whole. for clarity, we note that weaker contestants tend to have lower scores and we expect them to offer correct responses to easier items. we also expect them by definition to either leave difficult multiple-choice questions items blank or possibly choose randomly amongst some or all of the options on that item subset. the effect of correct guesses made by weaker contestants is twofold. each correct guess increases the candidate’s score, but simultaneously makes the item in question appear less difficult than expected because the observed frequency of correct responses is increased by correct random guesses. the above rationale underpins the process in a tailored analysis (andrich, marais & humphry, 2012). while the focus of the article is essentially on the treatment of missing values in the olympiad test, the first analysis was to discuss aspects of the test and the testing process that impact on the theoretical coherence of the test and the testing process. these aspects include the expected versus empirical difficulty, the fit of the test as a whole, the targeting of the test and investigation of specific items. a second analysis investigates the handling of missing data. the ranks of the item locations that represent the empirical difficulty levels as determined by the rasch analysis were compared with the ranks of the expected difficulty levels as assigned by the designers of the tests. test validity for the test as a whole, the summary statistics, including item and person means and standard deviations, point to the appropriateness of the test. the fit of the data to the model, reflected in the chi-square statistics, and the test reliability as reflected in the person separation index (psi) provide evidence of the robustness of the test. the psi, specific to latent trait models such as the rasch model, contrasts the variance among the ability estimates of persons tested in the data relative to the error variance within each person (andrich, 1982). the index provides a measure of internal consistency by providing an indicator of the separation of persons relative to the difficulty of the item. the equivalent in traditional test theory is the kuder-richardson 20, or cronbach’s alpha, which provides a measure of the internal consistency of the items, rather than a measure of person consistency relative to items (andrich, 1982). for measurement in the psychosocial sciences, as with the physical sciences, appropriate targeting of an instrument as a whole implies that maximum information can be expected in the functioning of the items over the persons who are to be assessed. targeting is an implicit goal of the test designers, in seeking to select k useful items as the instrument for their purpose (here using k = 20 items to select top-end mathematics talent on the basis of high total scores). in the rasch model, the item location mean is set by convention at zero as a reference value on the logit scale. the comparison of the item mean with the observed person location mean provides a summary post hoc indication of the appropriate targeting of the test for the observed contestant group. the closer the person mean to the item mean, zero, the more accurately measures of ability can be obtained. for individual items, the fit residual is the standardised sum of all differences between observed and expected values summed over all persons. if a fit residual is over 2.5, or less than –2.5, the item is regarded as possibly misfitting. in the application of the rasch model, the item fit is investigated and where anomalies are found these residuals are further investigated. the sign of the item fit residuals relates to the notion of item discrimination. in the sense used here, discrimination is the rate of increase in the probability of a correct response to a specific item with respect to the underlying person ability level. the rasch model suggests expected frequencies of item scores for each ability level. if the observed pattern of item performance against ability levels exaggerates the frequency of zero scores among contestants of lower ability, and also exaggerates the frequency of correct scores for high ability contestants, the overall item-fit residual will be negative (smaller than expected), pointing to overly high discrimination. overly high discrimination indicates that persons of high ability may be obtaining special advantage. this special advantage may indicate that a second dimension (possibly language proficiency) that is positively correlated with the intended construct gives learners of high ability an undue advantage (masters, 1988). this possible second dimension is to be investigated. the further analysis compared the two approaches to missing data; the first simply regarded the missing data as incorrect (by default) and entered zero (zm). this approach implies a judgement was made that the reason the contestant did not answer the question was that the correct answer was unknown to the contestant, hence the data was mnar and for a reason directly related to the construct knowledge being tested. the second approach was to reserve judgement (assume that the data may be mar, perhaps related to the time factor or a reluctance to offer a wrong answer) and rather to assign a value using maximum likelihood estimates of the parameters and of the corresponding expected values for the missing data elements. in this second approach the missing values are treated as simply absent (mm). guessing and tailored analysis it is inevitable, where the item difficulty far exceeds the proficiency of the candidate, that there may be guessing. here a tailored analysis is conducted to adjust for possible guessing on each of the data sets, zm and mm. this approach is based on the understanding that ‘guessing is a function of the difficulty of the item relative to the proficiency of the person’ (andrich et al., 2012, p. 1). andrich et al. (2012) propose a strategy to ‘remove those responses most likely to be affected by guessing’ (p. 3), making particular removals for each person based upon their pattern of scoring. these removed responses are then treated as missing responses. a tailored analysis was conducted with the cut-off at 0.20 or one-fifth, based upon the frequency for a randomly selected option from the set of five options offered for each of the multiple-choice questions. this cut-off is applied to the estimated probability of correct response, obtained from the rasch model. for each person all the difficult items with estimated probabilities of a correct response below 0.20 are eliminated, hence rendered missing, whether correct or incorrect. identifying best contestants the highest performing contestants in each of the two modes, zero for missing (zm) and missing as missing (mm), and then in each of the further two tailored analyses, zero for missing tailored (zmt) and missing as missing tailored (mmt) are ranked from highest to lowest. contestants whose rankings on all four treatments fall exclusively within particular ranges are coded. for example, contestant p may be ranked 3rd in the zm ranking, 6th in the zmt ranking, 4th in the mm ranking and 7th in the mmt ranking. this contestant would be coded a, as all four rankings fall into the top 25. contestant q may be ranked 39th in zm ranking, 47th in zmt ranking, 13th in mm ranking and 22nd in mmt ranking. this contestant would be coded b as all four rankings fall within the top 50. the contestants with all four rankings in the top 100 would fall into category c. this process is continued for contestants’ rankings falling within 150 and within 200 (see table 2). table 2: cumulative counts of contestants in rank groups on four analysis routines. the count of 14 includes only the contestants who achieved rankings exclusively in the top 25, 25th inclusive, in all four statistical treatments. likewise, the count 27 includes all contestants who achieved ranking in the top 50; it therefore includes the count of 14 who achieved rankings exclusively within the top 25 (see table 2). table 3 shows the number of contestants who obtained the highest total score of 19, down to the total score of 12, in the vertical column. along the top row are the number of missing items for each group. for example, for the group whose total score was 16 (looking along the row), 13 contestants had no missing values, three contestants had one missing value, two contestants had two missing values and one contestant had three missing values. it appears from this analysis that negative marking did not dissuade the contestants from answering questions whose correct answer was not immediately obvious. table 3: counts of contestants by total score and missing answers. micro-studies to further investigate the handling of missing data in the context of the samo we conduct two explanatory micro-studies for illustration purposes. in the first we compare three categories of contestants who would fall, according to this system, outside of the top 100. we identified three other categories, some 25 contestants who had 14 out of the 20 items correct and none incorrect (category f), some 39 contestants who had all 13 items that they attempted correct (category g) and some 26 contestants who had 12 out of 13 or 13 out of 14 (category h). contestants in these categories would fall out of the top 100, on one, two or three of the rankings but be included on others. one contestant from each of the categories f, g and h is described. a second micro-study identifies two contestants each of whom obtained 14 correct answers; the first answered 14 correctly and six incorrectly and the second answered 14 correctly and omitted the other six items. this comparison raises questions about the effect of time on assessing mathematics ability. it also raises questions about the proportion of correct to incorrect answers. does the contestant who answers everything they attempt correctly have greater potential than the contestant who answers almost a third of the test incorrectly? results throughout this investigation we conduct a comparison between the two data sets, the first case where the zeroes are given for missing information, denoted as zm, and the second case where missing values are dealt with through maximum likelihood estimation, denoted as mm. the results are presented in terms of both the test as a whole and the component items and on the performance of the contestants. the following analytic categories are discussed: expected and empirical item difficulty, summary statistics on the test as a whole, targeting of the test, individual item statistics and the guessing factor. two additional categories that might have been reported, namely differential item functioning and local independence, which may have been important for checking the robustness of the data, have not been covered here. the contestants are discussed in relation to their overall performance and in relation to the varied rankings based on the different analytic treatments, namely zm and mm, and in pertinent cases the tailored equivalents, mmt and zmt. test design the notional (or expected) difficulty of items from the perspective of the test designers was found to differ from the empirical difficulty outcomes. in 50% of the cases the items were at the expected levels (see table 4). table 4: comparison of intended and empirical difficulty, under zm and mm methods. the reasons for the differences are explored elsewhere (engelbrecht & mwambakana, 2016). some of the reasons for the unexpected differences may include curriculum coverage, language issues, lack of exposure to the particular problem-solving strategy required for a specific problem or an element of surprise not obvious at the time of setting the paper. associated with the change in item difficulty order, there will be some difference in most contestants’ scores when using the weighted scores, rather than using unweighted scores (as shown in table 1). summary statistics and targeting of the test a comparison between the two versions of the data set is made, the first with zero scores for missing information (zm) and the second case in which all missing values are dealt with through maximum likelihood estimation of person locations (mm). the high total chi-square and the extremely low chi-square probability indicate that there is poor fit to the model (see table 5). the psi indicates rather moderate reliability with slightly improved reliability when the missing values are estimated through the standard procedure of the rumm software. the moderate psi indicates a limited spread of person locations along the scale. this limited spread may be an indication of a fairly homogeneous group taking the test, a consequence of common selection from a previous first round of testing serving as a screening mechanism. table 5: test fit to the model. we note here that the large number of data cases, over 4000, inevitably exacerbates the statistics for any misfit. by artificially reducing the size of the data set to a manageable number, the fit statistics were found to be more acceptable. nevertheless, we use the fit statistics, prior to any statistical correction improvement (see table 5). the statistics suggest that the test is moderately well targeted, with a person location mean of –0.804 (standard deviation 0.868) for the zm case and a person location mean of –0.645 (standard deviation 0.942) for the mm case (see table 6). the standard deviation in both cases indicates a spread of items that is acceptable. the optimal situation for obtaining maximum information is for the person mean to be aligned with the item mean. in the case of this test, the mean of –0.804 suggests the test is slightly on the difficult side for this set of contestants. the person mean in the mm case is located closer to the item mean, zero, and, given an argument for regarding missing data as missing rather than zero, could be regarded as the more accurate test statistic. table 6: summary statistics. the person-item threshold distribution is depicted graphically in figure 1. the spread of items is fairly good, with slightly better fit for the mm analysis. there is, however, a cohort of contestants for whom there are no items within their proficiency range. about 500 contestants, roughly an eighth of the complete set, have locations below the easiest item, item 16. the implication here is that all of these contestants have a less than 50% chance of answering any specific item correctly. an explanation of this phenomenon is given in the discussion. figure 1: person-item distribution thresholds for zm (a) and mm (b) analyses. item difficulty the person-item maps (see figure 2, zm, figure 3, mm) are graphical pictures of both item difficulty and person proficiency aligned on the same scale. items are depicted on the right-hand side of the graph. on the left-hand side, the estimates of learner proficiency are located. item 10 and item 12 are distinctly more difficult than other items. the next three items, 5, 18 and 20, form a second but less difficult cluster. items 7, 8, 9, 11 and 15 form a cluster around the mean zero location. the remainder of the items spread from –0.2 to –1.8 on the logit scale, indicating relatively easier items. from an investigation of each cluster, it may emerge that particular problem-solving skills are required in addition to mastery of the topic area. figure 2: person-item map for zm data with mean item location at zero. figure 3: person-item map for mm data with mean item location at zero. the estimated item difficulty locations differ across the missing data treatments. however, the group of the five most difficult items is common but in a slightly different order (see table 7). similarly, the five easiest items are in slightly different orders and likewise both middle quartiles involve common items but with internal order changes. the range of the mm analysis is slightly narrower, with the easiest item apparently slightly harder. table 7: item difficulty locations and standard error: zm, mm and mmt methods. a tailored analysis, by adjusting for guessing, provides a better estimate of item difficulty and of item fit (andrich et al., 2012). for this reason, we report for particular items the statistics provided by the tailored analysis mmt, as derived from mm. these results are presented in the three columns on the right of table 7. individual items item 10 (see figure 4), one of the most difficult items, required an understanding of three concepts: the area of a circle, ratio and probability. a feature of this question was that the correct option was not the most frequent choice by even the top contestants. this selection can be explained by the first distractor being very seductive (most frequently chosen), while being incorrect. we report here on the statistics after applying a tailored analysis. figure 4: item 10, with item characteristic curve, and multiple choice distractor plot. for each item discussed, there is a description of the item, together with its item characteristic curve and the multiple-choice distractor plot. for further explanation see dunne et al. (2012). item 16, contrary to expectation, was found to be the easiest item empirically (see figure 5). the easiness of the item can be explained by the strategy of some contestants to, rather than use mathematical techniques, simply extend the pattern to the 81st term. figure 5: item 16, with item characteristic curve, and multiple choice distractor plot. item fit the item fit residual is a statistic for assessing the extent to which an aggregate of the person-item residuals deviates from its expected value, zero. when fit residuals are greater than 2.5 or less than –2.5, we regard them as exceeding criterion levels for misfitting. in the initial data analysis six items were overdiscriminating (1, 3, 6, 14, 17, 19 with negative item residuals) and six items were underdiscriminating (2, 7, 10, 11, 15, 18). with further analysis (see the section on the tailored analysis), some of these items conformed to the model. item 2 (see figure 6) initially exhibited large overdiscrimination (–7.883); however, after the tailored analysis, while still showing some overdiscrimination, the fit was better (–3.871). the percent topic is a complex construct as explained by parker and leinhardt (1995). the problem was that of ignoring the referent to which the percentage ratio is being applied. we might also posit here that additive reasoning is applied in this problem rather than multiplicative reasoning. figure 6: item 2, with item characteristic curve, and multiple choice distractor plot. item 6 (see figure 7) on the initial analysis was highly overdiscriminating (–6.269). with the tailored analysis, the discrimination was reduced to –2.772. this item requires an application of algebra that is perhaps unfamiliar to many contestants. in addition, the requirement to explore the numbers in a number block may not have been previously encountered. figure 7: item 6, with item characteristic curve, and multiple choice distractor plot. guessing and tailored analyses as stated previously, additional procedures were administered that would provide a more accurate picture of the difficulty of the items, by firstly eliminating item 11 (found to be faulty), and secondly by conducting a tailored analysis to adjust for guessing. table 8 shows a comparison of the summary statistics across the two approaches, the zm, and the associated tailored analysis, zmt, and the mm, and the associated mmt. note that the person location means of the tailored analyses being closer to zero indicate that the test is better targeted in this approach. the reason for this better fit is that where an item is judged to have a low probability of being answered correctly by a specific person, that data point is omitted and regarded as missing. the reported person separation index improves in this situation (see table 8). table 8: comparison of statistics across tailored analyses. with the application of the tailored analysis there was a better fit for some of the items, although some items emerged as worse fitting (see table 9). table 9: comparison of tailored analysis and mm case analysis from perspective of mm. further analysis, which explores possible reasons for some items improving with the tailored analysis and others not, as well as investigating local independence and the extent of guessing, is in process. selection of contestants the higher performing contestants, expected to have some chance of success in the final round, were clustered into five groups, as explained earlier, according to their performance in the four statistical treatments zm, zmt, mm and mmt. the number of contestants in each group is given in table 10. table 10: counts in contestant ranking categories within four analyses. the practice applied in samo 2012 was to select the top 100 contestants according to their performance when the zm approach is used. the analyses applied in this study seems to indicate that in order to select the most deserving 100 for the third round, it may be better to consider all contestants that fall into categories a to e. three other categories were also considered, comprising candidates who were not in any of the categories a to e. thus we have category f, those contestants with a score of 14 out of 20, with no errors, category g, those contestants who scored 13 correct, with no errors, and category h, where contestants scored 11 or 12 correct with no errors, or obtained 11, 12 or 13, with only one error (table 11). all of these contestants fall outside the top 100, but within the top 155 (see table 11). table 11: counts in further rank categories. it is with the above reasoning that a closer micro-analysis was conducted on three contestants falling into the above three categories to use for illustrative purposes (see table 12). table 12: comparison of three individuals, selected in top 100 under some criterion. fay scored 14/20 in the test. she qualifies for the top 100 if we use the zm approach but does not when any of the other approaches is used. grant only answered 11 items but answered no items incorrectly. using the mm or mmt approach he qualifies for the top 100 easily but he does not when we use either zm or zmt. henry attempted 13 items and answered 12 correctly. again, when we apply either mm or mmt criteria he qualifies for the top 100 easily but not when we use zm or zmt. in a second micro-analysis, we compare the results of two contestants. adri answered 14 items, all of them correctly. her score for the test according to the system that was used, that is, taking the weighting and penalties into account, is 68%. bets attempted all 20 items and 14 of her answers were correct. after weighting and penalties, her final mark was 64% for the test. in table 13 we compare the two contestants. the items in table 13 have been ordered according to the empirical difficulty (location) when analysed with missing values regarded as missing. table 13: item comparison of two contestants with a common score 14 (70%). we see from table 13 that although adri and bets answered the same number of items correctly (14), adri obtained a final mark of 68% and bets obtained a mark of 64%, because of the penalties for incorrect answers. it is also clear that most of adri’s missing answers were of high empirical difficulty. this pattern indicates that her missing answers could be classified as mnar: she deliberately avoided answering the difficult questions of which she was uncertain. bets lost most of her penalty marks in the more difficult items, where she possibly guessed but selected answers incorrectly. there is a high probability that bets might have guessed item 20 correctly. the fact that she answered item 4 incorrectly is something of an anomaly. discussion and recommendations the results of the rasch analysis show that the test is fairly robust and adequate for the purpose. drawing from both professional judgement and from the outcomes of the analysis we conclude that the test is appropriate for engaging contestants’ interest and enthusiasm; it is appropriately targeted with both easy items to encourage contestants at the lower end and challenging items at the difficult end which will discriminate between the top contestants. it appears that the time allowed was adequate and that most participants managed to reach the end of the test, though we do not know how the participants approached the test as a whole. the missing 7.2% of 4141×20 data points may have been due to time constraints. for the present purposes we acknowledge that the missing data may be impacted by time constraints and this fact should be considered in the handling of missing data. negative marking and weighting definitely impact the results. negative marking increases the count of missing data elements. weightings of item contribution to the final score needs serious reconsideration since the weighting is based on the anticipated rather than on the empirical location. we recommend no a priori weightings of item contribution to the final score. in the rasch analysis even empirical weighting is redundant, because item location ensures contestants who can handle difficult items are credited for that ability. in the rasch framework it is possible to construct open-ended items for which a higher maximum score than the value 1 can be allocated by a suitable memo. some aspects of the analysis may be informative for future test design. the cluster of contestants, about 500, for whom the test appears too difficult should perhaps have been omitted from the second round testing. a closer investigation of these 500 contestants indicate some anomalies in the system. a deviation from the normal selection procedure was made on account of providing opportunities to learners who were exceptional at their school but who did not meet the cut-off. the question arising is whether the positive benefits for participating outweigh possible knocks to confidence. samf currently have programmes in place for learners with potential to receive more tuition in problem-solving strategies earlier on in the olympiad cycle, in an attempt to address lack of familiarity with problem-solving strategies in cohorts such as these. the most important question that was addressed in this study is whether this testing process adequately supports the selection of the top 100 contestants who will go through to the third round of the olympiad. this question does not have a simple answer and the only way of addressing this issue would be to monitor contestants’ performance in the third round. such a study is recommended. our analysis suggests that some contestants may deserve to qualify for the top 100, for example contestants who answered all items attempted correct, subject to some lower cut-off. from our analysis it can be recommended that rather than only taking the top 100 in the existing scoring approach (zm), it may be advisable to consider all contestants who ended up in the top 150 or 200 in all four statistical applications. these contestants can be thought of as yielding consistent evidence for their selection. from the analysis it appears that the test is functioning adequately. the different approaches have slightly different effects on the difficulty levels of the items, but have a greater effect on the selection of contestants. the various analyses resulted in different subsets of contestants falling within the top 100. the reason for the differences rests on different views of underlying proficiency and therefore the differing approaches to missing data. a rationale may be made for the selection of one data set; however, a composite arrangement may offer advantages. the empirical evidence for the ranking must be critically examined as other factors may contribute to ranking, for example non-coverage in some schools of related curriculum elements. acknowledgements we would like to express our gratitude to the south african mathematics foundation (samf) for providing access to 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(2005). constructing measures: an item response modeling approach. new jersey: lawrence erlbaum associates. microsoft word 68-79 anesh & brij.doc 68 pythagoras, 70, 68-79 (december 2009) using an inductive approach for definition making:  monotonicity and boundedness of sequences    deonarain brijlall  school of science, maths and technology education  university of kwazulu‐natal  brijlalld@ukzn.ac.za aneshkumar maharaj  school of mathematical sciences  university of kwazulu‐natal  maharaja32@ukzn.ac.za    the  study  investigated  fourth‐year  students’  construction  of  the  definitions  of  monotonicity  and  boundedness  of  sequences,  at  the  edgewood  campus  of  the  university  of  kwazulu‐natal  in  south  africa.  structured  worksheets  based  on  a  guided problem solving teaching model were used to help students to construct the  two definitions. a group of twenty three undergraduate teacher trainees participated  in the project. these students specialised in the teaching of mathematics in the further  education  and  training  (fet)  (grades  10  to  12)  school  curriculum.  this  paper,  specifically, reports on the investigation of students’ definition constructions based on  a learning theory within the context of advanced mathematical thinking and makes a  contribution  to  an  understanding  of  how  these  students  constructed  the  two  definitions.  it was  found  that despite  the  intervention of a structured design,  these  definitions were partially or inadequately conceptualised by some students.  currently, many schools of education from different south african universities (e.g., witwatersrand, nelson mandela metropolitan university, kwazulu-natal) are revising the presentation of differential and integral calculus modules for undergraduate pre-service mathematics fet trainee-teachers. this is to address the learner-centred approach which underpins curriculum 2005. we report on an investigation based on the use two of worksheets on which students worked individually and later in groups to construct definitions, followed by the analysis of written responses and interviews. wu (2003) argued that pre-service development of teachers in grades 6-12 require courses which consolidate, mathematically, those topics which do not stray far from the high school mathematics curriculum. in particular, “...they should revisit all the standard topics in high school from an advanced standpoint, and enliven them with motivation, historical background, inter-connections and above all, proofs” (wu, 2003, p. 8-9). it was shown that many of the concepts dealt with in the real analysis module strengthen the ideas on which the high school mathematics is based (brijlall, 2005). such concepts are indicated in the next section. the above context led us to formulate the following research question: how does the implementation of a structured worksheet design influence students’ construction of definitions in real analysis? in particular we look at the construction of the definitions for monotonicity and boundednes of infinite real sequences. the structured design used an examples and non-examples approach. in answering this question we focus on sorting, reflecting and explaining, generalising, verifying and refining, and extension of generalisation (cangelosi, 1996). deonarain brijlall & aneshkumar maharaj 69 background knowledge base for teachers one of the expectations of the norms and standards for educators (doe, 1999) is that the teacher be well grounded in the knowledge relevant to her/his occupational practice. she/he has to have a welldeveloped understanding of the knowledge appropriate to the specialism. many mathematics teachers find themselves in a position requiring them to implement the syllabus, which includes certain topics they are unfamiliar with. according to adler (2002), teachers with a very limited knowledge of mathematics need to develop a base of mathematical knowledge. they need to relearn mathematics so as to develop conceptual understanding. taking this into account we attempted to make certain that trainee-teachers leave with a base of knowledge relevant to their occupational needs. mwakapenda (2004) concurs when stating that a significant concern in school mathematics is the development of an understanding of mathematical concepts. we show via a few examples the direct relationship of the concepts with topics from the school fet syllabus as suggested by national curriculum statement (doe, 2003). the following are examples of the types of problems that are covered: open and closed intervals the notion and notation of open and closed intervals play an important role throughout the fet curriculum. for example the solution to the inequality x2 – x < 6 uses an open interval x  (-2; 3). the question solve for  in 2 1sin  for   [00; 3600] uses a closed interval. graduate students confront many other types of open and closed sets. the module questions the openness/closedness of ℚ, ℤ, ℕ and ℝ. this should yield a better understanding of the number system as well. sequences and series the learning outcome in the national curriculum statement (2003) prescribes the investigation of number patterns culminating in arithmetic and geometric sequences and series. learners in the 2006 curriculum would at grade 12 be expected to interpret recursive formulae (e.g., tn+1 = tn + tn-1). when proving 2 irrational an approach using recursive formulae is employed. this exposes the teacher trainee to recursive formulae, were not covered in the old syllabus. undergraduate students therefore need to be well versed in sequences and series. the realisation that a real sequence is a function whose domain is the set of naturals is fundamental. deeper understanding into the concepts of convergence and divergence of both sequences and series are dealt with in the real analysis course. this will aid student teachers when dealing with the convergence of geometric series in grade 12. students are also allowed the opportunity to investigate ideas and tasks to acquire a wider perspective to arithmetic, geometric, linear, quadratic and cubic sequences which are taught in grade 11. monotonicity a sub-skill in learning outcome 2 (doe, 2003) for grade 12 is to identify the intervals on which the function xay  , a > 0 and its inverse x a y log increase or decrease. these ideas are consolidated when studying monotonicity. a monotonic function (or monotone function) is a function which preserves the given order. this concept first arose in calculus and was later generalised to a more abstract setting. formally, a function f is called a monotonically increasing function on [a; b] if )()( yfxf  whenever yx  , ];[, bayx  , and dually one can define a monotonically decreasing function on an interval. a function is called monotonic if it is either monotonically increasing or decreasing. boundedness for this study, a bounded set is regarded as a set which has a greatest element and a least element. a bounded sequence is a sequence whose set of values is bounded. formally, a sequence  1nnx is using an inductive approach for definition making 70 bounded if it is bounded above and below. a sequence  1nnx is bounded above if and only if there exists a real number m such that mx n  for all n . a sequence    1nnx is bounded below if and only if there exists a real number m such that mx n  for all n . theoretical basis piaget, cited in bowie (2000), expanded and deliberated on the notion of reflective abstraction which has the following two components: (1) a projection of existing knowledge into a higher plane of thought, and (2) a reorganisation of existing knowledge structures. he proposed this to be a major factor for the development of mathematical cognition, and distinguished three types of abstractions: empirical abstraction, pseudo-empirical abstraction, and reflective abstraction. reflective abstraction refers to the construction of logico-mathematical structures by a learner during the process of cognitive development (dubinsky, weller, mcdonald, & brown, 2005a). the two features of this concept are: that it has no absolute beginning but appears at the very earliest ages in the coordination of sensori-motor structures, and that it continues on up through higher mathematics to the extent that the entire history of the development of mathematics from antiquity to the present day may be considered as an example of the process of reflective abstraction. we define the following four concepts that are used in the apos theory of conceptual understanding (bowie, 2000): o action: an action is a repeatable physical or mental manipulation that transforms objects o process: a process is an action that takes place entirely in the mind. o object: the distinction between a process and an object is drawn by stating that a process becomes an object when it is perceived as an entity upon which actions and processes can be made. o schema: a schema is a “more or less coherent collection of cognitive objects and internal processes for manipulating these objects” in our analysis of students’ definition making we used the following five kinds of mental constructions associated with reflective abstraction as explained by bowie (2000) and dubinsky et al. (2005a; 2005b): o interiorisation: the ability to apply symbols, language, pictures and mental images to construct internal processes as a way of making sense out of perceived phenomena. actions on objects are interiorised into a system of operations o coordination: two or more processes are coordinated to form a new process o encapsulation: the ability to conceive a previous process as an object o generalisation: the ability to apply existing schema to a wider range of contexts. o reversal: the ability to reverse thought processes of previous interiorised processes. this study focused on advanced mathematical thinking required for the concepts of monotonicity and boundedness of infinite real sequences. this falls under the domain of apos theory. methodology the method adopted four stages: design of worksheet, facilitation of group-work, capture of written responses, and interviews. the first two stages were influenced by social constructivism. in the worksheets a guided structure was used to facilitate definition making in a collaborative manner. this was based on our assumption that the construction of knowledge is better facilitated in a social context which provides support. learning in a social context is recommended by social constructivists such as von glasersfeld (1984), cobb (1994), confrey (1990) and steffe (1992). steffe (1992) argued that reflective ability is a major source of knowledge in all levels of mathematics. this implies it is important for students to talk about their thoughts to each other and the facilitator. deonarain brijlall & aneshkumar maharaj 71 the students involved were undergraduate teacher trainees at the edgewood campus of the university of kwazulu-natal. they pursue a module on real analysis in their final year. this module, which included elementary topology of the real line, involves the learning of concepts in set theory, relations and functions, cardinality, countability, denseness, convergence and other related ideas on elementary topology. a group of twenty three fourth-year undergraduate teacher trainees participated in the project. these students specialised in the teaching of mathematics in the fet school curriculum. the research instruments used were worksheets, observation of classroom activity, and interviews. instrumentation design of worksheet worksheets were designed in accordance with ideas postulated by a guided problem solving model suggested by the work of cangelosi (1996). his work modeled how meaningful mathematics teaching could be planned with the aim of simultaneously addressing the cognitive and affective domains. the model, which is illustrated in figure 1, has the following three levels or phases: o inductive reasoning (conceptual level) o inductive and deductive reasoning (simple knowledge and knowledge of a process level), and o deductive reasoning (application level). however, note that there is always interplay between inductive and deductive reasoning. they are continuously present and constantly following each other in mathematical thinking. for example, in an inductive process very often a preliminary ‘generalising’ step is reached. a conclusion or the finalisation of an inductive part is the beginning of the deductive part. therefore generalising at each of the different levels implies that the deductive mode of reasoning comes into play. in creating constraints for the examples and non-examples in the guided worksheets implemented in this research we kept in mind the characteristics of boundaries as suggested by mason and watson (2004). boundary examples are those examples that “distinguish having and not having a specified property” (mason & watson, 2004, p.9). they suggested that if students are only offered well-behaved examples, or examples which have additional, but irrelevant features, then the reason for careful statements in a definition might pass them by. in choosing items for the examples and non-examples the conditions in the definitions of monotonocity and boundedness were considered. to illustrate this, the sequence     11 nn was given as an example of a sequence that is bounded above. however, it is a bounded sequence since it is also bounded below. for the design of the worksheets inductive learning activities were used to construct the concepts of monotonicity and boundedness of real sequences. these activities had the following stages: o sorting (examples and non-examples) and categorising, which target action in apos theory. o reflecting and explaining the rationale for categorising, which target process in apos theory. o generalising by describing the concept in terms of attributes (that is, what sets examples of the concept apart from non-examples). o verifying and refining (the description or definition is tested and refined if necessary), which both target object in apos theory. to determine the monotonicity of a sequence require constructing knowledge of a process. here an algorithm had to be formulated which required the calculation of successive terms of a sequence, and then comparing them. using an inductive approach for definition making 72 figure 1: a guided problem solving teaching model data collection procedures facilitation of group-work the fourth year class of 23 students were presented the worksheets and engaged with the activities individually for approximately fifteen minutes. we believed that this would prepare the students to make constructive contributions in the group-work context which was to follow. when constructing the concept of montonicity they worked in seven groups (five comprising of three members and two with four). each group, after discussing and reaching a collective decision, documented their thoughts for presentation to the class. capture of written responses each student was given a guided activity sheet. when they were in groups they were provided a separate worksheet which required the collective group response to the activities. the following five instructions appeared on the worksheets: (a) complete each worksheet on an individual basis. (b) you are now required to form groups of three or four. (c) now discuss your findings within the group to reach consensus. (d) write down a collective response and elect a leader to discuss with class. (e) finally conclude findings as a class with tutor. these worksheets were then collected by the lecturer for analysis of student thinking. interviews with group leaders during the analysis of the written responses it became necessary to interview the group leaders. based on the written responses, questions were formulated to clarify the written responses of certain groups. interviews with the groups were audio recorded and transcribed. this was used for the analysis and discussion section of this paper. instruments for learning and tools for data collection monotonic sequences the following extract from the worksheet, modeled on the construction of a concept at the inductive reasoning level (cangelosi, 1996), indicates the task based on examples and non-examples which the students engaged with: sorting the following infinite real sequences are called monotonically increasing: 1.   12 nn 2.    112 nn 3.          12 1 n n 4.    212 nn inductive reasoning level o construct a concept o discover a relationship o willingness to try, appreciation inductive and deductive reasoning level o simple knowledge o knowledge of a process (algorithms) o willingness to try, appreciation deductive reasoning level o application o solve problems o willingness to try o appreciation deonarain brijlall & aneshkumar maharaj 73 the following infinite real sequences are not monotonically increasing: 1.    12 nn 2.          1 12 2 n n 3.         1 2 1 n n 4.     11 nn reflecting and explaining after interrogating the above examples and non-examples of monotonic increasing infinite real sequences, explain why one would categorise them as such. generalising the description of monotonically increasing infinite real sequences now write out a statement which you would adopt to describe (define) an arbitrary infinite real sequence  1nnx : bounded sequences the following extract from the worksheet, modeled on the construction of a concept at the inductive reasoning level (cangelosi, 1996), indicates the task based on examples and non-examples which the students engaged with: sorting the following are examples of infinite real sequences which are bounded below: 1.   12 nn 2.    112 nn 3.          12 1 n n 4.    112 nn the following are examples of infinite real sequences are not bounded below: 1.           12 1 n n 2.    121 nn the following are examples of infinite real sequences which are bounded above. 1.    12 nn 2.          1 12 2 n n 3.         1 2 1 n n 4.     11 nn the following are examples of infinite real sequences are not bounded above 1.   12 nn 2.    112 nn reflecting and explaining after interrogating the above examples and non-examples of bounded (above or below) infinite real sequences, explain why one would categorise them as such. generalising the description of bounded above/below infinite real sequences 1. now write out a statement which you would adopt to describe (define) an arbitrary infinite real sequence  1nnx as bounded above: 2. now write out a statement which you would adopt to describe (define) an arbitrary infinite real sequence  1nnx as bounded below: using an inductive approach for definition making 74 verifying and refining check whether the following are bounded above/below infinite real sequences by applying the above definition. 1.  12log nn 2.    12 n n 3.           12 1log n n pre-knowledge at this stage in the course students covered fundamental concepts on set theory, methods of proof and logic, relations, and basic ideas on topology of the real line. hereafter, they were engaged in issues relating to real infinite sequences. they covered the definition of a real sequence and its convergence as: o a real infinite sequence   1nnx is a function :f ℕ  ℝ defined as nxnf )( o a real infinite sequence   1nnx converges to l if for every 0 , there exist a natural number n, such that n > n implies  lx n . post-knowledge during the guided problem solving activity they developed the following definitions: o a real infinite sequence   1nnx is called monotonically increasing (decreasing) if nn xx  1 ( nn xx  1 ) for all n ℕ. o a real infinite sequence   1nnx is bounded above (below) if there exists a real number m (m) such that mx n  ( mx n  ) for all n ℕ. a real infinite sequence   1nnx is monotone or monotonic if it is either increasing or decreasing and bounded if it is bounded above and below. the aim hereafter is to prove the theorem “every monotonic bounded real infinite sequence is convergent”. it is much easier to show that convergence implies boundedness. that the converse is not necessarily true can be demonstrated by the bounded sequence    11 nn which is not covergent. the need for monotonicity allows for the truth of the converse. this theorem then will be treated as a final object (dubinsky et al., 2005a; 2005b), since the construction of the two concepts discussed in this paper applied finite processes and the resulting generalisations culminated in the formation of objects. we end at this theorem as a last step and thus envisage it as a final object. findings and discussion table 1 summarises the seven group responses in reflecting and explaining, and generalising the concept monotonically increasing sequences. characterisation of coded categories is as follows: o none was used for no response o inadequate codes an incorrect or unclear response o partial codes gaps in description o complete codes a mathematically correct response. deonarain brijlall & aneshkumar maharaj 75 table 1: results on group constructions for monotonically increasing (n = 7) number of group responses stages none inadequate partial complete reflecting and explaining 1 0 5 1 generalising 1 2 3 1 it was observed that five of the seven groups when reflecting and explaining constructed a partial understanding of this concept. examples of such constructions stated by two of the groups were as follows: group e: the sequence always increase, which imply that term (tn + 1) minus tn will give a positive value and in monotonic increasing infinite real sequences there will be no variable in the denominator. the correct use of the symbols tn + 1 and tn to explain an increasing sequence implies that group e viewed such sequences as an object. however, the exclusion of variables in the denominator indicates an incomplete understanding. this misconception arose because the four illustrative examples excluded such cases. during the interview the group leader was asked why the group referred to “no variable in the denominator”. group e leader: because i think from our observations this monotonical don’t have a variable denominator but in the nonmonotonical there is … that’s why. it was pointed out to the respondent that the sequence     11 nn , which is a non-example, had no variable in the denominator. to this he responded: “the other thing is, if i can divide this thing by n, it means that the value of the term will decrease.” it seemed that this group, after working with only the examples generalised their definition for monotonicity. in the case of the nonexample they forced a variable in the denominator to satisfy their definition. here, the mathematical reasoning was faulty. note that in the above sequence if each term is divided by the replacement value for n, the sequence does not decrease. group f: a monotically increasing sequence is a sequence that approaches ∞ from the negative side as n → ∞. in other words, as n → ∞, xn increases in value to ∞. note that a monotonically increasing sequence could converge to a finite limit. the response of group f was also coded as such due to the phrase from the negative side. to clarify this, the following question was posed during the interview: interviewer: what do you mean by “from the negative side?” group f leader: what we meant was the value was becoming more and more positive as n approaches infinitely. even if the sequence started with the first term as say -5 or something like that … it er … if you had to analyse it as a graph and unfortunately in maths i do that all the time, i analyse everything as a graph and then i look at sort of limits and the limit of that as you move along … from the left or from the negative side per say … that’s were we got it from. the striking feature of this response is the visual aid adopted to respond to the question. it seems that the replacement values of n were used as the x-coordinates of a point on the cartesian plane, while the resulting values for the terms were used for the corresponding y-coordinates. in the context of apos theory the conceptual understanding is at the level of a schema. the cognitive objects are the ordered pairs and the visualisation of these graphically were the result of internal processes for manipulating these objects. using an inductive approach for definition making 76 table 2 summarises the seven group responses in reflecting and explaining, and generalising the concept boundedness of sequences. characterisation of coded categories is as for table 1. table 2: results on group constructions for bounded below/above (n = 7) number of group responses stages none inadequate partial complete reflecting and explaining bounded above 0 5 2 0 bounded below 0 5 2 0 generalising bounded above 0 4 1 2 bounded below 0 4 1 2 a striking observation is that the majority of the responses were categorised as inadequate. such responses for bounded sequences under reflecting and explaining led to inadequate generalisations. this indicates that during the process of constructions of concepts, inadequate reflecting and explaining is likely to produce inadequate generalisations. group b was the exception to this assertion. their response for reflecting and explaining is: a sequence xn is bounded if {xn / nn} is a bounded set. for a sequence to be bounded below it has x1 as it’s minimum value where it is a lower limit. for a sequence to be bounded above it has x1 as it’s maximum value where it is an upper limit. the response regarded the first term of a sequence as a determining factor in concluding the type of boundedness of a sequence. the following question was posed at the interview: interviewer: why did you all include the first term of the sequence as being important in concluding the type of boundedness of the sequence? group b leader: they figured if they chose the first term of the sequence that must be the lowest one … the first value of the sequence will be highest or lowest, and the sequence will either go up/down, increase/decrease … . if your sequence is increasing then, then your first number will be your lower bound and if decreasing the first number will be your upper bound. note that this group used monotonocity to reflect and explain the concept of boundedness. their explanation is limited since it does not apply to the sequence     11 nn . in generalising the description of bounded above this group’s response was: if mx n  for n n, m is a constant (independent of n) we say that the sequence   1nnx bounded above and m is called an upper bound. this group correctly generalised the dual of this statement for the concept bounded below. these responses were in accordance with the definitions of bounded above/below as discussed in the section on post knowledge. further the use of terminology such as upper/lower bound and upper/lower limit was spontaneously introduced by the students in group b. despite inadequately framing their responses (for the reflecting and explaining processes) in the construction of the concept of boundedness, their generalisation displayed that they conceptualised sequences as objects. this supports the claim of an interplay between inductive and deductive reasoning within the framework of the guided problem solving teaching model (see figure 1). group e gave a partial response in reflecting and explaining the concept of boundedness, yet were able to correctly generalise. this is more likely to occur than the exception which was illustrated above during the analysis and discussion of the response by group b. the written response by group e for reflecting and explaining the concept of boundedness was: deonarain brijlall & aneshkumar maharaj 77 group b: a sequence   1nnx is bounded above .,1  nxxn it is then bounded below .,1  nxxn to gain insight into this response we asked the following during the interview: interviewer: do you consider your written responses as complete? group e leader: i don’t think so … i think what we were trying to do there … we were just trying to abbreviate it … but by sort of taking the bounded above if and only if the nth term is less than a particular limit for all sort of natural number values or for any term basically. to gain further clarity we then asked: interviewer: what you are saying … that x1 is the first term all the time that is involved in your definition … so if for instance the sequence     11 nn , … , now in this case it is bounded above. is it? group e leader: yes. that one … is 1, -1, 1, -1, … and we kind of could not explain that in terms of our definition. during the interview it became clear to this group leader that the group’s definition for that stage was incomplete. the use of an appropriate counter-example, in this case     11 nn , assisted the group leader to realise the flaw in the definition that they constructed. such counter-examples support the processes associated with reversal construction during reflective abstraction in apos theory. table 3 summarises the seven group responses during the verifying and refining stages leading to further abstraction in constructing the concept of boundedness. table 3: results on verifying and refining concepts bounded below/above (n = 7) number of group responses sequence none incorrect correct  12log nn bounded above 5 1 1 bounded below 2 1 4  12 nn bounded above 5 1 1 bounded below 2 1 4           12 1log n n bounded above 3 0 4 bounded below 4 2 1 it was observed that six out of the seven groups when identifying the boundedness of a sequence believed that it is bounded either above or below, and not both. for example, group e when discussing the second sequence in the table wrote: group e: we note the misuse of notation in line 3, despite the correct conclusion of the sequence being bounded below. no investigation into the sequence being bounded above was done. during the interview the following was posed:   . 2 1.. 22 1. 1 1 belowboundedis xxn n n n n       using an inductive approach for definition making 78 interviewer: the question asked that you look at both bounded above/below. you looked at bounded below for the sequence  12log nn . why did you all not look at bounded above? group e leader: i think we kind of assumed that for each particular function unless it is periodic it is not going to be bounded below and above. we observe that this group did not find it necessary to investigate the other case of boundedness, if the sequence was not periodic. group b was the only group that investigated both the types of boundedness for each sequence. their response for the second sequence in table 3 was: group b: it is bounded below by 2 and not above. this was a concise and apt response. note that group b displayed understanding when generalising the concept of boundedness and also conceptualised sequences as an object (see the analysis and discussion for table 2). in dubinsky’s stages of reflective abstraction the processes of interiorisation, coordination, encapsulation and generalisation were demonstrated by group b. this seems to intimate that these processes are pre-requisites for successful construction of concepts. conclusion the structured worksheets, based on the examples and non-examples approach for constructing the concepts of monotonocity and boundedness of sequences, had a positive impact. this was so because it encouraged group-work, which fostered an environment conducive to social constructivism and reflective abstraction. the findings (as demonstrated by the analysis and discussion of the responses of group b) showed that students demonstrated the ability to apply symbols, language, and mental images to construct internal processes as a way of making sense of the concepts of monotonocity and boundedness of sequences. on perceiving sequences as objects students could apply actions on these objects which were interiorised into a system of operations. the verifying and refining stages in the construction of the two concepts required a conceptualisation of these concepts as objects. this conceptualisation enabled the formulation of new schema which we expect to be applied to a wider range of contexts. the concept of boundedness of sequences should now lead to the construction of the definitions of supremum (least upper bound) and infimum (greatest lower bound) of sequences. acknowledgement the authors thank the research office at the university of kwazulu-natal for making available a competitive research grant for this project. references adler, j. 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can be opened with acrobat and adobe reader 5.0 and later.) >> /namespace [ (adobe) (common) (1.0) ] /othernamespaces [ << /asreaderspreads false /cropimagestoframes true /errorcontrol /warnandcontinue /flattenerignorespreadoverrides false /includeguidesgrids false /includenonprinting false /includeslug false /namespace [ (adobe) (indesign) (4.0) ] /omitplacedbitmaps false /omitplacedeps false /omitplacedpdf false /simulateoverprint /legacy >> << /addbleedmarks false /addcolorbars false /addcropmarks false /addpageinfo false /addregmarks false /convertcolors /converttocmyk /destinationprofilename () /destinationprofileselector /documentcmyk /downsample16bitimages true /flattenerpreset << /presetselector /mediumresolution >> /formelements false /generatestructure false /includebookmarks false /includehyperlinks false /includeinteractive false /includelayers false /includeprofiles false /multimediahandling /useobjectsettings /namespace [ (adobe) (creativesuite) (2.0) ] /pdfxoutputintentprofileselector /documentcmyk /preserveediting true /untaggedcmykhandling /leaveuntagged /untaggedrgbhandling /usedocumentprofile /usedocumentbleed false >> ] >> setdistillerparams << /hwresolution [2400 2400] /pagesize [612.000 792.000] >> setpagedevice microsoft word 46-55 nyaumwe.docx 46 pythagoras, 69, 46-55 (july 2009) in‐service mathematics teachers’ effective reflective actions  during enactment of social constructivist strategies  in their teaching    lovemore j. nyaumwe  marang centre for maths & science education, university of the witwatersrand  lovemore.nyaumwe@wits.ac.za  david k. mtetwa  university of zimbabwe  dmtetwa@education.uz.ac.zw    this  study  documents  two  case  studies  of  in‐service  teachers  whose  reflective  actions  during teaching belonged to the effective category. stratified sampling was used to select  the  in‐service  teachers  whose  reflective  actions  during  teaching  achieved  effective  reflection category in the first round of assessments. the sampled in‐service teachers were  jointly observed by two researchers whilst teaching high school mathematics classes in the  second and third rounds of assessment visits to determine their teaching actions whilst  enacting effective reflective actions. classroom observations were followed by post lesson  reflective interviews. the in‐service teachers’ effective reflective actions during teaching  were noted as aligning learners’ prior knowledge with activities to develop new concepts,  sensitivity to learners’ needs, using multiple pedagogical methods, and causing cognitive  conflicts that facilitated  learners’ reflections on the solutions that they produced. these  findings provide insight into theorising in‐service teachers’ reflective actions that informs  reform  on  appropriate  enactment  of  social  constructivist  strategies  in  mathematics  classrooms.    background of the study the results of the third international mathematics and science study (timms) gave rise to a constant search for ways of improving secondary school mathematics learners’ achievement throughout the world (julie, 2004). this search has given rise to raging debates on what pedagogical methods facilitate learner understanding and retention of mathematical knowledge. these debates are guided by teachers’ orientations that belong to either the absolutist or problem-solving paradigm. in the absolutist paradigm, mathematics is viewed as a universal static subject whose concepts reside outside human senses. teachers holding absolutist views demonstrate mechanistic applications of axioms and theorems that they expect learners to accept as indisputable truths. a criticism that is often levelled against absolutist methods is that they transmit inert mathematical knowledge that learners passively access in restricted contexts. passive learning of mathematical concepts has potential to limit learner applications of the concepts in a variety of contexts that are different from textbook questions. this criticism has necessitated the advocating of problem-solving instructional reforms in many countries. the problem-solving reforms are informed by social constructivist perspectives. social constructivist reform encourages classroom environments that provide learners with opportunities to engage in dynamic mathematical activities that are grounded in rich problem-solving mathematical tasks (nctm, 1991). teaching in such classrooms encourages learner construction of mathematical knowledge through active cognitive and social engagements using the experiential world lovemore nyaumwe & david mtetwa 47 (von glasersfeld, 1995). teacher roles when enacting problem-solving strategies shift from dispensers of mathematical knowledge to coordinating, facilitating, guiding, and providing resources. teaching and learning mathematics in such classrooms ceases to be learner and teacher convergence on pre-existing truth, to divergence and broadening on what is knowable, doable and viable (breen, 2005). based on vygotsky’s (1978) co-construction theory, the social constructivist perspective advocates that mathematical knowledge is not built by individual learners, but in a wider social context which is linked to learners’ environment and cultural activities. effective teaching in the social constructivist perspective is based on creating opportunities for learners to experience, discuss, discover and socially construct mathematical knowledge using their contexts. such teaching involves learner active engagements on physical and mental activities that involve interpreting, constructing and reflecting on their decisions in ways that enhance learners to create viable mathematical knowledge through personal experiences, observations, discoveries, and interactions with other learners and the local environment. this study is focused on in-service teachers’ effective reflective actions as they enact some social constructivist tenets in their teaching. to achieve effective reflection category, the in-service teachers integrate problemsolving and absolutist teaching strategies at appropriate stages of lesson development in order for learners to develop both conceptual and procedural understanding of mathematical concepts and process skills. success in complementing problem solving with absolutist teaching strategies requires in-service teachers’ effective reflections. reflection is the means through which in-service teachers assess the effectiveness of their instructional decisions through a critical analysis of what happened, why it happened and what they could do differently to improve their teaching performance (galvezl-martin, 2003). dewey (1933) noted that during reflection teachers examine their teaching performance by critically analyzing what, how and why they did what they did in their teaching with an ultimate goal of improving learner understanding. effective reflections during teaching enhances in-service teachers to consciously think about pedagogical actions at the time they are being enacted, interpret learner actions, and appropriately shape successive teaching using multiple pedagogical approaches in order to promote learner understanding (schon, 1987). the process of formulating effective reflective actions during teaching renders the enterprise of teaching a complex one (even, 2005). the complexity of teaching arises through involvement of in-service teacher self-critical through questioning their interpretation of learners’ actions in ways that necessitate them to frame and reframe their assumptions about learners’ learning capacities and problematising their teaching. making effective reflective actions during teaching that enables in-service teachers to balance and respond to learners’ cognitive, psychological and social needs is demanding. it calls for ability to apply singly or a combination of qualitatively distinct kinds of reflections such as technical, deliberate and critical reflection to given instructional situations as they unfold in the classroom (hatton & smith, 2006). in-service teachers’ effective reflective actions during their teaching can facilitate learners to construct mathematical knowledge in ways that enhances them to apply the learnt concepts in solving routine and non-routine problems. this study was in response to richardson’s (2003) request for more theory building in social constructivist pedagogy. this request was based on the fact that social constructivism is a theory of learning and that scanty literature is available on social constructivist teaching. high school mathematics classroom contexts were used in attempting to achieve the goal of theory building on social constructivist pedagogy. high school contexts were chosen for this study due to in-service teachers’ concerns that mathematical concepts at this level were too abstract to develop them from learners’ everyday contexts. analysing in-service teachers’ ability to critically reflect on their decisions and actions during teaching makes it possible to use concrete examples from a lesson to assess effective reflective actions drawn from their teaching practice. the assessment facilitates theorising how the effective in-service teachers succeeded in enacting social constructivist approaches in their teaching. in pursuing the goal of the study, the following research question was posed: what are the effective reflective actions that in-service mathematics teachers on full-time teaching practice of their programme are capable of achieving when implementing social constructivist strategies in their teaching? answers to the research question may provide insight that can inform teacher professional development programmes on enacting social constructivist strategies that are emphasized in mathematics curricula reforms. in-service mathematics teachers’ reflective actions 48 conceptual framework from a theoretical perspective teaching mathematics is a complex enterprise that sometimes involves teachers to make effective and ineffective reflections. schon (1987) portrayed reflection that is embedded in action. he concluded that teachers frame and reframe their teaching, test various interpretations that they infer from learning contexts and modify their teaching actions accordingly. schon (1987) perceived teacher reflection as taking place during and after teaching calling them reflection-in-action and reflection-on-action respectively. reflection-in-action refers to the process of enacting pedagogical strategies, interpreting and analyzing learner outcomes with a view to improve teaching at the time it is taking place (schon, 1987). in-service teachers perform reflective actions during teaching that are informed by their theories-in-use. after teaching a lesson in-service teachers engage in reflection-on-action. reflection-on-action takes place when an in-service teacher mentally reconstructs an instructional arena long after teaching a lesson in order to analyse actions and their outcomes (schon, 1987). effective reflection involves wise choices of resources, pacing and sequencing content, creating conducive classroom environments that promote learner participation, challenging, and extending learner thinking (even, 2005). teacher effective reflections enable learners to develop conceptual understanding of the process skills that enable learners to individually or collectively develop a repertoire for making constructions that can concur with viable mathematical knowledge (davis, 1990). this is possible because teacher effective reflections involve learner engagement in constructive, self-regulated, goaloriented, situated, collaborative, and cumulative process of building mathematical concepts, and making meaning out of them (schelfhout, dochy, janssens, struyven, & gielen, 2006). in-service teacher engagements in reflection facilitate their effectiveness in enacting social constructivist tenets. the social constructivist theories have a basis in philosophical and psychological backgrounds. for instance, ahmed in wheatley (1991) noted that effective learning of mathematics can be achieved through discourse, experimenting, discovering, inventing, discussing and reflecting. in proposing learning through interactions, observations and negotiations vygotsky in ferguson (2005) proposed the zone of proximal development (zpd). zpd theorizes that group learning has potential to contribute more learners’ understanding than what learners are capable of constructing as individuals. the argument of social learning is that cognition and intelligence are not properties of individual learners but arise from interactions with other learners in the learning environment. active construction of mathematical knowledge in social contexts facilitates the development of conceptual understanding and transferability of the concepts to applications in a variety of contexts. to facilitate learners’ social interactions in mathematics classrooms, learning is made problematic through provisions of authentic problem-solving tasks existing in the learners’ environment. such tasks can intuitively be approached by learners using trial, error, and success. enactment of social constructivist strategies in the classroom requires authentic learning environments that facilitate learner building of new mathematical knowledge from their prior knowledge. such learning facilitates learners’ applications of mathematical concepts in a variety of contexts in the classroom and their environments. absolutist methods of demonstrating mathematical concepts for learners to reproduce them restrict learners’ success to solving routine problems. complementing absolutist approaches with problem-solving approaches is an ideal strategy that has potential to maximize learners’ mastery of procedural and conceptual mathematical knowledge and skills. for instance, the complementarities may proffer effective learning when concepts that are abstract or difficult to connect and derive from learners’ environment are demonstrated by in-service teachers. complementing problem-solving approaches with absolutist methods requires effective reflections at appropriate stages of a lesson that attains different reflection categories in the process. this is the essence of this study to assess effective reflective actions that some in-service mathematics teachers were capable of achieving when enacting social constructivist strategies in their teaching during full-time teaching practice of their programme. design of the study the in-service teachers who participated in this study were enrolled in a two year full-time programme that upgrades their content and professional knowledge to teach mathematics at high school (16  yearlovemore nyaumwe & david mtetwa 49 old learners) level. the in-service teachers were on a four-week school attachment part of their bsc ed programme to practice implementation of social constructivist pedagogies. in-service teachers’ learning to enact social constructivist pedagogical strategies in their teaching is a situated practice because it takes place in a particular location shaped by a unique set of personal, institutional, and social factors. in-service teachers’ effective reflective actions during teaching that this study focused on are those belonging to the good reflection category. case studies of two in-service teachers, attached at different schools, who achieved effective reflection categories in the first round of assessment visits by faculty members, were assessed by the two researchers in the second and third rounds of assessment visits. this means that stratified sampling was the method used to select the two in-service teachers who provided data for this study. given the sampling criteria, a sample of two out of a cohort of 14 in-service teachers studying mathematics as a major study is normal. power, clarke and hine (2002) argued for small sample size of high level teacher reflection because achievement of effective category of reflective actions during teaching is difficult to attain for most in-service teachers. this category of reflective actions during teaching is hard to achieve because in-service teachers are usually pre-occupied with processes of enacting new pedagogical reform that they sometimes overlook their process of learning how to teach. this observation makes the sample large enough to provide data that can be used to theorise effective reflective actions of in-service teachers implementing reform on social constructivist teaching strategies. in-service teachers’ ability to make effective reflective actions during teaching in response to learners’ needs can be detected through assessments of lessons and post lesson reflective interviews that produce texts from illustrative episodes and elaborations of their responses to learners’ actions and needs. two researchers who were also mathematics education lecturers assessed the in-service teachers’ teaching. the assessments were conducted after reaching agreement on what constitutes effective and non-effective reflective actions during teaching. three agreed common areas of focus to make valid and uniform assessments by the researchers were social constructivist tenets of learner interactions, pedagogy and student learning. assessments on learner interactions focused on (a) learner-learner interactions, (b) learner-in-service teacher interactions, (c) learners’ explorations, (d) learners’ discussions and construction of mathematical knowledge, and (e) learners’ reporting of group activities. on pedagogy the assessments focused on the tenets of (a) multiple use of teaching methods, (b) clarity of instructions, (c) logical progression of activities, and (d) assessment of learners’ understanding. on student learning the following tenets were assessed (a) use of learners’ prior knowledge, and (b) suitability of activities for learner construction of mathematical knowledge. during the joint assessment the two researchers made detailed field notes on each of the two in-service teachers’ reflective actions during teaching. they provided evidence from a classroom episode of conclusions that they made on instructional and reflection competencies achieved by an in-service teacher. post lesson reflective interviews were held after assessment of teaching. their purpose was to enable the in-service teachers to elaborate some of the decisions that they made during teaching. to increase the validity and reliability of interpreting the teaching and field notes produced, the assessed lessons were video taped. the video tapes validated the researchers’ interpretations by providing concrete evidences from lessons that were used to ascertain teaching actions and contexts during post lesson reflective interviews. the post lesson reflective interviews were audio taped and transcribed so that the two researchers could use the transcriptions to assess the effective reflective actions shown by an inservice teacher. audio and video tapes facilitated consensual agreement on effective reflective actions during teaching exposed by the in-service teachers during post-lesson reflective discussions. a common understanding of effective and ineffective teaching practices provided insight on the extreme ends of a continuum on which reflective actions during teaching could be scored. realizing that typical teaching practice combine both effective and ineffective decisions, gave rise to satisfactory reflective actions during teaching as falling in the middle of the effective to non-effective decision continuum. whilst effective reflective actions during teaching are relatively easy to identify from learners’ responses to instructional decisions and actions, it is not relatively easy to identify ineffective reflective actions during teaching. this is so because reflection is an abstract concept that cannot be identified as out rightly ineffective but can be unsatisfactory. based on this argument, the tacit nature of reflection necessitated that scoring of in-service teachers’ reflective actions during teaching be scored on the continuum unsatisfactory, satisfactory and effective categories. an example of an unsatisfactory reflection category in-service mathematics teachers’ reflective actions 50 is ignoring learners’ misconception in favour of right or wrong response. in the satisfactory reflection category an example is teaching that does not cater for learners’ differentiated needs. an example in the effective reflective actions during teaching category is relating content to the existent reality of learners and other subjects that they study. results the results from each of the two case studies are presented separately in order to provide cross-case analyses that can show commonalities and differences between them. verbatim researchers’ extracts from classroom episodes are presented first, followed by post lesson reflective interview intercepts. pseudonyms are used for the in-service teachers in order to protect their identities. monica taught the concept of integration as a limit of a sum using the trapezium rule to estimate the area under a curve. she used the discovery method in which pairs of learners were asked to find the area under the curve y = x 2 between x = 0 and x = 1. first, learners calculated the definite integral of xx d 1 0 2 and found it to be 3 1 or 0,33. later, the learners looked for a method that could produce the same area. the learners debated alternative methods to the definite integral approach. monica moved round the classroom to assess learners’ progress and approaches they were using. she gave hints to struggling groups that provoked insight into recalling relevant prior knowledge that helped them to approach the task. after 15 minutes of pair-work, the learners were asked to stop their group-work in order to discuss the ideas that they generated in their groups with the rest of the class members. during class discussion learners were picked to come to the board to contribute a step towards the solution of the problem. the first learner chosen drew the graph of y = x 2 and divided the interval 0  x  1 into five equal sub-intervals. learners struggled on what to do next. another learner volunteered to go to the board and calculated the width of each sub-interval. the width was found it to be 5 1 . a fourth learner calculated the successive heights of the rectangles in the interval y = x 2 between x = 0 and x =1. these were found to be 2 5 1       , 2 5 2       , 2 5 3       , 2 5 4       . another learner multiplied the width ( 5 1 ) by the height of each rectangle in the interval. the total area of the rectangles in the interval was found to be 0,24. monica asked the learners to explain why the solution was different from 0,33 that was obtained using the definite integral approach. the learners assessed the validity of the new solution, reviewed the steps and method used but could not identify flaws in them. monica drew the attention of the learners to the rectangles of the sub-intervals on the graph in order for them to see some triangles whose areas lay outside the area calculated. this helped the learners to deduce the reasons why the solution from the definite integral was larger than the one obtained from adding five rectangles in the interval. later the learners were asked to divide the interval into ten equal parts and find the sum of the areas of the rectangles formed. this area was found to be 0,285 which is closer to 0,33 than 0,24. the learners were asked to think abstractly in order to get a more accurate area than the ones they got from the physical graph on the board. one learner suggested that the more the number of rectangles in the interval, the thinner the widths, and the better the rectangles approximated the area under a curve between the given limits. this generalisation was deduced from the fact that thinner rectangles formed smaller number of triangles that lay between the curve and tops of the rectangles in the interval. this means that the tops of the rectangles were as close as possible to the curve which minimised the area between the curve and the rectangles that is not included in a sub-interval. as the number of rectangles got very large (goes to infinity), the width of the rectangles in a sub-interval got small to coincide with the tangent of a curve. to link the physical and the abstract methods, monica extended learners’ generalisation to an algebraic algorithm. this was done for the interval ax 0 when the interval is divided into n equal parts. this lovemore nyaumwe & david mtetwa 51 formed sub-intervals of width n a and n – 1 rectangles. the heights of the rectangles in the interval were 2       n a , 2 2       n a , 2 3       n a , …, 2 )1(        n an . the areas of the rectangles in the sub-intervals were calculated thus: 2 22 2 2 2 2 2 2 )1( ...94 n an n a n a n a n a n a n a n a  = 3 3 n a {1 + 4 + 9 + … + (n – 1) 2 }. the right hand side of the generalisation (sum of squared numbers) was summed to )132( 6 )32( 6 1 2 3 33 3 3 nn annn n a  . when n is very large (as n tends to the infinite), n  , 3 )132( 6 3 2 3 a nn a  , since 03  n and 01 2  n . this general form concurs with the definite integral of 3 1d 1 0 2  xx for the case when a = 1. this general part enabled learners to realise the need for as many divisions as possible on a sub-interval for a more accurate approximation. this realisation prompted one learner to ask the question “when does a tangent of a curve at a point coincide with sides of rectangles approximating the curve”. learners theoretically visualised the answer to the question when there were infinitely many rectangles under the interval of a curve. during the post lesson reflective interview monica reflected on her interactions with the learners in this way: i let learners to figure out on their own what method to use. i wanted them to think eh, eh, eh and design their own approaches. as was expected, the learners started off from the definite integral that was familiar to them before exploring a different method. i mixed demonstrations and… active learner activities in order for the learners to recall previously learnt concepts such as definite integrals, summation of squared consecutive numbers, as well as make selective choices on the concepts to use (post lesson reflective interview with monica, july 2006). another in-service teacher, cecil, was assessed teaching the concept of probability of two independent events. for instance, he asked learners to calculate the probability of getting a sum of 10 when two dice are thrown. he correctly assumed that learners were not familiar with the concept of dice as they were not commonly used in their environment. he showed the learners a die and asked them to discuss its properties. after the discussion, the learners pondered on how to represent the outcomes of the two thrown dice and how to get a sum of 10 from the possible numbers thrown. after ten minutes of learners’ discourse on a plan to get the totality of the outcomes from which to draw the sample space that answers the question, cecil gave the learners some clues. he showed them a possible outcome event after throwing the two dice once. learners were later asked to write all the possible ordered pairs of events when two dice are thrown together. at first the learners wrote the possible combinations as random numbers without being mindful of the systematic pattern that could ensure that all possible outcomes are noted. some clues were given for learners to realise that the events from the two dice can be represented by the adjacent sides of a square, one die representing the length and the other die the width. the learners labelled the adjacent sides of the square 1 to 6 representing the possible outcome on each of the two dice when thrown. the ordered pairs of the point of intersections of perpendicular and horizontal lines passing through the six numbers on the face of each die were marked as coordinate points. from the constructed outcome table, learners were asked to identify the ordered pairs that could provide a sum of 10. the learners identified this sample space as (4, 6), (5, 5) and (6, 4). the learners discussed whether (4, 6) and (6, 4) were different. the concepts of permutations and combinations helped the learners to understand that the order of the numbers was important as they were drawn from different dice. in-service mathematics teachers’ reflective actions 52 during the post lesson reflective interview cecil reflected on the lesson as follows: dice are not common in the learners’ environment … that’s why i showed them a sample. i gave learners opportunities to figure out for themselves how to construct an outcome table and to identify a sample space with combinations of … numbers adding up to 10. i used guided discovery to enable learners to draw the outcome table in order to show the sample space of pairs adding up to 10. this necessitated a combination of demonstrations and learner explorations that i used in the lesson (post lesson reflective interview with cecil, july 2006). discussion the results of this study indicate that the in-service teachers’ effective reflective actions during teaching resulted from aligning learner prior knowledge with new content, integrating problem-solving and absolutist teaching methods at appropriate stages and probing learners’ thinking. the integration of instructional approaches from two diverse paradigms complemented the strengths and weaknesses of each approach. the in-service teachers in this study based their choices of teaching strategies and reflective actions on their assumptions and conceptions of teaching and learning mathematics based on their concerns for learner understanding. the assumptions and instructional conceptions that they held about effective implementation of social constructivism approaches revealed the mental maps that determined their teaching actions. similar to conclusions made by (smith, 2001) mental maps determined an inservice teacher’s instructional actions more than pedagogical theories. mental maps on how students learn mathematical concepts and skills facilitated in-service teachers’ integration of absolutist and constructivist teaching approaches. the in-service teachers integrated multiple teaching methods to maximize learner processing of the concepts of integration and outcome tables that they were learning. for instance, monica deduced the trapezium rule from learner prior knowledge of concrete areas of rectangles observed on a graph, to abstraction using calculus concepts such as definite integral and summation of series. the mental maps that the in-service teachers constructed of how learners could comprehend mathematical concepts and procedures facilitated their appropriate reflective actions to decide when to guide learners or when to let them struggle. for instance, the guidance that cecil gave learners on dice activated them to make connections between their prior knowledge on coordinates and the concepts of finding the probability of getting a sum of 10 when two dice are thrown. structured feedback was given at the opportune moments to facilitate learners to discuss the sample space for getting a sum of 10 in meaningful ways. on the other hand, monica’s effective reflective actions during teaching were portrayed through multiple representation of content that facilitated learner understanding. she used graphical representations in order for learners to see the small triangles that were lying outside the rectangles that influenced the estimated area under a curve. visualisations of the outlying areas facilitated the learners’ use of theoretical discussions that extended their understanding from physical observations to abstraction where the concept of infinite sub-divisions was used to obtain a more accurate area. combining physical representations of mathematical generalisations with their theoretical algebraic explanations facilitated learners to gain both conceptual and procedural understanding of the trapezium rule in ways that facilitated their understanding the rule gives an approximation rather than an accurate area in an interval of a curve. monica also made effective reflective decisions on when to refer to a drawn graph and when to allow learners to think abstractly in order to formulate and theorise generalisations. this allowed learner explorations and collaboration on making deductions through struggling with mathematical tasks. learner successes were possible because monica appropriately aligned mathematical tasks with learner prior knowledge. as a result of the alignment she probed learner thinking without reducing the complexity of the task (henninsen & stein, 1977). appropriate alignment of learner prior knowledge and new concepts facilitated their reviewing of previously covered concepts and applying them in new situations. as explained by moll (2002) learning is a process of recollection where new concepts grow from existing ones. the use of prior knowledge made learners develop cognitive conflicts when they noticed discrepancies in the solutions produced by procedures familiar to them. for instance, in monica’s class, learners recalled their knowledge of definite lovemore nyaumwe & david mtetwa 53 integrals to calculate the area under the curve y = x 2 between x = 0 and x = 1 and found it to be 0,33. when they used five rectangular sub-intervals of the region to estimate the area, they found it to be 0,24 which was different from the previously found area of 0,33. the differences in the definite integral and rectangular estimation solution created cognitive conflicts that facilitated learners to reflect on the methods that they used and checking for possible computational errors or any inconsistencies that might have influenced the solutions to be different. reflection in pursuit of effective and manageable balance between possibly conflicting solutions is necessary for learners to assess the viability of their solutions in the light of constraints of the question (davis, 1990; schelfhout et al., 2006). when the learners found no inconsistencies or computational errors in the solution strategies used, they were prompted to think of possible explanations to account for the differences noticed. monica probed learners at opportune times to guide their resolving of the cognitive conflict created. the probing enabled the learners to refocus their attention on the curve showing the interval divided into sub-sections. the refocus facilitated learners to notice that there were some overlapping triangles between the top of each rectangle in a sub-interval whose areas were not calculated. this observation prompted the learners to generalise that the more the sub-sections dividing the interval, the more accurately the areas of the possible rectangles approximate the area under a curve. to reinforce and formalise learners’ intuitive generalisations, monica developed the learners’ ideas theoretically using an algebraic approach. the algebraic approach facilitated learners to clarify the cognitive conflict that was created by the discrepancies in the solutions 0,24 and 0,33 and linking their informal generalisation with a formal mathematical generalisation. the algebraic algorithm also enhanced learners to connect observations on a graph with their abstract notions. this approach enabled learners to develop deep understanding of the theory of infinity from a theoretical perspective to a practical one when applied to physical situations of infinitely many rectangles fitted in a curve until a slope of the curve (tangent) becomes a side of a rectangle. for instance, a learner who asked, “when does a curve and a side of a rectangle coincide?” was engaged in deep thinking of the possibilities of infinitely many rectangles and trying to visualise them on a physical diagram. learners’ questions, cues, or actions influenced the interactions between the learners and the in-service teachers. the two in-service teachers’ instructional strategies involved enactment of discovery teaching methods. the methods were enacted through posing problematic questions and letting learners pondering on how to solve them. the discovery methods were implemented in order to facilitate learners to build new concepts from their prior knowledge. to enable learners to recall relevant prior knowledge that was useful to build new ones, the learners were often asked to answer questions in pairs. pair-work designs were used based on an understanding that social learning can promote argumentation that can generate multiple perspectives on how to solve mathematical problems. argumentation in the in-service teacher’s mathematics classes enhanced learners’ explorations, making connections and generalisations that engaged them in recalling their prior knowledge that was used at appropriate stages of concept development. for instance, monica’s learner involvement in the development of the trapezium rule enabled the learners to think deeply, reason logically to connect concepts from a physical graph in order to construct an algebraic formula of the trapezium rule. discourse in social learning settings facilitated learners to review their understanding of some mathematical processes, modify them in light of the reasoning or evidence provided by peers and extend them to new understanding which was often at a deeper level. for instance, the debates on whether (4, 6) and (6, 4) represented different events when two dice are thrown invoked learners’ reviewing their knowledge of permutations and combinations, their generalised forms and applications in probability problems. effective reflective actions during in-service teachers’ teaching were influenced by their alertness to events occurring in a classroom and sensitivity to learners’ actions (verbal or non-verbal) and questions. for instance, monica’s sensitivity to a learner who asked the question on tangent of a curve coinciding with sides of rectangles. listening to learner concerns is an essential social constructivist tenet for three purposes of evaluative, interpretive and hermeneutic listening. evaluative listening involves assessing the viability of a learner’s thought processes, while interpretive listening involves assessing learners’ sensemaking in order to determine their understanding or sources of possible misconceptions. hermeneutic listening involves a dialogue that interrogates the taken-for-granted and the prejudices that frame learners’ perceptions and actions (maoto & wallace, 2006). in-service teachers’ sensitivities to how learners process new information in order to understand it influenced their reflective actions category during teaching. in-service mathematics teachers’ reflective actions 54 conclusion although conclusions were made based on two case studies presented in this study, it seems possible to explore these conclusions further on a larger sample in other contexts. the in-service teachers’ effective reflective actions during teaching were shown in integrating problem-solving and absolutist teaching approaches in order to meet learners’ needs, listening carefully to learners’ ideas, deciding appropriate times to provide information or when to let learners struggle and orchestrating discussions among learners. implementing multiple pedagogical strategies in response to learners’ different learning styles and rephrasing learners’ explanations to concur with formal algebraic mathematical language also showed effective reflections during teaching. as also observed by even (2005), effective reflective actions during teaching facilitated the in-service teachers to make wise choices of resources, pedagogical strategies, deciding the pace of instruction, challenge and extend learners’ thinking. this study again highlights some of the challenges that in-service teachers experience when implementing innovative social constructivist teaching practices. in-service teachers’ implementation of new pedagogical strategies in real classroom contexts showed potential to narrow their theory-practice gap of social constructivist teaching approaches and integrating them with absolutist teaching strategies in order to maximize learner understanding. findings from this study provide teacher teachers and in-service teachers with insight to debate, critique, reflect, and extend theories on designing in-service teacher programmes that offer opportunities to build personalized knowledge and skills of how to effectively implement social constructivist approaches in their teaching. references breen, c. 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(1991). constructivist perspectives on science and mathematics learning. science education, 75(1), 9 – 21. abstract introduction methods results and discussion conclusion acknowledgements references about the author(s) robert prince centre for educational testing for access and placement, university of cape town, south africa vera frith numeracy centre, university of cape town, south africa citation prince, r. & frith, v. (2017). the quantitative literacy of south african school-leavers who qualify for higher education. pythagoras, 38(1), a355. https://doi.org/10.4102/pythagoras.v38i1.355 original research the quantitative literacy of south african school-leavers who qualify for higher education robert prince, vera frith received: 20 oct. 2016; accepted: 26 may 2017; published: 31 oct. 2017 copyright: © 2017. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract there is an articulation gap for many students between the literacy practices developed at school and those demanded by higher education. while the school sector is often well attuned to the school-leaving assessments, it may not be as aware of the implicit quantitative literacy (ql) demands placed on students in higher education. the national benchmark test (nbt) in ql provides diagnostic information to inform teaching and learning. the performance of a large sample of school-leavers who wrote the nbt ql test was investigated (1) to demonstrate how school-leavers performed on this ql test, (2) to explore the relationship between performance on this test and on cognate school-leaving subjects and (3) to provide school teachers and curriculum advisors with a sense of the ql demands made on their students. descriptive statistics were used to describe performance and linear regression to explore the relationships between performance in the nbt ql test and on the school subjects mathematics and mathematical literacy. only 13% of the nbt ql scores in the sample were classified as proficient and the majority of school-leavers would need support to cope with the ql demands of higher education. the results in neither mathematics nor mathematical literacy were good predictors of performance on the nbt ql test. examination of performance on selected individual items revealed that many students have difficulty with quantitative language and with interpreting data in tables. given that ql is bound to context, it is important that teachers develop ql practices within their disciplinary contexts. introduction many students leaving the school system in south africa, while prepared for leaving school, are to some extent unprepared for higher education. there is therefore great inequality in the experiences of and outcomes for students in the higher education sector. according to scott, yeld and hendry (2007, p. 42) ‘the educational factor to which poor performance is perhaps most commonly ascribed across the higher education sector is student under-preparedness for standard undergraduate programmes’. this ‘under-preparedness’ is to a large extent the result of difficulties experienced in the area of academic literacies: the real key to whether a student will pass or fail relates to the literacy practices she brings with her to the university from her school and home environments, and the extent to which these have commonalities with the literacy practices of her chosen discipline. (mckenna, 2009, p. 8) the importance of quantitative literacy (ql) for higher education is widely recognised (see for example, steen, 2004) and there is also an increasing awareness that many academic disciplines make complex quantitative demands that are often very different from those that are the focus of traditional mathematics courses. this ability to work with quantitative information in academic contexts is one of the academic literacies and presents particular difficulties for many students. results from the national benchmark tests project (nbtp) illustrate this clearly. for example, in 2014, of the 76 693 prospective applicants to higher education who wrote the nbtp ql test, only 11% performed at the ‘proficient’ ql level. the remaining 89% were expected to experience academic challenges arising from their low levels of ql proficiency. just over 40% performed at the ‘basic’ level, which means that they would be severely challenged academically in higher education (centre for educational testing for access and placement, 2015, p. 26). one of the goals of the nbtp is to provide lecturers and curriculum developers in higher education with information about the capabilities of students ‘to inform the nature of foundation courses and curriculum responsiveness’ (griesel, 2006, p. 4). this is because the higher education sector recognises that university teaching needs to address the ‘articulation gap’ (scott et al., 2007, p. 42) between the level of many students’ quantitative (and other) literacies and the demands of higher education. however, it is recommended that teachers and curriculum advisors in schools also make use of the diagnostic information provided by the nbtp, in order that they can better prepare students for higher education. a second goal of the nbtp is to ‘assess the relationship between higher education entry level requirements and school-level exit outcomes’ (griesel, 2006, p. 4). the national senior certificate (nsc) and national benchmark test (nbt) assessments have two complementary purposes. while the nsc largely determines whether school-leavers are ready to leave the school system, the nbts are based on the assumption that prospective higher education students are ready to leave the school system with a higher education pass, and the tests attempt to determine to what extent these prospective students are ready for the demands of higher education. nsc candidates have to take either mathematics or mathematical literacy as subjects and both of these subjects are cognate to the nbt ql domain, but there are substantial differences between them. the nature of these nsc subjects is probably widely understood, but the nature of ql for higher education (the nbt ql domain) may be less familiar. examples of the ql demands of higher education curricula are outlined by frith and prince (2009), frith and gunston (2011) and prince and simpson (2016). the relationship between performance in the nbt ql test and subsequent higher education performance has been investigated by prince (2016). since ql involves dealing with and communicating quantitative information in various academic contexts, it is not only in the school subjects of mathematical literacy and mathematics that ql is developed. teachers in many other subjects, for example geography, the sciences or economics, have a very important role to play in developing this vital literacy by, for example, expecting learners to interpret a variety of representations of quantitative information, by using (and expecting learners to use) correct and precise language when expressing quantitative ideas and by assessing learners’ ability to practise and express quantitative reasoning in context. in this article, we provide an overview of the results of the nbtp ql test written by a large sample of successful school-leavers across south africa who were intending to apply to enter higher education institutions in 2015. these learners all achieved nsc results that would allow them to enter higher education. we also report on the associations between performance on the nbt ql test and the nsc subjects mathematics and mathematical literacy. we further analyse the performance of these students on subgroups of items and on a selected number of key items. this analysis is not intended to give a comprehensive exposition of the ql competencies relevant to higher education, but is intended to alert school teachers and curriculum specialists to the requirements of higher education and to provide some examples of areas where lack of student competence will have detrimental effects on their success in higher education, if not addressed. quantitative literacy assessment in the nbtp higher education south africa commissioned the national benchmarks tests project in 2005 with its main aim being to assess the academic proficiency of prospective students wishing to enter higher education. the tests assess proficiency in ql, as well as academic literacy (al) for all students and mathematics (mat) for those students intending to enter courses or programmes that have a significant mathematical component. these tests are designed by academics in higher education to provide information complementary to that provided by the nsc, to assist with selection and placement of students into appropriate courses and programmes. another function of the nbtp is to provide information to inform curriculum development in higher education. the nbts provide a measure of students’ readiness for higher education and the competencies that are assessed in the nbts are regarded as key areas in which students entering higher education should have minimum levels of proficiency. the content and competencies assessed in each domain are described by griesel (2006). these tests are ‘constructed to provide information about the level of a test-taker’s performance in relation to clearly defined domains of content and/or behaviours (e.g. reading, writing, mathematics) that requires mastery’ (foxcroft, 2006, p. 9). minimum benchmark scores for three different proficiency levels are established through a rigorous standard setting process. the proficiency levels are thus defined in terms of the percentage of the content specified in the test construct that a student has mastered. this is done according to the judgement by higher education academics as to the requirements of their disciplines, with reference to each item in the test. associated with each proficiency level there are recommendations for the kind of educational provision that is appropriate for a student whose performance is at that level. in particular, the nbtp ql test aims to measure the levels of proficiency in ql of school-leavers who are aspiring to enter higher education. the construct informing the design of this test is outlined by frith and prince (2006, p. 28). this test is written during or at the end of their grade 12 year, which is considered to be a time when students could realistically be expected to be ready for the ql demands of higher education study. in practical terms the nbtp ql test assesses students’ ability to competently interpret and reason with quantitative information that is presented in a variety of modes. for example, the test specifications include that they must understand and use a range of quantitative terms and phrases, read and interpret tables, graphs, charts, diagrams and texts and integrate information from different sources. the test also assesses the ability to apply quantitative procedures in various situations, to do simple calculations and estimations which may involve multiple steps and to formulate and apply simple formulae. students are also required to identify trends and patterns in various situations, interpret two-dimensional representations of three-dimensional structures and to reason logically. the questions are designed to assess ql practices and do not assume that students have the knowledge of any particular school subject. theoretical framework for the nbtp quantitative literacy test in this section, we will outline the theoretical considerations about the nature of ql that underpin the construct of the nbtp ql test. there has been an ongoing debate about what constitutes ql, especially in england and australia (where it is usually referred to as ‘numeracy’) and in the united states (where it is usually called ‘quantitative literacy’). one aspect of this debate is to do with the relationship of ql with mathematics. hughes-hallett (2001) sums up the difference as follows: mathematics focuses on climbing the ladder of abstraction while quantitative literacy clings to context. … mathematics is about general principles that can be applied in a range of contexts; quantitative literacy is about seeing every context through a quantitative lens. (p. 94) some authors prefer to conceptualise ql as a social practice (street, 2005; street & baker, 2006), in line with many al practitioners. in the south african school context, the subject mathematical literacy (which comprises the same kinds of competencies as ql) is defined as ‘a subject driven by life-related applications of mathematics’ (department of education, 2003, p. 9). many authors focus on the aspect of ql that has to do with thinking critically about the use of numbers in society (johnston, 2007) and some prefer to think of it as part of multiple academic literacies (chapman & lee, 1990). however, all the definitions of ql stress that it is fundamentally concerned with mathematics and statistics used in context: at the very least then, the definitions garnered from this debate would agree that numeracy is to do with ‘using maths in context’ and that to be numerate is to have the ‘capacity to use maths effectively in context’. (johnston, 2002, p. 4) it follows that ql cannot be taught as a generic skill and that learned rules will not be sufficient to enable the solution of ql problems. thus, in almost all of the questions in the nbtp ql test, writers have to apply quantitative methods and reasoning within a realistic (mostly relevant to higher education) context. there is a wide range of both the contexts used for the items and the kinds of competencies required by them, in order that the test has both face validity and relevance for all disciplines in higher education. the definition of ql that is the foundation of the construct of the nbtp ql test, is strongly influenced by the definition of numerate behaviour underlying the assessment of numeracy in the adult literacy and lifeskills (all) survey (gal, van groenestijn, manly, schmitt & tout, 2005, p. 152) and the new literacies studies view of literacy as social practice (kelly, johnston & baynham, 2007; street, 2005; street & baker, 2006): quantitative literacy is the ability to manage situations or solve problems in practice, and involves responding to quantitative (mathematical and statistical) information that may be presented verbally, graphically, in tabular or symbolic form; it requires the activation of a range of enabling knowledge, behaviours and processes and it can be observed when it is expressed in the form of a communication, in written, oral or visual mode. (frith & prince, 2006, p. 30) the construct informing the test design is based on the idea that each item can be described in terms of three dimensions of what it assesses: the main mathematical and statistical ideas, the underlying reasoning and behaviours (competencies) and the level of cognitive complexity. the construct does not specify the contexts for the items, but rather that there is a range of different kinds of contexts. this is necessary for face validity and because familiarity (or unfamiliarity) of the context could affect the manner in which a candidate responds to it. mathematics and mathematical literacy assessment in the national senior certificate with the introduction of the nsc in 2008, one national set of grade 12 examination question papers was introduced. after a review of the nsc curricula, the curriculum and assessment policy statement (caps) was introduced and implemented in 2012 in grade 10, with this cohort being the first to write the nsc examinations based on the caps in 2014. the scoring of the nsc assessments, administered by the department of basic education (dbe), is norm referenced and therefore the rating codes associated with them cannot easily be used to assess whether candidates meet a certain standard in a subject or domain. for the nsc, the final subject score is made up of the course mark and the examination mark and then the scores are ‘standardised’ or ‘normed’ to the five-year rolling average score for each subject. while a candidate may perform well compared to the norm, they may still fail to meet a particular standard in the domain being tested. the achievement scale for nsc subjects is shown in table 1. the descriptions against the rating codes are not benchmarks or standards, but rather descriptive categories of what a percentage score range means in terms of a candidate’s test achievement. table 1: the achievement scale for the national senior certificate. on completing the nsc, a candidate can qualify for higher certificate, diploma or degree study. the criteria in table 2 are used to determine these entry requirements. table 2: criteria for higher certificate, diploma and degree study. all nsc candidates must write the examinations for either mathematics or mathematical literacy, which are both cognate with, but not the same as ql, as can be seen from their descriptions in the ncs caps documents. the nscs caps document for the subject mathematics defines mathematics as: a language that makes use of symbols and notations for describing numerical, geometric and graphical relationships. it is a human activity that involves observing, representing and investigating patterns and qualitative relationships in physical and social phenomena and between mathematical objects themselves. (dbe, 2011a, p. 8) the caps document claims that studying mathematics will develop a student’s ability to think logically, critically and creatively and to be able to solve problems, in order to obtain a better understanding of the world around us. this focus on problem-solving and critical thinking in order to understand real-world phenomena has strong similarities with the definition of ql, but the main focus of the subject is in fact on learning the discipline of mathematics itself in order to ensure ‘access to an extended study of the mathematical sciences and a variety of career paths’. on the other hand, the ncs caps for mathematical literacy states that the competencies it develops should: allow individuals to make sense of, participate in and contribute to the twenty-first century world – a world characterised by numbers, numerically based arguments and data represented and misrepresented in a number of different ways. (dbe, 2011b, p. 8) it suggests that these competencies, which include the ability to reason, solve problems, interpret information and use technology, should be developed by exposing learners to both elementary mathematical content and authentic real-life contexts. this exposure is intended to enable the learner to be a ‘self-managing person, a contributing worker and a participating citizen in a developing democracy’ and an ‘astute consumer of the mathematics reflected in the media’. the emphasis on using mathematical knowledge and skills in context is what makes this subject similar to ql, but in higher education the contexts are academic disciplinary contexts, not necessarily everyday life-related contexts, as emphasised in the nsc caps document for mathematical literacy. this brief overview of the nsc mathematics and mathematical literacy subjects and the method of assessment used for the nsc is important in order to reveal the complementary nature of the information derived from standardised benchmark or criterion-referenced assessments, such as the nbts and a norm referenced assessment, such as the nsc. this complementary information about student competence is particularly useful for making decisions about whether students should be placed in extended or regular degree curriculum structures and for providing information to inform teaching and learning in higher education. methods the nbt ql test results were analysed for a large sample (n = 7 464) of school-leavers from across south africa who wrote one version of this test in 2014, indicating that they were intending to apply to higher education institutions for study in 2015. the sample only includes data for those who then went on to write the nsc and who obtained a result that allowed them to progress to some kind of higher education. since these are therefore prospective higher education students, for the sake of brevity we will from now on refer to them as ‘students’ (see table 5 for some demographic characteristics of the sample). the structure, administration and scoring of the nbtp quantitative literacy test there are 50 multiple choice items in the ql test, which are selected in accordance with the specification table (frith & prince, 2006, p. 32). this specifies the proportions of items that should address each of the competencies, mathematical and statistical ideas and levels of cognitive complexity deemed to be representative of the ql demands of the first year of higher education and defined in the test construct. the nbts are administered at test centres under standardised conditions by specially trained invigilators. the items in the ql test comprise two out of the seven sections of the nbtp academic and quantitative literacy test which is administered in paper and pencil format. thirty minutes is allocated for the completion of each of the 25-item sections that make up the ql test. calculators are not used, but students are only required to calculate with simple numbers, for example, with fractions that can easily be simplified by cancellation. many questions can be answered by estimation. students writing the tests record their responses on mark-reading (bubble answer) sheets which are then scanned using optical scanner technology. responses are dichotomised (either 1 for right or 0 for wrong). the unidimensional three parameter item response theory (irt) model (yen & fitzpatrick, 2006) is used to determine a student’s ability and generate a score for the candidate on a scale of 0% to 100%. results for different versions of the ql test are linked and equated using the stocking and lord method (holland & dorans, 2006) to ensure that a candidate’s score is independent of the version of the test that they wrote. the benchmarks for the nbtp quantitative literacy test table 3 provides the score ranges of the three main ql proficiency categories for degree study, and the recommendations for appropriate responses by institutions for students whose scores place them in these categories. the scores defining the proficiency categories are established at standards-setting workshops by panels of diverse south african academics who teach courses relevant to the domain. this process is carried out using the ‘modified angoff’ method (hambleton & pitoniak, 2006) and to date has been led by a senior psychometrician from the educational testing service (ets), princeton, new jersey. thus, the proficiency categories are not described in terms of particular ql knowledge and competencies, but in terms of academic teachers’ expectations for student performance on the test items, based on their knowledge of their own curricula. table 3: national benchmark tests project quantitative literacy test benchmarks for degree study, set in 2012. it has also been useful to differentiate between different levels of support that would be most appropriate for students with scores in the ‘intermediate’ category and so this level is divided into ‘upper intermediate’ and ‘lower intermediate’ bands, as shown in table 4. this differentiation is not done through the standards-setting workshops but is effective for pragmatic reasons, as the majority of scores are in the ‘intermediate’ category. table 4: national benchmark tests project quantitative literacy test degree intermediate benchmarks and how they should be interpreted. measuring performance on subgroups of items in addition to the analysis that is routinely done to generate scores for nbt results, for this article each item in the ql test was assigned to a discrete subgroup of items based on the main competency that the item was designed to assess, as follows: computing: interpreting problem statements and calculating or estimating (e.g. calculating areas or percentages). knowing: recalling simple facts and applying them (e.g. mean and median of a distribution). translating: identifying alternative representations (e.g. identifying the graphical representation of a relationship described verbally). using data: deriving information from data representations (e.g. reading required value/s off charts or tables). reasoning: reasoning and synthesising (e.g. identifying, reading off and calculating with more than one appropriate value from tables or charts; reasoning about rates). extrapolating: predicting and visualising (e.g. recognising patterns or predicting terms in a sequence; visualisation in three dimensions). performance on each subgroup was then calculated separately so as to obtain more detailed diagnostic information about students’ competencies. for this analysis, the scores used were not generated using the three parameter irt model as for the overall nbt score, but merely using the percentage of the sample who answered each item correctly. measuring performance on individual items in order to examine how students responded to individual items, the proportion of students who chose each alternative answer was recorded. this was done for the whole cohort and separately for the students whose scores for the whole test were in each of the proficiency bands. statistical packages the statistical package r (r core team, 2016) was used to do the data analysis and the r package ggplot2 (wickham, 2009) was used to create the graphical representations. ethical considerations ethical clearance for this research was obtained from the research ethics committee of the faculty of higher education development at the university of cape town. this included approving the consent declaration signed by prospective students writing the nbt, which allows the use of their results for research purposes and assures anonymity in the use of these data. results and discussion the results are presented for a large sample of school-leavers who wrote one version of the nbtp ql test and who qualified to progress to some kind of higher education. these results are therefore not representative of all nsc candidates, but only of those who were qualified to enter higher education. we first present some information about the characteristics of the students in the sample. we then present the overall distribution of scores on the entire ql test for the whole sample as well as for nsc mathematics candidates and nsc mathematical literacy candidates separately. this will be followed by some results showing the relationship between performance on nsc mathematics or mathematical literacy and on the nbt ql test. in addition, we report on the performance of the whole sample on subgroups of items in the nbt ql test, defined in terms of the competencies they assess. finally, we discuss the performance of the students in the different proficiency bands on individual example items, in order to highlight some particular areas of difficulty they experienced. characteristics of the sample some characteristics of the students in the sample are shown in table 5. approximately 60% of the students were african and the majority did not have english as their home language. english home language speakers however formed the largest language group, comprising 40% of the students. there were considerably more female students than males in this sample (about 60% and 40%, respectively). the preponderance of female students is also generally observed in the larger cohorts of all nbtp test writers. table 5: demographic characteristics of the total sample of national benchmark tests project quantitative literacy test candidates (n = 7 464). distributions of scores for the whole test the data in table 6 and shown in figure 1, show the descriptive statistics for the distributions of nbt ql scores for nsc mathematics candidates, nsc mathematical literacy candidates and for the whole sample. figure 1: distributions of national benchmark test quantitative literacy scores for national senior certificate mathematics, national senior certificate mathematical literacy candidates and for the whole sample. table 6: descriptive statistics for the distributions of the quantitative literacy performance of the whole sample (n = 7 464) and of subsets defined by national senior certificate mathematics subject written. the most obvious observation to be made is that the nbt ql scores of the nsc mathematical literacy candidates are considerably lower than those of the nsc mathematics candidates. the median score for the whole sample (44%) is approximately in the middle of the ‘lower intermediate’ proficiency band, while the median score for nsc mathematics candidates is within the ‘lower intermediate’ band, but the median for nsc mathematical literacy candidates is well within the ‘basic’ band. the third quartile for all the distributions illustrated in figure 1 are below the top of the ‘intermediate’ band, showing that in all cases less than 25% of the scores are in the ‘proficient’ category. table 7 and figure 2 show the same comparisons but in terms of distribution of scores according to proficiency bands defined by the benchmark scores. figure 2: percentage distribution of national benchmark test quantitative literacy scores by proficiency category for the whole sample, national senior certificate mathematics and mathematical literacy candidates. table 7: numbers and percentage distribution across quantitative literacy performance categories for the total sample and of subsets defined by national senior certificate mathematics subject written. the majority of the scores are in the ‘basic’ and ‘lower intermediate’ bands and less than 15% of the scores are classified as ‘proficient’. the most striking difference is seen when comparing the distributions for those students who wrote nsc mathematics and nsc mathematical literacy: the proportion of scores that are ‘basic’ is more than twice as large for mathematical literacy as for mathematics candidates (over 60% and about 30%, respectively). less than 1% of the scores for mathematical literacy candidates fall in the ‘proficient’ category. from this it is clear that the nsc mathematical literacy subject does not prepare students for the quantitative demands of higher education study. it cannot however be concluded that the somewhat better performance on the nbt ql test of students who did nsc mathematics can necessarily be ascribed to their having taken this subject, as ql is also developed in other subjects such as in the physical, earth and life sciences. comparison of performance in nsc subjects and nbtp quantitative literacy test in this section, we will further explore the relationship between performance in the nsc mathematical subjects and in the nbt ql test. table 8 contains the descriptive statistics for the distribution of marks achieved by the nsc mathematics and mathematical literacy candidates. these distributions are illustrated in figure 3, where they are juxtaposed with the distributions of the nbt ql scores obtained by the students in these two subsets of the sample. figure 3: distributions of national senior certificate mathematics and national senior certificate mathematical literacy scores and of national benchmark test quantitative literacy scores for national senior certificate mathematics candidates and national senior certificate mathematical literacy candidates. table 8: descriptive statistics for the national senior certificate mathematics and mathematical literacy results for the whole sample (n = 7 464). in figure 3 we see that in general the results for nsc mathematical literacy are higher than for mathematics, with median values of 66% and 60% respectively, with greater variability shown by the mathematics results. as we have already seen, the comparison of the nbt ql results for the same two subsets of the sample shows the reverse, with the nbt ql scores obtained by the nsc mathematical literacy candidates being considerably lower than those of the nsc mathematics candidates. in figure 4 and figure 5 the relationship between the nsc mathematics results and the nbt ql scores is explored further. figure 4: national senior certificate mathematics and national benchmark test quantitative literacy scores (n = 6 271). figure 5: percentage distribution of national benchmark test quantitative literacy scores by proficiency band for different national senior certificate mathematics levels achieved. figure 4 shows a scatter plot of the nbt ql scores of the 6 271 nsc mathematics candidates plotted against their mathematics results. even though the correlation coefficient is 0.63, it is clear that the nsc mathematics result is not a good predictor of nbt ql score. there is a wide range of nbt ql scores associated with any particular nsc mathematics result, with this range being wider the higher the mathematics mark. for example, for students who obtained over 80% (‘outstanding’) for mathematics, the nbt ql scores range from less than 25% to nearly 100%. even a low mark for mathematics is associated with quite a large range of possible nbt ql scores. for example, for those who obtained mathematics marks less than 30% (‘not achieved’), the nbt ql scores range from about 15% to 75%. from the fact that most of the points lie to the right of the dashed line (made up of points where the mathematics result and the nbt ql results are equal) we can see that in general the nbt ql scores are lower than the mathematics results (which can also be seen by comparing the box plots in figure 3). the same data (as in figure 4) is used to produce the chart in figure 5. here the nsc mathematics results are classified into the levels (1 to 7) as defined in the curriculum statement (dbe, 2009, p. 5) and the nbt ql scores are classified according to proficiency band. for each nsc mathematics level the proportion of nbt ql scores in each proficiency band is illustrated. for those students who obtained a level 1 nsc mathematics result (‘not achieved’), just over 75% obtained a ‘basic’ nbt ql score. this proportion decreases fairly linearly as the nsc mathematics levels increase, so that only about 5% of the students with nsc mathematics level 7 (‘outstanding’) obtained nbt ql scores in the ‘basic’ band. however, it is most noteworthy that, even for those who obtained level 7 results for nsc mathematics, less than 50% obtained a ‘proficient’ nbt ql score, while the proportion of ‘proficient’ scores for the level 6 (‘meritorious achievement’) mathematics results is even smaller, at just under 25%. thus, based on the nbt ql scores, the majority of even the best nsc mathematics performers will require some kind of additional support to cope with the quantitative demands of higher education study. in figure 6 and figure 7 the relationship between the nsc mathematical literacy results and the nbt ql scores is explored (in the same way as it was for nsc mathematics in figure 4 and figure 5). figure 6: national senior certificate mathematical literacy and national benchmark test quantitative literacy scores (n = 1 193). figure 7: percentage distribution of national benchmark test quantitative literacy scores by proficiency band for different national senior certificate mathematical literacy levels achieved. figure 6 shows a scatter plot of the nbt ql scores of the 1 193 nsc mathematical literacy candidates plotted against their mathematical literacy results. even though the correlation coefficient is 0.68 it is clear that for all but two students the nsc mathematical literacy result is higher than the nbt ql score by far. as for the nsc mathematics candidates there is a fairly wide range of nbt ql scores associated with any particular nsc mathematical literacy result, with this range being wider the higher the mathematical literacy mark. for example, for students who obtained over 80% (‘outstanding’) for mathematical literacy, the nbt ql scores range from about 25% to about 80%. the proportions illustrated in figure 7 reveal that the vast majority of nsc mathematical literacy candidates with results lower than level 6 have ‘basic’ nbt ql scores, indicating that they are highly unlikely to cope with degree-level study without extensive and long-term support in ql. given that only a minimal proportion of those with the highest nsc mathematical literacy marks achieved ‘proficient’ nbt ql scores, it is safe to conclude that practically speaking, nearly all students who have taken nsc mathematical literacy will need some kind of supplementary support to cope with the quantitative demands of higher education. performance on subgroups of nbtp quantitative literacy items each nbt ql item was assigned to a single subgroup based on a judgement of the main competency which the item was designed to assess. this results in six discrete subgroups of items, with the numbers of items in each subgroup as shown in table 9. the distributions of scores for these subgroups are shown in figure 8. figure 8: distributions of the scores of the entire sample for each subgroup of items based on the main competency they assess. table 9: classification of the national benchmark test quantitative literacy items according to the main competence required. in all subgroups except ‘extrapolating’, the upper quartile is at or below the top of the ‘upper intermediate’ band, and the median is at or below the top of the ‘lower intermediate’ band, indicating that at least three-quarters of the scores are not in the ‘proficient’ band, and at least 50% are not in the ‘upper intermediate’ or ‘proficient’ bands. the subgroups on which the performance was the weakest are ‘computing’ and ‘knowing’. for ‘computing’ the median is the lowest, at 40%, and about a quarter of the scores are below 30%. this reflects students’ poor number sense (as calculators were not used in the test session and many items required estimation) as well as difficulty with interpreting problem statements, especially in some cases where the context may have been unfamiliar. the strongest performance was on the subgroup ‘extrapolating’ which contains several items that require predicting future terms in a given sequence. these presumably draw on skills that are better developed at school. the median scores for ‘reasoning’, ‘knowing’, ‘translating’ and ‘using data’ are all about 50% and near the top of the ‘lower intermediate’ band. in figure 9 the mean scores for each subgroup are compared for the nsc mathematics and mathematical literacy candidates. figure 9: mean scores (and confidence intervals) for each subgroup of items based on the main competency they assess reported separately for the national senior certificate mathematics and mathematical literacy candidates. the mean scores for the subgroups for the nsc mathematics candidates follow the same pattern that is seen in figure 8. this is expected, since these candidates make up 84% of the total sample. as expected from the comparison of the overall scores for the nsc mathematics and mathematical literacy candidates (see figure 3) the scores are lower for the mathematical literacy candidates in all the subgroups. however, these mean scores deviate from the pattern in figure 8. on the whole the mean score for each subgroup is about 15 percentage points lower, but for the ‘knowing’ subgroup the mean score is even lower, and for the ‘reasoning’ subgroup it is only 10 percentage points lower. diagnostic information obtained from examining performance on individual items the examples described in this section illustrate how a detailed examination of the proportions of students who chose different alternative answers in certain items can be used to gain rich insights into students’ abilities. in these items some of the alternative (incorrect) answers are obtained through applying common misconceptions or fallacious thinking, so examining the patterns of responses can be revealing. however, it is difficult to report this kind of information without also describing precisely the structure and content of the items, and for security reasons this is undesirable. thus, we will try to illustrate how diagnostic information relevant to higher education studies is revealed, without publicising the actual test items. example 1: interpreting percentage values in a table a table from a reading that was given to first-year medical students and which contains both numbers and percentages is shown in table 10. in a table like this, one has to identify which percentages add up to 100% in order to understand the meaning of the percentage values. using this approach, a student can identify the ‘whole’ of which the percentage is expressing a fraction. in this way they can identify that, for example, the percentage at the top of the third column means: ‘23.1% of the homicides of people under 15 years in age were females’, not ‘23.1% of under 15 year old females were homicides’ or ‘23.1% of the female homicides were under 15’ or some other variation of the relevant phrases. table 10: homicide and suicide by gender and age (preliminary nmss data, first quarter 1999). figure 10 shows the results for an item in the nbtp ql test that refers to a table with similar data to table 10. it shows the percentages of students who chose each of the alternative answers in each of the proficiency categories and the choices made by the total cohort. students were classified into proficiency categories based on their performance on the whole ql test. figure 10: for a question requiring interpretation of a percentage in a table, the proportions of students who chose each alternative answer, for the total cohort and for each proficiency band. less than half of all students could identify the correct description of the meaning of a particular percentage value in the table (alternative d). only about 60% of those scoring overall in the ‘upper intermediate’ category and about 40% in the ‘lower intermediate’ category were able to do this. in the bottom two categories more than half of the students chose alternative a (58% and 54% in the ‘basic’ and ‘lower intermediate’ categories respectively). this was equivalent to their choosing ‘23.1% of the female homicides were under 15 years of age’ in the above example, showing that they identified the correct row and column headings, but could not identify what the ‘whole’ was that the percentages were a ‘part’ of. students in higher education will have to interpret tables of data and percentages in many disciplines and most lecturers will assume that they can understand these representations. the data in figure 10 shows that many students will have trouble interpreting the language used to describe percentages and with making sense of data in tables, especially when it includes percentages. example 2: converting to square units figure 11 shows the results for an item that required students to say how many square millimetres (mm2) there are in 2 cm2 (with the fact that there are 10 mm in 1 cm given). the alternative answers were a 20, b 40, c 200 and d 400. only one-third of all students and less than two-thirds of those with scores in the ‘proficient’ category answered this correctly. students with scores in the bottom two categories preferred alternative a (the answer 20), indicating that they treated all units mentioned as linear, ignoring the references to square units. even those in the ‘upper intermediate’ category were almost as likely to choose alternative a as the right answer (between 35% and 40% of them). alternative d (answer 400) was chosen by between 10% and 20% in all proficiency bands. in this case students were aware that squaring the numbers was appropriate, but applied it inappropriately to the value 2, as well as in order to convert the units. similarly, alternative b (answer 40) was chosen by between 10% and 20% of the students with scores in the lower three proficiency bands, indicating that they were aware of the need for squaring, but not aware of which numbers to square in order to convert the units. these results indicate that students are unable to think flexibly about square units in the metric system, or that many of them do not read questions carefully (interpreting mm2 as mm and cm2 as cm). figure 11: for a question requiring converting to square units, the proportions of students who chose each alternative answer, for the total cohort and for each proficiency band. the pattern of performance on this question shows that most students are not able to do a simple conversion from linear to square units, which is a competency that they would most likely be expected to have in many scientific and technical courses in higher education. example 3: proportional reasoning and integrating data from different sources given values in a chart for the proportion of the population in several provinces that is, say, over 15 years of age, as well as the values of the total population for those provinces, one might ask which province had the greatest number of people over 15 years of age. if in addition the proportions are all similar, but the total populations are significantly different, one can conclude that the province with the largest overall population also has the largest number of people over 15 years of age. the results for a task similar to the one just described are shown in figure 12. alternative a is the correct answer and alternative d represents the greatest proportion (the province with the tallest bar in the chart). the correct answer (the province with the greatest number of people over 15 years old) was chosen by only 40% of all students and by only 75% of those whose scores were ‘proficient’. the majority of students with scores in the ‘intermediate’ categories (and surprisingly, a slightly smaller proportion in the ‘basic’ category) chose the largest proportion (alternative d), in effect answering the question ‘which of the provinces had the greatest percentage of people over 15 years of age?’. figure 12: for a question requiring proportional reasoning, the proportions of students who chose each alternative answer, for the total cohort and for each proficiency band. the proportion of all students who could in fact answer correctly by using the proper reasoning about proportions was probably smaller than 40%, as it was possible to arrive at the correct answer by answering the question equivalent to ‘which of the provinces had the greatest number of people?’ not ‘which of the provinces had the greatest number of people over 15 years of age?’ (by using the data for the total populations and not checking that the proportions of people over 15 were similar in the four cases). the pattern of performance on this task illustrates the difficulty that many people have with reasoning with proportions, understanding the distinction between absolute and relative quantities and the language used to make this distinction. these difficulties are further analysed by frith and lloyd (2016). it appears also that many students also did not recognise that the question required them to integrate information from two different data representations. example 4: calculating percentage of percentage and integrating data from different sources given data for the percentage distribution of the population in the provinces of south africa and the proportion of the population that is over 15 years of age in each province (for example in a stacked bar chart), a question might ask what percentage of the total population is over 15 years of age and lives in a particular province. this requires combining the percentage of the population that lives in the given province and the percentage of that provincial population that is over 15 years old, by calculating a percentage of a percentage. figure 13 represents the results for a task similar to this one, where alternative a represents the correct answer, alternative b represents the percentage of the total population that is in the province, alternative c is the percentage of the population of that province that is over 15 years. the vast majority of students (70%) selected alternative c, which means that they effectively answered the question ‘what percentage of the population in the province is over 15 years?’ rather than ‘what percentage of the total population is over 15 years and lives in the province?’. even more than a quarter of the students with scores in the ‘proficient’ category selected this incorrect alternative. figure 13: for a question requiring calculating a percentage of a percentage, the proportions of students who chose each alternative answer, for the total cohort and for each proficiency band. the pattern of performance on this question illustrates the extent to which students struggle to interpret the precise language used to describe percentages and, as in the previous example, that many of them are unable to integrate information from two different data representations. conclusion in this article, the results are presented for a large sample of school-leavers from across south africa who wrote one version of the nbtp ql test in 2014 and who obtained nsc results that qualified them for entry into higher education in 2015. nearly 70% of the scores for the test are in the lower two proficiency bands, with 36% in the ‘basic’ band and 13% in the ‘proficient’ band. this suggests that the majority of candidates aiming to enter higher education are in need of some kind of supplementary support for developing their ql, while more than a third will require extensive support. this support could be in the form of foundation courses, supplementary tutorials integrated into disciplinary curricula or online provision. teachers and lecturers should be mindful of the assumptions they make about students’ ql competencies and should at all times attempt to make the implicit literacy practices of their disciplines more explicit. students who wrote nsc mathematics perform considerably better on the nbt ql test than those who wrote nsc mathematical literacy. nearly two-thirds of those who wrote nsc mathematical literacy have scores in the ‘basic’ category, compared to 30% of the nsc mathematics candidates. less than 1% of those who wrote nsc mathematical literacy have a ‘proficient’ score, indicating that in general the school subject mathematical literacy does not prepare students to cope with the ql demands of higher education. comparison of the nbt ql scores and the nsc mathematics results reveals that the nsc mathematics result is a very poor predictor of the ql score. for any particular nsc mathematics score there is a large range of nbt ql scores, but this range is particularly large for the lower nsc mathematics scores. in general, the performance on the nbt ql test is lower than for the nsc mathematics. a similar weak relationship is observed when the nbt ql scores are compared to the nsc mathematical literacy scores. in addition, the difference in overall performance is much greater (with a median score for the nbt ql test approximately half of the median for nsc mathematical literacy). the scores on subgroups of items classified according to the main competencies they were designed to assess are also considered. three-quarters of the candidates score below the ‘proficient’ level in all of these subgroups except the one mainly requiring recognising patterns or predicting terms in a sequence, a competency that is probably quite well developed in school mathematics. the weakest performance is in the subgroup that requires candidates to interpret problem statements and calculate or estimate answers. this could reflect difficulties experienced with interpreting the problem statements (the majority of students are not first-language english speakers), but probably also points to students’ poor number sense and dependence on calculators, as most questions in this subgroup require estimation. this kind of analysis of the scores on subgroups of items provides diagnostic information that can inform the design of interventions to address the specific needs of students in terms of the competencies to be developed. due to the context-bound nature of ql, teachers and lecturers need to use this information to develop their own curriculum solutions in their own disciplinary and classroom contexts. where large numbers of students are not first-language english speakers, and the possibility exists that language difficulties contributed to poor performance, it should possibly be assumed that some form of al intervention will also be required in order to help students cope with the language demands of their quantitative studies. in the last part of the article, close examination of patterns of performance on examples of individual items are presented to illustrate how diagnostic information can be derived from this kind of analysis. these examples show that many students have difficulty with quantitative language, especially the language used to describe percentages and absolute and relative quantities. students generally also have difficulties with interpreting data in tables, especially percentages, and many struggle with proportional reasoning. these are all examples of concepts and competencies essential for practising ql in many academic disciplines. in a society where many arguments and the understanding of many situations and problems draw on quantitative data, competencies like these are also essential for effective and critical citizenship. it would therefore be essential to ensure that students develop these competencies in all appropriate contexts in school and also as graduate attributes in higher education. these results stress that there is a lack of alignment between the exit level outcomes from schooling and the expectations of higher education in terms of ql. higher education institutions need to recognise this fact and modify curricula accordingly. at the same time it would be productive if the school sector could give more attention to the development of this vital literacy, which is needed not only for higher education, but also for critical citizenship. acknowledgements we thank the nbt project team at the centre for educational testing for access and placement at the university of cape town who provided the opportunity to conduct this research, with the goal of contributing to the nbt project’s purpose of assessing the relationship between entry level proficiencies and school-level exit outcomes. competing interests the authors declare that we have no financial or personal relationships that might have inappropriately influenced us in writing this article. authors’ contributions r.p. and v.f. contributed equally to the conceptualisation and execution of this research and the production of the manuscript. references centre for educational testing for access and placement. 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(2006). item response theory. in r.l. brennan (ed.), educational measurement (4th edn., pp. 111–153). westport, ct: greenwood/praeger. microsoft word 64 front cover final.doc 52 pythagoras 64, december, 2006, pp. 52-61 reflections on the role of a research task for teacher education in data handling in a mathematical literacy education course vera frith and robert prince faculty of higher education development, university of cape town vfrith@maths.uct.ac.za and rprince@maths.uct.ac.za the introduction of the subject “mathematical literacy” in the further education and training band from 2006 has created an urgent need for large numbers of teachers to be educated about the nature of mathematical literacy and to become effective teachers of it. in this paper we report on an attempt to contribute to this goal through a curriculum component in an “advanced certificate in education” course. this curriculum component on data handling was structured around a research task which required the teachers on the course to practise mathematical literacy in a context where there is close linkage with other vital competencies, such as verbal reasoning, writing and computer literacy. this approach articulates well with the kind of teaching envisaged by the curriculum statement for “mathematical literacy”. we report on an initial analysis of the teachers’ reflections on their experience of the curriculum-embedded research task on this course, the manner in which this task contributed to their understanding of mathematical literacy as a practice and themselves as practitioners. note: we will use the term “mathematical literacy” (in quotes) to denote the school subject, and mathematical literacy, with lowercase letters, to denote the practice of mathematical (or quantitative) literacy. introduction although “mathematical literacy” is being introduced as a “subject” in the new south african further education and training (fet) curriculum, we argue that mathematical (quantitative) literacy should be understood in the context of the social practices in which it is embedded. the term practice is used to include “both what people do and the ideas, attitudes, ideologies and values that inform what they do” (baynham and baker, 2002: 2). the promotion of any definition of mathematical literacy will implicitly or explicitly promote a particular social practice (jablonka, 2003). the introduction of mathematical literacy as part of the new school curriculum has the potential to contribute to the transformation of south african schooling and society by contributing to the development of individuals’ ability to participate fully and critically as citizens (national department of education, 2005a). one of the consequences of discriminatory educational policies in south africa in the past is that improvements in the general quality of the mathematical literacy practices in south african society are urgently needed. we believe the new curriculum is very welcome as a means towards promoting this goal. in the medium to long term, the creation of a more mathematically literate society will contribute towards the level of mathematical, scientific and technological achievement in the country. the introduction of the new subject of “mathematical literacy” necessitates the training (or retraining) of teachers to implement this complex curriculum appropriately. in the same way that a teacher’s mathematical content knowledge may be the most important predictor of learning in the mathematics classroom (kenschaft, 2005; ball, bass and hill, 2004), so the development of mathematical literacy by school learners is likely to be most strongly affected by the availability of teachers who are highly mathematically literate themselves. as part of the preparation for the implementation of the new curriculum, the four higher education institutions in the western cape collaborated in developing a number of different mathematical literacy advanced certificate in education (ace) courses for delivery in 2004. these courses were completely new and the experiences of both the teachers and the teacher educators on these courses should therefore be instructive for the design of future courses, which will no doubt be necessary. in this paper we will discuss aspects of the design and the experience of one part of one of these ace courses. the part of the course we will focus on consisted of a brief introduction to mathematical literacy in the context of the new curriculum, followed by a detailed vera frith and robert prince 53 exploration of the topic of data handling. this will be referred to as the “data module” in this paper. in this paper we will first discuss the nature of mathematical literacy and the approach which sees it as a practice within a social context. we will briefly examine the various roles played by an inservice teacher learning to teach the unfamiliar curriculum for “mathematical literacy”. we will then outline the design of the “data module” of the ace course in the light of these ideas. we will describe the role of a research task in developing the teachers’ mathematical literacy practice. in order to discuss the contribution of the research task to this development, we examine the teachers’ written reflections on their experience of the task. this is an initial investigation that is part of a larger, more long-term research programme investigating aspects of teacher education for mathematical literacy. mathematical literacy as contextualised practice this paper argues for an approach to mathematical literacy (and other literacies) which sees them as practices embedded in particular social contexts (archer, frith and prince, 2002). baker, clay and fox (1996: 3) refer to “the collection of numeracy practices that people engage in – that is the contexts, power relations and activities – when they are doing mathematics.” mathematical literacy should not be seen as only a set of identifiable mathematical skills that can be learnt without reference to the social contexts where they might be applied. according to jablonka (2003: 78), “any attempt at defining mathematical literacy faces the problem that it cannot be conceptualised exclusively in terms of mathematical knowledge, because it is about an individual’s capacity to use and apply this knowledge. thus it has to be conceived of in functional terms as applicable to the situations in which the knowledge is to be used.” this emphasis on the use and application of knowledge implicitly assumes the importance of the associated quantitative thinking and reasoning. we take the view that mathematical literacy events can be described in terms of the contexts, the mathematical and statistical content and the underlying reasoning and behaviours that are called upon to respond to a situation that requires mathematical literacy practice. there is a subtle but important difference between this view, which sees mathematical literacy as a practice, and the view expressed in the south african national curriculum statement grades 10-12 (general) mathematical literacy (national department of education, 2003: 9), which defines mathematical literacy as a “subject”. however, the emphasis on problem solving in real contexts is fundamental to both definitions. the debate about the meaning of the term ‘mathematical literacy’ (also known as ‘numeracy’ or ‘quantitative literacy’ in different countries) and its relationship to ‘literacy’ and to ‘mathematics’ is exemplified by the various articles in the book mathematics and democracy: the case for quantitative literacy (steen, 2001). the articles in this book reinforce the idea that mathematically literate practice, as opposed to mathematics, is always embedded within a context. hughes-hallett (2001: 94) summarises the difference between mathematical (quantitative) literacy and mathematics as follows: “mathematics is about general principles that can be applied in a range of contexts; quantitative literacy is about seeing every context through a quantitative lens”. in thinking about ‘context’, usiskin (2001) warns against the use of contrived ‘real-life’ examples masquerading as ‘reality’ in the mathematics classroom. teaching “mathematical literacy” requires the use of contexts which are real for those involved, and which need to be understood as clearly as the mathematics that is being applied. very often mathematical literacy events (the situations where mathematical literacy is practised) require the content, contexts and reasoning that are usually associated with the use of statistics (hughes-hallett, 2001). thus data handling is arguably the most important component in the “mathematical literacy” curriculum and one which distinguishes it from what is being called the core “mathematics” curriculum (national department of education, 2005b). we adopt the following definition of mathematical literacy, in which all three approaches to the description (contexts, content and reasoning) are embedded: mathematical literacy is the ability to manage situations or solve problems in practice, and involves responding to quantitative (mathematical and statistical) information that may be presented verbally, graphically, in tabular or symbolic form; it requires the activation of a range of enabling knowledge, behaviours and processes and it can be observed when it is expressed in the form of a communication, in written, oral or visual mode. reflections on the role of a research task for teacher education in data handling in a mathematical literacy education course 54 this definition has been informed by the discussions and definitions implicit in the frameworks used by the timms (mullis et al., 2003), pisa (programme for international student assessment, 2003) and all (adult literacy and lifeskills, 2002) studies, but it draws most heavily on the latter. the definition of mathematical literacy which we have adopted implies that being mathematically literate requires the ability to express quantitative information coherently in a verbal and visual form. kemp (1995) argues that mathematical literacy includes the ability to communicate clearly and fluently and to think critically and logically. in dealing with quantitative or mathematical ideas in context, students should be able to interpret information presented either verbally, graphically, in tabular or symbolic form, and be able to make transformations between these different representations. the transformation of quantitative ideas into verbal messages is the area where a student’s ability to write coherently about quantitative ideas will be exercised. mathematical literacy also requires the ability to choose the appropriate form for the expression of a quantitative idea, and to produce a text that expresses that idea. thus the practice of mathematical literacy must include the ability to put together a document for a particular purpose in a particular context. the teacher as learner in the ace course just as mathematical literacy as a domain can be usefully conceptualised as a social practice, so has learning itself been extensively described using a social practice perspective (lave and wenger, 1991; wenger, 1998). in particular this approach has been found useful for investigating teacher learning (graven, 2004; 2005). thinking of teacher learning as taking place within a community of practice, throws the focus strongly onto the teacher’s sense of identity and the changes in this identity that the educational programme brings about. graven (2004) also highlights the central role played by the learner teacher’s confidence. by examining the teachers’ reflections on their experience we also emphasise the importance of these factors. we have argued that mathematical literacy can be thought of as a practice within a social context. similarly mathematical literacy as experienced by the learners in educational interventions will be a different but related practice, which with careful management by the educator, could be closely enough related to the practices in broader society to allow for significant transfer of practices into the world outside of the educational setting. this transfer is facilitated by educational practices that mirror the mathematical literacy practices of society as closely as possible and assignments where learners will be required to practise mathematical literacy outside of the educational context. this is one of the reasons why, for example, a research task such as the one described in this paper is so important. although we are focusing in this paper on the development of the teachers’ mathematical literacy practices themselves, it is worth pointing out that the teachers were on the ace course to develop their own mathematical literacy practices and to learn how to teach the subject “mathematical literacy”. in both cases they were subject to the dichotomy between classroom and “real-world” practices, and the corresponding confusion of identities. so not only were they hoping to transfer their own mathematical literacy practices from the ace classroom to their lives as citizens, as professionals and as “mathematical literacy” teachers in particular, but they were also hoping to transfer learning about the practice of teaching from the ace classroom to their own work practice in school. the teachers in the ace classroom were expected to maintain a dual identity as learners and as reflective teachers contemplating implementing a new curriculum, but without the opportunity to do so until after the course was completed. their identity as learners was further complicated by the fact that they are in practice already teachers, but not of “mathematical literacy”. so in a sense they were experiencing inservice teacher education and in another sense it was pre-service education, given that the new curriculum had not yet been implemented in the schools. so in engaging with the research task in the course, the teachers had to play and/or reflect upon various roles (identities). for example, in doing the research task they were themselves learners within the context of the course, but were also modelling the school learners they would be teaching in the future, by carrying out an assignment identical to one that might be expected of grade 10 or 11 learners. at the same time as developing their own mathematical literacy, they were expected to reflect upon their role as teachers in the classroom, both in their current practice and in their future practice as “mathematical literacy” teachers. vera frith and robert prince 55 the “data module” of the mathematical literacy ace course the part-time mathematical literacy ace course, which lasted for two years, consisted of five modules, each lasting for the equivalent of one semester (about 14 weeks). four of the modules were devoted to the “learning outcomes” of the school “mathematical literacy” curriculum. thus there was one module on “data handling”, one on “numbers and operations”, one on “functions and algebra” and one on “space and shape”. the teaching and assessment of the material for the fifth module, which was on “curriculum and assessment”, was integrated throughout the four other modules. the students on the course, who were practicing teachers, were required to attend scheduled three-hour classes once a week. there were also additional classes and computer laboratory sessions for those needing extra assistance. students were also expected to do homework and complete assignments for assessment in their own time. there was a final written examination for each of the four modules that were based on the “mathematical literacy” curriculum learning outcomes. the first semester of the ace course was devoted to data handling (which we call the “data module” for convenience). the mathematical and statistical content for this course was the analysis, representation and interpretation of data. for the design of the data module we drew upon our experience in designing curricula for and teaching on quantitative (mathematical) literacy courses for first year university students on a variety of study programmes (archer, frith and prince, 2002; frith, jaftha and prince, 2005). the principles that guided our curriculum design (consistent with seeing mathematical literacy as contextualised social practice) were: • that material should be context-based and make use of real relevant intrinsically motivating contexts, wherever possible; • that curriculum tasks should require the exercise of several related competencies, such as writing and using computers, not just mathematical skills; • that the production of a (mainly verbal) product as an outcome of mathematically literate practice is important (as well as the understanding and interpretation of existing information); • that students’ confidence should be promoted; • that co-operative learning should be emphasised. the 14 three-hour classroom sessions were run as workshops with limited presentation of course content at the blackboard. students worked in groups and engaged with the course materials while lecturers acted as facilitators. some sessions were conducted entirely in the computer laboratory so that students could develop proficiency in using a spreadsheet program. the first three sessions provided an initial orientation to the nature of mathematical literacy and to the subject statement and assessment standards for the subject “mathematical literacy”. the rest of the first semester was devoted to the study of learning outcome 4: data handling from the subject statement (national department of education, 2003). at the beginning of the course the 33 teachers (of whom 29 completed the ace course successfully) wrote an extensive pre-test, designed to assess their mathematical literacy and reveal areas of strength and weakness. the use of this test for similar purposes was described by prince and mcauliffe (2005). for those students whose results on this test indicated the need for additional instruction in specific mathematical and statistical concepts, extra workshops were provided. these workshops were intended to provide these teachers with a sense of competence and confidence to play an active and constructive role in the main classroom sessions. for this module formative assessments were used and a summative assessment (final examination) was written at the end. the research task in the data module structuring the curriculum of the “data module” around the execution of an independent research task provided an ideal environment for the development of the teachers’ mathematical literacy practice by realising the curriculum design principles outlined above. a research task of this nature is a mathematical literacy event which has many affordances, such as enriching the understanding of sampling, bias and uncertainty, developing data handling techniques, use of language and technology, and promoting quantitative reasoning. similar affordances provided by graphical representations are explored in prince and archer (2005). it was assumed that by learning about data handling within the context of completing an authentic task, the relevance of the curriculum content was demonstrated and the motivation of learners was promoted. it also allows reflections on the role of a research task for teacher education in data handling in a mathematical literacy education course 56 for the various literacies (for example, writing and use of technology in the form of calculators and spreadsheet applications) to be used in a way that is less contrived and closer to a ‘real-world’ experience. although the research task was done by the teachers individually mostly in their own time, the course provided extensive scaffolding exercises and opportunities for cooperative work with other students. the task was identical to one presented using a similar curriculum structure in a grade 10 textbook (bowie, frith, prince and schauerte, 2005) and similar to the research task in the companion grade 11 text book (bowie, frith and prince, 2006). the task was presented at the beginning of the data module and students were informed that they would be expected to work consistently on the execution of the task, in step with the scaffolding provided throughout the course. an extract from the statement for the research task is shown in figure 1. research task for this project you will work on your own and do your own survey. in this chapter you have seen two examples of the kind of survey you should do. all the steps that you have to do to complete a survey are covered in the units in this chapter. suggested example of a survey: purpose: to find out about young people’s attitudes towards living in south africa. questions: • age, gender, grade at school, etc. • how positive do they feel about their future in south africa? • how important do they think it is to vote in an election? • do they think south africa has a lot to offer young people? • how proud are they to be south african? • how important is it to them that south african sportsmen and women should win in international competitions (such as the olympics)? figure 1: abridged extract from a grade 10 textbook presenting the “research task” teachers were also free to devise their own research questions, relating to aspects of the lives of the school learners they were currently teaching. for example, one teacher researched learners’ living conditions and another researched learners’ attitudes towards nutrition and exercise. the scaffolding exercises were structured to provide support in parallel with the execution of this task, in a similar manner to the way support was provided in the textbook. we believe that it is more effective to provide scaffolding as the task unfolds (rather than before it is attempted), as transfer of knowledge is more likely to take place if the connections are made more explicit in this way. for example, the first session provided a framework for understanding the research process as a whole, while during the second session students worked together to pilot and refine their survey questions for the project. at this stage they were expected to conduct their survey and gather their data. during sessions two to seven, students were expected to reflect on the relevance to their research of the techniques for data analysis and representation covered in the workshop sessions, and to apply these techniques to their own data (including the use of spreadsheets). at the same time they were told that they should be working on writing up their report. in the ninth session they were assisted by a language expert (a lecturer from the language development unit at the university) to peer-edit each other’s writing and to provide constructive criticism, before revising their draft reports for final submission at the end of the data module. in structuring the “data module” this way, the intention was to provide as authentic as possible an opportunity to experience and become more expert in mathematical literacy practice. the intention of the research task was that the teachers would: • become proficient at analysis, representation and interpretation of data. • develop an appreciation for the processes that comprise quantitative research and the manner in which they could influence the research results. • recognise through experience the importance of using relevant contexts as a vehicle for learning “mathematical literacy”. • work cooperatively with peers and other educators. • communicate their findings effectively in the form of a written report. • reflect on the implications of their experiences for their mathematical literacy practice. the reflection task in the data module for the discussion of the effectiveness of the research task in achieving the intentions listed vera frith and robert prince 57 above, in this paper we will focus on the teachers’ reflections on their experience of this task. as part of the research task (“project”) statement, students were also asked to reflect on their experience of the process of completing the task. to assist them to structure their thoughts, they were given the following guiding questions (which are also part of the “project” task in the school learners’ textbook referred to above): • which part of the project did you find the easiest to do? • which part was the hardest? • what would you do differently next time? • is there any part of the process of doing a survey that you will need to get help with before you do a similar survey again? • did you enjoy doing the project? explain why or why not. • describe the main thing that you think you learned from doing this project. of the 33 teachers on the course, 29 included written reflections with their project submission. teachers’ reflections on the research task (the “project”) parts of the “project” teachers found easiest: most teachers said that the easiest part of the project was the actual collection of the data by administering their questionnaire to the learners at their school. some cited the choice of research question as the easiest aspect and several others found the mechanical processes of data analysis (such as drawing up frequency tables or calculating statistics) the easiest part. parts of the “project” teachers found difficult: more than half the teachers described their difficulty in designing the questionnaire for their survey. comments like the following were fairly common: “(the hardest part was) setting and changing the questions to the questionnaire to make it appropriate for my survey’s statement and to minimize ambiguity.” many teachers also reflected on the difficulties they had experienced with the analysis and interpretation of their data. in some cases they were aware of difficulties in deciding which kinds of analysis and representations to use, or difficulties in actually performing this analysis: “the analysis of the data was difficult for me as i was not always sure which tools to use to present the data and if the choices i made was the appropriate ones.” about one third of the teachers reflected specifically on their difficulty with the writing of the report and the kinds of reasoning that this required. the following is typical: “the analyzing and the reporting of the findings were difficult as this needed a reflection on the data provided and to comment on what was evident there. the writing of the report was also difficult because it needed an unbiased opinion and the facts stated had to be always based on the information provided.” parts of the “project” teachers would do differently: the most frequent observations under this heading were to do with improving the questionnaire design and changing the size and nature of the sample, either to make the task more manageable or to improve the quality of the information obtained. several teachers also commented on the need for better planning and time management, particularly to allow for more input from and interaction with peers. some mentioned the need for a more unified approach to the whole research process and others observed that they would use a spreadsheet for their data analysis. the following quotes highlight some of these issues: “based on my question, i would get an idea of what it is that i would like to discuss in the report and ensure that the questionnaire provides me with information to be able to do this. in this way each part of the activity is not seen as isolated or independent of the other, because the questionnaire is eventually going to impact on what you are able to write about in the report.” and, “the second (thing i would do differently) is to get constant feedback from fellow class mates so that i can face the next task with more confidence.” parts of the process where teachers would need more help: most teachers who responded to this question said that they would need help with the questionnaire design and/or the writing of the report. thus it appears that they are least confident about the ability to communicate quantitative ideas in writing. a few teachers also indicated a lack of confidence in their ability to analyse and present their data graphically in an appropriate way. a few students identified the need to improve their ability to interpret the data: “(i would need more help with the stage where) you have collected the data and need to make statements about the findings. the session on writing the report focused on the reflections on the role of a research task for teacher education in data handling in a mathematical literacy education course 58 actual writing of the interpretations. how to make those necessary interpretations is important and needs more developing.” teachers’ enjoyment of the task: more than two thirds of the teachers said that they had enjoyed the research task, in spite of some having reservations about time pressure and anxiety about whether they “were on the right track”. the following quotes illustrate how the context of the learning task can influence the affective reaction to it: “i did enjoy doing the survey. the challenge of designing an applicable questionnaire was exhilarating. seeing the results emerging, analyzing it, seeing a pattern in the learners’ responses and then writing up the results kept me curious.” and “… i feel a sense of achievement. i started something which i took control of fully and have completed it to the best of my ability.” teachers’ opinions about the main things they learnt from doing the project: teachers mentioned a large variety of topics under this heading, such as report writing, data analysis techniques, questionnaire design and the value of collaboration, for example: “i learned mostly how to work with people including learners and my fellow students and lecturers.” however, the most frequently mentioned topics were to do with an appreciation of the processes that research entails, and/or the need for planning and time management, for instance, “i now have a fairly good idea of what research entails. i actually collected the raw data which we only see in books. we started a process at the very beginning, processed the data and carried the process through to the end.” it is telling that for some teachers the most significant learning was not about data handling or the research process, but arose from the context of the research itself. these teachers cited as most important insights they had gained about their school learners, which arose either from their actual research results or from the experience of their engagement with the research task. reflections like the following illustrate how relevant the research task was to the teachers’ real working environments: “i learned to be more tolerant and patient towards the learners. i now consciously listen to their reasons for absenteeism”; “young people are extremely willing to be co-operative and helpful. they are enthusiastic to help adults on condition that their assistance is acknowledged and they don’t feel exploited.” discussion we discuss the teachers reflections in the light of each of the stated intentions of structuring the data module around the execution of a research task. these intentions were listed above at the end of the section under the heading “the research task in the data module”. becoming proficient in the analysis, representation and interpretation of data: although the analysis and interpretation of data was a major component of the material covered in the classroom sessions and some teachers cited it as the easiest aspect of the project, there were still many teachers who reflected on the difficulties they had experienced with the analysis and representation of their data. other teachers commented not so much on the data analysis as on their difficulties with the more subtle task of deriving meaningful quantitative information from their representations of the data. these difficulties reveal a worrying lack of competence with tasks which are fundamental to the achievement of learning outcome 4 data handling (national department of education, 2003), but which cannot be assumed to be common knowledge even for a person with a reasonable level of mathematical education. this points to the need for extensive training of teachers in data handling and interpretation, especially as this is a new component in the mathematics curriculum as well. developing an appreciation for the processes that comprise quantitative research and the manner in which they could influence the research results: a number of students discussed their growing awareness of the interdependence of the components of the research process. in grappling with their difficulties with questionnaire design, they developed an awareness of the impact of the research instrument and method on the quality of the data that could be collected, and hence the conclusions that could be drawn. this kind of insight is valuable both for their own development as mathematically literate practitioners, and for making them more able to promote critical thinking about data in the classroom (which is heavily emphasised in the “mathematical literacy” fet curriculum). an appreciation for the dependence of the quality of the data and the conclusions drawn from it on the research vera frith and robert prince 59 processes (such as possible bias in sampling) is one of the learning outcomes, which we believe is best learned in practice. when they reflected upon the research process, in terms of their own engagement with it, one of the strongest themes to emerge was to do with time management. teachers developed an appreciation for the importance of drafting and redrafting questions and text using input from other people: “… my concern was the process that i had to follow. but when i started interacting with my fellow colleagues, the whole picture became more clear. i started rephrasing my research questions, and once this was done, working through the process step by step became easier. the lectures on analytical and graphical presentations of data added more clarity. hence the knowledge that i gained in class helped me further to develop my understanding of my own research project.” the teachers’ reflections on the process make it very clear that they feel that it takes considerable time to communicate with peers and to assimilate and practise applying the new competencies learnt in the course. the reflections generally create the impression that the research task was an effective vehicle for this, although even more time would have been appreciated. the implication of this is that “crash courses” for “mathematical literacy” teachers will probably be less effective at developing teachers’ mathematically literate practice. recognising through experience the importance of using relevant contexts as a vehicle for learning “mathematical literacy”: most of the teachers reported having enjoyed the research task. one of the intrinsically motivating factors was their interest in the information they were gathering itself. so for these teachers the task had intrinsic value, not just as an assignment for the course, but as a task that illuminated and enriched their teaching experience. some teachers also made statements that revealed that completing the research task gave them a sense of worth and an identity as a researcher. these experiences highlight the power of using real, relevant and intrinsically interesting contexts as a vehicle for developing mathematical literacy and promoting learners’ confidence. working cooperatively with peers and other educators: just under half the teachers made some comment reflecting their appreciation of the value of collaborative work with other teachers (and in some cases their own school learners) and of gaining input from the lecturers during the research process. “peer assessments of our projects helped a lot, because you can make the changes as you go along. other people’s different perspectives allows you to broaden your own perspective and enriches yourself to complete a project with success.” in our opinion, this aspect of their experience of doing the research task may be one of the most valuable in terms of preparing them to be effective facilitators of learning in the “mathematical literacy” classroom, where they will be expected to motivate school learners to work collaboratively and to facilitate this kind of learning. communicating findings effectively in the form of a written report: more than half the teachers described their difficulty in designing the questionnaire for their survey. this highlights the importance of the “literacy” component of mathematical literacy. about one third of the teachers reflected specifically on their difficulty with the writing of the report and the kinds of reasoning that this required. the general level of difficulty experienced in writing about data was also reflected in the quality of the reports, which indicated that teachers were probably not adequately prepared to support school learners with this kind of writing task, which is a fundamental component of “mathematical literacy”. further research will focus on the insights that can be derived from a close study of the actual research reports submitted. conclusion in designing a curriculum for “mathematical literacy” teacher training, it is useful to frame mathematical literacy as contextualised social practice. some of the implications of this framework for the curriculum are that mathematical and statistical content should be taught through learners’ engagement with realistic, relevant contexts; that critical thinking and communication are important elements and that collaborative work should be encouraged. the development of positive attitudes and confidence should also be promoted. we maintain that structuring the curriculum of the data handling component of a “mathematical literacy” course (for teachers or for school learners) around the execution of a scaffolded research task provides an effective environment for the realisation of these principles. this view is supported by the results of our analysis of the reflections written by the reflections on the role of a research task for teacher education in data handling in a mathematical literacy education course 60 teachers about the research task in the “data module” of the mathematical literacy advanced certificate in education course. acknowledgement we are very grateful to our late colleague, stella clarke, who assisted us with facilitating the writing component of the research task, reported in this paper, at a difficult time in her life. we are also grateful for the insights and inspiration we have received from her. references all. 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(1998). communities of practice: learning, meaning and identity. new york: cambridge university press. mathematics is not only real, but it is the only reality. that is that entire universe is made of matter, obviously. and matter is made of particles. it's made of electrons and neutrons and protons. so the entire universe is made out of particles. now what are the particles made out of? they're not made out of anything. the only thing you can say about the reality of an electron is to cite its mathematical properties. so there's a sense in which matter has completely dissolved and what is left is just a mathematical structure. – martin gardner abstract introduction problem statement mathematics education in the south african context research design and methodology findings and discussion conclusion acknowledgements references about the author(s) surette van staden department of science, maths and technology education, faculty of education, university of pretoria, south africa puleng motsamai northern cape education department, curriculum and assessment services, south africa citation van staden, s., & motsamai, p. (2017). differences in the quality of school-based assessment: evidence in grade 9 mathematics achievement. pythagoras, 38(1), a367. https://doi.org/10.4102/pythagoras.v38i1.367 original research differences in the quality of school-based assessment: evidence in grade 9 mathematics achievement surette van staden, puleng motsamai received: 19 feb. 2017; accepted: 21 july 2017; published: 31 oct. 2017 copyright: © 2017. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract this non-experimental, exploratory and descriptive study, using a qualitative case study approach, aims to investigate whether there is evidence of variance in the quality of school-based assessment (sba) in grade 9 mathematics. participants were purposefully selected from five schools in a district in the northern cape in south africa. after questionnaires were completed, individual face-to-face semi-structured interviews were conducted with participants from the participating schools. documents were collected and analysed to corroborate or contradict data from the questionnaires and interviews. lack of adherence to policy, variation in classroom practice and inconsistent monitoring and moderation practices were identified as themes of possible sources of variation in sba. an analysis of the interviews and document analysis revealed that most of the heads of department and principals lacked in-depth knowledge and understanding of their roles and functions in making sba reliable, credible and valid. this was not only due to a lack of capacity to perform such functions, but was also due to a lack of effective induction and training by the district and provincial offices. findings from the current study point to the necessary role that a periodic evaluation of sba may play to ensure its effectiveness, credibility and reliability as part of successful assessment practices in a mostly developing context. introduction assessment is at the heart of the teaching and learning process (chisholm, 2004). at the dawn of democracy in south africa, the department of education (doe) replaced traditional assessment methods such as tests, examinations and year marks with continuous assessment in order to redress the focus on traditional examinations of the past. continuous assessment constitutes school-based assessment (sba) and examinations. sba encompasses all forms of assessment that are conducted by the teacher and teachers develop their own assessments (black & wiliam, 2010; poliah, 2010). gipps (1994) is of the view that sba has the potential to be a more valid form of assessment as it covers a wide range of curricular outcomes. however, due to the subjective nature of sba that weakens its design, it opens itself to lower levels of reliability, and reduced validity and credibility of learner performance (poliah, 2010; reyneke, meyer & nel, 2010). south african learners across all grades continue to perform poorly in mathematics when compared to their counterparts globally, nationally and regionally. in international studies such as the trends in mathematics and science study, south africa performs along with other poor performing participating countries in mathematics. similarly, in the national assessments such as the annual national assessment (ana), results show a very low performance in grade 9 mathematics specifically. the main research question that guided this study was: what evidence is there in teachers’ classroom assessment practices that points to possible variation in the quality of sba? in order to address this question, issues of adherence to policy, classroom practice, monitoring and moderation practices and learner performance in sba and external assessments will be discussed. problem statement post 1994, south africa made major changes in assessment policies and practices. traditional assessment practices such as tests and examinations were re-conceptualised to accommodate types of continuous assessment. continuous assessment therefore currently comprises sba and internal examinations. yet, fleisch (2008) points out that assessment was initially underdeveloped and did not form a key element in the initial training and support within education when implementing the new curriculum. kanjee (2007) further elaborates that assessment was the most neglected aspect of government’s efforts to transform the education system, and was the area that received the most criticism. the south african doe then presented assessment policies and practices in the form of guidelines. because of these guidelines, assessment is most likely to be interpreted and applied differently by teachers of the same subject and the same grade, which in this case is grade 9 mathematics. additionally, there are currently no common external assessments in grades below grade 12 in the south african education system. the problem with the weighting of sba is its quality, reliability, validity, and credibility. long, dunne and de kock (2014) confirm that there are no measures and systems in place in the south african education system to ensure that sba is reliable, valid and credible in the general education and training (get) band. despite the statutory body (known as umalusi) that ensures quality assurance, quality is ensured at grade 12 level only. there are no agreed standards across provincial does, across districts within the same provincial doe, or across schools within the same district (poliah, 2003). from the work of poliah (2010), it is evident that there is room for variation in the scoring of assessment tasks among teachers, particularly when the assessment tasks are not the same. mathematics education in the south african context a significant amount of research has taken place in mathematics content and teaching internationally and in south africa (dunne et al., 2002; mullis et al., 2011; setati, 2002; shalem, sapire & sorto, 2014). however, mathematics education research in south africa has mainly focused on curriculum and pedagogy, and has been dominated by cognition of how learners acquire mathematical understanding. post 1994, the introduction of curriculum 2005 saw mathematics being replaced with the learning area mathematical literacy, mathematics and mathematical sciences (doe, 2002). mathematical literacy, mathematics and mathematical sciences represented a major shift in the philosophy of mathematics and mathematics education, and thus demanded a major philosophical shift of both teachers and learners (graven, 2002). graven (2002) identified three major shifts: the approach to teaching mathematics: emphasis is placed on a constructivist, learner-centred and integrated approach to the teaching and learning of mathematics. this way of teaching moves away from the performance-based approach to the competence-based approach. the nature and content of mathematics. the role of mathematics education. the rationale for mathematical literacy, mathematics and mathematical sciences is focused on constructing mathematical meaning in order for learners to understand and make use of that understanding. specific outcomes (sos) for mathematical literacy, mathematics and mathematical sciences indicate changes in the content of school mathematics. however, vithal and volmink (2005) argue that mathematical literacy, mathematics and mathematical sciences poses a serious challenge in terms of both content and pedagogy, which are essential foundational competencies. the ongoing implementation challenges in the revised national curriculum statement (ncs) (doe, 2002) resulted in the development of the curriculum and assessment policy statements (caps). the rationale for the implementation of caps addressed four main concerns, namely: (1) complaints about the implementation of the ncs, (2) teachers who were overburdened with administrative duties, (3) different interpretations of curriculum requirements, and (4) the underperformance of learners (moodley, 2013). moreover, learning areas are now known as subjects. in mathematics for the senior phase (grades 7 to 9), there is too much content, which is combined with reduced time allocation. in the ncs, the time allocated for mathematics for grades 7 to 9 was five hours of contact time; however, this has been reduced to four and a half hours in caps (department of basic education [dbe], 2012). a conspicuous feature is that there is ‘linear progression’, which means that certain topics and concepts must have been dealt with in previous grades before teachers can teach new concepts in the present grade. this approach suggests that sequencing and pacing poses a threat in the classroom should learners not have been taught those concepts in previous grades. it also means that the educator has to teach the specific content that was supposed to have been taught previously in order to proceed with what has been prescribed for that particular lesson or week allocated to that content. several studies have reported a number of shortcomings in the teaching and learning of mathematics in south africa. one of the challenges, according to makgato and mji (2006), is that not all schools in the south african education system offer mathematics in the further education and training (fet) band. moreover, many of those schools that offer mathematics do not have the necessary facilities and equipment to provide effective mathematics teaching and learning. the current picture depicts a south africa where success in school mathematics is not randomly distributed across the population, with some groups systematically doing better than others (reddy et al., 2012). adler (2002) explains that mathematics needs to become more meaningful for learners, and one way of establishing meaning is by embedding mathematical problems in real world contexts. this practice would invite more learners to continue with mathematics, and thus reduce the inequalities in mathematics performance that we currently see when comparing learners from varying socioeconomic backgrounds. there are a number of long standing, unresolved and unaddressed questions where mathematics instruction and assessment are concerned, as stated by schoenfeld (1992). these challenges may be caused by the following reasons: learners do not know which needs are met by the mathematics topics introduced or how these are linked to known concepts. links to the real world are weak, generally too artificial to be convincing, and applications thereof are stereotypical. there are few experimental practices and modelling activities provided. learners have little autonomy in their mathematical work and often merely reproduce activities. (adapted from united nations educational, scientific and cultural organisation, 2012, p. 21). there is a body of evidence that suggests that one of the challenges in mathematics education is that mathematics teachers teach mathematical concepts in isolation. simply put, mathematical concepts are regarded as ‘stand-alone’ concepts and are taught separately from each other. more than two decades ago, schoenfeld (1992) recommended to policymakers that lessons should come in large coherent chunks, and take between two and six weeks to teach. furthermore, lessons should be motivated by meaningful problems and be integrated with regard to subject matter, for instance simultaneous use of algebra and geometry, rather than having geometry taught separately from algebra. this strategy will dissuade teachers who do not feel comfortable teaching certain topics and concepts from skipping such topics and concepts. geometry, in particular, in the get band, as indicated by usiskin (2012), is a section of the curriculum that mathematics teachers do not feel confident teaching. there is a small body of research that suggests that learners in the senior phase (grade 7–9) are not taught geometry in the fet (grade 10–12) mathematics curriculum. at the fet phase, geometry was optional, and higher institutions of learning, universities for instance, did not calculate this section in the admission point system. currently, geometry is a compulsory component of mathematics in the fet band, and, as such, learners in the senior phase are introduced to the content area space and shape in order to prepare them for the fet band. school-based assessment school-based assessment is a process of measuring learners’ achievements against the defined outcomes conducted by the teacher (maile, 2013). many researchers define sba as classroom assessment, formal assessment and formative assessment. as sba is an ‘engine of educational change’ that should inform teaching, it forms an integral component of teaching and learning in the classroom. sba is being practised in many countries; however, umalusi found that teachers the world over experience challenges in finding their roles in assessments (umalusi, 2010). school-based assessment has its own challenges such as different schools that are not equally effective, and teachers’ subjective judgements that are frequently accused of being biased. in the south african context, the weighting of sba varies considerably across the education system (as stipulated in the national protocol for assessment for grade r to 12, dbe, 2011a), which poses additional challenges, as will be discussed in later sections. school-based assessment is further made up of informal and formal assessments (dbe, 2011a). informal assessments are mainly formative and prepare learners for formal assessment. sba’s informal assessment role is to ensure, among others, that basic mathematical concepts are mastered to improve teaching and learning. regular informal activities such as homework and classwork, coupled with regular feedback, provide information to learners and teachers, and may help the teacher to gauge what learners’ performance will be in the formal assessment. learners should be familiar with the type of tasks used for formal assessment and should also be given the opportunity to master mathematical concepts (davison, 2007). quality assurance in school-based assessment school-based assessment, when defined as teachers’ own assessment tasks in the classroom, is an important tool, but when it serves as a component of national educational benchmarking, it needs to be rigorously controlled and quality assured (poliah, 2014). quality assurance in sba can be conceptualised as all of the quality control measures put in place in keeping with the required standards (adler, 2002). maxwell, field and clifford (2006) explain that these quality control measures are important to address issues of validity, reliability, fairness, authenticity, as well the quality of marking of these assessment tasks. in grade 9 mathematics, the forms of assessment available are tests and internal examinations, investigations, assignments and projects (dbe, 2013; world bank, 2008). the latter three of these assessment tasks are completed by learners under uncontrolled conditions, for example at home, or even at a library. according to the european network for quality assurance (daniel, kanwar & uvalić-trumbić, 2009), institutions should have policies and procedures in place for quality assurance; south africa is no exception. in the south african context, the doe developed mechanisms in order to address quality assurance in sba after the reliability and validity thereof were questioned. it has to be noted that while efforts are made to put policies and acts in place, these do not ensure compliance or standardisation across the system. the doe promulgated a number of policies and acts, such as the national protocol on assessment grade r–12, general and further education training on quality assurance act no. 58 of 2001, curriculum 2005, the revised ncs (grade r–9), the assessment guidelines in get (grade r–9), common assessment tasks in grade 9 of the get band, and the caps. however, these documents provide inadequate guidelines and are silent on the internal quality assurance processes that schools need to apply to ensure standardisation among schools (dbe, 2011a; maile, 2013; wilmot, 2005). thus far the focus on the system is on umalusi as a statutory body to ensure that assessments are quality assured at the exit points of the system. in terms of the south african education system, the exit points are at the end of the get and fet bands, and are grades 9 and 12 respectively. umalusi (cited in poliah, 2014) reports that there is huge disparity in the quality of sba from one school to another across education districts at grade 12. the significance of the current study could point to similar disparities in grade 9 in mathematics, thereby extending umalusi’s findings beyond the evidence found for grade 12 learners. findings from the current study would aim to inform the necessity of a periodic evaluation of sba to ensure its effectiveness, credibility and reliability as part of successful assessment practices in a mostly developing context. adler (2002) finds that a lack of assessment guidelines leads to variations, which may include: the marking standards of teachers (which may be too high or inflated) (maile, 2013; poliah, 2010). types of uncontrolled assessment tasks such as investigations, assignments and projects in mathematics. poliah (2010) highlights the fact that some teachers use homework as part of sba. the degree of guidance and assistance given to learners. torrance and pryor (1998) are of the opinion that learners are strategically guided with instructions and assistance for deeper understanding and discussion. this is done to close the gap between their current level of understanding and the desired goal. research design and methodology this study was exploratory, non-experimental, descriptive and interpretative in nature and formed part of a larger study (motsamai, 2017). the approach to empirical research adopted for this study was one of a qualitative case study. this approach was chosen because the aim was to capture in-depth views of the participants in order to make meaning and draw conclusions (guba & lincoln, 1994; onwuegbuzie & leech, 2007). the participants’ questionnaires were mainly used to ascertain participant profiles, backgrounds and experience. in some cases, the questionnaires were incorrectly completed; however, these were corrected together with the participants. face-to face individual semi-structured interviews were conducted, recorded and analysed. documents collected, such the grade 9 mathematics sba tasks, with their memoranda, moderation and monitoring reports were used to triangulate the data obtained from the questionnaires and interviews in order to corroborate or contradict data. participants and study context the larger study (motsamai, 2017) was conducted in five different schools that offer grade 9 mathematics in the john taolo gaetsiwe district in the northern cape province in south africa. the schools were drawn from rural, semi-rural, township and former model c schools offering grade 9. schools are sparsely scattered and distant from each another, an important characteristic of the northern cape. in each of the five schools, a grade 9 mathematics teacher, mathematics head of department (hod) and school principal were selected for participation. tables 1–3 summarise the profiles of the teachers, hods and school principals as obtained from data from the questionnaires. table 1: profiles of the participating mathematics teachers. table 2: profiles of the participating mathematics heads of department. table 3: profiles of the participating school principals. the sample included five teachers, five hods and five school principals from the participating schools. here, the authors will additionally report on the qualifications and mathematics teaching experience of the participants. schools were named school a to e, with teachers a to e, and hods a to e. all participants appear to be qualified to be appointed as teachers. however, data show that there are only two teachers who are adequately qualified to teach mathematics in grade 9. the remaining three teachers have three-year senior primary teachers’ diploma and secondary teachers’ diploma respectively. the senior primary teachers’ diploma qualification does not have a specialisation in any school subject. according to spaull (2011), senior primary teachers’ diploma has a ‘primary phase’ qualification and specialises in either mathematical literacy (40%) or mathematics (36%). however, these teachers obtained an advanced certificate in education with specialisation in natural science, technology and mathematics. the teacher with a secondary teachers’ diploma qualification specialised in physical science. one hod has a secondary teachers’ diploma with specialisation in mathematics and teaches mathematics in grades 10 to12. a striking feature that emerged from the data is that of an appointed hod who does not possess any mathematics qualification nor has ever taught mathematics in his teaching career. the other hod possesses a senior primary teachers’ diploma qualification only and did not teach mathematics at the time of data collection. in school e, the hod obtained an honours degree with a management qualification. according to the data presented above, only one hod reported to be highly qualified in mathematics. this hod teaches grade 12 mathematics, however, additionally conducts afternoon mathematics classes for learners in grades 4 to 12. data on the mathematics qualifications of school principals were not sought as this data were not relevant for this study. teachers’ teaching experience of mathematics varies between 5 months and more than 10 years. it should also be noted that most of the management staff are acting in their positions and do not hold these positions permanently. there were exceptions, where for example in school c, the permanently employed principal and hod have more than 10 years of managerial experience. methods of qualitative data collection the research data in this investigation are drawn from the three main sources: questionnaires, semi-structured interviews, and document analysis. questionnaires questionnaires were chosen specifically for this study as the responses would determine whether the participants’ biographical data had any association with their implementation of sba and assessment policy and practice. the questionnaires were completed by all of the participants in their own time prior to the interviews. the use of questionnaires was to obtain biographical data of the participants. biographical data included information such as gender and age, languages used for assessment, primary language of participants and the learners, participants’ mathematics qualifications and experience, training on assessment principles, policies and practice, among others. there were three sets of questionnaires: one for the teachers, the other for the hods and the last set was for the school principals. interviews face-to-face semi-structured interviews were conducted after the questionnaires were completed. interviews were conducted mainly in english; however, participants’ primary language was allowed to ensure that participants fully expressed their ideas and opinions. procedures to be followed were explained to participants and all indicated that they did not have problems with being recorded. in the interviews, participants were probed to explain their interpretation, experiences and insights with regard to each of their responses when it came to the concepts of sba, quality, quality assurance, moderation and learner performance in sba. heads of departments and principals were asked about their roles in ensuring quality and credible sba tasks and learner performance. all audio transcripts were recorded, transcribed and stored in a safe place. document analysis document analysis was required in this study so that the data in the key documents could be compared, examined and interpreted in order to elicit meaning and gain understanding (creswell, hanson, clark & morales, 2007; taole, 2013). documents such as the grade 9 mathematics sba tasks with their assessment tools (marking tools) and moderation reports were collected and analysed in order to corroborate or contradict data obtained from the questionnaires and interviews (mcmillan & schumacher, 2010; mouton, 2001). table 4 illustrates the different data sources that are applicable to each of the themes that are discussed for the purposes of this study. table 4: themes and supporting data sources. ethical considerations ethical clearance (sm14/05/01) and permission to conduct the study were obtained from the university’s ethics committee. permission to conduct the research was also granted by the northern cape department of education and the schools where this research was conducted. since one of the authors is a district official, this study was conducted in a different district in order to minimise power over the participants. all the participants were informed of the purpose and the rationale of the study, namely that we wanted feedback in order to understand their views, experiences and their perceptions of the quality of sba in grade 9 mathematics. participants were also informed that participation is voluntary and participants may withdraw during the study if they so wished to do so. participants who agreed to take part in the study were assured of anonymity and confidentiality. schools’ and participants’ names were maintained by use of pseudonyms such as school a, b, c, d and e, teacher a, b, c, d and e, hod a, b, c, d and e and principal a, b, c, d and e. all data collected were kept in a secure place. findings and discussion in order to address the research question, evidence of variations of sba will be described in terms of adherence to policy, classroom practice, monitoring and moderation and learner performance as themes that strongly emerged from the semi-structured interviews. school-based assessment in terms of adherence to policy assessment in the south african context comprises sba and the end of year formal examinations. the national protocol on assessment, the national policy pertaining to the programme and promotion requirements of the ncs and the caps further state that for the grades below grade 12, the end of year examinations are to be set internally. the national protocol on assessment requires every subject teacher to submit an annual assessment plan to the hod and the school management team in order to draw up a school assessment plan (dbe, 2013). the assessment plan should assist in the smooth running of the assessment activities and also in regulating sba. in addition, the national protocol on assessment requires that learners and their parents receive the term’s assessment plan at the beginning of each term to improve parental involvement. however, evidence from the interviews points to the fact that none of the participating schools had assessment plans, except one school, which appeared to have cycle tests in place. great variation in adherence to this policy is observed across the participating schools in this study. the weighting of sba across the grades and subjects is stipulated in the national protocol on assessment for grades r to 12 (dbe, 2011a). this protocol takes the form of guidelines, which are open to varied interpretations. the policy states that sba in the get band carries more weight than in the fet band. the policy further divides the weighting of the get into grades 1 to 8, the sba of which is 100%, and grade 9, the sba of which is 75% and the weighting of examination is 25%. the weighting also varies across various subjects. mathematics and home language carry the most weight as a learner has to obtain a minimum of 40% (level 3) in mathematics (dbe, 2011b) in order to be promoted to the fet band. although the assessment policy provides clear guidelines regarding the number of assessment tasks and forms of assessment to be used, it is silent on the quality of these tasks. the subject educator determines what and how to assess content, skills, and knowledge in mathematics. the quality of these assessment tasks therefore depends on how each individual grade 9 mathematics educator interprets them. however, the weighting and quality of the mathematics percentage or level of these assessments may paint a misleading picture for the parents and learners as the percentage or level may not be a true reflection of mathematical knowledge, skills and understanding. in terms of the national policy pertaining to the programme and promotion requirements (dbe, 2011b), where the promotion and progression requirements of learners are stipulated, there is evidence of variation in interpretation and implementation. this policy stipulates that learners should achieve a minimum of level 3 (30% to 49%) in mathematics and a minimum of level 4 (50% to 59%) in home language in order to be promoted to the next grade. these levels are made up of the sba mark (40%) and the end of year examination mark (60%). this study reveals that the focus in schools is more on learners’ mathematics mark than on their home language mark (motsamai, 2017). according to the national policy pertaining to the programme and promotion requirements, learners who do not meet the minimum levels for promotion should be progressed to the next level on the condition that such learners have spent four years in the phase, which is known as ‘the age cohort’. progressions should only be approved by the circuit manager; however, the evidence presented in this study shows that, prior to the circuit manager progressing learners who did not meet the minimum requirements, the mathematics teachers had already inflated the learners’ scores. the recording of assessment scores is, in many cases, inflated. one participant acknowledged that all of the previous grade 8 learners, who at the time of the study were in grade 9, had not achieved between 30% and 49%. the participant further explained that the grade 8 mathematics scores were adjusted to a level 3 by the grade 8 mathematics teacher. this practice translates to non-adherence to the national policy pertaining to the programme and promotion requirements policy. some of the participating school principals admitted that they did not fully understand the national policy pertaining to the programme and promotion requirements; as a result, they had varying interpretations and implementations of the policy. this practice could give learners and their parents the false impression that the learners have met the minimum promotion requirements. variation of classroom practice teachers are given a greater responsibility in designing quality sba tasks. however, the guidelines on how to develop quality, reliable, credible and valid sba mathematics tasks are problematic as these guidelines are largely generic in nature with limited specification to mathematics. from their responses, it appeared that the participants were not adequately trained to develop quality sba tasks. coupled with inadequate academic qualifications, it stands to reason that the development of assessment tasks is well beyond their capabilities, especially in the absence of support. as guidelines are open to interpretation and implementation, evidence emerged from the data suggested that different teachers in different schools developed grade 9 mathematics sba tasks that varied in quality. according to caps, different forms of assessment in mathematics included test and assignment (term 1), and test, mid-year examinations and investigation (term 2). however, most teachers and hods admitted that they do not know the difference between the different forms of assessment in mathematics. teacher a said he finds it very challenging to develop an assignment and an investigation. as a result, there was no evidence in any of the schools that participated in the study of any assignment or investigation being performed. learners in the five schools were only assessed in one form of assessment, which took the form of a test. this evidence could suggest teachers’ lack of knowledge in devising alternative forms of assessment or lack of adequate in-service training and support to empower teachers to develop a repertoire of assessment skills. another finding that emerged from the study was the difficulty in interpreting and implementing bloom’s revised taxonomy of cognitive levels. this finding is in line with long et al. (2014) that bloom’s taxonomy of cognitive levels is problematic to interpret and implement. when analysing documents such as sba tasks that were collected, evidence that emerged was that most teachers could only test learners on level 1 and level 2 questions for the sba. teacher a, teacher b and teacher c added a few level 3 and level 4 questions from the past ana question papers. however, such questions were taken verbatim. varying explanations emerged from the responses of the participants. for instance, teacher a explained that he could not differentiate between level 3 and level 4 questions, while teacher b and c believed that adding questions from the past ana question papers would standardise their sba tasks. however, lack of expertise in the development of sba tasks was demonstrated by the low levels of cognitive demand and poor questioning. the caps does not provide clear sba task specifications. as such, the policy is restricted on the uniformity and weighting of the forms of assessment. monitoring and moderation monitoring and moderation are the two processes run concurrently to ensure quality assurance. monitoring has always been done by the hods in the form of class visits to ensure curriculum coverage, as well as ensuring that the assessment programme is unfolding according to plan. however, when probed into how monitoring is done and its frequency, all participants admitted that monitoring is not conducted at their schools. although there is a moratorium on class visits by the teacher unions, informal and impromptu class visits are conducted at school c. hods also cited the many roles they play and the many workloads they have as factors that inhibit the process of monitoring. this study revealed that monitoring is non-existent. moderation is one of the most important processes in ensuring the quality, credibility, reliability and the validity of assessment which result in improved learner performance. heads of departments should use a moderation protocol obtained from the provincial doe. evidence from the documents obtained and analysed suggest that hods appeared to be confused in using the monitoring and moderation tools. while some hods seem to use both the monitoring and moderation tools to moderate sba tasks, others use either the monitoring or the moderation tool for moderation. all hods claimed that they never received any training on moderation. they added that the northern cape doe district officials gave them the moderation protocol document without any training. hod b seems to be frustrated and confused with the origin of the moderation protocol, as she received the moderation protocol from a colleague, who also did not know its origin. the moderation reports of school d were not collected as the teacher who has been assigned to moderate mathematics sba tasks was not available at the time of data collection due to his studies. in addition, teacher d did not possess a copy of such a report. in school c, no moderation reports were collected as the process unfolds differently. one hod admitted not conducting pre-moderation in which the sba tasks are moderated before they are written. post-moderation is being conducted by means of the marking. in one case, the hod leads mathematics teachers through a marking process as she has a rich experience in marking and moderating ncs mathematics. in school a, b and e, the moderation protocol is being used as a checklist, to check spelling and grammatical errors only. when perusing some of the sba tasks, glaring errors were found such as content of grade 7 being covered and incorrect use of mathematical symbols. there are neither constructive comments nor follow-up on any verbal comments. hods’ lack of expertise and experience may have contributed to lack of guidance in terms of producing quality assessment tasks. moreover, the monitoring tool had been used as a substitute for the moderation protocol. hod a claimed that pre-moderation is done hurriedly, where teachers request his signature and the school’s stamp, without him going through the sba tasks. most hods agreed that sba tasks are submitted without a memorandum. there were inconsistencies in terms of post-moderation. teachers claimed that they select marked scripts to be moderated. in some instances, teachers confessed that due to exhaustion, pressure and large classes, they do not mark all learners’ sba tasks. the study found that moderation is not rigorous and is inconsistent. the study also revealed that learner performance is not a true reflection of their potential. evidence in the recording sheets shows that learner sba marks were tampered with and were inflated (motsamai, 2017, p. 165). learner performance in school-based assessments school-based assessments are developed and marked by subject teachers at school or classroom level. in almost all of the schools selected for the study, the learners had been performing relatively well in their sba as compared to external assessments. when probed regarding the reason for the higher performance of learners in the sba, the participants gave varied reasons. it would appear that teachers often explained questions to their learners during tests, which may have led the learners to the answers. the hod at school b said, ‘teachers are explaining questions … telling them what the question wants’. teacher a recounted having shared a similar experience: ‘teachers explain questions to the learners in class’ (motsamai, 2017, p. 126). the forms of classroom-based assessment associated with mathematics made it appear that the learners performed well. most of the forms of assessment, such as assignments, investigations and projects, were done under uncontrolled conditions and, in some cases, in groups. the principal at school e expressed his views in saying, ‘good performance because of group work like assignments, assistance and all the like’ (motsamai, 2017, p. 126). principal e further elaborated that, ‘with the help of the parents, because some of the work learners are doing at home and parents will assist and that it’s sometimes higher’ (motsamai, 2017, p. 126). it seems that the high learner performance was often due to a lack of curriculum coverage in the teaching and assessment of certain topics with which the teacher and learners may have felt comfortable and that one concept that learners proved to understand well was repeatedly being asked during tests. the hod at school b, for instance, stated that: the teachers are asking the same questions. you find that question 1 is the same as question 2 and is based only on one concept. a lot of marks come from one concept. (motsamai, 2017, p. 126) the teacher at school c alluded to the fact that learner performance at her school was high due to the fact her learners were familiar with her style of questioning. at school a, this familiar style of questioning was not the case as sba was handled differently. the teacher at school a reported that he got a lot of his test questions from past ana question papers and refused to explain questions to his learners. in school c, for instance, the school principal scrutinised all of the sba marks and compared them to examination marks. according to the principal, if there was a wide mark variation between the sba and an examination mark, the teacher was called in to explain how the wide variation had occurred (motsamai, 2017, p. 127). the principal further elaborated that the mark variation between the two sets of marks was usually 5% or less. as a result, although the learners’ sba performance was higher than their examination performance, this gap was kept to a minimum. this practice might be associated with the fact that a few of the staff members at the school, including the mathematics hod, were involved with the ncs marking processes and were therefore able to filter this knowledge down to other grades (motsamai, 2017, p. 127). there was an overall agreement by the participants that their learners performed better in sba because the standard of sba, as well as the quality of the questions, was much lower than that of external assessments. the participants hold the view that good performance in mathematics at school level is a result of the quality of sba questions that are lower than the external assessments (such as the anas). based on this evidence, the reliability, credibility, validity and quality are questioned. in 2014 in the northern cape, only 9.6% of the grade 9 mathematics learners achieved acceptable levels in the anas. in addition, in the john taolo gaetsiwe district, where this study was conducted, 9.3% of its grade 9 learners achieved acceptable levels, yet these were below the national benchmark of 10.4%. when asked about their learners’ performance, the participants admitted that their learners were not performing well in the anas (motsamai, 2017, p. 130). while percentage comparisons across the ana results are not recommended, patterns observed in the northern cape provide some indication that good performance in sba tasks can be misleading. the ana was introduced as a national measurement tool by the dbe in 2011 and 2012 for grades 1 to 6 and 9, respectively, as outlined in the education sector plan, action plan to 2014: towards the realisation of schooling 2025 (dbe, 2013). the main purpose of the ana is to enable a systemic evaluation of educational performance, through which learners’ skills and their achievement may be measured. these nationally standardised assessments measure the skills and knowledge that learners are expected to have acquired as a result of teaching and learning based on the mathematics and languages curriculum. it would appear that most of the participants shared the same findings as those of pournara (2015) in terms of the difficulty of ana question papers over the years. the hod at school c confirmed this, stating that ‘2013 ana question paper was a bad, bad one, but 2014, it was a little bit better’ (motsamai, 2017, p. 130). whereas the principal of school a complained that the 2014 mathematics ana ‘was a disaster. in english they are performing, but in maths… it was horrible’ (motsamai, 2017, p. 128). moreover, the teacher at school e expressed her views in saying, ‘ana 2014 was the easiest’ (motsamai, 2017, p. 128). the teacher at school a reported that only one learner passed the 2014 mathematics ana, which was corroborated by the hod at school a, who expressed his anger in saying, ‘no learner passed; 0.1% … round it off, it is 0%!’ (motsamai, 2017, p. 128). additionally, at school d, the teacher lamented the fact that ‘with ana, it was very, very bad … no one passed. it was 0%’ (motsamai, 2017, p. 128). the teachers in schools a and d only found out after the fact that learners’ performance in mathematics was dismally low. this was due to the fact that they were not teaching at their respective schools or had been on sick leave for a long duration, respectively. when asked about the reasons for the poor performance of learners, the participants offered varied reasons for the poor mathematical performance in the grade 9 anas. however, all of the participants were unanimous that the standard of the questions in the ana was too high. the hod at school c had strong feelings about the anas: ‘ana is too difficult. there’s a question that is, according to my knowledge, is not part of the syllabus … they are asking them about exponents of grade 11’ (motsamai, 2017, p. 129). the teacher at school b further added, ‘our learners are scared of any papers with the departmental logo’ (motsamai, 2017, p. 129). from the four schools selected, the general challenge when answering ana questions was the question of language. the participants found that learners who did not speak the same language as that in which they were being tested tended to have problems in interpreting the mathematics questions. the principal at school b stated, ‘there is nothing wrong with ana, it’s just that our learners cannot interpret the questions’ (motsamai, 2017, p. 129). the hod at school b expressed his frustration: it is the language problem … the standard of language is too high … learners do not understand the language. reading is a problem. with the word sums, out of 30 learners, at least two will get 30%. (motsamai, 2017, p. 129) differences in home language and language of the test seem to be exacerbated by the complexity of mathematical language that grade 9 learners have not mastered either. the teacher at school a gave this account: performance is lower … only two people passed mathematics in grade 8 last year. i have 174 grade 9 learners; it means 172 of them can’t do mathematics. they are in grade 9 because of departmental policy. there are only five level 7 learners in my class … there are a lot of learners in my class who cannot have the ability to do maths. (motsamai, 2017, p. 130) this study has revealed that, according to the responses from the interviews, poor curriculum coverage added to the poor performance of learners in ana. the ana is written during term 3 and, according to the participants, only covered term 1 and 2’s work instead of the required curriculum for terms 1, 2 and 3 (motsamai, 2017). in examining the ana paper, this claim seems to be true. teacher views on this issue further speak to time that is wasted on revising work and drilling learners to obtain higher scores without ensuring that learners understand the work. conclusion this study sought to analyse evidence of variation in the quality of sba from the perspective of principals, hods and teachers. this is an important topic as the management, monitoring, moderation, and implementation of sba filter down from the principal through to the teachers and, eventually, to the learners. this study was able, using a small case study sample, to confirm what has long been suspected in the education system: sba is not as effective as it could be. themes highlighted in the current study that point to possible sources of variation include lack of adherence to policy, variation in classroom practice and inconsistent monitoring and moderation practices with differences in learner performance when sba tasks are administered compared to national, external assessments. while the results of this study are not generalisable, they provide insight into this topic, and provide a starting point for further research on the matter. an analysis of the interviews and the document analysis revealed that most of the hods and principals lacked in-depth knowledge and understanding of their roles and functions in making sba reliable, credible and valid. this was not only due to a lack of capacity to perform such functions, but was also due to a lack of effective induction and training by the district and provincial offices. sba is supposed to be used as formative assessment, should be used throughout the year as assessment for learning, and should provide feedback to teachers to inform and guide their teaching. school-based assessment has been deeply problematic since teachers vary in how they construe mathematical concepts. findings from the current study confirm the views of stiggins (2004) that current assessment systems are harming learners due to a failure to balance the use of standardised tests and classroom tests. poliah (2010) posits that learners obtain high marks due to the quality of question papers at schools. teachers set papers that are not of the required standard, which pass through the hands of the hods, yet are not properly moderated. the absence of proper moderation is problematic in itself and could disadvantage further attempts to ensure valid and reliable assessment (maile, 2013). moreover, fleisch (2008) argues that many get mathematics teachers are uncertain of what is expected of them. any change in the curriculum and assessment policies would require intensive training to be made available to all of the stakeholders: school principals, hods and teachers. sufficient time for training and exposure to sba should be provided to all teachers. the feedback gathered from stakeholders such as teachers and hods should provide the relevant information to the ministry in terms of their attempt to decipher and make the necessary changes and modifications to the existing assessment policies and guidelines. according to talib, naim, ali and hassan (2014), the cascade model is not always the best model to be used as information withers and is lost during training. the cascade model proved to have failed to prepare district officials, school principals, hods and teachers for the complexity involved in the implementation of the assessment policy, particularly the sba component (dichaba & mokhele, as cited in talib et al., 2014). in a developing context, the main challenge in assessment is to find strategies that will be fair to all learners from diverse backgrounds and to provide quality, reliable, credible and valid results. findings from the current study clearly point to the fact that the effectiveness of sba depends on a variety of issues pertaining to teachers and learners. with constant curricular changes being made, it is imperative for sba to be evaluated from time to time. acknowledgements the authors acknowledge the national research foundation for providing bursary funding for this study to be undertaken. competing interests the authors declare that we have no financial or personal relationships that might have inappropriately influenced us in writing this article. authors’ contributions s.v.s. was responsible for the introduction, problem statement, literature review, discussion of results and conclusions. p.m. was responsible for the compilation and analysis of all data referred to in the article. references adler, j. 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(2005). designing sampling strategies for qualitative social research: with particular reference to the office for national statistics’ qualitative respondent register. survey methodology bulletin-office for national statistics, 56, 53. world bank. (2008). curricula, examinations, and assessment in secondary education in sub-saharan africa. world bank working paper no. 128. africa human development series. washington, dc: world bank. available from http://hdl.handle.net/10986/6372 abstract introduction theoretical model of teacher change the study: methodology, participants and procedures findings and discussion conclusion acknowledgements references about the author(s) piera biccard department of curriculum and instructional studies, college of education, university of south africa, pretoria, south africa citation biccard, p. (2019). the professional development of primary school mathematics teachers through a design based research methodology. pythagoras, 40(1), a515. https://doi.org/10.4102/pythagoras.v40i1.515 original research the professional development of primary school mathematics teachers through a design-based research methodology piera biccard received: 11 sept. 2019; accepted: 28 nov. 2019; published: 17 dec. 2019 copyright: © 2019. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract this article sets out a professional development programme for primary school mathematics teachers. clark and hollingsworth’s model of teacher change provided the theoretical framework necessary to understand teacher change. a design study allowed for increased programme flexibility and participator involvement. five volunteer primary school teachers teaching at south african state schools were involved in the programme for a period of one year and their pedagogy, use of mathematical content and context developed during the programme. twenty lessons were observed over the year-long period. an observation rubric that specifically focused on mathematical pedagogy, use of context and mathematical content scale guided the researcher to gauge global changing teacher practices. teacher growth was evident through their professional experimentation and changes in their personal domain. the design features emanating from the study are that teachers be given opportunities to experience reform tasks (e.g. model-eliciting tasks) in the role of learners themselves and teachers should be encouraged to use contextual problems to initiate concept development. more mathematical detail in lesson planning is also necessary. furthermore, teachers need appropriately designed resource materials to teach in new ways. it is recommended that professional development includes teachers engaging collaboratively in solving rich tasks. this study adds to the growing body of knowledge regarding teacher development programmes that focus on how teachers change their own classroom practices. keywords: mathematics teacher professional development; primary school mathematics teachers; in-service training; design-based research; model-eliciting tasks. introduction the need to develop mathematics teacher practices is known worldwide (borko, 2004; jaworski, 1998; koellner, seago, & jacobs, 2018). professional development (pd) environments in which teachers will ‘take up’ (adler, 2005) new paradigms and practices are necessary to make a difference in classrooms. often, teachers take part in pd workshops but are not able to integrate new ideas into their current practices (clark & hollingsworth, 2002; richardson & placier, 2001). cai et al. (2017, p. 234) explain that research findings should be of the correct ‘grain size’ for teachers to be able to use them in their own lesson planning. although brown (1992) reminds us that interventions should be able to transfer to everyday classrooms; interventions are only meaningful if they result in changes in mathematics classrooms (wilson, 1998). professional development programmes that have been successful include those that focus on: representations, explanation and communication (hill & ball, 2009; kilpatrick, swafford, & findell, 2001), engagement in a variety of activities (porritt & earley, 2010), learner centeredness (polly et al., 2015) and a strong connection to classroom contexts and practices (chigonga & mutodi, 2019). from a mathematical models and modelling perspective, researchers propose that teachers should have the same experiences that they want their learners to learn from (schorr & lesh, 2003). materials and tasks also affect learning in pd programmes (ferrini-mundy, burril, & schmidt, 2007; hill, 2004) while pd that focuses on reflective thinking (artzt, armour-thomas, curcio, & gurl, 2015) shows positive results. elements such as learner centeredness, engaging in a variety of activities and teacher reflection have always resonated with me as the researcher as important aspects of successful mathematics classrooms. merchie, tuytens, devos and vanderline (2018), in their evaluation of the effectiveness of pd initiatives, set out that extended engagement, collective learning and active participation are hallmarks of effective pd interventions. this study wanted to ascertain if these pd features could be translated into a successful pd programme for a local context, and furthermore if teachers changed their classroom practices. the teachers involved in this study were grade 5 and grade 6 teachers in south african primary schools. all teachers had at least three years of teaching experience and were teaching in well-resourced suburban schools. all schools and classes were mixed in terms of cultural background, ethnicity, gender and ability. the language of teaching and learning in all schools was english. however, for about one-third of the learners in the classes, english was not their home language. only one teacher specifically trained to be a mathematics teacher. the challenges and economic effects of poor-quality mathematics education in south africa are well documented (spaull, 2013) and development of mathematics teachers is considered a national priority. however, how to effectively change teacher actions and decisions at a classroom level is less apparent. schoenfeld (2011) set out that teachers make decisions based on three interlinking aspects: their resources (which includes various forms of knowledge), their orientations (which includes their beliefs) and their teaching goals. this pd programme set out to resource teachers by providing both knowledge, classroom materials and activities to support teacher professional experimentation and decision-making. the aim of the development programme was to facilitate teacher decision-making towards increased use of contextual problems at the outset of teaching a section of work, more robust mathematical discussions and increased learner activity during mathematics lessons. this study concurs with the ideas of chirinda and barmby (2017) that design-based pd interventions should allow teachers to construct knowledge through being actively engaged in the pd. this pd ran in three cycles, and included teachers being actively involved and not simply listening to presentations or lectures. in a review essay that critically examined mathematics teacher change, goos and geiger (2010, p. 505) found that a common thread in the literature on teacher change was the role of ‘productive tensions in creating opportunities for mathematics teacher change’. the study reported on here proposed that model-eliciting activities may provide productive tension since these types of open problems were new to the teachers involved. engaging collaboratively with this productive tension may catalyse professional experimentation and possible changes in teachers’ classroom practices. model-eliciting tasks were used as a ‘springboard’ for the pd sessions. modelling problems are open tasks where learners are presented with a real-world problem where the instruction explicitly requests that a model for thinking about the problem is generated (see figure 2 as an example). model eliciting tasks have been found to be beneficial both to learner mathematical competencies (maaß, 2006) and for teacher professional development (doerr & lesh, 2003). borromeo ferri and blum (2013) lamented that very few studies involving primary school teachers were evident in the literature. in south africa specifically, limited research exists on mathematical modelling in studies involving both learners or teachers, although the results are very positive (see biccard, 2013; durandt, 2018). teacher change should be conceptualised holistically through interventions, resources and change processes within the classrooms. however, teacher development programmes should empower teachers to determine their own growth and change paths (frid & sparrow, 2009). teachers need to be in the driving seat of how change will take place in their classrooms for this change to be meaningful to them. deficit type narratives (adler & sfard, 2017) are not particularly useful in change discourse; rather change should be seen as teachers understanding and appreciating the flexibility of their teaching knowledge and practices (wilson & cooney, 2002). in this study, teachers were not expected to reproduce any of the modelling tasks or pd activities in their classrooms; they were requested to use what was useful to them from the pd programme to transfer to their own teaching and classrooms. in this study, clark and hollingsworth’s (2002) adaptation of guskey’s (2002) alternative model is employed to understand global changes and growth in teachers’ classroom practices. where guskey advocates that changes in teacher actions precede changes to their belief systems, clark and hollingsworth provide a more authentic model for how teacher professional development can take place through an interconnected model. the model proposes the concurrent activation of four domains: the external domain, the domain of practice, the domain of consequence and the personal domain (see figure 1). the research question guiding this design study was: what changes are evident in primary school mathematics teachers’ classroom practices when participating in a design-based professional development programme? this leads to the sub-question: what are ‘model-eliciting activities’ and how do they exemplify a design-based pd? the idea around change and growth in teachers is complex. the word change may denote any change (positive or negative) while growth suggests improvement in teaching practices. in this study the term ‘change’ was used since i did not necessarily pre-empt a positive change in teachers due to the professional development intervention. in the next section, a model to understand change from teacher professional development is set out. figure 1: model of teacher change. theoretical model of teacher change clark (1998 in clark & hollingsworth, 2002) set out that guskey’s model could be more useful if it is seen as a cycle with multiple entry points rather than a linear process. guskey’s model proposes that teachers must first experience a change in their classroom practices before they will generate changed beliefs. philipp (2007) suggests that supporting teachers to jointly change their beliefs and actions may be more important than worrying about which comes first (beliefs or actions). clark and hollingsworth’s (2002) model (see figure 1) explains that teacher change and growth is a process of enactive and reflective relationships between four inter-related domains. the model provides a more holistic, flexible and integrated model for understanding teacher change through intervention studies. according to the change environment, the external domain initiates the change cycle by providing new information or stimulus for teachers. in the case of this study, a professional development programme that would introduce teachers to model-eliciting activities as a springboard for discussing mathematics teaching is the external stimulus. the external domain then promotes a change in the domain of practice through enactment of professional experimentation, that is, the teacher may try something different in the classroom. through professional experimentation, certain salient outcomes are evident (e.g. increased learner participation or enhanced learning); this is the domain of consequence (the classroom). these consequences of a teacher’s experimentation may result in changes to their personal domain (knowledge, beliefs, attitudes or orientations). clark and hollingsworth (2002) conceptualise the interconnectedness between the four domains as comprising two inter-related processes of enactment (solid arrows) and reflection (dashed arrows). reflecting on the domain of practice may change the personal domain while the enactment of the personal domain will result in changes to the domain of practice. clark and hollingsworth acknowledge the complexity of teacher learning and that teacher change can take place through ‘multiple growth pathways’ (p. 950). what may be deduced from their model is that the external domain (professional development for example) does not have an immediate effect on the domain of consequence. external sources of information or external stimulus catalyse changes in the domain of practice or the personal domain. through reflecting on these changes in the personal and practice domain, salient outcomes in the domain of consequence (the classroom) may be evident. the professional development programme designed in this study makes provision for both teacher enactment of change (through classroom observation) as well as reflection (on the various pd activities). this article focuses on the enactment of change in teachers’ own classrooms. a guiding principle in this study was to acknowledge that teachers learn in ways that they find most useful (clark & hollingsworth, 2002) and that teachers need to be in control of the decisions they make in a classroom, if teacher change is meant to be significant (richardson, 1990). what follows in the next section is an explanation of the methodology as well as setting out the procedures in more detail. the study: methodology, participants and procedures methodology the methodology for the study is an intervention study, which can be seen as a type of design study, which is a common type of design study (cobb, jackson, & dunlap, 2015). design-based research (dbr) typically takes place in three phases (bakker, 2004). a planning phase where the local instruction theory is formulated (gravemeijer & cobb, 2006). the second phase is the actual teaching experiment where designing, redesigning and ongoing analyses of the instructional activities take place (gravemeijer & cobb, 2006). the third phase, the retrospective analysis, sets out to contribute to a local instruction theory. although dbr is not common in professional development of mathematics teachers (cobb, confrey, disessa, lehrer & schauble, 2003; sztajn, wilson, edgington, myers, & dick, 2013), its flexibility is of value when dealing with teachers who already are in service and have a bank of experiences and beliefs regarding mathematics teaching. cobb and steffe’s describe dbr as (in cobb et al., 2003, p. 9; addition by author) ‘a sequence of teaching sessions with a small number of learners [teachers] aiming to create a small-scale version of a learning ecology so that it can be studied in depth and detail’. the aim of the pd programme was to present teachers with model-eliciting tasks and to use teachers’ first experiences with these tasks in the pd programme (both as participants in model-eliciting group work and by observing learners solving the same tasks) to springboard discussions on more learner-centered and more problem-centred approaches. participants and procedures five mathematics teachers volunteered to be part of the pd programme. the teachers indicated that they were interested in learning about new ways of teaching mathematics. these teachers were teaching grade 5 and grade 6 learners (10 to 12 years old). they were teaching at three schools that formed a convenience sample. the schools were suburban government primary schools that were well resourced. the classes were of mixed ability, cultural background and gender with around 33 to 35 learners in each class. the study ran for a period of one year, although actual contact with the teachers was for a period of nine months. the relevant stakeholders and gatekeepers gave their permission for the study to take place. the pd ran in three cycles (see figure 3) with classroom observation visits taking place before the sessions of researcher-teacher intervention commenced. during session 1 of each cycle, teachers would take positions as learners themselves where they solved a mathematical modelling problem as a group. an example of a modelling task (figure 2) given is the one used for the third task in the study. figure 2: an example of a modelling task. figure 3: design-based research cycles. during session 2 of each cycle, the teachers would observe small groups of learners solve the same problem. the teachers were asked to become critical reflectors during this session. during session 3 of each cycle, i considered the discussions in session 1 and session 2 and provided support and scaffolding that the teachers indicated they required. pd through dbr requires both engaging teachers in one setting (the pd setting) while trying to reorganise their practices in another setting (the classroom) (cobb, zhao, & dean, 2009). session 3 of each cycle was adapted and designed as the programme unfolded and as i gained a better understanding of the dynamics of the pd. observing and recording holistic changes in a mathematics teacher’s lessons is not a simple task. for this purpose, the rubric created by fosnot, dolk, zolkhower, hersch and seignoret (2006) was used during classroom observations since classroom observations can be used to gauge general teacher quality (merchie et al., 2018). it includes a pedagogy scale, use of context scale and mathematical content scale. the design concept for this study is congruent with that of fosnot et al. (2006). their framework is set within the theoretical work of realistic mathematics education where ‘mathematics is a constructive, cognitive activity of making meaning in a social world’. in order for teachers to teach mathematics in this way, we engaged in-service teachers in experiences that involved action, reflection, and conversation within the context of learning/teaching. we took the perspective that teachers need to construct new gestalts, new visions of mathematics teaching and learning. to do this they need to be learners in an environment where mathematics is taught as mathematizing, where learning is seen as constructing. (p. 7) for validity purposes, an existing observation framework was used to understand the observed lessons and to document the teacher’s development through the programme. the rubric is too extensive to repeat here, so a summary is presented in figure 4. the coding protocol for this study is included in brackets. figure 4: classroom observation rubric. these formulations allowed me to consider teacher development across three interrelated domains that could capture some elements of teacher change in a mathematics classroom. the rubric is detailed enough to ensure consistency in its interpretation and use. the rubric focuses on teacher actions within the classroom and the repercussions for learning through these actions. this pd followed a similar approach to that of cobb et al. (2009) in that i did not work directly with teachers in their classrooms, but by presenting collaborative professional development sessions the mobility of the innovation and how it can be supported was the primary focus. there was also no expectation for the teachers to implement model-eliciting tasks in their classrooms, but to make changes that they felt would improve the teaching and learning in their own classrooms. the study reported on here proposed that model-eliciting activities (as the external stimulus) would catalyse professional experimentation and possible growth in teachers’ personal domains. teachers were asked to teach lessons that would meet the requirements of the curriculum and content that they needed to teach for that day. the research was not to interfere with their normal teaching schedule. ethical considerations the research was granted ethical clearance by the overseeing university (reference no. desc_biccard2012). permission to conduct the research was also given by the overseeing provincial department of education. principals of the schools also granted permission for the study to take place. participants took part voluntarily in the study and were assured confidentiality. participants signed informed consent documents and were allowed to withdraw from the research at any point. findings and discussion the findings of some of the other areas of this study have been partially disseminated (see biccard, 2013, 2018; biccard & wessels, 2015, 2017) which is consistent with design research in that ‘within a larger study, several sub-studies often take place’ (mckenney & reeves, 2012, p. 15). the findings presented here set out the global development in the classroom practices of the five volunteer teachers. twenty lessons were observed through the programme and the development of the teachers gauged using the rubric in each case by the same researcher. a brief summary of the lessons is presented (see table 1) as well as the overall rubric score (p1, p2, p3 for the three ratings on the pedagogy scale; c1, c2, c3 for the three ratings on the use of context scale and m1, m2, m3 for the three ratings on the mathematical content scale). these results reflect the enactment of the pd on the teachers’ domain of practice through professional experimentation. table 1: summary of baseline lessons. teaching by telling was observed as the dominant pedagogy during the baseline lessons. the learners were mostly seated and silent during these lessons. only one lesson involved pupils working in groups to calculate averages followed by a whole class discussion. however, this was to present answers and not to facilitate learner constructions. this lesson was identified as showing ‘signs of change’ (p2). during the weeks that followed the baseline observations, teachers were involved in cycle 1 of the pd programme (see table 1). the design principle of teacher-as-learner and moving problems to the beginning of a lesson were the focus of this cycle. during session 2, the teachers worked as a group to solve the airplane modelling problem. their initial responses revolved around how the problem was unsuitable to grade 6 learners because no methods were given nor suggested and that the question was too open. however, when they observed how groups of grade 6 learners approached and solved the problem, they were surprised that learners (who were learners at one of the schools) were able to organise the problem and model an answer. during the final session for this cycle, i provided the teachers with printed cards that had either a contextual (word) problem or a decontextualised problem. teachers had to ‘match’ the contextual problem with its more traditional partner. we then discussed when and how giving learners a contextual problem at the outset (before showing methods) could assist with conceptual development. an example of the type of matching problem is provided in figure 5. figure 5: example of card matching activity. teachers discussed how two different types of questions on the same concepts could either alienate students or be more inclusive: teacher e: these [referring to number-only problems] are all memory, they must remember what to divide and when. researcher: and if i don’t remember? teacher a: then it’s over. teacher c: but that [problem-centred questions] they can still figure out – it’s words. teacher a: they don’t have to remember that you have to do that, and then that… teacher e: you see, i am still telling them…i have to go to where they work it out for themselves…that is where i am struggling. (biccard, 2013, p. 191) what is evident here is that teacher change must be approached holistically to include teachers’ personal domain since this is a strong enabler of changing a teachers’ domain of practice to develop improved classroom outcomes. i visited the teachers at the beginning of the next school term; table 2 reflects a summary of the lessons observed. table 2: summary of cycle 2 lessons. during the follow-up lessons in cycle 2, two lessons demonstrated ‘signs of change’ where teachers allowed ideas from learners to guide the discussion while the question and answer sessions were more in-depth and focused on connecting the mathematical ideas. two lessons facilitated learners’ own constructions, which presents some evidence of professional experimentation in the domain of practice. these lessons saw learners working in groups trying to construct meaning and not working individually and silently as in the first series of lessons. in the one lesson, groups had to use blocks to explain to the rest of the class what of 18 ‘looked like’ while the second lesson involved learners working in pairs to calculate equivalent fractions of area models. the inverse operations lesson was still traditional in nature. two lessons were still dominated by teachers specifying methods while three lessons had learners think about methods. the time allocated for student interaction (teacher a and teacher c) in the lessons allowed for students to think reflectively about their working with the manipulatives. the first session of pd that followed these lessons had teachers solve the second modelling task in groups. some teachers struggled with calculating the ratio for the task (e.g. how to increase a 4 cm length to 7 cm) and once again felt that learners would need more guidance. teachers also worked through a variety of other proportional reasoning problems in a variety of contexts. the shift from additive reasoning to multiplicative reasoning was the main discussion of this session. the summary of proportional research (van de walle et al., 2010, p. 350) was also presented to them. during the second session, the teachers observed groups of learners solving the same task. the learners were able to use a variety of techniques with varying degrees of success. teachers observed both additive and multiplicative reasoning. the teachers did, however, note that the task allowed learner thinking to be more visible. during the lesson observation, i also noted that teachers’ lesson planning was not very detailed. the lesson plans were mostly populated with dates and topics but no further elaboration of concept development. i, therefore, presented a session on thinking about the mathematical goal of the lesson and hypothetical learning trajectories as an important aspect of lesson thinking and lesson planning. i wanted teachers to see that the mathematical goal of the lesson not as the lesson title, for example adding fractions, but rather as what mathematics learners would need to learn, for example writing fractions in equivalent forms. i visited the teachers again and the lessons are summarised in table 3. table 3: summary of cycle 3 lessons. the third cycle lesson observation showed a move away from teachers’ teaching by telling toward a greater focus on asking learners to work collaboratively. lessons now included questions that stimulated learner thinking rather than seeking specific answers. the external domain appeared to have activated a change cycle in the other three domains (domain of practice, domain of consequence and the personal domain). three lessons during cycle 3 showed ‘signs of change’ while two allowed for learners’ own constructions to guide the lesson. in the three that showed signs of change, learners worked in groups and discussed their ideas regarding multiplication and division word problems, factor trees and 3d shapes that they identified around the school grounds. in the two lessons coded as p3, one lesson saw learners cutting paper pizzas and chocolates to share between numbers of learners and in the other lesson, learners had to construct their own 3d prism before moving onto formal terminology, diagrams and concepts. learners were spending more time working with each other and verifying their work with each other. teacher b’s and teacher e’s lessons included questions that stimulated student thinking rather asking for specific answers. lessons in this cycle also reflected many more mathematics moments due to the increase in student activity. the professional experimentation results in a change to the domain of consequence (outcomes) and changes to the personal domain. it supports guskey’s (1986, 2002) principle that teachers have to experience learner success in their classrooms to change their beliefs about teaching. although teachers were asking learners to present their ideas, no explicit connections between the ideas was made by the teachers. the final sessions of pd once again started with the teachers solving a modelling task (see figure 2). teachers found this problem challenging and were starting to talk about the various ways in which the learners would approach it. they expressed concern that learners would not manage to work with so much data. although the groups of learners did produce models for the problem, they were not able to work with the busy, steady and slow aspects of the data but rather aggregated the data. teachers indicated throughout the pd that they needed assistance in finding contextual problems that covered the curriculum topics at the correct level. this is consistent with borromeo ferri and blum’s (2013) quantitative findings on the barriers teachers cited to implementing model-eliciting tasks. we discussed how teachers could use the resources that they do have (e.g. textbooks) to facilitate mathematical learning where learners not only follow set methods but are allowed to first tackle problems and think about them. we discussed using textbook problems at different stages of the lesson. teachers looked at different examples of typical textbook problems and discussed if they should always be used to conclude the lesson. they shared examples of how, at times, it may be beneficial to use the textbook problems at the beginning of the lesson or as pair work during the main part of the lesson and to use a whole class discussion afterwards. during this final session, i provided urls to open online resources that teachers could refer to as well as book titles (e.g. mathematics in context or the shell centre for mathematics) that teachers could use. at the end of this session, i asked teachers to reflect on their own teaching. their responses show that they are starting to consider their roles as facilitators more critically. they stated the following when asked which aspects of their teaching they still wanted to improve. these responses indicate changes in their personal domain. to have the patience not to give groups who are struggling the answer, but to guide them patience with their methods/ideas – not to tell them how to work out the answer. talk less and listen more to facilitate more and control less. (biccard, 2013, p. 257) i visited the teachers at the end of the term and the summary of lessons is presented in table 4. these observations took place either just before or just after the schools’ mid-year exams. teachers were revising or catching up work, before the winter vacation. table 4: summary of final lessons. in the final lessons, two showed signs of change while three focused on learners’ own constructions. the signs-of-change lessons included pair work on word problems with a follow-up whole class discussion while the second lesson involved groups of learners working collaboratively on ordering decimal numbers. for the three learner construction lessons, one involved a model-eliciting problem and two involved learners’ own constructions on profit models and learners constructing their own nets for 3d shapes. teacher a and teacher d appeared to have folded back to more traditional teaching for this lesson. summary the baseline lessons and cycle 2 lessons were predominantly calculation-based lessons. during cycle 3 lesson observations, three teachers used some other contexts as starting points (e.g. word problems) while two teachers presented truly problematic situations where no known procedures could simply be applied to the problems (sharing pizzas and constructing 3d objects). in the final lesson observations (almost a year after the start of the programme) three teachers presented truly problematic situations but only two of these teachers guided the classroom discussion to connect learner understandings. it was during this cycle that one of the teachers presented the model-eliciting problem based on finding the winner of an athletics event. the teacher noticed that this problem (presented before moving onto typical textbook problems) assisted her learners in understanding concepts related to ordering decimal numbers. this teacher tried a mini-experiment in her classes (professional experimentation). she had one of her classes work on the textbook problem first and then complete the model-eliciting problem. she found that this class struggled with the textbook problem while those that did the model-eliciting problem first did not. teachers’ initial lessons were typical mathematics lessons in terms of focusing on bare numbers, procedures or skills. during cycle 2 observations, some of the teachers started exploring mathematical moments in their classes by allowing learners to enter the mathematical discussion through self-directed activities and group work. by cycle 3 observations, two teachers had their classes explore mathematical ideas through their own constructions (e.g. pizza sharing lesson). by exposing mathematical ideas through contexts, teachers in this study were able to capitalise on the ‘mathematics moments’ in their classrooms. of the 20 lessons observed, only 4 reached the third level on the pedagogy scale. changing teacher practices is a complex endeavour and the time needed for paradigm shifts in mathematics teaching cannot be underestimated (guskey, 2002; wilson & cooney, 2002). teacher change is also not a linear process of teachers moving from point a to point b to point c but moving around between the various stages of change or development based on their knowledge, context and goals at the time. teachers take up certain aspects from the pd to try out in their classrooms when they feel professional experimentation is appropriate. the lesson observation rubric focused on three big areas (pedagogy, context and content) where the potential for mathematisation can be gauged in a classroom. the three elements of the rubric are interrelated in terms of classroom practice. ‘teaching by telling’ was often associated with bare numbers emphasising procedural mathematics (with the teacher sticking to a predesigned script) while a pedagogy of facilitating learner constructions necessitated the use of truly problematic situations and a focus on the underlying structure of the problem. in the latter lessons, teachers were using learner involvement to a greater extent, although not always making explicit connections between learner ideas. teachers’ evolving pedagogy involved using more contexts and more focus on learners’ own constructions during the lessons. this is consistent with the findings of cobb et al. (2009). the teachers also increased the physical and visual material used in the latter lessons. however, this in itself does not signify change if the materials are used superficially. there is a danger that teachers may use added materials because they think they are expected to in terms of their participation in pd. conclusion teachers are the most ‘critical layer of the school system in terms of efforts to change what happens in schools’ (smith & southerland, 2007, p. 397). however, the notion of teacher change is a complex concept that makes professional development of mathematics teachers more a design process than an implementation process. the formulation of design principles in this study (teacher as learner, bringing contextual problems forward, more detailed lesson planning, using a variety of representations and resourcing teachers) may allow others to reflect and select or design their own principles in their own settings. this study sought to contribute to the micro theory level by proposing activities such as modelling tasks for teacher professional development and did not necessarily seek replication of tasks in the classroom. in an attempt to move away from deficit-type narratives regarding teachers, the dbr approach in this study did not prescribe which practices teachers should take up since there are ‘many different ways for teachers to create powerful learning environments’ (schoenfeld & the tru project, 2016) in their classrooms. rather richardson’s (1990, p. 14) suggestion that meaningful change will take place if teachers have control over what they ‘adapt, adopt or ignore’ from the pd was used for this study. the modelling activities became springboards for discussion, and may have created avenues for conceptual change in teachers. the two-tiered process (working on a model-eliciting problem themselves followed by observing learners solve the same problem) appeared to assist teachers in making some changes in their own classes. the modelling problems may have supported the change environment since professional experimentation led to some personal domain changes. the processes of enactment and reflection facilitate changes in one domain being translated to a change in another domain (clark & hollingsworth, 2002). this study was limited to five teachers in one region of south africa, teaching grade 5 and grade 6 mathematics that is congruent with one of the known limitations of dbr – designing to scale (cobb et al., 2015). further research is needed in how to implement dbr at scale. the study did not focus on teachers implementing specific model-eliciting problems in their classrooms. rather, it sought to work with teachers in terms of where they were and how they chose to develop their own pedagogies. teachers appeared to have knowledge needs (what is modelling, why present problems at the outset, etc.) and orientation needs (how do learners learn mathematics in meaningful ways?). teacher change in this study was modest and since this is also not a longitudinal study, it is not known if teachers maintained some of the changes evidenced during the pd intervention time period. since ‘it is what teachers think, what teachers believe, and what teachers do at the level of the classroom that ultimately shapes the kind of learning that young people get’ (hargreaves, 1994, p. ix), there is a need for ongoing research in mathematics teacher professional development that focuses on how teachers can develop their own pedagogy, use of context and mathematical content in their own classrooms. acknowledgements competing interests the author declares that she has no conflicting interests in producing or publishing this article. authors’ contributions i declare that i am the sole author of this article. funding information this research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors. data availability statement data sharing is not applicable to this article as no new data were created or analysed in this study. disclaimer the views and opinions expressed in this article are those of the author and do not necessarily reflect the official policy or position of any affiliated agency of the author. references adler, j. 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(2002). mathematics teacher change and development: the role of beliefs. in g.c. leder, e. pehkonen, & g. torner (eds.). beliefs: a hidden variable in mathematics education (pp. 127–147). dordrecht: kluwer. webb pythagoras 60, december, 2004, pp. 13-19 13 eastern cape teachers’ beliefs of the nature of mathematics: implications for the introduction of in-service mathematical literacy programmes for teachers lyn webb and paul webb department of science, mathematics and technology education, university of port elizabeth email: lyn.webb@upe.ac.za; paul.webb@upe.ac.za various studies have shown that what teachers consider to be optimal ways of teaching mathematics is influenced by their beliefs about the nature of mathematics, and that it is advantageous to determine teachers’ conceptions of the nature of mathematics before developing curriculum interventions. with the imminent introduction of mathematical literacy in the fet phase in south africa this study provides a snapshot of beliefs of teachers in the eastern cape, south africa. various methods were employed to stimulate teachers to both reflect on their beliefs and to make them explicit. a questionnaire was administered to 339 in-service teachers in urban and rural areas of the eastern cape. a sample of ninetyfive of these teachers completed a second questionnaire based on videotapes of lessons recorded during the timss (1995) study that they had viewed. these teachers also ranked their own teaching on a continuum ranging from traditional to constructivist approaches and provided explanations for their ranking. a further sub-sample of thirty-six teachers participated in individual interviews, which explored their perceptions of the nature of mathematics and their own teaching practice. in order to investigate whether these beliefs were mirrored in practice, four teachers were observed and videotaped in their classrooms. the data generated by this study suggest that the participating teachers’ espoused beliefs of the nature of mathematics tended to be innovative, and correlated with innovative views of teaching and learning; however these views were often not reflected in their practice. the implications that the apparent inability of teachers to translate their beliefs into practice have for the introduction of a contextual, problem-based mathematical literacy curriculum for teachers is explored. introduction in response to the proposed introduction of mathematical literacy in south african schools in 2006, the eastern cape department of education (ecde) has been proactive in proposing that higher education institutions (heis) in the eastern cape develop and deliver an advanced certificate in education (ace) in mathematical literacy. a consortium of seven heis has consulted and collaborated and the first cohort of teachers (identified by the ecde) has been registered at each of the institutions. the teaching style for mathematical literacy, as expressed in the national curriculum statement of the department of education (doe, 2003), is through contextual problem solving. this represents a change from traditional instructional practices employed in eastern cape classrooms, especially at the further education and training (fet) level for grades 10, 11 and 12, where outcomes based education has not yet been introduced. recent research publications, e.g., chapman (2002), hart (2002), llinares (2002), lloyd (2002), philippou and christou (2002), and wilson and cooney (2002), draw strong links between teachers’ beliefs and teachers’ practice and suggest that it is advantageous to determine teachers’ conceptions of the nature of mathematics before developing curriculum interventions. this study investigated a sample of eastern cape teachers’ beliefs about the nature of mathematics; how such beliefs were linked to these teachers’ beliefs about teaching and learning; and whether espoused beliefs were mirrored in classroom practice, in order to ascertain whether these attitudes and practices were in line with those required for the introduction of a contextual, problem solving curriculum for mathematical literacy in south africa. theoretical framework thompson (1992) maintains that teachers differ a great deal in their beliefs about both the nature of mathematics and how they view the teaching and learning of mathematics. teachers’ conceptions of the nature of mathematics range from seeing mathematics as an absolute, fixed body of knowledge to viewing the discipline as a fallible and expanding human invention. schoenfeld (1985) concludes that most students view eastern cape teachers’ beliefs of the nature of mathematics: implications for the introduction of in-service mathematical literacy programmes for teachers 14 mathematics as a body of knowledge to be memorised, despite the fact that teachers often emphasise the importance of understanding the subject. schoenfeld (1985) also notes that learners experience neither understanding nor a perception of utility of the subject in practice. these findings are supported by morar’s (2000) and stoker’s (2003) studies of teachers in south africa who, despite professing beliefs in a constructivist paradigm, used traditional approaches that led learners to see mathematics as a subject to be memorised. views of the nature of mathematics can be seen on a continuum, from an ‘absolutist’ viewpoint in which mathematical truth is unquestionable, certain and objective at one pole, to a ‘fallibilist’ viewpoint, in which mathematical knowledge can be seen as a social construction and is therefore fallible (as it can be revised and corrected), at the other pole. learning mathematics can also be represented on a continuum from mastery of skills to problem-solving, with the opposite poles of a continuum for teaching mathematics being represented by the notions of the teacher as instructor or the teacher as facilitator. (lerman, 1986). ernest (1989) maintains that there are three philosophies of mathematics, i.e., instrumentalist, platonist and problem-solving views of mathematics. in the instrumentalist view mathematics is seen as a set of unrelated but utilitarian rules and facts. it is an accumulation of facts, rules and skills that are to be used in the pursuance of some external end. the platonist view of mathematics is that of a static, but unified body of certain knowledge. mathematics is therefore discovered not created. the problem-solving view sees mathematics as a dynamic, continually expanding field of human creation, a cultural product, which is constantly being revised and constructed. ernest (1989) views mathematics as “a process of inquiry and coming to know, not a finished product, for its results are open to revision” (p. 250) and sees the above three philosophies as a hierarchy, with instrumentalism at the bottom and problem-solving at the top. he also links teachers’ views of the nature of mathematics with their models of teaching and learning and maintains that teachers’ conceptions of the nature of mathematics form their philosophy of teaching and learning mathematics, despite the fact that they may be unable to articulate their beliefs fully, as they are often implicitly held (ernest, 1989). ernest (1989) theorises that the teacher who views the nature of mathematics as a problemsolving activity should act as a facilitator in the classroom, regard learning as an active construction of understanding, and possibly even see learning as an autonomous problem-posing and problem-solving activity. however, in his research he admits that there was a discrepancy and discontinuity between teachers’ espoused beliefs and enacted practices, and suggests that the cause might be attributed to the negative effects of some social and educational contexts of teaching. this investigation was an attempt to explore the views of eastern cape teachers concerning the nature of mathematics and to ascertain whether there was any correlation between their espoused views of learning and how they teach mathematics in their classrooms. the possible effect of the often deprived social and physical contexts in which many teachers in the eastern cape operate, on their ability to transfer new and appropriate beliefs into practice, are viewed as challenges for the effective development of any meaningful professional development programme in mathematical literacy that aims at meeting the demands of the national curriculum statement (doe, 2003). sample the sample of teachers in this study was drawn from almost exclusively black schools in east london, king william’s town, kokstad, lusikisiki, port elizabeth, queenstown and umtata in the eastern cape. the participating teachers were registered for under-graduate (bed) and postgraduate (bed hons) programmes in science and mathematics education at the university of port elizabeth. all data were gathered during lecture contact sessions. method lerman (2002) cautions that a number of research studies start from the premise that espoused beliefs and teachers' practice are consistent, but that this is frequently not the case. wilson and cooney (2002) highlight the methodological weakness in presuming that questionnaires and interviews will reveal beliefs, or systems of beliefs, that determine a teacher’s actions in the classroom. they note that there appears to be a shift away from purely ‘penand-paper’ or quantitative techniques towards a wider acceptance of qualitative research designs. because of these caveats a mixed mode methodology was employed in this study. data were generated by means of a ‘traditional’ pencil and paper likert type questionnaire that was lyn webb and paul webb 15 administered to 339 teachers. the questionnaire was adapted from an amalgam of reliable tests (mcginnis, 1997; raymond, 1997; schoenfeld, 1985; sibaya & sibaya, 1998; warren & nisbet, 2000), but the vocabulary was simplified and amended as english was not the respondents’ mother tongue. although the data were viewed critically and sceptically with lerman’s caveat in mind, they were valuable in identifying positive correlations between the scores obtained in terms of teachers’ views of the nature of mathematics, and their views about teaching and learning in this subject. these quantitative data were then complemented and supplemented by qualitative findings. ninety-five of the participating teachers viewed the third international mathematics and science study (timss) videos of an american lesson and a japanese lesson and wrote critically comparative essays about the teaching practices that they had seen. they also wrote reflective essays on their own teaching practice and rated their teaching on a scale from 1 (highly traditional) to 10 (highly innovative). individual semi-structured interviews were conducted with thirty-six teachers and videotapes were made of four teachers in their classrooms to assess whether there was a link between their espoused beliefs and their enacted practices. the data sets were combined in an attempt to allow for reliable and valid inferences to be drawn. results questionnaire the initial questionnaire was designed in order to ascertain a teacher’s beliefs about the nature of mathematics (e.g. mathematics changes all the time, it is dynamic; mathematics is a set of rules, etc.); about learning mathematics (e.g. mathematics is a problem-solving activity; learning mathematics mostly involves memorising, etc.); and about teaching mathematics (e.g. mathematics teaching should be learner-centred; learners learn mathematics best when the teacher lectures to them, etc.). the resulting data were plotted on a continuum and graphed. the two ends of the continuum of the nature of mathematics were an absolutist stance (that mathematics is an abstract, unrelated collection of facts, rules and skills that is static, certain and unrelated to the real world) as opposed to a fallibilist stance (that mathematics is practical, connected, dynamic, creative and problem driven). as regards learning mathematics, the two ends of the continuum were deemed to be that learners learn individually and passively through memorisation as opposed to learners learning actively through discussion, co-operative learning and problem-solving. as regards teaching mathematics, teacher-centred lecturing of unconnected topics (where mastery of skills was emphasised) was balanced with co-operative, learner-centred methods (where understanding was emphasised). all three graphs were skewed towards innovative beliefs, something that could have been caused by the fact that the teachers had just recently been exposed to constructivist teaching philosophies during their courses at the university of port elizabeth (upe). however, there was no statistically significant difference between the data generated by those teachers starting their second year of study and those starting their first year of study. there was a statistically significant positive correlation of (0.60; p < .05) between participating teachers’ views on the nature of mathematics and their views on how mathematics is learned. their beliefs concerning the nature of mathematics and the teaching of mathematics also correlated positively (0.59; p < .05) as did their views on how mathematics should be taught and how it is learned (0.71; p < .05). these data suggest that the participating teachers who viewed the nature of mathematics through fallibilist eyes believed that learning takes place actively and co-operatively and that the subject should be taught in a learnercentred way. those who viewed the nature of mathematics as instrumentalist believed that learning takes place through mastering algorithms and skills and through teacher-dominated instruction. these correlations indicate that if institutions can guide teachers towards fallibilist beliefs this could result in teaching styles that promote problem-solving through real-life activities and constructivist learning as required in teaching the new mathematical literacy curriculum. rating of their own teaching practice ninety-five teachers rated their own teaching practice on a continuum from 1 (highly traditional) to 10 (highly innovative). the mean score was 5,49 with the minimum score being 3 and the maximum score being 8. there are a number of reasons for this rather neutral response. it may be that teachers realised that, despite their espoused beliefs, their own teaching methods were still traditional in approach. it may be that they did not fully understand the tenets of constructivist learning, or perhaps they were not sure what constructivist teaching might look like. eastern cape teachers’ beliefs of the nature of mathematics: implications for the introduction of in-service mathematical literacy programmes for teachers 16 classroom visits four teachers, whose questionnaire scores fell towards the fallibilist end of lerman (1986) and ernest’s (1989) continuum of understanding of the nature of mathematics, were observed in their classrooms. apart from their espoused fallibilist beliefs, these particular teachers were also chosen because their schools were situated in port elizabeth and were therefore easily accessible. what became immediately clear during the classroom observations was that although in the questionnaire the teachers had professed to fallibilist beliefs about the nature of mathematics, their classroom practice was not learner-centred nor did they act as classroom facilitators. on the contrary all seemed to teach in ‘instructor’ mode, where the emphasis was on skills mastery and only correct performance was acceptable. there was no observed attempt to use learners’ prior knowledge or to work from the concrete to the abstract. although learners sat in groups there was no group interaction or whole class discussion. questions posed required one-word answers that were chorused from the class. as such, there was little evidence that their espoused beliefs, as reflected in the questionnaire, were enacted in their classroom practices. interviews and responses to the timss video clips each teacher interviewed (n=36) stated that their views about the nature of mathematics, teaching and learning had changed over the period of their bed studies at upe as a result of being exposed to new ideas and concepts. prior to the commencement of their studies they had neither been challenged to describe mathematics, nor to reflect on their current practice. during the interviews, the participating teachers appeared to realise that there was a discontinuity between their own beliefs and classroom practices. in order to throw some light on the reasons for this mismatch they were asked why they felt this was the case. the reasons they expressed for not being able to implement effective teaching and learning strategies included deficient initial teacher training and the fact that they had no recourse to appropriate resources and on-going professional development to make their methods more successful. other reasons offered were lack of control in the classroom, moving out of their comfort zones, syllabus and time constraints and lack of confidence in their mathematics content knowledge and pedagogical content knowledge. however, the teachers’ (n=95) responses to the timss video clips suggested that there might be other reasons contributing to the discontinuity between beliefs and practice. from the analysis of the data generated by the respondents’ critical comparison of the timss teacher practices it appears that they recognised that their own teaching practice was similar to that presented in the ‘american model’. they were able to reflect on the shortcomings evident in both the american and the japanese models, but were not able to discuss meaningfully in their essays methods that would have been more appropriate, and probably more successful, than the ones shown in the videos. to a large degree the participating teachers were initially convinced that the american teacher was using more constructivist methods than the japanese teacher. they commented in their essays that the most obvious indicator of constructivist practice that they observed was the fact that the american teacher’s students were seated in groups and she moved among them giving individual attention. this was in contrast to their observations in their essays that the japanese teacher stood in front of straight rows of students for the majority of the lesson. this suggests that the teachers in this study judged constructivist teaching and learning mainly by means of physical trappings. their knowledge of constructivism appeared to be based on the forms, and not the practices, of the theory. however, through later discussion in lectures they were able to unpack the concept of problemsolving and active learning – with the teacher as a facilitator and not an instructor of algorithms. on reflection, the participating teachers were able to recognise and discuss in groups during a lecture that the american example relied on skills whereas the japanese lesson emphasised understanding a single problem in depth. the fact that many participating teachers were initially unable to distinguish between the exterior manifestations of constructivism (e.g. moving around the class and having individual teacherlearner dialogue) and were not able to recognise a contextual, problem-solving approach that draws on prior knowledge, as was demonstrated in the japanese lesson, possibly throws some light on why discontinuities exist between teachers’ classroom practices and their espoused belief in constructivist approaches to teaching and learning. this has important implications for the design and implementation of any ace mathematical literacy programme for teachers that requires a problem-solving and contextual approach. lyn webb and paul webb 17 discussion the teachers in this study consistently expressed innovative views about the nature of mathematics and how this subject should best be taught and learned. however their practices did not mirror their espoused beliefs. these data support stoker’s (2003) contention that teachers in the eastern cape, south africa, have learned the rhetoric but have not internalised the reality of constructivist approaches to teaching and learning. stoker’s (2003) conclusion is supported by brodie (2001), who has shown that south african teachers can change considerably regarding mathematical knowledge and knowledge of pedagogy, but that they have difficulty in changing their teaching practice towards methods of engaging learners in a learner-centred approach. lerman (1986) and ernest (1989) have also found this to be the case in other parts of the world and both mention the gap between espoused and enacted beliefs. hoyles (1992) maintains that inconsistencies between beliefs and practices are accentuated when teachers are faced with an innovation, which in this case could be the introduction of a contextual, problem-solving approach to teaching a new subject. she introduces the idea of ‘situated beliefs’, where situations produce beliefs through activities that are dependent on context and culture. this concurs with ernest’s (1989) view that espoused beliefs pass through a filter of social constraints before they appear as enacted practices. ensor (1998) supports this notion, purporting that beliefs are not stable across contexts, and that differences in social situations result in multiple positioning of teachers. she suggests that the repertoires of knowledge and skills that one acquires could be called ‘beliefs’, which in turn are foregrounded and backgrounded according to the context in which the person is operating at the time. the notion of ‘situated beliefs’ (hoyles, 1992), coupled with the compelling arguments of chapman (2002), hart (2002), llinares (2002), lloyd (2002), philippou and christou (2002) and wilson and cooney (2002), who draw strong links between teachers’ beliefs and practices, suggest a two pronged approach for continuing professional development of mathematical literacy teachers. this involves identifying teachers’ beliefs and understandings as well as identifying the social contexts and constructs that prevent espoused beliefs from being reflected in classroom practice. ernest (1989) states that changes in the key beliefs of teachers concerning the nature of mathematics, and the nature of mathematics teaching and learning, are connected to reflection and autonomy on the part of the teachers themselves. this study has shown that the teachers who participated in this study in the eastern cape do harbour beliefs about the nature of mathematics that tend towards fallibilism, and that there is a correlation between their views on the nature of mathematics, and how it is best taught and learned. this could be as a result of their being encouraged to reflect on their teaching practices during their studies. however, there is a clear disparity between their espoused beliefs and the enactment of these beliefs. as noted above, this is not unique to the eastern cape, as research has shown that this disparity occurs internationally (ernest, 1989). furthermore, there seems to be a degree of consensus among researchers that the disparity occurs because of social constructs (ensor, 1998; ernest, 1989; hoyles, 1992; lerman, 1986). the lack of depth of understanding of constructivist tenets by the teachers participating in this study, the possible constraints preventing teachers from enacting their beliefs, and their particular social and educational contexts in the eastern cape may be, or may not be, unique. however, what is clear is that the current paucity of data on these issues suggests that the contexts in which south african teachers find themselves, and how these contexts may possibly affect their ability to actualise their espoused beliefs, requires further research. conclusion this study provides a snapshot of eastern cape teachers’ beliefs. it is encouraging to note that their beliefs tend towards constructivist philosophies and progressive views on teaching and learning. the national curriculum statement (doe, 2003) defines mathematical literacy as a subject that will be taught through contextual problem-solving. if teachers hold fallibilist views of the nature of mathematics they will be open to new curriculum interventions that encompass a problem-solving approach as they believe that it optimises learning. interventions that aim at encouraging teachers to reflect on their beliefs and practices and to become autonomous agents of change, like the ace for mathematical literacy that is being developed in the eastern cape, are of vital importance in the changing south african educational landscape. however, if this type of intervention is to be successful, the educational and social constructs that prevent espoused beliefs from being reflected in classroom practice need to be identified by further research and addressed in eastern cape teachers’ beliefs of the nature of mathematics: implications for the introduction of in-service mathematical literacy programmes for teachers 18 order to enable improved classroom practice. unless this is achieved, it is possible that the new constructivist orientated mathematical literacy curriculum for fet learners will not meet the expectation of producing far reaching and transformative consequences in south african education. acknowledgements appreciative acknowledgement is made to the south african–netherlands research programme on alternatives in development (sanpad) for financial support for this study, which forms a subset of a larger sanpad research initiative on science and mathematics teachers’ perceptions of the nature of science. references brodie, k., 2001, “changing practices, changing knowledge: towards mathematics pedagogical content knowledge in south africa”, pythagoras 54, pp. 17-25 chapman, o., 2002, “belief structure and inservice high school mathematics teacher growth”, in leder, g., pehkonen, e., & törner, g., eds, beliefs: a hidden variable in mathematics education?, pp. 177-194, netherlands: kluwer academic publishers department of education, 2003, national curriculum statement grades 10-12 (general), mathematical literacy, pretoria: government printers ensor, p., 1998, “teachers’ beliefs and the ‘problem’ of the social”, pythagoras 46/47, pp. 3-7 ernest, p., 1989, “the impact of beliefs on the teaching of mathematics”, in ernest, p., ed., mathematics teaching: the state of the art, pp. 149-151, london: falmer press hart, l., 2002, “a four year follow-up study of teachers’ beliefs after participating in a teacher enhancement project”, in leder, g., pehkonen, e., & törner, g., eds, beliefs: a hidden variable in mathematics education?, pp. 161176, netherlands: kluwer academic publishers hoyles, c., 1992, “illuminations and reflections – teachers, methodologies and mathematics”, in proceedings of the sixteenth pme conference 3, pp. 263-286 lerman, s., 1986, alternative views of the nature of mathematics and their possible influence on the teaching of mathematics, phd dissertation: university of london lerman, s., 2002, “situating research on mathematics teachers’ beliefs and on change”, in leder, g., pehkonen, e., & törner, g., eds, beliefs: a hidden variable in mathematics education?, pp. 233-243, netherlands: kluwer academic publishers llinares, s., 2002, “participation and reification in learning to teach: the role of knowledge and beliefs”, in leder, g., pehkonen, e., & törner, g., eds, beliefs: a hidden variable in mathematics education?, pp. 195-210, netherlands: kluwer academic publishers lloyd, g., 2002, “mathematics teachers’ beliefs and experiences with innovative curriculum materials. the role of curriculum in teacher development”, in leder, g., pehkonen, e., & törner, g., eds, beliefs: a hidden variable in mathematics education?, pp. 149-160, netherlands: kluwer academic publishers mcginnis, j.r., 1997, “development of an instrument to measure teacher candidates’ attitudes and beliefs about the nature of and the teaching of mathematics and science”, paper presented at annual meeting of national association for research in science teaching, oakbrook morar, t., 2000, “evaluating the contribution of 2 key teachers to the systemic transformation of educational support and the professional development of their colleagues in the eastern cape province”, paper presented at annual saamse conference, port elizabeth philippou, g. & christou, c., 2002, “a study of the mathematics teaching efficacy beliefs of primary teachers”, in leder, g., pehkonen, e., & törner, g., eds, beliefs: a hidden variable in mathematics education?, pp. 211-232, netherlands: kluwer academic publishers raymond, a., 1997, “inconsistency between a beginning elementary school teacher’s mathematics beliefs and teaching practice”, journal for research in mathematics education 28(5), p 550 schoenfeld, a.h., 1985, “students’ beliefs about mathematics and their effects on mathematical performance: a questionnaire analysis”, paper presented at the annual meeting of the american educational research association, chicago sibaya, d. & sibaya, p., 1998, “black university students’ beliefs about the nature of mathematics”, journal of education and training 19(2), pp 1-11 stoker. j., 2003, an investigation of mathematics teachers’ beliefs and practices lyn webb and paul webb 19 following a professional development intervention based on constructivist principle, unpublished phd dissertation: curtin university of technology thompson, a.g., 1992, “teachers’ beliefs and conceptions: a synthesis of the research”, in grouws, d., ed., handbook of research on mathematics teaching and learning, new york: macmillan publishing company third international mathematics and science study, 1995, national center for education statistics. institute of education sciences, washington dc: department of education warren, e. & nisbet, s., 2000, factors in primary school teachers’ beliefs about mathematics and teaching and learning mathematics, paper presented at annual mergai conference, fremantle, western australia wilson, m., & cooney, t., 2002, “mathematics teacher change and development. the role of beliefs”, in leder, g., pehkonen, e., & törner, g., eds, beliefs: a hidden variable in mathematics education?, pp. 127-147, netherlands: kluwer academic publishers “that arithmetic is the basest of all mental activities is proved by the fact that it is the only one that can be accomplished by a machine.” schopenhauer (1788-1860) abstract introduction literature review theoretical framework methodology findings discussion implications conclusion acknowledgements references about the author(s) eunice k. moru department of mathematics and computer science, national university of lesotho, roma, lesotho makomosela qhobela department of science education, national university of lesotho, roma, lesotho citation moru, e.k., & qhobela, m. (2019). social science students’ concept images and concept definitions of anti-derivatives. pythagoras, 40(1), a484. https://doi.org/10.4102/pythagoras.v40i1.484 original research social science students’ concept images and concept definitions of anti-derivatives eunice k. moru, makomosela qhobela received: 07 apr. 2019; accepted: 21 aug. 2019; published: 13 nov. 2019 copyright: © 2019. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract the purpose of the reported study was to investigate the social science students’ concept images and concept definitions of anti-derivatives. data were collected through asking students to answer 10 questions related to anti-derivatives and also by interviewing them. the theory of concept image and concept definition was used for data analysis. the results of the study show that the students’ definitions of anti-derivatives were personal reconstructions of the formal definition. their concept images were coherent only to a certain extent as there were some conceptions of some ideas that were at variance with those of the mathematical community. these were more evident when students solved problems in the algebraic representation. some students did not know which integration or differentiation methods they should apply in solving the problems. the significance of such findings is to enable the mathematics educators to pay attention not only to the use of signs and symbols representing mathematical concepts but also to their semantics. keywords: anti-derivatives; concept definition; concept image; social science students; integration. introduction calculus is an important branch of mathematics in a number of social and natural science disciplines. in the social sciences anti-derivatives are required in tackling problems in marginal analysis and optimisation problems. in the natural sciences students need to understand anti-derivatives to deal with rates of change for concepts such as velocity, acceleration and rate of flow. however, students fail to understand some of the basic concepts of calculus (brijlall & ndlazi, 2019; maharaj, 2014; metaxas, 2007; grundmeier, hansen & sousa, 2006). for example, students have problems in relating fundamental concepts and their definitions with associated procedures. this disconnect suggests that students have difficulties understanding functional and operational relationships between the mathematical notations and the calculus concepts they represent. what is desirable is that mathematical procedures be understood conceptually, but this is not necessarily the case since many students continue to learn mathematical procedures by rote application of an algorithm (moru & qhobela, 2013). a general anti-derivative, which is sometimes referred to as an indefinite integral, is an essential concept in calculus because it serves as a basis for many real-life problems and when students continue further into calculus courses, they frequently encounter anti-derivatives more than derivatives (jones, 2013). like other calculus concepts the learning and understanding of anti-derivatives involves significant use of signs and symbols. but these symbols continue to lack meaning for students (ferrer, 2016; sengul & katranci, 2015). to accomplish proper understanding of this concept various researchers have tried different ways to improve its teaching and learning. tall and vinner (1981) have suggested investigating students’ concept images and concept definitions of mathematical concepts. as they suggest, understanding a mathematical concept involves forming the concept image of it. brijlall and ndlazi (2019) have suggested developing the genetic decomposition of a concept (that is suggesting ways of how the concept might be learnt) so that instructors may organise their teaching of the concept for learning to take such a route. the current study was therefore conducted with the purpose of documenting university students’ concept images and concept definitions of anti-derivatives, an important idea, applicable in the social sciences. this was with the intention to find ways to improve the teaching of the concept. we believe that one cannot remedy what they have not diagnosed. in the mathematical context the terms anti-derivative and indefinite integral may be used interchangeably; in this study the choice of using anti-derivative as opposed to indefinite integral is because everyday etymology suggests that the prefix ‘anti’ in the term ‘anti-derivatives’ implies the reverse process and product of derivatives. this interpretation poses problems in the mathematical context because our everyday meaning of ‘reverse’ means going back to where one started. thus this would show if students would use the everyday meaning or the scientific meaning where the reverse processes of integration and differentiation may be used depending on the task at hand and their products would be either integrals or derivatives. the other reason for this choice was because the term ‘derivative’ has a close relationship with the word ‘anti-derivative’ as it constitutes the part of it after the prefix ‘anti’. having highlighted some difficulties that students have in relation to the topic of study, studies related to students’ understanding of this important calculus concept have not yet been conducted in lesotho, where the current study took place. this could probably be because in lesotho students encounter calculus for the first time at university or other tertiary institutions, of which there are not many; hence, the idea is not very popular. the findings of this study will not only be useful to the national university of lesotho where the study was conducted, they will also be useful to researchers elsewhere in terms of either similarities or differences in findings and their implications. this study addressed the following research questions: research question 1 (rq1): what definitions of anti-derivatives do second-year social sciences students give? research question 2 (rq2): what connections do students make between a function and its anti-derivative in algebraic representation? research question 3 (rq3): how do students solve problems involving differentiation and integration? the foundation of this study involved the characteristics of a specific calculus concept, the anti-derivative. the theoretical framework by tall and vinner (1981) on concept image and concept definition was the main lens through which the students’ understanding of the concept was analysed. this theory was complemented by some parts of the literature review. literature review the foundation of this study involved characteristics and understanding of a very specific calculus concept. the review of the literature includes the type of difficulties that the students encounter in understanding the concept of anti-derivative together with possible causes for such. students’ difficulties in understanding anti-derivatives (or indefinite integral). in the studies of hall (2010), and pino-fan, gordillo, font, larios and castro (2017), students were explicitly asked to define an indefinite integral and an anti-derivative, whereas in the study by metaxas (2007) students were provided with a set of words that they could use to describe the indefinite integral. the words that were provided for the description were: object, process, tool and concept. hall (2010) found that everyday language affected the students’ understanding of the word ‘indefinite’ in indefinite integral with the everyday meaning being ‘less precise’, a meaning that is not correct in the mathematical context. when asked what an indefinite integral is, a student gave the following response: ‘an indefinite integral, i mean, it’s more … it’s more open ended, less precise, obviously’ (hall, 2010, p. 12). the student is aware that in finding an indefinite integral, the limits of integration are not used. the problem is that in this case instead of using the correct mathematical explanation, inappropriate terminology that is non-mathematical is used. in the study of pino-fan et al. (2017) some students described an anti-derivative as ‘the inverse of derivation’ and other described it as a procedure that allows one to find the ‘original function’ from which a derived function comes (p. 41). metaxas’s (2007) exploration of students’ classification of their understanding of indefinite integral was in the context in which students had been introduced to the definition of an indefinite integral as: ‘indefinite integral of a function f continuous in an interval [a,b], is defined to be the set of all the anti-derivatives of f in [a,b]’ (p. 266). the student who was interviewed said that an indefinite integral is a process and this was interpreted as a conception that shows procedural understanding. this is a reflection that in both cases students had their own personal reconstruction of what an indefinite integral is. the definition provided in the study of metaxas (2007) implies that one indefinite integral (a general anti-derivative) contains a number of particular anti-derivatives. students in the metaxas study were also asked if is equal to. during the interview one student said that the two indefinite integrals are not equal. the student’s interpretation seems to be in alignment with the idea of initial value problems where the constants of particular anti-derivatives of the two functions would differ by 1 (one). if written in the general form the answer would be given as . on the surface the two integrals seem to produce the same function but in essence the c stands for an infinite number of constants which may not necessarily be equal. in the study by kiat (2005), when students were asked to find some students gave the answer as instead of . thus, by omitting a constant of integration, a procedural error was committed. in our view, it could be that students were not aware that an integral (a general anti-derivative) is constituted by a set of particular anti-derivatives as the c stands for varying constants. thus, is one of the anti-derivatives with c = 0 and writing it down alone ignores the other anti-derivatives with different constants. on the other hand there is a possibility that the students just forgot to write the c especially when a procedure was just learned as a set of rules without meaning attached. giving incorrect answers due to calculation errors, failure to use relevant rules and to link derivatives to their anti-derivatives and inability to use appropriate integration techniques in solving indefinite integral problems by students were dominant in the studies by maharaj (2014), sengul and katranci (2015), ferrer (2016), brijlall and ndlazi (2019), ndlazi and brijlall (2019). in the study by maharaj students made some structural errors as they could not differentiate objects from processes from the way the integrals were represented. a process was defined as the ability to perform operations that will enable the student to find the integrals of functions while an object was taken as the visualisation of strings of processes as a totality and performing some written or mental actions on the internal structure of the given function to enable integration. in sengul and katrachi, the choice of strategy to use was a serious problem. ferrer asserts that learners’ difficulties in solving integrals were attributed to the inherent mathematical knowledge and skills acquired by the students from basic mathematics. in brijlall and ndlazi’s studies, students committed errors when using symbols of integration and basic differentiation rules. students also had problems with the syntax of symbols as in the case of maharaj. theoretical framework the theory of concept definition and concept image by tall and vinner (1981) and vinner (1991) was used for data analysis. this theory was complemented by parts of the literature that were directly related to the concept of study. these include: the formal definition of anti-derivative, its relationship with indefinite integral, and the relationship between differentiation and integration. concept image and concept definition tall and vinner (1981) describe the formal concept definition as the form of words used to specify the concept as accepted by the mathematical community. they suggest that this may be learnt by an individual in a rote fashion or more meaningfully learned and related to a greater or lesser degree to the concept as a whole. it may also be a personal reconstruction of the definition by the student which may differ from the formal concept definition. the term concept image is described as the total cognitive structure that is associated with the concept, which includes all the mental pictures and associated properties and processes (tall & vinner, 1981, p. 152). part of the concept image that is activated at a particular time is termed the evoked concept image. at different times conflicting images may be evoked. cognitive conflict or confusion occurs when conflicting aspects are evoked simultaneously. vinner (1991) contends that to know a concept definition by heart does not guarantee understanding of the concept. he says that to understand a concept means to have a concept image of it. the definitions of anti-derivative provided in some calculus textbooks include a list of connected conditions: ‘an anti-derivative of a function f is a function f such that f′(x) = f(x)’ (haeussler, paul & wood, 2008, p. 624); ‘a function f is called an anti-derivative of f on an interval i if f′(x) = f(x) for all x in i’ (stewart, 2009, p. 274); ‘a function f is an anti-derivative of f on an interval i if f′(x) = f(x) for all x in i’ (larson & edwards, 2012, p. 398). these authors go on further to show that any two anti-derivatives (particular) of a function differ only by a constant and that an indefinite integral is the most general anti-derivative. larson and edwards (2012, p. 399) state that ‘the term indefinite integral is a synonym for anti-derivative’ and to find a particular anti-derivative one has to solve the initial value problem. these definitions mean that if we have a function f, it qualifies as an anti-derivative of another function f if when differentiated it gives f as the result. in order to check if f is an anti-derivative of f, we simply have to differentiate f. if we get f, then we can conclude that f is the anti-derivative of f. this further suggests implicitly that if we integrate f we will get f, which is not necessarily the case. this implied mathematical meaning as a symbolic representation is , where c is the constant of integration (haeussler et al., 2008). therefore, if we integrate (integration operator ∫) a function f, then differentiate (differentiation operator ), the result will be what we started with. however, if we differentiate f and then integrate, we will not necessarily go back to exactly where we started because f(x) + c is an expression for a family of functions rather than the single function, f(x), that we started with. although the researchers agreed with all the given definitions, the students were exposed to the definition by haeussler et al. (2008); hence, that is the one that the sample of the study was expected to use. methodology research design this study involved two methods of data collection to document students’ understanding of anti-derivatives. questions on anti-derivatives and interviews to illuminate the students’ questions responses were used. these methods of data collection were important because they could both enable the researchers to get qualitative data that could answer the research questions by studying students’ responses to get direct answers or make inferences. the sample the participants in this study were 117 second-year university social science students whose major subjects were economics and statistics. the sample group was taught algebra and calculus in their first and second years by the first author. the calculus topics covered during their second year included: integrals, techniques of integration and the relationship between integration and differentiation with initial value problems. students were also aware that the indefinite integral was used as a general anti-derivative. the students had encountered problems in which the terms anti-derivative and integral (indefinite) had been used interchangeably in their calculus textbooks. they also had solved application problems such as: marginal cost, marginal revenue, marginal propensity to save, marginal propensity to consume, and initial value problems. students knew for example that the cost function is the anti-derivative of the marginal cost function obtained through the process of integration and to check this they had to differentiate to obtain the marginal cost function. the students had not solved problems in which the integration and differentiation operators were paired. they encountered this for the first time during the study. they were also not familiar with the integration of expressions that cannot be broken down into elementary functions. the learning took place through lectures, class discussions and tutorial sessions. the participants had used mainly the prescribed calculus textbook in the social sciences (haeussler et al., 2008), which was recommended on the basis of relevance to the course of study. the questions ten questions (see table 1) were constructed by both authors and validated by two mathematics educators from universities outside lesotho. the comments from these experts were discussed to reach agreement and the consensus suggestions were used to revise the questionnaire. table 1: the questions, their justification and research questions their data would answer. interviews interviews were used to ensure validity and provide richer elaborations of the questionnaire responses. this also provided a form of methodological triangulation (seale, 1999). the semi-structured protocol was constructed by the authors and external assessors were involved until an agreement was reached. the first author conducted the interviews to maximise the reliability of the responses since she had already established a rapport with the students as their lecturer. the interviews were audio-taped to preserve data in a ‘raw’ form in that it removes the selective effect of researchers (seale, 1999). the interviews were conducted with a stratified sub-sample of participants (n = 12). four interviewees were chosen from each of the first two popular categories of responses and two from each of the two least popular categories of responses based on the analysis of the questions (see table 2). interview questions were aligned with and supplemented each category of response from the questionnaires. for example, the first interview question was constructed by considering the four categories of responses (an integral, integration, reverse and original function). the same procedure was applied to questions 1, 2, 3, 4 and 9. the other interview questions were asked to find out if the students could successfully solve problems involving the procedures used in finding anti-derivatives and their relationship with their description of what an anti-derivative is. on average each interview lasted one hour. table 2: students’ definitions of anti-derivative. data analysis data was analysed using the theory of concept image and concept definition. the theory was complemented by some parts of the literature. for example, in analysing the students’ definition of anti-derivatives (see table 2) students’ answers were compared with the formal definition of the anti-derivative as accepted by the mathematical community which was already known to the authors beforehand (a priori knowledge). if the answers did not match the formal definition, they were taken to be students’ personal definitions. in forming the classifications of the personal definitions the authors looked for the common words that were used by the students. such words seemed to be related to the language that in most cases is used in the context of anti-derivatives. such words include: reverse, integration, integral and original function. these personal definitions, although they contained familiar terms within the context of anti-derivatives, were only classified after reading students’ responses (posteriori knowledge). the authors did not have any prior knowledge of how students would define anti-derivatives through their personal reconstructions of the formal definition they had encountered in class. in analysing data that relates a function to its anti-derivative, two possibilities were anticipated by the authors: (1) that a function has one anti-derivative or (2) that a function has many anti-derivatives. these two categories were anticipated because there is a possibility that students could take the general anti-derivative as one function if they are not aware that the c in the expression is an arbitrary constant. in analysing data for the third research question, students’ definitions of anti-derivatives and the way they related the processes of integration and differentiation were used as a major lens. the given students’ definitions were checked if they matched some of their explanations or the way they solved the given problems represented in algebraic form, thus checking students’ concept images (the total cognitive structure) of anti-derivative. comparison with results found by researchers elsewhere was also made. ethical considerations students were asked if they were willing to take part in the research by the first author who taught them. they were made aware that any information they give will be treated with great confidentiality. this is also evidenced by the fact that their names are not used in the research and they were aware of this. they were also told that they are free to withdraw from the research activity if they felt uncomfortable. but because of the good relationship that existed between the first author and the students, they showed great interest in taking part in the research. perhaps another factor that contributed to students’ willingness to participate is that they were told that the information that they give will also help their instructor to improve her teaching not only for their benefit but also for the classes that will follow. the permission to conduct the study was also obtained from the head of department of mathematics and computer science as the course is offered by the mentioned department. findings the presentation of results in this section is categorised into the assertions and evidence aligned with the research questions. the assertions are accompanied by supporting evidence composed of summaries of the questions’ responses and specific in-depth quotes from the student interviews from different performance groups and discussion. rq1 assertion: students provided four categories of definitions of anti-derivatives that are personal concept definitions from the 117 students’ responses about the definition of anti-derivative, 111 yielded four categories: an anti-derivative is the integral of a function (67 students – 60% of students including interviewees s41, s59, s96 and s100); an anti-derivative is the integration of a function (34 students – 31% of students including interviewees s12, s42, s48 and s101); an anti-derivative is the function that undoes (change, reverse) the derivative (7 students – 6% of students including interviewees s25 and s52); an anti-derivative is the function that when derived gives the original function (3 students – 3% of students including interviewees s66 and s67). one student defined an anti-derivative as a function, there were no responses from two students and three students’ responses were unclassified. table 2 illustrates the actual responses of some students that were coded into the four categories. the given definitions are students’ own personal reconstruction of the formal concept definition; similar results were obtained in the studies of metaxas (2007), hall (2010) and pino-fan et al. (2017) when students were asked to give the definition of an indefinite integral (a general anti-derivative). in the first category, when students say that an anti-derivative is an integral it shows that they are aware that the two concepts are related. integrals are either definite or indefinite and the latter is used synonymously with the term anti-derivative whereas the definite integral represents a number. the discussion with s41 from the first category went as follows. r: what do you mean when you say that an anti-derivative is an integral of a function? what is an integral? s41: the integral is the inverse of the derivative. r: can you show me what you mean by this. s41: she then writes [talking through the steps] [as she writes she says:] if f(x) is equal to 2x, then its derivative is 2 but the integration of 2 is going to be 2x. r: is it what you mean by inverse? s41: yes. s41 associates the term anti-derivative with two terms, integral and inverse (of the derivative). we believe that this emanates from the use of the prefix ‘anti’ and also from the fact that differentiation and integration are referred to as reverse processes. in her working she starts by talking about the derivative, then shows how one can get the indefinite integral from the derivative. this shows her concept image of inverse which matches her personal concept definition. her working shows that the process of integration was performed twice. although the relationship between 2x and 2 is shown and explained clearly, when integrating the integral 2x the c is left out and the second integration was not relevant as per the discussion. in the second category, s42 and s101 use the term integration while s12 refers to the sign, an integration operator, which has the same fundamental meaning as integration. these showed that the students’ concept images of anti-derivative had the process of integration as part of their cognitive structure. in the third category the words undoes and reverse are explicitly used in the definition. these words are commonly used when the two processes of integration and differentiation are discussed, whether in textbooks or in teaching. in the last category the students use the word derived for finding the derivative of a function. this category is aligned with the way one would check whether or not a function is an anti-derivative of another. if a function f is differentiated and the result is another function f, then f is the anti-derivative of f. in comparison with the findings of hall (2010), where students’ definitions of indefinite integrals were characterised by the use of everyday language, in this study that was not the case. in the current study students used words that are commonly associated with the description of indefinite integral, either explicitly in the explanations or implicitly in the formal definition when relating a function to its anti-derivative. thus the concept images of students in the reported study consist of mental structures that are more coherent in the mathematical sense than those displayed by students in the study of hall. the same argument of coherence of definitions in the mathematical sense could be made for the study of pino-fan et al. (2017) as the categories of the definitions of students in their study are similar to categories 3 and 4 of the reported study: inverse (taken to be synonymous with reverse) and original form. rq2 assertion a: students related a function to its anti-derivative with multiple interpretations the two questions that required students to explicitly relate a function to its anti-derivative are question 2 and question 7. question 2 read: ‘does a function have one anti-derivative? explain’ while question 7 read: ‘what is the relationship between the functions g(x) = xn and , if any?’ in responding to question 2 the expectation was that the students would say that a function has many anti-derivatives, as taught in class. they were further expected to say that the function , is a set of anti-derivatives of g(x) = xn. in responding to question 2, 79 students (69%), including s12, s25, s52, s59, s96 and s101, said that a function has many anti-derivatives. of these respondents, 51 were aware that a function has many anti-derivatives that differ by a constant while 22 students, including s12 and s96, said that a function will have many anti-derivatives depending on the order of differentiation of the function. six students did not explain their answers. another 32 students (28%), including interviewees, s42, s48, s66, s67 and s100, said that the function has only one anti-derivative, by which they meant a general anti-derivative. four students (3%) including interviewee s41 gave unique responses and two students did not respond. the response that a function has many anti-derivatives does not mean that all the students had the same interpretation, as interview excerpts from s12 and s25 illustrate. their questionnaire responses are first presented: s12: it depends on how many times it has been derived. if it has been derived twice, then it will be anti-derived twice. s25: no. it is because any anti-derivative has the constant c which represents any number so function has many anti-derivatives. s12’s interview: r: when asked if a function has only one anti-derivative, you say that it depends on how many times it has been derived. can you explain what you mean? s12: i mean if a function has been derived twice as in y′′, then you will integrate it twice to get to the original function y. r: can you explain what you mean. s12: [to demonstrate what she means, she wrote:] [as she wrote she said:] if y double prime is equal to 3x squared, then you have to integrate. the answer will be x cubed plus c which is y prime. to get y you will have to integrate and if you are given the value of c, y will be x to the power four over four. r: can you show me on what you have written which functions are the anti-derivatives of the others? s12: this integral is the derivative of y double prime. r: which one? s12: this (y prime) function is the anti-derivative of y double prime. r: so the anti-derivative of y double prime is y prime, and that of y prime is … s12: [interrupting] it is y. r: does this c stand for any particular number? s12: it stands for any particular number if we are not told how to find the c. r: can it be y′ = x3 + 2 or y′ = x3 – 3 or y′ = x3 – 1? s12: yes madam. r: so is x3 + c one anti-derivative? s12: yes it is one. s12’s concept image shows that a function has many anti-derivatives but these anti-derivatives are counted based on the number of times in which a function is integrated to its original form. the example she gives is that of a function which has two anti-derivatives because it has been integrated twice to get back to y. she accepts the existence of particular anti-derivatives but as she explains such anti-derivatives may exist only when one is provided with the conditions that will enable the calculation of c which she takes to represent one number at a time. her written work seems to indicate a problem in handling the c when integrating. she does not manage to write its integral as cx + k, where k is a constant. she seems to think that if you are given the value of c the other part disappears or becomes zero as in the case of differentiating a constant. thus she may confuse the processes of integration and differentiation, a situation which occurred in kiat’s (2005) study. s41 like s12 no longer refers to the anti-derivative as the integral sign, integration operator, but as the integral. she also relates an anti-derivative with the last category, back to the original form. thus the interviews further reveal that the categories of descriptions and anti-derivatives overlap. s25 was among the students who said that a function had many anti-derivatives but his reasoning was different from that of s12 as indicated by his interview: r: can you explain what you mean when you say that a function has many anti-derivatives? s25: taking x2 + c as an example, c can be anything. let’s say we have x2 + 3, x2 + 4 and x2 + k, any constant; my understanding is that x2 + 3 is different from x2 + 4. therefore, those anti-derivatives are different. r: if you were to differentiate each of these anti-derivatives what would you get? s25: they will be the same. r: what will they be? s25: they will be 2x. r: is this why you say a function has many anti-derivatives? s25: yes madam. the excerpt shows that s25 gave the interpretation of anti-derivative as taught. which also matches his personal concept definition of a reverse process. he takes an indefinite integral to be a set of anti-derivatives that differ by a constant as reflected by his questionnaire response. thus his concept image of anti-derivative was coherent. the multiple interpretations of anti-derivatives given by students, though different, were the same for each student. this means that there were no conflicts that were experienced between the individual students’ evoked concept images. rq2 assertion b: students related a function to its anti-derivative the same way this is the second assertion made in trying to answer the second research question on how students related the function to its anti-derivative. s42 was among the students who said that a function has only one anti-derivative, the general, consistently. s12 also consistently related a function to its anti-derivative by saying that it depends on the number of times the function had been differentiated. s42’s interview about her question 2 response illustrates this point. s12’s response to question 2 has already been discussed and reference will be made to it when discussing both of their responses to question 7. s42: yes. but it has different constants. (the only anti-derivative). s42’s interview: r: [referring to earlier discussion] can i say is the anti-derivative of f(x) = 3x? s42: no! r: if i have the integral , what can i do to this integral in order to get 3x? s42: we can differentiate. r: what about the derivative of , what is it? s42: it is 3x. r: is the anti-derivative of 3x? s42: no! the anti-derivative is . s42’s conception of anti-derivative is that of a general anti-derivative. she said that is the only anti-derivative of 3x because when we differentiate it we get 3x. this is a reflection that she accepts an anti-derivative to be of the form f(x) + c. this concept image matches her personal definition that an anti-derivative is an integration because when finding the integral the constant of integration is always included. she takes an integral and integration as the same; however, the first is the mathematical object whereas the second is the process. some responses to question 7 had one dimension while others had two dimensions. one-dimensional responses included: f(x) is the integral of h(x) (29 students including s100), f(x) is the anti-derivative of h(x) (24 students including s12, s52, and s41), f(x) is the integration of h(x) (7 subjects including s67), h(x) is the derivative of f(x) (1 subject, s48) and f(x) is the integrant of h(x) (3 subjects). two-dimensional responses included h(x) is the derivative of f(x) and f(x) is the integral of h(x) (17 students including s42 and s96), f(x) is the integral of h(x) and h(x) is the differential of f(x) (8 students including s25), they are reverse functions (4 students including s59), they are inversely related (4 students) and they are power functions (3 students). categories such as ‘f(x) is the opposite of h(x)’ and ‘no relationship’ fell under individual responses given by the remaining 17 students. questionnaire responses of s12 and s42 show consistency of interpretation. s12: f(x) is the anti-derivative of h(x). s42: h(x) is the derivative of f(x) and f(x) is the anti-derivative (integration) of h(x). s12 was asked to explain her response in the interview: r: here you say that f(x) is the anti-derivative of h(x), what do you mean by this? s12: what i meant is that we got this one [h(x)] when we differentiated and we got this one [f(x)] when we integrated. in this case she says that there is one anti-derivative because the integration or differentiation had been performed once. this is consistent with her understanding that the number of anti-derivatives of a function depends on the number of times a function had been differentiated. s42 had earlier described an anti-derivative as the general anti-derivative only. her response reflects the same earlier interpretation. she had also referred to anti-derivative as integration of a function and she emphasises this conception by enclosing the word integration in brackets after writing anti-derivative. since in question 5 the students obtained h(x) from f(x) through the process of differentiation, this shows that the students’ understanding of anti-derivative is that the function is an anti-derivative of the other if it yields the function in question when differentiated; this meaning resonates with the definition of an anti-derivative of a function. rq3 assertion: students failed to apply their concept definitions of anti-derivative and their concept images of integration and differentiation in solving the problems the previous sections have shown that students’ descriptions associated anti-derivatives with integration and differentiation. when relating functions to their anti-derivatives, the two processes also played a major role. this is in alignment with the literature review about anti-derivatives. this section presents findings of how students’ conceptions may have influenced the way the students solved problems involving integration and differentiation. before we present the findings of how such problems were solved we begin by giving the findings of how the students related the two processes explicitly in question 4. the question required the students to show how differentiation and integration are related. in responding to question 4, four categories of correct explanations were identified as: reverse (39 students; 33%), opposite (16 students; 14%), inverse (12 students; 10%) and back to original form (8 students; 7%). of the remaining students, 14 (12%) gave individual responses, such as opposite as reverse, opposite as negative and vice-versa, 23 (20%) gave wrong responses such as ‘differentiation is the gradient while integration is the area’ (s24) and 5 (4%) did not respond to the question. some errors made include confusing the processes of integration and differentiation, the incorrect use of integration and integration operators, and giving the descriptions of how some techniques (e.g. power rule) of the two processes are individually performed without relating the processes. most of students’ descriptions used words such as inverse, reverse and back to original form which resonated with the categories of descriptions of anti-derivative in table 2. thus students’ concept images of anti-derivatives and associated processes of integration and differentiation were evoked at different times and did not have some conflicts. this is not only because they were evoked at different times but also because their interpretations were the same when evoked. students’ responses to question 9 follow. this question required students to find (a) (b) and (c) . the expected answers for the questions were (a) x2 + c, (b) ex2 + c and (c) . table 3 gives a summary of the type of responses students gave. table 3: summary of responses to question 9. responses to questions 9(a) and 9(b) were reasonable, but responses to question 9(c) were problematic because students wanted to break the expressions down into elementary functions that did not exist. this may be because for questions 9(a) and (b) the differentiation operator was to be applied before the integration operator and the majority of students managed to find the derivative of as . a few students wrote the answers to these questions without showing any working. most students were confined to the manipulations of symbols to get to the results. s42 was among the students who wrote the correct response and it appeared she used the idea that differentiation and integration are reverse processes. the interview with s42 revealed that this however was not necessarily the case. in her responses to other parts of the questionnaire she had shown that differentiation and integration are opposites of each other. r: how did you get ? s42: i integrated. r: what did you get as you integrated? s42: i got . i let u = x2 and du = 2xdx. so . r: but the question does not require you to find . s42: yes but when you differentiate it you get . when s42 finds that the 2x she expects to exist is not available in order to make the manipulation possible, she introduces it so that the expression can be of the form , which would be easier to integrate. the student seems to be stuck in procedures and she does not go beyond the manipulation to understanding that if the two processes are opposites or reverses of each other clearly using their operators together depending on which is applied first takes one back to where they started. s42 was further asked some questions that required her to reflect on her actions: r: in responding to question 4 you said that differentiation and integration are the opposites of each other, what do you mean by this? s42: if we take the function f(x) = 3x, if we integrate it we get the integral. to go back to the same function we get the derivative and the process is called differentiation. r: can you tell me what the result would be here without working, ? s42: . r: now with that knowledge what do you think the result here, , would be? s42: [silent] r: what encouraged you to work out ? s42: because i knew how to work it out. r: what about here, ? s42: i just wrote the answer because i did not know how to work it out. s42 had made connections between the processes of integration and integration. she managed to solve the problems successfully when differentiation was carried out first before integration. but, when integration precedes differentiation she makes the connection between the two processes beyond the algebraic manipulation. this appears to mean that the correct answer was obtained with the understanding that integration and differentiation are opposites of each other but could not give reasons beyond saying she could not work the problem out. in addition, the integrals were written without a constant of integration, which might also be a sign that the order in which the integration and differentiation operators are presented has no significance in terms of the concept they signify or it could be that the student at this moment just forgot to write the constant of integration. s12 was among the students who said that integration is the opposite of integration; however, her explanation is still clouded by language issues. she wrote: s12: because when we differentiate we subtract the constant to complete the differentiation. so since integration is the opposite of differentiation, we add instead of subtract. s12 persistently uses the words subtract and add to show that the two processes are opposites of each other. everyday language gets in the way of her explanation. what she means is that when differentiating the constant of integration the result is zero, thus the constant disappears so she uses the everyday word subtract because by subtraction we mean take away and when integrating we add thus we see the constant coming back. thus she shows a coherent concept image about the way the two processes are related. the only problem is how to explain her actions using proper mathematical language. these findings on language issues are similar to those of hall (2010). for example, one student said that ‘definite integral’ is the opposite of ‘indefinite integral’. this meaning is also true in everyday life but not in the mathematical context. her (s12) written work to question 10(c) which has the same external structure as question 9(c) follows: (c) r: how did you get your answer here? s12: maybe i should have differentiated. r: but the integration operator comes before the differentiation one. s12: but i think i prefer differentiation first before integration. r: why do you prefer differentiation first? s12: because i think it is going to be a little bit simpler. s12 divides by the derivative of the power x3 as 3x2. she then cancels it and leaves x2 and applies the quotient rule. none of the algebraic representation and manipulation is correct. this shows that as in the case of s42 she knows that the two processes are opposites of each other (in the mathematical sense) verbally but she cannot use this understanding in solving problems that involve algebraic symbols. although she says that she would have preferred differentiation before integration her work shows that she cannot perform the procedure for differentiating successfully. her response to question 10 (b) further illustrates this point: the first step shows that s12 seems to be applying the power rule inappropriately. she takes it to be the power function. she gets by bringing down the x3 and reducing its power by one to get x2. she seems to have problems in relating the symbolic external structure of the function, the syntax, with the appropriate technique. she takes eu to be of the form xn, she tries to apply integration by parts on the wrong expression and this does not help because even here her choice of integration technique does not seem to make sense. this seems to resonate with the view that symbols in themselves have no meaning until the learner attaches meaning to them (sengul & katranci, 2015). so applying the conception that integration and differentiation are opposites of each other here seems to fail as it seems to have strong association with symbolic manipulation and there is no connection made to the concept. hence, it is difficult for her to go beyond this stage of manipulation rather than the way the concepts are related in terms of thinking. thus her concept image of the two processes of differentiation and integration is not coherent. maharaj (2014) also found that in solving questions on integrals, students got incorrect answers due to lack of proper mental structures. using the theoretical framework of the reported study this finding could be equated to having an incoherent concept image of integrals as the associated properties and processes to the differentiation and integration concepts in the students’ minds could not make sense in solving problems, as indicated by their obtaining incorrect answers. discussion the study has shown that students gave personal definitions of anti-derivatives that overlapped with the formal definition as reflected in their interviews. these findings are different from those of hall (2010) as highlighted earlier. while some students accepted the existence of both the general and the particular anti-derivatives, others accepted the existence of the general anti-derivative only. some questions about anti-derivatives were given similar responses but with multiple interpretations. the word anti-derivative was also used synonymously with some other words which did not necessarily carry the same mathematical meaning with it but did have a connection of some sort. these are words such as: integration, integral, inverse and back to original function. integration is a process used in finding anti-derivatives, an anti-derivative is used synonymously with indefinite integral, the prefix ‘anti’ in anti-derivative is associated with the word inverse or reverse and since integration and differentiation are reverse processes, it means that applying the process of integration to an integrand gives back the function that one started with. in some cases the students interpreted the word anti-derivative the same. students persistently showed that a function has one anti-derivative or many in the interviews. some students failed to use their conceptions of how integration and differentiation are related in solving problems where differentiation and integration operators were paired. the main problem was that students could not go beyond the manipulation of symbols to relating them to the concepts they represent by taking them as reverse processes and sometimes there was no need to carry out any algebraic manipulations but just to write the correct answers. while some of students’ conceptions were unique to the context of the study (integral and integration) some were shared across contexts (inverse and back to original function) as in the study of pino-fan et al. (2017), which could be a sign that such interpretations were related to the nature of the concept and not necessarily the contexts in which the studies took place. these findings are also similar to those of maharaj (2014) and brijlall and ndlazi (2019), where students got incorrect answers due to failure to recognise the structure or syntax of symbols. we concur with this observation because in our view once the structure of the function as an object is not realised it would be difficult to associate it with the appropriate technique of either differentiating or integrating. with regard to how students interpreted an anti-derivative and the processes of differentiation and integration, there seemed to be no conflicts between students’ evoked concept images. as highlighted earlier this is not a conclusion made only due to the fact that the concept images were evoked at different times. it is also because when such concept images were evoked they were not at variance with the ones already evoked in the preceding stages. implications the purpose of teaching is to promote understanding; questionable understanding means questionable teaching. this may be true as in the case of the way question 5, question 6, and question 7 in the reported study are written. the restriction, n ≠ −1, should have been written in each of those questions so that unnecessary generalisation is not made by students. in class students had been taught that when integrating a function whose power is negative one (–1) the power rule of integration does not apply as the denominator becomes zero, hence the integral becomes ln|x| + c. such errors created by lack of precision when asking questions by educators may lead to improper understanding of a concept. meaningful understanding can be achieved if appropriate representation, connections between mathematical ideas and procedural operations are made. thus the implications to be suggested from this study should be that mathematics teaching needs to consider the semiotics and stress the explicit consideration of the connections between sign, idea and process and the syntax-semantics relationship. these include: asking students to explain their answers, discussing the meanings of technical terms in class, putting emphasis on the relationship between syntax and semantics, and stressing the association between concept labels (words) and associated experiences and operations. implementing these ideas has the potential to close the gap between the teachers’ (including the mathematical community) and students’ interpretation of the same mathematical idea. what remains a challenge for future research is investigating the complex manner in which mathematical knowledge is constructed. conclusion it is evident from the study that students do sometimes make their own reconstruction of mathematical ideas taught. the mental structures that students have about the concept anti-derivative is not fully coherent as a result. there are a number of instances where students failed to solve problems on integration and differentiation even though they had related the two concepts correctly verbally. this shows that making connections between symbols and the concepts they represent is still a challenge that needs further attention. although there has been a claim about data triangulation, when dealing with a very large group of students it is not possible to interview each and every one of them because of the enormity of the exercise. because of this the choice of 10 students for interviews cannot be claimed to be a fair representation of the whole class. if a different group was chosen for interviews the result would perhaps have been slightly different. this is one of the limitations of the reported study. the second limitation is that of the time that elapsed between administering of questions to students and the conducting of the interviews. the researchers had to go through responses to questions given in order to choose the interviewees. this on its own would also affect the results as through curiosity students might have discussed their responses with their peers and might also have checked their textbooks to see if the answers they had given were correct. while this has been a limitation the interview excerpts do show that students still had problems with solving problems given regardless of time that had elapsed. thus it could be that some students did not bother to check if they were correct based on the fact that they were not going to be given any marks for the responses they gave for the posed questions. acknowledgements we would like to whole-heartedly thank larry yore and anthony essien for their powerful and thoughtful comments in the writing of this article. competing interests the authors declare that they have no financial or personal relationships that may have inappropriately influenced them in writing this article. authors’ contributions e.k.m. contributed in writing the article, reviewed the literature, collected data and analysed data. all these were done by holding a series of discussions with m.q. where necessary. funding information this research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors. data availability statement data sharing is not applicable to this article as no new data were created or analysed in this study. disclaimer the views and opinions expressed in this article are those of the authors and do not necessarily reflect the official policy or position of any affiliated agency of the authors. references brijlall, d., & ndlazi, n.j. 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(1981). concept image and concept definition in mathematics with particular reference to limits and continuity. educational studies in mathematics, 12, 151–169. https://doi.org/10.1007/bf00305619 vinner, s, (1991). the role of definitions in the teaching and learning of mathematics. in d.o. tall (ed.), advanced mathematical thinking (pp. 65–81). dordrecht: kluwer academic publishers. https://doi.org/10.1007/0-306-47203-1_5 juter pythagoras 61, june, 2005, pp. 11-20 11 limits of functions – how do students handle them? kristina juter kristianstad university college, sweden email: kristina.juter@mna.hkr.se this paper aims at formulating and analysing the development of 15 students in their creations of mental representations of limits of functions during a basic mathematics course at a swedish university. their concept images are sought via questionnaires and interviews. how do students respond to limits of functions? the data indicate that some students have incoherent representations, but they do not recognise it themselves. introduction the notion of limits of functions is an important part of calculus. it is the foundation of many other important concepts. if students do not understand what limits are about, how can they understand concepts as, for example, derivatives and integrals? students can experience cognitive difficulties when they are learning about limits of functions, for example understanding the formal definition or the rules for deciding limit values (cornu, 1991; szydlik, 2000; tall & vinner, 1981; williams, 1991). it can be hard for them to grasp the full meaning of the compact formulation with which they are faced. it is important for teachers to know the different, and perhaps unexpected, results students can get from taking a course in calculus. students in the same class can form totally different mental representations from the same books and lectures. if we know how these different representations are formed, then we are better equipped to provide a rewarding learning environment for the students. in this paper i will discuss the development at the first university level of 15 swedish students’ learning about limits. the results are however relevant for other countries as well since the difficulties encountered by students are similar. a browse through some south african universities’ course descriptions indicates that courses at basic level comprise mainly the same topics as the swedish courses do. the questions posed in the paper are: how do the students’ mental representations of limits of functions develop during their first semester of mathematics? how do the representations change, if they change, during this time? are the mental representations of the high achieving students different from the rest of the students? answers to these questions provide an image of the students’ development of the notion of limits of functions. theoretical framework in this section i present a theoretical framework for the results of the study. it starts with a discussion about concept formation followed by a presentation of results from similar studies from other countries. concept image tall (1991) states that there are many kinds of minds. different kinds of minds are needed in the development of mathematics and no one kind of mind is always superior to another. one way to distinguish these different minds is to look at the way the learners are thinking of a concept and its formal definition. tall and vinner (1981) introduce two notions called concept image and concept definition. the concept image is the total cognitive structure associated with a concept. it can be a visualisation of the concept or experiences of it or both. the concept image is individual and in different contexts the same concept name can evoke different concept images. for this the term evoked concept image is used (tall & vinner, 1981). this is not automatically all that an individual knows about a concept. the concept definition is a form of words used by the learner to define the concept. the concept definition can be learnt by heart from a book or it can be a personal re-construction of a definition to fit in with the person’s mental structure. a personal concept definition can differ from the formal concept definition, which is a definition accepted by mathematicians in general. vinner (1991) describes a model with two “cells”. one contains the definitions of a concept and the other contains the concept image. if a concept definition is memorised without meaning, the concept image cell is empty. a concept can have several concept images that are evoked at different times. after some time, two or more concept image cells can merge when the student limits of functions – how do students handle them? 12 understands the relation between the representations. time can also have the opposite effect, the student hangs on to one concept image and forgets about the others or only gets access to some images in certain situations. when a problem is at hand and the aim is to solve it, there should be some kind of contact with the definition. it can be that the person first consults the concept image and then the definition to solve the problem, or there can be several changes between the cells. another scenario is when only one cell is consulted. if it is the image cell that is the only one, it might be that the person’s everyday life experience is misleading. in most cases the problem can be solved, but in an unfamiliar situation it might not be enough. then the lack of the formality of the definition can be the reason for failure. abstraction, objects and processes reflective abstraction (dubinsky, 1991) is defined as the construction of mental objects and actions upon them. these actions become processes by interiorisation. the objects are created by encapsulation of processes. a concept can be thought of both as an object and as a process depending on the current context (dubinsky, 1991; sfard, 1991). a scheme is a network of processes and objects. the schemas overlap and form a large complex web in the individual’s mind. according to dubinsky, there are five different ways of construction in reflective abstraction, two have already been mentioned: interiorisation: construction of mental processes in order to understand a perceived phenomenon. coordination: construction of a process by coordination of two or more other processes. encapsulation: construction of an object through a process. generalisation: ability to apply an existing schema on a greater range of phenomena. reversal: the individual is able to think of an existing internal process in reverse to construct a new process. generalisation is not as cognitively hard as abstraction since abstraction often means that the individual has to make a re-construction of the mental representation (dreyfus, 1991). crucial properties of the object at hand must be recognised and isolated from the object so that the properties are applicable in other situations. if visualisation is possible it can be a great help. the images can make the important structures and relations clear in a global manner. when an individual is generalising, he or she has a foundation of examples and experience from which to build. it is an extension of what is already there. abstraction includes the possibility for both synthesis and generalisation. to synthesise is to create a whole from parts. when this is done, it often becomes more than its parts together. many small disjoint parts get linked together and suddenly more things fall into place. this is a rewarding feeling and once an individual has passed this process, he or she cannot undo it. soon all the little important pieces of the synthesising process are forgotten and the individual takes the product for granted, that is, the understanding of the notion. there are cognitively different kinds of generalisations to do. tall (1991) describes three of them. expansive generalisation is when existing mental representations are unaltered and the new knowledge is attached as a complement. reconstructive generalisation is when existing mental representations have to be changed in order to make sense. disjunctive generalisation is when the individual is rote learning pieces that he or she should learn and just adds it to what is already in the mind. no integration occurs. it becomes impossible to get an overview of the notion at hand. learning and understanding when a process or an object is mentioned we refer to it by a mental representation (dreyfus, 1991). some learners prefer visual thinking and have pictures in their mental representations while others use symbols or examples to be able to think of notions. one concept can be represented in more than one way and there can even be conflicting representations that are evoked at different times depending on the context. if the representations are not contradicting they can merge into one when the individual is able to see the connections. a coordination or a synthesis can be the result. if they are incoherent a conflict may arise. the more connections between the mental representations, the better the individual understands the concept. then he or she can go from one to another depending on the demands of the task. knowledge of a concept is, according to dubinsky (1991), the individual’s tendency to bring to mind a scheme in order to be able to handle, organise or make sense of a problem situation. it can be hard to keep mathematical knowledge apart from mathematical construction. in an attempt to do so, we can observe individuals solving problems. such observations will not kristina juter 13 explicitly reveal the objects, processes and schemas, but can indicate how knowledge is created. a part of learning is, according to dubinsky, applying reflective abstraction to existing schemas to create new ones that provide an understanding of the concepts in question. does this imply that learning cannot take place until at least very simplistic schemas are made? otherwise the result can be that the students only know how to solve routine tasks and are unaware of the actual range of a concept. they can also create schemas that are incoherent and wrong. periods of confusion are necessary when new knowledge is to be implemented with existing knowledge and this confusion is a part of the learning process (cornu, 1991). it is the confusion that creates a need for order and the students start to adjust their schema. james hiebert and thomas p. carpenter (1992) define understanding of a mathematical concept to be something an individual has achieved when he or she can handle the concept as a part of a mental network. the degree of understanding is decided by the strength of the connections in the net and of the number of accurate connections. vinner (1991) states that to understand a concept is not the same as to be able to form its definition formally, but if a person has a concept image he or she can understand the concept. in this paper i will regard understanding and knowledge of a concept as hiebert and carpenter (1992) and dubinsky (1991) do. the limit concept the limit concept causes difficulties in teaching and learning (cornu, 1991). it is a complex concept but there is also a possible cognitive problem in the distinction between the definition and the actual concept. students can remember the definition but it does not necessarily mean that they understand the concept. vinner (1991) claims that mathematical definitions represent the conflict between the structure of mathematics and attainment of the concepts via cognitive processes. mathematics is deduced from axioms and definitions. new concepts arise from previous ones in a logical manner. this is not always the way mathematics is created though. in the beginning students can meet the notion of limit in an informal intuitive way where the tasks involve situations where they can easily see the outcome. this creates a feeling of control and the students think they know what the concept is about even if they could not solve a more demanding task where they would have to master the full meaning of the definition (cornu, 1991). one problem is the quantifiers “for every” and “there exists” in the ε – δ definition: “for every ε > 0 there exists a δ > 0 such that |f(x) – a| < ε for every x in the domain with 0 < |x – a| < δ ” in everyday language the quantifiers can have a slightly different meaning compared to the ones used in mathematics. if the students have such conceptual obstacles they can get into trouble later on. the conceptions of an idea before any teaching occurs are called spontaneous conceptions by cornu (1991) and they can be hard to get rid of even after teaching. one reason for this is the fact that when students solve problems, they tend to go to natural or spontaneous reasoning rather than scientific theories. it is as in vinner’s (1991) model with the cells, where the individual only refers to the image cell. the obstacles start a process of re-constructing existing knowledge that is based on spontaneous conceptions and this can lead to misunderstandings. monaghan (1991) found that the students in his study saw the word limit as more specific than, for example, approaches and other verbs. the students had problems with the vagueness of some expressions and this led to confusion. cornu (1991) talks about four epistemological obstacles in the history of the limit concept. two of them are the notions of infinitely large and infinitely small respectively and the question whether the limit is attainable or not. there are also metaphysical obstacles that students of today struggle with, such as the abstractness of the limit concept and the feeling of lack of rigor caused by the notion of infinity (cornu, 1991; tall, 1992). if the obstacles are located, the teaching can be altered accordingly, not by excluding the difficult passages but by supporting and helping the students to overcome them. tall (1991) describes the generic extension principle. it is when an individual is in a situation where he or she only meets examples with a specific property. if no counter-examples are present, the mind assumes that the property is valid in every context even if it is not explicitly stated. one example is if students only see examples of convergent sequences that do not reach their limits, then they might assume that convergent sequences never do. these results indicate that there are several aspects to consider in a learning situation about limits. the study in this paper is conducted to reveal how swedish students cope with such situations. limits of functions – how do students handle them? 14 the study the students and the course are described, followed by the methods and instruments used to answer the questions: how do the students’ mental representations of limits of functions develop during their first semester of mathematics? how do the representations change, if they change, during this time? are the mental representations of the high achieving students different from the rest of the students? the sample 112 students participated in the study and 29% of them were female. they were aged 19 and up. they were enrolled in a first level university course in mathematics. the course was divided into two sub-courses. both of them dealt with calculus and algebra. the courses were given over 20 weeks, full time (10 weeks for each course). the students had two lectures and two sessions for task solving three days per week. each lecture and session lasted 45 minutes. thus the total teaching time for each course was 90 hours. the notion of limits of functions was presented in the first course before derivatives. it was taught again in the second course in different settings such as integrals and series. the first course had a written examination and the second had a written examination followed by an oral one. the marks awarded were ig for not passing, g for passing and vg for passing with a good margin. i neither taught the students at any stage nor did i know any of them. methods different methods were used to collect different types of data. the students were confronted with tasks in different ways at five times during the semester, called stage a to stage e. the students got a questionnaire at stage a at the beginning of the semester. it contained easy tasks about limits and some attitudinal queries. the scope of these and subsequent tasks is described in the instruments section. the students were also asked about the situations in which they had met the concept before they started their university studies. the attitudinal data is not presented in this paper. after limits had been taught in the course, the students got a second questionnaire at stage b, with more limit tasks at different levels of difficulty. the aim was for the students to reveal their habits of calculating, their ability to explain what they did and their attitudes in some areas. the students were asked if they were willing to participate in two individual interviews later that semester. 38 students agreed to do so. 18 of these students were selected for two individual interviews each. the selection was done with respect to the students’ responses to the questionnaires so that the sample would as much as possible resemble the whole group. the gender composition of the whole group was also considered in the choices. the first session of interviews was held at the beginning of the second course, at stage c. each interview lasted about 45 minutes. the students were asked about definitions of limits, both the formal one from their textbook and their individual ways of defining a limit of a function. they also solved limit tasks of various types with the purpose of revealing their perceptions of limits and they commented on their own solutions from the questionnaires to clarify their written responses where it was needed. the students got a third questionnaire at the end of the semester, at stage d. it contained just one task. two fictional students’ discussion about a problem was described. one reasoned incorrectly and the other one objected and proposed an argument to the objection. the students in the study were asked to decide who is right and why. a second interview was carried out at stage e after the examinations. each interview lasted for about 20 minutes. the students commented on the last questionnaire and in connection to that, the definition was scrutinised again. the quantifiers “for every” and “there exists” in the ε – δ definition were discussed thoroughly. 15 of the 18 students were interviewed at this point. field notes were taken during the students’ task solving sessions and at the lectures when limits were treated to give a sense of how the concept was presented to the students and how the students responded to it. tasks and results from other parts of the study are described in more detail in other papers (juter, 2003; juter, 2004). instruments at stage a the students solved some easy tasks about limits of functions. example 1: f(x) = 12 2 +x x . what happens with f(x) if x tends to infinity? the tasks did not mention limits per se, but were designed to explore if the students could investigate functions with respect to limits. kristina juter 15 at stage b the tasks were more demanding. some of the tasks were influenced by szydlik (2000) and tall and vinner (1981). three tasks had the following structure: example 2: a) decide the limit: 1 2 lim 3 3 + − ∞→ x x x . b) explanation. c) can the function f(x) = 1 2 3 3 + − x x attain the limit value in 2a? d) why? example 2 is what i regard to be a routine task. there were also non-routine tasks. a solution to a task was presented to the students. it could be incomplete or wrong and the students were to make it complete and correct. there were two such tasks. at stage c, which was the first set of interviews, the students were asked to comment on statements very similar to those used by williams (1991) in a study about students’ models of limits. the statements the students commented on are the following (translation from swedish): 1. a limit value describes how a function moves as x tends to a certain point. 2. a limit value is a number or a point beyond which a function cannot attain values. 3. a limit value is a number which y-values of a function can get arbitrarily close through restrictions on the x-values. 4. a limit value is a number or a point which the function approaches but never reaches. 5. a limit value is an approximation which can be as accurate as desired. 6. a limit value is decided by inserting numbers closer and closer to a given number until the limit value is reached. the reason for having these statements was to get to know the students’ perceptions about the ability of functions to attain limit values and other characteristics of limits. the students were given the statements to have something to compare to their own thoughts. there were other tasks designed to make the students consider the formal definition to clarify what it really says and tasks about attainability, for example: example 3: is it the same thing to say “for every δ > 0 there exists an ε > 0 such that |f(x) a| < ε for every x in the domain with 0 < |x – a| < δ” as “for every ε > 0 there exists a δ > 0 such that |f(x) a| < ε for every x in the domain with 0 < |x – a| < δ”? what is the difference if any? as indicated before, at stage d the students got a task with a description of two students arguing over a solution to a task. the problem argued about was whether the limit value 0 lim →x 10000 )cos( 2−x exists or not. the task for the students in the study was to decide which argument, if any, was correct and to explain what was wrong in the erroneous argumentation. at stage e, the second interview, the students’ written responses to the task at stage d were discussed. example 3 was also brought up again in connection with the task at stage d. results four of the 15 students’ responses are described at the five stages a, b, c, d and e above. the descriptions are summaries of fuller ones. martin, tommy and frank had similar responses, filip, leo and dan had similar responses, louise, mikael, dennis and oliver had similar responses and julia and emma had similar responses. one student’s description from each group of similar responses is presented in table 1. the number in brackets after each name in the table indicates the number of students with similar responses. anna, john and david had responses which did not match any of the above. pseudonyms have been used to retain student anonymity. the digits at stage c indicate the students’ preferred choices from the six statements. the bold and larger digits are the students’ choices of statements most similar to their own perceptions of limits. the first points at stage e are the students’ explanations of what a limit is and the last points are about example 3 where the students explain the difference between the correct and incorrect definition connected to the task from stage d. table 2 presents the students’ marks. the first letter or letters is the mark for the first course and the second is the mark for the second course. the numbers after the marks are the numbers of times the students needed to take the examination to pass. the students’ connections of limits to other concepts are dominated by derivatives and various o limits of functions – how do students handle them? 16 t table 1. students’ developments through the course. time tommy (3) leo (3) mikael (4) julia (2) a -links to derivatives -solves easy tasks well -links to nutrition, physics and biology -solves easy tasks well -links to prior studies, problem solving, physics -unable to solve easy tasks -links to prior studies -solves easy tasks well b -limits are attainable in problem solving -cannot state a definition -solves routine tasks and explains -limits not attainable in problem solving -cannot state a definition -solves routine tasks and explains -limits are not attainable in problem solving -cannot state a definition -solves tasks and explains well -limits are attainable in problem solving -can state a definition -solves tasks and explains fairly well c -limits are attainable in problem solving -limits are not attainable in theory -a limit of a function is how the limit stands with respect to another function, no motion -3, 4, 5, 6 -cannot state or identify the definition -not a real limit if attainable by function -problems in solving tasks -links to derivatives and curves -claims to master the notion of limits -limits are not attainable in problem solving (hesitates) -limits are not attainable in theory -thinks of limits in pictures -3, 4 -cannot state or identify the definition -solves tasks well -links to derivatives and number sequences -claims to master the notion of limits fairly well -limits are attainable in problem solving -limits are not attainable in theory -thinks logically rather than explicitly defining -2, 3, 4, 5 -cannot state but can identify the definition after investigation -problems in solving tasks -links to prior studies and graphs -claims to master the notion of limits in problem solving but not the definition -limits are attainable in problem solving -limits are attainable in theory -it comes closer and closer to a as x comes closer and closer to a -1, 3, 5 -cannot state but can identify the definition (claims that both def in ex 3 state the same, ε and δ come in pairs) -solves tasks well -links to derivatives, graphs and number sequences -claims to master the notion of limits d -identifies the error -identifies the error, makes other error -identifies the error, makes other error -identifies the error e -a limit is a point to tend to but never to reach, motion -links to integrals -cannot identify the definition -does not understand the quantifiers’ roles -a function tends to a value as the variable controlling the function changes, it never reaches the limit -no links -cannot identify the definition -does not understand the quantifiers’ roles -if the x-value tends to a value then the function tends to a value connected to the x-value -links to taylor expansions, series and number sequences -can identify the definition -can explain the difference between the definitions in ex 3 -if the function goes closer and closer to a number as x goes closer and closer to a value -links to continuity -can identify the definition after investigation -can explain the difference between the definitions in ex 3 fairly well kristina juter 17 applications in areas other than mathematics at the beginning of the course. in the middle the connections concern derivatives and functions and at the end taylor expansions, integrals and series. the results reflect the content of the course quite well. table 1 indicates that tommy is able to solve routine tasks but not non-routine tasks at stage b. by this i mean that he solved all the routine tasks correctly but did not present any correct solutions to the non-routine tasks. most students were able to handle the easy tasks at stage a at least partially. at stage b, where the students got the non-routine tasks, there was a clearer distinction between high achievers and the rest of the students. at stage c, there is no longer a clear distinction. the two women with highest marks managed the problems while the two men with highest marks did not. the low achievers did not show such a gender distribution and it is probably just a coincident that the high achievers did. there is overall a higher rate of correctly solved problems of all kinds among the high achievers as could be expected. the students had a hard time formulating a definition of a limit of a function depending on x as x tends to a number at stage b. only julia managed to do this in a meaningful manner. most of the other students just reformulated the words in the task to something that did not say much. the six statements at stage c revealed that among students with marks g and ig the most frequently selected statement as most similar to their own perception was number four, which says that limits are never attained. statement three was the most frequently selected statement among the other students. statement three was not selected as most similar to their own perception by any of the students with marks g and ig. the students seem confused about functions’ abilities to attain limit values. most students’ concept images are incoherent at this point. theoretical situations are treated differently from problem solving situations. despite their lack of capability to state a definition that describes the limit of a function, many of the students still felt as if they had control over the concept. three of the students, all without the mark vg, thought that they did not have control over the concept of limits at stage c. the other students were rather certain that they had control over the concept, despite the fact that almost none of them could formulate or in some cases even recognise the definition. only emma is able to formulate the formal definition. from the seven students with marks g or ig it was only anna who recognised the correct definition in example 3. among the eight others there were five who recognised the correct definition, but among them, emma and julia stated that they both said the same thing. dan, who could not recognise the definition, thought so too. david had also indicated that the two definitions in example 3 meant the same thing at stage e, but now both julia and emma were able to explain the difference of the definitions and thereby also the meaning of the quantifiers. emma explained her solution to example 3 like this: “…in the real definition, δ depends on ε.” “…it is the independent variable which so to speak forces the function value in and not the other way around.” many students did not show such security in their explanations. the quantifiers were still a mystery to them. dan reasoned about example 3 like this: “…in most cases i think you had been able to do it this way too, it does not seem totally unreasonable.” [about the wrong definition] “…you can go the other way, but you have to in a way check as you go along.” [check if it is possible to make ε arbitrarily small for the δ chosen in the wrong definition] students with marks g or ig had trouble understanding the difference between the two definitions through the course. it was not easy for them to see connections between the formal definition and the example with the fictitious students arguing. most students with higher marks were, after some thinking, able to tell the difference at stage e. they showed a better understanding (dubinsky, 1991; hiebert & carpenter, 1992) in that they could explain the roles of the different components used in the definitions in connection to each other. the question whether limits are attainable seems to be confusing, as other researchers have also established (cornu, 1991; williams, 1991). most table 2. students’ marks. name mark name mark filip g2/ig dan vg1/g1 martin g1/g2 mikael vg1/g1 tommy g1/g2 david g1/vg1 anna g1/g2 julia vg1/vg1 john g1/g1 dennis vg1/vg1 frank g1/g1 emma vg1/vg1 louise g1/g1 oliver vg1/vg1 leo vg1/g1 limits of functions – how do students handle them? 18 of the students’ responses to questions and tasks about attainability presented in this paper are incoherent. this study shows that the students interpret the definition as stating that the limits can not be reached by the function. when they solved problems on the other hand, they could see that limits sometimes are attainable for functions. discussion the three questions posed by this study are here answered with the results and theoretical framework presented. the students’ developments are discussed, followed by the changes in their concept images. the question whether high achievers’ results differ from low or average achievers’ results is discussed along with the two first questions. the students’ developments the students seem to be very pragmatic when they learn about limits. they focus on the problem solving and not so much on the theory, with the result that the theory is somewhat vaguely represented in their minds. yet they are almost all confident about their ability to grasp the concept. this lack of awareness can perhaps be made explicit for them if the students are put in challenging and explorative situations more often.1 it is impossible to know what the students are actually thinking (asiala et al., 1996). all i can do is to see what they can accomplish and what they say and do. after reading results from other studies (cornu, 1991; monaghan, 1991; tall, 1991; tall, 1992; vinner, 1991) i was prepared to see students having difficulties, not least with the definition, but i did not expect to see so much confusion through the course. looking at the students as a group there is a positive development in their achievements. they get better at problem solving and understanding the theory, but some of them do not get as far as to fully synthesise (dreyfus, 1991) the concept during the course. parts of the concept and its applications are known, but not enough for it to become an entity. many students could briefly explain what a limit is, but when it came to explaining the partial transposition of ε and δ in example 3 there were serious problems in most cases. some students claimed that the transposition did not make any difference since the ε and δ come in pairs. this 1 some students actually told me that tasks like the non-routine tasks in the study would be good to have in the course to stimulate discussion. view is independent of the students’ marks. it is hard for them to understand the role of the quantifiers as cornu (1991) also found. dan, for example, does not take the words “for every” into account when he says that you can check if there is an ε which can be made arbitrarily small for every choice of δ in the wrong definition. the students do not seem to focus on theory in the first place since so few can explain what the definition says throughout the semester. many have a split concept image which is particularly obvious in the attainability question. the incoherence in the limits’ attainability shows that the representation of the definition is not evoked in the problem situation (tall & vinner, 1981; vinner, 1991). a reason for this can be that since the students thought that there is no point in dealing with limits if the limits are attainable, they did not consider them to be limits if they were. it can be a case of the generic extension principle (tall, 1991). the functions in most of the examples the students meet do not attain their limits. many of the students think that the definition says that functions can not attain limit values. there is a big difference in how high achievers perceive this part of the definition from how low and average achievers do. the results from the six statements in the instruments section show that the latter group of students is more likely to perceive limits as unattainable. cottrill, et al. (1996) argue that the problem of learning limits can be dependent on difficulties in creating mental schemas rather than the formal definition. the results of this study rather imply that the students have problems creating mental schemas because of the formal definition. changes the representations that the students form change through the duration of the course. some changes are rearranging entire parts of the concept image. some changes involve just smaller modifications of, or attachments to, the concept images. then there are parts that even though they can be wrong, are left unaltered. in some cases the students do not give up their first representations so readily. one example is the students who think that limits are unattainable for functions and stick by this apprehension through the entire course despite counter examples. williams (1991) mentions one possible reason for this. suppose a student is in a situation where his or her experience so far has been consistent with the theory as he or she perceives it. then the student encounters an example or something else that does not fit in with kristina juter 19 the theory. then, instead of adjusting his or her interpretation of the theory, the student might just regard the example as a minor exception. it can be hard to change a part of a mental representation. it can affect the rest of the representation in a way that requires a further modification. this can be a reason for leaving parts as they are even if the individual is aware of the error. but changes did come to pass in the study. tommy for example stated at stage c that there is no motion in limits. at stage e he had changed his mind and altered his concept image. he had also gone from perceiving a limit as a comparison between functions to an unreachable point. john had also changed his concept image. he went from focusing on the distance between the function and the limit to seeing the limit as a border. those changes need not make any difference in the rest of the students’ mental representations since they all can be true. but they alter the way the concept is thought of. perhaps the mental representations are not changed, but have stimuli that evoke different parts of the concept image (tall & vinner, 1981). despite the problems, most students still had a sense of control over the concept of limits. the sense of control probably comes from successful problem solving in the course (cornu, 1991). the students can have a concept image that works in the situations they have been in so far. when they are put in a different kind of situation their concept images can be altered. as long as the students’ concept images are challenged in some way, they will still have the opportunity to alter, refine and expand them. if they had a good concept definition compatible with the formal definition, the adjustments would probably be successful (vinner, 1991). anna is one example of this. at stage c she was solving a problem where she had a graph of a function and she had to decide a limit value at a discontinuous point. right and left limit values were both three but the functions value at the point of discontinuity was two. anna correctly stated the right and left limit values, but hesitated when she had to give the total limit value. she first said two but was uncomfortable with it since she felt as if she had contradicted herself. her concept image and concept definition were in conflict with each other. anna’s concept definition, saying that functions never reach their limits, made her say that the right and left limits are three. her concept image, saying that the functions value at the point should be the limit value if it is attainable, made her want to say it is two. after a while she got to look at the formal definition and then she was able to correctly solve all parts of the task. the material presented here has been further analysed and more results have been published (juter, 2003; juter, 2004). among the results is the impact of algebra on the students’ learning of limits. the more we know about the students’ abilities and perceptions, the better we can do our jobs as mathematics teachers. references asiala, m., brown, a., devries, d., dubinsky, e., mathews, d. & thomas, k., 1996, “a framework for research and curriculum development in undergraduate mathematics education”, cbms issues in mathematics education 6, 1-32 cornu, b., 1991, “limits”, in tall, d., ed., advanced mathematical thinking, pp 153-166, dordrecht: kluwer academic publishers cottrill, j., dubinsky, e., nichols d., schwingendorf, k. & vidakovic, d., 1996, “understanding the limit concept: beginning with a coordinated process scheme”, journal of mathematical behaviour 15, pp 167192 dreyfus, t., 1991, “advanced mathematical thinking processes”, in tall, d., ed., advanced mathematical thinking, pp 25-41, dordrecht: kluwer academic publishers dubinsky, e., 1991, “reflective abstraction in advanced mathematical thinking”, in tall, d., ed., advanced mathematical thinking, pp 95126, dordrecht: kluwer academic publishers hiebert, j. & carpenter, t. p., 1992, “learning and teaching with understanding”, in grouws, d. a., ed., handbook of research on mathematics teaching and learning, pp 65-97, new york: national council of teachers of mathematics juter, k., 2003, learning limits of function, university students’ development during a basic course in mathematics, licentiate thesis: luleå university of technology, department of mathematics juter, k., 2004, “limits of functions – how students solve tasks”, in bergsten, c., ed., proceedings of madif 4, the 4th swedish mathematics education research seminar, pp 146-156, malmö, sweden monaghan, j., 1991, “problems with the language of limits”, for the learning of mathematics 11, 3, pp 20-24 sfard, a., 1991, “on the dual nature of mathematical conceptions: reflections on processes and objects as different sides of the limits of functions – how do students handle them? 20 same coin”, educational studies in mathematics 22, pp 1-36 szydlik, j., 2000, “mathematical beliefs and conceptual understanding of the limit of a function”, journal for research in mathematics education 31, 3, pp 258-276. tall, d., 1991, “the psychology of advanced mathematical thinking”, in tall, d., ed., advanced mathematical thinking, pp 3-24, dordrecht: kluwer academic publishers tall, d., 1992, “students’ difficulties in calculus”, in proceedings of working group 3, the 7th international congress on mathematical education, pp 13-28, québec, canada tall, d. & vinner, s., 1981, “concept image and concept definition in mathematics with particular reference to limits and continuity”, educational studies in mathematics 12, pp 151169 williams, s., 1991, “models of limit held by college calculus students”, journal for research in mathematics education 22, 3, pp 219-236 vinner, s., 1991, “the role of definitions in the teaching and learning of mathematics”, in tall, d., ed., advanced mathematical thinking, pp 65-81, dordrecht: kluwer academic publishers parallelism to martin gardner “lines that are parallel meet at infinity!” euclid repeatedly, heatedly, urged. until he died, and so reached that vicinity: in it he found that the damned things diverged. piet hein abstract introduction situating this study research methodology findings and discussion conclusion and implications acknowledgements references about the author(s) qetelo m. moloi department of primary education, tshwane university of technology, pretoria, south africa anil kanjee department of primary education, tshwane university of technology, pretoria, south africa nicky roberts centre for education practice research, university of johannesburg, johannesburg, south africa citation moloi, q.m., kanjee, a., & roberts, n. (2019). using standard setting to promote meaningful use of mathematics assessment data within initial teacher education programmes. pythagoras, 40(1), a493. https://doi.org/10.4102/pythagoras.v40i1.493 original research using standard setting to promote meaningful use of mathematics assessment data within initial teacher education programmes qetelo m. moloi, anil kanjee, nicky roberts received: 27 apr. 2019; accepted: 09 nov. 2019; published: 12 dec. 2019 copyright: © 2019. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract within initial teacher education there is increasing pressure to enhance the use of assessment data to support students to improve their knowledge and skills, and to determine what standards they meet upon graduation. for such data to be useful, both programme designers and students require meaningful and comprehensive assessment reports on students’ performance. however, current reporting formats, based on percentages, are inadequate for providing meaningful qualitative information on students’ mathematics proficiency and, therefore, are unlikely to be used for interventions to improve teaching and enhance learning. this article proposes standard setting as an approach to reporting the assessment results in formats that are meaningful for decision-making and efficacious in subsequent interventions. mathematics tests, developed through the primary teacher education (primted) project, were administered electronically on a convenient sample of first-year and fourth-year primted students (n = 1 377). the data were analysed using traditional descriptive statistical analysis and the objective standard setting method. the two reporting formats – one using a percentage score and the other using standards-based performance levels – were then compared. the study identified important distinguishing features of students’ mathematical proficiency from the two reporting formats, and makes important findings on the specific knowledge and skills that students in south african initial teacher education programmes demonstrate. we conclude that reporting assessment results in standards-based formats facilitates differentiated interventions to meet students’ learning needs. furthermore, this approach holds good prospects for benchmarking performance across universities and for monitoring national standards. keywords: standard setting; meaningful reporting; objective standard setting; performance levels; performance standards. introduction the centrality of assessment to effective and deeper learning in higher education has been asserted in a number of research studies (coates & seifert, 2011; filetti, wright, & william, 2010; ping, schellings, & beijaard, 2018; russell & markle, 2017; sambell, mcdowell, & montgomery, 2013). while there has been some research conducted on the value, and use, of feedback from assessment by students at university level (sadler, 1998), no corresponding research has been reported on meaningful formats of reporting results of tests and examinations in ways that can help lecturers improve teaching and help enhance learning by students. arguing that assessment remains one of the final frontiers of change, coates (2015) notes that despite the substantial improvement within many parts of the higher education sector, assessment systems have not changed for a century, and that student knowledge and skills are still most commonly measured in traditional ways. however, there has been recognition of the limitations of reporting assessment results in raw scores, including percentages, and the tendency to aggregate scores of exclusive and independent constructs (coates & seifert, 2011). these limitations present a challenge to the users of assessment results because the extent to which lecturers and students can interact with assessment results, in ways that can help them learn better, hinges heavily on the meaningfulness, the consistency and comprehensiveness of the assessment reports that they receive. to address this challenge, a new project was established aimed at changing the dominant culture from one in which assessment data is only utilised for reporting, promotion and certification to one in which assessment data is regarded as a rich source of information for use in improving teaching and learning. meaningful reporting and utility of assessment data is a central requirement for programme coordinators of bachelor of education programmes who are collaborating in the assessment workstream of the primary teacher education (primted) project. the aim of the primted assessment workstream is to: advocate for higher education institutions’ participation in common assessment approaches. encourage collaboration on teacher assessment approaches towards developing teacher competence in relation to improving the teaching and learning of mathematics. develop a teacher competency assessment framework (related to mathematics), for foundation and intermediate phase, for student teachers graduating from initial teacher education (ite) programmes. this is not a trivial task as there is, as yet, no agreement about the knowledge needed for teaching mathematics. even in countries such as the united states, where there are claims of teaching standards that have been developed, and teachers are assessed against such standards using a variety of assessments, review of released assessment items reveals that there is still ‘lack of agreement over what teachers need to know’ (hill, shilling, & ball, 2004, p. 12). in the south african context such agreement is being sought through the collaborative work of three primted workstreams (that focus on mathematical thinking, number sense and geometry and measurement). teams of professionals from across south african universities are collaborating on assessing the mathematics knowledge of student teachers when they enter the bachelor of education (b.ed.) programme in their first year, and again when they exit at fourth-year level. the purpose of agreeing on such a common assessment – and reporting on its findings – is to allow programme designers and lecturers to reflect on and improve their programmes over time. the article reports on observations made in the primted assessment workstream (with a specific focus on mathematics). firstly, the shortcomings of current reporting practices are presented. the article explores and illustrates the value of the standard setting approach for improving the reporting and use of assessment data. next, the article presents the process followed in identifying performance levels (pls), performance level descriptors (plds), and cut-scores to determine students’ levels of proficiency (kanjee & moloi, 2016). this is followed by a presentation of available results to demonstrate how a standards-based framework can be used to establish and report the levels at which students are functioning in order to: (1) identify specific learning needs of students for developing appropriate interventions to address these needs and (2) determine the knowledge and skills with which students graduate upon completion of their ite programmes. the article concludes by outlining implications for enhancing the use of assessment data to improve ite programmes for mathematics teachers in south africa and proposing the next steps in developing a teacher competency assessment framework for universities participating in the primted. situating this study there has been an international shift in focus in teacher education towards an ‘intensification of standardisation’, which in many countries has resulted in high stakes summative assessment. summative assessment is then expected to fulfil multiple functions: providing evidence of a teacher candidate’s achievement at a given point against the standards, guiding and informing jurisdictional decision-making around teacher preparation, licensure and advanced certification, and furnishing evidence of teacher effectiveness and readiness to teach (allen, 2017; deluca & klinger, 2010). allen (2017) argues that focusing on summative assessment is revealing of what is valued and privileged – and what is not – in teacher education. in contrast – although not as dichotomies – formative assessment in teacher education begins with students understanding what it is they are intending to learn, offering feedback to students, student goal setting and keeping track of their own learning and includes formative use of summative tests (brookhart, 2017). as such there is a growing requirement within the field of ite to design standardised tests which can be meaningful to both programme designers and to students. yet in a systematic review of what, how, and why teacher educators learn (ping et al., 2018), a focus on assessment and how to use it to improve teacher education practice appears to be missing. it is therefore unsurprising that in south africa, reporting on teacher knowledge for mathematics has been limited and – as evident in the studies discussed below – reporting on assessment has been in relation to mean attainment of percentage of teachers meeting a benchmark. in one empirical study that involved practising teachers, venkat and spaull (2015) analysed the southern and eastern africa consortium for monitoring educational quality (sacmeq) data collected from a representative sample of learners and their teachers (n = 401) to determine levels of teachers’ content knowledge. the authors used the curriculum and assessment policy statement grade level and content strands to classify the sacmeq test items to better align to the south african curriculum, and 60% as a minimum benchmark for mastery at the grade 6/7 level. venkat and spaull reported that 79% of south african grade 6 mathematics teachers were classified as having content knowledge levels below grade 6. there was even less research available about student teachers in ite programmes. a scan of literature relating to ‘initial teacher education’ and ‘assessment’ in south africa revealed a dearth of studies with this focus. a recent systemic review of south african studies pertaining to ite for primary school, and particularly foundation phase (baxen & botha, 2016), identified these main thematic areas: diversity and social justice, theory and practice, curriculum concerns, identity and subjectivity, language issues and policy and transformation. the type, utility and reporting on assessment is conspicuous by its absence, and is not mentioned even as a subsection to any of the themes. baxen and botha (2016, p. 10) describe the ite research landscape in south africa as ‘nascent’ with the research projects reviewed considered ‘isolated and uncoordinated’. taylor (2018) refers to a 2010 council on higher education report which described the state of the ite sector as uncoordinated and of unknown quality of performance. deacon (2012) called for the establishment of benchmarks – to diagnose what mathematics and english the students entering the b.ed. programmes possess. the initial teacher education research project (iterp) considered intermediate phase courses in five universities and found that the quality of ite in south africa, in terms of the knowledge and skills possessed by first-year entrants, the content that is offered and the exit competencies of final-year students, was questionable in relation to mathematics and language courses. bowie and reed (2016, p. 116) reported that all five lecturers in the iterp study noted that on entering a b.ed. programme some student teachers are no more proficient in mathematics and english than the grade 4 to 6 learners they are preparing to teach. in their review of primted assessment results for first-year students (n = 1 117) across seven universities, alex and roberts (2019, p. 68) reported that the ‘majority of the first-year students (71%) do not meet the minimum benchmark of 60% for knowledge of the mathematics content at primary school level’. all of these studies – pertaining to both practising teachers and those in ite programmes – simply report a deficit in mathematical knowledge for teaching. the mean results and percentage of those attaining a minimum ‘benchmark’ provide quantitative evidence of a dire situation but do not offer qualitative information on what may be done or how best to intervene. typical, and additional, forms of reporting assessment results traditionally assessment results are reported largely in raw scores which are mainly presented as percentage correct responses obtained by students in a test or examination. in any typical report emanating from end-of-year examinations or large-scale assessments, results are generally presented as mean percentage scores by subject areas, often accompanied by some reporting scale that provides an interpretation of the scores. for example, 60% and greater is considered as mastery level (alex & roberts 2019; venkat & spaull 2015). in some instances (for example, as in fonseca, maseko, & roberts, 2018), additional information is also reported on the performance distribution of students taking the tests, as well as by sub-domain and cognitive levels for the subject tested (see figure 2 and figure 5 discussed in a later section). this information may be useful for monitoring student progress or identifying students that meet required levels of performance. however the utility of this information for identifying specific learning needs of students, or determining the specific sets of knowledge and skills that students master, is limited (bond & fox, 2007; montgomery & connoly, 1987). raw scores on their own cannot be used qualitatively to indicate what students were able or not able to do in a test (bond & fox, 2007; moloi & kanjee, 2018). raw scores, including percentages, are sampleand test-dependent. on the one hand, in the same test, a sample of relatively more proficient students will achieve higher percentage scores than their less-proficient counterparts. on the other hand, the same sample of students will achieve lower scores in a more difficult test than on a relatively easier one. furthermore, the use of percentage scores is predicated on the assumption that the effort of improving one’s score on the lower end of the proficiency continuum is the same on the upper end. yet, studies based on the latent test theory show that it is more difficult to improve from a score of, say, 90% to 95% than from 20% to 25% even though the interval (5%) is the same in both cases (bond & fox, 2007). these weaknesses render the use of percentage scores in reporting performance generally unreliable as a measure of either student proficiency or progression over time. as a result, performance standards which are seen to be more stable and provide detailed information on what students know and can do are increasingly used (glass & hopkins, 1984). our focus this article explores and proposes standard setting as an approach to reporting the results of assessment in formats that are meaningful for decision-making and efficacious in subsequent interventions. the premise of the article is that, unless assessment data is analysed and reported in meaningful formats, it is unlikely to be used for interventions that will lead to improvement in teaching and to enhancing learning by students. importantly, this article does not seek to examine what teachers of primary school mathematics are required to know, and hence the underlying constructs and possible assessment standards. such research is being conducted across multiple mathematics workstreams within primted. rather this article seeks to demonstrate how existing assessment data may be used to feed into the broader research agenda, and offer reporting templates that are both meaningful and usable for participating universities. value and use of standards-based reporting of assessment results setting performance standards involves determining a cut-score that separates test-takers into identifiable categories on the basis of what they know and can do, or what they do not know and cannot do, in a test. along the continuum from not knowing or not being able to do anything at all in a given test, on the one end, to the other extreme of knowing or being able to do everything, what score is necessary for performance to be judged as ‘acceptable’? to determine the necessary score, one has to establish ‘a set of content standards and a set of test questions intended to reflect that content as a starting point to a process that will lead directly to setting performance standards’ (barton 1999, p. 19). cizek and bunch (2007, p. 13) define standard setting as ‘a process of establishing one or more cut-scores on a test for purposes of categorising test-takers according to the degree to which they demonstrate the expected knowledge and/or skills that are being tested’. the standard setting process involves technical analysis of student responses as well as expert inputs from teams of professionals and members from relevant stakeholder groups who serve to validate the technical results (tiratira, 2009). performance standards specify the amount of knowledge or skills that a test-taker must demonstrate, on a given continuum separated by cut-scores, to satisfy the demands of a particular outcome. an important feature of performance standards is pls and plds. zieky and perie (2006, p. 3) describe pls as general policy statements that indicate the official position on the desirable number and labels of categories to be used in classifying students according to their knowledge and skills in a particular subject and grade. plds are defined as detailed descriptions of ‘the knowledge, skills, and abilities to be demonstrated by students who have achieved a particular performance level within a particular subject area’ (zieky & perie, 2006, p. 4). morgan and perie (2005, p. 5) affirm that plds are ‘working definitions of each of the performance levels (that) … define the rigor associated with the performance levels.’ at the university level, plds can provide detailed information on the specific knowledge and skills with which students enter and leave higher education programmes. at individual student level, lecturers can identify students who meet, or do not meet, expected levels of knowledge or skill, categorise them according to need, and decide on appropriate interventions for each category. not only can performance standards help focus interventions to address specific individual needs, but they can also ensure that students and lecturers share a common understanding of what is expected in student performance (california department of education, 2007, p. 2). clear knowledge of what is expected may encourage students, lecturers and universities to exert more effort to achieve it. performance standards also help improve the precision with which reporting to relevant stakeholders can be made. clearly and precisely communicated student progress reports which are based on fair, valid, and easy-to-understand standards demystify expectations and increase trust between the concerned institution and its clients (california department of education, 2007, p. 2). while different methods of developing performance standards and corresponding plds have been explored, a common conclusion from research has been that each method has its own strengths and weaknesses and, therefore, the best standard setting method may not exist (tiratira, 2009). notwithstanding possible pitfalls related to the complexities and subjectivity inherent in different methods of setting performance standards, cizek and bunch (2007, p. 36) and bejar (2008, p. 3) emphasise the critical role of communicating and using clear standards to enhance decision-making among all key stakeholders. the purpose of the study the purpose of this study was to explore the use of the standard setting approach for reporting of primted mathematics results so as to enhance their use by lecturers to identify and address specific learning needs of students. in this regard, the following research questions guided the study: what reporting formats provide meaningful information to identify the support first-year b.ed. students require for improving their mathematics knowledge and skills? what reporting formats provide meaningful and valid information on the mathematical knowledge and skills that final-year students graduate with upon completion of their ite programme? research methodology the study involved testing student teachers’ knowledge and skills in mathematics, followed by a standard setting exercise to select the number of pls, establish the plds and calculate the cut-scores that categorised students according to identified levels of performance (cizek & bunch, 2007), and the analysis and reporting of results. adopting a cross-sectional design, the same primted assessment was administered at both first-year and fourth-year levels in this study. sample a purposive sample for this study was drawn from eight universities that participated in the administration of the primted mathematics test to either their first-year or fourth-year students or both in 2018. a total of 1377 students from the eight universities participated in the testing, of which 85% were students in first year and 15% were in fourth year (table 1). the students’ mathematics test data was subsequently obtained from the primted coordinators. in order to maintain confidentiality and to protect the identities of the participating universities, no demographic data is given. table 1: student sample participating in the 2018 primted assessment. test design and administration as it is not the direct focus of this article, we only briefly describe the design and administration of the primted mathematics test. fonseca et al. (2018) describe the test design pilot, refinement process and how it was administered online across universities. further details as well as exemplar items are presented in alex and roberts (2019). for this article we simply highlight some key features of the test, understanding of which will help to interpret the two report formats – percentage score reporting and standards-based reporting – which we contrast later in the article. we now know that the ‘content knowledge for teaching mathematics consists of more than the knowledge of mathematics held by any well-educated adult’ (hill et al., 2004, p. 27). the overarching construct that the primted test began to assess was ‘mathematical knowledge for teaching: knowledge teachers require to teach primary mathematics well’. alex and roberts (2019) argue that mathematical knowledge for teaching includes knowledge of: (1) the content (know how to do the mathematics itself), (2) why the content makes sense, (3) how to represent concepts using multiple representations, (4) how the particular aspect of content connects to other topics and grades, and (5) at what stage children are ready to learn this content. the primted mathematics test was a first attempt at designing a common assessment framework for use across ite programmes. the development of the test was done in the absence of agreed assessment standards (which are being developed across the primted mathematics workstreams). as such, we do not claim that the primted mathematics test assesses all aspects of mathematical knowledge for teaching in primary schools, but the above description is used to elaborate on the test intention and its long-term objective. in this article we simply refer to ‘mathematical knowledge and skills’ to describe the underlying construct that the current primted test begins to assess. fonseca et al. (2018) explain that the primted mathematics items were classified as lower cognitive demand, higher cognitive demand or pedagogy items. the cognitive demand levels applied the stein, grover and henningsen (1996) framework on tasks, where ‘lower cognitive demand’ items were considered to be routine procedures; the ‘higher cognitive demand’ items involved moves between representations, required insight, connected across topic areas or had no obvious procedure or starting point (venkat, bowie, & alex, 2017). an example of the distinction between lower and higher cognitive demand is offered in figure 1. figure 1: exemplar items for lower and higher cognitive demand. the third category of items related to ‘mathematical pedagogy’ and were specifically phrased to solicit analysis of a student’s work or of a common error in mathematics (alex & roberts, 2019, p. 65). this category of knowledge is described as ‘pedagogical content knowledge’ by shulman (1986, 1987), and is thought to ‘include familiarity with topics children find interesting or difficult, the representations most useful for teaching an idea, and learners’ typical errors and misconceptions’ (hill et al., 2004, p. 12–13). an example item to assess pedagogy is presented in figure 2. figure 2: exemplar pedagogy item. the primted mathematics test was administered online to first-year and fourth-year students and was monitored by at least one mathematics lecturer at each university. the instrument comprised 50 multiple choice mathematics items which covered four of the mathematics topics prescribed in the curriculum and assessment policy statement document for intermediate phase (dbe, 2011). these include numbers, operations and relationships, patterns, functions and algebra, space and shape (geometry) and measurement. data handling was excluded as this topic forms a very small component of the foundation and intermediate phase curriculum at primary school level. the answers to all items were either ‘multiple choice’ or ‘single answer’. no partial marking was necessary. each item was marked correct (1) or incorrect (0) using online marking. ethical considerations this study was undertaken following the ethical processes requiring voluntary, informed consent for educational research and under the university of johannesburg’s protocol number of 2017-072. this ethics application was for the primted assessment process as a whole and included all participating universities. first-year and fourth-year b.ed. students were invited by their mathematics course coordinators to write the primted mathematics test, as a voluntary part of their b.ed. programme. they were told that the assessment was not ‘for marks’ but rather to be used as diagnostic assessment. on commencement of the online test, each student could choose to ‘opt in’ to have their data from the test included in a wider research study and for publication, or to ‘opt out’ and have their data excluded. they were assured that all data would be anonymised. development of performance levels and performance level descriptors a standard setting session was convened at which mathematics lecturers from the participating universities defined the four pls (see figure 3), established the plds and participated in the process of determining the cut-scores. the objective standard setting (oss) method (stone, 1995) was used to determine cut-scores. the choice of the oss method was informed by our research that indicates this method as being both cost-effective and providing curriculum-rich information for meaningful reporting of assessment results. interested readers can refer to moloi and kanjee (2018) for general procedures of standard setting and to stone (2001) for specific technical procedures of the oss method. the information gathered from the standard setting session was used to conduct an intensive audit of the knowledge and skills that characterise students at each performance level to establish the plds. this was intended to be illustrative of process and did not involve all of the participating university lecturers. it is acknowledged that further discussion relating to these plds is required for common agreement on these to be reached. a summary of the illustrative primted mathematics test plds is given in figure 4. figure 3: generic performance level definitions. figure 4: illustrative primted mathematics performance level descriptors. for relative ease and accuracy of reporting we proposed the use of four reporting levels defined as not achieved, partly achieved, achieved and advanced. the process of the development of pls, their policy definitions as well as their generic descriptions has been presented in detail in moloi and kanjee (2018). the definition of pls includes information on the implications of the pl in terms of student progression and necessary intervention (figure 3). figure 4 provides a summary of the primted plds with illustrative knowledge and skills that characterise a student’s mathematics proficiency at each pl. the distinctions made in the opening clause of each level descriptor draw on the components of mathematical proficiency as advocated by kilpatrick, swafford and findell (2001, p. 5): conceptual understanding: comprehension of mathematical concepts, operations, and relations. procedural fluency: skill in carrying out procedures flexibly, accurately, efficiently, and appropriately. strategic competence: ability to formulate, represent, and solve mathematical problems. adaptive reasoning: capacity for logical thought, reflection, explanation, and justification. while these are used as intertwined strands of mathematical proficiency, the analysis of items likely to be correct at each performance level, together with the analysis of expert contributions to the oss process, was suggestive of these distinctions. these plds are only intended to be illustrative. as noted above, a detailed process of engagement with all mathematics workstreams on this process is still required to ensure common understanding and utility of these kinds of descriptors. in figure 4 the plds for the not achieved level, that is, the level below the partly achieved level, have been omitted. typically, students who function at this level are characterised by scores that fall below the lowest cut-score and tend to provide sporadic responses that follow no discernible pattern of proficiency. the hierarchy of pls means that the knowledge and skills at a given pl are cognitively more demanding than those at a lower pl and less demanding than those at a higher pl: not achieved < partly achieved < achieved < advanced in terms of cognitive demand. secondly, a student who displays knowledge and skills that are characteristic of a given pl is expected to also have a high probability of displaying knowledge and skills at lower pls but is unlikely to display knowledge and skills at higher pls. for instance, a student who functions at the achieved level is expected to also demonstrate the knowledge and skills at partly achieved level but is unlikely to display advanced level knowledge and skills. analysis analysis was conducted at four levels. firstly, the primted mathematics test was investigated for appropriateness of ‘targeting’, which is an indication of whether the spread of item difficulties appropriately matched the abilities or proficiencies of the students (wright & stone, 1979). tests that are not properly targeted have large measurement errors and low reliability indexes (bond & fox, 2007, p. 43) and this may affect standards that are set from such tests. secondly, the ratings of items from the standard setting process were analysed, using the oss method (stone, 2001) to determine cut-scores that separate different pls. thirdly, using the cut-scores, students were categorised based on their test scores into one of the four pls. this categorisation indicated the proportions of first-year and fourth-year students who functioned at each pl. lastly, statistical analyses were conducted, using descriptive tests, t-tests and chi-square tests to determine significant differences where comparisons of scores were made. the results are presented as examples of both mean score reporting and performance-based reports to illustrate the benefits and effects of using each reporting format to improve teaching and enhance learning. findings and discussion the findings of the study are presented and discussed in this section. firstly, the item map was generated to determine the validity of the primted test as an instrument that will produce consistent information about the test-takers’ proficiency when other relevant factors remain unchanged (newby, 2014). next the cut-scores derived from using the oss method are presented. thirdly, practical implications of applying the standards-based approach to address each research question are presented. in order to illustrate the value of this approach results of mean score reporting are also provided for each question addressed. finally, a summary of the comparisons and their implications is presented. primted item targeting the item map that provides a visual representation of how well the primted test item difficulties or cognitive demands matched the proficiencies or abilities of the test-takers is presented in figure 5. figure 5: primted mathematics item map. as reflected in figure 5, the easiest mathematics item was item 8 with item 22, item 1 and item 26 relatively more difficult but overall on the easy end of the cognitive spectrum (although there were still a few students who had a 50-50 probability of answering these items correctly). it is worth noting that there were up to 24 test-takers, represented by the two dots at the bottom end of the map, for whom all the items were too difficult, that is, they had less than 50% chance of answering any of the items correctly. item 50, item 38 and item 47 were on the difficult end of the spectrum but again there were still a few students who had a 50-50 probability of answering these items correctly. there were up to 60 test-takers, represented by the six dots at the top end of the map, for whom there was more than 50% probability of answering all the items correctly. the rest of the test items, from item 23 on the easy end to item 45 on the difficult end, were well matched to the abilities of the test-takers. on the whole, the item map shows that there was a reasonable match between person abilities and item difficulties, suggesting that the primted math test was well targeted to the population of the test-takers. according to bond and fox (2007), optimal information about the proficiency of test-takers is obtained when the test is neither too easy nor too difficult for them. cut-scores and their implications using the oss method, the cut-scores that mark transition from one performance level to the next were calculated to be: 40% for the partly achieved level, 53% for the achieved level and 68% for the advanced level. these cut-scores were used to delineate the pls and devise the reporting format to provide information on the student’s level of functioning as well as the knowledge and skills that the student had mastered. in the next section, the key research questions are addressed. question 1: what reporting formats provide meaningful information to identify the support first-year b.ed. students require in terms of improving their mathematics knowledge and skills? figure 6a and figure 6b present a percentage-based primted mathematics report for first-year students from two universities, university a and university b. the information presented in the report comprises: (1) sample size, mean scores and standard deviations, (2) students’ score distributions by deciles, (3) student mean scores by content domain and (4) student mean scores by cognitive demand and pedagogy categories. figure 6: (a) first-year student 2018 performance for university a; (b) first-year student 2018 performance for university b. the results of the independent sample t-test reveal that the mean scores of students in university b are significantly higher than those of students in university a (t(310) = 6.19, p < 0.00). moreover, the mean scores of the content sub-domain areas of the two universities were also consistently higher for university b than for university a. the scores in the two universities differ in three other important aspects: variability, skewness and content cognitive demand. firstly, university b scores are more variable or spread out (sd = 18.24) than those of university a (sd =12.67) signalling greater diversity in mathematics proficiency. secondly, university b scores are skewed more to the right or towards higher scores signalling the presence of high performing outliers among the test-takers. thirdly, students in university b were more proficient than their counterparts in university a on items that placed greater cognitive demands on the test-takers. while the aforesaid quantitative descriptive features of relative performance across the universities underscore the distinctiveness of the two universities, they do not, however, provide qualitative information that can be utilised to address specific learning needs of either individual students or categories of students with distinct learning needs. typically, reports of this kind constrain end-users to employ one-size-fits-all interventions with all known shortcomings of such approaches (van geel et al., 2019). reflection on the relative strengths relating to particular topics (sub-domains of mathematics arranged by topic in the curriculum and assessment policy statement) are also evident. from figure 6a and figure 6b the content knowledge, measured in mean score percentages, of students in university b were consistently higher than those of their counterparts in university a. this was the case in the overall mathematics scores as well as in the various sub-domains that constitute the subject. notable was the observation that in both universities the highest mean percentage (whole number) scores were in the sub-domains of measurement, whole numbers and geometry, ranging between 39% for measurement and 45% for geometry in university a, and between 60% for whole numbers and 65% for measurement in university b. for both universities, the fact that geometry and measurement were scored relatively higher than other sub-domains may require further investigation as lecturers participating in the team reported that students tend to have more challenges in these sub-domains, particularly in geometry. in addition, across both universities the lowest performance was in rational numbers and algebra. mean scores in these two sub-domains ranged between 28% and 38% in university a and between 46% and 53% in university b. this is an observation of great concern because rational numbers and algebra do not only constitute the majority of the content in secondary school mathematics, but algebraic expressions and rational numbers find a lot of application in the other sub-domains. typically, for both universities, student mean scores were higher for items that made lower cognitive demands than items of a higher cognitive demand. figure 6a and figure 6b show that the respective mean scores (rounded to whole numbers) were 43% and 32% for university a compared to 65% and 47% for university b. for both universities the lowest mean scores were in items classified as ‘pedagogy’, namely 26% in university a and 32% in university b. however, given that the students were in their first year of a four-year course, the low scores in pedagogy may not be of particular concern but they do point to knowledge deficits that need to be addressed. to adequately equip the student teachers, the percentage score report shows that both universities seem to face a singular challenge of establishing a deeper understanding in the fundamental mathematics content knowledge, although the challenge is much more acute in university a. however, the raw score report does not provide necessary detail on the specific challenges that each student faced – whether it be in a specific content sub-domains, different cognitive demands or pedagogic methods. for effective and targeted interventions, both lecturers and students would be better equipped if detailed information is provided on what students know, and can or cannot do, in the subject of concern. as can be observed from figure 6a and figure 6b on the one hand, and figure 7a and figure 7b on the other hand, there are markedly distinct features that distinguish the two reporting formats. figure 7a and figure 7b present user-friendly information on how students were distributed across the pls in terms of their mathematics proficiency, as well as information on the knowledge and skills that students who function at a particular pl demonstrate. also, the reports provide an overall picture of student performance in the form of pie charts. test results reported in this format are more likely to be used by lecturers to intervene to improve the performance of their students. also, this report provides information that the student can use to make informed decisions on how they can improve their performance in specific areas of knowledge and skills. figure 7: (a) first-year student 2018 performance for university a, (b) first-year student 2018 performance for university b. figure 7a indicates that about a quarter (24%) of first-year students in university a were functioning at either the achieved or advanced level in mathematics which, according to the appropriate plds, are levels characterised by a predominance of conceptual over procedural reasoning. in university b it was just more than half (56%) of the first-year cohort who were functioning at either the achieved or advanced levels. in terms of the definitions of the pls, these were students who could be expected to succeed and progress to the next level in mathematics with minimal support. in contrast, 41% of students in university a were functioning at the not achieved level compared to 21% in university b. this category of students demonstrates very limited knowledge of the requisite mathematics knowledge and skills and is unlikely to succeed without intensive support. overall, these results indicate that lecturers in university a would have to focus more on supporting students with basic mathematics knowledge and skills than their counterparts in university b. the typical standards-based report that we recommend in figure 7a and figure 7b comprises a number of uses and benefits to both lecturers and students. firstly, the lecturer who receives this cohort of students, for example mid-year or at the beginning of the second year of study, is provided with a detailed performance profile of the cohort of students and information on what each student knows and can do. working from the report, the lecturer will be able to design and plan targeted interventions that address the specific learning needs of the various categories of students. secondly, provided that information is available, lecturers are easily able to monitor individual students’ progress over time. thirdly, lecturers can address issues of equity within their classes and ensure that additional and adequate resources and support are provided to those students that have the greatest need, that is, students at the not achieved and partly achieved levels. fourthly, because each student will know the performance level at which they function, they would be better able to plan and pace their own learning from an informed position in terms of the specific areas of knowledge and skill that they need to focus on. studies that have contrasted the use of marks or test scores with textual comments in feedback to students have confirmed that students tend to focus on test scores rather than paying attention to comments that direct them to what to do in order to improve their performance (nicol & macfarlane-dick, 2006). consequently, giving feedback in the form of test scores tends to militate against students becoming self-regulated lifelong learners when compared to using descriptive feedback that outlines how students should improve their own learning (nicol & macfarlane-dick, 2006). the effect of these benefits of standard setting formats of reporting should be improved and effective teaching that is likely to enhance students’ learning. question 2: what reporting formats provide meaningful and valid information on the mathematical knowledge and skills that final-year students graduate with upon completion of their ite programme? the mathematics results of fourthand final-year students’ performance for university c and university d have been summarised in figure 8a and figure 8b. the percent-score results show that at the point of graduation, the mathematics mean score of students from university c ( = 51.95% (15.12)) is significantly higher (t(179) = 3.27, p < 0.00) than that of their counterparts in university d ( = 43.79% (17.39)) and further interrogation of the distribution of scores shows that university d has more outlier students of outstanding mathematical proficiency than university c. the outlier students of outstanding proficiency from the apparently weaker university d, in terms of mean scores, account for the equal mean scores ( = 41%) on items that make high cognitive demands on students in both universities. figure 8: (a) fourth-year student 2018 performance for university c, (b) fourth-year student 2018 performance for university d. in both universities the highest mean percentage scores of students were in the sub-domains of geometry, measurement and whole numbers, ranging between 54% for whole numbers and 69% for geometry in university c and between 48% for whole numbers and 53% for geometry in university d. a general observation regarding the results for both universities is that students graduated and commenced teaching with far lower knowledge and skills, in terms of their mean scores in the primted test. these scores differ substantially from the minimum of 60% that venkat and spaull (2015) applied for determining mastery levels in the sacmeq data, and which was also adopted by alex and roberts (2019). this was the case for both content knowledge and the pedagogical content knowledge where mean scores were 36% for university c and 32% for university d. although detailed analysis of the mean scores distribution serves to unravel inequalities in performance across the universities, it still does not provide information on what the test-takers know, can or cannot do in the subject. the standards-based report, for the same institutions, as shown in figure 9a and figure 9b addresses this shortcoming. a review of figure 9a and figure 9b indicates that 51% of students from university c and 73% of students from university d may graduate while still functioning at the not achieved and partly achieved levels in mathematics. figure 9: (a) fourth-year student 2018 results by performance levels for university c, (b) fourth-year student 2018 results by performance levels for university d. in terms of the pl definitions and plds, these students demonstrate either very limited or partial understanding of the mathematics knowledge and skills required to teach mathematics, and show more evidence of procedural fluency than conceptual understanding. for instance, they can do the four basic operations arithmetically, but they experience difficulty when they have to work with algebraic symbols. approximately half of the fourth-year students in university c, compared to 27% in university d, function at either the achieved or advanced level in mathematics. in terms of the pl definitions, they demonstrate either sufficient or comprehensive knowledge or skills to teach mathematics. in terms of plds, they show evidence of ‘procedural fluency’, ‘conceptual understanding’, ‘adaptive reasoning’ and ‘strategic competence’. for instance, they operate equally well with numbers and symbols, can solve complex problems that involve more than one variable in mathematics and can support their viewpoints with valid reasons. this detail on the students’ proficiency and thus their readiness to teach mathematics is very explicit in the plds of the standards-based reporting format. more importantly, the standards-based report provides a clear indication of: (1) how many students can be considered to have graduated from a specific ite programme with the requisite knowledge and skills, and thus are likely to effectively teach mathematics upon entering the teaching profession, (2) whether the specific ite programme is fulfilling its mandate to prepare students with the requisite knowledge and skills to teach in their subject areas, and (3) the specific areas that need improvement, within an ite programme, in order to better prepare students for entering the teaching profession. this information is missing from the mean score reports, and may result in misleading interpretations on how well ite programmes are functioning. the key findings from this study were that the traditional use of mean scores to report assessment results lacked necessary information for identifying what students know and can do and this compromises the feasibility of meaningful and targeted interventions for improving teaching and enhancing learning. furthermore, this study has shown that the use of pls to report student performance provides meaningful reporting that facilitates its use by students, lecturers and university management to enhance student learning as well as monitor student progress over time and the impact of the ite programmes on producing teachers with the requisite content knowledge and skills to effectively teach their subject areas. in this regard, the study addresses a key challenge confronting the higher education sector that coates and seifert (2011) identified – that research contributions over the last decade have mainly focused on providing criteria on how to evaluate the impact of such reporting rather than on what needs to be done differently to overcome the weaknesses of aggregating and reporting assessment results in raw scores. conclusion and implications the iterp study reported on by bowie (2014) was an important step towards drawing attention to the lack of common frameworks and assessment of mathematics content knowledge of new primary and secondary school teachers in south africa. its findings demonstrated the need for the primted intervention. the subsequent articles emerging from the primted assessment workgroup provided a common assessment framework which confirmed the widespread deficits across a wider range of ite programmes. the overall results of this study cohere with previous reports (alex & roberts, 2019; bowie 2014; fonseca et al., 2018) in that they make explicit the current deficits in mathematical knowledge for teaching which are evident at both first-year and fourth-year levels in ite programmes for primary teachers. more importantly, this study – which draws on the same primted assessment data – not only highlights the limitations inherent in current reporting approaches, but also proposes an alternative for enhancing the reporting and use of student’s assessment results. specifically, this study has extended the reporting on assessment results beyond mean percentage scores to information-rich, qualitative descriptions of the typical knowledge and skills that characterise student teachers of mathematics in ite programmes. by enhancing the quality of reporting through the use of performance level reporting, the current study does not only help to fathom the depth of the deficit in mathematics knowledge among student teachers, but it also contributes a response to deacon’s (2012) call for the establishment of benchmarks to diagnose knowledge gaps in student teachers of mathematics. proactively, institutions will be able to use performance level reports to develop relevant interventions to address specific learning needs of students in mathematics classes, set improvement targets for themselves, and also assist individual students to do the same. the use of pls in reporting will also provide evidence-based benchmarking across institutions which in turn will create a basis for professional collaboration among staff from the different institutions. the primted project provides both the data that can be used among universities that have diverse histories, and the opportunity for exploring ways of monitoring the effectiveness of teacher training. in particular, our study opens vistas for effective ways of reporting assessment results using standards-based formats to overcome the weaknesses inherent in traditional formats of reporting results. to further the primted research agenda, the primted assessment workstream will initiate a process to seek agreement on the plds. we envisage a process of detailed engagement with the professional standards emerging from across the mathematics workstreams. this includes consideration for the assessment framework and test items. refining the standards, the assessment instruments used to measure these standards and the related plds will require collaboration – working across universities and across the primted workstreams. we think such a collaborative process will contribute to securing meaningful buy-in from university lecturers on the preferential use of standards-based reporting. while this article offers an illustration of how this may be done, the university lecturers themselves will be engaged further on the performance level descriptions and the report format. we consider such buy-in and meaningful engagement to be essential for the effective use of the information that flows from such reports to help students learn better. the implications of adopting a common standards-based reporting format does not only hold good prospects for professional collaboration among universities and the lecturers, but it also creates opportunity for establishing benchmarks for monitoring the effectiveness of teacher development in south africa. with common standards-based benchmarks it will be possible to provide inter-institution monitoring and support so that there will be common expectations on the knowledge and skills that teacher trainees are armed with when they start their teaching careers. this area forms the basis for further research within the primted project. finally, there could be value in providing standards-based reports to schools and thus allowing schools and mentors of newly graduated teachers to use the reports for supporting the students after they enter the teaching profession. acknowledgements competing interests we declare that we have no financial or personal relationships that might have inappropriately influenced our writing of this article. authors’ contributions this article was developed collaboratively by the three authors. q.m.m. did the initial literature review, led in technical aspects related to standard setting and prepared the manuscript to comply with publication standards. a.k. led the conceptualisation of the article, prepared the data for analysis, analysed the data and guided the writing of the manuscript. n.r. described the primted test design process and prior reporting on the test outcomes, framed the test theoretically in relation to how the instrument was developed, reviewed the level descriptors and led in the quality assurance of the manuscript. funding information this publication was made possible with the support of: (1) the teaching and learning development capacity improvement programme (tldcip) which is being implemented through a partnership between the department of higher education and training and the european union, and (2) the assessment for learning niche area, located within the faculty of 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(2006). a primer on setting cut-scores on tests of educational achievement. educational testing service. retrieved from https://www.ets.org/media/research/pdf/cut_scores_primer.pdf feza & webb 36 pythagoras 62, december, 2005, pp. 36-47 assessment standards, van hiele levels, and grade seven learners’ understandings of geometry nosisi feza and paul webb department of science, mathematics and technology education, university of port elizabeth email: nosisi.feza@nmmu.ac.za and paul.webb@nmmu.ac.za a number of researchers in mathematical education assert that the instruction in geometry offered in south african schools is inadequate and that traditional teaching strategies do little to promote teachers understandings of their learners’ levels of mathematical thought. van hiele specifically states that the inability of many teachers to match instruction with their learners’ levels of geometrical understanding is a contributing factor to their failure to promote meaningful understandings in this topic. this study investigated whether a sample of grade seven learners in previously disadvantaged primary schools met both the assessment criteria for geometry as stated by the south african revised national curriculum statement and the implied van hiele thinking levels. the data generated suggest that none of the 30 learners who participated in this study had attained these requirements and that language competency in general is a barrier to the attainment of higher levels of understanding amongst this group of secondlanguage learners. it is suggested that not only van hiele levels and assessment standards, but also learners’ cultural background and their specific use of words in the vernacular context, need to be taken into consideration by teachers when developing learning programmes. possible strategies to meet these requirements are suggested. introduction internationally there appears to be widespread realisation that teaching in mathematics in general has failed to overcome a number of cognitive barriers to learner understanding (carpenter & fennema, 1988; kempa, 1993; peterson, 1998). likewise, a number of researchers in mathematical education assert that the instruction in geometry offered in south african schools is inadequate in terms of providing learners with the necessary skills needed to operate at the level of axiomatic thinking required for most high school courses (mcauliffe, 1999). mansfield and happs (1996) note that traditional teaching strategies do little to promote teachers understandings of their learners’ levels of mathematical thought and van hiele (1986) specifically states that the inability of many teachers to match instruction with their learners’ levels of geometrical understanding is a contributing factor to their failure to promote meaningful understandings in this topic. a comparison of the assessment standards of the south african mathematics curriculum with van hiele descriptors suggests that in terms of geometry south african learners who have completed primary school should have reached van hiele’s thinking level two, i.e., they should be “able to describe and represent the characteristics and relationships between 2-d shapes and 3-d objects in a variety of orientations” (department of education, 2002: 6). this study investigates whether a sample of grade seven learners who had recently completed their intermediate phase of schooling in previously disadvantaged primary schools meet both the assessment criteria for geometry as stated by the intermediate phase revised national curriculum statement (department of education, 2002) and the requirements of van hiele thinking at level two. this has been done in order to get an indication of the probable (but not generalisable) level of learners’ understanding in geometry at the end of their primary schooling and, as such, to raise teachers’ awareness of what may be reasonably expected of children who have come through previously disadvantaged south african primary schools. background and significance it has become apparent since the release of the de lange report in 1981 that the problems surrounding science and mathematics education contribute significantly to the current south african national crisis in education. after the installation of the nationalist government (and consequent adoption of apartheid as a national policy) in south africa in 1948, a system of ‘bantu education’ was introduced for black south africans (samuel, 1990). bantu education was for black people and was to be largely based in the nosisi feza and paul webb 37 bantustans or homelands where ‘natives’ would be prepared for a life in reserves (davies, 1986), with different and inferior curricula – usually with no science or mathematics offerings (hartshorne, 1992). the impact of that system can be seen even today and the children participating in this study all had teachers who are products of bantu education. these teachers were under-qualified to teach mathematics and were teaching in under-resourced schools, mainly in rural areas of the eastern cape. this study should make a contribution to our understanding of these children’s understanding of geometry after a period of learning in this context and, hopefully, provide some indicators of what is needed in order to successfully teach geometry to children in similar schools and social conditions. van hiele and the development of geometrical thinking van hiele (1986) hypothesised five sequential levels of geometric reasoning. they are labelled visualisation, analysis, informal deduction, formal deduction, and rigour, which describe the characteristics of the thinking process (burger & shaughnessy, 1986). the model suggests that, assisted by appropriate instructional experiences, the learner moves sequentially from the initial or basic level (visualisation), where space is simply observed and the properties of figures are not explicitly recognised, through the sequence listed above to the highest level (rigour), which is concerned with formal abstract aspects of deduction (fuys & liebov, 1997). at level one of the van hiele hierarchies the analysis of geometric concepts begins (visualisation is at level zero). for example, through observation and experimentation learners begin to discern the characteristics of figures. these emerging properties are then used to conceptualise classes of shapes. learners at this level cannot yet explain relationships between properties, interrelationships between figures are still not seen, and definitions are not yet understood. at the level of informal deduction (level two) learners are able to establish the interrelationship of properties both within figures (e.g., in a quadrilateral, opposite sides being parallel necessitates opposite angles being equal) and among figures (a square is a rectangle because it has all the properties of a rectangle). thus they can deduce properties of a figure and recognise classes of figures. class inclusion is understood and definitions are meaningful. informal arguments can be followed and given but the learner at this level does not comprehend the significance of deduction as a whole or of the role of axioms. empirically obtained results are often used in conjunction with deduction techniques. formal proofs can be followed, but learners do not see how the logical order could be altered nor do they see how to construct a proof starting from different or unfamiliar premises. at level three learners understand the significance of formal deduction as a way of establishing geometric theory within an axiomatic system. they are able to see the interrelationship and role of undefined terms, axioms, postulates, definitions, theorems, and proof. learners at this level can construct, not just memorise, proofs; they accept the possibility of developing a proof in more than one way. the interaction of necessary and sufficient conditions is understood; distinctions between a statement and its converse can be made. level four (rigour) is the highest van hiele level. at this level learners can work in a variety of axiomatic systems, that is, non-euclidean geometries can be studied, and different systems can be compared. geometry is seen in the abstract. what is important in terms of pedagogy is that, as wirszup (1976) suggests, people at different levels of mathematical understanding speak, use and understand terms differently, and that teachers often use terms that can only be understood by learners who have progressed to the third or fourth van hiele level. consequently, when trying to communicate with learners who operate at lower levels, their intentions may be completely misunderstood. a major purpose for distinguishing learners’ levels of understanding is to recognise obstacles that they may experience in the learning process, and to allow teachers to develop strategies which will enable children to progress in terms of conceptual development (bishop, 1997). austin and howson (1979) confirm that there is a major difference in mental preparation for mathematics learning between a learner whose language makes use, in some recognisable form, of international greek-roman terminology, and its prefixes (pre-, post-, anti-, sub-, co-, mono-, etc.), suffixes (-ation, -or, -ant, -ise, etc.) and roots (equ, arithm, etc.), and a learner whose language contains neither these items nor translation equivalents of them. van hiele levels and the revised national curriculum statement the intermediate phase assessment standards for geometry as expressed in the south african revised national curriculum statement (rncs) documents require that learners are able to name assessment standards, van hiele levels, and grade seven learners’ understandings of geometry 38 shapes, describe and/or classify shapes using properties, and construct shapes correctly in order to attain learning outcome three, i.e., that: the learner is able to describe and represent the characteristics and relationships between 2-d shapes and 3d objects in a variety of orientations and positions. (department of education, 2002: 6) at van hiele level zero (fuys & liebov, 1997) the learner identifies, names, compares and operates on geometric figures according to their appearance. similarly the rncs (department of education, 2002) assessment standards, which are guided by van heile levels, are characterised by the naming and visualising of shapes and objects in natural and cultural forms. as such, both van hiele and the rncs assessment standards characterise this level by recognition of the shape as a whole. van hiele level one is characterised by the analysis of figures in terms of their components and their relationships, a stage which allows learners to discover properties/rules of a class of shapes empirically. the characteristics of the rncs’s assessment standards are the definition of shapes and objects in terms of properties such as their faces, vertices and edges. the characteristics of both the van hiele level and the assessment standards are concurrent in that they define shapes and objects using their properties. at the informal deduction level (van hiele level two) learners logically relate previously discovered properties/rules by giving or following informal arguments such as drawing, interpreting, reducing, and locating positions. this fits well with the rncs assessment standards which state that learners must be able to provide informal arguments such as drawings, interpretations, and the reducing and locating of positions. the first three van hiele levels (levels zero to two) cover all the assessment standards of the intermediate phase as stated in the rncs (department of education, 2002). therefore the exit level outcomes for learners in the intermediate phase of the south african curriculum can be related to the expectations of van hiele level two. methodology this study aimed at eliciting learners’ understanding within the notion of respecting and recognising the uniqueness of each individual. open and flexible semi-structured interviews were used in order to allow the interviewer to understand how individuals experience their lifeworld (welman & kruger, 1999). bless and smith (1995) support this technique by stating that it helps to clarify concepts and problems and it allows for discovery of new aspects of the problem by investigating in detail some explanations given by the respondent. teachers and schools all of the six schools that participated in this study had relatively common backgrounds in terms of the type of mathematics teachers on their staffs. similarities between the teachers were drawn in terms of the type of mathematics education programmes they had attended, the level of their qualifications in mathematics education and the institution that had provided the educational programmes. the schools participating in this study were divided into three socio-geographic groups: urban schools, peri-urban schools, and rural schools. each socio-geographic group was represented by two schools with each school contributing five learners to the study, i.e., a total of 30 learners. initial inspection of the interview data revealed the learners could be grouped into three van heile categories according to the criteria described earlier. the first category, learners who are strictly on thinking level zero of van hiele levels, came mostly from rural schools with two from periurban and one from an urban school. there were ten learners in this group. the second group of learners were located in a band from thinking level zero but progressing to van hiele level one. this group of 15 learners were mostly from urban schools; nine came from urban schools, three from peri-urban schools, and three from rural settings. the third group consisted of five learners, all from the same school in a periurban setting. this third group had attained level one of van hiele’s hierarchy. selected learners all learners were interviewed, but three learners (each as the most representative individual from each of the three categories described above) were selected for in-depth analysis and description. each selected participant’s background is given below. learner a is a girl from a village school in the ex-transkei area. four teachers from this school have completed a degree or a diploma in mathematics and science education with the department of science, mathematics and technology education (smate) at the university of port elizabeth (upe). one of the teachers was nosisi feza and paul webb 39 also involved in a diploma (fde) offered by smate as part of the open society science and mathematics initiative (ossmi) that was initiated to train key teachers especially for rural schools. this teacher upgraded his fde to a bachelor of education degree in mathematics and science through upe and is now a principal of this school. the school is under-resourced, having twelve classrooms and a very small office with muddy floors for the principal. there is no electricity or water in the school. learner b is a girl who attended a citytownship school. two teachers from this school had attended non-formal mathematics education programmes offered by upe for in-service teachers. one of these teachers was awarded a british council bursary to study at leeds university in england. this teacher has subsequently been involved in running workshops for other mathematics teachers in his own and surrounding schools and has been promoted to deputy principal. compared to the other schools participating in this study the school was relatively well resourced as it had a computer room with computers, a photocopier, water, and electricity. learner c is a boy who attended a township school in a small town situated between the exciskei and transkei. six teachers in this school had obtained their mathematics and science degree or diploma from upe through smate. the learner’s mathematics teacher has upgraded her qualifications to the bed level and had been running a number of workshops for teachers in her area and had recently been promoted to head of department for mathematics in her school. the school was well maintained but lacked resources such as computers and a photocopier. manipulatives osborne and gilbert (1979) suggest a shared external focus of pictures and ‘hands on’ manipulatives provide a comfortable focus for conversation ranging widely over the rich prior experiences of the learner. manipulatives were used during the interviews conducted in this study in an attempt to reveal a range and variety of learner understandings. the manipulatives used in this study consist of a picture of a convent school in lady frere in the eastern cape which allowed the recognition of shapes in complex situations, picture cards of 2-dimensional shapes from threesided shapes to eight-sided shapes. the shapes were made out of thin coloured cardboard. construction sticks were also used to allow learners to construct shapes and also to probe learners’ understandings of concepts of shapes through constructions during the interview. these techniques allowed follow-up questions to be asked which explored the reasons for initial responses (carr, 1996). in this way the use of the manipulatives helped the researcher match learners’ thinking with van hiele levels and the rncs assessment standards. interview protocol and approaches as this study explores the learner’s interpretation of geometrical concepts, the general framework of activities using manipulatives and questions were prepared prior to the interview. the interview began with a request for the learner to respond with his or her own understanding of a focus concept by discussing the manipulatives as they were successively revealed, a method championed by carr (1996). as it is impossible to remember all of the complex ideas explored during a lively interview, and attempting a written record takes attention away from careful listening and responding (carr, 1996), these interviews were recorded on an audiotape recorder and a video recorder. the audio-tape recorder was used to record verbal communication between the interviewer and the interviewee. then the video recorder was used to take pictures of constructions of geometrical shapes constructed by the interviewee as a response to some questions asked by the interviewer. fuys et al. (1988) point out the value of video recording these activities as many characteristics are not expressed verbally and may be missed during the interview process. data collection data collection and initial data analysis occurred simultaneously as the researcher interacted with the participants. data analysis included triangulation of on-site observations of learners’ geometrical constructions using the manipulatives provided, the interviews held with the learners during this process and the video and audio-tape recordings of the sessions. although data collection and initial analysis happened at the same time, synthesis across data sources only occurred when data collection for the entire study was completed. both xhosa and english was used during the interviews because second language translation and code switching (using two different language within the conversation) on the part of both teacher and learners occurred regularly (bantwini, england, feza, foster, lynch, mgobozi, peires, & webb, 2003). only learner c used english assessment standards, van hiele levels, and grade seven learners’ understandings of geometry 40 exclusively throughout the interview. the other two learners employed code-switching, with both the interviewer and the participants slipping between xhosa and english during interview. one of the major difficulties of research in which the language of the people under study is different from that of the report is gaining conceptual equivalence or compatibility of meaning between the researcher and the participant (deutscher, 1968; whyte & braun, 1968; sechrest, fay & zaidi, 1972; temple, 1997). vulliamy (1990) asserts that one of the factors to be considered during translation is the researcher’s knowledge of the language and the culture of the people under the study, in this case the learners. in this study translation of the data was done by the researcher who is fluent in xhosa and who shares the same culture and language of the participants. as participants become tired if they are involved for a lengthy period, the interviews were confined to thirty minutes for each learner. results it was evident from the initial 30 interviews that the learners were clearly not at the same levels of verbal expression, mathematical terminology, or understanding of geometry. the selection of the learner episodes that are used to exemplify the participating learners’ levels of understanding was done by first grouping the 30 learners into three appropriate van hiele thinking levels as described under ‘methodology’ above. ten learners were strictly at level zero, 15 between zero and one, while five learners had fully attained level one (table 1). a ‘representative’ learner for each group was chosen according to the richness and clarity of the responses that they gave during their initial interviews and also in terms of their apparent suitability for representing the other learners in the category i.e., they gave answers that were clear examples of the criteria used to categorise the responses in terms of van hiele levels. these learners’ responses are presented as three episodes. the episodes that follow provide examples of verbal expression (their words), mathematical terminology, and apparent level of understanding for each learner. in the presented episodes r stands for researcher and l a, l b, and l c stand for learners a, b and c respectively. episodes with learner a learner a did not recognise any shapes in the picture of the convent school. as such the following questions were asked: r: do you know the meaning of the word shape? l a: yes i do. r: what do you call this? [showed the learner a rectangular piece of cardboard] l a: right angle r: can you show me other shapes in the picture besides the one you have mentioned? l a: pointed to a square but did not name it. when learner a was shown picture cards of squares, rectangles, and triangles and asked to name them, she kept quiet. when shown each card one at a time, again she kept quiet and just shrugged her shoulders and gave no verbal response. during construction activities learner a was able to construct a rectangle using the construction sticks provided. the following questions were asked, and these answers given, during the construction process: r: what must happen to the angles of your rectangle as you construct it? l a: i do not know angles but i know that there should be four corners. r: what must happen to the sides of your rectangle? l a: two sides should be equal and other two should also be equal. van hiele level school type level zero level zero→ 1 level 1 rural school 7 3 peri-urban school 2 3 5 urban school 1 9 table 1. distribution of number of learners at different van hiele levels per school type (n=30) nosisi feza and paul webb 41 when asked to construct a square, learner a again constructed a rectangle. then the following questions were asked: r: tell me every step you do while constructing the square. l a: i take two equal sticks and other two equal sticks. then the two equal sides should look at one another. when asked to construct any triangle, learner a constructed an equilateral triangle and explained that it should have three sides. when given a selection sheet of different shapes and asked to identify rectangles, she pointed to all the rectangles and stated that she recognises a rectangle by four corners. when asked to identify parallel lines, learner a showed understanding that lines are parallel if the distance between them is equal. this was evident by the fact that she gave an example of power lines but did not use the mathematical word for the concept. when asked to identify parallel lines in a selection sheet, she did not respond. learner a was presented with a number of shapes that had been sorted by number of sides and angles into quadrilaterals, triangles, pentagons, hexagons, and octagons. she was asked to guess the rule by which the shapes had been sorted. in response she suggested that the shapes are sorted according to similar colour. she went further by saying that “the shapes look alike”. when the researcher again asked the learner to describe the sorting procedure and to give names to the shapes in each group, her response was “only triangles – other shapes are not the same”. the researcher then showed the learner a group of quadrilaterals and asked the following: r: how could we place these shapes into groups that belong together? l a: [silent, she was sorting the shapes.] r: how are you sorting the shapes? l a: [silent, she sorted squares together, rectangles together, parallelograms and rhombi together, and trapeziums together.] r: [pointing a group of squares.] if you were talking with your friend over the phone and you wanted to describe these pieces, what could you say about them? l a: the sides are equal like a box. questions on quadrilaterals followed: r: when you sorted the first set of shapes, do you remember that you had a group of triangles, and one of the quadrilaterals or four-sided figures, and five-sided ones, and sixsided ones. where would all the shapes on the table belong? l a: four-sided shapes. although she referred to the quadrilaterals as foursided shapes, a response reflecting van hiele level one, it was the only time she did this, and it is possible that she copied the researcher’s words when she posed the question. the following inclusion activity questions were asked and the following responses were received. r: [the researcher picks up a square from a selection of figures sorted into squares, rectangles, parallelograms, trapeziums, triangles and polygons.] so i could move this square to the quadrilateral or foursided shapes – a square is a special kind of a quadrilateral. what makes it special? l a: [says nothing.] r: can we move the square to the rectangle group? l a: yes. r: why? la: because it has four corners. r: someone yesterday said that a square is a special kind of a rectangle with equal sides. do you think she could be right? l a: yes, because it has four corners. r: what do you think she said when she put a square in a parallelogram pile? l a: [silent.] r: would she have put a rectangle in the parallelogram pile? what would she have said? l a: no. r: would it be possible to call every square a rectangle? la: yes, because you can put squares together they make a rectangle. r: would it be possible to call every rectangle a square? l a: no, because a rectangle has two equal sides but the square has four equal sides. this learner has not attained sufficient mathematical vocabulary in order to express herself coherently. her conceptual understanding also appears to be under-developed because she used sorting according to colour, and sorting according to a “look like” basis. there was no use of properties at all in her statements, her definition of shapes confirmed this as she defined a square as assessment standards, van hiele levels, and grade seven learners’ understandings of geometry 42 something with four sides like a box, and did not seem to be aware that a box is a 3-dimensional object not a 2-dimensional shape. the learner could only recognise a square, could not name shapes, could not describe or classify shapes using properties, could not construct shapes correctly and, as such, was considered not to have progressed beyond level zero in terms of shapes of objects or have achieved the assessment standards or have attained learning outcome three from the rncs which states that: the learner will be able to describe and represent characteristics and relationships between two-dimensional shapes and three-dimensional objects in a variety of orientations and positions. (rncs, 2002: 48) in terms of language as an obstacle in learning geometry learner a’s responses show that her development is blocked by language. when she has to name shapes she becomes quiet but when she is asked to construct these shapes she does not struggle. therefore the fact that she cannot name them does not mean that she does not know them because she constructs the shapes using correct properties. episodes with learner b learner b recognised the shapes from the picture and selection sheets and named them correctly with the exception of parallelograms. in this case she simply looked at them and shook her head. she constructed a rectangle, defined it as a shape with four corners and went further by mentioning that two opposite sides are equal (not two pairs of opposite sides). she constructed a square and defined it as a shape with four equal sides and constructed a triangle, defining it as a shape with three sides. learner b showed understanding of the concept of parallel lines and also gave an example of electric power lines. when identifying parallel lines in a selection sheet she identified them as “equal lines”. during sorting activities the following conversation took place between the researcher and the learner: r: [the researcher showed the learner a collection of cardboard cut-out polygons.] these shapes came from several different boxes but they got all mixed up. this is how someone tried to put them back in groups, which belong together. [she then sorted just the first four shapes according to equal number of sides. picking up a square she said,] “can you guess where this one belongs?” l b: [silent.] r: can you describe how the pieces were sorted and what their names are? l b: triangles are together. r: [the researcher showed the learner a group of quadrilaterals.] how could we place these into groups of things that belong together? l b: we can group them into squares, rectangles, and rhombuses. r: how are you sorting these shapes? l b: according to shapes. r: [she pointed squares.] if you were talking with your friend over the phone and you wanted to describe these pieces, what could you say about them? l b: a square looks like a box, and has four corners. the learner also described a rectangle as a longer square but could not describe a parallelogram. she recognised squares and said “they look like a box and have four corners”. the researcher showed the learner a group of quadrilaterals and continued the conversation as follows: r: when you sorted the first set of shapes, do you remember that you had a group of triangles, and one of the quadrilaterals or four-sided figures, and five-sided ones, and sixsided ones. where would all the shapes on the table belong? l b: four-sided group. r: [she picked up a square.] so i could move this square to the quadrilateral or four-sided group. a square is a special kind of a quadrilateral. what makes it special? l b: [silent.] r: can we move the square to the rectangle group? l b: yes. r: why? l b: they all have four corners. r: someone yesterday said that a square is a special kind of a rectangle with equal sides. do you think she could be right? l b: [silent.] nosisi feza and paul webb 43 the researcher probed for a response but learner b kept quiet until the researcher suggested that someone put the rectangle in the parallelogram pile and asked what the learner thought the reason for that was. the learner’s response was that parallelograms look like rectangles and have four corners. then the researcher continued. r: would it be possible to call every square a rectangle? could i move a rectangle to the square group? l b: yes, because it has four corners. r: would it be possible to call every rectangle a square? l b: yes, because it has four corners. when asked to guess the rule used by the researcher when sorting using parallel lines, the learner described parallelograms as straight shapes and trapeziums as shapes made from triangles. to her, angles were corners. it is possible that language may have constituted a further barrier to learning. at school, a rectangle is called “uxande” in xhosa, whereas at home “uxande” is used to describe a rectangular house. because the word “angle” is not used in vernacular xhosa, this may have created difficulties in seeing a difference between a square and a rectangle as each has four corners. learner b’s thinking differs from learner a’s in that she used the correct names for shapes and recognised all except for the parallelogram. her concept of shapes is moving towards level one of van hiele’s levels. however, the language that she used suggested that there was still a need for further development to meet all the requirements of level one. her definitions and descriptions suggested that she had an adequate understanding of shapes and their properties, but that she lacked the vocabulary to express her understanding, therefore she struggled to define the sorting of shapes and was not able to find words to describe what she saw and did. learner b defined shapes on a ‘look-like something’ basis. when she defined a trapezium she described it as “a shape made from a triangle”. to her the word ‘parallel’ is synonymous to ‘straight’. thus it appeared she lacks vocabulary rather than understanding and this was possibly the factor that prevented her from being categorised at van hiele level one. van hiele (1986) notes that each level of thinking has its own language standard that needs to be attained. the rncs assessment standards require learners to describe and classify shapes using properties and this learner lacked the vocabulary to do this. the rncs assessment standards also require learners to describe the relationship between shapes using properties of rotations, reflections, and translations. this learner identified the number of corners as the common element for all quadrilaterals, but made no reference to rotation, reflection or translation in her argument. episodes with learner c learner c recognised all the shapes (octagon, hexagon, and rhombus) in the picture of the convent school and was able to name them correctly. he also recognised triangles, squares, rectangles, and parallelograms from picture cards and named them correctly. during construction activities, this learner constructed a rectangle correctly. the researcher asked follow-up questions as follows: r: what must happen to the angles of your rectangles as you construct it? l c: all four angles must be 90 degrees. r: what must happen to the sides of your rectangle? lc: two opposite sides should be equal. when asked to construct a square, learner c took four stick lengths and mentioned that all four sides of a square should be equal. then he was asked to construct a triangle. he constructed a triangle stating that it must have three sides. when asked to observe a selection sheet of shapes, learner c pointed to rectangles and stated that they are rectangles because they have two opposite sides that are equal. during parallel line activities this learner showed an understanding of the concept and gave a relevant example of electric power lines. when asked to guess the researcher’s rule in sorting activities, learner c suggested that sorting had been done according to sides. when describing sorting he mentioned that shapes are grouped into “quadrilaterals, triangles, hexagons, and hotagons” (in the place of octagons). the researcher then showed the learner a group of quadrilaterals and asked: r: how could we place these [showing the learner different quadrilaterals) into groups of things that belong together? l c: rectangles together, squares together and rhombuses together. r: how are you sorting the shapes? l c: according to shapes. when learner c was asked to describe a square to a friend who could not see the shape, he said “a square is a shape with four equal sides” and assessment standards, van hiele levels, and grade seven learners’ understandings of geometry 44 described a rectangle as “a shape with two pairs of parallel sides and two pairs of equal sides”. a parallelogram was described as “a shape with two angles that are less than 90 degrees and two angles that are greater than 90 degrees”. he also noted “a parallelogram also has four sides”. the researcher continued with quadrilateral activities in the following manner: r: [the researcher showed the learner a group of quadrilaterals.] when you sorted the first set of shapes, do you remember that you had a group of triangles, and one of the quadrilaterals or four-sided figures, one of five-sided figures, and one of six-sided figures. where would the shapes on the table belong? l c: four sided group. r: [she picked up a square.] so i could remove this square to the quadrilateral or four-sided group – would you accept it if i say a square is a special kind of a quadrilateral. if you do, what makes it special? l c: [silent.] r: can we move the square to the rectangle group? l c: yes. r: why? l c: because a square has four sides. r: someone said yesterday that a square is a special kind of a rectangle with equal sides. do you think she could be right? l c: [silent.] r: what do you think she said when she put a square in a parallelogram pile? l c: [silent.] r: might she have put a rectangle in the parallelogram pile? what would she have said? l c: a parallelogram is a slanting rectangle. r: would it be possible to call every square a rectangle? could i move this rectangle to the square group? l c: no. r: why? l c: the square has four equal sides. r: would it be possible to call every rectangle a square? l c: no. r: why? l c: because the rectangle has two long sides and two short sides. the researcher sorted a number of quadrilaterals according to parallelism (without including any squares or rectangles) and asked the following questions: r: look at my sorting. can you guess my rule? how did i sort the shapes? l c: the rule is shapes with two pairs of parallel lines together and shapes with one pair of parallel lines together. therefore the shapes are grouped in parallelograms and trapeziums. during the entire interview this learner responded fluently in english. he recognised shapes and used correct mathematical terminology when naming the shapes. when naming the shapes he used his hands to indicate the shape he was mentioning by pointing at the picture and drawing the shape in the air. he used the correct terms for each shape, except the octagon which he named “a hotagon” which may be either a tongue fluency problem or a sound based problem (bantwini et. al., 2003). learner c defined shapes using angles, sides, and parallelism, showing that he had a good grasp of these geometric concepts. his constructions of shapes showed that he had an understanding of accuracy as he measured by comparing the sticks before constructing the shapes. however, this learner did not appear to notice any relations between shapes when sorting them, as he sorted the shapes using their individual properties without commenting on any common properties. this suggests that his level of thinking was at van hiele level one, i.e., that he had attained both levels zero and one, but has not progressed from level one towards level two. he still defined shapes in an isolated way and did not use the properties of rotations and the vocabulary as required by the rncs assessment standards. discussion the episodes described above with the grade seven learners, who had been identified as being representative of the three categories, suggest that none of the 30 learners had attained the requirements of the rncs assessment standards or could be categorised at van hiele level two. burger and shaughnessy (1986), as well as dickson, brown and gibson (1984) assert that many learners in the middle years of schooling have severe misconceptions concerning a number of important geometric ideas. de villiers and nosisi feza and paul webb 45 njisane (1987), and govender (1995), in their studies in south african schools indicate that grade 12 learners and high school learners in general are still functioning more at concrete and visual levels than at an abstract level in geometry, in spite of the fact that the national school exit examination requires a clear understanding of underlying abstract processes. de villiers (1987) notes that this, and the fact that the transition from concrete to abstract levels of thinking, poses “specific problems to second language speakers”. de villiers (1987) also notes that, “since success in geometry also involves the acquisition of the technical terminology”, there is little wonder that our learners perform so poorly. findings in this study also suggest that language competency in general is a barrier to the attainment of higher levels of understanding amongst this group of learners, over and above van hiele’s notion that each level of geometrical thinking has its own language (fuys & liebov, 1997). this raises the issue of how to overcome language as a barrier for learners who speak english as a second language, a universal feature in multilingual societies. what is important in terms of pedagogy is that people at different levels of mathematical understanding speak, use and understand terms differently, and that teachers often use terms that can only be understood by learners who have progressed to the third or fourth van hiele level (wirszup, 1976). consequently, when trying to communicate with learners who operate at lower levels, teachers’ intentions may be completely misunderstood. as such it is crucial that geometry teachers investigate their learners’ understanding in order to be able to provide meaningful learning experiences at their particular level of development. one way of enabling teachers to use van hiele levels and the rncs assessment standards to establish their learners’ levels of geometrical understanding is to provide experience of van hiele levels in preand in-service training opportunities, by engaging them in activities that require classifying of answers by van hiele level, and by challenging them to match these responses to the rncs assessment standards. not only van hiele levels and assessment standards, but also learners’ cultural background and their specific use of words in the vernacular context, need to be taken into consideration by teachers when developing learning programmes. a possible strategy to attain all the requirements noted above is an inquiry approach where teachers and learners engage in conversations that allow teachers to ascertain prior knowledge, language use, cultural frameworks and levels of understanding of learners in terms of a particular topic, and which in turn allow learners to discern the direction further study will take (lindquist, 1987). finally, the fact that the three different learners interviewed were from three different schools, yet all mentioned telegraph lines as examples of parallel lines suggests that their teachers possibly rely on limited knowledge from textbooks or training courses that they have not adapted or elaborated for themselves. this raises the question of whether we are exploring learners’ understandings of geometry during interviews or merely hearing what the learners think school mathematics is all about or what their teachers would expect them to say. the relationship is complex, but it seems reasonable to propose that teachers relying on limited knowledge would 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(1976). breakthroughs in the psychology of learning and teaching geometry. in j. l. matin (ed.), space and geometry: papers from a research workshop (pp. 75-97). colombus, ohio: eric. “we teach best what we most need to learn.” richard bach 12 pythagoras 62, december, 2005, pp. 12-22 the role of technology in fostering creativity in the teaching and learning of mathematics balarabe yushaua, andile mjib and dirk c. j. wesselsb king fahd university of petroleum & minerals a , and university of south africa b email: byushau@kfupm.edu.sa, mjia@tut.ac.za, and wessedcj@unisa.ac.za the paper looks at interrelationships between creativity and technology in the teaching and learning of mathematics. it suggests that a proper use of various technologies especially computers in the teaching and learning of mathematics has the potential of helping learners to develop their creativity. the technologies can provide an atmosphere under which mathematical skills can be extended beyond the ability to calculate or reproduce problems and enable learners to investigate, analyse and interpret problems at hand. furthermore, with computers learners can use an experimental approach to deal with mathematical problems, which can lead to conjecture, pattern finding, examples and counter examples. in fact, if used effectively, computational aids can help in improving learners’ intellectual ability and hence mathematical achievement while fostering the requisite creativity not found in the traditional approach. introduction anyone who presumes to describe the roles of technology in mathematics education faces challenges akin to describing a newly active volcano – the mathematical mountain is changing before our eyes, with myriad forces operating on it and within it simultaneously (kaput 1992: 515). despite their side effects, technological advancements have produced a multitude of machines and gadgets with fascinating functions, features, and options. such machinery and gadgetry are aimed at changing human lives and improving our varied endeavours. the impact of these technological innovations is glaring. in communication terms, the world is now a global village, where the internet, satellites, and mobile telephones are commonly used. the latter years of the past century have brought a vast improvement and unprecedented growth in information dissemination. transportation has been vastly improved, now it takes a few hours to travel thousands of kilometres. similar improvements have taken place in the health and educational sectors. education is one sector that has most benefited from current technological advancement. with this development, time and space are no longer barriers to education. as a matter of fact, the concept of distance learning has been revolutionised to what is now known as e-learning, blended learning or web-based learning programs. in what is now popularly known as distributed learning, people use a wide range of computing and communications technology to provide learning opportunities beyond the time and place constraints of the traditional classroom. however, the real impact of technological advancement is not the superficial, glitzy or flashy attributes of these machines. the true transformation technology has brought is in its support of the human day-to-day activities. all this has had an impact on real human cognition. it has affected the way we think and in some respects altered the way we think. it should be remembered that all advancements are a product of human creativity. this has led to arguments about what accelerates what. is it human creative thinking that generates technological advancement or is it the other way round? perhaps more pertinently, how do people think and engineer these technological breakthroughs? how is their thinking different from others? what are the factors that trigger this kind of creative thinking? what are the roles of nature and nurture in creative thinking? how can we promote this kind of “special” thinking in our classrooms in general and in the mathematics classroom in particular? in this paper we examine the interrelationship between creativity and technology in the teaching and learning of mathematics. it is argued that technology, especially computer technology, has the potential for creating a conducive environment and an atmosphere that can engage learners in genuine mathematical activities that can foster creativity. furthermore, these technologies can equip teachers with the necessary tools for creative teaching that can assist in promoting creativity in their learners. to buttress these points, this paper is divided into seven subsections. after the introduction, the second section briefly discusses balarabe yushau, andile mji and dirk wessels 13 educational technology. the third section deals with creativity in the teaching and learning of mathematics, while the fourth section gives four areas in which creativity can be promoted in the mathematics classroom. the fifth section discusses the role of technology in developing a quasiempirical view of mathematics. the sixth section reflects on the potential of technology in fostering creativity in the teaching and learning of mathematics. the last section offers a concluding remark. educational technology the website www.dictionary.com defines technology as: • the application of science, especially to industrial or commercial objectives. • the scientific method and material used to achieve a commercial or industrial objective. based on this, educational technology can be defined in two ways. first, it has to do with the instruments used in classrooms for the achievement of educational objectives. second, it concerns the methodology and materials used in the application of instructional principles for the achievement of educational objectives. from these definitions, educational technology encompasses many everyday and familiar objects that may be taken for granted in the teaching and learning context. examples of these are books, blackboards, charts, maps and models. in fact, it has been argued that rewarding a child with a smile and saying well done when the child performs an act that conforms to the teacher’s intention is employing educational technology (heimer, 1972). the argument is that the teacher acts as an instructional tool to carry out techniques designed to achieve the teacher’s goal for the child’s behaviour. implicit in the above definitions are media borne communication and instructional materials such as films, filmstrips, slides, television, calculators, computers, etc. (heimer, 1972; carnegie commission, 1972). the computer and the internet/web have revolutionised the world of educational technology because they have fully integrated the features of other media for easier and more user friendly instructional use. the dominance of the computer is in fact so apparent that instructional aids can now be categorised as computer-related and non computer-related material. educational technology has made available different training methods that have hitherto been difficult for human instructors to use. as a matter of fact, the introduction of technology into the educational system has been considered a major revolution in education (see carnegie commission, 1972). in particular, the role of technology in mathematics education is so tremendous that it is difficult to describe, while the changes are so rapid that they are difficult to follow (kaput, 1992). however, both advocates and sceptics of technology are keenly interested in what mathematics education research has to say about it in the teaching and learning of the subject (cangelosi, 1996). as papert (1990) rightly notes, “technology is not the cause of anything either positive or negative, but rather should be thought of as a tool that society can use to shape the environment” (in edwards, 2001: 221). however, the real challenge is how to best utilise these technologies to achieve our educational objectives. in other words, how can we use technology to serve as a catalyst for positive change and as an accompaniment to the teaching and learning situation? in this case, technology should be used for “enrichment and improvement of the conditions in which human beings learn and teach” (carnegie commission, 1972: 89), and not as an end in itself. when used in this sense, research has shown that the benefits of using technology such as computers, graphing calculators, etc. are immeasurable and incomparable to traditional teaching approaches (see carnegie commission, 1972; dunham & dick, 1994; roblyer, 1989; kadiyala & crynes, 1998; corcoran, 2000; krska, 2001; setzer, 2001). creativity in the teaching and learning of mathematics the creative act is often portrayed as a mysterious and even mystical process, more akin to divine inspiration than to mundane thought… however, with the advent of contemporary cognitive science, psychology has come much closer to appreciating the mental processes that must participate in the creative act. (simonton, 2000: 152) creativity is a complex phenomenon that is difficult to define (standler, 1998; meissner, 2000: 151). consequently, many experts from different disciplines have resorted to a descriptive approach. simonton (2000) describes creativity as “one of the special ways that human beings display optimal functionality.” quigley (1998: 1) simply puts it as “…the ability to produce something effective and novel.” while standler (1998) resorts to giving the difference between creativity and intelligence visthe role of technology in fostering creativity in the teaching and learning of mathematics 14 à-vis the creative person and the intelligent person. according to him intelligence is the ability to learn and to think, while a creative person does things that have never been done before. a tacit implication in this definition is that most creative people are intelligent but the converse is not always the case. it has been argued (jacob, 1996; de villiers, 2004) that creativity can be categorised into two distinct types, (1) flash out of the blue and (2) process of incremental revisions. in ‘flash out of the blue’, creativity arrives in a sudden warm embrace, leaving one with a giddy sense of inspiration, vision, and purpose which results in a moment of clarity that is both inexplicable and undeniable (jacob, 1996). in the ‘process of incremental revisions,’ creativity is hard work, where one starts with a vague creative seed and spends countless hours of revision and rethinking to hammer out a work through blood, sweat, and tears, but mostly through frustration (jacob, 1996). four points have been cited by ripple (1989, in edwards, 2001: 222) as the basic assumptions that underlie our understanding of creativity: (a) it is a conceptually identifiable phenomenon, (b) it can be measured, (c) it has distinctive characteristics and developmental patterns, and (d) its development can be facilitated through education and training programmes (edwards, 2001: 222). using these assumptions, research has progressed in the area of creativity in four directional fronts: (1) the cognitive process involved in the creative act, (2) the distinctive characteristics of the creative person, (3) the development and manifestation of creativity across an individual lifespan and (4) the social environments most strongly associated with creative activity (simonton, 2000). researchers have reported that creative people possess some special qualities that perhaps help them in their unique way of thinking. in particular, schmalz (1988) has identified four key individualities that are common among the most creative mathematicians. these are: sustained attention, commitment, detachment from results, and growth in inner confidence. other researchers have pointed out that creative personalities are usually restless, rebellious, courageous, diligent, arrogant, and independent (cangelosi, 1996; meissner, 2000). they have a willingness to take risks, overcome obstacles, tolerate ambiguity, do not like to go in a conventional way, have a desire to shake things up, and are dissatisfied with the status quo (sriraman, 2004). creative people tend to prefer situations with a high degree of disorder and irregularity, as they derive their joy by regularising and organising ambiguous things into a unified and meaningful whole. their thinking mode tends more toward introversion and intuition (edwards, 2001). cangelosi (1996) has reported that in the mathematics classroom, mathematics creativity is displayed by learners who think divergently. these are learners who generate ideas, conjectures, algorithms, or problem solutions. he describes divergent thinking as atypical reasoning that is different from the ‘normal’ way of thinking. it occurs in situations where ‘unanticipated and unusual’ responses are anticipated and accepted. in a study conducted in south africa, nakin (2003) for example, reported a link between creativity, divergent thinking and effectively learning geometry. all these point to the fact that motivation, engagement, imagination, relative freedom, independence of thinking, relative originality and flexible thinking are necessary for fostering creative thinking (cangelosi, 1996; meissner, 2000). however, an alarming fact for educators is the rate at which the enthusiasm of young children for mathematics disappears step by step as they grow older (freudenthal, 1973; meissner, 2000). some concerned educators are even asking the question: do people’s mathematical intuitions and skills increase or decrease with school knowledge? (noddings, 1994: 99). it is known that children are generally highly creative, with vivid imaginations; they learn by exploring, risking, manipulating, testing, and modifying ideas until they reach school going age (paul & kathy, 1990). as they enter school, their divergent thinking gradually changes to its antithesis − convergent thinking. convergent thinking is reasoning that produces predictable responses for most people (cangelosi, 1996). this type of thinking results in a steady decline in curiosity and creative activity during the school years. in a study, westby and dawson (1995) examined elementary school teachers’ impact on learner creativity and found that teachers’ ratings of their favourite learners were negatively correlated with creativity. the findings indicated that teachers’ perceptions of learners as creative were only a positive trait as long as the learners were also easy to manage. therefore, learner creativity was only fostered to the degree that learners could learn to channel their creativity into acceptable behaviours. if not, teachers were more likely to extinguish creative behaviour than encourage it (edwards, 2001: 222). balarabe yushau, andile mji and dirk wessels 15 it appears therefore that, in order to enhance creative thinking, a necessary condition is the openness to new ideas and the willingness to encourage the exploration of the unknown. and hence, there is a need for more curious learners who dare to ask why rather than a docile lot who always say: yes i understand. teaching approaches that project mathematics as a rule-based subject are not conducive to creative thinking. typically, learners’ curiosity is stifled in such instances and the most creative minds are discouraged. for more creative learners such classrooms are ‘torture zones’ because they cannot express themselves. it should be remembered that creative people are unique in their ability to achieve anything. this means that they hardly function optimally under restricted conditions or when things have to be done in accordance with confining rules. it can therefore be argued that creativity is incompatible with mathematics teaching that does not allow learners ‘free expression’. promoting creativity in a mathematics classroom many studies have attempted to unveil how creative mathematicians create their mathematics. a more recent study is by sriraman (2004). as a result, many models and theories have been developed. for a more elaborate review on the new trends, perspectives and theories of creativity, see sriraman (2004). it has been argued that to promote creativity in our classrooms, polya’s vision of trying to make learners do mathematics following a heuristic approach akin to professional mathematicians has been considered as outstanding (sriraman, 2004). many researchers have suggested ways that teachers can nurture creativity in their learners (chambers, 1972; paul & kathy, 1990; cangelosi, 1996; meissner, 2000; sriraman, 2004). we shall summarise and synthesise these in the following four points. relative freedom and independence a necessary requisite for divergent thinking is relative freedom and independence. to develop this thinking, mathematics teachers should learn to relax the rigidity of following only one way of doing mathematics. they should respect the unusual questions learners ask, and their unusual ideas and solutions, for children will see many relationships that their parents and teachers miss. teachers should encourage this exploration and provide opportunities and an atmosphere for free expression. they should also give credit for selfinitiated learning. the classroom atmosphere should promote divergent thinking. overly detailed supervision, too much reliance on prescribed curricula, failure to appraise learning resulting from a learner's own initiative, and attempts to cover too much material with no opportunity for reflection interfere seriously with such efforts (paul & kathy, 1990). learners should be given more freedom to do their work, and they should be made to believe that it is really their work not the teacher’s. challenging problems the literature suggests that most creative individuals tend to be attracted to complexity, of which most school mathematics curricula has very little to offer. classroom practices and math curricula rarely use problems with the sort of underlying mathematical structure that would necessitate students’ having a prolonged period of engagement and the independence to formulate solutions. (sriraman, 2004: 32) children’s potential to solve problems in unprecedented ways is enormous. it has been shown that even children who seem to be mathematically weaker or slower than others can construct powerful mathematical concepts and generalisations provided the tasks they are presented with remain challenging and are not made easier, and the inquiring nature of the mathematics classroom is maintained (murray, olivier & human, 1992). however, teachers have to provide opportunities for creative expression, creative problem solving and constructive response to change. we need challenging problems to trigger creative thinking and so challenge the learners’ potential. challenging problems are stimulating for learners. however, they can be boring if there is no interest in the contents or if they are too demanding. therefore, the problems must be connected with the individual daily life experiences of the learners. we must meet their fields of experiences and their interest areas. the learners must be able to identify themselves with the problems and their possible solutions. the problems must be fascinating, interesting, exciting, thrilling, important, and thought provoking (meissner, 2000). encouraging/facilitating learners’ efforts as learners are faced with challenging problems, there is need for the teachers to guide them towards the solution without too much interference. learners should be guided and the role of technology in fostering creativity in the teaching and learning of mathematics 16 encouraged, otherwise, they can get frustrated and discouraged. in a study conducted by chambers (1972), it was found that individual encouragement was the best teacher behaviour to develop creativity in the learner (in edwards 2001: 222). teachers should encourage curiosity, exploration, experimentation, fantasy, questioning, testing, and the development of creative talents. at the same time they should provide adequate guidance and avenues for learners to develop important abilities to explore and visualise a problem, to invent their own or to modify given techniques, to listen and argue, to define goals and to cooperate in teams. learners should be prepared for new experiences, and helped to develop creative ways of coping with them. teachers should create an environment that will first further individual and social components of creativity, like motivation, curiosity, selfconfidence, flexibility, engagement, humour, imagination, happiness, acceptance of self and others, satisfaction and success. learners should be taught to appreciate and be pleased with their own creative efforts. furthermore, learners should be exposed to examples of creative production (for instance through historical accounts of mathematical inventions and discoveries, teachers can model divergent thinking in their learners). similarly, heuristic activities such as brainstorming and openended question sessions should be facilitated. in addition, discussions in which ideas for consideration are examined regarding purpose, structure, advantage, and disadvantages should be encouraged (see polya, 1957). teachers should be aware of the natural curiosity of their learners, and they should be encouraged to discover their environment. they themselves must describe and order or classify their observations, their activities, their questions, and their results. the teacher is only the guide through the "mathematics jungle". teachers must meet learners’ fields of interest and interweave these fields with relevant mathematical content. as teachers, we must guide the intuitive, unconscious, global, spontaneous and visual approaches smoothly into teamwork discussions to further arguing and local logical thinking. on this path, guess and test procedures are not only allowed, they are necessary intermediate steps to gain a more conscious and systematic overview for the given problem (see polya, 1957). the aim is to show learners that their ideas have value by listening to their ideas, and encouraging them to test them and communicate them to others. enough time for brainstorming, exploration and experimentation creative individuals are described as having a greater degree of personal openness, an internal locus of self-evaluation and the ability to toy with materials for considerable lengths of time (edwards, 2001, sririman, 2004). therefore, when learners are faced with challenging problems enough time and opportunity should be provided for them to learn, think, and discover without threats of immediate evaluation. constant evaluation, especially during practice and initial learning, discourages learners from using creative ways to learn. we must accept their honest errors as part of the creative and learning process. south africa has the potential to put the above proposal into practice. an example is the learning situation described by de villiers (2004), which is supported and integrated with a subject-typified computer-based program, such as geometer’s sketchpad. this type of dynamic learning environment provides a basis for effective learning and the accompanying thrust for creativity and divergent thinking. the role of technology in developing a quasi-empirical view in mathematics the importance of technology in mathematics and mathematics education is examined in de villiers’ (2003 and 2004) articles on quasi-empirical methods in this area. he argues that technology (in this case the computer) with the dynamic mathematics programs (geometer’s sketchpad) is an immensely powerful tool that could be used to ‘attack’ and ‘shake’ an epistemological perspective, like a one-eyed axiomatic-deductive view about geometry. he convincingly shows that the opportunity to do investigations, explorations and the formulating of conjectures is much wider. taking experimentation and quasi-empirical methods as synonyms (2004: 2), he elaborates on the different functions of quasi-empirical methods which are: conjecturing, verification, global refutation, heuristic refutation, and understanding. when these functions are active, the participant’s intuition is activated. this plays a leading role in one’s struggles with problem solving experiences in geometry and mathematics in general. in explaining this de villiers (2004: 4) cites hofstadler in relation to mathematical modelling, who points out: probably the main reason these ideas are only now being discovered is that the style of exploration is entirely modern; it is a kind of experimental mathematics, in balarabe yushau, andile mji and dirk wessels 17 which the digital computer plays the role of magellan’s ship, the astronomer’s telescope, and the physicist’s accelerator. …. and perhaps after it (this era) we will witness a flurry of theoretical work to back up these experimental discoveries. de villiers adds: “… the main advantage of computer exploration of topics … is that it provides powerful visual images and intuitions that can contribute to a person’s growing mathematical understanding …” (2004: 5). with such technology one may be tempted to draw the conclusion that the ‘gap’ (if any) between experimentation and creativity in mathematics is very small. using himself as an example, de villiers (2004) has often described his own experiences of working in this mode and struggling to ‘see’ the solution. he has shown how working consistently has enabled him to make breakthroughs and to prove conjectures. he was successful because he focused on solving the problem and used all the problem solving strategies in his armoury. dealing in this way and at this level with problem solving strategies made him creative. without the use of the computer and geometer’s sketchpad, it is rarely possible to reach this level of constructive engagement and to solve problems creatively. for de villiers the ideal situation is when quasiempirical methods and deductive methods complement each other in everyday research mathematics. both are constituent and very necessary for the ‘making’ of new mathematics. with the help and mediating role of dynamic mathematics, creativity will flourish in such an environment. several conclusions can be derived from de villiers’ retrospection and metacognition. the most important is the natural display of the four attributes of highly creative personalities in mathematics: sustained attention, commitment, detachment, and inner growth of confidence as identified by schmalz (1988). in our last experiment (yushau, mji, bokhari & wessels, 2005) using matlab to teach trigonometric functions, all the four qualities were vividly observed in many learners. spending considerable time working in the lab with a high level of attentiveness, experimenting with many things, generating different types of graphs of a “strange” function that they could not have dreamt of prior to that experience, they could see the effect of changing parameters, the meaning of “translation”, “reflection”, “identity”, and various forms of symmetry, etc. as the learners visualised these concepts, their confidence increased. creativity and technology in the teaching and learning of mathematics several researchers have examined the degree to which computers may be used to facilitate creativity; however, the results are inconclusive (edwards, 2001: 223). in this section we argue that if used effectively, technologies, especially computers, can provide a learning atmosphere that can facilitate creativity in learners. a major force that seems to facilitate the emergence of creative personalities is the naturenurture relationship. as teachers we have very little or no control over the natural factors. however, a lot can be done in mathematics classrooms that can nurture the creative potential of our learners. the good news is that studies have confirmed that creativity comes more from environmental than hereditary factors (simonton, 2000, edwards, 2001). the importance of the environment on an individual’s ability to be creative stems from the inherent nature of a human being to explore his surroundings and try to make novel associations between things. if the environment is conducive for making such connections, then creative abilities will be facilitated, otherwise they will be hindered (edwards, 2001: 222). it is our belief that technologies can help teachers to develop an environment that can facilitate creative behaviour in their learners. one of the unique features of the computer as a teaching and learning tool is visualisation. through visualisation problems can be looked at from a macro perspective, and it allows for conceptualisation of the problem as a whole. it has been argued by edwards (2001: 221) that eminent creativity comes from the ability to think visually instead of verbally. this has been demonstrated by examining many of the great minds of the world, and “several of these historical figures had great powers of visualisation and thus, the importance of nonverbal thought was recognised as an important component of creativity”(edwards, 2001: 221). dreyfus (1993) observes that during the last 30 years, mathematics as an activity has become more experimental and more visual. in line with this development, the computer is a unique tool that has the potential to enhance both the visual and the experimental features. for instance, the powerful visualisation capacity of the computer is unprecedented and incomparable with traditional teaching aids. abstract concepts that have proved difficult for teachers to explain or for learners to grasp using traditional teaching approaches or aids can now easily be produced and understood by the role of technology in fostering creativity in the teaching and learning of mathematics 18 using the powerful animation and graphical display capabilities of computers. with this, learners’ reasoning and manipulative powers are facilitated especially by computer graphics. learners can also work with visualisation and modelling software to simulate concepts or ideas that are similar to real life situations. not only will this increase the experimentation, exploration and understanding of the learners, but it also increases the likelihood of transferability of knowledge from school to real life settings (dreyfus, 1993; brandsford, brown & cocking, 1999), therefore making mathematics sensible to learners. in a study on the influence of visualisation, exploring patterns and drawing generalisations, nixon (2003) reported that her learners indicated visual representation in a computer screen as beneficial to their understanding compared to diagrams in books. it is well known that the more senses we engage in the teaching and learning process the more we understand and retain knowledge. through multimedia, educational technology has the capability of appealing to our senses of sight, hearing, and touch. so it has the ability to widen and enrich the content and scope of our educational experiences. technology provides greater flexibility and variety in the organisation of instruction, and gives learners a more self-reliant role in their own education. this allows learners to become more active agents in managing and ensuring the success of their education – invariably sustaining their attention and commitment to mathematics. technology can assist teachers in developing a creative learning situation that takes cognisance of individual learning differences. also, it empowers learners and avails them of all the tools necessary for promoting creativity. there is no doubt that one of the greatest assets a teacher can possess is access to and technical know-how of technology because of its versatility. for instance, the preparation of teaching material for presentation in the classroom takes up most of teachers’ time. with the help of technology, the teacher can effectively address the challenge to organise mathematics instruction so that it attracts and develops the abilities of the greatest number of learners possible (nctm, 2000). multimedia, for example, merge text, graphics, audio, and video into one entity, and hence makes it possible for learners to use many of their senses in one lesson (see hale, 2004). this brings about the possibility of accommodating many learners with different learning styles in an unprecedented way — hence making learning mathematics more meaningful. furthermore, it is common knowledge that people use and follow different ways of collecting and organising information into useful knowledge. some accomplish their best learning through interaction with their peers. others accomplish this through lone study and contemplation. certain individuals on the other hand prefer to learn a skill by manipulating concrete objects, watching, listening, or by reading an instruction manual (cross, 1976). issues such as time constraints, the teachers’ experience and so on, make it extremely difficult for the teacher to cater for these individual differences. this situation sometimes results in learning difficulties for some learners. to address this, some teachers resort to more or less prescriptive teaching, where the rules and mechanics of teaching are followed, while others use creative teaching, which approaches situations in an unprecedented way. now computers provide learners with access to instructional programs designed with bigger budgets, more expertise, and greater talent than would be possible in the realm of human teaching. this enriches and supplements the classroom instruction that is already available to learners, hence providing them with alternative modes of instruction for the same subject. furthermore, this increases independent learning which is good for creativity. paul and kathy (1990) define creative learning as a natural healthy human process that occurs when people are curious and excited. good learning on the other hand requires learners to follow skills such as recognition, memory and logical reasoning, which are the abilities frequently assessed in tests of intelligence and scholastic aptitude (paul & kathy, 1990). however, creative thinking and learning involve the abilities to sense problems, inconsistencies, and missing elements, fluency, flexibility, originality, and elaboration and redefinition (paul & kathy, 1990). it has been shown that learners prefer to learn in creative ways rather than by just memorising information provided by a teacher or parents and that they also learn better and sometimes faster when a creative approach is used (paul & kathy, 1990; simonton, 2000). however, these are abilities that are rarely developed in our classrooms despite “good” teaching intentions. to promote these, mathematics should be viewed differently – as a science of pattern rather than as a set of rules. in this regard learners should be given more control over what they learn in order for them to explore and appreciate these patterns. they should be actively involved in the learning process and the pattern searching for knowledge to be meaningful. balarabe yushau, andile mji and dirk wessels 19 in a typical classroom, technology saves time and provides easier and clearer illustrations than those a teacher would make. as a matter of fact, there are relatively very few teachers that have the time or artistic talent to produce illustrations “by hand with chalk, overhead transparency pens, or marking pens that can compete with those generated with a computer” (cangelosi, 1996: 202), or even graphic calculator. this can be seen in the case of three-dimensional objects. such objects are difficult to draw on the chalkboard and it is much more difficult for learners to visualise them. with the help of computers and graphic calculators learners themselves may creatively draw three dimensional objects, thus saving teachers’ precious and limited time. it has been shown that working with the appropriate computer software can pack a large amount of graphing experience into a relatively short amount of time with the result that learners deal with more graphs in class than learners typically experience in an entire series of algebra courses (see kaput, 1992; yushau, 2004), therefore, leaving them with more time and opportunity for creative work. motivation is considered to be a driving force for most human endeavours. in fact motivation has been a major research topic in the area of the psychology of teaching and learning (perry, menec & struthers, 1996). bell (1978) outlines four general reasons for people to be motivated to learn in and outside of school. these are: “to create things, to make things work, to obtain recognition, and to find personal satisfaction” (1978: 33). if learners are to be motivated and their enthusiasm enhanced, it is important that instruction be flexible enough to create room for creativity to prosper. technology has the potential of making this a possibility and as a consequence it develops the high level of motivation necessary for creativity. for instance, the intrinsic features of computers such as immediate feedback, animation, sounds, interactivity, and individualisation are more likely to motivate learners to learn than any other media (yang & chin, 1996). the spontaneous attention of children watching good movies, cartoons or doing computer games is apparent. it has been shown that if these technologies are properly used teachers can motivate learners to learn extrinsically and intrinsically (cox, 1997), which are both necessary for creativity. on the side of the learner, the creative potential seems to require certain exposure to diversifying experiences that help weaken the constraints imposed by conventional socialisation and challenging experiences that help strengthen a person’s capacity to persevere in the face of obstacles (simonton, 2000: 153). there is no doubt that using a variety of technological tools, such as calculators, computers, and hands-on materials, under the guidance of a skilful teacher, creates a rich mathematical learning environment. such an environment helps in exposing and preparing learners for diversified experiences – experimentation and exploration (beal, 1998). this is the exposure that is required and necessary to nurture creativity (de villiers, 2004; nixon, 2003). one of the factors that limit learner creativity in mathematics is their inability to recognise and connect mathematical structures and objects in different situations. in this respect, technology can help learners uncover shared and unshared patterns of a class of mathematical objects. for instance, the multiple representation for a function (tabular, graphical, symbolic), is much easier to grasp using technology. it has been demonstrated by yushau (2004) how one can display all these representations of one-dimensional functions in one setting in matcad. in that setting learners need only change a parameter to see changes in all these aspects. this in a way exposes learners to different sides of the ‘mathematical coin’ and allows them to see mathematics from different (and seemingly unrelated) angles. such exposure helps them to visualise, explore and deeply understand mathematical concepts in a spectacular way (cangelosi, 1996), invariably fostering their mathematical creativity. it is this exposure that informs learners that mathematics is not a linear subject and that there are a variety of ways of tackling problems. it also removes pervasive beliefs that the only way of tackling mathematics is by following rules, which in fact kills creativity. learning is an active process. however, many commonly used teaching strategies put learners in passive and receptive roles. this results in situations where learners have very little, if any, control of the learning environment (bell, 1978). technology has the ability to enrich the content of learners’ learning experiences, provide greater flexibility and give learners a more self-reliant role in their own education. as a result, learners become more active and participating agents in their education, which will open the door for creativity. creativity is more or less a solitary business (standler, 1998). similarly, learning is more effective and efficient when instruction can be tailored to the unique needs of each learner. with the aid of technology, especially computers, the role of technology in fostering creativity in the teaching and learning of mathematics 20 instruction can be flexible and adaptable to individual needs. also learner-teacher interaction and learning are significantly more learner-centred, thereby, creating room for learners’ optimal functionality and creativity. today's learners will live and work in the twenty-first century, in an era dominated by computers, by worldwide communication, and by a global economy. jobs that contribute to this economy will require workers who are prepared to absorb new ideas, to perceive patterns, and to solve unconventional problems (steen, 1989). under this dispensation, there is no better gift that a learner can get from school than empowering him/her with the necessary tools to face this challenge. it has been established that use of technology can empower learners to be critical thinkers, and better problem solvers (kaput, 1992; roblyer, 1989). in this way, their creative capabilities may be enhanced. thinking mathematically is considered (by many people) to be critical for everyday life skills. people use mathematical skills daily to identify problems, look for information that will help them solve problems, consider a variety of solutions, and communicate the best solution to others. however, the connection between the mathematics learned at school and the mathematics used in daily life is often not made. to bridge this gap, mathematics classrooms should provide practical experiences in mathematical skills that are a bridge to the real world. also, they should allow explorations that can develop an appreciation of the beauty and value of mathematics (beal, 1998). again the use of technology is a key for bridging this gap. this may be accomplished by providing learners with a variety of challenging real life problems that are fascinating, interesting, exciting, thrilling, important, and thought provoking – a wonderful asset for fostering creativity (see papert, 1980). this will help the learners to visualise and appreciate mathematical concepts and to look for creative solutions to real-world problems. although computers, as argued, have the potential to foster the creativity of learners in many different ways, three points are summarised by edwards (2001: 226) as necessary for accomplishing this. first, computer programs should allow for the destructuring of thought to facilitate the building of new creative concepts based on old patterns of thinking. second, the idea that a computer can aid in that process provides evidence that the highly structured environment of the computer can be used to facilitate creativity and not just to limit it. last, the idea of allowing people to determine the time spent considering options, and the general focus of control being with the individual, seems to be especially important to the facilitation of creativity. conclusion the main focus of this paper is on technology in general and the computer in particular. in the course of this review, we have seen that technology is a powerful tool that could be used in the teaching and learning context to make a difference to our traditional teaching methods. it has the capacity to “amplify” and “organise” our thought, teaching and learning process (pea, 1987). if used effectively computers are a proven medium for fostering creativity and divergent thinking in the classroom (edwards, 2001). they have the potential to provide a teacher with an enabling situation to create a rich and challenging learning environment that can foster the creative potential of learners. on the other hand, computers can give learners the opportunity to explore their own creative potential. therefore, the question of whether a child can learn and do more mathematics with a computer … versus traditional media is moot, not worth proving. that computational aids overall do a better job of converting a child’s intellectual power to mathematical achievement than do traditional static media is unquestionable. the real questions needing investigation concern the circumstances where each is appropriate. 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(2005). predictors of success in computer aided learning of mathematics. in g. m. lloyd, m. r. wilson, j. l. wilkins, & s. l. behm (eds.), proceedings of the 27th annual meeting of the north american chapter of the international group for the psychology of mathematics education [cd-rom]. eugene, or: all academic. “if you are seeking creative ideas, go out walking. angels whisper to a man when he goes for a walk.” raymond inmon abstract introduction literature review theory methodology findings discussion and conclusions acknowledgements references about the author(s) anthea roberts schools development unit, university of cape town, cape town, south africa kate le roux academic development programme, university of cape town, cape town, south africa citation roberts, a., & le roux, k. (2019). a commognitive perspective on grade 8 and grade 9 learner thinking about linear equations. pythagoras, 40(1), a438. https://doi.org/10.4102/pythagoras.v40i1.438 original research a commognitive perspective on grade 8 and grade 9 learner thinking about linear equations anthea roberts, kate le roux received: 15 june 2018; accepted: 05 mar. 2019; published: 28 mar. 2018 copyright: © 2019. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract concerns have been expressed that although learners may solve linear equations correctly they cannot draw on mathematically valid resources to explain their solutions or use their strategies in unfamiliar situations. this article provides a detailed qualitative analysis of the thinking of 15 grade 8 and grade 9 learners as they talk about their solutions to linear equations in interviews. the article stems from a study that describes whether learners use mathematically endorsable narratives to explain and justify their solutions. sfard’s theory of commognition is used to develop a framework for analysis of their discourse. the findings show that all learners use ritualised rather than explorative discourse, characterised by applying strict rules to operations with disobjectified entities. the only mathematical objects they produce endorsed narratives about are positive integers. thus they do not meet the relevant curriculum requirements. nevertheless, the analytic tools – adapted from sfard specifically for the study of linear equations – give a particularly nuanced account of differences in the learners’ ritualised discourse. for example, some learners used endorsed narratives about negative integers, algebraic terms and the structure of an equation when prompted by the interviewer. there is not sufficient evidence to suggest that any learners are in transition to explorative discourse. however, the article shows that learner discourse is a rich resource for teachers to understand the extent to which learners are thinking exploratively, and offers suggestions for how their thinking can be shifted. this is an opportunity for teacher professional development and further research. keywords: school mathematics; discourse; algebra; linear equations; commognition. introduction this article is located in the enduring and stubborn problem of poor mathematics performance among the majority of school learners in south africa (adler & pillay, 2017; graven, 2014; spaull, 2013). mathematics education research has, necessarily, focused on different aspects of this complex problem, for example wider socio-economic factors affecting performance (e.g. setati, 2008; taylor, 2009), school and classroom management (e.g. fleisch, 2008; taylor, 2009), teacher practice and professional development (e.g. adler, 2011; gardee & brodie, 2015; venkat & adler, 2012), and learner thinking (e.g. brodie, 2010; gcasamba, 2014; pournara, hodgen, sanders, & adler, 2016). this article focuses on learner thinking in the context of solving linear equations in grade 8 and grade 9. linear equations as a topic is productive for exploring learner thinking in these grades as learners transition from arithmetic to algebraic thinking. at this juncture, learners are expected to make a shift from seeing the equal sign as signifying an operation, to seeing this symbol as having relational significance and to using the notion of equivalence. in the south african school curriculum the topic is located in the content area ‘patterns, functions and algebra’ (department of basic education [dbe], 2011). the curriculum document describes the expected trajectory for learners’ conceptual understanding of functions and algebra between grade 7 and grade 9 as a transition from ‘a view of mathematics as memorised facts and separate topics to seeing mathematics as interrelated concepts and ideas represented in a variety of equivalent forms’ (dbe, 2011, p. 21). the ‘concepts and skills’ for solving linear equations include using additive and multiplicative inverses, and ‘substitution in equations to generate tables of ordered pairs’ (dbe, 2011, p. 91). the study on which this article is based emerged from the practice of the first author who, in her experience in teacher professional development, found that neither she nor teachers could explain learners’ incorrect solutions to linear equations. in addition, the marks assigned in conventional written assessments suggested that some learners could produce the correct answers to familiar-looking linear equations, yet interviews with learners, conducted to inform the design of a professional development course for teachers, provided a more complex assessment: when learners were required to explain their strategies it emerged that they were not using mathematically valid resources to solve these equations, and could not apply their strategies in unfamiliar situations. the most common procedure observed in these interviews involved shifting entities across the equal sign. there was some evidence of applying additive and multiplicative operations, but virtually no evidence that teachers encouraged learners to use relational equivalence as a resource for solving linear equations. a review of the literature – presented in detail in the next section – suggests that the observation of learners’ strategies in this context is not idiosyncratic. research has pointed to the prevalence of the ‘change side-change sign’ technique (kieran, as cited by hall, 2002, p. 12), and the erroneous application of the distributive property (seng, 2010, p. 153). indeed, this literature suggests that, across contexts, learners tend to manipulate symbols in an instrumental way without knowing the reason for the procedure, rather than demonstrating relational thinking (compare skemp, 1976), by recognising, using and explaining connections between mathematical structures and demonstrating which manipulations are useful to perform for particular problems (hoch & dreyfus, 2004; linchevski & livneh, 1999; mason, stephens, & watson, 2009). the question could be posed: if learners produce correct answers to problems, why does it matter that they achieved their solutions by merely following an instrumental rather than a relational approach? mathematics education researchers recognise the central role of relational thinking in the transition from arithmetic to algebra. mason et al. (2009) argue that, while learning manipulation is part of mathematical sense-making, purely instrumental thinking restricts a learner to the particular and hence to rote learning. relational thinking, on the other hand, is essential for generalising and abstracting. sfard (2008) – who uses concepts such as explorative and ritualised discourse to describe different forms of thinking – suggests that both the how of solving a problem, that is, the procedure, and the when and why of using the procedure are hallmarks of using mathematically valid resources to think about mathematical objects in a way that is individualised, independent, and flexible. her concern is that the thinking of learners who can solve problems without the when and why can only be restricted and rigid. learners thinking in this way can only reproduce what others say or do, and remains heavily reliant on the situation and available mediational means. sfard further suggests that imitative, procedural manipulation is a part of learning school mathematics, but that if the teaching focuses only on such, learners will not have the opportunity to transition further to the valued relational thinking. certainly, the description of the school curriculum in this introduction suggests that the relational thinking regarded as important in the literature, is valued. this article proceeds from the argument that decisions about teaching linear equations that promote relational thinking require detailed understanding of learner thinking on this topic. thus, the aim of this article is to provide a theoretically informed description of the how, why and when of learner thinking when solving linear equations. to achieve this, we adopt the theory of commognition (sfard, 2008). we describe the discourse of learners who knew how to solve familiar linear equations to search for evidence – based on sfard’s conceptualisation of mathematical thinking – that they are using mathematically valid resources to think about linear equations. firstly, we describe each learner’s communication using sfard’s concepts of explorative and ritualised discourse. explorative discourse is the most sophisticated form and is characterised by narratives about mathematical objects that are endorsable in terms of mathematical axioms, definitions, and theorems. ritualised discourse on the other hand involves the learner following, or imitating, strict rules that are determined by an authority. secondly, for those learners whose discourse is classified as ritualistic, we delve deeper to ask whether there are any differences in the talk, gestures and writing of these learners, and whether some of them might be in transition to explorative discourse. we use these detailed descriptions of learner discourse about linear equations to make some tentative recommendations on how teachers could use the curriculum topics to support students in the transition from ritualised to explorative discourse. literature review most studies of learner thinking in the transition from arithmetic to algebra and more specifically about linear equations have drawn on cognitive theories of learning that can be located within an ‘acquisitionist’ paradigm of learning (sfard, 2015, p. 130). researchers have used theoretical concepts such as structure sense and relational versus instrumental thinking to characterise learner thinking. defining structure sense as the ability to ‘recognize mutual connections between structures, … and recognize which manipulations it is useful to perform’ (hoch & dreyfus, 2004, p. 50), hoch and dreyfus (2006) found that the majority of grade 10 learners in their study did not use structure sense as a resource when solving algebraic problems. those that used structure sense made fewer errors, but there was no correlation between learners’ structure sense and their manipulation skills. the latter finding suggests that learners might know how to solve problems but not the why and when aspects. indeed, linchevski and livneh (1999) argued that not having a sense of the structure of the number system underpinned an absence of – in skemp’s (1976) terms – relational understanding in algebra, that is, not having the conceptual structure that allows for independent decisions when solving problems. focusing on learner thinking about the equal sign, alibali, knuth, hattikudur, mcneil and stephens (2007), asquith, stephens, knuth and alibali (2007) and essien and setati (2006) identified an absence of relational understanding of the equal sign as impeding flexibility in the solving of algebraic equations. the studies reviewed here used the concepts of structure and of relational and instrumental thinking to characterise learner thinking in algebra. in particular, they associate the former with relationship-rich, flexible, independent, more accurate problem solving. some studies of learners’ algebraic thinking from within the acquisitionist paradigm have focused on describing the detail of how learners solve problems. the detail of learner errors is commonly described using the notion of misconception, that is, the use of a concept that, ‘although systematic and invariant across contexts, differs from the way this concept is used by experts’ (sfard, 2008, p. 16). focusing on linear equations and algebraic expressions, hall (2002) and seng (2010) showed that learners across geographic and socio-economic contexts present the same misconceptions. as noted, errors included the ‘change side-change sign’ technique (kieran, as cited by hall, 2002, p. 12) and erroneous application of the distributive property (seng, 2010). focusing on the equation 4m = 2m, de lima and tall (2008) noted that learners shifted symbols in a manner they described as using the notion of human embodiment: symbols were picked up and moved around. the detail of learner thinking offered by the research reviewed here can inform the development of teaching strategies to counter them, although these strategies could favour an instrumental rather than structural orientation to the concept. the studies of thinking about linear equations reviewed here have commonly focused on linear equations in various forms, such as ax + b = c, ax + b = cx + d and 4m = 2m (e.g. barahmand & shahvarani, 2014; de lima & tall, 2008; filloy & rojano’s, 1989; herscovics & linchevski, 1994). there is general agreement that learners respond differently to equations where there is a variable on one side of the equation, as opposed to equations where the variable occurs on both sides. however, researchers differ in their characterisation of the form of the latter that marks the transition from arithmetic to algebra. this debate lends weight to the point made by herscovics and linchevski (1994) that a study aiming to describe what learners do when solving linear equations must include a variety of equation forms. the different forms identified in this research have informed the methodological design of our research in this article. the studies located within the acquisitionist paradigm reviewed here contribute to the mathematics education community’s understanding of learner thinking when solving linear equations, by characterising learner thinking as instrumental or relational, identifying the importance of relational thinking for mathematical participation, and describing some detail of how learners think about linear equations. they have also made contributions to the methodological aspects of investigating learner thinking when solving these equations. yet, despite this extensive body of research, there is still a need for nuanced descriptions, not only of the how of learner thinking, but also the when and why of their strategies that help us to understand how learners might transition from instrumental to relational thinking. sfard (2008) argues that such descriptions require additional data, and the focus of this article is on the thinking of 15 learners who knew that they had solved an equation correctly, but had difficulty explaining why. such descriptions also need ‘penetrating theories’ (sfard, 2008, p. 22) and we use sfard’s theory of commognition, located in a ‘participationist’ view of learning (sfard, 2015, p. 130), for a nuanced description of these learners’ thinking. indeed, the usefulness of commognition to ‘penetrate’ the detail of mathematical thinking is suggested by the growing use of this theory (see special issues edited by tabach & nachlieli, 2016; sfard, 2012). commognitive studies focusing on learner thinking have included the shift from arithmetic to algebra (caspi & sfard, 2012), functions (clark, 2014; gcasamba, 2014; nachlieli & tabach, 2012; tabach & nachlieli, 2011) and geometry (sinclair & moss, 2012). our literature review did not reveal any commognitive studies focusing exclusively on learner thinking about linear equations. a number of researchers have used commognition to study change in learner discourse over time (heyd-metzuyanim, 2015; nachlieli & tabach, 2012). given the scope of the master’s thesis on which this article draws (roberts, 2016), we report only on learner thinking at a particular moment in the learning process. furthermore, viirman (2011), berger and bowie (2012) and bogdanova (2012) all used the theory of commognition to study the discourse of teachers or lecturers or to develop courses for teachers. the studies reviewed here suggest that the detailed understanding of learner thinking at a particular moment that this article offers could inform teacher professional development, with follow-up research to explore shifts in learner thinking over time. achieving our aim of presenting the empirical detail of the how, when and why of learner thinking about linear equations in this article requires an explanation of (1) the detail of the key inter-related concepts we use from sfard, and (2) how we put these to work in a framework for analysing learner discourse specifically about linear equations. we explain this detail next. theory overview sfard (2008) argues that mathematics is a discursive activity. unlike the objects of many school subjects, mathematical objects – numbers, variables and functions are examples – are not accessible to our senses. learners of mathematics therefore construct these abstract objects through their discourse and we speak of discursively constructed objects. sfard (2008) describes mathematics as ‘autopoietic’ (p. 129) in the sense that it is ‘a system that contains the objects of talk along with the talk itself’. she identifies this as the feature that makes mathematics difficult to learn: familiarity with ‘what the discourse is all about’ (p. 130) is needed for participation in the discourse, but paradoxically this familiarity only comes through participation. to describe a learner’s discourse in sfard’s terms we use the following key concepts, which are described in the rest of this section: keywords, visual mediators, narratives and routines, as well as the mathematical objects that are referenced. keywords, visual mediators and narratives in sfard’s (2008) terms keywords are words that signify numbers, variables and functions. visual mediators are the visible objects that act to communicate relationships and operations with mathematical objects. in mathematics these are mostly in the form of symbolic artefacts such as numerals, tables, algebraic expressions, equations and graphs. these symbolic artefacts are created to communicate relationships and operations with mathematical objects. keywords and visual mediators are used to produce narratives (sfard, 2008). a narrative is any text, spoken or written, that is ‘framed as a description of objects, of relations between objects, or processes with or by objects’ (sfard, 2008, p. 300). endorsed mathematical narratives are those that the mathematical community accepts as valid; definitions, proofs and theorems are examples of endorsed narratives in scholarly mathematics. this characteristic of narratives is a feature that we exploit in this article in order to categorise learners’ discourse: we consider whether learners use keywords and visual mediators to construct endorsable narratives pertaining to the solution of linear equations. we do this by constructing narratives based on what is perceptible in their discourse during interviews (sfard, 2017). routines regularities in the use of keywords and visual mediators, and their use in narratives, are referred to as routines (sfard, 2008, p. 138). a routine may be a procedure such as ‘taking’ a number across the equal sign, or a practice such as generalising, endorsing or rejecting a narrative. routines are governed by rules, which may be at the level of object (about the behaviour of objects) or a metalevel rule (metarule) that defines the pattern of learner actions. for example, the ‘change sides-change signs’ metarule (kieran, as cited by hall, 2002, p. 12) informs the routine in which learners solve linear equations by reorganising the position of algebraic terms. sfard (2008) argues that school mathematics is dominated by metarules that define the how, rather than the when or why. in this article we consider learners’ use of and awareness of the latter when solving linear equations, so that we can describe their thinking using the concepts of explorative and ritualistic discourse. sfard (2008) identifies rituals and explorations as different types of routines. the notion of mathematical object is key to distinguishing between these two routines, and we turn to this next. mathematical objects and realisation trees we adopt sfard’s (2008) particular use of the concept of mathematical object here, noting that the term is used in various – and in some cases contested – ways in mathematics and mathematics education. not all the ‘objects’ in a learner’s discourse are necessarily mathematical objects in sfard’s sense as described in this section. thus in this article we reserve the use of ‘object’ for cases that meet her definition and refer to other ‘objects’ as entities. since from a commognitive perspective mathematical objects are discursively constructed, a key feature of sfard’s (2008) notion of a mathematical object is the relationship between signifier and realisation. sfard suggests that, when solving a linear equation algebraically, a learner will proceed from signifiers in the form of symbols or words. each signifier would have particular significance for the learner. that significance produces a response (written or spoken), which is the realisation. therefore, a signifier mediates meaning between one entity and another. the chain of signifiers and their realisations is referred to as a branch, and the final realisation is the solution to the problem. sfard calls this chain a realisation tree. a signifier could lead to different realisations for different people. for example, some learners could talk about the term ‘–3x’ as a mathematical object and operate on it in a mathematically endorsed way, but for others ‘3x’ and ‘–’ could be separate visual mediators and would thus signify separate operations that would not be endorsable. relational thinking could produce more than one way to solve a problem and each method would produce a different branch on the realisation tree. sfard (2017) uses these concepts to describe a mathematical object, that is, ‘a mathematical signifier together with its realization tree’ (p. 43). these concepts are illustrated in figure 1, the realisation tree for the solution to the linear equation 2x + 7 = 13. figure 1: realisation tree for the solution to the linear equation 2x + 7 = 13. the realisation tree for the solution to the linear equation 2x + 7 = 13 in figure 1 has four branches. for the algebraic branch on the far right, the linear equation 2x + 7 = 13 is a signifier that leads to the realisation ‘2x + 7 – 7 = 13 – 7’. thus, ‘2x + 7 = 13’ and ‘2x + 7 – 7 = 13 – 7’ are a signifier-realisation pair. a signifier-realisation pair is referred to as a node (sfard, 2008, p. 165). the signifier-realisation pairs for this branch ultimately lead to the realisation that x = 3. however, the same linear equation 2x + 7 = 13 could also signify a realisation of the ordered pair (3;13) in a graph, table or flow diagram, shown in the remaining three branches of the realisation tree. in summary, the realisation tree is ‘a hierarchically organized set of all the realisations of a given signifier’, together with their realisations (sfard, 2008, p. 301). in deciding whether a learner’s discourse about linear equations is objectified, that is, whether the learner is acting with mathematical objects, we need to consider whether the story or narrative they construct to justify the relationship between a realisation and its original signifier is endorsed or not. in table 1, the first realisation ‘2x + 7 – 7 = 13 – 7’ is justified by applying the additive inverse of ‘+7’ to the expression on either side of the equal sign. this justification would be part of the narrative, which is endorsable because equivalence is preserved. table 1 shows the endorsable iterative relationship between signifiers and realisations for the algebraic solution of the equation. table 1: signifier-realisation-narrative framework for the solution of 2x + 7 = 13. sfard (2008, p. 166) notes that the realisation tree is a ‘personal construct’ because it represents the learner’s discourse. by constructing realisation trees from each learner’s talk about linear equations in interviews, we obtained visual representations of the discursive objects they constructed. this allowed us to identify whether their routines were ritualised or explorative, as explained next. ritualised and explorative routines explorations are the most sophisticated form of routine. explorative discourse is characterised by narratives about mathematical objects that are endorsable in terms of mathematical axioms, definitions and theorems. in this article we look at the realisation tree analytically to determine whether the learner can use different ways to solve the equation. this is a hallmark of whether the learner is working with a mathematical object. according to sfard (2008), rituals are characterised by strict rules that are determined by an authority (the teacher or textbook). the discourse of rituals is limited to justifying how to do something, but not when to do so or why it works. since mathematics is an autopoietic system, learners first imitate others, which makes rituals an acceptable interim phase in the learning process. this implies that their routines have been discursively mediated by the teacher. ideally, the learner will gradually gain an understanding of the why and when, which is one of the markers of the transition from the discourse of rituals to explorative discourse. explorative discourse for these learners may be described as the production of algorithmic realisations, where the narratives are intradiscursive – they are produced by manipulating existing narratives (sfard, 2008). in explorative discourse, a learner’s word and mediator use signify numbers, algebraic terms and functions as mathematical objects in their own right. the trajectory whereby the learner shifts towards the attainment of explorative discourse is complex. sfard (2008) identifies four phases in the use of words within the discourse; passive use, routine-driven use, phrase-driven use and objectified use (sfard, 2008, p. 182). the passive use of mathematical terminology generally pertains to an initial encounter with a key word or phrase. once the learner starts using the terminology in the context of mathematical routines, it becomes routine-driven. as the learner’s confidence with the terminology expands, it becomes phrase-driven and used as part of constant phrases. ultimately the learner would use words as nouns, appropriately across contexts, and thus in an objectified way. a learner would be initiated to explorative discourse through others’ – mainly teachers and textbooks – objectified use of words and visual mediators. table 2 outlines the criteria for evaluating discourse and shows the distinguishing features of explorative and ritualised discourse, which we have adapted from sfard (2008, p. 243) for the focus on linear equations. table 2: features of explorative and ritualised discourse about linear equations. methodology studying learner thinking when solving linear equations from a commognitive perspective involves investigating learners’ discourse about mathematical objects. the study of their verbal and non-verbal communication in interviews is thus productive for the purpose. selection of learners the interviews used in this study were initially conducted by the first author in the development of a short course for teachers. twenty-two grade 8 and grade 9 learners at two schools were selected to take part. there were no specific criteria for the selection of the schools, other than that the short course was being developed for their teachers. following gripper (2011), they were selected on the basis of their marks for their written assessments, which suggested they knew how to solve familiar linear equations. it was thus possible to investigate the learners’ sense of the when and why, characteristics of explorative discourse. in order to understand the learners’ interview responses in more detail, the research study that forms the basis of this article was conceptualised in order to investigate the learners’ discourse in the interviews. of the original 22 learners interviewed, 11 grade 8 learners and 4 grade 9 learners agreed to have their interviews analysed for the purposes of the research study. ethical considerations permission to conduct the interviews for the purposes of teacher professional development was covered under the memorandum of understanding between participating schools and the organisation providing the support. ethical approval for the research study was given by the ethics review committee of the faculty of humanities at the university (reference ednrec20180909). written permission to use the interviews for research purposes (including the publication of findings) was obtained from learners, their parents, the principals of the two schools and the provincial education department. consent forms focused on issues of informed consent, anonymity, confidentiality and recognisability, and harm. the interviews the linear equations used in the interviews (table 3) were selected from formal written assessment tasks that the learners had completed for their teachers. by implication, we had no control over the actual equations presented in each task. we note that for equation 2, the grade 8 equation ‘2x + 8 = −3x – 2’ contains more negative integers than the grade 9 equation ‘6x – 12 = 2x + 4’. the curriculum prescribes that from grade 7 learners solve problems involving addition and subtraction of integers, and recognise and use the commutative and associative properties of addition and multiplication for integers (dbe, 2011, p. 16). table 3: list of interview questions, with motivation. for each set of questions in the assessment tasks set by the two teachers, the instruction was ‘solve the following equations’. at the start of the interview learners were asked to explain what they understood by this instruction, following which they were asked what an equation was and to explain the meaning of the equal sign. some equations in the assessments were modified for further investigation of learners’ level of exploratory discourse. these modifications, as well as a linear equation with no constant term, were based on forms of linear equations used in other research studies, for example with the variable appearing more than once (filloy & rojano, 1989; herscovics & linchevski, 1994) and equations with no constant term (barahmand & shahvarani; 2014; de lima & tall, 2008). table 3 lists the equations and related questions used in the interviews. during the interviews learners were encouraged to point at entities in the written solutions. they were asked to elaborate on their responses and, where necessary, were provided with prompts and scaffolding. for example, after explaining how they solved the equation 2x + 7 = 13, learners were asked whether the equation 2x + 7 = 6 + 7 was another representation of the same thing. the following excerpt illustrates how the interviewer used questioning to prompt a grade 8 learner, sheena (pseudonym), in different ways: line 1 interviewer: okay and without doing the next step can you tell me what 2x is equal to just by looking at that? line 2 sheena: yes. line 3 interviewer: what’s it? line 4 sheena: 2x. line 5 interviewer: 2x? line 6 sheena: oh, sorry it’s 6. line 7 interviewer: why’s it 6? line 8 sheena: because 2 times 3 is 6 [points to the 2 and the x of 2x] and 6 plus 7 is 13 [points to 2x + 7 on left-hand side] and 6 plus 7 is thirteen [points to 6 + 7 on right-hand side]. the interviewer’s prompt in line 1 is a question aimed to encourage sheena to use the concept of equivalence to solve the problem. this differs from the prompt in line 5, where the interviewer revoices sheena’s response in question form, to signal that she should rethink her response in line 4. the prompt in line 7 asks sheena to provide an explanation. the interviews were audioand video-recorded, with the video recorder focused on each learner’s script, and related gestures and not their faces. the interview recordings were transcribed into excel spreadsheets by the first author. what was said and done was ‘re-presented’ (setati, 2003, p. 294) in adjacent columns of the spreadsheet, to enable an analytic focus on a learner’s word use and the visual mediators that accompanied this. analysis process the analysis was organised into two levels. at level 1 a learner’s words and visual mediators were identified and used to reproduce the routines and narratives, and ultimately their realisation trees (with each recorded in a column of the spreadsheet). the data in these columns allowed us to perform the level 2 analysis; it provided the evidence for us to classify a learner’s discourse using the seven features of explorative versus ritualised discourse (figure 1). level 1 analysis: operationalising the tools of the discourse figure 2 illustrates the progression from transcript to realisation tree in the level 1 analysis, as described in this section. figure 2: steps in the level 1 analysis from transcript to realisation. to briefly illustrate how we use sfard’s (2008) concepts to analyse the detail of a learner’s discourse, we focus in this section on a short excerpt from the interview with, sheena, based on her initial solution to the equation 2x + 8 = −3x – 2 (extract 1). sheena was awarded full marks for her written solutions to equations 1, 2 and 3. we focus on the discourse of all 15 learners, including sheena in the presentation of the findings: extract 1 interviewer: okay. okay, talk to me about the next sum [refers to the equation 2x + 8 = 3x – 2]. sheena: [silence]. um … here was … here is variables on this side and here is numbers without, so you first had to um take … put it on the right spot. [points to +8 on left-hand side of line 1, then −8 on right-hand side of line 2] interviewer: right. sheena: so this was minus … negative three [points to –3x in line 1] so we had to make it plus, so it was 2x plus 3x. and this stays negative two, but this must be −8 [points to +8 on the left-hand side of line 1] because you’re putting it on this side [points to right-hand side]. interviewer: okay. why do you do that, hey? do you know? sheena: why do i do what? interviewer: why do you change the sign when you take it over? sheena: because that’s how i was taught. words: this level of analysis draws on a functional perspective of language in mathematics, as used by morgan (1998) and sfard (2008). from this perspective, language use in mathematics is not neutral, but performs a particular function in mathematical discourse. we categorised a learner’s language use according to its linguistic features, and the related meaning of these features in the context (morgan, 1998, sfard, 2008), as shown in table 4. table 4: linguistic features and their associated meanings. in extract 1, sheena uses a combination of mathematical and colloquial words to reference entities. she refers to ‘variables’ but refers to constants as ‘numbers without’. when she says ‘you first had to put it on the right spot’ she uses an adverb of time (‘first’) to suggest she is following steps. the material process verb ‘put’ together with the adverbial phrase of place (‘on the right spot’) points to how she, as the subject of the sentence and the doer of action, is using spatial arrangement to reorganise entities as part of her procedure for solving the equation. indication of her obedience to authority lies in her use of high modality verbs such as ‘must’ and ‘had to’. this obedience to authority of the teacher is confirmed in her reason for ‘changing the sign’, where she makes a direct reference to how she had been taught. visual mediators: the visual mediators, that is the discursive prompts used in the interviews to communicate about the operations and relationships (sfard, 2008), are restricted to symbols, integers and operational signs in algebraic expressions and equations. in extract 1 sheena makes frequent reference to ‘sides’ of the equation as she reorganises the entities so that the variables are on the left-hand side and the constants are on the right-hand side. each term is a discursive prompt that visually mediates the position of the entities. sheena frequently points to terms in the equation (referencing them as ‘this’), with a focus on the operational signs. her talk indicates that the sign is an independent visual mediator because whether she ‘makes it a plus’ or it ‘stays negative’ is affected by where she ‘puts’ the term. narratives: for this analysis narratives were categorised either as descriptions of entities and relations between them, or as narratives about actions with or by entities. we used a learner’s discourse in the transcript to re-present the narratives. for example, the statement ‘but this must be negative 8 because you’re putting it on this side’ was re-presented as ‘“+8” becomes “−8” when it is shifted to the other side of the equal sign’. where appropriate, the source of the narrative was included. for this narrative the source is ‘spatial arrangement’, which is not an endorsed narrative in mathematical discourse. routines and realisation trees: the realisation routine is the set of sequential steps for solving an equation; each step in the solution process is a realisation based on signifiers from the previous realisation. we remind the reader that the visual mediators are signifiers that mediate meaning between one entity and another. sfard (2008) posits that truly explorative thinking would be marked by a learner’s ability to access more than one way to find the solution to the equation. this has been illustrated in the realisation tree in figure 1. in this study all learners only had one branch, which was to be expected, given that the preferred teaching strategy for solving equations is the algebraic algorithm. thus our focus when looking at learners’ realisation trees in the analysis is on how the visual mediators signify meaning between one entity and another for the learners. the learners’ written solution, speech and gestures were used to interpret the realisation routines, which were then used to construct the realisation trees – the visual representations of the learners’ personal constructions of the ‘mathematical objects’. in figure 3 we illustrate sheena’s discursive construction of the solution of the equation 2x + 8 = −3x – 2 as a realisation of a narrative about the functions f(x) = 2x + 8 and g(x) = −3x – 2. figure 3: the re-presentation of sheena’s realisation routine and tree for the solution to the equation 2x + 8 = −3x – 2. sheena’s written solution, speech and gestures were used to describe her realisation routine (figure 3, column 2), which was to perform a number of actions with the entities, based on the visual mediators (circled in column 3), with each of these sub-nodes labelled 1-a, 1-b, etc. column 4 shows her realisation tree, with all nodes and sub-nodes included. some might argue that sheena’s realisation tree (figure 3) illustrates a mathematically endorsable solution to the equation. while it can be argued that her narrative about the actions for obtaining realisation 1 are mathematically endorsable, we find no evidence of explorative discourse in her explanation. the first step in her procedure is an action with 3x, which she described as having to ‘put it on the right spot’ (sub-node 1-b). her action with 3x was separated from her action with the negative sign, which was to ‘make it a plus’ (sub-node 1-c). nodes 1-d and 1-e illustrate a similar procedure for the completion of realisation 1. we conclude that her realisation routine is thus not endorsable. level 2 analysis the level 2 analysis uses the perceptible features of a learner’s discourse about linear equations recorded at level 1 (word and mediator use, narratives and realisations) to classify the routines using sfard’s (2008) seven features of ritualised and explorative discourse (table 2). the detailed list of indicators for these features is available in roberts (2016, p. 91). due to space limitations, in this article we illustrate how we utilise the level 1 analysis of sheena’s solution for 2x + 8 = −3x – 2 presented thus far, for the first two features of discourse, namely the degree of objectification and the extent to which the narrative is endorsed. we note, however, that whether a learner’s discourse is explorative or ritualistic at the time of the interview can only be determined after the full analysis. the level 1 analysis suggests that sheena uses mathematical and non-mathematical words as part of phrases in her routines, and hence in a phrase-driven way (indicator 1.2.1), as illustrated in the statement ‘here is variables on this side and here is numbers without’. her use of material process verbs in the active voice and pronouns indicates that she performs actions with and by mediators as disobjectified entities (indicator 1.2.2). the adverbs of place indicate the spatial arrangement of these entities (indicator 1.2.2) and the material process verbs suggest they actively move to different sides of the equation, thus disrupting horizontal equivalence (indicator 1.2.3). as expected, her realisation tree has only one branch, an algebraic branch (indicator 1.2.5). the level 1 analysis suggests that sheena’s narrative about how entities ‘move’ horizontally relies on the spatial arrangement and is not endorsable (indicators 1.2.3, 2.2.1). in terms of vertical equivalence, we note that some of her actions and narratives for obtaining realisations are endorsable, but the overall the solution is not (indicator 1.2.4). the indicators identified here, together with others in the full analysis of sheena’s interview, lead us to conclude that her discourse at this moment is ritualised. validity according to maxwell (1992), ‘validity’ in qualitative research refers to the relationship between the research account of what happened and the real-life situation (in this case, the interview). for ‘descriptive validity’ (maxwell, 1992, p. 285) in this research, speech and gestures in the recordings were accurately transcribed at a level of detail appropriate for the research focus, and transcripts were refined during the analysis through revisiting the recordings and engagement between the authors. the ‘theoretical validity’ (maxwell, 1992, p. 291) of sfard’s (2008) theory of commognition in this research is borne out by its use by a growing group of researchers, as reviewed in this article. in addition, in this article we define and illustrate the use of sfard’s tools specifically to study thinking about linear equations. since this is a small-scale qualitative study, the findings are not generalisable to any other group of learners or topic in mathematics. however, the analytic tools provide a rich description of the learners’ discourse, allowing the broader community to decide to what extent the findings may or may not apply in other contexts. findings the starting point for the investigation reported in this article was the set of findings in table 5, which indicates the learners’ scores on the set of equations used in the research. these findings suggest that many of the learners know how to solve these equations. the table shows that not all learners performed perfectly. this was useful for the study because, while initially we hypothesised that the learners whose solutions attracted full marks would be thinking exploratively, our findings showed that this was not the case. most learners lost marks through computational errors. table 5: grade 8 and grade 9 learner scores on equations from their written assessment task. in this section we present the results of applying sfard’s (2008) commognitive framework to each learner’s discourse in the interview. we structure these results according to sfard’s seven features of explorative and ritualised discourse (table 2). the study indicates that learners’ discourse when talking about their solutions to the linear equations was ritualised, not explorative. however, the detailed analysis showed subtle differences in the ritualised discourse across learners. to illustrate these differences we draw on a selection of learners in reporting on the findings of the study. note that although we only report on a selection of findings, we analysed the discourse of all 15 learners, including sheena. feature 1: degree of objectification of learners’ discourse there are some findings that show similar responses from the learners in the study. one of these is the way learners talk about positive integers (whole numbers) compared to the way they talk about negative integers, variables and algebraic terms. all learners talk about positive integers as mathematical objects in their own right, which indicates objectified word use. however, how they talk about negative integers and algebraic terms suggests differences in their thinking. another finding is that all learners use words in a routine-driven way in the context of their routines. this is illustrated by carla whose discourse is typical of many grade 8 learners. when solving the equation 2x + 8 = −3x – 2, her mediator and word use for the sum ‘–2 – 8’ points to routine-driven, disobjectified use of mediators and words on one line of an equation: ‘i moved this … −3x over but made it a … a positive number … because of the … the negative and positive … the opposite’. carla uses the reference pronoun ‘this’ for the keyword ‘3x’ and talks about the material action of personally ‘moving’ this entity ‘over’ (indicators 1.2.2 and 1.2.3). she also uses the colloquial word ‘opposite’ instead of talking about inverse operations as the curriculum specifies (indicator 1.2.1). there is little difference between the discourse of the grade 8 and grade 9 learners in this respect. kabelo (grade 9) explains part of his routine for solving the equation 6x – 12 = 2x + 4: ‘then i took this negative 12, then i put it on the right-hand side. then it became positive.’ on enquiry from the interviewer about why he ‘put it’ on the right-hand side, he explains that ‘it doesn’t have a x, and this one [points to 4 on right-hand side in line 2] doesn’t have a x’. his discourse is characterised by routine-driven actions with entities and his narrative is not endorsed. however, there are nuances between learners in the degree of objectification evident in their talk. in the methodology section, we suggested that sheena’s (grade 8) use of nouns, material process verbs in the active voice, pronouns, and adverbs when solving the equation 2x + 8 = −3x − 2 points to phrase-driven use of keywords. this, since she uses phrases constantly in her explanations (indicator 1.2.1) and performs actions with and by mediators as disobjectified entities (indicator 1.2.2). yet in her interview we also identify instances of objectified talk about integers (indicator 1.1.2). for example, in this narrative she does not split the integers into separate entities, and the relationship between them is endorsable (indicator 2.1.2): ‘−2 times 6 is −12. and −1 times 1 is −1’. similarly, kabelo (grade 9) describes a routine in a mathematically endorsed way using mathematical objects: ‘−2 times x is equal to −2x. and −2 times +3 is equal to −6’. there are also nuances in learners’ perception of the equal sign. we report on perceptive differences among three grade 8 learners. erin relates the equal sign to a calculation: ‘if you calculate something, that (points to equal sign) will indicate that you get your answer’. fatima’s use of adverbs of place shows how the equal sign is a spatial organiser for entities (indicator 1.2.3): ‘um … the constants on the one side and the variables on the other side.’ nadia, on the other hand, uses gesture to suggest that the left-hand side and right-hand side of the equation are equivalent: ‘it means that (points to left-hand side) is equal to that (points to right-hand side)’. although she has not expressed herself in words, her explanation relating to horizontal equivalence between the two functions is endorsable (indicator 1.1.3). of the four grade 9 learners three explain that the equal sign signified an equation. the discourse of the fourth learner, emily, shows that she sees the horizontal equivalence imposed by the equal sign: ‘the one side must … must have the same value as the other side’. she is also the only learner in the study whose solution to the equation 4m = 2m is endorsable. overall, we note that no learner in the study uses horizontal or vertical equivalence as a signifier or source of the narrative, except when prompted to do so (indicators 1.2.3, 1.2.4 and 1.2.5). an example of this is how, through prompting, tumisho (grade 9) explains that there is equivalence between ‘2x’ and ‘6’ in the equation 2x + 7 = 6 + 7: interviewer: can you, just looking at this [waves hand across both sides of the equation 2x + 7 = 6 + 7] … what’s … what’s 2x equal to? tumisho: um … 2x … um … equals to um … i think equals to 3. interviewer: 2x? tumisho: yes interviewer: and what’s x equal to if 2x is equal to 3? tumisho: [silence] interviewer: can you see that 2x is equal to 6? [points simultaneously to ‘2x’ on left-hand side and ‘6’ on right-hand side] tumisho: yes. so x … [points to ‘2x’] … so x will be 3 because 2 times 3 will get 6. we also presented learners with different versions of the equation ‘2x + 7 = 13’, namely ‘2x + 7 = 6 + 7’ and ’13 = 2x + 7’ to explore whether learners use horizontal equivalence as a resource for solving linear equations, which bears further discussion. some learners use horizontal equivalence to show that ‘2x’ and ‘6’ are equivalent in the equation ‘2x + 7 = 6 + 7’, and to see that ‘2x + 7 = 13’ and ‘13 = 2x + 7’ are equivalent equations and thus that the solution is the same for both (indicator 1.1.3). william’s reasoning is similar to sheena’s when explaining that ‘2x’ was equal to ‘6’ in the equation ‘2x + 7 = 6 + 7’: ‘that six represents that 2x, so … it equals two times three’. many learners revert to finding the solution for ‘2x + 7 = 6 + 7’ (indicator 1.2.4): ‘if i carry the seven over … and then the difference between seven and … negative seven is nought’. when repeating the calculations, these learners often make computational errors because the variable is not on the left-hand side. none of them question that their solution is different to that of the equivalent equation. these findings suggest that teacher prompts could potentially shift learners to think exploratively. where learners do not respond favourably to such prompts, as was the case with learners who reverted to finding the solution, teachers could possibly change their teaching strategies to encourage a shift to explorative thinking. when solving the given equations in the interview no learner produces endorsable narratives that reference vertical equivalence. rather, shakira’s narrative in which her mediator and word use interrupts equivalence is typical of all learners: ‘and then you have 2x. i took the x down … and i … move that [points to ‘2’ of ‘2x’ on left-hand side] over to there [points to right-hand side of next line]’. in this excerpt the adverb of place ‘down’ indicates material movement (‘took’) of ‘the x’, her gesture shows where she materially ‘moved’ ‘2’ to the right-hand side of the next line, and the adverb of time ‘then’ refers to the sequence of actions (indicator 1.2.3). to determine whether learners use vertical equivalence as a resource, learners were asked about the significance of substituting the solution into the original equation. it is curious that no learner shows familiarity with the notion of substitution, as this is a topic in the curriculum. therefore it was not productive to analyse their responses, but we note that substitution in relation to solving equations could encourage relational thinking in learners – and particularly in the development of their thinking about horizontal and vertical equivalence. finally, in terms of degree of objectification, we note that the realisation trees for all learners only have one branch, an algebraic branch (such as sheena’s in figure 3). thus, the equation does not signify more than one realisation for any learner (indicator 1.2.5). based on our evidence presented in this section we conclude that the discourse of all learners is predominantly disobjectified (indicator 1.1). this is not a criticism, because, as noted, learner orientation to the solution of linear equations is to the algebraic procedure. feature 2: endorsed narratives we have argued that the only mathematical objects students regularly talk about are positive integers, thus it is to be expected that their endorsed narratives are limited. all learners produce endorsed narratives about operations with positive integers (whole number arithmetic) (indicator 2.1.3). for example, explaining his calculation ‘6x − 2x’ in the equation ‘6x – 2x = 12 + 4’, zahir says: ‘then i minus um … 2 from … from 6, and it becomes 4’. although zahir writes the solution as ‘4x’, he only talks about the whole numbers. besides the endorsable narratives about positive integers, learners produce narratives that are not endorsed, about actions with disobjectified entities (indicator 2.2.1). they also use spatial arrangement and visual appearance as the source of the narrative (indicator 2.2.2). in the example that follows, erin uses spatial arrangement on the signifier ‘2x = 6’ to realise . she performs separate actions on ‘2’ and ‘x’: ‘you put the x at the bottom of 6 and bring over the 2 because … you must get a equal sign to x’. however, as with words and visual mediators, there are differences in the degree to which their narratives are endorsable. some learners structure their thinking by talking about the relationship between objects in a phrase-driven way. for example, sheena says: ‘negative two times three will give you negative six’. others produce endorsable narratives about the nature of integers, variables and equations (indicator 2.1.1). an example is tumisho’s narrative about the equation ‘2(4x – 5) – (3x + 6) = −2(x + 3)’. in response to an interview prompt about an arithmetic error, he points to the negative sign in the term ‘–(3x + 6)’ and explains: ‘there is a one there that you can’t see’. yet, he still produces the unendorsed realisation ‘–3x + 6’ (indicator 2.2.1). this example illustrates that although some learners recall previously endorsed narratives about mathematical objects, they do not necessarily link such narratives to endorsed actions with these objects. this is a feature of ritualised discourse. no learner uses endorsed narratives about objects as the source of their realisations (indicator 2.2.2), unless prompted to do so (as in tumisho’s case here). based on the evidence presented here, we argue that the extent to which learners’ narratives are endorsable is not characteristic of explorative discourse. feature 3: closing condition the analysis shows that no learner sees the closing condition as being to produce an endorsed narrative about the original equation. for all learners the closing condition is that the final realisation should have the appearance ‘x = [some number]’ (indicator 3.1). this is particularly evident in their solution of the equation that does not have a constant term. for example, initially nadia (grade 8) gives her solution to the equation ‘2x = −3x’ as ‘–1x’. she justifies her solution: ‘i think three minus two is one, right? … because the signs are different’. but later she changes her mind: ‘i don’t think you should put x there, because you’re trying to solve x’. nadia’s need to have an integer as a solution makes her discard the variable. there are numerous similar narratives in the data set about the closing condition for this equation. to investigate further the interviewer prompted learners to think about ‘zero’ as a solution. few learners could use the prompts to conclude that the variable would equal zero, with most wanting the solution to be a positive integer. in the words of zaahir (grade 9), ’there must be a constant or something’. however, prompting proved to be a good strategy to get some learners to think in an explorative way. we present an excerpt from the interview with joshua (grade 8) as a typical example of this. his initial response when asked to solve the equation 4m = 2m was ‘i don’t think i can, miss. because i never … my miss didn’t give us this sums yet miss’. the interviewer then prompted him to consider zero as a solution. parts of the interaction are presented below: interviewer: er … can m be zero? joshua: this m? [points to left-hand side]. which one, miss? interviewer: mm. which m? [silence] do the two ms have the same value? joshua: i don’t think so, miss, because this is a higher number [points to 4m] than this [points to 2m]. interviewer: oh okay. so how does that affect the value of m? initially, joshua is confused about substituting zero for m, but when that is clarified, the interview ends on a positive note: interviewer: and if m was zero? joshua: then the whole sum is zero, miss. interviewer: yes … so could m be zero? joshua: yes, miss then it come to the same values. interviewer: on both sides? joshua: yes, miss. feature 4: for whom the routine is performed there is no evidence that the learners’ discourse is characterised by internal persuasion based on mathematical properties of the object. rather, the learners’ regular use of adverbs of time and high modality verbs suggests that they solve the equations for, and with others (indicator 4.2). typical is sheena’s talk, which contains both the adverb of time ‘always’ and the high modality auxiliary verb ‘must’, suggesting that visual appearance is the source of the narrative about their actions: ‘the last sum must always end with the variable’. who are the ‘others’ whose authority drives learners? there is almost no reference to the teacher as the authoritative source of their narrative, except for zahir who, when faced with the equation ‘4m = 2m’ explains: ‘my ma’am never gave me a sum like this before. there must be a constant or something’. however, learners defer to other people as the authority. although they didn’t know the interviewer prior to the interview, many learners see her as a source of authority for solving the equation ‘4m = 2m’. this is particularly evident when she provides scaffolding by asking whether ‘m’ could be zero, a cue that many assume to mean that the solution had to be zero. yet only two learners could endorse this narrative. feature 5: by whom is the routine performed? the analysis shows that no learner solves the equations independently, using ‘thoughtful imitation’ (sfard, 2008, p. 249). rather, learners follow the metarules of others (indicator 5.2.2). sheena’s obedience to a rule is suggested by the high modality verb ‘must’: ‘the negative twenty-one must change to a positive twenty-one’. learners also use non-mathematical sources such as visual mediators and spatial organisation as the source of narratives (indicator 5.2.1). some learners defend their realisations in an endorsable way, but only when prompted by the interviewer (indicator 5.2.1). feature 6: level of flexibility this feature of the learners’ discourse was analysed using the alternate representations of the equation ‘2x + 7 = 13’, namely ’13 = 2x + 7’ and ‘2x + 7 = 6 + 7’. the aim was to determine whether their concept of equation could accommodate these modifications. as indicated, a minority of learners use horizontal equivalence flexibly to show that ‘2x’ equals ‘6’ in the equation ‘2x + 7 = 6 + 7’. most often, though, learners revert to the routine for finding the solution (indicator 6.2). feature 7: level of correctibility the measure of correctibility is related to learner errors, so evidence for this feature depends on whether learners make errors in their solutions. these errors are limited to arithmetic errors (identified in the teachers’ assessment) and errors relating to the structure of the equation. nadia, carla, kabelo and emily all correct arithmetic errors, although nadia and emily miss some of these errors. in addition, both nadia and alison have errors relating to horizontal equivalence that they do not correct, even with prompting. as an example, we present alison’s solution to the equation ‘2x + 8 = −3x – 2’, for which she obtained full marks in the assessment task: alison’s realisation from the signifier ‘5x = −10’ is ‘’, yet when asked what the value of x is, she simply responds: ‘where’s x miss? i think the x fell away’. discussion and conclusions the research in this article emerged out of a concern – expressed in the literature and experienced in the first author’s practice – that learners who may achieve above average assessment scores when solving linear equations have difficulty using mathematically valid resources to explain why or when to use a particular strategy. in this article we use sfard’s (2008) ‘penetrating’ theory of commognition, located in a participationist view of learning, to understand these aspects of thinking about linear equations. the analysis presented here is limited in the sense that it focuses on learner discourse at one moment in time rather than over time (compare heyd-metzuyanim, 2015; nachlieli & tabach, 2012) and it does not include a study of the related teacher discourse (compare heyd-metzuyanim & graven, 2016; sfard, 2017). however, our use of sfard’s concepts specifically for the study of discourse about linear equations (presented in the detailed table 2) is productive for drilling down to identify the similarities and also some nuanced differences in the discourse of 15 learners. the evidence presented in this article suggests that the discourse of the learners – both grade 8 and grade 9 – can be described as ritualistic rather than explorative. the only mathematical objects learners regularly produce are endorsed narratives about positive integers. mostly they produce unendorsed narratives about disobjectified entities, including actions with these entities. no learner talks about the properties of number or axioms (like inverse operations) as a source of the narrative for solving linear equations. rather, they perform their routines for and with others using the rules of others, visual appearance and spatial organisation as the source of their narratives. their realisation trees illustrate that they often use non-mathematical entities as visual mediators. for all learners the closing condition is the appearance of the solution. in addition, learners have little or no flexibility. the limited evidence available points to a reasonable level of correctibility regarding arithmetic operations, but not regarding the structure of their solutions. the analysis of these interviews, conducted in the eighth month of the school year, indicates that curriculum requirements for grade 8 and grade 9 are not met. by this time learners should, especially at grade 9 level, have progressed from ‘a view of mathematics as memorised facts and separate topics to seeing mathematics as interrelated concepts and ideas represented in a variety of equivalent forms’ (dbe, 2011, p. 21). while the curriculum specifications are not in question, we agree with gcasamba (2014) who suggests that the curriculum guidelines may promote such ritualistic discourse. for example, the teaching guidelines section of the curriculum document provides teachers with examples of linear equations of different complexity levels and ‘steps’ for solving each (dbe, 2011, p. 94). these may promote reliance on the visual appearance of a problem for the selection of the routine. the findings in this article – presented using a commognitive approach to thinking – are consistent with the views of linchevski and livneh (1999) and hoch and dreyfus (2004) that learners regarded as not having what they call structure sense rely on other people’s routines rather than on the properties of number or functions to solve linear equations. it also supports hoch and dreyfus’s (2006) claim that there is no correlation between learners’ structure sense and their manipulation skills. indeed, these findings show that for learners whose routines for solving linear equations yield above average scores on written assessments, the mathematical objects they work with are restricted to positive integers. the description of how these learners spatially arrange disobjectified entities as part of their routines resonates with how de lima and tall (2008, p. 4) use the concept of embodiment to describe how learners move terms of an equation. while the findings in this article resonate with the findings of other studies, the analytic tools have produced a particular nuanced account of differences in the ritualised discourse of learners that other studies have not shown. firstly, although most learners use colloquial words or talk about disobjectified entities, some use keywords in a phrase-driven way to produce phrases that are mathematically endorsable. secondly, while most learners only produce endorsed narratives about positive integers, some also produce endorsed narratives about positive and negative integers, algebraic terms and the structure of equations. thirdly, we have argued that all learners use spatial arrangement and physical appearance as sources of narratives and perform routines for and with others. yet the prompting that took place in the interviews led to some learners producing endorsed narratives as sources and showing internal persuasion. furthermore, some learners have the flexibility to see the structure of the equation as a signifier for their realisation, whereas others rely solely on the routine. these observed differences do not provide sufficient evidence to argue that some of these learners are in transition from ritualised to explorative discourse, that is, that they are ‘thoughtful imitators’ (sfard, 2008, p. 249) in the sense that they have thought about their routines and tried to make sense of things for themselves. yet some learners respond positively to prompts, and then try to make sense of their routines using endorsed narratives about mathematical objects. thus, these prompts could be a possible lever for shifting learners towards explorative discourse. this leads to recommendations for practice and research, which we turn to next. since teachers’ discourse has been shown to impact on learners’ discourse (heyd-metzuyanim & graven, 2016; tabach & nachlieli, 2011), teachers’ understanding of these nuances in learners’ discourse could be used as points of engagement in teacher professional development. there is also potential in this process to explore how the prompts used as a research tool in this study might be taken into the classroom and used to shift learner thinking. prompts that encourage learners to explain their thinking, revisit their solutions, and invite alternative approaches such as substitution might be used in whole class and small group classroom interactions. we also recommend a focus in this professional development on links between the algebraic format of linear equations and other representations of functions like flow diagrams, tables and ordered pairs. in this article we have attributed the single-branch realisation trees of the learners in this study to the commonly used algebraic approach used in teaching. yet the school curriculum does prescribe other approaches, for example using tables of ordered pairs. we argue that the establishment of these links in the classroom creates opportunities for learners to shift to explorative thinking about linear equations, especially when confronted with unfamiliar problems. we believe there is scope to twin such professional development recommended here with a wider research study, using the analytic tools presented in this article, that explores learner discourse on linear equations over time. acknowledgements data for the study on which this article is based were made possible by the schools development unit at the university of cape town. we also wish to acknowledge the centre for higher education development and the research office at the university of cape town for writing support for the article. we acknowledge the personal communication from anna sfard during the development of the study on which this article is based. competing interests the authors declare that they have no financial or personal relationships that may have inappropriately influenced them in writing this article. authors’ contributions a.r. was the researcher, and k.l.r. supervised the research. the writing of this 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(2011). discourses of functions: university mathematics teaching through a commognitive lens. in m. pytlak, t. rowland, & e. swoboda (eds.), proceedings of the seventh congress of the european society for research in mathematics education (pp. 2103–2112). rzeszów, poland: university of rzeszów. about the author(s) anthea roberts schools development unit, university of cape town, cape town, south africa kate le roux academic development programme, university of cape town, cape town, south africa citation roberts, a., & le roux, k. (2019). erratum: a commognitive perspective on grade 8 and grade 9 learner thinking about linear equations. pythagoras, 40(1), a519. https://doi.org/10.4102/pythagoras.v40i1.519 note: doi of original article: https://doi.org/10.4102/pythagoras.v40i1.438 erratum erratum: a commognitive perspective on grade 8 and grade 9 learner thinking about linear equations anthea roberts, kate le roux published: 05 dec. 2019 copyright: © 2019. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. in the version of this article published earlier, the date of publication reflected the incorrect year. the publication date, under the heading ‘dates’, should have been 28 march 2019 instead of 28 march 2018. this correction does not alter the study’s findings of significance or overall interpretation of the study results. the publisher sincerely apologises for any inconvenience caused. microsoft word 64 front cover final.doc 6 pythagoras 64, december, 2006, pp. 6-13 mathematical literacy as a school subject: failing the progressive vision? iben maj christiansen school of education and development, faculty of education, university of kwazulu-natal christianseni@ukzn.ac.za the national curriculum statement (ncs) for mathematical literacy (ml) is part of a progressive agenda for increased democracy and social justice. it claims that the new school subject mathematical literacy will provide learners with awareness and understanding of the role that mathematics plays in the modern world. however, the analysis developed in this paper indicates that the superficial engagement with complex applications of mathematics implied by the ml ncs is not likely to live up to its claim. in addition, we do not understand enough about the connections between mathematical, technological and reflective knowledge/knowing/competencies to know how to facilitate the awareness and understanding that is part of the vision of the ml ncs. introduction internationally, ‘mathematical literacy’ refers to the competence of individuals. some writers see mathematical literacy as a narrowly defined competence, which can be demonstrated on word problems or even ‘pure’ calculations (bynner and parsons, 1997; basic skills agency, 1997). at the other end of the spectrum we see strong links to a critical or democratic competence (frankenstein, 1990; skovsmose, 1994; christiansen, 1996b; vithal, 2003; kibi, 1993; povey, 2003).1 one aspect thereof is using mathematics as a tool in gaining insights into oppression, inequalities, and exploitation (see in particular mellin-olsen, 1987; frankenstein, 1981; fasheh, 1996)), another is to become aware of the effects of applying mathematical models in society (niss, 1984; booßbavnbek and pate, 1989; niss, 1990; blomhøj, 1999; skovsmose, 1994; christiansen, 2000; blum and niss, 1989), and a third component has to do with mathematics as a ‘gate keeper’, i.e., access to further education, discourses of power, etc. (povey, 2003; jungwirth, 2003). in south africa, ‘mathematical literacy’ (ml) also refers to a school subject, about to be implemented. the ml curriculum justifies itself in two ways; one is through claims of utility, the other is through claims that it will “provide learners with awareness and understanding of the role that mathematics plays in the modern world” (department of education, 2003). elsewhere, i have interrogated the first justification 1 for an overview and discussion of different views on mathematical literacy, see jablonka (2003). (christiansen, 2007). in this paper, i interrogate the second one. the ml ncs adheres to a proclaimed progressiveness in stating that it is part of a larger agenda for improvement of living conditions, social justice and democracy (chapter 1, general to all the ncss). in specific reference to critical citizenry: to be a participating citizen in a developing democracy, it is essential that the adolescent and adult have acquired a critical stance with regard to mathematical arguments presented in the media and other platforms. the concerned citizen needs to be aware that statistics can often be used to support opposing arguments, for example, for or against the use of an ecologically sensitive stretch of land for mining purposes. in the information age, the power of numbers and mathematical ways of thinking often shape policy. unless citizens appreciate this, they will not be in a position to use their vote appropriately. (department of education, 2003) in this paper, i set out to investigate the claim of the ml ncs that it provides learners with awareness and understanding of the role of mathematics in the modern world. this gives rise to three questions, namely: (1) what do we know about the role of mathematics in ‘the modern world’? and does the ncs for ml reflect this? (2) what do we know about learners becoming aware of this role? iben maj christiansen 7 (3) what are the issues of a curriculum serving this purpose? i will address the two first questions in turn, but for the sake of brevity only raise some concerns relating to the third question. the role of mathematics in society the national curriculum statement for mathematical literacy writes about the role played by mathematics as if it can be assumed that this role is well known, not the least to the educators. but what do we know about this role? some general philosophical considerations have been given to it, but the literature still only contains sketches of an understanding. we know that mathematics can be a powerful modelling tool. it is in modelling complex phenomena that mathematics assists us in developing insights we otherwise could not have had. some of the examples listed in the ncs belong in this category. for instance, a ‘dialogue’ between medical sciences (including community health science) and mathematics can lead to the development of a model for aids. complex mathematical models, based on natural geography and geo-physics, have been developed for the interplay between our impact on the environment, global warming and the depletion of the ozone layer. statistical models can be used to indicate the extent to which two factors can be said to be correlated. on the other hand, this also means that mathematics can change the discourses around these issues. because the models are not transparent to most people, they dramatically change the discourse surrounding these issues: the modelling process could create a specific way of interpreting the problem; it could cause a limitation on the group of people who have the possibility of participating in the discussion of problem-solving, and, as a consequence of a rationalistic and calculatoric [sic] way of arguing, a specific type of solution could become created. (skovsmose, 1990: 776) the prestige of mathematics can emerge from theories which formulate results and connections in the language of mathematics, and thus provide these with a (false) sense of objectivity. on the other hand, those norms to which mathematics is contributed can reduce the importance of theories which cannot be formulated in a strict mathematical language. (skovsmose, 1984: 41-44, my translation) this of course rests on the false acceptance of an expert-ideology (kemp, 1980). the ways in which mathematics together with the expert ideology transforms language games have been demonstrated elsewhere (christiansen, 1996a; 1997). in relation to models of the depletion of the ozone layer, i found that: because models are used as arguments, they can be challenged, and this creates the possibility of shifting the attention from what is predicted to how the prediction was obtained. thus, the debate about which course of actions to take can be substituted by a debate about the certainty of predictions, the quality of the model, and the limits to the underlying theory and assumptions. this is one way in which the original problem and the form of argumentation could be altered, transformed into something else. as mentioned earlier the introduction of quantitative arguments could alter the perception of the original problem – how to act in response to the ozone depletion – in other ways. quantitative arguments are an integral part of hypothetical reasoning creating scenarios which can indicate the graveness of the problem. but the possibility of choosing between scenarios can also transform the problem towards a discussion of how much cfc can be emitted. the models become tools in operating closer to critical points. (1996a: 69). i refer to this transformation of a fundamentally political problem into a question which can be addressed through technical means as the technocratic transformation (christiansen, 1996a). it was also evident in other examples. perhaps it is these roles of mathematics to which the ncs refers, when it states that “mathematical literacy provides learners with an awareness and understanding of the role that mathematics plays in the modern world”? if so, we are lacking analyses of the ways in which mathematics influences decisions and political debates in south africa. it may be that the expert ideology is less prevalent here than in the european countries from which the philosophical considerations above arise. paola valero (1999) makes this point in relation to latin america: mathematical literacy as a school subject: failing the progressive vision? 8 [...] decisions are made based [...] also on personal loyalty [...], political convenience, power of conviction through the use of language or violent, physical imposition. in this political scenario and ‘rationality’, mathematics does not necessarily constitute a formatting power that greatly influences decision making. (valero, 1999) however, let me, for the purpose of this paper, assume that mathematics does indeed play the same role in south africa as in denmark, where the analyses of skovsmose and myself originate. paradoxically, some of the examples given in the ncs refer to fairly advanced mathematical models, but the learners are not expected to engage with the models on a level which would lead to substantial insights – neither in the phenomena modelled nor in the technocratic transformations and other alterations of discourses generated through the use of mathematics and science. for instance, learners should be able to interpret graphs of temperature against time of day during winter over a number of years to investigate claims of global warming (department of education, 2003) anyone who has engaged with issues of global warming knows that it requires substantial engagement with various issues to even begin to investigate claims of global warming. for instance, the increased urbanisation has meant that places which 100 years ago where fairly rural are now highly urbanised which does increase the temperature enough to blur any trends due to global warming. more importantly, the complex nature of the interplay between atmosphere and surface environment means that global warming may make certain areas warmer and others colder. a complex mathematical model based on a deep understanding of natural geography, physics, etc. can assist our understanding of this complex phenomenon. but this is disguised by the simple level of engagement which is expected from the ml learners. thus, it serves nothing in terms of engaging with global warming or becoming aware of the real uses of mathematics as a modelling tool; it is simply an exercise in reading graphs, in the disguise of global warming issues. in this respect, it is not inviting insights into the complex role and function of mathematics in society.2 2 in addition, it assumes little of south african ml learners’ ability to engage deeply with relevant issues. in that way, it is bound to construct the learners opting for ml as the less able. by referring to ability as constructed, i do not mean to imply that all learners can do in all fairness, i should mention that there are a few positive exceptions where the learners are directed to engage with the extent to which the models can say something about the underlying cause and effect, as in • does a positive correlation between mathematics marks and music marks necessarily mean that facility in mathematics is dependent on musical aptitude? (assessment criteria 12.4.2) • does a positive correlation between pollution levels and tb infections necessarily mean that pollution causes tb? (12.4.2) but perhaps the ncs is more directed towards using mathematics as a tool to obtain insights into society? however, given the transformative agenda of the ncss, so strongly emphasised in the general chapter 1, what is striking is the nature of most of the examples chosen. many are concerned with investments, profit margins, number of employees, choosing best car hire or cell phone purchase, but also include investments in different currencies, installing an imported washing machine, tracking weight loss, etc. there is no encouragement to compare incomes and benefits, living conditions, access, effect of education on future income levels, etc. in other words, the curriculum appears to be directed at creating educated consumers in a capitalist economy, which remains unchallenged; there are few attempts at directing teachers’ and learners’ attention to truly transformative issues. when the ncs states that the aim is to “heal the divisions of the past and establish a society based on democratic values, social justice and fundamental human rights”, it does so without explicitly challenging the divisions created by capitalism or the artificial hierarchy of knowledge resulting in the theory/practice division. in addition, there is no exploration of the reproduction of capital and its human consequences. and so the ncs reflects an adherence to status quo and thus a contradiction of its proclaimed aims (cf. also dowling, 1998: 19). it is thus set on a path of contributing to the reproduction of social inequalities. in that sense, the ncs fails to engage mathematics to obtain the same; when they enter grade 10; cultural capital is very much at play (bourdieu, 1983/2004). i am simply referring to the continuous construction and reconstruction of learners’ ability as a complex of ‘expected’ behaviours, ‘needs’, etc. (dowling, 1998). thus, it should also be clear that i am not simply referring to the construction of learners opting for ml as the less able in terms of not being ‘able’ to complete the mathematics course for grades 10-12, but as a broader set of expectations about learners’ ability to engage critically and analytically with any type of issue. iben maj christiansen 9 insights into society in general, not simply into the role of mathematics in society. facilitating learners’ awareness of the role of mathematics in society the ncs for ml is, naturally for an outcomesbased curriculum, mostly concerned with the competencies which learners should be able to demonstrate at the end of their education. there is mention of what learners should or could engage with to become more aware of the role of mathematics in society. all fourteen of these examples (excerpts of the 57 outcomes statements and assessment criteria) are about critical investigation or discussion of the use of mathematics. for instance: • critical awareness of how data can be manipulated to prove opposing views should be developed (chapter 2, outcome 4) • criticise numerically-based arguments (10.1.1) • … linking the discussion to the way mathematics [sic!] can be used to argue opposing points of view (12.1.3) • critically interpret tables and graphs that relate to a variety of real-life situations (10.2.3) • identify possible sources of bias in gathering the data (11.4.3) working with such examples could facilitate learners’ insights in some of the ways in which mathematics is used in society, and thus ultimately in the function of society itself. such examples could have the potential to address the ways in which mathematics may alter discourses, etc. however, it is also possible to meet these outcomes or assessment criteria with rather simple examples, without the potential to engage the complex issues discussed in the previous section. furthermore, even if the ‘right’ examples are chosen, will engaging with them necessary lead to exemplary insight into the use of mathematics in society? there is an existing body of research in mathematics education which addresses how learners can learn mathematical modelling and thereby experience the power of mathematics in addressing complex issues (see for instance the ictma publications such as matos et al. (2001)). it seems evident that in order to truly understand how mathematics can be used and what the effects thereof are, learners must engage in modelling of complex phenomena themselves (christiansen, 2001a), and engage in critical reflection thereon (cf. christiansen, nielsen, and skovsmose, 1997). this is a pedagogic challenge, which we have only really begun to address within the past 20 to 30 years (mclone, 1984; cf. burghes, huntley, and mcdonald, 1982 for early contributions). substantial progress has been made over recent years, indicating the value of interchanging work with the entire model process with work on specific sub-processes (blomhøj and jensen, 2002). one of the real difficulties being that modelling of this nature requires substantial mathematical competencies as well as in-depth understanding of the area being modelled. in an attempt to obtain the same outcomes with less difficulty, a pedagogic recontextualisation has taken place, in the development of modelling tasks suited for the classroom. some of these have proven quite powerful in engaging learners in discussions of the role of mathematics in society.3 ole skovsmose has developed a terminology to talk about the different types of competencies/knowledge/knowing involved in this type of activity: (a) mathematical knowledge itself. (b) technological knowledge, which in this context is knowledge about how to build and how to use a mathematical model. … (c) reflective knowledge, to be interpreted as a more general conceptual framework, or meta-knowledge, for discussing the nature of models and the criteria used in their construction, applications and evaluations. (skovsmose, 1990: 765, see also; 1994) what remains uncertain is how the various competencies are linked, and to what extent mathematical and/or technological knowledge is a prerequisite for reflective knowledge. we have examples of people criticising the results of mathematical models being applied in society in ways which did not necessarily require substantial insight in the model or the underlying modelling 3 cf. the project developed by henning bødtkjer, mikael skånstrøm, morten blomhøj, helle alrø and ole skovsmose described in chapter 7 of alrø and skovsmose (2002), the examples in skovsmose (1994), and the three cases discussed in christiansen (1996). vithal (2003) discusses the implementation of project work with younger learners with the same intentions, but not engaging modelling to the same extent. though julie (1991) mentions this aspect as part of the people’s mathematics programme in south africa, this programme has focused more on curriculum and materials development, and julie (1993) has indeed criticised the programme for not inviting working with real life problems and for failing to deal with mathematical models that are used to regulate society. it is therefore of less relevance to the particular focus of this paper. as vithal (2003: 33) suggests: “perhaps … people’s education … was not primarily intended and therefore not developed as a substantial educational theory or movement but rather as a political (op)position or programme around which to organise mathematics educators to oppose apartheid education.” mathematical literacy as a school subject: failing the progressive vision? 10 process (christiansen, 2000). however, the question remains to what extent feeling comfortable with mathematics was an essential enabler nonetheless. we cannot know this with any degree of certainty, but we can be sure that the knowledge of mathematics as well as the technological/modelling competency – as any other tool – effects the relation between the subject and the object, and thus by necessity will influence the reflective knowledge. it is likely to both facilitate and limit the reflective knowledge in various ways. in education as well there are no easy answers. it is equally certain that the classroom community in which the facilitation of reflective and/or technological knowledge is expected to take place, with its divisions of labour, power relations and rules, will affect the activity. a comparison between three classrooms working with realistic modelling documented that the organisation of the classroom practices and the objectives of the activities determined the extent to which the students engaged with the deeper issues or constructed virtual realities in which to work (christiansen, 1997; 2001b). however, the absence of reflective knowing was obvious; generally, only when it was put on the agenda by the educators did it surface. an experiment using project work with south african grade 6 learners confirmed that learners do not easily engage in critical reflections on the roles and effects of using mathematics on the basis of specific cases, even when they are related to projects of their own choice (vithal, 2003). though most of these experiences are from other political, social and cultural contexts, they do illustrate that learners’ awareness of the role of mathematics in society is not straight forward to facilitate. through engagement with authentic problems, classroom practices may still eventually come to frame a reflective practice within which the relation between science, technology, mathematics and society can be addressed in a way which is not possible in the work-related or disciplinary practices. but we do not understand enough about how these practices, with their recontextualised critical mathematics discourse, are developed and sustained, to know how to ensure the vision of the ncs. combining epistemological access and social empowerment? there were two main reasons to construct an ml school subject for south africa. the one was to reach the 200,000 learners leaving grade 12 every year without mathematics and the 200,000 additional learners who fail mathematics yearly (see parker (2004) for an overview of national performance in matric mathematics). the failure of south african learners in international comparison surveys/tests only added fuel to this. since there are strong indications that adults who are innumerate are seriously disadvantaged in their employment possibilities (bynner and parsons, 1997), widespread innumeracy is of both economic and social concern (locally, nationally and globally). the other reason was to teach learners competencies and knowledge which would be in line with the overall intentions of the national curriculum. the intentions of the ml ncs (as stated in chapter 1, common to all the ncss) are proclaimed for improvement of living conditions, social justice and democracy, in other words directed to overcoming the apartheid legacy. mathematics, as a school subject, plays an important role in maintaining the theory/practice hierarchy (cf. christiansen, 2007). it also serves to maintain a class distinction, which of course is related. it is a well-known sorting tool of learners into those who master the decontextualised, selfreferential discourses, and those who do not. thus, the introduction of ml as a school subject was in part driven by a vision of a non-esoteric mathematics with real use value, which could still provide reasonable access to further education, etc. this obviously raises the important question: is it possible to combine access to existing areas of privilege (epistemological access) and at the same time further social empowerment of learners? elsewhere (christiansen, 2000), i have engaged this discussion much further, and it is a wellknown dilemma in mathematics education (ernest, 1991). but it is of specific relevance to the ml ncs, which in a sense is offered as a local solution to the dilemma. just as the mathematics curriculum (cf. the discussion of the mathematics curriculum in naidoo and parker (2005)), the ml ncs is a political hybrid product. though it states that “[t]he approach that needs to be adopted in developing mathematical literacy is to engage with contexts rather than applying mathematics already learned to the context” (chapter 3, ‘contexts’), it has an obvious focus on mathematical skills and concepts throughout. it is using claims of utility to justify itself, yet its content is distinctly mathematical. thus, the organising principles of the content remain invisible and inaccessible to the learners. this iben maj christiansen 11 positions them as objects rather than cognising subjects (cf. dowling, 1998: 46). thus, the curriculum is likely to contribute to the reproduction of social inequalities rather than promote social justice. i substantiate this point further in a forthcoming paper (christiansen, 2007). here, it suffices to point out that the ml ncs has not given sufficient attention to the complexity of this problem, and therefore also in this respect fails the progressive vision. conclusion the role of mathematics in society may vary from country to country. the south african ml ncs assumes that mathematics (through the use of mathematical models) does indeed play a role in arguments and the shaping of political society. my first point is to argue that when this happens, transformations of discourses are likely to take place. however, none of this can be assumed known by teachers, in particular since these uses of mathematics often remain hidden to the public. and indeed the examples suggested by the ncs do not engage in any substantial way with these uses of mathematics. this is the second point of the paper. the third is that there is not enough research to give us insights in the connections between mathematical, technological and reflective knowledge/knowing/competencies. to facilitate an understanding and awareness of the role of mathematics in society, it is argued that more drastic measures are required. this raises its own issues, as a discourse belonging to mathematics education and critical applied mathematics must now be recontextualised to the classroom. we know too little about how, if at all, this can be achieved. finally, i point out that the ncs for ml has not given sufficient attention to the potential tensions between facilitating epistemological access and social empowerment along the lines of the aforementioned awareness and understanding. schooling exists as an institution in the intersection between recontextualised disciplines, representing a general epistemic interest, and political life, representing specific epistemic interests. though disciplines and society influence each other, they are still opaque to each other, partly because of the increased specialisation in the division of labour (cf. otte, 1994: 130). school can be seen as an attempt to intermediate between these practices, but because of the opacity, this attempt results in the consolidation of schooling as yet another societal institution with distinct practices.4 while this restricts the apprenticing into both societal/political/work practices and mathematics, it can also offer an opportunity to reflect on this interplay. it is a worthwhile question to investigate further if this opportunity can be realised, and how. thus the answer to the title is that this particular ml ncs may, but it is still possible to imagine one that may not. acknowledgements thank you to carol bertram, diane parker, vaughn john, morag peden, and wayne hugo for useful input on the first draft of this paper. thank you to the two reviewers, who forced me to make some of my points clearer and more specific to the south african context. references alrø, helle, and skovsmose, ole. 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(2004). mathematics and mathematics teaching in south africa: challenges for the university and the provincial department of education. in r. belfour, t. buthelezi and c. mitchell (eds.), teacher development at the centre of change (pp.119136). durban, south africa: semi. povey, hilary. (2003). teaching and learning mathematics: can the concept of citizenship be reclaimed for social justice? in l. burton (ed.), which way social justice in mathematics education? (pp. 51-64). westport, connecticut: praeger. skovsmose, ole. (1984). kritik, undervisning og matematik. københavn, denmark: lærerforeningens materialeudvalg. skovsmose, ole. (1990). reflective knowledge: its relation to the mathematical modelling process. international journal of mathematical education in science and technology 21(5), 765-779. skovsmose, ole. (1994). towards a philosophy of critical mathematics education. vol. 15, mathematics education library. dordrecht: kluwer academic publishers. valero, paola. (1999). deliberative mathematics education for social democratization in latin america. the zentralblatt für didaktik der mathematik/international reviews on mathematics education (1), 20-26. vithal, renuka. (2003). in search of a pedagogy of conflict and dialogue for mathematics education, mathematics education library. dordrecht: kluwer academic publishers. article information authors: mdutshekelwa ndlovu1 affiliations: 1centre for pedagogy, faculty of education, stellenbosch university, south africa correspondence to: mdutshekelwa ndlovu email: mcn@sun.ac.za postal address: private bag x1, matieland 7602, south africa dates: received: 30 jul. 2013 accepted: 16 nov. 2014 published: 12 dec. 2014 how to cite this article: ndlovu, m. (2014). the effectiveness of a teacher professional learning programme: the perceptions and performance of mathematics teachers. pythagoras, 35(2), art #237, 10 pages. http://dx.doi.org/10.4102/pythagoras.v35i2.237 copyright notice: © 2014. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. the effectiveness of a teacher professional learning programme: the perceptions and performance of mathematics teachers in this original research... open access • abstract • introduction • conceptual framing of rme as a tpl approach    • rationale for using a learning theory as a framework for the design of tpl    • criticisms of the rme approach    • principles of an rme pedagogy       • activity principle       • reality principle       • level principle       • intertwinement principle       • interactivity principle       • guidance principle • research methodology    • research approach    • the sample for this study       • instruments • results    • quantitative results of the questionnaire    • quantitative results of the achievement test • qualitative results    • relevance of tasks to the reality principle       • relevance of tasks to the level principle       • relevance of tasks to the activity principle       • relevance of tasks to the interaction principle       • relevance of tasks to the guidance principle       • relevance of geometrical content on transformations to the senior phase       • relevance and adequacy of teaching and learning materials used       • suggested improvements in future sessions • discussion • conclusion    • limitations of the study       • ethical considerations • acknowledgements    • competing interests • references • appendix 1    • sample worksheet tasks abstract top ↑ the purpose of this article is to report on an investigation of the perceptions and performance of mathematics teachers in a teacher professional learning (tpl) programme based on realistic mathematics education (rme) principles, which included a topic on transformations, undertaken by the researcher. forty-seven senior phase (grade 7–9) teachers took part in the mixed-methods study in which they answered a questionnaire with both closed and open-ended items. fifty teachers took an achievement test at the end of the programme. the tpl programme used the rme approach in the design and delivery of mathematical tasks intended to enhance teachers’ mathematical knowledge for teaching. the sessions were conducted in a manner that modelled one way in which rme principles can be adopted as a teacher professional development strategy. the significance of the study is that continuing tpl is acknowledged to contribute to improvement in teaching and learning to address the concern about unsatisfactory learner achievement in mathematics. the responses suggested that the majority of teachers experienced the sessions positively in relation to all but one of the six rme principles. the teachers reported that they took an active part both as individuals and in small groups and expressed their willingness to adopt the type of activities and materials for their classrooms, which is an essential first step in guskey's first level of evaluation of a teacher tpl programme. the teachers’ average performance in an achievement test at the end of the topic was 72% which was indicative of modest learning gains at guskey's second level of tpl effectiveness. introduction top ↑ the poor performance of south african grade 9 learners in the annual national assessments for mathematics in 2012 (department of basic education, 2012) highlighted, amongst other factors, the need for quality teacher professional development or learning programmes. guskey (2000, p. 16) defines professional development as processes and activities designed to enhance the professional knowledge, skills and attitudes of educators so that they might in turn improve the learning of students. the focus in this study was to make a contribution towards teachers’ classroom practice in relation to shulman's (1986) concepts of pedagogical content knowledge, subject matter knowledge and curriculum knowledge so that the teachers in turn could potentially improve the learning gains of their students. this study adopted clarke and hollingsworth's (2002) perspective of teacher change as growth or learning in which teachers are themselves learners who work in a learning community. when teachers come together for a contact professional development session they form a learning community. sowder (2007), however, notes that a lack of effective, job-embedded professional development for teachers can be observed in the field of continuing teacher professional learning (tpl) for practising teachers. limited by time constraints, the impact of short-term lectures, meetings or workshops on teachers’ behaviour and students’ learning is limited (hellmig, 2009). roesken (2011) acknowledges that these constraints typify the traditional in-service programmes that simplify professional development to special events at some days during the school year. in their rushed nature, traditional approaches do not provide appropriate learning opportunities for teachers because they are designed in a hit-and-run fashion that compels them to focus more on outputs than on processes and outcomes. traditional approaches have also been criticised for being change-oriented or deficit models that see teachers as in need of fixing thus depriving them of their own agency (krainer, 2002, as cited by roesken, 2011). however, contemporary views suggest that teachers cannot be developed (passively) or changed, but that they develop actively and what we need do is to provide opportunities for them to change and grow and to own the change processes (day, 1999) in the hope that changes in practice could lead to changed student behaviours and possibly student learning (guskey, 2002) and ultimately student achievement (outcomes) in national and international benchmark assessments. with fewer and fewer mathematics teachers being trained and more and more qualified mathematics teachers leaving the profession, tpl opportunities offer a reasonable prospect of addressing the imbalances in student achievement in the long term. as part of a probable solution this study sought to articulate the design and implementation of a realistic mathematics education (rme)-informed tpl programme and to investigate, at guskey's (2000) first (participants’ reactions to training) and second (participants’ learning) levels, the effectiveness of the programme in which the researcher was involved. the rme approach originated in the netherlands and has been trialled in other countries. examples include the mathematics in context high school textbook series in the united states (us) (romberg, 2001), the manchester metropolitan university's rme pilot project in the united kingdom (ul) (dickinson & hough, 2012), the indomath project for the in-service education of junior secondary teachers in indonesia (hadi, 2002) and the shanghai ‘teacher action education’ programme in china (cheung & huang, 2005). the netherlands has previously performed well in international benchmark tests such as the trends in international mathematics and science study (timss) and virtually all its mathematics textbooks are now rme based (van den heuvel-panhuizen, 2010). the traditional approaches to in-service teacher education have probably not achieved their intended goals due to a myriad of factors. the significance of the study is that continuing tpl is acknowledged to contribute to improvement in teaching and learning (goodall, day, lindsay, muijs & harris, 2005), which tallies with the objectives of the intervention programme in this study conducted in the western cape in south africa. this study mainly sought to carry out level 1 and level 2 evaluations of the initial contact or workshop session. guskey's (2000) first level evaluates the participants’ reactions to whether the experiences were enjoyable and useful. this level is the most familiar and most widely used. the second level measures participants’ learning in relation to gains in knowledge or skill. conceptual framing of rme as a tpl approach top ↑ rationale for using a learning theory as a framework for the design of tpl the curriculum materials used for mathematics in-service teacher education for the senior phase in this study were specifically designed to allow for a realistic mathematics education (rme) approach, which originated from the freudenthal institute in the netherlands (van den heuvel-panhuizen, 2000). rme is a neo-constructivist approach, which stresses that mathematics learning should, in freudenthal's (1977) view, be connected to reality, stay close to children's experience and be relevant to society, in order to be of human value. the focal point should not be on mathematics as a value-free or context-free system, but on the activity or the process of mathematisation (freudenthal, 1991). treffers (1987) elaborates on mathematisation as consisting of both horizontal (solving problems set in a real-life situation, going from the world of life to the world of symbols) and vertical (reorganisation from the mathematical system itself, finding shortcuts and discovering connections between concepts and strategies and applying these strategies) dimensions. clarke and hollingworth (2002) support models of professional development and growth that incorporate key features of contemporary learning theory; an rme approach is one such potential model. contemporary views of staff development also promote constructivist approaches in the delivery of professional development programmes (beck, czemiak & lumpe, 2000) on grounds that for teachers to effectively adopt new teaching strategies they need to experience a professional development programme that builds their understanding of those new strategies through explicit activities in order to bestow familiarity with them. this is also in accord with jaworski's (2006) view of teaching as learning in practice or llinares and krainer's (2006) view that aspects of teaching are inextricably connected with learning. jaworski (2006, p. 187) emphasises in particular that ‘teaching develops through a learning process in which teachers and others grow into the practices in which they engage’. conceiving teachers as learners has been neglected as ‘we fight shy of using learning in relation to adults’ (lerman, 2001). the design of the intervention in this study sought to explicitly model to teachers (adults) as learners how they could enact in their own classrooms a neo-constructivist approach to mathematics teaching embodied in the rme model. for example, the uptake of rme principles from a professional learning programme into classroom practice might become possible if the programme helps them modify their self-efficacy beliefs about their ability to enact the principles in their teaching through ‘vicarious experiences and contextualised practice’ (posnanski, 2010) with those principles. schoenfeld (2000) thus rightly attests that teacher knowledge leads to growth and change of teacher knowledge and hence to issues of teacher learning and professional development. schoenfeld (2006, p. 485) reiterates that some of the most interesting approaches to professional development are those that take the notion of teacher learning seriously. the rme approach adopted in this study attests to the seriousness with which tpl was viewed. criticisms of the rme approach however, note was made of the following criticisms of an rme approach directed to primary mathematics education as summarised by van den heuvel-panhuizen (2010). firstly, critics of rme argue that students do not get enough opportunity to practise, yet the emphasis in rme is practice with understanding and coherence, not isolated drill. secondly, rme is accused of abolishing traditional digit-based algorithmic calculations in favour of new whole number-based written calculation. thirdly, it is alleged that rme only involves word problems, yet these have always been an object of suspicion in rme. finally, rme is accused of teaching students as many different calculation strategies as possible, which confuses them, yet it argues that students must have an understanding of the numbers with which they calculate and be able to use, if possible, shortened smart strategies. as a result of these criticisms the main didactical principles of rme which follow have been called ‘didactical blunders’ by opponents of rme. principles of an rme pedagogy in the present study the workshop sessions for teachers were designed and conducted in accordance with van den heuvel-panhuizen's (2000, 2010) six principles underpinning rme pedagogy. i now briefly discuss these six principles. activity principle this principle refers to the interpretation of mathematics as a human activity (freudenthal, 1991) in which students are treated as active participants in the learning process. the transfer of ready-made mathematics directly to students is considered to be an ‘anti-didactic inversion’ (freudenthal, 1973, 1983, 1991) that does not work (van den heuvel-panhuizen, 2010). rather, students are confronted with problem situations so as to develop all sorts of mathematical tools and insights, formal or informal, by themselves (cheung & huang, 2005). amongst their 10 principles for effective professional development designed to improve the individual teacher's practice, clarke and clarke (2005) also recommend using teachers as participants in classroom activities to model desired classroom approaches so as to project a clearer vision of the proposed changes. as such in this study rather than being lectured to, participants were given activities (tasks) to work on by themselves. for example, in the module on symmetry and transformations participants were given materials (scissors, rulers, erasers, small frameless mirrors, 1cm grid paper, coloured pencils or pen and pencil, drawing pins and matchboxes) and workbooks with classroom activities to work on. reality principle the reality principle emphasises that rme is aimed at having students be capable of applying mathematics (van den heuvel-panhuizen, 2010). rather than commencing with certain abstractions or definitions to be applied later, students start with rich contexts demanding mathematical organisation, that is, contexts that afford horizontal and vertical mathematisation (freudenthal, 1991). thus, like many progressive approaches to mathematics education, rme strives to enable students to use or apply their mathematical understanding and tools to solve experientially real problems. cooney (2001), for example, supports a realistic approach when asserting that activities in and of themselves are neither good nor bad; rather, it is the context that makes them effective. the course material in this study was designed in a manner that allowed participants to mathematise everyday experiences of transformations. for example, investigating reflectional and rotational symmetry in matchboxes, playing cards, spanners and nuts, clock faces, flowers and geometric transformations in ndebele, zulu, xhosa and zambian art. that is, there was a strong emphasis on students ‘making sense’ of the subject as also suggested by hough and steve (2007). see sample mathematical task items in appendix 1. level principle the level principle underlines that learning mathematics means that students pass through various levels of understanding (van den heuvel-panhuizen, 2010). in other words, student activities should first start from the (informal) situation level closely bound to problem contexts so that domain-specific situational knowledge and strategies can be used (cheung & huang, 2005). the second or referential level encompasses the use of concrete mathematical models representing mathematical objects. in streefland and freudenthal's formulations this is a level of ‘model of’ in reference to the concrete models’ close connection to the situations described in the problem (van den heuvel-panhuizen, 2003). the third or general level is a transitional level in which relationships are analysed through general mathematical models that can be dissociated from the problem contexts. in streefland and freudenthal's terminology such dissolute models are referred to as ‘models for’ where the focus is more on paradigmatic (or typical examples of) solution procedures that can be used to solve new problem situations (van den heuvel-panhuizen, 2003). the fourth or formal level allows pure cognitive thinking or higher level of formal mathematical reasoning, reflection and appreciation (cheung & huang, 2005). one of the enduring strengths of the level principle, thus, is that it guides growth in mathematical understanding, from the concrete or enactive, to the iconic and, ultimately, to the symbolic representational forms espoused by bruner (1960). amongst the 10 principles for effective professional development that they cite, clarke and clarke (2005) assert the need to recognise that change is a gradual, difficult and often painful process. thus, change or learning from one level of understanding to another requires scaffolding by more knowledgeable others or the sequencing of instruction in such a way that new learning carefully builds on previous knowledge. for example, the symmetry and transformation activities in this study did not end with concrete models, but extended to triangles, quadrilaterals, irregular shapes, calculations or determinations of axes of reflectional symmetry, angles and orders of rotational symmetry and even the use of congruency and similarity to prove simple circle geometry theorems thus signifying some progression from everyday experiences to ‘models of’ (horizontal mathematisation) to ‘models for’ and to higher levels of mathematical reasoning or proof (vertical mathematisation). intertwinement principle this principle means that mathematical domains such as number, measurement and data handling are not considered as isolated curriculum chapters but as heavily integrated (van den heuvel-panhuizen, 2010). students are given tasks involving rich problems in which they can use various mathematical tools and knowledge both within and across topics in a subject. this principle aligns with shulman's (1986) knowledge of the curriculum (kc). for example, in this study geometrical transformations (rotations, reflections and enlargements) were integrated with analytic geometry when coordinates were involved and were also linked to cases of triangle congruency and to circle geometry to solve simple riders as alluded to earlier. a major strength of the intertwinement principle is that it engenders a more coherent experience of the mathematics curriculum. interactivity principle this principle signifies that the learning of mathematics is not only a personal activity but also a social activity (van den heuvel-panhuizen, 2010). to that end, learners should be afforded opportunities to share their strategies and inventions with each other. by discussing each other's findings, students can get ideas for improving their strategies (van den heuvel-panhuizen, 2000). for the simple reason that the whole class approach has been the hallmark of traditional methods, in this study i privileged collaborative group work to emphasise to teachers how it could solve problems often experienced in overcrowded classrooms. included amongst clarke and clarke's (2005) 10 principles for effective professional development is the logic to afford teachers opportunities for support from peers and critical friends and to discuss problems and solutions of learning difficulties as a group. moreover, interaction can evoke both individual and collective reflection, which can scaffold students to higher levels of mathematical understanding. in concurrence, krainer (2002, p. 283, as cited by roesken, 2011) asserts that an increased competence in reflection raises the quality of action and the knowledge of views of others enlarges the view of one's own situation. guidance principle this principle means that students are provided with a ‘guided’ opportunity to ‘reinvent’ mathematics by ‘striking a delicate balance between the force of teaching and the freedom of learning’ (freudenthal, 1991, p. 55). this implies that in rme pedagogy teachers are expected to play a proactive role in students’ learning and that educational programmes should contain scenarios that have the potential to work as levers to reach shifts in students’ understanding (van den heuvel-panhuizen, 2010). in this study the delivery mode privileged a problem-based learning approach in which the instructional materials guided the participants to work in small groups but were also individually accountable for the completion of their own homework tasks. the assessment tasks were incorporated to ensure that teachers developed what adler (2005) refers to as mathematics (knowledge) for teaching. the facilitator encouraged participants to seek assistance from colleagues first and foremost and consult the facilitator as a last resort. the facilitator remained available all the time to anticipate participants’ difficulties and to help groups in meaning negotiation and collective self-reflection on the effectiveness of their problem-solving strategies. the significance of the guidance principle is that teachers must be able to foresee where and how they can anticipate the students’ understandings and skills that are just coming into view in the distance (van den heuvel-panhuizen, 2003). the didactic approach to be used for the sessions was explained to the teachers in advance and teachers were encouraged to identify its strengths and challenges. with that understanding, the following research questions guided the study: what were the senior phase mathematics teachers’ reactions to the usefulness of the rme-based tpl programme? how did the participants perform in an achievement test at the end of the topic on transformations? research methodology top ↑ research approach a concurrent mixed-methods approach was adopted using a semi-structured questionnaire with both closed and open-ended items and an achievement test with assorted items on transformations. the research design took the form of a survey and an achievement test to elicit the quickest responses with the least strain on teachers and yet be informative enough for subsequent sessions later in the year. the sample for this study a convenience sample of 47 (out of 53) senior phase mathematics teachers that participated in the workshop sessions responded to the feedback questionnaire. table 1 shows the demographic information relating to the teachers. from table 1 it is observable that the majority of the teachers (57%) were teaching senior phase mathematics (grades 7, 8 and 9). just over a third (38%) taught both the senior phase classes and further education and training (grades 10–12) classes. furthermore, the majority of teachers (51%) had at least 10 years of teaching experience. at least 45% of the teachers had an advanced certificate in education (ace) or higher and the majority (57%) of the teachers were female. eight teachers left their highest professional qualifications column blank, which could indicate some sensitivity about disclosing qualifications. table 1: demographic data. fifty out of 53 teachers took the achievement test on transformations at the end of the tpl sessions. instruments the main instruments for the study were: a questionnaire consisting of eight likert-type items intended to elicit answers from both closed and open-ended questions and an open-ended item (see figure 1). an achievement test at the end of the two-week tpl sessions during which the topic was presented. figure 1: feedback questionnaire. results top ↑ quantitative results of the questionnaire table 2 shows the numerical results of the questionnaire for the eight closed-ended question parts. on the whole there appeared to be an overwhelming consensus about the suitability of the content, activities, didactical approaches and the teaching materials, with the exception of whether the mathematical tasks were challenging enough. table 2: item total statistics. the results show that most of the items or principles received approval as having been in evidence in the presentations of geometrical transformations. the cronbach's alpha reliabilities for all item deletions all fell within an acceptable range of between 0.7 and 0.8, including the overall value. two items, in respect of the level and the guidance principles, had cronbach's alpha values higher than the overall value, suggesting that their deletion would improve the reliability of the scale. however, the differences were not statistically different so the items were maintained for completeness of the reporting. quantitative results of the achievement test table 3 shows how teachers performed in the various categories of transformation concepts. the symmetry of quadrilaterals was best understood whilst the symmetries of irregular shapes (in sample task item 2, see appendix 1) were the least understood. the main problem from these items was that many teachers would determine only a limited number of lines of symmetry where reflectional symmetry was concerned. the order of rotational symmetry was a challenge to some students where rotational symmetry was concerned. furthermore, where both rotational and reflectional symmetry co-existed it was difficult for some teachers to recognise both. the second most misunderstood concept or process was the construction of a triangle and a parallelogram (plane shapes). table 3: distribution of marks per sub-topic. figure 2 is a box-and-whisker plot obtained from imathas.com's boxplot grapher (http://www.imathas.com/stattools/boxplot.html) showing the overall performance of the 50 teachers who wrote the test. figure 2: box-and-whisker plot of the overall performance. the five-number summary of the distribution was: a minimum mark of 40%, a lower quartile mark of 63%, a median mark of 74.5% (compared to a mean mark of 72.1%, meaning scores were skewed to the left by -0.412), an upper quartile mark of 82% and a maximum mark of 96%. the distribution had a standard deviation of 13.34 and a standard error of 1.887. the mean was still lower than would be expected of teachers teaching these concepts but the variance was understandable given that some teachers were not necessarily qualified to teach mathematics. qualitative results top ↑ relevance of tasks to the reality principle some responses gave evidence of awareness of the importance of linking mathematics to the everyday life experiences of learners (reality principle, as exemplified in sample task item 1 in appendix 1), as the following examples show: teacher 13: especially for the fet phase where learners do not have a grasp of linking it into everyday life. teacher 15: very good examples, the examples is very practical and relates to real-life situations. teacher 42: they were challenging, especially with transformation you ended up taking a tracing paper so that you can make sure if your transformation is correct. relevance of tasks to the level principle whereas some participants felt the tasks were easy, others felt that they were challenging or at least would be challenging enough for their learners. tasks that are challenging should scaffold learners to move from one level of understanding to the next (level principle, as exemplified in task item 2 in appendix 1); they need not be too difficult. the following example responses attested to the level principle: teacher 1: it start with self-discovery and it end with problems of a ‘higher order’ level. it allows critical thinking. there were some stage that i felt so ‘stupid’ but eventually i got it right. specifically the reflection in y = −x and y = x and the rotation. teacher 23: plus or minus 20% challenging. but i am currently teaching grade 9 and 10 maths (for 32 years!!). the ones i found challenging were module 30: rotations and enlargement; similarity activities. teacher 45: it took a lot of different cognitive skills. e.g. when you have to do the rotation of 180° clockwise and anticlockwise. relevance of tasks to the activity principle the level of participation by participants as individuals (activity principle) as well as in small groups (interaction principle) was perceived as in the following sample responses: teacher 15: i took active part as an individual in my group, i had some particular time where i was explaining to my colleagues. this took part for the whole group where one member would be explaining to us where we were not understanding. teacher 9: i found the answers first then consulted. teacher 37: as an individual i have to draw making the diagrams, answers then as a group we make comparisons of our answers. relevance of tasks to the interaction principle the importance of the interaction principle at work was illustrated by the following responses, amongst others, which show how collaborative work helped the participants overcome their challenges: teacher 8: at times i differed with group members about some answers. we explained to each other and learned from each other. teacher 10: i as a teacher struggle with some of the concepts and the team mates and facilitator made it clear to me. thanks! teacher 13: although we spoke different home languages and afrikaans is my mother tongue my peers helped me by explaining the meaning of difficult words and concepts and formulas. relevance of tasks to the guidance principle although most of the guidance principle was built into the materials and the overall approach, some of it was evident in the following sentiments: teacher 23: i have picked up/been exposed to a lot of new ideas how to present my lessons, especially on grade 9 and 10 level. be more practical in the class! more constructions. my own worksheets in class must be clear and well planned. teacher 30: if something wasn't so clear to us we consulted each other in groups or with the facilitator. teacher 39: the material can also be used in the classroom for own lessons as it contains adequate scaffolding. relevance of geometrical content on transformations to the senior phase the overwhelming majority of the teachers who attested to the relevance of the topic content to the senior phase described such relevance in varied ways, as the following examples show: teacher 19: it includes topics outlined in the work schedule and it had many activities which will assist learners in problem solving. teacher 23: it was relevant because it is on the syllabus of grade 8 and 9. it also emphasises the starting point for this topic. teacher 34: it is relevant in the senior phase but looking at the fet (grades 10, 11 and 12) maths the transformation is no longer done in caps. relevance and adequacy of teaching and learning materials used the majority of participants reported that the materials they used were adequate for the tasks and some declared their readiness to adopt some of the activities and materials used: teacher 17: everything we needed to use was available to us. the tracing paper were available to do the construction, instrument box to draw circle and triangle. teacher 4: the problem we encounter in our school have large numbers and we don't have some of the equipment to demonstrate these types of transformations (e.g. mirrors) line of symmetry same as with patterns. i learnt a lot from this class because everything was demonstrated to us. teacher 8: it was adequate and appropriate because the chapter of transformation were covered and we had all the material needed to complete the activities. the material needed are things you have in your class. baturo, cooper, doyle and grant (2007) argue that effective teacher education tasks should enable translation to classrooms (at the technical level), enhance the success of student learning (at the domain level) and facilitate transfer to other topics (at the generic level). the latter level coincides with the intertwinement principle, which was largely built into the materials. suggested improvements in future sessions many participants felt that the time allocated for the topics covered was inadequate. an almost equal number apiece were (1) happy with the delivery mode as it was, (2) needed more whole class facilitator explanations and discussions or (3) would have preferred their home language (predominantly afrikaans) to be used for instruction and materials. some even felt they were competent enough to be engaged as facilitators. the examples below show some typical recommendations: teacher 39: we need more time … was squashed!!! i was not able to complete the activities of any day except for the last day (friday) (that was when i concentrated mostly on myself.) this was not good in my opinion as we were at the course to share and enrich each other. teacher 8: i am happy with the workshop as it is. such workshops are good for us. you also need to do them even for grade 11 and 12 as well in future. topic we can be covered again in grade 9 is finance and conversion table (metric + imperial). teacher 40: well only worked small group; but maybe there could have even more of a plenary where we could hear from the class as a whole. teacher 11: notes in afrikaans will be much appreciated. you can go through us with the answers. more explaining. teachers forget things teacher 3: would enjoy being part of your presenters team – i am a 100% educator. discussion top ↑ whilst the quantitative results showed an approval of the rme-based teacher professional learning opportunity the teachers received, the qualitative aspects of the data gave some details about the specific instances in which the teachers perceived the contact session to have fulfilled their expectations at guskey's level 1 of professional learning effectiveness. all the main principles of rme were in evidence in the teachers’ feedback with the exception of the intertwinement principle, which was largely built into the materials. the participants overwhelmingly felt that they were actively engaged both as individuals (activity principle) and as small groups (interaction principle). they also felt that the activities were based on real-life experiences that could be of interest to their learners (reality principle) and challenging enough to their learners and to some of the participants (level principle). they felt that the materials used helped them to understand the geometry of transformations much better and they thought that they received enough guidance from the curriculum materials used (guidance principle). however, they felt that there could have been more whole-class discussions to tie up the loose ends. this was not surprising for teachers who came from a background of traditional teacher-centred approaches that still dominate the overcrowded, under-resourced mathematics classrooms in disadvantaged communities. in line with their varied levels of experience and expertise, the teachers varied widely in their levels of mathematical self-efficacy, some feeling overwhelmed because of teaching the subject for the first time or inadequately supplied with curriculum materials whilst a few others felt on top of the game and wished they could be involved as resource persons. that was an exciting prospect, which should open up the possibility of transformative models for use in sustained, collaborative, professional development (johnson & marx, 2009). the teachers performed modestly on the achievement test, showing some learning gains at guskey's level 2 as well. the varied levels of understanding were however shown by the analysis of variance statistics, which also confirmed the varied levels of experience and expertise in mathematics teaching at the senior phase level. surprisingly, some teachers did not even know where they could obtain topic-relevant materials such as tracing paper in order to make the teaching of transformations more hands-on, especially rotations, enlargements and reflections that are not along the x-axis and y-axis, which seemed to challenge some participants. lessons drawn from the workshop activities were principally that tasks should revolve around learners’ experiential world in order to be of interest and relevance to them (reality principle). there was considerable positive attitude expressed towards activity-based learning (activity principle) and collaborative learning (interaction principle). tpl programmes should thus relentlessly model the teaching strategies intended for teachers to adopt in their classrooms. as windschitl (2002) observes, many teachers cannot implement constructivist ways of teaching mainly because they have not experienced them themselves either during their own schooling or their own initial training. conclusion top ↑ the purpose of this study was to investigate in-service teachers’ perceptions of the relevance of a teacher professional learning programme modelled on rme principles. a major positive outcome of the study was that many teachers felt better prepared to adopt some activity-oriented tasks in their classrooms in the teaching of not only transformations but geometry and mathematics in general. unless teachers can commit themselves to instituting change in the ways they teach the subject in their classrooms little change can come about. given the endemic shortage of qualified mathematics teachers, it is not surprising that many teachers of mathematics are not specialist mathematics teachers especially in the senior phase and thus require constant support. of interest too was the fact that teachers are sometimes alert to syllabus changes that bring about discontinuity in mathematical content that is taught in different phases of the schooling system. the recent removal of transformation geometry from the curriculum and assessment policy statements for the further education and training phase (grades 10–12) has caused some teachers to question the future relevance of the topic to the senior phase. ‘if it is not going to be examined later at the national senior certificate, why bother?’ seems to be a rational question in a system increasingly driven by high stakes examinations. these are questions that curriculum planners have to ponder to convince teachers about the wisdom of including topics that the latter have historically not been comfortable with in the first instance. finally, the fact that teachers in this study engaged with mathematical content meaningfully helped to address some of their classroom mathematical knowledge needs. although a few still scored marks below 50% the average mark of 72% was presumably modest enough to inspire confidence in their feelings of self-efficacy in transformation geometry (guskey's levels 1 and 2). further follow-up research is needed to explore the impact of the tpl programme on organisational teacher support (guskey's level 3), implementation support and monitoring (guskey's level 4) and ultimately on student learning outcomes (guskey's level 5). limitations of the study this study was limited to teachers’ perceptions of the effectiveness of a contact session of a teacher professional learning programme based on rme principles and scores obtained in an achievement test. it therefore does not cover the full spectrum of the entire lifespan of the programme which included class visits and another contact session later in the same year. ethical considerations permission was granted by the western cape education department to conduct this study and ethical clearance was obtained from stellenbosch university's research ethics committee. the participants in this study signed letters of consent and were advised of the objectives for the research, which primarily sought to improve future in-service training. the participants were assured of confidentially and anonymity. their names were neither required in their questionnaire responses nor were they going to be used in the analysis of the achievement test results. acknowledgements top ↑ i acknowledge that my involvement in this study was part of my research duties at the stellenbosch university centre for pedagogy (suncep). i am also grateful to my colleagues, ramesh jeram (for allocating me the responsibility of the geometry modules covered in this study) and cosmas tambara (for helping me in the administration of the questionnaire). competing interests i declare that i have no financial or personal relationship(s) that might have inappropriately influenced me in writing this article. 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(2010). reform under attack-forty years of working on better mathematics education thrown on the scrapheap? no way! in l. sparrow, b. kissane, & c. hurst(eds.), shaping the future of mathematics education: proceedings of the 33rd annual conference of the mathematics research group of australasia (pp. 1–25). fremantle: merga. available from http://www.merga.net.au/publications/counter.php?pub=pub_conf&id=863 windschitl, m. (2002). framing constructivism in pracatice as the negotiation of dilemmas: an analysis of the conceptual, pedagogical, cultural, and political challenges facing teachers. review of educational research, 72(2), 131–175. http://dx.doi.org/10.3102/00346543072002131 appendix 1 top ↑ sample worksheet tasks sometimes there are good and practical reasons to make an object symmetrical. what type of symmetries do you find in each of the following four playing cards: 8 of clubs, 10 of diamonds, queen of spades, and king of hearts. why is it convenient when a playing card is symmetrical? what sort of symmetries occur in this nut and spanners? investigate each of the figures below for line (reflectional) or rotational symmetry by showing the following, where applicable (for line symmetry check your answer by folding or using the provided mirror, for rotational symmetry use the tracing paper provided): line axis (or axes) of symmetry, the centre point of rotation the angle of rotational symmetry, the order of symmetry. reviewer acknowledgement open accesshttp://www.pythagoras.org.za page 1 of 1 reviewing is an important part of scholarly work, making a substantial contribution to the field. reviewers’ comments serve two purposes, guided by two inter-dependent objectives: • pythagoras wishes to publish only original manuscripts of the highest possible quality, making a meaningful contribution to the field. reviewers advise the editor on the scholarly merits of the manuscript to help evaluate the manuscript and to decide whether or not to publish it. reviewers are encouraged to reject a manuscript if it is scientifically flawed, merely sets out observations with no analysis, provides no new insights, or is of insufficient interest to warrant publication. • pythagoras is committed to support authors in the mathematics education community. reviewers help the author to improve the quality of their manuscript. reviewers are encouraged to write their comments in a constructive and 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1. log into the online journal at https:// pythagoras.org.za 2. in your ‘user home’ [https://pythagoras. org.za/index.php/ pythagoras/user] select ‘edit my profile’ under the heading ‘my account’ and insert all relevant details, bio statement and reviewing interest(s). 3. it is good practice as a reviewer to update your personal details regularly to ensure contact with you throughout your professional term as reviewer to pythagoras. please do not hesitate to contact us if you require assistance in performing this task. publisher: publishing@aosis.co.za tel: +27 21 975 2602 tel: 086 1000 381 the quality of the articles in pythagoras and the credibility and reputation of our journal crucially depend on the expertise and commitment of our peer reviewers. no. of manuscripts processed in 2018 (outcome complete) 35 accepted without changes 0 accepted with minor changes (to the satisfaction of the editor)1 2 accepted after major revisions (re-submit, then re-review)2 10 rejected after review − not acceptable to be published in pythagoras 3 rejected without review − not acceptable to be published in pythagoras3 20 no. of articles currently in review 4 alwyn olivier anita campbell anass bayaga antonia makina balarabe yushau barbara posthuma bhekumusa khuzwayo carol bohlmann carol macdonald cerenus pfeiffer clement dlamini cyril julie david mogari dirk wessels duan van der westhuizen gerrit stols helena miranda ingrid mostert jenefer golding judah makonye kakoma luneta karen junqueira lekwa mokwana luckson kaino marc north marie joubert marthie van der walt maureen ledibane mdutshekelwa ndlovu michael murray mogege mosimege faaiz gierdien nosisi feza nyna amin olatunde osiyemi patrick barmby peter pausigere piera biccard rajendran govender richard alexander satsope maoto shaheeda jaffer sibawu siyepu suela kacerja tulsi morar verona leendertz wajeeh daher willy mwakapenda zulkardi zulkardi acknowledgement to reviewers 1. accepted after one round of review, with ‘minor’ changes as specified by reviewers and editor. 2. accepted after two or more rounds of review, with ‘major’ changes specified by reviewers and editor. 3. all submissions undergo a preliminary review by the editorial team to ascertain if it falls within the aims and scope of pythagoras, is of sufficient interest to our readers, offers substantially new knowledge, and is of sufficient quality to be sent for review. http://www.pythagoras.org.za https://pythagoras.org.za https://pythagoras.org.za https://pythagoras.org.za https://pythagoras.org.za https://pythagoras.org.za/index.php/pythagoras/user https://pythagoras.org.za/index.php/pythagoras/user https://pythagoras.org.za/index.php/pythagoras/user mailto:publishing@aosis.co.za article information authors: magret courtney-clarke1 helena wessels2 affiliations: 1independent educational consultant2research unit for mathematics education, university of stellenbosch, south africa. correspondence to: magret courtney-clarke postal address: po box 661, swakopmund 9000, namibia dates: received: 27 aug. 2013 accepted: 02 apr. 2014 published: 30 june 2014 how to cite this article: courtney-clarke, m., & wessels, h. (2014). number sense of final year pre-service primary school teachers. pythagoras, 35(1), art. #244, 9 pages. http://dx.doi.org/10.4102/ pythagoras.v35i1.244 copyright notice: © 2014. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. number sense of final year pre-service primary school teachers in this original research... open access • abstract • introduction    • the namibian context    • number sense    • the development of number sense and the role of the teacher • theoretical framework • research design • ethical considerations • findings    • conceptual understanding    • procedural fluency    • strategic competence    • adaptive reasoning       • productive disposition • discussion • conclusion • acknowledgements    • competing interests    • authors’ contributions • references abstract top ↑ number sense studies have indicated that the development of number sense should be the focus of primary school mathematics education. the literature review revealed that learner performance is linked to teacher subject knowledge and that teachers’ confidence in doing and teaching mathematics influences the way they teach and their willingness to learn mathematics. this study was motivated by the poor performance of namibian primary school learners in both national and international standardised assessment tests and explored the number sense of 47 final-year primary school pre-service teachers (psts) in namibia. the data in this mixed method research design were obtained from a number sense questionnaire, a written computations questionnaire, a mental calculations questionnaire and the mcanallen confidence in mathematics and mathematics teaching survey (mcmmts). six psts, randomly selected from the 47 participants, were interviewed to determine their use of number-sensible strategies. the overall results of this investigation revealed that the final year primary school psts demonstrated limited number sense and possessed very few of the indicators of number sense. unexpectedly, the confidence survey showed that they were confident in their ability to do and to teach mathematics. this study exposed one reason for the low standards of performance of namibian learners in mathematics and the lack of improvement over the last few decades. it indicates a need for teacher training institutions to identify the mathematics that teachers should know and the ways in which teacher understanding of subject content has to be transformed to enable them to develop the number sense of primary school learners. introduction top ↑ the skills and dispositions to make sense of numerical situations in everyday life underpin the ability to understand, critically respond to and use mathematics in different social, cultural and work contexts. these skills and dispositions are related to number sense, which is increasingly seen as an important outcome of primary school mathematics education in the 21st century.the term ‘number sense’ reflects the changes in the perception of the role and nature of arithmetic in primary school mathematics. people with number sense have a disposition to make sense of numerical situations, look at problems holistically and use numbers flexibly to do mental calculations, produce reasonable estimates of numerical quantities and use figures to support an argument (sowder & schappelle, 1989). this study explored the likelihood that the lack of number sense of primary school teachers might be a related to the poor performance of namibian primary school learners in both national and international standardised assessment tests. the namibian context in namibia, the identified skills shortage fields in the professional careers (e.g. engineers and accountants), the technical careers (e.g. computer technicians) and the trade careers (e.g. electricians) require appropriate numeracy skills or a high level of mathematical proficiency (links, 2010). however, educational statistics reveal both a shortage of grade 12 graduates with higher-level mathematics and low numeracy levels at the end of primary school. in 2012, only 3% of grade 12 mathematics enrolments were at higher level (directorate of national examinations and assessment, 2013); the grade 5 and grade 7 standardised achievement test results showed that most of the participating learners performed in the below-basic and basic categories (kavishe, 2013). spaull (2011) classified 53% (adjusted for dropout rates) of namibian grade 6 students as functionally innumerate based on the 2007 southern and eastern africa consortium for monitoring educational quality iii results.both kavishe (2013) and spaull (2011) linked low primary school learner performance to poor mathematics teacher competence. the grade 6 mathematics teachers’ average score was below the southern and eastern africa consortium for monitoring educational quality average learner score (spaull, 2011) and the standardised achievement test results suggest that teachers were challenged in delivering in the standardised achievement test competency areas (kavishe, 2013). this indicates that mathematics education in namibian primary schools is largely ineffective in developing the basic numeracy skills required for further studies in mathematics. mathematics teaching in namibian primary school classrooms is still characterised by drill, practice and rote learning (clegg & courtney-clarke, 2009). mathematics psts are a product of how and what they were taught in schools and will, without an appropriate education at tertiary level, carry the same practices forward to their own teaching at school. number sense the notion of number sense embodies the most important concepts, skills and attitudes that learners should acquire in primary school as a foundation for further studies in mathematics and the development of quantitative literacy. number sense is complex and can best be described by the skills and understandings that a person with number sense demonstrates: a concrete sense of quantity, including multi-digit whole numbers, fractions and decimals and their different representations, an understanding of the relative magnitude of an amount, an ability to compare quantities using benchmark numbers such as 0, or 100, automatic recall of basic facts, procedures and strategies and the application of this knowledge to operations and to solve problems, flexibility in performing mental calculations and the ability to judge the reasonableness of an answer.mcintosh, reys and reys (1992) define number sense as a person’s general understanding of number and operations along with the ability and inclination to use this understanding in flexible ways to make mathematical judgements and to develop useful strategies for handling number and operations. (p. 3) a firm understanding of numbers and the number system is central to learning mathematics (hiebert, carpenter & moser, 1982; sowder & wheeler, 1989). reys, reys, mcintosh, emanuelsson and yang (1999) in their study of the number sense of learners concluded that ‘it was the consistently low performance of students across all countries that reminded us of the common international challenge this topic provides’ (p. 68). mathematical activity is present in all cultures, even without formal schooling; dehaene (2001) hypothesises from a cognitive neuroscience perspective that number sense qualifies for a biologically determined category of knowledge. his hypothesis is based on studies of the brain as well as experiments with animals and infants to establish basic numerosity: i propose that the foundations of arithmetic lie in our ability to mentally represent and manipulate numerosities on a mental ‘number line’, an analogical representation of number; and that this representation has a long evolutionary history and a specific cerebral substrate as single, analog representation. (dehaene, 2001, p. 17) dehaene’s theory of an evolutionary, biological base of number sense fits observations about the intuitive understanding of numbers and counting of children under the age of six (jordan, 2007). it is through development and education that this central representation becomes connected to other cognitive systems and conceptual structures, which need to be internalised and coordinated. case (1989) observed that the degree of number sense varies between individuals, groups and cultures and is influenced by the type and amount of mathematical activity in which young children engage. the development of number sense and the role of the teacher children who lack a sound foundation for mathematical understanding will experience difficulties with mathematics in later years and this will become ever more difficult to remediate as time passes (chard et al., 2008). intervention programs have demonstrated that instruction including number sense activities, with focus on mental computation or estimation, can improve mathematics performance significantly (gersten, jordan & flojo, 2005; gersten & chard, 1999; hing, 2007; jordan, 2007; markovits & sowder, 1994).conceptual understanding as a foundational aspect of number sense is highlighted in the definition of number sense as ‘a well-organised conceptual network of number information that enables one to relate numbers and operations to solve problems in flexible and creative ways’ (markovits & sowder, 1994, p. 23). teaching for number sense should therefore focus on learners and their solution strategies rather than on the right answers, on thinking rather than the mechanical application of rules and on learner-generated solutions rather than on teacher-supplied answers (reys et al., 1992). this is very challenging for teachers who, like the teachers in namibia, have traditionally emphasised mastery of algorithms and repeated practice with little focus on conceptual understanding of mathematics concepts, which includes the understanding of numbers and the number system. children who do not grow up in an environment that exposes them early to quantitative thinking and analysis will not acquire number sense without formal explicit instruction. these include children from low-income families, minority groups, children from homes where both parents work, non-english speakers and children living in poverty (gersten et al., 2005). in namibia, early interventions and formal explicit instruction that is aimed at developing number sense might have to be geared towards the entire system as children in developing countries ‘are frequently at the same levels as children who in the developed world are seen as needing special help’ (reubens, 2009, p. 2). this puts the spotlight on the teachers and their competence to develop learners’ number sense. teachers should be confident in their knowledge and understanding of how the number system works, be able to use this understanding in flexible ways to make mathematical judgements and have a repertoire of strategies for handling numbers and operations (national council of teachers of mathematics, 1991); this means that teachers should possess number sense themselves. this argument is supported by other researchers. ma (1999) compared a teacher to a guide who takes learners from their current understanding to further learning and prepares them for future travel. to be able to do this, the teacher needs a sound understanding of the key mathematical concepts for the learners’ grade level as well as an understanding of how those concepts connect with learners’ prior and future learning. such knowledge includes an understanding of the ‘conceptual structure and basic attitudes of mathematics inherent in the elementary curriculum’ (ma, 1999, p. xxiv) as well as an understanding of how best to teach the concepts to children. this is what ball, thames and phelps (2008) termed specialised content knowledge, an ‘uncanny kind of unpacking mathematics’ (p. 400) that makes particular features of content visible to and learnable by all children. trafton (1989) indicates that number sense should not be treated as a new topic, but rather that ‘number sense needs to be an ongoing, informal emphasis in all work with numbers’ (p. 75). this is reflected in the notion of big ideas, which are statements that focus on mathematical ideas as networks of interrelated concepts so that ‘mathematics can be represented as coherent and connected enterprise’ (national council of teachers of mathematics, 2000, p. 17). for example, one big idea for early number sense is ‘numbers are related by a variety of number relationships’. hill (2008), in a study that explored the relationship between teacher subject matter knowledge and instruction, came to the ‘inescapable conclusion … that there is a powerful relationship between what a teacher knows, how she knows it, and what she can do in the context of instruction’ (p. 497). researchers into the number sense of teachers in various countries, for example australia (kaminski, 1997), the united states (johnson, 1998 in tsao, 2005), taiwan (tsao, 2004, 2005; yang, reys & reys, 2009) and kuwait (alajmi & reys, 2007) have concluded that the majority of primary school teachers have limited number sense and very little understanding of what number sense is. for example, teachers tend to rely on standard algorithms and avoid mental calculations and estimation; they do not use benchmarks to reason about the effects of operations; they cannot flexibly apply numbers and operations to computational situations; they hold some of the same misconceptions as their learners, especially in the domain of rational numbers. if we want to improve learners’ number sense, the development of number sense should become a focus of pre-service primary school teacher education (yang et al., 2009) however, not only content knowledge and skills play an important role. teachers’ self-confidence affects the way in which they approach problems, the connections they are able to make between related concepts and their repertoires of strategies (ball, 1990). graven (2004) hypothesises that confidence in being able to learn mathematics is a resource that enables teachers with little mathematics training to learn the mathematical competencies required for teaching mathematics. attempts to improve psts’ content knowledge should therefore be coupled with a safe and supportive environment in which these students can develop confidence and positive feelings towards the subject. one cannot develop number sense in others if one does not possess a certain measure of number sense oneself; this was aptly put by greeno (1991) who stated that if someone ‘is to serve as an effective guide to newcomers in an environment, it is essential that the guide himself or herself should be a comfortable resident of the environment’ (p. 55). theoretical framework top ↑ more recently, the focus has shifted to defining number sense in terms of its characteristics as summarised by kalchman, moss and case (2001): fluency in estimating and judging magnitude, the ability to recognise unreasonable results, flexibility when mentally computing, the ability to move amongst different representations and to use the most appropriate representations and the ability to represent the same number or function in multiple ways depending on the context or purpose of this representation.in this study the characteristics of number sense were regarded within the framework of ‘mathematical proficiency’. according to kilpatrick, swafford and findell (2001), mathematical proficiency has five strands: conceptual understanding refers to an integrated and functional grasp of mathematics. conceptual understanding is reflected in the ability to represent mathematical situations in different ways and to apply the most useful representations in different situations. procedural fluency refers to the knowledge of procedures, when and how to use them and the skill in performing them flexibly, accurately and efficiently. it also includes the knowledge of ways to estimate the results of a procedure. rapid, automatic access to facts and procedures enables a learner to pay attention to the interrelationship between concepts. strategic competence refers to the ability to formulate, represent and solve mathematical problems. proficient problem-solvers are flexible in their approach to solving non-routine problems, form mental representations of problems, detect mathematical relationships and devise new solution methods where needed. strategic competence allows flexibility in performing mental calculations. adaptive reasoning refers to the capacity for logical thought, reflection, explanation and justification: ‘in mathematics, adaptive reasoning is the glue that holds everything together, the lodestar that guides learning’ (kilpatrick et al., 2001, p. 129). adaptive reasoning includes informal explanation and justification and intuitive and inductive reasoning based on pattern, analogy and metaphor. adaptive expertise requires an understanding of how and what procedures work and how these can be modified to fit the constraints of a problem. it requires the ability to deal with a problem holistically, to know what to do without the need to articulate a procedure before beginning and to monitor and regulate the process of problem-solving. productive disposition is a tendency to see mathematics as sensible, useful and worthwhile and seeing oneself as capable to learn, understand and do mathematics. these strands are not independent; they represent different aspects of a complex whole and have to be developed together. this is exemplified by proficiency in estimation and mental calculations, important indicators of number sense. kilpatrick et al. (2001, p. 215) argue that computational estimation integrates all strands of mathematical proficiency. it requires a flexibility of calculation that emphasises adaptive reasoning and strategic competence, guided by a person’s conceptual understanding of both the problem situation and the mathematics underlying the calculation and fluency with computational procedures. good estimators rely on mental calculations to find an approximate but satisfactory answer. they demonstrate a deep understanding of numbers and operations; they are flexible in their thinking and use a variety of strategies when solving problems. a disposition to make sense of a situation to produce reasonable answers instead of wild guesses is a prerequisite of competent estimators. mental calculation and estimations skills should be developed continuously throughout the primary school years and beyond in order for number sense to develop. to become proficient, learners need ‘to spend sustained periods of time doing mathematics – solving problems, reasoning, developing understanding, practicing skills – and building connections between their previous knowledge and new knowledge’ (kilpatrick et al., 2001, p. 133). research design top ↑ the focus of this project (courtney-clarke, 2012) was to explore the skills, knowledge, strategies and confidence related to number sense of namibian final year pre-service primary school teachers.in this study a concurrent mixed methods design was used. questionnaires and tests were administered to collect quantitative data on pre-service primary school teachers’ number sense. concurrent with this data, qualitative data was collected through semi-structured interviews. the role of the qualitative data was to clarify and understand the quantitative data. the results of the two methods were integrated in the interpretive phase. number sense is an elusive construct and its operationalisation requires the expertise of an experienced professional researcher. the number sense questionnaire, mental calculations questionnaire and the written computation questionnaire were based on instruments developed by yang in 1997 (reys & yang, 1998). according to yang (in hing, 2007), the split-half reliability of the number sense test is over 0.80 for both grade 6 and grade 8 learners and the cronbach’s alpha reliability coefficient is 0.80. yang’s number sense test has been used in research with both learners and teachers (alajmi & reys, 2007; hing, 2007; reys et al., 1999, reys & yang, 1998; tsao, 2004, 2005) and in the absence of similar reliable instruments of southern african origin, we opted for using yang’s tests. some of the selected items were adapted to make them more accessible to the expected competency levels of namibian primary school teachers in the following ways: • the whole number range was reduced to numbers less than 10 000 (e.g. 9135 ÷ 61 instead of the original 913 582 ÷ 6183). • the number of questions in the whole number domain was increased by changing decimals to whole numbers (e.g. the lengths of strings measuring 54.125 m and 29.85 m in the original question, were changed to 541 m and 298 m respectively). • contexts, such as names and units, were localised (e.g. ‘wang’ and ‘lin’ were replaced by ‘sarah’ and ‘leonard’ and cubic feet were changed to litres). the reliability of the adapted tests was not established. the number sense questionnaire was central to the research. the 27 selected items reflected the strands of number sense as a mathematical proficiency, namely the conceptual understanding of order, properties and representations of numbers, procedural fluency in calculating and in estimating and judging magnitude, strategic competence in carrying out mental calculations and detecting mathematical relationships, adaptive reasoning in recognising unreasonable results and in representing the same number or function in multiple ways depending on the context or purpose of this representation. the mcanallen confidence in mathematics and mathematics teaching survey (mcmmts) (mcanallen, 2010) was used to determine whether the psts were confident in their ability to understand, do and teach mathematics, that is whether they possessed a productive disposition towards mathematics. the survey consists of 25 items, 12 relating to personal self-efficacy and anxiety (factor 1) and 13 relating to teaching self-efficacy and anxiety (factor 2). the reliability analysis of the mcmmts (mcanallen, 2010) yielded the following: factor 1 (personal mathematics self-efficacy and anxiety factor) had a cronbach‘s alpha of 0.923 and factor 2 (teaching self-efficacy and anxiety factor) had a cronbach‘s alpha of 0.952. certain words and phrases were replaced with terminology used in the namibian context (e.g. the word ‘student’ was replaced with ‘learner’). to triangulate and validate the findings from the quantitative data, a semi-structured interview schedule was used to investigate what understandings, procedures, strategies and reasoning the students used in answering six questions selected from the number sense questionnaire. all the interviews were transcribed from the audiotape and the results coded into four categories: initial answers, approaches (rule-based or number-sense based), language use and confidence. the research was conducted with 47 final-year lower primary and upper primary mathematics psts enrolled at a university education campus in namibia that attracts students from all regions of the country. the use of purposive sampling of the total number of primary pre-service students was based on considerations of expense, time and accessibility. to increase the reliability and validity of the adapted instruments and the semi-structured interview a pilot study was conducted with local lower primary teachers. the pilot study proved to be essential in refining the instruments and the data collection process, for example by changing some unfamiliar phrases, adding practice examples and timing the mental calculation questions through the use of a powerpoint presentation. the data analysis followed the path of the concurrent triangulation model. the data from the four quantitative instruments were analysed using the predictive analysis software (spss, 2009). descriptive statistics (mean, median, standard deviation and graphical representations) were obtained for the number sense questionnaire, the written computations questionnaire and the mental calculations questionnaire. correlation analysis was performed on the results to establish a possible relationship between the student’s written computation and mental calculation skills and their number sense and their confidence in mathematics and mathematics teaching and their number sense. ethical considerations top ↑ this investigation was compliant with all ethical requirements set by the tertiary institution where the project was registered. approval for the research was obtained from the tertiary institution where the data were collected. participation was voluntary and information about the project was shared with participating psts. the commitment to keep all results confidential and report on the research and findings anonymously was communicated to all participants. findings top ↑ in analysis of the collected data, five main findings, based on the theoretical framework of mathematical proficiency, arose. conceptual understanding the analysis of performance on the three number domains of whole numbers, common fractions and decimal fractions (table 1) showed that the primary psts are more competent and comfortable working with whole numbers than with either common fractions or decimal fractions. performance on the common fractions domain was particularly poor on the number sense and the mental calculations questionnaires. however, some psts were proficient in applying paper-and-pencil methods to calculations with fractions as the standard deviation of 29% on the written computations questionnaire shows. table 1: descriptive statistics for the number sense questionnaire, written computation questionnaire and mental calculations questionnaire by number domain (%). these results were reflected in the mean of 22.7% obtained in the number sense domain ‘knowledge of and facility with numbers’ and in the interview data for this domain. we illustrate the main findings with a few examples: 1. only 32% of the respondents could spot the incorrect number sequence in question 17, which tested the knowledge of orderliness of whole numbers. which of the following number sequences is wrong? a. 3000; 3500; 4000; 4500 b. 7600; 7700; 7800; 7900 c. 6097; 6098; 6099; 7000 d. 8080; 8090; 8100; 8110 2. in question 20, less than half the psts could find the fraction that was represented by the shaded area. this question involved multiple representations of fractions on two levels: finding the correct part of the whole and then simplifying the answer. 3. question 5 required the recognition of the relative size of common fractions and was answered correctly by 25% of the psts. question 5 was included in the interview with six randomly selected students. all of the respondents found this question challenging; as one of them put it, ‘a very tricky one’. only one chose the correct answer, and went on to explain her reasoning: ‘because ... mmm ... you divide the two ... the one into the seven will be a remainder, but if i take the first expression, then the three can go into the eight like two times ... that will be less than one over eight ... that will be bigger than seven over ...’. this attempted explanation is an example of the difficulty the psts had in verbalising their thinking and their understanding, pointing to a lack of deep conceptual understanding of the mathematics that they have to teach. procedural fluency despite the focus in namibian schools on the application of procedures and standard algorithms to obtain a correct answer, the mean of the written computation questionnaire was only 39.3%. however, a high standard deviation of 18% indicated that some psts were competent in performing written computations.the analysis of the performance of the four operations in the written computation and the mental calculations questionnaires (table 2) showed that the psts are more competent in performing addition and subtraction than multiplication and division. performance on division was particularly poor with a mean of 18.1% on the written computation questionnaire and 10.6% on the mental calculations questionnaire. table 2: descriptive statistics for the written computation and mental calculations questionnaires by operation (percentages). domain (%). the pearson correlation coefficient of r = 0.457, p < 0.01 (two-tailed) between the two questionnaires was substantial and positive, indicating than there is a significant relationship between written and mental calculation proficiency. the analysis of the written computation questionnaire and interview data shed more light on this relationship as the following examples illustrate: 1. the preferred method in the written computation questionnaire to answer the question 995 + 872 + 838 + 809 was step-wise column addition, that is adding the first two numbers, then the third to the answer, and so on. this indicated that psts were not confident in their ability to mentally add four one-digit numbers correctly. 2. only 11 psts arrived at the correct answer to the question: find the value of □, if 45 × □ = 2700. most attempted to solve the problem by repeated addition: 45 + 45 = 90; 90 + 45 = 135, but failed to keep track of the addition and left the question uncompleted. it was inferred from both the quantitative and the qualitative data that many psts were not fluent in their basic facts knowledge and weak in their application of standard procedures in the rational number domain. strategic competence the very poor performance on the mental calculations questionnaire with a mean of 23% and standard deviation of 15% indicated that psts demonstrated a low proficiency to calculate mentally at a grade 6 level. figure 1 illustrates the performance on the different number domains. figure 1: distribution of scores for the three number domains on the mental calculations questionnaire. mental calculations require a combination of memory of basic facts and strategies (kilpatrick et al., 2001), that is a combination of procedural fluency and conceptual understanding. the poor performance in the whole number domain on the mental calculations questionnaire indicated that the psts lacked basic facts knowledge and strategic competence to calculate mentally. some examples are: 1. only six out of the 47 psts found the correct answer to 232 − 98. this indicated that the majority were trying to work this out by mental application of the algorithm, which is difficult because if it involves ‘borrowing’. the ‘count-up’ strategy would have yielded the correct answer very quickly: 98 + 2 + 132 = 232. 2. 512 ÷ 4 received one correct answer. the preferred strategy of repeated addition of 4 would indeed take a very, very long time. here application of the fact that 4 × 25 = 100 would have been one possible efficient strategy. 3. 36 × 50 was answered correctly by only two psts. perhaps some psts tried to add 50 thirty-six times; others might have tried to do long multiplication mentally. the obvious strategy here would have been to multiply by 100 and then divide by 2. 4. 4 × 3 was answered correctly by four psts. the correct answer could have been obtained simply by ‘double and double again’. the interview data showed that the respondents relied on the mental application of an algorithm to work out the answer to 541 – 298 and admitted that the size of the numbers made it difficult to find the answer to the subtraction question mentally. the problem ‘how many minibuses will be needed to transport 170 learners, if one minibus can transport 15 people?’ was solved by repeated addition of 15: ‘i did like fifteen, fifteen, fifteen, fifteen, fifteen ... then i go thirty, thirty.’ number sense-based strategies of estimation and mental calculations were not available to the respondents. the lack of number sense-based strategies coupled with the lack of conceptual understanding and basic facts fluency made the mental calculations questionnaire very challenging for the psts. the correlation between the mental calculations questionnaire and the number sense questionnaire of 0.540, p < 0.01 (two-tailed), indicated that the relationship between mental calculation and number sense is positive, substantial and significant, supporting the claim that mental calculations and number sense develop together and ‘one cannot exist without the other’ (griffin, 2003, p. 306). adaptive reasoning estimation is regarded as an indicator of number sense. it requires knowledge of place value, basic facts and properties of operations as well as an ability to compare numbers by size, compute mentally and work with powers of ten (sowder & wheeler, 1989). tsao (2005) found that pre-service primary school teachers in his study felt uncomfortable providing estimates. this may be true for namibian psts as well: the overall mean for the 13 estimation questions on the number sense questionnaire was only 40%.table 3 shows an analysis of the 13 estimation questions by number domain and operations. table 3: analysis of estimation questions on the number sense questionnaire. estimation skills were apparent in questions where rounding small whole numbers to add or subtract was appropriate, but not in questions involving multiplication and division or rational numbers. this was confirmed during the interviews. some examples are: 1. not a single respondent used estimation to place the decimal comma in the multiplication problem ‘place the decimal comma in the calculation 534,6 × 0,545 = 291357’. of the three respondents who obtained the correct answer, one could not give a reason and the other two used the rule for addition of decimal numbers by rewriting the question in column form, aligning the decimal commas. an incorrect answer of 29,1375 was arrived at by using the rule for multiplication of decimals, but then not reflecting upon or questioning the reasonableness of the answer. 2. when asked ‘a minibus can transport 15 people. how many minibuses would you need to transport 170 learners?’, answers of ‘5’, ‘6’ and even ‘16’ were given. these were guesses more than justified estimates. not a single respondent made use of the number fact 10 × 15 = 150 to arrive at a reasonable estimate. productive disposition the pearson correlation coefficients between the total scores on the number sense questionnaire and the total scores obtained on the mcanallen confidence in mathematics and mathematics teaching survey overall, factor 1 and factor 2, were all positive, substantial and significant, namely r = 0.553, p < 0.01; r = 0.478, p < 0.01 and r = 0.502, p < 0.01 respectively. this indicates the psts with greater confidence in their ability to do and their ability to teach mathematics also displayed a greater degree of number sense.despite the poor performance on all the tests and the demonstration of a lack of confidence in answering number sense questions during the interviews, the psts were very confident in both their ability to do mathematics (factor 1) and in their ability to teach mathematics (factor 2) with an overall mean score of 3.78 out of a possible score of 5. this result was both surprising and disturbing, because this investigation revealed that both the psts’ language competence and subject matter knowledge were insufficient for effective teaching of mathematical concepts and procedures. the test results did not demonstrate the psts ability to do mathematics; however, the mean score on factor 1 was 3.57. we would like to put forward two conjectures: firstly, that the confidence that the psts displayed in their ability to do mathematics was, in their minds, not linked to the performance on the questionnaires, but to their performance in their university mathematics course. secondly, and more worryingly, these primary psts might exhibit confidence in their own incorrect knowledge. the mean for factor 2, confidence in the ability to teach mathematics, was 3.97 − substantially higher than the mean for factor 1. possibly the psts with their lack of classroom experience, assume that anyone who has completed secondary school mathematics is competent to teach at lower primary level and that teaching mathematics consists of explaining sets of rules and procedures. more research is needed to find an explanation for these contradictory findings. discussion top ↑ the overall results of this study revealed that the final year primary psts demonstrated limited number sense and possessed very few of the indicators of number sense as a mathematical proficiency. the absence of deep conceptual understanding was demonstrated by their inability to move amongst different representations of numbers, to recognise the absolute and relative values of numbers and to explain their thinking. they lacked procedural fluency particularly in the domain of rational numbers and the operations of multiplication and division. they had little or no access to a variety of flexible strategies to solve problems and calculate mentally. they showed a tendency to guess answers, instead of arriving at estimates through reasoning and reflection and paid little attention to unreasonable results. although the interviewed respondents demonstrated a lack of confidence, the confidence survey showed that these primary psts judged themselves as capable to learn, understand, do and teach mathematics.during the interviews respondents had to explain their answers. one must wonder whether primary school learners would be able to follow the often confused and incoherent explanations. apart from language problems, the respondents’ struggle to explain and reason indicated that they lacked conceptual understanding, which is a prerequisite for number sense. however, they could recall rules or algorithms reasonably fluently, if not always correctly. this concurs with the taiwanese studies by tsao (2005), who found that only high ability students demonstrated the use of number sense characteristics in their explanations, and yang et al. (2009), whose study showed that less than one-third of the 280 psts used components of number sense to explain their answers. although the mean on the number sense questionnaire of 38.1% indicates that namibian primary psts’ performance on the number sense questionnaire was poor compared to that found in similar international studies (tsao, 2004, 2005), the namibian teachers are not an exception. as yang et al. (2009) concluded, breaking the shackles of rotely applying algorithms and promoting greater development and application of number sense components in mathematical problem solving is an international challenge. (p. 400) developing number sense as a mathematical proficiency implies a shift away from teaching a collection of procedures and rules to developing a mathematical disposition that engages children at an early age in mathematical thinking and on laying the foundation for learning concepts that are more advanced. this study has revealed that the psts do not have the specialised content knowledge (ball, 1990) to serve as ‘guides’ in the environment of numbers and operations. indeed, there were indications that they also lack common content knowledge as their inability to use basic facts and procedures fluently in written computations showed. however, the research showed that the primary psts are confident in both their ability to learn and to teach mathematics. this indication of a productive disposition is a resource that enables teachers to learn the mathematical competences required to teach mathematics (graven, 2004) and may be a way forward to move namibian teachers from teaching procedures to teaching conceptually and developing number sense, both their own and that of the learners. although the findings were limited to a sample of only 47 final-year psts of one university campus and cannot be generalised across all teacher training or similar institutions, similar shortcomings in in-service primary mathematics teacher knowledge and its implications were identified in the reports reviewed in the introduction. more research is needed in the field of mathematics education in namibia, possibly in collaboration with other southern african countries that experience similar problems. the mathematics education departments at teacher training institutions should research and identify the mathematics that teachers need to know and the ways in which teacher understanding of subject content needs to be transformed so that they can teach children to develop mathematical proficiency. academic language proficiencies required for mathematics teaching to ensure that conceptual and procedural discourse can take root in primary school classrooms should form part of this research. conclusion top ↑ the mathematical proficiency characterised by conceptual understanding, procedural fluency, strategic competence and adaptive reasoning in the domain of numbers and operations forms the fundamental building block of the entire mathematics curriculum (kilpatrick et al., 2001). this study has shown that the final year primary psts are not proficient in number sense themselves and are therefore ill-equipped to promote number sense in the learners that they will teach. in addition, we may infer that many in-service teachers do not possess the knowledge, skills and understanding to implement a focus on number sense.the lack of a sound foundation in the domain of numbers and operations may be the root cause of the low standards of performance of namibian learners in mathematics at all levels and the lack of improvement over the last decade or more. this leads to the conclusion that teacher training institutions need to re-assess the mathematics education curriculum and include programmes to develop the number sense of psts. this investigation has explored broad issues around number sense. it should serve as a starting point for mathematics educators to discuss the challenges of developing a numerate society in namibia. acknowledgements top ↑ the authors are grateful to professor der-ching yang and dr rachel mcanallen for providing us with the number sense tests and the mcanallen confidence in mathematics and mathematics teaching survey respectively and their permission to use and adapt their instruments for the purposes of this research. competing interests the authors declare that they have no financial or personal relationship(s) that may have inappropriately influenced them in writing this article. authors’ contributions the article presents the results of a study in which m.c.-c. conducted the research whilst h.w. 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(2004). exploring the connections among number sense, mental computation performance and the written computation performance of elementary pre-service teachers. journal of college teaching and learning, 1(12), 71–90. tsao, y-l. (2005). the number sense of pre-service elementary school teachers. college student journal, 39(4), 647–679. yang, d.c., reys, r.e., & reys, b.j. (2009). number sense strategies used by pre-service teachers in taiwan. international journal of science and mathematics education, 7(2), 383–403. http://dx.doi.org/10.1007/s10763-007-9124-5 article information author: duncan samson1 affiliation: 1education department, rhodes university, south africa correspondence to: duncan samson postal address: po box 94, grahamstown 6140, south africa dates: received: 16 apr. 2012 accepted: 26 aug. 2012 published: 26 oct. 2012 how to cite this article: samson, d. (2012). pictorial pattern generalisation: tension between local and global visualisation. pythagoras, 33(3), art. #172, 9 pages. http://dx.doi.org/10.4102/ pythagoras.v33i3.172 copyright notice: © 2012. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. pictorial pattern generalisation: tension between local and global visualisation in this original research... open access • abstract • introduction • theoretical background    • visualisation and figural apprehension    • local versus global visualisation    • enactivism and knowledge objectification • methodology    • ethical considerations    • reliability and validity • findings and discussion    • an illustrative vignette       • part 1       • part 2 • concluding comments • acknowledgements    • competing interests • references • footnotes abstract top ↑ this article engages with the notion of local and global visualisation within the context of figural pattern generalisation. the study centred on an analysis of pupils’ lived experience whilst engaged in the generalisation of linear sequences presented in a pictorial context. the study was anchored within the interpretive paradigm of qualitative research and made use of the complementary theoretical perspectives of enactivism and knowledge objectification. a crucial aspect of the analytical framework used was the sensitivity it showed to the visual, phenomenological and semiotic aspects of figural pattern generalisation. a microanalysis of a vignette is presented to illustrate the subtle underlying tensions that can exist as pupils engage with pictorial pattern generalisation tasks. it is the central thesis of this article that the process of objectifying and articulating an appropriate algebraic expression for the general term of a pictorial sequence is complicated when tension exists between local and global visualisation. introduction top ↑ the use of number patterns, specifically pictorial or figural number patterns, has been advocated by numerous mathematics educators as a didactic approach to the introduction of algebra and as a means of promoting algebraic reasoning and supporting the fundamental mathematical processes of generalisation and justification (e.g. de jager, 2004; mason, graham, pimm & gowar, 1985; pegg & redden, 1990; walkowiak, 2010). number patterns presented as a sequence of pictorial terms have the potential to open up meaningful spaces for classroom exploration and discussion. however, despite the potential richness of such pictorial contexts, potentially meaningful pattern generalisation activities carried out in the classroom often become degraded to simple rote exercises in which the given pictorial sequence is simply reduced to an equivalent numeric sequence. as such, the generalisation process becomes a somewhat superficial or mechanistic exercise using set algorithmic methods. whilst such an approach may well be successful in arriving at the correct general formula, the potential for genuine mathematical exploration offered by the context of the question is largely lost, with the generalisation process becoming ‘an activity in its own right and not a means through which insights are gained into the original mathematical situation’ (hewitt, 1992, p. 7). as thornton (2001) remarks, the danger with such an approach is that the focus becomes ‘… the development of an algebraic relationship, rather than the development of a sense of generality’ (p. 252), the result being little more than disconnected algebraic formulation. this disconnection from the original context becomes particularly problematic when importance is placed on the justification or validation of the general rule, since algebraic expressions arrived at through this process become ‘statements about the results rather than the mathematical situation from which they came’ (hewitt, 1992, p. 7). this is unfortunate since not only can generalisation of pictorial patterns lead to different but algebraically equivalent expressions of generality, thereby opening up excellent opportunities to engage with the notion of algebraic equivalence, but, as orton (2004, p. 114) points out, justifying pattern generalisations provides pupils with legitimate and valuable experiences of proof en route to more formal mathematical proofs. number patterns presented in the form of a sequence of pictorial terms are thus far more than simply a visual representation of a given numeric pattern. in essence, the use of a pictorial context aims to exploit the visual decoding of the pictorial sequence to give meaning to the symbolic expressions constructed. two critical aspects of this process are the ability not only to grasp in a meaningful way the perceived underlying structure of the pictorial context, but also the ability to use this structure to articulate a direct expression for the general term of the sequence. theoretical background top ↑ visualisation and figural apprehension as duval (2006, p. 116) succinctly notes, there are many different ways of seeing. consider the simple geometrical figure composed of a number of line segments (figure 1). figure 1: a simple geometrical figure. pupils visually interacting with the figure have the potential to perceive it in a number of different ways. in the same vein, a single pupil may be able to perceive the figure in multiple ways. if we take figure 1 as an example, the figure could be perceived as comprising two overlapping hs. alternatively, it could be seen to comprise four overlapping squares, where the ‘lids’ of the top two squares and the ‘bases’ of the bottom two squares are missing. a third possibility is for the figure to be perceived as comprising three vertical lines with two shorter interconnecting horizontal lines. yet another possibility is for the figure to be seen as a ‘+’ symbol contained within two sets of vertical lines, one on either side.if we now take the same image but place it in the context of a pictorial sequence, then the visual scenario is profoundly altered (figure 2). figure 2: a simple pictorial sequence. the middle diagram in figure 2 is identical to the image shown in figure 1. however, the contextual setting is no longer simply that of a geometrical figure. there is now a tacit suggestion that the image is part of a larger context, that it is part of a sequence of visually or structurally similar images. given this new context, and the associated yet implicit sense of sequential growth, the middle diagram in figure 2 could be perceived as a vertical line on the left followed by two sideways t-shapes. alternatively, it could possibly be perceived in terms of a horizontal ‘backbone’ with vertical lines extending off it in two directions, upwards and downwards. the previous four visualisations are of course still possible, but the added context provides additional/alternative perceptual features to be made apparent. thus, by modifying the context, different ways of perceiving the figure are brought forth. on a note of clarity, use of the word ‘feature’ is not meant to imply that such features (or structures) are intrinsically contained within the image, simply waiting to be extracted or noticed by an observer. rather, such features are seen to co-emerge from the interaction of a perceiver and the given figural context. a further dimension is added to the visual image shown in figure 2 when the pictorial sequence is used as a referential context for finding an algebraic expression for the number of lines in the nth diagram (or term) of the sequence. although the context remains the same as that represented in figure 2, the process of generalisation provides a further layer of complexity as it necessitates not only the perception of the figure within the context of a sequence of similar figures, but it requires the perception of generality, the notion that the figures in the sequence have a related structure. finally, there is a requirement that this perceived generality must be articulated in such a way that it can be written in the form of an algebraic expression. a critical aspect of the visualisation process thus relates to the usefulness or meaningfulness of the perceived structure of the image in terms of the extent to which this perceived structure supports or hinders the process of generalisation. drawing on the nomenclature used by fischbein (1993), figures such as the image shown in figure 1 could be said to contain figural properties. what one sees in the image is a result of the gestalt laws of figural organisation (helson, 1933; katz, 1951; wertheimer, 1938; zusne, 1970, pp. 111−135). images such as that shown in figure 2 could be said to contain both figural properties and conceptual qualities. what one sees in the image is still based on the gestalt laws of figural organisation, but this is further influenced by the additional conceptual qualities of the image that have been added by virtue of the image being contextualised, in this case within a sequence of similar images. the critical point here is that an underlying tension is likely to pervade visual strategies applied to pictorial pattern generalisation tasks as a result of the relationship between the figural properties and conceptual qualities of the given images. local versus global visualisation visual approaches to pictorial pattern generalisation can be divided into two broad categories. the first category, which i previously termed local visualisation (samson, 2011a), incorporates those visual strategies that are characterised by the foregrounding of the local additive unit – that is the structural unit which is added to (or inserted into) a given pictorial term in order to form the next term in the sequence. this focus on the structural additive unit represents an iterative or recursive process of visual reasoning. the second category, which i previously termed global visualisation (samson, 2011a), incorporates those visual strategies characterised by a more holistic or global view, where each term of the pictorial context is seen in terms of a generalised structure that does not make use of the iterative addition of the additive unit. by way of example, consider figure 3, which shows term 3 and term 5 of a pictorial sequence1. figure 3: term 3 and term 5 of a typical pictorial sequence. using a local visualisation, one could reason that to get from one term to the next requires the addition of three matches in the form of a backwards c-shape, the visual additive unit. the visual deconstruction of the pictorial context based on this local visualisation could be based on either (1) an initial starting match and n multiples of three matches in the form of a backwards c-shape (the additive unit), yielding the general formula tn = 1 + 3n, or (2) a 4-match constant followed by (n − 1) multiples of the visual additive unit, yielding the general formula tn = 4 + (n − 1) × 3. both of these scenarios are illustrated in figure 4. figure 4: local visualisation of term 5. by contrast, global visualisation is characterised by a more holistic view that does not make use of the iterative addition of the additive unit. rather, each pictorial shape is visualised in terms of a generalised structure. by way of example (see figure 5), one could subdivide each term of the given pictorial context into an upper row of n horizontal matches, a lower row of n horizontal matches, and a central row of n + 1 vertical matches, thus yielding the general formula tn = 2n + (n + 1). alternatively, each term could be seen to contain n overlapping squares, each made up of four matches, thus giving a count of 4n matches. however, this would result an overcount because there are n − 1 overlaps. correcting for this overcount gives the final formula tn = 4n − (n − 1). these two different global visualisations are illustrated in figure 5. figure 5: global visualisation of term 3. duval (1998, p. 41) makes the pertinent point that most diagrams contain a great variety of constituent gestalts and subconfigurations. critically, this surplus constitutes the heuristic power of a geometrical figure, since specific subconfigurations may well trigger different visual generalisations. thus, within the context of figural pattern generalisation, the processes of visualisation and generalisation are deeply interwoven, and a complex relationship is likely to exist between different modes of visualisation. one of these underlying tensions is that between local and global visualisation, and it is this particular tension that forms the focus of this article. enactivism and knowledge objectification the broader study (samson, 2011b) of which this article forms a part centres on two key theoretical ideas, enactivism (maturana & varela, 1998; varela, thompson & rosch, 1991) and knowledge objectification (radford, 2003, 2008). the manner in which these complementary theoretical lenses combine to provide a rich tool for analysis is described elsewhere (samson & schäfer, 2011), but a brief overview is presented here.the basic tenet of enactivism is that there is no division between mind and body, and thus no separation between cognition and any other kind of activity. within an enactivist framework, there is a purposeful blurring of the line between thought and behaviour (davis, 1997, p. 370), and cognition is thus viewed as an embodied and co-emergent interactive process. from this theoretical stance, as davis (1995, p. 4) points out, language and action are not merely outward manifestations of internal workings, but should rather be seen as ‘visible aspects of … embodied (enacted) understandings’. radford’s (2008) theoretical construct of knowledge objectification resonates strongly with an enactivist theoretical framework since it foregrounds the phenomenological and semiotic aspects of figural pattern generalisation. knowledge objectification thus represents an ideal theoretical construct to critically engage with pupils’ whole-body experience and expression whilst they explore the potentialities afforded by a given pictorial pattern generalisation task. knowledge objectification is a theoretical construct to account for the manner in which learners engage or interact with a given scenario or context in order to make sense of it en route to a stable form of awareness (radford, 2006, p. 7). knowledge objectification is premised on two notions. firstly, semiotic means such as gestures, rhythm and speech are not simply epiphenomena, but are seen as playing a fundamental role in the formation of knowledge (radford, 2005a, p. 142). secondly, in order to study the process of knowledge production one needs to pay close scrutiny to multiple means of objectification, for example words, linguistic devices, gestures, rhythm, graphics and the use of artefacts, where ‘…meaning is forged out of the interplay of various semiotic systems’ (radford, 2005b, p. 144). it is through this multi-systemic, semiotic-mediated activity that the objects of perception, or rather the objects of knowledge, progressively emerge – a process of ‘concept-noticing and sense-making’ (radford, 2006, p. 15). importantly, from an enactivist stance, use of the word ‘object’ by no means suggests that these ‘objects’ are pre-existing properties inherent in the environment. rather, the ‘objects’ of perception are brought forth through the co-determination of knower and known, the co-evolution of individual pupils and their surroundings. methodology top ↑ the broader study (samson, 2011b) of which this article forms part is oriented within the conceptual framework of qualitative research, and is anchored within an interpretive paradigm. the study aims ultimately to gain insights into the embodied processes of pupils’ visualisation activity when engaged in figural pattern generalisation tasks through an in-depth analysis of each pupil’s lived experience. a mixed-gender, high-ability grade 9 class of 23 pupils constituted the research participants for the broader study (samson, 2011b). from this group of 23 pupils, seven research participants were identified as preferring a visual mode when solving pattern generalisation tasks. these seven research participants were individually provided with a linear pattern presented in a pictorial context and were required to provide, in the space of one hour, multiple expressions for the nth term of the sequence. tools such as paper, pencils and highlighters as well as appropriate manipulatives such as matchsticks were provided. participants were asked to think aloud whilst engaged with their particular pattern generalisation task, and the researcher also prompted the participants to keep talking or provide further explication as and when necessary. each session was audio-visually recorded and field notes were taken. audio-visual recordings were analysed with specific reference to how participants made use of multiple means of objectification en route to a stable form of awareness. these means of objectification included the use of words, linguistic devices, metaphor, gestures, rhythm, graphics and physical artefacts. these processes of ‘coming to know’ were carefully scrutinised through multiple viewings of the audio-visual recordings of each research participant. the data analysis was guided by an enactivist methodological framework in which the researcher and research environment are seen to co-emerge (reid, 2002). this interdependence of researcher and context was characterised by a flexible and dynamic process of investigation (trigueros & lozano, 2007). the iterative and reflexive process of co-emergence was built up over time through the use of multiple perspectives and the continuous refinement of methods and data analysis protocols. audio-visual data was examined repeatedly in different forms (e.g. video and transcript) and in conjunction with additional data retrieved from field-notes and participants’ worksheets. in addition, nodes of activity that seemed particularly interesting were identified and meticulously characterised with reference to the various semiotic means of objectification in the form of descriptive vignettes. ethical considerations before the broader study started, formal permission to conduct the research was obtained from the principal of the school in question. anonymity of both the school and the research participants was assured, and appropriate pseudonyms are used throughout the text when referring to research participants. in addition, only those pupils who agreed to participate in the study through voluntary informed consent formed part of the research sample, and participants had the freedom to withdraw from the study at any stage without explanation. in the case of participants who were audio-visually recorded, written consent was obtained from each research participant as well as from each participant’s parents or legal guardians. from a more philosophical standpoint, there is also an important ethical consideration stemming from the enactivist theoretical underpinnings of this study. in enactivist terms we need to be sensitive to the notion that ‘… our actions have the potential to alter the worlds and possibilities of others’ (simmt, 2000, p. 158). furthermore, an enactivist stance compels us to see each person’s certainty as being ‘… as legitimate and valid as our own’ (maturana & varela, 1998, p. 245). sensitivity to both of these ethical considerations was maintained throughout the study. reliability and validity in terms of reliability and validity considerations, not only the appropriate choice of the figural pattern generalisation questions themselves, guided by pertinent literature, but also the nature of their presentation were of critical importance. a literature review and previous research experience (samson, 2007) suggested that linear sequences of the form ax ± c (c ≠ 0) would be most appropriate in terms of eliciting rich data.it has been shown that patterning tasks presented with consecutive terms encourage a recursive strategy (hershkowitz et al., 2002; samson, 2007) and thus tend to draw attention away from global structural features that could potentially co-emerge through the interaction of a perceiver and the given figural context. since not all patterning tasks could be presented unambiguously using a single term, it was decided to use two non-consecutive terms for all questions. this purposeful decision was thus not intended to encourage global visualisation per se, but rather to ensure that the potential for global engagement with the pictorial context was not discouraged by virtue of terms being presented consecutively. in addition, data collection and analysis protocols were sensitive to the enactivist underpinnings of the study, and thus made use of multiple data sources and approaches to data handling (as previously outlined). this process acted as a form of triangulation, which sought to ensure validity. findings and discussion top ↑ participants were presented with two non-consecutive terms of a linear pictorial sequence rather than a series of consecutive terms. nonetheless, during their engagement with the presented pictorial terms, all pupils created physical instantiations of specific terms of their pictorial sequence through the process of drawing. this drawing process was in many ways a two-edged sword. in some instances, the physical process of drawing led to the emergence of structural commonalities or regularities. this supported the generalisation process where these regularities were algebraically useful (i.e. where the generality of what was noticed in the phenomenological realm could be readily expressed using algebraic symbolism). however, the physical process of drawing often led to attention being focused on the recursive nature of the step-by-step process of construction, thereby foregrounding local considerations rather than allowing for a more holistic or global apprehension. this often resulted in an underlying tension between these two different modes of visualisation. an illustrative vignette the following vignette attempts to capture and characterise the tension between local and global visualisations as evidenced by the generalisation activity of one of the research participants (terry, a high-ability grade 9 pupil). part 1 terry was presented with two non-consecutive terms of a typical pictorial sequence (see figure 6). figure 6: pictorial pattern presented to terry. after staring at the two terms for a few seconds, he remarked:it seems like it’s basically just adding on the same sort of thing again, every time, and then just finishing it off with that [indicating the rightmost > shape]. (samson, 2011b, p. 131) after making this remark he carefully drew term 4 in a very structured manner. he began by drawing the three matches of the leftmost triangle. thereafter he drew the middle section of the structure in a very rhythmic fashion: 4,5,6 … 7,8,9 … 10,11,12 … 13,14,15. interestingly, instead of drawing each group of three matches in the flowing form of a backward c-shape (top match, then vertical match, then bottom match) which would have been slightly more economical, he instead methodically drew each group of three matches by first drawing the top horizontal match, then the bottom horizontal match, and then finally the vertical match. to check that he had drawn the correct number of matches for term 4, he then carefully counted the four ‘squares’ in the diagram before adding on the two oblique matches on the far right. after completing the middle section of the diagram, he drew a series of inverted v-shapes across the top of the structure (rhythmically drawing them in pairs from left to right) and finished the diagram by drawing a series of v-shapes along the bottom of the diagram, once again in rhythmic pairs from left to right (figure 7). figure 7: terry’s drawing procedure for term 4. based on this drawing procedure, terry was able to arrive at the formula 7n + 5: so you started off with your little triangle [indicating the leftmost < shape] so that’s obviously +2, then you finish it off with a little triangle again [indicating the rightmost > shape], plus another 2, so it’s +4; and then however many things in between, just work out how many it is for that [indicating the 7-match additive unit] (…) well how many it is for the top triangle, bottom triangle and then plus that one [indicating the vertical match connecting the top and bottom triangles in the 7-match additive unit]. (samson, 2011b, p. 132) at this point terry wrote down the formula 7n + 4. however he quickly realised that he had not taken into account the first vertical match from the left: shape 3 is term 3, it’s got 3 of these little squares like that [pointing to the three central squares of term 3] and shape 5 has 5. so then you’ve got these 2 [indicating the leftmost < shape] so you start off, there’s +2, there’s another 2 [indicating the rightmost > shape] plus 4, then you’ve already got [points to the leftmost vertical match and realises he has missed it out] – oi! (…) with the front triangle [indicating the triangle formed from the leftmost < shape and the leftmost vertical match] it’s the full triangle that you’re starting off with, so it’s 1, 2, 3 matchsticks. with the end one you’ve already got the (…) base of the triangle coming from the previous square. (samson, 2011b, pp. 132−133) terry thus gave his final formula as 7n + 5 which he subsequently altered to 3 + n(3 + 3 + 1) + 2 as being more representative of how he was visualising the pictorial context (figure 8). figure 8: two different formats of terry’s initial visualisation. both expressions represent a local visualisation since they are based on a recursive addition/insertion of the 7-match additive unit. in the altered version of the formula, the 7-match unit is further subdivided into a ‘top triangle’, a ‘bottom triangle’ and a vertical line. an important distinction here is that terry is not seeing each term as being holistically composed of multiple 7-match units enclosed between a triangle on the left and a > shape on the right. rather, the 7-match unit is seen as an additive structural unit in that it is recursively or iteratively added to (or inserted into) an existing term in order to construct the next term in the sequence, as evidenced by terry’s remark ‘it’s basically just adding on the same sort of thing again, every time.’an interesting aspect of terry’s discussion is his frequent reference to squares in the pictorial terms. he makes express reference to the fact that the nth term in the sequence would contain n ‘little squares’. in addition, when he initially drew term 4 he did so by drawing the central structure first and then checking that he had drawn the correct number of matches by quickly counting the four squares. however, these squares do not feature anywhere in either of the two versions of his initial formula. in fact, the horizontal matches from these squares are seen to form part of an upper and lower triangle. thus, after visually deconstructing the diagram into triangles, the squares become negative space as the matches that originally formed them have been apportioned to different component parts. nonetheless, terry continued to refer to them as a helpful structural unit. there are two possible reasons for this that are worth considering. firstly, the ultimate aim of the patterning task from terry’s perspective is to arrive at an algebraic expression for the general term through a process of visualisation. it is possible that this goal had an unconscious influence on the visualisation process since some visualisations would be algebraically more useful than others – for example, squares would overlap and a correction would thus be necessary for the resulting overcount. a second possibility is that there is a tension between local and global aspects of the pictorial context. local considerations focus on the additive unit by virtue of attention being focused on the step-by-step process of constructing the next term from the previous one. it is possible that this local focus obscured a more global outlook where the structural unit of a square could be properly incorporated into the general expression. part 2 after silently and motionlessly staring at the two printed pictorial terms for a few minutes he made the following comment:what i’m trying to do now is almost use the squares. so now instead of having that sort of backwards c, actually have a full-on square (…) that gets connected to another square [gesturing to the right with his pencil], that then, just got to take out that one [indicating the overlapping match between two squares], and gets connected to another square [making multiple gestures further to the right with his pencil]. (samson, 2011b, p. 134) whilst terry was explaining his strategy, he made use of a number of crucial semiotic means of objectification. the first of these was his gesturing to the right whilst saying the words ‘that gets connected to another square’. this indexical or deictic gesturing was specifically related to term 3, which terry had in front of him. he thus used the gestures to signify existing physical structures in the particular diagram he was looking at. his second set of gestures, accompanying the words ‘and gets connected to another square’ mark a transition from existential signification to what sabena, radford and bardini (2005, p. 134) refer to as imaginative signification. this second set of gestures moves from indicating materially instantiated aspects of the pictorial term to miming an ongoing sequence of connected squares, squares that are not yet materially present. we thus see a progressive distancing from the physical referent. another important aspect of terry’s objectification process is his use of the words ‘another square’. these words serve an important generative action function in terms of objectifying the generality of the interconnecting squares through an imaginative conception of iterative potential action. this linguistic device supports the process of objectification by allowing the recursive addition of squares to be ‘… repeatedly undertaken in thought’ (radford, 2000, p. 248). however, and critically important in terms of the local-global visual tension, the words ’and gets connected to another square’ foreground an iterative process. thus, whilst supporting an important generative action function in terms of objectifying the generality of the structural unit of the square, they also tend to focus attention on the recursive nature of step-by-step construction, thus drawing attention away from a more holistic view of the overall general structure. terry then went on to draw term 4. interestingly, the order in which he drew the various lines did not seem to correlate with his description of overlapping squares. instead, his drawing process seemed to suggest a subdivision into a triangle at either end, two rows of horizontal matches, a row of vertical matches, and v-shapes at the top and bottom. after completing his drawing of term 4, terry sat staring at it for just over a minute before commenting: ‘i had something and now i’ve, i had something else but now i’ve lost it.’ it thus seems that his initial idea of using overlapping squares came from a flash of insight that has since receded. it is possible that this may have been at least partially precipitated by terry’s drawing of term 4 in a manner which did not mimic his initial visualisation of overlapping squares. in this instance, it is possible that the drawing process itself obfuscated the visual apprehension. however, another interpretation of the data could suggest that the drawing process actually reflects a competing, albeit unconscious, visualisation of the pictorial terms, thus suggesting an underlying visual tension between two different apprehensions of the pictorial context. terry then came up with the formula 3 + n(4 − 1) + 4n + 2. the ‘3’ at the beginning of the formula represents the starting triangle on the far left whilst the ‘+2’ at the end of the formula represents the > shape at the extreme right of each term. terry described the n(4 − 1) portion of his formula as representing ‘each square minus the one that’s being taken up by either the previous one or the next one’ whilst the 4n is required for ‘the triangles above and below it’. although he specifically refers to ‘squares’, these structural features are not reflected in his present visualisation. although his initial visualisation was suggestive of overlapping squares, he has in essence reverted to a previous visualisation in which the central structure is seen not in terms of overlapping squares but rather in terms of a series of backward c-shapes (figure 9). figure 9: terry’s change in visual apprehension to a local visualisation. a possible explanation for this reversion to an earlier visualisation is that terry’s focus on the recursive nature of the construction process supported a local generalisation, but not a global one. a global generalisation or visualisation would entail seeing the structure in a holistic manner as being composed of a series of n overlapping squares. since four matches are needed for each square, the n squares would require a total of 4n matches. however, this would lead to an overcount since overlapping would mean that some matches would in effect have been counted twice. to correct for this, we would need to subtract n − 1 matches from the tally since n overlapping squares would have n − 1 overlaps. however, terry’s constant focus on a recursive, step-by-step process of construction is incompatible with this global view. to proceed from one term to the next would require the addition of a square and the removal of the overlapping match each time. the addition of a whole square each time thus becomes a redundant process if the overlapping match is immediately removed, since the process could be accomplished in a far simpler manner by just adding on three matches in the form of a backward c-shape each time, thereby avoiding the unnecessary removal of the overlapping match. it is this focus on a stepwise process of construction that is likely to have contributed to the initial visualisation of overlapping squares being transformed into a visualisation of backward c-shapes.at this point i asked terry what had happened to his initial idea of focusing on the squares: i don’t know, i had something … i was busy looking at it and something hit me and then i lost it. i noticed something to do with n minus, open brackets n minus 1, and then that in brackets [i.e. (n − (n − 1))], that had something to do with it, but i cannot for the life of me remember what it was. (samson, 2011b, p. 137) terry’s reference to his noticing something to do with (n − (n − 1)) does not initially seem to make any sense as the expression simplifies to +1. however, it retains an interesting remnant of his initial visualisation in which there are n overlapping squares with n − 1 overlaps. at my suggestion he continued to pursue his initial idea. after staring at the diagrams for about half a minute he commented: i think i might have found it … so what i’m trying to do is now, is almost separate it so you’ve got, you just put all the squares together (…) and then take out this extra match right at the end [pointing in turn to each of the three overlaps between the four squares in term 4]. (samson, 2011b, p. 137) this marks the crucial moment when terry changes from a local to a global visualisation (figure 10) and is thus able to make sense of, and articulate, his initial fleeting visualisation. after trying to incorporate (n − (n − 1)) into his general expression, he eventually abandoned it and came up with the final formula 2 + 4n − (n − 1) + 4n + 2: that works, then you’ve got your 2 that starts it off [indicating the leftmost < shape], your 2 that finishes it off [indicating the rightmost > shape], you’ve got your four for each square, then the − (n − 1) (…) for each square there’s an extra line except for the first (…) then + 4n for each triangle above and below it. (samson, 2011b, p. 137) figure 10: terry’s final global visualisation of overlapping squares. concluding comments top ↑ the vignette serves to illustrate the subtle underlying tensions that can exist as pupils engage with pictorial pattern generalisation tasks. within the context of figural pattern generalisation, the processes of visualisation and generalisation are deeply interwoven. pattern generalisation rests on an ability to grasp a commonality from a few elements of a sequence, an awareness that this commonality is applicable to all the terms of the sequence, and finally being able to use it to articulate a direct expression for the general term. there are thus two important aspects of this notion of generalisation, namely, (1) a phenomenological element related to grasping the generality, and (2) a semiotic element related to the sign-mediated articulation of what is noticed in the phenomenological realm (radford, 2006, p. 5). although competing visualisations have been shown to cause tension, the crucial aspect relates to the process of coming to realise how the visualisation is regular, and how this regularity can be expressed in an algebraically useful manner. thus, although both local and global visualisations can be useful in their own particular way, it is likely that the process of objectifying and articulating an appropriate algebraic expression for the general term is complicated when tension exists between these two modes of visualisation. an awareness of and appreciation for these subtle tensions has the potential to provide an added depth of engagement for a sensitive practitioner. acknowledgements top ↑ the financial assistance of the national research foundation (nrf) towards this research is hereby acknowledged. opinions expressed and conclusions arrived at, are those of the author and are not necessarily to be attributed to the nrf. this work is based on research supported by the firstrand foundation mathematics education chairs initiative of the firstrand foundation, rand merchant bank and the department of science and technology. any opinion, findings, conclusions or recommendations expressed in this material are those of the author and therefore the firstrand foundation, rand merchant bank and the department of science and technology do not accept any liability with regard thereto. competing interests the author declares that he has no financial or personal relationships which may have inappropriately influenced him in writing this article. references top ↑ davis, b. 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(1970). visual perception of form. new york, ny: academic press. footnotes top ↑ 1. on a note of clarity, the expressions ‘shape’ and ‘term’ are used synonymously and interchangeably throughout the text. both expressions refer to the independent variable (i.e. the position of the term) in a sequence. article information authors: dalene swanson1,2 peter appelbaum3 affiliations: 1educational research and engagement, nelson mandela metropolitan university, south africa 2faculty of education, university of alberta, canada 3department of curriculum, cultures, and child/youth studies, arcadia university, united states correspondence to: dalene swanson postal address: po box 32180, summerstrand, port elizabeth 6019, south africa dates: received: 21 aug. 2012 accepted: 21 oct. 2012 published: 04 dec. 2012 how to cite this article: swanson, d., & appelbaum, p. (2012). refusal as a democratic catalyst for mathematics education development. pythagoras, 33 (2), art. #189, 6 pages. http://dx.doi.org/10.4102/ pythagoras.v33i2.189 copyright notice: © 2012. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. refusal as a democratic catalyst for mathematics education development in this original research... open access • abstract • on mathematics and democracy • on development • on globalisation • on citizenship • moses and gus reprise • acknowledgement    • competing interests    • authors’ contributions • references abstract top ↑ discussions about the connections between mathematics and democracy amongst the general populace have not been explicitly well rehearsed. a critical relationship with democracy for mathematics education may involve directing and redirecting its purposes. but, we ask, what if the ‘choice’ to not participate in experiences of mathematics education, or in its (re)direction, were itself also a critical relationship with mathematics education? what if this refusal and disobedience to the evocative power of mathematics were a democratic action? we argue that consideration of mathematics education for democracy and development must take seriously specific acts of refusal that directly confront the construction of inequality common in most development contexts. globalisation and development discourses, via citizenship and nationalism, construct relationships with learners and mathematics education in very specific ways that delimit possibilities for egalitarianism and democratic action. but, might such action not be recognised, not as refusal to participate per se, but as a refusal to participate in mathematics education’s colonising and/or globalising neo-liberal gaze? in arguing for the opening of a position of radical equality, we introduce jacques rancière to mathematics education theory, noting that for rancière emancipation is the intentional disregard for ideological narratives such as the ones produced by mathematics education discourses. thus, we provoke serious reconsideration of the assumptions behind most school improvement and professional development projects, as well as mathematics education policies and practices framed by globalising development discourses, and in the process we challenge our colleagues to consider ‘refusal’ not as deficit or failure, but as a critical position of radical equality in relation to mathematics education. on mathematics and democracy top ↑ discussions about the connections between mathematics and democracy amongst the general populace have not been explicitly well rehearsed, other than to either assume that mathematics has nothing to do with anything political, being neutral in form and practice, so it has nothing to do with democracy, which is something political; or that it is implicitly democratic. this latter assumption is underpinned by a conflation of capitalism with principles of democracy, so that if mathematics has economic utilitarianism, usually via technology and development, then it is per se democratic in nature. skovsmose and valero (2001, p. 39) reference this last position as an assumption of mathematics education’s ‘intrinsic resonance’ with democracy. in contrast, arguments for its ‘intrinsic dissonance’ have been relayed by mathematics educators conscious of the role mathematics plays in the social domain as a discourse of power, and of its deleterious effects on the lives of many people. this dissonance has been manifest either through lack of access to the discourse and practice of mathematics in the light of its gatekeeping role to avenues of employment, further education and social ‘advancement’ within the hierarchies of discourses in the social domain, or because of their alienation from it and the strong negative emotions it often arouses. skovsmose and valero (2001) note in this regard:mathematics has a power that escapes the boundaries of rationality and argumentation, and through its applications, it has become one of the forces of social reflexive modernization. ... as a subtle and implicit force, disguised and protected by the ideology of certainty, ... the destructive power of mathematics has escaped the suspicions of citizens, scientists, and social scientists. (p. 41) skovsmose and valero (2001) also reference a third position: mathematics education’s ‘critical relationship’ with democracy, in the sense that it can go both ways. in this sense, they state that: a mathematics education that is committed to democracy cannot simply rest on the intrinsic qualities of mathematics or the conceptual constructs of the discipline itself. instead, many social, political, economic, and cultural factors have to be seen as constantly directing and redirecting its development. (p. 43) we ask: what if the ‘choice’ to not participate in experiences of mathematics education, or in its direction or redirection, was in itself also a critical relationship with mathematics and mathematics education? what if this refusal and disobedience to the evocative power of mathematics were a democratic action? the scene is set through a ‘critical rhizomatic narrative’ rendering of deliberations and dialogues on the ‘construction of disadvantage’ through the discourse and practice of school mathematics in a post-apartheid south africa (swanson, 2005). in her research journey, explicated through narrative methodology, dalene encounters moses, a south african teacher and (post)graduate student, at an international education conference. during the conversation, moses asks about dalene’s research and what her research intention is (swanson, 2005, 2007b). after listening to her explanation for a whilst, and hearing her use the word ‘empowerment’ on several occasions, he responds politely, but directly (in a recognisably cultural ‘south african’ manner) with: empowerment ... empowerment ... empowerment! they tell us we have a lack, that we are supposed to be here’, [he gesticulates a movement suggesting ‘progress’], and that we need to be here, and then here. they tell us that we are disempowered and inform us what must be relevant for us to be empowered. i don’t feel disempowered, but i am told that i am disempowered and what i have to be to be empowered. (2005, pp. 217–218, 2007b, pp. 17–18) we (peter and dalene) are revisiting those aspects of our past research that have been gathering momentum as we each pursue questions of mathematics and democracy. for dalene, the full weight of moses’ response and the power of its implications for positions of activism at the time, including for her own activism associated with mathematics education and democracy, felt tremendous. as much as moses’ statement was profoundly provocative, it concomitantly provided a powerful self-learning opportunity and opened up a critical, revelational space for self-interrogation. mathematics education research was propelled by this conversation between researcher and teacher into questions about the epistemic locations of our activism and the ontology of knowing not only what ‘truth’ is, but what might be accepted, at least for the moment, as fundamentally ‘right’ and ‘just’. dalene asked herself, ‘who authors this?’ raising for us now, as we look back on such research moments, how critical to democratic action authorship might be. how interwoven with authorship and the feelings of authorship are the principles of power and the referents for defining or judging the ethics of action? (swanson, 2007b). peter thinks back to gus, a strong and interested student who slowly opted out of a path to ‘success’ once his grades on assignments had assured him of a passing grade in mathematics (appelbaum, 2008). the choice to slip out of engagement enabled gus to spend more time on his painting and more time with his girlfriend: when gus first stopped coming to class every day, i asked him if anything was wrong. he told me absolutely not. he really enjoyed our class, but he had other things to do right now. … i was bewildered. here was possibly the strongest student in the class, someone who so easily caught on to every idea in our course, and he was simply opting out. i pushed him on this, and he told me it had nothing to do with me, and he hoped to take the next semester of mathematics with me as his instructor, but his grades in the first part of the semester were good enough to let him coast right now and not end up with lower than a ‘c’ in the course, which would be good enough. but gus should be a mathematics major, i thought! this class should be the first priority for him, because he is so good at it. gus, though, was happy as an art major, and his painting was his first priority. both moses and gus represent a potentially vital aspect of the democracy we wish to bring to the mathematics education conversation: refusal. sometimes ‘refusal’ takes the form of rejecting labels or categories assigned by others. in other contexts, it is enacted as a rejection of values and beliefs. at still other times it might be termed ‘disobedience’. much of mathematics education has been demonstrated to serve a disciplining or formatting and pacifying function in which students implicitly learn to do what is expected of them (skovsmose, 2012): to solve assigned problems, to memorise standard algorithms, to complete tasks. yet democracy has at its core the idea of the rights of the individual to pursue their own destiny and forms of happiness, an idea that might entail disagreeing with a dominant social paradigm or the will of an authority (such as a teacher or a government set of learning objectives). and so we ask, what is moses’ location, and what are the set of stimuli producing this articulation of his position on ‘empowerment’? who is ‘telling’ moses that he is disempowered? what is the source of these messages? why is he personalising this perspective on disempowerment? is it so coercively embedded in the fibre of social context and the dominant discourses in the social domain that, even as he contests it, it carries the authorial voice of the ‘deficit’ meta-narrative in such a way that it holds the production of meaning ransom, even as it precedes any verbal articulation of it? (swanson, 2005, 2007b). we ask, too, why gus so easily dismisses the potential of mathematics, given the ease with which he excels and the apparent enthusiasm with which he displays creative experimentation in class with others. in what ways is painting and spending time with one’s girlfriend more obviously satisfying in the immediate sense than school mathematics, and, in this way, how might we condemn mathematics for not living up to its promise in this situation? when dalene first had her conversation with moses, she was humbled by his contestation of the extraneous gaze (semiotically recruited through her articulation of ‘empowerment’), which produced and reified positions of disempowerment for him and ‘his people’ in ways that categorised, objectified, essentialised and homogenised communities, oppressively holding them to these descriptions through the language of disempowerment (swanson, 2005, 2007b). when peter first experienced the bewildering dismissal of mathematics by gus who had so relished the opportunity to ask new questions and invent new mathematical investigations, he, too, was humbled by the contestations of the extraneous gaze (semiotically recruited through his articulation of ‘engagement’), which produced and reified positions of disempowerment for gus and ‘students like gus’ who ‘can’t appreciate the potential of mathematics’ for their pursuit of happiness and pleasure. in contrast to both moses and gus, the youths who worked with brazilian artist vik muniz on an exhibit inspired by the sculptures of giacometti knew that the kind of people who commonly visit museums are those who have the power to show them what they can never have. their exhibit turned the tables for once, as the artists (the children) became the ones producing desire: the original sculptures were of people holding various things up with their hands. the youths’ sculptures of themselves holding invisible items, combined with the placement of those items in a black bag, deprived visitors of the knowledge of what each child was holding (magill, 2000). the contrast is one not of sculpture versus mathematics, but of the location of the refusal. the children in salvador, brazil, located their refusal as a characteristic of the relationship between themselves as creators of knowledge and their audience as people desiring to know. was moses perhaps working to distance himself from the community in which he taught, so that he was not painted with the same brush of ‘poverty’? at face value, he certainly did not fit the mould of ‘impoverishment’ or ‘disadvantage’ in the stereotypical sense (except through the construction of ‘race’, whose historical imperatives have tied it to the latter constructions, especially within the apartheid context). he was a teacher, studying for his master’s degree. yet, as if to belie these distinctions, he made remarks in general conversation to the effect that he was ‘just a soweto boy’! why did he feel he needed to do this, dalene questions in her mind, and how did it relate, via the processes of positioning and posturing, to his remarks on disempowerment? was he speaking from a position of privilege or disadvantage? (swanson, 2005, 2007b). what was peter hoping that mathematics could ‘give’ gus that painting or social interactions would not? how is mathematics typically implicated in fantasies of empowerment, and how are these fantasies tied up with our cultural stereotypes of mathematically based work-lives and status? in what ways might we read gus’s choices as a seizure of self-determination via the disempowering act of refusing mathematics? how might we characterise the attitudes and present experiences of the youths who worked with vik muniz, and what these types of experiences of manipulating desire and challenging expectations of knowledge might promise in contrast to the expectations that teachers of mathematics hold for students of mathematics? were moses and gus resisting being positioned as ‘disempowered’, or are these legitimate ‘disempowered voices’ speaking back to the referential gaze that produces constructed ‘disempowerment’? (notice how our language eludes us ... constructing even as we attempt to deconstruct it.) in the interests of a critical analysis that addresses power as a characteristic of social relationships, and that makes available the referential gaze that produces the relativities of power relationships that produce it, is it then permissible to ask if moses or gus might be ‘entitled’ to make their calls as insiders? or should we interpret their refusals as having more to do with resistance to a perceived subject position ... a desire to be an outsider to ‘disempowerment’ rather than a socially enforced insider through the localising discourse on mathematics? on development top ↑ conceptualisations of ‘the learner’ are most often driven by dominant western educational discourses that normalise competition and draw on individualistic ideological investments globally. these prevailing discourses enable life opportunities for individuals within certain valued groups whilst delimiting opportunities for others. in so doing, they reify dominant cultural formations over localised ones, and these dominant discourses become the master print for entry or denial of access. life opportunities are, however, beyond a question of mere ‘access.’ normalised assumptions inhabit questions of what is valued, what is conserved and what is foreclosed in terms of being and imagining within other frames of reference. the ways in which these ideological assumptions impact on the recognition and validation of indigenous, generational or localised ways of knowing and being, and how they permit or enclose imaginative possibilities for communities to be otherwise, are all interconnected and relate directly to the false promise of the ends of freedom and egalitarianism, and misconception of well-being through the instrumental and material means of techno-scientific and economic ‘progress’ (swanson, 2010). increasing neo-liberalisation of institutions and the global modernisation agenda has set the terms of global economic and social participation, by increasing the monitoring and regulation of individuals, groups and targeted communities. such measures serve to perpetuate the global neo-colonial project. the current conception of development, framed as it is as ‘economic progress’ within the neo-colonial project, has become a truth that tolerates little resistance, that excludes a range of other possible meanings and ways of engagement, and that attempts to silence alternative voices. the more discourses on development become increasingly foreclosed in these terms, the greater freedom and the possibilities of freedom and egalitarianism, as framed by globalising development discourses, become enclosed (swanson, 2010, 2012). the issue of position and empowerment percolates in mathematics education discourse particularly when one considers the role of mathematics education in programs of ‘development’, and the role of ‘development’ programs in supporting the mathematics education of learners in countries and communities targeted by development agencies or with governance framed by development discourses. development as a concept presumes a need for development on the part of the targeted communities. in this sense, any development program situates the communities that are ostensibly aided as ‘lacking’ and in need of assistance. at the same time, political discourses within developing countries often frame the needs of their citizens in terms of deficit and economic lack. many observers of such communities, including some members of the communities themselves, would agree with such characterisations. yet the assumption is that the community members need the wisdom of the experts in the agency (or their development government) as well as the financial and material support. a radical assumption of equality would work against such assumptions, suggesting that mathematics education with and for development would begin with a refusal of help in favour of a program of collaboration. we begin with the acknowledgement that people are simply ‘where they are’ because they are, in this cultural, political, historical, etc. moment, where and how they are, with others who are also ‘where they are’. people are by their very nature incapable of being elsewhere. as rancière (2009) notes, people are not ‘unable’ because they ignore the reason for being there; they are ‘unable’ because ‘being unable’ means the same thing as being there or here. those branded ‘in need of mathematics’ for themselves, for development, or for any reason, are tautologically defined as needing mathematics simply because of whom and where they are. development in this sense is little more than a mechanism of defining inequality. development with rather than for or in spite of begins with the radical assumption of equality and the presumption that all participants contribute important components of a successful mathematics education project. a first principle of such a mathematics of and with development would understand that the development project cannot declare in advance what the outcomes will be for others that will benefit, but instead must begin prior to such setting of objectives, in the coming together of participants to discuss the range of possibilities. in order to break the cycle of expertise from outside, the initial meetings need to create spontaneous activities that work to avoid the presumption that some collaborators need to be led by the others in defining purpose, methods of working and forms of knowledge. likewise, development governments who speak on behalf of their citizens, thereby structuring a patriarchal relationship with them, reproduce the inequality of that relationship. such governments who claim to know, on behalf of their citizens, what is ‘good’ for them, reinforce and reproduce the deficit positions. this performs the inequality as natural. this is a subtle shift from well-known theorisations of ‘critical mathematics education’ that attribute a ‘critical competence’ to learners (skovsmose, 2009), in which it might still be imagined that this competence is ‘given’ by authorities, to the more provocative stance of a priori equality. rancière (2009) does not directly address the function of mathematics per se in his essay on the method of equality. we propose, however, that mathematics could be included in what he refers to as art or literature: mathematics is one form of activity in which descriptions become narratives larger than themselves, reasons for inequalities. in fact, dowling (1998) speaks of the recontextualising gaze of mathematics, drawing on the work of bernstein here, where this gaze is cast across other social events and practices and is recontextualised, through its dominance, to the terms of mathematics. here mathematics becomes the judge of those events and practices, colonising them in the process. others have suggested the fabricating power of mathematics, in which ‘facts’ and ‘realities’ are the result of mathematical action (lawler, 2010; skovsmose, 2012). emancipation, for rancière, is an intentional disregard for ideological narratives, rather than, as frequently taken in these sorts of discussions, a lack of awareness of the functioning of narratives in establishing common sense. in the same way that we might understand art or literature as triggering passions by which new forms of balance or imbalance uproot icons of faith, emblems of power, or poetic hierarchies, mathematics might in a new form of development project perform this radical manoeuvre of equality, refusing the distinction between those who know and those who do not, those who have and those who do not. indeed, democracy might well be defined as the exercise of power by the unqualified or unauthorised, the un-entitled. mathematical ‘knowledge’ in this exercise of power matters less than the claim to mastery presumed in any claim to knowledge. on globalisation top ↑ development is advanced through discourses of globalisation and the imprinting of universally ‘good’ goals or ‘best practices’ onto local, particularised settings. ‘globalisation’ refers to the way in which capitalism has spread across the world, changing the trade agenda, displacing indigenous companies and communities in favour of multinational corporations (myers, 2003). the effects of globalisation are not just commercial, since what is at stake are the national cultures and political bodies underpinning and supported by resident industries. the more capitalism spreads, the more it works to dissolve the efficacy of national domains, dissipating local traditions and values in favour of ‘universal’ ones. the only way to offset the increased homogeneity and to assert the worth of the particular against the global is to cling with ever greater tenacity to one’s specific ethnic or indigenous fantasies (myers, 2003, p. 106). here, too, we find that local and global must be identified and integrated within each other, as ‘glocalisation’ in an effective creolisation of culture (appelbaum, 1995), if there is to be an enactment of radical equality.development seems to presume that mathematics will have a salutary effect on a community and in the process on democratic life. mathematics education as a component of development projects would be charged with transforming the taste of a people, taking the community by surprise, and promoting the ‘higher’ mathematised culture over the ‘lower’, ‘uneducated’, rural (ignorant) masses. partly, the need for development is generated by a culture of fear propagated in the south african context by discourses of educational crisis. media and government agency communications raise alarm about the dire circumstances of south african students’ poor performance in mathematics. south africa’s participation in the timms-r study, in which south africa ranked 48 out of 48 nations competing, was used to beat the drum of despair and galvanise south africans into panic mode over fears for the economy, drawing attention away from issues of corruption, fraud and poor governance. almost every major multinational or other major corporation in south africa ‘benevolently’ gives money to schools and universities to bolster their mathematics and science in the ‘hopes’ of an ‘improvement’ for the ‘economy’s sake’ (which makes their social responsibility portfolios look impressive). the more the stakeholders of the education industry in south africa perform ‘crisis’ over the ‘dismal mathematics results in south african schools’, the narrower, more prescriptive, more functionalist and instrumental the curricula, methods and discourses around mathematics education and achievement become. the crisis serves the interests of the managerialist mindset of the newly neo-liberal post-apartheid south africa and, more significantly, the urgent need for revolutionary educational programmes. as rancière (2012) describes these revolutionary educational programmes, they tend to fit one of two paradigms: the education of the public by artistic impregnation, as in a theatre-of-the-people, by the people, for the people; or as a poetic celebration of the local spirit, in which the light shining down from the outside onto the locality merges with the sap rising from the ground up. attempts to meld the two paradigms end up sounding like jules michelet in the 19th century: ‘feed the people from the people’ (rancière, 2012, p. 13). but this relationship of people to people, in which the development worker-educator takes the place of the travelling salesman, turning mathematics and forms of mathematics education into commodities that people will buy into (a focus of desire), ends up having nothing in common with the local passions if it is constructed to begin with as something to be desired. rancière notes how seductive it is to imagine a theatre of the people that operates like a ‘mirror in which the people observe their own actions’ through a ‘performance without separation, in which the engaged citizen writes and enacts his own victories’ (cited in hallward, 2009, p. 146). in this way, we can see the pull to engage with mathematics education as a theatre of the people, in which increasingly mathematically literate citizens employ mathematics and its forms of question and response as a tool of reflection on democracy and empowerment. however, as hallward (2009, p. 157) writes, ‘rancière knows as well as anyone that the theatre is never more theatrical than when it finds new ways of blurring, without eliminating, the boundaries with the nontheatrical.’ and so it would go with mathematics education as a theatre of the people: one would need to abandon those features of the theatrical that run counter to the radical equality at the heart of such democratic action (for example, those forms of mathematics as a theatre of democracy in which the incorporation of those who have no part is controlled in some way by the supervision of ‘appropriately managed’ institutions) if one were to have any hope for such a theatre to enable democratic interaction. examples of such practices to be abandoned include the scripting of lessons and the assignment of particular objectives from outside, the creation of purposes for the project, or the crafting of plans that leave no room for local improvisation or adaptation. on citizenship top ↑ common progressive and utilitarian rhetoric on the ‘importance’ of mathematics learning in schools often advocates ‘good citizenship’ and vocational advancement. echoing the languages of michelet after the french revolution, a ‘successful citizen’ would be someone with access to the power of mathematics to ‘know the world’. this is because of the description of the nature of mathematics as ‘one way of trying to understand, interpret, and describe our world’ (ministry of education, province of british columbia, 2007, p. 13). yet the politics of such ‘coming to know’ is most commonly denied, so that the ability of mathematics to enable its knowing subjects to ‘describe our world’ is purportedly divorced from subjective influence and human interference: mathematics has great utilitarian worth here, but is untainted by the messiness of politics and human vulnerability. ‘failure’, in these terms, is therefore constructed, ironically, as a condition of being an unknowing mathematical subject, of refusing to participate in the theatre of democracy. as we noted above, we want to problematise this rhetoric, and consider whether the refusal to participate in this way might be a more radically democratic act than acquiescence. we do not mean, however, to celebrate mathematical ignorance, but instead to refuse the presumption of ignorance on the part of so many potential ‘citizens’ that accompanies this common rhetoric.in this common rhetoric, a citizen’s purpose and worth is defined by their access to mathematical numeracy: ‘numeracy … [is] … required by all persons to function successfully within our technological world’ (ministry of education, province of british columbia, 2007, p. 11) so that someone without access is a problem to the state and a ‘failed’ citizen. our position using rancière’s perspective refuses this construction, and the implications of such a construction. in the common rhetoric, undemocratic forms that produce inequity, often in the name of ‘necessity’ or ‘common sense’ underwritten by modernisation discourses, work against mathematics education as a force of democratisation. for example, access to mathematics in institutions and programmes must of necessity be differentiated to satisfy the socio-economic and political requirements of the nation state. not all citizens are enabled to meet criteria of minimal mastery or to excel at mathematics. it is not for nothing that mathematics is most often the most divisive subject on the school curriculum (dowling, 1998). standardised testing, streaming or tracking systems in schools for mathematics and pronounced differentiated teaching practices in this subject, as well as other gatekeeping controls, ensure that a differentiated hierarchy of access is produced that emulates, assists, (re)produces, and is (re)produced by the hierarchy within capitalist relations of production. the high status of mathematics in the ‘social division of labour of discourses’ within schools and society, makes it a high-stakes game to play, and its ‘strong grammar’ (bernstein, 2000) provides it with significant cultural value for those with the luck and privilege to have access to it as knowing subjects and citizens. thus we reject the notion of ‘successful citizenship’ constructed accordingly along the lines of privileged access to mathematical culture and referenced in terms of ‘innate capacities’ and ‘ability’; this discourse ensures that the privileged access is hidden, normalised, and often even justified under the auspices of being ‘democratic’ (swanson, 2007a). moses and gus reprise top ↑ ‘glocal’ refusal as mathematics education takes shape is a positive rather than a negative enunciation: the stance of radical equality enables in action rather than constrains. it should not be surprising that so much school mathematics in most contexts is met with forms of resistance. the psychoanalytic notion that such resistance might be a positive sign of imminent intrapersonal insight aside (appelbaum, 2008), it might be time for mathematics education to recognise refusal as a pervasive emotional experience that could be understood in generative terms instead of the usual and elaborate production of research and professional recommendations for erasing, co-opting, or circumnavigating resistance as a problem to be solved. imagine a meeting of moses and gus, two people with very different backgrounds, motivations, dreams and fears; they agreed to join us in a new social enterprise: a mathematics education project as a people’s theatre of democracy. this ‘development’ programme has at its heart the axiom of radical equality, and the notion that mathematics can bring people together as well as provide personal satisfaction; mathematics will be the central resource for strong and equitable communities characterised by creativity, joy, can-do attitudes, and the courage to act on their convictions (appelbaum, 2011, 2012).we turn for inspiration to the rise of people’s mathematics in the 1980s in south africa (adler, 1988; bopape, 1998; swanson, 2005). although it never really gained traction, and it could be considered a very narrow, functionalist, arithmetical and impoverished form of mathematical content, in that moment there was a realisation that school mathematics in its presumptuousness of neutrality and superiority was at the same time colonising, and could be associated directly with the apartheid state and its intentions. in that moment of refusal, of disobedience to the norms of schooling in mathematics, mathematics became culpable of inequality. the emphasis on collectives in and out of school generated new conceptions of overlapping communities of mathematics education that would mutually buttress each other’s efforts. and the mandate to create new curriculum materials that were informed by a critique of apartheid assumptions laid a strong foundation for dreams of a new democracy. today, in the new south africa, globalising neo-liberalism has pacified resistance, and the refusal of the people’s mathematics movement is left to lie as a memory for instigation another day. acknowledgement top ↑ competing interests we declare that we have no financial or personal relationship(s) which might have inappropriately influenced our writing of this article. authors’ contributions d.s. 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(2012). the owl spreads its wings: global and international education within the local from critical perspectives. in y. hébert, & a.a. abdi (eds.), critical perspectives on international education (pp. 333−347). rotterdam: sense publishers. article information authors: helena wessels1,2 hercules nieuwoudt2 affiliations: 1research unit for mathematics education, university of stellenbosch, south africa2school of natural sciences and technology education, north-west university, south africa correspondence to: helena wessels postal address: private bag x1, matieland 7602, south africa dates: received: 06 apr. 2012 accepted: 04 apr. 2013 published: 02 may 2013 how to cite this article: wessels, h., & nieuwoudt, h. (2013). teachers’ reasoning in a repeated sampling context. pythagoras, 34(1), art. #169, 11 pages. http://dx.doi.org/10.4102/ pythagoras.v34i1.169 copyright notice: © 2013. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. teachers’ reasoning in a repeated sampling context in this original research... open access • abstract • introduction • statistical reasoning in repeated sampling tasks       • professional development of in-service teachers in statistics    • research design and methodology       • purpose of the research       • research approach       • research context       • participants       • data generation and analysis       • trustworthiness       • ethical considerations • findings and discussion    • analysis of candy bowl task questions       • question 1       • question 2       • question 3    • analysis of post intervention questions       • question 1       • question 2       • question 3    • implications for teaching    • limitations of the study • conclusion • acknowledgements    • competing interests    • authors’ contributions • references abstract top ↑ the concepts of variability and uncertainty are regarded as cornerstones in statistics. proportional reasoning plays an important connecting role in reasoning about variability and therefore teachers need to develop students’ statistical reasoning skills about variability, including intuitions for the outcomes of repeated sampling situations. many teachers however lack the necessary knowledge and skills themselves and need to be exposed to hands-on activities to develop their reasoning skills about variability in a sampling environment. the research reported in this article aimed to determine and develop teachers’ understanding of variability in a repeated sampling context. the research forms part of a larger project that profiled grade 8–12 teachers’ statistical content and pedagogical content knowledge. as part of this larger research project 14 high school teachers from eight culturally diverse urban schools attended a series of professional development workshops in statistics and completed a number of tasks to determine and develop their understanding of variability in a repeated sampling context. the candy bowl task was used to probe teachers’ notions of variability in such a context. teachers’ reasoning mainly revealed different types of thinking based on absolute frequencies, relative frequencies and on expectations of proportion and spread. only one response showed distributional reasoning involving reasoning about centres as well as the variation around the centres. the conclusion was that a greater emphasis on variability and repeated sampling is necessary in statistics education in south african schools. to this end teachers should be supported to develop their own and learners’ statistical reasoning skills in order to help prepare them adequately for citizenship in a knowledge-driven society. introduction top ↑ one just needs to open a newspaper, listen to the radio or watch television to realise that numbers and quantitative procedures are important in the world we live in. every day we are flooded with statistics and have to rely on sensible quantitative reasoning to make decisions based on information in the media and in the workplace. these decisions concern most aspects of our lives and determine issues such as our health, prosperity, safety and much more; we have to be statistically literate to cope in our world. being able to reason statistically empowers people by giving them tools to think for themselves, to ask intelligent questions of experts, and to confront authority confidently. these are skills required to thrive in the modern world. (steen, 2001, p. 2)statistics in most countries in the world, including south africa, is not a separate subject in school curricula, but is included as a content area in the mathematics curriculum. statistical thinking however differs from mathematical thinking. cobb and moore (1997) explain that this difference results from the focus in statistics on variability and the all-important role of context: statistics requires a different kind of thinking, because data are not just numbers, they are numbers with a context. in mathematics, context obscures structure. in data analysis, context provides meaning. (p. 801) variability can be defined as the quality of an entity to vary, including under uncertainty. bakker (2004a) elaborates on the relationship between variability and uncertainty: ‘uncertainty and variability are closely related: because there is variability, we live in uncertainty, and because not everything is determined or certain, there is variability’ (p. 14). the interchangeable use of the terms variation and variability often causes confusion. reading and shaughnessy (2004) elucidate the difference between the two terms by defining variability as ‘the [varying] characteristic of the entity’ and variation as ‘the describing or measuring of that characteristic’ (p. 202). for instance, when learners write a test one can expect their marks to vary without being certain about how the marks will vary (the degree of dispersion of the marks). in order to describe the degree of dispersion, one needs some measure of how widely the marks vary in relation to, for example, the mean of the marks. variability is omnipresent in the world around us and affects all facets of our lives: ‘it is variation that makes the results of actions unpredictable, that makes questions of cause and effect difficult to resolve, that makes it hard to uncover mechanisms’ (wild & pfannkuch, 1999, p. 235). variability is fundamental to statistical thinking and reasoning and the presence of variability in the world necessitated the development of statistical methods to make sense of data (cobb & moore, 1997; franklin et al., 2005; moore, 1990; shaughnessy, 2007; watson, 2006; wild & pfannkuch, 1999). it is self-evident that the development of statistical thinking and reasoning should be one of the major goals in statistics education and should take into account the omnipresence of variability in data (franklin et al., 2005). the south african school mathematics curriculum requires learners to be able to ‘predict with reasons the relative frequency of the possible outcomes for a series of trials based on probability’ (department of basic education, 2011c, p. 36). wessels and nieuwoudt (2011) point out that the wording of the learning outcome and assessment standard quoted does not explicitly state the kind of skills needed for the statistical thinking and reasoning necessary in the statistical problem solving process. the curriculum and assessment policy statement (caps) documents for mathematics in the senior phase and further education and training phase do not use the words variation or variability in the context of data handling and probability (department of basic education, 2011a, 2011b, 2011c), nor do they refer to the essence thereof in statistics. this is a serious shortcoming as variability is fundamental to and hence plays an all-important role in the statistical process (franklin et al., 2005). watson, kelly, callingham and shaughnessy (2003, p. 1) state that ‘statistics requires variation for its existence’. data handling instruction without the inclusion of activities to develop learners’ understanding of the role of variability is therefore questionable, and without doubt undesirable. a strong case could therefore be made for the pertinent inclusion of the fundamental statistical idea of variability, together with certain ideas of describing and measuring variation, in the south african mathematics curriculum. learners specifically need to develop the ability to reason about variability in samples and acquire a sense for expected variability to be able to predict results: ‘when random selection is used, differences between samples will be due to chance. understanding this chance variation is what leads to the predictability of results’ (franklin et al., 2005, p. 21). exposure to appropriate activities to develop this understanding should therefore be frequent and well planned: [t]o improve students’ feel for the expected variability in a sampling situation, students need considerable hands-on experience in first predicting the results of samples, and then drawing actual samples, graphing the results, comparing their predictions to the actual data, and discussing observed variability n the distribution. (shaughnessy, ciancetta & canada, 2004, p. 184) research on teachers’ statistical reasoning has up to now received little attention in south africa (wessels & nieuwoudt, 2011). many teachers lack a sound background in and proper understanding of statistics in general, and the ideas of variability and variance in particular; even if they did take statistics courses as part of their teacher education, such courses traditionally tend to be procedurally rather than conceptually inclined (nieuwoudt & golightly, 2006, p. 109), leaving many teachers lacking the critical proficiency to apply their statistical knowledge in practical settings (wessels & nieuwoudt, 2011). as a result of such ‘incomplete’ statistical learning experience, teachers do not feel confident about their own statistical reasoning (shaughnessy, 2007) and lack pedagogical statistical knowledge. we concur that teachers have to understand how different fundamental statistical concepts are scaffolded in a child’s mind to be able to prepare statistically literate learners who will be able to be critical consumers of the data they are deluged with every day (wessels, 2009, p. 4). opportunities should therefore purposely be created for teachers to develop their own reasoning about variability, and specifically about variability in sampling situations, to equip them better for the development of learners’ awareness of variability present in samples or data in general. statistical reasoning in repeated sampling tasks top ↑ data sets tell stories and these stories can usually be found in the variability in the data (shaughnessy, 2007). variability is not only present in the data, but occurs from one entire sample to another. research on reasoning in repeated sampling situations mainly developed around tasks where repeated samples were taken from a known mix of differently coloured candies (sweets) in a bowl (kelly & watson, 2002; reading & shaughnessy, 2000; shaughnessy et al., 2004; shaughnessy, watson, moritz & reading, 1999; watson & shaughnessy, 2004). an example of a task used to explore the way learners reason about the variability of data in a sampling context, is the lollie task. the lollie task or candy bowl task (figure 1), evolved from the gumball task, an item in the 1996 national assessment of educational progress in the united states (shaughnessy et al., 1999; torok & watson, 2000). in the task, learners are asked to predict how many reds would be in a handful of ten candies pulled from a bowl of candies of known mixed colours. figure 1: the lollie task. learners then have to predict how many reds are likely to be pulled from the bowl if this experiment was repeated five times, each time returning the candies to the bowl and mixing them up. more important than the actual predictions is learners’ reasoning about their predictions. different versions of the task had been administered to thousands of learners from grade 3–12, mainly in the united states and australia (kelly & watson, 2002; reading & shaughnessy, 2000; shaughnessy et al., 2004; shaughnessy, 2007; torok & watson, 2000). reading and shaughnessy (2000, 2004) categorised learner responses in interview tasks according to measures of centre as high or low, and measures of spread as wide, narrow and reasonable. some learners predicted, for the expected numbers of reds pulled from the mix of 20 yellow, 30 blue and 50 red candies in the bowl, all low numbers (all below 5) or all high numbers (all above 5), whereas others predicted a wide (range ≥ 8) or a narrow (≤ 1) range of numbers. learners tend to give more reasonable predictions if they get the chance to draw their own samples from the candy mix (shaughnessy, 2007). a ‘demonstrated questionnaire’ of the candy task was developed for use in south africa with grade 6, 8 and 10 learners (reading, wessels & wessels, 2005). based on the system of observed learning outcomes (solo) taxonomy (biggs & collis, 1982), a conceptual model for the categorisation of reasoning in repeated sampling tasks emerged in the research on variability in sampling situations (canada, 2004; kelly & watson, 2002; shaughnessy, 2007; shaughnessy et al., 2004). the solo taxonomy is a neo-piagetian framework for the analysis of the level of sophistication or complexity of a response on a specific task (biggs & collis, 1982, 1991). according to this model, learner responses in the candy bowl task display four distinctive patterns of reasoning, following a progression from iconic, to additive, to proportional and finally to distributional reasoning (shaughnessy, 2007). iconic reasoning, such as relating personal stories and using physical circumstances, is usually evident in younger learners’ responses (kelly & watson, 2002). examples of iconic reasoning do not refer to the actual contents of the candy bowl or the proportions of the candy mix in it. such responses might refer to luck: ‘maybe they are lucky and will get all the reds’ or the physical act of pulling out the candies: ‘they might get more reds because their hand could find them’ (shaughnessy, 2007). additive responses are characterised by reasoning where no acknowledgement is given to the role of proportions in the mixture; reasoning is just about absolute numbers or frequencies of reds in the candy mix. implicit proportional reasoning focuses on ratio, percentage or probability of reds whilst referring back to the original composition of the mixture. explicit proportional reasoning involves reasoning about sample proportions, population proportions, probabilities or percentages. finally, distributional responses give evidence of reasoning about centres as well as the variation around the centres. shaughnessy et al. (2004) categorise responses of secondary school learners in repeated sampling tasks in a chance setting into only three broad groups: additive reasoning, explicit and implicit proportional reasoning, and distributional reasoning. these authors regard proportional reasoning as the cornerstone of statistical inference and call for more opportunities for learners to improve their proportional reasoning skills. they furthermore emphasise that the power of proportional reasoning in statistical situations needs to be identified much more explicitly in order for our students to evoke the connections of proportional thinking to statistical settings. (p. 4) professional development of in-service teachers in statistics in the past decade, statistics started to play a more important role in the school mathematics curricula in many countries and necessitated professional development initiatives for teachers involved in the teaching of statistics. well-designed professional development materials can serve more than one purpose: not only to support teachers’ statistical content and pedagogical content knowledge but also to research how they understand fundamental statistical ideas (shaughnessy, 2007, p. 998). many of the professional development and curricular activities all over the world were focused on grade 1–9 teachers and were also coupled with research projects on the statistical thinking and reasoning of mathematics teachers (burrill, franklin, godbold & young, 2003; canada, 2004; friel & bright, 1998; makar & confrey, 2002; shaughnessy, barrett, billstein, kranendonk & peck, 2004; shaughnessy & chance, 2005; watson, 2006). in south africa however, professional development in statistics was mostly aimed at grade 10–12 teachers (north & scheiber, 2008; zewotir & north, 2011; wessels & nieuwoudt, 2011). grade 1–9 teachers however also need training and support as they have to prepare learners for grade 10–12 mathematics and statistics. little evidence can be found in the literature of professional development initiatives to promote understanding of fundamental ideas in statistics education for grade 1–9 teachers or of research on their statistical thinking and reasoning about these ideas (wessels, 2009; wessels & nieuwoudt, 2011). embedded in the professional development initiatives in statistics should be opportunities for teachers to conduct statistical investigations themselves to develop their own statistical literacy as well as appropriate competent statistical thinking and reasoning skills (shaughnessy, 2007). to facilitate the transfer of the knowledge acquired through their own engagement in such statistical investigations, teachers should be supported in the classroom by researchers, advisors and colleagues after such professional development. research design and methodology purpose of the research the research described here constitutes part of a larger research project aimed at the improvement of statistics teaching in grade 8 and 9 in the secondary school. the first phase of the project comprised the profiling of grade 8−12 mathematics teachers with regard to their statistical knowledge for teaching to determine their professional development needs in statistics education. the teacher profiling information provided a basis from which a professional development intervention to broaden their knowledge of statistics and of the teaching of statistics could be designed (wessels & nieuwoudt, 2011). of particular interest in phase 2, the intervention, is teachers’ reasoning in a repeated sampling context. research approach throughout the project, investigations of an explorative nature were undertaken from an interpretivist view with the aim to understand what and how teachers made sense of their learning experiences. for the first part of the project a qualitative–quantitative multi-method research design was used (wessels & nieuwoudt, 2011); a qualitative design was employed for the second part of the study. research context building on research described in the literature (burrill et al., 2003; canada, 2004; makar & confrey, 2002; shaughnessy et al., 2004; shaughnessy & chance, 2005; watson, 1998, 2006) and the analysis of the profiling questionnaire (wessels & nieuwoudt, 2011) used in the first part of the project, a series of eight professional development workshops in statistics were developed. the two main sources used in the developing process were a sequence of modules for pre-service teacher training in statistics in the united states developed by canada (2006) and a professional development programme in statistics for in-service teachers in australia developed by watson (1998). a problem-centred approach was used as point of departure for the series of workshops, focusing on statistical knowledge for teaching, which consists of content knowledge and pedagogical content knowledge in statistics. this model of statistical knowledge for teaching is based on the construct of mathematics knowledge for teaching developed by ball, thames and phelps (2008), and includes: • knowledge of statistics content and of relationships between statistical ideas (specialised content knowledge) • knowledge of how students understand statistics concepts and develop statistical thinking and reasoning (knowledge of content and students) • knowledge of how the statistics content should be facilitated and taught (knowledge of content and teaching) • knowledge and interpretation of the curriculum • knowledge of the use of technology to develop statistical thinking and reasoning (technological pedagogical content knowledge) • the development of competent statistical thinking and reasoning skills of teachers. seven of the workshop sessions lasted two hours each; a four-hour session on the use of the computer data exploration software tinkerplots® (konold & miller, 2005) in a computer laboratory was also included. the first six workshops were presented twice a week, one on a weekday with a repeat on saturdays for teachers who could not attend during the week because of full schedules. during the workshops teachers’ statistical knowledge as well as statistical thinking and reasoning skills were developed through rich learning experiences that included all components of the statistical investigation process: posing problems, collecting data, analysing them, drawing conclusions and making predictions. the following topics were addressed in the workshops (wessels, 2009): • data generation, representation and analysis of single categorical data sets to develop the language, argumentation skills and mindset for exploratory data analysis as preparation for the comparison of multiple data sets later in the programme. • the influence of variability in measurement of a single person on interpretation of findings for the whole class. • identifying trends in data amidst variability. • chance as a precursor to probability. • repeated sampling from a known and then an unknown population. • probability. • using the internet and statistics education software in the development of statistical thinking and reasoning. • informal inferential reasoning – comparing data sets of the same and different sizes. participants the research was conducted in a large city in the gauteng province of south africa in 23 socio-economically and culturally diverse schools (wessels & nieuwoudt, 2011). mathematics teachers from all 23 schools where the profiling questionnaires were distributed were invited to attend the series of eight workshops. fourteen grade 8–12 teachers (13 women and one man) from eight schools regularly attended the series of workshops. the target group for the professional development workshops initially was grade 8 and 9 teachers, but a number of grade 10−12 teachers requested to be included. this article describes and explores these grade 8–12 teachers’ reasoning on one of the tasks they completed at the beginning of the workshop on repeated sampling. none of the teachers had had any previous exposure to repeated sampling in a probability setting. most of the previous research on variability in a repeated sampling context dealt with the statistical reasoning of students and pre-service teachers in such situations (canada, 2004; reading et al., 2005; shaughnessy et al., 2004; torok & watson, 2000; kelly & watson, 2002). the study described in this article focused on variability reasoning in a repeated sampling context of a different population, namely of in-service teachers. data generation and analysis to answer the research question of how teachers reason in a repeated sampling context, participating teachers engaged in a set of four tasks about probability distributions developed by canada (2004). these tasks were set midway through the series of professional development workshops, before the two workshops focusing on repeated sampling in a probability context. the version of the candy bowl task comprised three questions shown in figure 2 (zawojewski & shaughnessy, 2000, p. 259). figure 2: the candy bowl task. following shaughnessy (2007), the model of four distinctive types of reasoning according to level of sophistication or complexity of a response was then used as point of departure to analyse the teachers’ responses to the questions. according to this model, participant responses in the candy bowl task were expected to display iconic, additive, proportional or distributional thinking in repeated sampling tasks. corresponding codes and code descriptors developed and refined by reading and shaughnessy (2000) and shaughnessy et al. (2004) were used in the analysis of the data. trustworthiness validity of the candy bowl task as an instrument was established in earlier research studies (kelly & watson, 2002; reading & shaughnessy, 2000; shaughnessy et al., 2004; shaughnessy, 2007; torok & watson, 2000). for the purpose of phase 2 of the study project reported here, coding reliability was established by double coding of responses by an independent researcher with experience of the analysis of responses in the candy bowl task. the two coders had a 95% agreement on the coding of all responses and, after discussion, consensus was reached on the other 5%. ethical considerations approval to conduct the research was obtained from the tertiary institution where the project was hosted. information about the project was shared with all schools and participating teachers. teachers were invited to participate in the workshops; therefore participation was voluntary. teachers could at any time end their participation in the intervention. the researcher communicated the commitment to all participants and principals to keep results of the research confidential and report on the research and findings anonymously. data, results and findings have strictly been used for the stated research purpose only. findings and discussion top ↑ findings for each part of the candy bowl task are discussed in turn, supported by examples from teachers’ responses to illustrate aspects of their understanding of variation. analysis of candy bowl task questions question 1 first part: ‘how many red candies do you think you might get?’ this question aimed to determine whether teachers expected variation when taking a sample. in this task two teachers expected something other than six candies in the handful of candies drawn from the bowl, with responses ‘any number, more red than yellow’ and ’± more than half more than yellow’. three teachers acknowledged variability and indicated a range rather than one number of candies with responses such as ‘4–6’ and ‘0 ≤ red ≤ 10’. more than half of the teachers (57%) however answered that they expected six red candies to be drawn (table 1). this result concurs with other research findings on learner and pre-service teachers’ reasoning about variability in the candy bowl task (shaughnessy et al., 2004; watson & shaughnessy, 2004). a probable reason for this focus on centre and not possible variability could be that the teachers’ experiences with and understanding of theoretical probability in their mathematics and statistics education or in professional development fostered limiting constructions and hindered their understanding of variability in a sampling context (shaughnessy et al., 2004). their understanding of what mathematics is could also have prompted a single value point answer, not considering possible variability present in repeated samples. they might even have considered possible variability but felt that they had to give a single point answer. table 1: responses to question 1: expected number of reds. second part: ‘why do you think this?’in this second part reasons for expectations were teased out. the responses were categorised with codes and code descriptors distinguishing between iconic, additive, proportional and distributional reasoning (shaughnessy et al., 2004). results are summarised in table 2 and discussions are elucidated with examples of participants’ reasoning. table 2: summary of responses to question 1: reasons for expected number of reds. iconic reasoning: no reason or a vague reason is given, or participant refers to physical circumstances. three teachers responded with iconic reasoning: ‘the candies are all mixed, you put your hand in and take any 10 candies. you cannot feel whether they are red or yellow’ and ‘can’t be more than 10 or less than 10’.additive reasoning focuses on frequencies and not relative frequencies. no acknowledgement is given to the role of proportions in the mixture; reasoning is just about absolute numbers or frequencies of reds in the candy mix. three teachers responded with additive reasoning, for example ‘20 more red candies than yellows’ and ‘possibility of getting more reds than yellows’. implicit proportional reasoning attends to ratio, percentage or probability of reds whilst referring back to the original composition of the mixture. one teacher’s response showed implicit proportional reasoning, referring to the composition of the mixture with a fraction: ’will be red, that is the grouping in the container’. it is incorrect to say that are red, but the proportion of yellow to red candies is 40:60, which relates to 2:3 and might have prompted this assumption. this comment could however be indicative of centre as is more than half. explicit proportional reasoning involves explicit reasoning about sample proportions, population proportions, probabilities or percentages. six teachers used explicit proportional reasoning for this question: ‘60% is red candies and 40% is yellow’ and ‘ratio for red and yellow is 6:4’. distributional reasoning involves reasoning about centres as well as the variation around the centres. none of the teachers used distributional reasoning. an example of distributional reasoning is: ‘the number of red candies will be 6, but also spread out around 6.’ in summary: six out of the 14 participants responded to question 1 with iconic reasoning or additive reasoning. seven teachers employed implicit proportional reasoning or explicit proportional reasoning. only three considered a range of possible outcomes for a handful of ten candies drawn from the bowl. of the latter three teachers, one backed up her expected number of reds (4–6) with iconic reasoning (‘the candies are all mixed, you put your hand in and take any 10 candies. you cannot feel whether they are red or yellow’). the second teacher’s expectation for the number of red pulled from the bowl was ‘10 ≤ red ≥ 10’; she supported her choice with iconic reasoning: ‘can’t be more than 10 or less than 10’. it can be argued that this teacher was referring to a range, but because of teachers’ busy schedules they were not available for individual interviews and therefore responses had to be analysed as presented. the third teacher also expected a range, answering ‘±6’, but used explicit proportional reasoning: ‘ratio 100:10 and red, yellow’. none of the teachers used distributional reasoning, showing that their ability to reason about variability in a repeated sampling context may not be well developed. one of the reasons for this situation may be a lack of exposure to variability in repeated sampling situations before the intervention alluded to in the group interview. too much emphasis on a procedural rather than a conceptual approach when engaging with the ideas in previous training could also be at the root of the observed phenomenon. question 2 suppose you do this several times (each time returning the previous handful of 10 candies and remixing the container). do you think this many reds would come out every time? why do you think this?question 2 aimed to determine whether teachers expected variability in samples. responses to question 2 were analysed according to the codes and descriptors developed by shaughnessy et al. (2004) and are summarised in table 3. table 3: codes for question 2: reasons for expectations in repeated trials. one teacher did not respond to this question. two teachers were of the opinion that the same number would come out every time – their responses were categorised on level 0. ten of the 13 teachers indicated that the number of reds pulled out will not be the same every time. these responses however were on different levels because of the reasons given. of these ’no’ answers, eight showed level 1 reasoning: ‘just a chance’, and ‘we have red and yellow candies – chance that there is many yellow or red’. one teacher’s answer gave implicit indication of variation with no further specific mention of the distribution and was categorised as a level 2 response: ‘not a chance that you will draw the same number of reds every time’. one teacher’s response was considered level 3. she answered that the number of reds pulled would be ’more or less’ the same each time and motivated her answer with ‘ratio 6:4’. one of the teachers responded that the number of reds would not be same every time, her reason being: ‘random, can be red or yellow. average colours will indeed be more red than yellow’. the reasoning does not explicitly use proportions but may be indicative of a consideration of variability and uncertainty and might be considered an intermediary stage to proportional thinking. question 3 suppose that six classmates do this experiment, each time returning the previous handful of 10 candies and remixing the container. write down the number of reds that you think each classmate might get. why did you choose those numbers?this question aimed to elicit teachers’ predictions for the results of repeated samples and responses were analysed using a four-point rubric as shown in table 4 (shaughnessy et al., 2004). table 4: codes for question 3: reasons for expectations. of the inappropriate predictions, none was narrow or high, whereas two each were wide or low. eight teachers gave a reasonable prediction for the number of red candies to be pulled in repeated trials. reasonable predictions are spread in a more normative way around the centre, such as 2, 4, 6, 5, 8, 7. the results in table 5 show that half of the teachers giving an appropriate range used additive reasoning whilst the other half displayed proportional and distributional reasoning. conversely, all teachers using proportional and distributional reasoning suggested a reasonable range for the number of reds pulled from the mix. this result concurs with the research of watson and shaughnessy (2004) who point out that in their research participants using explicit proportional reasoning ‘were more likely to suggest a reasonable amount of variation around the expected mean of the samples’ (p. 108). table 5: summary of responses to question 3: ‘write down the number of reds that you think each classmate might get. why did you choose these numbers?’ in a group interview after completing the task, teachers admitted that they had not been exposed to any activities on variation in a repeated sampling context prior to the professional development workshops. in summary, analysis of teacher responses shows that although only one teacher employed distributional reasoning in one of the tasks, varying levels of reasoning – from iconic, to additive, to proportional reasoning – were found across the different tasks. to assess if teachers had a better understanding of variability after the intervention and were able to apply this understanding to related tasks, several tasks were given to them two weeks after the two sessions on variability. these tasks (called post intervention tasks) included questions concerning a 50:50 (black:white) spinner, the tossing of a die, the wait times at two movie theatres and the comparison spelling test scores of two classes. results of only the first three questions about the spinner will be presented here because the questions concerning the spinner were similar to those in the candy bowl task, except that they focused on probability. instead of sampling from a mix of candies, teachers this time had to predict the number of times a 50:50 (black:white) spinner would land on black in 50 spins and six sets of 50 spins. the task, used by canada (2004) in his research with pre-service teachers as participants, consisted of three questions (figure 3). figure 3: the spinner task. only eight of the original 14 teachers who participated in the candy bowl task attended the session in which the spinner task was completed. analysis of post intervention questions question 1 question 1 concerned the teachers’ expectations for the number of times the arrow would land on black. three teachers displayed iconic reasoning: ‘not sure’, backing the answer up with ‘it can be any number’. five teachers showed explicit proportional reasoning: ‘between 10 and 40 times’ because ‘the probability is 50%’. another teacher showing explicit proportional reasoning answered ’40–60% of the times’, stating that physical causes such as not lining up the spinner in the same spot for each spin, or not applying the same amount of force, would account for the variation, question 2 this question focused on the comparison between two samples of 50 spins. three teachers said that the results of the second set of 50 spins would be more or less the same as the first set, whereas five teachers implicitly or explicitly referred to possible variation between the two sets. question 3 question 3 required a list of and a motivation for the expected outcomes for six sets of 50 spins. one teacher misunderstood the question and gave the expected outcomes for six spins with an unreasonable spread (from 1 to 6). four teachers gave a reasonable spread for the expected number of spins (between 20 and 30), one gave a wide spread (between 15 and 35) whereas another one gave an improbably large spread of 18–45. the teacher who misunderstood the question gave an incomplete reason for her expectations. of the other seven, one teacher displayed explicit proportional reasoning, bordering on distributional reasoning: ‘big variation where it will land. therefore i have made my choice between 40–60% of the total’. the other six teachers all showed distributional reasoning: ‘it is more or less half of 50’ and ‘it gives me the 50% chance, but with some variation’. these results of the post intervention questions clearly show an increased awareness of variation in multiple trials and a shift from iconic reasoning to proportional and distributional reasoning after the intervention on variability. the results of this study point to teachers’ lack of familiarity with and understanding of variability in a repeated sampling context before the intervention. analysis of the post intervention questions shows that teachers’ understanding of the concept of variability grew during the intervention and that 75% of the eight teachers were able to transfer this increased understanding to related tasks. implications for teaching a number of crucial statistics concepts are under-emphasised in the south african mathematics curriculum (wessels & nieuwoudt, 2011). bakker (2004a, p. 273) states that ‘the most fundamental key concepts in statistics are variability and uncertainty’ and points out that a sense of the variability present in a certain event creates the need to consider a sample or a distribution of the data. yet, the words ‘variability’ and ‘uncertainty’ are not mentioned in the caps for mathematics in grade 10–12 even once (department of basic education, 2011b, 2011c). appropriate learning experiences, aimed at facilitating teachers’ profound understanding of the mentioned and other fundamental statistical ideas, need to be developed to afford teachers the opportunity to engage meaningfully and purposely with the ideas in ways that will support them in their decision making regarding the teaching and learning of the ideas in their classes.reasoning about centre and spread in a data set needs to be scaffolded carefully through well-chosen tasks and rich discussions, supported by a classroom culture conducive to independent thinking and shared understanding. although reasoning about repeated trials in the face of probability is mentioned in the curriculum (department of basic education, 2011c, p. 36), no mention is made of the crucial underlying concept of variability in repeated sampling. another crucial aspect, proportional reasoning, is regarded as the ‘cornerstone of statistical inference’ (shaughnessy et al., 2004, p. 184) and connects statistical reasoning in data and probability (watson & shaughnessy, 2004). proportional reasoning is likewise not mentioned in connection with statistics in the curriculum. the development of learners’ notions of these important concepts is dependent on teachers’ in-depth and in-breadth knowledge and well-developed thinking and reasoning skills about these concepts. it is therefore crucial that teachers’ awareness and understanding of variability and uncertainty in specific situations, such as repeated sampling and probability situations, is developed through well-planned activities and discussion opportunities that can be provided during professional development experiences. teachers find the development of reasoning about variability in samples and distributions challenging (bakker, 2004b) and might avoid teaching it. most teachers who participated in the study had experienced a traditional education in statistics, emphasising procedural rather than conceptual competence, and had typically not been exposed to activities that could build their understanding of and reasoning about repeated sampling (wessels & nieuwoudt, 2011). research about variability in a repeated sampling environment that included an intervention (canada, 2006) has shown an improvement of participants’ descriptions of what they expected (description) as well as of reasons for their expectations (causality). after this intervention, participants increasingly appreciated how variation occurred in multiple trials whilst reasons for their expectations improved, and progressively emphasised proportional reasoning coupled with a realisation of what is likely in the presence of variation. the results of our study concur with the results reported by canada. canada emphasises that to be effective, teacher education programs need to include an environment where teachers ‘can learn in a similar way that they themselves will aim to teach’ (p. 61). teachers must get the opportunity to draw real samples and discuss differences and similarities in the distributions to develop their own skills about this topic so as to enable them to facilitate the development of proportional reasoning skills and a sense for expected variability in learners. the value of proportional thinking in repeated sampling situations also needs to be made explicit (shaughnessy et al., 2004; watson & shaughnessy, 2004). limitations of the study the research question of this study focused on teachers’ reasoning in a repeated sampling context. the sample was limited by the fact that only 14 volunteering teachers participated. a larger sample would have yielded a better picture of teachers’ reasoning with repeated trials. the focus of the article however is not to generalise, but to summarise what can be learnt from the responses of these teachers. the fact that teachers’ time to participate was limited made individual interviews impossible – a fact that curtailed conclusions from their responses. reasons for their responses could not be probed in depth; for example, in question 1 of the candy bowl task, one teacher gave the range ‘1, 2, 4, 5, 6, 7’, leaving out 3 as a possible number of reds to be pulled from the candy bowl. did she intentionally break the sequence to impart that chances are that some numbers will not be drawn? she responded with a range (4–6) to question 3, backed up by iconic reasoning, responded with explicit proportional reasoning on question 2 and then said she guessed the six possible numbers for reds in successive trials. it is only possible to determine the realistic level of her reasoning through an in-depth interview. the question can also be asked about what notions of proportionality teachers held at the beginning of this research project. the importance of variability as a fundamental concept in statistics necessitates more research about teachers’ and learners’ reasoning about variability, and especially their reasoning about variability in a repeated sampling context. interview tasks to probe participants’ thinking more thoroughly should be an integral part of such a study. conclusion top ↑ shaughnessy (2007) points out that beliefs and conceptions about outcomes of repeated trials do not easily change, and emphasises the importance of including empirical experiments and simulations in data handling and probability instruction. it is crucial to be involved in such activities on a regular basis. the following citation is just as true for teachers as for students: beliefs and conceptions about data and chance are very difficult to change, and research has suggested that empirical experiments and simulations must be systematically built into instruction over a longer period of time in order to change the patterns of students’ intuitive conceptions. (p. 976) a greater emphasis on variability and repeated sampling is necessary in statistics education in south african schools to develop learners’ statistical reasoning skills and prepare them adequately for citizenship in our knowledge-driven society. pre-service teacher education programs as well as professional development experiences in statistics of in-service teachers need to include ample opportunities for developing competence with regard to proportional thinking and an appreciation of the role of variability and uncertainty in statistics in order to equip them for this task. to this end, purposeful and effective opportunities should be created for teachers to engage collaboratively with activities and relevant materials to gain such competence and statistics teaching proficiency. acknowledgements top ↑ the authors are grateful to dan canada and erna lampen for their helpful feedback on earlier drafts of the manuscript. the financial assistance of the national research foundation and the north-west university is hereby gratefully acknowledged. any opinions, findings and conclusions or recommendations are those of the authors and do not necessarily reflect the views of the supporting organisations. competing interests the authors declare that they have no financial or personal relationship(s) that may have inappropriately influenced them in writing this article. authors’ contributions h.w. (university of stellenbosch/north-west university at time of the research) initiated and conducted the research and drafted the manuscript. h.n. (north-west university) contributed to the conceptualisation, planning and execution of the research, as well as the analysis of the data and the finalisation of the manuscript. references top ↑ bakker, a. (2004a). design research in statistics education: on symbolizing and computer tools. utrecht: cd-beta press.bakker, a. (2004b). reasoning about shape as a pattern in variability. statistics education research journal, 3(2), 64–83. available from http://www.stat.auckland.ac.nz/~iase/serj/serj3(2)_bakker.pdf ball, d.l., thames, m.h., & phelps, g. (2008). content knowledge for teaching: what makes it special? journal of teacher education, 59(5), 389–407. http://dx.doi.org/10.1177/0022487108324554 biggs, j.b., & collis, k.f. 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(1999). statistical thinking in empirical enquiry. international statistical review, 67(3), 223–265. http://dx.doi.org/10.1111/j.1751-5823.1999.tb00442.x zawojewski, j., & shaughnessy, j. (2000). data and chance. in e. silver, & p. kenney (eds.), results from the seventh mathematics assessment of the national assessment of educational progress (pp. 235–268). reston, va: national council of teachers of mathematics. zewotir, t., & north, d. (2011). opportunities and challenges for statistics education in south africa. pythagoras, 32(2), art. #28, 5 pages. http://dx.doi.org/10.4102/pythagoras. article information authors: catherine harries1 julia botha1 affiliations: 1school of health sciences, university of kwazulu-natal, south africa correspondence to: catherine harries postal address: private bag 7, congella 4013, south africa dates: received: 13 aug. 2012 accepted: 30 july 2013 published: 10 sept. 2013 how to cite this article: harries, c., & botha, j. (2013). assessing medical students’ competence in calculating drug doses. pythagoras, 34(2), art. #186, 9 pages. http://dx.doi.org/10.4102/ pythagoras.v34i2.186 copyright notice: © 2013. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. assessing medical students’ competence in calculating drug doses in this original research... open access • abstract • introduction    • dosage calculation in practice    • the cognitivist paradigm versus the sociocultural paradigm    • lack of focus on teaching therapeutics    • the calculation process    • the sociocultural perspective and best teaching practices    • our teaching    • objectives and research questions • research design    • materials and setting    • design and procedure    • statistical analysis    • reliability    • validity    • ethical considerations • findings • discussion • conclusion • acknowledgements    • competing interests    • authors’ contributions • references abstract top ↑ evidence suggests that healthcare professionals are not optimally able to calculate medicine doses and various strategies have been employed to improve these skills. in this study, the performance of third and fourth year medical students was assessed and the success of various educational interventions investigated. students were given four types of dosing calculations typical of those required in an emergency setting. full competence (at the 100% level) was defined as correctly answering all four categories of calculation at any one time. three categories correct meant competence at the 75% level. interventions comprised an assignment with a model answer for self-assessment in the third year and a small group tutorial in the fourth year. the small groups provided opportunities for peer-assisted learning. a subgroup of 23 students received individual tuition from the lecturer prior to the start of the fourth year. amongst the 364 eligible students, full competence rose from 23% at the beginning of the third year to 66% by the end of the fourth year. more students succeeded during the fourth than the third year of study. success of small group tuition was assessed in a sample of 200 students who had formal assessments both before and after the fourth year tuition. competence at the 75% level improved by 10% in attendees and decreased by 3% in non-attendees, providing evidence of the value of students receiving assistance from more able same-language peers. good results were achieved with one-on-one tuition where individualised assistance allowed even struggling students to improve introduction top ↑ dosage errors are a cause of morbidity and mortality worldwide. drug dosage calculations require basic mathematical literacy skills, but competence is inadequate amongst both medical students and qualified, practising doctors. dosage calculation in practice studies amongst doctors, nurses and paramedics confirm that many healthcare providers are not sufficiently competent in calculating drug doses (coben, 2010; conroy et al., 2008; eastwood, 2009; mcmullan, jones & lea, 2010; simpson, keijzers & lind, 2009). medical errors are a worldwide health concern, with dosage administration errors a key finding in the united kingdom. consequently, teaching the skills to calculate a dose correctly is an area that needs to be addressed in undergraduate courses (coben, 2010; national patient safety agency, 2009; simpson et al., 2009). dosage calculation skills have not traditionally been taught to medical students. one explanation for this is that they require proportional reasoning skills; mathematics taught in school is thought to transfer to situations encountered outside school (stasz, 2001). consequently, it is assumed that these proportional reasoning skills are acquired at school, although it is not expected that the specific context of dosage calculation will have been encountered there. however, research suggests that mathematical knowledge transfer does not happen readily, partly because of differences between formal knowledge, which may involve mathematical algorithms, and contextual knowledge, which often involves contextual cues as part of the processing (stasz, 2001). the cognitivist paradigm versus the sociocultural paradigm historically, medical and mathematics teaching have operated within a normative or cognitivist paradigm. within this paradigm, a skill is considered to be separate from the person acquiring the skill (or measuring the acquisition of the skill) as well as from the context within which it is learnt. it is thus expected that the skill will transfer readily from one context to another (stasz, 2001). accordingly, the context within which the skills will be needed is not considered relevant. the objectification of skills and knowledge in this way has been criticised because it fails to consider the unique ability of the human ‘to continually shape and be shaped by their social contexts’ (roth & lee, 2007, p. 4). because the relevance of context is ignored, cultural diversity is not taken into account. it also fails to consider the unique ability of the human to interpret and represent experiences and it ignores or presumes its subjects’ interpretation of situations (cohen & manion, 1980). consequently, the role of the relationship between teacher and student is regarded as limited to an opportunity for skills to be passed from the teacher to the student. the control over what is learnt, and how this is done, is with the teacher, the student not seen as needing a voice or requiring an opportunity to be heard. the teacher assumes a student is available to acquire skills if they have been delivered, that the student has then acquired these skills, or will if they make sufficient effort, and has sufficient ability to reinforce learning. the responsibility for the acquisition of the skills, once the teacher has delivered them, is believed to lie with the student. in contrast, according to the vygotskyan sociocultural view of learning, skills are learnt naturally within the context of social activity and within a culture using artefacts (such as language or, for dosage calculation, vials, syringes and treatment guidelines) developed within that culture. accordingly, the skill cannot be understood without reference to the context, namely the development of skill with the socio-historically developed tools involved that mediate thinking, for example the language and equipment relevant to the activity (rogoff, 1990, p. 35). these contextual factors give the skill its meaning and importance. this assists with motivation to acquire the skill. they provide pragmatic cues which improve performance and assist with retrieval of the skills in future use. within the context and culture of social activity there are relationships with other participants. vygotsky’s theory emphasises development as a process of learning to use the intellectual tools provided through social history (roth & lee, 2007). social interaction is expected to foster learning through the guidance provided within relationships with people who are skilled in the use of those intellectual tools (rogoff, 1990). accordingly, in a one-on-one relationship a novice will learn from an expert who guides as they participate in a meaningful activity. key to the learning process within this paradigm is the relationship between novice and guide who engage together in dialogues, which may be tacit or non-verbal. through these, a guide is able to hear the novice’s voice and to be heard and to vary the amount of control needed to ensure the novice moves towards expertise. within this paradigm, a student is recognised to be more or less available for learning and factors such as anxiety, motivation and distractibility are taken into account. the dialogues allow a guide to be responsive, in other words to provide a balance between support and challenge to keep the student optimally receptive to learning. dialogues allow the guide to give just the right amount of challenge so that the activity is slightly beyond the learner’s competence, within the learner’s zone of proximal development. this vygotskyan concept refers to the range of tasks that can be completed: the lower level of this zone corresponding to the level of skill that the learner can reach independently whilst the higher level refers to the potential skill level that can be completed with the assistance of the expert (rogoff, 1990, p. 14). learning is thought to occur through the relationship, which builds a bridge from the mind of the teacher to that of the learner so that the learner can borrow the perspective of the teacher to develop his own skills (rogoff, 1990, p. 19). consequences of the prevailing positivist paradigm to students is that (1) the context particular to dosage calculation has not been provided, hampering retrieval, (2) although proportional reasoning skills may have been acquired at school, these skills may have been taught in the early school years only, and half-forgotten as the students focused on more abstract mathematics, and (3) skills may not have been adequately taught or acquired leaving students incompetent in this area of mathematics, but without a guiding relationship within which to communicate the need for support, possibly leading to anxiety. aversion towards mathematics is a common phenomenon amongst students in higher education (taylor & galligan, 2006) and such anxiety might lead to increased defensiveness, which might prevent them from seeking help at medical school and later from checking answers with colleagues in a clinic situation. lack of focus on teaching therapeutics an approach that may remediate some of the problems of earlier learning within a positivist paradigm is that of problem-based learning (pbl), in which students are presented with a problem that forms the starting point and focus of learning. students use a problem-solving routine interactively to identify what their learning needs are in order to manage the problem; they then use self-directed learning to meet these needs and then meet to discuss and summarise what they have learnt. a well-designed pbl programme is expected to improve the integration of prior learning with new information, the transfer of skills to a real-life setting and the elaboration of knowledge for better understanding and retention. an established result of pbl research is increased student motivation (albanese & mitchell, 1993). there has been widespread adoption of forms of this approach amongst medical schools worldwide since its introduction at the mcmaster university school of medicine in the late 1960s. a problem-based learning medical curriculum has been followed at our (the authors’) school of medicine in kwazulu-natal since 2001, with several modifications since 2006 to include an instructor-led basic science module at the beginning of the first year (to provide cognitive scaffolding) and more clinical bedside teaching in the fourth year (tufts & higgins-opitz, 2012).according to the pbl approach, our students, in small groups and guided by a facilitator, develop common learning goals from a paper case related to a medical theme and report back. students are expected to reach these goals individually, with support from useful literature, structured learning sessions, tutorials, practicals and clinic or hospital visits. to some extent, the problem-based learning approach has remedied students’ lack of exposure to context because the paper cases provide students with some of the contextual factors of real-life health problems. however, the paper cases focus on diagnosis and less attention is paid to the details of treatment, including the technicalities of prescribing, such as calculating the correct dose. the failure of curricula to emphasise prescribing and ‘teaching the skill to treat the patient’ is a cause of concern internationally (hogerzeil et al., 2001). as a therapeutics department, we aimed to improve prescribing competence by offering a programme of therapeutics learning opportunities relevant to the medical themes being discussed, largely in the form of large group sessions and assignments because these fit most readily into complicated timetables. in light of the inadequate dosage skills amongst doctors and students (simpson et al., 2009; wheeler et al., 2004) we (the authors) felt that dosage skills training should be introduced to the medical curriculum. we believed we were best placed to provide the context relevant to dosage calculations and, like huijser, kimmins and galligan (2008), that students ‘would learn better if the mathematical skills required for dosing were taught as an integral part of our programme, rather than separated and remediated in a content vacuum‘ (pp. a-24). accordingly we set about including dosage calculation training in some of our therapeutic learning opportunities so that students could have opportunities to access prior mathematical skills as well as to determine and remediate missing mathematical skills. for example, within the endocrine theme, where students are introduced to the body’s hormones, one of the students’ pbl cases focused on a patient who developed hypercalcaemia (abnormally high levels of calcium in the blood) as a result of overproduction of the hormone involved in controlling the level of calcium in the blood (parathyroid). our learning session initially focused on which medicines would be used to manage this condition and how these medicines worked. in this case, an appropriate option is the administration, by intravenous infusion, of the medicine pamidronate. in order to bring in the dosage calculation skills students needed to manage this case, students were provided with dosage information and asked to consider how this drug should be reconstituted from a powder to a solution, what volume of this solution should be added to a litre of normal saline and what drip rate should be set in order for this infusion to be given over 6 hours. the calculation process in contrast with medical students, nursing students have historically received dosage calculation skills training. common practice is to teach, according to rule-based strategy, a formula known as ‘the nursing rule’ (hoyles, noss & pozzi, 2001, p. 13), explained by nurses as ‘what you want over what you’ve got, times the volume it comes in,’ or in its written form: we illustrate the use of this formula in solving the following example:problem: a child is to be sedated with 1.5 mg of midazolam. a vial of midazolam has 15 mg in 3 ml. how many millilitres of midazolam are required? solution: what you want = the amount of drug prescribed = 1.5 mg. what you’ve got = the amount of drug dissolved in the formulation = 15 mg. the volume it comes in = 3 ml so when the formula is applied: shortcomings associated with using formulae are recognised. coben et al. (2010) contrast memorising a formula with developing a competence where, when presented in a particular context with a prescription with a specified dose, an ampoule with a particular strength and volume and a choice of syringes with which to draw it up, a student can manipulate these to produce the correct prescribed dose. (p. 4) a formula may be forgotten when it is needed in practice or the vague terms involved may lead to the inclusion of incorrect values; for example, ‘what you’ve got’ may be understood as the concentration rather than the amount of drug and this wrong information would lead to calculation of the wrong dose. various educational interventions have been implemented and found to improve dosage skills amongst nursing students. in comparison with traditional lectures, where formulae (such as the nursing rule) were taught, significantly higher sustained learning was reported with methods that built on students’ existing mathematical problem-solving skills and focused on avoiding conceptual errors (gillies, 2004; koohestani & baghcheghi, 2010). with this in mind, we avoided the mechanistic use of formulae. vergnaud’s (1982) model of proportionality includes a multiplicative structure involving direct proportion between two measure spaces. determination of an appropriate volume of a drug in a solution could be characterised by this structure, the measure spaces being the mass of the drug and the volume of the solution. according to this model, the correct volume of a drug could be obtained either by rule-based, functional or scalar approaches. the first two approaches involve manipulating figures across measured spaces, whilst the last (that is the scalar approach) involves working only within measure spaces by adopting various strategies. nurses, although generally taught rule-based strategies, most commonly adopt scalar strategies in a clinical setting (hoyles et al., 2001). scalar approaches to solving proportionality problems are more flexible and generalisable to the workplace, preserving the meaning of the quantities and their relationship by keeping variables separate (nunes, schliemann & carraher, 1993). in view of these findings, we adopted a scalar approach to solving dosage problems in our large group sessions to help keep students in touch with the meaning of the quantities. we called it the ‘two column stepwise method’ in which ‘whatever was done’ to one column (one measure space: mass) ‘was done’ to the other (the second measure space: volume). in other words the same arithmetical calculation was applied to each measure space. corresponding with the unitary method (one of the scalar strategies described by vergnaud (1982), the initial step is to reduce the quantity in the first space to unity (one). this is illustrated in figure 1 using the problem and solution below. figure 1: illustration of the ‘two column stepwise method.’ problem:a child is to be sedated with 1.5 mg of midazolam. a vial of midazolam has 15 mg in 3 ml. how many millilitres of midazolam are required? solution: in order to get from 15 mg to 1.5 mg the first ‘step’ is to get to 1 mg by dividing the number by itself (in this case 15). the same divisor is then applied to the second column (volume). next, the resultant numbers in both columns are multiplied by 1.5. after this, students are required to be pragmatic and to check, in a different way, that the answer ‘makes sense’. this means appraising whether the result obtained is feasible. for example, in the above example a different scalar approach can be used to see if the answer ‘looks’ sensible: 1.5 mg is a tenth of 15 mg, so a tenth of 3 ml is needed and a tenth of 3 ml equals 0.3 ml. the ‘two column stepwise method’, followed by the pragmatic check, was used in all large group sessions, report backs and model answers. the sociocultural perspective and best teaching practices in our efforts to design appropriate and effective dosage calculation training we were cognisant of the fact that the students and medical school operate within the context of a newly democratic post-conflict society in which continuing poverty and inequality exist. within this society there are two needs affecting curriculum decisions: one is the focus on equity and social justice to redress historical inequality and the other is the drive to develop skills for further learning and participation in a globalised and increasingly technological workplace (vithal, 2012). teaching within the sociocultural paradigm, which requires students to have a voice and to participate in the learning process whilst still being guided by the societal need for competent prescribers, would meet these needs. accordingly, we aimed to improve learning from the sociocultural perspective, within the constraints of the teaching interactions we could readily provide for students, namely the large group session and assignment for self-assessment. one of the goals relevant to large group sessions would be to provide meaningful tasks that closely resemble those that must be undertaken in the workplace as a focus for joint attention. although resource and safety restraints precluded the use of real tasks, we developed paper dosage calculation problems of patients receiving drug treatment for conditions relevant to the current theme, using formulations and dosage regimens that would be encountered in practice in order to preserve sufficient meaning for learning to occur and be transferable to a clinic setting, as advocated by coben (2010, p. 10). we also took guidance from the seven hallmarks of good practice in undergraduate education, as described by chickering and gamson (1987). these were developed from 50 years of student and teacher experience and research and are widely regarded as a gold standard for measuring the quality of undergraduate education. these guidelines include encouraging student-faculty contact (improving student-teacher communication would be an important part of this contact), developing cooperation and reciprocity amongst students, participating in active learning, providing prompt feedback, communicating high expectations, respecting diversity and emphasising the necessity of dedicating time to the acquisition of a skill (‘time on task’). these guidelines fit well with the sociocultural paradigm because the seven principles would be expected to occur within the social dialogues between a learner and a teacher or a more competent learner working together on a meaningful activity. our teaching after we included dosage calculation training in our large group sessions and assignments, with a particular focus on real-life contextual examples, we assessed student competence as the years progressed. this allowed us to evaluate the impact of ‘time on task’. in an effort to improve student dosage competence further we aimed to enrich the interaction between learners and experts (whether teachers or more capable students) in our interventions. we gave individual students one-on-one tuition (strengthening student-teacher contact) and offered a tutorial that allowed for peer-assisted learning (developing cooperation and reciprocity amongst students). objectives and research questions our teaching provided the opportunity to examine the impact of three of chickering and gamson’s principles (1987, p. 3) and consequently our research questions were: what is the effect of (1) time on task, (2) peer-assisted learning and (3) closer student-teacher contact on dosage calculation competence amongst medical students? research design top ↑ materials and setting we examined the responses of medical students at our school of medicine in kwazulu-natal to dosage calculation problems included in examinations during their third and fourth years. we also examined case notes, written responses to paper problems and interview transcripts collected during the individual tuition of 23 of these students who were selected from those who were not able to dose competently by the end of their third year.most of the students started medical school immediately after leaving school and the median age of our group was 21 years with a range of 18–34, as a few students already had other degrees. there is considerable diversity in terms of both culture and educational background, with students drawn from homes with eleven different languages other than english. the majority of students come from government schools, some of which were historically disadvantaged in terms of resources and access to skilled teaching. other students come from advantaged privately funded schools, both in south africa and neighbouring african countries. design and procedure medical students at our school of medicine were exposed to various different dosage calculation training methods during their third and fourth years of study. paper problems involving formulations of medicines and regimens included in the south african standard treatment guidelines (department of health, 2008) were used as training and assessment material. the calculations were of four different types, namely the determination of a drip rate and three calculations where the concentration of drug was expressed in three different ways: in units of mass per volume (e.g. midazolam labeled as 5 mg/ml), as a ratio (e.g. adrenaline 1:1000) and as a percentage (e.g. 1% lignocaine).every assessment comprised four questions, one of each type. these were randomly selected from a bank of appropriate questions. students were deemed competent at the 100% level if they got all four calculations correct at any one time. at least three of the four questions correct meant they were competent at least at the 75% level. after a baseline test at the beginning of third year, students were assessed repeatedly and the cumulative number of students attaining competence by their final assessment in their third and fourth year was measured. in order to investigate the effect of ‘time on task’ advocated by chickering and gamson (1987), competence acquired in the third year was compared with that achieved in the fourth year. in the third year, students were given an hour-long introductory lecture and an assignment, followed a week later by a report-back session and a model answer for self-assessment. in fourth year, in an effort to improve ‘peer contact’ as advocated by chickering and gamson (1987, p. 3), they were offered a further lecture followed by a tutorial in which they worked together through examples. it was hoped that this peer-assisted teaching strategy would help to overcome language barriers, as students would be able to seek assistance from more able same-language peers. a sample of 200 of these students were given formal assessments both before and after the extra tuition. the change in calculation competence was compared between those who attended and those who did not. in an attempt to test the value of increasing ‘contact between students and faculty’ (chickering & gamson, 1987, p. 3), 23 students were offered individual tuition prior to the start of the fourth year. after giving written consent, each was asked to work through dosage calculations that they had previously answered incorrectly. if they faltered, the researcher would provide just sufficient information or explanation to enable the student to continue and would make a note of the difficulty experienced by the student. the interview transcripts, written calculations and case notes were analysed to determine common key problems experienced by the students. the influence of one-on-one teaching on student competence was assessed retrospectively by comparing their results for assessments immediately before and after this tuition. there were two subgroups in this group of 23 students. the first (n = 13) was sampled from those who in previous tests had omitted questions or made errors where the student’s reasoning could not be followed. the sample was stratified across the range of student ability. the second subgroup (n = 10) was a convenience sample from the poorest performing students. there were 38 such students all of whom were invited but only ten chose to attend. details of the training offered to the different groups of students are provided in table 1. table 1: training offered. statistical analysis epi-info version 3.5.3, a public domain statistical software package for epidemiology (centre for disease control and prevention, 2011), was used to perform chi-squared tests and risk ratio calculations to investigate demographic factors, including race and english as a home language, as predictors of performance, as well as to compare change in calculation competence between group tutorial attendees with those who did not attend these sessions. statistical significance was set at a level of 5% or less (i.e. p < 0.05). qsr nvivo 8, a software package for qualitative data analysis (qsr international, 2008), was used to analyse the material related to the students given individual teaching sessions to determine the key problems they experienced. reliability in order to ensure the reliability of results, our questions were based on a tested instrument to assess dosage competence. this instrument comprised questions from an australian study (simpson et al., 2009). we modified this instrument slightly where necessary to reflect dosage formulations and regimens used in south africa. validity in order to ensure that our measurement of student ability would reflect their ability as future doctors to dose their patients appropriately, we aligned our teaching and assessment materials as closely as possible with situations that will be encountered when prescribing and administering a drug. we used the prevailing prescribing regimens and formulations recommended by the texts that guide prescribing in south africa, the south african standard treatment guidelines (department of health, 2008) and south african medicines formulary (university of cape town division of clinical pharmacology, 2010). students were provided with pictures and explanations of materials and equipment peculiar to the task of dosing drugs, such as vials, ampoules, powders requiring reconstitution, administration sets and infusion solution bags. the assessment questions selected were typical of those encountered in an authentic emergency setting. ethical considerations ethical approval (reference number be185/09) for this mixed-methods study was obtained from the university’s biomedical research ethics committee. after being given written information, each participant gave their written informed consent to the written publication in a research journal of the overall findings from their data. findings top ↑ of the 364 students eligible for the study, 82 were found to be competent at the 100% level at the beginning of their third year. of the 282 remaining, 58 (20.6%) were competent by the end of third year. by their last test in fourth year, 99 of the remaining 224 (44.2%) had attained competence. thus, more students became fully competent during the fourth than during the third year. likewise in the third and fourth year respectively, 29.6% and 54.9% of students became newly competent at the 75% level (table 2). at each screening stage, students who did not achieve full competence but who attained competence at the 75% level were most commonly prevented from achieving full success by a mistake with a calculation involving concentrations expressed as a ratio or percentage (each between 31% and 43% of the time). mistakes with mass per volume and drip rate questions which prevented full competence were made in between 6% and 22% of cases. of the 364 students, 51% were english speaking whilst the rest spoke an african language at home. the average age was 21 years with an age range of 18–34 years. over half (59%) were female students. although race and home language were predictors of performance at baseline, both associations had been lost by the final assessment. neither age nor gender was associated with competence either at the beginning or the end of the study. table 2: number and percentage of students competent over time (n = 364). in the subgroup of 200 students who had formal assessments both before and after the fourth year group tutorial, 83 were attendees and 117 non-attendees. attendees performed better overall before tuition, 30% and 55% of them scoring 100% competency and 75% competency respectively. equivalent figures for non-attendees were 21% and 53%. attendance made no difference to students’ ability to achieve 100% (0% and 3% change for attendees and non-attendees respectively (table 3). however, attendance had a significant influence on the numbers of students achieving 75% competency (10% and -3% changes for attendees and non-attendees respectively, p < 0.05). there was no statistical difference in improvement for any of the four different types of question between the attendees and non-attendees. table 3: percentage of students competent before and after peer-teaching intervention (n = 200). regarding the one-on-one teaching, in the group with the range of abilities (n = 13), three students did not attend the final assessment. the remaining 10 all finally achieved 100% − four having started at 25%, four at 50% and two at 0% (table 4). table 4: dosage results for stratified sample of students given one-on-one teaching. in the subgroup comprising the very weakest students (n = 10), three students did not complete the final assessment. there was an improvement amongst the remaining seven, most of whom started with 0%. the marks of three improved from 0 to 50% and one each from 0 to 25%, 0 to 75%, 0 to 100% and 25 to 50% (table 5). table 5: dosage results for sample of very weak students (n = 10; 3 not included in final assessment). during the interviews, it was apparent that students most frequently had difficulties with the concentration of a solution expressed as a ratio or percentage. the next most commonly encountered problem was unit conversion, for example converting from milligrams to grams or micrograms to milligrams.the group comprising very poor students had the additional difficulty that, despite our efforts, they still did not understand that the concentration of a solution is actually giving information about the quantity or mass of a substance relative to the volume, in other words two variables. this group was also overly reliant on calculators. they lacked an ability to simplify numbers or see patterns and had no insight into an unrealistic number generated by the calculator. discussion top ↑ more students became competent, at both the 100% and 75% level, during the fourth than during the third year of study. this could be due, in part, to the fact that by fourth year students had had more calculation practice. there is some evidence of this in that a small number of students improved to the 100% level even though they did not attend the group tuition. in other words, repeated learning and assessment over two years allowed students to benefit from spending ‘time on task’, consistent with the findings of chickering and gamson (1987, p. 3). celebi, weyrich, kirchoff and lammerding-koppel (2009) found an improvement in prescribing skills even amongst a control group of students and attributed this to a training effect as students gained practice with the assessment method. amongst nurses, inclusion over three years in drug calculation learning opportunities, which gradually increased in difficulty, and repeated assessment were found to be successful strategies (elliott & joyce, 2005). for students achieving at the 75% level of competence but failing to attain full competence, calculations involving concentrations expressed as a ratio or percentage presented the most difficulty. such calculations have also been found to present the greatest difficulty in other studies (simpson et al., 2009; wheeler et al., 2004).there was significant improvement in those students who attended the group tuition. besides affording students dedicated time to focus on calculations away from competing priorities, the small groups provided more intimacy as there were fewer students present than in lectures. this gave students the opportunity to learn from more competent peers who may have guided them through difficult steps and clarified areas of misunderstanding. this finding is in line with the advantage of peer-assisted learning proposed by chickering and gamson (1987). in a group of nursing students, guidance from mentors and informal opportunities to engage with slightly more competent peers were also found to be helpful (penman & white, 2006). help and support from peers also afforded an opportunity for theorisation of learning in a more accessible language as, because there are 11 official languages and limited personnel within these language groups with appropriate teaching skills, teaching and assessments are entirely conducted in english. it is interesting to note that, by the end of our study, not speaking english at home had been lost as a risk factor for poor calculation competence. one-on-one teaching markedly improved competence. all randomly selected students achieved 100% and even in the group selected from the poorest students, most improved. our success with individual tuition is consistent with vygotsky’s theory of learning: when the informal, disorganised concepts of the learner meet with the formal, methodical reasoning of the teacher this allows for co-regulation between the teacher and the student (fogel, 1993). this state of inter-subjectivity between the student and the teacher enables the teacher to guide the student forward toward a better understanding of a situation (kolikant & broza, 2011). students who participated in the one-on-one interventions also had good attendance at the group tuition, so their results are a reflection of both group and individual tuition. it may be that the competence they developed during the individual tuition motivated them to attend the group tuition. during the one-on-one interviews, it was apparent that students’ most common problems related to selecting the appropriate units to express a concentration given as a ratio or percentage. this was in line with findings from the written assessment, where such questions were more problematic for students achieving at the 75% level of competence but failing to attain full competence. in the randomly selected group of interview students, once this was clarified they were able to develop competence and master the calculations. the next most common problem, namely that with unit conversion, was also improved with individual teaching. the group comprised of the very weakest students had additional conceptual problems that benefitted in particular from the one-on-one tuition. this was consistent with jackson and de carlo (2011), who found that nursing students with conceptual deficiencies required more rigorous remediation. similarly, kolikant and broza (2011) contend that low-achieving students need appropriate interventions to improve their conceptual understanding. the very weak group struggled with the concept of the strength of a solution and the idea that a ratio or percentage can describe the relationship between the amount of drug in a solution and the volume of that solution. this suggests that they had difficulty handling intensive quantities, which are measured by a relation between two variables. concentration (an intensive quantity) is measured in terms of units of mass per units of volume. reasoning about intensive quantities has been shown to be difficult for school children who lack opportunities in school to develop this understanding (nunes, desli & bell, 2003). although one-on-one teaching may seem like a luxury in a world where cost-cutting exercises make it a rarity, it does have benefits beyond the obvious; huijser et al. (2008) noted that ‘meanings negotiated during one-on-one consultations are not a one-way street, but rather part of a mutual learning experience during which valuable insight is gained’. they noted that individual teaching provides the opportunity for teacher responsiveness not only to the learner receiving the individual tuition but, by allowing misconceptions to be identified, to the group as a whole. likewise, once our individual teaching had revealed that understanding related to intensive quantities could not be taken for granted, we changed our overall practice. we now include visual representations of particles in solution in our large group learning. we also make the analogy that when a large cup of sweet tea is poured into smaller cups it maintains the same sweetness (concentration), although the amount of sugar and volume of liquid in each cup is less. once students are reminded of their understanding of intensive quantities in this way, the idea that drug concentrations describe the relationship between two quantities (or vergnaud’s two measure spaces) is introduced. the teacher now describes a hypothetical 1% solution, asking students to imagine 1 g of powdered active ingredient dissolved in 100 ml of water and then poured into each of one hundred 1 ml ampoules. attention is drawn to the fact that the concentration has not changed; it remains 1%. the very weak students were also overly reliant on calculators. this may have contributed to their lack of insight into an unrealistic result generated by a calculator. mcmullan et al. (2010) found that young nurses who had relied on calculators since school had a false sense of security, resulting in more conceptual errors. gillies (2004) has also expressed concern that unquestioning learning of a formula as well as over-reliance on a calculator could prevent students from monitoring their calculations thoughtfully causing them to blindly accept results, abandoning common sense and resulting in the acceptance of clearly unreasonable answers (gillies, 2004). conclusion top ↑ our study found that student competence improved with repeated exposure and training using the scalar ‘two column stepwise method’. group tuition helped struggling students to improve further, confirming the value of providing opportunities for learners to focus jointly on authentic paper problems with teachers or more capable peers. although time consuming, one-on-one tuition achieved good results with individualised assistance, allowing even the very poorest students ultimately to improve. the insight gained from these interactions allowed us to provide responsive subsequent learning interactions. accordingly we modified our programme to incorporate interventions focused on the understanding of intensive quantities. acknowledgements top ↑ we would like to thank elizabeth nicolosi for her assistance with capturing the data. competing interests we declare that we have no financial or personal relationships that may have inappropriately influenced us in writing this article. authors’ contributions c.h. and j.b. 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(1982). cognitive and developmental psychology and research in mathematics education: some theoretical and methodological issues. for the learning of mathematics, 3(2), 31–41. vithal, r. (2012). mathematics education, democracy and development: exploring connections. pythagoras, 33(2), 1–14. http://dx.doi.org/10.4102/pythagoras.v33i2.200 wheeler, d.w., remoundos, d.d., whittlestone, k.d., palmer, m.i., wheeler, s.j., ringrose, t.r., et al. (2004). doctors’ confusion over ratios and percentages in drug solutions: the case for standard labeling. journal of the royal society of medicine, 97(8), 380–383. http://dx.doi.org/10.1258/jrsm.97.8.380, pmid:15286190, pmcid:pmc1079557 abstract introduction on sites and the participating teachers on toolkits and their design on applying conversation analysis and ethnomethodology research framework methods of data analysis site-related issues concluding remarks acknowledgements references about the author(s) faaiz gierdien department of curriculum studies, faculty of education, stellenbosch university, stellenbosch, south africa charles smith school of science and mathematics education, university of the western cape, cape town, south africa cyril julie school of science and mathematics education, university of the western cape, cape town, south africa citation gierdien, f., smith, c., & julie, c. (2019). keeping sites in sight: conversations with teachers about the design of toolkits peculiar to a continuous professional development initiative. pythagoras, 40(1), a475. https://doi.org/10.4102/pythagoras.v40i1.475 original research keeping sites in sight: conversations with teachers about the design of toolkits peculiar to a continuous professional development initiative faaiz gierdien, charles smith, cyril julie received: 20 feb. 2019; accepted: 12 nov. 2019; published: 05 dec. 2019 copyright: © 2019. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract the aim of this article is to shift the notion of ‘sites’ as places of work peculiar to continuous professional development (cpd) to a theoretical level, independent of, yet intimately connected to, their physical meanings, for example universities and schools. most cpd initiatives have to contend with at least one of these two sites, in which university-based mathematics educators and school teachers can have different and at times overlapping ways of talking about the same mathematics. using research on number and operations, non-visually salient rules in algebra and algebraic fractions, and analytic tools and notions peculiar to conversation analysis and ethnomethodology, the authors identify and analyse site-related issues in the design of particular problem sets in grade 8 and grade 9 toolkits and related conversations between a mathematics educator and participating teachers. the article concludes with the implications of ‘keeping in sight’ ways in which universities and schools talk and work when it comes to designing and discussing toolkits. keywords: continual professional development; mathematical toolkits; productive practice; algebraic fractions; visual syntax of algebra; equivalence; conversation analysis; ethnomethodology. introduction continuous professional development (cpd) initiatives involve different workplaces, that is, sites. university-based mathematics educators collaborating with in-service teachers designing and using toolkits that have research knowledge inscribed concern the issue of sites, namely the university and the school. both universities and schools have their peculiar ways of knowing, working and talking or conversing, that is, discursive practices. as a way to begin to understand activities related to cpd, we refer to three physical sites, namely the university (site a), the school or school classroom (site b), and a venue either on or off campus (site c), where the participating teachers and mathematics educators meet and interact. the work of mathematics educators entails contributing to and drawing on mathematics education literature, that is, research knowledge. on the other hand, the schooling system with its associated policy documents (department of basic education [dbe], 2011) provides guidelines and details on the intended and implemented curricula (julie, 2013), and structure the work of schoolteachers, for example. in this unfolding process of mathematics educators interacting with teachers in the activities of the cpd initiative, mathematics educators and teachers can have differing ways of talking, understanding and working with respect to the teaching and learning of the mathematics. moreover, at an individual level, mathematics educators and schoolteachers may have site loyalty or preference, meaning that they believe that ways of knowing, talking and working related to their workplaces, for example the university, should be preferred or taken into consideration. during conversations, both groups may invoke non-present others, for example teachers may refer their learners as ‘our kids’ or ‘the child’ to make a point when considering certain questions, or mathematics educators may mention ‘boundary objects’ (cobb & mcclain, 2006). for example, the curriculum assessment policy statement (caps) policy document has details on cognitive levels for questions, which structure what needs to happen in classrooms. in this article, we examine how a particular cpd initiative designs and uses toolkits as a way to work with teachers. we analyse three data incidents that occurred at site c. these involved interactions between the two discursive practices, namely the mathematics educator (i.e. the first author) and a group of teachers where the focus was on examining and conversing about the design of particular grade 8 and grade 9 problem sets, which are part of what we call ‘toolkits’. the grade level choices had to do with the mathematics educator’s involvement in the cpd initiative. the toolkit content served as conversational anchors (roth, 1998, p. 186) between the teachers and the mathematics educator. the data incidents, presented as data excerpts, are in the form of selected turns of transcribed audiotaped conversation excerpts between the teachers and the mathematics educator. based on this, we reviewed and applied particular conversation analysis tools and ethnomethodology notions, later on, to answer the research question. the research question that drove this article was: what are the site-related issues when it comes to analysing toolkits-based conversations peculiar to a cpd initiative between mathematics educators and a group of teachers? site-related issues stemming from the university or the school were present in the toolkits, as well as in the conversational exchanges associated with them. in all cpd initiatives, university-based mathematics educators, especially, need to be aware of what it takes to get better at ‘doing interaction’ (ten have, 1990, p. 24) when conversing with teachers, in addition to designing toolkits that have inscribed research knowledge related to school mathematics. answers to this question have implications for cpd initiatives. a review of different strands of literature pertinent to the data incidents, as well as data excerpts, follows. on sites and the participating teachers the teachers involved in the cpd initiative work in high schools or at sites located in socioeconomically challenging environments in the greater cape town area. in relation to curriculum reform, some commentators have described teachers working in these schools as ‘just too badly educated themselves’ (paton, 2016). clearly, the teachers are referred to in derogatory ways. to date we know little about what it takes to work with these teachers (setati, 2005). we do know of pragmatic ways of working of teachers in schools in more affluent environments in relation to curriculum reform in south africa (harley & wedekind, 2004). in addition to visiting and interacting with the teachers in their classrooms, the cpd initiative organises workshops and institutes at venues that are on or off campus (site c). workshops have a time limit of one and a half hours after school and occur four times during the academic year, whereas at institutes, being residential, teachers stay over at a venue and workshops last a day and a half. at site c workshops, mathematics educators and teachers work through and discuss different issues that are of mutual concern. workshops take the form of a structured programme consisting of a general discussion involving all the teachers, followed by sessions in which teachers from the different participating schools meet and discuss tasks related to their particular grade levels. sites are repositories of various kinds of knowledge and have ways of working that are associated with them. for the university-based mathematics educators, knowledge involves competence in the design of scientific investigations that assist in the focused understanding of phenomena. for example, the investigation is into the design of toolkits aimed at engaging with the participating teachers. for the teachers, on the other hand, there is the concern of becoming competent in promoting student learning given complex contextual constraints informed by the ‘daily grind’ (lortie, 1975). differentiating between, say, universities and school classrooms as sites with different offerings is thus a convenient way to present our argument. site c interactions offer the possibility of mutual engagement, theoretically and practically between mathematics educators and teachers. in other words, site c is a place or venue where the two groups converse and engage around mathematics content, informed by their respective perspectives related to their workplaces, that is, sites, with toolkits as conversational anchors. on toolkits and their design the toolkits associated with the cpd initiative and the data incidents are versions of curriculum materials (brown, 2009; davis & krajcik, 2005), professional development tools (clark-wilson & hoyles, 2019; collopy, 2003), and supplementary materials that need to ‘accompany the various textbooks’ that teachers have in their classrooms, that is, in site b (kindt, 2011, p. 176). they are programmatic in their design, meaning that they relate to the different mathematics in the various grade levels. a toolkit consists of a set of tools – 35 or so problem sets – that span the school mathematics content of a particular grade level, with ‘productive practice’ as a main design feature. productive practice aims at enabling learners general ways of working in school mathematics through ‘“deepening thinking”-like problems whilst practising’ (julie, 2013, p. 93; may & julie, 2014; okitowamba, julie, & mbekwa, 2018; smith, 2015). kindt (2011) concludes his discussion of principles of practice with the notion of ‘productive practice’ (p. 175). in this respect he comes up with several recommendations regarding the design of productive practice problem sets or exercises. those applicable to the problem sets related to the data excerpts of this study are: vary the practice formats and activities as much as possible. challenge the students to reason logically (for example by using coherent strings of problems). pay attention to verbal readings and writing of algebra rules or formulas. (kindt, 2011, p. 176) a component of productive practice relevant to the data incidents of this study is ‘deepening mathematical thinking’ (dmt) (watson & mason, 1998), which focuses on an engagement with the mathematics. deepening thinking-like problems have inscribed a design that aims at expanding learners’ understanding of mathematical objects as found in the content areas of the intended mathematics curriculum (julie, 2013), namely the policy documents per grade level (dbe, 2011). based on this, the problem set questions are phrased in ways where learners are required, for example, to comment on whether mathematical statements are always, never, or sometimes true, and to comment on concepts embedded in procedures for finding the lcm (lowest common multiple) in the case of simplifying an algebraic fraction, to name but a few. we thus view the toolkit design as ecologically relevant in terms of what the teachers have to teach in their classrooms in terms of cognitive levels; for example, see figure 1. figure 1: problem set a and set b from the grade 8 toolkit. a grade 8 problem set on ‘always true, sometimes true, never true’ the design of problem set a (see left-hand column in figure 1) requires the learner or teacher to deepen their mathematical thinking or reasoning around the concept of number and the procedure or operation of subtracting a number, which requires interpreting the minus sign. the design exemplifies cognitive level 3, namely ‘complex procedures’ in the content area of number, operations and relations (dbe, 2011, p. 157), making it ecologically relevant. in the case of the integer –4’, the minus sign has a unary function: it is a ‘structural signifier’ (sfard, 2000). when subtracting a negative integer from 5, the minus sign has a binary function (vlassis, 2004, p. 472): there is the procedure or operation of subtraction, which amounts to the difference between 5 and –4. in this instance, the answer will be greater than 5. here the minus sign is an ‘operational signifier’ (sfard, 2000): it signifies the procedure or the operation of subtracting, as well as being a negative integer. the dmt is about becoming aware of the minus sign being a structural signifier as well as an operational signifier (vlassis, 2004), for example 5 – (–4). for mathematics educators and designers, this always true, sometimes true, never true question is about drawing attention, in a productive way, to the multidimensionality of the minus sign (vlassis, 2008, p. 560) and integers. below we elaborate on how this minus sign and its meaning are used in conversations. in the design of problem set b there is a scramble related to equivalence (gattegno, 1974), in which it is possible ‘to replace one item by another’ (p. 83), for example y0 can be replaced with 1 (see right-hand column in figure 1). this problem set requires the learner or teacher to ‘unscramble’, that is, to replace and to distinguish between 1 and 0 when it comes to the algebraic expressions involving exponent laws, concepts associated with the equality symbol (kieran, 1981) and meanings of the minus sign. this design stems from teachers’ concerns that their learners have difficulty in the content area of patterns, functions and algebra (dbe, 2011, p. 157). it exemplifies cognitive level 3, namely ‘complex procedures’. inscribed in the design are a ‘visual syntax of algebra’ (kirshner, 1989), the non-visually salient rule y0 = 1 (kirshner & awtry, 2004, p. 230), and operational and symmetry meanings of the minus sign, when it comes to verbal readings and writing of algebra rules or formulas (see kindt, 2011, p. 176). ‘visual syntax of algebra’ refers to different written out lines of procedures that involve manipulating algebraic expressions, fractions or equations. the non-visually salient rule, as in problem set b, is one that is not obvious to the learner. this design in the form of an ‘always true, sometimes true, never true’ prompt aims at dmt of the learner or teacher with respect to apprehending the syntactic structure of y2 = 2y – 3 and 2y – 2 – y2 = y0, that is, the way these two ‘elementary algebra’ equations ‘look’ and the connections between them (kirshner, 1989, p. 274). when checking to see if y2 = 2y – 3, then 2y – 2 – y2 = y0 is always, sometimes or never true, there is the procedure: 2y – 2 – (2y – 3), which is equal to 1. in this procedure or operation, the second minus sign in front of the brackets, has a ‘symmetric role’ (vlassis, 2004, p. 472). when the concept of the distributive property is applied, there is an ‘inversion’ (p. 472) in the meaning of the minus sign: it becomes a ‘plus’, as in 2y – 2 – 2y + 3, which equals 1. the teacher or learner thus has to decide on y2 + 1 – y2, that is, 1 in relation to y0, which is a non-visually salient rule. in other words, there is no obvious route to the solution (dbe, 2011, p. 157). when speaking with the teachers regarding problem set b, there is the likelihood of ‘indexical’ expressions or words for ‘transposing’ and viewing equations or equivalence relationships (gattegno, 1974) that have symmetric balance (kieran, 1992). for example, ‘bringing it over’ is indexical of the procedure of ‘transposing’ when it comes to checking whether the mathematical statement is always, sometimes or never true. similarly, in the case of ‘you must do the opposite’ which, as a procedure, is also a verbal or semantic instance of transposing. we elaborate on these issues below. a grade 9 problem set on simplifying an algebraic fraction the design in figure 2 aims at dmt through verbal or semantic readings, and writing of procedures related to syntactic algebra rules for finding the lcm (lowest common multiple) indicated in lines a to f, when simplifying the algebraic fraction (see figure 2). this design goes beyond ‘simplifying algebraic fractions using factorisation’ (dbe, 2011, p. 24), because all the procedures are given in lines a to f. by way of background, the inclusion of algebraic fractions has historical roots in the incorporation of abstract algebra in the school curriculum (matz, 1980). some may view this algebraic fraction in one variable as the ‘wrong algebra’ because it is not referentially rich (national council of teachers of mathematics [nctm], 2000). the design should, however, be viewed in light of fostering ‘a computational theory of algebraic competence’ (matz, 1980). in this regard, ‘write down what was done in the steps numbered a to f’ serves as a lexical support system (kirshner, 1985). as mathematics educators, we view this lexical support system exercise as ecologically relevant because teachers in their classrooms (site c) are familiar with the abstract, base rules (matz, 1980) of finding the lcm. more interestingly, lines a to f are also instances of visually moderated sequences. taken from davis (1984), a visually moderated sequence is ‘a visual cue v1, which elicits a procedure p1 whose execution produces a new visual cue v2, which elicits a procedure p2, … and so on’ (p. 35). differently put, the visual cue in line a elicits the procedure, namely factorises the denominator x(3x 2), using the concept of the distributive property, which leads to line b. in line b, there is an explicit site a input, namely a new visual cue of multiplying by 1, the identity element for multiplication. in the ‘visual syntax of algebra’ this becomes . one criticism related to line b is that finding the lcm as a set of procedures addresses structural algebra purely as mathematical method (italic in original, kirshner & awtry, 2004, p. 253). however, in line b, the procedure takes on a ‘mathematical’ method where the mathematics is in the form of, . each of the last three is an algebraic or structural equivalence relationship (liebenberg, sasman, & olivier, 1999) of 1, the identity for multiplication. another reason for line b has to do with potentially sustaining learners’ epistemic engagement with this syntactic version of the identity for multiplication, as part of procedures for finding the lcm. the latter is a generic ‘base rule’ (matz, 1980, p. 95) when it comes to algebraic competence in the domain of algebraic fractions. on its own, line b is not enough, hence the lexical support system, namely ‘write down what was done’. additionally, in the procedures in line d and line e there is the symmetric role (vlassis, 2004) of the second minus sign in the case of the distributive property. when the brackets are removed, this minus sign is an operational signifier related to the operation of subtracting. the minus sign is thus inverted and becomes a ‘plus’, in other words –2 becomes + 4 (see line e). this last sentence reflects site c, that is, a school classroom discourse with respect to the minus sign appearing ‘outside the brackets’. the design differs from the usual site c rules for finding the lcm format, which entails, for example, finding a common denominator. figure 2: visually moderated sequences showing procedures for finding the lcm when simplifying an algebraic fraction. a way to simplify is given in figure 2. further comments on ways the teachers or mathematics educators refer to x’s as variables and ‘terms’ in figure 2 are necessary. in all the lines, the x’s have ‘symbolic value’ (matz, 1980, p. 131) that is suspended from arithmetic, meaning that there is no need to substitute numerical values for the x’s. if the algebraic fraction or expression were to represent a rational function, then the x’s as variables can take on the particular real number values, excluding 0 and (see line a). however, when commenting on the steps in simplifying this algebraic fraction, learners or the teachers in this case have to ‘relax arithmetic expectations’ (p. 131) involving well-formed answers, that is, instances where the answer is a number. all the operations involving the x’s are thus ‘suspended operations’ (matz, 1980, p. 131). in fact, lines a to f deal with the ‘surface structure’ (kieran, 1992) of the factorised ‘terms’ in the denominators shown in lines a to f. the x’s are thus not ‘variables’ that take on numerical values. in the mathematics curriculum, variables have many uses and different meanings (rosnick, 1981; usiskin, 1988), which we will not go into here. this can be ambiguous to site c – classroom teachers – because there are instances in the mathematics curriculum where the x’s do take on numerical values, that is, they become variables. for an interesting discussion on the role and power of ambiguity in mathematics as a discipline, see byers (2007, p. 78). on applying conversation analysis and ethnomethodology conversation analysis is a research tradition that focuses on the order, organisation and orderliness of social action, particularly those social actions that are located in everyday interaction, in discursive practices, and in the sayings, tellings and doings of members of society (psathas, 1995, p. 2). teachers and mathematics educators, as members of society, have their respective discursive practices, that is, peculiar ways of knowing, talking and working. conversation analysis has elaborate and detailed notations for voice inflections, emphases and pauses, and so forth, when it comes to doing line-by-line transcriptions of audiotaped conversations, which we will not focus on. instead, we refer to conversation analysis and use its analytic tools in an ‘applied’ sense because these offer ‘bottom up’ value when it comes to identifying site-related issues in the data excerpts (see below). in particular, conversation analysis is concerned with ‘meaningful human conduct across settings and modalities (visual, auditory) of production and understanding’ (pomerantz & fehr, 2011, p. 166), which are core features of all cpd initiatives. historically, conversation analysis and ethnomethodology are connected in their ‘broad contours’ (maynard & clayman, 2003, p. 176), with the latter preceding the former (heritage, 1984, 2009; lynch, 2000; maynard & clayman, 2003; ten have, 1990). ethnomethodology is about the study of methods, which can be spoken or written, that people or participants within a given linguistic community (ethno) use to establish and maintain intersubjective understanding based on their ‘practical sociological action and reasoning’ (garfinkel, 1967, p. 1). the data excerpts reflect many instances where the teachers and the mathematics educator talk or reason, that is, where they share their practical understandings and methods, peculiar to their places of work, in relation to the design of the problem sets. in an applied sense, the conversation analysis analytic tools, namely turn transition relevance place (ttrp) (lynch, 2000, p. 530), and repairs and epistemic order (heritage, 2009, pp. 305–309; pomerantz & fehr, 2011; ten have, 1999, pp. 111–121), in addition to the em notion of indexicality, are useful in terms of answering the research question. a recent study highlights the analytic value of applying ‘ethnomethodological approaches’, which include the mentioned analytic tools in the case of the mathematics classroom, that is, site b interactions (ingram, 2018). in the line-by-line transcriptions of conversation excerpts, a turn viewed as a ttrp (lynch, 2000, p. 530) can be used to identify and to analyse understanding between the teachers and the mathematics educator. in any ongoing conversation sequence, a turn is thus a transition ‘place’ where participants are trying to understand one another, through dialogue. this ‘place’ comes about dialogically and is dependent upon the exhibited understandings between the teachers and the mathematics educator, for example. a ttrp as an analytic object in a conversation sequence is essentially ‘local and situated’ (lynch, 2000, p. 530), meaning that it depends on how recipients respond, and how the current speaker reacts to the recipients’ utterances to the current turn as it unfolds. the various turns in the data excerpts are thus ‘places’ that reveal ways of talking and knowing peculiar to a site, which could be site a, the university, or site b, a school classroom. in this sense, site-related issues can be identified in the utterances of the mathematics educator or the teachers, whether they meet at a university, a school or elsewhere. repairs are ‘places’ in the conversation where either the mathematics educator or the teachers fix, modify or correct what they are saying (pomerantz & fehrer, 2011, p. 171). related to ‘repairs’, epistemic order (heritage, 2009) refers to ‘places’ in the conversation where either the mathematics educator or the teachers position or orient themselves relative to the mathematics inscribed in the design of the problem sets, based on their respective discursive practices. indexicality, in ethnomethodology parlance, refers to expressions or words whose sense cannot be determined without knowledge of the purposes of the user or the context of use. what the teachers and the mathematics educator say, as transcribed in the data excerpts, are reflexively related to the context, namely a cpd workshop (site c) meeting. this means that interactions between the mathematics educator (i.e. the first author) and the teachers have to be analysed in terms of a site c context at quite a local level of interaction, for example what occurs before and after the interaction, or ‘before and after an individual turn within an interaction’ (ingram, 2018). in a seminal discussion on indexicality, barnes and law (1976, p. 228) conclude that it is an essential and irreparable feature of all talk, scientific or otherwise. this includes ambiguity in the ways mathematics educators, mathematicians (byers, 2007) and teachers talk in general and about the same mathematics. byers notes the power of ambiguity and ambiguity in mathematics as a discipline. a mathematics educator and a teacher, for example, may use ‘minus’ and ‘subtract’ interchangeably. the phenomenon of indexical expressions or words is truly unavoidable (garfinkel, 1967). although indexical expressions or words (indexicals) ‘are of enormous utility, they are awkward for formal discourse’ (p. 5). a teacher may speak of ‘any number’ and not necessarily use ‘objective expressions’ (garfinkel, 1967, p. 5), for example real numbers or negative integers (see figure 1). teachers may not use ‘transpose’, but say ‘you bring it over’, a procedure related to solving an algebraic equation. in the case of the symmetry of an equation, they may say ‘minus y squared on that side and on this side’ and not use more objective expressions like ‘perform the same operation on both sides of the equality sign’ when dealing with equivalence (gattegno, 1974). another example is that of mathematics educators who view 1 – the identity for multiplication – as having a numerical equivalence as well as an algebraic or structural equivalence (see figure 2). teachers, on the other hand, do not speak in such formal ways about ‘the same mathematics’. the meaning and understandability of any indexical expression or word, rather than being fixed by some abstract definition, depend upon the environment in which it appears (maynard & clayman, 2003, p. 183), for example a school or university. ethnomethodologists do not treat indexical expressions or words as a nuisance to be remedied (maynard & clayman, 2003, p. 183). instead they view them as ‘resources’ for more objective expressions or words (ingram, 2018). research framework to answer the research question we need a research framework (niss, 2007) shown in figure 3. its rows and columns serve as a way to integrate the different strands of literature. in the top row (level 1) there is a general description of the empirical data incidents. level 2 and level 3 are the analytic levels, which contain incomplete details on the conversation analysis and ethnomethodology analytic tools and notions and the literature related to grade 8 and grade 9 problem sets. figure 3: skeletal outline of the research framework. methods of data analysis the unit of analysis, ‘site-related issues’, has two interconnected data sources (see level 2 and level 3 in figure 3). the first is in the design, that is, the wording of the particular grade 8 or grade 9 problem sets (see figure 2 and figure 1). the second is selected turns of transcriptions of conversation excerpts peculiar to the problem sets which served as the conversational anchors. in the case of the latter, we applied – in a selective way – the interchangeable conversation analysis and ethnomethodology tools and notions, namely turn transition relevance place (ttrp) or turn, repairs, epistemic order and indexicals (indexical expressions or words). this enabled us to identify and to analyse the unit of analysis and thus how and where deeper mathematical thinking occurred or not. as stated earlier, a ttrp can signal a repair or an epistemic order, and can exemplify indexical words or expressions used either by the mathematics educator or the teachers. by examining the ttrp or turns, it became possible to find ‘places’ in the transcriptions where the teachers and the mathematics educator talked about or commented on site a inputs inscribed in the design of each of the three problem sets. evidence of ‘keeping sites in sight’ are thus the turns or repairs in the conversation excerpts where the mathematics educator attempts to dmt in relation to the design of the problem sets based on teacher responses. to follow the analysis, the reader has to view the relevant problem sets above together with the numbered turns of the transcriptions of conversation excerpts. in the presented excerpts below, me stands for mathematics educator and t1, t2, t3, … for the different teachers. we analysed the three problem sets and associated conversation excerpts separately. ethical considerations the research ethics committee of the university of which this particular study is a part cleared the project (registration number 11/9/33). the project was also approved by the western cape education department through a memorandum of understanding between the university and the western cape education department. site-related issues grade 8: problem set a the 29 turns of transcriptions (see table 1) were selected because they contain evidence of the following interconnected site-related issues: indexical expressions and words related to signs as they appear in (1) number and (2) operations. we divided the transcriptions into turns, each of which reveal different ‘places’ where the mathematics educator tries to ‘keep sites in sight’. site a evidence is in the design of the problem set and in ways the mathematics educator talks to the teachers in the conversation excerpts. on the other hand, site b evidence is in ways the teachers speak about and understand the design or wording of the problem set. table 1: transcript related to grade 8: problem set a. in table 1 the mathematics educator orients the two teachers (t1 and t2) to the design of the problem set. in turn 3, the teacher (t1) mentions ‘add and subtract’. this is indexical because ‘add and subtract’ can only refer to ‘numbers’. turn 4 is a repair in which the mathematics educator aims to have the teacher modify or correct, that is, deepen her mathematical thinking in relation to an answer to ‘if we subtract a number from 5, the answer is less than 5’, being ‘always, sometimes or never true’. turn 4 and turn 6 are repairs on the part of the mathematics educator, which also reflect an epistemic order with respect to number and the minus sign (turn 6). ‘signs’ are indexical because they could refer to positive or minus signs as operational or structural signifiers. in turns 9–11 in particular, the teacher (t1) refers in an indexical way to two meanings of the minus sign. for example, in the case of ‘subtract’, the minus sign is an operational signifier, implying the binary operation of subtraction. ‘a negative’ and ‘a negative number’, on the other hand, can refer to a negative integer or a real number, in which case the minus sign is a structural signifier, assuming a unary operator meaning, for example 4. turn 10 and turn 12 are further repairs and an epistemic order in which the mathematics educator aims at dmt with respect to the ‘site a’ designer intentions, namely always, sometimes or never true. dmt happens in turn 13 where the teacher uses ‘minus’ to mean the binary operation of subtraction (‘5 minus negative 4’). in turn 13 the teacher (t1) moves from the indexical ‘a number’ to making a formal distinction by specifying a negative integer. in this turn ‘minus’, that is, the minus sign, is an operational signifier. turn 14 is the repair in which t2 corrects, that is, fixes, an earlier response as in turn 11. in turns 15–22, turn 17 can be viewed as a ttrp in which the mathematics educator directs the teachers’ attention to the site c boundary object – the caps policy document – and its relevant cognitive level 3 details, namely ‘complex procedures’. this is an instance of ‘keeping sites in sight’, in which the mathematics educator brings into view how and where the site a-designed problem set focusing on the two meanings of the minus sign connects with ‘complex procedures’. turn 20 reveals further instances of repairs in which the teacher modifies his or her thinking on number and the meanings of the minus sign as being either an operational or a structural signifier. the teacher uses ‘minus’ and ‘negative’ interchangeably. for example, turn 20 (‘it can be minus 5’), (‘it can be a negative number’), and (‘negative 4’) in which the minus sign has a structural meaning. in turn 20 ‘minus’ also has an operational meaning (‘5 minus negative 4’). in turn 20 to turn 29, there are thus further instances of epistemic order, in the form of repairs with respect to number and the minus sign. turn 20 reveals evidence of t1 clarifying the indexicality of ‘any’ number by giving the examples of positive and negative integers. in this turn t1 becomes more aware of two meanings of the minus sign, which this respondent did not consider before (see turn 14). t1 does not mention negative integers or real numbers explicitly, meaning that a resulting site-related issue is to use current indexical articulations as a ‘resource’ for a more formal discourse on meanings of the minus sign and responses to the particular dmt question. in summary, ‘keeping sites in sight’ from the analysis above is about the mathematics educator becoming aware of the indexical expressions and words in the conversation exchanges at the level of the turns and repairs, in addition to aligning the productive practice or dmt inscribed design in the problem set with the site c caps policy document guidelines on cognitive levels. grade 8: problem set b problem set b requires the learner or teacher to unscramble – distinguish between 1 and 0 when it comes to the equivalence relationships involving exponent laws – concepts associated with the equality symbol and meanings of minus and the minus sign (see table 2). the selected 28 turns of transcribed conversation excerpts contain evidence of the following three interconnected site-related issues: indexical expressions and words with respect to (1) the non-visual salience of y0 = 1, (2) transposing and performing the same operations on both sides of the equality sign as instances of equivalence and (3) meanings of the minus sign as it appears in (1) and (2). table 2: transcript related to grade 8: problem set b. in turns 1–12, the two teachers begin to unscramble the problem set with its key focus on the non-visual salience of y0 = 1. based on preceding conversation details not given, turn 1 is a repair, in which the mathematics educator asks the teachers, in words, if y2 – y2 = y0. in this turn, minus is used as an operational signifier, that is, it has to do with the operation of subtraction. as is evident from turn 2 and turn 3, the teachers understand the question. ‘minus’ is an indexical word. a more objective word corresponding to the meaning of ‘minus’ that could have been used is ‘subtract’. the point to note is the indexicality around the use of ‘minus’, based on seeing the minus sign in the design of the problem set. this repair is aimed at having the teachers modify, that is, deepen their mathematical thinking with respect to the zero, which in this case is an exponent (see turn 6), by contrasting it with subtracting two equivalent algebraic terms, for example y2, which also gives 0, which both teachers answer correctly. turns 8, 10 and 11 are evidence of the teachers becoming aware of how the inscribed design of the problem set scrambled equations and exponents. of interest in turns 13–24 are the indexical expressions and words the teachers use in arriving at the correct answer, prompted by the repair in turn 20 (see also turn 21), namely that the mathematical statement in the problem set is always true. turn 13 is a repair in which the mathematics educator ‘keeps in sight’ the design of the problem set by having the teachers modify, that is, deepen, their mathematical thinking. in the case of ‘you take it over,’ ‘you bring it over’ ‘take over,’ and ‘to get rid of a thing on the one side, you must do the opposite’ (turns 14, 17 and 22), the more objective word is ‘transpose’. in turn 14 the use of ‘minus’ has both an operational meaning – the operation of subtraction – and a structural meaning as in ‘minus y squared’. in this turn the teacher also uses ‘minus’ and ‘negative’ interchangeably. in one instance, t1 means the integer –1 and, in another instance, the operation of subtraction. in turn 1 the mathematics educator does the same, that is, uses ‘minus’ instead of the more objective ‘subtract’. this also happens in turn 21. the indexical words and expressions, ‘you’re going to minus on that side and on this side’ (turn 21), are evidence of the symmetry of an equation. more objective words are ‘you’re going to perform or do the same subtraction operation on both sides of the equality sign’. in the case of ‘the same formula, the same equation or whatever’ (turn 16), the more objective expressions and words are ‘equivalent expressions on either side of the equality sign, that is, an identity or, more comprehensively, an instance of equivalence (gattegno, 1974, p. 83). in turns 22–26, the mathematics educator ‘keeps sites in sight’ by pointing out how the design of the problem set connects with the cognitive level on ‘complex procedures’ specified in the caps document. teacher (t1) reiterates how and where the non-visually salient rule (y0 = 1) is inscribed in the design of the problem set, which t2 points out in turn 26. grade 9: simplifying an algebraic fraction we selected 21 turns of transcriptions of conversation excerpts (see table 3) because they reveal evidence of the following site-related issues: (1) indexicality around 1 as the identity element for multiplication, (2) the meaning of variable and (3) how finding the lcm (lowest common multiple) when simplifying the algebraic fraction relates to (1) and (2). table 3: transcript related to grade 9 algebraic fractions problem set. in the beginning the teachers together with the mathematics educator orient themselves to the design of the problem set. central to this design is a lexical support system for finding the lcm (see grade 9 problem set), which is evident from reading what the teachers say in all the different turns. turn 1, in the form of a question, is an epistemic order that aims at dmt with respect to numerical and algebraic equivalences in the case of 1 as the identity for multiplication. in turn 4, the mathematics educator’s response, there are indexical expressions that correspond to the more formal ‘algebraic equivalence of 1 as the identity element for multiplication’. in turn 6 the mathematics educator asks the teacher to give reasons for his answer. turn 6 and turn 8 exemplify further evidence of indexical words related to 1 as the identity for multiplication. turn 8 is, in fact, a repair wherein the teacher modifies, that is, deepens his thinking around the identity for multiplication. in turn 8 a teacher recognises the ‘kgv’ or lcm from the visually moderated sequences (see figure 2), based on his site c experiences. the turn 10 reference to ‘variable’ deserves attention. in the visually moderated sequences (lines a to f), related to simplifying this algebraic fraction, the x’s have ‘symbolic value,’ meaning there is no need to substitute numerical values for them. in the problem in figure 2, there is no need to substitute numerical values for the x’s. elsewhere in the mathematics curriculum, ‘variables’ do take on numerical values. this is ambiguous when it comes to site c teachers. in turn 13 and turn 15 the indexical ‘minus’ can more objectively be replaced with ‘subtract’. also evident in turns 13–17 are ways the teacher, following the lexical support system design of the problem set, attempts deepening her mathematical thinking by reasoning on the surface structure of algebra; for example, she refers to terms and common denominator. turn 18 is a repair in which the mathematics educator modifies the conversation by inserting ‘equivalence’ as a way to point to the algebraic equivalence of 1. turn 20 and turn 21 are repairs in which the mathematics educator ‘keeps sites in sight’ by using equivalent fractions examples to explain algebraic equivalence. also, in turn 20 he uses the pervading indexical ‘minus’ instead of ‘subtract’. at this point, as learned from barnes and law (1976), it is clear that indexical expressions and words are an essential and irreparable feature of all talk. turn 21 is an epistemic order in which the teacher makes clear the importance of dmt around 1 as the identity for multiplication, that is, line b. concluding remarks the purpose of this article was to explore site-related issues by focusing on shifting sites, beyond their physical meaning, in the case of cpd. we started by sketching three physical sites or workplaces, namely the university (site a), the school classroom (site b), and a venue off or on campus (site c), each having attendant ways of knowing, talking, and working with respect to mathematics. site c is significant because it is a place for mutual engagement between mathematics educators and teachers around mathematics. data excerpts emanating from site c focused on toolkit problem sets tested on an ‘always, sometimes, never true’ basis in the case of number and operations, visual and nonvisual salience of algebraic transformations involving exponent laws, and semantic or verbal comments on written procedures for finding the lcm when simplifying an algebraic fraction. these design features of the problem sets served as conversational anchors with the participating teachers. in the design of each of these problem sets, there was evidence of site a and site b inputs. applying the conversation analysis analytic tools of turn transition relevance places, repair, epistemic order, and indexicality from ethnomethodology enabled us to find evidence of sites based on site-related issues in the selected data excerpts. in these instances, the notion of a site took on a theoretical meaning, independent of a physical site where the conversations occurred. in other words, meetings with the teachers could, in fact, have occurred perfectly neutrally at a university or at a school. answers to the research question – ‘what are the site-related issues when it comes to analysing toolkits-based conversations peculiar to a cpd initiative between mathematics educators and a group of teachers?’ – have implications. at a general and practical level, university-based mathematics educators have to ‘keep sites in sight’ when designing toolkits. for example, the toolkits need to incorporate policy document cognitive levels and there needs to be a focus on these cognitive levels when interacting with teachers. policy documents as boundary objects have a stronger link to what happens in classrooms, compared to toolkits that university-based mathematics educators bring to teachers. this is an important way to narrow the distance between university and school. simultaneously, when ‘doing interaction’ (ten have, 1990, p. 24), that is, using the designed toolkits with the teachers, mathematics educators have the task of anticipating ‘sites moments’ and issues peculiar to conversations across two discursive practices, namely university-based mathematics education and school mathematics teaching. the indexical expressions or words that emerged from the analysis have implications for cpd work. in the two grade 8 problem sets there were those associated with meanings of the minus sign and the word ‘minus’ as they appeared in number, operations, transposing and a symmetry view of equations, and the equality sign and equivalence. in the grade 9 case, there was the analogy between numerical and algebraic equivalences when simplifying the particular kind of algebraic fraction. what should be noted in all these instances is the power of ambiguity and the ambiguity. for example, ‘bring it over’ or ‘take it over’ have a kinaesthetic imagery for the more objective or mathematically correct word ‘transpose’. finally, to be effective, university-based mathematics educators in all cpd initiatives have the task of keeping sites ‘in sight’ whenever they interact with teachers. if not, they are bound to ‘lose sight’ of the realities of teachers and their classrooms. in turn they need to gain insights around being mindful of how the university and the school relate, each with its peculiar ways of talking and working around the same mathematics. acknowledgements competing interests we declare that we have no financial or personal relationships that may have inappropriately influenced us in writing this article. authors’ contributions the article arose from an ongoing continuous professional development project in which c.j. is the principal investigator, r.s. a research fellow and f.g. a collaborator. f.g. was responsible for data collection and analysis. f.g. contributed to the design of the toolkits of the project and to the writing process. r.s. also contributed to the writing process. funding information this research is supported by the national research foundation (nrf) of south africa under grant number 77941. standard bank supported the work with grade 8 and grade 9 teachers. data availability statement data sharing is not applicable to this article as no new data were created or analysed in this study. disclaimer any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the nrf or standard bank. references barnes, b., & law, j. 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(1998). questions and prompts for mathematical thinking. derby: association of teachers of mathematics. article information author: erica d. spangenberg1 affiliation: 1department of science and technology education, university of johannesburg, south africa correspondence to: erica spangenberg postal address: p o box 5584, krugersdorp west, 1742, south africa dates: received: 30 may 2012 accepted: 04 nov. 2012 published: 06 dec. 2012 how to cite this article: spangenberg, e.d., (2012). thinking styles of mathematics and mathematical literacy learners: implications for subject choice. pythagoras, 33(3), art. #179, 12 pages. http://dx.doi.org/10.4102/ pythagoras.v33i3.179 copyright notice: © 2012. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. thinking styles of mathematics and mathematical literacy learners: implications for subject choice in this original research... open access • abstract • introduction and background • thinking styles • the nature of mathematics and mathematical literacy • research design    • purpose of the study    • research methods and procedures       • sample       • data collection: document analysis, interviews and questionnaires       • data analyses       • ethical considerations       • reliability       • validity • main findings    • findings from the document analysis    • findings from the interviews    • findings from questionnaires • discussion • conclusion • acknowledgements    • competing interest • references • appendix 1    • teachers’ interview questions • appendix 2    • sample of learners’ thinking style questionnaire abstract top ↑ in this article i report on research intended to characterise and compare the thinking styles of grade 10 learners studying mathematics and those studying mathematical literacy in eight schools in the gauteng west district in south africa, so as to develop guidelines as to what contributes to their subject choice of either mathematics or mathematical literacy in grade 10. both a qualitative and a quantitative design were used with three data collection methods, namely document analysis, interviews and questionnaires. sixteen teachers participated in one-to-one interviews and 1046 grade 10 learners completed questionnaires. the findings indicated the characteristics of learners selecting mathematics and those selecting mathematical literacy as a subject and identified differences between the thinking styles of these learners. both learners and teachers should be more aware of thinking styles in order that the learners are able to make the right subject choice. this article adds to research on the transition of mathematics learners in the general education and training band to mathematics and mathematical literacy in the further education and training band in south africa. introduction and background top ↑ this article focuses on the characterisation of the thinking styles of grade 10 mathematics and mathematical literacy learners in eight schools in gauteng west, south africa. since 2006, learners have had the choice to study either mathematics or mathematical literacy in grades 10–12 (department of education [doe], 2003a).a subject is defined as ‘a specific body of academic knowledge’ where ‘knowledge integrates theory, skills and values’ (doe, 2003c, p. 6). in this article i refer to a subject as a particular area of study that schools offer, for example accounting, english, mathematics or mathematical literacy. learners study mathematics from grade 4 to grade 9; mathematical literacy is a new subject which can only be studied in grade 10 to grade 12. initially, parents and teachers guide learners in their subject choices, based on factors such as future career, language, socio-economic background, interests and achievements in the lower grades (spangenberg, 2008). however, in the higher grades, learners’ own thinking styles influence their preference for the different subjects (borromeo ferri, 2004). research reveals that thinking styles play an important role in teaching and learning (borromeo ferri, 2004; cilliers & sternberg, 2001; grigorenko & sternberg, 1997; moutsios-rentzos & simpson, 2010; sternberg, 1990; sternberg & wagner, 1992a; zhang, 2006). in particular, sternberg and grigorenko (1993) found that certain thinking styles correlated positively to a learner’s success in a variety of academic tasks, whereas other thinking styles tended to correlate negatively to success in the same tasks. van der walt (2008) also noted that a learner’s thinking style is a factor that influences the effective learning and teaching of mathematics and could predict achievement of mathematics in school. therefore, i argue that learners’ thinking styles could affect their choice to study either mathematics or mathematical literacy. by establishing which thinking style is associated with learners in each of the two subjects, one should be able to guide them, their parents and teachers in making more informed decisions with regard to the choice between mathematics or mathematical literacy as a subject. a study that characterises and compares the thinking styles of grade 10 learners taking mathematics and those taking mathematical literacy is new to south africa, although there have been investigations into thinking styles at both teacher and learner levels (cilliers & sternberg, 2001, de boer & bothma, 2003). in addition, moutsios-rentzos and simpson (2010) conducted a study in greece on the thinking styles of university students, zhang (2006) asked ‘does student–teacher thinking style match/mismatch matter in students’ achievement?’ in hong kong, and borromeo ferri (2004) conducted an empirical study on mathematical thinking styles of 15–16-year-old learners in germany. all these researchers referred to the thinking styles inventory of sternberg and wagner (1992b); i too have used the precepts of sternberg’s theory in order to characterise and compare the thinking styles of learners taking mathematics and those taking mathematical literacy in south africa. this article reports on part of a broader study that i had previously conducted on the placement of grade 10 learners to provide advice for learners, parents and teachers in terms of choosing between mathematics and mathematical literacy (spangenberg, 2008). the question arising from the above discussion is: which thinking styles are associated with learners taking mathematics and those taking mathematical literacy? hence, i conducted a literature inquiry with regard to the thinking styles and the nature of mathematics and mathematical literacy. thinking styles top ↑ a style is a particular procedure or manner by which something is done or a specific way or tendency unique to a person (soanes, 2002). in particular, zhang and sternberg (2000) define a thinking style as ‘a source of individual differences in academic performance that are related not to abilities but how people prefer to use their abilities’ (p. 469). whereas a learning style refers to a way of approach to learning (kolb, boyatzis & mainemelis, 1999), a thinking style refers to a particular act, idea, tendency or way of thinking about the execution of a task in the learning process (sternberg, 1994). thus, a learning style is how a learner receives information, whilst a thinking style is how a learner processes information and reflects on ideas in their mind. for cilliers and sternberg (2001, p. 14) a thinking style is a ‘preference’ for using abilities in certain ways during processing.the theoretical basis for this study is based on sternberg’s theory of mental self-management. sternberg and wagner (1992b) developed a thinking style inventory consisting of 13 thinking style dimensions divided into five categories, namely functions, forms, levels, scopes and learning (sternberg, 1990). functions refer to the ‘basic types of thinking styles’, including legislative (preference for creativity), judicial (preference for judging) and executive (preference for implementing rules and instructions) thinking styles. forms are ‘general ways’ in which learners ‘approach their environments and the problems the environment presents’ including hierarchic (preference for having multiple prioritised objectives), anarchic (preference for flexibility), monarchic (preference for focusing on only one goal) and oligarchic (preference for having multiple equally important targets) thinking styles. levels refer to the ‘amount of engagement individuals prefer in a given activity’ including local (preference for details and the concrete) and global (preference for general and the abstract) thinking styles. scopes are ‘stylistic variables which divide learners into two basic personality types, including internal (preference for working alone) or external (preference for working in a group) thinking styles. finally, learning explains ‘the methods and rules by which learners solve problems’ including liberal (preference for novelty and originality) and conservative (preference for conformity) thinking styles (richmond, krank & cummings, 2006, p. 59). each of these thinking styles has its own characteristics, as represented in table 1. table 1: thinking style dimensions. sternberg (1990, p. 368) noted that a teacher may not appreciate a learner’s ability and may view him or her as ‘slow’ or ‘behind’ because of a difference in thinking style between the learner and the teacher. conversely, van der walt (2008) argued that learners’ thinking styles may contribute to their inability to solve mathematical problems, even though they have the necessary knowledge. however, research conducted by sternberg (grigorenko & sternberg, 1988, 1997; sternberg, 1990, 1994, 1997) has revealed that teaching and learning can improve if teachers give more attention to thinking styles. both learners and teachers bring their own individual characteristics and thinking styles to the learning environment (zhu, 2011). these thinking style preferences lead to learning style preferences and in turn determine learners’ dominant cognitive modes, that is, the ways in which they communicate and receive information. more specifically, ‘cognitive functions are accommodated when teaching activities are constructed to comply with a learner’s preferred mode of thinking’ (de boer & bothma, 2003, p. 1). the nature of mathematics and mathematical literacy top ↑ in south africa, mathematics and mathematical literacy relate to each other, but differ in terms of their nature and aims. in particular, mathematics ‘enables creative and logical reasoning about problems in mathematics itself’, which ‘leads to theories of abstract relations’ (doe, 2003c, p. 9). on the other hand, mathematical literacy equips and sensitises learners with an understanding of the relevance of mathematics in real-life situations (doe, 2003b, p. 9). its purpose is to apply mathematics to make sense of the world. mathematical literacy was specifically introduced as an intervention to improve numeracy skills of south african citizens in response to poor performance in mathematics in the past (bansilal, mkhwanazi & mahlabela, 2012).learners who can think in terms of ‘symbolic representation or abstract conceptualization – thinking about, analyzing, or systematically planning’ and in terms of the concrete reality, including mathematical modelling and more applied mathematics, should achieve in mathematics, whereas learners who can only think ‘through experiencing the concrete, tangible, felt qualities of the world, relying on our senses and immersing ourselves in concrete reality’, and not in terms of abstraction, should achieve in mathematical literacy (kolb, boyatzis & mainemelis, 1999, p. 3). moreover, the grade 10–12 mathematics syllabus, as set out in the national curriculum statement (doe, 2003c) is a purely academic subject which focuses more on content that the learners have to deal with, memorise and reflect on, as opposed to mathematical literacy which is a practical subject where learners learn practical skills that will enable them to find concrete solutions to numeric, spatial and statistical problems associated with the everyday challenges of life. in mathematics attention has to be paid to specific details and, as bohlmann and pretorius (2008, p. 43) claim, ‘the conceptual complexity and problem-solving nature of mathematics make extensive demands on the reasoning, interpretive and strategic skills of learners.’ mathematics is an abstract, deductive discipline that is required in the scientific, technological and engineering world. according to venkat (2007): emphasis is laid on abstract rather than concrete concepts, on intra-mathematical connections rather than mathematics-real-world connections, on rigour and logic rather than interpretation and critique, and on knowledge itself, as well as applications of knowledge. (p. 77) in comparison to mathematics, which is more abstract in nature, most definitions of mathematical literacy focus on the concrete dimension of mathematics with the context determining the content to be learned. learners use real-life situations to gain new knowledge; thus, a learner who tends to process information in a concrete way should achieve in mathematical literacy. gal (2009) pointed out that mathematical literacy focuses on the relevance of learned knowledge to everyday life and linked it to diverse real-world contexts, whilst frith and prince (2006) stated that people are mathematically literate if they have the ability to express quantitative information in a verbal and visual form. mathematical literacy creates a consciousness about the role of mathematics in the modern world and is therefore driven by practical applications. the subject develops the ability and confidence of learners to think numerically in order to interpret daily situations (doe, 2003b). according to zhang (2002, p. 179) ‘students who reasoned at a higher cognitive developmental level tended to use a wider range of thinking styles than students who reasoned at a lower cognitive developmental level’. therefore, learners taking mathematics, which focuses on ‘creative and logical reasoning’ (doe, 2003c, p. 9), likely utilise different styles of thinking when reasoning than learners taking mathematical literacy. ideally, mathematical literacy learners should be able to reason by communicating, either verbally or in written form, because mathematical literacy uses everyday language for ‘practical relevance and applications’, which may be easier for learners to understand, whereas mathematics uses highly technical mathematical language for ‘further math learning’ (graven & venkat, 2007, p. 69). in particular, venkat, graven, lampen, nalube and chitera (2009) noted that mathematical literacy promotes ‘thinking as communication’ which ‘consists of acts such as asking questions, hypothesizing, finding counter-arguments and drawing conditional conclusions within a situation’ (p. 48). in contrast, mathematics learners should use higher levels of visual-spatial reasoning and abstract thinking. hence, to achieve in mathematics, the ‘use of symbols and notations’ and ‘mental processes that enhance logical and critical thinking, accuracy and problem solving’ and ‘mathematical problem solving’ (department of basic education [dbe], 2011b, p. 8) should be emphasised as the content is in an abstract and generalisable form. mathematics deals with concepts as ideas or abstractions which learners have to bring together to solve a mathematical problem to enable them ‘to understand the world’ (dbe, 2011b, p. 8). in contrast, mathematical literacy deals with ‘making sense of real-life contexts and scenarios’ and ‘mathematical content should not be taught in the absence of context’ (dbe, 2011a, p. 8). table 2 presents the differences between grade 10 mathematics and grade 10 mathematical literacy with regard to the content as prescribed by the national curriculum statement (doe, 2003b, 2003c). table 2: differences in content between grade 10 mathematics and mathematical literacy. due to the differences in the natures of mathematics and mathematical literacy, it is expected that learners require different thinking styles to achieve in these subjects. the identification of the different thinking styles will contribute towards informing teachers, parents and learners, in an objective manner, about the choice of either mathematics or mathematical literacy as a subject. research design top ↑ the research question is intended to characterise and to compare the thinking styles of learners taking mathematics and of learners taking mathematical literacy. as noted above, learners’ thinking styles can influence their subject choices. the lack of guidelines from the doe with regard to the placement of grade 10 learners in either mathematics or mathematical literacy convinced me to research for a practical solution to guide learners, parents and teachers to make informed subject choices and, thus, to adopt a pragmatic philosophy that is concerned with ‘what works’ and ‘what provides solutions’ in an authentic situation (creswell, 2003, p. 11). i utilised both quantitative and qualitative techniques in the study because a combination of the two techniques provides a more in-depth knowledge of the theory and practice (creswell, 2003). purpose of the study the purpose of this article is to establish which thinking styles are associated with learners studying either mathematics or mathematical literacy, so as to develop guidelines that will contribute to the subject choice of either mathematics or mathematical literacy by grade 10 learners, and eventually to better performance in the two subjects. i established teachers’ perceptions regarding the differences in thinking styles between learners selecting mathematics and those selecting mathematical literacy as a subject through a qualitative technique whilst i compared the thinking styles of learners quantitatively. the following research questions were addressed in the qualitative approach: • which thinking styles of learners are you using to advise learners on their choice between mathematics and mathematical literacy? • which characteristics would you attribute to learners who have chosen mathematics? • which characteristics would you attribute to learners who have chosen mathematical literacy? in order to compare the thinking styles of grade 10 learners taking mathematics and those taking mathematical literacy, the following hypothesis was interrogated in the quantitative approach: • there are significant differences between a learner’s thinking style dimensions and the subject they choose, either mathematics or mathematical literacy. research methods and procedures sample a convenience stratified sampling technique (creswell, 2003) was used to select teachers and learners from secondary schools in a single district in south africa, namely gauteng west. the area was chosen because i worked in the area and had easy access to schools. before sampling, the population of 32 secondary schools in the district was divided into types: there were nine afrikaans-medium suburban schools, three english-medium suburban schools, three rural schools and 17 township schools, all of which were heterogeneous in respect of learners studying mathematics and mathematical literacy. thereafter, the population was sampled within each stratum; i chose eight schools: two afrikaans-medium suburban schools, one english-medium suburban school, one rural school and four township schools. these schools were selected to ensure that all types of school in the district were represented proportionally and because i had good working relations with them. the sample also included eight mathematics teachers and eight mathematical literacy teachers, one teacher from each subject (mathematics and mathematical literacy) from each participating school. the teachers were selected on a voluntary basis and they granted me the right to interview them. all the grade 10 mathematics and mathematical literacy learners from each selected school were included in the sample, a total of 1046 grade 10 learners. moreover, i could only utilise a naturally formed group, namely learners in a classroom setup, for this research, which justifies a convenience sample (creswell, 2003, p. 162).the 16 teachers who were interviewed were selected on the basis that they had at least one year’s teaching experience in grade 10 mathematics and/or mathematical literacy, thus ensuring that they had the necessary knowledge and experience to teach these subjects. the teachers differed in age and both men and women were included in the sample. the teachers participated on a voluntary basis. in terms of the quantitative phase of the study, 1046 learners completed the questionnaire on thinking style dimensions, indicating their choice of either mathematics or mathematical literacy. of these, 56.2% (588) selected mathematics. data collection: document analysis, interviews and questionnaires both a qualitative and a quantitative design were used to collect the data through document analysis, interviews and questionnaires (creswell, 2003). document analysis of the content for grade 10 mathematics in comparison with the content for grade 10 mathematical literacy, as prescribed by the national curriculum statements of south africa, was conducted to establish whether different thinking styles would be demanded in mathematics and mathematical literacy. this document analysis was used to supplement the data obtained from the other methods (bell, 1995). unfortunately, access to learners’ written work and assessment documents was denied due to the integrated quality management system at the schools.i followed a qualitative approach during the first phase of the research by conducting one-to-one interviews with teachers through a semi-structured questionnaire (see appendix 1). the aim was to ascertain their perceptions regarding the characteristics and differences in thinking styles between learners selecting mathematics and those selecting mathematical literacy as a subject. during the second phase of the research, i used a quantitative research method: a survey (structured questionnaires) amongst learners. the aim of this phase was to compare thinking styles of a mathematics learner compared to those of a mathematical literacy learner. the questionnaire (see appendix 2) was based on an existing standardised instrument, the thinking style inventory of sternberg and wagner (1992b), which aimed to determine the different strategies used by learners to solve problems, execute tasks or projects and make decisions. the questionnaire consisted of 13 thinking style dimensions divided into five categories. for each characteristic (e.g. self-management function: legislative) there were eight questions on a 1–7 point likert scale, with 1 = not at all well and 7 = extremely well. thus, for 13 thinking style dimensions, there were 104 questions. scores were then averaged over each characteristic. the characteristic associated with each of these appears in table 1 (sternberg, 1994). data analyses tesch’s protocol of data analysis (creswell, 1994) was used to analyse the data from the interviews for the qualitative inquiry. firstly, each interview was audio recorded and transcribed. secondly, the transcriptions were read to obtain a holistic perspective, after which relevant answers were separated from irrelevant answers. thereafter, saldana’s (2009) method of coding was used. according to this method, coding is a ‘heuristic exploratory problem-solving technique without specific formulas to follow’ (saldana, 2009, p. 8) where a code in qualitative inquiry refers to a ‘word or short phrase that symbolically assigns a summative, salient, essence-capturing, and/or evocative attribute for a portion of language-based or visual data’ (p. 3). after coding was applied to the data, codes sharing the same characteristics were grouped into sub-themes. similar sub-themes were then grouped together to form concepts or themes (saldana, 2009). table 3 summarises the themes, sub-themes and codes emerging from the data qualitative analysis. table 3: themes, sub-themes and codes. the statistical package for the social sciences, edition 15 (spss 15) was used in collaboration with the statistical consultation service at the university concerned, to conduct the quantitative data analyses. the quantitative data analysis included univariate descriptive measures, namely frequencies and percentages and cross-tabulations of the categorical variables and descriptive statistics of the scale variables (specifically the thinking style dimensions). inferential statistics, namely independent sample t-tests (for the scales variables) and chi-squared tests of independence (for the categorical variables), were used in order to identify significant differences between the thinking styles of learners taking mathematics and those taking mathematical literacy. a significance level of 0.05 was assumed throughout. the internal reliability of each of the thinking style dimensions was determined by the cronbach α coefficient.to analyse the research question, descriptive statistics for each of the group variables (the independent variable being mathematics or mathematical literacy and the dependent variable thinking style) were used. for the purposes of classifying participants into thinking style categories, each participant’s highest score for a given category was chosen to represent the category. for example, if a learner’s scores on the 7-point likert scale on average were for legislative = 3.2, judicial = 6.2 and executive = 4.7, they would be categorised as judicial thinkers. this process was performed on all five thinking style categories. ethical considerations the ethical committee of the education department at the university concerned granted ethical clearance for the study and permission was obtained from the gauteng doe, gauteng west district and the schools in gauteng west to conduct the research.all the participants’ contributions were recognised by proper referencing. the rights and interests of the participants were protected and sensitivity was shown towards them based on common trust (mouton, 2001). furthermore, all information supplied was treated with confidentiality and the outcomes of the research made available on request. tape recordings and data were kept under lock and key and were destroyed after completion of the research study. sternberg granted permission for the usage of the thinking style inventory of sternberg and wagner (1992b) and the intellectual property rights were recognised. furthermore, data obtained was personally analysed by means of statistical verified methods and procedures (eiselen, 2006). reliability the questionnaires were shown to colleagues for comments and responses, to ensure that the constructs were clearly conceptualised. consequently, the questionnaires were amended with regard to timeframes, language, terminology, readability and clarity and piloted with one class group of 30 learners at a school that was not part of the sample before they were administered to the eight schools in the sample. the purpose was to ensure coherency and consistency of the questions. the questionnaires were administered under examination conditions.the internal reliability of each thinking style dimension was determined by using the cronbach α coefficient, after which descriptive statistics, namely averages and standard deviations, of each dimension were used. the cronbach α coefficient is recommended for large samples where items are not scored right or wrong and was thus suitable for this study of 1046 learners. a score of 0.7 and higher was assumed as reliable for this study. the internal consistency of the five thinking style categories is presented in table 4. table 4: internal reliability of the thinking styles dimensions. some of the items on the original thinking style inventory could be ignored, namely the judicial, oligarchic, anarchic, global, local, internal and conservative thinking styles, because of the findings from the reliability analyses. however, due to the standardisation of the instrument, no item was deleted and the dimensions, as identified by sternberg and wagner (1992b), were calculated. validity the characteristics and differences of the thinking styles of the mathematics and mathematical literacy learners were measured by means of the thinking style inventory of sternberg and wagner (1992b), which is an existing standardised instrument that had already complied with all validity aspects. therefore, no items were omitted and the dimensions as identified by sternberg and wagner were calculated. main findings top ↑ findings from the document analysis the basic principles of numeracy laid out in the general education band develop in grade 10 mathematics, using more symbolic methods, such as numeric sequences and series (doe, 2003c). in comparison, grade 10 mathematical literacy does not include number systems, numeric or geometric patterns, but focuses on using numbers within contexts relevant to daily life, such as profits and losses, budgets, loans, commission and banking (doe, 2003b). furthermore, grade 10 mathematics includes mathematical modelling, linear, exponential and quadratic equations, linear inequalities, products and factorisation, trigonometry, coordinate geometry and euclidean geometry, which do not appear in the grade 10 mathematical literacy curriculum (doe, 2003c), as displayed in table 2.from the above, it is evident that there is more mathematical content in the grade 10 mathematics curriculum than in the grade 10 mathematical literacy curriculum. hence, learners who choose mathematics are likely to do many things with a ‘hierarchic thinking style’ (sternberg, 1990, p. 369). mathematics requires that learners be able to think in terms of ‘symbolic representation or abstract conceptualization’ (kolb, boyatzis & mainemelis, 1999, p. 3). thus, learners should have a preference to create, discover and design (‘legislative thinking style’) (sternberg, 1990, p. 38). this viewpoint is also supported by bohlmann and pretorius (2008, p. 43), who claimed that ‘the conceptual complexity and problem-solving nature of mathematics make extensive demands on the reasoning, interpretive and strategic skills of learners’. grade 10 mathematical literacy focuses more on contexts relevant to daily life. thus, learners who prefer to work with other people, focus on the outside and are interdependent, with an ‘external thinking style’ (sternberg, 1990, p. 38) should rather choose mathematical literacy. findings from the interviews from the 16 personal interviews conducted with teachers, i could find no evidence that teachers consider the thinking styles of learners when they advise learners on their choice between mathematics and mathematical literacy. rather, teachers indicated that they use three other methods. in the following protocols the names of teachers are pseudonyms to protect their identity. all the protocols are from spangenberg (2008, pp. 229–242).firstly, learners’ marks obtained in grade 9 are used as an indication of which subject to take. matle mentioned that he is guided by ‘the mark that the learner obtained in grade 9’. bana noted that ‘if you performing poor in natural sciences you can see that you are not going to do maths. you are going to do maths literacy’. samuel added: [i]f he gets 60% in grade 9 and above then basically you are allowed to choose maths. anything less than 60% you wouldn’t have a choice … that is like a policy in our school at the moment in time. (spangenberg, 2008, p. 229) secondly, tests guide teachers when advising learners in their subject choices. jack stated that ‘we give them aptitude test in terms of mathematics … if he pass … we just place him or her’ and mary alluded to ‘some test that they do to test their ability to do maths as it test their ability to do maths lit … we set an internal maths paper and we use that as a guide’. lastly, subject combination packages and future careers are also indicators of the subjects learners should take. rosen claimed that ‘if a learner chooses a subject package, then the package makes provision for him in a certain direction’, adding that for the science then we have included the pure maths in that package and if it has accounting we have include the pure maths in the accounting package and in all the other courses whereby the learner has the commerce fields or learning areas we have included the maths literacy in that package. jack referred to four streams: the first steam is for maths and science. the second one is maths and accounting. the third one is maths literacy and economics. the fourth one is maths literacy and history … this learners who are doing maths literacy, most of them, they must consider this career opportunities of law, human resources, those that are not attached with mathematics. bana explained that: i look at his or her ambitions, whether which career does he or she want to follow. if she wants to be a scientist, then i say okay maths is good for you. if you want to be a lawyer i say maths literacy is good for you. in addition, sam argued that ‘some of the learners choose it because of the career that they want to go in’. teachers could clearly distinguish between the characteristics of each type of learner. from their observations and perceptions, teachers described learners who choose mathematics, firstly, as being interested in the subject. sam explained that it will depend … on the interest of the learner. maybe if he likes working with numbers, he can choose the mathematics … but the learners that choose maths … you see that these learners are interested in the subject. these learners are perceived as self-disciplined and diligent. santa mentioned that ‘the learner must be dedicated’, supported by shisha that ‘they tend to be the more conscientious student’. christine argues further that mathematics learners are motivated and focused. she noted that ‘i would choose a learner who’s able to focus … to concentrate’. zane added that these learners are hardworking by commending that ‘most of them are quite conscientious workers, enthusiastic workers’ and carmen supported that ‘those that choose mathematics tend to be those that are very hard workers’. teachers also described mathematics learners as having the ability to memorise in a logical manner. geoff claimed that mathematics is for the brainy … that there are those that are very intelligent and as a result they need to do mathematics, because they can think far to their ability and then mathematical literacy is like it’s made for those who are less able to do literacy … but those who are very intelligent, they have to do mathematics. shisha add that ‘they are able to work logically’ and noted that ‘they understand theorems and them they are able to apply them immediately’. lastly, teachers described learners taking mathematics as independent workers, able to work on their own, thus displaying an ‘internal thinking style’ (sternberg, 1990, p. 38). sam stated that mathematics learners are ‘very independent … they can work on their own … they are very disciplined learners who are taking maths seriously’. on the other hand, teachers characterised mathematical literacy learners as having good general knowledge. samuel noted that for mathematical literacy learners ‘the most important thing is the knowledge of the outside environment that they are in’. he further continued that these learners are socially adaptable, by arguing that how you can be able to adapt in your everyday life and how you adapt in the environment you in … you don’t have to be a good academic learner to be a good social person being … you just need to be well equipped to handle everyday experiences, have the grasp or basically knowing what’s happening around you and being interesting. ilze described mathematical literacy learners as entrepreneurial, perceiving a mathematical literacy learner as ‘a child who can stand on his own feet, a child who wants to start his own business’. she further added that these learners have an interested in life and people, noting that ‘you learn him about the life’ and ‘more interested in the human being’. shisha mentioned that mathematical literacy learners are able to express themselves, thus displaying an ‘external thinking style’ (sternberg, 1990, p. 38) by claiming that ‘a learner must be open-minded. usually we’re using the open-minded, especially the history learners, because they are used to expressing themselves’. in contrast, however, other teachers described these learners as lacking discipline. hannah perceived a mathematical literacy learner as ‘a guy who does not have discipline’ and bana added that ‘they are not willing to learn. they are not willing to participating in classes’. hannah continued to describe a mathematical literacy learner as having a short attention span, by noting that ‘he disappears in class, he loses concentration. he does also not have the ability to concentrate’ and carmen argued that mathematical literacy learners lack an interest in mathematics, by stating that mathematical literacy learners are ‘those that for any reason don’t like mathematics. they haven’t enjoyed mathematics’. zane agreed that a mathematical literacy learner is one who just only doesn’t want to choose maths, but he is forced to do something in the maths field now, because it is compulsory. so, he chooses maths literacy. lastly, teachers described mathematical literacy learners as having a fear of mathematics. rosen mentioned that ‘some learners will choose it because of their fear for pure maths’ and matle agreed: ‘fear … many learners they’ve got this mentality that mathematics is a difficult subject … there is a possibility that i will fail.’ it is, however, important to note that the above-mentioned findings were based on teachers’ perceptions regarding the characteristics and differences in thinking styles between learners selecting mathematics and those selecting mathematical literacy as a subject. the findings should thus not be generalised to all grade 10 mathematics learners or grade 10 mathematical literacy learners findings from questionnaires statistical analysis was undertaken to investigate whether differences between learners’ thinking style dimensions and which subject they are studying (mathematics or mathematical literacy) were significant. the precise means (x) and standard deviations (sd) pertaining to the thinking style dimensions (dependant variable) of learners taking mathematics and mathematical literacy (independent variable) (according to the questionnaires constructed for this study) are indicated in table 5. table 5: descriptive statistics pertaining to the thinking style dimensions of learners taking mathematics and mathematical literacy. the findings support the hypothesis about the comparisons between learners’ thinking style dimensions and which subject they are studying (mathematics or mathematical literacy). table 6 shows the value of the test statistics in terms of the null hypothesis (t), the p-value for each case and the degree of freedom (df) where the variances were accepted. table 6: the independent sample test of learners taking mathematics and mathematical literacy and their different thinking styles. learners studying mathematics and those studying mathematical literacy differed significantly at a 95% level in terms of eight of the 13 sternberg thinking style dimensions, namely legislative (p = 0.000 < 0.05), executive (p = 0.000 < 0.05), judicial (p = 0.000 < 0.05), hierarchic (p = 0.000 < 0.05), anarchic (p = 0.043 < 0.05), local (p = 0.000 < 0.05), internal (p = 0.001 < 0.05 and liberal (p = 0.002 < 0.05).it was found that mathematics learners (average = 5.1949) are more likely than mathematical literacy learners (average = 4.8973) to like following instructions, to do whatever they are told to do and to prefer fixed structures (executive thinking style). mathematics learners (average = 5.3095) are also more likely than mathematical literacy learners (average = 5.0713) to design and do things in their own ways (legislative thinking style). furthermore, mathematics learners (average = 4.6395) are also more likely than mathematical literacy learners (average = 4.4194) to be critical, in the sense that they like to judge people and to evaluate things (judicial thinking style). although the mathematics learners on average obtained a slightly higher score than the mathematical literacy learners in terms of each of the three above styles, it was found that the legislative thinking style (average = 5.3095) on average measured the highest for the mathematics group, followed by the executive thinking style (average = 5.1949), and lastly by the judicial thinking style (average = 4.6395). even though these findings appear contradictory, it is important to note that, according to sternberg (1990), the mind performs each of the legislative, executive and judicial functions, but one of these tends to be more dominant in a person. in terms of the four forms of cognitive self-management distinguished by sternberg (1990), namely hierarchic, anarchic, monarchic and oligarchic, the two groups only differed significantly in terms of the first two: mathematics learners (average = 5.1173) are more likely than mathematical literacy learners (average = 4.8001) to do many things at the same time and to set priorities pertaining to what to do, at what time to do it and how much time and energy to spend on it (hierarchic thinking style). in contrast, mathematical literacy learners (average = 4.6441) are more likely than mathematics learners (average = 4.6256) to do one thing at a time and spend almost all their energy and resources on it (monarchic thinking style). in terms of the two levels of self-management, namely local and global, it was found that mathematics learners (average = 4.7901) are more likely than mathematical literacy learners (average = 4.5342) to find detail, specifications and concrete examples to be important (local thinking style). as far as the scope of self-management is concerned, described by sternberg (1990) as being either internal or external, it was found that a mathematics learner (average = 4.7010) is more likely than a mathematical literacy learner (average = 4.4961) to prefer to work alone, to focus inward and to be independent (internal thinking style). lastly, in terms of the distinction made by grigorenko and sternberg (1997) between two ways of learning, namely liberal and conservative, it was found that a mathematics learner (average = 5.5013) is more likely than a mathematical literacy learner (average = 4.8691) to do things in new ways and to deviate from traditions (liberal thinking style). the findings did not correspond with those of sternberg and grigorenko (1993), who found that the judicial and legislative styles correlated positively to academic achievement, whereas the executive, legislative, oligarchic and liberal styles tended to correlate negatively to academic success. it is, however, important to note that thinking styles are interrelated (garcia & hughes, 2000, p. 413). one must take into account that thinking style interrelationship is complex, since it is influenced by many variables, such as education, subject, age and gender (sternberg & wagner, 1992a), which should be researched in depth. discussion top ↑ this study characterised and compared the thinking styles of grade 10 learners taking mathematics and those taking mathematical literacy. it could not, however, find any evidence that teachers use thinking styles of learners to advise learners on their choice between mathematics and mathematical literacy in the gauteng west district.both the interviews with teachers and the survey of learners revealed differences between the two groups as far as characteristics and thinking styles are concerned. even though the teachers’ reflections on the difference between mathematics learners’ and mathematical literacy learners’ characteristics suggest broad distinctions, these are not clearly mirrored in the learners’ responses. furthermore, the differences between mathematics and mathematical literacy, as identified from the curriculum analysis, may not directly link to pedagogy and thinking styles. however, learners’ thinking styles could be taken into consideration, as a guideline, when advising learners regarding their subject choices. both learners and teachers should be more aware of thinking styles if they are to make the right subject choice and thus minimise switching between subjects. if learners understand their thinking styles and how these match either with mathematics or mathematical literacy, they are more likely to select the appropriate subject. also, the pressure on teachers who have to deal with larger classes due to subject changes later during a year will decrease and they will not have to re-teach subject content to learners which switch subjects. based on the findings of this research, further research is required to develop a quantitative instrument to capture the backgrounds and thinking styles of grade 9 learners to enable schools to provide learners with the necessary information to make an informed choice. in particular, the following information should be included in an instrument of this nature: • age • number of times retained in a grade • perceptions of the quality of tuition they received in mathematics in the past • grade 9 marks in mathematics • subject choice in grades 10−12 (excluding mathematics of mathematical literacy). the findings also suggest that a shorter edition of the thinking style inventory (sternberg & wagner, 1992b) should be used. in particular, only the legislative, executive, judicial, hierarchic, anarchic, local, internal and liberal dimensions of the inventory need to be measured, as this study found differences in these thinking styles between learners taking mathematics and those taking mathematical literacy. given the localised nature of this study, namely a single district in a single province in south africa, the findings obtained should be confirmed through similar studies of this nature in other provinces and districts. in this way, a better understanding of the differences, both cognitive and non-cognitive, between mathematics learners and mathematical literacy learners can be obtained. the ways in which empirical realities manifest are much more complex than the broad groupings pointed to in the literature and curriculum analysis in this article. further research in this regard should also be conducted. conclusion top ↑ the article focused on the characterisation and comparison of thinking styles of learners studying mathematics and those studying mathematical literacy. this was an extract from a broader study on the placement of grade 10 learners in either subject (spangenberg, 2008). the aim was to establish which thinking styles are associated with learners studying these subjects, so as to develop guidelines that will contribute to the subject choice of either mathematics or mathematical literacy by grade 10 learners, and eventually to better performance in the two subjects. it was found that there is a relationship between learners’ thinking style dimensions and which subject they are studying. mathematics learners are more likely than mathematical literacy learners to execute instructions, to design and do things in their own way and to be critical. where mathematics learners are more likely to do many things at the same time and to set priorities pertaining to what to do, at what time to do it and how much time and energy to spend on it, mathematical literacy learners are more likely to do one thing at a time and spend almost all their energy and resources on it. this information about the thinking styles of learners could be used to help place learners more appropriately and possibly reduce the number of learners who make inappropriate choices. also, this will ease pressure on teachers who have to deal with larger classes due to subject changes later during a year and may have to re-teach subject content. more learners will gain university exemption. in support, borromeo ferri (2004, p. 2) argued that thinking styles should ‘not be viewed as being unchangeable, but they may change depending on time, environment and life demands’. in conclusion, now that access to education and the right to learning have been established for most learners in south africa, the time is ripe to set key priorities for the country’s future. there is an urgent need to increase the number of learners with sufficient and well-established mathematical knowledge and skills, and so enable them to progress in the short, medium and long term to higher education, the business world and industry. there is a great demand for teachers in mathematics and mathematical literacy to equip learners with the necessary knowledge and skills. acknowledgements top ↑ i am grateful to the participants and the authorities of the participating schools for their cooperation. a special word of thanks to my supervisors for their professional assistance, guidance and support, statkon at the university of johannesburg for professional consultation and analyses of the data, and the gauteng department of education for permitting the research in the schools. also, many thanks to my colleagues who read the draft manuscript and made comments. the 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(2011). thinking styles and conceptions of creativity among university students. educational psychology, 31(3), 361−375. http://dx.doi.org/10.1080/01443410.2011.557044 appendix 1 top ↑ teachers’ interview questions 1. which criteria do you use to advise a learner in choosing between mathematics and mathematical literacy? 2. which method(s) or criteria does your school use to place grade 10 learners in either mathematics or mathematical literacy? 3.1 which other factors influence the placement of learners in mathematics or mathematical literacy? 3.2 can you motivate why you made that statement? 4. which thinking styles of learners do you use to advise learners on their choice between mathematics and mathematical literacy? 5. which characteristics would you attribute to learners that have chosen mathematics? 6. which characteristics would you attribute to learners that have chosen mathematical literacy? 7. is there anything that you wish to add with regard to the placement oflearners in mathematics or mathematical literacy? appendix 2 top ↑ sample of learners’ thinking style questionnaire circle the number that best describes the way you do things. use the following code: abstract introduction mathematics as a language learning and teaching both english as a second language and mathematics the theoretical model conclusion acknowledgements references about the author(s) maureen ledibane department of english, north-west university, south africa kotie kaiser department of english education, north-west university, south africa marthie van der walt department of mathematics education, north-west university, south africa citation ledibane, m., kaiser, k., & van der walt, m. (2018). acquiring mathematics as a second language: a theoretical model to illustrate similarities in the acquisition of english as a second language and mathematics. pythagoras, 39(1), a347. https://doi.org/10.4102/pythagoras.v39i1.347 review article acquiring mathematics as a second language: a theoretical model to illustrate similarities in the acquisition of english as a second language and mathematics maureen ledibane, kotie kaiser, marthie van der walt received: 04 aug. 2016; accepted: 30 july 2018; published: 29 oct. 2018 copyright: © 2018. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract mathematics has been defined by researchers as a ‘second or third language’ and, as a result, it should be taught as a second language. results of the literature reviewed from the theories on the teaching of mathematics and english as a second language, as well as on mathematics learning and english as a second language acquisition, have resulted in the emergence of four themes, which are similar to the ones on the teaching and learning of both mathematics and english as a second language; these are: comprehensible input, language processing and interaction, output, and feedback. in this article, the themes are illustrated in a theoretical model and discussed to show how english as a second language and mathematics can be acquired simultaneously. (english as a second language in the south african context is referred to as english as a first additional language.) introduction many researchers have described mathematics as a language in itself (esty, 1992; setati, 2002). it, therefore, stands to reason that mathematics should be taught by teachers and acquired by learners in the same way as any other second or third language (garrison & mora, 1999). in south africa, where the majority of learners have to learn through the medium of english as a second language, mathematics teachers have to teach and scaffold two languages, in effect, in their classrooms. this might be one of the reasons why language is often regarded as one of the challenges or barriers to learning in mathematics classrooms, especially in south africa (howie, 2003; reddy et al., 2011). this article traces the parallel theories relating to the teaching and learning of mathematics and english as a second language (esl) in order to integrate those theories into one coherent theoretical model for teaching and learning. the acquisition of mathematics and esl, therefore, can be accommodated in one process and need not be two separate teaching and learning processes. furthermore, due to the fact that questioning is an integral part of teaching and learning in mathematics classrooms (brualdi, 1998; rosenshine, meister, & chapman, 1996; sutton & krueger, 2002), the model also incorporates the functions of questions, questioning techniques and teacher strategies that can be used simultaneously for the acquisition of both esl and mathematics. the model focuses on four crucial processes of language acquisition, namely comprehensible input, language processing and interaction, output, and feedback. mathematics as a language setati (2002) describes mathematics as a language as it uses notations, symbols, terminology, conventions, models and expressions to process and communicate information. furthermore, esty (1992) defines mathematics as a language, because, like other languages, it has its own grammar, syntax, vocabulary, word order, synonyms, conventions, idioms, abbreviations and sentence and paragraph structures. the language that is specifically used in mathematics classrooms, classified as mathematical discourse, includes aspects summarised in figure 1. figure 1: types of language in mathematics. mathematics educators are cautioned to pay more attention to language learning because, firstly, language learning is often an expected outcome of mathematics education and, secondly, there is evidence that language learning and mathematics learning are intimately related (barwell, 2008). in fact, the academic language involved in mathematics has been referred to as a third language for english language learners since research has shown that native english-speaking learners learning academic language face many of the same challenges as learners learning esl and, as a result, they should be paired during group work activities (biro, chatzis, roper, & sehr, 2005). the next section therefore discusses the relationship between mathematics and esl in as far as their teaching and learning are concerned. learning and teaching both english as a second language and mathematics learning a second language is not a separate process that has no impact on mathematics learning (barwell, 2008). in other words, the learning of mathematics in multilingual classrooms depends to a large extent on the acquisition of english as a second or third language. this interdependency of the learning of esl and mathematics therefore allows for certain acquisition processes to take place simultaneously. the important role of language in mathematics learning is succinctly captured by harrison (2014, par. 12): ‘language is the cement that allows us to build upon prior knowledge learning. if language is weak, so too is the ability to learn’. similarly, thompson and rubenstein (2000) argue that language plays at least three crucial roles in our classrooms: we teach through the medium of language. it is our major means of communication. learners build understanding as they process ideas through language. we diagnose and assess learners’ understanding by listening to their oral communication and by reading their mathematical writings. the next section discusses the conditions for and the theories on the teaching and acquisition of esl, and also on mathematics teaching and mathematics learning. these theories include a combination of second language acquisition (sla) teaching and learning theories, and also their similarities to the principles of realistic mathematics education (rme). parallels in teaching mathematics and english as a second language in the 1960s, mathematics education in most parts of the world and in the netherlands was dominated by a mechanistic teaching approach (van den heuvel-panhuizen & drijvers, 2014). this means that learners sat passively in mathematics classrooms while teachers demonstrated how problems are solved. also, teachers asked closed questions that were followed up by learners’ answers and teachers’ feedback, engaging learners in the initiate-response-evaluate discourse in mathematics classrooms. similarly, the mechanistic teaching approach to esl with regard to the audio-lingual method, emphasising the spoken language, became popular in the middle of the 20th century. it involved a systematic presentation of the structures of the second language, moving from simple to complex, in the form of drills that learners had to repeat. it was influenced by a belief that the fluent use of a language was essentially a ‘set of “habits” that could be developed with much practice’. much of this practice involved ‘hours spent in the language laboratory repeating oral drills’ (yule, 2010, p. 190). in reaction to the mechanistic approach to mathematics teaching, freudenthal, a mathematician who became interested in mathematics education, propagated a method of teaching mathematics that is relevant for learners. his method included carrying out thought experiments to investigate how learners can be offered opportunities for guided reinvention of mathematics and, in this way, contributed to the development of the rme theory (van den heuvel-panhuizen & drijvers, 2014). the main characteristics of rme are problematisation, construction and reflection. the teacher is the activator in the process of problematisation and the tutor in the process of construction, ‘taking learners’ informal strategies as a starting point for the interactional development of mathematical concepts and insights’ (van eerde, hajer, & prenger, 2008, p. 33). similarly, in recent years, contemporary language teaching has moved away from dogmatic practices of ‘right’ or ‘wrong’, becoming much more eclectic in its attitudes, and more willing to recognise the potential merits of a wide variety of methods and approaches. as a result, ‘the interest in the contribution of the learners in the teaching/learning dichotomy was resurrected, accommodating the learning strategies that learners employ in the process of language learning’ (griffiths & parr, 2001, p. 248), through methodologies such as task-based instruction (tbi). according to powers (2008), in tbi, teachers prepare lessons that are constructed according to the language required to perform specific tasks. this means that learners learn language structures through induction as they focus on task completion and meaning. their interaction during the tasks facilitates transfer of information they have previously learned and incorporates it with new information they receive as they perform the task. the three characteristics of rme (problematisation, construction and reflection) correlate well with the three phases in a task-based language lesson as described by ellis (2003), namely the ‘pre-task phase, the during-task phase and the post-task phase’. during the pre-task phase, learners are provided with examples of similar problems and they are given time for strategic planning, so that they can plan how they will solve the specific problem or perform the task. in the during-task phase, learners are scaffolded so that they can discuss the problem or task while using the appropriate discourse and it allows them to take linguistic risks. the post-task phase allows learners to reflect on the task, so that they can develop the metacognitive strategies of planning, monitoring and evaluating in the process. in a similar vein, moschkovich (2002) proposes the following three perspectives for bilingual and esl learners to ‘communicate mathematically’, both orally and in writing, and to participate in mathematical practices: acquiring vocabulary: for esl learners to communicate mathematically, they should acquire vocabulary, usually referred to as mathematical discourse. acquiring vocabulary is emphasised in learning mathematics as it is the central issue that second language learners are grappling with when learning mathematics (moschkovich, 2002). learners can only communicate mathematically if they have acquired the vocabulary, which comprises the different types of languages shown in figure 1. constructing meanings: the second perspective describes mathematics learning as constructing multiple meanings for words rather than acquiring a list of words. learning mathematics, therefore, involves a shift from everyday terms to more mathematical and precise meanings, referred to as ‘mathematical register’ (moschkovich, 2002, p. 194). however, everyday meanings and learners’ home language can also be used by the learners as resources to communicate mathematically. participating in discourse: from this perspective, learning to communicate mathematically involves more than learning vocabulary or understanding meanings in different registers and, according to (moschkovich, 2002), it is seen as using social, linguistic and material resources to participate in mathematical practices. the integration of these theories will be discussed in terms of four crucial processes of language acquisition, namely comprehensible input, language processing and interaction, output, and feedback, in the teaching and learning of both english and mathematics. comprehensible input ellis (1986) defines input as the language that learners are exposed to. he further explains that it is possible for the input provided by the teachers and interlocutors to be ‘comprehensible (i.e. input that learners can understand) or incomprehensible (i.e. input that they cannot understand)’; when it is incomprehensible, it becomes ‘the impetus for learners to recognise the inadequacy of their own rule system’ (gass, mackey, & pica, 1998, p. 301). garrison and mora (1999) in their study on latino mathematics learners recommend the use of krashen’s (1994) comprehensible input formula i + 1 in the teaching and learning of mathematics. krashen’s input hypothesis on second language acquisition claims that: an important condition for second language acquisition to occur is that the acquirer understands (via hearing and reading) input language that contains structure ‘a bit beyond’ his or her current level of competence. (krashen, 1981, p. 100) this formula is used in the theories on and principles for english sla and mathematics learning with regard to the roles of input, together with teacher strategies applied. the formula i + 1 is recommended because it provides comprehensible input for english language learners to build mathematical concepts based on the principle of teaching the unknown from the known. the role of input from the definitions of input, one can outline the roles of input in language acquisition and mathematics learning as explained below. firstly, input plays a very important role in as far as language learning and acquisition is concerned. it provides the data that the learner must use to determine the rules of the target language. in the same way, the researchers of the universal grammar view input as a trigger that interacts with an innate system or the native language to promote learning (gass & mackey, 2006). therefore, input forms the positive evidence that learners use as they construct their second language and mathematics grammars. this role of input has therefore resulted in many researchers describing the type of input learners receive in esl classrooms as ‘foreigner talk’ (gass & mackey, 2006, p. 5). secondly, input, in the form of grammar rules (for both english and mathematics), information from mathematics textbooks, and knowledge from the language teachers and interlocutors, and also from the learners, provides the stepping stone for any form of learning to take place. it is up to the teachers and interlocutors to decide what they should do with all the input that they have to make it comprehensible for learners to learn and acquire the languages (i.e. esl and mathematics). lastly, seliger (1983) explains how the role of input gives credit to learners for successful acquisition to take place. the findings in the study showed that: learners referred to as high input generators maintained high levels of interaction in the second language, both in the classroom and outside, and progressed at a faster rate than learners who interacted little, referred to as low input generators. (p. 262) this is also supported by cummins (1991, p. 85) when he states that ‘appropriate input is clearly essential for development of all aspects of proficiency’. teachers, at the beginning of a lesson, write down the vocabulary and symbols on the board. they discuss the definitions and representations (in mother tongue if necessary). the learners then have a reference to the meaning of the words and terminology as well as how to use it. even though krashen in his input hypothesis does not credit the role of learners in as far as input is concerned when he states that ‘comprehensible input is the only causative variable in sla’ (brown, 2007, p. 297), many researchers with an increasing interest in social constructivist analyses of language acquisition focus on the characteristics of successful language learners. they have come up with learning strategies that successful learners apply with regard to input to acquire language, including mathematics language, by making it comprehensible, and thus crediting learners’ role with regard to input. learning strategies the learning strategies, according to brown (2007), include: meta-cognitive strategies: metacognitive is a term used in information-processing theory to indicate an ‘executive function’, and it includes strategies that involve planning for learning, thinking about the learning process as it takes place, monitoring one’s production, and evaluating learning after an activity has been completed; this evaluation includes self-monitoring, self-evaluation, advance organisers, and delayed production. cognitive strategies are more limited to specific learning tasks and involve more direct manipulation of the learning material itself; these include repetition, resourcing, translation, grouping, note-taking, deduction, and others. socio-affective strategies have to do with social-mediating activities and interacting with others, for example, cooperation ad asking questions for clarification. these also relate to output. learning can be constrained by learners’ or teachers’ belief systems and attitudes towards mathematics and the nature of mathematics, and how it should be learned. these inform learners’ decisions to avoid or embrace challenges; and these may influence the learners or teachers, attributing failure or success to cognitive (in)abilities rather than to effort. the content should therefore be meaningful to learners, and that links it to the reality principle of rme. the reality principle in realistic mathematics education: the reality principle can be recognised in rme in two ways. firstly, it expresses the importance that is attached to the goal of mathematics education, including learners’ ability to apply mathematics in solving ‘real-life’ problems. secondly, it stresses the point that mathematics education should start from ‘problem situations that are meaningful to learners and that offer them opportunities to attach meaning to the mathematical constructs they develop when solving problems’ (van den heuvel-panhuizen & drijvers, 2014, p. 523). likewise, in tbi, as pointed out in ellis (2006), a focus on form approach is valid as long as it includes an opportunity for learners to practise behaviour in communicative tasks, thus providing learners with opportunities also ‘to apply mathematics in solving real-life problems’. the grammar taught emphasises not just form, but also the meanings and uses of different grammatical structures. as krahnke (cited in powers, 2008, p. 73) points out, ‘connecting tasks to real-life situations contextualises language in a meaningful way and provides large amounts of input and feedback to assist learners in the learning process’, especially in the second step of acquisition, namely language processing and interaction. language processing and interaction: even if input is understood, according to ellis (1986), it may not be processed by the learner’s internal mechanisms. this is what krashen means when he states that ‘comprehensible input is not a sufficient condition for second language acquisition’ (ellis, 1986, p. 159). it is only when input becomes intake that sla takes place. input is the second language data that the learner hears; intake is that portion of the second language that is assimilated and fed into the inter-language system (ellis, 1986) and, as a result, intake ‘is the subset of all input that actually gets assigned to our long-term memory store’ (brown, 2007, p. 297). interaction is, therefore crucial in the acquisition process of any language. the role of interaction the important role of interaction is revealed in the study conducted by wong-fillmore (1983, cited in cummins, 1986) on hispanic learners in esl classrooms, which showed that learners learned more english in classrooms that provided opportunities for reciprocal interaction with teachers and peers. this reciprocal interaction can be achieved, according to gass and mackey (2006), when second language learners are presented with input that they do not understand, as that will force them to ‘negotiate meaning by using confirmation checks, clarification requests, and comprehension checks, in order to change it into comprehensible input, thus making it the result of modified interaction’ (brown, 2007, p. 305). the combination of input and interaction, using forms of negotiation, makes input and interaction major players in the process of acquisition (brown, 2007). long, cited in brown (2007, p. 305), in his interaction hypothesis, posits that comprehensible input is the result of modified interaction. similarly, ellis (2006, p. 100) states that ‘input-based feedback models the correct form for the learner (e.g. by means of a recast)’ and ‘output-based feedback elicits production of the correct form from the learner (e.g. by means of a clarification request)’. these different types of interactions, referred to as modifications or negotiations, are applied by teachers and interlocutors to make input comprehensible to the learners in esl and mathematics classrooms. confirmation checks: a confirmation check is defined by long as ‘any expression … following an utterance by the interlocutor which are designed to elicit confirmation that the utterance has been correctly heard or understood by the speaker’ (cited in gass & mackey, 2006, p. 7). it can be used for learners to receive comprehensible input. clarification requests or paraphrases: a clarification request is any expression designed to elicit clarification of the interlocutor’s preceding utterances (gass & mackey, 2006). it can be applied by saying an incorrect utterance in a rising intonation for the learner to reflect on the answer provided and come up with the correct utterance. for example, if the learner says denominator, instead of numerator, the teacher could say, denominator with a rising intonation, and that could result in the learner reflecting on the wrong answer and saying the correct answer, numerator. comprehension checks: a comprehension check is an attempt ‘to anticipate and prevent a breakdown in communication’ (gass & mackey, 2006, p. 8). it can be used also in the form of a yes or no question by the teacher to check if the learner understands the meaning of one of the utterances spoken for communication to continue, for example: given the right-angled triangle abc, with æa = 90°, and ab = 3 cm, and bc = 4 cm, use pythagoras’ theorem to find the length of the hypotenuse. do you know the pythagoras theorem? in response to the learner’s negative answer, the teacher will draw a right-angled triangle and show learners what they mean by the word ‘hypotenuse’, thus assisting them in how to calculate the values of the hypotenuse of any right-angled triangle. this exercise would enable the learners to find the length of the hypotenuse in the problem initially given. recasts: another form of negotiation for the learners’ feedback is recasts, defined as ‘utterances that rephrase a child’s utterance by changing one or more sentence components (subject, verb, or object) while still referring to its central meaning’ (gass & mackey, 2006, p. 8). recasts involve the teacher’s reformulation of all or part of a learner’s utterance minus the error (gass & mackey, 2006). in response to the learners’ incorrect answer, the teacher may repeat the correct answer for the learner to identify the error that they have committed so as not to make the same error in the future. from the examples of negotiations and modifications given, one could say that these negotiations or modifications alert the learners to the mistakes they have made in their utterances, thus providing them with opportunities to ‘focus their attention on language and the correct mathematical concepts; ‘to search for more input in their future utterances; and to be more aware of their hypotheses about language and mathematics’ (gass & mackey, 2006, p. 12). also, modifications in interactions, according to long (cited in menezes, 2013, p. 405), are consistently found in successful sla; therefore, they should be applied in esl as well as in mathematics classrooms. when these are applied by the teachers and interlocutors, they provide learners with opportunities to process their utterances and responses mentally before they can produce them, and also help them to reflect on their learning process, thus enhancing their acquisition and learning. this is confirmed by cummins (2000, p. 74), when he states that ‘bics [basic interpersonal communication skills] and calp [cognitive academic language proficiency] both develop within a matrix of social interaction’. as learners respond to the modifications and interactions discussed, they are actively involved in their own learning, and ultimately produce output. the activity principle in realistic mathematics education the activity principle emphasises that learners should be treated as ‘active participants in the learning process since mathematics is best learned by doing mathematics’ (van den heuvel-panhuizen & drijvers, 2014, p. 523). this is strongly reflected in freudenthal’s interpretation of mathematics as a human activity. learners should not be passive listeners but active participants in mathematics classrooms, and this can be achieved if learners are taught learning strategies such as metacognition to think about their learning process so as to make input comprehensible. similarly, tbi is based on krashen’s language acquisition hypothesis. krahnke, cited in powers (2008, p. 73), explains that the theory asserts that the ability to use language is gained through exposure to and use of it, thus discouraging learners from being passive and to rather be active participants in the learning situation. krahnke, as cited in powers (2008, p. 73) goes on to explain that ‘tbi develops communicative competence including linguistic, sociolinguistic, discourse and strategic competence’, thus processing the information used during specific tasks through understandable input to provide students with linguistic and sociolinguistic competence in a systematic, step-by-step process (cited in powers, 2008, p. 73), relevant to the level principle of rme. the level principle in realistic mathematics education the level principle underlines that learning mathematics means that learners pass various levels of understanding: from informal context-related solutions, through creating various levels of shortcuts and schematisations, to acquiring insight into how concepts and strategies are related (van den heuvel-panhuizen & drijvers, 2014). particularly for teaching operating with numbers, this level principle is reflected in the didactical method of ‘progressive schematisation’ where transparent whole number methods of calculation gradually evolve into digit-based algorithms (van den heuvel-panhuizen & drijvers, 2014, p. 523). similarly, in his interaction hypothesis, long (1985, 1996) explains in detail how input is made comprehensible, thereby picking up where krashen left off. he posits that comprehensible input is the result of modified interaction (brown, 2007), and it includes various types of interactions, such as clarification requests, paraphrases and comprehension checks for learners to interact and process the language and integrate knowledge from different domains in order to make sense of their own learning. the intertwinement principle in realistic mathematics education the intertwinement principle means mathematical content domains such as number, geometry, measurement and data handling are not considered as isolated curriculum chapters, but as heavily integrated (van den heuvel-panhuizen & drijvers, 2014). learners are offered rich problems in which they can use various mathematical tools and knowledge. this principle also applies within domains. for example, within the domain of angles, triangles, sines and cosines, quadrilaterals are taught in close connection with each other. in other words, different sections of mathematics should not be taught in isolation, but as a unit showing relationships between one another so as to make sense to the learners (van den heuvel-panhuizen & drijvers, 2014). similarly, content-based instruction, a modified form of tbi is defined by brinton, snow and wesche (1989) as the concurrent study of language and subject matter, with the form and sequence of language presentation dictated by content material. in other words, content and language are not taught in isolation or separately, but always within a meaningful context. content-based instruction can take place at all educational levels, and it refers to total immersion (approximately 90% of school time in the second language), or it can refer to content-based themes in language classes (cenoz, 2015). output reading and listening are not enough for learners to learn the language; therefore, teachers should provide learners with vast opportunities to try out and produce language using pair and group work activities. the three major functions of output in sla, according to swain (2005), emphasise the role of output in language production. these are similar to the conditions of mathematics learning as listed by van eerde et al. (2008): learners become self-informed through their input. there should be ample opportunities for language production. language learners need feedback on their utterances. firstly, learners become self-informed through their output. this condition claims that learners while attempting to produce the target language may notice their erroneous attempts to convey meaning. this prompts them to recognise their linguistic shortcomings, thus becoming self-informed about their output. output helps the learners to ‘try out’ one’s language: to test various hypotheses that are forming. speech and writing can offer a means for learners to productively reflect on language and mathematical language in interaction with peers. furthermore, swain (cited in gass & mackey, 2006) also suggests that output provides an opportunity for learners to test hypotheses about the target language, and modify them where necessary. also, for modified output to be useful, most interaction researchers suggest that it is necessary for learners to notice the relationships between their initially erroneous forms, the feedback they receive and their output, since it is possible for learners’ perceptions to differ according to the type of feedback they receive and the focus (swain, cited in gass & mackey, 2006). similarly, the second condition for mathematical language development emphasises language production. in other words, the condition for output in mathematics learning is the promotion of active participation of pupils, giving them the opportunity to construct and verbalise their mathematical solutions, promoting classroom discussions and asking for clarifications and justifications (van eerde et al., 2008). output is defined as ‘the process of producing language in the form of speaking and/or writing’ (brown, 2007, p. 293). similarly, gass and mackey (2006, p. 13) define it as ‘the language that learners produce’. krashen has been criticised by other researchers for disregarding the function of learners’ output in sla when he says that ‘output is too scarce to make any important impact on language development’ (brown, 2007, p. 298). de bot (1996, p. 529), argues that ‘output serves an important role in second language acquisition … because it generates highly specific input the cognitive system needs to build up a coherent set of knowledge’. as a result, interaction research, according to gass and mackey, focuses on output that has been modified, and therefore modified output promotes learning since it stimulates learners to reflect on their original language. this is done by utilising a number of communication strategies. communication strategies learners use a variety of communication strategies to request assistance, to modify the output produced with the feedback they receive from their interlocutors, and thereby produce modified output. these are strategies that learners use as ‘potentially conscious plans for solving what, to an individual, presents itself as a problem in reaching a particular communicative goal’ (brown, 2007, p. 137). these include: avoidance strategies include message abandonment, leaving a message unfinished because of language difficulties, and topic avoidance, avoiding topic areas or concepts that pose language difficulties (brown, 2007). learners use these strategies by changing the topic or pretending not to understand it because it is too difficult for them to express. compensatory strategies are used for compensation for missing knowledge, and these include code-switching, circumlocution, appeal for help and non-linguistic signals like miming, among others. memory strategies include creating mental linkages, applying images and sounds, reviewing well and employing action. for example, in order for learners to remember how they learn sections in geometry, they can simply look at the shape of their desks and use that mentally to remember the formulae for calculating the area of rectangles, squares, parallelograms, etc. cognitive and metacognitive strategies were discussed under the learning strategies section. affective strategies include lowering your anxiety, encouraging yourself, and taking your emotional temperature. social strategies include asking questions, cooperating with others and empathising with others. learning strategies ‘relate to input’, whereas communication strategies ‘relate to output’ (brown, 2007, p. 132). these strategies help learners to interact meaningfully in the course of their learning. the interactivity principle in realistic mathematics education the interactivity principle signifies that learning mathematics is not only an individual activity, but also a social activity (van den heuvel-panhuizen & drijvers, 2014). it encourages teachers to make full use of group work and whole-class discussions to provide learners with opportunities to share ideas and strategies on how they solve mathematical problems and, in this way, produce output. as learners share their ideas, ‘they evoke reflection, which enables them to reach a higher level of understanding’ (van den heuvel-panhuizen & drijvers, 2014, p. 523). similarly, proponents of tbi (ellis, 2003) agree that communicative activities used during pair and group work are appropriate vehicles and that language learning activities should directly reflect what learners ‘potentially or actually need to do with the target language’ (swan, 2005, p. 377). also, the role of the teacher in the tbi classroom is to supply task-related vocabulary where necessary, offering recasts or acting as interlocutors, ‘casting the teacher’s role as a manager and facilitator of communicative activity rather than an important source of new language’ (swan, 2005, p. 391). for esl and mathematics learning to take place successfully, learners need support in the form of feedback from their teachers, interlocutors, adults and peers, so as to perfect acquisition and the learning process. this brings us back to the third condition of mathematics learning, namely that language learners need feedback on their utterances (van eerde et al., 2008). feedback the important role of output has resulted in many researchers claiming that output provides the forum for receiving feedback (gass & mackey, 2006). in other words, when learners produce first language utterances, they rely on the interlocutors’ feedback to see if they are on the right track in terms of language acquisition and mathematics learning. this is also emphasised in rme’s guidance principle the guidance principle in realistic mathematics education the guidance principle refers to freudenthal’s idea of ‘guided re-invention’ of mathematics. it implies that ‘rme teachers should have a proactive role in learners’ learning and that educational programmes should contain scenarios that have the potential to work as a lever to reach shifts in learners’ understanding’ (van den heuvel-panhuizen & drijvers, 2014, p. 523). to realise this, the teaching and the programmes should be based on coherent long-term teaching-learning trajectories. according to gravemeijer (2009, p. 114), the principle means that ‘learners should be provided with the opportunity to experience a process similar to the process by which a given piece of mathematics was invented’. the role of the teacher in this case is to revise the sections taught in previous classes that will enable them to perform the tasks before giving them the actual problem to solve, thus providing them with scaffolding. for example, a question like find the sum of 49 + 58, should not require learners to crack their heads with adding the two numbers, but they should simply think of the internalised method of factorising common factors by applying the distributive law taught in class, and thus group like and unlike terms by expanding 49 into 40 + 9 and 58 into 50 + 8, and find the sum of like terms 40 + 50 = 90, and that of units 9 + 8 = 17 to get the answer 107. in doing so, the learners would be outsourcing guidance or scaffolding by remembering what their teachers taught them before and also what they learnt from mathematics textbooks. the role of the teacher in this case is to revise with the learners the previous sections taught that are required for them to perform the tasks in the example given before giving them this exercise, thus providing them with scaffolding. similarly, in esl classrooms, vygotsky’s theory on the zone of proximal development (zpd) stresses the fact that learners acquire language in the social world and, as a result, individuals learn best when working together with others during joint collaboration. it is through such ‘collaborative endeavours with more skilled persons that learners learn and internalise new concepts, psychological tools, and skills’ (shabani, khatib, & ebadi, 2010, p. 237). the fact that learners need input from a more knowledgeable other in vygotsky’s zpd also links to the qualification of the type of input specified in krashen’s input hypothesis i + 1 (krashen, 1981, 1994), namely that learners should be exposed to input at a level higher than their current level of language proficiency. likewise, ellis (2006, p. 102) believes that ‘corrective feedback is important for learning grammar in esl and that it is best conducted using a mixture of implicit and explicit feedback types that are both input-based and output-based’. to provide scaffolding, teachers can provide learners, for example, with the vocabulary that is required to perform a particular task, or design fill-in-the-gap conversations to practise as they simulate the task and to give them the means by which they can produce appropriate output. for learners’ utterances and mathematics language to be perfected, learners need support or what is termed ‘scaffolding’, in the form of feedback from the teachers, interlocutors, peers and adults, for them to be in a position to reflect on and correct the mistakes made and to perfect the acquisition and learning process. scaffolding is understood as the assistance learners get from others (teachers, relatives, classmates) and it enables them to perform learning tasks (menezes, 2013). feedback from the teacher on pupils’ contributions should not be immediate, but ‘delayed to promote contributions from different pupils and horizontal interaction between pupils’ (van eerde et al., 2008, p. 36). in this instance, vygotsky’s theory of zpd is also applicable in mathematics classrooms. the theory, as a result, puts emphasis on the role of feedback. teachers and parents (as the more knowledgeable others) are therefore advised to offer learners this assistance and support for successful learning and language development to take place in both esl and mathematics classrooms, using the following strategies: strategies for scaffolding in a study in which a teacher was encouraged to employ seven strategies in a multilingual classroom, the results showed that the strategies used in table 1 promoted pupils’ language development (smit & van eerde, 2013). table 1: strategies for scaffolding language and examples for each strategy. similarly, biro et al. (2005) encourages mathematics teachers to help english language learners to develop and practise academic language for learning mathematics using scaffolding strategies, such as having learners restate other learners’ comments, using graphic organisers or gestures, correcting errors and providing positive feedback, providing handouts to help learners structure and guide their work, among others. a constructivist approach to teaching and learning (summarised below) should also be applied for assessment in english sla and mathematics language learning classrooms to take place in such a way that learners are able to reach the intended outcomes discussed. using a constructivist/open-ended approach to teaching and learning a constructivist approach to teaching and learning, according to mahlobo (2009), is characterised by the use of open-ended tasks or questions, and it encompasses the following: the learners: take the initiative in solving mathematical problems and do not depend on the teacher. determine their own approach when solving problems. express their own ideas more frequently when solving mathematical problems. modify other learners’ ideas. can stimulate the exploration of concepts and ideas and facilitate creative and critical thinking processes. (p. 38) this article has elaborated on the teaching of both mathematics and esl, and also on the learning of mathematics and esl acquisition. therefore, the theoretical model in figure 2 shows the similarities between the theories on the teaching of both esl and mathematics, and also on the learning of mathematics and esl acquisition. from the literature reviewed, the themes that emerged, namely comprehensible input, language processing and interaction, output, and feedback, are in response to the research question, namely: what are the theories that underpin the effective questioning techniques and strategies to promote esl acquisition? figure 2: a theoretical model to illustrate the simultaneous acquisition of a second language and mathematics. the theoretical model this article has elaborated on the teaching and learning of both mathematics and esl, and also on the learning of mathematics and esl acquisition. as a result, the theoretical model in figure 2 shows the similarities between the theories on the teaching of mathematics and esl (tesl) and english second language acquisition (esla) and mathematical proficiency. these are indicated through different venn diagrams. from the literature reviewed, the themes that emerged, namely comprehensible input, language processing and interaction, output, and feedback, are indicated. these processes are listed next to the relevant venn diagrams and in a circle in order to indicate the cyclical nature of language acquisition and the interactivity and dynamics of the different processes. the movement from one process to another is stimulated by scaffolding of the teacher and output by the learners in the form of different strategies. in addition, the inner circle shows that esla is taking place throughout the processes, while the outer concentric circle shows that mathematics teaching, together with mathematics learning, produces mathematical proficiency in all four of the processes discussed. also, the cyclical arrows show movement created by questions, questioning techniques and teacher strategies, thus moving learners from input to language processing and interaction, from language processing and interaction to output, from output to feedback, and vice versa. comprehensible input according to krashen’s input hypothesis in esla (krashen, 1981, 1994), as well as van eerde et al. (2008)’s first condition for mathematics learning, learners in esl and mathematics classrooms should be provided with comprehensible input at level i + 1, that is, input that is challenging to the learners and not input that is very easy at level i + 0, or difficult at level i + 2. furthermore, for learners to be able to understand what is taught in mathematics classrooms and achieve mathematical proficiency, according to moschkovich (2002)’s first perspective, provisions should be made for learners to acquire vocabulary referred to as ‘mathematical discourse’. furthermore, learners should be given real-life problems that are meaningful to them, according to rme’s reality principle (van den heuvel-panhuizen & drijvers, 2014), for them to understand what is taught in mathematics classrooms (ledibane, 2016). the functions of questions (fq), the questioning techniques (qt) and teacher strategies (str) indicated, together with learning strategies, when used in these classrooms, will assist learners to comprehend the input provided. these should also be used in mathematics classrooms. language processing and interaction for learners to be able to process and interact using language, according to long’s interaction hypothesis (long, 1985, 1996), comprehensible input has to be modified using the different types of interactions. similarly, moschkovich (2002)’s second principle emphasises constructing meaning, implying that everyday meanings and learners’ home language can be used for mathematical formulations and concepts for learners to acquire mathematical proficiency. furthermore, rme’s level and the intertwinement principles (van den heuvel-panhuizen & drijvers, 2014) underline that learning mathematics means that learners should be taught in such a way that they see the concepts taught as inter-related, and not isolated from each other (ledibane, 2016). the functions of questions (fq), the questioning techniques (qt), and teacher strategies (str) indicated, together with types of interactions or modifications, when used in these classrooms, will provide learners with opportunities to process and interact using the language. even though the interactions are used in esl classrooms, they can also be used in mathematics classrooms for learners to achieve the intended outcomes. output for learners to produce output, according to swain’s output hypothesis in esla, and van eerde et al (2008)’s second condition for mathematics learning, learners should be provided with opportunities to produce output. also, to achieve mathematical proficiency, according to moschkovich (2002)’s third perspective, learners should be given opportunities to participate in discourse. similarly, rme’s activity and interactivity principles (van den heuvel-panhuizen & drijvers, 2014) emphasise that learners should be treated as active participants in mathematics classrooms, and therefore group work activities should be used in these classrooms for learners to be provided with opportunities to produce output (ledibane, 2016). the functions of questions (fq), the questioning techniques (qt) and the teacher strategies (str) indicated, together with a combination of learning and communication strategies referred to as strategies-based instruction that assist learners to try out and produce language in esl classrooms (brown, 2007) are also captured. all these should be used in both esl and mathematics classrooms. feedback for learners to do well in acquiring both esl and mathematical discourse, they should be provided with feedback on their utterances. this is emphasised in vygotsky’s zpd theory on feedback and also in van eerde et al. (2008)’s third condition for mathematics learning. this is also stated in rme’s guidance principle (van den heuvel-panhuizen & drijvers, 2014), where teachers are encouraged to provide scaffolding or support in the form of feedback on learners’ utterances (ledibane, 2016). once again, the quality of the feedback can be linked to krashen’s input hypothesis i + 1, as the purpose of the input is to improve proficiency, and therefore needs to be at a level beyond the learners’ current proficiency. the functions of questions (fq), the questioning techniques (qt) and teacher strategies (str) indicated, together with scaffolding strategies, will provide learners with feedback on their utterances in esl and mathematics classrooms. in addition, the reflection processes described in the visual representation are cyclical in nature as the teachers provide input throughout the learning situation when they also reflect on the input provided to learners and on the output produced. similarly, the learners reflect on what is taught and also on what they bring into the learning environment by applying meta-cognitive processes to speed up the process of producing language and acquiring it in the long run. hence, the arrows on both sides show reflections throughout the learning process as teachers and learners reflect on the input provided and output received by using meta-cognitive knowledge (e.g. declarative, procedural and conditional; person, task and strategy variables) as well as the self-regulated processes (e.g. planning, monitoring, evaluation). conclusion current research on second language acquisition and mathematics learning shows that learners go through similar processes when acquiring both subjects, namely they need to actively use comprehensible input, to process language through interactions, to produce new linguistic elements in meaningful contexts and to receive feedback to integrate new knowledge into their existing knowledge systems (krashen, 1981, 1994; long, 1985, 1996; moschkovich, 2002; swain, 2005; van den heuvel-panhuizen & drijvers, 2014; van eerde et al., 2008; vygotsky, 1978). the researchers on second language acquisition and mathematics learning have brought good news for mathematics teachers, specifically in grade 10–12 classes, who are struggling to bear the burden of teaching both language and mathematics in their classrooms. when basing their planning and presentation of lessons on the proposed model, mathematics teachers will not necessarily have to carry that burden. if they carefully plan the strategy instruction (from both the fields of esl and mathematics), using the functions of questions, questioning techniques and teacher strategies associated with moving learners from one acquisition process to the next, they can teach both mathematics and english in such a way that learners acquire these simultaneously. however, very few mathematics teachers have been trained as language teachers as well, so they will have to be trained in order to create awareness of the similarities and useful functions of questions, questioning techniques and teacher strategies associated with the acquisition of a second language. the hope is, therefore, that this model could assist teacher educators in the pre-service and in-service training of grade 10 mathematics teachers who have to teach through the medium of a second language, and prepare learners to perform well in the final grade 12 mathematics examination papers. acknowledgements competing interests the authors declare that we have no financial or personal relationships that might have inappropriately influenced us in writing this article. authors’ contributions m.m.l. and k.k. conceptualised the article, with major contributions from m.v.d.w. the data was collected and analysed by m.l. all three authors were involved in the interpretation of the theories and the framework was designed with major contributions from k.k. and m.v.d.w. references barwell, r. 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(1978). mind in society. boston, ma: harvard university press. yule, g. (2010). the study of language. cambridge: cambridge university press. https://doi.org/10.1017/cbo9780511757754 about the author(s) divan jagals school for professional studies in education, faculty of education sciences, north-west university, south africa marthie van der walt school for science and technology education, faculty of education sciences, north-west university, south africa citation jagals, d., & van der walt, m. (2019). corrigendum: metacognitive awareness and visualisation in the imagination: the case of the invisible circles. pythagoras, 40(1), a464. https://doi.org/10.4102/pythagoras.v40i1.464 note: doi of original article: https://doi.org/10.4102/pythagoras.v39i1.396 corrigendum corrigendum: metacognitive awareness and visualisation in the imagination: the case of the invisible circles divan jagals, marthie van der walt published: 13 feb. 2019 copyright: © 2019. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. in the author list of this article published earlier, marthie van der walt’s first name and orcid were unintentionally misprinted as ‘martha’ and ‘https://orcid.org/0000-0002-6057-8352’. the author’s correct first name is ‘marthie’ and the orcid is ‘https://orcid.org/0000-0002-0465-6600’. the authors sincerely regret this error and apologises for any inconvenience caused. barnes 42 pythagoras 61, june, 2005, pp. 42-57 the theory of realistic mathematics education as a theoretical framework for teaching low attainers in mathematics hayley barnes university of pretoria email: hayley.barnes@up.ac.za this article recounts the process embarked on and reasons for selecting the theory of realistic mathematics education (rme) as the theoretical framework in a study carried out with low attaining learners. in the study an intervention for low attaining grade 8 mathematics learners was implemented in an attempt to improve the understanding of the participants with regard to place value, fractions and decimals, and to identify characteristics of this type of intervention and potential design principles that could be applied in similar interventions. in this article, the theoretical framework for the intervention is discussed and theoretical (rather than empirical) reasons for selecting the theory of realistic mathematics education (rme) for use with low attainers are put forward. from a literature review that looked at the teaching and learning of mathematics to learners who fall into the category of performing below the required standard, five common aspects emerged. once these aspects had been identified, a theory in mathematics education was sought that encompassed these five aspects. the theory of rme was subsequently selected as the theoretical framework to drive the design and implementation of the intervention and is being suggested as a possible way forward for working with low attaining learners. low attainers many terms or descriptions are used in the literature to refer to learners in this category. these include terms such as: remedial, disadvantaged, special needs, under-achievers, slow-learners and low achievers (e.g. denvir, stolz & brown, 1982; haylock, 1991; swanson, hoskyn & lee, 1999; kroesbergen & van luit, 2003), which are used in schools to refer to children with undefined problems. for this article, the term low attainer has been chosen to refer to learners who do not meet the required standard of mathematics performance as set out by the school. this implies that the observable performance of the learners is described, without implying a cause (denvir et al., 1982). teaching and learning mathematics (with specific reference to low attainers) from a critical review of the literature on mathematics interventions and programmes for learners with mathematical difficulties (e.g., baroody & hume, 1991; dockrell & mcshane, 1992; mercer & miller, 1992;), learning disabilities (e.g., cawley & parmar, 1992; swanson, et al., 1999; dunlap & thompson, 2001; geary & hoard, 2001), special educational needs (sen) (e.g., daniels & anghileri, 1995; kroesbergen & van luit, 2003; magne, 2003) and low attainers (e.g., hart, 1981; denvir et al., 1982; trickett & sulke, 1988; haylock, 1991), there appears to be considerable evidence that arithmetic computation and basic mathematics skills are the dominant domains. the definition of mathematics provided in the new revised national curriculum statement for grades r-9 in south africa (department of education, 2002: 1) broadens the scope of mathematics far beyond this. the definition states that: mathematics is a human activity that involves observing, representing and investigating patterns and quantitative relationships in physical and social phenomena and between mathematical objects themselves. through this process, new mathematical ideas and insights are developed. mathematics uses its own specialised language that involves symbols and notations for describing numerical, geometric and graphical relationships. mathematical ideas and concepts build on one another to create a coherent structure. mathematics is a product of investigation by different cultures – a purposeful activity in the context of social, political and economic goals and constraints. the document also outlines the interrelated knowledge and skills included in the scope of mathematics, and stresses the importance of mathematical literacy to enable persons to hayley barnes 43 "contribute to and participate with confidence in society" (department of education, 2002: 2). the teaching and learning of mathematics can enable the learner to: • develop an awareness of the diverse historical, cultural and social practices of mathematics; • recognise that mathematics is a creative part of human activity; • develop deep conceptual understandings in order to make sense of mathematics; and • acquire the specific knowledge and skills necessary for: ▫ the application of mathematics to physical, social and mathematical problems, ▫ the study of related subject matter (e.g. other learning areas), and ▫ further study in mathematics" (department of education, 2002: 4). in this context, daniels and anghileri (1995) identify the fundamental aim of teaching mathematics as, to equip learners with the strategies, skills, knowledge and most importantly the confidence to use their mathematics to solve problems that learners will encounter throughout their lives. if mathematics teaching does not result in providing learners with these skills, then an important part of their preparation for life is missing and they have been denied access to a basic human right (doe, 2002). also, denvir et al. (1982) categorise mathematical aims under three broad headings, that is: useful: as a tool for the individual and society, e.g. social competence, vocational skills. cultural: as part of our culture of which all pupils should have knowledge and experience. pleasurable: as a potential source of enjoyment. they add that the aims for low attainers do not differ from those stated above, although the priorities may differ depending on the needs of the learner. if the experiences in the classroom are not resulting in the learner gaining in any of these categories above, there remains little justification for keeping learners in the mathematics classroom. with the pending implementation of mathematical literacy in south africa (doe, 2002), which will result in all learners needing to pursue this subject until they leave school, we need to confirm that we do indeed have sufficient justification for keeping all learners learning mathematics. we need to ensure that even the low attainers will profit from the scope and aims of mathematics as outlined above. although i acknowledge that in practice computation has been interpreted as a prerequisite to any other mathematical knowledge (parmar & cawley, 1991), by continually focusing too much on this domain, are we allowing low attaining learners the full benefit of the definition and scope of mathematics? daniels and anghileri (1995: 23) suggest the following in response to this rhetorical question: to bring sen pupils to an understanding of the relationships and patterns that constitute mathematics itself, they will need to be involved with practical tasks, applying mathematics to “real-life” problems, exploring and investigating their findings and discussing their thinking with peers and teachers. the rest of this article suggests ways in which we can address this, but first examines possible characteristics and causes of low attainment in mathematics. possible characteristics and causes of low attainment kroesbergen and van luit (2003) draw on the work of goldman (1989), mercer (1997) and rivera (1997), and offer some general characteristics of learners who have difficulty in learning mathematics. these include: memory deficits, inadequate use of strategies for solving mathematics tasks, and deficits in generalisation and transfer of learned knowledge to new and unknown tasks. in this regard haylock (1991) adds the following to this list: reading and language problems, perceptual problems and poor spatial discrimination, social problems and mathematics anxiety. this is not to say that all low attainers exhibit most or even many of the characteristics outlined above, but that these are general observations from research within this field. in their book entitled, low attainers in mathematics 5 16: policies and practices in schools, denvir et al. (1982) offer the following list as likely causes of low attainment: physical, physiological or sensory defects; emotional or behavioural problems; impaired performances due to physical causes such as tiredness, drugs and general health; attitude, anxiety, lack of motivation; inappropriate teaching; too many changes of teachers (lack of continuity); general the theory of realistic mathematics education as a theoretical framework for teaching low attainers in mathematics 44 slowness in grasping ideas; cultural differences, english not first language; impoverished home background; difficulty in oral expression or in written work; poor reading ability; gaps in education, absence from school, frequent transfers from one school to another; immaturity, late development, youngest in the grade; low selfconcept leading to a lack of confidence (denvir et al., 1982: 19). they further subdivide these factors into three categories, which include: factors beyond the control of the school, factors partly within the school's control and factors that are directly within the control of the school. the causes, which they then identify as controlled by the school, include: • inappropriate teaching methods or content; • lack of suitable materials; • lack of responsiveness to learner's problems or lack of teacher's time to reflect on the learner's difficulties and plan suitable work; • a teacher's lack of detailed knowledge of the mathematics being taught, including a knowledge of which skills, concepts, etc are involved; • a teacher's inability to motivate and involve learners and organise work efficiently. (denvir et al., 1982: 21) also, feuerstein (1980) has suggested that many different reasons, ranging from genetic to environmental factors, explain low cognitive performance. abel (1983) takes the standpoint that environment rather than innate ability may be a key factor in learners’ performance in mathematics. referring to research reported by ginsberg, klein and starkey (1998) and gouws (1992) as examples, reusser (2000) proposes that there is convincing evidence that most observed failures and low performances in mathematics are due to insufficient teaching-learning environments and not due to genetic factors at all. he also states that learning difficulties that have a neuropsychological diagnosis are "substantially reinforced and shaped by environmental influences such as insufficient measures taken by the instructional and educational support systems" (reusser, 2000: 1). baroody and hume (1991) agree and make a case that most children who experience learning difficulties are recipients of instruction not suited to how children think and learn. this in turn puts the onus on the curriculum and instructional techniques (the environment) as opposed to the learner. in my opinion, these possible characteristics and causes identified in the preceding paragraphs, suggest that low performance or attainment in mathematics is something that can be "treated". in most cases, it is not an incurable condition that learners are born with, but something that develops as a result of the type of instruction learners receive and the teaching-learning environment (reusser, 2000) within which they experience mathematics. the implications of this for the inquiry i carried out were: that the instructional approach and teaching-learning environment to be applied in the intervention became central to the literature review and the subsequent choice of a theoretical framework. improving teaching and learning mathematics for low attaining learners i therefore agree with abel (1983), baroody and hume (1991) and reusser (2000) and work on the assumption that the environmental aspects of the mathematics teaching and learning can affect a learner’s performance. in order to identify the environmental aspects that might make a difference, literature by experts in the field of mathematics education and more specifically low attainment in mathematics was further reviewed. this was done to ascertain whether or not there were any common aspects that could be recognised within the literature. aspects suggested by various experts are foregrounded (using italics) in the paragraphs below and the common aspects that emerge are summarised in the final paragraph of this sub-section. in their book entitled secondary mathematics and special educational needs, daniels and anghileri (1995) examine the benefits of environmental aspects such as appropriate practical work, problem solving, games in the mathematics classroom, group work, co-operative learning, reciprocal teaching and the active participation of learners during lessons. they also stress the point that learning needs to be relevant to the lives of the low attaining learners in order for it to be meaningful. this does not however mean that all mathematics problems should be based in real-life contexts, as puzzles, games, patterns and brainteasers can also be used. i here want to refer to specific aspects that are relevant for creating conducive learning environments; for instance, denvir et al. (1982) encourage teachers to embrace the role of experimenters and to try out ideas developed by themselves and their colleagues. in doing so, they hayley barnes 45 encourage teachers to observe the low attainers in order to gain some insight into their "strengths and weaknesses, present state of knowledge, and to probable causes of the low attainment…" (1982: 50). this allows the teacher to plan suitable work for individuals that can be extended, adjusted or abandoned, depending on how effective it turns out to be. they in turn warn against continued emphasis on computations (arithmetic skills) at secondary school and motivate this with the indication from research that learners in the 12 to 15 age range show little improvement in their performance in this regard (hart, 1981). denvir et al. (1982) also propagate the value of learners discussing their work as well as the advantages of engaging in problem solving with low attainers. due to the poor memory for facts and procedures that many low attainers appear to have, the research discourages the use of instrumental instruction that relies heavily on memory, and instead encourages more emphasis on relational understanding. in doing so, they refer to the work of skemp (1971; 1989) relating to understanding. skemp (1971; 1989) differentiates between relational and instrumental understanding. on the one hand, he suggests that instrumental understanding is "rules without reasons" in that learners may possess the necessary rules, and ability to use them, without actually comprehending why or how that rule works. often learners will need to memorise more and more of these rules in order to avoid errors and this type of understanding therefore encompasses a "multiplicity of rules rather than fewer principles of more general application" (1989: 5). relational understanding, on the other hand, involves integrating new ideas into existing schemata and understanding both "what to do and why". although lower ability learners may need more substantial support than other able learners in constructing their own meanings and connections, this building up of a schema (or conceptual structure) becomes an intrinsically satisfying goal in itself and the result is, once learnt, more lasting. skemp (1989) uses an analogy of a stranger in a town to differentiate between the two types of understanding. one could have a limited number of fixed plans that take one from particular starting locations to particular goal locations in the town. he provides this as an example of instrumental understanding. on the other hand one could have a mental map (schema) of the town, from which one can produce, when needed, an almost infinite number of plans to guide one from a starting point to a finishing point, provided only that both can be imagined on the mental map (relational understanding). the work of haylock (1991) is significant because it discusses factors associated with low attainers, drawing on classroom-based research, and proposes a strategy for teaching learners in this regard. although focused on learners who are between 8 and 12 years old, haylock's book on teaching mathematics to low attainers can still be considered relevant for lower secondary learners (aged between 13 and 15). haylock's work foregrounds the following main themes: • the development of understanding as opposed to the learning of routines and procedures, • the importance of tending to language development in teaching mathematics, • the need to specify realistic and relevant objectives for the learners, • the aspect of numeracy and the basic ability to use a calculator effectively, • the use of small group games and finally, • the need to identify "purposeful activities in meaningful contexts" (1991: 5). haylock is of the opinion that it is necessary to maintain a balance between providing learners with success through the attainment of set objectives while also providing them with activities in meaningful contexts that they find relevant and purposeful. baroody and hume (1991) suggest that in order for mathematics instruction for low attainers to improve, it needs to focus on understanding, encourage active and purposeful learning, foster informal knowledge, link formal instruction to informal knowledge, encourage reflection and discussion and include socratic teaching (which involves a combination of the aforementioned elements). parmar and cawley (1991: 1) challenge the "routines and passivity that characterise arithmetic instruction for children with mild handicaps". they suggest that more approaches, which encourage learners to be active, productive learners and allow them the opportunity to demonstrate the extent of their thinking and creativity, are needed in special education classes. looking through the aspects above that pertain to the teaching of low attainers mentioned in this sub-section, one that appears repeatedly is the aspect relating to a greater involvement on the part of the learner in the learning process (i.e. the learner being more active). it is suggested more the theory of realistic mathematics education as a theoretical framework for teaching low attainers in mathematics 46 than once that in order to do this, learners need to be engaged in more meaningful or purposeful contexts, such as problem solving and games. other aspects referred to by more than one scholarly source are: the need to focus on the development of understanding and the importance of discussions, both between learners themselves and with the teacher. using these common aspects from the literature, and drawing on my own experience as a mathematics educator, a list of five aspects to include in the instructional approach to use in the intervention, was compiled. the following section outlines and examines these aspects in more detail. relevant environmental aspects in an instructional approach for low attaining learners in the previous section, the process that was used to identify the aspects explained in this section was illustrated. a clear demarcation between these aspects is however not intended, as they do overlap on a number of features. the five identified aspects to be focused on in the instructional approach of the intervention are: • more focus on relational and conceptual understanding as opposed to learning by rote and memorisation (instrumental understanding) • creating meaningful learning contexts that actively involve learners • greater emphasis on problem solving and less emphasis on computation and arithmetic skills • the importance of social interaction in the learning process (i.e. group work, reciprocal teaching, games, etc.) • the importance of language development and discussion with and between learners in teaching mathematics. more focus on understanding as demonstrated by skemp's (1971) differentiation between relational and instructional understanding, a chasm may exist between what learners are able to do and what they in fact understand. knowing what to do in a specific situation, but not necessarily understanding why it works, may limit the transfer of that procedure or skill. the increasing number of procedures that learners need to commit to memory in mathematics often results in learners in secondary school becoming confused or partly remembering and trying to apply procedures they have never fully understood (daniels & anghileri, 1995). understanding on the other hand promotes remembering and enhances transfer owing to the reduced number of bits of knowledge that need to be simultaneously held in the short-term memory (hiebert & carpenter, 1992). the understanding that comes from making connections, seeing how things fit together, relating mathematics to real situations and articulating patterns and relationships also carries with it a satisfaction which can further motivate low attaining learners (haylock, 1991). also relating to this point are the fundamental misconceptions that learners might have and the necessity to reveal these in the learning process in order to facilitate further understanding (hart, 1981; daniels & anghileri, 1995). adapting to a teaching and learning style that encourages understanding therefore also requires the study of learner errors that occur while solving mathematical tasks (reusser, 2000). this observation and analysis of errors provides a powerful means for analysing learner understanding as well as being a valuable source of information when used as diagnostic tools (booth, 1984; resnick et al., 1989). rather than being seen as indicators of failure, errors should be viewed as "learning opportunities and as challenges to clarify conceptual misconceptions" (reusser, 2000: 21). involving the learner through the use of meaningful contexts it is a common understanding that most people are less resistant to learning something new when they can see the purpose or meaning of it. this is equally important for children at school, especially with regard to mathematics. many people in fact currently hold an instrumentalist view of mathematics, which ernest (1988) proposes: …is the view that mathematics, like a bag of tools, is made up of an accumulation of facts, rules and skills to be used by the trained artisan skilfully in the pursuance of some external end. thus, mathematics is a set of unrelated utilitarian rules and facts. (1988: 10) in order to not restrict low attaining learners to this view but to instead meet the challenge of giving learners a full experience of what mathematics is, as defined by the revised national curriculum, we need to seriously consider the purposefulness of activities that we require learners to engage in. hayley barnes 47 when committed to a task that makes sense to them, there is a good chance that low attainers will surprise us with what they can achieve in mathematics (haylock, 1991). for this purpose, the teacher should take on the roll of learning facilitator and assist in piquing the curiosity of the learners in order to actively engage them in the task. active involvement can be regarded as any situation that creates questions or cognitive conflict in children's minds and that further encourages them to rethink their views (baroody & hume, 1991). denvir et al. (1982) suggest that low attainers may learn incidentally when they become involved in an absorbing activity and actively participate in the "struggle". they also add that through this activity children may learn because they spot inconsistencies in their thinking, which they then try to resolve. de korte (1995) lists learning as being "situated" as one of the major features of effective learning processes in mathematics. by this, he means that "learning essentially occurs in interaction with social and cultural context and artefacts, and especially through participation in cultural activities and practices" (1995: 41). greater emphasis on problem solving as already mentioned, many mathematics interventions currently focus on improving computation skills of low attaining learners. from a number of observations made during school visits, denvir et al. (1982) concluded that some of the children who do not master arithmetic skills at primary school spend most of their secondary school repeating this computation with very little success. compounding this is the fact that problem solving is often seen as an activity that is considered unsuitable for low attainers as, amongst other reasons, there are so many other skills to be practised that no time is left for such a luxury (and here clearly views differ on what is regarded as luxury and necessity)! another reason cited for this is that the basic mathematical knowledge of low attainers is so weak that they will not be able to apply it to the solution of problems. this raises the question as to the usefulness and purpose of this basic mathematical knowledge if it cannot be used when required to solve a problem! as noted by the cockcroft report (1982: para. 249): mathematics is only 'useful' to the extent to which it can be applied to a particular situation, and it is the ability to apply mathematics to a variety of situations to which we give the name 'problem solving'. some of the benefits of the problem solving approach for low attaining learners as identified by trickett and sulke (1988) include "better ability and willingness to question, to transfer and apply their mathematics, and to sort out even quite difficult problems" (as cited by daniels & anghileri, 1995: 66). however, the understanding and solving of even simple mathematical word problems is a complex process that requires skilful interaction of at least three kinds of knowledge: linguistic, situational and mathematical (reusser, 2000). learners who are therefore severely lacking in the relevant types of knowledge and skills may instead adopt coping strategies that bypass the logic of mathematical sense-making activities. such learners in turn need the guidance of "effective pedagogical settings" (2000: 23). this includes presenting problems in contexts that are more familiar, realistic and therefore also meaningful to the learner, while also providing the necessary instruction and strategies to help low attainers to analyse, reflect and practice the overall required sequences in understanding and solving different types of problems. social interaction as part of learning cobb and bauersfeld (1995) identify two general theoretical positions on the relationship between social process and psychological development. while one favours the social and cultural processes (collectivism), the other gives priority to the individual autonomous learner (individualism). one of the most well known theories relating to collectivism is that of vygotsky (1979 as cited in cobb & bauersfeld, 1995) where "mathematical learning is viewed primarily as a process of acculturation" (1995: 3). individualism on the other hand is exemplified by neo-piagetian theories, where the focus is on the individual, autonomous learner as he or she takes part in social interactions. while there appears to be an apparent opposition between these two views, both social and cognitive processes have their place in the learning of mathematics. cobb and bauerfeld (1995: 7) cite the following quotation from saxe and bermudez (1992): an understanding of mathematical environments that emerge in children's everyday activities requires the coordination of two analytic perspectives. the first is a constructivist treatment of children's mathematics; children's mathematical environments cannot be understood apart from children's own the theory of realistic mathematics education as a theoretical framework for teaching low attainers in mathematics 48 cognizing activities…the second perspective derives from socio-cultural treatments of cognition….children's construction of mathematical goals and sub-goals is interwoven with the socially organised activities in which they are participants. (1992: 2-3) without getting further into these theories, it suffices to say that social interaction remains an integral part of learning. interactions with both peers and teachers can enhance learning through creating opportunities for learners to share understandings and verbalise thought processes (daniels & anghileri, 1995). some suggested forms of this are group work, reciprocal teaching, sharing of strategies and games. schoenfield (1985) supports the use of small group work for the following reasons: opportunities for teacher assessment, an opportunity for learners to practice collaboration, less secure learners can watch more capable peers struggle, and decision making in a group facilitates the articulation of reasoning and knowledge. palinscar and brown (1988) share an additional instructional procedure for small groups that they refer to as “reciprocal teaching”. this mode of cooperative learning assumes the form of a discussion between the members of the instructional group and the teacher (or another facilitator which could also be a learner) who acts as a leader and a respondent. four strategies are used to direct the discussion. the leader first frames a question to which the group members respond. a piece is then read and the leader summarises the gist of that piece. the group then comments and elaborates on the leader's summary and any necessary points are clarified. finally, the leader prepares to move onto the next portion of text by making predictions about the upcoming content. reciprocal teaching is underpinned by the premise that expert-led social interactions can provide a major impetus to cognitive growth (following along the lines of vygotsky). it therefore plays an important role in learning and has been used by palinscar and brown (1988) as a strategy for collaborative problem solving. dockrell and mcshane (1992) differentiate between learners being able to use a strategy and knowing when to use it. they hold the view that children are often unaware of the effectiveness of a strategy in relation to a particular problem and therefore do not make adequate use of it. however, when learners are encouraged to share their strategies and receive feedback that indicates the positive effect of the strategy, they tend to increase their use of it. the authors also argue that there is a dynamic relation between a knowledge base and strategies. they suggest that: strategies often play a vital role in establishing a knowledge base, but once acquired, the role of strategies may become less important within the domain, because the relevant knowledge is available for retrieval. in cases of learning difficulties, it is often the case that the acquisition of knowledge is an issue. thus, the use of strategies becomes a critical factor. strategies require a knowledge base that provides the appropriate information on which the strategy can operate. in considering strategy training it is important to consider, as a first step, whether or not the child's knowledge base contains the information required for successful execution of the strategy. (dockrell & mcshane, 1992: 188) in the extensive meta-analysis of interventions for students with learning disabilities carried out by swanson et al. (1999), they classify studies within the analysis into two general approaches, namely direct instruction and strategy instruction. strategy instruction includes verbal interaction between the teacher and the learners and the learner is viewed as a collaborator in the learning process. the teacher also provides individual feedback and makes use of verbal modelling and "think-aloud" models to solve a problem. from their first tier of analysis it was concluded that: "strategy instruction produces larger effect sizes than those studies that do not use such procedures" (1999: 220). sharing of strategies can therefore be included as an important aspect that can contribute to effective learning taking place in the teaching of low attainers. games are often regarded as primary school activities or something that can be used to fill up time or as an end of term activity. the united kingdom in particular has recognised the powerful environments created through a game; so much so that it has recently incorporated games that enable assessment into its national curriculum assessment. some of the benefits of games are that they provide the opportunity for learners to practise and consolidate routine procedures and number skills in a motivating environment that is neither threatening nor monotonous (daniels & anghileri, 1995). they also enable learners to hayley barnes 49 develop problem solving strategies and aid in the acquisition and development of concepts. the opportunity is also created for teachers to observe their learners' thinking strategies and to interact with learners on a less formal level (ernest, 1986; haylock, 1991; daniels & anghileri, 1995). the importance of language development and discussion the effect of language on the learning of mathematics is a widely researched and debated topic not only internationally but also in south africa (e.g. howie, 2002; setati, 2002). while there is no magic formula or solution as to how this issue should be addressed, specifically with regard to low attainers, it nonetheless remains a pertinent issue when designing programs or interventions for these learners. poor language skills such as reading, writing and speaking are often associated with low attainment in mathematics and, in addition to that, mathematics has its own set of language patterns, symbols and vocabulary. a major part of developing an understanding of mathematics involves learning to handle these and make connections between symbols and their corresponding terminology and meaning (haylock, 1991). daniels and anghileri (1995) stress that speech and written language are the tools of mathematical dialogue. the development of some aspects of mathematical thought may be constrained through a lack of access to these tools. as dockrell and mcshane (1992) point out, when solving a problem it is crucial that the learner first understands the problem before planning and executing a method for solving it. understanding is based on the child's cognitive and linguistic skills; planning a method involves constructing a mathematical representation of the problem; carrying out the plan involves executing the mathematical procedures that have been selected…difficulties can arise in the comprehension of the problem, the construction of the mathematical model, or in the execution of strategies in solving word problems. however, it seems to be the complexity of the text of the word problem and the availability of a suitable basis for its mathematical representation that are the major determinants of performance (1992: 139). both the phrases "complexity of text" and "mathematical representation" in the quotation above relate to use of language, in different senses however. the first relates to the written and spoken language of, for example, english. the second refers to mathematics as a collection of symbols, notation and terminology and how these all connect. difficulties in either (and in many cases both) of these will indeed affect learners' performance and possible development in mathematics. it would be useful if interventions aimed at assisting low attainers could therefore include components that can be used to diagnose and address the complexity of mathematics as a language and language as a tool for mathematics. this process can be assisted by the use of discussions in the classroom where learners are encouraged to verbalise their understanding, thoughts, solutions and ideas on the problems and tasks presented to them. this is not a simple task, however. in research carried out by baxter, woodward and olson (2001), it was indicated that whole-class discussions are often dominated by verbal, capable learners, while the low attainers tend to remain passive. when they do in fact respond, their answers are typically simple and at times incomprehensible (ball, 1993; chard, 1999, as cited in baxter et al., 2002). baxter et al. (2002) report on the results and dilemmas that emerged during a year-long case study they carried out that focused on ways to include these learners in class wide discussions of problem solving. one of the major dichotomies they allude to is that remedial environments that bring together only low achieving learners are not likely to result in rich, learner-centred discussions, while regular education classrooms may not provide the most optimal solution to the problem. they therefore suggest the use of small group work and sharing of strategies (without necessarily identifying the best solution) as possible interventions to alleviate the problem. this section has presented the five aspects and has examined each of them in more detail. knowing that these aspects were to be the focus of the instructional approach in the intervention, a theoretical framework was sought that would accommodate all of them. the domain-specific theory of realistic mathematics education (rme) from the freudenthal institute in the netherlands was selected as the most appropriate theory to accomplish this task and the theoretical underpinnings of rme are provided in the section below, followed by an explanation of why rme was selected for working with these low attainers. the theory of realistic mathematics education as a theoretical framework for teaching low attainers in mathematics 50 the theory of realistic mathematics education (rme) realistic mathematics education has its roots in hans freudenthal's interpretation of mathematics as a human activity (freudenthal, 1973; gravemeijer, 1994). to this end, freudenthal accentuated the actual activity of doing mathematics; an activity, which he proposed should predominantly consist of organising or mathematising subject matter taken from reality. learners should therefore learn mathematics by mathematising subject matter from real contexts and their own mathematical activity rather than from the traditional view of presenting mathematics to them as a ready-made system with general applicability (gravemeijer, 1994). these real situations can include contextual problems or mathematically authentic contexts for learners where they experience the problem presented as relevant and real. the verb mathematising or the noun thereof mathematisation implies activities in which one engages for the purposes of generality, certainty, exactness and brevity (gravemeijer, cobb, bowers & whiteneack, as cited in rasmussen & king, 2000). through a process of progressive mathematisation, learners are given the opportunity to reinvent mathematical insights, knowledge and procedures. in doing so learners go through stages referred to in rme as horizontal and then vertical mathematisation (see figure 1). horizontal mathematisation is when learners use their informal strategies to describe and solve a contextual problem and vertical mathematisation occurs when the learners' informal strategies lead them to solve the problem using mathematical language or to find a suitable algorithm (treffers, 1987). for example, in what we would typically refer to as a "word sum", the process of extracting the important information required and using an informal strategy such as trial and error to solve the problem, would be the horizontal mathematising. translating the problem into mathematical language through using symbols and later progressing to selecting an algorithm such as an equation could be considered vertical mathematisation, as it involves working with the problem on different levels. the traditional formal and authoritarian approach to teaching mathematics that has dominated in south african classrooms for a number of years has not afforded learners many opportunities to make use of horizontal mathematisation. mathematics lessons are often presented in such a way that the learners are introduced to the mathematical language relevant to a particular section of work and then shown a few examples of using the correct algorithms to solve problems pertaining to the topic before being given an exercise or worksheet to complete (venter, barnes, howie & jansen van vuuren 2004). the exercises or worksheets are usually intended to allow learners to put the algorithms they have been taught into practice and may even contain some contextual problems that require the use of these algorithms. according to the rme model depicted in figure 1, this type of approach places learners immediately in the more formal vertical mathematisation process. the danger in this is that when learners have entered that process without first having gone through a process of horizontal mathematisation, a strong possibility exists that if they forget the algorithms they were taught, they do not have a strategy in place to assist contextual problem mathematical language describing solving algorithm figure 1 representation of horizontal and vertical mathematisation horizontal mathematisation ( ); vertical mathematisation ( ) source: adapted from gravemeijer, 1994. hayley barnes 51 them in solving the problem. as pointed out in the literature in the previous section, this is especially prevalent with low attainers. this experience can be equated to someone being shown and told what is on the other side of a river and being expected to use what is there for their own benefit. however, they are not given or shown the bridge that assists one in crossing to the other side in order to make proper use of what is there. the horizontal mathematisation process provides this bridge. this section began with an overview of the theoretical underpinnings of rme. the two subsections below endeavour to expand on this synopsis by briefly introducing two important tenets of the theory of rme, namely: the role of developmental research in continually developing and refining the theory, and the instructional design principles that the theory encompasses. developing realistic mathematics education the rme theory is one that is constantly "under construction", being developed and refined in an ongoing cycle of designing, experimenting, analysing and reflecting (gravemeijer, 1994). developmental research plays a central role in this process and, in contrast to traditional instructional design models, focuses on the teaching-learning process, focusing in specifically on the mental processes of learners (rasmussen & king, 2000). cyclic processes of thought experiments and instructional experiments form the crux of the method of developmental research and serve a dual function (see figure 2). they both clarify researchers' learning about learners' thinking and address the pragmatic affairs of revising instructional sequences (gravemeijer, 1999). instructional sequences are designed by the curriculum developer who starts off with a thought experiment that imagines a route learners could have invented for themselves. the lesson is implemented and the actual process of learning that takes place in relation to the anticipated trajectory is analysed. this analysis can then provide valuable information in order to revise the instructional activities. rme instructional design principles gravemeijer (1994, 1999) identifies three key heuristic principles of rme for this process of instructional design, namely: • guided reinvention through progressive mathematisation • didactical phenomenology • self developed or emergent models guided reinvention through progressive mathematisation the principle of guided reinvention requires that well-chosen contextual problems be presented to learners that offer them opportunities to develop informal, highly context-specific solution strategies (doorman, 2001). these informal solution procedures may then function as foothold inventions for formalisation and generalisation, a process referred to as "progressive mathematising" (gravemeijer, 1994). the reinvention process is set in motion when learners use their everyday language (informal description) to structure contextual problems into informal or more formal mathematical forms (armanto, 2002). the instructional designer therefore tries to compile a set of problems that can lead to a series of processes that together result in the reinvention of the intended mathematics (doorman, 2001). the idea is not that learners are expected to reinvent everything on their own but that freudenthal's concept of "guided reinvention" should apply (freudenthal, 1973). this should in thought exp. thought exp. thought exp. thought exp. thought exp. instruction exp. instruction exp. instruction exp. instruction exp. figure 2. developmental research, a cumulative cyclic process (gravemeijer & cobb, 2002). the theory of realistic mathematics education as a theoretical framework for teaching low attainers in mathematics 52 turn allow learners to regard the knowledge they acquire as knowledge for which they have been responsible and which belongs to them. with guidance, the learners are afforded the opportunity to construct their own mathematical knowledge store on this basis. the word "realistic" in the rme theory does not indicate however that everyday contexts need to be continuously sought or used to motivate learners to reinvent the mathematics. rather, the contexts selected for use in the process of instructional design should be experientially real for learners, relevant and challenging in order to act as a catalyst for progressive mathematisation (freudenthal, 1973; gravemeijer, 1994; treffers, 1987). the principle of didactical phenomenology this principle was advocated by freudenthal (1973) and implies that in learning mathematics, one has to start from phenomena meaningful to the learner, and that implore some sort of organising be done and that stimulate learning processes. according to treffers and goffree (1985) this principle should fulfill four functions: • concept formation (to allow learners natural and motivating access to mathematics), • model formation (to supply a firm basis for learning the formal operations, procedures, and rules in conjunction to other models as the support for thinking), • applicability (to utilise reality as a source and domain of applications), • practice (to exercise the specific abilities of learners in applied situations). the principle of emergent or self developed models this third principle for instructional design in rme plays an important role in bridging the gap between informal and formal knowledge (gravemeijer, 1994). in order to realise this principle, learners need to be given opportunities to use and develop their own models when solving problems. the term "model" is understood here in a dynamic, holistic sense and learners enhance their models by using their former models and their knowledge about mathematics. as a consequence, the symbolisations that comprise the model and those rooted in the process of modelling can change over time. learners therefore progress from what is termed a "model-of" a situated activity to a "model-for" more sophisticated reasoning (gravemeijer & doorman, 1999 as cited in kwon, 2002). this is quite different from the former (and in many instances still current) practice in south africa, where learners are presented with a model or algorithm by the teacher and then given repeated opportunities and problems to practise using that model. why rme for low attainers in the preceding sections in this article, literature on the teaching and learning of mathematics to low attaining learners was examined and common environmental aspects that could be incorporated into the instructional approach of an intervention were identified. the theory of realistic mathematics education was then proposed as a possible theory to drive the design and implementation of such an intervention. the theoretical underpinnings of rme were subsequently outlined. this section expands on these underpinnings and the identified aspects in order to substantiate the choice of rme. to facilitate this argument, rme is discussed in relation to three other global trends in mathematics education in order to highlight some of the unique features, which make it the recommended theory for working with low attainers. rme in relation to other global innovations in mathematics education treffers (1987) identifies three global trends in mathematics education, which he refers to as the arithmetical, structural and empirical trends. the didactical approach of the arithmetical trend (also known as “new math”) is similar to that of drill and practice instruction in the past with the main objectives being the teaching of certain arithmetic routines, notations and rules and the transfer of knowledge. the influence of the arithmetical trend on rme includes, amongst others, the inclusion of puzzles, practice games and ideas about learning basic operations. the mathematical activity in the structural trend is mainly directed towards the construction of formal mathematical structures and aims less at the relationships with the reality of everyday experience. the approach is best expressed by the work of dienes and makes use of "imagined" reality and "artificial surroundings" as a basis for mathematical analysis and exploration of mathematical structures. treffers (1987) presents the shortcoming of this approach as being the large hayley barnes 53 gap between the constructed world in which the mathematics takes place and everyday reality. this makes it almost impossible to connect the two. in spite of this criticism, influences from the structural trend are visible in the work of rme, for example, in the use of arrows and "machines" in the basic operations, in the approach to problems of reasoning via arrow diagrams and the attention paid to structuring aids such as number lines, charts, grids, diagrams and graphics. in contrast to the structural trend, the empirical trend takes it subjects for mathematics study almost exclusively from the biological, physical or social reality, which means that the starting point for mathematical activities lies within "the neighbourhood of the child's everyday experience" (treffers, 1987: 10). the lack of a mathematical source of inspiration and strict methodological structure sometimes results in a badly organised collection of activities though, and it becomes problematic to ensure that children are not repeating the same experience at different stages of their school life (biggs, 1971 as cited in treffers, 1987). some similarities between rme and the empirical trend include the use of charts, graphs and materials, the connection with actuality and the attention paid to the measuring aspect of number in early mathematics education. one of the main differences between the two, however, is that while rme draws on everyday contexts, the use of "imagined" realities is also subscribed to, which is not the case in the empirical approach. the main purpose for presenting this background has been to indicate how elements of global trends, such as these, have influenced the development of the theory of rme. as previously mentioned though, the main thrust of rme is that of viewing mathematics as a human activity (freudenthal, 1973) and the subsequent central element of mathematisation (treffers, 1987). this central element of rme is now further investigated and discussed in relation to the other three global trends. mathematisation for low attainers treffers (1987: 247) describes mathematising as "…the organising and structuring activity in which acquired knowledge and abilities are called upon in order to discover still unknown regularities, connections, structures." furthermore, mathematising is directed towards: the acquisition of factual knowledge, the learning of concepts, the attainment of skills and the use of language and other organising skills in solving problems that are, or are not, placed in a mathematical context. (1987: 52-53) this process or activity alone already accommodates most of the aspects suggested for inclusion in the instructional approach of the intervention for low attainers. to place the instructional approach within one of the other three trends would not allow all five of the suggested aspects to be included. to take this a step further, let us look more closely at the differentiation treffers (1987) makes between horizontal and vertical mathematisation, as referred to previously. in his words, in general one can say that ‘horizontal mathematisation’ consists of a schematisation of the area that makes it possible to attack the problem by mathematical means. the activities that follow and that are related to the mathematical process, the solution of the problem, the generalisation of the solution and the further formalisation, can be described as ‘vertical mathematistion’. (1987: 71) treffers admits that an exact distinction is hard to make but that the distinction is meaningful in that it demonstrates how activities such as constructing, experimenting and classifying also fit into the process of mathematising along with the more common ones of symbolising, generalising and formalising. making a schematic comparison between the other three trends and rme, in relation to the use of horizontal and vertical mathematisation, is also a helpful way of table 1. classification by treffers of inclusion of horizontal and vertical mathematisation in four different mathematics education trends. trends mathematising horizontal vertical mechanistic (arithmetic) empiricist + structuralist + realistic + + the theory of realistic mathematics education as a theoretical framework for teaching low attainers in mathematics 54 demonstrating why rme is being suggested as the domain-specific theory for use with low attainers. in this regard, treffers (1987) presents the above classification in table 1. in the mechanistic (or arithmetic) trend, no real phenomenon is used as a source of mathematical activity, little attention is paid to applications and the emphasis is on rote learning. this results in weaknesses in both horizontal and vertical mathematisation. the empiricist trend places a strong emphasis on horizontal mathematisation in that the emphasis is on environmental rather than on mental operations. formal mathematical goals do not feature as a high priority and there is little pressure for learners to pass to a higher level, thus demonstrating the weakness with relation to vertical mathematisation. in structuralist instruction, where mathematical structures are emphasised, the vertical component is dominant. this is evident in this approach in that the principal part of the mathematical activity operates within the mathematical system. instead of real phenomena, embodiments and materialisations of mathematical concepts or structures or structural games are used to create a concrete basis for learners from which to work and real phenomena subsequently do not function as models to support operating within the mathematical system. in realistic mathematics instruction, however, careful attention is paid to both components. as treffers (1987) puts it, this means that the phenomena from which the mathematical concepts and structures arise are implicitly used both as source and domain of application. this, according to the tenet of the theory, creates for the learner the possibility of concept attainment by orienting himself to a variety of phenomena, which benefits the building of formal mathematical concepts and structures and their application (1987: 251). from the literature reviewed in relation to low attainers, it appears that a lot of the teaching and learning in this domain has tended towards the mechanistic (arithmetic) and structuralist trends. the focus of the instruction and assessment has therefore been in the vertical component of table 1, which could explain the dominance of instrumental rather than relational understanding. the major activities in this component are symbolising, formalising and generalising. as low attainers often struggle with these, they may have experienced repeated failure with continued emphasis on this component. misconceptions may also be hampering them within the vertical component and may have developed due to a lack of adequate exposure to constructing, experimenting and classifying, which lie in the horizontal component. in order to rectify this, it therefore seems necessary to select an instruction theory that will pay careful attention to both components. learners are thereby also afforded more opportunities to bridge the gap between their informal understanding and formal knowledge. this is not a once-off or linear process, however, and should be viewed as a continual cycle. the desired outcome is that learners acquire the cyclical strategy of moving between horizontal and vertical mathematisation in order to assist them in improving their understanding and subsequently their performance in mathematics. from the discussion above, it should be clear that rme provides more of a focus on relational and conceptual understanding as opposed to rote learning. in order to do this, meaningful learning contexts are created (which can be from everyday situations or "imagined" reality) that facilitate the process of progressive mathematisation. this means that learners are actively involved in solving problems and constructing their own meaning and understanding. by continual use of horizontal and vertical mathematisation, learners are using mathematical symbols and language interchangeably and hence tending to the importance of language development. one of the general principles of progressive mathematisation, that has not yet been mentioned, is that of "interactivity" (treffers, 1987). according to this principle, learners are confronted with the constructions and productions of their peers, which: …can stimulate them to shorten their learning path, to help themselves up on procedures of others, to become aware of the drawbacks or advantages of their own productions, and that copying others' work slavishly will not aid their own progress. in brief, the learning process is part of the interactive instruction where individual work is combined with consulting fellow students, group discussion, collective work reviews, presentation of one's own productions, evaluation of various constructions on various levels and explanation by the teacher. (treffers, 1987: 249) hayley barnes 55 this principle satisfies the importance of social interaction, an aspect that was earlier identified as being relevant. the central theme of rme, mathematising, therefore adequately incorporates all the aspects suggested for inclusion in an instructional approach for low attainers. conclusion in this article, the choice of the term “low attainer” was explained and related terminology mentioned. primary sources of literature in this domain were identified and consulted in order to present some general characteristics and causes of low attainment. these sources were also examined for common environmental aspects and practices to be included in the suggested instructional approach of an intervention for low attaining learners. these aspects were listed and explained. the theory of realistic mathematics education (rme) was then suggested as the theoretical framework to drive the design and implementation of such an intervention. the instructional approach suggested for low attainers is therefore embedded in this domainspecific theory of rme. the theoretical underpinnings of rme were then discussed and rme was examined as an instructional approach to teaching mathematics in relation to three other global trends in this domain. through this comparison, it was shown how rme is able to satisfy all the aspects suggested in the instructional approach for low attainers. it is therefore recommended that the theory of rme be considered in the design and implementation of interventions with low attainers. references abel, t. m., 1983, “women and mathematics: research vs achievement in education”, paper presented at the 9th annual midyear conference of the american educational research association (aera), tempe armanto, d., 2002, teaching multiplication and division realistically in indonesian primary schools: a prototype of local instruction theory, unpublished doctoral dissertation: university of twente, the netherlands baroody, a.j. & 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vuuren, s., 2004, mpumalanga secondary science initiative learner progress research project. pretoria: centre for evaluation and assessment “facts do not cease to exist because they are ignored.” aldous huxley article information authors: yael shalem1 ingrid sapire1 belinda huntley2 affiliations: 1school of education, university of the witwatersrand, south africa2mathematics department, st john’s college, south africa correspondence to: ingrid sapire postal address: division of curriculum, school of education, private bag 3, wits 2050, south africa dates: received: 07 sept. 2012 accepted: 22 mar. 2013 published: 22 may 2013 how to cite this article: shalem, y., sapire, i., & huntley, b. (2013). mapping onto the mathematics curriculum – an opportunity for teachers to learn. pythagoras, 34(1), art. #195, 10 pages. http://dx.doi.org/10.4102/ pythagoras.v34i1.195 copyright notice: © 2013. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. mapping onto the mathematics curriculum – an opportunity for teachers to learn in this original research... open access • abstract • introduction • the data informed practice improvement project    • curriculum standards – teacher knowledge and interpretation       • curriculum mapping       • accounting – an opportunity to learn from the examined curriculum       • the project process       • the curriculum mapping activity       • sample of items       • data analysis       • validity and reliability       • ethical considerations • findings    • overall alignment results    • increased confidence and improved judgement    • mapping icas items and own test items    • gaps between the intended and the enacted curriculum • conclusion • acknowledgements    • competing interests    • authors’ contributions • references • footnotes abstract top ↑ curriculum mapping is a common practice amongst test designers but not amongst teachers. as part of the data informed practice improvement project’s (dipip) attempt to de-fetishise accountability assessment, teachers were tasked to investigate the alignment of a large-scale assessment with the south african mathematics curriculum. about 50 mathematics teachers from grade 3–9 worked in groups together with subject facilitators from the gauteng department of education and a university postgraduate student or lecturer who acted as group leader. the first project activity, curriculum mapping, provided a professional development opportunity in which groups mapped mathematical assessment items to the assessment standards of the curriculum. the items were taken from three sources: the 2006 and 2007 international competitions and assessments for schools tests and from ‘own tests’ developed by the groups in the last term of the project. groups were required to analyse the knowledge base underlying test items and to reflect on what they teach in relation to what the curriculum intends them to teach. they used a protocol (mapping template) to record their responses. this article deals with the question of how to transform data collected from large-scale learner assessments into structured learning opportunities for teachers. the findings were that through the curriculum mapping activity, groups became more aware of what is intended by the curriculum and how this differs from what is enacted in their classes. the findings were also that the capacity of groups to align content was better when they worked with leaders and that with more experience they gained confidence in mapping test items against the curriculum and made better judgments in relation to curriculum alignment. involving teachers in the interpretation of both public assessment data and data from their own classroom activities can build their own understanding of the knowledge base of test items and of the curriculum. introduction top ↑ as a policy lever for benchmarking standards and for monitoring performance, the south african department of basic education has embarked on a number of national initiatives to collect learner assessment data. a variety of international and local large-scale systemic assessments have been conducted in the country. to date the data from these systemic assessments, the test items as well as the test results, have been used by mathematical and language experts, economists and statisticians at a systemic level and predominantly for benchmarking. teachers have not participated in the production of this evidence nor has the opportunity for developing teachers’ interpretive skills of such data been taken up. the question is how to transform data collected from large-scale learner assessments into structured learning opportunities for teachers. this article deals with this question. merely having another set of data in the form of benchmarking, targets and progress reports that ‘name and shame’ schools leads to resentment and compliance but not to improvement of learning and teaching (earl & fullan, 2003; mcneil, 2000). in south africa, kanjee (2007) sums up the challenge: for national assessment studies to be effectively and efficiently applied to improve the performance of all learners, the active participation of teachers and schools is essential. … teachers need relevant and timeous information from national (as well as international) assessment studies, as well as support on how to use this information to improve learning and teaching practice. thus a critical challenge would be to introduce appropriate policies and systems to disseminate information to teachers. for example, teacher-support materials could be developed using test items administered in national assessments. (p. 493) it appears that in using annual national assessments the south african department of basic education is aiming to provide teachers with timeous information from national assessments to guide planning and monitor progress (department of basic education, 2010). what is not clear is how the department is planning to support teachers on how to use this information to improve learning and teaching practice. very little attempt has been made to involve teachers in data interpretation and not enough emphasis has been placed on the potential value of the data available from these systemic evaluations for informing teaching and learning practices. international research has engaged with the question of how to use assessment data beyond benchmarking (earl & fullan, 2003; earl & katz, 2005; katz, earl & ben jaafar, 2009). in thinking about this question, katz, sutherland and earl (2005) drew an important distinction between two very different kinds of practices in benchmarking: ‘accounting’, which is the practice of gathering and organising of data, and ‘accountability’, which refers to teacher-led educational conversations about what the data means and how it can inform teaching and learning. katz et al.’s (2005) distinction is very important and is in line with elmore’s (2002) and hargreaves’s (2001) important arguments. hargreaves (2001, p. 524) argues that the future of collegiality may best be addressed by (inter alia) taking professional discussion and dialogue out of the privacy of the classroom and basing it on visible public evidence and data of teachers’ performance and practices, such as shared samples of student work or public presentations of student performance data. elmore (2002) claims that teachers can be held accountable for their performance only if they have a deep sense of the demands made upon them. although this may seem obvious, the challenge lies in identifying what counts as making accountability standards explicit. literature on professional development programmes for teachers shows that piecemeal forms of intervention are not effective (borko, 2004; cohen & ball, 1999; earl & katz, 2005; elmore & burney, 1997; katz et al., 2009). a broad consensus seems to emerge around the following claims: firstly, that teachers require continuous interactive support over a substantial period of time. secondly, that teacher learning should be focused on specific (and few in number) educational objects and guided by an expert who is acting as a critical friend. thirdly, that within the current emphasis on accountability, professional conversations by teachers, in support networks (broadly referred to as ‘professional learning communities’), can provide teachers with a productive opportunity to cultivate a sense of ownership of what the data means, specifically in relation to their current practices. the data informed practice improvement project top ↑ working with teachers on interpretation of learner assessment data was the central goal of the data informed practice improvement project (dipip), phase 1 and phase 2 (shalem, sapire, welch, bialobrzeska & hellman, 2011). the dipip project provided a context for professional conversations in which mathematics teachers, together with university academics, graduate students and department-based subject advisors, discussed assessment data. in these discussions, groups were dealing with information from the assessment data that could be used to think about reasons for learners’ errors, map the test items to the national curriculum statements (ncs), read and discuss academic texts about mathematical concepts (e.g. the equals sign) and learner errors related to these, develop lesson plans, and reflect on videotaped lessons of some teachers teaching from the lesson plans.the positive outcomes of research done on the efficacy of professional learning communities served to inform the approach used in this project (brodie & shalem, 2011). the term ‘professional learning communities’ generally refers to structured professional groups, usually school-based, providing teachers with opportunities for processing the implications of new learning (timperley & alton-lee, 2008). commonly, professional learning communities are created in a school and consist of school staff members or a cross section of staff members from different schools in a specific area of specialisation. the groups in the dipip project were structured differently and included teachers and practitioners with different knowledge bases and role specialisations (see below). as professional learning communities, the groups worked together for a long period of time (weekly meetings during term time at the wits education campus for up to three years from 2007–2010), sharing ideas, learning from and exposing their practices to each other. in these close-knit communities, teachers worked collaboratively on curriculum mapping and error analysis, lesson and interview planning, test setting and reflection. to provide the basis for a systematic analysis of learners’ errors, test items and learner achievement data of an international standardised multiple-choice test, the 2006 and 2007 international competitions and assessments for schools (icas), was used.1 for the curriculum mapping activity the groups were tasked with investigating the alignment of the icas tests items (not learner achievement data) with the curriculum at the time, that is, the ncs for mathematics (department of education, 2002). this article focuses on the nature and outcome of the curriculum mapping activity, presenting findings on how the curriculum mapping activity provided groups of practitioners an opportunity to engage with and reflect on the curriculum, and discussing in what ways and to what extent this activity succeeded. curriculum standards – teacher knowledge and interpretation there are two main different forms of curriculum. the first is skills based and presents a collection of statements (outcomes and assessment standards). the second is content based and its form foregrounds the conceptual structure of the intellectual field from which it selects specific subject matter. research in south africa has shown that an outcomes-based curriculum provides weak signals to teachers about coverage, sequence and progression. upon the findings of several national investigations, the ncs was replaced with a content-based curriculum (department of basic education, 2011). there is hope that the provision of a curriculum that gives better signals on content and forms of learning will enable teachers to implement that curriculum more effectively. we argue that it is one thing to design a better curriculum, but it is a very different matter to achieve teachers’ understanding of what standard is required for the grade they are teaching and what content they should focus on in their teaching.teachers’ understanding of accountability demands, their consent and their readiness to accept change are interrelated processes (shalem, 2003), but policymakers often assume that curriculum standards make policy requirements sufficiently clear to teachers. in practice, ‘reading’ the curriculum requires an application of teacher knowledge. shulman (1986) refers to three categories of teacher knowledge, namely, pedagogic knowledge, content knowledge and pedagogic content knowledge. in order to properly interpret the curriculum, teachers are expected to draw on their subject matter knowledge and to contextualise curriculum standards within their learning environment, taking into account the needs of their learners. in terms of shulman’s categories of knowledge, this means that teachers need to draw on both their content knowledge and pedagogic content knowledge in order to interpret and apply the curriculum. more specifically, ball’s sixth domain of teacher knowledge, ‘horizon knowledge’, is useful here. it refers to teacher knowledge ‘of how mathematical topics are related over the span of mathematics included in the curriculum’ (ball, thames & phelps, 2008, p. 403). following the work by ball et al., one can say that in order to set high expectations for their learners, in addition to their specialised mathematical knowledge which straddles six domains of mathematical knowledge for teaching, teachers need to understand the sequence and progression of the mathematics they teach. teachers need to understand what the curriculum aims to achieve in an earlier grade and in what ways the topics they teach connect to the conceptual development of the same concept in a later grade. in this way, teachers could better understand the standards required of the curriculum. international empirical research shows that curriculum statements about assessment standards, together with results of various standardised assessments, do not, in themselves, make standards clear (darling-hammond, 2004; katz et al., 2009). empirical research in south africa has identified misalignment between the demands of the curriculum, teaching and assessment (reeves & muller, 2005). classroom research suggests that many teachers simply ignore important aspects of the ncs and continue to teach poorly what they taught before (brodie, jina & modau, 2009; chisholm et al., 2000; fleisch, 2007; jansen, 1999). there are a variety of reasons for this overarching finding. research in south africa gives primacy to two inter-related explanations: poor teacher knowledge, in particular subject matter knowledge, and poor signalling of the (ncs) curriculum (taylor, muller, & vinjevold, 2003). the argument is that the outcomes-based curriculum provided teachers with very weak signals as to what content should be made available to learners and how this should be done. many teachers in south africa lack strong content knowledge to ‘design down’2 tasks, activities and assessments from the outcomes specified in the curriculum (south african qualifications authority, 2005). ncs curriculum standards were generally weak, both in content and progression, and therefore provided a weak guide for teachers (muller, 2006; reeves & muller, 2005; shalem, 2010). reeves and mcauliffe (2012) found in their study of ‘topic sequence’ and ‘content area spread’ in mathematics lessons that ‘for most learners mathematics was not presented in a coherent and composite manner over the school year’ (p. 28). when the curriculum specifies content as a list of topics, it does not elaborate sufficiently on the topics conceptually, and does not relate the topics to one another adequately. furthermore, lists of skills of what learners must do without any specific content attached to these skills allow for too much variation (low reliability) in the types of textbooks that are produced, in the criteria used by schools to select textbooks, in teachers’ professional judgement of what counts as an achievement, in the kind of tasks teachers design and in ‘curriculum coverage’. the main point here is that ‘lists of statements’ do not necessarily show what concepts are key to a field, what activities are worthwhile and what texts are worthwhile (shalem, 2010, p. 91). taken together, these explanations suggest that teachers struggle to interpret the curriculum (brodie, shalem, sapire & manson, 2010). in this article we add a third explanation. we propose that even if standards are sequenced and well explicated by examples, they do not disclose to teachers what instructional practice should look like or what constitutes acceptable coverage and cognitive demand of curriculum content. curriculum standards intend to transmit criteria to teachers of what, when and how to teach mathematical content, but transmission this through telling teachers is not enough. teachers need to be involved, we argue, in a practice that will require them to use the curriculum standards so that they understand what they mean and how they are related to their existing practice. criteria, says cavell (1979), are embedded in practice. we ‘find’ them in the way we do and say things. it is in the way we speak or in the way we do things that we make relevant connections, and thereby show that we understand the way a concept is related to other concepts, or its criteria (shalem & slonimsky, 1999). put differently, by doing knowledge-based professional work, teachers, we argue, are given an ‘epistemological access’ (morrow, 1994) to the form in which the curriculum is designed, and more specifically to the content that it privileges. according to ford and forman (2006), this kind of professional development work requires a relational framework between three fundamental constitutive disciplinary resources: disciplinary material (working with ‘the material aspects’ of a specific intellectual field or with a set of propositional knowledge limited to the field), collectivity (using the norms of the intellectual field to produce proofs and grounds for judgement), and disciplinary procedure (following procedures to evaluate claims made about the natural or the social world) (p. 4). taking part in curriculum mapping activities, we argue, provides such an opportunity for teachers’ professional development. curriculum mapping curriculum literature distinguishes between the intended, enacted and examined curriculum (e.g. stenhouse, 1975). in broad terms, this distinction refers to the differences and connections between what the official curriculum document intends, including the academic literature teachers use to decide what to emphasise when they teach a mathematical concept in a specific grade (the intended curriculum), what teachers do in their classrooms (the enacted curriculum) and what is assessed in order to determine achievement and progress (the examined curriculum). ‘curriculum alignment’, the idea that informs ‘curriculum mapping’, describes what counts as a productive educational environment. biggs’s (2003) premise is that when a teacher covers the content of an ‘intended curriculum’ at the appropriate cognitive level of demand and their learners perform well on high quality tests (the examined curriculum), they have created a productive learning environment (the enacted curriculum), aligned to the demands intended by the curriculum. the corollary of this is that if the quality of a learning environment is judged from the high results of the learners, all things being equal, it can be said that the results of the learners demonstrate that they have studied key content of the subject (curriculum coverage) and that they are able to use the content to answer a range of questions (cognitive level of demand). this is an important insight for understanding the role of curriculum knowledge in teacher practice and the significance of having experience in curriculum alignment.teachers can become more familiar with the requirements of the curriculum and in this way improve the conditions for achieving curriculum alignment (burns, 2001; jacobs, 1997) through their involvement in curriculum mapping activities. curriculum mapping is defined as a ‘tool for establishing congruence between what is taught in the classroom and what is expected in state or national standards and assessments’ (burns, 2001). the idea was formulated by english in the 1980s (in burns) and in its common form includes a type of calendar on which teachers, in grade-level groups, record time-on-task in each of the topics they teach and the order in which they teach the topics. over time this practice developed to include teachers’ records of the ways they taught and assessed the topic. in this common form, curriculum mapping is focused on the relation between the enacted curriculum and the intended curriculum. accounting – an opportunity to learn from the examined curriculum it is current practice by policymakers to strengthen accountability by using large-scale assessments (the examined curriculum). the policy idea here is that providing teachers with a range of assessments at different levels of cognitive demand in relation to key subject matter content, education departments hope to use the examined curriculum to make curriculum standards explicit. to be considered rigorous, of high quality and valid, large-scale assessments need to be shown to be aligned to the curriculum (brookhart, 2009; case, jorgensen & zucker, 2008; mcgehee & griffith, 2001). however, and this is the argument of this article, if teachers do not have opportunities to participate in analysing the content of these learner assessments, more specifically to profile the test items or to examine the curriculum standards that they articulate, their mathematical content and its alignment with the curriculum standards of the grade they are teaching, and which mathematical concepts or skills are needed in order to find the solution to a test item, teachers will not be able to fully gauge the requirements of the intended curriculum. we believe that an opportunity for teachers to learn is missed here. by working with test items (the examined curriculum) and thinking about the links between the content present in the test items in relation to the content present in the curriculum standards (the intended curriculum), the teachers in the dipip project were doing a different and more unusual form of curriculum mapping. the use of test items as an artefact to focus teachers’ thinking when they interpret the curriculum addresses the main challenge faced when interpreting any curriculum document, that is, to identify ‘what’ has to be covered as well as ‘the level’ at which the selected content needs to be taught. in curriculum terms this refers to curriculum coverage in a specific intellectual field at levels of cognitive demand appropriate for specific grades. by structuring professional conversations around curriculum mapping of test items, the mapping activity intended to provide the teachers with a relational framework, one in which they enact dimensions of expertise that are commonly excluded from them (curriculum mapping). through this we hoped to enable the teachers to gain a deeper and more meaningful understanding of the curriculum (the ncs), which, as we have shown above, is a skills-based curriculum that is, in best case scenario, opaque, especially in the case of teachers with weak subject matter knowledge. it is to the design of the activity that we turn next. the project process there were two rounds of the mapping activity in our project: the first in february − may 2008 (round 1) and the second in august − mid-september 2010 (round 2). about 50 mathematics teachers from grade 3 to 9 worked in groups. the initial selection of teachers for participation in the project was guided by the gauteng department of education. in particular, teachers from ‘better performing schools’ that had participated in the icas tests were selected. however, as a few teachers dropped out of the project, they were replaced by other mathematics teachers selected from schools with easy access to the wits education campus (shalem et al., 2011). the group membership was highly stable and over the three year period, a total of 62 teachers participated. the teachers were divided into 14 groups, two groups for each grade. each group consisted of 3–4 teachers, a subject facilitator from the gauteng department of education and a graduate student or a university staff member as a group leader. two points must be emphasised here. firstly, the group leaders were selected for their mathematics classroom experience or alternatively for their involvement in initial teacher education and in-service teacher development. at different points of time during the duration of the project, before the introduction of a new activity, the leaders were trained by a mathematical education expert. their role was very important in the project, which is borne out by the findings (see below). secondly, since all the activities were conducted in groups, reporting on dipip activities relates to the groups and not to individual teachers. although results about individual teachers’ mapping would be more desirable, from a research perspective, the idea of professional learning communities and this methodological criterion are in conflict. notwithstanding, the results reported in this article are statements arrived at through group discussion in relation to specific activities and reflect the consensual decision made by the group.the icas tests were not designed especially for the south african curriculum. these tests, which are used in many countries in the world, were used in south africa in good faith that they represented an ‘international’ mathematics curriculum. the mapping activity that the groups were given to complete was not done on this test by any expert or department official. the gauteng department of education treated the test as generally valid for the mathematical content of the grades tested. our curriculum mapping confirmed this assumption. table 1 shows our analysis of the content coverage in the icas 2006.3 the table lists the number of items per curriculum content area for each of the grades studied in the project (consistent with the ncs topic weighting). the curriculum mapping activity in round 1, all 14 groups mapped the 2006 icas test items. the groups met once a week for about two hours for 14 weeks, working with their group leaders. groups were expected to map a minimum of 20 items in round 1. in round 2, 11 of the groups mapped icas 2007 items or items from tests that they had set themselves (hereafter referred to as ‘own tests’). groups worked without group leaders in round 2. this was done in order to see the extent to which they could manage the task on their own.a modified curriculum document was prepared for use by the groups (scheiber, 2005). the tabulated curriculum enabled the groups to navigate and refer to the ncs document more easily, to look at and compare the content and contexts across the different grades. the tabulated curriculum has a landscape page setup and matches the assessment standards for each grade, across the page, using numbers. this makes it easy to compare the assessment standards across grades and to see at a glance how concepts are built up in each grade. figure 1 gives an illustrative example of the mental arithmetic assessment standard strand from grade 1 to 5. in the full tabulated document, assessment standard strands from grade 1 to 9 are given across the page. the groups were given a template (see figure 2), which structured their conversations and guided the process by which they arrived at a consensus, which was recorded in the template as the ‘group response’. the template was given to the groups in order to focus the conversation around what the icas test assessment data (the examined curriculum) means, how it aligns with the conceptual demands of the ncs (the intended curriculum), and how it fits with teachers’ professional knowledge and experience (the enacted curriculum) (brodie et al., 2010). for each test item the groups needed to: • identify the mathematical concept or concepts being tested by the icas item • find the relevant assessment standards relating to the concepts • justify the choice of the assessment standard • state when or if the content is taught and whether it is taught directly or indirectly.4 table 1: number of items per curriculum content area for each grade in the icas 2006 test. figure 1: exemplar assessment strands (mental arithmetic, grade 1–5) in tabulated format. the template required the groups to think about and decide which mathematical concepts or skills are needed in order to find the solution to a test item. the group’s decision was to be based on which assessment standard(s) its members linked the test items to. the groups needed to be sure that their selection was appropriate, and to do this they were asked to give strong motivation for their decision. this selection was not confined to the ncs of the grade they were analysing, but was related to several grades. this was possible as the group worked with all the assessment standards across grade r–9. the last section of the activity gave the groups an opportunity to examine the alignment between the intended and the enacted curriculum, comparing the content coverage assumptions made by the test designers and what, in fact, they cover in the classroom. in the last column of the template (figure 2), the groups needed to report on their teaching practices. this allowed the gap and the congruence between the intended and the enacted curriculum (based on groups’ reporting) to be made explicit to the teachers in their groups. in this way, the icas test was used as an artefact (a concrete textual item), which mediated between the intended curriculum and the groups’ professional knowledge and experience of the enacted curriculum. the upshot of this is that by being involved in professional work (the work that normally mathematics and curriculum experts do when they align the examined with the intended curriculum), teachers, in their respective groups, reported that they came to understand the demands of the ncs curriculum for the first time (brodie et al., 2010). figure 2: curriculum mapping activity template. sample of items of the 402 icas items that groups mapped in round 1, 140 items were selected for analysis. from the first half of each grade level test 10 items were chosen and 10 items were chosen from the second half of the test. this gave 20 items per grade. the selected items included all the content areas tested. all 82 of the items mapped by the groups in round 2 were selected for analysis. data analysis a mathematics education expert was employed to map the sample of the icas test items. the expert’s mapping was validated by a project manager and based on this agreed mapping, groups were considered to have ‘misaligned’ items with the curriculum standards when their mapping was different from the validated mapping. coding of the alignment was recorded in spreadsheets. coding of the remaining data involved recording (using spreadsheets) groups’ comments on content taught ‘directly’, ‘indirectly’ or ‘not at all’. coded data was analysed quantitatively, finding observable trends and relationships evident in the sample. examples of groups’ explanations from the template were recorded to exemplify quantitative findings. we refer to some of these examples in the findings. for the purposes of reporting on the analysis, we combined the following sets of groups: grade 3–6 and grade 7–9. validity and reliability a protocol (the mapping template shown in figure 2) was used for the recording of the groups’ responses in the curriculum mapping activity. the protocol was discussed amongst colleagues in the project management team. findings were reported on at local and international conferences where these could be discussed to enhance quality. the following are points to be noted as a possible validity threat:• since only one group of grade 7–9 mapped items in round 2, comparisons cannot be made for these grades with round 1 mapping. • one of the groups (grade 7) attained full matching with the expert. this may have skewed the data. • round 2 was a slightly abridged version of the curriculum mapping, due to time constraints, in which the teachers only mapped curriculum content and did not report on when and how they taught this content. ethical considerations approval for this study was granted by the department of education and at an institutional level by the university ethics committee. informed consent was obtained from all of the teachers, university staff and students who participated in the professional development project meetings. confidentiality and anonymity of participants was maintained through the use of classified group names (e.g. grade 3 group a, which is denoted as g3ga). findings top ↑ in total, in round 1, the grade 3–6 group mapped 246 and the grade 7–9 group mapped 156 icas 2006 items. in round 2, five of the 11 groups mapped their own tests. altogether these five groups mapped 27 items. the other six groups mapped 55 icas 2007 items. in total, in round 2, the grade 3–6 group mapped 57 items and the grade 7–9 groups mapped 25 items. in sum, 402 items were mapped in round 1 and 82 items were mapped in round 2. the group responses recorded on the mapping templates formed the data set on which the analysis presented in this article is based.we first present the overall results of the curriculum mapping, in which we look at the alignment quality of the groups’ mapping. we then discuss the groups’ reporting on content ‘taught’ (directly and indirectly) and content ‘not taught’, which yields insight into the relationship between the intended and the enacted curriculum. overall alignment results two overall results can be noted. firstly, the overall agreement between the experts’ (indicated as ‘alignment’) and the groups’ mapping was relatively high: 83% in round 1 and 67% in round 2. these percentages represent the average of the groups’ correctly aligned assessment standards to test items compared to the experts’ agreed curriculum alignment of the test items. this is an indication that the curriculum mapping was generally successful, particularly in round 1. the mean misalignment was 26% higher in round 2. it is important to remember that in round 1 groups worked with group leaders whilst in round 2 they worked without group leaders, which may have contributed to the increased misalignment in the round 1. secondly, the mapping alignment of grade 7–9 groups was found to be stronger than grade 3–6 groups. it stayed at the same level of accuracy (around 80%) in both rounds. increased confidence and improved judgement we investigated what specifically in round 1 may have contributed to differences between groups’ strength in mapping in round 2. we compared the means of the groups who had mapped different numbers of items in the two rounds. we found that groups that had mapped more items in round 1 achieved higher mean alignment in round 2 than groups that had mapped fewer items. put differently, groups that gained more experience of mapping in round 1, as measured by their mapping more items, consistently showed higher alignment percentages in round 2 (see figure 3). for this comparison we could only use data from the grade 3 to 6 group and we compared each paired grade group individually. since only one group for each of the grades in grade 7–9 did the curriculum mapping activity in round 2 (see the discussion on validity), comparisons were not possible here.this finding suggests that with more experience groups gain confidence in mapping test items against the curriculum and make better judgments in relation to curriculum alignment. mapping icas items and own test items higher misalignment was found in round 2 of ‘own test’ items than of icas 2007 test items in the grade 3–6 group. this finding is counter-intuitive: one would have thought that groups would be more familiar with content in a test that they had drawn up themselves. this might suggest that the groups designed tests that included mathematical content about which they were not entirely confident. alternatively, it could be that the ‘own test’ items required a different way of thinking when aligning to the assessment standards of the curriculum, so the groups’ familiarity with the task made it easier in round 2 to align icas test items but not ‘own test’ items. a third explanation may be that when groups designed their own tests, because they were expected to select a misconception and to design the test items around it, they may have focused their attention on the misconception rather than on the level of content required. it seems they had difficulty embedding the concepts at the appropriate level. figure 3: comparison between round 1 and round 2 in terms of mean alignment for grade groups that mapped more items or mapped fewer items. taking all these findings together, we suggest that the curriculum mapping activity gave the practitioners (we are especially interested in the experience of the teachers in the group) an opportunity to understand the selection of the mathematical content for the icas tests, per grade, as well as its level of cognitive demand. since groups were working with curriculum standards ranging across several grades, teachers were given an opportunity to analyse what assessment standards (and related mathematical conceptual knowledge) in the south african curriculum learners would need to have achieved in order to answer a test item correctly. analysing the items conceptually, which groups needed to do in order to identify the mathematical concepts being tested by the icas item, alerts the groups to conceptual progression. gaps between the intended and the enacted curriculum the data about the intended curriculum and the groups’ reflections on practice are drawn from the groups’ reporting in round 1, specifically from the information the group included in the last column of the curriculum mapping template (see figure 2). the findings for this section refer only to the mapping activity of the icas 2006 test items (see the discussion on validity). teachers identified the mathematical content in an icas 2006 test item and after aligning it to the ncs they needed to consider it in relation to their own teaching. this served to make explicit to the teachers the difference between the intended and the enacted curriculum. the highest percentage (46%) of the icas test related content was reported to be taught ‘directly’. next highest (26%) of the icas test related content was reported as ‘not taught at all’. the lowest percentage (24%) was reported as being taught indirectly.5quotes taken from grade groups’ recorded responses are given as examples of teachers’ reporting on: • direct teaching: ‘when we teach bonds’ (g4ga); ‘taught in term 1, as mental calculations’ (g5ga); ‘decimal fractions are taught in the first term of grade 7’ (g7ga). • indirect/linked teaching: ‘should be covered in all problem-solving activities across all grades’ (g6ga, g8gb); ‘place value is taught in the first term. link it with money, mass and capacity’ (g5gb); ‘should be done specifically as the concept of symmetry but it can be done via the theme e.g. special me (body parts-left and right sides of the body)’ (g3gb). • content not taught at all: ‘number patterns – grade 1 onwards. not focused on in grade 8. knowledge is assumed’ (g8gb); ‘the concept of odd and even numbers – assume that it has been taught in earlier grades’ (g5gb); ‘… pictorial representation of patterns is a neglected area, thus making it difficult for them to conceptualise what is required’ (g3gb); ‘rotation is not taught in grade 4, but is in the curriculum for grade 5’ (g4ga, g4gb). figure 4: grouped grades’ reporting on ‘when i teach it’ (round 1). figure 5: grouped grades’ reasons given for content not taught. table 2: summary of mathematical content reported ‘not taught’. it is interesting to note (see figure 4) that the grade 3–6 group identified more content in the tests which they said they did not teach because it was at a higher level than they were expected to teach (topics included irregular shapes, rotational symmetry, tessellations, reflections, rotations and probability). the opposite was true for the grade 7–9 group.we further investigated the reasons given by the groups for content ‘not taught’ (see figure 5) in their completed mapping templates. our investigation gave rise to two categories of mapping referred to as ‘mapping downwards’ and ‘mapping upwards’. mathematical content of an item that according to the curriculum should be covered in a lower grade than in the test was classified as ‘mapped downwards’ – reported as: ‘this [different perspectives of geometric solids] is not specifically covered in grade 8, it is formally required to be taught first in grade 6’ (g8gb). or, pointing to the problem-based nature of an item, groups reported: ‘although the number range falls within the scope of the phase, we do not teach this, because the way in which the problem is presented is beyond the scope of the phase’ (g3ga). mathematical content of an item that according to the curriculum should be covered in a higher grade than in the test was classified as ‘mapped upwards – reported as: ‘we would use this task [modelling involving area] as an extension for the stronger learners’ (g8ga). most commonly, explanations for ‘not teaching’ fell into one of these categories. it is also interesting to note that more content in the primary school (grade 3–6) than in the secondary school (grade 7–9) was classified as ‘not taught’ for reasons of being at a higher than expected level for the grade (according to the intended curriculum identified in the ncs). table 2 shows that most items reported as ‘not taught’ were at the grade level of the test, according to the intended curriculum. few of the items reported as ‘not taught’ were mapped up or down, relative to the grade level according to the intended curriculum. in the grade 3–6 group, 44% of the content reported as ‘not taught’ was at the expected grade level. in the grade 7–9 group, 76% of the content reported as ‘not taught’ was at the expected grade level. in both grade groups, the content reported ‘not taught’ at own grade level included mathematical content across all five of the ncs content areas. the grade 3–6 group mapped up some data, measurement and geometry items and they mapped down some number and data items. the grade 7–9 group did not map down any items but they mapped up some number, geometry and measurement items. some of the specific neglected areas reported by the teachers include (quotes are taken from grade groups’ recorded responses): • icas items that included irregular shapes, rotational symmetry, tessellations, reflections, rotations and probability were reported as beyond the scope of grade 3 (g3ga, g3gb). • the pictorial representation of a pattern was reported as not taught at the grade 3 level. patterns are usually taught as a horizontal sequence of numbers: ‘… pictorial representation of patterns is a neglected area, thus making it difficult for them to conceptualise what is required’ (g3gb). • reasoning logic was reported as not taught in grade 8: ‘never! not mathematics’ (g8ga). • finding fractional parts of whole numbers was reported as not taught using learning aids in grade 5: ‘we never use manipulatives to determine fractional parts of whole numbers’ (g5ga). our data analysis shows a direct relationship between the teachers’ perception of the enacted curriculum – content taught ‘directly’ – and the degree of success in aligning the international test item to the curriculum. figure 6 shows higher percentages of misalignment of item content reported as ‘not taught’ (compared to content reported as ‘ taught’) for both groups, but particularly in the lower grades. content reported as ‘taught directly’ is, for the best part, aligned better. this difference in alignment could be an indication of teacher content knowledge. it could be that in the higher grades teachers are teaching more work with which they are not sufficiently familiar, yet they do teach it because it is required of them. it is also possible that the teachers in the lower grades may have reported more openly on content ‘not taught’. the teachers in the higher grades may leave out content with which they are not familiar but they do not report on it. alternatively, the items may include content that the teachers expect should be taught earlier and thus do not report on it. in conclusion, by allowing them to analyse if and when the content specifications of the intended curriculum are covered in practice, the mapping activity gave the teachers an opportunity to reflect on their practices. we argue that the mapping activity enabled them to develop their pedagogical content knowledge (shulman, 1986). by examining curriculum coverage, cognitive demands and alignment, the teachers were involved in ascertaining the match between an international test, the south african curriculum and their professional knowledge and experience. this should have, at least to some extent, developed their understanding of the sequence and progression of the mathematics they teach and hence their grasp of ball’s sixth mathematical knowledge domain (ball et al., 2008). figure 6: misalignment of content taught and content not taught. conclusion top ↑ the findings of this research suggest that through the curriculum mapping activity, teachers became more aware of what is intended by the curriculum, in particular of the discrepancy between what they understand is intended by the curriculum and report is enacted in their classes. in relation to the intended curriculum, the teachers were able to report on when they teach and do not teach content and on ‘neglected content area in schools’, which shows that such activities can build teachers’ awareness of the presence or absence of curriculum content in their classes. the findings also suggest that teachers’ curriculum mapping ability is stronger when they are more familiar with and hence have greater confidence in doing the activity. when content was reported as ‘not taught’, a higher level of misalignment was generally seen, which indicates familiarity with mathematical content does affect the quality of curriculum mapping. the differences between alignment in round 1 and round 2 indicate that teachers’ capacity to align content was better when they worked with well-selected and trained group leaders. this research supports the claim that teachers can benefit a great deal from being involved in interpreting large-scale assessment tests. curriculum mapping, using the format of a structured interface, creates a ‘defensible focus’ (katz et al., 2009) for this kind of professional development. the analysis above shows that the structured process of interface enabled teachers to actively engage in curriculum translation of the disciplinary material embedded in curriculum statements. working in groups, the teachers learned about the form of the curriculum by using an artefact (the test items) to engage the curriculum content and standards. involving teachers in the interpretation of both public assessment data and data from their own classroom activities can build their understanding of the knowledge base of test items and of the curriculum. acknowledgements top ↑ we acknowledge the funding received from the gauteng department of education and in particular would like to thank reena rampersad and prem govender for their support of the project. we would also like to acknowledge the pivotal role played by karin brodie (project leader, phase 1 and phase 2) in the conceptualisation of the project. the views expressed in this article are those of the authors. competing interests the authors declare that we have no financial or personal relationship(s) which might have inappropriately influenced our writing of this article. authors’ contributions y.s. (university of the witwatersrand) was the project director, developed the theoretical framework for the article and contributed to the analytical part of the article. i.s. (university of the witwatersrand) was a project coordinator, was responsible for the data analysis and contributed to the theoretical and analytical parts of the article. b.h. (st john’s college), a mathematics education expert, was a group leader on the project and was involved in the data coding; she also contributed to the writing and development of the article. references top ↑ ball, d.l., thames, m.h., & phelps, g. (2008). content knowledge for teaching: what makes it special? journal of teacher education, 9(5), 389–407. http://dx.doi.org/10.1177/0022487108324554 biggs, j.b. (2003). teaching for quality learning at university. (2nd edn.). buckingham: open university press/society for research into higher education. borko, h. 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(2008). reframing teacher professional learning. an alternative policy approach to strengthening valued outcomes for diverse learners. review of research in education, 32(1), 328–369. http://dx.doi.org/10.3102/0091732x07308968 footnotes top ↑ 1.the icas test is designed and conducted by educational assessment australia (eaa). in the gauteng province of south africa, 55 000 learners across grade 3–11 in both private and public schools (3000 in total) wrote the icas tests in 2006, 2007 and 2008.2.‘design down’ was one of the imperatives of the ncs. this meant that teachers had to start with curriculum specifications and design lesson plans through which these specifications would be delivered in their classes. 3.coverage is designed in a similar way in the icas 2007 tests.4.‘indirectly’ means through an assignment, a project or homework task, or linked to another content area. 5.it should be noted that the grade 3–6 group reported certain items taught both directly and linked (for example) so the total of the ‘when i teach it’ percentages for this group goes slightly over 100%. this was not the case in grade 7–9. article information authors: mark jacobs1 duncan mhakure2 richard l. fray3 lorna holtman4 cyril julie5 affiliations: 1electrical engineering department, faculty of engineering, cape peninsula university of technology, south africa2numeracy centre, university of cape town, south africa 3department of mathematics and applied mathematics, university of the western cape, south africa 4postgraduate studies, university of the western cape, south africa 5school of science and mathematics education, university of the western cape, south africa correspondence to: mark jacobs postal address: po box 1906, bellville 7535, south africa dates: received: 21 feb. 2013 accepted: 23 mar. 2014 published: 21 may 2014 how to cite this article: jacobs, m., mhakure d., fray, r.l., holtman, l., & julie, c. (2014). item difficulty analysis of a high-stakes mathematics examination using rasch analysis. pythagoras, 35(1), art. #220, 7 pages. http://dx.doi.org/10.4102/ pythagoras.v35i1.220 copyright notice: © 2014. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. item difficulty analysis of a high-stakes mathematics examination using rasch analysis in this original research... open access • abstract • introduction    • examinations    • number patterns and sequences in the curriculum    • the focus of this research    • procedure and methods    • item analysis • findings • discussion • conclusion • acknowledgements    • competing interests    • authors’ contributions • references abstract top ↑ the national senior certificate examination is the most important school examination in south africa. analysis of learners’ performance in mathematics in this examination is normally carried out and presented in terms of the percentage of learners who succeeded in the different bands of achievement. in some cases item difficulties are presented – item refers to the subsection of each examination question. very little attention is paid to other diagnostic statistics, such as the discrimination indices and item difficulties taking into consideration partial scores examinees achieve on items. in this article we report on a study that, in addition to the usual item difficulties, includes a discrimination index of item difficulties taking into account partial scores examinees achieved. the items, considered individually, are analysed in relation to the other items on the test. the focus is on the topic sequences and series and the data were obtained from a stratified sample of the marked scripts of the candidates who wrote the national senior certificate examination in mathematics in november 2010. rasch procedures were used for the analysis. the findings indicate that learners perform differently on subsections of topics, herein referred to as items, and that focusing on scores for full topics potentially mask these differences. mathematical explanations are attempted to account for difficulties learners exhibit in these subsections, using a hierarchy of scale. the findings and our analysis indicate that a form of measurement-driven testing could have beneficial results for teaching. also, for some items the difficulty obtained from the work of examinees runs counter to the commonly perceived wisdom that an examination ought to be structured in such a way that the less difficult items are at the start of a topic. an explanatory device anchored around the construct of ‘familiarity with problem types through repeated productive practice’ is used to account for the manifested hierarchy of difficulty of the items. introduction top ↑ in most countries schooling culminates with learners having to write examinations in various subjects in order to obtain a certificate to matriculate. in south africa the matriculation or the final national senior certificate (nsc) examination is a high-stakes examination and the outcome is a public event in which results are announced by the minister of basic education and published in national newspapers. the nsc results can offer access to higher certificate programmes, diploma courses and the much sought-after degree courses at higher education institutions in the country. mathematics or mathematical literacy is taken by all learners as a qualifying subject in the nsc examination. mathematics is seen as the gatekeeper subject to access many degree programmes as mathematical literacy has only been differentially accepted by higher education institutions and not at all in most cases to degree programmes requiring mathematics. a point score system is used at most universities in south africa and the mathematics score is usually weighted higher (it is doubled) in the score.in this article we are interested in the extent to which learners experience difficulty with the topics in such high-stakes examinations. in particular, we analyse learner performance in sequences, series and the accompanying finance, growth and decay applications as evidenced by their performance in the high-stakes ncs mathematics examination. examinations it is well known that there are different kinds of examinations serving different purposes. for example, there are criterion-referenced, norm-referenced, formative and summative examinations. in terms of the procedures employed for the construction, the mechanisms used for assessing the responses of examinees, the methods of quality control of assessment of examinees’ responses (known as moderation) and the major beneficiaries of the examination, a simple classification along the lines of external or internal and examinee or system as the major beneficiary can be made. table 1 presents such a classification for different examinations operative in the south african schooling system. table 1: classification for different examinations operative in the south african schooling system. in table 1, internal and external refer to whether the teacher or a group of teachers who teach mathematics at the institution have the responsibility for the particular procedure of the examination process or not. if the responsible teacher is totally responsible but there is some form of external oversight as is the case, for example, with the teacher-constructed end-of-year (grade) examination where oversight is exercised by curriculum advisors, then this is indicated by ‘±’. with respect to beneficiaries, the examinee is the major beneficiary if the level of achievement has consequences for the examinee in terms of what worth it is to them. so, for example, attaining a level 4 (50% – 59%) pass in mathematics in the nsc examination will allow access to a bachelor’s degree if the examinee also performs at pre-determined levels in some other subjects. where the system is the beneficiary, an examination is essentially a mechanism to supply bureaucrats and politicians with information regarding the effectiveness of the entire system and has no direct consequence for examinees in terms of how it will impact their immediate progression to, say, a next grade or obtaining a level of achievement that will allow access to various forms of further studies. as is clear from table 1 the nsc examination is one of a few high-stakes examinations which is entirely independent of those teaching the subject at hand on all the processes involved in the examination process. in this sense it is on par with examinations for entry into high-status professions such as the board examination to become a chartered accountant. number patterns and sequences in the curriculum in mathematics, curriculum developers both in national and international arenas have identified the teaching and learning of number patterns as one of the main aims of mathematics. vogel (2005, p. 445) argues that ‘the analysis of number patterns and the description of their regularities and properties is one of the aims of mathematics’. according to devlin (1996) the question of ‘what constitutes mathematics?’ has been the subject of much discussion in the 19th century. however, the definition of mathematics accepted by mathematicians only emerged three decades ago: ‘mathematics is the science of patterns’ (devlin, 1996, p. 3). given this definiton, there is no doubt of the importance of the teaching and learning of number patterns within the south african national curriculum.in this study we selected to focus the analysis on the topics of sequences and series with their organic relationship with number patterns and the closely related applications to finance, growth and decay. in the curriculum and assessment policy statement, the topics are labelled ‘number patterns, sequences, series’ and ‘finance, growth and decay’ (department of basic education, 2012, p. 9). these two topics account for about 27% of the marks allocated in the first paper of the nsc examination for mathematics. the learning of patterns, sequences and series does not start in grade 12. it commences in the foundation phase where it is stated: ‘in this phase, learners work with both number patterns (e.g. skip counting) and geometric patterns (e.g. pictures)’ (department of basic education, 2011, p. 9). the topic continues through the intermediate phase and senior phase. in grades 10–12, at the further education and training phase, the exploration of number patterns is expanded and emphasis shifts to include goals that will allow learners to be able to: • investigate number patterns leading to those where there is a constant difference between consecutive terms, and the general term is therefore linear • investigate number patterns leading to those where there is a constant second difference between consecutive terms, and the general term is therefore quadratic • identify and solve problems involving number patterns that lead to arithmetic and geometric sequences and series, including infinite geometric series (department of basic education, 2012, p. 12). in sum, the teaching and learning of number patterns plays a significant role in mathematics in schools. firstly, mason (1996) argues that learning number patterns develops learners’ abilities in expressing generality, which is key to studying algebra and abstract mathematics. herbert and brown (1997, p. 126) express the same sentiments as mason: ‘from generalizing the pattern, students understand the power of algebraic thinking’. secondly, recent lines of research (lee, bull, ng, pe & ho, 2011) show that generating additional numbers in sequences enhances learners’ arithmetic knowledge of computation and that proficiency in number patterns greatly helps in achieving good performance in algebra. research has also shown that algebra has foundations in number patterns and that most of the challenges experienced by learners when learning algebra are as a result of lack of proper foregrounding in number concepts (carraher & schliemann, 2007). lastly, if number patterns are taught using socio-cultural contexts then learners will develop notions that there should be patterns or norms in our daily life activities, and that the disruptions to these patterns should signify new challenges whose solutions need to be found. not only is the topic of patterns, sequences and series of importance for school mathematics from a historical perspective, it also underpinned newton’s work and its extension by lagrange on the difference formula to determine polynomial functions fitting given sets of numbers (see e.g. cuoco, 2005). in addition many non-polynomial functions, such trigonometric functions can be represented as power series, as for example, a taylor series expansion of sin x: the topic thus stretches its tentacles deep into post-school mathematics. the focus of this research the issue pursued in this article therefore is the difficulty learners find with sequences, series and the accompanying finance, growth and decay topics as evidenced by their performance in the high-stakes ncs mathematics examination. analysis of learner performance in this examination is normally done at a global level. at this level feedback is provided of performance in an entire question, which normally deals with a topic as given in the prescribed curriculum document. the procedure followed to determine the difficulty of a question is of a classical test theory nature in which the emphasis is on the percentage of examinees who answered a question correctly. no consideration is given to subsections of a question and scant attention is given to partial scores a candidate obtains.the research reported here is underpinned by the assumption that insightful information about learners’ proficiency in high-stakes mathematics examinations can be gained by analysing test scores, taking into account scores obtained in subsections of questions on a topic. it would therefore be a form of measurement-driven testing leading to teaching that is directed towards achievement in these kinds of examinations, the antithesis of psychological and curriculum-driven testing. procedure and methods the study reported here is part of a larger study in process dealing with learners’ ways of working in high-stakes school examinations in mathematics. this larger study focuses on a 12% random sample of the scripts of examinees in the western cape who wrote the 2010 ncs mathematics examination. the sample was stratified according to former (pre-1994) cape education department schools, department of education and training and house of representatives schools, new schools established post-1994 and independent schools as well as the eight school districts in the western cape. for logistical reasons of not having to go through all the scripts in the province to obtain individual scripts, scripts from an entire school (examination centre) were selected. this rendered 1959 scripts. from this collection of scripts a similar random sample of 1122 scripts from the school were selected for the study reported here. item analysis analysing items is an investigation of ‘the performance of items considered individually either in relation to some external criterion or in relation to the remaining items on the test’ (thompson & levitov, 1985, p. 163). for this article the focus is on the latter. the statistics given are the item difficulties, the rasch item difficulties and the discrimination coefficients. these are the usual diagnostic statistics, which are calculated to determine how a particular test or subset of a test is functioning in order to ascertain the appropriateness of the test for a particular cohort. the item difficulty is the percentage of examinees who were awarded full marks for an item. the rasch measure is the item difficulty obtained by applying a rasch partial credit analysis to determine how both items and examinees are ranked in order of the difficulty level of the items. in this article we are not discussing the technicalities of rasch modelling. there exists a rich corpus of literature pertaining to this and dunne, long, craig and venter (2012), for example, provide a comprehensive description of such technicalities. see dunne et al also for a fuller description of the rasch measurement theory where it is used to support both classroom-based and systemic assessment of mathematics proficiency. in particular, they show how systemic assessment using this form of analysis can lead to more informed teaching in the classroom.for our purposes we use the rasch measurement to analyse within items, that is, subsections of items, taken from sequences and series. for this paper rasch analysis was done using the winsteps version 3.65.0 software. as alluded to above ‘the rasch measure for items is the item difficulty’ (linacre, 2008, p. 362). we report the measure for the items. ‘the unit of measure used by rasch for calibrating’ (linacre, 2008, p. 362) items is obtained is by logs-odds scaling (ln odds = , where p = percentage answered correctly) and reported in logits. the discrimination coefficient is the point-polyserial correlation between the examinee’s score for an item and their total score excluding the score for the item that is correlated. it is the pearson product-moment correlation and gives an indication of whether higher scores correspond to higher achievement for the test or a coherent subsection thereof. as a rule of thumb, high discrimination coefficients indicate that such items indicate higher performing examinees. in essence the discrimination coefficient is a measure that identifies a good item: an item that will discriminate between high and low scorers. the advantage of using the discrimination coefficient instead of other forms of discriminative measures (such as the discriminative index) is that the former can be used with less than the full group of examinees (backhoff, larrazolo & rosas, 2000). this is applicable in the case of this paper as a selection of the scripts of the total number of examinees who sat for the mathematics 2010 paper 1 was used. diagnostic statistics were computed for each item, by which we mean the subsection of each question. the statistics reported are the difficulty level, the discrimination coefficient and the rasch item difficulty. these are the usual diagnostic statistics reported for large scale international tests such as, for example, timss (mullis, martin, gonzales & chrostowski, 2004). findings top ↑ table 2 presents the diagnostic statistics for the sample of examinees’ scripts selected for this study. table 2: diagnostic statistics for the selected scripts and items. items were identified as ‘not attempted’ when there was no indication on the script that the examinee had answered the question. the examinee either only wrote down the item number but the script indicated no further work with it or the item did not appear at all. this is distinguished from a ‘0 (zero)’ mark where there were indications that the question was attempted or answered and a mark of ‘0’ was awarded for the item by the markers. by indicating the ‘not attempted’, we are following the customary format of reporting diagnostic statistics as used in large-scale testing such as is done in the timss reports (mullis et al., 2004). reporting the ‘not attempted’ gives a sense of completeness. table 2 shows the item difficulty order for both the item difficulty and the rasch difficulty measure. the order slightly differs for the two measures of difficulty. this is to be expected since for the item difficulty only the examinees who scored full marks were taken into account to determine the index. the partial credit model used in calculating the rasch measure takes into account all marks for an item. for example, item 2.2.1 had an allocation of 3 marks and an examinee could be awarded a mark of 0, 1, 2 or 3. the rasch procedure used the mark the examinee was assigned and the item difficulty index calculation does not take partial marks into reckoning. the person-item map (figure 1) represents the rasch measures in order of difficulty. the right-hand side is a hierarchical ordering of the difficulty of the items with the most difficult item at the top and the easiest item at the bottom. the number of examinees who had success at a particular level of difficulty is on the left. in essence the right-hand side gives an indication of the number of examinees who had at least a 50% chance of succeeding on items of similar difficulty. as is indicated in the figure, the ‘#’ indicates 11 examinees and ‘.’ indicates from 1 to 10 examinees. so, for example, there were, at most, 142 learners who had at most a 50% chance of succeeding on items of a similar kind to item 2.3.2. these same learners had a less than 50% chance to succeed on items of the kind above 2.3.2 (items 2.2.1, 3.1, 3.2, 7.1, 7.2.1, 7.2.2 and 7.2.3). they had a more than 50% chance of succeeding on items of the kind below 2.3.2 (items 2.1, 2.2.2 and 2.3.1). figure 1: person-item-map. julie (2012) developed an organisational scheme to cluster items into four zones of difficulty (see table 3). this clustering is around the mean rasch measure and the standard deviation. these zones and the items appearing in them, based on the item map, are presented in table 4. table 3: clustering of items in zones of difficulty. table 4: item difficulty and rasch difficulty measures for selected items. the moderately high and moderately low zones are volatile and it can be expected that items will, over time, transition between the zones (julie, 2012). with this in mind, the hierarchy of the items as per the classical item difficulty index and the rasch difficulty measure are the same. discussion top ↑ for this quantitative study the only data we had were the scores as reflected on the scripts of the learners. we saw a particular pattern in terms of difficulty according to the model postulated by julie (2012). as an organiser we looked at the nature of the question and tried to generate tentative explanations for the placement of the item in those particular zones of difficulty. the explanations are anchored around three constructs: familiarity with problem types through repeated productive practising, procedural complexity and interpretive complexity. familiarity with solution of problem types is akin to the east asian perspective that repeated practice is a necessary condition for the development of ‘cleverness and creativity’ (julie et al., 2010, p. 366). productive practising (selter, 1996) is the notion that repeated practice is not just senseless drill for automated responses to cue-based questions. it includes drilling for mastery as well as activities for developing learners’ mathematical thinking skills of the constructs that are being drilled. it is thus different from the kind of drill in which, for preparation for the nsc mathematics examination, learners only work through previous examination papers. watson and mason (1998, p. 20), for example, pose the question ‘is it always true that operating the same way on both sides of an equation produces an equivalent equation?’ as a task to develop the skills of explaining, justifying, verifying and refuting when learners are engaged with developing procedural fluency with the techniques for solving polynomial equations. procedural complexity is ‘the number of decisions and operations and sub problems’ (watson & de geest, 2012, p. 223). interpretive complexity refers to the learners’ comprehension of a problem statement. it is not restricted to ‘word problems’ or contextdriven tasks. it encompasses instruction-type statements such as ‘show that expression a = expression b’, where expression a and expression b are mathematically equivalent. this kind of instruction-type statement might be interpretively more complex than ‘simplify expression a’. therefore, procedural complexity and interpretive complexity are in part an outcome of familiarity.the zone of low difficulty: the items in this zone are 2.3.1 and 2.2.2. they are essentially arithmetical in nature and examinees only have to do calculations with numbers. item 2.3.1 is the easiest. for this item, examinees have to identify that the sequence is arithmetic, find the common difference, identify the first and last term and recall or copy the appropriate formula, do the necessary substitutions and then the calculations. a brute force procedure by writing down all the terms up to 101 and counting the number of terms could also have been followed. these are solution methods that are well-practised over an extensive number of years in different grades. a similar explanation holds for item 2.2.2. this lends credence to a position that the regular practising over extended periods of particular problems and their solution routines contributes towards success in examinations on these kinds of problems. the zone of moderately low difficulty: the items in this zone are 2.1 and 2.3.2. item 2.1 deals with the sigma notation as a compact form for representing a sum. examinees have to do expansion through substitution of a sufficient number of values in order to recognise the kind of sequence, recall or copy the formula for the sum of a geometric sequence and determine and identify the given elements needed for substitution in the formula. it cannot be discounted that the learners could have been made aware of searching for certain cues to identify the sequence for these problem kinds. for example, in this case it might be: ‘if the letter indicating the terms is an index of the expression after the summation sign, then the sequence is geometric.’ brute force in terms of ‘write down all the terms and add them’ can also not be dismissed. as with item 2.3.1 and item 2.2.2, once the preliminaries are settled, only arithmetic calculations must be carried out. for these items, there is an increase in procedural complexity. item 2.3.2 has an additional interpretive complexity. meaning has to be assigned to ‘even numbers’ and even ‘remove’. examinees have to decide which terms will have even numbers and construct a new series with these terms. a challenge is to decide whether it is the terms that are even or the terms in even-numbered positions. it is contended that these complexities of the items place them in a higher difficulty category than those in the zone of low difficulty. the zone of moderately high difficulty: there are six items in this zone. item 2.2.1 is unusual for this kind of geometric series problem. normally examinees would not expect to find the value of a variable. further, the problem calls for deep understanding of a convergent geometric series. to complicate matters (and raise the procedural complexity) it is so that an inequality must be solved. problem unfamiliarity, procedural and interpretive complexity thus plausibly account for problems of this kind having a moderately high difficulty level. item 3.1 requires understanding of a quadratic sequence in terms of second differences. usual expectation is that second differences will be used to determine a quadratic function. familiarity with problem type and interpretive complexity account for the item’s moderately high level of difficulty. item 3.2 is contingent on item 3.1 and the second highest percentage (29%) of the sample cohort did not attempt this item. for item 7.1, the number of years is the only concept that is linked to what learners normally encounter. the language used is very compact. the learners must be able to transform the standard formula to accommodate the quarterly interest rate and doubling of an unknown sum. the item is descriptive and general rather than specific in terms of what is given. this item is both procedurally and interpretively complex and 27% of the sample cohort did not attempt it. that only 9% of the sample cohort did not attempt item 7.2.1 is indicative that a high level of familiarity with the context inherent in the problem is needed. there are, however, subtleties, delayed payment of the first instalment after the securing of the loan and the number of months that elapsed between 01 february and 01 july, for example, which require a sophisticated level of comprehension to correctly determine the elements needed for problem solution. furthermore, there is the issue of superfluous information. for the item under discussion the r450 is not required and this goes against the grain of learners expecting that all information must be used. we contend that interpretive complexity accounts for the moderately high level of difficulty of this item. interpretive complexity is also at play with item 7.2.2. it requires that a new principal amount be established as a result of interest accrued up to the end of july. the procedural complexity is also increased because the learners have to make the time period (n) the subject in the compound interest formula, which makes it a reversal problem. learners normally find reversal problems difficult due to limited practice with such problems. the zone of high difficulty: this zone has only one item, 7.2.3. it is interesting to note that prior questions do not give any scaffolding for dealing with this question and the learner has to start this item afresh. it requires careful interpretation and dealing with sub-problems, which bring procedural complexity into play. these complexities can plausibly account for this item being experienced as the most difficult by examinees, with 40% of the sample cohort not attempting it at all. conclusion top ↑ the above findings indicate that the difficulty of items, as evidenced by the performance of the examinees in the high-stakes nsc mathematics examination, should not only be deduced from the perspective of topics in the curriculum in which learners did not perform satisfactorily. examinees perform differentially on the sub-questions of a topic. this overall finding echoes that of mayberry (1983). she found that prospective teachers attained different van hiele levels for different quadrilaterals on a test designed to measure the van hiele levels of understanding.from the perspective of classroom teaching and preparing grade 12 learners for the nsc, the explanatory framework suggests that more attention should be accorded to the development of familiarity through productive practising. such a focus would contribute tremendously, we contend, to improvement in learners’ handling of complex layered questions like question 7. a final issue that needs careful thought and deliberation is the structure of the mathematics examination. common wisdom dictates a structure that is topic based, such as in the case of this article: three questions (2, 3 and 7) dealt with the topic at hand. a further element of the organisation of examination papers is that items perceived as less difficult are placed at the beginning of the subsection. however, the rational choices examiners (including teachers as examiners of in-school designed high-stakes examinations) make might not always parallel those of learners, as exhibited here by the actual responses of learners. for example, item 2.2.1 was found to be more difficult than item 2.2.2 by the examinees. this indicates that examiners should give more consideration to item placement in an examination. in the design of a high-stakes examination the assumption is that the placement of less difficult items early in the examination paper will lessen the anxiety load accompanying high-stakes examinations. consideration should therefore be given to place less difficult items at the start regardless of the topic. such a reconfiguration might lead to examinees having more courage and willingness to tackle and engage with items higher up a hierarchy of difficulty levels. the analysis of the difficulty levels of items as manifested by examinees’ responses to items in a high-stakes mathematics examination can provide valuable insights for curriculum interpretation, the design of examinations and for teaching. in the quest for enhanced performance in high-stakes examinations, thoughtful consideration of the outcomes of such analysis has the potential to positively contribute to this quest. however, such analysis and the dissemination of the findings should be done fairly near to the conclusion of the examination. this will ensure that all major stakeholders can use the outcomes for planning a reasonable time before a next round of teaching for enhanced performance in high-stakes examinations commences. acknowledgements top ↑ this work is supported by the national research foundation under grant number 77941. any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views the national research foundation. competing interests the authors declare that they have no financial or personal interest(s) that may have inappropriately influenced them when writing the article. authors’ contributions c.j. (university of the western cape) conceptualised the research and collected the data. with l.h. (university of the western cape), he did the quantitative analysis. m.j. (cape peninsula university of technology) and d.m. (university of cape town) led the interpretation of the results. the mathematical aspects of the article were attended to by r.l.f (university of the western cape). all authors contributed towards the construction of the final version of the article. references top ↑ backhoff, e., larrazolo, n., & rosas, m. (2000). the level of difficulty and discrimination power of the basic knowledge and skills examination (exhcoba). revista electrónica de investigación educativa, 2(1), 1–16. available from http://redie.uabc.mx//contenido//vol2no1/contents-backhoff.pdfcarraher, d.w., & schliemann, a.d. (2007). early algebra and algebraic reasoning. in f.k. lester (ed.), second handbook of research in mathematics teaching and learning (pp. 669–705). charlotte, nc: information age publishing. cuoco, a. (2005). mathematical connections: a companion for teachers and others. washington, dc: mathematical association of america. department of basic education. (2011). curriculum and assessment policy statement. mathematics. grades 1–3. pretoria: dbe. available from http://www.education.gov.za/linkclick.aspx?fileticket=ehgepqzxz7m%3d&tabid=671&mid=1880 department of basic education. (2012). curriculum and assessment policy statement. mathematics. grades 10–12. pretoria: dbe. available from http://www.education.gov.za/linkclick.aspx?fileticket=qpqc7qbx75w%3d&tabid=420&mid=1216 devlin, k. (1996). mathematics: the science of patterns. the search for order in life, mind, and the universe. new york, ny: scientific american library. dunne, t., long, c., craig, t., & venter, e. (2012). meeting the requirements of both classroom-based and systemic assessment of mathematics proficiency: the potential of rasch measurement theory. pythagoras, 33(3), art. #19, 16 pages. http://dx.doi.org/10.4102/pythagoras.v33i3.19 herbert, k., & brown, b.r. (1997). patterns as tools for algebraic reasoning. teaching children mathematics, 3(6), 340–344. julie, c. (2012). the stability of learners’ choices for real-life situations to be used in mathematics. international journal of mathematical education in science and technology, 44(2), 196–203. http:/dx.doi.org/10.1080/0020739x.2012.703337 julie, c., leung, a., thanh, n.c., posadas, l.s., sacristan, a.i., & semenov, a. (2010). some regional developments in access of digital technologies and ict. in c. hoyles, & j.-b. lagrange (eds.), mathematics education and technology − rethinking the terrain (pp. 361–83). new york, ny: springer. http://dx.doi.org/10.1007/978-1-4419-0146-0_17 lee, k., bull, r., ng, f.s., pe, l.m., & ho, r.h.m. (2011). are patterns important? an investigation of relationships between proficiencies in patterns, computation, executive functioning, and algebraic world problems. journal of educational psychology, 103(2), 269–281. http://dx.doi.org/10.1037/a0023068 linacre, j.m. (2008). winsteps rasch measurement [computer software]. chicago, il: winsteps.com. mason, j. 1996. expressing generality and the roots of algebra. in n. bednarz, c. kieran, & l. lee (eds.), approaches to algebra: perspectives for research and teaching (pp. 65–86). dordrecht: kluwer academic publishers. http://dx.doi.org/10.1007/978-94-009-1732-3_5 mayberry, j. (1983). the van hiele levels of geometric thought in undergraduate preservice teachers. journal for research in mathematics education, 14(1), 58–69. available from http://www.jstor.org/stable/748797 mullis, i.v.s., martin, m.o., gonzales, e.j., & chrostowski, s.j. (2004). timss 2003 international mathematics report: findings from iea’s trends in international mathematics and science study at the fourth and eighth grades. chestnut hill, ma: timss & pirls international study center. available from http://timssandpirls.bc.edu/timss2003i/mathd.html selter, c. (1996). doing mathematics while practising skills. in c. van der boer, & m. dolk (eds.) modellen, meting en meetkunde. paradigma’s van adaptief onderwijs [models, measurement and geometry. paradigms for adaptive education] (pp. 31–43). utrecht: panama/hvu & freudenthal institute. thompson, b., & levitov, j.e. (1985). using microcomputers to score and evaluate test items. collegiate microcomputer, 3, 163–168. vogel, r. (2005). patterns – a fundamental idea of mathematical thinking and learning. zdm: the international journal on mathematics education, 37(5), 445–449. http://dx.doi.org/10.1007/s11858-005-0035-z watson, a., & de geest, e. (2012). learning coherent mathematics through sequences of microtasks: making a difference for secondary learners. international journal of science and mathematics education, 10, 213–235. http://dx.doi.org/10.1007/s10763-011-9290-3 watson, a., & mason, j. (1998). question and prompts for mathematical thinking. derby: association of teachers of mathematics. article information authors: piera biccard1 dirk wessels1 affiliations: 1department of curriculum studies, stellenbosch university, south africa correspondence to: piera biccard email: pbiccard@yahoo.com postal address: 652 16th avenue, rietfontein 0084, south africa dates: received: 06 mar. 2011 accepted: 16 june 2011 published: 16 sept. 2011 how to cite this article: biccard, p., & wessels, d. (2011). development of affective modelling competencies in primary school learners. pythagoras, 32(1), art. #20, 9 pages. http://dx.doi.org/10.4102/ pythagoras.v32i1.20 copyright notice: © 2011. the authors. licensee: aosis openjournals. this work is licensed under the creative commons attribution license. issn: 1012-2346 (print) issn: 2223-7895 (online) development of affective modelling competencies in primary school learners in this original research... open access • abstract • introduction • affective modelling competencies    • aspects of affect and belief    • beliefs    • beliefs and mathematics • modelling and models • modelling competencies    • cognitive, meta-cognitive and affective modelling competencies • research design    • research method       • participants       • contact sessions       • tasks       • data collection and data processing       • reliability       • validity       • ethical considerations • results • discussion • acknowledgements    • competing interests    • authors’ contributions • references abstact (back to top) learner affect and beliefs about mathematics are complex and multifaceted aspects of mathematical learning. traditional teaching and learning approaches in mathematics education often result in problematic beliefs about mathematics. since beliefs influence what learners learn and how they deal with learning mathematics, it is essential that the roles of beliefs and affect in mathematics classrooms are carefully examined. in solving modelling problems, learners andteachers take on new roles in the classroom: learners are placed in an active, self-directing situation in which they solve real-world problems. when learners engage in modelling tasks, they display and integrate cognitive, meta-cognitive and affective competencies. a modelling approach therefore allows one to detect learner beliefs in an authentic learning environment. will this environment lead to students having more positive and productive dispositions towards mathematics?this article presents partial results of a study documenting the development of modelling competencies in learners working in groups over a period of 12 weeks. through a design research approach, 12 learners working in groups solved three modelling problems, and transcriptions of learner interactions, questionnaires and informal interviews revealed that learner beliefs improved over this short period when exposed to modelling tasks. the results are encouraging, and may providemathematics education with an avenue to develop more positive learner beliefs in mathematics. introduction (back to top) beliefs are a stable, affective response to mathematics (mcleod, 1992), which encompass an individual’s understandings and feelings that shape the way they conceptualise and engage in mathematical activities (schoenfeld, 1992). this means that beliefs as part of the affective domain do not function independently from the cognitive or meta-cognitive domains. if beliefs shape learner understanding in learning mathematics, then they should be considered an important part ofmathematics education research. in fact, cobb (1986, p. 4) states that beliefs are ‘a crucial part of the assimilatory structures used to create meaning’. this study accepts that beliefs ‘are largely cognitive in nature and are developed over a relatively long period of time’ (mcleod, 1992, p. 579), and since the ‘cognitive and affective domains intersect in the area of beliefs’ (schoenfeld, as cited in mcleod, 1992, p. 590), this is a significant realm of research for understanding and improving mathematical learning. this statement assists in understanding that beliefs influence what is learnt by learners in mathematics classrooms as well as how it is learnt. according to walberg’s (1984) theory of educational productivity, the following nine factors are considered to be ‘potent, consistent and widely generalisable’ (p. 20), and if optimised could develop learners’ affective, behavioural and cognitive domains. the nine factors, divided into three groups, are as follows: • learner personal factors: the learner’s ability; age and stage of maturation; motivation and willingness to persevere. • instructional factors: the amount of time spent engaged in learning; the quality of the instructional experience including psychological and curricular aspects. • environmental factors: the learner’s home; the learner’s class social group; the peer group outside the school; and the learner’s use of out of school time. in the main study (biccard, 2010) learners were involved in collaborative work on modelling tasks. this meant that one factor from each of walberg’s groups was directly affected by the programme. learners were highly motivated during the programme; they spent more time on quality tasks, whilst the grouping focused on shared ideas and opinions. this was a change from the traditional approach to mathematics tasks that they had experienced. traditional teaching methods in mathematics classrooms have led to learners holding common negative beliefs about mathematics (schoenfeld, 1992). this is because a traditional teaching approach focuses on drilling specific procedures – that are not necessarily meaningful to learners and which they do not alwaysunderstand or relate to anything else in their lives. traditional teaching is entrenched in what ernest (1988) called an instrumentalist view of mathematics, which sees the subject as well defined and as a correct set of procedures that need to be practiced and memorised. traditional teaching methods also have learners working individually, which according to cobb (1986, p. 8) means they lose valuable feedback mechanisms that they would experience working with peers, such as dealingwith inconsistencies, expressing concepts and comparing procedures. interaction assists learners to build a socially rich view of mathematics. cobb (1986) therefore suitably defines learner beliefs about mathematics as their ‘attemptedsolutions to problems that arise as they interact with the teacher and their peers’ (p. 8). cobb further clarifies that learner beliefs are not simply communicated from teacher to learner, but that students construct beliefs that fit into the classroom norms that they have experienced. in this sense, cobb explained learner beliefs as both social and cognitive, developing in response to the learners’ experience of classroom mathematics. changes in classroom experiences may hinge aroundtasks set for the classroom. the hypothesis of this study is that interaction around good-quality tasks develops more positive beliefs about the value and use of mathematics in learners. in the main study, the development of modelling competencies in primary school learners was documented over a period of 12 weeks. three main areas of competence – cognitive, meta-cognitive and affective – were considered. within the affective domain, it was decided to determine if a modelling perspective on learning mathematics could result in more positive beliefs being held by learners. traditional classrooms tend to encourage rote learning and drilling of abstractconcepts; very often problems are routine and removed from anything that resembles real-life problems. to change pupil perceptions and beliefs, problems set in real life are necessary. modelling problems present real situations that need learner engagement and structuring. very often modelling tasks result in a product that is different to traditional exercises (lesh & doerr, 2003a, p. 16), often a letter or report to someone (a client) who has a real need for the problem to be modelled. thismeans that learners may experience mathematics as useful and connected to real-life problems and to real people who make decisions based on models. in terms of the main study, the focus was on changing the quality of not only the teaching experience but what was considered to be the ‘learning experience’.modelling tasks require that learners work in groups, and much more time is needed to complete them because of their complex and real nature. this study considered whether well-formulated modelling tasks in teaching could positively transform learner beliefs about mathematics. during the 12-weekperiod that learners were involved in the modelling programme, positive results to support this hypothesis were found, which are presented in this article. affective modelling competencies (back to top) aspects of affect and belief mandler’s view on affective factors (mcleod, 1992) is that learner emotions play a role when a sequence of actions cannot be carried out. learner emotions also play a role when a sequence of actions can be played out, but negative emotions may be of a more lasting nature. mandler proposes three aspects of affective experience.firstly, learners do hold certain beliefs about mathematics, about how it should be taught and about the social context (often solitary and competitive) in which mathematical learning takes place. secondly, according to mcleod, interruptions and blockages are an inevitable part of learning mathematics and learners will experience both positive and negative emotions – more noticeably during tasks that are novel. thirdly, learners will develop a positive or negative attitude basedon a repetition of certain emotional responses. this article specifically focuses on learner beliefs in the affective domain. it will be shown that learners involved in modelling tasks within a group environment encounter a positive emotional response that may lead to building more positive beliefs about mathematics. goldin’s (2008, p. 188) view of affect is that it should be ‘regarded as an internal representation system’, and ‘to see affect as representational is to hypothesize that emotions encode information’. he adds that it is ‘detailed, context-dependent, rapidly changing information essential to the doing of mathematics’. to uncover learner-affective internal representations is not an easy task, but whilst modeling, learners do leave ‘auditable trails of documentation’ (lesh & doerr, 2003a, p. 31).these are the many written and verbal products of learner interactions when modelling, which allow glimpses into what learners are thinking even on an affective level. this trail of documentation that learners produce whilst modelling allows a certain rendering of learner-affective factors not always possible with more traditional teaching approaches. as explained by lesh and doerr (2003a), problem-solving leads to significant forms of learning. this learning can also includeaffective learning or an awareness of the use and value of mathematics through tasks that would encourage learners to think mathematically. lesh and doerr further clarify that thinking mathematically involves constructing, describing, explaining, and making patterns in complex systems (2003a, p. 16). this suggests that the content of learners’ mathematics classroom experiences may have an impact on the beliefs that learners would hold as a result of these experiences. the relationship between experiences and beliefs is a premise for this study. if emotion encodes information, then all learning would have an affective element and all learning experiences would allow learners to formulate certain beliefs about mathematics. this means that learning through tasks such as modelling may enablelearners to develop more positive beliefs about their abilities to interact successfully with mathematics, and more positive beliefs about the role of mathematics in real-life situations. beliefs definitions of ‘beliefs’ lead to discussion of knowledge and the nature of knowledge. beliefs are considered to be the acceptance that something is true based on evidence. this means that learners hold certain beliefs if some evidence generated those beliefs. beliefs that mathematics is about learning and memorising rules means that learners were given this evidence in their classrooms. beliefs are different to knowledge, in that they include an affective component (speer, 2005). uncontested knowledge can be tested, whilst personally accepted knowledge which includes beliefs has a personal judgement facet to it. beliefs may better be conceptualised as knowledge that has been reflected on from a personal perspective. when learners become personally involved in their learning and reflect on this learning, then a formation of beliefs takes place. learners develop beliefs about the role-players in the learning situation. they develop beliefs about themselves and their abilities and beliefs about the relevance of the subject matter to their lives and real problems in general. learners also develop beliefs through the role and nature of content in the mathematics classroom experience. ernest (1998) reminds us that since the time of plato, mathematical knowledge has been thought of as secure and infallible. this means that in a traditional classroom, mathematical knowledge is handed over from teacher to learner as absolute truths that learners have to memorise and learn to apply. from a modelling perspective, learners produce, sort out and develop conceptual tools that describe or explain a mathematically significant system (lesh & doerr, 2003a, p. 9, 2003b, p. 532). when learners experience the former in their classrooms, their beliefs about mathematics may be very different to when they experience modelling in mathematics classroom activities. the learners involved in this study often used phrases such as ‘your method’, ‘my method’ or ‘what did you do?’, which showed more ownership of the problem than just memorising of rules. österholm (2009, p. 158) asked how knowledge of the relationship between the diameter and circumference of a circle is ‘less’ affective than beliefs about mathematics teaching. it may be that the latter is personally reflected knowledge and this adds the affective component, whilst the former is not personally reflected knowledge but knowledge that is accepted as true. it may be that knowledge is unlikely to be disputed, whilst beliefs are not consistent for all people: for any belief that may exist, a diametrically opposing view may also exist. wilson and cooney (in speer, 2005, p. 365) propose a different perspective than insisting on definitions for beliefs. in a study on teacher beliefs they stated that concerns over precise definitions of beliefs are not as important as issues of understanding teacher knowledge. schoenfeld (2011, p. 52) establishes that beliefs are ‘very powerful explanatory constructs’. the aim of this article is to understand and provide an explanation for the development of learner beliefs relating to mathematics during a modelling programme, and not so much to provide new definitions for beliefs. answers to the questions posed in a pre-modelling and post-modelling sessions indicate learner beliefs, and their responses are reported on. if learner beliefs do change as a result of modelling tasks, then it can be inferred that modelling provides learners with the evidence, experience and repetition they need to modify some of their beliefs. furthermore, it can be inferred that if beliefs do change, then modelling provides learners with a means for personal reflection that leads to improved beliefs about mathematics. beliefs and mathematics in his work on problem-solving, schoenfeld (1987, p. 190) focused and elaborated on three related but distinct meta-cognitive categories: knowledge about own thought processes, which includes how accurate learners are in describing their own thinking; self-regulation, which includes how well learners keep track of what they are doing and how well they use the input from observations to guide their problem-solving actions; and beliefs and intuitions, which includes what ideas aboutmathematics learners bring to their work in mathematics and how these shape the way they do mathematics. this article focuses on the third category of learner beliefs and intuitions about mathematics. schoenfeld (1992, p. 359) lists typical learner beliefs about the nature of mathematics: 1. mathematics problems have one and only one right answer. (s1) 2. there is only one correct way to solve any mathematics problem – usually the rule that the teacher has most recently demonstrated to the class. (s2) 3. ordinary learners cannot expect to understand mathematics; they expect simply to memorise it, and apply what they have learned mechanically and without understanding. (s3) 4. mathematics is a solitary activity, done by individuals in isolation. (s4) 5. learners who have understood the mathematics they have studied will be able to solve any assigned problem in five minutes or less. (s5) 6. the mathematics learned in school has little or nothing to do with the real world. (s6) these six categories relating to learner beliefs were used to code the data for this study. although schoenfeld generated these from work on problem-solving, whilst traditional teaching pervades schools, these categories are still relevant and can be used to code transcriptions of learner work whilst modelling. goldin (2007, p. 289) states that affective competencies refer to a person’s ability to use affect positively during a mathematical activity, and that future research should be devoted to studying how learners change global affective structures regarding mathematics. goldin’s concept of a global affective structure was also included as this concept extends the discussion on beliefs. global affect refers to complicated affective structures which include emotions, beliefs, values, stories, memories and one’s history. goldin (2007, p. 289) further set out three essential affective structures which were also used to code data: • mathematical integrity (g1): this relates to learners being committed to truth and understanding in mathematical activities. this also relates to thelearners’ willingness to acknowledge that they do not understand something, their decision to take action and the nature of the action that the learner decides to take (debellis & goldin, 2006, p. 138). • mathematical self-identity (g2): this relates to the learners’ personal sense of self in relation to mathematics. goldin (2007, p. 291) explained that this isdeveloped over time, incorporates a feeling of ownership or non-ownership regarding mathematics, and involves a set of current possibilities relating to who the individual is relating to mathematics. • mathematical intimacy (g3): this can be associated with the concept of ‘flow’, which describes an experience that is engrossing and enjoyable to theextent that it is worth doing for its own sake (csikszentmihalyi, 1999, p. 824). csikszentmihalyi (1999) describes some of the conditions that make flow in an activity possible: it should allow one the feeling of being separate from the routines of everyday life, be so demanding that little attention is left for irrelevant stimuli, have clear goals every step of the way (either set by the person or the activity itself), provide immediate feedback on what is being done, and enable one to balance skillswith the challenge at hand. beliefs seem to form the underlying core of these affective structures, although these structures are more holistic than learner beliefs. these affective structures wereused for coding the data to trace affective competency development. both schoenfeld’s and goldin’s categories could be used to analyse learner beliefs whilst they are engaged in a modelling task. it may be valuable to use existing concepts and categories to code data on modelling: schoenfeld’s categories focus specifically on typical learner beliefs, whilst goldin’s categories were used to focus on more widespread affective facets of learning. modelling and models (back to top) a modelling task results in learners producing a model to solve a task. the explanation of a model is taken from lesh and doerr (1998), who state that a modelis a scheme that describes a (real-life) system, that assists in thinking about that system, making sense of it or making predictions. a model consists of elements, relationships, operations that describe how the elements interact, and patterns or rules that apply to the preceding relations or operations. a model focuses on the underlying structural characteristics of a real-life system being described. therefore a model can be a description, explanation or prediction of a real-life situation. from the above, it is apparent that modelling is very different to traditional mathematics tasks. modelling requires a larger domain of cognitive, meta-cognitive and affective competencies than traditional tasks. the process that learners go through when modelling is displayed in a normative modelling cycle in figure 1. modelling is more complex than traditional tasks (chamberlin & moon, 2005; lesh & doerr, 2003a). in a traditional classroom learners remain at a single node of the cycle – working mathematically, whilst the teacher or textbook is responsible for the rest of the cycle. from this experience learners could develop negative beliefsabout what mathematics is, how it is learnt and how it is relevant to real life. once learners are placed in a directorship role of the entire cycle, their beliefs about mathematics should start to change. initially their negative beliefs leave themfrustrated and they try to solicit a traditional ‘didactical contract’ (brousseau, 1997, p. 39) by asking the teacher to show them the method or solution. blomhoj and jensen (2007) describe a key to modelling as learning to cope with feelings of ‘perplexity due to too many roads to take and no compass given’ (p. 49). this type of perplexity is not easily overcome. as learners gain experience of modelling, they are able to develop their modelling competencies. by modelling, learners are responsible for wider aspects of problem-solving, which changes their experiences, and this may result in a change in affective competencies. modelling tasks also provide learners with evidence that mathematics is useful and that they can use mathematics to solve problems. with modelling tasks, groups of learners negotiatemeaning with each other and create a model for the situation in their own way without any predetermined, memorised methods. figure 1: a modelling cycle. modelling competencies (back to top) cognitive, meta-cognitive and affective modelling competencies in the main study (biccard, 2010) the nodes of the cycle were termed modelling competencies and explored as learners modelled three problems over a period of three months. there are, however, other competencies necessary for learners to model successfully. included are not only cognitive competencies, but important domainsof meta-cognitive and affective competencies. the following cognitive modelling competencies were considered in the main study (see figure 1): understanding, simplifying, mathematising, working mathematically, interpreting, validating, presenting and arguing. meta-cognitive competencies that were included were: planning and monitoring, a ‘sense of direction’ (treilibs, burkhardt & low, 1980, p. 52) and using informal knowledge (mousoulides, sriraman& christou, 2007, p. 39). from the realm of affective competencies, beliefs were selected. competence is taken from weinert’s definition (henning & keune, 2007, p. 225), as the sum of available abilities and skills and the willingness of a learner to solve aproblem and to act responsibly concerning the solution. modelling competencies encompass all the abilities and skills a learner uses in solving a modelling problem. a learner employs these abilities and skills together with certain beliefs, and these beliefs affect the way in which the learner may successfully use available abilities and skills. mcleod (1992, p. 578) also identified beliefs as a concept in the affective field that is more stable than emotions or attitudes. this was an indication thatbeliefs could be considered a competency. beliefs were included as a competency so that an awareness or development/change in learner beliefs could be monitored together with the other competencies that were identified. since the instrument did not use likert-type scales but a qualitative open coding, the development of learner beliefs as a natural progressing entity could be documented before and after a modelling programme. beliefs as an element of affect play an important role in learning and are a consequence of learning experiences. the relationship between teaching approaches and beliefs has been established (spangler, 1992). since this cycle is not easily broken, the focus of research should then be on using the teaching approach-beliefs cycle andnot working at breaking the cycle. if the relationship between teaching and beliefs about mathematics exists, then changing instruction to a modelling approach may lead to more positive beliefs about what mathematics is, how it is learnt and how it is relevant to real life. beliefs are a stable, affective construct developed over a long period of time. however, harel and lesh (2003) show that during 90 minutes of modelling, the stages oflearner thinking are strikingly similar to those described by van hiele (1986, cited in harel & lesh, 2003) or piaget (1958, cited in harel & lesh, 2003). these ‘compact versions of developmental sequences’ (harel & lesh, 2003, p. 381) assisted in hypothesising that learner beliefs may also develop over a relatively short period of time whilst they are involved with modelling tasks. research design (back to top) research method a design research paradigm was followed for the main study. the design phase included the preparation of tasks and instruments. the teaching experiment involved contact sessions between the researcher and the learners, whilst ongoing analysis of data allowed for changes to be made to the design of subsequent modelling sessions. bakker (2004) gives a detailed description of the three phases of the design research approach, that is, design, the teaching experiment and retrospectiveanalysis. participants twelve grade 7 learners were selected for the study and worked in three groups with four learners in each group. these learners attend a well-resourced suburban primary school. the groups comprised equal numbers of girls and boys and included various population groups. the groups were purposively selected and comprised two groups of learners who were ‘weak’ in mathematics (groups 1 and 3) and one group who were ‘strong’ in mathematics (group 2). the selection wasbased on learners’ school results. part of the results of the full study included comparing the development of modelling competencies in weak and strong learners. learner ages ranged from 11 to 13 years. they worked for 12 weeks solving three modelling problems. contact sessions learners met with the first author weekly after school hours for sessions of approximately one hour. these sessions were audio-recorded and transcribed, whilst the sessions where learners presented their solutions were video-recorded. the researcher also kept a book of field notes. all learners’ written work was kept andanalysed. informal interviews with groups were recorded and transcribed. each task required approximately four sessions, which included a report-back session. all groups completed all three tasks. these learners had not been exposed to modelling problems before. the groups worked autonomously whilst the researcher moved between the three groups, acting as soundboard and listener, posing questions from time to time. tasks the tasks used were sourced from existing modelling literature and all related to proportional reasoning. box 1 provides a précis of the three tasks the groups worked on. box 1: the three modelling tasks. data collection and data processing learners completed a pre-modelling questionnaire before they started working on the modelling tasks. the questionnaire included questions on what learners believemathematics is all about, what they enjoy about mathematics and where they find mathematics relevant in everyday life. learners answered another questionnaire after the 12-week session. all contact sessions were audio-recorded and transcribed. learners had to complete a number of progress documents each week. their informal written work was also scrutinised. at the end of each task learners presented their solutions to the other groups involved in the study. data were coded for all the competencies identified for the main study, and specifically according to schoenfeld (1992) and goldin (2007) for affective competencies. findings in terms of learner growth in affective modelling competencies are presented in the ‘results’ section. reliability the tasks used in this study were sourced from existing literature by experienced researchers in modelling, which meant that the responses and models produced bylearners were consistent with existing research. however, this study also considered the development of modelling competencies and did not consider the tasks in isolation. transcriptions were made by the first author on the same day as the session was held, and could therefore be used in conjunction with the first author’s field notes. the study involved three groups, and the results being consistent across the three groups added to the reliability of the study. the findings of the study are notcontrary to those in other studies, and as such the reasonableness of the inferences (bakker, 2004) is continuous. validity direct participant transcriptions added to the validity of the study, as did the recorded data and hard copies of transcriptions. the use of two coding systems (those of schoenfeld [1992] and goldin [2007]) for the data also added to the validity of the study, as did the use of numerous data sources (learner work, audio-recordings of modelling sessions, video-recording of presentation sessions, questionnaires and informal interviews). ethical considerations permission for the study was granted by the gauteng department of education, the participating school as well as the affiliated university. parents of learners selected for the study were briefed as to the aims and objectives of the study and gave consent for their children to be involved. learners participated in the studyvoluntarily. the aims and objectives of the study were discussed with learners and they were allowed to ask questions. the name of the school and those of the learners were not divulged. the recordings, transcriptions and learner work were stored away safely by the first author. results (back to top) the first part of this discussion relates to schoenfeld’s categorisation (s1–s6) of learner beliefs, whilst in the second part goldin’s affective structures (g1–g3) are used to mould the discussion. the selected excerpts from transcriptions highlight a healthy view of the nature of mathematics and an awareness of the use and valueof mathematics. since beliefs are built over a long period of time, it may be possible that the tasks provided learners with evidence that may begin a change in their beliefs about mathematics. learner ideas about what constitutes a mathematical task form part of their beliefs. group 2 had the following discussion after a heated argument in task 3, that can be related to beliefs about mathematics and the nature of mathematical tasks (in the following protocols, letters denote the names of different children): learner s: ‘it’s incorrect!’ learner t: ‘nothing is incorrect with this, ok.’ learner s: ‘i am really good at maths.’ learner t: ‘yes, but with these problems ...’ (biccard, 2010, p. 124) here learner t is moving towards changing beliefs about mathematics problems, but learner s remains unchanged, holding that ‘i am really good at maths’. this shows that learner t may have been becoming more aware of a difference between these problems and traditional tasks. in learners’ written reports after each session, they had to write down where they needed assistance. group 3 wrote this after the first session of task 1: what we need help with? [learner e] to get the thing right, to plus it or to divide it or whatever. [learner m] this shows that the learners expected to apply some mathematics that they knew, without really understanding what they were doing (s3). by task 3, they were discussing how to find a ratio, with the focus on their own understanding of the task (development of s3): learner g: ‘we need to find something related to that.’ [pattern measurement and real measurement.] learner e: ‘we need to measure this which is 27, 4 … now we need to find something that will give us ... something that is …’ learner m: ‘related … we are trying to see our answer.’ learners were informally interviewed during task 2, when they were asked by the researcher about their experience of working in groups as opposed to working individually (s4) and both groups responses are given in box 2. by the end of task 3, the following comments were made by group 3, showing a gradual change in their beliefs about the nature of mathematical tasks (development of s5): the thing about the beginning of a task is that we are always so confused. [learner g] that’s impossible. look, i don’t think we are going to get a direct answer at the very beginning of a task. [learner g] no, we’ll never do that, unless we have a time machine. [learner w] (biccard, 2010, p. 125) the problems involved a large amount of reading, which is not common in traditional classrooms. since real problems are set in a context that needs to be understood, often through reading, learner response to reading the supporting material for each task allowed s6 to be gauged. during task 1, the learners were unsure what all the reading material was about, as one indicated by saying ‘i don’t think the teacher meant for us to read this stuff’. this shows that they may not have believed that mathematics problems involved much reading. however, by task 3 they did not question the supporting material for each task, but simply referred to it when they needed to: learner s: ‘this problem is a little more serious than the last. oh well, what the heck, let’s give it a try. how do we start the problem a?’ learner a: ‘ok. read this.’ learner j: ‘do you think we have to make this outside also?’ learner a: ‘no, that is a photo frame.’ learner j: ‘read what we are asked to do. just read it again …’ evidence was found, which includes elements from s1-s6, that learners believe that mathematics must be done neatly and that whole number answers aremore ‘correct’ than messy decimals. an incident showing that learners viewed mathematics as a neat, systematic endeavour is noted at a later stage, and is also mentioned as it directly affected group 3’s modelling competencies. it was noticed that group 3 spent a lot of time rewriting a list of items from a catalogue for task 2 (catalogue problem), and lost valuable time doing this. the next week, they were questioned about their decision: researcher: ‘why did you write the prices over? when they were here?’ [on the instruction sheet provided] ‘was there a reason?’ learner m: [a little embarrassed] ‘we didn’t want to mess the page.’ (biccard, 2010, p. 124) during task 2, group 2 noticed that they had reached an answer of a whole number. this led them to think that they must be on the right track. they still assumed that neat, round numbers were more correct than decimal numbers, and learner j exclaimed ‘[b]ut ours and yours adds up to r200 [exactly], there is a connection!’ (biccard, 2010, p. 124). by task 3, this group worked with a decimal of up to 5 digits without thinking that it was incorrect. when looking at goldin’s essential affective structures, it was found that modelling elicits the affective structures he describes. mathematical integrity as an affective structure includes an awareness of one’s limitations in understanding and acknowledging them and working towards removing them (goldin, 2007). this structure is strongly affective and confirms learner beliefs in mathematics and their experiences of it. it was found that learners were unafraid to acknowledge theirconfusion or frustration with these tasks – which is largely kept silent during traditional instruction. if it is evident, the teacher feels responsible for clearing up the confusion due to the didactical contract (brousseau, 1997) that exists. with modelling tasks learners had to support and provide scaffolding for each other. the following condensed excerpt illustrates the need for mathematical integrity (g1) that modelling elicits (task 1): learner s: ‘we are really stuck.’ learner j: ‘i don’t get it.’ learner j: ‘what about the adding 20 method?’ learner s: ‘that was your method. no, that was a random thing.’ researcher: ‘can you explain that to me?’ learner j: ‘no, i can’t back it up.’ learner j: ‘we are trying to see the significance between it.’ [foot size and height] learner s: ‘you mean a relationship?’ learner j: ‘maybe it can be a pattern?’ the input from learners corresponded with mathematics self-identity (g2) are listed in box 3. learners were slowly starting to place themselves, their thoughts and values into the mathematical arena, which resulted in a much fuller mathematical experience. the following highlights mathematical intimacy (g3), which includes a ‘vulnerable interaction involving one’s sense of self during mathematical activity’ (goldin,2007, p. 291); the final question on the questionnaire asked for any comments the learners wished to offer, and a number of learners stated: ‘when are we going to do this again?’ the following comment by one learner fits into csikszentmihalyi’s concept of flow: ‘i did enjoy the problems because you got to go way out of the ordinary and when you’re done you feel so fantastic’. affective structures and beliefs are brought to light whilst learners are involved in modelling tasks, which allow one to view learner beliefs and affective structures formed over the entire span of learner experiences with mathematics. modelling therefore fulfills two different but mutually important roles: it serves as a mirror in which learners reflect not only their mathematical knowledge and skills but also their affective orientations, and also suitably fills the role of generating anenvironment within which to build more positive beliefs about mathematics. box 2: learner responses when asked about their experience of working in groups as opposed to working individually. box 3: the questions and answers from learners in group 2 on mathematics self-identity. discussion (back to top) goldin’s view (2007, p. 293) that ‘mathematical modelling activity itself brings about not only cognitive development, but development of the affective system’ is accepted by the authors by virtue of the results of this study. learner affective competencies seem to be ‘far more complex than one might infer from just the surface consideration of emotional feelings’ (goldin, 2007, p. 281). the results point to the idea that learners can undergo ‘global affective changes’ (goldin, 2007, p. 293). what is important to note is that a relatively short modelling course facilitated a shift in some learner beliefs. mcleod’s (1992) view that what is needed is a change in the curriculum that fosters learners’ negative beliefs is also accepted. modelling allows for such a change or modification to both the curriculum and teaching strategies. possibly, as suggested by wilkins and ma (2003), recognition of negative trends is a central first step for teachers in attempting to reverse negative learner beliefs. as such, these results are concurrent with the findings of wilkins and ma (2003), in that exposing learners to challenging mathematics may curb the development of negative beliefs about mathematics. returning to walberg’s (1984) model of productivity (p. 21), modelling affects at least one factor in each of the three main areas. with regard to learners’ personal factors, modelling increases learner motivation. modelling also makes significant changes to the teaching factors by improving the quality and amount of time spent on the learning experience, and affects environmental factors by making changes to the traditional classroom environment and interaction. modelling allows mandler’s affective aspects (mcleod, 1992) to come into view. within the modelling sessions learners did hold certain beliefs about mathematics, and some of these beliefs showed signs of positive change towards the end of the programme. learner blockages were evident – and so were their routes through these blockages; it was their use of informal methods through these blockages that built more positive beliefs about mathematics. in order to maintain or consolidate these positive attitudes, repetition of these tasks is necessary. modelling as a teaching approach in mathematics classrooms will lead to learners using their own (often invented) methods and allow learners to mathematise situations in ways that are meaningful to them. learners are not expected to follow a rehearsed method or a set procedure (that they often do not understand). the model that results from engaging in a well-formulated modelling task is constructed by learners and shows their understanding of the problem situation. in perrenet and taconis’ (2009) study, that showed significant shifts in the beliefs of university students, the students were asked to explain their own shift in beliefs to point out differences between their secondary school and university mathematics experiences. according to these students, the difference was in the nature of the problems that they were given to solve. they said that school problems were standard closed problems, whilst those at university were more challenging and complex. goldin (2007, p. 294) affirms that teacher-guided modelling produces long-term effects that have cognitive and affective consequences. modelling as an environmental factor allows learners to work in groups, collaborating with each other. from a socio-constructivist view, affective processes are considered an essential part of problem-solving and learning (op’t eynde, de corte & verschaffel, 2006) so they form a key part of a modelling environment and are also developed through this environment. from a social-interactionist lens, individuals construct meaning through social interaction but, more importantly, social interaction is the process that forms learner conduct and is not simply a setting for this conduct to take place (yackel, 2000). social interaction therefore forms learner beliefs about mathematics. this too was evident in the study. goldin’s (2007) view that ‘adequate foundations for studying and enhancing mathematical learning and development must incorporate the affective domain – not as an architectural add-on, but as a structurally essential building block’ (p. 281) provides encouragement to further research affective factors in mathematics learning. acknowledgements (back to top) the financial assistance of the national research foundation (nrf) towards this research is hereby acknowledged. opinions expressed and conclusions arrived atare those of the authors and not necessarily to be attributed to the nrf. the input and suggestions of the reviewers are acknowledged and the authors would like to thank them for their contribution. competing interests the authors declare that they have no financial or personal relationships which may have inappropriately influenced them in writing this article. authors’ contributions this article presents partial results from a larger study in which p.b. conducted the research whilst d.w. supervised the study. this article from the main study wasplanned as a team. the manuscript was exchanged several times between authors. d.w. gave valuable insights as the article developed and unfolded. references (back to top) bakker, a. 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(2000). creating a mathematics classroom environment that fosters the development of mathematical argumentation. paper prepared for working group 1: mathematics education in pre and primary school of the ninth international congress of mathematical education, tokyo/makuhari, japan. abstract introduction purpose of study literature review methodology discussion of data and results learners’ reflections on the home language versions of the task conclusion acknowledgements references about the author(s) nkosinathi mpalami department of mathematics, natural sciences, and technology education, faculty of education, university of the free state, phuthaditjhaba, south africa citation mpalami, n. (2022). complexities of translating mathematics tasks from english to learners’ home languages. pythagoras, 43(1), a560. https://doi.org/10.4102/pythagoras.v43i1.560 review article complexities of translating mathematics tasks from english to learners’ home languages nkosinathi mpalami received: 25 may 2020; accepted: 31 aug. 2021; published: 18 jan. 2022 copyright: © 2022. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract mathematics education remains problematic in south africa’s schools. however, some mathematics educators are deliberately using learners’ home languages in tasks to assist learners to understand mathematics. research-based evidence shows that learners’ home languages when used as a resource have a potential to enhance learners’ understanding of mathematics. this article addresses the issue of translating mathematics tasks from english to learners’ home languages, a field that is less common in mathematics education studies. the study shows that there are complexities associated with such translation which all stakeholders in education should bear in mind. the article does so by referring to a study where a grade 11 mathematics educator in a multilingual class tried to use learners’ home languages in tasks with an aim to enhance learners’ understanding of linear programming concepts. the study was conducted in township school in gauteng province. ethical clearance was given by the gauteng department of education. data were collected through observations and were analysed qualitatively. the situated sociocultural perspectives guided the study. the findings show that during the translation process, the educator went as far as translating mathematics technical terms. such translation distorted the meaning of the original task and therefore made it hard for learners to comprehend concepts as envisioned. the recommendation is that the translation should not be left to individual mathematics educators but rather there should be a broader approach of having mathematics tasks translated from english into other official languages and such tasks be distributed to all schools throughout the country. professional translators must also be contracted to do such a job. keywords: multilingual classrooms; linear programming; home languages; mother tongue; mathematics tasks; translated tasks. introduction this article addresses the issue of translating mathematics tasks from english language to learners’ home languages. the article shows that the work of translating mathematics tasks is challenging, and care must be taken when such an activity takes place. recent reports indicate that south african learners’ performance in mathematics continues to be undesirable. when assessing the state of mathematics and science education, the world economic forum (wef) report (2017/18) mentions that south africa is ranked 128th out of 137 countries in the global competitiveness rankings (http://reports.weforum.org/pdf/gci-2017-2018-scorecard/wef_gci_2017_2018_scorecard_eosq130.pdf). furthermore, in the year 2018, out of the 800 843 learners who sat for the national senior certificate examination, 21% obtained a bachelor’s pass which was significantly low. according to the trends in international mathematics and science study (timss) 2002 and timss 2011, south african students are ranked among the lowest (saal, van ryneveld & graham, 2019). this crisis calls for immediate intervention by all stakeholders in education. while there could be multiple solutions to this problem, learners’ understanding of key mathematics concepts in their home languages remains critical (nhongo & tshotsho, 2019). when arguing from a ‘translanguaging’ perspective, nhongo and tshotsho (2019, p. 58) point out that ‘it is an undoubtable fact that a learner understands better when concepts are introduced to him or her in his or her first language’. it could be reasoned that translating mathematics tasks into learners’ home languages is a positive move towards helping african learners whose mother tongue is not english or afrikaans in south africa to gain a better understanding of mathematics. this implies that all ‘linguistic resources’ in multilingual classrooms must be used ‘flexibly and contingently to facilitate effective communication’ (costley & leung, 2020, p. 2). this point again drawn from translanguaging studies reinforces the notion that people communicate effectively when they interact in their home languages rather than when they talk in the language of others. purpose of study the study was set to explore the complexities associated with translating mathematics tasks from english to learners’ home languages, and how such translated tasks were used in a multilingual classroom. stein, grover and henningsen (1996, p. 460) define a mathematics task as ‘a classroom activity, the purpose of which is to focus students’ attention on a particular mathematical idea’. in the teaching and learning of mathematics, a teacher plays an important role in selecting or creating appropriate tasks for learners. the commonly used mathematics tasks are readily available in english. mathematics educators have a responsibility to assist learners to unpack the demands of such tasks. one way of doing that might be using multiple learners’ home languages. prior to a democratic south africa, the only legal languages used for instruction were afrikaans and english which inevitably advantaged the speakers of those languages. however, in this era teachers are at liberty to use any official language to help learners understand mathematical concepts (planas & setati-phakeng, 2014). while ‘code-switching’ might be regarded as a common practice in multilingual classrooms (setati, 2005b), the work of translating tasks is rare and limited in the field of mathematics education. tesseur and crack (2020) point out that translation in general is not a common practice in a variety of contexts such as in education. this article hopes to contribute some insights in the field of mathematics education. the teacher in the study used translated tasks in a lesson with an aim to make linear programming concepts accessible to african learners. the research question the research question that guided this study was: what are the complexities associated with translating mathematics tasks from english into learners’ home languages? statement of problem while translation may be narrowly viewed as ‘linguistic equivalence between two languages’, cultural contexts must be considered during the process of translation (chibamba, 2018). translation studies scholars concur that multilingual settings necessitate the process of translation (marais, 2014; nord, 1997; tesseur & crack, 2020). these studies, however, have not provided guidance as to how translated tasks might be used for teaching purposes. the south african department of education (dbe) (2011) stresses the need for educators to use learners’ home languages in class to help them comprehend subject matter. what remains unclear is how such languages must be used during instruction. if the official documents remain silent on this issue, then it becomes challenging for individual educators to know how they should begin using such languages. when drawing from setati (2008), barwell (2016) reminds us that the use of language in class should never be taken for granted as it has a potential to either ‘privilege or marginalize’ some learners. that is why it is imperative for instructional documents to spell out clearly how educators must make use of multiple languages in mathematics lessons. in this article, a class is regarded to be multilingual if participants (learners and educators) are potentially capable of drawing on more than one language as they interact during a lesson (setati & barwell, 2006). given that south africa is a rainbow nation, it then follows that several schools are either bilingual or multilingual. even though african learners have their own home languages, researchers have established that they continue to be taught mathematics in either english or afrikaans (setati, 2005; webb & webb, 2008). it is also known that most educators in this country are fluent in at least two languages (adler, 2001). adler argues that educators in south africa are faced with this ‘dilemma’ frequently: to either pay more attention to mathematical language or to focus on mathematical concepts. barwell (2003) poses this question: how best should mathematics educators use multiple languages in instruction? we could further ask how such use must support and enhance learners’ understanding of mathematical concepts. rationale unlike many other mathematics topics, linear programming has a lot of word problems that require learners to deeply understand the demands of each task so that they can form multiple representations (ed. luneta, 2018; mpalami, 2013; purwadi, sudiarta & suparta, 2019) as steps leading to a solution. tshuma (2020) highlights the role language plays in mathematics education by either making mathematics accessible to learners or hindering their advancement in mathematics. learners must master both the ordinary language used in tasks and the mathematical register thereof. according to sepeng (2014, p. 15), ‘mathematical word problems include pure mathematical tasks dressed up in a real-world situation’. then learners’ duty remains mainly to analyse and successfully find solutions. unpacking the demands of each word problem is linguistically challenging for learners, especially for those who study mathematics in a language that is not their own (moschkovich, 2009; setati, 2005a; tshuma, 2020). added difficulty is because most word problems use mathematical language, as will be seen later in this article. the translation effort in this study was based on the premise that if learners understand tasks in their home languages, then they might be in a better position to solve them successfully. however, chibamba (2018) cautions that translation is a challenging process in that no translation will possibly be congruent to its original text. in this study, a mathematics task was translated from english into several home languages that existed in the class, and those languages were isizulu, isixhosa, sepedi and sesotho. literature review the situated sociocultural perspective to learning and teaching informed this study. moschkovich (2002, p. 197) argues that: a situated-sociocultural perspective can be used to describe the details and complexities of how students, rather than struggling with the differences between the everyday and the mathematical registers or between two national languages, use resources from both registers and languages to communicate mathematically. in this study, it was helpful to understand how the educator used learners’ home languages in tasks as linguistic resources. because learning mathematics is inherently social and cultural (moschkovich, 1999), it was important to understand how the translated versions made it easier for learners to participate in class. moschkovich (2002) outlines four constructs embedded within a situated sociocultural framework. those are ‘practices, bi/multilingualism, code switching, and discourses’. these four intertwined notions might help researchers to understand interactions that take place in multilingual mathematics lesson. as an analytical lens, the four dimensions were used to provide analysis of the translated task and how such translation might have impacted on its use during instruction. the notion of ‘practices’ assisted in focusing on the content under discussion. some of the mathematical practices are abstracting, generalising, explaining, and connecting claims to representations. the code switching dimension assisted in explaining the differences and similarities that exist between the two versions of the same task because the translation process was mainly between two languages, namely english and a home language. the multilingualism dimension shed light on the nature of the languages that existed in this particular grade 11 class. the discourse dimension made it possible to analyse words that were used in the translated tasks whose meaning could be situated in the community and culture where such language is spoken. contentions about the choice of language of instruction in south african schools were raised in the early 2000s by researchers (setati, 2003). setati (2005b) highlights the point that despite the efforts that south africa’s government is putting in place to give african languages the status they deserve, many parents continue to prefer english as a medium of instruction for their children. this is no surprise because as setati puts it ‘mother tongue instruction’ is viewed as of less importance among speakers of african languages. they associate it with the historic apartheid regime where african languages were considered inferior and not suitable for use in teaching and learning. setati further states that english remains the language of socio-economic and educational development in south africa. of course, this is not unique to africa: sahr (2020:54) when doing research with german students concludes that ‘speaking english is clearly connected with “americanness” and whiteness’. setati (2008) has termed this powerful status of english a ‘cultural model’. barwell (2016, p. 36) concurs with setati and argues that the very same ‘cultural model of english as an international language is political in nature, as it privileges a particular language and people who are able to use it effectively’. the cultural model of english helps us understand why some african parents want their children to learn english and to be taught subjects like mathematics in english. it is the very same cultural model of english that influences educators’ daily decisions regarding the language they choose to use in mathematics teaching. when referring to their study carried out in the eastern cape, webb and webb (2008, p. 29) remind us that even educators themselves share a similar view about the use of learners’ home language (isixhosa) during instruction. they therefore argue that educators ‘expressed the fear that by using code switching they are depriving their learners of opportunities associated with the use of english’. however, some scholars who advocate for ‘translanguaging’ pedagogy support the use of learners’ home languages for teaching and learning (aoyama, 2020; garcía, 2009; sayer, 2013). translating mathematics tasks therefore remains pertinent for african learners. methodology this study employed qualitative approaches (cohen & manion, 1994). a grade 11 mathematics educator who was teaching in a multilingual classroom agreed to participate in the study. he agreed that data be collected at the time when he was teaching linear programming. he deliberately utilised african languages in tasks to enhance learners’ understanding, a trait that made him relevant to this study. this was an ‘exemplifying case study’ (bryman, 2008) whose focus was to capture and explore circumstances and conditions of an individual teacher and his ways of teaching a particular mathematics concept. according to mouton (2001, p. 149) case studies are ‘studies that are usually qualitative in nature and that aim at providing an in-depth description of a small number of cases’. so, the assumption here is that by focusing on a ‘case’ one is likely to acquire a deeper understanding of the phenomenon. the mathematics tasks that the educator used in each lesson formed the main source of data in the study. such tasks were collected and later analysed. however, for this article the focus is mainly on the task used in lesson 5. it is in this lesson where the educator used translated versions of the task. the teacher asked his colleagues from the language department to help with the translation of the task from english to learners’ home languages. the translated tasks were then taken to his mathematics teachers for validation. the tasks were piloted with 11 grade 12 learners representing the four ethnic groups. this was to ensure that such tasks were suitable for use in class. during the lesson, learners worked on the task in groups as follows: two sepedi groups, two sesotho groups, one isixhosa group, and one isizulu group. the gauteng department of education granted ethical clearance to carry out this study. discussion of data and results in this section, the analysis of each version of the task is done separately to pinpoint to the complexity in each translated task. complexities of translating tasks table 1 presents the translated tasks together with the original task taken from the textbook. learners were given the tasks to work on in their respective groups. their discussions were based on both the english version and the translated version. the educator moved from one group to another listening to how learners were reasoning about the task. however, while the educator was able to understand some of the home languages his main language is isizulu. table 1: mathematics task in original english and translation. the task could be referred to as a ‘real-life mathematical word problem without real context’ (sepeng, 2014). in part (a) the cost prices for cows are not realistic. the cheapest beef cow might not necessarily cost r1000 in south africa. however, this should never affect learners’ ability to solve the problem. the task has realistic mathematical demands irrespective of its unrealistic context. it must be mentioned here that the educator’s home language is isizulu which then suggests that the isizulu version might be the best translation compared to the others. the following discussion pays attention to each translated version. the discussion is not meant to expose the teacher’s weaknesses, but rather to understand the complexities associated with translating mathematics tasks. task in isizulu the two variables x and y are mentioned in the english task. in mathematics, x and y are symbols that represent various unknowns. in the task, x stands for a certain number of dairy cows and y for beef cows. it is worth noting that in the isizulu version x and y were translated into words. the teacher wrote the word esingazazi [unknown] for x and then translated y as ezingadalulwanga [anonymous]. this type of translation makes these two variables look like they were words that had to be translated. however, in (a)(iii) the translation for both x and y is the word ezidaluliwe (anonymous). such an inconsistency has the potential of complicating the task and confusing learners. again, in the isizulu version the amount r1000 is not written in figures like in the english version. it is translated as inkulungwane yamarandi (one thousand rands). but other amounts are presented both in words and numerals. this inconsistency might be confusing. however, in the isizulu version the technical word inequality is not translated and rather used in context. the use of this technical term in this manner might indicate to isizulu learners that they must construct constraints (inequalities) and later represent them on the cartesian plane to show a feasible region. the use of the word inequality might as well have lowered the complexity of task and made it easier to comprehend. the isizulu translation in this case seems to be more personal than the original version. for example, in part (a)(ii) the task reads: ‘singajabula uma ungasibhalela amanye ama inequalities …’ [we can be happy if you can write other inequalities …]. being ‘glad’ was never part of the original task. the original task reads: ‘write down any other inequalities …’. this incident indicates that translating mathematics tasks is complex. this type of translation might limit learners’ ability to comprehend mathematics content. the personal style of translation could be entrenched in the zulu culture, especially the way a teacher would relate to a group of zulu learners. such a practice does not exist in the english language. in general, that would be a polite way a person gives instructions to those he respects. this personal style therefore serves as a way of encouraging learners to actively participate in doing the task. however, the shortcoming of this style is that it increases the number of words in a task and therefore might make it hard to establish the actual demands of the task. task in isixhosa in the isixhosa version the word ezinga [unknown] was used for both x and y during translation (see (a)(i)). however, in part (a)(iii), the same word is not used for x and y. when looking at the two sections part (a)(iii) is clearer than section (a)(i). for instance, part (a)(i) reads: umlimi uthenge iinkomo zobisi ezinga. inye ixabisa iwaka elinamakhulu amabini aneshumi elinesihlanu (r1250) [the verbalization of 1250 is not linguistically correct here]. waze wathenga iinkomo zenyama ezinga, inye ixabisa iwaka leerandi (r1000). angachitha kangangamawaka amahlanu (r5000) yenza umlinganiselo weenkomo zobisi nowenkomo zenyama. section (a)(iii) on the other hand reads as follows: ingaba iinkomo zobisi nezenyama zingasinika owuphi umlinganiselo? chaza. perhaps there was no need to use the word ‘ezinga’ in section (i). another problem in section (a)(i) is that a mathematical term – inequality – was translated. the technical term inequality was translated as umlinganiselo [unequal things]. the term ‘umlinganiselo’ has a connotation of inequality, but it does not necessarily have the same meaning as inequality in mathematics. this implies that it might be hard for mathematics learners whose home language is isixhosa to interpret the task and realise that they must write an inequality. it is pertinent to note that in as much as the translation referred to the word ‘umlinganiselo’ consistently for inequalities in parts (a)(i) and (a)(ii), it was again used in section (a)(iii). however, the term inequality does not appear in the original version in section (a)(iii). this means the isixhosa version here is different from the original version. when the translated version is totally different from the original one, then the two are more likely to inevitably yield different results. in section (b), the original version demands graphical representation of inequalities on the xy-plane. but the isixhosa version on the contrary does not mention the word ‘umlinganiselo’, which then suggests that the two versions call for dissimilar outcomes. the literal translation of ‘umlinganiselo’ in english is measurement. this meaning is not equivalent to that of an inequality. this reinforces the idea that translating technical terms is unnecessary. task in sepedi unlike in the other two translations (isizulu and isixhosa), the sepedi version demonstrates some degree of consistency in sections (a)(i) and (a)(iii) for the use of the expression tše sa tsebjego (unknown) for both x and y. because x and y are both unknowns, this translation is correct and meaningful. in section (a)(i) the word tekatekanyetšo [measurement] is used for the word inequality, and that refers to the concept of measurement when used loosely. this word ‘tekatekanyet š o’ is not a proper translation for the mathematical term inequality. the translation of the word inequality has been extremely inconsistent in this language. for instance, in (a)(i) the word inequality is translated as tekatekanyetšo, while in (a)(ii) the word inequalities is translated as ditšhitišo [obstacles or constraints], and in (b) it is translated as mo di kopanago [where they meet]. this inconsistency has a potential to confuse the actual the meaning of the task. the three stated terms for inequality do not even mean the same thing. each term has its own distinct meaning. these terms do not even refer to inequality in mathematics. again, the translation of technical terms in this language was not necessary. task in sesotho in the sesotho version, the variables x and y were not translated in both parts (a)(i) and (a)(iii). yet it can be said that the translation makes a lot of sense. section (a)(i) reads as follows: ‘rapolasi o reka dikgomo tsa lebese … le likgomo tsa nama …’ [a farmer buys dairy cows … and beef cows …]. this is done without alluding to the number of cows (x and y). it is important to realise that when solving the problem, learners might still refer to the use of any two variables (not necessarily x and y): one for dairy cows and the other for beef cows. in the sesotho version, the translation for the term inequality is consistently referred to as kgaello [shortage] throughout the task. the word kgaello in sesotho means shortage of a certain quantity. as such the translation does not necessarily match the original use of the term (inequality). the use of the term kgaello for inequality distorts the original meaning of the task. perhaps the technical terms should have been left intact during translation. learners’ reflections on the home language versions of the task as part of the reflections on the lesson proceedings, learners were requested to mention their views on the use of the translated tasks. in general, learners stated that since it was their first encounter, they found it challenging to do the translated task. learners felt that the task in their home languages was unclear and they therefore would prefer the english version. it was hoped that the teacher would probe further on learners’ comments. conclusion the findings in this article point out that the process of translating mathematical tasks is a complex activity. the process of translating a mathematics task to learners’ languages was done by unprofessional translators (teachers in the language department). that compromised the quality of such translation. the teachers who helped the participating teacher in this study with translation are not acquainted with mathematics as a learning area. as a result, they translated variables and technical terms, which distorted the meaning of the original task. this could be avoided by engaging professional translators and qualified mathematics educators in the process. in a study where a translated task was used in class, sepeng (2014, p. 20) found that there was no significant difference in the way learners engaged with the task in both english and isixhosa. this suggests that it might be necessary to carry out more studies on translating mathematics tasks. the lesson drawn from this study points to a need for a deliberate effort to translate mathematics tasks even from other topics to help learners have access to mathematics content in their mother tongue. however, such translation must be done by professional translators to ensure a more robust process. again, such translation must not be left to individual educators but should be done at the national level so that all schools could be given such tasks as part of instructional resources. acknowledgements i would like to thank the teacher who participated in this study and the gauteng department of education for the permission to carry out this study. competing interests the author has declared that no competing interest exists. author’s contributions the author has contributed alone to this work. ethical considerations ethical clearance was obtained from the gauteng department of education. funding information this study was partially funded by the canon collins trust scholarship. data availability data sharing is not applicable to this article as no new data were created or analysed in this study. disclaimer the views and opinions expressed in this article are those of the author and do not necessarily reflect the official policy or position of any affiliated 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(2008). introducing discussion into multilingual mathematics classrooms: an issue of code switching. pythagoras, 67, 26–32. https://doi.org/10.4102/pythagoras.v0i67.71 mathematics for teaching: what is it and why is it important that we talk about it? jill adler marang centre for mathematics and science education, university of the witwatersrand email: adlerj@educ.wits.ac.za introduction i begin this paper1 with a brief discussion of what can be considered systemic ‘problems’2 in mathematics education before moving on to the research problem i am currently investigating. this broad introduction will help locate the quantum3 research project – a study of the mathematical work of teaching – that forms the substantive part of this paper. i will describe the research we are doing, its questions and objects. i will discuss some initial findings, what this research suggests for mathematics educators in the tertiary and school sectors, and so why it is important that we (communities of mathematics educators, and mathematicians) talk about it. systemic problems in mathematics education mathematicians often ask me: “what are the major problems in mathematics education? what progress has been made by mathematics education researchers in solving these?” these are typical of the way a mathematician would ask questions about research in mathematics, where the field advances through the increasing solution of outstanding problems.4 education, and by 1 this paper is based on a plenary talk presented at the south african mathematical society (sams) conference, potchefstroom, november 2004. i would like to take this opportunity to thank amesa (the association for mathematics education of south africa) for the honour bestowed on by the invitation to deliver a plenary address at the 2004 sams conference. the paper presentation will be published in the september 2005 notices of sams, the society’s newsletter. 2 i use ‘problem’ here to indicate an area of inquiry, that which needs systematic and informed investigation, an area of social life about which we seek understanding and improvement. 3 quantum is the name given to an r&d project on mathematics qualifications for teachers in south africa. the development arm of quantum completed its tasks in 2003. quantum continues as a collaborative research project. zain davis, from the university of cape town, has been pivotal in developing the theoretical orientation and approach to data. current co-investigators include diane parker from the university of kwazulu natal, and lyn webb from the nelson mandela metropolitan university and mercy kazima who is a post doctoral fellow at wits. 4 i acknowledge professor hyman bass, president of the international commission for mathematical instruction (icmi) and professor deborah ball, for many interesting and engaging conversations about the work of mathematicians and its contrast with mathematics education. implication, mathematics education, is a very different research field. problems are not as well defined, nor are they solved once and for all. there is, however, an established and thriving field of mathematics education research with advances in knowledge about the teaching and learning of mathematics, and how to research and improve these complex domains of social life. in this paper i describe this research as a study of ‘problems’ intentionally to enable discussion between mathematicians and mathematics educators on a significant shared ‘problem’. without detracting from the contribution of small-scale investigations into teaching and learning mathematics, there are identifiable systemic problems in mathematics education.5 at the most basic level, we have yet to understand how to make mathematics learnable by all children. with pressure on higher education institutions for better throughput, this issue is one that now concerns us at all levels. it is interesting to note here that there are currently discussions between the imu (international mathematical union) and the icmi (international commission for mathematical instruction) on a joint international study of what is called the “pipeline problem”, and our shared concern that fewer people are taking up advanced study of mathematics. this threatens the development of the discipline itself, as well as the provision of scientists and engineers, not to mention the provision of mathematically well qualified teachers for our schools. we have yet to understand what constitute the most effective interventions into key points or areas in the system. is this at school/institutional level, in classrooms, or at the level of individual learners? are the learning problems we see at all levels a function of curriculum? pedagogy? language? or again a combination of all three? 5 a number of papers in mathematics education were presented at the sams 2004 conference each of which is a small-scale study that relates in some way to common research thrusts, questions and problems in the wider field. the abstracts of these talks are available on the sams website: www.cam.wits.ac.za/sams/ 2 pythagoras 62, december, 2005, pp. 2-11 jill adler thirdly, and this is my own area of interest, we do not yet know enough about the mathematical preparation and ongoing support that enables mathematics teachers to do a capable and skilful job. at this juncture of massification (where all in society are expected to be mathematically literate), we face an enormous challenge of providing large numbers of adequately and appropriately trained mathematics teachers, at a time when fewer are choosing teaching as a profession.6 we face a particular and new problem of scale of provision, and this again is a significant concern that mathematicians and mathematics educators are likely to share. the question currently being investigated in the quantum project, one i regard as also systemic, is what mathematics teachers (at different levels) need to know and know how to do, in order to teach well. while we obviously need to know this if we are going to effectively prepare large numbers of teachers across the system, the question emerges out of two additional concerns. firstly, the curriculum teachers are now (in post 2000) expected to teach does not match their prior learning (in school or in teacher education). not only are there new orientations to teaching mathematics (and other subjects), in mathematics there are new topics, topics teachers currently in practice have not learned e.g. data handling, applications and modelling. secondly, and this is the more difficult area to conceptualise and understand, the mathematics that is used in teaching the curriculum is not synonymous with doing mathematics in other domains of practice (e.g. engineering, nursing, business). the kind of mathematical problem solving teachers do as they go about their work in school classrooms is less clear, yet critically important to understand. mathematics teaching as mathematical problem solving7 a few examples, what are usefully called problems of practice, will help to illuminate what is implied by the notion of mathematics teaching as a special kind of mathematics problem solving. i will begin with one area that is well known – error analysis. mathematics teachers constantly deal with student errors and/or misconceptions. here are two well known learner errors of interpretation of -3(x + 2). • -3(x – 2) = -3x – 6 6 see adler, ball, krainer, lin and novotna (2005) for further discussion of this issue and its implications. 7 i am indebted to deborah ball and hyman bass for this insight and productive analysis of the mathematical work of teaching. • -3(x – 2) = (x – 2) – 3 the mathematical roots of these errors are quite different. a teacher who faces these in class, needs to do on the spot analysis of the nature of the error, and its mathematical entailments, as well as what it means to engage learners productively to shift their thinking. at face value, the first response could be a slip, and if so, easily corrected. not so the second response. another example that illuminates learner thinking in school is taken from a pilot grade 9 examination set and carried out across many ieb (independent examinations board) schools in the early 1990s. learners were asked to read the following discussion between a teacher and two learners, and then to decide and explain which of the two answers was correct. teacher: expand (x + 2)2 lindi: x2 + 4x + 4 chris: x2 + 4 an alarming and surprisingly large number of learners from across the spectrum of poor to well resourced schools answered as follows: “both are right. if you use foil,8 then you get the answer lindi gave. if you use exponents, then you get chris’ answer”. the interesting thing about both examples above is that while these errors are well known, they reoccur. learners present these kinds of responses with consistency and conviction. moreover, each of them reflects a troubling absence of mathematical reasoning. and so the question: what do teachers need to know and know how to do (mathematical problem solving) in order to deal with ranging learner responses (and so some error analysis), and in ways that produce what is usefully referred to as “mathematical proficiency”,9 a blend of conceptual understanding, procedural fluency and mathematical reasoning and problem solving skills? i take a third and final example here from a grade 8 maths lesson,10 and a classroom where mathematical proficiency, particularly mathematical reasoning, was evident as a goal. as part of a sequence of tasks related to properties of triangles, 8 this is a heuristic often taught to remember binomial products. you multiply the fisrt terms, then the outer terms, the inner terms and the last terms. 9 see kilpatrick et al (2001) for an interesting illumination of the notion of mathematical proficiency and how it is an interweaving of five competencies: procedural fluency, conceptual understanding, adaptive reasoning, strategic competence and productive disposition. this notion is important as it cuts across attempts to polarize procedural fluency and conceptual understanding as if it were possible to me mathematically effective with either competence on its own. 10 this example comes from an earlier study (adler, 2001) focused on teaching mathematics in multilingual classrooms. 3 mathematics for teaching: what is it and why is it important that we talk about it? a teacher gave her grade 8 classes the activity in box 1. box 1. if any of these is impossible, explain why, otherwise draw it. • draw a triangle with 3 acute angles. • draw a triangle with 1 obtuse angle. • draw a triangle with 2 obtuse angles. • draw a triangle with 1 reflex angle. • draw a triangle with 1 right angle. learners worked on their responses in pairs. the teacher moved across the classroom, asking questions like: explain to me what you have drawn/written here? are you sure? will this always be the case? this elicited different learner responses. below are three learner responses to the second item: draw a triangle with two obtuse angles. some learners reasoned as follows: “an obtuse angle is more than 90 degrees and so two obtuse angles gives you more than 180 degrees, and so you won’t have a triangle because the angles must add up to 180 degrees.” some said that it is impossible to draw a triangle with two obtuse angles, because you will get a quadrilateral. and they drew: one learner and his partner reasoned in this way: “if you start with an angle say of 89 degrees, and you stretch it, the other angles will shrink and so you won’t be able to get another obtuse angle.” they drew: in the many diverse contexts where i have presented the study and this particular episode, much discussion is generated in relation to the mathematical status of the three responses, and their levels of generality, together with argument over what can be expected of learners at a grade 8 level. what constitutes a generalised answer at this level? is joe’s response a generalised one?11 these are mathematical questions, and some of the mathematical work this teacher did on the spot as she worked to engage learners and their reasoning. in addition, the design of the task required the ability to work with multiple representations of mathematical notions in such a way that these could first be anticipated and then elicited. the teacher then needed to be able to judge the mathematical worth of learner productions which in turn would require being able to relate different responses to each other in relation to mathematics. further mathematical demands for the teacher that are embedded in this task are the ability to work with definitions, relative to the community she is working with – grade 8 learners – and the ability to use language carefully to carry useful mathematical explanations. ball & bass (2000) and ball, bass & hill (2004: 59) describe these mathematical practices as elements of the specialised mathematical problems teachers solve as they do their work i.e. as they teach. they go on to look across these elements, and posit a more general feature. “unpacking”, they suggest, may be one of the essential and distinctive features of “knowing mathematics for teaching”. they contrast this with mathematics and “its capacity to compress information into abstract and highly usable forms” and posit further that “mathematicians rely on this compression in their work”. because teachers work with mathematics as it is being learned, they work instead with “decompression, or unpacking, of ideas” (emphases in the original). there is growing support for the notion that there is specificity to the way teachers need to hold and use mathematics in order to teach mathematics – and that this way of knowing and using mathematics differs from the way mathematicians hold and use mathematics. both mathematics and teaching are implicated in how mathematics needs 11 joe’s response is a dynamic one, based on visualising an angle changing size and the effect this can have on the triangle. the interesting thing here is that the kinds of technological tools (like sketchpad) where a drag function enables this kind of exploration of related properties of all figures, was not available to joe. his classmates found it difficult to follow his explanation. i noted earlier in the paper, the possible cognitive effects of learning geometry with tools that promote such visual and dynamic thinking. 4 jill adler to be held so that it can be used effectively to teach.12 unpacking or decompressing is a compelling description of the distinctiveness of the mathematical work that teachers do. the questions we need to now ask, and this will be the focus of the remainder of this paper, are: how/where is this specialised (practice-based) mathematical knowledge learned/taught? where and how are prospective teachers provided with opportunities for learning to unpack mathematics, and so develop pedagogically useful mathematics? researching the mathematical work of teaching: the quantum project phase 1 the above questions are in the foreground in the quantum research project currently underway. we are investigating whether and how unpacked mathematics is evident in mathematics teacher education programmes. in phase 1 (which has been completed, and so can be reported) we studied formal mathematics teacher education programmes across higher education institutions in south africa, and focused on the following specific questions: what mathematical (and teaching) knowledges are being assessed in mathematics teacher education programmes? is specialised (unpacked) mathematical knowledge assessed? where? how? as we set up the study, a first goal was to work across institutional sites. for practical and financial purposes, we restricted the survey to five of the nine provinces in south africa, working across both urban and non-urban contexts, and also in those provinces where we knew such programmes were on offer. both in south africa, and internationally, the dominant empirical domain of studies in the context of mathematics teacher education are single cases (adler et al., 2005; adler, 2005; krainer and goffree, 1999).13 our interest in an across site empirical sample was not for the basis of comparison, nor to identify good or better practice. rather, it was with the intention of building a comprehensive and robust description of how and what mathematics was being privileged across contexts of practice and so insight into a general, as well as particular, construction of what 12 see adler & davis (2004) for more detailed elaboration of the specificity of mathematics for teaching. 13 see alexander (2000) for an interesting challenge to arguments of single case studies of teaching or classroom practice as being necessary for insight, thick description and authenticity. he argues convincingly that culture and pedagogy can be held in dynamic interaction and not necessarily fragmented in larger and cross-cultural empirical studies is currently valued as mathematical knowledge for teaching. the first task was to identify all such courses/programmes across the five provinces. we focused on mathematics-specific in-service qualifications, and particularly advanced certificates of education (aces) designed for upgrading teachers’ mathematics and teaching competencies. sixteen such programmes came to light across thirteen different institutions spread across the provinces. we collected factual information on each course in each diploma so as to be able to identify whether they were courses in mathematics per se, mathematics education, or general education. we asked for details of average annual student intake, as well as the departments and faculties which taught these various courses. two institutions were offering an honours degree focused on mathematics education, that is, a programme one level higher than the ace. we included the courses within these in our study. of the eleven institutions offering aces, seven were offering a qualification for teachers across grades 7 to 12, that is, across the senior phase (sp grades 7 to 9) and further education phase (fet grades 10 to 12). one institution focused on sp only, and three on fet only. the average number of students in each cohort in each institution was ±50, with four institutions taking in between 50 and 150 students. in two institutions, mathematics courses comprised 80% of the qualification, the remaining 20% being in general education courses. in most, the split tended to be 50% mathematics, and 50% mathematics education courses. in one, all courses combined or integrated mathematics and mathematics education. the courses were predominantly run and taught by mathematics teacher education staff. in seven institutions, some courses were offered from mathematics departments. there are two interesting observations to make here. first, we were surprised by the relatively large numbers of practising secondary school mathematics teachers upgrading their qualifications by studying in ace programmes across the country. this phenomenon of largescale formalised in-service teacher education at the secondary level marks out a specificity to mathematics teacher education in south africa at present, a function of the legacy of apartheid education. the numbers were a further incentive for pursuing the study of the mathematics privileged in these programmes. second, these programmes were designed specifically for teachers and were being contributed to by some 5 mathematics for teaching: what is it and why is it important that we talk about it? example 1a: from a calculus module 1. …. 2. …. example 1b: from an algebra module 1. in solving the equation ax + b = cx + d we do things to both sides of the equation that can be “undone” (if we want). (a) make a list of the things we do and explain how they could be undone. (b) you have to be carefully about one of these steps, because, depending on the value of a and b, you might do something which results in something meaningless. explain. example 2a solve for x: x² – 2x = -1 example 2b here are a range of solutions to the equation x² – 2x = -1 presented by grade 10 learners to their class explain clearly which of these solutions is correct/incorrect and why (a) explain how you would communicate the strengths, limitations or errors in each of these solutions to the learners. (b) what questions could you ask learner 5 to assist her to understand and be able to formulate a more general response. learner 1: x = 1 because if x² – 2x = -1 then x² = 2x –1 and x = √ 2x – 1 x can’t be 0 because we get 0 = √-1 x can’t be negative because we get the square root of a negative x = 1 works because we get 1 = 1 and no other number bigger than 1 works learner 2: x = 1 because if x² – 2x = -1 then x(x – 2) = -1 and so x = -1 or x – 2 = -1 which leaves us with x = 1 (because x = -1 does not hold true) learner 3: x = 1 because if x² – 2x = -1 then x² – 2x +1 = 0 and this factorises to get (x – 1)(x – 1) = 0; so x = 1 learner 4: x = 1. i drew the graphs y = -1 and y = x² – 2x. they intersect in only one place, at x = 1. learner 5: x = 1. i substituted a range of values for x in the equation. and 1 is the only one that works. 6 jill adler mathematics departments in some institutions. it was equally interesting to see that there were cases where all courses, mathematics and mathematics education, were taught by mathematics teacher education staff either in schools of education, or in specialised centres for mathematics education within science faculties. this phenomenon could throw additional light on mathematics for teaching, and its emergence in courses in mathematics teacher education: on whether and how it is shaped by the wider mathematical/teaching discourses and practices of course presenters. this initial survey was then extended to include an analysis of formal assessment tasks across courses. we found sufficient similarity in the kinds of courses across institutions to enable us to select four diverse programmes for detailed analysis. three of these were ace programmes. focusing in on assessment tasks was a function of our drawing on the work of basil bernstein (1996). we selected bernstein’s sociological theory of pedagogy to assist in the construction of a principled gaze onto this complex terrain.14 according to bernstein (1996) any pedagogy transmits evaluation rules. evaluation condenses meaning, and transmits the criteria by which learners’ displays of knowledge are judged. we thought that as a first phase in quantum’s research, it would be illuminating to examine the formal evaluation tasks in each of the courses in each programme. these would reveal, at least partially, the kinds of mathematical and pedagogical or teaching competencies that teachers in these courses were expected to display and so too the kind of mathematical knowledge privileged in these courses. in addition, we hoped the evaluation tasks would reveal whether unpacking of mathematics was valued, and if so, in what ways. i have illuminated the notion of unpacking by discussing a few examples of the kinds of problems teachers solve as they go about their teaching. on the previous page there are examples of formal assessment tasks that illuminate what it might mean to assess unpacking as part of the mathematical competence teachers need. examples 1b and 2b are assessment tasks that require demonstration of the kind of mathematical problem solving teachers face. these are examples of unpacking tasks. in contrast, examples 1a and 2a are tasks that require the reproduction of some mathematical technique, and may or may not be a 14 much of this theoretical development has been led by zain davis, and is published as a quantum working paper, see davis et al. (2003) reflection of the learners’ (teacher’s) ability to unpack the mathematical ideas related to these tasks. the research sites, coding scheme and results as already mentioned, we focused in on four sites. sites 1 and 2 offered aces for senior secondary teachers, with 80% of courses in mathematics per se. the remaining course credits were in general education courses. site 3 offered an ace for secondary teachers, and here each course combined mathematics and mathematics education. from the assessments it appeared that the course was geared more to senior phase than further education teachers. site 4 offered an honours programme, where the mathematics courses are explained as having a pedagogical eye, and vice versa. the mathematics education courses were expected to have a strong mathematical eye. before beginning systematic coding and analysis we had noticed that across the courses and tasks in sites 1 and 2 there was a prevalence of tasks like examples 1a and 2a above. in other words, a learner would be able to recognise quickly that a mathematical procedure or technique was to be displayed. there were also tasks that asked for more, for explication, or justification as well as a display of a problem response, as in examples 1b and 2b. and there were tasks that were recognisable for not being obviously mathematical, focusing instead on teaching. we then developed a simple coding structure at the start: m to indicate a mathematical object of acquisition in the task, and t to indicate a teaching object. most tasks could be categorised as either m or t. each was then assessed as to whether the task object entailed unpacking, and if so, categorised as u. u was indicated in relation to either of the objects m or t, when more was demanded in the form of explication, justification, reflection, i.e. some form of unpacking. some tasks summonsed mathematical and/or teaching contexts, but the mathematical or teaching knowledge entailed was either assumed to be known (and so the display not up for evaluation) or alternatively, the task involved a mathematical or teaching idea, but the object to be acquired was unclear or obscured. these were labelled m or t. a list of codes and description of indicators follows: m indicates a task that requires reproduction of a mathematical object, e.g. a definition or carrying out a previously learned procedure. that is all. as the constructivists have revealed, such a display cannot be necessarily associated with its 7 mathematics for teaching: what is it and why is it important that we talk about it? underlying concepts, nor with associated mathematical practices. mt indicates a task as in m above, but a teaching context is summoned, though its purpose is either irrelevant or obscure. mu indicates a task that requires a reproduction of unpacked mathematics, e.g. demonstrating a grasp of the mathematical structure underlying the notion or mathematical object. mut indicates unpacked mathematics as above, but the task requires the unpacking to be explicitly related to tasks of teaching e.g. student errors, diverse responses, partial explanations, mathematical practices, e.g. reasoning, defining. mt indicates that both or either mathematics and teaching contexts are summoned, but the object of evaluation is not clear, and so the production of the legitimate text is not clear. mt indicates a task that requires reproduction of some aspect of knowledge of teaching, and a mathematical context is summonsed, though the mathematical knowledge entailed is assumed to be known. mtu indicates a task that requires reproduction and reflection on an aspect of knowledge of teaching and the mathematical context used. a mathematical context is summonsed, though the mathematical knowledge entailed is assumed to be known. tu indicates a task as above, but no specific mathematical context/object is called up or used. t indicates reproduction of some aspect of knowledge of teaching, where, for example, the text produced could rely on memory alone. table 1 contains the analysis of each of the tasks or a set of tasks within an assignment, across selected courses in these four sites. the sites are numbered 1 to 4 in the first column. the tasks are referred to as evaluative events. (an) stands for assignment n, (en) indicates this was an examination. the category or typing of each event is indicated in the particular column by a 1. the end column contains some content information on the course within which these assignments occurred. mathematical and teaching practices privileged in mathematics teacher education the categorisation of task types reveals most starkly that the mathematical knowledge privileged in mathematics courses in ace programmes (sites 1, 2 and 3) is compressed mathematics – the ability to demonstrate mastery of procedures and underlying concepts (though the display in no way guarantees underlying conceptual understanding). a similar privileging was evident in assignment tasks in the mathematics courses in the remaining institutions in the survey that offered ace upgrading programmes for secondary mathematics teachers. the topics in these courses across institutions included calculus and linear algebra, site no. m mt mu mut mt mt mtu tu t content of course 1 (ace) 25 25 50 mathematics – from precalculus, algebra, to calculus, linear algebra 2 (ace) 93 7 mathematics calculus and linear algebra 3 (ace) 30 10 10 10 30 10 mathematics and mathematics methods 4a (hons) 8.5 58 8.5 0 25 two mathematics courses (functions, geometry). 4b (hons) 12.5 12.5 50 25 four mathematics education courses (connecting; expressing; reasoning; assessing maths) table 1: categorisation of tasks in course assignments across sites, expressed as a percentage of the total number of tasks in each site/set of courses. 8 jill adler and were offered largely by lecturers in mathematics departments. the problem facing mathematics teachers currently in practice, as these courses suggest, is that they don’t know enough mathematics. an interesting contradiction here is that one of the major difficulties facing secondary teachers in schools today are the huge gaps in their learners’ mathematical knowledge. they talk of continuing struggles with “the backlog”. in this context, unpacking or decompression become more important, and indeed more demanding as teachers need to be able to trace back mathematical ideas and their antecedents with their learners. overall then, what was observed across these ranging ace programmes is the persistence and dominance of compressed mathematics in formal assessment. yet, the courses of which they are part were specifically designed for teachers. the courses are not part of mainstream mathematics courses, and so are not bound by mathematical goals, say, for undergraduate mathematics students. moreover, ace programmes, typically, are managed by mathematics teacher educators, most of whom would assert that to teach mathematics well, it is not enough to be able to do pieces of mathematics. it is, however, interesting to observe that alongside the dominance of compressed formal evaluations, there are instances in the assignments in sites 1 and 3,15 and in an examination in site 2 where a demonstration of unpacked or decompressed mathematics is required. here is evidence of some sense of the value of unpacked mathematics for teachers. the question, of course, is why are these relatively rare in formal assessments? in site 4, that site where a higher-level programme is offered, there is a far wider range of tasks, and indeed some interesting issues in the mathematics courses which have input from both mathematics teacher education lecturers in the school of education, and lecturers in the school of mathematics. we were intrigued by the assessment tasks in the courses on functions and geometry and the ways in which formal evaluation emerged. the functions course evidences struggles over how mathematics is or is not in the foreground in formal assessments in courses where there is greater integration of the pedagogic on the one hand, and mathematical processes on the other. 15 there are additional interesting points about the analysis of assessments in site 3 that have been discussed elsewhere (see adler & davis, forthcoming; adler, 2005) and are beyond the focus of this paper. this stands in contrast to the geometry course, where most of the assignments demand an unpacking of mathematics, but without an explicit eye on mathematics teaching, hence the percentages reflected in the table above. of course, hard conclusions are inappropriate without a further examination of what and how evaluative events punctuate the flow of mathematics in classroom practice within these courses, and so whether there is more evidence there of unpacking as a valued mathematical practice in courses. if this is the case, then a further question to pursue is why formal evaluation then condenses mathematical meaning to produce the privileging of compressed mathematics we have seen. these are being explored in phase 2 of the study. in sum, the analysis reveals more the absence, rather than the presence of unpacked mathematics for teaching in these across-site evaluation tasks, despite the courses of which they are part being specifically designed for teachers. why is this so? and what does this suggest for the preparation and ongoing support of mathematics teachers? what might this mean for each of the communities of mathematics educators and mathematicians, all of whom are involved, even if indirectly, in the preparation of mathematics teachers. why is mathematical unpacking absent in mathematics teacher education assessment, and what does this imply? i conclude this paper with some speculative comments on why this is so. firstly, and perhaps most simplistically, the absence of unpacked mathematics could be a function of this falling between what are typically subject (i.e. mathematics) and method (i.e. teaching) courses, as these are often offered by departments that remain quite insulated from each other. it is conceivable that lecturers in mathematics departments do not see this kind of mathematical problem solving as falling within the competencies that they are developing and assessing. at the same time, lecturers in mathematics education might not see this as falling within the scope of their courses, and expect this mathematical work to be done in the mathematics components of these programmes. hence the title of this paper and my view that it is important that we (our two mathematics communities) talk about this. more significantly, however, an explanation of this absence probably lies in the nature of this mathematical work, that it is not yet well enough understood and as a result it remains hard to teach 9 mathematics for teaching: what is it and why is it important that we talk about it? and so is hard to assess. we have argued elsewhere (adler & davis, 2004) that this kind of mathematical problem solving needs more careful and systematic description. the route to improved understanding is through further detailed study of the actual practice of mathematics teaching. from observation and work with many in teaching, it is obvious that some teachers have learned ‘on the job’ to solve the mathematical problems they face, i.e. in their practice. at the same time, from years of experience in mathematics teacher education, both pre and inservice, i am aware that many practising teachers would be hard pushed to spontaneously answer questions like: “what mathematical question would you ask a learner who does …” or “what are three possible, different solution strategies for …”. the question for us all then is: is this kind of knowledge teachable? the analysis presented above reveals that across selected formalised programmes for mathematics teachers, and particularly at the level of the ace, this kind of mathematical knowledge is not being assessed. while this does not necessarily mean it is not being taught, the empirical question remains, can it be taught, and then will it lever up the benefits we would want: more effective mathematical preparation and ongoing support for our teachers. my goals in this paper have been to make a case for understanding mathematics teaching as a particular kind of mathematical problem solving, and then to show that in some of our mathematics teacher education programmes, teachers are not being assessed on this kind of mathematical knowledge. we know that assessment reflects what is valued. that teachers are not expected to demonstrate this kind of competence also suggests that they are not provided with opportunities to learn this kind of mathematics in their formal education and training. it is my view that this ‘problem’ of the mathematical preparation and ongoing support of teachers is one that both communities of mathematicians and mathematics educators need to work on together. at a time when there is so much to do to foster and improve mathematics teaching and learning across all levels in south africa, it is encouraging that our relevant communities are working together.16 the practices of 16 the establishment of the south african mathematics foundation (samf) reflects the increasing areas of co-operation between amesa and sams, as do the invitations by each of sams and amesa to the other to deliver plenary addresses at their respective mathematicians and mathematics educators are very different, and often it is difficult for us to understand each other. through more conversation and joint activity, we will hopefully learn to communicate across our different questions, orientations and priorities towards goals we share, and the common challenges we face. references adler, j. (2001). teaching mathematics in multilingual classrooms. dordrecht: kluwer academic publishers. adler, j. (2005). researching mathematics teacher education: the quantum project and its progress. in c. kasanda et al (eds.) proceedings of the 13th annual conference of the southern african association for research in mathematics, science and technology education (pp. 11-24). windhoek: unesco namibia. adler, j. & davis, z. (2004). opening another black box: researching mathematics for teaching in mathematics teacher education. paper presented at aera, april, 2004, mimeo. adler, j., ball, d., krainer, k., lin, f.l. & novotna, j. (2005). reflections on an emerging field: research on mathematics teacher education. educational studies in mathematics, 61 (pages not yet known). alexander, r. (2000). culture and pedagogy. oxford: blackwells. ball, d. & bass, h. (2000). interweaving content and pedagogy in teaching and learning to teach: knowing and using mathematics. in j. boaler (ed.) multiple perspectives on mathematics teaching and learning (pp. 83-104). westport: ablex publishing. ball, d., bass, h. & hill, h. (2004). knowing and using mathematical knowledge in teaching: learning what matters. in a. buffgler & r. lausch (eds.) proceedings for the 12th annual conference of the south african association for research in mathematics, science and technology education (pp. 51-65). durban. saarmste. bernstein, b. (1996). pedagogy, symbolic control and identity: theory, research, critique, london: taylor & francis. conferences, and so too the context in which this paper was developed. 10 jill adler 11 davis, z., adler, j., parker, d. & long, c. (2003). elements of the language of description for the production of data. working paper 2, university of the witwatersrand, school of education, quantum research project. kilpatrick, j., swafford, j. & findell, b. (eds.) (2001). the strands of mathematical proficiency. in adding it up: helping children learn mathematics (pp. 115-118). washington: national academy press. krainer, k. & goffree, f. (1999). investigations into teacher education: trends, future research, and collaboration. in k. krainer, f. goffree & p. berger (eds.), on research in mathematics teacher education. european research in mathematics education iiii. osnabrück: forschungsinstitut für mathematikdidaktik. “let go of your attachment to being right, and suddenly your mind is more open. you’re able to benefit from the unique viewpoint of others, without being crippled by your own judgement.” albert einstein abstract introduction theory of image functions literature review part 1: the image functions intervention and its theoretical evaluation part 2: a first attempt at determining the image functions intervention’s viability discussion and conclusions acknowledgements references about the author(s) christiaan venter department of mathematics and applied mathematics, faculty of natural and agricultural science, university of the free state, bloemfontein, south africa citation venter, c. (2020). learning the function concept by exploring digital images as functions. pythagoras, 41(1), a524. https://doi.org/10.4102/pythagoras.v41i1.524 original research learning the function concept by exploring digital images as functions christiaan venter received: 03 dec. 2019; accepted: 10 july 2020; published: 31 aug. 2020 copyright: © 2020. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract despite the function concept being fundamental to mathematics, an adequate understanding of this concept is often lacking. this problem is prevalent at all levels of education and is reported in many countries. this article reports on a new pedagogical strategy based on exploring photographs or digital images as functions. the objective of this study was to determine if the image functions intervention (ifi) could be considered sound and viable with respect to the learning of the function concept. a two-part approach was used. the first part used the action-process-object-schema (apos) theory as a theoretical framework for evaluating the ifi. the second part implemented the ifi with a group (n = 27) of undergraduate students, followed by qualitative analysis of questionnaire data in order to look for indications of participants experiencing broadened thinking with respect to the function concept. the theoretical evaluation confirmed the ifi to be sound and providing opportunities for addressing common conceptual difficulties of the function concept. the qualitative analysis provided evidence of the participants’ expanded concept images. it was concluded that the designed intervention, the ifi, is sound and viable and shows promise with respect to increased understanding of the function concept. keywords: function concept; image functions; apos theory; genetic decomposition; mathematics. introduction that the function concept is fundamentally important to mathematics can be accepted as a commonly shared opinion. as stated by selden and selden in harel and dubinsky (1992, p. 1), ‘the function concept, having evolved with mathematics, now plays a central and unifying role’. and, more recently, ‘[t]he concept of function is central to students’ ability to describe relationships of change between variables, explain parameter changes, and interpret and analyze graphs’ (son & hu, 2015, p. 4). o’shea, breen and jaworski (2016, p. 279) reiterate that ‘[f]unctions are central to present day mathematics’ and elaborate that ‘going beyond calculus, functions are widely used in the comparison of abstract mathematical structures’. despite the high value attached to an adequate understanding of functions and the function concept, a full and nuanced comprehension is not common among undergraduate students (carlson, jacobs, coe, larsen & hsu, 2002, p. 353) or secondary school students (sajka, 2003, p. 229). doorman et al. (2012, p. 1243), in working with secondary school students, confirm the difficulty in learning the function concept and in particular state that ‘[f]unctions have different faces, and to make students perceive these as faces of the same mathematical concept is a pedagogical challenge’. this challenge is ongoing despite more than 50 years of research, producing ‘a vast literature on teaching and learning the function concept’ (dubinsky & wilson, 2013, p. 84). that it remains such a challenge can partly be understood in the light of the difficulties evident in the history of the development of the function concept. the concept is said to be an epistemological obstacle (sierpinska, 1992, p. 28) as the difficulties associated with it have been prevalent and persistent over a long time and are still commonly observed. the other reason could be attributed to what dubinsky and wilson (2013, p. 86) highlight as the little attention that has been paid to research that applies theoretical analyses (which is plentiful) to develop ‘pedagogical strategies for helping students overcome these difficulties’. simply put: (1) the concept of function is a difficult concept and (2) we have not been getting sufficiently practical in designing appropriate interventions, instructional treatments and didactical designs. some work has been done in getting practical, but, seemingly, more is needed. ayers, davis, dubinsky and lewin (1988) and breidenbach, dubinsky, hawks and nichols (1992) considered the use of simple programming environments to provide practical activities in creating and using functions. tall, mcgowen and demarois (2000) considered the use of the ‘function box/machine’ as a strong cognitive root to anchor the different ideas connected with the function concept. reed (2007) researched the effect of having students actively engage with the history of the concept of function. salgado and trigueros (2015) based their design and activities on models and modelling. this article reports on an intervention, the image functions intervention (ifi). the ifi explores digital images considered as functions. the intervention was firstly theoretically evaluated and secondly used with a small group (n = 27) of undergraduate students, followed by qualitative analysis. one aspect where this intervention differed from the implementations mentioned earlier was in that participants were meant to use the intervention in a self-directed manner. this was achieved by making use of e-learning authoring software in conjunction with a learning management system, making it possible for a participant to complete the ifi at their own pace. therefore, participants could use the ifi with no lecturer involved. the need for such a self-directed intervention arose firstly from time constraints with respect to direct contact time with students and secondly from the advantage of not needing teachers or lecturers to first become acquainted with the underlying ideas and content of the intervention. with the function concept in mind, akkoç and tall (2005, p. 7) point out that even in the face of a specific design, the outcome might not be achieved. they discuss a course that was designed to make the function concept foundational and an organising principle, but instead ‘many students focus on the individual properties of each representation without connecting them together’. in order to increase the probability of a design being successful, it should be based on research and theory. salgado and trigueros (2015) provide a good example of such a design informed by the action-process-object-schema (apos) theory. their design makes use of models and modelling. they first motivated their use of modelling by referring to research showing how modelling can raise motivation and interest, assist in identifying specific learning difficulties and facilitate learning and concept construction. thereafter a genetic decomposition (defined in the theoretical framework section) was constructed from which activities could be designed. in light of the understanding of what reasonable design implies, this article will use a theoretical evaluation guided by the apos theory, supplemented by the literature on the learning of the function concept, to determine if it would be reasonable to expect that the ifi could improve understanding of the function concept. the following research question was formulated: can the ifi be considered theoretically sound and viable with respect to the learning of the function concept? the research question was addressed in two parts. part 1 dealt with a theoretical evaluation of the ifi. keeping in line with the methodology of the apos theory, the current literature as well the researcher’s own experience were incorporated to create a genetic decomposition (defined in the theoretical framework section) of the function concept (dubinsky, 2000, p. 2; maharaj, 2010, p. 42). this genetic decomposition (gd) proposed the appropriate mental structures at the action, process and object levels that a student could potentially need in learning the function concept. this gd was used as a guide to evaluate the soundness of the ifi. the activities of the ifi were required to link with the function-related mental structures proposed in the gd. furthermore, it was examined if and how the ifi’s activities were addressing the common conceptual difficulties (dubinsky & wilson, 2013, pp. 85–86) associated with the function concept, such as ‘what constitutes a function’ and ‘confusion between univalence and injectivity’. part 2 dealt with a first attempt at testing the viability of the ifi. the ifi was implemented within a classroom setting with 27 participants. participants then completed a short questionnaire to gather some qualitative data on their experience of the ifi. the analysis of this data looked for indications whether the ifi could have value by broadening the participants’ thinking with respect to the function concept. this article has the following structure. following this introduction will be a section providing the theory of image functions, thus indicating how photographs or digital images can be considered as representing functions. that will be followed by a literature review which will firstly discuss the conceptual difficulties associated with the function concept in the context of secondary school and undergraduate level mathematics. secondly, the literature review will discuss the apos theoretical framework. it will end with the genetic decomposition of the function concept that was set up for this study. the literature review section is followed by a section that explains the details and workings of the ifi and then also proceeds to evaluate the soundness of the ifi from a theoretical perspective. this then concludes part 1 of addressing the research question. part 2 follows and addresses the first implementation of the ifi and the qualitative analysis that was used as an initial viability check. finally, the last section will bring together part 1 and part 2 and draw some conclusions. theory of image functions consider the photographs or digital images in figure 1. in figure 1a, the photograph of the horse consists of a finite number of pixels, or picture elements. this is easy to see in the zoomed image in figure 1b where we can distinguish individual elements of the eye of the horse. to each position in the image, a unique colour, figure 1a, or shade of grey, figure 1b, is assigned. therefore, we can interpret these images as functions. figure 1: (a) photograph of a horse. (b) zooming in on the horse’s eye, viewed as a greyscale image. defining the image function an image function, f(x,y), is a function with both x and y being positive integers (gonzalez & woods, 2017). any combination of such an x and y will form an ordered pair that will denote the position of a particular pixel in the image. corresponding to each ordered pair is a unique colour. typically, the different colours are represented using the rgb (red, green, blue) colour space. any specific output of an image function is then an ordered triple providing the specific combination of red, green and blue. typically, a scale of 256 different shades of red are used and the same for green and blue (gonzalez & woods, 2017). if we then let the first shade be represented by 0, the last shade would then be represented by 255. using these typical values, 2563 combinations of red green, and blue are possible. for example, the triple (255,0,0) will be bright red as it contains the full complement of red and zero contributions of green and blue. (255,255,0) is bright yellow, (0,255,0) is bright green and (57,229,212) would be called turquoise by some. if we only consider the possible outputs where the three components of each triple are equal, we end up with what is commonly referred to as a greyscale image, where outputs are shades of grey. for example, (0,0,0) is black, (255,255,255) would be white and (30,30,30) would be a dark grey. the image in figure 1(b) is an example of a greyscale image. as the three values in each triple will be equal, the outputs for greyscale images each consist of a single number that represents the light intensity at a particular pixel. an example: consider figure 1b, the greyscale image of the horse’s eye. domain: this image has exactly 51 rows and 91 columns. the domain of this image function, is the set of ordered pairs: is the set of positive integers. range: the word range can refer to two different concepts, namely the codomain and the image of the function, so care should be taken in using it. the codomain for a greyscale image is easily specified as the set . this is then the set of shades of grey from which any greyscale image could be ‘choosing’. when the term range is referring to the image of the function, it will consist of all shades of grey actually present in the particular ’picture’. here then the image of the function and the picture-image of the function are the same set. the picture set would normally have repeated values or colours and would thus be a different multiset from the function image. consider more aspects of image functions. surjectivity and injectivity of image functions an image function would seldom be surjective. with colour images using the rgb colour space, we have a total of 2563 = 16 777 216 unique elements in the codomain and most often many of these colours would not be present in the image. being closer to surjective is normally desirable when it comes to images, as this would generally mean the image has higher contrast. greyscale images typically have (only) 256 unique elements in the codomain; thus, being surjective has a much higher probability than in the case of colour images. it is clear that most images would not be injective either because it is highly probable that different pixels have exactly the same colour or shade of grey. existence of the inverse function as for all functions, the inverse will exist if the function is injective. in the previous paragraph, we saw that it is highly improbable for an image function to be injective and consequently it is highly improbable for the inverse to exist. with the high resolution of modern cameras, it is quite common for digital images to consist of millions of pixels. for greyscale images of such high resolution, it would then be impossible to have an inverse, as greyscale images only have 256 output options available. even for colour images with 16 777 216 possible output options, it will still happen often that at least two pixels will have the same colour. therefore, the probability of the inverse existing is small. continuity consider any point (x0,y0) in the domain of our image f(x,y). then we can show that: therefore, the image is continuous on its domain. proof: let ∈ > 0. let 0 < δ < 1. if then f(x,y) = f(x0,y0) because the domain of f is a subset of z × z. therefore, as the limit exists at any point in the domain and the limit is equal to the function value at that point, the function is continuous at any point in its domain. differentiability a digital image is not differentiable at any point, yet a discrete derivative in the form of a difference quotient plays an important role in image processing. applications where sudden changes such as steps, ramps, edges, lines or isolated dots need to be identified or accentuated often rely in part on some discrete implementation of a derivative. from calculus, we know that the derivative of a constant is 0, which translates to the important requirement of derivative-based filters to give back a small or even zero response in a homogeneous region of an image. see for example gonzalez and woods (2017) for more on the implementation of derivative filters and, for example, shrivakshan and chandrasekar (2012) for more on edge detection techniques through the use of derivative filters. literature review in this section, the focus will firstly be on the conceptual difficulties associated with the function concept as reported on in the literature. secondly, the theoretical framework for this study will convey a background on the apos theory and also portray the preliminary gd of the function concept. all these aspects are needed to guide the evaluation of the activities of the newly designed ifi. conceptual difficulties of the function concept the concept or notion of a function is in its essence quite abstract but is often understood at a level where much of the abstract nature is not truly comprehended or might even be entirely lost. a student might for example directly equate the function concept to the existence of a formula (dubinsky & wilson, 2013; sierpinska, 1992; vinner & dreyfus, 1989). one of the prominent indications of a lack of depth in the understanding of the function concept is the restrictiveness applied to what constitutes a function. if a student starts to fixate on particular types or certain representations, they lose much of the richness of the function concept. being able to recognise a certain formula or graph as (representing) a function is of course a necessary skill, but not sufficient in providing the student with the correct concept aspects and cognitive reasoning to be able to grasp and utilise higher mathematical concepts. for example, something as immediate as the inverse of a function, concepts such as limits, derivatives and not forgetting ideas that are even more abstract, such as topological homeomorphism and category theory, remain out of reach. thompson (1994, p. 39) argues that a fundamental difficulty is students’ lack of connections between the various representations of the same function. what is it that is being represented? thompson names this ‘something’ the ‘core concept of function’, that which is left unchanged when moving between the different representations. what it boils down to is that students sit with an inadequate or erroneous function concept image. according to tall and vinner (1981, p. 151), the concept image constitutes the ‘total cognitive structure that is associated with the concept’. this entails all definitions, properties, ideas, theorems and examples that a student has grouped under the heading of function over their mathematical career so to speak. although a student may know the formal definition of a function, when exposed to a problem, the full concept image will be utilised to solve the problem. doorman et al. (2012, p. 1245) also consider the concept image important and had as one of their specific goals the overcoming of a ‘too-limited’ function concept image. they investigated a new learning arrangement incorporating a computer tool to foster the transition from an operational to a structural view of functions. within the particular setting, they report some success in overcoming difficulties with integrating operational and structural aspects and providing an explorative environment with respect to the aspects of covariation (doorman et al., 2012, p. 1262). as the function concept is fundamental, yet misunderstood, the suggestion is that students should be introduced to the idea in such a manner that the resulting concept image will be as rich and accurate as possible. it is in these respects that the exploration of digital images considered as functions, or image functions for short, could be particularly useful. much research has been done in confirming the difficulty with the understanding of the function concept and addressing this difficulty. recent research includes that of chimhande, naidoo and stols (2017), which confirms that the difficulty is prevalent at school level. they showed that the mental constructions were typically at the action level of understanding, which is the lowest level according to the apos theory (arnon et al., 2014). doorman et al. (2012) explored the use of computer tools in aiding the transition to a structural view of function, that is, the object level of understanding. makonye (2014) also provides a theoretical analysis focusing on the use of multiple representations to foster a nuanced concept image through approaches where the function concept is kept embedded in students’ reality as far as possible. other research focuses on specific aspects of the function concept and not on functions in general. bansilal, brijlall and trigueros (2017) explored pre-service teachers’ understanding of injections and surjections through an apos study. they found that most participants were at the action level. maharaj, brijlall and govender (2008) explored the use of instructional design worksheets in advancing pre-service mathematics students’ understanding of the concept of continuity of single-valued functions. they found most participants were able to construct internal processes to make sense of continuity. this article discusses a new pedagogy. this refers to the new approach reported on in this article, which used the field of digital image processing to gather ‘mathematics for teaching’ (hoover, mosvold, ball & lai, 2016, p. 4). the specialised mathematical knowledge concerning image functions was used in a novel approach to the problematic teaching and learning of the function concept. theoretical framework working with constructivist ideas, essentially that learning is built upon previous learning, dubinsky and mcdonald (2001), as well as others before them such as breidenbach et al. (1992), formulated the apos framework (arnon et al., 2014) for modelling the learning of mathematical concepts. using the apos framework, the development of the function concept can be modelled where the conceptualisation passes through stages in a non-linear way, generally starting with actions (a), then processes (p), objects (o) and finally mental schemas (s). the non-linear here refers to the notion that the learning does not exclude the possibility of moving along different paths between the stages. for example, when busy conceptualising the function concept at the process stage, it might be useful or necessary to rethink and expand on one’s conceptualisation at the action level. at the action stage of understanding, an external stimulus such as an expression or an equation is needed to proceed, and the student cannot yet work with the concept entirely in the mind. in terms of image functions, we shall see in activity 1 of the ifi that explicit instructions are given to create a new image, one pixel at a time, by shading cells in a grid. at the process stage, the student can manage to construct the concept in the mind and also think about the underlying actions that make up the process, without actually performing any of these actions. in terms of image functions, the function as a process is realised when one can imagine how the possibly millions of pixels all get their respective colours (values) at the same moment when the photograph is taken. the number of pixels determines the domain, and which colours they potentially can get determines the codomain. when the student is at the object stage, the process has been encapsulated. the student is then able to think about other actions and processes that might be performed on this object. the student has then gone from an operational view to a structural view. in terms of image functions, the object level makes it possible to think about transformations, like contrast stretching, that can be applied to images. images can be mentally grouped according to certain criteria and as such form sets of functions. at the schema stage, a student will be able to move freely between considering and using the concept as an action, process or object (arnon et al., 2014, p. 30; asiala et al., 1996, pp. 7–8). when considering which transformations would potentially enhance an image, the thinking is mainly at the object level, but when it comes to planning and performing the transformation, the thinking must be at the process and action levels. the apos theory can help us understand how the learning takes place by explaining what we see when participants are trying to ‘construct their understanding of a mathematical concept’ (dubinsky & mcdonald, 2001, p. 1). the proposed mental structures needed for learning a mathematical concept are captured in what the apos theory refers to as the gd. this gd is the theoretical blueprint against which the intervention of this article, the ifi, was measured. the genetic decomposition of the function concept genes are the building blocks of life and so to determine a gd of a mathematical concept is to break down the learning of the concept into its imagined building blocks. the word imagined is used here as, in following the apos theory, the breakdown is, among other things, dependent on the researcher’s own knowledge (dubinsky, 2000, p. 2; maharaj, 2010, p. 42). the researcher would use personal experience, completed research and observations to imagine and create a set of necessary mental structures and mechanisms at the action, process and object level. these structures and mechanisms are what someone who is learning the concept could need and use along the path of conceptual understanding (arnon et al., 2014). having the mental structures available makes it possible to judge at which level of conceptualisation a particular person is, with respect to a specific mathematical concept. keeping to the analogy of building, if the gd describes the progressive structures of the mathematical concept (the building), then the support needed to reach these structures would be described as the scaffolding. part 1 of addressing the research question of this article deals with the evaluation of the intervention, that is, to ensure that the intervention is appropriate and sound. it must be appropriate and sound on two fronts: (1) addressing the mathematical content in line with the gd and (2) as scaffolding to support the student’s ‘construction of knowledge and skill’ (bakker, smit & wegerif, 2015, p. 1048). from arnon et al. (2014, p. 27) we get the formal definition: a genetic decomposition is a hypothetical model that describes the mental structures and mechanisms that a student might need to construct in order to learn a specific mathematical concept. necessarily we then need to define what a mental structure is. again, from arnon et al. (2014, p. 26): a mental structure is any relatively stable structure (something constructed in one’s mind) that an individual uses to make sense of mathematical situations. the gd given in table 1 is based on the decomposition given in arnon et al. (2014, p. 29). extensions and expansions are based on the researcher’s own experience complemented by current literature on the topic. table 1: preliminary genetic decomposition of the function concept. the gd given in table 1 conveys the mental structures of the function concept at the action, process, object and schema levels. furthermore, it also describes the mechanisms of progression, namely interiorisation, encapsulation and activity. in the apos theory, these mechanisms are the proposed means by which one can transition from one level to the next level of conceptualisation (arnon et al., 2014, p. 16). based on a literature review, table 2 provides further criteria that can indicate when and when not a student’s understanding of the function concept can be considered at the action, process or object stage of conceptualisation. alongside the gd of table 1, these indicators were also used in the theoretical evaluation of the soundness of the ifi. in table 2, reference is made to the activities of the ifi that are linked to the particular conceptual indicators given in brackets, for example a2 or p3. details can be found as part of the individual discussion of each ifi activity in the next section. table 2 provides indicators only up to the object level to align with the content and intentions of the ifi. table 2: indicators and counter-indicators of apos level attainment with respect to the function concept. as was discussed in the introduction, what follows now is a two-part approach. in the first part, the ifi will be introduced and evaluated to see if it is theoretically sound. the second part will report on the first implementation of the ifi and the subsequent gathering and analysis of qualitative data with the purpose of looking for any indications that the ifi might be viable with respect to the learning of the function concept. part 1: the image functions intervention and its theoretical evaluation the ifi was created using e-learning authoring software which delivered it as a package that could be uploaded into the learning management system (lms) used by the local higher education institution. this allows a student to complete the ifi on a computer at their own pace without any lecturer involvement. as a general context, the ifi deals with finding a missing student of which one recent photograph was available on the student’s facebook page. this photograph, however, was taken in low light conditions and as a result needs some processing before it will be helpful in finding the missing student. this theme runs like a story throughout the intervention. this theme was chosen as students are familiar with the context, they can easily understand the contingency relationships involving the variables that are present and they are generally interested in the type of context (donovan & bradsford, 2005, p. 359; eggleton, 1992). besides this storyline, the ifi conveys the theory of image functions as was discussed in the previous section. the theory is interwoven with reflective questions and specific activities. these activities are meant to keep in line with the following three principles: activities directly link with the mental structures determined in the gd of the function concept given in table 1 and the conceptualisation indicators in table 2 (salgado & trigueros, 2015, p. 107). activities address the categories of conceptual understanding. dubinsky and wilson (2013, pp. 85–86) grouped the most common conceptual difficulties associated with the function concept, as they had found in literature, into categories. activities form an experiential base for the aspects of the function concept to be studied (dubinsky & wilson, 2013, p. 90). the principles given above will be used as the criteria for the theoretical evaluation of the ifi. each of the ifi’s four activities will be discussed and evaluated with respect to the first of the three design principles given above. the sub-section following this one will evaluate the ifi with respect to the other two principles. activity 1 description: the student is asked to draw an 8 × 8 grid on paper as in figure 2a. the student must then take a pencil and shade the blocks at positions b2, b3, c1, d1, d5, d6, d8, e1, e4, f2 and f3. accurate shading leads to the result given in figure 2b. figure 2: (a) an empty 8 × 8 grid. (b) result obtained at the successful completion of activity 1. students are asked to reflect on the activity by letting them provide answers to questions pertaining to uniqueness aspects of functions and asking questions to let them think about the choices that can be made with respect to input and output. questions asked as a part of this activity: now that you have created your image, would you say that the colour at any specific position in your image is unique? if you choose a position in the image, what will you find there? are you free to choose any position in the image? once you’ve chosen a position in the image, can you then also choose which colour will appear there? so is it true that the colour is dependent on the position? to keep the activity and the ifi interactive, students are required by the software to first answer the questions before being able to proceed. the question formats vary between multiple choice and typing an answer. from this activity, there is a natural flow in letting the student discover that a photograph or digital image can be interpreted as a mathematical function. connecting to the gd and the conceptualisation indicators (cis): this activity aims to connect the repeated action of assigning a shade of grey to a specific position in the grid to a function value that is assigned to specific input. this activity is therefore aimed at helping students to construct the function concept as an action, which is the first stage of understanding according to the apos theory. by repeating the shading action, an attempt is made at facilitating the interiorisation mechanism as was described in the gd. the instructions of this activity connect with the external prompts (see the cis) or explicit steps needed by a student whose understanding is at the action level. the reflective questions of this activity connect with the ci criteria that the student will know the definition of a function and will as such be led to discover that an image can be interpreted as representing a function. activity 2 description: students visualise some function outputs by using the colour editor in the commonly available software paint. students are prompted to try different combinations of red, green and blue and they can then see what those combinations look like. this is an easy-to-use experiential playground where students are asked to try out for example negative values or non-integer values or values larger than the maximum used in the colour scheme and then observe what the software does to these inputs. the software automatically adjusts an inappropriate input in a default manner. if a negative value is typed, the value is changed to the absolute value. if a number larger than 255 is typed, it is automatically changed to 255. if a non-integer is typed, it is changed to the last digit that was typed. connecting to the gd and the cis: this activity allows students to engage with the idea of the range of an image function practically. at a later stage, students must also think about the domain by realising that the image only has a finite number of rows and columns. the concept of the domain of a function is thus a quite practical ‘thing’ with respect to image functions. this activity connects to the gd mechanism of interiorisation to assist in going from an action understanding to a process understanding. furthermore, it connects to the cis by attempting to broaden the student’s understanding of the definition of a function, by emphasising the necessity of the domain and range. this is necessary for a process understanding of the function concept. activity 3 description: students are brought back to the image function formed in activity 1, but now both the rows and columns are depicted by positive integers as in figure 3. figure 3: the function regarded as a process. reflective questions are used in this activity to assist students in constructing the function concept as a process: does the choice of the first block you coloured or the sequence in which you coloured the blocks make any difference to the final image? what is the difference between how you formed this image and how an image would be formed by your cell phone’s camera? does your camera also take one input at a time and ‘fill’ in the colour for that position? after a student completes these questions, a description is given comparing the point-by-point, successive way in which they shaded each individual block to what will happen inside a camera. inside the camera all the ‘blocks’ get shaded simultaneously. the function is then a process of taking the entire domain at once and filling it with the range. students are again led to think about the specifics of the domain and the range: thinking about the inputs does f(2,3) make sense? does f(2,3.561) make sense? what conclusions can be drawn about the set of allowable inputs? what do we call the set of inputs for a function? thinking about the outputs: in a grayscale image can the value at any position possibly be described as ‘blue’? in a colour image, can the value at any position possibly be described as ‘dark grey’? they are also led to think about the uniqueness property and the aspects of being injective and surjective: for the image function in figure 4, the codomain is {red, purple, green, blue, black, pink, orange, white, yellow}. what is the range of this image function? is this image onto? is this image one-to-one? figure 4: will this function be injective? finally, thinking about the inverse process is also introduced here, for example questions such as: yellow is a colour that is present in this image (figure 4). if you now make yellow the input, what will be the output? do you think the image function shown will have an inverse? connecting to the gd and the cis: as described in the cis, at the process level a student should be able to mentally construct the function as the complete transformation of the domain to the range. this idea is captured in how the camera captures all of the pixels’ values simultaneously. this activity also explicitly deals with the aspects of the reversal of the function process and the existence of an inverse and confronts students with a fuller grasp of the definition of a function while at the same time having no formula. having a function without a formula was also addressed in activity 1, but as it is one of the common conceptual difficulties associated with the function concept (dubinsky & wilson, 2013, p. 85), it is valuable to address it again. activity 4 description: with this activity, the aim is to assist students in constructing the function concept as an object. the story in which the intervention is set is brought to a peak here. the student sees how the knowledge of image functions, together with simple contrast stretching, is used to enhance the photograph discussed at the start of the intervention. it is enhanced to such a degree that sufficient information can be gathered from the image to assist in identifying the place where the photograph was taken. the contrast stretching seen in this activity is achieved through function composition. as an introduction, the activity lets the student explore contrast stretching with pen and paper. the student is asked to draw the 5 × 4 grid as pictured in figure 5a. figure 5: (a) low contrast image. (b) increased contrast after function composition. the number of vertical lines, v, in each cell (pixel) can be considered the colour of that cell. figure 5a is then an image function, say f(x,y). the student is then asked to draw a second 5 × 4 empty grid and then fill in its values by applying the function g(v) = 4v – 4 on the original image of figure 5a. a new image function h(x,y) is thus created through function composition. we obtain figure 5b through the function composition . for example, . this composition stretches the contrast to such an extent that we can now see the number 5 or the letter s present in the image. the 5 (or s) was of course already present in the original image, but it was difficult to distinguish it from its background. it was difficult to distinguish due to the low contrast of the original image. once the introductory contrast stretching is completed, the student is brought back to the photograph associated with the missing person’s case. this photograph is seen here in figure 6a. figure 6: (a) low contrast image. (b) increased contrast after function composition. students are initially asked to describe in their own words what is wrong with the photograph, or consequently the (image) function in figure 6a. how can a function be ‘wrong’, or for that matter be described? as an object, the function acquires global properties such as having low contrast, therefore a small average difference between adjacent pixels across its domain. this activity lets a student realise that a low-contrast image can be improved by regarding it as one single thing – an object – that can be transformed by another function. function composition is used to transform the original image function, f(x,y), into a new and improved image function g(f(x,y)). connecting to the gd and the cis: if we carefully design the transformation function, g, we can obtain the desired results and again describe the new function as a whole. a function then becomes a noun and a noun can be described by adjectives. we might say that the new function is beautiful, it is clear, it has high dynamic range, it is smooth or, as was planned for the image of the intervention, it has improved contrast as can be seen in figure 6b. this ability of seeing the function as a whole and not as something that you do, but rather something that can be acted on, is described in the gd at the object level. the function composition used in this activity fulfils the role of a process acting on the function process. this is described as encapsulation in the gd and is also a requirement in the cis. through acting on the function as object and reflecting on properties of the function such as contrast or brightness, a student could start to realise that image functions are a type of function, similar to how exponential or trigonometric or linear functions are different types of functions. this connects to the ci of being able to think about and convey the global properties of a type of function. furthermore, a student can be led to grasp that different photographs of for example the same scene or person form a set of image functions. mentally constructing a set with functions as its elements is a further indicator from table 2 of being at the object level of conceptualisation with respect to the function concept. expanded theoretical evaluation of the image functions intervention with respect to the first principle the ifi follows the apos general trajectory in the sense that information and in particular the activities are ordered to first let students retouch on the function concept as an action, then move on to a process and then finally to the function as an object. however, this trajectory is not the only possible path through the levels. as was discussed in the theoretical framework, in general the learning can pass through the apos stages in a non-linear way (arnon et al., 2014). in the discussion of the four activities before, it was already indicated how the activities aim to guide the student to construct the desired mental constructs that emerged in the gd and link to the cis indicators of table 2. to expand on the previous evaluation, we now track the gd while taking into account the ifi’s underlying path through the apos stages: action, process and finally object. action at the action level, a student is expected to take an element from the domain and find its corresponding value from the range. this is what is required in activity 1. repeating the actions (asiala et al., 1996:7; dubinksy & mcdonald, 2001:3) together with encouraged reflection about the actions and the involved sets (expanded on in activity 2) triggers the necessary interiorisation mechanism to lead the student to a process understanding of function. process at the process level, a student must now be able to capture the creation of the image of the function mentally (in the sense of the range) as a whole. possibly infinitely many function evaluations can be imagined taking place simultaneously in the mind. dubinsky and mcdonald (2001, p. 3) describe that at the process conception, the individual can think about infinitely repeating the same kind of action, as no external stimuli – such as following the steps of a formula – are still needed. activity 3 assists here by juxtaposing the point-by-point creation of a photograph by individual actions with the actual chemical or electric process that is going on inside a camera to form the photograph all at once. inside the camera, the entire film or sensor is illuminated and so all pixels get their values at the same time. the action of one ‘element of light’ reaching one point on the film or sensor to make one specific colour at that point or pixel can be imagined to be repeated simultaneously for all the millions of pixels in the eventual photograph. object at the object level, a student must now be able to grasp a function as one static entity. activity 4 aids the encapsulation of the former dynamic transformation process to a ‘thing’, to become a noun with associated adjectives and properties. the encapsulation of functions at the process level by performing actions or other processes on these functions is reported to be key to transitioning to the object level of understanding (asiala et al., 1996). yet, this type of encapsulation is missing from experience when it comes to functions (asiala et al., 1996, p. 8). in the ifi, this is exactly the type of encapsulation emphasised by means of the function composition in activity 4. actions and processes are applied to the image function in order to achieve the specified goals. as the intention with the intervention was only to look to improve understanding up until the object level, the schema level is not discussed in this analysis of the intervention. evaluating the image functions intervention with respect to the second and third principles in keeping to the second and third design principles, the ifi has to form a basin wherein the conceptual difficulties and other required concept aspects associated with the function concept can be explored in a familiar context. in the ifi, that context is created using photographs. to see if the ifi is compliant, we will look in turn at the most commonly occurring function concept difficulties (dubinsky & wilson, 2013, pp. 85–86) and other function aspects: what constitutes a function? using image functions encourages the realisation that a formula is not necessary to have a function. vinner and dreyfus (1989) categorised students’ definitions of a function into six categories: correspondence, dependence relation, rule, operation, formula and representation. ultimately all six of these categories can be valuable viewpoints when dealing with various functions, function properties and applications. however, if a student focuses too much on the idea of a formula or an equation for a function, the construct could easily be cemented in their thinking that only formulas, or rules or correspondences having known formulas or equations, can be considered as functions. this kind of restrictiveness is quite common (breidenbach et al., 1992; dubinsky & wilson, 2013; sierpinska, 1992; vinner & dreyfus, 1989). from the start of the intervention, the student is put in a state of disequilibrium by introducing an everyday thing, such as a photograph, as a function. this state of disequilibrium, in the style of piaget (wadsworth, 1978, p. 80), is necessary here as the student’s concept image of functions has been formed and re-enforced over a number of years already. therefore, by throwing the student off balance, room is created for the restructuring of the concept image. it is the suggestion of this article that the ifi’s reintroduction to functions via image functions can challenge the fixation and restricted thinking in terms of formulas and equations. this is necessary to develop beyond the action level. students also often focus on the symbols representing the variables, instead of the quantities they are representing. in the ifi, working with the image functions is done without the need for symbols. familiar or intuitively understood terminology such as row, column and grey level are used. vinner and dreyfus (1989, p. 361) also showed that any seemingly irregular behaviour such as discontinuity, a split domain or ‘[t]he idea that the graph of a function has to have a stable character’ are erroneous ideas that students often use to disqualify some rules or graphs as functions. the ifi, by working with image functions, again has the advantage over using linear functions, for example in that images change character easily across the two-dimensional domain. this is even more apparent when the object conception of function has been reached after activity 4. therefore, students will not be left with the idea that a function needs to act ‘nicely’ in any way to be considered a function. in contrast, carlson and oehrtmann (2005, p. 2) mention the case of students thinking that constant functions are not functions ‘because they do not vary’. also see bakar and tall (1991). confrey and smith (1991) refer to the constant function as an example of a ‘monster’ function and confirm that students exclude it as a function because they expect a function to ‘covary’. in working with the ifi’s activities, it occurs naturally that portions of the image will have the colour or grey level stay constant over smaller or even large areas. a student can thus discover intuitively that an image function and, by extension, all other functions are allowed to display seemingly non-regular behaviour over certain portions of their domains. univalence and injectivity in working with the image functions of the ifi, the often-problematic univalence property is made practical and simultaneously important. in addition, the confusion that is often seen (harel & dubinsky, 1992) between the univalence property and the function being injective is addressed in a tangible way. in terms of image functions, it is evident that at any specific input (pixel) the photo has only one output (colour) and as such exhibits the univalence property. furthermore, if we determine that for the particular photo, the colour at any pixel does not occur at any other pixel, we have determined that the photo or image function is one-to-one, thus injective. reflections to evoke these realisations with students are delivered through activity 3 of the ifi. multiple representations the activities of the ifi provide opportunities to ask new questions about multiple representations of functions. the multiple representations idea garners plenty of attention in textbooks as well as research (carlson & oehrtman, 2005; confrey & smith, 1991; stewart, 2015, p. 10). however, according to thompson (1994, p. 39), students still miss that which stays unchanged between these representations: the ‘core concept’. how would one represent an image function differently than by means of the photograph? setting up a table that states explicitly the colour that belongs at each position can illuminate the fact that the representation itself is not the function. students can be led to realise that similar to an equation or a graph, the purpose of a table or the photograph is to tell us the output that belongs to any specific input. why do we then have or use multiple representations? we only use them if they help us to understand or analyse the function. a blind person could get no value from a graph of a function, but could there maybe be a way to listen to a function? could an audio representation be created, and would it be helpful? in validating the design from a theoretical viewpoint, we are moving the intervention towards a correct organisation of the knowledge to optimise deep understanding. this is key in developing sufficient expertise to solve problems that flow directly from the topics involved as well as related problems (donovan & bransford, 2005, p. 16). naturally, the theoretical validation will need to be followed by empirical study to provide practical validation. this is the content of research currently in progress. for this article, qualitative data were gathered by means of a questionnaire, but not to validate the gd as is typical in the apos theory (arnon et al., 2014; dubinsky, 1991). rather, the purpose was to investigate the viability of the ifi. the next section reports on the first implementation of the ifi and the consequent use of a questionnaire and the qualitative data analysis. part 2: a first attempt at determining the image functions intervention’s viability this section deals with the first practical implementation of the ifi. it forms part 2 of addressing the research question as was given at the end of the introduction of this article. part 2 investigated the viability of the ifi by looking for indications that the ifi could have value through broadening the participants’ thinking with respect to the function concept. achieving this objective would give us proof of principle, that is, determining if the ifi is sensible and worthwhile to investigate further. proof of principle, together with qualitative analyses indicating what the content of an intervention should entail, assists in determining the need and validation for further testing (pressley, graham, & harris, 2006, p. 7). a questionnaire was used to gather qualitative data on the participants’ experience of the ifi. a qualitative method was used here to allow the exploration of participants’ perceptions and allow for unexpected feedback on the ifi. sampling and data collection the ifi was implemented in a classroom setting with a group of 27 students in a first-year calculus course. this is not the intended method by which the intervention will be implemented, as it was designed to be a self-directed mini module where a participant follows their own pace and can actively engage with the various activities of the intervention. however, to gauge the initial reaction of participants to the material and activities, it was decided that a classroom setting, together with a questionnaire at the end, would be sufficient and still enable us to achieve the objective of a proof of principle. this group was chosen for convenience but fulfilled the minimum criteria of having prior knowledge of the function concept. from casual observation, the group had male and female members and these members were from at least three different ethnic backgrounds. the questionnaire was handed out to all 27 participants and it was made clear that participation was voluntary and would be anonymous. no personally identifying information was asked, as this was not deemed necessary for a proof of principle endeavour, thus simply testing if the ifi could be viable. the questionnaire consisted of three grammatically closed, but conceptually open, questions (worley, 2015, p. 19) as this still allowed any elaboration the participant might wish to provide. data analysis the method of content analysis with emergent coding was used (maree, 2016, p. 111). the responses to the three questions were searched for any indications relating to the objective of the questionnaire. after reading the responses a few times, four themes were identified: what constitutes a function? students commonly struggle with misconceptions with regard to what can and what cannot be regarded as a function (dubinsky & wilson, 2013, pp. 85–86). more discussion on this topic was given as part of the theoretical evaluation of the ifi in part 1. functions are connected to real life from reading through the participant responses, it seems that many were almost surprised to find functions being used in such an everyday type of topic as photographs. this theme does not represent an improved understanding of the function concept but can possibly assist in making the topic interesting to participants. this interest can increase their motivation, which is key to effective learning (eggleton, 1992, p. 1). connecting the function concept knowledge to everyday experience also assists later retrieval and application (donovan & bradsford, 2005, p. 364). domain and range this theme relates to the understanding that a function requires a set of allowed inputs and, associated to each input, a unique output. the outputs form a set as well. from the gd, we saw that clear understanding of domain and range are required to construct the function concept at the action level and to interiorise the actions to start constructing at the process level. function inverses at the process level, being able to reverse the function and form the inverse function is required (breidenbach et al., 1992). each response given to the three questions was read and reread in order to judge if it contained any indication that the participant experienced a broadened understanding of the function concept. the indication was then classified as belonging to one of the four themes. validity it was not the intention of the questionnaire to deliver a generalisable result. the questionnaire was part of the effort to establish a proof of principle. proof of principle can be interpreted as looking for proof that the ifi can have value, at least in some settings with some participants. to increase the credibility of the qualitative data analysis, colleagues were asked to independently verify themes, occurrences and results. results of the qualitative data analysis in the data analysis section, four themes were discussed that emerged from repeated reading of the participants’ responses. the responses were then analysed individually within the structure provided by the four themes. this was done while keeping in mind that we were looking for indications that the participant experienced a broadened understanding of the function concept. in the analysis to follow, these three questions (q) were asked: q1: having completed the ifi, have you realised or learnt something in particular of the function concept? q2: is there an aspect of the function concept that is now clearer to you? q3: is there some aspect of the function concept that you might have thought about in some way before, but now realise that you were wrong about in some sense? theme 1: what constitutes a function? the analysis illustrates that participants are showing an expansion of their ‘concept image’ (tall & vinner, 1981, p. 151) related to what can also be considered a function. consider the following responses: ‘yes, i realised functions have a broader meaning and that it forms a big part of our technological lives.’ (participant 1, answering q1,) ‘yes, that all images are also functions.’ (participant 13, answering q1,) ‘yes a function can be determined in different ways.’ (participant 21, answering q1,) ‘yes, that not all functions are graphed on cartesian planes.’ (participant 18, answering q3,) theme 2: functions are connected to real life again we see participants’ concept images expanded. here it relates to a realisation that functions can be useful and specifically useful outside of mathematics itself. some participants realise that functions can be part of their lived experiences. consider the following representative responses: ‘yes, it can be used for various purposes.’ (participant 2, answering q1) ‘functions can be used for a lot of purposes out side of maths [sic].’ (participant 5, answering q1) ‘yes, i understand that math is used everywhere.’ (participant 6, answering q1) ‘yes, how unclear images been processed to have clear pictures about something and i did not know that how functions are used in life.’ (participant 22, answering q1) ‘yes. i didn’t know functions can relate to real life experiences and applies to images and is involved in biometric scanners.’ (participant 19, answering q3) ‘yes, i have learned that functions can be used for more than calculating or predicting change.’ (participant 25, answering q3) theme 3: domain and range a few participants gave some indication of increased understanding of the aspects of input and output. clarity on input and output, as relating to domain and range, would assist in transitioning from an action level to a process level of function conceptualisation. consider the following responses: ‘yes there is a input and a output.’ (participant 4, answering q1) ‘yes, there is an input and output.’ (participant 9, answering q1) ‘inputs and outputs of a function.’ (participant 3, answering q2) theme 4: function inverses some participants reported to have gained increased clarity on the aspect of function inverses. to understand the function concept at the process level, one needs to be able to reverse the actions of the function and then progress to formulating the inverse function or deciding if the inverse function will exist. the following responses are representative: ‘yes, the differentiation between inverse functions.’ (participant 13, answering q2) ‘the derivatives or inverse of functions.’ (participant 15, answering q2) ‘yes, inverse of a function.’ (participant 3, answering q3) discussion and conclusions in this article, our objective was to determine if the ifi could be considered a sound and viable tool with respect to the learning of the function concept. this objective was addressed in two parts. the first part was a theoretical evaluation of the ifi. this evaluation strongly depended on a gd of the function concept. not only can a gd serve as the basis for designing a learning intervention, it also serves to make the consequent analysis more reliable (arnon et al., 2014, p. 38). through empirical study, the success of the apos theory has been shown through using the gd not only to describe the mental constructions of participants but also to design ‘effective instruction’ (weller et al., 2003). the required gd was determined, showing the necessary mental structures at the action, process, object and schema levels. connecting with the gd, indicators were set up (see table 2) which could be used to judge the conceptual levels of a participant. consequently, the activities of the ifi were shown to keep true to the gd and the indicators of table 2. this was done by describing each activity and explicitly making connections between aspects of each activity and the requirements in the gd and the indicators of table 2. important to note is that the ifi provides opportunities for encapsulation of processes through the function composition used in its contrast stretching activities. in the apos theory, encapsulation is the mechanism by which conceptualisation evolves from the process to the object level (arnon et al., 2014). asiala et al. (1996) reported that this type of encapsulation, namely the encapsulation of functions conceived of at the process level, is necessary to transition to the object level of function conception. asiala et al. further reported that this type of encapsulation is mostly lacking in our experience with functions. besides adhering to the gd and providing encapsulation opportunities, the ifi was also shown to create opportunities to address some prominent conceptual difficulties associated with the function concept, specifically ‘what constitutes a function’, ‘univalence and injectivity’ and ‘multiple representations’. from the theoretical evaluation, it was concluded that the ifi is theoretically sound with respect to the design principles used in this study. part 2 of addressing the research question entailed the implementation of the ifi in a classroom setting, with subsequent qualitative analysis of a questionnaire given to participants. the aim was to determine if the ifi could be viable by looking for indications that participants experienced any broadening of their thinking with respect to functions, thus broadening their concept images. the ifi was put to the test in a classroom setting with 27 first-year calculus students. from the results of the analysis, we saw that at least some participants reported gaining new insight on inputs and outputs of functions and the inverses of functions. furthermore, it seems safe to conclude that for at least some participants, their ideas concerning what constitutes a function have been broadened. however, actual improved understanding of the function concept cannot be independently verified as it can be based only on the participants’ own reporting. this is partly the fault of the questions of the questionnaire. they were intended to be sufficiently inviting to lead participants to provide rich responses from which true improved understanding could be judged. this did not happen to a sufficient extent. seemingly, the questions were formulated to be too closed-ended. on a positive note, we can conclude from the results that the ifi has the potential to enrich the concept image (tall & vinner, 1981, p. 151). the concept image is closely related to the schema level of the apos theory in the sense that it is the construct one will utilise when confronted with solving an actual problem (dubinsky & mcdonald, 2001, p. 3). concluding that the concept image of participants can be enriched is based on the many instances of participants making a new connection between functions and everyday or real life and losing some of the restrictiveness regarding what constitutes a function. this enriched concept image gives us sufficient reason to claim proof of principle, thus concluding that the ifi is viable. the theoretical evaluation of the ifi, together with the proof of principle that was obtained, led to the conclusion that there is sufficient indication that the use of the ifi can have merit and is therefore worthwhile to explore further. subsequent qualitative and quantitative research will aim to independently verify not only if the gd is a true predictor of the mental constructions of participants, but also if the ifi can actually manage to improve participants’ understanding of the function concept. acknowledgements competing interests i declare that no competing interests exist. authors’ contributions i declare that i am the sole author of this article. ethical consideration ethical clearance for the research conveyed in this article was obtained via the university general/human research ethics committee (ghrec), reference number ufs-hsd2019/0006/1505. funding information this research received no specific grant from any funding agency in the public, commercial or not-for-profit sectors. data availability statement the data that support the findings of this study are available from the corresponding author, upon reasonable request. disclaimer the views and opinions expressed in this article are those of the author and do not necessarily reflect the official policy or position of any affiliated agency of the author. references akkoç, h., & tall, d. 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(2015). open thinking, closed questioning: two kinds of open and closed question, journal of philosophy in schools, 2(2), 17–29. https://doi.org/10.21913/jps.v2i2.1269 article information author: michael mhlolo1 affiliation: 1faculty of humanities, central university of technology, south africa correspondence to: michael mhlolo postal address: private bag x20539, bloemfontein 9300, south africa dates: received: 13 june 2013 accepted: 04 nov. 2013 published: 29 nov. 2013 how to cite this article: mhlolo, m. (2013). the merits of teaching mathematics with variation. pythagoras, 34(2), art. #233, 8 pages. http://dx.doi.org/10.4102/ pythagoras.v34i2.233 copyright notice: © 2013. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. the merits of teaching mathematics with variation in this original research... open access • abstract • introduction • background to the problem    • the theory of variation and its application to mathematics teaching       • contrast (c)       • separation (s)       • generalisation (g)       • fusion (f) • discernment unit as the analytical framework • methodology    • data presentation and analysis       • discernment unit 1       • discernment unit 2       • discernment unit 3       • discernment unit 4       • ethical considerations • summary    • implications for theory and classroom practice • acknowledgements    • competing interests • references abstract top ↑ there is a general perception that the south african curriculum statements for mathematics create polarity between the ‘old’ and the ‘new’, which does not benefit both the teachers and the learners. the new curricula demand a radical shift from the traditional teacher-led approaches that teachers are familiar with, yet does not provide a model of what it might mean to teach for conceptual understanding. this article aims to provide such a model by examining the potential of teaching with variation, which is viewed as an important mathematics teaching and learning style. proponents of the theory of variation claim that how teachers make available the object of learning to their students has been neglected yet it has a critical influence on learners’ learning. this is important for educators as they struggle to make sense of the seemingly contradictory requirements of the new curriculum. in this article a discernment unit comprising four variation patterns is used as a tool to analyse a seemingly rich teacher-led approach to teaching that was observed in one south african grade 11 mathematics classroom. the results of the analysis and implications for theory and practice are then discussed. introduction top ↑ teaching approaches associated with the ‘old’ curriculum are characterised as content driven, teacher centred, examination focused and transmission based (brodie & pournara, 2005) and are generally discouraged as they are considered to be inhibitive to learners’ deep understanding of mathematics. in contrast, the ‘new’ is characterised as learner centred and arguments in its favour claim that learners can benefit immensely from its judicious application. yet these promises have been questioned by critics who doubt that such approaches are appropriate in all cultural and resource contexts. south africa bears testimony to this doubt as numerous reports have shown how teachers struggle to implement such approaches (adler, 2009; brodie & pournara, 2005; long, 2005; vithal & volmink, 2005) and how learners subsequently end up without the envisaged knowledge and skills (schollar, 2004). one reason cited for the ‘lack of fit’ between the espoused and implemented curriculum was that there has been too much emphasis on the philosophical level where the different ‘isms’ of different schools of thought are full of conflicts which are difficult to resolve in order to arrive at a consensus (ling, 2012). in south africa, vithal and volmink (2005) made similar observations as they posited that such (new) reforms are driven largely by conjecture, stereotype, intuition, assertion and a host of untested assumptions, rather than by research. mayer (2009) contends that [o]ur field would be better served by trying to figure out research-based answers to how learning and instruction work rather than by engaging in high-level philosophical arguments about which ‘ism’ is the best. (p. 197) given that the ‘new’ is not often seen, even in well-resourced western countries in which school reform movements have been promoting these ideas for many years, brodie and pournara (2005) suggest that the next important steps, particularly for research and teacher education, are to critique, adapt, modify and complete these visions in ways that enable teachers and learners to achieve new ways of working in their classrooms. this article aims at contributing to these renewed efforts towards understanding what it might mean to teach mathematics in the traditional way whilst creating opportunities for learners to develop deep understanding of the subject. the article examines the potential of variation theory as an alternative lens through which teaching in the ‘traditional’ way could possibly be modified and adapted into classrooms that aim to teach for conceptual understanding. background to the problem top ↑ in a previous article (mhlolo, venkat & schäfer, 2012) we analysed 20 lesson transcripts to examine the quality of four teachers’ instructional representations of mathematical ideas and the potential therein to create opportunities for learners to make connections amongst relatable mathematical ideas. this was premised on the view that making connections in mathematics was critical for a deep understanding of the subject. lesson transcripts were coded using a three-point scale: faulty connections (level 0), weak connections (level 1) and the strongest connections (level 2). the results showed how one teacher’s representations fell into level 2 in 61% of the cases and into level 0 in only 0.5% of the cases as compared to the other three, whose representations fell mostly into lower levels. this suggested that this teacher might have created more opportunities for learners to develop higher quality mathematical connections than did the other teachers. this was despite the researchers’ observation that the teacher’s lessons were generally content driven, teacher centred and transmission based, approaches that are not consistent with the ‘new’ curricula. logically, these two opposite observations cannot be seen to be simultaneously valid; hence, this paradoxical observation triggered this interest to search for an alternative lens through which such ‘traditional’ teaching approaches could be viewed positively. this article is premised on the view that the use of variation in teaching mathematics might provide such a lens. the next question one might then ask is: what does the theory of variation say? the theory of variation and its application to mathematics teaching variation is about what changes, what stays constant and the underlying rule that is discerned by learners in the process (leung, 2012). the use of variation in learning and awareness was initially proposed by marton and booth (1997). the theory of variation was subsequently developed by marton and tsui (2004) as a generic learning theory; over the years researchers have refined and applied it over a broad spectrum of learning areas. with specific reference to mathematics education, [a] variation interaction is a strategic use of variation to interact with mathematics learning environment in order to bring about discernment of mathematical structure. (leung, 2012, p. 435) so a mathematics pedagogy that is rooted in variation is one that purposefully provides learners with the means to experience variation through strategically designed activities in order to create a mathematically rich learning environment (leung, 2010) that allows learners to discern the object of learning. the ‘object of learning’ is a special term in variation theory and does not mean the same as ‘learning objectives’, which point to the end of the process of learning. instead the object of learning points to the beginning of the process of learning and generally refers to the focus of a teaching situation, the thing to which learning is directed, that is, ‘what is to be learnt’ (ling, 2012). it is defined by its critical features that must be discerned in order to constitute the meaning aimed for (marton & tsui, 2004). so as a pedagogic approach, a pattern of variation is a useful tool for structuring teaching to make the learning of the object of learning possible. marton (2009) proposed four kinds of awareness brought about by different patterns of variation; these are discussed in more detail below. contrast (c) marton (2009) describes the awareness brought about by experiencing the difference (variation) between two values as contrast. an observation made by ling (2012) was that teachers had a tendency to place much emphasis on the use of examples to show similarities. however, according to variation theory, relying solely on similarities is not sufficient. contrast therefore pre-supposes that to know what something is you have to know what it is not, that is, to discern or learn whether something satisfies a certain condition or not (leung, 2012). for example, when teaching learners what a triangle is, the teacher should also show learners what a triangle is not by comparing it with other polygons (e.g. quadrilaterals, pentagons, hexagons and circles). separation (s) let us take the same example of polygons and assume that the triangle that is initially encountered when the teacher contrasts it with other polygons (non-examples) is a scalene triangle. because the learner has only encountered a scalene triangle they cannot yet distinguish this particular triangle (scalene) from other triangles. separation is premised on the view that everything has a multitude of features, each of which gives rise to a different understanding of that thing. similarly, a scalene triangle has a multitude of features some of which (e.g. shape) just help us to understand it as different from other polygons. if we want others to see a scalene triangle not just as a triangle but in a particular way that distinguishes it from other triangles, then we must focus on certain features that are critical to a certain way of seeing it, known as its ‘critical features’ (ling, 2012). in this sense learning is seen as a function of how learners’ attention is selectively drawn to the critical aspects of the object of learning. the view is that deliberate attempts to systematically vary certain aspects and keep certain aspects constant may help people to discern ‘new’ aspects of an object and construct ‘new’ meanings that might not have been apparent before. hence, according to leung (2012), separation is an awareness awakened by a systematic ‘refined contrast’ obtained by purposely varying or not varying certain aspects in an attempt to differentiate the invariant parts from the whole (p. 435). so when the learner suddenly becomes aware of the scalene triangle through a systematically and deliberately focused variation pattern, we say the scalene triangle is separated from other triangles, and it now has a separate identity. if we do not vary the kind of triangles, we have not helped learners to separate scalene triangles from triangles in general. generalisation (g) both leung (2012) and chik and marton (2012) posit that generalisation is a verification and conjecture-making activity checking the general validity of a separated out pattern, which is often a goal of mathematics exploration. ling (2012) points to links between separation and generalisation in that a pattern of variation can lead to either a separation or generalisation depending on the focus of variation or object of learning. according to ling, in deciding whether the pattern of variation leads to separation or generalisation, we must make clear what the object of learning is. in general, when the focus is on what is subordinate, it is separation: it separates the specific or particular from the general. when the focus is on what is superordinate, the awareness is of generalisation: it can be said about or is applicable to all. fusion (f) fusion takes place when the learner’s attention is focused on several aspects of an object, a concept or a phenomenon that vary at the same time. the view from a variation theory perspective is that if the learners can only discern individual critical features but fail to achieve the stage of fusion (the simultaneous discernment of all of the critical features and the relationships amongst them), then they may not be able to understand the object of learning thoroughly, and will not be able to apply such knowledge to solve new problems (ling, 2012). fusion therefore integrates critical features or dimensions of variation into a whole under simultaneous co-variation. it is important to note that besides synchronic simultaneity when learners focus on different aspects of an object of learning at the same time, diachronic simultaneity also plays a critical role in fusion by connecting variation experiences gained in previous and present interactions (leung, 2012). this follows the view that learners always bring what they have met before (previous knowledge) to bear on what they are learning now; hence, simultaneous co-variation can be explained in terms of both synchronic and diachronic simultaneity. discernment unit as the analytical framework top ↑ leung (2012) proposed an idea of a discernment unit that stands for a unit of a pedagogical process driven by the four kinds of awareness (c, s, g and f) brought about in a variation interaction. according to leung, in a pedagogical situation, these four types of variation interaction act together in a concerted way to bring about discernment, as shown in figure 1. according to leung (2012), the circular arrows and the dotted rectangle indicate that a mutually enhancing interaction between contrast and generalisation is at work to bring about awareness of dimensions of variation or critical features. leung posits that the process of mathematical understanding is sequenced by a chain of such variation interactions in which simultaneity and focus of attention play critical roles (figure 2). each of the discernment units in figure 2 represents a developing mathematical concept that is fused together by a process of contrast and generalisation driven by separation, as shown in figure 3. this pedagogical model (figure 3) shows how a lesson or series of lessons consists of a sequence of variation interactions that increases in sophisticated levels of contrast, reflecting the evolution of an idea from primitive stage to a more formal mathematical stage. methodology top ↑ the analysis focused on discernment units as described by leung (2012). these units were selected from five lesson transcripts of a grade 11 teacher who was teaching number sequences. the demarcation of a discernment unit combined ideas from leung, who posits that it is like a function of what is being varied, and andrews’s (2009) idea of a lesson episode, which he defined as that part of a lesson where the teacher’s didactic intent remained constant. the tables summarising each discernment unit were modified from tong’s (2012) study, which used similar summaries of lessons on reading. in tong’s study it was possible to use one table to summarise each of the different variation patterns. because of the nature of mathematics, leung’s proposition was that in a pedagogical situation, the four types of variation interaction act together in a concerted way to bring about discernment. consistent with this view, separation of each variation pattern would not have been productive in this article; hence, i modified the approach and combined summaries of the four variation patterns in one discernment unit. so within these discernment units, i looked at which aspects were fixed, what was varied and how it was varied, and what was thus available for discernment by the learners. because discernment is about how learners responded to the teacher’s variation activities, learner responses were critical as evidence of discernment in this analysis; hence, they are labelled as learner 1 or chorus 1 for easier reference in the discussion. data presentation and analysis discernment unit 1 at this stage of the lesson the teacher had just worked through the logic behind the general term for all linear number sequences, which shows that to get any term an we start from the starting point a1 then add or subtract the constant difference repeatedly (n ± 1) times. this had been summarised on the board as an = a1 + (n ± 1)d. from this general rule an explicit rule for a specific linear sequence could then be deduced. the lesson then progressed thus:teacher: suppose we are required to find the explicit rule for: [teacher writes three sequences on the board] (a) 4; 7; 10; 13; … (b) 2; 5; 8; 11; … (c) -10; -7; -4; -1; … chorus 1: we are still working sir. teacher: [after a pause] ok, ok, anybody who got it, the explicit rule [three learners are called to work a, b, and c on board]. learner 1: (a) tn = 3n + 1 learner 2: (b) tn = 3n – 1 learner 3: (c) tn = 3n – 13 teacher: alright, ok, ok, ok, ok. i want us to observe a pattern here. what have you observed as the constant difference in each of the sequences that i have given you? chorus 2: its 3. teacher: ok, between any two consecutive numbers there is a difference of 3. what have you observed about the explicit rule in each of the three cases? learner 4: the three rules all start with 3n then blablabla. teacher: ok, so if you were given the sequences (d) 3; 5; 7; 9; … and (e) -4; -2; 0; 2; … and asked to find the explicit rule, what would you say? chorus 3: tn = 2n then blablabla for both of them. teacher: ooooh yes! interesting, isn’t it? anybody with questions so far? comment: let us recall that contrast is an awareness brought about by experiencing difference. in this discernment unit one can see an awareness that was brought about by a contrast (c) variation in that the teacher provided two different sets of sequences {a; b; c} and {d; e} to enable learners to discern that a linear sequence with a common difference of 3 is not the same as that with a common difference of 2. once this distinction was made by the learners, one can see a ‘refined contrast’ or separation (s) as the teacher selectively draws learners’ attention to the sameness of the set of sequences {a; b; c}. through systematically varying the sequences and not varying the constant difference of 3 learners were enabled to separate the constant difference of 3 as a critical factor that identifies a linear sequence of the form tn = 3n + c. similarly, within the sameness of the set of sequences {d; e}, the learners could also separate and discern the constant difference 2 as a critical factor that identifies a linear sequence of the form tn = 2n + c. let us recall that generalisation is a verification and conjecture-making activity enabling learners to check the general validity of a separated out pattern. in this case after observing the links between the common difference and mx value of the explicit rule, learners were then enabled to generalise (g) that tn = 3n + blablabla for the first set of sequences {a; b; c} and that tn = 2n + blablabla for the set {d; e}. in terms of fusion, which is brought about through co-variation, it is important to recall that this kind of awareness (fusion) can result from either a synchronic simultaneity or a diachronic simultaneity. in this discernment unit we see more of the latter in that by varying the sequences with a common difference of 3 first and then later varying those with a common difference of 2 the teacher opened up opportunities for learners to integrate (hence fusion, f) through diachronic simultaneity. this enabled them to make the links between the common difference and the mx value of the explicit rules for all linear sequences. discernment unit 2 at the end of discernment unit 1, the teacher asked if learners had any questions: learner 5: can i please ask a question? the thing nee; you see i just want to find out why isn’t that you guys to find the general term why can’t you write tn = blablabla + 3 [constant difference]? why do you have -1 instead? that is my question. teacher: ok, ok alright i see what you mean. let’s use the following examples and see what is happening. [writes on the board] (f) 4; 7; 10; 13 … (g) 1; 4; 7; 10; 13 ... (h) 7; 10; 13; … teacher: i now want you to work out the explicit formula for these three sequences. learner 6: (f) tn = 3n + 1 learner 7: (g) tn = 3n – 2 learner 8: (h)tn = 3n + 4 teacher: ok, let’s observe something here. if you look carefully you will notice that i have been playing around with the same figures to generate three different sequences. what is it that i have not changed? chorus 4: the common difference is the same its 3 sir in all the three cases. teacher: ok, and we have agreed that when this common difference is 3 then the explicit rule will be tn = 3n + blablabla. so what is it that i have changed? learner 9: i think you have just changed the starting point for each of those sequences. teacher: ok, and what do we observe in the second part of our explicit rules? chorus 5: the second part is different each time. teacher: this seems to suggest some link of some sort between this starting point and the plus blablabla of this explicit formula. who would like to try and explain this relationship to us? [pointing to a learner]: can you come to the board and use these two examples (3; 7; 11; 15 … and 3; 5; 7; 9 …) to explain what you have seen. learner 10: [explaining how to get the second part -1 of the explicit rule tn = 4n – 1 for the sequence 3; 7; 11; 15 …] in this sequence, 3 (first term) minus 4 (common difference) gives us -1. even here [pointing tn = 2n + 1 for the sequence 3; 5; 7; 9 …] the 3 (first term) minus 2 (common difference) gives us 1.it always works. chorus 6: hoooo [and clapping their hands]. teacher: ok, someone has made an observation here to say alright, this 2 [pointing to the 2 in tn = 2n + 1] where is it coming from, it is the difference between any successive terms. then to get this 1 he simply says 3 [a1] minus 2 [common difference] which gives us 1. ok does it always work? ok let’s see if it works here. [pointing to 1; 4; 7; 10 …] so according to him the explicit rule here should be tn = 3n – 2? chorus 7: agreed! yes! yes! true! [learners go on to prove the rule by generating 1; 4; 7; 10 …] comment: in this discernment unit, one can see why leung (2012) posited that variation interaction can be teacher designed or learner initiated and that it should not always be assumed that learners will always understand an object of learning in the same way as the teacher intended. for example, learner 5 asked why the idea of constantly adding on the common difference was not coming out in the explicit rule. according to ling (2012) this is mainly because the two possible ways of seeing the same thing had not been placed in the foreground and so variation was not effective in bringing out the contrast between the ‘add on’ of the common difference in the recursive and the (+ c) in the explicit rule. literature confirms that learners usually see linear sequences through recursion (blanton, 2008). this therefore suggested that this (+ blablabla) part of the explicit rule was a critical feature that the teacher needed to make the object of variation. the teacher through ‘refined contrast’ variation then steered the learners’ awareness and hence opened up opportunities for learners to separate (s) the critical links and discern that the second part (+ c) of the explicit rule was equivalent to term one minus the common difference i.e. (a1 – d). because learner 5 and perhaps others had now gained a new way of seeing this part of the explicit rule it can be argued that a contrast variation was experienced. according to ling (2012, p. 86), contrast (c) can also be brought about when learners experience variation between their prior knowledge and the new way of seeing the same thing, as intended by the teacher. when the teacher probed: ‘does it always work?’, one can see how the teacher is using generalisation (g) as a verification and conjecture-making activity in which learners check the general validity of their separated out pattern (chik & marton, 2012). in this discernment unit one can see fusion (f) as the learners’ attention is focused on several aspects of the explicit rule under both synchronic and diachronic simultaneity. table 1: summary of discernment unit 1. discernment unit 3 after the discussion on the connections between the recursive rule and the explicit rule the teacher then posed another question:teacher: let us go back to the three examples that we started off with. in each case i want us to determine the 10th term using any of the rules that we have seen so far [three learners provide answers for sequences a, b, and c]. learner 11: (a) 31 learner 12: (b) 29 learner 13: (c) 17 teacher: which of the two rules recursive or explicit did you use? chorus 8: explicit sir, it’s much easier and faster. teacher: ok, let us explore this issue further. is there any other way that we could have calculated or located the 10th or any other term in these sequences? learner 14: sir, i think we could also use graphs to find the answers. chorus 9: aaaaaah! how can you draw graphs here? teacher: can somebody explain how? learner 15: instead of writing tn = 3n + 1 sir we could write y = 3x + 1 where y = tn and x = n then we can make a table of values connecting the term numbers (x) and their values (y) then plot the values on the x and y plane [cartesian]. chorus 10: hoooo [whilst clapping their hands. the class then makes the following table of values for the sequence: y = 3x + 1]: teacher: i hope you are making sense of what is happening here. because we have now run out of time, i want us to go home and try to make tables first then draw the graphs of the same tasks (a), (b) and (c) then we will discuss your findings tomorrow.comment: within this discernment unit one can see how the teacher has contrasted (c) the different representations of linear sequences starting with the recursive rule, through the explicit rule, through tables for the corresponding values of x and y. let us recall that contrast can also be brought about when learners experience variation between their prior knowledge and the new way of seeing the same thing, as intended by the teacher. we notice this when the learners suddenly realised that tn = 3n + 1 could be written as y = 3x + 1, which they were familiar with. in this discernment unit the teacher also makes deliberate attempts to systematically draw learners’ attention to the critical features of each representation, thereby helping the learners to separate (s) and hence discern certain advantages and disadvantages of working with each representation. for example, the recursive rule shows both the starting point and the constant difference of the linear sequence clearly, which the explicit rule does not. however, the explicit rule has an advantage in that it enables one to get straight to any term value without having to go via the previous term, which is the limitation of the recursive rule. this is evidenced in the learners’ responses when they were asked to determine the tenth term of each of the sequences. their responses (chorus 8) showed that it was easier for them to use the explicit rule to get the tenth value. probing the learners for another method that could be used to determine the tenth term enabled the learners to again discern the links between the explicit rule, tn = 3n + 1, and y = 3x + 1 (its equivalent form when written in terms of x and y), which they were more familiar with. this was critical in that this formed the foundation for learners to generalise (g) why such number sequences are called linear. the table then gives a visual connection between the term number (input) and its value (output) in a way that makes it easier for learners to identify how terms change from one to the next. this visual link between the input and the output might not have been discerned by the learners if the teacher had not contrasted the recursive and the explicit rules simultaneously (fusion, f). table 2: summary of discernment unit 2. table 3: summary of discernment unit 3. discernment unit 4 the following day learners brought their work on graphs. the teacher had drawn the graphs on an a0 size of paper and pinned it on the board for the classroom discussion (see figure 4): teacher: ok, let us proceed from where we left. what can you say about the graphs that you drew? chorus 11: they are all straight line graphs. they are parallel lines. they are increasing graphs. teacher: ok, good observations. with many sequences it may be easy to notice how terms change from one to the next if the terms are listed numerically. however, listing the terms does not show other characteristics. for example, when a pattern in a number sequence is found by adding or subtracting the same number every time then the sequence is called an arithmetic or linear sequence because as you can see from the graphs all arithmetic sequences make straight line graphs – something which we could not have seen from listing the numbers. but why are the lines parallel? chorus 12: sir, just by looking at them we can see that they are parallel. teacher: yes i agree, but how else can we tell they are parallel? learner 16: these lines, sir, have the same gradient. teacher: ok, somebody has made an observation that these lines have the same gradient. ok, do you agree? if so how do you determine that gradient? [there is some discussion with some learners pointing to the 3x or 3n in the explicit rule, others were counting the blocks to determine the , i.e. the change in y over the change in x. eventually the class reaches a consensus] chorus 13: the gradient is the same as the common difference when the terms are listed numerically. teacher: now let us come back to the question raised earlier that why do we have -1 in the explicit rule instead of the constant difference. in the recursive rule the common difference is added on to the previous term each time so it is visible every time we calculate a term value. in the explicit rule this common difference is considered as the rate of change, i.e. as we move from t1 to t2 and so on, the numbers increase by 3. hence if we wrote the arithmetic sequence in the standard function form y = mx + b, it is the m value which is equal to 3 and not the b. so the b is not standing for the common difference hence we cannot say tn = blablabla + 3 as you suggested (pointing to the learner who had asked earlier). who can tell us what the b is standing for in this case? learner 17: sir, it is standing for the point where the graph cuts the y-axis. teacher: what other term do we use for that? learner 18: the y-intercept, sir. teacher: how is this linked with the starting point of each sequence? remember in the recursive rule we have said we always start at a1. can you locate the three starting points for the three graphs that you have drawn? chorus 14: [the learners locate the points a, b and c as the starting points] teacher: can you see that in each case to move from each starting point (a, b and c) to the point where the graph cuts the y-axis we have to move 1 step back along the x-axis. it’s like getting to a point we can call a0. let us remember that each time we move one step on the x-axis we actually go up or down three steps (common difference) along the y-axis. this is the reason why in the standard form y = mx + b; the b value or a0 can always be found by a1 (starting point) minus d (common difference). simply put when given the starting point in a linear sequence move backwards by the common difference and you have the b-value of the explicit formula. that is exactly why (learner’s name) method always worked. comment: from this discernment unit one can see contrast (c) in the learners’ changed way of seeing different representations of a linear sequence. one can also see separation (s) in the manner in which learners were enabled to discern the links between critical features of a linear sequence when represented in a different way. from the teacher’s discussion, one can observe that by representing all three sequences on one graph, a fusion (f) is foregrounded and a number of mathematical connections are triggered. so in this discernment unit through the teacher’s varying and not varying, learners were enabled to discern and generalise (g) that all the sequences they had worked with so far could be represented graphically by straight lines that are parallel (same gradient), that are all ‘increasing’ (gradient is positive in all the three cases), and whose slope m is exactly the same as the common difference, d. it was also possible for learners to discern that the graphical representation, especially when all the sequences had been drawn on the same cartesian plane, became more like a ready reckoner: it allowed quick interpolation of values. knowing one of the two values (x or y) allowed learners to interpolate the other value by following across (or up) to one axis and then down (or across) the other axis to read the unknown data that lies within the known data range but was not measured initially or given in the initial listing of terms. for example, just by moving up the vertical line (x = 1), they were able to read off (chorus 14) the starting points (y values at a, b and c) for each of the sequences that had been represented on this graph. by moving up the vertical line x = 10, they were also able to read off the tenth values of the different sequences without much effort. in fact it can be argued that the way tasks were structured through variation opened up space for deep mathematical reasoning that would have enhanced learners’ higher order thinking skills and understanding of number patterns. consistent with leung’s model as shown in figure 3, one can see how number patterns were presented from the primitive listing of terms to the more formal and precise way through the use of recursive, explicit and graphical representations. it can therefore be argued that these forms of variation (contrast, separation, generalisation and fusion) provided a hierarchical system of experiencing processes through forming concepts (gu, huang & marton, 2004). watson and mason (2006) concluded that such results created by learners become tools for more sophisticated mathematics, and are a significant component of their mathematical progress. figure 1: a discernment unit driven by types of variation interaction. object of learning. figure 2: a pedagogical time sequence on the understanding of a mathematical object of learning. figure 3: a model of mathematics pedagogy based on variation. figure 4: graphs of the three arithmetic sequences. ethical considerations the department of education granted approval to proceed with this study under permit t-728 p01/02 u-848. at the institutional level, the university ethics committee granted approval under protocol 2007ec007. at the school level, the researcher received informed consent from teachers and parents of the learners who participated in the study. summary top ↑ this article was triggered by the paradoxical observation that a grade 11 teacher taught number sequences in a manner that would have been described generally as teacher led, content focused and transmission based, yet the instructional activities seemed to create opportunities for learners to make deep mathematical connections. despite the fact that this approach might have been described pejoratively as inhibitive to learners’ deep understanding, the researcher hypothesised that something could be learnt from this teacher’s instructional practices. borrowing leung’s (2012) framework of a discernment unit based on the theory of variation, the analysis was aimed at examining the extent to which learners were provided with opportunities to develop mathematical concepts. although the results are exemplified through four discernment units, all the five lessons on number patterns that were observed for this teacher showed that indeed all the characteristics of a strategic use of variation were at the centre of the teacher’s instructional activities. the discussion and illustrations thereof highlight how opportunities were created for learners to discern a number of important mathematical ideas. starting from a primitive interpretation of a linear sequence through recursion, that is, an = a1 + (n ± 1)d, one can notice how the learners moved to more sophisticated stages through establishing the links between the common difference and the coefficient of n in the explicit rule, the links between the first term and common difference to determine the b value of the explicit rule, and then eventually justifying and linking these relationships with gradients and y-intercepts on graphs of the same sequences. this way it can be argued that learners were forming a hierarchy of gradually refined concepts of number patterns through the teacher’s approach of varying and not varying certain things. it can be concluded that the ways in which strategies of variation are orchestrated in a class can foster meaningful learning. the four discernment units as outlined fully illustrate this point. these lessons were generally teacher led; there was nothing ‘novel’ about the tasks, they were just standard textbook tasks, but the manner in which the teacher varied the activities and kept some constant indeed created opportunities for learners to discern a number of important mathematical ideas. table 4: summary of discernment unit 4. implications for theory and classroom practice in terms of contributing to theory, the warning inherent in many theoretical analyses is that if we focus on the insights on teaching and learning rather than arguing about the differences between the theories at the philosophical level, then we will indeed find that many of the teaching approaches, strategies and designs suggested are similar and compatible (ling, 2012). through the activities within the four discernment units one can see how a pedagogy based on variation provides such a harmonising framework in which a predominantly teacher-led teaching approach supports deep understanding of mathematical concepts. in terms of contributing to curriculum development, this article contributes to harmonising the simplistic and bogus dichotomies that have been set up in the curriculum, which need to be addressed in future policy documents. the south african curriculum statements create uncalled-for polarity between the ‘old’ and the ‘new’ curriculum, but what is not acknowledged is that the ‘new’ is not often seen, even in well-resourced countries where school reform movements have been promoting these ideas for many years. with regard to teacher education, both literature and empirical evidence suggest that teacher educators have a legal, professional, moral and civic obligation to provide their student teachers with models of teaching that work in their contexts. from that perspective, it can be argued that this article could be of benefit to teacher education in that it explores a model of teaching mathematics that has been tried and tested elsewhere and locally and which might be considered for further trials with student teachers. in terms of classroom practice, vithal and volmink (2005) posit that waves of curriculum change often result in the implementation of an eclectic mixture of approaches. the ‘new’ curriculum demands a radical shift from the traditional teacher-led approaches that teachers are familiar with, yet it does not provide a model of what it might mean to teach for conceptual understanding. teaching with variation could provide teachers with this much-needed bridge between the ‘old’ that they are familiar with and the ‘new’ that is espoused. this is important for educators as they struggle to make sense of the seemingly contradictory requirements of the ‘new’ curriculum. with reference to how learners could benefit from such studies, empirical evidence shows that lack of insight into the pedagogical theories underpinning the reform movement may cause confusion even amongst experienced teachers, which leads to learners having neither conceptual nor procedural knowledge (schollar, 2004). the implication for schools and educators seems clear: mathematics education programmes in schools should incorporate both teacher-led and learner-centred approaches in their instruction practices. acknowledgements top ↑ i acknowledge the department for international development for funding the phd study from which this article is drawn. competing interests i declare that i have no financial or personal relationships that may have inappropriately influenced me in writing this article. references top ↑ adler, j. 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(2004). classroom discourse and the space of learning. mahwah, nj: lawrence erlbaum associates. mayer, r.e. (2009). constructivism as a theory of learning versus constructivism as a prescription for instruction. in s. tobias, & t.m. duffy (eds.), constructivist instruction. success or failure? (pp. 184–200). london: routledge, taylor & francis group. mhlolo, m.k., venkat, h., & schäfer, m. (2012). the nature and quality of the mathematical connections teachers make. pythagoras, 33(1), art. #22, 9 pages. http://dx.doi.org/10.4102/pythagoras.v33i1.22 schollar, e. (2004). primary mathematics research project. johannesburg: eric schollar & associates. tong, s.y.a. (2012). applying the theory of variation in teaching reading. australian journal of teacher education, 37(10), 1–19. http://dx.doi.org/10.14221/ajte.2012v37n10.3 vithal, r., & volmink, j. (2005). mathematics curriculum research: roots, reforms, reconciliation and relevance. in r. vithal, j. adler, & c. keitel (eds.), researching mathematics education in south africa: perspectives, practices and possibilities (pp. 3–27). cape town: hsrc press. watson, a., & mason, j. (2006). seeing an exercise as a single mathematical object: using variation to structure sense-making. mathematical thinking and learning, 8, 91–111. http://dx.doi.org/10.1207/s15327833mtl0802_1 abstract introduction the study methodology findings discussion and conclusions acknowledgements references about the author(s) million chauraya department of applied education, midlands state university, zimbabwe karin brodie school of education, university of the witwatersrand, south africa citation chauraya, m., & brodie, k. (2018). conversations in a professional learning community: an analysis of teacher learning opportunities in mathematics. pythagoras, 39(1), a363. https://doi.org/10.4102/pythagoras.v39i1.363 original research conversations in a professional learning community: an analysis of teacher learning opportunities in mathematics million chauraya, karin brodie received: 13 dec. 2016; accepted: 06 sept. 2018; published: 31 oct. 2018 copyright: © 2018. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract the growing perception of professional learning communities as an effective professional development approach needs to be supported with knowledge of how such communities create learning opportunities for teachers. activities in professional learning communities are underpinned by collegial conversations that foster learning, and in this article we analysed such conversations for learning opportunities in one professional learning community of mathematics teachers. data consisted of audio-recorded community conversations. the focus of the conversations was to understand the thinking behind learners’ errors, and teachers engaged in a number of activities related to learner errors and learner reasoning. our analyses show how opportunities for learning were created in identifying the origins of learners’ errors as well as learners’ thinking underlying their errors. results also showed that the teachers had opportunities for learning how to identify learners’ learning needs and in turn the teachers’ own learning needs. the teachers also had opportunities for deepening their own understanding of the conceptual meaning of ratio. the learning opportunities were supported by the following: having a learning focus, patterns of engagement that were characterised by facilitator questioning, teacher responses and explanations, and sharing knowledge. such mutual engagement practices in professional learning communities resulted in new and shared meanings about teachers’ classroom practices. our findings also show the critical role of a facilitator for teacher learning in professional learning communities. introduction teacher learning in professional learning communities is generally accepted as a teacher professional development approach that can significantly impact teachers’ mathematical knowledge and practices (e.g. brodie & borko, 2016b; horn, 2010; koellner & jacobs, 2015). in south africa the move towards professional learning communities as enunciated in the integrated strategic planning framework for teacher education and development (departments of basic education and higher education and training, 2011) marked a shift in teacher professional development policy, necessitated by the realisation that standalone workshops were not producing results in terms of improving teachers’ knowledge, practices and learners’ performance (bertram, 2011; mccarthy & oliphant, 2013; nel, 2015). the move towards adopting professional learning communities as an alternative requires knowledge of what teachers can learn, and what supports learning in such communities. research on professional learning communities has focused mainly on understanding what and how practising teachers learn in such communities and the extent to which such learning might improve teachers’ practices (brodie, 2014; brodie & borko, 2016b; chauraya & brodie, 2017; goldsmith, doerr, & lewis, 2014; horn & little, 2010; koellner-clark & borko, 2004). most studies highlight the significance of inquiry-focused conversations for learning in a professional learning community, but very few have analysed in detail the learning opportunities created in such conversations. this article addresses this gap by presenting analyses of the conversations of a professional learning community and how they provide learning opportunities. the questions addressed in the article are: what teacher learning opportunities are created in the conversations of a professional learning community? what are the characteristics of the practice of a professional learning community that support learning opportunities? teacher learning in professional learning communities in this article we regard a professional learning community as ‘a group of teachers sharing and critically interrogating their practice in an on-going, reflective, collaborative, inclusive, learning-oriented, growth-promoting way’ (stoll & louis, 2007, p. 2). in our study the professional learning community consisted of five mathematics teachers in one high school and the first author as facilitator. the community engaged in a number of developmental learning activities (see brodie & shalem, 2011, for more details), which involved conversations about their understandings of learner errors in mathematics. our analyses of these conversations in this article sought to understand the learning opportunities that were created. the theoretical framework that guided the study was situated learning theory (lave, 1991; lave & wenger, 1991; wenger, 1998), which regards learning as situated in human activities, inseparable from the environments in which knowledge is used, and occurring through social processes that involve the negotiation of meanings with others (buysse, sparkman, & wesley, 2003). according to the theory learning is situated in communities of practice, including professional learning communities, which we regard as a special kind of community of practice (brodie & borko, 2016a) that occurs in the practices of such communities (wenger, 1998). for wenger the practice of a community of practice has three dimensions or characteristics, which are a source of coherence for the community: joint enterprise, mutual engagement and a shared repertoire. the joint enterprise is the shared basis for action with colleagues, or the ‘negotiated response to their situation’ (wenger, 1998, p. 77) which the community strives to resolve or understand. in a professional learning community the joint enterprise is akin to a learning focus, which is an issue of concern, or a problem of practice that the teachers inquire into so as to develop new understandings that deepen their professional knowledge and practice (katz & earl, 2010). in this study the learning focus for the professional learning community was understanding learner errors in mathematics. the second dimension of practice is mutual engagement which refers to active collaboration involving such practices as sharing understanding, joint reflection, collective development of new meanings and joint decision-making (katz & earl, 2010). according to wenger (1998) mutual engagement fosters coherence in a community of practice and is a source of new meanings and understandings for the community, also called the shared repertoire. the shared repertoire, as the third dimension of the practice of a community of practice, ‘includes routines, words, tools, ways of doing things, stories, gestures, symbols, genres, actions or concepts that the community has produced or adopted in the course of its existence, and which become part of its practice’ (wenger, 1998, p. 83). in a professional learning community the shared repertoire consists of the meanings that are developed and ways of dealing with, or resolving, situations that are jointly negotiated and adopted as part of the community’s practice. the three dimensions of practice are presented in figure 1, which shows how having a shared learning focus and engaging in collective action supports mutual engagement activities which in turn lead to the development of new meanings and ways of addressing problems of practice. the three dimensions shape each other, hence the double arrows. in this article we adopted these three characteristics of practice as one of the two frameworks or analytical tools that we used in analysing the conversations of the professional learning community. figure 1: dimensions of practice in a community of practice. our goal in using this framework was to understand the teachers’ participation in the professional learning community as a practice that supported learning opportunities in the teachers’ conversations, which were the data that we analysed in this article. teacher knowledge in a professional learning community another aspect of teacher learning in a professional learning community that was of interest to us was what teacher knowledge could have been learned in the conversations of the community. to analyse this knowledge we used pedagogical content knowledge and content knowledge, which are seen as part of teachers’ mathematical knowledge for teaching (ball, hill, & bass, 2005; ball, thames, & phelps, 2008; hill et al., 2008). mathematical knowledge for teaching is defined as the knowledge that mathematics teachers need to know in order to teach the subject well, and includes teachers’ knowledge of: the content of mathematics, the different representations of the mathematics so that it is accessible to learners, learners’ mathematical thinking, and how to make in-the-moment decisions that support learners’ learning. so our second analytical tool consisted of pedagogical content knowledge and content knowledge as part of teachers’ mathematical knowledge for teaching. such knowledge is part of teachers’ pedagogical content knowledge and, as we will show, is related to their content knowledge. the two analytical frameworks presented above have a close relationship with each other that enabled us to analyse both the teachers’ participation in the conversations and what they learned in those conversations. the joint enterprise, mutual engagement and shared repertoire framework enabled us to analyse the teachers’ participation as a practice of the community. the mathematical knowledge for teaching framework enabled us to analyse the knowledge implicated in the conversations. for us the two frameworks are related in that in each conversation episode the mathematical knowledge was part of the learning focus or joint enterprise, and through active participation in the conversation, there were opportunities for deepening the teachers’ knowledge and developing new meanings about the teachers’ mathematical knowledge for teaching. inquiry in professional learning communities is usually supported by a facilitator or ‘critical friend’, who participates in the professional learning community and supports learning by asking for critical reflection on issues, probing for justification, challenging assumptions, pushing for deeper thinking and interpretations, reminding participants of what has been achieved, and guiding the direction of the ongoing conversations (brodie, 2016; katz, earl, & jaafar, 2009). a facilitator thus plays a fundamental role in supporting teacher learning and in this article we also analysed the facilitator’s role in the learning opportunities created in the professional learning community. research findings on teacher learning in professional learning communities research on teacher learning in professional learning communities shows that learning can occur in some important dimensions of teachers’ knowledge and practice. in some communities teachers learned to analyse mathematical problems in greater depth (e.g. borko, jacobs, eiteljorg, & pittman, 2006). learning was also observed in teacher reflection on classroom practices in ways that could result in changes in practice (horn, 2010). some studies have shown how teachers learned about their learners’ thinking in making errors and developed better understandings of learners’ thinking (e.g. brodie, 2014; kazemi & franke, 2004). other studies have shown changes in teachers’ knowledge of the curriculum (brodie, shalem, sapire, & manson, 2010), and the teaching practices (tam, 2015). in all the studies, teacher learning was attributed to participation and collaboration in teacher learning communities. in south africa findings from a teacher professional development project have shown that conversations in professional learning communities provided opportunities for developing both content knowledge and pedagogical content knowledge, that content knowledge and pedagogical content knowledge conversations often triggered each other (marchant & brodie, 2016), and that the different project activities supported talk on different foci: learners, mathematics and practice (chimhande & brodie, 2016). teacher participation in the professional learning communities supported the deepening of teachers’ mathematical knowledge (brodie, 2014; liebenberg, 2016), shifts in teachers’ talk about learner errors (brodie, 2014), shifts in teachers’ practices (chauraya & brodie, 2017; molefe, 2016) and shifts in teachers’ identities (chauraya, 2016). other studies involving teacher learning communities in south africa have shown that teachers developed confidence that was central to their learning (graven, 2004). the findings cited above highlight how teacher communities can support teacher professional learning in terms of understanding and addressing problems of practice, understanding learners’ thinking, developing new teaching practices, and deepening knowledge of mathematics and the curriculum. the studies also highlight the significance of collaboration, community and inquiry to learning in such communities. our article contributes to this growing knowledge base by presenting findings of analyses of learning opportunities created in conversations about learner errors. the study methodology the research reported in this study involved a school-based professional learning community of mathematics teachers, which was a pilot to phase three of the data-informed practice improvement project. the project establishes and supports professional learning communities of mathematics teachers in high schools in the johannesburg area. the underlying goal of the project is to develop teachers’ pedagogical content knowledge in order to support changes in both their teaching practice and their content knowledge. to address this goal, the project has developed a set of activities that supports teachers’ learning about how to engage with learner errors in ways that provide access to mathematics for both learners and teachers (brodie, 2013). in the study, five mathematics teachers in one township high school, together with the first author of this article as facilitator, worked as a school-based professional learning community on the professional learning activities. the professional learning community met once a week for two hours after school, during term time. the professional learning activities were: analysing the errors that learners made on tests with a view to understanding the learners’ reasoning behind the errors, interviewing selected learners on errors they had made to understand their reasoning in more depth, identifying learners’ learning needs in the form of critical mathematical concepts, reflecting on teachers’ own understanding of the mathematics concepts, designing and teaching mathematics lessons in those concepts, and reflecting on videotapes of the lessons that were taught with the aim of understanding how the teachers engaged with learner errors in their teaching. a critical concept is a mathematical concept that underlies a number of key areas of the curriculum, and learner difficulties with the concept can contribute to patterns of errors linked to the concept. the critical concept in this article is ratio, and it was identified as critical for learners through analysis of the learner errors that were observed on test items (see chauraya, 2013). the community analysed learner errors on a number of items on ratio, and we present one here that we selected as representative of the analyses. the data for this article consisted of episodes of conversations on learner errors on the topic of ratio, and learner interviews that were conducted by one of the teachers on this item. learners were selected for interviews according to the errors they made, and which errors the community identified as requiring more detailed information about the possible reasoning behind the errors. each teacher interviewed between two and four learners. in this article we focus on one teacher’s interview with one learner. these data were collected through audio recordings that were later transcribed for analysis. data analysis data analysis involved three stages. the first stage involved identifying the conversation episodes for analysis from the lengthy conversations of the community. the second stage involved summarising the selected episodes for presentation and analysis. the last stage involved the actual analysis of each episode using the analytical frameworks presented earlier in the article. in identifying the conversation episodes for analysis in this article we adopted the critical incidents approach which is a qualitative data analysis technique mainly used in analysing classroom events in order to understand teacher actions and reflections in instructional situations (angelides, 2001; yang & ricks, 2012). a critical incident in school situations is regarded as a commonplace event that occurs in the everyday life of a classroom but is justified as critical on the basis of its significance to the goals of the activity and the researcher’s interpretation of the event (angelides, 2001). the criticality of an incident depends on the ‘justification, the significance, and the meaning given’ by the researcher (angelides, 2001, p. 431). thus a researcher looks at the underlying trends, motives and structures in the incident and ascertains whether it is critical or not depending on what they are interested in before analysing the teaching or learning effect of the episode (yang & ricks, 2012). in adopting the critical incidents approach we regarded a critical episode as a conversation focused on one of the following themes: making sense of learners’ reasoning through analysis of learner interviews, understanding learners’ errors and identifying the critical concept for learners, and the teachers’ own understanding of the critical concept. these themes were deliberately chosen in line with the goals of the data-informed practice improvement project which included developing teachers’ knowledge of learner errors and how to work with errors in their teaching. in presenting and summarising each selected episode we identified only those conversation turns that contributed to the theme of the episode. a turn is an utterance or comment by a participant in a conversation. turns that were not relevant to the theme of the episode were disregarded. having identified the relevant turns in each episode, we then categorised these turns according to what was being said and the purpose that each turn served. for example a question by the facilitator was categorised as ‘facilitator question’, and then according to whether the question served to challenge the teachers’ knowledge or probe for deeper meaning. thus our presentation summarises the selected conversation turns in each episode in tabular form, showing the nature and frequency of particular categories of conversation turns, including examples of such turns. as indicated earlier, in the actual analysis we used two frameworks or analytical tools. the dimensions of practice in the community as depicted in figure 1 enabled us to analyse: what formed the focus of each conversation episode (the joint enterprise), how the teachers engaged in the conversation (mutual engagement), and the nature of the developing new meanings or practices (the shared repertoire). the mathematical knowledge for teaching framework enabled us to identify the nature of mathematical knowledge in each conversation episode and any new meanings that could have been developed about that knowledge. ethical consideration the relevant university research ethics committee cleared the project from which the data and findings reported in this article were drawn. the ethics clearance number was 2009ece171. the gauteng department of education approved the project through a letter detailing the conditions for the conduct of the project in schools in the province. to maintain anonymity all the names of participants in this article are pseudonyms and the name of the school is not mentioned anywhere in the article. findings the episodes that we analyse below were drawn from the conversations about learners’ errors on the following ratio test item: divide r800 in the ratio 5:3. the item was in a test given to grade 10 and grade 11 learners. the most common observed errors on this test item were as follows: individual inquiry into learner reasoning: learner interviews as part of the analyses of learners’ errors, each teacher interviewed selected learners on specific errors that they had made. learner interviews were intended to support the teachers to understand learner errors more deeply, in particular what was both valid and not valid in the learners’ reasoning. the interviews were also the initial step for the teachers in trying to make sense of the learners’ errors and were central to conversations about learner errors. table 1 shows an extract of one teacher’s interview with a learner on the error: . all the teachers’ and learners’ names are pseudonyms. janeth is the teacher in these extracts. table 1: extract of a learner interview. interviewing learners about their reasoning was not part of teachers’ practice prior to the professional learning community. in preparation for learner interviews, the professional learning community had discussed the need to ask and follow up on questions that would elicit the substantive learner reasoning behind the errors. the teachers role-played interviews and learned how to distinguish between questions that tried to guide the learners to the correct answer and questions which served to access the learners’ reasoning. in the episode in table 1 the teacher started off with a general question ‘can you explain how you solved this problem?’ after the learner explained his strategy, the teacher probed further for the learner’s reasoning, which the learner struggled to provide – he kept talking about his strategy. even the teacher’s question ‘why did you do that?’ produced the answer ‘because i wanted to get the final answer’. our data is replete with such examples, with learners struggling to make their reasoning explicit and teachers struggling to support them to do this. the teacher was responsive to the learners’ difficulties, and shifted her question to ‘but this idea of dividing, where did divide come from?’ this question supported the learner to say that he was guided by the word ‘divide’ in the problem, which shows the teacher where some of the difficulty lies. this interview extract shows how the learner interviews were learning opportunities for the teachers to learn how to elicit, in more detail, learners’ thinking behind observed errors. through the interviews the teachers also had opportunities to begin to understand why learners make errors in mathematics, and how the errors are reasonable for learners (brodie, 2014). this is an important aspect of teachers’ pedagogical content knowledge, which they can work with in their teaching as they respond to learners’ errors and learners’ learning needs. however in the interviews the teachers often missed opportunities for getting more insights into the learners’ understandings of some mathematics concepts. in the episode above the teacher did not ask the learner to explain his understanding of the ratio ‘5:3’, thereby missing the opportunity to get the learner’s understanding of the concept. collective inquiry into learners’ reasoning: community conversations in the professional learning community initial conversations about learner errors focused on understanding the learner errors, giving the possible reasons for the errors, and identifying the critical concepts for the learners, also known as the learners’ learning needs. the learner interviews served to provide evidence for the learners’ thinking in making errors. understanding learners’ reasoning in errors formed the learning focus or joint enterprise for the community in these conversations. the conversation presented in table 2 shows how the teachers mutually engaged in trying to make sense of the learners’ thinking in making the observed errors. table 2: understanding a learner error. in our analysis below we use an ordered pair notation, for example (t5, sibanda); the first value ti refers to conversation turn number and the second value refers to the pseudo names of the participants. the episode shows certain patterns of interactions in the community conversations which were also evident in other and similar conversations. the episode was characterised by the facilitator’s questions that served to initiate and probe for deeper thinking, the teachers’ interpretations, explanations and justification of learner errors, and elaborations by the facilitator as a way of summarising the meanings being developed. these patterns of interaction characterised the mutual engagement by the teachers in the conversations. the conversation drew from the learner interview presented earlier on, which illustrates how learner interviews were useful data that supported the joint enterprise. the larger part of the episode consisted of turns in which the teachers interpreted the error and proposed reasons for the error. interpreting the error involved identifying what the learners did in making the error. explaining the error involved suggesting the possible reasoning behind the error, and in this case the explanation was that the learners took ‘divide in the ratio …’ to mean ‘divide by …’. one teacher’s comment that this was correct according to the language suggests that she was beginning to see the reasonableness of the error from a learner’s perspective, which is an important step for teachers in learning how to view errors as integral to learning mathematics. however, the same teacher’s suggestion that the error could be justified in terms of learners forgetting what they had been taught was an example of blaming learners for errors, something that shows limited understanding of the nature of errors in mathematics learning. this episode illustrates how conversations about learner errors created opportunities for learning how to interpret learner errors and explain the thinking behind the errors, as well as identifying learners’ learning needs from their errors – all examples of opportunities for developing pedagogical content knowledge mutual engagement through practices such as facilitator questions and teachers’ responses and explanations which supported opportunities for learning about learners’ thinking in making errors. the teachers’ explanations of the error served as examples of developing new thinking or understanding about learner errors, which could be part of the teachers’ developing shared repertoire. the facilitator’s central role in this episode was that of initiating and probing through questions, as well as clarifying and building on the teachers’ contributions, all of which supported mutual engagement in the conversation. in the next section we show how the conversations shifted to the teachers’ subject content knowledge of ratio. collective inquiry: the teachers’ understanding of ratio one of the errors was attributed to learners not understanding what it meant to divide a quantity in a given ratio. in the following episodes we analyse how the facilitator shifted the conversation from the identification of learners’ learning needs to teachers’ learning needs. two critical episodes in the conversation focused on the teachers’ understanding of ratio as a concept and the conceptual meaning of the steps in the algorithm for dividing quantities in given ratios. both episodes were on the teachers’ subject content knowledge of ratio. table 3 shows a summary of the critical turns in the conversation and how we categorised the turns according to how they contributed to the theme of the episode. table 3: teachers’ understanding of the ratio concept. the facilitator’s questions served to support the teachers to share their individual understandings of ratio. the teachers’ responses showed an algorithmic understanding of ratio, in which they described ratio in terms of the steps in the algorithm for dividing a quantity in a given ratio. the ensuing funneling questions by the facilitator were a way of getting the teachers to the concept, upon realising that the teachers were not likely to give the conceptual meaning. in the last turns of the episode the facilitator elaborated the concept further as a way of supporting the teachers’ understandings. although there was no evidence that the teachers learned the conceptual meaning of ratio, the episode was an opportunity for the teachers to develop a new shared understanding of the concept of ratio, thus deepening their subject content knowledge. the data above show that in an effort to shift the teachers’ algorithmic understanding to an understanding of the concept of ratio, which includes the algorithm, the facilitator deliberately used funneling questions. we acknowledge the limitations of funneling questions, and in this case there was no evidence that this move by the facilitator resulted in deeper learning of the concept by the teachers. the conversation episode summarised in table 4 focuses on the teachers’ understandings of the conceptual meaning of the steps in the algorithm for dividing a quantity in a given ratio. table 4 presents a summary of the conversation and our categorisation of the critical turns. table 4: teachers’ understanding of the ratio algorithm. the facilitator’s challenging questions and the teachers’ expressions of their own understandings were the main forms of mutual engagement in this episode. the teachers’ responses to the initial question by the facilitator showed knowledge of the steps in the algorithm, but they could not explain the conceptual meaning of the algorithm as a whole. the teachers’ difficulties in explaining the algorithm could be attributed to how they normally teach the algorithm. mathematics teachers in south africa often teach the algorithm without worrying about its conceptual meaning, and knowledge of the steps in the algorithm is considered to be sufficient and relevant for the curriculum and learners’ examination needs (mogari, 2014). in not regularly thinking about the concepts behind the algorithm, the teachers come to see the algorithm as the concept. brodie and sanni (2014) give similar examples in which nigerian teachers’ pedagogical content knowledge and knowledge of the curriculum constrained conceptual reflection on their content knowledge. the episode culminated with the facilitator sharing his own knowledge of the algorithm, a move that was necessitated by his realisation that the teachers could not explain the conceptual meaning of the algorithm as a whole. this move by the facilitator was necessary for purposes of building professional knowledge, an important aspect of learning in a professional learning community and a practice that was also evident in other conversations that we analysed. after the facilitator’s explanation, remarks by mandla and bongiwe, while not necessarily showing their own conceptual understanding of ratio, indicate that they saw the conversation as developing their knowledge of teaching the ratio algorithm differently. in terms of their mathematical knowledge for teaching, the episode was an opportunity for the teachers to deepen their understanding of the algorithm, and hence their pedagogical content knowledge. we acknowledge that there was no evidence to indicate that they had learned this knowledge, but two teachers’ expressions of their willingness to work with this knowledge in their teaching could be indications of new and shared understandings that could influence their teaching. discussion and conclusions in this article we have analysed the teachers’ participation as a practice of the community and the nature of the mathematical knowledge that they could have learned in the conversations of the community. these analyses enabled us to respond to both the research questions that we set in the article. the episodes presented above show how a focus on understanding learners’ errors in the topic ratio developed into conversations about the teachers’ mathematical knowledge for teaching (ball et al., 2008), and these foci constituted the joint enterprise for the community. mutual engagement was evident through the teachers’ responses to the facilitator’s questions that served to challenge and probe the teachers’ understandings as a way of pushing for the development of new understandings. in the episodes we presented there were opportunities for the teachers to: understand learners’ thinking behind errors, interpret, explain and justify learner errors, thus becoming more aware of the origins of learner errors, and identify learners’ learning needs, from a variety of evidence. these opportunities indicate the potential for professional learning communities to develop new and shared understandings, or a shared repertoire for the teachers. our findings show that having a clearly defined, shared and understood joint enterprise, and mutually engaging with others in seeking to develop new understandings, are significant to the creation of learning opportunities in professional learning communities. such learning opportunities have the potential to develop a shared repertoire that can contribute to teachers’ knowledge and practice. the three dimensions of the practice of a community of practice (wenger, 1998) presented in figure 1 enabled us to analyse and characterise the patterns of participation in these conversations. for us these forms of participation typified the community conversations as a developing practice in which learning opportunities are created and supported. our analyses of the conversation episodes presented in this article show that the conversations were opportunities for developing the teachers’ knowledge of learners’ reasoning in making errors and the reasonableness of learners’ errors, and their own conceptual understanding of ratio and the algorithm for dividing quantities in given ratios. in the conversations the teachers showed that they had an algorithmic understanding of ratio as evidenced by their responses to the facilitator’s questions, which is consistent with how the topic is normally taught in mathematics classrooms where teachers emphasise knowledge necessary for examinations, which in mathematics is mainly procedural fluency (mogari, 2014). teachers have a tendency to limit their understandings to what they are used to teaching at the expense of conceptual meanings of topics (brodie & sanni, 2014). although there is no evidence to suggest that the teachers did learn about the conceptual meanings of ratio and the ratio algorithm, the conversations were opportunities for the teachers to deepen their own knowledge of the topic through negotiation of meanings with others in the community (buysse et al., 2003). we therefore maintain our argument that an opportunity for learning the conceptual meaning of ratio was created and if this supports teachers to teach in a new way, it may support gradual development of their conceptual understandings. in short, the episodes were opportunities for deepening the teachers’ knowledge of the subject, knowledge of their learners, and ways of teaching the topic ratio. these are important aspects of teachers’ mathematical knowledge for teaching which can support productive teaching of the subject, and this indicates the potential for professional learning communities to support new understandings for teachers in knowledge that is important for their practice. the facilitator was key to the learning opportunities provided in the community. the conversations show instances where the facilitator-initiated conversations challenged teachers for deeper thinking or understanding, guided the focus of each conversation, and provided knowledge when necessary. for example, in the conversations, the link between learners’ learning needs on the topic of ratio and the teachers’ own knowledge was raised by the facilitator. the teachers were not able to work from their learners’ learning needs to identify and reflect on their own learning needs without a strong intervention from the facilitator. these interventions by the facilitator have been found to be fundamental for the effective functioning of professional learning communities and underscore the critical role of facilitators in such communities (katz, earl, jaafar, elgie, & foster, 2008). our arguments resonate with those of other researchers who have highlighted the power of professional learning communities to influence teacher learning. we have shown how conversations in professional learning communities can create learning opportunities for the participating teachers in significant aspects of their mathematical knowledge for teaching. the critical role of a facilitator in professional learning communities was shown by the way in which he initiated the teachers’ reflection on their own understandings, thereby creating opportunities for them to develop new meanings. we have also illustrated how learning in professional learning communities is supported by a joint enterprise, mutual engagement, and the development of a shared repertoire (wenger, 1998). we argue that this article contributes to knowledge of how learning opportunities can be created in professional learning 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(2012). how critical incidents analysis support chinese lesson study. international journal for lesson and learning studies, 1(1), 41–48. https://doi.org/10.1108/20468251211179696 ten years of democracy: translating policy into practice in mathematics and science education kgabo masehela human sciences research council email: kmasehela@hsrc.ac.za background this paper provides a 10-year (1994 – 2004) review of the state of mathematics and physical science education (sme) in south africa with respect to participation and performance, and its relationship with policy implementation. the framework for this paper is guided by two broad questions: • which key policy initiatives were conceptualised, developed and disseminated in the last ten years? • how has the system performed in respect to the participation and performance rate in mathematics and science education since 1994? through interviews, we also provide an overview of the voices of stakeholders (subject advisors, teachers and senior officials). the interviews were conducted in order to illuminate the challenges in translating policy to practice. any adjudication of government policies in respect to physical science, mathematics and technology education (smte) should be read within the assumption that within ten years (1994 –2004) there would have been less dramatic changes in the performance of what was one of the most divided education systems in the world. in this paper, the ten-year period is divided into two periods, 1994 –1999, and 1999 – 2004. the period between 1994 – 1999 is described by policy analysts as the policy formulation era. this is a period that was marked by the establishment of a unified, democratic and accountable system of government. the second period (1999 – 2004), described as the policy implementation era, is marked by dissemination of government policies. the era of policy design and development 1994 – 1999 south africa has produced formal policies and legislation as part of education reform. these include: white papers (doe 1995), national education policy act (doe, 1996a), south african schools act and national norms and standards for school funding (doe, 1996b), employment of educator act (rsa, 1998), language policy and admission policy (1996), as well as the human resource development strategy (dol, 2001b). the reconstruction and development programme (rdp) document laid a firm foundation for the design of all these policies. its goals with respect to sme were: • an appropriate mathematics, science and technology education that is essential to stem the waste of talent. • rewriting the schooling curriculum and raising the quality of learning and teaching through in-service education of teachers (inset) and upgrading the professional competence of teachers. following the rdp document, the white paper on education and training (doe, 1995) outlined ‘the most direct way’ of redressing the imbalances of the past around issues of sme. these papers indicated that special initiatives were needed to prepare learners for subjects in short supply and noted the following, • only one in five black learners choose physical science and mathematics in standard 8 (grade 10), and the trend of performance in the senior certificate examinations has been low overall, with a particularly dismal matriculation exemption rate among learners taking these subjects at higher grade. • fewer black learners with science and mathematics qualify for normal entry to higher education. south african policies aim to bring equity and redress in education. however, under south african contextual conditions and reality, to what extent could these policies be implemented? are there some doubts whether these policies are achieving their intended objectives? have these policies increased the participation and performance rate in mathematics and science? to what extend have they increased? this will be revisited later in the paper. pythagoras 61, june, 2005, pp. 21-30 21 ten years of democracy: translating policy into practice in mathematics and science education the era of policy implementation 1999 – 2004 curriculum innovations and development the introduction of curriculum 2005 in february 1997 marked a major school reform initiative in education. this new curriculum, based on a constructivist epistemology, initiates learners into what stenhouse (1976) calls ‘worthwhile activities’ ranging from hands-on activities, group discussions, project work, etc. while the conceptual design of c2005 seemed plausible, the implementation proved to be a challenging exercise. the first problem in implementation was the failure to estimate the extent to which resources would be a constraining factor. the second shortcoming was the impracticality of providing in-service training to all teachers in the general education and training (get) phase of schooling. a review of the curriculum acknowledged deficiencies in design and content. this resulted in a streamlined curriculum in terms of learning areas that also reduced use of complicated terminology. in a sense, the revised national curriculum statements (rncs) overcame the fundamental flaws of c2005. human resource development strategy initiatives in mathematics and science were initially set out by the rdp document and white paper on education and training (1995). south africa’s president reiterated the centrality of these subjects in two state of the nation addresses (2000 and 2001). following this, the ministers of labour and education jointly launched the human resource development (hrd) strategy for south africa in june 2001. a practical initiative coming from the strategy was the creation of the 102 dinaledi schools (‘creating tomorrow’s stars’), established as an attempt to increase the participation rate. in line with the redress agenda, resources were concentrated on a set of schools – rather than all the schools – that would become quality sites of science and mathematics teaching (kahn, 2003). the hrd strategy, which also has its origins in the rdp, is an attempt to operationalise president mbeki’s ideas made in his 2000 and 2001 state of the nation addresses. in terms of the hrd strategy indicator five: mathematics and science results (dol, 2001b: 25) ‘the problem in mathematics and science has not to do with numbers who passed on higher grade or who obtained university exemptions’. the strategy notes that ‘generally, mathematical and science literacy are extremely poor in the entire schooling system’. in order to arrest the chronic situation the former minister of education, prof. kader asmal, tasked his deputy minister, mr. mosibudi mangena, with a clear utilitarian prescription – amongst others – to increase the rate of participation of black learners, especially females. following the department of education’s strategy to improve mathematics and science, the deputy minister released the names of 102 dedicated mathematics and science high schools as part of a national strategy to improve mathematics and science in schools (bot and masehela, 2003). provincial departments were requested to draw their own strategic plans for mathematics, science and technology education for the next six years (2003 – 2008). the provincial plans had to include their vision, mission and commitment to restructuring get and further education and training (fet) in terms of professional development, provision of resources, new curriculum implementation, delivery, administrative and community support. all these had to be designed in line with the national hrd strategy for mathematics and science. the department of education earmarked more than r400-million for the promotion of science and mathematics (mega boost for maths and science, 2001). this financial injection allowed the project to supply schools with resources such as books, satellite television and laboratory equipment, and mathematics and science educators were offered special training sessions. an audit of the 102 dinaledi project schools indicates that from a total of 307 mathematics and science teachers, 188 (64.6%) have obtained a grade 12 qualification and 177 (56.7%) a diploma as a professional qualification. a few teachers have obtained honours (17, or 5.6%) and the eastern cape has the highest number of educators (5) with masters degrees. it is in the light of this background that we can see the growing demand being made on teachers and learners for significant change in the teaching and learning of mathematics, science and technology. participation and performance rate it is now critical to take stock of how these policies were translated into practice in terms of participation, performance and quality higher grade (hg) passes by examining trends since 1996. participation in this paper refers to learners enrolling for matric while performance refers to the quality of passes, particularly in hg. south africa has in the last years witnessed a drop in the number of grade 12 candidates. table 1 22 kgabo masehela provides a picture of the enrolment trends since 1996. table 1 shows a decrease in enrolment data between 1996 and 2003. although the number of candidates passing has been on the increase, it is worrying that the number of candidates enrolling for grade 12 has decreased significantly by approximately 80 000. according to van der berg (2004), to judge matriculation results in context we should not only look at those who write the matriculation examination, but also those who should have been in matric. if we consider the 18year-old or matric aged cohort in 2003, 985.000 learners should have written the matric examination. instead 440.267 sat for the examination and 322,492 passed. 58% of african candidates do not reach matric, and only 5.2% of african candidates achieved results that should have given them university entry. the decrease noted in the table is what kozol (1996) calls the ‘human wastage’ and this is a challenge to south africa. for example, 330.717 passed the examination from a total of 467 985 in 2004 and 137.268 learners failed. given this data, where do failing learners go? what is government doing to absorb these learners into formal employment? will these learners be allowed to repeat if most of them are not over age? will they enrol for adult basic education and training (abet) programmes, or will they undertake supplementary examinations? these issues require further investigation. table 2 shows that the total enrolment of mathematics and science learners increased between 1996 and 2004 from 218.225 to 276 094 in mathematics and from 122.278 to 161.214 in physical science. the total number of candidates who passed mathematics also increased from 108.910 in 1996 to 156.795 in 2004. similarly, physical science increased from 74.110 to 119.543. the increases in the matric pass rate between 2000 and 2004 has been linked to falling candidate numbers (cf. table 1) and weaker candidates or ‘at risk’ grade 11 learners being filtered to grade 12 therefore leaving a pool of generally stronger candidates, a greater proportion of learners with the potential to pass (de souza, 2003). of particular significance (cf. table 2) is the fact that between 1997 and 2004, candidates writing mathematics hg dropped from 68.451 to 39.939. similarly, physical science had a drop from 76.086 to 55.969. however, the overall pass rate increased as from 2000. in mathematics the pass rate rose from 45% in 2000 to 59% in 2003 while it rose from 69% to 80% in physical science. with respect to standard grade (sg), on the other hand, the number of candidates who registered for mathematics and science increased especially in 2000 (cf. table 3). table 1. enrolment trends and pass and failure rate 1996 – 2003. year candidates (n) % increase / decrease candidates passing (%) candidates passing with exemption (%) 1996 518 032 278 958 (53.8) 79 768 1997 556 246 7.4 261 400 (47.0) 69 007 1998 552 384 -0.4 273 118 (49.3) 69 891 1999 511 474 -7.7 249 831 (48.8) 63 715 2000 489 941 -4.2 283 294 (57.8) 68 626 2001 449 371 -8.3 277 206 (61.7) 67 707 2002 443 821 -1.2 305 774 (68.9) 75 048 2003 440 267 -0.8 322 492 (73.3) 82 010 2004 467 985 6.3 330 717 (70.7) 85 117 sources: doe 2004; edusource datanews 2001/2002 23 ten years of democracy: translating policy into practice in mathematics and science education table 3 indicates a decrease in the number of learners writing mathematics and science sg. as the number of learners writing mathematics increased from 149 510 in 1996 to 236 155 in 2004, the number of passes increased from 89.896 to 109.446. similarly, learners writing physical science increased from 52.252 to 105.245 and the passes increased from 30.306 (58%) to 78.025 (74%). due to structural changes brought about by formal economies changing, with less reliance on industries based on mining and agriculture to reliance on jobs in the financial sector, there is a need for candidates with quality higher grade passes in mathematics and science (national skills development strategy, 2001). the national strategy for science, mathematics and technology education and the subsequent establishment of the 102 focus schools attempts to increase the number of hg passes and eventually respond to the question of supply and demand in the economic sector. if the declining trend of learners writing mathematics and science is not arrested, how will this affect the south african economy? according to the national skills development strategy (dol, 2001a) high skilled jobs increased by 20% between 1970 and 1998. the united nations development programme (undp, 2001: 28) indicates that science and technology activities require a skilled workforce and human resources development is therefore important. the south african white paper on science and technology (dacst, 1996), on the other hand, indicates that any national system of innovation requires an table 2. mathematics and science participation and performance 1996 – 2004. mathematics hg year total enrolment total passed (sg/hg) % pass wrote hg males passed females passed total passed hg candidates 1996 218 225 108 910 49.9 65 223 12 817 9 599 22 416 1997 252 617 116 836 46.3 68 451 12829 9969 22 798 1998 298 195 124 005 41.6 63 899 11579 9581 21 160 1999 281 304 122 225 43.4 50 105 10660 9194 19 854 2000 284 017 128 142 45.1 38 520 10207 9120 19 327 2001 263 945 123 149 46.7 34 870 10084 9420 19 504 2002 260 989 146 446 56.1 35 465 10804 9724 20 528 2003 258 323 151 905 58.8 35 956 12564 10848 23 412 2004 276 094 156 795 56.8 39 939 13325 10818 24 143 physical science hg year total enrolment total passed (sg/hg) % pass wrote hg males passed females passed total passed hg candidates 1996 122 278 74 110 60.6 70 269 15 140 10 322 25 462 1997 141 278 91 538 64.8 76 086 15925 11 046 26 971 1998 168 632 108 896 64.6 79 019 16443 11 651 28 094 1999 160 949 102 896 63.9 66 486 13818 10 373 24 191 2000 163 185 112 164 68.7 55 699 13135 10 209 23 344 2001 153 847 105 552 68.6 48 996 13609 10 671 24 280 2002 153 855 117 529 76.4 50 992 13979 10 909 24 888 2003 151 791 121 947 80.3 52 080 14935 11 132 26 067 2004 161 214 119 543 74.2 55 969 15447 11 528 26 975 source: own calculations based on department of education 2004, sce database 24 kgabo masehela enabling framework for socio-economic development in the country. the figures as illustrated in the tables are a challenge to the attainment of the white paper’s enabling framework. this is because, despite substantial increments in school resources and a more equitable allocation of resources in mathematics and science, the overall output of candidates has not kept pace with the input. if the above policy pronouncements are not fulfilled, there is a need for structural reconfigurations. implementation of these reconfigurations and changes will also require urgent attention. performance by province table 4 provides provincial data in terms of participation and performance trends in mathematics and science between 1996 and 2004. with the exception of north west, all provinces experienced an increase in the number of candidates passing the matric examination. the table shows that the number of learners passing with endorsement has on average largely remained the same in many provinces. gauteng and limpopo experienced a significant increase in the number of endorsements between 1996 and 2004 and, on the contrary, eastern cape and north west experienced a decrease. gauteng, kwazulu natal, limpopo, mpumalanga and north west experienced a drop in the number of learners enrolling for mathematics hg. similarly, with the exception of free state, all provinces experienced an increase in learners enrolling for science hg. given the data in table 4, ‘what is the performance of african candidates in senior certificate examinations for mathematics and physical science?’ table 5 presents provincial results using the language proxy method (kahn, 2003). kahn (ibid.) provides data (table 5) for a number of african learners who passed with mathematics hg. these results respond to questions raised in parliament in 2000 namely, is it true that only 3 000 african learners obtain a mathematics hg pass? kahn (ibid.) provides an analysis of provincial results using what he terms table 3. mathematics and science hg/sg mathematics sg failed passed pass % pass over previous year 1996 149 510 89 896 59614 39.9% 1997 184 166 98 818 85348 46.3% 43.17% 1998 234 296 140 467 93829 40.1% 9.94% 1999 231 199 136 161 95038 41.1% 1.29% 2000 245 497 142 232 103265 42.1% 8.66% 2001 229 075 131 310 97765 42.7% -5.33% 2002 225 524 104 593 120931 53.6% 23.70% 2003 222 367 99 155 123212 55.4% 1.89% 2004 236 155 109 446 126709 53.7% 2.84% physical science hg sg failed passed pass % pass over previous year 1996 52 252 21 946 30306 58.0% 1997 65 192 18 610 46582 71.5% 53.71% 1998 89 613 27 188 62425 69.7% 34.01% 1999 94 463 32 095 62368 66.0% -0.09% 2000 107 486 31 605 75881 70.6% 21.67% 2001 104 851 34 753 70098 66.9% -7.62% 2002 102 863 24 244 78619 76.4% 12.16% 2003 99 711 17 768 81943 82.2% 4.23% 2004 105 245 27 220 78025 74.1% -4.78% source: own calculations based on department of education 2004, sce database 25 ten years of democracy: translating policy into practice in mathematics and science education 26 the language proxy method that he developed to identify african candidates taking mathematics and science. such candidates were identified only by their having taken an african language among their senior certificate examination subjects. there might have been african candidates who did not study an african language, but it was argued that they could not number more than 5%. following the proxy method, table 5 provides the disaggregated data by province in 2002. it may be observed from the table that limpopo had the highest number of learners taking mathematics and science compared with gauteng and western cape, for example, because proportionally limpopo has the highest african population of these provinces. the results presented here respond to the questions raised in parliament in 2000. performance by gender the education white paper 2 (1995) recommends that females take these subjects (mathematics and science), which are critical for the further development and growth of our nation. it proposes ‘to increase the number of girls in science streams’ through the equitable-school based funding formula. the period 1996 – 2002 has seen the female performance in mathematics improving substantially, with both the number of female candidates participating growing at a faster rate than male candidates and the gender gap in pass rates decreasing. the percentage of female candidates passing higher grade mathematics was higher in 2002 than that of male candidates. table 4: performance by province. dept yr candidates who wrote candidates who passed candidates who gained endorsement m cand m hg m hg pass s cand s hg cand s hg pass 1996 66809 32639 7061 24567 3118 1031 15402 6249 1552 2000 74563 37118 5332 42747 1440 1085 25989 2251 993 eastern cape 2004 63426 33915 5564 39958 2392 1535 23941 2431 1487 1996 35554 18153 4208 15050 2761 1165 9400 3939 1811 2000 29477 15538 3697 16888 1685 1109 10305 3799 2149 free state 2004 24731 19459 5480 12423 1768 1346 7937 2960 1681 1996 72959 42142 13810 38635 11355 5932 25298 13692 7010 2000 68202 46056 12896 44799 7332 5577 28964 8835 6180 gauteng 2004 71382 54808 15780 44821 9062 6710 28660 10852 6889 1996 86608 53397 20040 40832 16677 6978 19613 13261 6186 2000 96423 55128 15655 64075 11325 4709 32775 13208 5869 kwazulu/ natal 2004 110635 81830 20950 74932 9230 5356 39051 13516 6221 1996 126081 47569 9351 40800 18505 1517 20069 15942 2202 2000 95191 48886 11100 46651 8389 1452 24719 13592 2041 limpopo 2004 77774 54897 16273 39228 7647 2046 21733 13757 3337 1996 41731 19739 4332 15654 4534 1220 9947 5855 1194 2000 41115 21694 4762 21369 2446 894 13342 4866 1207 mpumalanga 2004 37091 22913 4640 19334 2186 1283 13159 3809 1436 1996 46349 32185 7611 18272 4466 1430 10727 6453 1982 2000 40098 23366 5057 22595 1880 1041 13220 4886 1200 north west 2004 37327 24221 4647 20822 1853 1314 12993 3255 1601 1996 7111 5194 1225 2606 484 279 1574 511 333 2000 7054 5019 892 2910 330 298 1705 354 256 n cape 2004 6723 5609 1259 2767 459 382 1616 505 386 1996 34830 17940 12130 18317 3323 2864 10491 3958 3192 2000 37818 30489 9235 21983 3693 3162 12168 3910 3449 western cape 2004 38896 33065 10524 22023 5093 4268 12124 4884 3937 source: own calculations from department of education 1996, 2001, 2004 reports kgabo masehela table 6 shows that the number of female candidates that enrol for mathematics is higher than those of males. in fact the number of females is much higher than the number of males in sg mathematics. however, as is shown in table 2, the male pass rate in both mathematics and physical science hg has become increasingly higher than that of females between 1996 and 2002. according to van der berg (2004) and perry (2003), 13.7% of females, as opposed to 10% of their male counterparts, failed the 2003 matric examinations. the above trend suggests that there are more females participating in mathematics and science, and whether these females come from african schools or former model c schools is a point that requires further research. it could be discerned from that the above data that if mathematics and science are about participation and performance, males will continue to have access to power, as they will enrol at tertiary institutions in fields such as engineering. and males will resultantly occupy positions and benefits that come with the subjects. table 5: performance by province (following the proxy method). math hg math sg phys sci hg phy sci sg proxy entry proxy entry proxy entry proxy entry 2002 172 246 3865 5189 207 285 1975 2782 wc 28 57 424 688 22 45 263 397 nc 380 493 9758 10137 1607 1736 4836 6393 fs 599 833 33131 34674 704 972 19608 20471 ec 4560 5029 45724 48159 6838 7332 19147 19670 kzn 861 999 15537 16445 2487 2703 8199 8757 mp 5546 5779 27670 28266 10784 11101 7744 8003 lp 1568 2451 17876 22972 2339 3498 9385 12276 gp 731 832 16686 17054 2212 2358 8017 8244 nw totals 14445 16719 170671 183584 27200 30030 79174 86993 source: kahn 2003 table 6: number of mathematics candidates and passes, average annual growth and pass rates by gender, 1996 and 2002. subject gender 1996 2002 average annual % pass pass growth rate rate 1996 2002 male 103056 122902 3.0% maths candidates female 111677 138087 3.6% male 48701 63299 4.5% 47.30% 51.50% maths passes female 42625 58518 5.4% 38.20% 42.40% male 34577 18867 -9.6% maths hg candidates female 30646 16598 -9.7% male 12817 10804 -2.8% 37.10% 57.30% maths hg passes female 9599 9724 0.2% 31.30% 58.60% male 5497 2831 -10.5% maths hg conversion sg female 3799 2156 -9.0% male 68479 104035 7.2% maths sg candidates female 81031 121489 7.0% male 30387 49664 8.5% 44.40% 47.70% maths sg passes female 29227 46638 8.1% 36.10% 38.40% source: edusource 2002 27 ten years of democracy: translating policy into practice in mathematics and science education translating policy to practice: the voices of teachers, subject advisors and senior officials the last section of this paper illuminates the impediments towards increasing the participation and performance rate in mathematics and science. subject advisors and district officials from gauteng were interviewed following meulenbergbuskens’ free attitude interview technique (1997). the technique involves asking one question to the interviewees in a focus group and then allowing respondents to unpack, debate, discuss, and (dis)agree on why south africa is unable to increase the participation rate of black, and in particular female students. as the interviewer, i merely asked clarifying questions to keep the conversation on track and at various phases of the interviews provided a reflective summary to focus their minds on the main question. telephonic interviews were also conducted with mathematics and science educators from free state and limpopo. the input from this group of educators was solicited because of their contact with the reality of the situation on the ground. five key themes emerged from an analysis of the interview data. these were: • resourcing – the environment at school level is not conducive to increasing the participation rate, and township schools especially lack resources. to this effect a typical response was ‘there is a lack of media centres, laboratories, and current, relevant books in schools’. • support of teachers at the classroom level – learning area specialists have to undertake regular classroom visits to support teachers in planning their lessons and teaching process. however, as data from the interview indicates, there is little support from learning area specialists at district level because ‘districts are understaffed, under qualified, or have people with no qualifications at all. • qualifications – learning area specialists and teachers (including primary school teachers) have to improve their qualifications in order to increase the performance rate in mathematics and science. district officials are increasingly faced with new developments, including the implementation of the revised national curriculum statement (rncs), and are therefore not able to deal with the professional demands at hand. • discouraging learners to continue with the subject – as prescribed by the mathematical literacy, mathematics and mathematical sciences (mlmms) requirements, teachers confirm that ‘learners cannot be encouraged to enrol for mathematics in grade 10, especially if they have failed (obtained less than 33.3%) grade 9’. secondly, teachers discourage learners from taking mathematics and science from grade 11 onwards if they have obtained less than 40% at hg in grade 10. ‘if learners obtain between 33.3% and 40%, they can still study at sg’. according to district officials ‘teachers are worried that a school’s performance will drop’ and eventually affect the ‘school image’. there is, therefore, a tight selection of learners, and this eventually contributes to learners’ repetition of the same grade, to eventually dropping out (cf. table 1). • intervention programmes – there are a number of mathematics and science intervention programmes that have been implemented, and as district officials indicate ‘an impact evaluation study is yet to take place’. for example, ‘saturday classes require learners to perform at a certain level, and lessons are often taught at a higher level’. there is, therefore, a mismatch between intervention programmes and the cognitive level of learners attending the programmes. furthermore, ‘there is no remedial programme in place to support learners in need of individual attention, especially learners in overcrowded classrooms’.1 conclusion this paper has outlined key policy and legislative initiatives designed and implemented during the two ministerial terms of 1994 – 1999 and 1999 – 2004. this paper has attempted to examine the extent to which these policies have been translated into practice with respect to increasing the participation and performance rate in mathematics and science. although the policy encourages learners to enrol for mathematics and science hg, 1 the recommended number of learners per class at secondary school is 35 (sasa, 1996), but many township classes have an average of 46. 28 kgabo masehela data indicates that the majority of learners enrol for these subjects at sg level. for example, 30.086 learners passed mathematics hg and 126.709 passed at sg in 2004. this reflects a number of issues, amongst them are: • a lack of confidence in the quality of mathematics and science education by both education administrators and policy makers. the system is not ready in terms of teacher qualifications and resources to enrol more learners in hg and learners are not confident of passing at this grade level. • misinterpretation of policies at school level by pushing more learners into sg. • policies have as yet not filtered down to classroom levels hence learners increasingly take these subjects at sg. although the government laid the groundwork for improvements in the education system in terms of policies, the number of learners passing mathematics and science higher grade has not increased. it could be deduced that results in the first ten years of democracy have not been gratifying. however, it may be too soon to make a judgment compared to what was produced before 1994. however, there is a need to set some shortterm goals by considering targets on year-to-year increments. the analysis appears to show that while a lot has been done in terms of policy designs and implementation, a lot more remains unachieved. however, an understanding is slowly but surely emerging that policies have been taken seriously and will yield better results in the near future. for now we may feel dissatisfied with the numbers but the context and where we come from should be important. following jansen and sayed’s (2001) argument, it is plausible to study these policies in the context of the government having laid the groundwork for long-term and sustainable improvements in education. finally, the dinaledi schools project presents just one case of government initiative in terms of operationalising government policies with respect to the hrd strategy. we recommend that these schools should not be pressurised to produce immediate results. in as much as it takes time for a plant to grow, it takes time for policies to produce immediate results. references bot, m. & masehela, k., 2003, “provincialisation of education, a review. july 2001 – december 2002”, edusource datanews 41, pp 11-37 carrim, n., 2001, “democratic participation and education reform”, in jansen, j. & sayed, y., eds., implementing education policies, the south african experience, cape town: university of cape town press dacst (department of arts, culture, science & technology), 1996, south african white paper on science and technology, pretoria doe (department of education), 1995, white paper on education and training, pretoria doe (department of education), 1996a, “national education policy act”, government gazette 370 doe (department of education), 1996b, “south african schools act”, government gazette 377 doe (department of education), 1997, “language policy and admission policy” government notice, 383 doe (department of education), 2001, national strategy for improving science, mathematics and technology education, pretoria dol (department of labour), 2001a, national skills development strategy, pretoria dol (department of labour), 2001b, human resource development strategy indicator five: mathematics and science results, pretoria de souza, c., 2003, “senior certificate exams 2002: plausible progress or passes below par?” edusource datanews 39, pp 1-13 jansen, j. & sayed, y., 2001, implementing education policies, the south african experience, cape town: uct press kahn, m., 2003, “for whom the school bell tolls: disparities in performance in senior certificate mathematics and physical science”, perspectives in education 30, 4, pp 149-156 kozol, j., 1996, savage inequalities. children in america’s schools. new york: harper perennial mega boost for maths and science, 2001, august 24, the star, p 7 meulenberg-buskens, i., 1997, “free attitude interview technique”, research for the future, pp 1-6. unpublished workshop manual, vista university perry, h., 2003, “female performance in the senior certificate examination: excellence hiding behind the averages”, edusource datanews 39, pp 14-27 29 ten years of democracy: translating policy into practice in mathematics and science education 30 stenhouse, l., 1976, an introduction to curriculum research and development, london: heineman undp (united nations development programme), 2001, united nations development programme van der berg, s., 2004, institute for justice and reconciliation economic transformation audit 2004: taking power in the economy. retrieved may 26, 2005 from url: http://www.capacitors.co.za/ta/ta/fullacademica rticles/fullacademicarticles/education.the%20c risis%20in%20schooling.van%20der%20berg. ijrta%20chapter%202%202004.pdf pointless quest a needle in a haystack may be difficult to find. your chance of ever finding one is small especially with haystacks of the ordinary kind, which don’t have any needles in at all. piet hein article information authors: pam lloyd1 vera frith1 affiliations: 1numeracy centre, academic development programme, university of cape town, south africa correspondence to: pam lloyd postal address: private bag x3, rondebosch 7701, south africa dates: received: 14 june 2013 accepted: 02 sept. 2013 published: 10 oct. 2013 how to cite this article: lloyd, p., & frith, v. (2013). proportional reasoning as a threshold to numeracy at university: a framework for analysis. pythagoras, 34(2), art. #234, 9 pages. http://dx.doi.org/10.4102/ pythagoras.v34i2.234 copyright notice: © 2013. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. proportional reasoning as a threshold to numeracy at university: a framework for analysis in this original research... open access • abstract • introduction    • the nature of quantitative literacy    • background to current research question    • the problematic concept • theoretical framework • method    • data collection    • preliminary observations    • creating the framework    • using the framework for analysis • results and discussion • conclusion • acknowledgements    • competing interest    • authors' contributions • references • footnote • appendix 1    • question 1    • question 2 abstract top ↑ there is a generally acknowledged need for students to be quantitatively literate in an increasingly quantitative world. this includes the ability to reason critically about data in context. we have noted that students experience difficulty with the application of certain mathematical and statistical concepts, which in turn impedes progress in the development of students’ critical reasoning ability. one such concept, which has the characteristics of a threshold concept, is that of proportional reasoning. the main focus of this article is a description of the development of a framework using an adapted phenomenographic approach that can be used to describe students’ experiences in the acquisition of the concept of comparing quantities in relative terms. the framework has also helped to make explicit the elements that constitute a full understanding of the requirements for the proportional comparison of quantities. preliminary results from using the framework to analyse students’ responses to assessment questions showed that many students were challenged by proportional reasoning. when considering the notion of the liminal space that is occupied en route to a full understanding of a threshold concept, about half of the students in the study were at the preliminal stage of understanding the concept and very few were at the threshold. introduction top ↑ it is commonly accepted that citizens need to be quantitatively literate (numerate) in order to participate effectively in an increasingly quantitative world (department of education, 2003; gal, 2005; jablonka, 2003; steen, 2001). at our university we provide several first-year interventions intended to assist students to develop appropriate quantitative literacy for their disciplines (frith, 2012). amongst these is a context-based semester course that caters mainly for students studying law and humanities. the curriculum uses contexts that have a social justice focus in which students are exposed to some of the social issues that are important in a society in transition and that we judge to be relevant to both law and humanities students (frith et al., 2010). an aim of the course is that students develop the ability to reason critically about quantitative data in such contexts. the focus of this article, which is part of a wider study, is on understanding how students experience the learning and application of a mathematical concept whose difficulty impedes progress in the development of this ability. the nature of quantitative literacy there are numerous definitions of quantitative literacy in the literature which emphasise various aspects of this complex concept. we adopt the following definition:quantitative literacy is the ability to manage situations or solve problems in practice, and involves responding to quantitative (mathematical and statistical) information that may be presented verbally, graphically, in tabular or symbolic form; it requires the activation of a range of enabling knowledge, behaviours and processes and it can be observed when it is expressed in the form of a communication, in written, oral or visual mode. (frith & prince, 2006, p. 30) the development of this definition was influenced by the definition of numerate behaviour underlying the assessment of numeracy in the adult literacy and lifeskills survey (gal, van groenestijn, manly, schmitt & tout, 2005) and the view of literacy and numeracy as social practice (prince & archer, 2008; street, 2005; street & baker, 2006). the idea that quantitative literacy is mainly concerned with mathematics and statistics used in context is fundamental to all its definitions. in our definition the statement ‘it requires the activation of a range of enabling knowledge, behaviours and processes’ refers to the full range of competencies necessary for quantitative literacy practice, including mathematical abilities, logical thinking and thinking critically about data in context. our definition also emphasises that responding appropriately to quantitative information in a text and communicating quantitative ideas are both essential components of numerate practice. responding appropriately to quantitative information includes the ability to think critically about the meaning of such information. most numeracy practitioners would agree that a disposition and ability to think critically about quantitative information is an essential aspect of numerate practice (best, 2008; jablonka, 2003; johnston & yasukawa, 2001; steen, 2001). for example, johnston (2007) defines numeracy as ‘a critical awareness that builds bridges between mathematics and the real world’ (p. 54) and for boersma and willard (2008) the term quantitative literacy refers to ‘the ability to reason critically with quantitative information’ (p. 1). the literature on critical thinking (e.g. facione, 2013) suggests that this critical awareness consists of mental abilities or cognitive skills, such as interpretation, analysis, inference, evaluation, explanation and self-regulation, and the disposition or habit of mind to use these mental abilities. according to facione (2013, p. 10), ‘[t]he ideal critical thinker can be characterised not merely by her or his cognitive skills but also by how she or he approaches life and living in general.’ writers about numeracy have also stressed the importance of this critical habit of mind as a necessary attribute of a numerate individual (dingman & madison, 2010; steen, 2001). the example presented in figure 1 illustrates the interplay of the three facets of quantitatively literate behaviour outlined above, namely the application of mathematical concepts in context, critical reasoning and communication. when presented with data about the take-up rates of the child support grant (csg) in the various provinces of south africa in 2005 and 2006, a spontaneous response by a quantitatively literate student could be the posing of the question: ‘which province was most successful in increasing the take-up rate of the csg from 2005 to 2006?’ figure 1: example of a data representation requiring critical interpretation. the process of answering this question involves reasoning about the size of the percentage change in each province by using information gained from the graphical representation and then communicating the result of this reasoning. the development of a framework to describe this ability in students is the focus of this article. background to current research question the initial aim of our research project was to determine to what extent we were effectively achieving the outcome that students are able to reason critically about data in context. however, once we began to consider our context-based learning materials, our classroom interactions and the students’ responses to assessment questions it became apparent that the students in these courses were hardly ever thinking critically in the way that we would like. reflection on why this was the case led to the realisation that too many of the students were still struggling at the level of understanding the mathematical concepts and techniques to allow for us to focus effectively on developing their critical awareness. most of the classroom time was used for students to learn the basic mathematical and statistical concepts, leaving little time for drawing on these concepts in the process of thinking more critically about the contexts. steen (2001) defines numeracy as: [a]n aggregate of skills, knowledge, beliefs, dispositions, habits of mind, communication capabilities, and problem solving skills that people need in order to engage effectively in quantitative situations arising in life and work. (p. 7) drawing on this definition, we argue that our students in fact still need to acquire the skills and knowledge before we can effectively facilitate the development of appropriate habits of mind, communication capabilities and problem-solving skills. it is argued, for example by bailin, case, coombs and daniels (1999), that the extent to which one is able to think critically in a certain context, even if one possesses the cognitive skills and disposition to do so, will be determined by one’s depth of knowledge and understanding of that context. in the case of quantitative literacy, this means that the critical awareness that is fundamental to numerate behaviour depends heavily on a sound knowledge and understanding of some basic mathematical and statistical concepts. we thus decided that it would be more productive to first focus our attention on how students attempt to make sense of the mathematical concepts that they find to be difficult, with the ultimate aim of establishing more effective ways of teaching them. the problematic concept through many years of teaching quantitative literacy courses we have repeatedly identified the concept of proportion and proportional reasoning as being problematic for first-year university students. consequently, the curriculum of the course recognises that fractions, percentages and proportions are concepts that require revisiting – in various forms, at intervals, and in different contexts, with attention being paid to appropriate language use. continual emphasis is placed on the use of the proportion concept in the comparison of quantities. students are regularly exposed to situations where comparing quantities in absolute terms will give a very different impression from that gained by expressing each quantity as a fraction of some relevant base value before making the comparison. for example, one country may spend vastly more on health care than another with a similar population size, but as a proportion of its gross domestic product it could be spending far less. we believe that being able to move beyond thinking about quantities in absolute terms is crucial to the development of critical reasoning: the understanding of the concept of comparing quantities in relative terms and being able to identify situations where it may be relevant to do so is a very important tool for thinking critically about quantitative information. ultimately we would like our students to spontaneously recognise when it would be appropriate to ask questions such as: yes, this quantity is bigger than the other, but is it bigger in relative terms?despite the fact that almost all of our students have a school background that includes 12 years of mathematics or nine years of mathematics and three years of mathematical literacy, we have found that, for many of our students, even a basic understanding of the proportion concept and its use is lacking. despite mediation during the course the problem is pervasive and persistent. in her work describing the challenges involved in understanding ratio by south african school learners, long (2009, p. 34) notes that the development of proficiency in concepts such as fraction, ratio and rate is not confined to the early grades but continues through high school and beyond. analysis of test items involving ratio, proportion and percentage revealed that, in general, grade 8 learners in south african schools do not have an operational understanding of these concepts and are unable to solve problems in which these concepts are embedded (long, 2009). this problem is not confined to south africa: research over a 30-year period by the rational number project (n.d.) in the united states of america (usa) has resulted in over 90 publications focusing on the theory and practice of the teaching and learning of fractions in schools. it has been found that the representation of ratios and proportions as percentages presents particular difficulties. a review of research conducted in schools in the usa into the inadequate performance with percentages revealed that ‘percent is a topic in which students have displayed inadequate performance, and in some cases utter confusion, for over 60 years’ (parker & leinhardt, 1995, p. 422). investigating the difficulty in the learning of percentage, these authors drew attention to the mathematical complexity of the concept and the different ways in which it is used – to indicate a part-whole relationship; as a number, fraction or decimal; as an operator; and as a statistic or function. they also highlighted that the language of percentage uses concise and elegant linguistic forms requiring attention to unstated relationships and that this language is often in conflict with everyday language. in addition, they suggested that percentage is poorly taught and there is evidence that many teachers are also confused about the concept (parker & leinhardt, 1995). one of the strategies that many of our students appear to have adopted in order to cope with difficult mathematical concepts is to apply learned rules and procedures without much attempt at understanding. parker and leinhardt (1995) identify a problem in the teaching of percentage by means of fixed procedures that leads to students approaching problems by attempting to manipulate given numbers by rule, rather than by reason. we have observed that most of our students, when explicitly asked to do so, are able to calculate fractions such as percentage change by means of a formula when given the relevant quantities. that is, they are able to produce the ‘ritual knowledge’ described as the ‘routine that we execute to get a particular result’ (perkins, 1999, cited in meyer & land, 2006, p. 10). however, the troublesome nature of proportion becomes apparent when students are asked to reason about change in a quantity as a percentage change, that is as a fraction, without doing calculations and then to describe in words, using appropriate language, their reasoning. behr, harel, post and lesh (1992) describe this kind of thinking as ‘qualitative reasoning’. we use the term ‘proportional comparison’ to describe the comparison of fractions that requires qualitative reasoning. mastery of proportional comparison can be demonstrated when a student can compare change in two quantities in relative rather than absolute terms without making use of a formula or numerical values and can then use appropriate language to describe this reasoning. typically, we have been interested in whether students are able to reason qualitatively about the effect that a change in the size of the numerator and/or denominator of a fraction has on the size of the fraction. an illustration of this is a question that we have asked students about the data for two of the provinces in the chart shown earlier in figure 1. since the data in the example is presented in graphical form it is possible, for example in the case of limpopo and north west, to decide, without doing any calculations, which province experienced the greater percentage increase in csg take-up rate. this is because both have experienced the same absolute change, but north west had a smaller take-up rate in 2005, resulting in a larger percentage increase for this province. in order to ascertain the extent to which students have developed the ability to reason with proportions we examined students’ answers to two assessment questions requiring the application of proportional comparison. in this article we focus on describing the development of the framework for elucidating the elements contributing to successful reasoning about proportional change. we then demonstrate the use of this framework to analyse student responses. theoretical framework top ↑ in developing our method of analysis we used the threshold concepts framework as described by meyer and land (2003) and a phenomenographic type of analysis (marton & booth, 1997).we find that the threshold concepts approach provides a useful guide for thinking about how students understand concepts that are the foundations of numerate practice in higher education. according to meyer and land (2003), [a] threshold concept can be considered as akin to a portal, opening up a new and previously inaccessible way of thinking about something. it represents a transformed way of understanding, or interpreting, or viewing something without which the learner cannot progress. as a consequence of comprehending a threshold concept there may thus be a transformed internal view of subject matter, subject landscape, or even world view. this transformation may be sudden or it may be protracted over a considerable period of time, with the transition to understanding proving troublesome. such a transformed view or landscape may represent how people ‘think’ in a particular discipline, or how they perceive, apprehend, or experience particular phenomena within that discipline (or more generally). (p. 1) these concepts are characterised as, amongst others, likely to be troublesome for the student; transformative, leading to a shift in perception and use of language; irreversible in that the change in perspective is unlikely to be forgotten; and integrative, in the way they link to other concepts in the discipline. so threshold concepts can be seen as gateways to thinking and practising and communicating authentically in a particular discourse. the time taken for the process of internalising a threshold concept and thus effecting a transition from one way of thinking to another will vary depending on how troublesome the concept is for a student. during this time of transition a student may experience uncertainty and a sense of being stuck between a limited, superficial understanding of the concept and a full understanding; a student may well oscillate between stages of understanding. this conceptual space that is occupied by a learner is described by meyer and land as the ‘liminal space’: difficulty in understanding threshold concepts may leave the learner in a state of liminality (latin limen – ‘threshold’) a suspended state in which understanding approximates to a kind of mimicry or lack of authenticity. (meyer & land, 2003, p. 10) prior to the emergence of the threshold concepts theory, research in school mathematics education identified proportional reasoning as being both difficult for many students and as taking a long time to develop (behr et al., 1992; cramer & post, 1993; karplus, pulos & stage, 1983). more recently it has been recognised that fractions, ratios and proportions are amongst the most challenging mathematical concepts to learn and to teach (lamon, 2007) and long (2011, p. 211) asserts that there is ‘agreement generally that proportional reasoning is a threshold concept’. given that many students entering higher education have not yet gained this threshold in school mathematics, we take the view that proportional reasoning is one of many mathematical and statistical concepts that often deny students access to numerate ‘ways of thinking and practising’ (meyer & land, 2003, p. 12) in academic disciplines. this view is supported by the work of researchers in the sciences who have identified proportional reasoning as a threshold concept in their discipline. ross et al. (2010) hypothesise that there is a ‘web of threshold concepts’ or ‘epistemes’, including proportional reasoning, that underlie difficult (in this case, biological) concepts. method top ↑ data collection data was obtained from the written assessments of students in the law course in the form of their responses to two assessment questions that required them to reason about percentage change. permission for use of student work was obtained from students via a signed consent form in accordance with the requirements of the research ethics committee of the centre for higher education development at our university. students were informed that consent was voluntary and that anonymity would be respected. the work of students whose consent was not given was not used in this study.the data was collected in two batches: we recorded each student’s written response to a particular question in each of two assessments that were six weeks apart. students regularly encounter contexts throughout the course where they are required to work with percentage change in various ways. students continued to be exposed to contexts involving percentage change between the two assessments; however, we made no specific changes to the curriculum as a result of their performance in the first assessment. the intention of the questions was to assess students’ ability to reason proportionally about data in previously unseen situations. the data in the assessment questions was presented graphically and students were instructed not to do any calculations but to explain their reasoning in a written response. in each case the reasoning required the comparison of the sizes of two fractions, the relative sizes of the fractions being apparent from the geometry of the graphical representation. students had not previously discussed similar questions in which the data was presented graphically. see appendix 1 for the actual questions. question 1 was set in the context of children’s rights, specifically dealing with csgs in south africa and the take-up rates of csgs in the different provinces. the chart used in this question is that shown in figure 1. although students had spent about five hours in class working with quantitative issues that arise in allocating resources for social services for children, including csgs, the issue of take-up rate had not previously been encountered in classroom materials. however, in other contexts students had calculated percentage changes in quantities that were themselves percentages and been introduced to the difference between percentage change and change in percentage points. in question 1, students were required to compare the percentage change in take-up rate for two specific provinces, north west and limpopo. careful observation of the graphs would show that the absolute change for both provinces was essentially the same, but for one province the percentage change was clearly calculated off a lower base. question 2 was part of a later assessment at the end of the course. the context here was the real disposable income of south africans and how this has changed over time. the data was again shown graphically, this time in the form of four time series, one for each of the different race groups1. the question itself required the consideration of the real disposable incomes of two of the groups, and how these had changed from 1960 to 2005. students had been exposed to the concept of real income but had not previously compared relative changes in real income. as before, the absolute change for each group was approximately the same, but one clearly came off a lower base. preliminary observations from a preliminary examination of student responses it was clear that most students had not reasoned about percentage change in the way we had expected. many students compared the absolute changes instead of the relative changes and based their conclusion on the former comparison. others based their argument on the comparison of only the absolute sizes of the quantities (without even considering the absolute change). in order to reason correctly about the relative changes, both of these elements must be taken into appropriate consideration. however, we observed that many of the students were focusing on only one or two elements of the necessary evidence and then jumping to a conclusion. they would also frequently not link the elements they identified in a logical way, and many did not use appropriate quantitative language to describe their observations.we also realised that in both questions there was a large number of students who were distracted by the context in which similar questions had previously been experienced. for example, in question 1 many students used arguments based on not knowing the population sizes of the different provinces. we suggest that the reason for this is that when they had previously been exposed to examples where they had to reason about proportions it was in situations where the quantities under consideration were percentages of a population. for example: in a tutorial, students studied a table in which the percentage of households with television was higher in mpumalanga than in kwazulu-natal. they were then required to discuss the following question: ‘does this mean that mpumalanga has more households with tv than does kwazulu-natal? explain.’ in the later question many students were distracted by the nature of the data representation. much of their previous exposure to the interpretation of line charts involved making observations about trends and rates of change, and so it seems that many of them assumed that this kind of thinking was required in order to answer this question. creating the framework we required a more precise description of students’ understanding of proportional comparison. the framework for identifying and describing the elements that are required when reasoning about proportional comparison emerged from further examination of the data and applications of phenomenography suggested by the literature.the phenomenographic method of analysis (marton & booth, 1997) enables the description of the variety of ways in which people experience a phenomenon. this is usually achieved by means of open-ended interviews in which students reflect on their understanding of the phenomenon. the transcribed interviews are studied with the aim of categorising the descriptions of experiences in a way that will provide a limited, but complete, set of categories that covers all the variations. an iterative process is then embarked on where the interview data is re-examined using the categories and the categories are modified to ensure consistency with the data. the process continues until stability is achieved. the categories can often be arranged hierarchically, and a participant’s experience may be described by one or more categories. we adapted this approach by examining the written responses to assessment questions with the view to observing and categorising the ways in which students experienced proportional comparison. the variation in the ways of understanding the concept was described by an iterative process in which we read the students’ responses and defined categories describing the different ‘objects of focal awareness’ (marton & booth, 1997, p. 112) as experienced by students in their written responses, refined these categories and then coded students’ responses according to these categories. this process is described below. for the 36 responses to question 1 and the 32 responses to question 2, we independently identified preliminary descriptive categories and then coded student responses according to these categories. on comparing our categories we found that they were in effect the same and that our initial codings using our respective preliminary categories were very similar. at this stage it became apparent that the main categories were essentially descriptions of the elements required in the reasoning towards a correct answer: a consideration of the size of the absolute change, the size of the base from which the absolute change occurred, then a comparison of sizes of the resulting fractions. these insights informed the development of new category descriptions that were general enough to be used to code the responses to both questions. this resulted in the framework outlined in table 1. using the framework and the coding key shown in the table we then independently re-coded the student answers. agreement on a final coding was reached after further discussion. the stability of the framework was confirmed when, after a few months’ break from the data and the framework, the coding exercise was repeated and only a few changes were made. table 1: coding key for question 1 and question 2. using the framework for analysis codes a, b, c and d are elements of a correct answer. responses whose reasoning exhibited all four elements as well as the correct conclusion that the fraction with the smaller base will yield the larger percentage were coded only as g. responses that showed incomplete reasoning were given one or more codes identifying the particular elements. in addition, responses of students who were distracted by the context in which they had previously been asked similar questions were coded as f and those whose reasoning was based on an incorrect understanding of fractions were coded as e. those responses that we found incomprehensible or unable to classify were coded only as h.we then further classified students’ responses in relation to the liminal space. land, meyer and baillie (2010) describe the variation in the understanding of threshold concepts in terms of a journey through the preliminal, liminal and postliminal stages of conceptual development. according to our categories, codes a, b and c refer to answers in which students did not recognise that proportions were involved and referred only to absolute quantities or absolute changes. when viewed using the threshold concepts framework, these responses indicate that students do not appear to have engaged at all with the concept of proportional comparison and their understanding of the concept can be regarded as being at the preliminal stage. codes d, e and f for question 1 (and d and e for question 2), sometimes in combination with codes a, b and c, represent understandings that are at various stages within the liminal space and g represents an answer indicating that, at least in terms of the specific context of the question, the student has demonstrated mastery of the concept and its associated language. in this case their understanding can be considered to be at the threshold or, possibly, postliminal. results and discussion top ↑ in this section we present the results of an analysis, largely in order to illustrate the usefulness of the framework. for this reason we do not discuss the implications of the results in great depth. a summary of the coding of the responses to the questions is shown in table 2. the total of the percentages is not 100%, as many responses had multiple codes. table 2: results of coding of responses to question 1 (n = 36) and question 2 (n = 32). in question 1, only 11% of students, and in question 2 only 3%, correctly reasoned that it is the data category with the lowest initial value that has the greatest percentage change, given that in both cases the absolute changes were roughly equal. these responses indicate students being ‘at the threshold’. table 3 shows a summary of the classification of responses as preliminal, liminal and ‘at the threshold’. table 3: classification of responses to question 1 (n = 36) and question 2 (n = 32). more than one third of the students (42% for question 1 and 34% for question 2) considered only the size of the absolute changes or did not consider change at all (and compared the size of the absolute quantities) and so did not recognise the need to compare fractions. we suggest that this demonstrates an understanding that is in the preliminal stage: the concept of proportional comparison did not ‘come into view’ (meyer & land, 2005, p. 384). in particular, in question 1, one quarter of the students argued that neither province had the greater change as the absolute change was the same for the two provinces (see table 2). on the other hand, in question 2, none of the students recognised that the absolute changes were essentially the same. also, for this question we included in the preliminal classification those answers coded f, responses that indicated a focus on irrelevant trends shown by the graphs. more than one third of the students’ responses in question 1, and only one quarter in question 2, showed an understanding that fractions needed to be considered. some of these responses also noted the size of the absolute change but did not follow through with the correct argument. these understandings were considered to be in the liminal stage. in the case of question 1, one quarter of the students were distracted by the context and, instead of arguing using the data presented in the graph, tried to argue that the population size of the province would influence the percentage increase in take-up rate. as previously noted, the context of question 2 appears to have been even more distracting, with half of the students including in their arguments irrelevant observations about the trends shown in the chart. in this more distracting question some students, who were initially considered to be at the threshold, appear to have returned to the liminal stage, illustrating the observation of land et al. (2010, p. xi) that ‘the acquisition of threshold concepts often involves a degree of recursiveness’, that is, a student’s journey through the liminal space is not necessarily direct. we observed that not all students for whom understanding in question 1 seemed to be at the threshold were (irreversibly) over the threshold and could therefore not be regarded as being in the postliminal stage. conclusion top ↑ one of the attributes of a quantitatively literate person is the ability to think critically about data in context. the outcomes of a quantitative literacy intervention at tertiary level that seeks to promote access to ways of thinking and practising in academic disciplines would include this ability. however, this particular outcome is unattainable if students are unable first to cross certain conceptual thresholds, one of which is that of proportional reasoning. the development and application of a framework for the detailed analysis of student understanding of proportional comparison has enabled a systematic description of the range of student experiences in the process of gaining a full understanding of the concept. the development of this framework has made explicit the elements required when reasoning about proportions. it has also enabled the classification of student responses in relation to the liminal space that is occupied en route to a full understanding of proportional comparison. our study has shown that, throughout the course, and despite repeated exposure to the concept, most of the students retained an incomplete understanding of proportional comparison and almost half of the students remained in the preliminal stage. the study confirms the tenaciously troublesome nature of the concept and has implications for our teaching practice and the curriculum. these issues are to be the focus of future research. acknowledgements top ↑ we wish to thank kate le roux and viki janse van rensburg for reading drafts of this manuscript and suggesting useful improvements. competing interest we declare that we have no financial or personal relationships that may have inappropriately influenced us in writing this article. authors’ contributions p.l. and v.f. 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(2010). threshold concepts: challenging the way we think, teach and learn in biology. in j.h.f meyer, r. land, & c. baillie (eds.), threshold concepts and transformational learning (pp. 165–177). rotterdam: sense publishers. pmcid:pmc3198851 south african institute of race relations. (2009). south africa survey online 2009/2010: employment and incomes. johannesburg: south african institute of race relations. available from http://www.sairr.org.za/services/publications/south-africa-survey/south-africa-survey-online-2009-2010/downloads/employment.pdf/ steen, l.a. (2001). the case for quantitative literacy. in l.a. steen (ed.), mathematics and democracy: the case for quantitative literacy (pp. 1–22). princeton, nj: national council on education and the disciplines and woodrow wilson foundation. available from http://www.maa.org/sites/default/files/pdf/ql/mathanddemocracy.pdf street, b. (2005). applying new literacy studies to numeracy as social practice. in a rogers (ed.), urban literacy. communication, identity and learning in development contexts (pp. 87–96). hamburg: unesco institute for education. street, b., & baker, d. (2006). so, what about multimodal numeracies? in k. pahl, & j. rowsell (eds.), travel notes from the new literacy studies (pp. 219–233). clevedon: multilingual matters. footnote top ↑ 1. race classification in south africa is used as a social (rather than a biological) construct in the measurement of success (or otherwise) of transformation in society. appendix 1 top ↑ question 1 the child support grant (csg) is a cash grant to the value of r190 per month per child as of april 2006. children are eligible for this grant if their primary caregiver and his or her spouse jointly have r800 per month or less in income and live in an urban area and formal house. those who live in rural areas or informal housing in urban areas must earn r1100 per month or less to qualify for this grant. in july 2006, the csg went to over 7.4 million eligible children aged 0–13 years. using the general household survey 2004, budlender calculated that some 8.8 million children are eligible for the csg. this is 65% of all children under the age of 14 years. using this eligibility rate, it is estimated that 84% of all eligible children access (or take up) csgs across the country in 2006. there is substantial evidence that grants, including the csg, are being spent on food, education and basic goods and services. consider the take-up rates for limpopo and north west provinces, as shown in the graph. without doing any calculations, say which province experienced the larger percentage increase from 2005 to 2006 in csg take-up rate. explain your reasoning. question 2 the data in the chart below is taken from the south africa survey, 2009/2010 (south african institute of race relations, 2009). answer the questions about this chart below. (disposable income refers to earnings after taxes have been paid.) consider the real disposable income per person for indians and whites. without doing any calculations, say which race group experienced the greater percentage increase in real income per person from 1960 to 2005. explain your answer. etten & smit 48 pythagoras 62, december, 2005, pp. 48-58 learning material in compliance with the revised national curriculum statement: a dilemma bertus van etten and kosie smit fontys university, the netherlands, and stellenbosch university email: n.van.etten@concepts.nl and jhs@sun.ac.za this paper reports on the challenges confronting a developer of mathematics learning material according to the revised national curriculum statement of south africa. the material development is based on the approach to learning embodied in realistic mathematics education in the netherlands. it is a real challenge to realise the learning outcomes and assessment standards in the south african curriculum and simultaneously to explore learners’ informal strategies to come up with their ‘own’ solutions to practical problems and thus to experience mathematics as real, relevant and enjoyable. introduction when a new curriculum is implemented, a clearly spelled out theory of learning should underpin the process. in spelling out the intended outcomes of the revised national curriculum statement (rncs), grades r – 9 (department of education, 2002), the creators of the new curriculum did in broad terms give a perspective on the theory of learning they endorse. this includes a statement such as: “the outcomes encourage a learnercentred and activity-based approach to education” (department of education, 2002: 1). understandably, a generic document such as this one cannot be expected to present a detailed theory of learning for a specific subject such as mathematics. however, as a result, there are considerable differences of approach among the people who have to concretise this curriculum on the ground – the government officials, material developers, textbook editors, teacher trainers, teachers, etc. in our opinion, realistic mathematics education, which has been developed in the netherlands over the past 35 years, offers an approach to the learning of mathematics that enhances the realisation of the rncs. this approach entails, among other things, a process of “guided reinvention” and “progressive mathematisation” (freudenthal, 1973; gravemeijer, 1994; 1998). drijvers (2003: 52) describes this as follows: according to the reinvention principle, students should be given the opportunity to experience a process similar to that by which a given mathematical topic was invented. thus a route has to be designed that allows the students to develop ‘their own’ mathematics. this process, however, needs guidance from the teacher, to help to further develop sensible directions, to leave ‘dead-end streets’ and to ascertain convergence towards the common standards within the mathematics community. the point of departure for this process is the informal strategies that students come up with that gradually develop into more formalised methods. as far as the selection of subject matter is concerned, gravemeijer explains that “the subject matter that is to be mathematised should be experientially real for the students” (gravemeijer, 1998: 277). “these may be everyday situations, but they may also be fantasy worlds in which the students can immerse themselves. and – last but not least – it may be mathematics that is experientially real” (gravemeijer, 1998: 287); “the core principle is that mathematics can and should be learned on one’s own authority and through one’s own mental activities” (gravemeijer, 1998: 277). only at the end of the mathematising process is a concept/definition/result refined to a more formal format, usually with guidance of the teacher. on the basis of this approach the authors have over the past few years been experimenting with learning material for the senior phase. we have chosen realistic mathematics education as our vehicle to meet the challenges in the mathematics classroom. the brave maths mouse project the first author has since 2004 spent two periods of three months each in project schools of the socalled brave maths mouse project to observe the activities in the senior phase mathematics classrooms of a few historically disadvantaged bertus van etten and kosie smit 49 schools. these observations have significantly influenced the opinions expressed in this paper. features of the schools, teachers and learners when the project was initiated, are summarised below. • the language of instruction is the mother tongue of the learners. • the level of numeracy and literacy of a large percentage of learners is well below the level accepted in the curriculum. many learners entering the senior phase have not achieved the outcomes of the intermediate phase. • the differences occurring in a class can be considerable. certain learners in grade 7 cannot read. in the same class one finds a learner who does not know what the answer to 18 divided by 6 is, while another is proficient in the use of tables, can read fluently and measure and construct accurately. • the teachers in general follow one strategy of instruction, viz. to show the learners a mathematical recipe and to let them practise it. the teacher does the talking, while the learners have little opportunity to discuss the material. • the learners are accustomed to waiting for instructions. they are unfamiliar with the idea of taking responsibility for their own learning. • learners do not have their own textbooks. much time is lost in copying from the black board. • there is a shortage of set squares, compasses and pocket calculators in the class. the teacher also does not have drawing instruments for the black board. • domestic circumstances often prevent learners from doing homework. our objective has been to develop learning material for grades 7 to 9 in order to realise the mathematics objectives of the curriculum that are relevant for these learners. our strategy with the learner modules is that each learner receives her/his own copy and has to interact with the text, individually as well as in a group context. the teachers receive guidance on their role as facilitators of the learning process and as guides towards the acquisition of more structured mathematical expertise. on the basis of our experience we try in this paper to illustrate the difficulties we have encountered in choosing learning material that complies with the curriculum and links with the level of knowledge of these learners and teachers. we present examples of learner material to illustrate and motivate our point of view. learning outcome 2: patterns, functions and algebra the curriculum specifies: in the senior phase the focus of learning outcome 2 is to formalise the rules generating patterns. the learner should continue to: • investigate numerical and geometric patterns to establish the relationships between variables; and • express rules governing patterns in algebraic language or symbols. (department of education, 2002: 63) one of the corresponding assessment standards for grade 9 specifies: we know this when the learner constructs mathematical models that represent, describe and provide solutions to problem situations, showing responsibility towards the environment and the health of others (including problems within human rights, social, economic, cultural and environmental contexts). (department of education, 2002: 74) for the author of a textbook it is not a straightforward task to translate this learning outcome and the accompanying assessment standards into concrete learning material. the choice of a context is a delicate matter that requires careful reflection (sethole, 2004). whether a certain context functions successfully within a given situation depends on a multitude of factors. while topics can be suggested as potential sources of meaningful problems (such as in the abovementioned assessment standard), the specific mathematics that has to be mastered should always be the first priority. we also have serious concerns regarding the speed at which learners are expected to transfer from informal to formal procedures. in this regard drijvers (2003: 42) emphasises that “the transition from informal to formal algebra is a level jump that students find difficult to make. the lack of time spent on the informal phase and on the students’ schematisation is responsible for this. too soon strategies are shortened, automatised and condensed into compact algebraic forms.” referring to the experience in the netherlands, drijvers (2003: 57) continues: “much attention is paid to the exploration of realistic situations at the referential level, to the process of horizontal mathematisation, to the translation of the problem situation into mathematics and to the development of informal problem-solving strategies. variables, learning material in compliance with the revised national curriculum statement: a dilemma 50 for example, are often labelled with words that have a direct relationship with the context from which they are taken.” example 1: a choice between cell phones the problem of which cell phone company to choose is presented in table 1. it seems as if this context works satisfactorily in a grade 9 class. there is discussion about seconds and minutes and about 60 cents and r0.60. what is the meaning of a basic monthly fee? with 100 minutes of airtime easyphone is cheaper, while telebel is cheaper with 500 minutes of airtime. somewhere in between there must be a switching point. a table and a graph are produced to investigate this further (table 2 and figure 1). this enables jolene to advise her dad. this problem has a context which is very appropriate for the specific learners. • the context fits into their field of experience. they are familiar with the concepts cell phone, basic monthly fee and airtime minutes. • for these learners the problem is challenging. they are determined to find a solution. • the mathematics involved in this context is relevant. • the problem fits into their learning process. the learners communicate in the language of the context (table 3). there is discussion about essential mathematical issues, such as: • the subdivision of the number lines along the horizontal and vertical axes; • whether a straight line is the correct graph; • how accurately the point of intersection can be read from the graph; • how the formula can be used to find the exact solution. it is a real challenge to find the correct equation and to solve it. one line of argumentation is: if 0.9 x airtime min = 75 + 0.6 x airtime min, then r75 is sufficient for 0.3 airtime minutes. hence 0.3 x airtime min = 75 and hence 3 x airtime min = 750. the sought-after number of airtime minutes is a third of 750. hence for 250 minutes of airtime the cost is r225 for both companies. here we do not have the routine execution of an algorithm, but rather the insightful manipulation of a formula. the learner understands throughout what it is all about. a number of examples of this kind gradually familiarise the learner with the essential matters relating to linear functions: the form of the formula and the role of the coefficients for the gradient and the y-intercept. context-rich mathematics for a developer of learner material in the spirit of telebel no. of minutes 0 100 200 300 500 800 1000 cost (in rands) 75 135 195 255 375 555 675 easyphone no. of minutes 0 100 200 300 500 800 1000 cost (in rands) 0 90 180 270 450 720 900 table 2. comparative table (linked to figure 1) jolene’s dad wants to buy a cell phone. she considers the catalogues of the companies telebel and easyphone. telebel easyphone basic monthly fee: r75 tariff: r3 per 5 minutes basic monthly fee: none tariff: 1.5 cents per second what advice can jolene give her dad on the choice of a cell phone? table 1. choice between cell phones bertus van etten and kosie smit 51 realistic mathematics education, a context is not simply attractive background decoration used to illustrate mathematics. a context is purposefully chosen to teach learners how to use mathematics. the use of mathematical subject terminology is (initially) avoided. the learner is allowed to think and speak in terms of notions within the context. the exact switching point can be read from the graph, or calculated by means of an equation. telebel: cost = 75 + 0.6 x airtime min easyphone: cost = 0.9 x airtime min for the turning point we have: 0.9 x airtime min = 75 + 0.6 x airtime min 0.3 x airtime min = 75 airtime min = 250 cost = 225 figure 1. comparative graph (linked to table 2) language of the context mathematical language airtime minutes on horizontal axis x-axis cost on vertical axis y-axis price per airtime minute gradient basic monthly fee y-intercept for 250 airtime minutes the two companies’ tariffs are the same, viz. r225 read the coordinate of a point of intersection from a graph or solve an equation. for more than 250 airtime minutes telebel is the cheapest. solving an inequality table 3. language of the context versus mathematical language learning material in compliance with the revised national curriculum statement: a dilemma 52 example 2: interest rates a formula to save money with 10% interest can be written as: capitaloldcapitaloldcapitalnew ×+= 1.0 it is not difficult to convince learners that this can be rewritten as: capitaloldcapitalnew ×= 1.1 with examples of this kind there is no need to use terminology such as (common) terms, variable, coefficient and distributive law. by using words/expressions to denote variables and by avoiding the convention of omitting the multiplication sign, we make it easier for the learner to understand the formula. in this way we gradually prepare learners in grades 7 and 8 to achieve the following assessment standard: we know this when the learner interprets and uses the following basic algebraic vocabulary in context: term, expression, coefficient, exponent (or index), base, constant, variable, equation, formula (or rule). (department of education, 2002: 79) in grade 9 mathematical sentences such as the following can then be comprehended: for all numbers a we have: a2a3a5 ×=×−× ! for which number a do we have: 18a312a5 +×=+× ? a clear distinction is made between a rewriting rule and an equation. learning outcome 1: numbers, operations and relationships there are learning outcomes that originate from the mathematical needs of earlier generations, but that are less relevant for the workaday life of someone who lives in the 21st century. calculation with fractions is such a topic and it creates problems for the learners. the curriculum for the senior phase prescribes calculations with fractions as follows: the range of numbers developed by the end of grade 9 is the set of rational numbers. in this phase, the learner should: • be given ample opportunity to solve a variety of problems, using an increased range of numbers and the ability to perform multiple operations correctly and fluently. (department of education, 2002: 62) the assessment standards for grade 7 specify, among other things: we know this assessment standard has been met when the learner estimates and calculates by selecting and using operations appropriate to solving problems that involve: addition, subtraction and multiplication of common fractions. (department of education, 2002: 70) in this regard moss (2005: 309) remarks: “in fact, the rational-number system poses problems not only for youngsters, but for many adults as well.” in everyday life calculations with fractions have become outdated. calculations with fractions are done in decimal form. in our opinion learners need not be troubled with calculations such as: 6 5 6 2 6 3 3 1 2 1 =+=+ we prefer to emphasise the concept that: a fraction is a number. in this regard it is then important to let learners reflect on phrases such as: you pay half of one half. we give 25% discount. take two, pay only for one. if you buy for r1.50, you only pay r1.00. take three, pay only for two. the important ideas are the fraction as a number, its position on the number line and the equivalent decimal notation. calculations with fractions are restricted to calculations with the equivalent decimal numbers. this is also relevant in everyday life. much attention should also be paid to the concept of a fraction as a ratio. learners can solve many practical problems by making use of a socalled ratio table. in a ratio table each column has to be an equivalent ratio. learners should be familiar with the fact that they can create new columns by multiplying a column with a constant or by adding two columns. they are allowed to use as many columns as they like. for example: a car uses 8 litres of fuel for every 100 kilometres. how far can one drive with a full tank of 60 litres? a calculation using a ratio table can be seen in table 4. the conclusion is that one can drive 750 km with a full tank of fuel. another example: 60 learners have written a mathematics test. 70% of them have passed. how many learners have passed? one version of a calculation using a ratio table is bertus van etten and kosie smit 53 shown in table 5. the conclusion is that 42 learners have passed. consider another example (for which a ratio table is presented in table 6): the floor plan of a school is drawn with a scale of 1 : 200. a passage is 12 cm long on the floor plan. what is the length of the passage in reality? negative numbers in school mathematics what is true for fractions is also true for negative numbers: calculations with negative numbers never occur in everyday life. however, learners encounter formulae and the accompanying graphs where negative numbers occur, and therefore they should be familiar with calculations using negative numbers. the assessment standards for learning outcome 1, grade 7 specify, among other things: we know this assessment standard has been met when the learner estimates and calculates by selecting and using operations appropriate to solving problems that involve multiple operations with integers. (department of education, 2002: 70) there is no realistic context for calculations with negative numbers. debt and temperature can be written as negative numbers. the cancelling of debt can then serve as an illustration of subtraction of a negative number. however, multiplication of negative numbers cannot be illustrated with debt as a context. minus five degrees times minus three degrees also does not make sense. example 3: a fictitious context of a witch the following idea of a fictitious context where negative numbers are applied is developed from a hint during an informal conversation with freudenthal in approximately 1975. it uses the context of a witch and goes a long way towards meeting the objections against the use of negative numbers. the witch has a cauldron with hot (positive) and cold (negative) blocks. the witch adds blocks to the cauldron (addition) and removes blocks from the cauldron (subtraction). learners become aware of the two-fold meaning of the plus and minus signs: addition and subtraction (add or remove blocks), on the one hand, and positive and negative (hot or cold blocks), on the other. by working within this context, learners should become familiar with rules such as: a positive number times a negative number gives a negative number. in the end learners should routinely apply these rules. they should not memorise the rules as a recipe, but should rather be able to explain the rules. no. of litres 8 40 20 60 no. of kilometres 100 500 250 750 table 4. a ratio table of fuel consumption no. of learners 100 10 20 30 40 50 60 no. that passed 70 7 14 21 28 35 42 table 5. a ratio table of learner performance length on the floor plan (in cm) 1 2 4 8 12 real length (in cm) 200 400 800 1600 2400 real length (in m) 2 4 8 16 24 table 6. a ratio table of length of a passage learning material in compliance with the revised national curriculum statement: a dilemma 54 learning outcome 3: space and shape (geometry) for mathematics tuition to learners of this age group and specifically for geometry teaching this proverb applies: i hear and i forget i read and i remember i do and i understand. for learners in our project geometry is meant to be a “doing” activity! the curriculum specifies, among other things: in this phase the learner draws and constructs a wide range of geometric figures and solids using appropriate geometric instruments. (department of education, 2002: 64) the assessment standards for grade 9 opt for a formulation with ample use of the language of mathematics: we know this assessment standard has been met when the learner uses transformations, congruence and similarity to investigate, describe and justify (alone and/or as a member of a group or team) properties of geometric figures and solids, including tests for similarity and congruence of triangles. (department of education, 2002: 83) we regard it as possible to realise this outcome without reference to the language of mathematics. in the next example the properties of a rhombus are conveyed through cutting and the identification of symmetries. this approach increases the chances of really comprehending the properties and not simply memorising them. the term congruence does not need to be mentioned at all. example 4: paper-folding and -cutting for this exercise you have to fold an a4 sheet of paper twice and then give it a straight cut near the double-folded corner (figure 2). the resultant quadrilateral is called a rhombus. the properties of the rhombus can all be deduced through the mental processes accompanying the practical activities of cutting and folding. • the four sides have the same length, since they all fit on the cut side. • the diagonals bisect each other, since they are the folded sides of the a4 sheet. • the diagonals are orthogonal to each other, because the a4 sheet has right-angled corners. • the diagonals bisect the angles, because these eight angles fit four on four after the cutting has been done. • therefore the opposite angles are also equal and the opposite sides are parallel. the utilisation of similarity should precede the learning of the property that two triangles are similar when two angles are equal. for example: example 5: similarity john's measuring-rod has a length of 2 metres. at a certain stage the measuringrod has a shadow of 1.25 m. the shadow of a streetlamp is then 3.9 m. show in the sketch how the shadow is formed by sun rays. similarity is an enlargement or a reduction, i.e. the same form in the same proportions. the triangles formed by sun beams, objects and shadows are similar and therefore the unknown height of the streetlamp can be calculated (figure 3). figure 2. paper-folding and cutting – a rhombus bertus van etten and kosie smit 55 learning outcome 5: data handling the curriculum specifies, among other things: in this phase the study of chance (probability) should include both single and compound events. through experimentation and the analysis of situations, the learner should recognise the difference between the probability of outcomes and their relative frequency in simple experiments. (department of education, 2002: 66) the assessment standards for grade 9 specify: we know this assessment standard has been met when the learner considers situations with equally probable outcomes, and determines probabilities for compound events using two-way tables and tree diagrams. (department of education, 2002: 91) before you reach the stage where learners can talk about equally probable outcomes, they should experience that probabilities as such can be determined. they are familiar with the context of gambling machines. you can create a situation of a gambling machine in class. example 6: a gambling machine you playfully introduce a gambling machine in class. instead of a real machine, you use three dice. the rules of this gambling machine are: • if you throw three sixes, you win r10. • if you throw two sixes, you win r5. • if you throw one six, you win r1. • if you do not throw a six, you do not win anything. • before every throw you pay r1. every group of learners in a class can, for example, throw 100 times. the results per group can then be collected for the class as a whole and can be represented in a bar graph (figure 4). after this it is appropriate to consider whether the player or the owner of the gambling machine figure 3. similarity – calculation of an unknown height gambling machine 0 100 200 300 400 500 600 700 0 sixes 1 six 2 sixes 3 sixes number of sixes n u m b e r figure 4. gambling machine bar graph learning material in compliance with the revised national curriculum statement: a dilemma 56 has the best chance of making a profit. the question should also be considered whether playing with three dice is a realistic simulation of other situations of chance, e.g. a certain kind of playing machine in a café or at a fair. mathematisation revisited the context plays an essential role in the teachinglearning process. it is not only the mathematical activities that form part of the learning process, such as the gathering of information and of processing it in tables and graphs, but also the translation from context to mathematics and from the mathematical solution to an answer to the original question. involvement with contexts after a context has been investigated (what does it deal with? to which question do we have to find an answer?), four activities must occur (table 7). in a scheme, these are represented in figure 5. we illustrate the scheme with a final example from the field of algebra. example 6: whales in south african waters whales appear regularly along the south african coastline. in 1995 there were an estimated 2000 whales in south african waters. since then the population has increased by 7% per year. an information board informs tourists that this boils down to a doubling of the population in 10 years. mathematisation 1 there are various ways in which this context can be translated into a mathematical model. one possibility is represented in table 8a. doing mathematics 1 the table must be completed. this can hardly be done without a calculator (table 8b). interpretation 1 after 10 years the (rounded) number of whales has indeed doubled. mathematisation 2 the given situation can also be described as a formula: 199507.12000 −×= yearwhalesofnumber mathematisation: the context is translated into mathematics doing maths: find a mathematical solution through mathematical activities interpret: translate the mathematical solution to the context verify: test whether the solution fits into the context table 7. the process of problem solving figure 5. scheme of learning mathematics in a context bertus van etten and kosie smit 57 doing mathematics 2 examine whether 400007.12000 10 ≈× ? this can be rewritten as the question: is 207.1 10 ≈ ? the time it takes for the population to double is not dependent on the initial number (here 2000), but on the percentage increase (here 7%). interpretation 2 after 10 years the number of whales is indeed more or less double the original number, because 96715.107.1 10 = . verification in both cases there has to be reflection on the meaning of the mathematical results within the context. when the table was filled in, decimal numbers were rounded to natural numbers. there cannot be 2289.8 whales. but a number of 3934 whales in 2005 is also not very likely. these calculations are approximations of the real situation. the numbers 7% and 2000 are also only estimations made by marine biologists. however, these estimations enable us to say something about the population. in a broader perspective, the proponents of realistic mathematics education describe the phases in the problem-solving process as horizontal and vertical mathematisation (freudenthal, 1991; gravemeijer, 1994; 1998). drijvers (2003: 53) explains that “horizontal mathematisation concerns organising, translating, and transforming realistic problems into mathematical terms, in short, mathematising reality. vertical mathematisation concerns reflection on the horizontal mathematisation from a mathematical perspective, in short, mathematising the mathematical activities and developing a framework of mathematical relations.” this formulation refers not only to the solution of individual problems, but more generally to the mathematical development established through the solution of and reflection on a range of carefully selected problems. problems with the rncs the current version of the rncs makes it difficult for a developer to choose mathematics learning material. the curriculum does not clearly indicate what balance should be established between the learning of formal mathematics, on the one hand, and learning to use mathematics, on the other. as a result, a developer of material has difficult choices to make. by including the language of mathematics in the curriculum for grades 1 to 9, time and energy are demanded for learning material • that is sometimes not relevant to everyday life, • that is mostly presented too early for the learner, • that can only be reproduced in the format of a recipe, • that bears no or very little intrinsic motivation. on the contrary, learning material with the focus on using mathematics year 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 increase no. of whales 2000 table 8a. mathematical model of whales in south african waters year 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 increase 140 150 160 172 184 196 210 225 241 257 no. of whales 2000 2140 2290 2450 2622 2805 3001 3212 3436 3677 3934 table 8b. completed table of whales learning material in compliance with the revised national curriculum statement: a dilemma 58 • can well be developed in such a way that learners experience mathematics as relevant for their lives outside the classroom, • can well be adapted to the level of thinking of learners at their stage of development, • can put the emphasis on the development of mathematical concepts, • can initiate intrinsic motivation well. we admit that, when realistic contexts are explored, learners are expected to read and interpret text consisting of a greater use of words/language. in many instances this poses a problem, since in the project schools learners’ reading skills are often under-developed. however, our experience has convinced us that the desire to solve realistic problems often acts as a stimulus for learners to also improve their reading proficiency. responses of teachers the following responses (translated from afrikaans into english) of three teachers in the project schools illustrate their experience of the learning material: the learning material poses real challenges to learners. although i sometimes think that the examples are too difficult, the learners time and again surprise me with sensible solutions. the fact that the presentation is within context, and that the examples link to their experience (cakes, cars, etc.), makes it more understandable to the learners. for me as an educator it was interesting to observe that the learners look forward to the mathematics lessons. in dealing with a module, all the learners were given the opportunity to discover mathematics, but also to discover themselves. the learner can see in the module what he can and cannot do. he could write down his own conclusions in his own module. the weaker learner could progress according to his own abilities, while the stronger learner could tackle the more challenging work on his own. this was really a challenge to the learners. i make bold to say that the old mathematics had a restraining effect on the learner’s route of discovery. the modules involve much reading. this creates the opportunity and motivation for learners to improve their reading. they are anxious to read. however, for the weaker learner the volume of reading matter can indeed be a problem. on the basis of our experience we make a plea that material developers are allowed to translate the curriculum for the senior phase into context-rich mathematics and to postpone much of the language of mathematics until grades 10, 11 and 12. references drijvers, p.h.m. (2003). learning algebra in a computer algebra environment. utrecht: freudenthal institute. department of education (2002). revised national curriculum statement grades r – 9 (schools); mathematics, gazette no. 23406, vol. 443. pretoria: department of education. freudenthal, h. (1973). mathematics as an educational task. dordrecht: reidel. freudenthal, h. (1991). revisiting mathematics education. dordrecht: kluwer academic publishers. gravemeijer, k.p.e. (1994). developing realistic mathematics education. utrecht: freudenthal institute. gravemeijer, k. p. e. (1998). developmental research as a research method. in a. sierpinska & j. kilpatrick (eds.), mathematics as a research domain: a search for identity. books i and ii (pp. 277-295). dordrecht: kluwer academic publishers. moss, j. (2005). pipes, tubes and beakers: new approaches to teaching the rational-number system. in m.s. donovan & j.d. bransford (eds.), how students learn, history, mathematics and science in the classroom (pp. 309-349). washington dc: the national academies press sethole, g. (2004). meaningful contexts or dead mock reality: which form will the everyday take? pythagoras, 59, 18-25. abstract introduction setting the context of the study mathematical problem-solving teaching professional development in mathematical problem-solving pedagogy the adult learning theory: andragogy the design-based research project data gathering and study design participants research instruments and data collection procedure data analysis ethical considerations findings and discussion teachers’ personal meaning teachers’ reflective inquiry teachers’ social participation final thoughts acknowledgements references about the author(s) brantina chirinda department of science and technology education, faculty of education, university of johannesburg, johannesburg, south africa citation chirinda, b. (2021). professional development for teachers’ mathematical problem-solving pedagogy – what counts? pythagoras, 42(1), a532. https://doi.org/10.4102/pythagoras.v42i1.532 original research professional development for teachers’ mathematical problem-solving pedagogy – what counts? brantina chirinda received: 03 feb. 2020; accepted: 29 apr. 2021; published: 25 aug. 2021 copyright: © 2021. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract problem-solving is of importance in the teaching and learning of mathematics. nevertheless, a baseline investigation conducted in 2016 revealed that mathematical problem-solving is virtually missing in south african classrooms. in this regard, a two-cycle design-based research project was conducted to develop a professional development (pd) intervention that can be used to bolster grade 9 south african teachers’ mathematical problem-solving pedagogy (mpsp). this article discusses the factors that emerged as fundamental to such a pd intervention. four teachers at public secondary schools in gauteng, south africa, who were purposively selected, participated in this qualitative research study of a naturalistic inquiry. teachers attended pd workshops for six months where pd activities that were relevant to their context were implemented. between the pd workshops, teachers were encouraged to put into practice the new ideas on mpsp. qualitative data were gathered through reflective interviews and classroom observations which were audio-recorded with teachers’ consent. data were analysed through grounded theory techniques using constant comparison. the findings from the study suggested that teachers’ personal meaning, reflective inquiry, and collaborative learning are factors fundamental to their professional growth in mpsp. the major recommendation from the study is that facilitators of pd must acknowledge these factors to promote teachers’ professional growth in mpsp. if pd processes and activities are relevant to teachers’ personal meaning, reflective inquiry, and collaborative learning, teachers find the pd programme fulfilling and meaningful. this study contributes to the pd in mpsp body of knowledge by having worked with teachers in an under-researched context of historical disadvantage. keywords: mathematics education; mathematical problem-solving pedagogy; personal meaning; professional development; reflective inquiry; collaboration. introduction historically, south african learners perform poorly in mathematics in national tests of achievement like annual national assessments (ana), in regional tests of achievement such as southern and eastern african consortium for monitoring education quality (sacmeq), and in international tests of achievement like trends in mathematics and science study (timss). of the approximately 270 500 grade 12 learners who wrote mathematics in the 2018 matric examinations, only 37% passed with 40% and above. these results indicate that south african learners’ performance in mathematics is inadequate and this crisis has existed for over 20 years (van jaarsveld & ameen, 2017). in this regard, the department of basic education (dbe) in south africa propounds problem-solving as another way for improving the teaching and learning of mathematics (dbe, 2011). the curriculum and assessment policy statement (caps), which is south africa’s intended curriculum, states that its objectives are to produce learners who are able to identify and solve problems and make decisions using critical and creative thinking. an analysis of the caps curriculum shows that its content centres mainly on the proficiency of mathematical facts, procedures, and skills and the term problem-solving refers mainly to solving problems as in performing calculations (see dbe, 2011). on the other hand, in doing mathematics, learners make explorations, construct arguments and formulate proofs and conjectures, and use problem-solving to evaluate and interpret these actions (schoenfeld, 2013), such that mathematics should be taught from a problem-solving perspective (nctm, 2007). in addition, learners’ active participation in the problem-solving process encourages robust understanding (schoenfeld, 2014). nonetheless, given the importance of problem-solving in the teaching and learning of mathematics, the baseline study that i conducted in march 2016 revealed that participant teachers were unclear on implementing mathematical problem-solving pedagogy (mpsp) and required support in implementing it (see chirinda & barmby, 2018). the baseline study investigated south african grade 9 mathematics teachers’ teaching strategies, views on mpsp, and the support they required in implementing mpsp. mpsp refers to the practices that mathematics teachers adopt to assist learners to understand and be able to do problem-solving. further findings from the baseline study were that problem-solving is virtually missing in south african classrooms. the findings from the baseline study are consistent with jagals and van der walt’s (2016) observation that mathematical problem-solving is limited in the south african implemented curriculum and the assessed curriculum (e.g. national examinations and formative assessments). in light of the findings from the baseline investigation, i developed a professional development (pd) intervention for grade 9 teachers’ mpsp. the pd intervention is a framework that can be adapted by the south african dbe and other mathematics education practitioners to support teachers of mathematics in the teaching of problem-solving. i revisit this later to discuss how i designed the pd intervention. mpsp is of importance since, besides learners’ preconceptions, teaching is widely known to influence learning. consequently, if teachers are to encourage learners to do problem-solving or to improve learners’ problem-solving skills, then teachers need to deepen their mpsp. mpsp can enrich teachers’ mathematical experiences because it broadens their views of what it means to do mathematics with learners. teachers’ mpsp can be grown by providing them with pd programmes that focus on the teaching of mathematical problem-solving. this kind of pd is commonly known to improve teachers’ confidence in implementing mpsp and their interaction with learners. subsequently, this improves learners’ mathematical problem-solving processes and abilities. however, there is no sufficient literature on research on the pd of south african teachers that focuses on mpsp. over the years, several longitudinal projects that focus on pd of primary and secondary mathematics teachers have been conducted in south africa (see biccard, 2019; brodie & borko, 2016; pournara, hodgen, adler, & pillar, 2015; venkat, 2019) and none of these projects has focused on the pd of teachers in mpsp. consequently, pd programmes that focus on teachers’ mpsp are under-represented in south africa. the study reported in this article emerged from my continuing efforts to fill in this existing gap in the literature on the pd of south african teachers in mpsp. the purpose of this study was to explore factors that impact teachers’ professional growth in mpsp. this study is essential because for pd to be effective, the factors that support teachers’ professional growth must be understood. specifically, i formulated the following research question: what factors are fundamental to a professional development intervention for teachers’ mathematical problem-solving pedagogy? setting the context of the study context relates to the factors external to teachers and learners that may affect the process of teaching and learning (lester, 2013). in this study, context refers to the many factors surrounding the teaching and learning mathematics environment. these include physical aspects (schools, classrooms, class sizes, and resources), economic and political factors, and the curriculum which comprises the objectives and quality of teaching and learning and teachers’ daily grind that encompasses the intended, implemented, and assessed curricula. the factors listed here are not exhaustive of the influences on mpsp in this context and there might be other factors equally important. before developing the pd intervention for mpsp, i conducted a context analysis to identify the areas that needed attention to generate some useful ideas for the development of the pd intervention. i conducted this study at public south african secondary schools positioned in informal settlements. informal settlements are the areas where people build their temporary housing structures without the approval of the local government. the south african government has often upgraded these settlements to proper residential townships by providing infrastructure. from the context analysis, it was evident that the curriculum was strictly centralised, overloaded and teachers worked under pressure to cover the prescribed material within the stipulated time. in terms of the teaching and learning environment, the analysis revealed that the classes were multilingual, large, overcrowded, and resource-deficient in terms of technology and furniture. multilingual classes are instances where learners speak more than two different languages in one class. most learners came from poor homes where there were neither desks nor electricity to do homework after school. mathematical problem-solving teaching this study focused on designing a pd for teaching mathematical problem-solving. problem-solving is when a problem-solver is confronted by a mathematical problem for which they have no known solution (schoenfeld, 2013) or are unaware of how to solve it promptly (newell & simon, 1972). several scholars agree that problem-solving is when a goal exists, and the problem-solver does not know how to attain it (lester, 2013). polya (1957) suggested a four-step process that a problem-solver experiences during the process of problem-solving. polya proposed that understanding the given problem prior to solving it is the first step of the problem-solving process. it involves learners familiarising themselves with the given problem and pinpointing the unknowns and the useful data (barmby, bolden, & thompson, 2014). polya’s second problem-solving step is devising the plan, where learners must gather all available information and ponder possible problem-solving strategies to apply. these strategies can be making a table or a chart, problem reformulation, guess and check, etc. the third problem-solving step is carrying out the plan where learners implement the selected problem-solving strategy. the final step is looking back at the answer and problem-solving strategy to assess if they make sense and fit with the original problem. during this step, the teacher requires learners to justify their actions and consolidate what they would have learned. polya’s steps are linear, but the problem-solving process is rarely sequential or straightforward as depicted by the model (reys et al., 2012). during the problem-solving process, learners rarely solve problems linearly but frequently skip or revisit steps and move back and forth among the steps. over the years, i have noticed that when presented with a problem, learners usually get better comprehension during the devising a plan stage and may need to look back at how they understood the problem or devised the plan; therefore, the process of problem-solving is seldom linear. in this study, polya’s (1957) four steps were used as a guideline during the workshops and implementation of the mpsp by the teachers. polya’s steps only provide guidance on how teachers can teach learners problem-solving, but do not provide ways teachers can identify learners’ difficulties in problem-solving or skills they require to do problem-solving successfully. mpsp is complex, as observed by mason (2016): in order for problem-solving to become an integral part of learners’ experience in school and university, all aspects of the human psyche, cognition, affect, behavior, attention, will and metacognition or witnessing must be involved. focusing on only one or two aspects is simply inadequate and very unlikely to lead to full-scale integration into learners’ ways of being in the world. (p. 109) the picture presented by mason (2016) is intricate in that teachers are on a daily basis faced with the intended curriculum, which forms part of the implemented curriculum that, in turn, is impacted by the examined curriculum. schoenfeld (2016, p. 20) recognises teachers’ daily grind and notes that ‘individuals do not work, or learn, in a vacuum’, but the context shapes the mpsp and the problem-solving process. in the south african context, learners are faced with aspects of the language of learning and teaching. english is the language of learning and teaching in south africa, yet 76% of the population has an indigenous south african language as their first language (statistics south africa, 2017). this implies that most south african learners are learning in a language that is not their mother tongue. polya (1957) did not incorporate language issues in his four-step problem-solving process, yet language and the process of problem-solving are not separate (chirinda & barmby, 2017). during problem-solving, learners need to understand the problem first before attempting to solve it, yet the language of learning and teaching is a challenge to most learners in the south african context. professional development in mathematical problem-solving pedagogy professional development involves the programmes implemented to enhance teachers’ knowledge and practices with the key objective of improving learner achievement (loucks-horsley, hewson, love, & stiles, 2010). pd in mpsp focuses on supporting teachers to be able to teach problem-solving effectively. several pd projects that focus on supporting teachers’ mpsp have been conducted around the world. in chile, the activando la resolución de problemas en las aulas (arpa) initiative pd programme, which focuses on integrating problem-solving into teachers’ practices, has been running since 2014 (felmer, perdomo-díaz, & reyes, 2019). in the arpa initiative pd programme, teachers attend workshops for a year where they participate in problem-solving activities and how to introduce them to their learners. the major finding from the programme was that a pd in problem-solving pedagogy should be flexible to accommodate different teachers, contexts, schools, etc. (felmer et al., 2019). in australia, widjaja, vale, groves and doig (2017) conducted a lesson study project with 10 elementary school teachers and numeracy coaches. widjaja et al. examined the participants’ pd from implementing structured mathematical problem-solving classes. the findings reported that the project had a positive influence on teachers’ mpsp and beliefs in mpsp. an after-school research programme, the informal mathematical learning, was conducted in the united states for three years. this particular project comprised a pd component that focused on assisting mathematics teachers from a local school district to improve their capability to design and enact mathematical problem-solving activities. findings revealed how the teachers gave guidance to learners on problem-solving (see weber, radu, mueller, powell, & maher, 2010). in israel, an in-service pd programme was conducted with 170 teachers who participated in a mathematical activity ‘solving problems in two different ways’ that focused on the notion of symmetry (see leikin, berman, & zaslavsky, 2000). the purpose of the pd programme was to increase participant teachers’ problem-solving expertise. findings were that teachers started implementing various problem-solving strategies in their lessons because they found them helpful to themselves and their learners. the above programmes are from around the world; however, in south africa there are few programmes that explicitly support teachers in teaching mathematical problem-solving. to address this identified gap, in the large project, i developed a pd intervention for teachers’ mpsp. the study reported in this article focuses on the factors fundamental to such a pd intervention. in the context of the study, the pd intervention was about supporting teachers who worked with mathematical word problems found in the caps curriculum. mathematical word problems are verbal or (con)textual descriptions of a problem situation (verschaffel et al., 1994) that incorporate mathematics tasks implanted in real-world situations. the adult learning theory: andragogy i adopted andragogy, the adult learning theory, as a theoretical framework for this study. in 1968, malcolm shephard knowles proposed a theory that distinguished adult learning (andragogy) from children learning (pedagogy). andragogy is the art and science of adult learning. it refers to any kind of adult learning (kearsley, 2010). knowles (1984) suggested five assumptions about adult learning: adults have a developed self-concept, previous experience, practical reasons for learning, and are ready to learn since they are internally motivated. the first assumption is that adult learners have an established sense of self (knowles, holton, & swanson, 2005). as human beings mature, they move from being dependent to being independent and form a complete entity called ‘self’. they become distinct from those around them and require a sense of independence in learning. consequently, they prefer a more self-directed learning approach rather than being teacher-led. in this study, teachers were contributors to their own pd experiences in mpsp: they were actively involved in the workshops’ activities and the implementation of the mpsp. thus, teachers had control over their learning. secondly, past experiences are important in adult learning (knowles et al., 2005). as people mature, they accumulate academic and social experience from which they can learn. accordingly, to develop teachers professionally, facilitators need to build on their existing knowledge. facilitators need to merge new ideas with teachers’ lived experiences. in this regard, i did not go into teachers’ contexts as a knowledgeable person on mpsp. instead, i conducted a baseline study and started the pd intervention from teachers’ views and experiences of mpsp. in this way, i created an environment where teachers felt that they had ownership of the pd intervention. thirdly, adults are ready to learn when there is a purpose and relevance, for example when it is about professional growth connected to their work. adult learning experiences must be scheduled so that they are concurrent with their readiness to learn (knowles et al., 2005). the baseline investigation disclosed that participant teachers were unclear on implementing mpsp and indicated that they required support in implementing it. this suggested that teachers were ready to participate in the pd intervention. fourthly, adults are interested in learning practical skills that help them to solve problems. as human beings mature, their orientation towards learning switches from subject-centeredness to problem-centeredness. teachers were interested in participating in the pd for mpsp because it would help them assist their learners in doing mathematical problem-solving. finally, adult learners are driven by internal motivation. adults develop their methods of motivation and are driven to learn for their reasons, for example for promotion or to bolster self-esteem. knowles (1984, p. 12) observed that as people ‘mature the motivation to learn is internal’. adults are motivated to learn if they believe that what they are learning is relevant to their life and profession (knowles et al., 2005). during the baseline investigation, teachers complained that they had experienced pd that was not usable in their practice. teachers’ exhaustion from low-quality and irrelevant pd programmes was my primary motivation to engage design-based research (dbr) as a research methodology. knowles (1984) also offered four principles that can be applied to adult learning education to make it more effective. the first principle is that adults require to be actively involved in the planning and evaluating of their learning. secondly, experience provides the foundation for the learning activities. thirdly, adults are captivated by learning content that is immediately relevant to their job or personal life. fourthly, adults are inclined towards problem-centred learning rather than content-oriented (kearsley, 2010). to this end, andragogy was relevant as a lens to design the teachers’ pd in mpsp which was self-directed, relevant, contextual, task-based, less theoretical, and more practical. the five assumptions and four principles of andragogy were also used as a lens in designing a semi-structured interview schedule that was used during the reflective interviews. the design-based research project sztajn, borko and smith (2017) observe that many researchers inadequately recount the design and content of their pd programmes. i avoid the identified pitfall by giving a detailed description of the pd programme for this study in this section. a dbr project was initiated in 2016 and focused on designing, implementing, evaluating, and redesigning a pd intervention that can be used to support grade 9 south african teachers in teaching mathematical problem-solving. the context of the study was a crucial aspect; therefore, i engaged dbr to develop a pd intervention that was relevant to the teachers’ context. dbr is a research design that utilises cycles and was appropriate for this study since, at the beginning, i did not have in mind an example of a pd intervention for grade 9 teachers’ mpsp but wanted to develop one that was sensitive to teachers’ context. after identifying the need to design a pd intervention to support grade 9 teachers in mpsp, the first phase involved a literature review, context analysis, and the baseline study. the second phase involved two cycles of design-implementation-evaluation-redesign of prototypes of the pd intervention. i designed the first prototype from the design principles generated from the literature review, context analysis, and the baseline investigation. i implemented the first prototype in the first cycle. next, i designed the second prototype from the design principles generated from the first cycle and i implemented it in the second cycle. from the design principles generated in the second cycle, i redesigned the second prototype to develop the third prototype. figure 1 illustrates the study’s research design. figure 1: the research design of the study. in the first cycle, three pd workshops were carried out with teachers on the last wednesday of the first, third, and fifth months of a six-month period. in the second cycle, the workshops were on the last friday of the first, third, and fifth months of the six-month intervention period. each workshop was three hours long and this amounted to nine hours of training for teachers in each cycle. the first workshop had five main activities, which were 30 minutes long, with a 30-minute break between the third and fourth activities. the activities were theoretically informed by the literature, the context, my personal preferences as a researcher, the adult learning theory, and the intended, implemented, and examined curricula that the teachers were dealing with on a daily basis. activity 1: teachers watched two short videos either from japan, singapore, or the united states on mpsp. this study supported teachers’ mathematical problem-solving pedagogy using polya’s (1957) problem-solving framework as a guideline; therefore, the videos clearly showed teachers implementing polya’s four problem-solving processes. the videos, although from different contexts, were to initiate discussions about mpsp, and inspire and model the teaching of mathematical problem-solving to participant teachers. i discussed the videos with the teachers, focusing on what mpsp entails and how to apply polya’s four steps of problem-solving in their practice. at the beginning of the workshop, i had provided teachers with polya’s how to solve it book that focuses on mathematical problem-solving. we used it as a guideline during the discussions. activity 2: teachers collaboratively solved at least two word problems relating to the topic they were teaching at the time and simulated how to teach these problems to their learners. teachers worked through the problems by going through polya’s (1957) problem-solving steps, that is, understanding the given problem prior to solving it, devising the plan, carrying out the plan, and looking back at the answer. teachers shared ownership of the problem, its solution strategy, and the final solution. teachers expounded on ways of introducing or posing these problems so that learners understood the given problems. the following are examples of grade 9 mathematics word problems that were used during the workshops: elizabeth is going to her sister’s wedding this morning. she began driving at 7 am on the n1 highway at a constant speed of 80 km per hour. at 7:30 am, elizabeth’s mother followed her and began driving along the same highway (n1) at a constant speed of 95 km per hour. at what time will elizabeth’s mother catch up with elizabeth? sipho built a circular patio that perfectly fits inside his father’s square yard measuring 14 meters on the sides. if he needs to plant a lawn in the four corners of the square, how many square meters of the lawn does he need to buy? the following is a demonstration of how discussions of this problem unfolded in the workshop. understanding the problem: the question requires us to find the amount of lawn needed to be planted on the four corners of the yard. this area is inside the square but outside the circle. the square’s sides are 14m long. the circle’s radius is not given, but we know the circle fits perfectly inside the square (see figure 2). figure 2: circular patio fitted in a square yard. devising a plan: we can find the area of the square and circle by using formulas. area of square = s2 and area of a circle is a = πr2. by finding these two areas, we can get the amount of lawn by subtracting the area of the circle from that of the square. carrying out the plan: side of square = 14m: the circle fits perfectly inside the square; therefore, the radius of the circle is half the side of the square, r = 7m: sipho must buy asquare – acircle = 196 – 154 = 42m2. looking back: estimation. the shaded region looks like a quarter of the square area therefore 42m2 is reasonable. activity 3: we discussed a structure of implementing the mpsp and came up with guidelines that we modified from polya’s (1957) problem-solving steps: the lesson starts with a problem (preferably word problem) in the textbook, or the teacher poses a problem related to those found in the caps curriculum. learners read the problem as individuals or in pairs and the teacher determines if all learners understand the problem. if some learners do not understand the problem at hand, the teacher assists learners to understand by using approaches like re-reading the mathematical problems carefully with the aim of understanding before solving them. i encouraged teachers to ask learners questions like: do you understand what the problem is looking for? do you know all the words? can you repeat the problem in your own words? this stage involves learners devising strategies to solve the given problems. we discussed with teachers how they could help learners to create a plan to solve a given problem. for example, during problem-solving teachers could ask learners questions like: what operation are you going to engage? do you need to draw a picture? a table? would you use an equation? this stage involves learners finding a solution to the problem independently, in pairs or as a group. i discussed with teachers how they could assist learners in carrying out the plan, persisting with a chosen plan, or if a plan does not work to discard it and choose another one. the teacher, during this process, circulates in the classroom, asking learners questions about their work, clarifying misunderstandings, giving suggestions, and helping or giving hints to learners who get stuck. moreover, the teacher may repeat the understanding process with the stuck learners if necessary. concurrently, the teacher looks for learners who have interesting solutions with the objective of asking them to explain their solutions in a certain order during the whole class discussions. after getting a solution to the given problems, i persuaded teachers always to help learners review their answers by reflecting on what would have worked and did not work and try to find alternative solution strategies. at this point, the class conducts a whole class discussion where learners present their ideas and solutions as well as listen to others. learners are required to look at the similarities and differences among the solutions being presented by their peers and to realise that there are multiple solution strategies to some given problems. the teacher sums up the solutions presented by learners, highlights the main points, and concludes the lesson or, if there is still time, learners solve another problem. activity 4: during the fourth activity, i supported teachers on implementing mpsp in a multilingual context. in the first cycle, we focused on code-switching, and in the second cycle we focused on translanguaging. code-switching is generally known as the utterance of two or more languages in the same conversation. translanguaging is ‘the flexible and meaningful actions through which bilinguals select features in their linguistic repertoire to communicate appropriately’ (velasco & garcía, 2014, p. 7) and, in this context, it involved learners receiving information in one language and repeating it in another. activity 5: in the last activity of the workshop, we discussed the importance of reflection. i coached teachers on how to continuously reflect on self during the implementation of the mpsp. i concluded each workshop by asking teachers to summarise what they were taking from the pd. between the pd workshops, teachers implemented the mpsp for two months. during the implementation of the mpsp, i conducted classroom observations and guided teachers as was required. following the initial and second implementation, the second and third workshops were conducted, where the objective was for teachers to listen to the audio-recordings of the observed lessons and reflect on how they had implemented mpsp. after the reflective process, teachers watched two more videos showing mpsp and collaboratively solved mathematical word problems similar to those they were currently teaching. teachers also simulated how to teach mathematical word problem-solving. i discussed with teachers their experiences with learners’ language during the implementation of mpsp and linguistic strategies appropriate to support learners in their context. finally, i coached teachers on how to reflect on their mpsp and classroom practices incessantly. data gathering and study design the study reported on in this article is a qualitative research approach of a naturalistic inquiry (salkind, 2010) which was conducted within a dbr project. the qualitative research approach permitted interaction with teachers in their natural settings, namely classrooms and schools (denzin & lincoln, 2011). the study was naturalistic in that i encapsulated and interpreted teachers’ experiences, thinking, and actions about a phenomenon, that is, problem-solving and mpsp (salkind, 2010). the five assumptions and four principles of andragogy were used as a lens in designing the pd programme for teachers’ mpsp, and the reflective interview schedule. participants thirty-one grade 9 mathematics teachers (19 female and 12 male) from 20 different south african public secondary schools were conveniently selected to participate in the baseline study. for the pd intervention, two schools (school a and school b) were purposefully selected out of the initial 20 schools based on their accessibility and representativeness of south african public secondary schools that are resource-deficient, multilingual, and have a high learner to teacher ratio. furthermore, during the baseline study, the grade 9 teachers of the two selected schools revealed that they misunderstood mpsp, were utilising traditional methods of teaching mathematics, and had shown interest in the project. two female teachers (with pseudonyms mary and bertha), aged between 31 and 40 years, from school a, participated in the first cycle. one female and one male teacher (ms n and mr m), aged between 25 and 30 years, from school b, participated in the second cycle. both teachers in the first cycle had bachelor’s degrees in mathematics education. mary had 13 years’ experience and bertha had 19 years’ experience in teaching secondary school mathematics. mr m had a diploma in mathematics education and ms n had a master’s degree in mathematics education. mr m had six years of experience teaching secondary school mathematics and ms n had one year of experience of teaching secondary school mathematics. research instruments and data collection procedure i employed a mathematics teacher open-ended questionnaire to conduct the baseline study to establish grade 9 teachers’ views on mpsp (see chirinda & barmby, 2018). i gathered data in both cycles using classroom observations and reflective interviews which i audio-recorded with participants’ consent. only data from classroom observations and reflective interviews were relevant for the study reported in this article. i conducted pre-observations before the implementation of the pd intervention and observations during the implementation of the intervention. i conducted pre-observations and observations while teachers were delivering lessons and recorded information on the spot, as it occurred. information was recorded with teachers’ permission on the observation comment card. the purpose of the observations was to observe teachers’ practices and establish the difficulties that occurred during implementation of the mpsp. the reflective interviews were conducted with each teacher on the lessons observed and were facilitated by playing the teachers selected audio recordings of the observed lessons. a semi-structured interview schedule was used to facilitate the reflective interviews. data analysis i engaged grounded theory data analysis techniques using constant comparison to analyse the data, such that the process was both ongoing and retrospective. grounded theory data analysis techniques involve systematic guidelines for analysing qualitative data to generate codes, themes, and theories grounded in the data (chamaz, 2014). after completion of data collection, i conducted the retrospective analysis by going through ‘comparative levels of analysis’ (chamaz, 2013, p. 295). comparative analysis permitted uninterrupted interaction with the data since i could continuously ask analytic questions about these data and emerging themes (chamaz, 2013). to develop major categories, i compared data with data, data with emerging codes, codes with codes, and raised noteworthy codes to provisional categories based on the frequency of occurrence. i then compared data and codes with the provisional categories to establish major categories (themes). i created codes, for example, by pinpointing significant statements on mathematical problems, problem-solving, and mpsp made by teachers during classroom observations, pd workshops, and reflective interviews. to increase the credibility of the findings, the codes and themes were verified by an independent coder who either validated the findings or eliminated codes or themes based on disconfirming evidence. ethical considerations permission to conduct research was granted by the overseeing university and gauteng department of education (gde) with reference numbers 2016ece002d and d2016/373aa. regarding informed consent, teachers freely participated in the study without implicit or explicit coercion and were given assurance that they could withdraw from the study at any time without being disadvantaged. i upheld issues of beneficence throughout the research by ensuring that the study was beneficial to mathematics teachers and mathematics education. the teachers were not subjected to any possibility of harm. i viewed teachers as autonomous beings and respected them throughout the study. i did not divulge participants’ information that they disclosed during the study to others or institutions. the audiotapes and transcripts were stored in password-protected files on a password-protected computer. the data or reports did not reveal the participants’ names or their schools. findings and discussion in this study, i was interested in identifying the factors fundamental to the pd intervention for teachers’ mpsp. the research question was: what factors are fundamental to a professional development intervention for teachers’ mathematical problem-solving pedagogy? the major findings from the study were that teachers’ personal meaning, reflective inquiry, and collaborative learning are factors fundamental to their pd in mpsp. in the next sections, i discuss the findings and present the themes with direct quotations to ‘give life’ to the data analysis and to make teachers’ voices audible. teachers’ personal meaning personal meaning is what the teacher desires, feels, thinks, considers to be truthful, and includes what meaning they give to certain routines or conflicts (elbaz, 1990). howson (2005, p. 18), referring to learners, explained that personal meaning relates to learners’ personal sense ‘relating to relevance and personal significance (e.g., “what is the point of this for me?”)’. this explanation is homologous to teachers in this study since personal meaning emerged as a factor fundamental to their pd in mpsp. teachers’ personal meaning in the previous pd programmes in a particular context during the pre-observations, all four teachers indicated that they had previously participated in several pd programmes and viewed pd as not improving their pedagogy or promotion. teachers had developed negative attitudes towards these pd programmes, which they indicated as short-term, often lasting not more than one day. teachers believed that these programmes were designed and conducted by pd facilitators without teaching experience in their contexts. teachers were frustrated that the pd programmes were unrelated to their profession and real classroom challenges. as noted by mary: ‘i have attended short workshops before, where facilitators spoke about other trivial things in maths. however, i really want to attend workshops based on different strategies for teaching mathematical problem-solving. these workshops should also teach me to design problem-solving activities that can shift the learners’ mindset to love mathematics.’ the above quote implies that teachers’ disposition towards past pd programmes impacted on their professional learning. this was not surprising because personal meaning is known to be a foundation for structuring one’s actions in the world. kilpatrick, hoyles and skovsmose (2005, p. 2) observe that ‘we may claim that an activity has meaning as part of the curriculum, while learners might feel that the same activity is totally devoid of meaning’. this is comparable to what teachers felt about the pd programmes that the school required them to attend. the reflective interviews revealed that the teachers viewed pd facilitators as authorities who came to impart new knowledge to them but seemed not to understand who they were, what context they were working in, what they experienced every day when they were working in overcrowded and multilingual classes, and how they were supposed to learn to enhance their practice. this was put across by mr m: ‘the school sometimes requires us to attend workshops to improve on mathematics teaching. unfortunately, these workshops do not cover what happens in our classrooms. the facilitators always speak of models that are helpful overseas and not to me.’ in this regard, i engaged dbr for this study, such that the teachers’ input assisted in developing a pd intervention that was relevant to their lived experiences and context. as the intervention progressed, the reflective interviews disclosed that the teachers had reshaped their personal meaning in professional learning and were interested in the pd intervention because it had relevance and personal significance to their mpsp and classroom practices. the pd intervention provided teachers with prospects of learning from their mpsp on a daily basis, with and from their colleague. a key factor in pd for teachers’ mpsp was giving teachers the opportunity and voice to pinpoint what practices would best support their strengths in mpsp. this observation suggests that, before implementing the intervention, the pd facilitators should first investigate the characteristics of pd in mpsp that teachers need through informal observations, surveys, focus groups, or discussion groups. this is the only way pd can effectively address teachers’ personal meaning in mpsp. if teachers find the pd programme irrelevant to their personal meaning and non-aligned to their concerns on mpsp, it rapidly disappears from their memory. teachers’ personal meaning in mpsp at the beginning of each cycle, teachers exhibited what i would term ‘traditional’ approaches to teaching mathematical problem-solving. from the pre-observation lessons, i could not see the teaching of problem-solving in mathematics classrooms (see chirinda & barmby, 2018). instead, learners listened attentively as the teachers demonstrated the algorithm to solve the given problems and then learners worked to practise the given algorithm until they acquired the skills. teachers’ reasons for adopting the traditional methods of teaching were that they felt that learners lacked basic problem-solving processes to do problem-solving in grade 9 and that the overloaded curriculum left no time for problem-solving. in the workshops, we discussed that when teaching problem-solving, it is not mandatory to get answers for the given tasks, but it is important to help learners to do problem-solving. initially, teachers did not recognise that doing problem-solving enhances learners’ mathematical understanding of problem-solving processes that can be applied to different problems in the future. in the interviews, teachers also revealed that they taught procedurally because they assumed that learners did not understand since they struggled with the language of teaching and learning (chirinda & barmby, 2018). after attending the pd intervention, classroom observations revealed that the teachers were no longer drilling procedures but implementing the problem-solving strategies they learned in the workshops. i discuss, in a vignette (figure 3), how mary discussed with her learners each step of polya’s (1957) four-step problem-solving processes while solving the following word problem: reverend joseph of the methodist church of south africa has passed on, and his wish before he died was that people should wear only white, red, and pink hats at his funeral. 171 women and 93 men attend the funeral. his wife made 13 dozen white hats, five dozen red hats, and three dozen pink hats for the guests. at the end of the funeral, 11 hats were not worn. how many hats were worn by the guests? figure 3: implementation of the mathematical problem-solving pedagogy by teacher mary. as the intervention progressed, i observed that teachers began to emphasise doing of mathematical problem-solving rather than focusing on learners finding the correct answers. learners were now given opportunities to participate in problem-solving and contribute meaningfully during lessons. teachers began to promote classroom discussions, by providing learners opportunities to discuss solutions or solution strategies in pairs or groups. learners were required to monitor their work instead of relying on the teachers’ guidance. during problem-solving, i observed that teachers frequently probed learners to be open-minded by asking them to explain their solutions and solution strategies. learners were given opportunities to present their solutions on the board so that their classmates could learn different solutions. teachers’ personal meaning in giving learners full ownership of problem-solving at the start of each cycle, teachers in the study did not believe that their learners could learn mathematical problem-solving because they did not have basic problem-solving processes from prior grades. participant teachers believed that they were not supposed to give learners full ownership of problem-solving, but rather demonstrate each procedure step by step and require learners to practise the procedures (chirinda & barmby, 2018). teachers reflected that they believed that they had the responsibility of re-explaining procedures in response to learners’ questions. they felt that learners were underprepared from previous grades and had difficulty grasping grade 9 concepts. subsequently, during pd workshops, i worked with teachers on how they could implement mpsp with underprepared learners. furthermore, i proposed that teachers explain to learners how to understand the given problem and how to devise problem-solving strategies. after that, the teachers could discuss the different problem-solving strategies applicable to the given problem, eventually solve it, and reflect on the solution. this illustration could then be followed by a class exercise for learners to practise the problem-solving processes with the teacher moving around as a facilitator assisting learners as was necessary. as teachers found relevance and personal meaning in implementing the new ideas on mpsp from the workshops, i observed that they gave learners full ownership of ps. for example, in one lesson, i observed ms n incorporating open-ended word problems that encouraged learners to develop a sense of ownership. the following is an example of a scenario that ms n gave to her learners and required them to pose problems: meghan’s old grandmother starts walking at 8am to go to the mall for shopping. meghan must clean the house; therefore, she follows grandma after a few hours. they both arrive at the mall at the same time. i observed the learners working in pairs trying to understand the given problem prior to solving it. i could hear most of the learners planning and formulating problems with excitement and looking back at the answers after finding solutions. nonetheless, a few learners wanted to be shown how to pose the problem; nevertheless, i observed that ms n identified meaningful ways to motivate learners to persist in problem formulation. this seems to suggest that ms n had established self-understanding in personal meaning in giving learners full ownership of doing mathematical problem-solving. teachers’ reflective inquiry reflective inquiry is the teacher’s act of looking back at the teaching and learning of what has transpired and re-constructing or re-capturing the events’ occurrences, emotions, and experiences (schon, 1987). in both the first and second cycles, i conducted the second and third workshops whose chief objective was for teachers to reflect on their implementation of mpsp collaboratively. teachers conducted the reflective process by reviewing and reflecting on the audio-recordings of the observed lessons. i also used reflective interviews to assist teachers’ reflective inquiry on the implemented lessons and learners’ problem-solving processes. during the workshops and reflective interviews, i noticed that as teachers intentionally and methodically reflected on their participation in the pd activities and implementation of mpsp, they made sense of their actions and extracted meaning from those actions. reflective inquiry assisted teachers to consciously improve their mpsp as put across by mr m during one discussion in the workshop: ‘in my class, i usually give learners problems to work on as individuals. when they get stuck, i quickly give them hints or show them how to get the answer before the bell rings. now i understand, i will not be giving learners opportunities to engage in problem-solving when i do this. from now on, i will not rush to provide learners answers. learners need to take their time. they need time to think about the problems and find ways of how to get answers by themselves.’ bertha also indicated this in one of the reflective interviews: ‘i often show my learners the meaning of a problem before they begin solving it. i now see that i am taking away from learners, the chance to learn to analyse problems on their own. going forward, i will let learners take the lead in understanding the problems of the day.’ the above excerpts seem to demonstrate that self-inquiry evoked meaningful learning in the teachers. reflective inquiry as a means of teacher pd is recommended by muir and beswick (2007), who advocate that to improve teachers’ classroom practice, emphasis should be placed on teachers’ learning and self-reflection. when teachers engage in reflective practice, they become cognisant of their own and others’ thinking and assumptions (drago-severson, 2007). in this study, this awareness resulted in participant teachers’ growth. reflective thinking shifted teachers away from routine thinking towards reflective action that involved learning from doing (dewey, 1916) and allowed teachers to build knowledge that guided their mpsp (schon, 1987). this finding seemed to imply that teachers’ reflective inquiry was a factor fundamental to their pd in mpsp. i observed that self-reflection during the workshops and reflective interviews forced teachers to confront and become critical of their personal meaning in mathematical problem-solving and mpsp. i also discovered that as teachers became critical of their personal meaning in what the teaching of problem-solving implied, they were able to self-correct and improve their mpsp. during the reflective interviews, i realised that teachers did not reflect in a vacuum, but that context was an important aspect. when teachers reflect, they adapt ideas to their contexts (zaslavsky, chapman, & leikin, 2003), and for this study, it allowed teachers to function better in their context and become more confident in implementing mpsp. in this context, self-reflection was an important aspect of pd that reshaped teachers’ personal meaning and promoted their professional growth. teachers’ social participation teachers were required to work collaboratively during pd workshops and the implementation of the mpsp. collaboration means teachers are working and learning together to address challenges they grapple with in their profession (robutti et al., 2016). introducing collaboration at the beginning of each cycle was challenging as teachers were used to working as individuals and not sharing expertise with their colleagues. teachers felt that their practice they had kept personal and invisible for a long time was now being made visible to their colleagues. after explaining the characteristics of collaboration, teachers in each cycle rose to the challenge. they accepted the joint responsibility of working on the word problems and other activities during the workshops and implementation of mpsp. it is well documented in the literature that collaboration brings teachers together to implement new teaching strategies and reflect on their practices (robutti et al., 2016). during these interactions, teachers can observe, emulate, and advance their cognitive functions (vygotsky, 1978). this implies that learning is located in practice and happens as one actively participates in activities (lave & wenger, 1991). this was true for this study when teachers worked collaboratively during workshops. as teachers worked together during the intervention, i observed that they were able to look back and reflect on their teaching. this observation suggests that collaboration assisted teachers to easily accept the new ideas on mpsp that they were learning in the workshops since they tried them out through the support of a colleague during the implementation stage. this can be noted from ms n: ‘it was a great privilege to work with my colleague during this programme. having real conversations with him before and after the observations of the lessons benefitted me professionally in that he is familiar with the subject, caps curriculum, and mathematics problem-solving teaching strategies.’ teachers observed each other’s successes, challenges, and failures. my observations during the workshops were that as teachers collaboratively solved the word problems and planned how to teach these problems to their learners, they reached a shared understanding of how to implement polya’s (1957) four steps of problem-solving processes in their lessons. in addition, i observed that when teachers worked on activities collaboratively during the pd workshops it resulted in the advancement of their mpsp. the reflective interviews gave insight into teachers’ perspectives about the process of collaboration during the pd intervention. teachers in both cycles embraced collaboration and valued the discussions they carried out with their colleague on how to teach specific mathematics problems to learners. teachers reflected that working cooperatively with a colleague strengthened feelings of positive interdependence between them. positive interdependence, which means the success of one teacher was determined by the success of their colleague, is acknowledged to deepen learning (drago-severson, 2007). in this study, i observed that positive interdependence reinforced teachers’ understanding of new ideas on mpsp that they learned in the workshops, which in turn improved learners’ problem-solving processes (see chirinda, 2018). the outcome of collaborative learning was not a surprise since it concurs with loucks-horsley et al.’s (2010) observations that teacher collaboration results in effective pd and improved teaching quality and learning outcomes. in the second cycle, ms n, a novice teacher with one year of teaching experience, reflected that collaborating with her colleague during the pd intervention had vastly improved her mpsp and eliminated feelings of loneliness. as a newcomer both in the profession and in the school, ms n explained that she had felt isolated. this implies that collaboration eradicates feelings of isolation that beginner teachers usually find themselves experiencing. ms n valued collaborating with mr m as she felt she had someone to work with and no longer felt isolated. this finding seems to imply that there is value in pairing new teachers with the experienced members of the teaching staff. the finding agrees with vygotsky’s (1978) notion of scaffolding, where the more experienced help the novice progress to higher levels of understanding. this was evidenced by ms n’s reflection that collaborating with mr m during the workshops made her understand that grade 9 learners normally have the same misconceptions in problem-solving. she elaborated that having discussions with mr m on using problem-solving to teach some word problems helped her see how she could address these misconceptions. as she indicated in this quote: ‘as a beginning teacher, the discussions i conducted with my colleague helped me understand several aspects on the teaching of problem-solving including misconceptions learners usually have in problem-solving and how to handle them.’ the above quote suggests that collaboration supported ms n’s professional growth. this growth resulted in her teaching being accessible and effective to learners as she was now able to identify and handle learners’ misconceptions in mathematical problem-solving. this finding suggests too that collaboration is a factor fundamental to pd intervention for teachers’ mpsp in the study. nonetheless, my observations were that teachers’ collaboration in the context seemed limited because of the prescribed curriculum and a tight school timetable. this observation agrees with cookson (2005), who perceives that an overloaded curriculum like in the context of this study limits teacher collaboration and hampers the possibilities of teachers working together. the reality that participant teachers did not have time to work together implies that collaboration should be embedded in the school curriculum so that teachers have enough time in the school day to meet regularly and focus on supporting each other’s professional growth. final thoughts this study explored factors that are fundamental to a professional development intervention for teachers’ mpsp. this study’s findings were that teachers’ personal meaning, reflective inquiry, and collaborative learning are the factors fundamental to their professional growth in mpsp. this study’s focus on teachers’ personal meaning in the previous pd programmes they had attended seemed to have effectively addressed their personal meaning in pd in mpsp. in addition, the pd facilitator’s ability to recognise teachers’ personal meaning in mpsp seemed to have facilitated professional growth in their mpsp. the reflective inquiry process, in turn, gave teachers insight into their personal meaning in the activities and processes of the pd intervention. clarke and hollingsworth (2002, p. 947) posit that ‘if we are to facilitate the pd of teachers, we must understand the process by which teachers grow professionally and the conditions that support and promote that growth’. in this study, teachers’ personal meaning, reflective inquiry, and social participation supported and promoted their growth in mpsp. the primary recommendation from the study is that facilitators of pd must take into account these factors if they are to encourage teachers’ professional growth in mpsp. in this article, i argue that if pd processes and activities are relevant to teachers’ personal meaning, reflective inquiry, and collaborative learning teachers find the pd programme fulfilling and meaningful. this study contributes to the pd in the mpsp body of knowledge by having worked with teachers in an under-researched context. a study in this context is valuable since research conducted in contexts of advantage is ubiquitous in the mathematics education field and few studies originate from countries and contexts of historical disadvantage (skovsmose, 2011). because of inadequate monetary resources, the study focused on teachers in one south african district. the limitation of focusing on one district is that i could not explore how a pd intervention for mpsp could be designed and implemented to support grade 9 teachers in other districts. however, the objective of this study was not necessarily to explore several participants but to design a pd intervention to support teachers’ mpsp. nonetheless, i recommend that further research can be done with a larger, more diverse sample of south african teachers to determine how personal meaning, reflective inquiry, and collaborative learning promote their growth in mpsp. i feel that there has not been enough critical examination of this aspect in the south african context. this study revealed that collaboration is a factor fundamental to pd intervention for teachers’ mpsp; however, further research needs to be done to establish how best collaboration can be conducted in contexts where teachers feel that there is not enough time to collaborate because of prescribed school curriculums and tight school timetables. acknowledgements patrick barmby for supervising my phd. this article emanates from my phd work. competing interests the author has declared that no competing interests exist. author’s contributions i declare that i am the sole author of this research article. funding information this research was made possible by the national research foundation 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(2003). professional development of mathematics educators: trends and tasks. in a.j. bishop, m.a. clements, c. keitel, j. kilpatrick, & f.k.s. leung (eds.), second international handbook of mathematics education (pp. 877–917). dordrecht: kluwer academic publishers. article information author: jayaluxmi naidoo1 affiliation: 1school of education, mathematics and computer science education cluster, university of kwazulu-natal, south africa correspondence to: jayaluxmi naidoo postal address: private bag x03, ashwood 3605, south africa dates: received: 23 sept. 2011 accepted: 17 june 2012 published: 21 aug. 2012 how to cite this article: naidoo, j. (2012). teacher reflection: the use of visual tools in mathematics classrooms. pythagoras, 33(1), art. #54, 9 pages. http://dx.doi.org/10.4102/ pythagoras.v33i1.54 copyright notice: © 2012. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. teacher reflection: the use of visual tools in mathematics classrooms in this original research... open access • abstract • introduction • the teaching and learning of mathematics in south africa • strategies to promote mathematics and science in south african schools • the use of visual tools in the classroom • theoretical framework • methodology    • introduction    • participants       • master teachers       • learners    • ethical considerations    • research instruments or methods    • reliability and validity • findings and discussion    • why do master teachers use visual tools?       • making mathematics easier to remember       • making mathematics more interesting and fun       • making mathematics more concrete, accessible and comprehensible    • how do master teachers use visual tools in?       • level 1: organising the learning environment       • level 2: exploring teacher-learner interaction       • level 3: the use of representational tools • conclusion    • limitations and recommendations for future work    • implications for professional development • acknowledgements    • competing interests • references abstract top ↑ research has shown that the use of visual tools in mathematics classrooms is beneficial, but what we do not know is how south african teachers negotiate the use of visual tools (e.g. diagrams, gestures, the use of colour, et cetera) in classrooms. research was conducted with six ‘master teachers’ to explore the use of visual tools. master teachers in this study are expert teachers identified by the kwazulu-natal department of education. they are experienced teachers with the potential to mentor new teachers. master teachers were asked to complete a questionnaire, and they were observed and recorded whilst teaching mathematics lessons. each master teacher was observed at least three times. all the video recordings were analysed, after which each master teacher was interviewed. after each master teacher interview had been analysed, one focus group interview was conducted with learners at each school. the study was undertaken within a qualitative, interpretive paradigm. the study was framed within schön’s theory of teacher reflection. the findings suggest that each master teacher incorporated the use of visual tools in order to make mathematical concepts easier to understand for the learners. for example, one master teacher used a stick with coloured rubber bands to teach rotation about a point; another master teacher used various colours and lines on an interactive smart board to teach number patterns and a third used hand gestures to demonstrate the direction of the gradient of a line. interview data suggest that the incorporation of such visual tools came about as a result of teachers’ reflecting in action. these findings are important for advancing teacher and curriculum development. introduction top ↑ although 18 years have passed since the first democratic elections in south africa, schools are still unequally resourced (chisholm & sujee, 2006; soudien, 2004). schools differ largely in terms of human resources (the number of personnel at each school), physical resources (infrastructure, desk, chairs, classrooms, etc.) and teaching resources (equipment, textbooks, black or white boards, etc.). mathematics teachers, especially, are sceptical about the feasibility of teaching the same curriculum within the same time frame to all learners (naidoo, 2006), regardless of the inequitable distribution of resources (adler, 2001; reddy, 2005). despite their frustration some teachers with limited resources nevertheless manage to assist their learners in grasping complex concepts in mathematics.this study seeks to explore the following two research questions: 1. why do master teachers use visual tools in mathematics classrooms? 2. how do master teachers use visual tools in mathematics classrooms? visual tools in this study refer to diagrams, pictures, transparencies, mathematics manipulatives, gestures, and the use of colour. the teaching and learning of mathematics in south africa top ↑ education reformers in south africa are concerned about mathematics teaching in south africa because of the apparent inability of south african learners to compete successfully with their peers from other countries in global mathematics tests (howie, 2003; reddy, 2005). this concern emanates from the reality that society requires mathematical knowledge in order to survive and prosper, and that south african society is far behind in attaining this knowledge. recognising that education can contribute to positioning students on an equal level (freire, 1985), the department of education (doe) in south africa has urged south african learners to become ‘critical citizens’ in a mathematically democratic society (doe, 2003a, pp. 1–7). the underlying premise is that if learners are well educated, especially in a gatekeeper subject such as mathematics, they will be able to move up the ‘social ladder’. all south african learners are required to select either mathematics or mathematical literacy as a school subject in grade 10 and to continue with one of these subjects until the end of their school life. the mathematics curriculum requires the teaching of pure mathematical concepts. in the further education and training phase, learners are exposed to mathematical experiences that provide them with many opportunities to develop their mathematical reasoning and creative skills (department of basic education, 2011a). the intention of the mathematics curriculum is to prepare learners for abstract mathematics, which they will encounter in further mathematical studies in higher education. the mathematical literacy curriculum focuses on teaching concepts that are required for ‘everyday life situations’. the educational goal of this subject is to enable the learner to become an independent individual capable of contributing to and participating in the workforce of a developing democracy (doe, 2003b). furthermore, according to the mathematical literacy policy document (department of basic education, 2011b), the teaching and learning of mathematical literacy ought to provide learners with opportunities to analyse and devise mathematical ways of solving ‘authentic, everyday problems’. strategies to promote mathematics and science in south african schools top ↑ due to the global demand for an improvement in mathematics and science, the doe in south africa promoted a national approach to improve participation and performance in mathematics and science education (reddy, 2005). part of this approach was the identification of dinaledi schools nationally. dinaledi schools are considered ‘star schools’ in south africa. they were identified after discussions in 2000 on ways to promote the participation and performance of black learners and female learners in mathematics and science. initially, 102 schools were identified (kahn, 2004). in december 2008 there were 500 dinaledi schools throughout south africa, 88 of which were located in kwazulu-natal (kzn). dinaledi schools were provided with resources such as calculators, computers, text books and educational wall charts (doe, 2008). to further the aims of this approach, the kzn doe also announced that 120 ‘master teachers’ would be appointed, and that an additional 2400 master teachers would be identified (makapela, 2007). selected teachers from each dinaledi school were invited to professional development workshops in mathematics, physical sciences and life sciences. from this cohort of teachers, master teachers in mathematics and science were trained and afterwards professionally supported by subject specialists employed by the doe. the focus of the support and training was not necessarily on the use of visual tools but on improving teachers’ content knowledge in mathematics and science, improving the teaching of mathematics and science and enhancing learner performance in mathematics and science. in south africa, master teachers serve the same purpose as a mentor or expert teacher. they are senior teachers with the potential to mentor new teachers. the teachers observed in this study were master teachers. the use of visual tools in the classroom top ↑ the master teachers in the study often used visual tools unknowingly in their classes, for example when they resorted to the use of gestures, colour, lines and symbols. additionally, this study indicated that the master teachers often used visual tools with the intention of assisting learners to grasp abstract concepts in order to support and improve mathematical conceptual knowledge development. this is supported by elia and philippou (2004), who claimed that visual tools play an important role in communicating mathematical ideas and supporting the process of reflection. it also confirms the fact that teachers’ tacit knowledge, professional development and beliefs concerning the teaching and learning of mathematics influence the way in which they teach mathematics (remillard, 2005). roodt and conradie (2003) showed that the use of different approaches to the same problem enriches both learners and teachers. good teachers often use symbols, colour, diagrams and gestures in the classroom as an alternative to the routine approach of ‘talk and chalk’ teaching. the use of colour and other visual tools creates an exciting and interesting mathematics classroom (naidoo, 2011a). more approaches which encourage learners to be active and allow them the opportunity to demonstrate the extent of their thinking and creativity are therefore needed (barnes, 2005). stokes (2000) suggested that the use of visual tools assists in uncovering the role that visual reasoning plays in solving problems in mathematics. this leads to interesting results in the teaching and learning of mathematics. visual tools may also be used as a starting point to achieve interactive and stimulating learning environments (breen, 1997). in these learning environments, learners are able to interact easily with abstract concepts. theoretical framework top ↑ in order to consider the two research questions, we first need to consider some of the theory related to teachers’ practices. this study was framed within schön’s theory of teacher reflection and is based on the premise that a good teacher continuously reflects on his or her teaching. reflection is a process of reviewing the experience of teaching in order to describe, analyse, evaluate and inform learning about teaching (preen, 2007). schön defined two types of reflection: reflection-in-action and reflection-on-action (schön, 1983, 1987). reflection-in-action refers to the teacher’s ability to reflect during a specific lesson rather than after the lesson. this presents a dynamic approach in the teaching and learning process. in contrast, reflection-on-action involves thinking about and reviewing the lesson after the lesson has been concluded. this allows the teacher the prospect of evaluating and commenting on the lesson. teachers ought to learn from their own practical experience. in doing so, they can either engage in shallow problem solving processes entrenched in traditional norms, or (preferably) engage in a deeper level of problem solving which is more meaningful and challenging (schön, 1983). through this reflective process, a good teacher recognises that teaching is not a display of knowledge but a process which includes identifying an area of learning and deciding on interventions that will foster learning in this area (ursano, kartheiser & ursano, 2007). teachers are required to know how different concepts are interconnected so that, as the mathematics lesson unfolds, they can rectify any misconceptions learners may have (ball, lubienski & mewborn, 2001). in addition, mathematics teachers should comprehend the progression of concepts across grades as well as the appropriate use of context. the choice of substantial and applicable contexts, based on anecdotal experience, is necessary when teaching mathematics to ensure effective teaching and learning. learning is seen as a way of developing knowledge within meaningful contexts (handal & bobis, 2004). for contexts to be meaningful, the learners ought to be able to relate to these contexts. essentially, social contexts that are conducive to learning ought to be created, because the process of learning is itself both social (putnam & borko, 2000; wentzel, 2002) and cognitive. methodology top ↑ introduction the purpose of this study was to explore master teachers’ use of visual tools in mathematics classrooms. data were collected by means of various research methods so that the research questions could be considered. these methods included administering a master teacher questionnaire (mtq), grade 11 mathematics lesson observations, video recordings of the observed lessons, master teacher interviews (mti), and a focus group interview (fgi) with grade 11 learners. during the lesson observations a video camera was focused on the master teacher. the camera followed the master teacher throughout the lesson and captured all visual tools used during each observed lesson. this study was located within an interpretive paradigm. participants master teachers forty-five master teachers teaching at forty-five different dinaledi schools in kzn, south africa were invited to participate in the study. the reason for selecting master teachers was that they were experienced teachers who had shown evidence of being able to assist learners to produce good results in the grade 12 mathematics examination. the teacher selection was based on the grade they taught, convenience and accessibility. twenty master teachers accepted the invitation. the selection of the master teachers for the pilot and main study was random. ten master teachers were selected for the pilot study. the remaining ten participated in the main study. a further selection was done before the final sample of six master teachers was chosen. the six master teachers taught at schools in different social milieus. the schools differed with respect to location, context and access to resources. there were three male teachers and three female teachers in the final sample. some information about the master teachers participating in the study is given in table 1. all six master teachers were observed teaching grade 11 mathematics in their own classrooms. each master teacher was observed and video recorded at least three times. table 1: the master teachers in the final sample. learners a focus group interview (fgi) was conducted with selected learners of each of the schools in the final sample of the main study. learner selection was based on the learners’ level of interaction during the observed lesson. each focus group comprised of learners that interacted frequently in class, learners that did not interact frequently and learners that did not interact at all. these interactions encompassed interactions with both the master teacher and other learners within the classroom. ethical considerations the research officer at the kzn doe was contacted via e-mail and a copy of the proposal was posted for perusal. once gatekeeper clearance was granted by the kzn doe, ethical clearance from the university’s research office was applied for. before the pilot and main studies were conducted, principals of the selected dinaledi schools were informed of the study and processes that would be followed. subsequently each master teacher and learner who participated in the study was provided with an introductory letter. this letter discussed and defined informed consent, the right to withdraw and confidentiality. it also provided each participant with the reasons and purpose of the research. each participant was required to provide their written consent. in the case of the learners, their parents or legal guardians were required to provide written consent. pseudonyms were used to protect the identity of the schools, teachers and learners. research instruments or methods the master teacher questionnaire (mtq) was a structured pen-and-paper instrument divided into three main sections. these sections included the school profile, the school infrastructure and the master teacher profile. before the pilot test was conducted, the mtq was peer reviewed with colleagues working within similar research areas. after minor editing, the mtq was pilot tested with ten master teachers. i went to each school to administer the mtq. i asked the master teachers to complete the mtq at home, because i wanted to provide them with enough time to reflect on what they did in the classroom. i collected the questionnaires after three to five days, after the teachers had informed me that they had completed them. after some additional minor adjustments during the pilot testing phase, the mtq was administered to the remaining group of ten master teachers in the main study. i analysed each mtq by using a thematic coding process, and selected a purposive sample for the study. some of the themes used for coding were: resources used, teaching strategies, school infrastructure, teacher professional development, qualification levels of the teacher, teaching experience of teacher and class size. the selection of the final sample of six master teachers was based on their in-depth responses and diverse teaching strategies as mentioned in their questionnaire responses. the second research instrument that i used was an observation schedule. the observation schedule included a checklist of common visual tools (i.e. diagrams, symbols, gestures, calculators, charts, pictures, mathematics manipulatives, and the use of colour) and the frequency of their use during each lesson. the observation schedule also had an additional blank space at the end of the schedule to add other visual tools that were used by each master teacher. part of the observation schedule afforded me the opportunity to describe how each visual tool was used during the lesson. after each lesson observation, i analysed the video recording and notes from each observation schedule. i looked at instances where the teacher used a visual tool to assist in the effective teaching and learning of mathematics. in some instances the teacher brought in a visual tool. maggie, for example, brought in an overhead projector transparency that she had prepared because of a class discussion during her previous mathematics lesson. this was evidence that she had reflected on her previous lesson (reflection-on-action). in other instances the teacher created a visual tool ‘on the spur of the moment’. for example, alan realised during his lesson on transformation geometry that learners were confused about the terms ‘reflection’ and ‘rotation’. he then used his body as a visual tool to explain the term ‘rotation’ (an example of reflection-in-action). these examples of the teachers reflecting on and in action helped me in my planning for the next lesson observation; i knew what i needed to be aware of. at the end of the three lesson observations, each master teacher was given a copy of the video recordings. this assisted the master teacher in preparing for the master teacher interview (mti). i used semi-structured interviews in the study. i asked each master teacher a few basic questions to start off the interview and then based the rest of the interview on the master teacher’s responses on their use of visual tools within the classroom. at least six video clips of specific visual tool use were shown during each interview. these clips were used to remind each master teacher of the visual tools he or she had used. this prompted the master teacher to talk about the reason for the choice of visual tool, how he or she had thought it would help the learners during that specific lesson and whether or not the visual tool had been planned before the lesson or had been developed whilst the master teacher reflected in action. after each mti had been analysed, focus group interviews (fgis) were set up with groups of learners at each of the six schools. the selection of the focus group participants was based on their level of interaction during each lesson. video recordings of lessons were shown to each focus group. i asked the learners to focus on specific clips from each observed lesson and discussions ensued. the discussions revolved around the learners’ interpretations of each master teacher’s use of visual tools. i used these discussions to assist me in answering the research questions. essentially table 2 encapsulates the data collection process. reliability and validity the instruments were carefully designed and pilot tested. prior to pilot testing the instruments were circulated and discussed with colleagues within the same research area. the language used was basic and appropriate for the master teachers as well as the learners. the participants were from all parts of kzn and hence it could be presumed that the responses of these master teachers were representative of the responses of master teachers with similar backgrounds and qualification levels. however, no broad generalisations are made. at every step of the research process, data were analysed and coded using thematic coding. the possibilities of lurking variables were investigated before final coding themes were identified. findings and discussion top ↑ why do master teachers use visual tools? the master teachers in the sample were selected because they used visual tools in most of their mathematics lessons. this information was obtained from the mtq. in order to address the first research question we will look at the results of the data collected. evidence obtained in the study suggests that the key reasons why master teachers used visual tools in mathematics classrooms were to ensure that the mathematics became:• easier to remember • more interesting and fun • more concrete, accessible and comprehensible. table 2: the data collection process. making mathematics easier to remember penny used concrete objects to make abstract mathematical concepts more relevant and memorable to the learners. she did not use expensive visual tools, but chose to use tools that were easily accessible to learners in her school. she used different colours on the board, charts, pictures and gestures. whilst teaching the parabola she used smiley faces and sad faces. she believed that when learners were exposed to these symbols in future lessons, they would remember, on reflection, what the symbols signified. she explained that: ‘… children remember it … i use it for that purpose expecting them to remember’ (mti, 25 june 2009).similarly, maggie used symbols that her learners had been exposed to in previous lessons or grades. she explained her reasons for using the symbol that represented a 90 degree angle as follows: ‘… the standard symbol that we use in geometry … it is something that is standard and you learn and remember from grade 8’ (mti, 05 august 2009). dean also used visual tools to assist learners to remember rules: ‘… if they [the learners] can remember the rule … by showing them [the learners] the rotation ... the learner sees ... how the position changes‘ (mti, 26 june 2009). alan’s learners were of the opinion that the use of visual tools helped them to remember the mathematics they were taught. this was evident in the following excerpt taken from the fgi with learners: learner 6: in the exams it is easier to remember stuff and to study. sometimes the rules are so close that one can easily misinterpret it for the next rule and by having a diagram we can actually pinpoint which is the exact rule. (fgi, 13 august 2009) making mathematics more interesting and fun in karyn’s lessons she wanted to prevent her learners from becoming bored or losing interest. she used her visual tools to assist her in this. this was evident from three separate responses:… here again they need to visualise it, to see it … if you just talk to them they might fall asleep. … it might be boring … i think if you point you can just get their attention … this is what i am talking about. … to make the lesson more interesting. (mti, 18 june 2009) in the excerpts above, karyn was referring to her use of gestures and different colours in her mathematics lessons. sam had a similar notion; he claimed that he used diagrams ‘... to make the lesson more interesting ... ’(mti, 02 july 2009). making mathematics more concrete, accessible and comprehensible with the concretisation of abstract mathematics concepts, mathematics becomes more comprehensible. for example, dean was teaching a section in transformation geometry where he wanted to show learners how to rotate two points 90° about the y-axis. he used a stick with coloured rubber bands when he demonstrated the direction the learners needed to rotate the points a and b. this was a tool that was easily accessible within the context of the school. using the stick with the coloured bands, dean indicated where the new points would be located after the transformation. dean demonstrated the rotation ‘on the spot’; he developed this concrete visual tool when he realised that his learners were having a problem with understanding the concept of rotation. dean wanted to make the mathematics he taught concrete and more accessible to his learners and so he reflected-in-action. by using a simple visual tool, he managed to make an abstract concept concrete so that the learners could come to grips with it. this was evident from the following learner comment: learner 5: if he [dean] hadn’t used the stick ... then we would have been confused, we would not have known which direction we were moving in [the direction of rotation]. (fgi, 25 august 2009) once he had completed his demonstration, dean prompted his learners to try to solve the initial task. evidence obtained in the study showed that they were able to solve the problem successfully. in another example that occurred in dean’s classroom, dean highlighted parts of a diagram on the board by using coloured chalk. he did this to make the mathematics more comprehensible and visible. this was evident from what dean later said during the interview: i think that it is beneficial … i highlighted only certain parts … so they [the learners] know exactly which parts are undergoing some transformation … instead of giving them just a set of notes … they can see exactly which part we are manipulating. (mti, 26 june 2009) alan also used tools that were inexpensive and easily accessible, such as paper, coloured chalk and his body. alan’s learners commended their teacher on using visual tools because they felt that visual tools helped them. this was evident in the following transcript taken from the fgi: learner 3:it becomes much easier to grasp, the diagram was … helpful. (fgi, 13 august 2009) how do master teachers use visual tools? in order to address the second research question, we will look at the results of the data collected. master teachers used visual tools in the mathematics classrooms as a scaffolding technique. scaffolding teaching and learning through the use of visual tools in mathematics classrooms recasts the relationship between what teachers teach and how they teach. it foregrounds the fact that the ways in which teachers teach and the ways in which learners learn are inextricable aspects of the classroom culture. the master teachers showed evidence of reflecting in action as well as reflecting on action when using visual tools in the classroom. for example, alan used his experience of what worked and what did not work when he prepared his lessons. he displayed evidence of reflecting on action:… i learnt … if something doesn’t work in the first class i will try something else … in the second class. (mti, 06 august 2009) the classroom observations, video recordings, the mti and the fgi provided evidence that, rather than using direct teaching strategies or the traditional approach to teaching mathematics (chalk and talk), each master teacher incorporated scaffolding techniques to support their learners’ development in mathematics (see also naidoo, 2011b). they did so without any training or urging from the researcher. anghileri (2006) distinguished three levels of scaffolding. the three levels are explained below. level 1: organising the learning environment level 1 scaffolding relates to the manner in which the teacher organises the learning milieu. this organisation may include the use of pictures, charts, seating arrangements and peer collaboration. for example, whilst teaching the different types of graphs (hyperbolas, exponential graphs and parabolas), penny decided to divide her class into smaller groups. penny provided each group with chart paper and asked them to answer different problems on the chart paper. she then asked each group to present their solutions. in penny’s classroom the learning environment was reorganised to accommodate active peer collaboration to scaffold the teaching and learning of mathematics. this was evident from her response in the mti:the aspect of group work helps those that are not picking up the concepts easily ... it helps to encourage peer learning … learners presented their work on charts which is visual as well. so they get to see it. (mti, 25 june 2009) penny used charts as a manipulative that encouraged learners to work collaboratively within a visual environment. the use of learner centred approaches to learning promotes the development of higher order skills such as critical thinking and problem solving (brush & saye, 2002). learners were also encouraged to work collaboratively with one another and discuss their ideas in front of the class. this strategy reinforces the importance of peer learning because it provides learners with the confidence to talk about mathematics. the strategy also gives learners the opportunity to become a part of a group where they feel supported and validated (dodge & kendall, 2004). penny knew from past experience that group work was important for teaching and learning, and she prepared this activity, which involved both group work and the use of a visual tool (a chart), before coming into the classroom. she planned in advance to create a group work activity and environment in order to ensure effective teaching and learning of graphs in algebra. during our discussion of her group work activity, penny showed evidence of reflecting-on-action. the next level of scaffolding revolves around the interactions within the classroom, specifically the interactions that occur between the teacher and learner. figure 1: teacher strategies for scaffolding learning at level 2. level 2: exploring teacher-learner interaction when teachers demonstrate, discuss and explain concepts to their learners, learners do not always find it easy to comprehend these concepts. to assist in alleviating this challenge, learners and teachers need to interact with each other, and there needs to be collaboration between all members of the learning community. scaffolding at level 2 includes different levels of teacher-learner interaction. this kind of interaction relies on the teacher reviewing and restructuring what is happening in the classroom and therefore requires the teacher’s reflecting-in-action. level 2 scaffolding may be represented diagrammatically as shown in figure 1.as can be seen in figure 1, scaffolding at level 2 incorporates two major aspects: reviewing and restructuring. reviewing consists of two subsets. the first subset involves the teacher looking, touching and verbalising what is required. this then progresses to parallel modelling. parallel modelling refers to instances where the teacher identifies learner misconceptions and misunderstandings, creates tasks that share characteristics with the learners’ problem and then solves the tasks in collaboration with the learners (anghileri, 2006). the second subset of reviewing involves the teacher using probing as well as prompting techniques. this is followed by the learners explaining and justifying their ideas. subsequently the teacher interprets the learners’ actions and discussions. there was evidence of prompting in dean’s classroom, as he used gestures to prompt learners to use the correct keys when they used the calculator. when questioned about the use of his gestures, dean showed evidence of reflecting-in-action. during the lesson he realised that the learners were having a problem with the use of the calculator, and so he immediately used gestures to demonstrate the correct sequence of keys to be used. in the mti he stated: it’s the force of habit, i am not aware that i am doing this [using gestures] … when i did this [he used the sign language of a bracket during the interview] then they [the learners] can see it … that’s how it is on the calculator so they know which key to press. (mti, 26 june 2009) during the reviewing process, learners ought to be encouraged to verbalise what they see and think. the learners need to be motivated to explain and justify their actions and comments. through interpretation of learner comments, prompting and asking probing questions, teachers have a higher probability of identifying misconceptions and misunderstandings in mathematics thinking and learning. this was evident in karyn’s classroom when she asked her learners to find the perimeter of a triangle. as she spoke to the learners she realised that they did not understand what was required of them. karyn explained her reasons for probing and the use of gestures as follows: to make sure that they know what is perimeter … if you use a word make sure that they understand what you are talking about. they always get confused between the perimeter and area. (mti, 18 june 2009) the above transcript shows evidence of karyn restructuring and reviewing her questions and discussion in class to ensure that what she was teaching became more comprehensible to the learners. this demonstrated that karyn was reflecting-in-action; she did this ‘on the spur of the moment’. as discussed earlier, parallel modelling occurs after touching, looking and verbalising. karyn used parallel modelling when she used examples similar to the ones learners had a problem with. she did this when she reviewed; she solved problems until the problem-solving process made sense to her learners. she formulated problems and examples from her learners’ comments. for example, whilst karyn was teaching analytical geometry she placed a question on the board for her learners to attempt. in order to assist her learners she prompted and probed learner responses by asking key questions. this is evident from the observation transcripts: karyn: what is the restriction for n and m? learner 1: they must be an equal distance apart. karyn: class, the suggestion is that they must be equal distance apart. how can we make sure of this? learner 2: calculate the distance of mp and np; they must be the same. (obs 1, 18 march 2009) karyn used deictic gestures and supported her learners’ understanding of tasks by operating from her learners’ ideas. once karyn was confident that her learners could work on their own, she allowed them to work independently. karyn reflected in action because the lesson developed as the learners interacted with her. in figure 1 two subsets to restructuring in the classroom are also shown. the first subset requires the teacher to provide meaningful contexts which then leads to the next level, where the problems are simplified. the second subset of restructuring tasks involves rephrasing what the learner says, which leads to the level of negotiating meanings. in my study i found that whilst the master teachers were in control, as is common with traditional approaches to teaching (chalk and talk), they also involved their learners in the discussion. they reviewed and restructured tasks to accommodate their learners’ needs. this was evident in maggie’s lesson on calculating areas, for example. she used her mathematical language to serve as a scaffold to the teaching and learning of mathematics. whilst this is what is expected in a mathematics classroom, the manner in which maggie used the register of mathematics stood out. she made the register of mathematics meaningful. maggie started by talking about the area of triangles. she then asked her learners to think about calculating the area of triangles (this is generally taught in earlier grades). she provided a meaningful context by reminding learners of the mathematics they had been exposed to previously. she spoke about perpendicular lines; the base of triangles, the heights of triangles and the vertex of the triangle. she punctuated her words with deictic gestures. this was also evident in penny’s classroom. when penny taught parabolas, she wanted to teach her learners a method for remembering the shape of the parabola. through discussion, penny and the learners negotiated to use facial gestures to signify the shape of the parabola. a smile signified that the coefficient of a in the function y = ax2 + bx + c was positive and a frown signified that the coefficient of a in the function y = ax2 + bx + c was negative. through her rephrasing of what learners had articulated and through negotiating meaning, she helped the learners to remember the shape of the parabola, based on the sign of the coefficient of a. whilst this is not something new, penny and her learners also negotiated the notion that the change in shape was prompted by a change in the value of the coefficient of a. in another example that occurred in alan’s classroom, alan used a sheet of paper to represent the cartesian plane; he negotiated and mediated meaning in his classroom to achieve his outcomes. he expected his learners to use this manipulative to transform, reflect and rotate coordinates about the x-axis and y-axis. he reflected-in-action and used the sheet of paper as a visual tool to facilitate understanding. after demonstrating, parallel modelling and probing, his learners comprehended the different transformations, according to the evidence in the video recordings of his lesson. when restructuring tasks, the teacher simplifies the problem or rephrases the learners’ comments with the aim of negotiating meanings and advancing understanding. meaningful contexts are created so that abstract situations become more accessible to the learner. during a trigonometry lesson, for example, karyn used the context of flying. this context was meaningful and useful to the learners as can be seen in the following: learner 4: we are doing trigonometry in maths and we did an example with an aeroplane so even in maths we are learning how to apply why they use trig and why we need trig. learner 8: when she [karyn] uses things like reality stuff then it makes me understand more better ... with real life examples it makes it [mathematics] easier to understand. (fgi, 21 august 2009) one of maggie’s learners commented on another example that the teacher had used: learner 1: as she [maggie] is going along she will draw the triangle and she will say like point a and then she would redo point a and point b and point c and then as she goes along explaining what she does she redraws the triangle basically with either the same colour or even with a different colour, she is actually redoing line ab just to emphasise that this is the line that you will use and these are your measurements. (fgi, 19 august 2009) level 3: the use of representational tools level 3 scaffolding refers to the use of representational tools with the aim of generating conceptual discourse within the classroom (verenikina & chinnappan, 2006). these tools assist in making abstract mathematics more accessible to learners. for example, alan used a sheet of paper to represent the cartesian plane. during his discussion with his learners he taught them how to transform shapes about the x-axis and y-axis via the use of this representational tool. the abstract concept of reflection was taught by using fold lines on the sheet of paper. alan’s choice of using the paper came as a result of his reflecting-in-action. during the mti and after being shown the video clip of him using this sheet of paper, alan said that whilst he was teaching this lesson he realised that learners did not understand the concept of reflecting shapes about the axes, so he decided to use a hands-on example to help them.dean used a stick with coloured rubber bands to make the abstract concept of rotation more accessible to his learners. during the mti dean maintained that he had used this visual tool ‘on the spur of the moment’. this comment illustrates evidence that dean reflected-in-action to advance the effective teaching and learning of mathematics. conclusion top ↑ from the data collected it was evident that the master teachers used visual tools to make the mathematics that they taught easier to remember, interesting and fun. additionally, the master teachers wanted to make the mathematics they taught concrete, accessible and comprehensible. in order to achieve their aims, each master teacher used different levels of scaffolding techniques during their lessons. this study also emphasised the importance of teacher reflection in using visual tools to assist learners in understanding mathematical concepts. visual tools were used in different contexts in order to advance the teaching and learning of mathematics. the evidence gathered in the study showed that most of the visual tools that were used were accessible and inexpensive. they were tools that could be obtained and used within any social context.the use of the visual tools were either planned before the lesson or created during the lesson. this study also illustrated that each master teacher successfully engaged their learners in the classroom by using visual tools. as a result of the use of visual tools, the learners were not merely passive recipients, but active participants. meanings were constructed in the classrooms through negotiation with the classroom community. there was constant interaction between the teachers and learners within the classroom community. this study also demonstrated that whilst scaffolding has become useful for teachers (verenikina & chinnappan, 2006), the purpose of scaffolding is to provide learners with a teacher-supported transition. this implies that after learners had looked at and listened to the teacher, as he or she illustrated a particular mathematical concept, they could then be required to perform the skill independently. however, whilst the use of scaffolding in mathematics is necessary, scaffolding is useless on its own. it is necessary that scaffolding be complemented by mathematical understanding, together with the ability to think, perceive and analyse mathematically (lewis, 2010). this mathematical understanding is achieved through constant teacher-learner interaction and visual tool use. this study further demonstrated that an effective teacher does not rely on the traditional ‘chalk and talk’ methods only, but is a facilitator who values the power of tangible teacher-learner interaction. through the use of visual tools, well planned interactions, reflecting in and on action, effective teaching and learning of mathematics may be promoted in any classroom regardless of context. however, it is imperative that the teacher chooses each visual tool carefully, taking into account the exact outcome he or she has in mind. limitations and recommendations for future work based on evidence obtained in this study, the use of visual tools in mathematics classrooms has proved to be beneficial; it is recommended that pre-service and in-service institutions provide teachers on-going support in this area. this study explored master teachers’ use of visual tools in one province of south africa; it is therefore recommended that further research be conducted which focuses on exploring the use of visual tools in other provinces in south africa. this study has provided valuable data; however, one limitation involves a technical aspect. it would probably have been beneficial to use two video cameras, one directed at the master teacher and one focused on the learners. this would have captured the learners’ responses and expressions, especially in situations that were considered critical moments in each lesson. this would also have captured relevant learner responses when the master teacher resorted to using visual tools whilst reflecting-in-action. implications for professional development the teaching and learning of mathematics through the use of visual tools is a useful strategy for teachers teaching in schools in different locations and contexts. this study foregrounds teacher reflection via the use of visual tools. to assist in promoting the use of visual tools, teacher educators ought to provide training to both in-service and pre-service teachers on the use of visual tools in the classroom with the aim of advancing the effective teaching and learning of mathematics. to do this teacher educators need to understand and value the importance of teacher reflection. acknowledgements top ↑ i would like to thank the university of kwazulu-natal for funding this study (doctoral research grant and competitive grant). the opinions expressed here are those of the author and do not necessarily reflect the position, policies or endorsements of the university. competing interests no financial or personal relationship(s) have inappropriately influenced the writing of this article. references top ↑ adler, j. 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(2002). are effective teachers like good parents? teaching styles and student adjustment in early adolescence. child development, 73 (1), 287−301. http://dx.doi.org/10.1111/1467-8624.00406 6201 bennie pythagoras 62, december, 2005, pp. 23-30 23 developing teachers, developing as a teacher: a story about a story kate bennie university of cape town email: kbennie@maths.uct.ac.za in this paper i reflect on my changing roles as a mathematics educator, that is, as a teacher educator and as a classroom teacher in a secondary school. this is a personal account of the challenge of translating my beliefs about mathematics teaching and learning into everyday classroom practice. the presentation i use is based on the work of rowland − the account is presented in the form of personal reflections on a story written about playing the two different roles of teacher educator and classroom teacher. i use the process of writing to try to make sense of my experiences and to explore the use of story as a research methodology. although the story is intensely personal, there are identifiable themes that run through the narrative, which i suggest may resonate with the experience of other mathematics educators. introduction this paper is a reflection on a story. the story is about two consecutive periods in my career in mathematics education − working as a teacher educator for a non-governmental organisation, followed by classroom teaching in a secondary school. the reflection is on my experience of change in moving from working with teachers to working as a teacher. this paper is, in fact, a story about a story for as rowland (1997: 19) argues, “reflecting is a way of storying”. in this paper the reflections themselves constitute a story about my attempts to make sense of my experience. background i spent the first four years of my career in mathematics education teaching in a secondary school. i was then given a three-year secondment to work as a project worker for a nongovernmental organisation, mathslearn.1 the general aim of the project was to improve mathematics education through the development of learner and teacher materials, teacher education and research. as a project worker, i was expected to be involved in all three components of the project. the work at mathslearn was informed by what came to be called the “mathslearn philosophy of teaching and learning”, the features of which are as follows: a socio-constructivist view of mathematics and the learning of mathematics; researchbased materials development; learner-centredness;2 1 the name of the organisation has been changed. 2 “learner-centred” in the mathslearn sense means that the teaching and learning process recognises and builds on the the promotion of both equity and excellence; and the belief that all learners can learn mathematics. national and international mathematics educators gave input into the development process and the work was trialed in selected project schools. the mathslearn project ended after three years and i returned to full-time classroom teaching, working with grade 10, 11 and 12 learners in a secondary school. this school has an excellent record of academic achievement in the national examinations and is well known for its efforts to work with learners from varying educational backgrounds. during my fourth year at this school, i presented a workshop at the association for mathematics education of south africa (amesa) national congress. in this workshop i shared ideas on the project work that my learners had been doing. a workshop participant posed the question, “i am interested in your move from being a teacher educator to a practising teacher at school level. have you been able to apply what you were advising teachers to do?” “no, not at all,” i initially replied. however, as i tried to explain my thoughts to the audience my answer changed. i ended my response: “yes, everything i have done in the past four years has been influenced by my experience at mathslearn.” this short exchange during the workshop, and my subsequent unease about my very different responses to the question, provided the spark for me to reflect on my experiences since leaving mathslearn four years earlier. existing knowledge of learners, and responds to their different learning needs. developing teachers, developing as a teacher: a story about a story 24 motivation my reasons for writing this paper are threefold. firstly, i feel that my experience of returning to classroom teaching after working in the field of teacher education is a rather unique one. the trend in the movement of educators appears to be in the opposite direction, from teacher to teacher educator. i feel that my reflections may shed some light on the complex relationship between teacher educator and classroom teacher. secondly, in writing this paper i have had an opportunity to explore the idea of writing a story as a method of research, as suggested by rowland (1997). lancy (1993: 169) notes that there is little tradition of personal accounts in educational research, but comments: in general i see this area growing in importance as we find that humans use story or narrative to organise their understanding and memory of events. i also see that our best access to teachers’ and students’ understandings of the meaning of classroom experience may be through life narratives, rather than through structured classroom observations and interviews. the view that meaning can be created through a narrative is based on a constructivist view of knowledge. crotty (1998: 58) describes “constructivism” as “the meaning-making activity of the human brain”. so, thirdly, from the constructivist perspective, the writing of this story could help me to make sense of my experiences of change. the structure of the paper the structure of this paper is based on rowland’s notion of reflections on a story (1997). the story about my experiences as a teacher educator and as a classroom teacher is written in the boxed sections of text. this narrative takes the form of descriptions of events and interactions i remember from a seven-year period. it also contains extracts from my writing as well as writing by my mathslearn colleagues. these extracts have been selected to illustrate my thinking while playing the two roles, teacher educator and classroom teacher. my reflections on this story are given in plain text, in between the boxed text. in presenting this paper for publication i have chosen to structure the narrative according to certain themes that emerged during the writing process. these themes, both personal and public, relate to wider issues in mathematics education. it should be noted that the decision to make use of this particular structure does not represent a conclusion in my reflections, but only one moment in the sense-making process. for me, the process of “storying” continues beyond the publication of this paper. the story my career choices as a reflection of my values during the third year of mathslearn the staff members are actively seeking work for the following year. i inform them of my intention to return to classroom teaching. a colleague asks, “so why go back to classroom teaching? why not continue working as a teacher educator and use all that you have learnt in the past three years?” the decision to return to classroom teaching in a secondary school was not taken lightly. my work as a mathslearn project worker was interesting, challenging and exciting. i visited schools every week, working with teachers in their classrooms on implementing the mathslearn approach to teaching and learning and trialing the mathslearn materials. i had the luxury of being able to view the school system as an outsider and to reflect on the work of a classroom teacher. i was frequently reminded of the daily stresses of being a full-time teacher in a secondary school. why, then, did i decide to return to this system? the decision to return to teaching was a purposeful one on my part, and was certainly not a last resort when i could not find other work. firstly, i definitely missed the challenge of teaching mathematics to teenagers, the daily interaction with these learners and the opportunity to get to know them as individuals. secondly, there were times during my work as a teacher educator that i felt ill-equipped to advise the well-qualified and resourceful teachers with whom i was working. who was i to advise teachers what to do, when i only had four years’ teaching experience? i felt that, if i was to have credibility as a teacher educator in the long-term, i needed to gain more classroom experience. does this mean that i was not convinced of the practicality of the mathslearn philosophy i had been advocating? i think not. i had seen the mathslearn approach work in classrooms that were certainly more difficult than the situations i was likely to encounter in my new classroom. i wanted to try out what i had been working on for three years with my learners. i saw my classroom teaching as an opportunity to “develop”, rather kate bennie 25 than to “test” the mathslearn philosophy of teaching and learning. thirdly, on being offered a secondment to mathslearn i was made aware that, as a practising teacher with only four years of experience, i was privileged to be given the opportunity to be part of this project. throughout my three-year period of employment i was conscious of my position in this regard, as well as my duty to continue to make a contribution to mathematics education when the project ended. by returning to teaching, i was taking what i had learnt back to the classroom and to other mathematics educators. my colleague at mathslearn continues the challenge: “well then, why not apply to work at a previously disadvantaged school? why go to a private school?” prior to my secondment to mathslearn i was employed at a secondary school. my job required that i not only teach mathematics, but that i also get involved in extra-mural activities. while i enjoyed these additional duties and could clearly see the value of this work, i gradually became frustrated by the time it was taking up. i felt that i did not have enough time to become the type of mathematics teacher i really wanted to be. my experiences while working at mathslearn had confirmed to me that i had made a good choice of career. now i wanted to find the environment in which i could do my work as a mathematics teacher to the best of my ability. my new appointment would provide the opportunity for me to focus on my teaching (without the pressure to get involved in extra-mural activities). further, the opportunity to work with small classes of learners of varying mathematical ability would enable me to implement the mathslearn ideas i so strongly believed in. the choices i made about my career beyond mathslearn were based on my values as an individual and as an educator: i value teenagers; i value classroom experience; i value finding personal fulfilment in my work; i want to be a good mathematics teacher; and i want to make a contribution to mathematics education in the country. by returning to teaching i was attempting to live in the direction of my values (mcniff, 2002: 13). perspectives on equity while acknowledging the importance of the distribution of financial resources and the provision of a curriculum that caters for different genders, cultures, language groups, etc. in addressing the issue of “equity” in education, mathslearn has developed particular views on the notions of “equity” and “diversity” and an accompanying teaching approach. this philosophy is based on the expectations we have of learners’ abilities to do mathematics and is aimed at ensuring that all learners are successful in mathematics for longer than has been the case in the past. (extract from a paper written by mathslearn staff) the term “equity” is used in a variety of ways in education, for example “equity as access”, “equity as multiculturalism” or “equity as support”. there is clearly a need to pay attention to these perspectives on equity in the context of south african schooling, for example through the provision of financial resources, infrastructure, appropriate curricula and support. yet the context is a complex one. my experience of working with teachers at some of the mathslearn project schools suggested that these so-called “previously disadvantaged” schools were staffed by the most talented and dedicated mathematics teachers i had yet encountered, and they performed their work under difficult circumstances. working with these teachers forced me to interrogate my assumptions around the notions of “disadvantage” and “advantage” in relation to schooling, as well as my ideas on the role of teacher education in achieving equity in different schools. the mathslearn philosophy of equity is a broad notion that extends to all learners in all schools, whether classified as “advantaged” or “disadvantaged”. the school at which i accepted a teaching post after the closure of mathslearn offered me an opportunity to work with learners from a variety of academic backgrounds. some of the learners were high achievers in mathematics, while others simply wanted a second chance at passing the subject. it was this feature of the school that was the deciding factor in taking the job as head of mathematics. in line with the mathslearn philosophy of equity, here was my opportunity to help all learners take mathematics for longer. did it matter that not all the learners at the school were from so-called “previously disadvantaged” backgrounds? working with learners drawn from a variety of schools and backgrounds, i was constantly reminded of the common difficulties all my learners had when learning mathematics. of course some of the learners had to overcome great hurdles, for example, of learning mathematics in a second developing teachers, developing as a teacher: a story about a story 26 language, but many of the learners had a poor understanding of mathematical concepts such as fractions and functions. in my new job at the school it soon became very clear to me that there was a great deal to be done to help all the learners achieve their potential, no matter what their background. furthermore, a number of the learners who enrolled at the school had been told by their previous schools that they were either not capable of doing mathematics, or were not capable of doing the subject on the higher grade. these learners, many of whom were from “previously advantaged” schools were being filtered out of mathematics. while many of these assessments were clearly correct, some i believe were not accurate. during my time at the school i witnessed a number of learners who, given another chance and additional support, proved that they did have the potential to succeed. this did not mean that they passed grade 12 mathematics with distinction, but the fact that they had persevered and passed the subject was a great personal achievement for each one. and a most exciting and rewarding victory to witness. our view of “equity” is that all learners can learn provided they are given appropriate instruction and time. (extract from a paper written by mathslearn staff) john enters grade 10 at the school, having skipped grade 9. he is determined to take mathematics, and we decide to give him a chance. very soon i notice large gaps in his mathematical knowledge. for example, john cannot add and subtract integers mentally. when he tries these calculations on his calculator, he usually gets the wrong answer. after some questioning i notice that he cannot order integers on a number line. following discussions with the school principal, i decide to work at a slightly slower pace with john. this leads to giving him different assessments to the rest of the class. none of the other learners seem to take much notice of this. john works hard during the year, and at the end of grade 10 he passes the same exam written by all grade 10 learners. john continues with mathematics in grade 11, but by the middle of the year his absenteeism from the school is high. when he does attend class he is aggressive and he repels my efforts to help him. he scrapes through the final exam, with the help of a private tutor. during his grade 12 year, john’s high level of absenteeism continues. he shows little interest in his studies and he fails mathematics on the standard grade. when john passed mathematics in grade 10 i sensed that he might turn out to be one of the school’s many success stories. within two years this “success story” had failed the subject. i began to ask myself, “can all children learn mathematics?” in my first year at the school i have a large class of grade 11 students of mixed mathematical ability and mathematical background. they are typical teenagers – boisterous and keen to test the tolerance level of their new mathematics teacher. jason stands out as a particularly difficult member of this class. he seems intent on displaying an aggressive attitude towards both his teacher and his classmates. noticing the negative attitude of many of the learners in this class to the subject, i try to vary the tasks, often dipping into the wealth of resources i have brought with me from mathslearn. on this particular day the learners are working on an investigation in algebra. predictably, jason refuses to work with a partner as instructed and grudgingly begins the task. gradually he makes progress and i am astounded by the mathematical skills i see being used before me: he is asking himself questions, investigating different possibilities and communicating his ideas with ease. i decide that this is one of the most exciting things i have seen in my classroom. a year later, with a few months to go to the final grade 12 exam, jason drops mathematics and takes another subject. “can all children learn mathematics?” yes, i still firmly believe that this is so. in line with the mathslearn philosophy of equity, john and jason were given time and support, yet they did not succeed. my explanation is that there are other issues at play here. we cannot teach learners mathematics in a vacuum, removed from the everyday pressures of being a teenager. in both john and jason’s cases it appears that there were other factors, beyond the mathematics classroom, kate bennie 27 that prevented them from achieving their potential in mathematics. using assessment to inform teaching and learning … assessment can be and should be a communication process between the teacher and the learners, a process that supports teaching and learning. it is a tool that fosters equity by providing the teacher with the necessary analysis of the learner’s needs. the needs, if not fulfilled, will in the long run filter them out of mathematics. if we want to give learners a fair chance to succeed in mathematics in the long run, and not only in the short term, we have to find ways to support them in the construction of some key mathematical notions.(extract from a paper written by mathslearn staff) using diagnostic assessment i was able to identify gaps in john’s understanding of number. i had the support of the school principal to use different assessments with different learners. diagnostic assessment and the use of assessment to inform teaching are integral to the mathslearn philosophy, but there were many times in my four years in the classroom that i asked myself, “is my approach learner-centred in the mathslearn sense? am i really working from what the learners know?” i often felt that i was not spending enough time doing diagnostic assessments and profiles of learners’ understandings to help me plan my teaching. there just seemed to be no time. i have a possible explanation for my feeling of dissatisfaction with my performance. thinking back i was, in fact, using diagnostic assessment, whether this was in the form of specific diagnostic tests at the beginning of a section, information i gathered from formative assessments, or from my observations of the learners and my interaction with them. perhaps trying to assess my learners’ understanding had become so much part of my daily teaching that i was unaware of actually doing it. perhaps the fact that i kept questioning my practice and feeling dissatisfied with my performance, means that i was constantly aware of the need to be learner-centred. “covering” the curriculum there is another possible reason for my dissatisfaction with my ability to implement the mathslearn philosophy. although i think i had an understanding of the needs of my learners, i often felt powerless under the time-constraints to respond to their needs. although some of my learners entered the school in grade 10, a large proportion only joined the school in their grade 11 or 12 year. i continue to ask myself, “given the need to prepare learners for the current matric exam, is it realistic and even possible to fill the gaps in learners’ conceptual understanding in a short time?” sam had completed twelve years of schooling and had passed standard grade mathematics in grade 12. she enrolls at the school to repeat her grade 12 year in the hope of improving on her performance. one of her aims is to pass mathematics on the higher grade. she is a remarkably determined and diligent learner. we meet once a week after school hours to work on the subject. during one of these sessions i notice that she has no concept of a number line and cannot order numbers on a number line. carter and richards (1999: 73) refer to “the universal issue/dilemma of time” and “the teachers’ belief that if they do not spend enough time “covering” the “curriculum” they will be damaging their students.” “covering” the curriculum certainly seems a safe route for the teacher: if she can place a “tick” in the correct space next to each mathematics topic in the work scheme, then it appears that she has done her work (and hence not “damaged” the learners). but i would argue that if teachers focus on completing the syllabus at the expense of conceptual understanding they may, in fact, be “damaging” their learners. sam had studied school mathematics for twelve years, yet she had no concept of the number line. she had passed standard grade mathematics in grade 12, but what does this result say about her understanding of the subject? the final grade 12 examinations are coming closer and closer. i lie awake at night, “will i finish the syllabus? will i finish the syllabus? will i finish the syllabus?” runs through my mind, over and over. although i had strong feelings about teaching for understanding, i was working within a system at the school that required me to complete the grade 12 syllabus each year, and to make a very good job of it, too. fortunately i was given a certain amount of flexibility in grades 10 and 11 to work on learners’ grounding in mathematics. but each year, i was faced with new learners who only entered the school in grade 12, many of whom had large gaps developing teachers, developing as a teacher: a story about a story 28 in their mathematical understanding. what happened to my dreams of implementing the mathslearn approach to teaching and learning? despite my attempts to be learner-centred in the mathslearn sense, there were times that i resorted to “chalk-and-talk” in grade 12. i am not proud of this, but the only redeeming feature i can find in this experience is that this approach was accompanied by a constant sense of unease about what i was doing. i could never accept this approach as “the way mathematics teaching is”. this was definitely not the type of mathematics teacher i wished to be. innovation in the classroom looking back i realise that i adopted certain strategies to cope with my frustration at having to “cover the syllabus” and at the same time keep the mathslearn vision alive. for example, i worked at a slightly slower pace than some of my colleagues, as i tried to build in some remediation of the problems i diagnosed along the way. so after a test, i did not proceed on to the next topic immediately, but dedicated a small amount of time to trying to work with learners on their difficulties. this was usually followed by a re-assessment. this meant that i did not have three weeks for revision before exams, as many of my colleagues had, but my explanation to them (and to my learners who expected this revision time), was that i had been revising all along. my experience at mathslearn had given me reservations about teaching a topic relatively quickly without remediating along the way. to me this was creating a need for lengthy revision before examinations. i was concerned that, by working too fast, i would be losing some learners along the way. i also had a sense that some learners tended to rely on revision as part of the teaching process: “why learn it now if there will be another chance later?” when i planned my teaching of a new topic, i tried wherever possible to revisit a topic studied earlier. so when teaching quadratic inequalities, i built in opportunities to revisit the theory of quadratics and solving quadratic and cubic equations. in my view, my learners were constantly being required to revisit their earlier learning. some learners did find this approach challenging. to them it was “not how maths is taught”. however, my experience of working with teachers and their classes at mathslearn had taught me that an innovation will only succeed if one perseveres. so rather than giving up on the approach, i tried to address their concerns and continued to work on the innovation. when working under pressure to complete the rather overloaded grade 12 curriculum, i struggle to find time to help learners make the links between different topics they are studying. this skill of integrating knowledge across topics is one that i would like my higher grade learners to develop, not only because it will assist them in their final grade 12 examinations, but also because it will be of great benefit in their further studies. in order to tackle this problem i have developed a project that challenges learners to reflect on all the mathematics they have studied in their 12-year school careers. this project-work is one component of learners’ cass portfolios. (extract from “looking for links in mathematical topics: project work for grade 12”, a workshop presented at the amesa congress, 2003) the project on links in mathematics became one of the highlights of my grade 12 teaching each year (it was, in fact, an idea i got from a teacher at a mathslearn project school). it gave me an opportunity to digress from the daily routine of “covering” the curriculum and to give learners a chance to apply their learning in different ways. it was also an opportunity to do some action research of my own, as i used my experiences of using the project to adapt the material each year. promoting a community of mathematics educators in general mathslearn staff have noticed a change in the focus of all six teachers at the school. discussions during window sessions3 now involve teachers’ descriptions of the mathematical strategies used by their learners rather than discussion of administrative details. (extract from a paper written by mathslearn staff) when i began my new teaching post at the school i had visions of sharing what i had learnt at mathslearn with my new colleagues. i thought i could continue with my work as a teacher educator. not only did i firmly believe in the mathslearn philosophy, but i had also come from a very stimulating environment in which i was able 3 a “window session” is the term used by mathslearn staff for a scheduled period during the school day in which all mathematics teachers are freed from their teaching duties to meet as a group to discuss and reflect on their work. kate bennie 29 to talk about our work with the project school teachers and my fellow project workers. for three years i had been challenged to reflect on what i was doing. i certainly hoped that this would continue. i am watching sue at work on a mathematics problem. i notice that she makes an error when she manipulates a fraction. i am fascinated by the thinking behind her approach, which emerges as i try to assist her. at the end of the lesson my first thought is, “i must share this experience with my colleagues”. i join the other teachers in the staff room at lunchtime. i listen to the conversation, but somehow i don’t find an opportunity to share this classroom event. as head of mathematics at the school i organised for all the mathematics teachers to meet once a week or once a fortnight. i hoped that we would be able to talk about what was going on in our classrooms. to encourage teachers to focus on the mathematics happening in their classrooms, i tried giving them some preparation for each meeting, for example, something specific to look out for in their classes. but the meetings seldom developed as i intended. they either became discussions of administrative issues or never took place at all. can i explain this? firstly, it seems that teaching has become a matter of “survival” for many teachers. focusing on strategies used by learners in the classroom is often not the first priority. ironically, my experiences at mathslearn meant that focusing on the mathematics going on in my classroom was probably the most exciting part of my work. this became my way of “surviving” the other pressures that went with my role as a secondary school teacher. i so desperately wanted to share what i observed with my colleagues. secondly, many of my colleagues at the school were highly experienced and good teachers. over the years they had refined their methods, and they saw no need to change their teaching practice. interestingly, as the implementation of the new curriculum in grades 10 to 12 approached, some of my colleagues became more interested in my experiences at mathslearn. it was only after three years at the school that i had an opportunity to workshop these ideas with the staff. how much influence did i have on the school’s teachers in the four-year period? the use of the grade 12 project described earlier provided an exciting opportunity for mentoring of my colleagues who used this project in their classrooms. one of the teachers subsequently copresented an amesa workshop on our use of the projects. however, in general i have a sense of making little progress in conveying the mathslearn philosophy to my colleagues. most of my opportunities to talk about the work of mathslearn were in one-to-one discussions with a few of my colleagues when we discussed our classroom practice. perhaps i planted a spark in the minds of these colleagues. however, i have no way of measuring the impact of my ideas on these teachers. there were many occasions during my time at the school when i missed what might be called the “mathslearn community”, that is, the opportunity to talk about and reflect on issues in education and in mathematics education in particular. but my involvement in amesa helped me to cope with this frustration. when mathslearn closed i was determined to continue my involvement in amesa. i think i saw this involvement as a way of giving something back to mathematics education after having benefited so much from my three years at mathslearn. but on reflection i realise that it was not just me giving something back, but me getting something from amesa. for here was a community in which the mathslearn ethos could be kept alive. some remarks as the “storying” continues kate, do you know what your problem is? you do not reflect on what you do. (extract from a discussion at mathslearn) my three years as a project worker at mathslearn were a challenge, both professionally and personally. when i began working at mathslearn i was thrown completely out of my comfort zone, and in that three-year period i never seemed to regain it. one challenge just led to another and i was on a constant learning curve. and what did i learn? i believe that, most importantly, my mathslearn experience taught me to reflect on myself and on my work. this “story about a story” is an illustration of how i have learned to reflect on my experiences and my values. does this “story about a story” have any value? can “storying” be regarded as research? from the constructivist perspective, the process of writing has certainly helped me to organise my understanding of my experiences as a teacher educator and classroom teacher. for if i was once again asked the question “i am interested in your move from being a teacher educator to a practising teacher at school level. have you been able to developing teachers, developing as a teacher: a story about a story 30 apply what you were advising teachers to do?” i would definitely reply “yes, everything i have done in the past four years has been influenced by my experience at mathslearn”. but this is just one story about one story. in writing the story about my experiences over a seven-year period, i have used either my most vivid memories or what i regarded as important at the time of writing. in preparing this paper for publication i have used themes to organise the story, for example, my values as an educator, perspectives on equity, and promoting a community of mathematics educators. however, each time i read my reflections on this story, i construct another, slightly different story. this process will continue. each reader of this paper will read a different story. one reader may identify with the themes used to structure the paper. for another reader, this narrative may be interpreted as a story about how teachers and teacher educators adopt different discourses. for another reader, this may be a story of how a teacher may retain certain core beliefs, but act differently in different settings. for yet another reader, this may be a story about… references bennie, k., (2003). looking for links in mathematical topics: project work for grade 12. in s. jaffer & l. burgess (eds.), proceedings of the 9th annual congress of the association for mathematics education of south africa, 2 (pp. 21-25). cape town, south africa. bennie, k., olivier, a. & linchevski, l. (1999). everyone can learn mathematics: addressing the fundamental assumptions and goals of curriculum 2005. in proceedings of the 5th annual congress of the association for mathematics education of south africa, 2 (pp.10-24). port elizabeth, south africa. carter, r. & richards, j. (1999). dilemmas of constructivist mathematics teaching: instances from classroom practice. in b. jaworski, t. wood & a. dawson (eds.), mathematics teacher education: critical international perspectives (pp.69-77). london: falmer press. crotty, m., (1998). the foundations of social research. st leornards: allen and unwin. lancy, d.f., (1993). qualitative research in education: an introduction to the major traditions. london: longman. linchevski, l., liebenberg, r., sasman, m. & olivier, a. (1998). assessment in support of planning teaching to improve learning. in proceedings of the 4th annual congress of the association for mathematics education of south africa (pp.47-52). pietersburg, south africa. mcniff, j. (2002). action research for professional development: concise advice for new action researchers [booklet]. retrieved july 19, 2005, from http://www.jeanmcniff.com rowland, s. (1997). reflections on a story, academic development, 3(2), 19-36. “learning is finding out what you already know. doing is demonstrating that you know it. teaching is reminding others that they know just as well as you. you are all learners, doers, teachers.” richard bach article information author: renuka vithal1 affiliation: 1school of education, university of kwazulu-natal, south africa correspondence to: renuka vithal postal address: howard college campus, 2nd floor francis stock building, mazisi kunene road, glenwood, durban, south africa dates: received: 05 july 2012 accepted: 20 nov. 2012 published: 18 dec. 2012 note: this article was re-published with the corrected received date of 05 july 2012. the article was submitted online on 22 october 2012. how to cite this article: vithal, r. (2012). mathematics education, democracy and development: exploring connections. pythagoras, 33(2), art. #200, 14 pages. http://dx.doi.org/10.4102/ pythagoras.v33i2.200 copyright notice: © 2012. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. mathematics education, democracy and development: exploring connections in this original research... open access • abstract • introduction • mathematics education as a concern for democracy and development • mathematics education, its distribution and educational possibilities • mathematics education pedagogy as democratic living and engagement with development • democracy, development and mathematics content issues • conclusion • acknowledgements    • competing interests • references abstract top ↑ mathematics education and its links to democracy and development are explored in this article, with specific reference to the case of south africa. this is done by engaging four key questions. firstly, the question of whether mathematics education can be a preparation for democracy and include a concern for development, is discussed by drawing on conceptual tools of critical mathematics education and allied areas in a development context. secondly, the question of how mathematics education is distributed in society and participates in shaping educational possibilities in addressing its development needs and goals is used to examine the issues emerging from mathematics performance in international studies and the national grade 12 examination; the latter is explored specifically in respect of the south african mathematics curriculum reforms and teacher education challenges. thirdly, the question of whether a mathematics classroom can be a space for democratic living and learning that equally recognises the importance of issues of development in contexts like south africa, as a post-conflict society still healing from its apartheid wounds, continuing inequality and poverty, is explored through pedagogies of conflict, dialogue and forgiveness. finally the question of whether democracy and development can have anything to do with mathematics content matters, is discussed by appropriating, as a metaphor, south africa’s truth and reconciliation commission’s framework of multiple ‘truths’, to seek links within and across the various forms and movements in mathematics and mathematics education that have emerged in the past few decades. introduction top ↑ there is no doubt that notions of democracy and development are highly contested in themselves and in education; hence, so too would be any exploration of their links to mathematics education. whilst recent research, theory and practice, has emerged in the literature about connections between mathematics education and democracy and the related issues of equity and social justice, arguably much less has been written about mathematics education and development and related aspects such as poverty.the four questions addressed in this article were inspired and have evolved from those first posed by skovsmose (1994) with respect to general education and democracy. they were later reformulated to focus more directly on mathematics education (vithal, 2003). both have been framed within mathematics education from critical perspectives. this article brings in a further dimension, that of development, and attempts to connect the triad of mathematics education, democracy and development, through the four key questions, using the case of mathematics education in south africa. this expansion enables a more explicit engagement with issues that are found across countries like south africa, with developmental features such as high levels of poverty and inequality, when considering connections between mathematics education and democracy. the article explores the following four questions: 1. can mathematics education be a preparation for democracy that includes a concern for development? 2. how is mathematics education being distributed in society and thereby shaping educational possibilities? 3. can mathematics education pedagogy be a space for democratic living and engaging development issues? 4. can considerations about democracy and development in mathematics education have something to do with mathematics content matters? this article begins, at most, a conversation about notions of development in deepening and broadening understandings of the many facets of mathematics education in a young democratic, post-conflict society. from the many gains and challenges emerging from post-apartheid south africa, a selection of issues are engaged in each of the questions being posed to reflect on the many aspects of mathematics education in a transforming society. mathematics education as a concern for democracy and development top ↑ the first question that is posed is: can mathematics education, by itself and as part of general education, provide an introduction and preparation for democracy, and teach about democracy in ways that contribute to a society’s development agenda? this question is discussed by drawing on the conceptual tools of critical mathematics education, in particular the formatting power of mathematics within a ‘developmental state’.mathematics grows as it addresses questions and problems arising from within its own self-referential system(s), but increasingly it also advances as a discipline as it is applied to a diversity of problems in society, from everyday life to warfare and to poverty. d’ambrosio (1994), points out the paradox in which mathematics is centrally implicated: in the last 100 years, we have seen enormous advances in our knowledge of nature and in the development of new technologies. … and yet this same century has shown us despicable human behaviour. unprecedented means of mass destruction, of insecurity, new terrible diseases, unjustified famine, drug abuse, and moral decay are matched only by an irreversible destruction of the environment. much of this paradox has to do with an absence of reflections and considerations of values in academics, particularly in the scientific disciplines, both in research and in education. most of the means to achieve these wonders and also these horrors of science have to do with advances in mathematics. (p. 443) much of the concern in the developed contexts of the north is with how rapid advances in science and technology are fundamentally changing those societies and how these changes might pose a threat to democracy because of how they might limit the capacity of the electorate to participate meaningfully in understanding and influencing decisions that affect their lives. another aspect of the concern is with the complexity of the science and technology and having to rely on experts (an ‘expertocracy’), in particular the capacity of politicians and decision-makers to fully grasp the implications. in developing countries like south africa, with ‘emerging’ or ‘young’ democracies, a much less literate electorate, lower levels of (science and technology) education amongst politicians and smaller pools of experts, this threat increases manyfold, especially with the global transfer and trade in science and technologies, which usually take place from developed to developing nations in view of their significant development challenges. the question of how mathematics participates or is recruited in this is explained by skovsmose (1994), through what he calls the ‘formatting power of mathematics’: [m]athematics produces new inventions in reality, not only in the sense that new insights may change interpretation, but also in the sense that mathematics colonises part of reality and reorders it. (p. 42) the issue here is not that mathematics itself does anything, but rather it is about how mathematics is used by people, institutions or ‘agencies’ through all types of applications that come to produce and result in a formatting of society. society today is increasingly mathematised. keitel (1993; keitel, kotzmann & skovsmose, 1993) has demonstrated the complexity of this through the relationship between social abstractions and thinking abstractions. as mathematics is applied to all sorts of processes, structures, problems and organisations, these in turn change and require further mathematisation. whilst an increasing amount of implicit mathematics is found in all areas of life today, it requires a more sophisticated mathematically literate person to question the applications within a democracy. paradoxically, at the same time less procedural mathematics is needed as technology takes over, for example the widespread availability of calculators. this has serious implications for the mathematics education provided across the education system to strengthen democracy and fuel development. for countries like south africa, if the notion of the formatting power of mathematics is accepted then it is imperative to educate those who will come to participate in that formatting power and address problems of development as well as those who will need to be able to react to it to ensure that fair, equitable and just solutions are found. it may be argued that a mathematics curriculum has an obligation to produce both ‘insiders’ and ‘outsiders’ to the formatting power of mathematics. the insiders are those who come to participate in the formatting (as high performers), be they ‘constructors or producers’ of mathematics or ‘operators or users’ of mathematics. the outsiders are those who must read and react to that formatting power as (mathematically literate or numerate) consumers of mathematics or as the marginalised of society (skovsmose, 2003; vithal, 2004a). the recently introduced new mathematics and mathematics literacy curricula for grades 10−12 in south africa could be said to roughly match these two requirements (discussed later). what is evident in this line of argument is that access to and competence in mathematics serve very different purposes. both the presence and absence of mathematics education has real consequences; it is neither neutral nor value free. mathematics is used in a multitude of ways in society: to predict, control, interpret, describe and explain within a particular cultural, economic and sociopolitical context. mathematics for those who will come to participate as the ‘formatters’ or ‘high performers’ in positions of power (as government or experts), especially in developing countries with scarce high-level human resources, needs to include critical engagement about development and poverty and integrate an ethics and social responsibility as part of the multiple goals of mathematics education. this is important since a democratic competence cannot be assumed and nor does ‘high-level’ or abstract mathematics necessarily produce an integrated critical mathematical literacy competence. this is because the thinking tools and language of mathematics do not by themselves provide the full means for criticising its applications in society. similarly, a mathematical literacy for the majority has to be more than a functional or practical literacy. it is expected to integrate a critical and democratic competence with a mathematical competence if a citizenry is to participate meaningfully in a young democracy and growing economy and be able to grasp the mathematical basis implicit in the decisions taken for or against them. mathematics needs to be responsive to a diversity of contexts, avoiding a ghettoisation of the mathematics curriculum and yet providing a mathematics that is inclusive of the consumers of mathematics and those who do not enter further and higher education or the world of work. it is not only necessary to know when a problem is dealt with using mathematics, whether the correct mathematics has been chosen, whether the mathematics has been correctly executed and whether the result can be relied on; it is also important to include reflections about how the use of mathematics to solve the problem relates to the broader (social, political, economic) context and its possible consequences, whether the problem could have been solved without mathematics and whether the evaluation or reflections could have been done differently (keitel et al., 1993; skovsmose, 1994). the developmental challenge for mathematics education is not confined to particular parts of the world. it is a global challenge. at the centennial symposium of the international commission of mathematics instruction (icmi), at which mathematics educators reflected on mathematics education of the past hundred years and attempted to map out directions for the future, setati (2008) proposed a significant new role in contributing to the acceleration and attainment of the united nations millennium development goals of eradicating poverty, promoting gender equality and universal primary education. she argued that mathematicians and mathematics educators need to work together – from different levels of the education system, in different aspects of research and practice, from different perspectives, and from different parts of the world – to address poverty, injustices, inequity, illiteracy and access to education. in proposing a shift to the developing world, she recommended icmi studies of mathematics education within contexts of poverty, multilingualism and multiculturalism and a focus on mathematical literacy. supporting development goals through general education in order to deepen and strengthen democracy is well understood, well known and accepted. however the key role mathematics education has in this, and how it can provide an introduction to democratic life and values and thereby to a better life for all, is arguably less so. within developing and poorer countries, mathematics education has an explicit and critical role. lubisi (2008) spells out the direct connection between mathematics education and notions of the ‘developmental state’ as it is being deliberated in south africa. for him, mathematics education is the key to empowering people with knowledge and skills that are necessary to reach the targeted economic growth rates, to create employment and to fight poverty. he argues that analyses of most of the skills areas of the economic sectors that are being targeted to ensure that this growth is achieved require mathematics. in particular, the shortage of artisans and technicians in south africa, that is, those who are involved in various kinds of applications of mathematics as the ‘users or operators’, has become evident and needs to be taken seriously. therefore, strengthening mathematics (and science) teaching in schools is important in order to reach development goals and the needs of economic growth. for developing countries, improving the basic conditions of peoples’ lives, including schooling and the quality of all aspects of mathematics education is crucial to sustaining democracy. mathematics education provides not only access to mathematical knowledge and skills, which is important for living in the 21st century, but in many countries, performance in mathematics determines access to jobs and further or higher education studies in a range of areas, from the natural and physical sciences to economics and technology. it is for this reason that mathematics is on the one hand regarded as a gateway subject to a large number of these high-status, high-paying professions, but on the other hand also functions as a gatekeeper for the many who fail to learn and perform at the requisite levels or are failed by the education system. in this respect, mathematics education functions implicitly to stratify society. how it does this is important to analyse if it is to be addressed so as to open more and better life opportunities for all students, whatever role they come to fill as producers, users or consumers of mathematics. as a high-stakes subject, it is not surprising that that there is much concern about who gets access to and performs well in mathematics and who gets excluded. whilst mathematics educators have arguably not focused on developmental challenges, research and literature on the nature and extent of such challenges is emerging from outside mathematics education. the world bank’s world development report on equity and development (world bank, 2006) offered the following south african narrative, which illustrates the complexity of engaging development contexts: consider two south african children born on the same day in 2000. nthabiseng is black, born to a poor family in a rural area in the eastern cape province, about 700 kilometers from cape town. her mother had no formal schooling. pieter is white, born to a wealthy family in cape town. his mother completed a college education at the nearby prestigious stellenbosch university. on the day of their birth, nthabiseng and pieter could hardly be held responsible for their family circumstances: their race, their parents’ income and education, their urban or rural location, or indeed their sex. yet statistics suggest that those predetermined background variables will make a major difference for the lives they lead. nthabiseng has a 7.2 percent chance of dying in the first year of her life, more than twice pieter’s 3 percent. pieter can look forward to 68 years of life, nthabiseng to 50. pieter can expect to complete 12 years of formal schooling, nthabiseng less than 1 year. … nthabiseng is likely to be considerably poorer than pieter throughout her life. … growing up, she is less likely to have access to clean water and sanitation, or to good schools. so the opportunities these two children face to reach their full human potential are vastly different from the outset, through no fault of their own. such disparities in opportunity translate into different abilities to contribute to south africa’s development. … as striking as the differences in life chances are between pieter and nthabiseng in south africa, they are dwarfed by the disparities between average south africans and citizens of more developed countries. consider the cards dealt to sven − born on that same day to an average swedish household. his chances of dying in the first year of life are very small (0.3 percent) and he can expect to live to the age of 80, 12 years longer than pieter, and 30 years more than nthabiseng. he is likely to complete 11.4 years of schooling − 5 years more than the average south african. these differences in the quantity of schooling are compounded by differences in quality: in the eighth grade, sven can expect to obtain a score of 500 on an internationally comparable math test, while the average south african student will get a score of only 264 − more than two standard deviations below the organisation for economic cooperation and development (oecd) median. nthabiseng most likely will never reach that grade and so will not take the test. (pp. 1–2) the above narrative is reproduced in detail to illustrate how mathematics education operates as a system to open or limit educational possibilities for students. it also points to the necessity for locating mathematics education research, practice, policy and theory within a broader landscape in relation to other aspects that impact and shape (mathematics) educational opportunities. it shows the need to link mathematics education to multidisciplinary development studies, addressing issues of concern to developing countries, and especially to social and economic development. south africa is a useful case as a young democracy of almost two decades within which four waves of curriculum reforms have occurred, and in which mathematics educators, mathematicians, and a range of other ‘stakeholders’ have had the possibility to participate in shaping the mathematics curriculum. it has been observed that there are two main imperatives driving and shaping curriculum debates in south africa: one is the post-apartheid challenge for greater equity and social justice, to redress decades of deliberate inequalities and to entrench and deepen democratic life; and the other is the global competitive and development challenge to provide opportunities to learn and access knowledge and skills to participate effectively in the internationalised and globalised economy of the 21st century (vithal & volmink, 2005). the development and democracy challenges for mathematics education are captured in the lives of nthabiseng and pieter. despite each successive minister of education effecting some or other official curriculum change since the advent of democracy, with incremental improvements in school infrastructure and resourcing, nthabiseng and pieter continue to experience very different and substantially unequal implemented and attained mathematics curricula. however, in recognising the formatting power of mathematics, this discussion points to the potential of mathematics education to transform both nthabiseng’s and pieter’s lives in contributing to strengthening both democracy and development imperatives of south africa. mathematics education, its distribution and educational possibilities top ↑ the second question is: how is mathematics education, in terms of mathematical knowledge, skills, values and attitudes, distributed in society and thereby shaping educational possibilities? this question is engaged through south african learners’ mathematics performance in the much-publicised international studies and the national grade 12 examination results. possible explanations for these are then explored in terms of the mathematics curriculum, its recent reforms and related teacher education challenges.one of the ways in which the distribution of mathematics education is made visible and public is through international studies of student mathematics performance and national tests and assessments. in the public imagination, shaped by the media, mathematics education is reduced to league tables of student mathematics performance scores. south africa’s repeated ranking at the very bottom of international studies of student mathematics performance and equally poor outcomes in the annual high-stakes national grade 12 matric examination results, when each are released, follow with endless speculation about the reasons and causes of south africa’s continued poor mathematics performance. the mathematics performance of nthabiseng and pieter and their consequent educational possibilities and life journeys allude to the deeply unequal conditions of schooling and opportunity to learn which have endured almost two decades since the advent of democracy. inequities in the quality of south african schooling and living conditions are reflected in the test and assessment outcomes. however, an aspect that has not received much public attention is whether these studies, tests and assessments do indeed offer an accurate account of the mathematics knowledge and skills of learners. if all learners are deemed to have some mathematical knowledge by virtue of having lived in particular communities or cultures, as ethnomathematics for example argues, what do the tests in fact reveal, if anything, about what mathematics learners know and understand? these large-scale assessments, which are costly to mount, are often driven as much by political imperatives as they are by educational ones and conducted within funding and other constraints. methodological issues about the language in which the tests are conducted, familiarity with the format of the test items and the reliance on only paper and pencil assessments are seldom discussed publically to qualify the outcomes and findings. even though these international studies have long come under criticism by mathematics educators and the use and misuse of their results cautioned against (e.g. kaiser, luna & huntley, 1999), they have been latched onto by governments, including south africa’s, and used to introduce wide-scale national testing regimes. in the uk, which has a history of national testing, studies based on these national tests to explain leaner performance demonstrate the caution with which these results need to be interpreted. cooper and dunne (2000) showed, by comparing test and interview data, that many children fail to demonstrate in tests mathematical knowledge and understanding that they actually possess. they showed learners’ confusion over the requirements of ‘realistic’ test items as compared to ‘esoteric’ items, and how this was particularly the case for children from working-class backgrounds. this is an important finding, relevant to the new south african school curriculum, which foregrounds applications and context in mathematics, as the newly introduced annual national assessments are being implemented at all levels of the education system. it raises serious questions about what the assessments really reveal and about whom or what. the point here is not to suggest that testing should not take place, but it is necessary to understand how such test outcomes have become a public window to the mathematics classroom and have come to generate a particular discourse about the distribution of mathematical knowledge to which politicians and policymakers are particularly responsive. the outcomes of such studies need to be qualified with reference to issues of methodology and contextualised historically, taking into account the sociopolitical, economic, urban–rural and cultural dynamics. in the trends in international mathematics and science study, which yet again confirmed south africa’s poor performance, reddy (2006) categorised learner performance scores according to the previously racially segregated schools and showed, not surprisingly, that despite their levels of resourcing, the former white schools (model c) performed only at the international average, whilst former african schools performed at half the average of the white schools. african schooling has come under intense scrutiny, has had a myriad of interventions and is being researched for its lack of effectiveness. white schooling, however, has remained outside the research gaze and has not been interrogated, for example, for its failure to exceed international averages given their disproportionate share of considerable resources and other advantages. much less is known about why pieter is not performing as well as sven, given comparable educational contexts and advantages. another very public lens on the distribution of mathematical knowledge and skills is the high-stakes annual grade 12 mathematics matric examination, written by some half a million students each year, which plays one of the most important direct roles in apportioning further educational opportunities. these results are released at the end of each year amidst much fanfare and commentary. we could well ask: what are the chances nthabiseng and peter would be amongst those to have studied and passed mathematics? issues of race and gender have been foregrounded in post-apartheid south africa in considerations of who gets access to higher education. analyses by kahn (2005) demonstrate a steady increase in the numbers of african students studying and passing mathematics at the higher grade (hg), which provides eligibility to access university. the 1991 figures have increased tenfold by 2005, with just under 10 000 african students successfully passing hg matric mathematics. this is, however, a miniscule proportion relative to both the numbers of african learners that enter schooling in grade 1 and those who make it to grade 12 each year. according to kahn (2005) ‘the white community generates science based skills at something close to saturation level’ (p. 142). pieter is assured of a pass in mathematics to enter university but what of nthabiseng? although overall gender differences in participation and performance in matric mathematics are not significant, when race is intersected with gender, major differences are found between african females and other female candidates. restricting analyses of performance to only gender has been found to mask large disparities in matric hg mathematics pass rates for african females, which in 2002 were found to be only a quarter of the pass rates of white females (centre for development and enterprise, 2004). it is possible to speculate that even amongst her white, indian and coloured sisters, nthabiseng has had the system odds stacked against her passing mathematics to secure entry into higher education. the issue of race, however, is a vexed one and the department of basic education, in releasing matric performance statistics, has in recent years reported on gender but not race data. notwithstanding the dangers and arguments against entrenching apartheid constructed racial categories, this analysis demonstrates that such information is crucial to developing appropriate and targeted interventions that can impact the most marginalised in society. being able to identify the nthabisengs of south africa for redress to ensure she does not continue to carry a disproportionate burden of apartheid’s damage. the south african national mathematics curriculum underwent a major reform for the grade 10−12 band in which all grade 12 students from 2008 wrote either a mathematics or a mathematical literacy examination. these changes were made to address two main problems that directly relate to the mathematical knowledge distribution and education opportunity challenges. the first problem was that although the numbers taking mathematics (whether hg or sg [standard grade]) in the period from 1995 to 2007 had risen to above 60%, much of this increase was due to an increase in sg mathematics enrolments but it was a pass in hg mathematics that typically gave entry to higher education opportunities. the numbers studying and passing the sought-after and much-needed hg remained low and appeared to have reached a ceiling that remained below 30 000. the mathematics taught at sg, which was largely procedural and excluded key topics that were found in the hg curriculum, did not provide the kinds of mathematical competencies needed for further higher education studies to fuel the high-level science, economic and technology needs of the country and sustain the supply of constructors or producers of mathematics or indeed even the users or operators of mathematics (skovsmose, 2003; vithal, 2004a). a second problem that contributed directly to inequities in the distribution of mathematical knowledge and skills was that at least 40% of matriculants each year did not take any mathematics at all and hence were not taught any mathematics in schools between grades 10 and 12. this meant they were not provided with even the competence to be consumers of mathematics that a citizenry in a democratic south africa of the 20th century should at least have acquired through schooling. mathematical literacy was intended for this group. one of the difficulties that the new mathematics and mathematical literacy curricula have faced is a lack of consensus and clarity about what each of these are, their relation to the previous hg and sg mathematics curricula and their relation to each other. as a new subject, mathematical literacy faces a particular difficulty in escaping the image of a practical or functional lower order mathematics (for those who were deemed incapable of or uninterested in doing mathematics) rather than being conceptualised as a different integrated contextualised competence. deriving from a broader ‘mathematics for all’ movement, it has been variously labelled within policy, theory, research and practice as numeracy, quantitative literacy (steen, 2001) and mathemacy (skovsmose, 1994), amongst others. this points to different ideological orientations, intentions and goals of such a mathematics, which extends from a concern with acquiring basic mathematics to a sophisticated critical integrated competence (vithal, 2004a). the most recent south african mathematical literacy curriculum policy describes some of the key elements as involving ‘the use of elementary mathematics’, ‘authentic real-life contexts’, decision-making and communication and ‘the use of integrated content and/or skills in solving problems’ so that they become ‘participating citizens in a developing democracy’ and ‘astute consumers of mathematics in the media’ (department of basic education, 2011b, pp. 8–10, [emphasis in original]). on the introduction of the new curricula, fears that large numbers of students would choose mathematical literacy rather than mathematics were not fully founded. the numbers who followed the new mathematics curriculum in 2008 was just over half of all matriculants. however, the more troubling observation since then is that the number choosing mathematics has decreased year on year. from a high of 300 000 taking mathematics in 2008 the number had decreased by 25% in 2011(department of basic education, 2011a). although many more students are deemed to be succeeding in mathematics, the number passing at 40% relative to those studying mathematics have stabilised at a low of about 30% whilst the number passing at 30% has hovered between 45% to 47% (department of basic education, 2011a). the new mathematics curriculum (department of basic education, 2011c), it would appear, is being differentiated by assessment rather than by content and levels of difficulty. these changes have, however, increased the numbers of students eligible for entry into university and opened another debate about their readiness to pursue and succeed in higher education programmes. which learners are allowed to do mathematics and the quality of mathematics education learners receive in school are shaped by many factors. although all secondary schools must now offer mathematical literacy and system-wide interventions have taken place for its implementation, the same has not been obtained for the delivery of the new mathematics curriculum, which is much more demanding than the previous sg curriculum and is much more application oriented, with several new areas and topics, compared to the former hg mathematics curriculum. the new mathematics curriculum has been implemented in a context in which only half of all secondary schools who previously offered mathematics offered it at the hg (centre for development and enterprise, 2004). this means that the opportunity to learn mathematics is limited in real terms for those learners who find themselves in schools that do not offer the new curriculum or, where it is offered, do not have appropriately educated and trained teachers to deliver it. we could ask: in which schools are pieter and nthabiseng likely to find themselves and what teachers are they likely to encounter? by all accounts, teachers are critical to the delivery of any curriculum. from this perspective the success or failure of the new mathematics curriculum hinges on the question of what further education and training provisions are being made at a system level for the substantial cohort of teachers that were teaching the 300 000 sg mathematics students in 2007 and were then required from 2008 to deliver a new, different and more demanding mathematics curriculum. the new official intended mathematics curriculum has been found to compare well internationally, regarded to be as difficult as or more so than the previous hg mathematics and as embodying best practices and knowledge about pedagogy and content that should go some way toward preparing students for the scientifically, technologically and mathematically advancing society of the 21st century. it is, however, in the implemented curriculum, at the school and at the classroom level, that the challenges are to be found. no doubt many different kinds and levels of resources and infrastructure are needed for the successful implementation of the new curricula; however, the main lever is the quantity and quality of competent and confident teachers who can deliver the new mathematics curriculum and thereby shape south africa’s democratic ideals and contribute to its development goals. to understand the extent of this challenge, of mathematics knowledge and skills of teachers, it is necessary to appreciate the historical legacy of mathematics teacher education from a system point of view. in south africa, through the deliberate underdevelopment of apartheid, the education system has inherited a substantial core of teachers with diplomas as opposed to degrees, and an uneven preparation in the core content knowledge of mathematics. this legacy remains intact and must be addressed for any radical break with the past and for substantial improvements in providing learners with adequate and appropriately qualified mathematics teachers to acquire the kinds the mathematical knowledge and skills the official curriculum promises. the magnitude of the task relates to both the issue of supply of new teachers and the continuous education of existing teachers. in the past decade, the number of students seeking to become senior secondary teachers of mathematics has not kept pace with demand as teaching is unable to compete with the status, remuneration and prestige of other expanding career options in science and technology, given the small pool of successful candidates in matric mathematics. this problem may acerbated the policies introduced in the late nineties to redistribute teachers, which resulted in a number of qualified mathematics teachers exiting the system. furthermore, there has been no systemic state intervention for upgrading mathematical content knowledge at these higher levels, for example, systematically targeting all former mathematics teachers who were only able to teach sg mathematics. more than a decade since the first mathematics teacher audit (arnott, kubeka, rice & hall, 1997), approximately 20% of grade 10–12 mathematics teachers are professionally unqualified and of those that are qualified, still only 21% have some university level courses (parker, 2010). there is also evidence to suggest that qualified mathematics teachers in the system are either not teaching mathematics or not teaching it at the level at which they are qualified (parker, 2010; peltzer et al., 2005), for a number of different reasons, showing that a limited and scarce resource is being poorly utilised. in south africa, every new minister of education since 1994 has introduced curriculum reforms, resulting in several waves of curriculum changes. in this context, the fragile and weaker parts of the system are more likely to become dysfunctional. teachers require time to be inducted into new or changing content and pedagogy. for teachers who may be struggling with mathematical content knowledge, forms of assessments and their associated pedagogical reforms, this may acerbate the problem especially in poorer and under-resourced schools and classrooms, the kind of classroom in which nthabiseng is likely to find herself. a vicious cycle persists if a curriculum reform is evaluated too soon, when it is more likely to show a dip in performance areas as the new curriculum is still bedding down; and if further changes are introduced before they are thoroughly understood and institutionalised. it is in this respect that stability in the official curriculum is crucial so that teachers are given a chance to interpret and give effect to the curriculum. in this context, it is assumed that other foundational infrastructure is in place, such as adequate and timely provision of core mathematical teaching and learning resources, for example, appropriate quality textbooks relevant to and necessary for each curriculum reform. attempting to increase and better distribute educational opportunities for many more learners to effectively break the glass ceiling of mathematics performance, particularly for those at the margins, in the most impoverished parts of the schooling system, requires a targeted, systemic and systematic long-term mathematics teacher development intervention, a stable curriculum policy environment, and, at the very least, a critical level of resourcing and schooling infrastructure for the mathematics education system to function. mathematics education pedagogy as democratic living and engagement with development top ↑ the third question is: can democracy in mathematics education refer to the very life of a mathematics classroom, learning democratic values, democratic attitude and democratic competence in a context that recognises and seeks to address issues of development? this question is explored through a discussion on pedagogies of conflict and dialogue, and of forgiveness.the new mathematics national curriculum policy reforms in all their different waves in post-apartheid south africa, take as their point of departure quite explicitly the new constitution and provide the imperative for teachers to explicitly make connections between mathematics and the real world. the question of whose world gets selected, by whom and for what purpose in a mathematics classroom then becomes important. teachers make selections of content, context and pedagogy and realise different kinds of actual and hidden curricula, for instance, in choosing to teach about mathematical modelling through the context of hiv and aids or some other development challenge or inequity in society. in his study on teaching and learning mathematics for social justice in an urban latino school, gutstein (2003) showed how mathematics can be taught in a way that develops learners’ sociopolitical consciousness and sense of agency, develops positive social and cultural identities through a classroom pedagogy that assists them to ‘read the world (understand complex issues involving justice and equity) using mathematics’ (p. 37), develops mathematical power in the ways in which they do and think mathematically, and thereby changes their disposition and orientation toward mathematics. much of the foundation for this kind of pedagogy was laid in the eighties and early nineties and has spawned a diverse literature in mathematics education describing and analysing activities and theoretical ideas that explored a political mathematics education (e.g. mellin-olsen, 1987) or critical mathematical literacy (e.g. frankenstein, 1987). it has continued in different forms and endured to the present in debates particularly about equity, quality and social justice in mathematics education (e.g. atweh, graven, secada & valero, 2011) an overtly political approach to mathematics education also has early roots in south africa. ‘people’s mathematics for people’s power’ was a part of the broader phenomenon of the people’s education movement that arose during the apartheid era, which viewed schools and classrooms, including the mathematics classroom, as important sites for the struggle against apartheid. a number of mathematics educators engaged these early ideas in their teacher education programmes at the time (adler, 1991; breen, 1986; julie, 1991). suffice it to say here, not surprisingly there was deep contestation and resistance as any historical account of the people’s mathematics movement demonstrates (vithal, 2003). however, as a society still struggling with deep inequalities and continuing injustices, the question of whether mathematics education can participate in moving us toward more humanitarian goals – democracy, equity, social justice, non-racism, non-sexism – is as relevant today as, or perhaps more so than, it was all those years ago. the new south african mathematics curriculum provides a policy space for such engagement but the question of teachers’ implementation of such a pedagogy remains open. it was with this explicit concern and ideological orientation that i undertook my doctoral study in the mid-nineties (vithal, 2003) and from which this section of the article draws in pointing to a pedagogy that emerged from empirical work, and which i reflect on and extend. i explored the question of what happens in mathematics classrooms when student teachers attempt to realise what could be called a social, cultural, political approach to the mathematics curriculum that integrates a critical perspective in practice. although the student teachers were introduced to diverse practices related to this broad approach, the dominant curriculum practice engaged by them was that of project work (vithal, 2004b, 2006). the particular conception of project work employed was one that is well developed within the scandinavian context (olesen & jensen, 1999) and widely implemented and researched in mathematics teaching and learning from primary to university education (christiansen, 1996; nielsen, patronis, & skovsmose, 1999; niss, 2001; vithal, christiansen & skovsmose, 1995). it specifically seeks to develop a critical perspective through an approach that is problem orientated, interdisciplinary and participant directed. by choosing exemplary problems of societal relevance to investigate, learners develop both knowledge and skills and the means for critiquing that very knowledge and skills. an in-depth study and detailed description of how one student teacher, sumaiya, taught mathematics in a grade 6 mathematics classroom of african and indian learners, in a school located in a predominantly indian suburb, enabled an interrogation of the theory and practices advocated in the literature that seek to construe mathematics classrooms and schools as spaces for enacting democratic life. the student teacher, class teacher and learners engaged in groups a range of projects that required enacting democratic life as they dealt with issues of development – from creating a mathematics newsletter and questioning the school’s use of their school funds and provision of facilities to the inherent inequalities and gendering of time for mathematics homework. sumaiya brought a deeply reflective perspective as she grappled with introducing what were considered radical ideas about teaching mathematics in which democratic approaches were engaged in dealing with real micro local problems of development selected by learners. the study generated a thesis of a pedagogy of conflict and dialogue embedding five dual-concept themes: freedom and structure, democracy and authority, mathematics and context, equity and differentiation, potentiality and actuality. these concept pairs capture the multifaceted and multidimensional nature of mathematics classrooms that choose to engage matters of democracy and development in this direct way. conflict and dialogue and the dual-concepts are themselves explained as being in a relationship of complementarity, a complex relationship of cooperation and opposition. drawing specifically on the interpretation offered by otte (1990), complementarity allows one to see concepts (such as object/content and tool) firstly as woven together, each presupposing the other, where the one cannot be defined or described without the other, and secondly as contradictory to each other, opposing each other, where the one does not directly show itself in the other. the mathematics classroom in this framing is seen as a functional whole, not only in the school but also linked to the broader societal setting in which it is located. it is a space fraught with conflict and contradictions, but also containing all the possibilities and hope for their engagement and potential for resolution. in a pedagogy of conflict and dialogue, in particular, not only are conflict as content and dialogue as tool in a relation of complementarity, but there are also complementarities within each. if mathematics classrooms are to be spaces not only for learning about democracy but also for enacting democratic life, then conflicts and crises of society will become part of classroom life and dialogue is needed, even as it is resisted. dialogue as an epistemic or didactic tool (mellin-olsen, 1993) to deal with different kinds of (knowing/knowledge) conflicts functions to provide a better understanding not only of each other but also with each other. for mellin-olsen, dialogue is not a search for consensus or compromises as much as it is a search for deeper insight with the partners of the dialogue. the disagreement or conflict has to be engaged in a way that does not destroy the dialogue, which creates a paradox: ‘confrontation and disagreement … have to be developed in a context of agreement and co-operation’ (mellin-olsen, 1993, p. 256). hence, such a mathematics education legitimates not only engaging with different kinds of knowledge conflicts but also learning to dialogue and about dialogue as a method of confrontation and cooperation. both conflict and dialogue are needed in a pedagogy that attends to issues of democracy and development from a critical perspective. in south africa, nthabiseng and pieter may well find themselves in the same mathematics classroom. however, they are arguably more likely to find themselves in a school where one or the other dominates: we could imagine a pedagogy of conflict without dialogue degenerating into anarchy and chaos or dictatorship. in current actual situations in south african schooling [more likely nthabiseng’s school] we have seen pupils’ expression of dissatisfaction with the school, its authority, structures and dimensions of differentiation, expressed through violent means and then curbed through enacting stronger forms of autocracy rather than democracy. … here, we can see how a pedagogy of dialogue is essential to a pedagogy of conflict especially if democracy, freedom, context and equity are to be valued in schools. we could also imagine a pedagogy of dialogue without a pedagogy of conflict reduced to benign endless rounds of entertaining, interesting safe talk and action. teachers in more advantaged schools [more likely pieter’s school] could talk about the inequalities and injustices brought about through apartheid in a pedagogy of dialogue without conflict. the pupils in this pedagogy could never really come to make connections between the apartheid past and the present, or question or act on the conflicts immediately around them. … a pedagogy of conflict and dialogue means therefore that each, conflict and dialogue, presuppose the other in a mathematics curriculum approach that seeks to focus on social, cultural, economic and political aspects of society. they are separate and each must be developed independently, conflict as content and dialogue as tool. but they are also connected, and therefore must be realised in relation to each other in a classroom. (vithal, 2003, p. 356) in a mathematics education that embeds a critical perspective, there is no doubt a level of risk and uncertainty that attempting a pedagogy of conflict and dialogue invokes, especially when the inequalities and injustices do not reside in some distant place or time but are embodied in the very students and teachers in the classroom and each is somehow seen as implicated as ‘victim’ or ‘perpetrator’. even though not all mathematics can be taught with reference to context, given its abstract nature, creating some spaces in the curriculum for critical societal issues of development, diversity, equity and social justice must be argued for. yet it is likely to be resisted in much the same way as people’s mathematics was, though for different reasons, by different parts of the schooling system. schools in general and mathematics classrooms in particular need to become spaces for learning about and through democracy or we risk repeating past failures. that so many members of the white community in south africa continue to claim today not to have known about the huge suffering of black people perpetrated in their interest can be analysed as a most serious failure of white schooling and mathematics education. that young white learners who come through white schooling today fail to appreciate their positionality and privileged inheritance from the injustices of the past continues to be a failure of white schooling. the ‘whiteness’ of white schooling is to be understood in terms of not only its demographics, but also its rituals, rules and traditions that enculturate its members into a culture of ‘whiteness’ and has remained largely outside the mathematics educational research gaze. much of the attention has been focused on the deficits of black education and much less on the pathologies of ‘whiteness’ in how it is being reproduced or transformed in post-apartheid south africa. not surprisingly a pedagogy of conflict and dialogue is less likely to be enacted in diverse settings, where it can be a dangerous if not painful path to tread, where white teachers and learners cannot escape being seen as ‘perpetrators’ and black teachers and learners cannot escape the feelings of ‘victimhood’ and suffering. it is in this respect that a pedagogy of conflict and dialogue must also integrate a pedagogy of forgiveness (waghid, 2005). a mathematics that reveals inequities and injustices of the past or present is likely to produce feelings of resentment and hate. in such contexts, as waghid (2005, p. 226) notes, ‘learning about forgiveness can become useful in enhancing pedagogical relation’ and when teachers and learners ‘cultivate forgiveness’, it becomes a way to ‘engender possibilities whereby people are attentive to one another’ and can engage ‘imaginative action’ to move forward. a pedagogy of conflict and dialogue for a mathematics education for equity and social justice invariably opens wounds so that the ‘truth’ can be known, even relived, and understood. each learns by being in the place and experience of the ‘other’. but if such a pedagogy is not to run the risk of deepening divides and difference then it must provide a means to heal. a pedagogy of forgiveness integrates into conflict and dialogue, a point of hope and creative action. the principle of hope, skovsmose (1994) argues, needs to be preserved in a critical mathematics education. it is not surprising that ‘forgiveness pedagogies’ have emerged and are being engaged and studied within educational settings in societies that have had histories of political conflict and trauma (zembylas & michaelidou, 2011) as part of processes of reconciliation. such pedagogies are needed in any mathematics education enacted with a consciousness for issues of development and deepening democracy. both a pedagogy of conflict and dialogue and a pedagogy of forgiveness take their bearing from south africa’s own post-apartheid processes of the truth and reconciliation commission (trc) (1998). in doing so, these pedagogies require the creation of spaces for ‘truth’ to be told so that reconciliation can occur, and mathematics by its power and status in society opens a unique and special way for such truths to be told. only then can dignity be reclaimed, compassion shown and respect and friendship built. critical, feminist and social justice mathematics pedagogies have sought to mobilise the power of mathematics knowledge and skills to overt political and social agendas. but in order for restoration, healing and peace to emerge, such pedagogies will have to attend not only to mathematics education pedagogies but also to mathematical knowledge itself. democracy, development and mathematics content issues top ↑ the fourth question is: can mathematics education, democracy and development have something to do with mathematics content matters? this question is discussed by appropriating, as a metaphor, south africa’s trc’s multiple notions of truth in seeking a similar framing for linking the various forms and movements in mathematics and mathematics education that have emerged in the last few decades.the myth that mathematics and mathematics education are neutral and value free has long been exploded. d’ambrosio (1994) implicates mathematics in both the beauty and the devastation brought by advances in science and technology and raises serious questions for mathematics education for the 21st century. ethnomathematics, critical mathematics education, mathematics for equity and social justice are areas of study and practice that have grown rapidly in the last few decades, forcing a re-examination of what constitutes and counts as mathematical knowledge, questioning how it has been and continues to be produced and legitimated, raising issues about who is recognised for its production and problematising mathematics curricula for their purpose, relevance and appropriateness for different groups in society they are intended to serve. these different orientations to mathematical content, which appear discrete and disconnected, may be brought into relation with each other by appropriating, as a metaphor, a framework from south africa’s trc (1998). the trc enabled south africans in post-apartheid society to confront the truth about apartheid by acknowledging, legitimating and validating multiple forms of ‘truth’. it may be similarly proposed that there are different forms of ‘truth’ constituting mathematics and that engaging more than one ‘mathematical truth’ in a mathematics classroom, especially in a pedagogy of conflict and dialogue and of forgiveness, is necessary for both learning and reconciliation to occur. the trc dealt with the complexity about what constituted truth and whose truth by developing a conceptual framework comprising four notions of truth: factual or forensic truth, dialogue or social truth, personal or narrative truth, and healing or restorative truth. each of these truths in turn provides a means for presenting recent challenges to mathematical content questions as multiple forms of ‘mathematical truths’. this framework is useful to engage questions of what is taken to mean and count as ‘mathematical truth’ and whose ‘mathematical truth’ is privileged in mathematics education. it allows for a way of bringing divergent notions of mathematics content that have emerged into a single framing that enables these to coexist. the first kind of truth recognised in the trc (1998, p. 111) was forensic or factual truth based on ‘objective information and evidence’ that could be corroborated. this was truths that could be validated through impartial objective procedures. they were considered ‘scientific truths’ which utilised ‘empirical processes’ and were also regarded as ‘legal truths’. in mathematics education, for many there is only one objective mathematics variously described as academic mathematics or school mathematics, a canonical mathematics knowledge, free of context or social, political or cultural bias and unambiguously identifiable and articulated in the official curriculum at all levels of formal mathematics education. ‘mathematical truths’ are in the main those truths that can be proved and there are universally agreed ways for establishing these truths. it is a powerful mathematics underpinning and manifest in much of the material, technological, scientific and social world today. this conventional mathematics is discernible through its own signs and symbols and its own discourse even when written in different natural languages. however, this mathematics has increasingly been challenged, paradoxically referred to as ‘western’ mathematics and seen as a product largely of western culture. it is regarded as a paradox since many nationalities and cultures have and continue to contribute to its development. for bishop (1990, p. 51), mathematics, ‘as one of the most powerful weapons in the imposition of western culture’, has participated in ‘the process of cultural invasion in colonized countries’ through at least three agents: trade and commerce, for example, units, numbers, currency; mechanisms of administration and government, for example, computation systems; and imported systems of education, for example, mathematical curricula for the elite few. despite being seen as most outside the influence and realm of the social or cultural, this mathematics is deeply implicated in the distribution and enactment of political power. this is also one of the major criticisms launched by ethnomathematics. powell and frankenstein (1997) outline the main goal of ethnomathematics as challenging the particular ways in which eurocentrism permeates mathematics education [in] that the academic mathematics taught in schools world-wide was created solely by european males and diffused to the periphery; that mathematics knowledge exists outside of and unaffected by culture; and that only a narrow part of human activity is mathematical. (p. 2) bishop (1988) concludes that mathematics must now be understood as a kind of cultural knowledge. … just as all human cultures generate language, religious beliefs, rituals, food producing techniques, etc., so it seems do all human cultures generate mathematics. (p. 180) although mathematics can be thought of as a cultural product generated by different cultures in different social, political, economic environments, this does not mean that the forms such mathematical knowledge take are completely indistinguishable from each other. bishop identified five fundamental activities that he argues are universal across all cultures that have been studied: counting, locating, measuring, designing, playing and explaining. all cultures have, for example, developed systems for counting, but how these are organised, the number words and symbols used differ and are tied to the contextual needs and conditions of different peoples. numeration systems in africa range from a few number words of some san people who live in desert areas to complex systems developed by those who have a long history of commerce, such as the yoruba of nigeria who have been urbanised farmers and traders for many centuries before colonialism and use a vegesimal system that requires both addition and subtraction to express a number, for example, 525 = (200 × 3) – (20 × 40) + 5. the new south african mathematics curricula explicitly recognise mathematics as a cultural product and this has relevance for how access is provided to academic mathematics and also for valuing the different mathematics that learners bring into the classroom by virtue of the knowledge and skills they acquire from their community and life experiences. d’ambrosio (1985), as a founder of ethnomathematics, has argued that there are many mathematics, of which academics mathematics is but one, and these mathematics are developed by different sociocultural groups – from engineering mathematics to the mathematics of basket weaving. for him, ‘mathematics … are epistemological systems in their socio-cultural and historical perspectives’ (d’ambrosio, 1991, p. 374): this is a very broad range of human activities which, throughout history, have been appropriated by the scholarly establishment, formalised and codified and incorporated into what we call academic mathematics. but which remain alive in culturally identifiable groups and constitute routines in their practices. (d’ambrosio, 1985, p. 45) a mathematics education trc, if it were to be held, would from the evidence led by ethnomathematicians point to a second kind of truth conceptualised by the trc, that is, social or dialogue truth. whilst ‘the first (truth) is factual, verifiable and can be documented and proved’, social or dialogue truth according to the trc, is ‘the truth of experience established through interaction, discussion and debate’ (p. 113). this kind of truth acknowledges the importance of participation, of listening carefully, and in which ‘all possible views could be considered and weighed one against the other’ (p. 113). the trc argues that social truths established through dialogue promote transparency and democracy as a basis for affirming human dignity and integrity. the process of establishing the truth is as important as the truth itself. if mathematics is understood as a cultural activity and product then it follows that different groups in society come to develop different kinds of mathematics to deal with problems and needs they face, whether or not they refer to this as conventional mathematical knowledge. moreover, the process by which a ‘mathematical truth’ comes to be established as a truth is as important as the truth itself, if we follow this notion of the trc truth. in this regard, it is possible to refer to two broad areas: the mathematics of traditional societies, of indigenous peoples in both the developed and developing worlds, and the mathematics of different social and cultural groups in societies of today, of adults and children. these could be deemed ‘dialogue or social truths’ constituting a form of ‘mathematical truths’ that have come to be established over time by particular peoples or communities. although mathematics as a category is often not found in traditional or indigenous cultures, those who study mathematics in such contexts draw on a range of methodologies and disciplines, such as anthropology, archaeology, history, linguistics, economics, art, literature and oral traditions. ongoing ethnomathematics research in particular has demonstrated that a wide variety of mathematical ideas are found in traditional cultures. these ideas have been elaborated through games, patterns, art, architecture, systems of time and money, logic, kinship relations, and practices and artefacts used in everyday and traditional life. in each culture or community, certain groups or individuals share a mathematical disposition and are in a sense custodians of mathematical ideas that evolve over time (ascher, 1991), making comparisons across cultures difficult. it is however by linking mathematical developments to broader social, cultural, historical and political changes that descriptions of the mathematics of different groups (joseph, 1991) and the ‘mathematical truths’ of different peoples may be valued and can be understood in an authentic and unprejudiced way. a large and growing body of research has also shown that mathematical knowledge is generated by different groups of adults and children in a wide variety of contexts outside formal schooling. studies involving dairy workers, carpenters, bookies, builders, fisherman, farmers, street vendors, shoppers, market sellers, dressmakers and many others have all been shown to develop efficient strategies for solving mathematical problems in their everyday life and work situations. informal mathematical concepts and skills have been observed in children across nationalities, social classes and cultures. however, the mathematical understanding that children acquire has been explained to be rooted in their social and cultural experiences and may not resemble those expected or required in mathematics classrooms. this ‘distance’ between school culture and different groups in society has been analysed not only with respect to learners from traditional or indigenous cultures, but also with respect to other marginalised groups within western society, such as women and the working class. differences and similarities between school mathematics and out-of-school mathematics have been documented. for example, school mathematics is predominantly written, whilst oral forms have been found to characterise out-of-school mathematics (nunes, schliemann & carraher, 1993). the third kind of truth set out in the trc framework, personal or narrative truth, is one in which each person is ‘given a chance to say his or her truth as he or she sees it’ (p. 112). it is a truth based on the lived experiences of the individual who is reporting, a form of truth that reflects the ‘constructed nature of meaning-making’ (dhunpath & samuel, 2009, p. x). this form of truth was recognised as recovering national memory that had been officially ignored, a ‘validation of the individual subjective experience of people who had previously been silenced or voiceless’ (trc, 2008, p. 112). in mathematics education, it has long been recognised that each person develops mathematics ideas, knowledge and skills by virtue of their individual thinking processes and schemas. more recently, it has been recognised that this knowledge is also acquired by an individual by virtue of the community in which that person lives, works and functions. whilst the former has derived largely from psychological perspectives, the latter has arisen from more sociological ones. a substantial and well-established body of research in the area of (socio) constructivism demonstrates how individuals make sense and meaning of new mathematical ideas in terms of the frameworks each person has and how each develops strategies for dealing with mathematics that they are confronted by. many of the studies of groups identified above have come out of in-depth research into how individual learners (adults and children), in varying contexts, reason and think mathematically and do mathematics. a key challenge for teachers of mathematics is to be able to discern the kinds of mathematical ideas, knowledge, skills and even attitudes each of their learners brings to school to provide access into school mathematics. each learner develops their own ‘mathematical truths’, by virtue of their personal life trajectory, conditions and opportunities for learning, which have to cope with differences and conflicts between mathematical practices in school and in out-of-school contexts, for example, when they migrate from rural to urban areas. the official south african mathematics curricula give due recognition to the unique experiences and knowledge of the individual but the question that remains is accessing the ‘mathematical truths’ of learners. the challenge is to avoid stereotyping students, for instance choosing a mathematics problem involving traditional zulu home building if there are african learners in a diverse classroom, even though none of the learners may have lived in or experienced a rural context. these personal ‘mathematical truths’ carried by each learner are also rendered invisible and not accessed or accommodated in national and large-scale mathematical assessments. a fourth kind of truth in the trc is healing and restorative truth. it is ‘the kind of truth that places facts and what they mean in the context of human relationships – both amongst citizens and between the state and its citizens’ (p. 114). in the trc it was not enough to establish what was the truth, as objective and factual, but it was equally important to see it as connected to how it was acquired and the purpose it was to serve. the role of ‘acknowledgement’ was highlighted as a form of affirmation of dignity by placing information on record and publically recognising it. if what counts as mathematical knowledge and truth in mathematics is broadened then it should be possible to admit and accept that there are different mathematics within and across societies. however if the power of mathematics as an abstract knowledge is to be maintained and owned by all, then the relations between academic, western or conventional mathematics and the different mathematical knowledges and practices of different groups and individuals have to be brought into dialogue with each other, to be connected and contextualised. by valuing different kinds of mathematics and ways of knowing (and doing) mathematics, different peoples are valued and respected. notwithstanding that the playing field of the different mathematics are not level, for mathematics to have a restorative power in situations of conflict, there has to be at the very least, recognition that there are different ways of knowing the world mathematically, which may be relevant, useful and appropriate in different contexts. the enormous power of academic mathematics to cast its gaze on almost any human activity today and re-present or appropriate it through its discourse gives healing and restorative ‘mathematical truths’ a particularly important place in mathematics classrooms. the legacy of colonialism and apartheid, which damaged the growth of indigenous knowledge systems, must be addressed both for its own sake to reclaim lost and hidden ‘mathematical truths’ and also because it provides possibilities for new knowledge, even if defined in terms of academic or western knowledge systems. the role of ‘acknowledgement’ in restoring dignity lies in the recognition that different cultures on every continent, in different periods of its history, have contributed mathematical knowledge. acknowledging multiple histories is part of healing. the hegemony of western or academic mathematics has been challenged for the ways in which conventional histories of mathematics have ignored, marginalised, devalued or distorted the contributions of peoples and cultures outside europe – of china, india, north africa and the arab world – to that mathematics that is referred to as academic or western mathematics. joseph (1991) points out that [s]cientific knowledge which originated in india, china and the hellenic world was sought out by arab scholars and then translated, refined synthesised and augmented at different centres of learning… from where this knowledge spread to western europe. (p. 10) however, eurocentric historiographies of mathematics have also been criticised from another perspective: for failing to acknowledge the independent histories of mathematics of peoples who have developed their own mathematics, particularly the indigenous peoples of different regions of africa, america and australia (ascher, 1991). a healing and restorative mathematics would therefore be one that recognises the rich mathematical histories of peoples not only in terms of conventional mathematics but on its own terms and its own forms, which may or may not be easily distinguishable as mathematics, and would be dignified by being given a proper space and engagement in mathematical curricula. recognising multiple ‘mathematical truths’, as well as the processes by which these truths come to be constructed, allows for improved possibilities for the critique of truths in mathematics to be found within mathematics. in particular, these varied forms of ‘mathematical truths’ have the potential to make visible and more explicit the formatting power of mathematics, which acknowledges each ‘formatter’, from the constructors or producers of mathematics to the consumers and those marginalised, because each kind of personal, social or academic ‘mathematical truth’ is part of a network of truths in mathematics and each is seen to have value. within this framework of ‘truths’, ‘mathematical truth’ as factual, objective, invariant and decontexualised may be deemed but one kind of truth within a framework of ‘truths’ that need to get expression in a mathematics classroom. it alludes to how conflicts and dialogues that take place in such classrooms need to be handled if mathematics education is not only about increasing knowledge and awareness of inequities and injustices, but also a means for forgiveness and healing. often mathematics is presented as a one and only truth, the most objective or neutral and this one truth is to be most valued whilst social or personal mathematical knowledge, skills and practices are subordinated or silenced. it is a healing and restorative ‘mathematic truth’ that gives meaning to a pedagogy of forgiveness. if these ‘mathematical truths’ are seen to be in a relation of complementarity with each other, then it is possible to acknowledge: firstly, that all kinds of truth in mathematics coexist, even though not all forms of mathematical truths find expression at any one moment in a classroom; secondly, that they need not be in harmony with each other because they exist in relations of cooperation and opposition; and thirdly, that this may be necessary for the growth and development of each. conclusion top ↑ the triad of mathematics education, democracy and development was explored with reference to the mathematisation of society through the first question. this societal focus drew attention to the formatting power of mathematics and the developmental challenges faced in a country like south africa, pointing to the potential powerful role of mathematics education in addressing these for both nthabiseng and pieter. the second question threw a spotlight on the mathematics education system. the distribution of mathematics education and its associated educational possibilities was brought into sharp relief through a discussion on international studies and the national grade 12 mathematics assessments and performance, and demonstrated how mathematics education becomes complicit in the inequities that are reproduced in society through mathematics curricula reforms and teacher education provisions. the third question moved the discussion into the school and classroom. it exemplified how mathematics classrooms can be places where democracy is learnt and development issues are engaged through a mathematics education pedagogy of conflict and dialogue that embodies forgiveness. the trc truth framework metaphor drawn on in the fourth and final question to elaborate mathematical content matter, shows how the very mathematisation of society can recognise different forms of ‘mathematical truths’ that can coexist and come to constitute a mathematical knowledge and a mathematics education that can be healing and restorative of the dignity of people. just as human beings are connected in complex relations of cooperation and contradiction, so too are our knowledge forms, including mathematics. acknowledgements top ↑ this article is based on my inaugural lecture, mathematics education, democracy and development: challenges for the 21st century, faculty of education, university of kwazulu-natal, durban, on 04 april 2008. competing interests i declare that i have no financial or personal relationship(s) that may have inappropriately influenced me in writing this article. references top ↑ adler, j. 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(2011). teachers’ understandings of forgiveness in a troubled society: an empirical exploration and implication for forgiveness pedagogies. pedagogies: an international journal, 6(3), 250−264. microsoft word 64 front cover final.doc pythagoras 64, december, 2006, pp. 29-36 29 concerns about the south african mathematical literacy curriculum arising from experience of materials development lynn bowie and vera frith marang centre for mathematics and science education, university of the witwatersrand, and numeracy centre, academic development programme, centre for higher education development, university of cape town lynn.bowie@wits.ac.za and vfrith@maths.uct.ac.za in this paper we reflect on our experience of developing mathematical literacy material for the further education and training (fet) band in south african schools, adult learners, university students and for participants in a youth development project. we use this experience to highlight some problems and concerns about the south african mathematical literacy curriculum for learners in the fet band and offer some cautions and suggestions. in particular we highlight the importance of the educational community in south africa developing a shared understanding of what mathematical literacy is. we discuss the importance of distinguishing between mathematics and mathematical literacy and of clarifying the role of technology in mathematical literacy. we explore the difficulties and importance of a proper understanding of the contexts used to teach mathematical literacy and argue that more attention needs to be paid to the integration of mathematical literacy with other school subjects. finally we raise some of the issues that a common final assessment task might have on the learning and teaching of mathematical literacy. introduction we believe the introduction of mathematical literacy into south african schools presents both exciting opportunities and enormous challenges. mathematical literacy has the potential to provide learners, who previously did not continue with mathematics beyond grade 9, with access to the kind of skills that are crucial in order for them to participate meaningfully in the modern world. for this reason we believe it is important that the mathematical literacy that is taught in schools is of a high quality. we believe that it is vital that, as we implement the mathematical literacy curriculum and identify the difficulties we face, we engage with constructive criticism of the curriculum so as to improve it and make it more likely to achieve the goals for which it was introduced. in this paper we reflect on our experience of developing mathematical literacy material for the fet band in south african schools, adult learners, university students and for participants in a youth development project. we use this experience to highlight some problems and concerns about the south african mathematical literacy curriculum for learners in the fet band and offer some cautions and suggestions. one of the key issues that has emerged for us is that it is vital that the educational community in south africa develop a clear and shared understanding of what mathematical literacy is. this is a “slippery concept, the subject of lively debate” (coben, 2003: 9) for at least the last decade particularly in australia and england (where it is usually called ‘numeracy’) and in the united states (where it is most often called ‘quantitative literacy’). this debate, which is comprehensively reviewed by coben (2003), not only concerns itself with the definition of the concept, but also its relationship to mathematics itself. hughes-hallet (2001: 94) expresses this distinction as follows: “mathematics focuses on climbing the ladder of abstraction while quantitative literacy clings to context. … mathematics is about general principles that can be applied in a range of contexts; quantitative literacy is about seeing every context through a quantitative lens.” this idea that mathematical literacy is mainly concerned with mathematics used in context is fundamental to all the definitions of mathematical literacy, whether it is seen as a social practice, form of literacy, a critical approach, or a behaviour (or even a set of skills). “at the very least then, the definitions garnered from this debate would agree that numeracy is to do with ‘using maths in context’ and that to be numerate is to have the ‘capacity to use maths effectively in context’” concerns about the south african mathematical literacy curriculum arising from experience of materials development 30 (johnston, 2002: 4). mathematical literacy is not the same thing as basic mathematics, which provides learners with decontextualised mathematical skills. some authors even claim that numeracy is “not less than or even part of mathematics, but something more than mathematics” (johnston and yasukawa, 2001: 280) being “the ability to situate, interpet, critique, use and perhaps even create mathematics in context…” (johnston and yasukawa, 2001: 279). the debate about the meaning of the terms ‘mathematical literacy’ (‘quantitative literacy’ or ‘numeracy’) also brings to the fore the framing of the concept as an ability, behaviour or a social practice, rather than a ‘subject’ or area of study. the focus is on defining what a mathematically literate person does, rather than what collection of topics, skills and contexts mathematical literacy could be thought to consist of. so definitions contain statements like the following: “numeracy: the ability to interpret, apply, and communicate mathematical information” (adult literacy and lifeskills survey website) and “quantitative literacy is the ability to identify quantitative relationships in a range of contexts” (hugheshallett, 2001: 94). evans (2000: 236) defines numeracy as social practice: “the ability to process, interpret and communicate numerical, quantitative, spatial, statistical, even mathematical, information, in ways that are appropriate for a variety of contexts, and that will enable a typical member of the culture to participate effectively in activities that they value” (cited in coben, 2003:10). the national curriculum statement offers the following definition, which frames mathematical literacy as a “subject” rather than a competency, a behaviour or a practice: mathematical literacy is a subject driven by life-related applications of mathematics. it enables learners to develop the ability and confidence to think numerically and spatially in order to interpret and critically analyse everyday situations and to solve problems. (department of education, 2003: 9) this definition makes it clear that there are three key elements of mathematical literacy: the “content” (i.e. the mathematics), the “contexts” (i.e. the life-related applications, the everyday situations, the problems) and the abilities and behaviours that a mathematically literate person will exercise (confidence, thinking, interpreting, analysing and solving). however what is less clear is exactly how those three elements should play out together in the mathematical literacy classroom. in the course of our work we have heard opinions from teachers, department of education officials, publishers, university lecturers and administrators about what they feel mathematical literacy will be. for some, mathematical literacy means teaching basic mathematics through “word sums”. for others it means that the mathematical literacy learning programme a teacher develops must be structured around contextual themes from which the underlying mathematics can emerge. in some quarters it is portrayed as adequate for the educator simply to keep a mathematical eye on the media and create mathematical tasks based on what emerges from these observations. for others, it brings to mind strongly the mathematising notion of realistic mathematics education. the central idea of realistic mathematics education is “that mathematics can best be learnt from starting from a concrete, realistic situation that appeals to students” (vos, 2002: 31). the problems in these realistic situations are mathematised by being “transferred to a more or less mathematical problem” (de lange, 1996: 69) which can then be analysed with mathematical tools. in contrast, for others mathematical literacy is simply a remodelling of the old standard grade mathematics curriculum. the subject assessment guidelines for mathematical literacy released by the department of education (september 2005) addresses the “content-context” debate head on: on the one hand, mathematical content is needed to make sense of real-life contexts; on the other hand, contexts determine the content that is needed. when teaching mathematical literacy, teachers should avoid teaching and assessing mathematical content in the absence of context. at the same time teachers must also concentrate on identifying in and extracting from the contexts the underlying mathematics or ‘content’. that is avoid teaching and assessing contexts without being deliberate about the mathematical content. (department of education, 2005b: 7) although this begins to address some of the confusion, in our experience, the learning outcomes and assessment standards in the mathematical literacy curriculum do not provide a lynn bowie and vera frith 31 framework that gives clear guidance on how this content-context interplay can be achieved. in addition it is in places quite ambiguous about the specific content topics and contexts that should be taught. it is also quite vague about whether learners are expected to learn to use technology or not. in this paper we will discuss our observations and concerns about the curriculum under four headings, which highlight four issues in the current mathematical literacy curriculum that we believe need to be addressed. issue 1: the mathematical literacy curriculum looks too much like mathematics we have already mentioned that the international debate around mathematical literacy includes a strong thread that discusses the distinctions between mathematics and mathematical literacy (hughes-hallet, 2001; coben, 2003). for example, in his paper “mathematics and numeracy: two literacies, one language”, lynn arthur steen (2001) reviews a us government report on what the workplace requires of schools. he notes that this report organises mathematical ideas into entirely different categories to traditional mathematics. these range from basic skills (like arithmetic, estimation, understanding chance) to thinking skills (e.g. reasoning, planning) to more advanced competencies (e.g. allocating resources, technology, organising information, applying technology). he notes: “algebra, geometry, trigonometry, and analysis sure sound different from resources, information, systems, and technology. these dramatically different perspectives on mathematics education – classical canon vs. modern employment – illustrate important differences between mathematics and numeracy. one conveys the power of abstraction, the other the power of practicality; one is organized by categories inherited from the past, the other focuses on the way knowledge is used in the information age; and one is encountered mostly in school, the other mostly in real life” (steen, 2001: n.p.). the mathematical literacy curriculum does not attempt such a re-framing. the learning outcomes for the mathematical literacy curriculum are: learning outcome 1: number and operations in context learning outcome 2: functional relationships learning outcome 3: space, shape and measurement learning outcome 4: data handling. the learning outcomes for the mathematics fet curriculum are: learning outcome 1: number and number relationships learning outcome 2: functions and algebra learning outcome 3: space, shape and measurement learning outcome 4: data handling and probability. thus the mathematical literacy curriculum is divided into four outcomes using the same contentbased divisions as the mathematics fet curriculum. this division frames the way educators will interpret the document. the intention of the curriculum authors was not that mathematical literacy learning programmes should be structured according to these divisions. in fact, in the learning programme guidelines, they are at pains to point out that “teachers should choose meaningful contexts to embed the content gleaned from the assessment standards in clusters across the learning outcomes where possible” (department of education, 2005a: 13) they also stress that “teachers should view the learning outcomes as integrated and connected.” in the subject assessment guidelines they point out that in a mathematical literacy examination “each question will integrate assessment standard from more than one learning outcome” (department of education, 2005b: 13). however, the framing of the curriculum by the division into four learning outcomes is powerful. it is difficult for educators who have been educated under a traditional mathematics curriculum (and many of whom may have also taught a traditional mathematics curriculum) to break away from the idea of doing some number work, some algebra and graphs, some geometry and some statistics. it is particularly difficult to envisage a new way of structuring a learning programme in a mathematical literacy classroom when the curriculum itself constantly pulls one back to the traditional framing. evidence of how difficult it is not to define the substance of a mathematical literacy class in purely mathematical terms can be seen in some of the curriculum support documents themselves. for example, the mathematical literacy subject assessment guidelines states that the grade 12 mathematical literacy examination will have a spread of questions that ensure that each of the learning outcomes is allocated 25% of the total marks. this allows the mathematical content to assume the importance of the major organising concerns about the south african mathematical literacy curriculum arising from experience of materials development 32 principle for assessment. in addition, in the learning programme guidelines (department of education, 2005a: 20) it is suggested that educators: “consider what forms of assessment will be best suited to each of the learning outcomes and assessment standards and list these for the three grades”, thus encouraging assessment practices that do not integrate assessment of the learning outcomes under the study of relevant contexts. another factor that made it difficult for us (and we believe will make it difficult for other educators) to view mathematical literacy, as defined by the curriculum documents, as different from “an easier version of mathematics” was some of the choices of mathematical content. for example, in grade 10 the curriculum states that, in terms of functional relationships we should include “linear, inverse proportion and compound growth in simple situations” and in grade 11 they expand this list to include “quadratic functions”. these are all functions which have traditionally been part of south african school mathematics. it is difficult to escape the conclusion that this is the reason they have been included in the mathematical literacy curriculum. on the other hand, the inclusion of functions like piece-wise defined functions or step functions, which provide mathematical models for a number of real-life situations (e.g. parking fees, stepped tariffs for water use), would have been more in keeping with the intentions of the curriculum. why are these then excluded from the list? in asking these questions around the inclusion of content (particularly content that looks like traditional school mathematics) we are not necessarily arguing that content topics that are currently included should not be included, but we are arguing for a need to provide a clear “mathematical literacy” motivation for their inclusion. as mathematical literacy is a new subject, we believe this will help teachers, students, materials developers and the developers of the final assessment papers, to develop a better shared understanding of what mathematical literacy is. it will help us see the key goal behind the inclusion of any specific content, and so enable us to focus activities or problems we use in the classroom to achieve that goal. in addition recording the reasoning behind curriculum decisions also provides a resource for future development of the curriculum. it seems clear that the mathematical literacy curriculum will have to be adapted and improved. it is being implemented for the first time in schools this year and the feedback from learners, teachers, parents, the business community, higher education and other interested groups should have an impact on how it plays out in practice. a motivation for why particular curriculum decisions were made allows us a framework to argue against or work with to further build the curriculum. this motivation will help to ensure that good ideas in the curriculum are retained, even if there are problems in the initial implementation. we have pointed out that the assessment standards in the curriculum are largely specified in terms of mathematical content. there is a real danger that this can be interpreted to imply that mathematical literacy is slightly toned-down standard grade mathematics “with word sums”. this type of misconception is heightened by statements in the curriculum (department of education (doe), national curriculum statement, mathematical literacy, 2003: 21, assessment standard 11.2.1) like “for example, interpret and critique quotations for two similar packages given by cell phone providers”. this is essentially intended to be about simultaneous linear equations. any real-life look at packages offered by cellphone providers is unlikely to end up in linear equations except perhaps through an extensive process of modelling using data about average patterns of usage and involving averaged tariffs. as most people would struggle to get access to this data one can only assume that the curriculum intends a “word sum” of the sort: “cellphone package a costs r40 per month and r2 per minute of call time, cellphone package b costs r50 per month and r1,75 per minute of call time – compare.” as this scenario bears no relation to reality, we would argue this kind of example is not mathematical literacy, but couching learning about linear equations in inauthentic supposedly real-life examples. these kinds of “pseudo-contextualisations need to be avoided at all costs. not only are they demeaning to adults and to youths, they fail to prepare them for participation in the varied discourses of the workplace” (fitzsimons, 2005: 38) this idea is also emphasised by usiskin (2001: 84) who points out that artificial word problems “are not applications, nor should they substitute for them”. if the mathematical literacy curriculum is to have credibility as a preparation for coping with the kinds of poorly-defined problems that make up the real demands of life and work, then inauthentic “applications” must be avoided. lynn bowie and vera frith 33 issue 2: computer technology “current technology has caused much of the increase in the need for quantitative literacy. without this technology, newspapers, financial institutions, scientific endeavors, and everything else that uses mathematics would not be the same” (usiskin, 2001: 82). thus, mathematical literacy increasingly requires the ability to make use of and to understand the role of computers in science, social science, professional and everyday life, and in the workplace. “the changing nature of workplaces and the ubiquity of computer-based systems for the automation and control of processes and the management of information, has brought about the need for employees at all levels to engage with these systems, to interpret their outputs and to make sense of the abstract models on which they are based” (kent, hoyles, noss and guile, 2004: 1). at its most fundamental level this knowledge includes the role and use of calculators and in the context of tertiary education and the workplace often includes the effective use of spreadsheets (frith, jaftha and prince, 2005). hoyles, wolf, molyneux-hodgson and kent (2002) investigated the mathematical skills required in the workplace by studying aspects of work in the engineering, financial services, health care, food processing, packaging, pharmaceutical and tourism sectors in the united kingdom. they report that their findings suggest that at all levels of the workforce “there is an inter-dependence of mathematical literacy and the use of it in the workplace” and that “it and mathematical skills are interdependent” (ibid.: 3). fitzsimons (2005: 29) reported on an extensive study of mathematical literacy practices in australian workplaces and also concluded that aspects which are significant in mathematical literacy include “integrated mathematics and it skills”, and “an ability to create a formula (using a spreadsheet if necessary)”. the importance of the use of appropriate technology is given recognition in the original subject statement (see for example national curriculum statement, mathematical literacy learning outcomes 10.2.2 on p. 22; 10.4.1 on p. 30) and in the subject assessment guidelines (p. 7); although it is de-emphasised in the core curriculum which is to the assessed in the first three years of implementation of the new curriculum. clearly we are aware that there are some schools that lack electricity and many that lack adequate computing facilities or computer literate teachers. however, learning to use appropriate technology is a crucial aspect of mathematical literacy and we believe that the department of education needs to offer a clear plan of how and when this problem of inequitable access to knowledge about technology will be addressed. this needs to be incorporated into the way the mathematical literacy curriculum is written and seen as developing. a large proportion of learners are currently disadvantaged by lack of access to and knowledge of computer technology (and in many cases even scientific calculators). if this problem is not addressed these learners will invariably encounter barriers to progress and the “digital divide” between the richer and poorer members of society will be propagated. there is a need for an investigation into the mathematical skills required in south african workplaces, similar to the ones referred to above, in england (hoyles et al., 2002), australia (fitzsimons, 2005) and the usa (steen, 2001). issue 3: where does understanding the contexts fit in? perhaps one of the key problems we came up against in developing mathematical literacy materials is that in order to mathematise a context one needs to have a good understanding of the context. this poses enormous challenges for teachers. mathematical literacy teachers will not only be required to understand mathematics, but also voting systems, mortgages, retirement funding, hiv/aids, global positioning systems, socially responsible trade (to name but a few of the contexts suggested in the current curriculum). similarly, mathematical literacy learners will have to develop a good grasp of these contexts and herein lies a host of problems. for example, the mathematical literacy curriculum has a focus on personal finances. the topics that learners are expected to deal with range from basic budgeting to compound interest to the effect of changing interest rates on mortgage repayments to comparing different retirement options. these are important skills and we feel that it is entirely appropriate to include them in the school curriculum. however, in planning the teaching of these topics we cannot assume that all learners in our country have an adequate experience of banks, let alone have an understanding of interest or of notions of risk and return on investments. if we expect the mathematical literacy learner to be able to use mathematics “to interpret and critically analyse everyday situations” (doe, national curriculum statement, mathematical literacy, 2003: 9) then they have to have enough familiarity with the concerns about the south african mathematical literacy curriculum arising from experience of materials development 34 situations or develop sufficient understanding of the situations in order to use their mathematical knowledge to analyse them. when we attempted to create material to deal with the example suggested in the curriculum for grade 12: “calculate the net effect of different interest offerings and bank charges when saving schemes are considered” (doe, national curriculum statement, mathematical literacy, 2003: 19, assessment standard 12.1.3), it became apparent that there were a large number of notions we should deal with if we really wanted learners to be able to do this in a way that would be useful for them. we felt that we should look at different kinds of bank accounts (transaction accounts, saving accounts), at features of these bank accounts (atm cards, debit cards, cheques, stop orders, debit orders, etc.) and at practical issues (writing a cheque, using an atm, reading a bank statement). all of this would require a large amount of teaching time to meet just one example listed under one assessment standard in the curriculum. teachers, under time pressure in the classroom, and aware of the need to teach what will be examined are going to be faced with a dilemma. we suggest that there are two ways to deal with this dilemma. the first is to “avoid teaching and assessing contexts without being deliberate about the mathematical content” (doe, national curriculum statement, subject assessment guidelines, 2005b: 7), and simply leave out the aspects of the context that are not mathematical. but this could leave us in a bizarre situation where we have a learner who can “calculate the effect of a fixed interest rate against probable variations in interest rates when buying a house or when choosing an investment” (doe, national curriculum statement, mathematical literacy 2003: 19, assessment standard 12.1.3) but who does not know how to write a cheque or how to use an atm safely! the second is to take seriously the notion of integration. the curriculum stresses that integration is seen as important: “integration is achieved within and across subjects and fields of learning. the integration of knowledge and skills across subjects and terrains of practice is crucial for achieving applied competence” (doe, national curriculum statement, mathematical literacy, 2003: 3). but realistically, simply stressing that integration is important is not sufficient to ensure that integration will happen. perhaps at some schools where there are exceptionally motivated teachers, the teachers will get together and find ways to create better learning opportunities for integration, but we believe that even they will struggle in the absence of curricula that pay more than lip service to it. we believe that the type of learning that can be achieved in mathematical literacy can be considerably strengthened through integration. however, we also believe that the place where serious attention needs to be paid to integration is at the curriculum level. if the design of curricula purposefully create spaces for integration then it will be more feasible for classroom teachers to act on this. for example, if in the life orientation and mathematical literacy curriculum understanding and managing personal finances was seen as a crucial skill to develop, then an extremely beneficial integrated learning experience could be developed. an integrated look at personal finances allows not only the time to investigate different bank accounts, charges and interest rates, but also to discuss attitudes to money, saving and risk, for example. in a similar way an integrated approach could mean that learners look at data gathering and analysis within the context of hiv/aids or substance abuse in a way that gives the learners a fuller and more personally meaningful understanding of the context they are mathematising. in addition, topics like hiv/aids, substance abuse, teenage pregnancy and poverty are not necessarily going to be perceived as theoretical contexts to be mathematised by learners. for many learners these are very real issues they are struggling with in their daily lives and it could be uncomfortable and irresponsible to treat them as mathematical tasks only. it is our belief that insufficient thought has been given to integration at the curriculum level and because of that we have missed an opportunity to provide a rich and meaningful learning experience for our learners. as a result of working on a project creating an integrated mathematical literacy, life orientation and communication curriculum for national youth service projects we conclude that, while integration is extremely difficult to achieve, the potential it has to enhance the learning of mathematical literacy, life skills and communication makes it worthwhile. in addition the integration ensures that mathematics, and the way the mathematics is taught, is truly aimed at enabling the learners to “interpret and critically analyse everyday situations” (doe, national curriculum statement, mathematical literacy, 2003: 9) lynn bowie and vera frith 35 issue 4: a single national assessment requires that the curriculum is more specific about content and contexts at the end of grade 12 mathematical literacy learners will be assessed largely through a common final assessment task. all mathematics teachers are familiar with the tension between providing quality mathematics education and covering the curriculum in a way that prepares learners for the final examination. this tension will no doubt play itself out in the mathematical literacy classrooms too. the implications of this for mathematical literacy as a school subject are too numerous and complex to do justice to in this paper. however, we do want to point out that our experience of developing material for fet mathematical literacy brought to light the fact that in many cases the curriculum does not specify precisely enough the content and, in some cases, the contextual knowledge, that learners will be assessed on. for example, the grade 12 assessment standard 12.1.3 (doe, 2003: 17) states that learners should “analyse and critically interpret a wide variety of financial situations mathematically, inclusive of …. critical engagement with debates about socially responsible trade.” socially responsible trade is the subject of vast debate, encompasses issues that range from environmental impact to global economic power relations. a critical engagement with the debates about socially responsible trade could (and does) form the basis of a year-long university level economics course. alternatively socially responsible trade could be dealt with at a superficial level. the mathematical literacy curriculum provides no indication of where on this spectrum mathematical literacy teachers need to place the activities they develop for learners on this topic. similarly in learning outcome 3, (doe, 2003: 24) we are told that learners need to be able to work with the “perimeters and areas of polygons”. our experience in working with prospective mathematical literacy teachers has shown that some interpret this to mean taking another look at the perimeters and areas of the various quadrilaterals that were dealt with in grade 9, whereas others feel this means they need to find a way to help learners deal with any polygon. in assessment standard 11.1.1 (doe, national curriculum statement, mathematical literacy, 2003: 15) the curriculum says that learners need to be able to “find ways to explore and analyse situations that are numerically based, by…working with complex formulae by hand and with a scientific calculator, for example: a acbb x 2 42 −±− = .” does this mean learners must be able to find the roots of a quadratic equation or could we use another formula here? in raising this issue and discussing these examples we are aware that we might sound irritatingly like the learner who continually asks “will this be in the test?” instead of focusing on learning and understanding. however, unfortunately the common final assessment task will have enormous implications for learners and so making sure that learners are able to deal with what is in it will have to be taken seriously by any mathematical literacy teacher. conclusion in this paper we have identified four issues that emerged for us in the process of developing mathematics literacy material: • we need to develop a shared understanding of what the subject “mathematical literacy” is and how it differs from the subject “mathematics”; • it is important that we address the issue of technology within the mathematical literacy curriculum; • if we want to provide a responsible and meaningful learning experience then the contexts used within mathematical literacy need to addressed from both mathematical and non-mathematical points of view. as we rethink the curriculum we need to address the issue of the integration of mathematical literacy with other subjects more seriously; • the fact that there will be a single national assessment for mathematical literacy at the end of grade 12 requires that the curriculum is clearer about what content and contexts learners need to be familiar with. we offer these issues for consideration by the broader mathematics education community because we believe that the process of developing a high quality mathematical literacy curriculum requires that all people involved in the implementation of the curriculum should reflect on and share their experience. through this process of sharing with and learning from each other we will not only strengthen what we are doing as we implement the current mathematical literacy curriculum, but we will also lay the basis for a strong revised mathematical literacy curriculum. concerns about the south african mathematical literacy curriculum arising from experience of materials development 36 references adult literacy and lifeskills survey website. retrieved october, 2006, from http://nces.ed.gov/surveys/all/faq_all.asp?faq type=1 coben, d. (with colwell, d., macrae, s., boaler, j., brown, m., & rhodes, v.) (2003). adult numeracy: review of research and related literature. london: national research and development centre for adult literacy and numeracy. retrieved february, 2006, from http://www.nrdc.org.uk/uploads/documents/doc _2802.pdf de lange, j. (1996). chapter 2: using and applying mathematics in education. in a. bishop (ed.), international handbook of mathematics education (pp. 49-97). dordrecht: kluwer. department of education, south africa. (2003). national curriculum statement grades 10 12 (general). mathematical literacy. retrieved february, 2006, from http://www.education.gov.za/content/document s/111.pdf department of education, south africa. (2005a). national curriculum statement grades 10 12 (general). learning programme guidelines mathematical literacy. retrieved february, 2006, from http://www.education.gov.za/ content/documents/737.pdf department of education, south africa. (2005b). national curriculum statement grades 10 12 (general). subject assessment guidelines mathematical literacy. retrieved february, 2006, from http://www.education.gov.za/ content/documents/754.pdf evans, j. (2000). adults’ mathematical thinking and emotions: a study of numerate practices. london: routledge/falmer, taylor & francis. fitzsimons, g.e. (2005). numeracy and australian workplaces: findings and implications. australian senior mathematics journal, 19(2), 27-40. frith, v., jaftha, j.j., and prince, r.n. (2005). interactive excel tutorials in a quantitative literacy course for humanities students. in: m.o. thirunarayanan, and aixa pérez-prado (eds.), integrating technology in higher education (pp. 247-258). maryland: university press of america. hoyles, c., wolf, a., molyneux-hodgson, s., and kent, p. (2002). mathematical skills in the workplace. final report to the science, technology and mathematics council, institute of education , university of london. retrieved february, 2006, from http://www.ioe.ac.uk/tlrp/technomaths/skills200 2/maths-skills-workplace-final-report.pdf hughes-hallett, d. (2001). achieving numeracy: the challenge of implementation. in l.a. steen (ed.), mathematics and democracy, the case for quantitative literacy (pp. 93-98). usa: the national council on education and the disciplines. retrieved february, 2006, from http://www.maa.org/ql/093-98.pdf johnston, b. (2002). numeracy in the making: twenty years of australian adult numeracy. an investigation by the new southwales centre, adult literacy and numeracy australian research consortium (no. 0868039446). sydney: university of technology, sydney. retrieved february, 2006, from http://www.staff.vu.edu.au/alnarc/publications/ 02johnston.pdf johnston, b., and yasukawa, k. (2001). numeracy: negotiating the world through mathematics. in b. atweh, h. forgasz, and b. nebres (eds.), sociocultural research in mathematics education. an international perspective (pp. 57-73). new jersey: lawrence erlbaum associates. kent, p., hoyles, c., noss, r., & guile, d. (2004). techno-mathematical literacies in workplace activity. international seminar on learning and technology at work, institute of education, london, march 2004. retrieved march 17, 2005, from www.ioe.ac.uk/tlrp/technomaths/kent-ltwseminar-paper.pdf steen, l. (2001). mathematics and numeracy: two literacies, one language. the mathematics educator, 6(1), 10-16. retrieved october, 2006, from http://www.stolaf.edu/people/steen/papers/twoli ts.html usiskin, z. (2001). quantitative literacy for the next generation. in l.a. steen (ed.), mathematics and democracy, the case for quantitative literacy (pp. 79-86). usa: the national council on education and the disciplines. retrieved february, 2006, from http://www.maa.org/ql/079-86.pdf vos, p. (2002). like an ocean liner changing course: the grade 8 mathematics curriculum in the netherlands, 1995-2000. den haag: cip gebegens koninkelijke bibliotheek. 27-35 samson final.doc pythagoras, 72, 27-35 (december 2010) 27 visual technology for the autonomous learning  of mathematics1    helmut linneweber‐lammerskitten  university of applied sciences northwestern switzerland  helmut.linneweber@fhnw.ch  marc schäfer & duncan samson  rhodes university, south africa  m.schafer@ru.ac.za & d.samson@ru.ac.za    this paper describes a collaborative research and development project between the university of  applied sciences northwestern switzerland and rhodes university in south africa. the project  seeks  to  establish, disseminate  and  research  the  efficacy  and use  of  short video  clips designed  specifically for the autonomous learning of mathematics. specific to the south african context is  our interest in capitalising on the ubiquity of cellphone technology and the autonomous affordances  offered by mobile learning. this paper engages with a number of theoretical and pedagogical issues  relating to the design, production and use of these video clips. although the focus is specific to the  contexts of south africa and switzerland, the discussion is of broad applicability.      the use of both videos and video clips as a medium of learning and teaching is nothing new to the mathematics classroom. in our view, however, many of these videos are too long. furthermore, they are often underpinned by predetermined specific outcomes and pedagogical imperatives. there is a lack of short, succinct video material that both teachers and learners alike can access swiftly and efficiently, and use autonomously. the visual technology for the autonomous learning of mathematics (vitalmaths) project has been established to fill this void (linneweber-lammerskitten & schäfer, 2010). students at the school of teacher education at the university of applied sciences northwestern switzerland and mathematics education students at rhodes university in south africa have been developing a bank of video clips that unpack a variety of mathematical concepts. these video clips are all very short (1-3 minutes long) and specifically make use of natural materials (as opposed to high-tech animations) to animate and develop mathematical ideas and processes. specific to the south african context, we are especially interested in making use of cellphone technology as the primary distribution platform for these video clips. the implementation of a research agenda around the use of the video clips is currently being developed. numerous groups of teachers and learners in both participating countries will participate in this research, and a broad spectrum of school contexts will be considered. the research questions surrounding the video clips are multifaceted but will focus on: o their efficacy, o their use by participating teachers and learners, and o their accessibility. 1 this paper is an extension and elaboration of our previous report on the vitalmaths project (linneweberlammerskitten & schäfer, 2010). visual technology for the autonomous learning of mathematics 28 this research agenda will in turn inform the design and production of the video clips through a cyclical and reflective process. the text in the video clips will also be translated into numerous languages, including indigenous languages, and it is anticipated that this will open up further interesting research arenas. this paper engages with a number of theoretical and pedagogical issues relating to the use of video clips as a medium for teaching and learning mathematics. although the focus is specific to the contexts of switzerland and south africa, the discussion will be of broad applicability. exploration as an aspect of mathematical competence among the implications that can be expected from the implementation of the national educational standards in switzerland, as well as a number of other european countries, the following three appear to be most important for mathematics education (linneweber-lammerskitten & wälti, 2008): o it will be necessary to find better ways to deal with heterogeneity – especially to provide more support for weaker pupils. o it will be necessary to give more attention to the non-cognitive dimensions of mathematical competency, such as motivation, sustaining interest and the ability to work in a team. o it will be necessary to deal with aspects of mathematical competence that were mostly neglected in the past – especially the ability and readiness to explore mathematical states of affairs, to formulate conjectures, and to establish ideas for testing conjectures. these necessities find resonance with similar implications which have arisen from the implementation of the revised national curriculum statement in south africa. this is true not only in terms of subjectspecific outcomes and assessment standards, but it is also echoed by the social transformation imperatives of the curriculum and the desired attributes of the kind of learner envisaged within the south african education system (department of education [doe], 2002, 2003). according to the national educational standards in switzerland, mathematical competence comprises the ability and readiness to investigate and explore, to formulate conjectures, and to establish ideas for testing conjectures. of importance is that readiness, and other such non-cognitive dispositions, are considered constituent moments of the underlying notion of competence. the concept framework of the national education standards in switzerland defines competencies as "…cognitive abilities and skills possessed by or able to be learned by individuals that enable them to solve particular problems, as well as the motivational, volitional and social readiness and capacity to utilise the solutions successfully and responsibly in variable situations” (weinert, 2001, as quoted in klieme et al., 2004, pp. 16-17). this definition is based on the notion that motivational, volitional and social readiness as well as capacity are not only seen as fostering factors for, but rather as constitutive parts of mathematical competence. this can be made clear by a terminological distinction between competence and skill: a competence is defined as the ability to successfully meet complex demands in a particular context. competent performance or effective action implies the mobilization of knowledge, cognitive and practical skills, as well as social and behavior components such as attitudes, emotions, and values and motivations. a competence – a holistic notion – is therefore not reducible to its cognitive dimension, and thus the terms competence and skill are not synonymous. (oecd, 2003, p. 2) this is a valuable notion of competence to keep in mind when reading the south african revised national curriculum statement for mathematics where the study of mathematics is seen to contribute to personal development “through a deeper understanding and successful application of its knowledge and skills, while maintaining appropriate values and attitudes” (doe, 2003, p. 9). it is also worth noting the emphasis placed on mathematical exploration: “competence in mathematical process skills such as investigating, generalising and proving is more important than the acquisition of content knowledge for its own sake” (doe, 2003, p. 9). in terms of attitudes and dispositions, the teaching and learning of mathematics aims to develop (a) the confidence and competence to deal with mathematical situations without being hindered by a fear of mathematics, (b) an appreciation for the beauty and elegance of mathematics, (c) a spirit of curiosity, and (d) a love for mathematics (doe, 2002, p. 4). finally, it is significant that the curriculum seeks to create lifelong learners who are both confident and independent (doe, 2002, p. 3). helmut linneweber-lammerskitten, marc schäfer & duncan samson 29 bearing in mind that these educational standards and curriculum statements are representative of the minimum levels of knowledge and skills achievable at each grade, it follows that the establishment of such aspects of competence as minimal standards for all learners can only be successful if appropriate measures are taken to fully integrate weaker learners in the learning experience, and to create an environment that is stimulating, motivating, interesting and encourages social competence. mathematical investigation and exploration presupposes a higher level of motivation and volition than any other aspect of competence, since it requires not only the ability and readiness for self-organised fulfilment of a job order, but also the motivation to start acting mathematically even if there is no external order do so. furthermore, it requires a curiosity and a will to go beyond the mere results obtained. however, there is much evidence to suggest that weaker learners, particularly when they feel unsure, tend to prefer explicitly formulated work orders which according to their opinion define the beginning and the end of a mathematical activity. motivating mathematical exploration through video clips the national educational standards in switzerland and the revised national curriculum statement in south africa, especially in light of the various aforementioned implications, present both a challenge and an additional strain for future teachers that should not be underestimated. learners with weaker mathematical ability – a group that in itself is far from homogenous – will require stronger assistance and support, while at the same time the average and high-ability learners should not be neglected. it will thus be necessary to establish additional instruments, new forms of supervision and teaching strategies on the one hand, and new learning environments and tools on the other to meet this challenge. in accordance with the mathe 2000 initiative in germany (wittmann, 1995), new learning materials for mathematics have been introduced in the last decade in almost the whole of the german-speaking part of switzerland. although the materials encourage learning environments for autonomous exploration, they still require support and motivational activities for weaker learners. this was the starting point and challenge for student teachers at the university of applied sciences northwestern switzerland – to search for the development of auxiliary means that could not only release teachers from the frontal introduction to mathematical themes on the one hand, but also to provide an opportunity for weaker learners to experience genuine and challenging mathematical activities. students produced a series of short-length silent video clips illustrating and developing a variety of mathematical concepts. this initiative led to the genesis of vitalmaths. the vitalmaths project seeks to establish and disseminate a freely accessible growing databank of short video clips designed specifically for the autonomous learning of mathematics. common design principles of these video clips are that they are short, not too difficult to produce, are aesthetically delightful, and are self-explanatory – i.e. they require minimal instruction and are ready to be viewed and used to record observations, considerations and reflections in written form. the video clips do not require too much mathematical knowledge, nor do they make excessive intellectual demands which could lead to frustration. however, they nonetheless encourage genuine mathematical exploration that transcends the mere mathematical content of the film by encouraging a desire to experiment, use trial-and-error, formulate conjectures, and generalise results. a dedicated website2 has been established to house this growing databank of video clips from which the video files can either be freely downloaded or streamed. autonomous learning through video clips the notion of autonomy is differently conceptualised in different contexts (davies, 1987). however, the nurturing of learner autonomy as a general educational goal has been widely recognised by the teaching profession (chan, 2001). broadly speaking, learner autonomy is seen as the ability or readiness of a learner to take charge of his or her own learning. this entails both a capacity and willingness on the part of the learner to act independently (and in co-operation with others) as a socially responsible person 2 http://www.ru.ac.za/vitalmaths visual technology for the autonomous learning of mathematics 30 (dam, 1995, as cited in chan, 2001, p. 506). as mccombs and whisler (1989) comment, in order to become autonomous, learners need to attach personal meaning to learning activities while at the same time they need to develop cognitive and metacognitive capabilities for regulating both affect and motivation. furthermore, inasmuch as the capacity and willingness for the learner to take such responsibility may not necessarily be innate, it is important to recognise autonomous learning as a developmental phenomenon (mccombs & whisler, 1989). autonomy represents an inner endorsement of one’s actions – a sense that one’s actions emanate from within and are one’s own (deci & ryan, 1987, as cited in reeve & jang, 2006, p. 209). teachers cannot directly provide learners with an experience of autonomy (reeve & jang, 2006), but rather they need to provide genuine opportunities that encourage, nurture and support autonomous learning. the vitalmaths project aims to provide just such opportunities through the medium of short video clips. as mousley, lambdin and koc (2003, p. 425) succinctly comment, “autonomy is not a function of rich and innovative materials themselves, but relates to genuine freedoms and support given to students.” thus, critical elements of the design principles of the video clips take into account both cognitive and non-cognitive dimensions. these video clips unpack a variety of mathematical concepts which are progressively developed while encouraging a desire to experiment, use trial-and-error, formulate conjectures, and generalise results. importantly, however, they also encourage and support non-cognitive dimensions of mathematical competence such as curiosity, motivation, a willingness to know, determination to overcome disappointment and frustration, and a willingness to try things again. the video clips are purposefully made from natural materials, as opposed to high-tech graphics animations, in order to support autonomous learning on two levels. firstly, in terms of cognitive access, the use of natural materials should allow for a more direct and personally meaningful engagement with the content of the video clips when compared with the additional abstract dimension associated with hightech graphics animations. secondly, learners will be able to personally source all the required material to explore identical or similar scenarios, thus encouraging hands-on mathematical exploration that will have personal meaning for each learner. feedback from learners and teachers will be used to continuously refine the design principles of the video clips through a cyclical feedback process. in addition, to further the autonomous learning imperative of the project, our aim is to make these video clips available in a number of different languages, including south african indigenous languages. mobile learning specific to the south african context is our interest in the use of cellphone technology as the primary distribution platform for these video clips. not only will cellphone technology facilitate access to these video materials, but it is likely to enhance and support the autonomous learning objective of the enterprise. there are a variety of mobile devices that have found application within the education arena  personal digital assistants (pdas), pc tablets, ipods, and some games devices. however, fuelled by the development of powerful telecommunication networks which support an ever increasing range of data access services, coupled with technological advances and steadily declining costs of cellphones themselves, cellphones have emerged as a viable option for mobile learning. the vitalmaths project aims to capitalise on the flexible and versatile potential of cellphones for mobile learning. selanikio (2008) makes the pertinent comment that “for the majority of the world’s population, and for the foreseeable future, the cell phone is the computer” (para. 28). this sentiment is echoed by ford (2009) in her pronouncement that “the cellphone is poised to become the 'pc of africa'” (slide 3). the challenge for teachers is thus “to capitalize on the pervasive use of cell phones by younger students for educational purposes” (pursell, 2009, p. 1219). the educational potential for mobile learning afforded by cellphone technology is diverse (kolb, 2008; prensky, 2005). within south africa a number of projects have already harnessed the ubiquity of cellphone technology to support the learning of mathematics. a number of these projects are briefly outlined here in order to contextualise the growing potential for mobile learning through cellphone technology within the south african education landscape. helmut linneweber-lammerskitten, marc schäfer & duncan samson 31 imfundoyami/imfundoyethu is a mobile learning platform that delivers mathematics education to grade 10 learners through the instant messaging service mxit. learners are able to access exercises and additional help. mathematics teachers log into a web-based management system from which they can distribute exercises to the learners, provide personal support, and monitor both results and learner activity (vosloo, 2009). the m4girls project aims to improve the mathematics performance of grade 10 female learners in rural schools in south africa. learners are provided with cellphones containing a number of mini videos (2-3 minutes in length), mobile episode animations, and games. all mathematical content is aligned with the south african mathematics curriculum (vosloo, 2008). mobitm is a proprietary product that can be accessed through most java-enabled cellphones. the product aims to provide mobile mathematics education to grade 10 to 12 learners, and the content is specifically tailored for the south african mathematics curriculum (vosloo, 2007). mobitm maths provides the user with access to tutorials (in the form of streamed videos), past examination papers with solutions, and an opportunity for learners to use closed chat rooms to form study or discussion groups (botha, 2007). dr math enables learners to access assistance with their homework assignments by interacting with tutors through mxit, a mobile instant messaging platform. in addition, dr math encourages drill and practice of basic skills through games and competitions via the mxit platform (butgereit, 2009; vosloo & botha, 2009). design principles of critical importance to the vitalmaths project are the design principles on which the video clips are modelled. discussing the design principles of educational software applets, underwood et al. (2005) highlight the observation that the design of technology tools plays a critical role in terms of how learners interact with the tools. this in turn can have a dramatic affect on the efficacy of the affordances offered by the technological medium. there are two distinct tiers to the design process, although they are by no means independent of one another. the first relates to the broader design principles of mobile learning in general, while the second relates specifically to the technical details of the video clips themselves. broad design principles of mobile learning herrington, herrington and mantei (2009) identify a number of broad design principles for mobile learning. of particular relevance to the vitalmaths project, specifically in relation to the autonomous learning initiative of the project, are the following five, each of which is briefly discussed in relation to the project: use mobile learning in non-traditional learning spaces the video clips are specifically designed to foster and support non-cognitive dimensions such as curiosity, motivation, and a willingness to know. it is thus envisaged that learners (and teachers) will be inspired to view these video clips in non-traditional learning spaces – waiting for a bus, travelling home on a mini-bus taxi, standing in a queue etc. employ mobile learning in contexts where learners are mobile when learners are inspired to explore the scenarios suggested in the video clips, they will be able to act immediately. their endeavours will not be frustrated by having to return to a computer laboratory, for example, to review the video clip. they will have instant access to the video clip on their own cellphone, and will thus be able to take advantage of the momentum of the given moment. make use of mobile learning both individually and collaboratively the video clips encourage learners to explore, in a hands-on fashion, similar scenarios to those presented in the videos. this can be done either individually or collaboratively, based on the autonomous decision of each learner. visual technology for the autonomous learning of mathematics 32 exploit the affordances of mobile technologies although the video resolution of a computer is generally greater than that of a cellphone, the affordances offered by mobile technology far outweigh this difference. the graphics resolution of most cellphones is not only sufficient for the purposes of the video clips, but the use of cellphones will ensure that the video clip files are as small as possible (between 500 and 800 kilobytes) which in turn will ensure widespread accessibility. employ the learners’ own mobile devices. finally, if the video clips are able to be downloaded onto each learner’s cellphone, not only will there be a heightened individual sense of ownership and autonomy, but this will provide support for the other broad design principles. technical details of video clips in terms of the design principles that relate specifically to the technical details of the video clips themselves, the video clip “rectangular products” is discussed as an example. the video clip makes use of geometric algebra (suzuki, 2009) to give elegant visual support of the distributive property. the film first presents the result that 26 × 38 = 988. the suggestion is then made that 26 and 38 can be expressed as 20 + 6 and 30 + 8 respectively. this scenario is then presented visually as shown in figure 1. figure 1: frame 25 from the video clip “rectangular products” the large 26 by 38 rectangle is then separated, using animated movement, into four smaller rectangles. the area of each rectangle is then calculated as shown in figure 2.3 figure 2: frame 35 from the video clip “rectangular products” 3 the reader will notice that the rectangles are not all perfectly aligned. that is because the video clips are not computer animations, but are produced using natural materials, in this case real paper cut-outs. helmut linneweber-lammerskitten, marc schäfer & duncan samson 33 the sum of the areas of the four rectangles is then shown to be the same as the original product, i.e. 26 × 38 = 600 + 180 + 160 + 48 = 988. the question is then posed as to whether a similar approach could be used to determine the product of 3-digit numbers or even the product of algebraic expressions. this video clip develops a particular mathematical theme in a progressive manner. however, it does so in a way that is purposefully not underpinned by specific pedagogical imperatives or predetermined outcomes. indeed, terms such as distributive property are purposefully avoided so as to ensure maximum flexibility of the material, and to broaden the context (e.g. grade-specificity) of its usefulness. teachers and learners are thus encouraged to use the video clip as autonomously as they desire. this wish is in turn supported by the broad and open philosophy embraced by the design principles on which the video is conceived. “rectangular products” purposefully refrains from giving specific instructions for repeating or extending the exercise with different numbers. however, it closes with probing questions that encourage extension and experimentation in a spirit of mathematical exploration. this video clip is short in duration (1 minute 48 seconds), small in terms of file size (646 kilobytes in mp4 format; 519 kilobytes in 3g2 format), but is nonetheless aesthetically delightful and self-explanatory. the mp4 format is suitable for pcs, ipods, iphones and many modern cellphones, while the 3g2 format is designed specifically for use on older or more basic cellphones. concluding comments the vitalmaths project aims to develop visual technologies for autonomous learning in mathematics and research their efficacy and impact in order to ensure sustained development in mathematics education in south africa and switzerland. in particular, the specific use and application of cellphone technology will be explored. it is anticipated that this innovation will have a significant positive impact for teachers in deep rural settings where access to mathematics resources is very limited. the project seeks to capitalise on the versatility and ubiquity of cellphone technology to provide a mathematics platform to store and disseminate short mathematical video clips that can be used both as teaching and motivational resources. since a large cross-section of teachers will be accessing these video clips it is important that the videos embrace a broad and open philosophy, and that they avoid pedagogical imperatives that render them useful only in narrow contexts. it is envisaged that these video clips will be used in the preparation of lessons, for personal conceptualisation of mathematical concepts, and as motivational and explanatory tools, with the emphasis lying on teachers and learners to use them as autonomously and independently as they wish. in order for the videos to be used as autonomously as possible, they need to be short, succinct, visually and intellectually appealing, relevant and mathematically inspirational. a growing databank of video clips has already been produced. feedback from teachers and learners relating to the efficacy, use and impact of these video clips is being used to continuously refine the design principles that inform their conceptualisation and production. continued research into the use and impact of these video clips seeks to develop a base for sustained growth and development, while at the same time contributing and participating in the academic discourse surrounding the use and development of visual technologies in the mathematics education arena. acknowledgement this work is based upon research supported by the firstrand foundation mathematics education chairs initiative of the firstrand foundation, rand merchant bank and the department of science and technology, as well as the swiss south african joint research programme. any opinion, findings, conclusions or recommendations expressed in this paper are those of the authors and therefore the firstrand foundation, rand merchant bank and the department of science and technology do not accept any liability with regard thereto. visual technology for the autonomous learning of mathematics 34 references botha, a. 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(1995). mathematics as a 'design science'. educational studies in mathematics, 29(4), 355374. doi: 10.1007/bf01273911 book review pedagogical narratives in mathematics education in south africa book title: the pedagogy of mathematics in south africa: is there a unifying logic? author: istván lénárt; patrick barmby; karin brodie; mellony graven; fritz hahne; stephen lerman; mogege mosimege; werner olivier; nicky roberts; anna rybak paul webb and nicky roberts (eds). isbn: 978-1-928341-11-6 publisher: mistra (mapungubwe institute for strategic reflection) and real african publishers, johannesburg, 2017, 282 pages, *r310 *book price at time of review review title: pedagogical narratives in mathematics education in south africa reviewer: charles r. smith1 affiliation: 1school of science and mathematics education, faculty of education, university of the western cape, cape town, south africa corresponding author: charles smith, smithcharlesraymond@gmail.com how to cite this book review: smith, c.r. (2020). pedagogical narratives in mathematics education in south africa. pythagoras, 41(1), a573. https://doi.org/10.4102/pythagoras.v41i1.573 copyright: © 2019. the authors. licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. introduction the pedagogy of mathematics in south africa: is there a unifying logic? is an important book for anyone interested in how to accelerate the slow improvements in mathematics from grade r to grade 12 that are underway in south africa. after reading the book i was reminded of this quote by bélanger (1997) which, and i paraphrase, reminds us that: [s]ocial development is synonymous with the capability of communities to take responsibility for their own development. in this regard, education is an indispensable tool for any country and its people to achieve social development in that it mediates young people to scientific concepts needed for social development. (p. 83) this captures the message the editors intended with this book. the purpose of the book is elucidated as: there is a well-worn tale of ‘the crisis in mathematics education’ in south africa. while recognizing the crisis, this book attempts to articulate less commonly known narratives as to how our situation is in fact improving. its specific focus is on mathematics pedagogy, which is interpreted in the broad sense of interaction between society and its values. within that broad interpretation, the various chapters seek elements that may point to the way to develop a unified south african mathematics pedagogy, rather than a systematic review of current ideas and knowledge pertaining to the issue. (p. 20) the central message of the book to fully appreciate this book, it is important to first understand its background and genesis. during the science forum south africa, which took place on 06–08 december 2017 at the council for scientific and industrial research (csir) convention centre in pretoria, mapungubwe institute for strategic reflection (mistra) launched its research report on the pedagogy of mathematics in south africa: is there a unifying logic? as one of the forum’s side events (university of johannesburg, 2018). this book investigated the best pedagogical methods for nurturing of mathematical talent among young people. it studied how the pedagogy of mathematics has evolved in south africa over time and assessed current approaches against best practice in successful domestic schools and in selected international case studies. it also commented on the impact of historical socio-economic and political relations on mathematics proficiency in our country. against this backdrop, this book presents an overview of the historical as well as the current position of teaching and learning of mathematics in south africa. it poses the question whether there is (or should be) a unifying logic informing the way mathematics is taught and learnt. chapters written by a number of eminent local and international mathematics teachers and researchers contribute ideas towards creating deeper understandings of mathematics, developing learners with productive mathematical identities, and ways of nurturing abstract reasoning. the sections focus the reader to understand the logic of the book: the introduction and overview are presented in chapter 1. chapter 2 is titled: setting the scene. this is followed by section 3: international frameworks as levers for change (chapters 3–5). section 4 is titled: what south africans are saying (chapters 6–14). the book concludes by discussing cross-cutting issues and lessons learnt. the different authors and their contributions will be elucidated in the course of this review. key issues that emerged were the importance of teaching mathematics in a way that links to learners’ concrete social environment, and the necessity for joint efforts on the part of government, unions and private partners. chapter 1 and chapter 2 provide the rationale, and this is elaborated in the other chapters. in addition, the book argues for the importance of teachers developing a deeper understanding of mathematics and of creating learners with productive mathematical identities, capable of making sense of mathematics in south africa’s diverse languages. in the south african context social pedagogy takes a critical stance on the historical, structural and systemic issues that beset the education of the marginalised south african population. thus, social pedagogy concerns itself with education, well-being and growth. this comes across in the opening chapters, which are written by the editors. in chapter 1 they note that: south africa emerged from years of colonial domination with its first democratic elections in 1994. the immediate task was to unite the country around the programme of eliminating the vestiges of colonialism and creating an equitable society. a critical element of that ongoing effort is extending access to education and ensuring quality educational services. (p. 20) in all, there are 15 chapters in the book which the reader may read sequentially or as self-contained papers, depending on the need for specific insights. the editors claim that the book is targeted at policy actors, policy implementers and other stakeholders in the education field. this suggests that education administrators, researchers as well as teachers are the intended audience. teachers may wish to read chapter 1, in order to get an overview of the central problem that is at the core of this book and interventions by various role players in the search for workable solutions. they may then selectively read other chapters in which they may have an interest. there is no need to read the chapters sequentially, besides taking cognisance of the division into sections in which chapters may address a given thread. more of this later in the review. critique of the title the title may be somewhat misleading for the lay person in that it may suggest to teachers that it is narrowly about teaching and learning in the mathematics classroom. the question ‘is there a unifying logic?’ is a quest by the editors to add to the debate surrounding the crisis that faces mathematics education in south africa. thus, by inviting a number of accomplished academics and researchers in the field of mathematics education to contribute chapters in the book, they seek to find ‘resonances’, in other words a common thread in the south african discourse aimed at solution-seeking interventions to improve the teaching and learning of school mathematics. in this regard they also pursue international perspectives as potential levers of change (chapter 3 by stephen lerman, chapter 4 by fritz hahne and chapter 5 by istván lénárt and anna rybak). however, given the suggested target audience, namely policymakers and policy implementers, the gap in the discourse is the voice of teachers. the editors missed the opportunity to highlight the challenges faced by teachers in terms of a very dense or overfull curriculum and its support mechanism in terms of a very prescriptive ‘pace setter’ or ‘annual teaching plans’. this flawed mechanism, although well intended, has the unintended consequence of reducing curriculum delivery to curriculum coverage (see a related comment by venkat on page 155: ‘the pace of teaching may well outstrip the pace of learning’). furthermore, the book is silent about the intended, delivered and assessed curricula in the south african context. despite this absence of the voice of the teachers the book is a compelling read for the serious student of mathematics education, be they an academic, administrator or classroom practitioner. the section called’ international perspectives as potential levers for change’ is included to provide lessons from international research that may assist south africans in their quest for transformation and improvement. this section draws on international perspectives on mathematics education and is followed by a section called ‘what south africans are saying’, which is authored by patrick barmby. this author worked both in an international as well as the south african context and has an in-depth understanding of issues at both levels. he especially focuses on the preservice and in-service training needs of teachers. the section that follows presents the voices of south african academics in the field of mathematics education and is complemented by a chapter on ethno-mathematics by mogege mosimege. he surveys this genre as it applies to the teaching and learning of mathematics in south africa. mosimege is arguing for acknowledging indigenous knowledge systems as well as cultural artifacts that may serve as starting points to introduce mathematical concepts. he provides a number of examples of how this could be done. there is a chapter on visualisation in the mathematics classroom by marc schäfer, a chapter by werner olivier on a techno-blended approach and two chapters on language issues – a chapter by paul webb and a chapter by lindiwe tshuma. mellony graven, in chapter 8, foregrounds homework as a practice that can support student learning. an important observation made is that visualisation processes are critically important in both teaching and learning of mathematics in actual classrooms. the issue of e-learning gained traction in both national and provincial education departments and the technology blended model (olivier, p. 205). the technology blended model described in this book offers many pointers to possible ways to make resources available offline. this is an important consideration given the challenge of internet access, especially in the deep rural areas of the country. karin brodie (chapter 9) speaks to the impact that teachers’ actions have on learning by the students. she also highlights other important areas of mathematics teaching such as: selecting rich mathematical tasks that support learners to develop key mathematical concepts, problem solving strategies and productive dispositions … and being sensitive to learners’ mathematical errors in ways that are generative for developing disciplinary knowledge. (p. 238) the chapters on language issues in the mathematics classroom make for interesting reading. the authors lean heavily on the works of researchers in this field especially the south african academics such as adler and phakeng (phakeng, 2016; phakeng & essien, 2016). in this regard tshuma, in chapter 14, takes the perspective that language is a resource and takes lessons from her doctoral research to highlight linguistic features that influence mathematics teaching and learning in the middle grades. the authors then suggest pedagogical strategies for improving the mathematics register, that is, the evolving vocabulary that teachers and learners use in the classroom, and to minimise lexical ambiguity in mathematics classrooms. related to this is mosimege’s chapter (chapter 10) making a plea to teachers to: regard their classrooms activity as part of the efforts of africans engaged in the formidable task of reclaiming their heritage and restoring african pride in attaining mathematical knowledge. (p. 235) the concluding chapter by webb and roberts considers cross-cutting issues and lessons learnt in order to answer their opening question, taken up in the title of the book: ‘is there a unifying logic?’ in this chapter they explore resonating themes from the preceding chapters that relate to the affordances and hindrances existing in the mathematics education landscape, as it is relevant to the teaching and learning of school mathematics. they define what they mean by a unifying pedagogy and then proceed to suggest three pillars on which this may be constructed. they encourage teachers to: work with a curriculum of engagement. a curriculum of engagement is premised on learners and teachers developing positive disciplinary identities and making connections with real life socio-cultural experiences, with one of the key factors being a teacher who believes that their learners can do mathematics. (p. 254) in the current project i am involved in, namely the local evidence driven improvement of mathematics teaching and learning initiative (ledimtali), we promote a similar notion of ‘mathematicalness’. this is an important ongoing task of teachers. they have to create learning environments where learners engage with each other and with mathematics in order to deepen mathematical thinking in relation to the problems they attempt to solve in the classroom. this needs to be ecologically relevant in terms of the sociocultural context of the learners. my conclusion after reading the book is that there are common threads or resonances that come through in the book. this especially relates to the conclusions that: educational change and improvement are incremental, and demand focused and sustained effort involving all stakeholders. teacher development and support is an essential element in the process. resource development and provisioning support the process. technology integration has the potential to enhance the process. lessons learnt in research need to be embedded in practice. references bélanger, p. (1997). literacy, education and social development. hamburg: unesco institute for education. university of johannesburg. (2018). edubrief. retrieved from https://www.uj.ac.za/faculties/facultyofeducation/documents/edu_brief_july_2018.pdf phakeng, m., & essien, a.a. (2016). adler’s contribution to research on mathematics education and language diversity. in m. phakeng & s. lerman (eds.), mathematics education in a context of inequity, poverty and language diversity (pp. 1–6). cham: springer. phakeng, m.s. (2016). mathematics education and language diversity: mathematics education and language diversity. in a. halai & p. clarkson (eds.), teaching and learning mathematics in multilingual classrooms (pp. 11–23). the netherlands: brill sense. microsoft word 03-13 engelbrecht et al final.docx pythagoras, 72, 3-13 (december 2010) 3 are obe‐trained learners ready for university mathematics?    johann engelbrecht, ansie harding & patrick phiri  university of pretoria  johann.engelbrecht@up.ac.za, ansie.harding@up.ac.za & patrick.phiri@up.ac.za    the 2009  intake of university students were  the first  to have received complete school  education within the recently implemented outcomes‐based education (obe) system. a  feature of the matriculation examination results of these students was the exceptionally  high grade 12 marks for mathematics. this paper addresses the question of how the 2009‐ intake of students performed at university with respect to general performance, general  attributes,  mathematical  attributes  and  content  related  attributes.  it  appears  that  these  students  are  better  prepared  with  respect  to  personal  attributes  such  as  confidence.  however,  in  many  instances  they  are  weaker  than  their  predecessors  with  respect  to  mathematical and content related attributes.  yet, there are positive indications that these  students adapt and improve over a semester. we make some suggestions on how to make  the transition from secondary to university mathematics somewhat smoother.  one of the concerns of the department of education (doe) after 1994, was that many learners did not develop the required problem solving skills and the critical reasoning ability during the learning process (doe, 2000). a new education system became a priority and in 1998 a new education system called curriculum 2005 was implemented. the key principles of curriculum 2005 include integration, holistic development, relevance, participation and ownership, accountability and transparency, learner centredness, flexibility, critical and creative reasoning, quality standards and international competitiveness (doe, 1997). not only did the obe system bring about changes in the approach but also a new curriculum. this meant a change at the very heart of the education system. one of the concerns regarding the new education system was to what extent these learners would be prepared for university mathematics. therefore the university preparedness of the 2009 intake of students received much attention. the first-year intake of the previous three years all had had partial exposure to obe – these students experienced obe for a few years but returned to the old curriculum in their final three years of schooling. the 2009 intake was the first group of students who had followed the obe curriculum for their entire school career. mathematical university preparedness in a recent study in ireland investigating students’ inability to cope successfully with the transition between secondary and university mathematics, hourigan and o’donoghue (2007) found that essentially there is a big difference between the nature of first-year students’ mathematics experience at preuniversity level and that which they experience at university in mathematics intensive courses. they also found that the unpreparedness of students caused permanent damage to students’ further mathematics careers at university. several researchers have reported on this problem. craig (2007) and hoyles, newman and noss (2001) reported on a number of studies dealing with the difference between university expectations and the wide spectrum of mathematical abilities of the new students. de la paz (2005) and hoyles et al. (2001) name, amongst others, the changing school curricula as one of the reasons for these changes. are obe-trained students ready for university mathematics? 4 the focus in this paper is specifically on this facet. changes in the school curriculum bring challenges for university lecturers and may result in the development and implementation of bridging courses as well as the need for changes to university curricula and assessment strategies (craig, 2007; wood, 2001). we have reported on the first stage of our project (engelbrecht & harding, 2008), in which we investigated the preparedness of the transition group (the 2008 intake of students, who had had partial exposure to obe). we found that the transition group was on par in most skills categories and performed even better in geometry, but was lacking in modelling skills. however, the results of the 2008 paper and results presented here are not directly comparable as the instrument used in 2008, namely the alternative admissions research project (aarp) tests then used for entrance purposes, is no longer used. we have reported preliminary results of the 2009 intake of students in engelbrecht, harding and phiri (2009). this paper expands on this report1 by including additional data and re-interpreting the findings. the 2009 intake of students the 2009 intake of students at university were the first to have written a grade 12 paper in the obe system. the matriculation mathematics examination results were surprisingly positive. more than 47% of those who wrote the examination passed mathematics compared to about 43% in 2008. of the learners who passed mathematics, almost half (63 000) obtained a mark of more than 50% in contrast to the typical 25 000 of previous years whose achievement was similar to that of mathematics higher grade (keeton, 2009). among the concerns raised were that the papers were too ‘easy’, and that too many learners achieved distinctions. the concerned mathematics educators (2009) group claimed that the final examination in mathematics was watered down and has therefore widened the gap between school and university for the top learner. the type of questioning was unchallenging for talented and competent learners and if this standard is going to be used as a benchmark for future examinations it will not adequately prepare young learners to study mathematics related courses at university level. (para. 2) the department of education appointed a panel of experts to evaluate the papers. the ministerial panel (2009) reported that those learners who scored 50% or more on the 2008 national senior certificate mathematics papers would historically have passed mathematics on the higher grade in the previous system at 40%. it was also found that there was a lack of differentiation between the 70 – 79% level and the above 80% level and that the paper did not provide enough questions at the “knowledge” level of the taxonomy. although the grade 12 results look good on paper there are reasons for concern. the question remains whether the obe system is sufficiently preparing the learners for university studies. in the national benchmark tests project, which assesses how much of the school mathematics curriculum has been mastered “a staggeringly low 7 percent of the students [2009 university entrants] who wrote the maths tests were found to be proficient. in other words, they would not need extra help to pass their exams. about 73 percent had ‘intermediate’ skills. the rest, 20 percent, had only the most basic skills and would need long-term, consistent attention” (smith, 2009, para. 7). the challenge faced by universities is clearly enormous, and the need for curriculum responsiveness is evident. several lecturers who taught first-year mathematics in 2009 reported on under preparedness of students. for example, huntley (2009) reported that 71 percent of first-year engineering students had passed in 2008 compared with just 35 percent in the 2009 mid-year examination. in addition, the fact that more grade 12 learners than ever before had passed in 2008, caused record enrolments in university mathematics courses. this had led to enormous strain on the university’s ability to teach. in short, although learners may have achieved the minimum requirements to allow them to study for a bachelors degree, this does not necessarily mean that they will be able to cope at university (roodt, 2009). 1 we re-publish here some of the initial data with the permission of the editors of suid-afrikaanse tydskrif vir natuurskap en tegnologie and pythagoras. johann engelbrecht, ansie harding & patrick phiri 5 mathematics in die obe system the obe system resulted in differences in the curriculum and approach. in mathematics some of the more difficult topics, including geometry, were moved to a third paper, an optional examination paper written by only 11 000 students in 2008 nationwide (keaton, 2009). certain topics were excluded from the new curriculum, for instance absolute value and some parts of trigonometry and logarithms. it is easy to determine the differences in content but much more difficult to determine the level of mathematical skills of the learners who passed grade 12 in 2008. we refer here to mathematical skills such as algebraic manipulation and graphical interpretation. in addition to these mathematical skills there are also a number of personal attributes, such as confidence and work ethics, that may have been either improved or compromised because of the different approach of the obe system. these attributes could also have an influence on success in mathematics at university. the research question addressed in this paper is: to what extent was the 2009 intake of students prepared for university mathematics with regard to performance, general attributes, mathematical skills, and content related skills? the sample this study is firstly based on a questionnaire completed by a number of experienced lecturers involved with the 2009 intake of mathematics students. it is secondly based on the results of two tests in a first semester calculus course, written by a group of first-year engineering students at a large university in gauteng, after five and ten weeks of lectures, respectively, hereafter referred to as semester test 1 and semester test 2. the course had an enrolment of 1282 students. we experienced some administrative problems, but managed to gather and analyse the data of 924 students for semester test 1 and 820 students for semester test 2, with an overlap of about 750 students. in this calculus course students attended four lectures of one hour each and one tutorial of two hours per week. the entrance requirement in mathematics for these students was level 6 (a mark of at least 70%) in grade 12 mathematics. we used the questionnaire to determine the experience of the lecturers with the 2009 intake and used these opinions as guidelines for further investigation in analysing semester tests 1 and 2. grade 12 marks distribution in 2007, 22% of students enrolling for this course achieved a distinction (more than 80%) in grade 12. in 2008 the corresponding figure was 24% and in 2009 it was 55%  dramatically higher than for the previous two years. note that these percentages include students repeating this course and that the actual percentages for new students were somewhat higher. in fact, in 2009, 72% of the new students enrolling for this course in 2009 achieved a distinction in mathematics in grade 12. figure 1: percentage frequency distribution of grade 12 performance 0 5 10 15 20 25 30 35 40 0 10 20 30 40 50 60 70 80 90 100 performance (%) 2007 2008 2009 percentage of students are obe-trained students ready for university mathematics? 6 figure 1 shows the percentage frequency distribution of the 2009 first-year-students’ grade 12 marks. it is clear that the grade 12 marks of the 2009 intake were exceptionally high relative to the previous two years. note that all the graphs (figures 1 – 10) are percentage frequency distributions, with the performance intervals as a percentage (0-100) on the horizontal axis, and the percentage of students in the relevant category on the vertical axis. on the horizontal axis the category 50, for example, refers to performance scores of 41 – 50 and 100 to scores of 91 – 100. note that the category 0 then refers to scores of exactly 0. general performance in university mathematics in figure 2 we compare the grade 12 marks of the 2009 group of students with their performance in semester test 1 and semester test 2.   figure 2: student performance in grade 12 and in semester tests 1 and 2 from figure 2 it is clear that there was a strong shift to the left from the grade 12 marks to marks in semester test 1. the marks for semester test 2 does not display the same strong shift to the left. in fact, the marks distribution for semester test 2, as well as for the june examination (see figure 3) show a slight shift to the right and is more similar to the grade 12 marks distribution. the pearson correlation coefficient between grade 12 performance and performance in semester test 1 is 0,37 and for semester test 2 it is 0,35, and are both significant (p < 0,001). this implies that grade 12 can still be considered as a good predictor of university success in mathematics but that students have to realise that their grade 12 performance will not easily be repeated in university mathematics. the percentage frequency distribution of marks in the june examination is given in figure 3. from this figure it is clear that students have recovered somewhat. however, we should not conclude too much from this graph, because students with a semester mark of less than 30% did not comply with examination entrance requirements and are excluded, reducing the group size for the june examination to 1144. figure 3: student performance in the june examination 0 5 10 15 20 25 30 35 40 0 10 20 30 40 50 60 70 80 90 100 grade 12 sem test 1 sem test 2 0 5 10 15 20 0 10 20 30 40 50 60 70 80 90 100 johann engelbrecht, ansie harding & patrick phiri 7 questionnaire to lecturers a questionnaire was completed by ten experienced first-year lecturers – some involved in first-year mathematics for engineering students, some in teaching the mainstream first-year courses including mathematics, actuarial sciences, financial mathematics and computer sciences and others in the precalculus extended programme. lecturers were asked to express an opinion on aspects in which the 2009 students were better off, were worse off and in which there was no change with respect to the following three facets: general personal attributes, general mathematical attributes and content related skills. general personal attributes lecturers were almost unanimous in their opinion that the 2009 intake of students had more confidence and were more willing to try. students had a positive outlook and had confidence in their abilities. it was noticeable that students were not prepared to blindly follow the method suggested by the lecturer, they wanted to “experiment” and do things “their” way. unfortunately this way of doing goes along with a lack of mathematical rigour and a way of writing that often makes sense only to the student him/herself. some lecturers felt that students had too much self-confidence and spoke about mathematics too loosely rather than writing it out carefully. when marking tests this personal characteristic of (excessive) confidence was also noticeable. as one lecturer remarked: “students write down everything that they can think about, without any coherence, and hope that somewhere there will be something that can earn them a mark.” this way of doing was not entirely new for 2009 students but probably more so than in the past. it is also possible that the noticeable confidence of the new students can be ascribed to their high grade 12 symbols for mathematics and not only to the influence of obe. our findings of increased confidence agree with that of adler, quoted by smith (2009, para. 23): “the kids are much better than they were, they are more confident, they are more aware of what they do and don’t know and they are more willing to try.” general mathematical skills there was agreement amongst lecturers regarding the deterioration in general mathematical skills. lecturers were unanimous regarding decrease in the specific skills of factual knowledge, algebraic manipulation and mathematical formulation. most lecturers felt strongly that there was a decrease in all these skills while the odd lecturer felt that there was an improvement in graphical manipulation and mathematical intuition. it appears that the self-confidence with which students started out with in the course was not justified and that this was not supported by the necessary mathematical skills. for example, in semester test 1 many students made mathematical errors such as e xx xx xx xx eexx 44ln 4lnln)4ln( 3ln ln3lnln 4ln44ln ln44ln3 3     according to lecturers there was not only a noticeable weakening of knowledge regarding the properties of logarithms and logarithm manipulation, but there was also an increase in elementary errors such as 22 42 41)41( 2 1 2 1 2 1     xx xx xx b a ba   are obe-trained students ready for university mathematics? 8 furthermore, there was a particular lack of knowledge and a lack of skills in trigonometry, leading to errors such as 42 1sin  xx there was particular concern regarding the poor ability of students to ‘write’ mathematics. students wrote little and appeared to be uncertain. often something that made sense was obscured between nonsensical writing. we distinguished between the following general mathematical attributes: algebraic manipulation, graphical manipulation and interpretation, concept application and interpretation, and basic factual knowledge. we grouped questions in the two semester tests into these categories and investigated students’ performance in each of these components. the performance of students in questions that mainly required algebraic manipulation in semester tests 1 and 2 are shown in figure 4. figure 4: performance in algebraic manipulation in semester test 1 and 2 in semester test 1 the distribution leans to the left. there are too many students that scored less than 50% for these questions. it is expected of students to be fluent in algebraic manipulation to ensure success in university mathematics. for semester test 2 there seems to have been a slight improvement in the skill of algebraic manipulation. the distribution leans somewhat to the right and although the marks are still poorer than the grade 12 marks, there was less reason for concern. whereas students seemed to be out of their depth in semester test 1 as far as algebraic manipulation was concerned, they seemed to have adapted and improved. in figure 5 we give the performance of students in questions in semester tests 1 and 2 that mainly required graphical interpretation. figure 5: performance in graphical interpretation in semester tests 1 and 2 0 5 10 15 20 25 0 10 20 30 40 50 60 70 80 90 100 sem test 1 sem test 2 0 5 10 15 20 25 30 0 10 20 30 40 50 60 70 80 90 100 sem test 1 sem test 2 johann engelbrecht, ansie harding & patrick phiri 9 it is clear that performance improved noticeably from test 1 to test 2. graphical interpretation has always been difficult for students but although it appeared to still be the case in semester test 1, the pleasing finding was the remarkable improvement in this category in semester test 2. performance in the other mathematical skills categories for the two semester tests is given in figure 6. possibly the best of these categories is the concept application category, especially in semester test 2, a surprising finding. it appeared that students’ comprehension was not poor at all but that technical manipulation was the stumbling block. a disturbing observation was that basic factual knowledge was not on par, neither in semester test 1 nor in semester test 2. is it perhaps a feature of the obe generation that they prefer to do but are not so keen to spend the required time on studying? figure 6: student performance in factual knowledge and concept application in semester tests 1 and 2 the correlation coefficients between student performance in each of the components and their grade 12 performance are given in table 1. table 1: correlation coefficients between university and grade 12 performance semester test 1 semester test 2 general results 0,37 0,35 algebraic manipulation 0,47 0,28 graphical manipulation and interpretation 0,39 0,21 concept application and interpretation 0,34 0,32 basic factual knowledge 0,27 0,22 these correlation figures are all significant (p < 0,001). the poorer correlation in basic factual knowledge supports the notion that many of the students simply did not study enough, and that this was not only the case for poor students. content related skills it was not surprising that a deterioration in preparedness was observed regarding topics such as absolute value, trigonometric functions and exponents and logarithms, as it was exactly in these topics that content decreased or was totally omitted in the new curriculum. it appears that students were not only inept with respect to general mathematical skills but also with respect to content related skills. in the empirical investigation we grouped questions that mainly concerned each of the content components and considered student performance in each of these components. are obe-trained students ready for university mathematics? 10 the correlation coefficients between student performance in semester test 1 in each of the components and their grade 12 performance are given in table 2. table 2: correlation coefficients between performance in components of semester test 1 and grade 12 general result of semester test 1 0,37 inequalities and absolute value 0,33 functions and graphs 0,36 trigonometry 0,38 exponents and logarithms 0,19 limits and continuity 0,46 the correlation coefficients between student performance in semester test 2 in each of the components and their grade 12 performance are given in table 3. table 3: correlation coefficients between performance in components of semester test 2 and grade 12 general result of semester test 2 0,35 differentiation  concepts 0,29 differentiation  technical 0,24 differentiation  applications 0,29 limits and asymptotes 0,25 continuity 0,25 all correlations are significant (p < 0,001) and of approximately the same size, except for the component of exponents and logarithms in semester test 1, which caught the lecturers slightly off guard. exponents and logarithms are in the school curriculum and lecturers assumed that students were equipped with the same level of knowledge as in previous years. this was not the case. students were initially inept in the properties of logarithms and manipulation with exponents and logarithms, more so than before. the result was that although this topic was included in the school curriculum, the teaching pace at university was such that students floundered with their shallow knowledge. on realising this, effort was made for allocating extra time and to giving more attention to logarithms and exponents. figure 7: student performance in inequalities and absolute values, logarithms and exponents and trigonometry in semester test 1 johann engelbrecht, ansie harding & patrick phiri 11 inequalities and absolute value together form a topic to which more time needs to be devoted, as is clear from figure 7. too many students performed poorly in this topic. as for trigonometry, although there is a group of students that has mastered the topic, there are too many that scored less than 50%. these students could probably be rescued through a slower presentation pace. the main thread of the mathematics course runs through functions, limits, continuity and differentiation – from the concepts to techniques to applications. we now follow this thread with respect to student performance. for functions, limits and continuity (the start of the topic) the performance in semester test 1 is disappointing (see figures 8 and 9). the level of knowledge in these topics was too low. as a result a decision was taken to spend considerably more time on this topic and to gain the time by moving some of the work to the second semester. figure 8: student performance in functions in semester test 1   figure 9: student performance in limits and continuity in semester tests 1 and 2 there was no improvement from semester test 1 to semester test 2 in the limits topic (see figure 9). it is clear that the topic is difficult for students. yet, student performance increased in the continuity category. as was seen in other categories, students initially struggled to find their feet but then improved in performance. the concepts involved in differentiation did not appear to be problematic for the majority of students (see figure 10), although there was a tail of students for which this finding was not true. this tail seemed to be increasing as the topic was deployed. the performance decreased through technical differentiation to applications. applications are by nature more difficult but the relatively poor performance in technical differentiation did not bode well for the follow-up topic of integration, which is far more complex. technical differentiation requires the rigour of applying rules, secured by repeated practice. these students did not seem to be geared for practice, and more practice, until it becomes second nature to do differentiation. 0 5 10 15 20 0 10 20 30 40 50 60 70 80 90 100 are obe-trained students ready for university mathematics? 12 figure 10: student performance in differentiation in semester test 2 discussion the change to a new education system at school level has been the most significant of the past few decades and it was to be expected that certain difficulties and discrepancies would be experienced. it was important to identify problem areas and address eminent issues. the interface between secondary school and university has to provide for a smooth transition and it would have been fortuitous if the new system decreased the existing gap. this study unfortunately shows that for mathematics, the new system has enlarged rather than decreased the gap. we do not put blame on anyone in this study. the national department of education involved higher education continually during the curriculum development process and the problems experienced at the moment are not necessarily someone’s “fault”. we are in a transition phase and need to address unexpected problems as they emerge. the secondary education structures as well as universities should devote attention to addressing the widening gap between secondary school and university mathematics. one cannot accept students for university courses if we know in advance that they are under prepared and will most likely experience difficulties. possible solutions to smoothen the discontinuity between secondary and university mathematics include o further research to make a detailed analysis of the problem o continued close cooperation between university and secondary education authorities o possible changes in the school curriculum o the possibility of an additional mathematics subject as another of the seven school subjects o raising of admission requirements at universities o placing more students in extended programmes at universities o extension of support programmes at universities o possible changes in the university mathematics curriculum at university level there is a realisation that the teaching pace should be slowed down and that the curricula in first-year mathematics courses need to be revised. content in first-year courses has gradually increased with time and this resulted in a quickening of the teaching pace. this trend should be halted and reversed. in some university mathematics courses the emphasis has also shifted to a more theoretical approach for which the new intake of students is clearly not ready. the 2009 intake of students thus forced both universities and schools to pause for stock taking. an outreach should be done from both sides for successful education systems at both levels. johann engelbrecht, ansie harding & 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(2009, september 18). when a stands for abysmal. sairr today. retrieved from http://sareporter.com/index.php?option=com_content&task=view&id=862&itemid=1 smith, j. (2009, august 15). obe education system produces confident illiterates. saturday star. wood, l. (2001). the secondary-tertiary interface. in d. holton (ed.), the teaching and learning of mathematics at university level. an icmi study (pp. 87-98). dordrecht, the netherlands: kluwer academic publishers. pyth 40(1)_contents.indd open accesshttp://www.pythagoras.org.za index i issn: 1012-2346 (print) issn: 2223-7895 (online) editor-in-chief alwyn olivier stellenbosch university (retired), south africa associate editors anthony essien university of the witwatersrand, south africa faaiz gierdien stellenbosch university, south africa michael murray university of kwazulu-natal, south africa rajendran govender university of the western cape, south africa editorial board jill adler university of the witwatersrand, south africa bill barton university of auckland, new zealand marcelo borba são paulo state university, brazil doug clarke australian catholic university, australia jeremy kilpatrick university of georgia, united states of america gilah leder la trobe university, australia stephen lerman southbank university, united kingdom frederick leung university of hong kong, sar, china liora linchevski hebrew university of jerusalem, israel john malone curtin university, australia andile mji tshwane university of technology, south africa willy mwakapenda tshwane university of technology, south africa john olive university of georgia, united states of america david reid university bremen, germany paola valero stockholm university, sweden renuka vithal university of the witwatersrand, south africa anne watson oxford university, united kingdom volume 40 number 1 december 2019 ‘semiregular tiling 3-4-6-4 (small rhombitrihexagonal)’ by r.a. nonenmacher. retrieved from https://commons. wikimedia.org/wiki/file:tiling_ semiregular_3-4-6-4_small_ rhombitrihexagonal.svg page i of ii information for authors and readers original research social science students’ concept images and concept definitions of anti-derivatives eunice k. moru, makomosela qhobela towards validation of a rational number instrument: an application of rasch measurement theory jeremiah maseko, kakoma luneta, caroline long comparison of geometric proof development tasks as set up in the textbook and as implemented by teachers in the classroom lisnet mwadzaangati the use of semiotic representations in reasoning about similar triangles in euclidean geometry ifunanya ubah, sarah bansilal using standard setting to promote meaningful use of mathematics assessment data within initial teacher education programmes qetelo m. moloi, anil kanjee, nicky roberts the standard written algorithm for addition: whether, when and how to teach it nicky roberts a commognitive perspective on grade 8 and grade 9 learner thinking about linear equation anthea roberts, kate le roux mathematics teachers’ levels of technological pedagogical content knowledge and information and communication technology integration barriers gabrielle de freitas, erica d. spangenberg keeping sites in sight: conversations with teachers about the design of toolkits peculiar to a continuous professional development initiative faaiz gierdien, charles smith, cyril julie the professional development of primary school mathematics teachers through a design-based research methodology piera biccard webs of development: professional networks as spaces for learning kenneth ngcoza, sue southwood book review book review: international reflections on realistic mathematics education stanley a. adendorff correction erratum: a commognitive perspective on grade 8 and grade 9 learner thinking about linear equation anthea roberts, kate le roux reviewer acknowledgement guidelines for authors ii 1 13 25 39 49 63 80 95 108 119 129 136 139 140 141 microsoft word 30-40 taxonomy-matters-berger.docx 30 pythagoras, 71, 30-40 (july 2010) taxonomy matters: cognitive levels and types of  mathematical activities in mathematics examinations    margot berger, lynn bowie & lovemore nyaumwe  marang centre for mathematics and science education  university of the witwatersrand  margot.berger@wits.ac.za; lynn.bowie@ wits.ac.za & nyauml@unisa.ac.za    we argue that there are intrinsic difficulties in using the subject assessment guidelines  for mathematics  (sagm)  for evaluating  the standard of a south african matriculation  mathematics examination and  for determining how  it aligns with  the curriculum. our  argument has two prongs. first, the sagm taxonomy conflates cognitive level with type  of mathematical activity; we contend that such a conflation leads to problems in assessing  the complexity of the examination item. second, the sagm taxonomy provides no space  for  important mathematical activities such as  justification and conjecturing despite  the  promotion  of  these  activities  in  key  curriculum  documents  (department  of  education,  2003). this absence may obscure weak alignment of the examination and the curriculum.  these arguments are  illustrated  through examples  taken  from  the 2008 department of  education  (doe)  and  independent  examination  board  (ieb)  grade  12  mathematics  examination papers. we also examine other well‐known taxonomies in order to discern  those  aspects  of  a  taxonomy  which  may  be  useful  for  evaluation  and  alignment.  we  conclude with the construction of a matrix that could provide an appropriate taxonomy  for the south african matriculation mathematics examination. it has activities on one axis  and levels of complexity on the other.   for any high-stake national school matriculation examination there is a need for a set of criteria which can be used to determine the difficulty level of examination items and how the examination aligns with the curriculum. this is necessary in order to assess the examination’s fairness as a tool for selection of learners for admission to further study or careers and as a measure of a nation’s mathematical competence. in south africa, the national department of education publishes subject assessment guidelines for mathematics (sagm) that outlines the criteria and expected weightings of learning outcomes stated in the national curriculum statement (doe, 2003). doe (2008a) also publishes a sagm taxonomy designed for use in constructing and assessing school leavers’ final mathematics examination. this paper argues that there are intrinsic difficulties in applying the sagm taxonomy. the argument is based on two major reasons. first, and in common with many other taxonomies, the sagm taxonomy (doe, 2008a) conflates cognitive level with type of mathematical activity. in the sagm taxonomy it is assumed that cognitive level increases with the type of mathematical activity; that is, memorisation has the lowest cognitive level, then routine procedures, then complex procedures, then problem solving. this assumption is questioned in this paper. second, the sagm taxonomy provides no space for key mathematical activities such as justification, conjecturing, and communicating mathematical ideas despite the national curriculum statement’s (ncs) promotion of these activities as important components of the curriculum (doe, 2003). absence of these mathematical activities within the sagm could lead to weak alignment of the examination and ncs. margot berger, lynn bowie & lovemore nyaumwe 31 the two arguments stated above are illustrated through examples taken from the 2008 doe and ieb grade 12 mathematics examination papers (doe, 2008b, 2008c; ieb, 2008a, 2008b). the arguments are also broadened and deepened by considering other taxonomies such as porter (2002), stein, smith, henningsen and silver (2000, 2009) and national assessment of educational progress (naep) (national assessment governing body (nagb), 2008). towards the end of the paper we theorise a desirable matrix that could provide an appropriate taxonomy for assessment in the ncs. such a taxonomy would have mathematical practices advocated by the ncs on one axis and different levels of complexity on the other axis. background at the end of 2008, south african grade 12 school learners wrote the doe national senior certificate (nsc) examinations for the first time. these examinations were based on the new ncs. the doe mathematics examination was written by about 300 000 learners and provoked considerable public interest in terms of its standard, partly because it was a new form of examination. up until the end of 2007, students taking mathematics as a school-leaving subject studied higher grade mathematics or standard grade mathematics. higher grade mathematics was of a higher academic standard than standard grade mathematics and it was generally considered a necessary preparation for further study in engineering or the sciences at tertiary level. under the new ncs system, higher and standard grade mathematics have been merged into one subject, mathematics. the concern of civil society and the tertiary education sector was to what extent this lowered the standard of higher grade mathematics and what mark could be considered as acceptable for entry into various university courses. two examination boards, doe and ieb, exist in south africa. the majority of schools in south africa are public institutions and mostly write doe examinations; private schools write ieb examinations. the standard of the 2008 mathematics examination, particularly the doe examination, has been the subject of a number of comments (concerned mathematics educators, 2009; dibetle, 2009; maree, 2009) and reviews (ministerial panel, 2009; umalusi, 2009) and is not the focus of this paper. however these reviews rely heavily on the sagm taxonomy (doe, 2008a). the application of this taxonomy to the examination, together with our exploration of alternative taxonomies, leads us to address the issue: how does the use of different taxonomies enable us to assess both the standard of a mathematics examination and the spread of different types of mathematical activities in that examination? different taxonomies within mathematics, a number of different taxonomies have been developed for various purposes. these purposes may be related to the cognitive levels and/or types of mathematical activities and tasks in an assessment (doe, 2008a, national assessment governing body, 2008), classroom tasks (stein et al., 2000) or alignment of curriculum with assessment (porter, 2002)1. although the purposes of some of these taxonomies are not specifically assessment, we believe that aspects of these different taxonomies suggest useful views of the relationship between taxonomies and assessment. we use them to highlight certain arguments about what features of taxonomy may be useful for setting and evaluating items in the nsc examinations. the sagm taxonomy is based on the 1999 timss mathematics survey (doe, 2008a). clearly we wish to examine this taxonomy since this is the official taxonomy which informs the setting of the grade 12 doe mathematics paper. the sagm taxonomy uses categories of knowledge, routine procedures, complex procedures and problem-solving, each with their own descriptors. it suggests the approximate proportion of the examination paper which should be allocated to each category. 1 in the appendix we present a summary version of these taxonomies. taxonomy matters: cognitive levels and types of mathematical activities 32 the porter (2002) taxonomy has five categories: memorisation, perform procedures, communicate understanding, solve non-routine problems and conjecture, generalise and prove. porter’s taxonomy is widely used and a respected tool in the usa (hess, 2006, porter, 2002) and some of its categories refer to particularly important mathematical practices. for example, categories such as “conjecture, prove, and generalise” are relevant to the ncs’s vision which states: “competence in mathematical process skills such as investigating, generalising and proving is more important than the acquisition of content knowledge for its own sake” (doe, 2003, p. 9). another important category of porter’s scheme not evident in the sagm taxonomy but again contained in the ncs’s critical outcomes is the need for learners to be able to communicate mathematically (doe, 2003, p. 2). porter’s category “demonstrate understanding of mathematical ideas” refers to a learner’s ability to communicate or explain mathematical ideas and so addresses this critical outcome. stein et al. (2000) designed a taxonomy consisting of four categories, namely, memorisation, procedures without connections, procedures with connections and doing mathematics. although this taxonomy’s primary purpose is for analysing classroom rather than assessment tasks, we believe it illuminates important distinctions in types of mathematical tasks. in particular we found the distinction they made between the categories “procedures with connections” and “procedures without connections” particularly useful. “procedures without connections” refer broadly to “tasks that require students to perform a memorised procedure in a routine manner” (stein et al., 2009, p. 1). “procedures with connections” refer broadly to “tasks that demand engagement with concepts and that stimulate students to make powerful connections to meaning or relevant mathematical ideas” (ibid.). sagm (doe, 2008a), stein et al. (2000) and porter (2002) all use some form of mathematical practice or activity as their taxonomical categories, for example, routine procedures, memorisation, nonroutine problems, procedures with connections, and conjecture, generalise and prove. furthermore, they merge these practices with cognitive levels. as we later demonstrate, this conflation of the type of the thinking (as described by the practice, for example, doing procedures or proving and so on) with level of thinking (as described by the complexity of the item) makes it very difficult to use these taxonomies as tools for measuring the standard of an examination. we will show that in order to have an effective taxonomy which can be used to measure both type and level of thinking, these two attributes need to be separated. in contrast, naep (nagb, 2008) focuses on complexity levels of items only (not mathematical practices). naep (nagb, 2008, p. 18) describe these levels as the “specific level of thinking” which the item is designed to measure. thus mathematical complexity level refers to an attribute of a task which measures the level of thinking required to complete the task. the naep taxonomy distinguishes between three levels of complexity: low, moderate and high. low complexity items require students to recall or recognise concepts and/or to perform routine procedures. moderate complexity refers to items for which the method of solution is not directly given; the learner needs to decide on how to approach the problem. more flexibility of thinking is required compared to the low complexity category. items with high complexity require that students use reasoning, planning, analysis, judgment, and creative thought. in addition they may be expected to generalise or to justify mathematical statements or construct mathematical arguments. placement of a task in an appropriate category using any of the above taxonomies can be tricky (stein et al., 2000). besides other considerations, the level of thinking required by a task depends on the familiarity of a learner with the mathematical content of the task (nagb, 2008). for instance, learners must have encountered a mathematical procedure many times in order to achieve the “routine procedure” cognitive demand level. those learners who are not familiar or lack experience with the tasks cannot successfully achieve this cognitive demand level. realisation that demands of a task in part depends on the experience of learners necessitates that due consideration be given to learner age and grade level when deciding the cognitive demand level of a task. furthermore, and as we demonstrate below, conflation of cognitive level with type of thinking required, is problematic. margot berger, lynn bowie & lovemore nyaumwe 33 applying the taxonomies to the nsc examinations we explored the four different taxonomies by applying them to each item in 2008 doe and ieb mathematics examinations papers and, when necessary, reaching consensus on a categorisation. from this taxonomical analysis, we chose exemplars (presented below) to illustrate our arguments. do the taxonomies provide a useful way of evaluating the standard of the examinations? the stein, sagm and porter taxonomies purport to reflect the cognitive demand of tasks. stein et al (2009, p. 1) define cognitive demand as “the kind and level of thinking required of students in order to successfully engage with and solve a task”. sagm and porter assume a similar conflation of kind and level of activity. in stein’s definition “level” would seem to imply that we can take cognitive demand as an indicator of difficulty, but the fact that it is coupled with “the kind of thinking” gives pause for thought. indeed our experiences of applying the different taxonomies to analysing the paper suggest that the conflation of the notions of “levels” and “kinds” of thinking makes this chain of deduction less straightforward. we illustrate the difficulty of applying the taxonomies when level and kind of thinking are conflated, through an example which focuses on the stein category “procedures with connections”. question 5.3 (doe, 2008b) states (the learners have been given that 8)1(2)( 2  xxf and have drawn this graph in question 5.1): 5.3 the graph of f is shifted 2 units to the left. write down the equation of the new graph. the fact that a learner has to make connections between different representations led us to classify this problem as a “procedure with connections” task in the stein taxonomy (a high level cognitive demand category). it is, of course, possible to do this task by recourse to a learnt algorithm (i.e. if a graph shifts 2 units to the left, replace x with x + 2). however as we cannot determine how a learner will do the task we have to rely on the descriptors of the category, which suggest that tasks that involve links between different representations are classified as “procedures with connections”. the fact that shifting graphs and its effect on the algebraic representation for a function is a stated outcome in the curriculum and that this kind of problem would thus be very familiar to learners, led us to classify this question as a routine procedure in the sagm taxonomy (a low level cognitive demand category). what we see here is a focus on two different aspects of the cognitive demand of the task. noting the connections between different representations that needs to be made seems to prioritise the “kind of thinking” that is needed. noting that the task is familiar seems to prioritise the “level of thinking” that is needed. in a related fashion, doe (2008a) refers to the sagm taxonomy of categories (memorisation, routine procedures, complex procedures and problem solving) as both taxonomical categories (p. 12) and cognitive levels (p. 13). again we suggest that the conflation of kind and level of thinking in sagm taxonomy makes it difficult to use the taxonomy to evaluate an examination and that it will probably also make it difficult to use the taxonomy to construct an examination. we illustrate our arguments using two sets of examples drawn from ieb and doe mathematics paper 1 (ieb, 2008a, doe, 2008b). each example shows how different items placed in the same sagm taxonomical category may have very different complexity levels. question 7(b) (ieb, 2008a) and question 1.3 (doe, 2008b) were both classified as “solving problems” in the sagm taxonomy and “non-routine problems” in the porter taxonomy. 7 (b) given 6 6 6 6 ...x      (1) write down x2 in terms of x. (2) hence determine the value of x. taxonomy matters: cognitive levels and types of mathematical activities 34 1.3 given x = 999 999 999 999, determine the exact value of 2 4 2 x x   . show all your calculations. we suggest that question 7(b) is of high complexity because its notation is unfamiliar; its method of solution is not direct and the learner needs to treat 6 6 6 6 ...    as an object rather than a process. that is, to solve the problem, the learner needs to start by squaring both sides of the equation, 6 6 6 6 ...x      . this leads to the equation 2 6 6 6 6 6 ...x       which is equivalent to 2 6x x  . solving 2 6 0x x   gives 3x  or 2x   . but 2x   since x is a positive square root. in contrast we suggest that question 1.3 is of moderate complexity. the expression 2 4 2 x x   should be familiar to many students. complexity lies in the need for the student to ‘see’ that factorisation and cancellation will transform the expression into a simple form, 2, 2x x  . to solve the problem, the learner just needs to substitute 999 999 999 999 into 2x  to get 1 000 000 000 001. these two examples illustrate how the porter and sagm taxonomies may obscure subtle differences in complexity between these two questions. in this example, stein’s categories of “doing maths” and “procedures with connections” do distinguish between the different nature of these two questions. question 7(b) is “doing maths” since it requires non-algorithmic thinking; in addition it requires that the learner understands the nature of certain mathematical concepts and processes (and the interplay between the process and the concept). question 1.3 is “procedures with connections”: its execution uses procedures and in so doing highlights the relationships between factorisation, canceling and substitution. we use one last example to further illustrate how taxonomies which conflate complexity and type of mathematical activity may obscure differences between levels of thinking in a particular item. we consider question 7(c) and question 4(c)(2) (ieb, 2008a). according to sagm taxonomy both questions are “complex procedures”. however we suggest that this classification obscures the different levels of thinking (complexity) required. question 7 (c) reads: 7(c) given: ( ) 3f x x , find a simplified expression for    11 1( ) .f x f f x x f x        in order to solve this problem, the learner needs to deal with each expression in turn. computing 1 f x       requires an understanding of how to work with functional notation; this understanding may be purely procedural. that is, to obtain 1 f x       , the learner substitutes 1 x for x in the expression 3x. evaluating   1 f x similarly requires procedural knowledge (substitute 3x for f(x)). finding the inverse function  1f x similarly is procedural. ultimately the learner needs to add these expressions together (a computational task). given the focus on the use of procedures to simplify several expressions, we regard this task as having moderate complexity. margot berger, lynn bowie & lovemore nyaumwe 35 in contrast, question 4(c)(2) below has high complexity: 4(c) given: 3 2 2y x x x    (1) find the equation of the tangent to the curve at the point where x = 1. (2) find the x-coordinate of another point on the curve at which the tangent is parallel to the one found in (1). in order to solve question 4(c)(2), the learner has to first solve question 4(c)(1), a routine procedure of moderate complexity. this yields the equation of the tangent as y = 4x + 5. to complete question 4(c)(2), the learner needs to first realise that the ‘new’ tangent has a slope of 4 (the slope of the ‘old’ tangent). this requires a conceptual understanding of what it means for one line to be parallel to another. the learner must then set the derivative of the original function equal to 4, i.e. 3x2 2x  1 = 4, a complex procedure requiring non standard thinking. the learner then needs to solve this equation to find x = 1 (already given in part 4(c)(1)) or 3 5x . thus 3 5x . this example illustrates how the sagm taxonomy is unable to distinguish between a “complex procedure” with high complexity and a “complex procedure” with moderate complexity. similarly, using the stein taxonomy both items would be classified as “procedures with connections”: question 7(c) illustrates the subtle relationship between notations and the relevant mathematical objects; question 4(c)(2) focuses on the conceptual connection between the notion of slope and derivative. with regard to porter, this example illustrates the difficulty of using this taxonomy. both question 4(c)(2) and question 7(c) require the use of non-routine procedures. so do we classify either of these as “perform procedure” or is it a non-routine problem? arguments may be made for either classification. application of the naep taxonomy (nagb, 2008) on its own is more reliable as a yardstick for measuring the standard of the examination. however it does not distinguish between different types of mathematical practices. for example, ‘easy’ examination tasks, such as conjecturing a simple pattern, is put in the same category as a familiar and routine procedure, no matter the different nature of the activities. do the taxonomies provide a useful way of distinguishing between different types of mathematical practices in an examination? porter’s taxonomy is attractive in that, unlike stein or sagm, it has specific categories for key mathematical practices such as communicating, understanding, conjecturing, generalising and solving non-routine problems. however the conflation of type and level of thinking again confounds the categorisation and leads to questions with very different levels of complexity being placed in the same category. we illustrate through two ‘comparable’ items, namely question 8(b)(3) (ieb, 2008b) and question 1.3 (doe, 2008c). see overleaf. using porter’s taxonomy, we classify both question 1.3 and question 8(b)(3) in the “conjecture, generalise, prove” category. question 1.3 can be successfully completed by a straightforward application of the distance formula and the converse of pythagoras’ theorem (low complexity in the naep taxonomy). however the route to the proof in question 8(b)(3) is not straightforward and to construct an entirely coherent and complete proof requires considerable insight (high complexity in the naep taxonomy). porter’s category “conjecture, generalise, prove” thus encompasses both low and high complexity items. actually, porter’s taxonomy is designed to measure the degree of alignment between curriculum, instruction and assessment, not to reflect a complexity rating. he uses a two dimensional matrix that combines topics and cognitive demand as a tool to describe the content of an intended curriculum, an assessment and of instructional practice separately. he then uses the matrices to calculate the degree of alignment between any two of these (e.g. between the intended curriculum and the assessment). thus the taxonomy m 36 8(b) (1) ( ( (2) (3) question 1. 1.3 prove quality of t curriculum disaggregate one could a and series” t matters: cog (i) find, corr x-axis. (ii) find, cor the x-axis study the ta the form y  based on th following se the inclinat prove the co 3 is: e that adc the assessme and/or instru e the assessm argue that the topic area ne equation inclination equation inclination gnitive levels rect to two d rrect to two d s. ables below w ax and y he informati entence: tions of the li onjecture tha c = 90o. ent is determ uction matrix ment in a qu e ncs (doe eeds to be set y = 3x y = 71,60 18, y = 5x y = 78,70 11, and types o decimal digits decimal digi which contai 1 x a  for a ion in the ta ines y ax at you formul mined by the x. this notion uite specific f e, 2003) sugg t at the cogni = 13 x ,40 = 15 x ,30 of mathematic s, the acute a its, the acute in the values  3, 4, 5a  . ables above, and 1 y x a  lated in (2). e closeness o n of alignmen fashion. for gests that a c itive demand equation inclination cal activities angle of incl e angle of in of the inclin , formulate x where a  of fit betwee nt means that example, us certain perce d level of “co y = 4x y = 76,00 14,0 ination of the nclination of nations of lin a conjecture 0 en the assess t porter’s tax sing porter’s entage of the njecture, gen 1 4 x 00 he line 2y  the line y  nes with equa e by comple sment matrix xonomy is in s notion of al e “patterns, s neralise, prov x to the 1 2 x to ations of eting the x and the ntended to lignment, sequences ve”. margot berger, lynn bowie & lovemore nyaumwe 37 although the above broad brushstrokes analysis is not how porter intended his taxonomy to be used, it illustrates why it is important to distinguish types of activity, such as “generalise, conjecture or prove” from level of complexity. the notion of using alignment as the criteria for assessing the fairness and spread of an examination is an appealing one. however the mechanisms by which to measure it and achieve it are anything but obvious. we could apply porter’s taxonomy. however his taxonomy has been applied chiefly in contexts where some version of the american nctm (national council of teachers of mathematics) standards is being implemented and nctm standards have different priorities compared to the ncs vision. furthermore, as we demonstrated above, his taxonomy is not useful for answering questions about the standard of an examination. a new taxonomy in order to measure both the standard of an examination and the alignment of the examination with the curriculum, we need a tool which distinguishes between key mathematical practices as well as between complexity levels of items. the sagm taxonomy does not fulfil requirements of such a tool. for example, tasks involving either reasoning or application of mathematics to real-world problems (activities prioritised in the ncs for mathematics) would both be classified as “solving problems”. similarly, sagm has no specific space for key mathematical activities such as justification, conjecturing, and communicating mathematical ideas despite these activities being regarded as important components of the nsc curriculum. building on the work of graven (2002) and parker (2006), bowie (2009) has argued that four orientations towards mathematics are implicit in the ncs for mathematics. bowie makes these orientations explicit as follows: first, mathematics is important for critical democratic citizenship. it empowers learners to critique mathematical applications in various social, political and economic contexts. mathematics is part of broader society and is important for all learners. second, mathematics is relevant and practical. it has utilitarian value and can be applied to many aspects of everyday life and in other disciplines. mathematics modelling is an important tool that can be used in the exploration of contexts. third, mathematics is created and organised using particular processes. these include investigating, conjecturing, generalising, justifying, proving, axiomatising and defining contexts in which concepts are constructed. fourth, mathematics is a body of knowledge and learners at school need to study a subset of it. this knowledge includes specific algorithms, facts, conventions, notations and forms of representation. bowie (2009) argues that, although the orientation towards mathematics as a body of knowledge predominates, the orientations towards mathematics as relevant and practical and the orientation towards mathematical practices receive particular attention. we suggest that a taxonomy that highlights these aspects is important both because it reflects the implicit ncs and because it is in line with international taxonomies, such as stein et al. (2009) and porter (2002). given the problems associated with using any of sagm, stein et al. or porter, we suggest that, for purposes of setting an examination paper, a taxonomy which distinguishes between low, medium and high complexity on one scale, and categories which derive from the different orientations of mathematics explicated by bowie, may be more useful. this can be constructed as shown in table 1. an elaboration of the above taxonomy is beyond the scope and space constraints of this paper and will be the subject of a future paper. its presence is to signal the possibility of distinguishing complexity (or difficulty level) from the type of mathematical activity in a taxonomy and of simultaneously using this taxonomy to align an examination with the intended curriculum. taxonomy matters: cognitive levels and types of mathematical activities 38 table 1: a new taxonomy categories low complexity moderate complexity high complexity solving real-world problems; mathematical modelling mathematical practices, such as conjecturing, generalising, justifying, proving. mathematical knowledge (facts, procedures, forms of representation. references bowie, l. 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(2009). report of the ministerial panel established to review the grade 12 nsc mathematics examination papers of november 2008. retrieved from http://www.mathsexcellence.co.za/papers/ report%20of%20panel%20of%20maths%20experts.pdf national assessment governing board. (2008). mathematics framework for the 2009 national assessment of educational progress. washington dc: us department of education. available from http://www.nagb.org/publications/frameworks/math-framework09.pdf porter, a. c. (2002). measuring the content of instruction: uses in research and practice. educational researcher, 31(7), 3-14. doi: 10.3102/0013189x031007003 stein, m. k., smith, m. s., henningsen, m. a., & silver, e. a. (2000). implementing standards-based mathematics instruction: a casebook for professional development. reston, va: national council of teachers of mathematics. stein, m. k., smith, m. s., henningsen, m. a., & silver, e. a. (2009). implementing standardsbased mathematics instruction, a casebook for professional development (2nd ed.). reston, va: national council of teachers of mathematics. umalusi. (2009). 2008 maintaining standards report. part 1: overview. pretoria: umalusi. available from http://www.umalusi.org.za/ur/research/umalusistandards09highres.pdf margot berger, lynn bowie & lovemore nyaumwe 39 appendix: summary of the different taxonomies sagm taxonomy knowledge  knowledge and use of formulae or algorithms. routine procedure  problems are not necessarily unfamiliar and can involve integration of different learning outcomes.  well-known procedure.  simple applications and calculation with many steps and which may require interpretation from given information.  identifying and manipulating formulae. complex procedures  mainly unfamiliar and involve integration of different learning outcomes.  no direct route to solution but involve higher level calculation skills and/or reasoning.  may be abstract and require fairly complex procedures. solving problems  non-routine, unseen.  interpreting and extrapolating from solutions obtained by solving problems based in unfamiliar contexts.  using higher level cognitive skills and reasoning to solve non-routine problems  breaking down problem into constituent parts.  non-routine in real context. porter taxonomy memorise facts/definitions/formulae  recite basic math facts.  recall math terms and definitions or formulae and computational procedures. perform procedures  use numbers to count, order, denote.  computational procedures or algorithms.  follow procedures/instructions.  solve equations/formulae/ routine word problems.  organise or display data.  read or produce graphs or tables.  execute geometric constructions. demonstrate understanding of mathematical ideas  communicate math ideas.  use representations to model math ideas  explain finding and results from data analysis strategies.  develop relationships between concepts.  show or explain relationships between models, diagrams or other representations. solve non-routine problems/make connections  apply and adapt a variety of appropriate strategies to solve non-routine problems.  apply mathematics in contexts outside of maths.  analyze data, recognise patterns.  synthesise content and ideas from several sources. conjecture/generalise/prove  determine the truth of a math pattern or proposition.  write formal or informal proofs.  recognise, generate or create patterns.  find a mathematical rule to generate a pattern or number sequence.  make and investigate math conjectures. taxonomy matters: cognitive levels and types of mathematical activities 40  identify faulty arguments.  reason inductively or deductively. stein taxonomy memorisation  reproduction of previously learnt fact, formulae, rules or definitions.  cannot be solved using a procedure.  are not ambiguous  exact reproduction of previously seen material.  have no connection to concepts or meanings that underlie the fact, rules, formulae or definitions. procedures without connections  algorithmic.  little ambiguity about what needs to be done.  have no connections to the concepts or meaning that underlie the procedure.  focused on correct answer rather than understanding.  require no explanations (or explanation solely on procedure). procedures with connections  focus students’ attention on use of procedures for purpose of developing deeper levels of understanding.  suggest pathways to follow that are broad general procedures that have close connections to underlying conceptual ideas.  make connections among multiple representations.  general procedures cannot be followed mindlessly. students need to engage with the conceptual ideas that underlie the procedures. doing mathematics  require complex and non-algorithmic thinking  require students to explore and understand nature of mathematical concepts, processes or relationships.  demand self-monitoring or self-regulation of one’s own cognitive processes.  require students to access and use relevant knowledge and experiences.  require students to analyse the task and actively examine task constraints that may limit possible solution strategies and solutions.  require considerable cognitive effort and may involve some level of anxiety due to unpredictable nature of solution processes. naep taxonomy low complexity  students to recall or recognise concepts or procedures.  items typically specify what the student is to do, which is often to carry out some procedure that can be performed mechanically.  the student does not need to use an original method or to demonstrate a line of reasoning. moderate complexity  items involve more flexibility of thinking and choice among alternatives than do those in the lowcomplexity category.  student is expected to decide what to do and how to do it, bringing together concepts and processes from various domains. for example, student may need to represent a situation in more than one way, to draw a geometric figure that satisfies multiple conditions, or to solve a problem involving multiple unspecified operations.  students might be asked to show explain their work but would not be expected to justify it mathematically. high complexity  students are expected to use reasoning, planning, analysis, judgment, and creative thought.  students may be expected to justify mathematical statements or construct a mathematical argument. items might require students to generalise from specific examples.  items at this level take more time than those at other levels due to the demands of the task, not due to the number of parts or steps. << /ascii85encodepages false /allowtransparency false /autopositionepsfiles true /autorotatepages /none /binding /left /calgrayprofile (dot gain 20%) /calrgbprofile (srgb iec61966-2.1) /calcmykprofile (u.s. web coated \050swop\051 v2) /srgbprofile (srgb iec61966-2.1) /cannotembedfontpolicy /error /compatibilitylevel 1.4 /compressobjects /tags /compresspages true /convertimagestoindexed true /passthroughjpegimages true /createjobticket false 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can be opened with acrobat and adobe reader 5.0 and later.) >> /namespace [ (adobe) (common) (1.0) ] /othernamespaces [ << /asreaderspreads false /cropimagestoframes true /errorcontrol /warnandcontinue /flattenerignorespreadoverrides false /includeguidesgrids false /includenonprinting false /includeslug false /namespace [ (adobe) (indesign) (4.0) ] /omitplacedbitmaps false /omitplacedeps false /omitplacedpdf false /simulateoverprint /legacy >> << /addbleedmarks false /addcolorbars false /addcropmarks false /addpageinfo false /addregmarks false /convertcolors /converttocmyk /destinationprofilename () /destinationprofileselector /documentcmyk /downsample16bitimages true /flattenerpreset << /presetselector /mediumresolution >> /formelements false /generatestructure false /includebookmarks false /includehyperlinks false /includeinteractive false /includelayers false /includeprofiles false /multimediahandling /useobjectsettings /namespace [ (adobe) (creativesuite) (2.0) ] /pdfxoutputintentprofileselector /documentcmyk /preserveediting true /untaggedcmykhandling /leaveuntagged /untaggedrgbhandling /usedocumentprofile /usedocumentbleed false >> ] >> setdistillerparams << /hwresolution [2400 2400] /pagesize [612.000 792.000] >> setpagedevice microsoft word 64 front cover final.doc pythagoras 64, december, 2006, pp. 45-51 45 teacher education for mathematical literacy: a modelling approach bruce brown and marc schäfer education department, rhodes university b.brown@ru.ac.za and m.schafer@ru.ac.za introduction the introduction of mathematical literacy into the further education and training (fet) curriculum in south africa has brought with it formidable challenges to teacher education in this field. this paper attempts to unravel some pertinent issues arising in the training of mathematical literacy teachers, using an approach based on mathematical modelling. it does this by discussing the design and implementation of an ace(ml), an advanced certificate in education, specialising in mathematical literacy teaching. a number of impressions were gained in the first implementation cycle of this course ending in december 2005. lessons learned from these impressions were used to adapt and improve the course during delivery and will be used to further improve following cycles of the course. specific impressions gained from two particular incidences in the course, together with a discussion on the theoretical underpinnings of the course form the core of this paper. the programme under consideration was developed by the first author while working at rhodes university mathematics education project (rumep). it was developed in response to an urgent request by the eastern cape department of education, to provide professional training in mathematical literacy teaching, for 1000 teachers, for the implementation of the new fet curriculum in 2006. a collaborative committee with representatives from all the higher education institutions in the eastern cape was constituted in order to facilitate this task. a curriculum subcommittee, chaired by the first author, developed the basic structure of the curriculum. this curriculum was accredited so that training of the first cohort could be started in the second half of 2004. each participating higher educational institution in the eastern cape developed the detailed design of its own programme in line with the basic curriculum structure. the final programme implemented by rumep for a cohort of 70 teachers is the focus of this paper. mathematical literacy the issue of what constitutes mathematical literacy is currently under debate. but a feature that does appear to distinguish mathematical literacy from mathematics is an emphasis on the use of mathematics in context (pisa 2003; steen, 2001). in particular, contexts that are common or relevant to the day-to-day life of the ‘ordinary’ person in society (duba, 2004; laridon, 2004). and it is the use of mathematics in these contexts, by this ‘ordinary’ person that is an important focus of mathematical literacy. that is, to be mathematically literate, it is important that a person be able to identify mathematics relevant to the context at hand and then be able to use this mathematics as one means contributing towards the achievement of one’s goals in the context. one important observation that arose from this is that it may not be particularly helpful to view mathematical literacy as a global attribute of a person. it may be better to take a more local approach and view mathematical literacy as relative to context and so deal with mathematical literacy in context, for different contexts. a similar distinction is commonly made for literacy (pisa, 2003) and can be seen implicitly in the fet curriculum documents (department of education, 2003) in their reference to the three broad contexts. • mathematical literacy for self-managing persons. the scope of this context is the use of mathematics for personal decision making and enrichment. a particularly important theme, that is identified as such in the curriculum documents, is that of personal finances. • mathematical literacy for contributing workers. this encompasses the interpretation and use of mathematics common in the workplace. particular examples of such mathematics are teacher education for mathematical literacy: a modelling approach 46 measurement, tables, schedules and graphs of productivity and performance. • mathematical literacy for participating citizens. here the focus is on civic participation, involving the use of mathematics to engage with social issues and to interpret and respond to statements made by different actors in society. the ability to critically analyse statistics is often important in this context. approach to teacher education we decided to adopt an activityand investigationbased approach that is characterised by hands-on experience of a modelling process that seeks to develop an understanding of mathematics in context. our course was broadly structured as follows: • first, teachers themselves needed to develop positive experience of the use of mathematics in context, as described above. this experience of being mathematically literate themselves in some contexts, would enable them to appreciate the final goal for the subject, of enabling their learners to become mathematically literate in these contexts. loosely, they may begin to know ‘what it feels like’ to be mathematically literate. • once teachers had developed their experience of mathematical literacy, they would reflect on the teaching and learning process that they had been through. in this way, they would be able to identify and conceptualise issues of teaching and learning important for the teaching of this subject. these issues would include practical matters such as planning, resources and assessment. they also included matters more difficult to formalise, such as processes of learning and how better to facilitate such processes. • teachers would also be asked to reflect on their experience of mathematical literacy ‘in action’ and so begin to develop their own, more formal, conception of what it means to be mathematically literate. a mathematical modelling approach to mathematical literacy when designing the teachers’ training programme, an approach to the subject was sought that was aligned with mathematical literacy’s particular focus on engaging with mathematics in ordinary contexts. in particular, we wished to encourage teachers to take an active view of mathematics, seeing it as relating naturally to life contexts and enabling them to make more effective decisions in such contexts. the process of identifying and then using relevant mathematics to analyse and make decisions about contexts showed many similarities to that of mathematical modelling. the major differences appeared to be that mathematical modelling is generally described using more advanced mathematics, in more technical contexts. but the basic approach of mathematical modelling could just as easily be applied to the use of elementary mathematics for modelling more everyday contexts (pisa, 2003). the standard approach to mathematical modelling (blum, 2002) may be very simply described as a cyclical process involving three components. • exploration of the context and formulation of the model and question. • analysis of the model in order to answer the mathematical question formulated. • interpretation and validation of the mathematical conclusion in terms of the context. this description of the modelling process is based on the view that the person doing the modelling possesses the mathematical skills and concepts needed to formulate and analyse the model. but we expected the majority of the learners in the mathematical literacy class to display a rather low level of mathematical competence. as a result, they would need to work to develop mathematical skills and concepts as well as learning how to relate their growing mathematical understanding to contextual situations. with this in mind, we decided to adopt the “open modelling” approach to teaching described by galbraith (1989) and carr (1989). that is, learners should be encouraged to formulate and solve contextual problems using the mathematics that they find accessible. as a result, the mathematical model to be analysed could be anything from an instance of a simple numerical or spatial pattern, to an abstract mathematical system formulated in symbolic terms. in order for teachers to build on this foundation and further develop learners’ mathematical competence, we included a fourth component in the cycle. • consolidation and extension of the mathematical skills and concepts developed in the other stages of the cycle. bruce brown and marc schäfer 47 this would allow learners to develop mathematical understanding based on their experience of the grounding context and motivated by the use of the efficacy of the mathematics in context. this approach is similar to that taken in “realistic mathematics education” advocated by the freudenthal institute (van den heuwel-panhuizen, 2000). they term the process of developing mathematical representations directly related to a grounding context “horizontal mathematicization”. while building on this mathematical foundation to develop and refine further mathematical tools and representations is termed “vertical mathematicization”. with the above issues in mind, a more detailed conceptualisation of each stage of this modelling cycle was developed. this included a number of features that would not necessarily be considered an integral part of modelling, but would be important when using modelling as a vehicle for teaching mathematical literacy. this conceptualisation may be outlined as follows: formulation: • exploration of the context. • formulation of the model – identification and representation of measurable quantities and possible patterns in the context. • formulation of the question in the mathematical terms of the model. • identification and estimation of missing data. analysis of the model in order to answer the mathematical question formulated: • explore the regularities and relationships in the model. • develop mathematical tools to manipulate the model. • identify any extra assumptions which would make it possible to use the tools to answer the question. • use the tools developed to answer the mathematical question posed. interpretation and validation of the mathematical conclusion in terms of the context: • provide interpretations of mathematical representations and operations developed in the analysis. • interpret regularities and relationships developed within the analysis. • interpret the mathematical conclusion in terms of the context and the problem to be addressed in the context. • validate the interpreted conclusion by checking to see that it appears reasonable and in context and can be justified by appealing to the context. • understand the effects of the assumptions needed for the mathematical analysis, on the validity of the conclusions obtained. • understand the effects of the assumptions needed for the practical formulation of the model and posing of the question, on the validity of the conclusions obtained. consolidation: • practise using the mathematical representations and skills developed when formulating and analysing the model. • identify and explore mathematical relationships in the model. • identify and explore relationships between the mathematics developed to solve this problem and any other mathematical systems known by the modeller. • develop more abstract mathematical systems to represent and analyse these higher order relationships. • identify other contexts that may be related in similar ways to the model and mathematical systems developed in this process. examples of the modelling approach two tasks from the course will be discussed as examples of the teaching approach adopted. both tasks were framed as assignments for a small business and so could be interpreted as falling in the broad context of “mathematics for contributing workers”. a) tiling a floor a simple example of the use of this modelling cycle in the course arose from the task of designing a tiled floor for a community hall. this task was initially framed as an assessment exercise for the course. teachers were asked to design a pattern using black and white tiles, by first designing a pattern for a block of tiles and then replicating this block to cover the entire floor. they were provided with the dimensions of the hall (21 m by 15 m), the teacher education for mathematical literacy: a modelling approach 48 dimensions of the block (3 m by 3 m) and the dimensions of the tiles (300 mm by 300 mm). for assessment, they were asked to draw the block they had designed (in a square outline provided), to count or calculate the number of each colour of tile in their block, and then to calculate the number of each colour of tile that would be needed to tile the community hall floor. we had expected that most teachers would enjoy developing a creative design and would be able to calculate the number of tiles needed. most teachers did indeed enjoy creating a design, but only half of the teachers on the course were able to calculate the number of tiles needed for the hall floor. in response to this, a simple contextual problem involving tiling a floor (with a single tile type) was posed for group work later in the contact session. • in the formulation stage, teachers were asked to explore and identify possible patterns in the context. in particular, to discuss the process of practically laying out tiles to cover the floor and to describe any patterns they could identify in this process that might be useful for counting the tiles. most groups saw that they could approach the tiling task by laying out tiles in rows or columns across the floor. some groups related this to the concept of area, but many did not. • in the analysis stage, groups were asked to use the patterns they had identified to count the tiles and they were able to do this successfully. they developed different counting strategies to count the tiles, but at the end of the process, most groups had identified the product of the number of rows and the number of columns as important. because the dimensions did not result in an integral number of rows and columns, different conclusions were reached, depending on how the remainders were handled. • in the verification stage, the groups agreed on the importance of the basic product (rows × columns). this was followed by an animated discussion where groups described and justified their procedures for tiling the remaining space and described how these influenced their final estimate. at the end of the session the estimates of most groups had been accepted as reasonable. a few groups had used the concept of area to make their estimates. in justifying their results, they generally talked about the number of times the area of a tile could divide into the area of the floor. some teachers were convinced by this, but others were sceptical. • after this a short consolidation was carried out. the facilitator described the view of measuring area as covering a region with a unit area and then counting the number of units used to form the cover. the teachers were interested that this view of measuring area related the tiling situation directly to the area measurement process. later in the course, a number of further consolidation tasks were carried out. these included calculating the area of a region by counting covering squares. they then spent time investigating how the measurement changed when the size of the unit square changed, and then practised doing this conversion in a number of cases. note that even though most of the teachers were able to solve the tiling problem using their different strategies, consolidation was necessary to enable them to better grasp the concept of area and develop related skills. the reasoning of most groups was strongly dependent on, and expressed in terms of, the context of tiling that particular floor. in the subsequent discussion, these groups did state that the product of rows and columns was a useful and efficient summary, but they did not interpret this product in terms of area. when queried about area, they tended to define area as length times breadth, but they did not relate this to the product that had been developed for tiling. the solutions of these groups could thus be seen as situated in the context, as opposed to applications of an abstract mathematical concept (area) to the context (collins, brown and duguid, 1989). in the consolidation work, we worked from teachers’ initial solutions to the problem to develop insight into ways that the concept of area, and the technical skills involved in calculating area, related to such tiling situations. these skills allowed them to solve similar tiling problems more efficiently in future. this was evidenced in their later assessment when most teachers were able to solve such tiling problems more efficiently using the mathematical skills that had been developed. bruce brown and marc schäfer 49 b) an investigation on designing parking lots one of the first investigations of the ace(ml) was a group task that asked the teachers to design the parking spaces for cars in a parking lot. teachers were provided with regulations for the minimum size of parking spaces of different orientations. they were also provided with a plan view showing the shape of the parking lot. the scale of the plan was given by specifying the actual width of the parking lot. achieving a fitting formulation for this problem was surprisingly problematical for most groups. because of the openness of the task, it was expected that there would be a number of different responses. this was indeed the case. but what was surprising was that many of these differences were due to the teacher’s view of the scale of the parking lot − a number of plans with different scales were drawn. most groups spent very little time formulating the problem and identifying precisely what information they had. instead they immediately began drawing a design. this corresponds well to the behaviour of students who were faced with unfamiliar problems, reported by schoenfeld (1992). approximately 60% of these students made a quick decision and then pursued their chosen direction with no concurrent attempt to judge the success or failure of their attempt. in this case, it appeared as if the design was started without an explicit decision about the scale of the plan. the teachers seemed to make an implicit judgement of the scale and then proceeded using this implicit scale, disregarding the information that had been given about the length of the side of the lot. the first attempt to correct the scales of the plans was unsuccessful. the facilitator asked these groups if a ‘real’ car would fit properly in their parking spaces. they all replied that it would, and demonstrated a car size that did fit their plan. in response, the facilitator demonstrated a car of a size consistent with the size of the parking lot given in the task, which was noticeably different from theirs. most groups questioned the size of this car, rather than the scale of their plan. the second attempt fared better. the facilitator asked the groups to calculate the scale of the plan by measuring the length of the side and comparing this with the specified length of that side. when asked to calculate the size of their parking spaces using this scale, the groups with very small parking spaces (less than half a metre wide) recognised a problem, but many of the other groups did not. but drawing a car to the calculated scale in one of their parking spaces, alerted the remaining groups to the inconsistency between their choice of scale and the size of the parking lot given in the task statement. that the teachers were working relative to an implicit scale was evident from the fact that groups did not draw parking spaces of arbitrary sizes in their lots. in each design, the sizes of the parking places drawn were consistent. possibly the teachers used a form of visual scaling, judging visually that the parking spaces were properly scaled according to a measure that they had intuitively chosen. in their response to the first question about the fit of a ‘real’ car, the inconsistent groups appeared to use the same visual scale for cars as they did for their parking lots, and so saw ‘real’ cars as indeed fitting their design. unfortunately, few groups used an intuitive scale that was fitting to the size of the lot specified in the task. most groups saw the size of their parking lot as much larger, or much smaller than the size given in the task. when making their designs, these groups did not appear to interpret the given plan as a representation of a real parking lot. they appeared to understand that they were working on a ‘problem’ and not a design of a real parking lot and, as a consequence, they did not attend to the size of the parking lot given in the task. in our work with the teachers, the interpretation of the plan as a representation of a real parking lot with dimensions as specified in the task, was crucial for the successful completion of the task. this interpretation could be seen as the process of relating mathematical objects or properties to objects and properties of a real situation. in this case: • interpreting the mathematical object (the plan) in terms of a real object (the parking lot), and • interpreting the dimensions of the plan as systematically related (by means of a fixed scale) to the corresponding dimensions of the real parking lot. even though this plan did not really relate to a real object, being prepared, and able, to make such interpretations would be important for using and drawing plans in life, where the plans would indeed relate to a real object. the process of interpretation seemed to be made more difficult for the teachers by their understanding that this was a teaching task and so the first level of interpretation was artificial – in terms of a real object that did not exist. this may have made it easier to ignore the second level of interpretation needed. such an issue could be a concern for the teaching of mathematical literacy, teacher education for mathematical literacy: a modelling approach 50 where most problem situations will not be real, even if they originate in real situations. impressions gained context and content – the need for process mathematical literacy involves the use of mathematics in context. to learn mathematical literacy, it is important to master the mathematics used, as well as to develop familiarity with the different contexts. but it is also necessary to develop the skills needed to be able to effectively relate mathematics and context. when training the teachers, we found it useful to explicitly identify such process skills, as these provided points of focus for identifying and developing these skills in their learners. as is evident in both examples, one such skill that was identified as important in teachers’ responses was that of interpreting mathematical concepts and skills in relation to a context. a number of other important skills were identified in the training, including that of mathematical idealisation, or idealising contextual patterns and relations to form mathematical structures. this skill is discussed in the pisa framework (pisa 2003) as an important part of the modelling process. in the mathematical literacy training, we found it useful to view mathematical idealisation as dual to that of interpretation. the modelling approach a) accessibility of the context based approach most of the teachers appeared to enjoy this approach and found it quite accessible. as described in the example, they worked well in their groups and were able to develop effective solutions to the problems set, even if these were not very efficient. as well as familiarising themselves with the context, this work also provided a good foundation for the understanding and skills developed in the consolidation stage. although initially most teachers elected to discuss the tasks between themselves, or with the facilitators, they were generally able to complete similar tasks successfully themselves. b) time needed to successfully apply this teaching approach, the teachers on the course needed to be afforded time to effectively engage with the contextual problem. as an illustration of this, the episode described in the second example took place over a two and a half hour period. at the end of this period, most groups had developed a rough plan of their design using the correct scale – they were working in the analysis stage. as can be seen from the first example, properly completing such a contextual problem and then mastering the mathematical issues that arise in the process requires considerable time. but once done, these issues did appear to be well mastered with the mathematics learned being well related to the grounding question. c) non-separateness of the modelling cycle it was noticed in facilitating these contextual problems that the components of the modelling cycle are not fully discrete. for the work at each stage was informed by the work done and the possibilities inherent in the other stages. that is, recognising patterns and developing the model was often guided as much by the possible mathematical structures known as it was by the characteristics of the context. also, the analysis of the model was often guided by intuitive considerations based on the context. the modelling cycle thus became seen as a useful analytical tool, yielding insight into both the teaching and solution processes, rather than a constraining structure that needed to be rigidly followed. d) need to balance contextualised work and consolidation most of the teacher groups successfully completed the tiling problem introduced in response to the initial assessment in the first example. but many used a straightforward counting argument that, while simple and yielding a valid estimate, was rather inefficient, especially for large areas. without the consolidation work, these groups would have probably continued to use this strategy in other problems of this type, particularly because they had successfully used it in the initial problem. the consolidation work enabled them to improve their mathematical understanding and to see how these skills could be used to more efficiently solve the contextual problem from which it flowed. in teaching the programme it became important to balance contextual work with consolidation. conclusion the modelling approach worked well with the majority of reasonably and well skilled teachers. but many of the teachers with weaker mathematical skills took considerably longer to master the contexts and skills developed in the activities described above. this suggests that a teacher’s level of mathematical skill is an important determinant of success in such a programme. but also that a lack of mathematical bruce brown and marc schäfer 51 skill is not the only barrier to success. properly identifying these factors and their interrelationships would be an interesting question for further research. due to the pressure of time on the programme, it was not possible to include consolidation work that was appropriate for the less skilled teachers, for all the contexts considered. the first objective of providing teachers with the experience of being properly mathematically literate, was thus not fully achieved for less skilled teachers. in consequence, the more reflective parts of the programme became less real to them and were partly memorised rather than evidenced through experience. references blum, w. (2002). icmi study 14: applications and modelling in mathematics education. discussion document. educational studies in mathematics 51, 149-171. carr, a. (1989). teaching mathematical modelling. in d. blane & m. evans (eds.), mathematical modelling for the senior years (pp. 66-71). parkville: the mathematical association of victoria. collins, a., brown, j.s.a., and duguid, p. (1989). situated cognition and the culture of learning. educational researcher, 18(1), 32-42. department of education. (2003). national curriculum statement grades 10–12 (general), mathematical literacy. pretoria: government printer. duba, t. (2004). mathematical literacy: an international perspective. paper presented at the shuttleworth mathematical literacy seminar, cape town, february 2004. galbraith, p. (1989). from applications to modelling. in d. blane & m. evans (eds.), mathematical modelling for the senior years (pp. 78–86). parkville: the mathematical association of victoria. laridon, p. (2004). mathematical literacy and the fet curriculum. paper presented at the shuttleworth mathematical literacy seminar, cape town, february 2004. pisa. (2003). assessment framework: mathematics, reading, science and problem solving knowledge and skills publications 2003. schoenfeld, a.h. (1992). learning to think mathematically: problem solving, metacognition and sense making in mathematics. in d. grouws (ed.), handbook of research on mathematics teaching and learning (pp. 334-370). new york: macmillan. steen, l. (2001). the case for quantitative literacy. in l. a. steen (ed.), mathematics and democracy: the case for quantitative literacy (pp. 1-22). usa: national council on education and the disciplines. van den heuwel-panhuizen, m. (2000). mathematics education in the netherlands: a guided tour. freudenthal institute cd-rom for icme9. utrecht: utrecht university. errors using inadequate data are much less than those using no data at all. – charles babbage book review book review: international reflections on realistic mathematics education book title: international reflections on the netherlands didactics of mathematics: visions on and experiences with realistic mathematics education author: marja van den heuvel-panhuizen (ed.) isbn: 978-3-030-20222-4 doi: https://doi.org/10.1007/9783-030-20223-1 publisher: springer open, cham, switzerland, 2019, r812.07* *book price at time of review review title: book review: international reflections on realistic mathematics education reviewer: stanley a. adendorff1 affiliation: 1department of general education and training, faculty of education, cape peninsula university of technology, cape town, south africa corresponding author: stanley adendorff, adendorffs@cput.ac.za how to cite this book review: adendorff, s.a. (2019). book review: international reflections on realistic mathematics education. pythagoras, 40(1), a525. https://doi.org/10.4102/pythagoras.v40i1.525 copyright: © 2019. the authors. licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. introduction initially i was reluctant to accept what i considered to be the daunting task of reviewing this academic text which exclusively shares encounters that deal with manifold authentic incidents of grappling with realistic mathematics education (rme) on a global scale. i still wonder why i said ‘yes’ eventually. my first exposure to realistic mathematics education started when i was working on the realistic mathematics education in south africa (remesa) project (see chapter 5) under cyril julie (university of the western cape) and heleen verhage (freudenthal institute, university of utrecht) at the turn of the century and later on with bertus van etten of fontys hogenscholen, university of fontys, on the brave maths project at stellenbosch university, south africa. that was the time when i ‘made acquaintance with rme’ (van den heuvel-panhuizen, p. 2) and in a sense became ‘part of the dutch rme community’ (van den heuvel-panhuizen, p. 1). having been exposed to the principles underlying rme also gave me ‘a new perspective on understanding mathematics education’ (van den heuvel-panhuizen, p. 3), similar to the experiences of some of the contributing authors. the purpose of this review, however, is not to share what i understand rme to be, but rather to make sense of, and express some impressions of the value of, this particular book in terms of the advancement of rme globally. the thesis of the book what is the central issue the book addresses? well, the title says it all. the title of the book is focused and unambiguous: international reflections on the netherlands didactics of mathematics: visions on and experiences with realistic mathematics education. these visions and experiences already come to life in the editor’s chapter 1. the global use and application of rme, and the unique and diverse experiences in various international communities, are clearly evident from this text. the editor’s chapter 1 is well written and summarises the overview of the salient aspects quite impressively. it consequently makes me wonder why an additional review of the book was sought. the section titled ‘making acquaintance with rme’ is compelling and absorbing and captures the global essence of rme. the way the book is organised seems to support this thesis. the chapters seem to be randomly ordered with particular aligned themes evident. the different points of departure of the authors are contained in keywords that appear in the titles of the chapters: in chapter 2 the emphasis is on ‘application’ of rme; chapters 3, 4, 12, 16 and 17 critically deal with the ‘influence’ and ‘impact’ thereof; chapter 5 and 10 share ‘reflections’ of particular experiences with rme; chapter 6 pays ‘tribute’ to rme; chapter 18 gives a historical perspective over ‘two decades’ of adapting rme to a particular context; chapter 19 discusses how rme is used in an ‘intervening’ sense in a particular context. apart from critically analysing the title initially, the attention intuitively shifts to who the contributing authors are. in this book you need to read the chapter first to get some idea of the authors in terms of credibility and qualifications. some brief personal information, and in addition a website link where more related information about the authors may be found, would be beneficial. this book, i believe, was not necessarily written with profit-making in mind, but mentioning the established reputation of contributing authors could be compelling in terms of purchasing a copy of the book. names – in no particular order – such as van den heuvel-panhuizen, janssens, julie, gierdien, niss and zulkardi are more familiar to me, but brief biographical information of the authors is needed to orientate myself as reader. as far as could be ascertained each chapter states the author’s place of work and gives a brief indication of who the authors are, together with their email addresses. there are some definite universal experiences related to rme that are common to all the different experiences and scenarios as captured by the different authors. at the same time the unique encounters the authors experienced when introducing, sharing and training educators in terms of rme are also evident. in addition to providing a global perspective, rme is dissected meticulously, to reveal its dynamic and intricate nature. having gained insight into the crux of each of the chapters has made me realise that rme is an all-encompassing theory of teaching and learning mathematics: it includes and applies the ideas of vygotsky’s social constructivism, skemp’s (1978) ideas related to relational versus instrumental teaching and learning of mathematics, the van hiele levels of geometry learning, sfard’s (2015) theory of commognition, of dealing with cognition and communication when learning and teaching mathematics, to name but a few. the search for didactical strategies other than those involving drilling and algorithmic procedures (arcavi, p. 83) compelled educationists to explore the merits of adopting and working with rme principles. this resulted in what de bock, van dooren and verschaffel in chapter 3 (p. 53) call ‘rme inspired changes’. they noticed in terms of the different mathematical domains ‘that in numbers and operations, the attention shifted from obtaining insight in the structure of number systems to linking numbers to quantities’. as a result, ‘numbers are no longer purely abstract entities, but objects that children learn to know and recognise in different forms.’ in addition, ‘with respect to operations, the emphasis shifted from discovering and accurately formulating the commutative, associative and distributive laws to linking operations to concrete and meaningful situations’. these are just some of positive outcomes reported that are rme inspired. critique of realistic mathematics education the critique lodged against rme is meaningfully analysed and reported on by the editor in chapter 1. whether horizontal mathematisation is over-emphasised in the rme context, as opposed to vertical mathematisation as claimed by critics, is debatable. it is understandable that a more balanced approach would be preferred to over-emphasising of applications. i agree that the ‘assumed lack of guidance’ that rme is accused of is indeed the ‘opposite of what rme stands for’ (p. 13). the rme principle of guided re-invention is proof that guidance is generally an inherent feature of rme. the debate that centres on the interpretation (p. 14) of the meaning of ‘realistic’ in rme terms is interesting. i agree that the ‘experiential or emotional worlds’ may not always refer to the child’s reality. what is crucial is the context used to enhance sense-making, understanding and conceptualisation. relevant here is the fantasy context used by freudenthal to explain addition and subtraction of integers that relates to the witch who brews potions, using wonder cubes to regulate the temperature inside the pot she uses (adendorff, 2012). the train of thought of the authors the train of thought of the contributing authors i find relatively easy to follow. although the authors are from different backgrounds and experiences, with discernible authentic contexts, the central theme pivotal to rme never fades or gets lost. the connections, related to rme principles, that tie together the various parts, events and arguments in the different chapters, are foregrounded. so the thread of the central theme, namely the reflective international reflections on rme encounters, is never lost as the reader goes from one chapter to another, and from one country context to another. the rme related contexts used by the authors allow the reader to construct meaning and also direct the reader’s attention to a specific train of thought and interpretation (samson, 2017). glossary i paged to the back of the book to look for a related term and found that there was no glossary. readers of this book will naturally come from various backgrounds and will have different levels of expertise. there are numerous rme terms that may not be familiar to educators, thus educators who are new to rme may find a glossary of terms at the end of the book quite useful. significance of the book this book not only aligns favourably to other books written on rme – such as that of koeno gravenmeijer (1994), titled developing realistic mathematics education, and marja van den heuvel-panhuizen’s (ed., 2001) children learn mathematics: a learning-teaching trajectory with intermediate attainment targets for calculation with whole numbers in primary school – but also gives diverse perspectives on the implementation and use of rme globally. the intended purpose of the book relates to the reflections of 44 countries outside of the netherlands on rme, which originated in the netherlands (p. v). the significance and influence of this book lie in the powerful testimonies shared by the contributing authors. the contributions of authors from the various countries provide ample evidence of the extent of the impact of rme on different communities worldwide. rme seemed to have served as catalyst for meaningful, deep and wide-reaching change. sun and he (chapter 10, p. 175) mention the fact that ‘rme exerted an effect on the latest curriculum reform in mathematics education in china in terms of policy-making, new textbook design and change in classroom teaching’. da ponte and brocardo (chapter 12, p. 209) talk of rme as having ‘a clear influence in portugal, in both research and in curriculum development’. zulkardi, putri and wijaja (chapter 18, p. 326) state that ‘learning from the successes in the united states and south africa in reforming mathematics using realistic mathematics education (rme), indonesia also used and adapted rme to improve mathematics education’. the intended purpose of the book clearly resides in its title and contributes to the advancement of rme as ‘a didactic approach or a domain-specific instruction theory for mathematics’ (zulkardi, putri & wijaja, p. 326). the rigour of the research and scholarship are evident throughout. the narratives and arguments provided are generally sound and relatively easy to follow. as indicated earlier, a comparison with earlier texts on rme places this book in the existing literature. i strongly recommend this book as a valuable resource. references adendorff, s.a. (2012). a critical analysis of a textbook activity: the teaching and learning of addition and subtraction of integers in the senior phase. in s. nieuwoudt, d. laubscher, & h. dreyer (eds.), proceedings of the 18th annual national congress of the association for mathematics education of south africa (vol. 1, pp. 70–80). potchefstroom: amesa. retrieved from http://www.amesa.org.za/amesa2012/volume1.pdf gerson, s., & gerson, s. (2006). technical writing: process and product. upper saddle river, nj: pearson prentice hall. gravemeijer, k. (1994). developing realistic mathematics education. utrecht: freudenthal institute. samson, j. (2017). why context matters in writing: building a relationship with the reader. the writing cooperative. retrieved from https://writingcooperative.com/why-context-matters-in-writing-f52ad075c07a sfard, a. (2015). learning, commognition and mathematics. in d. scott, & e. hargreaves (eds.), the sage handbook of learning (pp. 129–138). london: sage. https://doi.org/10.4135/9781473915213.n12 skemp, r. (1976). relational understanding and instrumental understanding. mathematics teaching, 77, 20–26. van den heuvel-panhuizen, m. (ed.). (2001). children learn mathematics: a learning-teaching trajectory with intermediate attainment targets for calculation with whole numbers in primary school. utrecht: freudenthal institute. microsoft word 16-31 gierdien.doc 16 pythagoras, 70, 16-31 (december 2009) musings on multiplication tables  and associated mathematics and teaching practices  faaiz gierdien  research unit for mathematics education, university of stellenbosch  faaiz@sun.ac.za      this paper is based on my reflections on a deceptively simple tabular representation of a  combined 12×12 multiplication table showing multiplier and multiplicand, starting at a  time when i taught mathematics full time at a primary (elementary) school through to my  present  teaching  of  mathematics  education  modules  to  prospective  teachers.  a  historically‐motivated framework on the importance of tables as expressions of complex  information  in  two‐dimensional  form  is  used  to  gain  insight  into  and  understand  multiplication  tables.  through  this  framework  it  is  shown  that  the  modal  practice  of  “knowing one’s tables”  in the primary grade  levels  is really about knowing sequenced  and separated  lists of whole number multiplications. in contrast, tabular multiplication  sequences in a combined multiplication table can, through appropriate teaching practices,  enable  the  discovery  of  multiple  relationships  beyond  multiplication  or  arithmetic,  resulting  in  significant  mathematics  that  spans  the  grade  levels.  implications  for  mathematics teacher education practice with  its current focus mathematical knowledge  for teaching, are considered.   beyond the so-called multiplication tables there is significant mathematics in a 12×12 tabular representation of the multiplication tables (see figure 1), which deserves attention. “what” this mathematics is has a great deal to do with “how” one approaches figure 1. on a similar point, dewey (1904/1964) writes extensively on dichotomies such as content and pedagogy, subject matter and method, and the implication of separating the two. put differently, “what” the mathematics in figure 1 is as content or subject matter has to do with how or the pedagogy or method that is used. this is the main argument for the title of the paper, namely mathematics and teaching practices. we know only too well the inert status of the multiplication tables in relation to the rest of the mathematics curriculum. the idea of the paper is therefore to throw multiplication tables into “fresh combinations” (whitehead, 1929/1951) with other branches of the mathematics curriculum. any curriculum associated with figure 1 is thus intimately connected to the type of teaching that accompanies it. in this article i reflect on some of the mathematics and teaching practices associated with figure 1 which i first noticed in burns (2000) while teaching mathematics full time at an elementary (primary) school. i became intrigued by the structure, pattern and symmetry in this multiplication table and the attendant history of tables as human inventions and have been using it since in prospective mathematics education modules and professional development opportunities for in-service teachers. current curriculum policy statements in south africa provide a certain orientation towards multiplication tables that is helpful but also inadequate (department of education (doe), 2003). in the primary or elementary grade levels, multiplication is mainly associated with the multiplication tables, hence there will be a focus on references to multiplication in the policy document (doe, 2003). policy statements contain no detailed vignettes that illustrate the power of tables with respect to multiplication tables to start with. for example, they suggest that multiplication should be introduced through “repeated addition” in faaiz gierdien 17 the first grade and then multiplication of whole 1-digit by 1-digit numbers in the second grade and whole 2 digit by 1-digit numbers in the third grade. from the fourth to the sixth grades learners should be encouraged to memorise multiplication fluently to at least 12×12. the point to note is that multiplication tables, through an understanding of tables per se is never revisited further along in the grade levels in terms of their potential to connect multiplication tables with other mathematics. moreover, confining the multiplication tables to learning outcome 1  numbers, operations and relationships  is a limiting conception of the possible power of tables as bearers of mathematical information. depending on the representations of multiplication tables that are being used and accompanying investigations as suggested in the policy document, it can be shown too, that they connect with learning outcome 2 patterns, functions and algebra, for example. such investigations can take us well into grade levels 11 through 12, i.e. the further education and training band starting with the multiplication tables in the lower grade levels.  1 2 3 4 5 6 7 8 9 10 11 12 1 1 2 3 4 5 6 7 8 9 10 11 12  row 1 2 2 4 6 8 10 12 14 16 18 20 22 24  row 2 3 3 6 9 12 15 18 21 24 27 30 33 36 4 4 8 12 16 20 24 28 32 36 40 44 48 5 5 10 15 20 25 30 35 40 45 50 55 60 6 6 12 18 24 30 36 42 48 54 60 66 72 7 7 14 21 28 35 42 48 56 63 70 77 84 8 8 16 24 32 40 48 56 64 72 80 88 96 9 9 18 27 36 45 54 63 72 81 90 99 108 10 10 20 30 40 50 60 70 80 90 100 110 120 11 11 22 33 44 55 66 77 88 99 110 121 132 12 12 24 36 48 60 72 84 96 108 120 132 144 figure 1: a combined multiplication table teaching investigations that are fruitful and beyond multiplication tables imply instructing students in ways that draw attention to the myriad of patterns in the rows, columns and diagonals in figure 1 and the questions we pose about them. this deweyan idea of the “what” and the “how” relates to a current focus in mathematics teacher education research, namely “mathematics for teaching” (adler, 2005; ball & bass, 2000; ball, bass, & hill, 2004). in a sense we can think of the “what” as mathematics and the “how” as teaching. when it comes to teaching, the mathematics for teaching literature alerts us to compression and decompression or unpacking of the mathematics in figure 1. for the purposes of this paper, teaching practices refer to teaching and learning situations exemplified by the uses and non-uses of mathematical skills such as specialising, generalising, representing, problem posing and recognising and extending patterns (doe, 2003) in relation to figure 1 or otherwise stated. when teachers teach the multiplication tables, they instruct their students in various ways. some might focus on a rote memorisation while others might want their students to search for patterns with respect to odd or even numbers when they are multiplied. yet others will want their students to see beyond the multiplication and will push them to ask what happens if? or how can we generalise a diagonal or a horizontal sequence?, thereby focusing on structure in their teaching. amongst others, mason’s (1998), brown and walter’s (1990) and olivier’s (1999; 2002) work exemplify these mathematical skills. it should be noted that mathematics for teaching with respect to figure 1 can also be connected to multiplication literature and other literature on ways of connecting arithmetic and algebra. these are instances of fresh combinations. the first implies paying attention to steffe’s (1988; 1994), izsák’s (2004; 2005) and clark and kamii’s (1996) work on the importance of fostering and identifying children’s musings on multiplication tables 18 multiplication schemes, through the use of multiple representations and particular mathematical skills involving investigations. the same holds true for the second, where we bridge the “gap” between arithmetic and algebra (blanton & kaput, 2003; nunn, 1919; olivier, 1999; 2002; sfard & linchevski, 1994). for instance, it can be shown that through particular investigations, algebra can come about through the reification or generalisation of arithmetical processes noticeable in possible recursive and functional relationships of the arithmetical equivalences in figure 1. if this is done we notice the emergence of mathematical or algebraic objects at another level, namely arithmetical processes becoming compressed as symbols. such type of investigations with respect to figure 1 are important for prospective and in-service teachers because they are about ways of seeking alignment with the policy statements on learning outcomes and mathematical skills and to mathematics for teaching. concerning prospective teachers with whom i am working presently, adler (2005, p. 5) asks where and how prospective teachers are provided with opportunities for learning to unpack mathematics, and so develop pedagogically useful mathematics, i.e. mathematics for teaching. as a first step, we need to loosen up earlier and present iterations of meanings and thereby subtly question settled dogmas such as knowing one’s tables. a historical perspective on tables, multiplication tables and related teaching practices the construction of figure 1 has to be viewed in light of the historical legacy of tables as bearers of information or data in two dimensions. the construction of tables and tabular formatting occurred in ancient iraq during the second millennium before the common era. it is the best documented period of mathematics in scribal schools (robson, 2000) and thus an appropriate period to look at for the purposes of this paper, which in part focuses on school mathematics. during this period, scribes developed the “tabular account” as an efficient way of sorting, recording and storing data (robson, 2003). some of the data they worked with were in the area of metrology, e.g. weights, areas, and volumes and capacities. here they used tables to do conversions between different weight, area and volume units, for example. this is of course not the only purpose of tables. a key feature of the tabulation that evolved then is the horizontal and vertical separation of quantitative and sometimes qualitative data such as the number of sheep in a flock, or areas and capacities, for example (robson, 2003). a table thus has rows and columns that are aligned. it could also be linear, i.e. having columns only that function more like columns of a newspaper than columns of a table. in such a case there are no relationships between the information or data in rows and in columns and we have a “list.” in such a case it would not be possible to find or express numerical or other relationships between and within columns and rows. this distinction between tables and lists must be borne in mind when we examine multiplication tables in present day schools. there is more to know about multiplication lists. we now know that some multiplication or arithmetical lists during this period in babylonia were “prosaic,” or verbose, i.e. in words, while others were terse. robson (1998) gives the example of a tablet of the 2 times tables (see figure 2): obverse reverse 2 times 1 2 times 12 24 times 2 4 etc. … times 3 6 … … … … … … … … … times 30 1 00 … times 40 1 20 times 11 22 times 50 1 40 figure 2: the 2 times tables taken from the babylonian period faaiz gierdien 19 these would be the 2 times tables in the sexagesimal place value system, explaining why 2 times 40 equals 1 20! these multiplication tables are verbose because of the use of words (“times”). in terms of teaching, students in scribal schools were required to memorise a list of a standard series of multiplication through repeated copying and revision. they would do this first as individual sections and then as long extracts from the whole list (robson, 2003, p. 31), which could be up to 1000 lines in total! robson (2003, pp. 32-36) shows the example of a fifteen times multiplication table, meaning that she questions our use of table. she notes that our collective blindness to document formatting has caused us to refer erroneously to the intrinsically tabular documents as “lexical lists” in sumerian and akkadian translations for example, while their list-like arithmetical counterparts are always known as “multiplication tables” (emphasis in original). it is therefore not surprising that in some of her research she calls tables a misnomer because the scribal students were in fact memorising lists of number facts (robson, 2002, p. 361). more intriguing is that she found a strong disinclination towards a truly tabular format in school arithmetic and thus in its teaching uses. here one might have expected the proliferation of tables! she notes that mesopotamian scholastic mathematics employed tables very rarely, preferring to express arithmetical and metrological equivalences as lists (robson, 2003, p. 42). truly tabular mathematical documents at the time have their debt to administrative and astronomical practices. in other words, such documents have their rows and columns aligned, i.e. they have a two-dimensional layout or format from which data such as the visibility of planets or stars and time of year or month can be retrieved because of the two dimensional format. in her more recent work, she observes that we underestimate the invention of tables as “truly powerful information-processing tools, cognitively distinct from well-organised lists” (robson, 2007, p. 421). in this sense figure 1 is not a list but a “truly tabular table” (robson, 2003, p. 42) because the reader’s cognition or attention can be directed to the myriad of relationships between the numbers in the rows, columns and diagonals. these turn out to be of a profound nature, which we can investigate depending on the problems or questions we pose about them. for example, we can investigate the numerical information imbedded in figure 1 beyond multiplication, through the use of the mathematical skills mentioned earlier, thereby opening up the reader’s recognition as to what the relationships could be. we should be careful not to dismiss the babylonian period in terms of its school mathematics as being simply a period of multiplication lists accompanied by teaching practices that focused on a rote memorisation of multiplication facts in sequence, and a laborious copying of long lists of multiplication in verbose and terse forms. instead, it is the cognitive distinction between columnar and row organisation in the construction of figure 1, which has its precedent in old babylonian tables stemming in part from tabular bookkeeping or accounting for instance, that we should appreciate (robson, 2004). to summarise, robson’s work points us to the need to differentiate between lists and tables or list-like and truly tabular representations, and hence multiplication lists and tables. the former are onedimensional, typical of newspaper columns, whereas tables essentially have a two-dimensional layout where numerical or other type of information is stored, recorded and from whence it can be retrieved. it is with this lens that we now turn our attention to multiplication tables and related teaching practice in schools. on modal multiplication tables teaching practice in schools the format or layout of multiplication tables in schools and the associated teaching practice impact on what mathematics students learn and in turn how they learn. in most schools, multiplication tables are represented as separate albeit well-organised lists of arithmetical equivalences for the different multipliers 1 through 12. a reason for 12 may well have to do with dozens, pennies and inches. from prospective and in-service teachers’ anecdotes and what i have observed, these representations are usually in the form of wall charts, as one teacher reported during a professional development opportunity connected to a mathematics teacher education project (figure 3): musings on multiplication tables 20 figure 3: a teacher’s explanation for the 12×12 multiplication tables (mt) in classrooms such wall charts are no doubt well-organised and in sequence, meaning arithmetic equivalences or “multiplication facts” are listed from 1 through 12. this is probably no different from what robson (1998; 2002; 2003) found in the case of scholastic mathematics in mesopotamia. through systematic and careful teaching, students’ cognition could be directed towards patterns and structures that are separate for each of the multipliers and multiplicands. these would be confined to the columns that one needs to read in a vertical manner, such as newspaper columns. this newspaper-like representation in wall charts does not afford students or teachers the opportunity to make cognitive distinctions between columnar and row organisation as is possible in figure 1. in the latter it is possible to pose questions about the rows and diagonals as will be seen later on in this paper. it is the two-dimensionality of tables per se that is missing in separately listed multiplication equivalences as in the wall charts described by the teacher. what is thus evident is that the layout of multiplication tables in schools impacts on teaching. sources of evidence for this claim are my own schooling and the anecdotal accounts given by prospective and practicing teachers in the mathematics education modules i teach for the past several years. most of my current students have never encountered a multiplication table like the one in figure 1. they describe multiplication tables in schools as separate lists for each of the different multipliers 1 through 12, including zero at times. in terms of teaching practices they paint pictures of moments where they have had to sit in corners and memorise the ‘difficult’ “seven times tables” and “twelve times tables.” they were surprised to see connections between the ten times tables and the two times tables through the distributive property in the case of the twelve times tables. thus, what we have in most schools are well-organised multiplication lists in sequence as opposed to truly tabular multiplication tables such as figure 1. in our teaching practice are we insisting that our students know their multiplication tables or their multiplication lists? by examining instances of teaching practice with regards to multiplication tables we gain a clearer sense of what students, whether children and/or prospective teachers are learning and also how they are learning. figure 4 shows a reproduction of an abbreviated version of commercially available worksheets on multiplication tables aimed at the middle grade levels: 1. 3×2 = __ 26. 6×2 = __ 2. 5×0 = __ 27. 2×11 = __ 3. 4×1 = __ 28. 10×5 = __   23. 3×10 = __ 48. 5×6 = __ 24. 7×2 = __ 49. 5×5 = __ 25. 4×4 = __ 50. 3×4 = __ figure 4: an abbreviated excerpt of commercially available multiplication tables such worksheets consist of columns of two digit multiplications where the answers have to be filled in. the levels of these sheets go from a through z. a perfunctory glance at these sheets tells the reader that there is a strong focus on answers and perhaps no intellectual purpose or overall goal in terms of children faaiz gierdien 21 filling in the sheets. the powerful influence of such commercial worksheets on the practice of teachers should not be underestimated. the mathematics imbedded in truly tabular multiplication tables escapes children, in addition to teachers. as a true multiplication table, figure 1 is a means to store or compress mathematics which can then be decompressed as length quantities in the case of rectangular areas on grid paper, for example. we now turn to a case where third grade students were taught their multiplication tables in such a way. when third grade students explore multiplication tables through rectangles during the early 2000s, i experimented with the idea of using figure 1 as a means to teach the multiplication tables to a group of third grade students by using rectangles which they had to shade in on grid paper. this approach ties in with the two-dimensional layout of figure 1, i.e. rows and columns. the instruction to the third grade students was that they had to search for equivalences such as 2×12=24, 12×2=24, 3×8=24, 8×3=24, 4×6=24 and 6×4=24 in figure 1. these they then had to shade as rectangles on grid paper. these equivalences appear on the inside of figure 1 in quite a structured way. the students attended a school connected to a mathematics education project in a university town in the midwest in the united states where i taught as a full time teacher. there was pressure from the school administration and parent community to teach the students their multiplication tables through rote memorisation. the way i went about examining the multiplication tables and two digit multiplication as elaborated so far, were thus not completely sanctioned by the school administration because it was found to be different. there was also no possibility of interviews with the students. the students’ written work is therefore the main data source. to do the multiplication as shown in the equivalences, i instructed the students to shade rows and columns of unit squares or blocks as they preferred to call them on the grid paper provided, to form rectangles. the grid paper was the centimetre paper as found in burns (2000). for example, 3×8=24 meant shading a row of 3 blocks and a column of 8 blocks to form a rectangle that has 24 blocks. in other words, there is a connection between rows and columns and multiplier and multiplicand. this meant that they had to coordinate rows and columns in the way they had to do their shading. such an action connected well with the two-dimensional format of the tabular multiplication table in figure 1 which shows multiplier and multiplicand. the equivalences referred to earlier are examples of the complex numerical information in a twodimensional form or layout typical of tables, as robson (2003) points out. at the time, i became intrigued by the tabular formatting and especially the intercolumnar relationships, as revealed by the spread of the equivalences. interestingly, they lie symmetrically with respect to the square numbers, e.g. 1, 4, 9, … etc. they might well be considered as lists of multiplication facts when viewed outside the tabular multiplication table in figure 1. also, i hoped the students would recognise 8 + 8 + 8 = 24 or 3 + 3 + 3 + 3 + 3 + 3 = 24 as “repeated addition”, as they coordinated their counting of shaded blocks in the rows and columns. in the end it was hoped that they would make a transition to multiplicative thinking from additive thinking. admittedly there was nothing revolutionary about the instructing the third grade students how and what to do as they went about counting the blocks. the two-dimensional form of the multiplication table in figure 1 and its imbedded equivalences have conceptual connections with steffe’s (1988; 1994) research on tracing third grade students’ psychological structures for multiplication out of their structures for counting. for the students’ multiplication schemes to emerge, they would have to think of a number such as 16 for example, through the notion of what steffe calls composite units. a child who has formed composite units can understand the number 16 simultaneously as one group of sixteen and as sixteen individual units. a child’s multiplying scheme involves the coordination of two composite units, which the tabular multiplication table (figure 1) ought to foster in addition to my teaching. sure enough there were cases where the students had the correct rectangles shaded on the grid paper in relation to the multiplication equivalences. examples of work where students did not coordinate their shadings of the blocks or rectangles warrant our attention because they reveal a struggle in their transition from additive to multiplicative thinking despite my explicit instructions. musings on multiplication tables 22 the third grade student’s work in figure 5 shows her emergent multiplicative scheme. it comes from the case where a group of third grade students was asked to shade equivalences such as 2×8 = 16, 8×2 =16, and 4×4 = 16. figure 5: a third grade student’s transition to multiplicative thinking the third grade student’s emergent multiplying scheme is related more to additive thinking in spite of the multiplication sign. for example, 4×4 = 16, represents counting  4 + 4  which equals 8. on the right side, the counting is 8+2 (=10) and 2+8 (=10). this third grade student’s connection between multiplication and the underlying row and column structure of unit squares or blocks is not there. there was no coordination in the shading and in the counting of the blocks as composite units that yield 16. what we see is what happened despite my instructions to the students on how to coordinate the rows and columns once equivalences have been identified inside figure 1. the finding illustrated here complements research that indicates that many students in the primary grade levels, whether they are in third or fifth grade, have trouble understanding the multiplicative relation between length and rectangular area measurement. for example, izsák (2004; 2005) found that fifth grade students, when representing rectangular areas on grid paper had problems because they had to “count the squares” or the blocks as in this case. his research was on how fifth grade students used their computational knowledge of whole number multiplication and connections between multiplication and discrete arrays to construct understandings of area. numbers in figure 1 can be interpreted as composite units, counting schemes, and understandings of area. these are not the only possibilities with respect to figure 1. the table offers the potential for the gradual development of subject matter other than might naively be expected because of the numerical information presented in a two-dimensional, compact format. it is to this that we turn next. if these are multiplication tables, why are we doing addition …? this question reflects a prospective teacher’s reaction to my attempt to broaden thinking around the multiplication tables using figure 1. the question came about when the following problem was posed to a class of prospective primary school teachers in one of the mathematics education modules that i taught: is there a way to find the sum of all the numbers in the 12×12 multiplication table, and can we extend our answer to an n×n multiplication table? the fact that the multiplication tables, as represented in figure 1 could be used for purposes other than multiplication was an unusual idea for the student, hence her question. my students had been made aware of the structure and symmetry in figure 1 and were aware of its arithmetic. even so, as revealed by this student’s remark, they saw the arithmetic as fragmented, i.e. multiplication as separate from addition. my aim with the question was to seek alignment in my teaching with learning outcome 2, i.e. patterns, functions and algebra and mathematical skills such as specialising and generalising, pattern recognition and pattern extension. from previous class sessions, my students had been puzzled by the question on how one would introduce algebra in the middle grades. from a historical perspective figure 1 is a tabular representation in which arithmetical information is stored, recorded and where it can be retrieved. in particular, it can be used to investigate connections between arithmetic and algebra because of its two-dimensional format and imbedded multiplication sequences. following nunn (1919) we recognise no harsh dividing line between arithmetic and algebra: faaiz gierdien 23 the difficulty of finding a precise difference between arithmetic and algebra (as these terms are commonly understood) is well known. it is due to the fact that the distinction between them consists not so much in a difference of subject-matter as in a difference of attitude towards the same subject matter. (p. 1) the mathematical skills mentioned earlier play a key role in doing these investigations. for most of the prospective school teachers, the arithmetic in figure 1 was obvious. it is to the problem posed to them earlier that we now return. the first row of the table below shows a shortened and generalised version of the numbers in row 1 of figure 1, i.e. the multiplication sequence 1, 2, 3 … the last row in figure 6 below is the sum of the consecutive multiplication sequence in row 1 of figure 1, i.e. the triangular numbers. the point is to find the functional relationship between the sum of a particular sequence in row 1 and the indicated sum in the last row. using equivalence, a particular relationship or pattern is recognised and extended, as indicated by the arrows. the middle rows, labeled multiply, show how this pattern is recognised by using equivalence. the last column is a generalisation of this pattern. through pattern recognition, pattern extension, specialising and generalising, it is possible to rewrite the arithmetical processes in the investigation to the point where the mathematical object emerges from the generalisation: row 1 of figure 1 1 2 3 4 5 6 … n multiply 1  2 11 2  2 12 3  2 13 … multiply (equivalence)  2 2  2 3  2 4  2 5  2 6  2 7 … multiply (equivalence)  2 11  2 12   2 13   2 14   2 15   2 16  …  2 1n sum of row 1 (triangular #s) 1 3 6 10 15 21 … 2 )1(  nn figure 6: generating triangular numbers by summing the numbers in row 1 in figure 1 we would like a functional formula that will connect the numbers in the first row of figure 6 with the last row as indicated by the various arrows. it is not always easy to find a function relationship (formula) by inspection, i.e. by just “looking” (olivier, 1999, p. 2). by “filling in the gaps” such as writing  2 11 and  2 12 and the equivalences shown in the second row in figure 6, we notice an emerging structure. the general case is arrived at by examining the special case of, for example, 5×3 = 15 as highlighted in figure 6. in an inductive way we notice that the sum of the numbers in row 1 of figure 1 is: 2 )112(12 12321   ... the sum for row 2 therefore is:           2 1)(1212 2 12... 3 2 12 the sum for row 3 therefore is:           2 1)(1212 3 12... 3 2 13 musings on multiplication tables 24 the sum for row 12 is:           2 1)(1212 12 12... 3 2 112 the sum of all twelve rows therefore is:   2 2 1)12(12 2 1)(1212 2 1)(1212 12 ...321 2 1)(1212 2 1)(1212 12... 2 1)(1212 3 2 1)(1212 2 2 1)(1212 1                                                     through induction we conclude that in an n×n tabular multiplication table as in figure 1, the sum of the numbers is 2 2 )1(       nn . the latter is a particular instance of reification where we end up with a mathematical object that captures particular arithmetical processes. in this way we have established a connection between arithmetic and algebra that stems from the multiplication table in figure 1. in summary, it is difficult to claim that the object 2 2 )1(       nn is dangling in the air, disconnected from arithmetical processes, as we have seen so far.1 square numbers next, we turn to the sequence of square numbers and investigate a way to find the sum of square numbers as shaded in the diagonal in figure 7: 1 2 3 4 5 6 . 2 4 6 8 10 12 . 3 6 9 12 15 18 . 4 8 12 16 20 24 . 5 10 15 20 25 30 . 6 12 18 24 30 36 . . . . . . . . figure 7: the square numbers in figure 1 this is in line with the prospective teacher’s concern with why we were doing addition when we were focusing on the multiplication tables. the square numbers can be expressed by the object n2. we thus pose the problem: how do we find the sum of sequence of square numbers, i.e. 12 + 22 + 32 + ... + n2? it turns out the triangular numbers are helpful in terms of finding a functional relationship between the position of the square number and the sum of the square numbers, as illustrated in figure 8. 1 there are other interesting connections. for example, we know that 13 + 23 + 33 + 43 + … + n3 = 2 2 )1(       nn . so we can confidently conjecture that the sum of the numbers in an n×n tabular multiplication table is 13 + 23 + 33 + 43 + … + n3. we leave it to the reader to “see” and explain 13 + 23 + 33 + 43 + … + n3 in figure 1! and we also know that 13 + 23 + 33 + 43 + … + n3 = (1 + 2 + 3 + 4 + … + n)2 = 2 2 )1(       nn … faaiz gierdien 25 position 1 2 3 ... n triangular numbers 1 3 6   2 1nn multiply 1  3 5  6 14 multiply (equivalence)  3 3  3 5  3 7 multiply (equivalence)  3 112   3 122   3 132   3 12 n sum of diagonal sequence 1 5 14 3 12 2 )1(   nnn generalising figure 8: generating a formula for summing the square numbers by following the reasoning in figure 8, we can establish a formula for the sum of the first n square numbers:      6 121 3 12 2 1 ...321 2222     nnnnnnn further reification of arithmetical processes is possible if we ask how we can algebraify the sequence shown in the shaded diagonals symmetrical with respect to the square numbers in figure 9: 1 2 3 4 5 6 . 2 4 6 8 10 12 . 3 6 9 12 15 18 . 4 8 12 16 20 24 . 5 10 15 20 25 30 . 6 12 18 24 30 36 . . . . . . . . figure 9: the sequence of numbers below and above the square numbers position 1 2 3 4 n multiply 1×2 2×3 3×4 4×5  1 nn multiply (equivalence) 1× (1+1) 2× (2+1) 3× (3+1) 4× (4+1)  1 nn diagonal sequence 2 6 12 20  1 nn generalising figure 10: generating a formula for sequence of numbers below and above the square numbers the table in figure 10 involves recognising, describing and representing patterns and relationships where eventually algebraic language and skills emerge. is there a way we can find an object for the sum of the multiplication sequence 2, 6, 12, ... as shown in the diagonal in figure 9? what we are doing here is addition in the multiplication tables, which my students found as puzzling as the previous investigation. musings on multiplication tables 26 diagonal sequence 2 6 12 20 ...  1 nn sum of diagonal sequence 2 8 20 40 ? figure 11: seeking a formula for the sum of diagonal sequences in figure 9 what is the functional relationship between the variables in figure 11, and how do we eventually generalise this relationship? from the previous exploration we noticed that the sequence of numbers in the shaded diagonal 2, 6, 12, …, which can be expressed as ).1( nn after several attempts, we discover that the relationship is between the position of the sequence and its sum, multiplied by a triangular number, as is the case with the square numbers that we saw earlier: position 1 2 3 4 n diagonal sequence 2 6 12 20  1 nn multiply ×1 × 6 8 × 12 20 × 20 40 ? multiply (equivalence) × 6 6 × 6 8 × 6 10 × 6 12 ? multiply (equivalence) × 6 412  × 6 422  × 6 432  × 6 442  × 6 42  n sum of diagonal sequence 2 8 20 40     6 421  nnn generalising figure 12: seeking a formula for the sum of the diagonal sequence 2, 8, 20, … the arithmetical processes are reified as the mathematical object    6 42 1   n nn in the lower right cell in figure 12. through similar processes of recognising patterns and generalising, the arithmetical processes can become mathematical objects for the next diagonal sequence and thus algebra. figure 13 shows the different diagonal sequences, and related objects that describes their generalisations. diagonal sequence: arithmetic mathematical objects: algebra addition of diagonal sequence 1, 4, 9, … 2n 6 )12)(1(  nnn 2, 6, 12, … )1( nn 6 )42)(1(  nnn 3, 8, 15, … )2( nn 6 )72)(1(  nnn 4, 10, 18, … )3( nn 6 )102)(1(  nnn 5, 12, 21, … )4( nn 6 )132)(1(  nnn  figure 13: a partial list of arithmetic processes reified as algebraic objects faaiz gierdien 27 a word of caution about the algebraic objects in figure 13 is necessary. sfard and linchevski (1994, p. 191) point out that eventually all mathematical conceptions like these are endowed with a “processobject” duality. for example, the algebraic expression, say 3(x + 5) + 1 may be interpreted in several different ways, for example: o it is a concise description of a computational process, a sequence of instructions: add 5 to the number at hand, multiply the result by 3 and add 1. o it represent a certain number. it is a product of a computation rather than the computation itself. even if this product cannot be specified at the moment because the number x is presently unknown, it is still a number and the whole expression should be expected to behave like one. o it may be seen as a function – a mapping which translates every number x into another. the expression does not represent any fixed (even if it is unknown) value. rather it reflects a change, it is a function of two variables. o it may be taken at its face value, as a mere string of symbols which represents nothing. it is an algebraic object in itself. although semantically empty, the expression may still be manipulated and combined with other expressions according to certain well-defined rules. what sfard and linchevski say about the expression 3(x + 5) + 1 is applicable to the algebraic expressions in figure 13. what we have in figure 13 are illustrations of how different numbers can be thought about and represented in various ways, according to learning outcome 1 and the emergence of algebra as generalised arithmetic, according to learning outcome 2 (doe, 2003, pp. 7-9). we notice emerging mathematics that bridges obvious arithmetic multiplication tables and algebra. the reality is that, during most of my teaching, my students found the experiences of mathematical skills such as generalising, specialising, representing, pattern recognising and extending, and problem posing hard to follow, despite being told that we want to investigate ways of introducing algebra. one speculation is that their own schooling was never focused on such mathematical skills. figure 13 is a summary of the compression or reification where we end up with mathematical objects in the column on the right hand side for each of the rows. on the other hand, it shows decompression of mathematical objects into arithmetical processes in the column on the left hand side in each of the rows. implications for mathematics teacher education practice the “unpacking” and “compressing” of the rows, columns and diagonals in figure 1 have implications for mathematics teacher education practice with its current focus on mathematics for teaching. in turn these are connected to the mathematical skills mentioned in the policy documents. as adler (2005) notes unpacking and compressing are distinctive descriptions of the mathematical work that teachers do as they do their work, i.e. as they teach. how and where teachers should be afforded opportunities to learn mathematical skills that are centred on multiplication tables, which are after all, the mathematics in the foundation phase? the power of tables first, teachers should be provided with opportunities to comprehend the central message on the power of tables, starting with a historiography of tables per se. robson’s research informs us that during babylonian times the tabular arrangements of the multiplication of numbers for the purposes of accounting and other metrological activities was a significant development. additions and subtractions were carried out mentally or by other means and were never found on tablets. surveyors needed multiplication, division and standardised constants for a number of tasks. this means that knowing your tables has to be viewed from a historical perspective because of the rich uses and opportunities that tables present us. the ahistorical notion of tables is common to traditional knowledge assumptions in the mathematics curriculum and related teaching (lampert, 1986a; 1986b). it is the expression of complex information in a two-dimensional form, as in figure 1 that will forever elude us if we insist on narrow ways of thinking musings on multiplication tables 28 about tables. such views of tables will never get us to envisage how arithmetical processes such as multiplication can be reified or compressed into mathematical objects commonly associated with algebra. here are instances of packing or compressing where arithmetic becomes algebra. in short, the complexity of the mathematics in figure 1 is astonishing depending on the problems we pose regarding its rows and columns. it is the cognitive gains of the tabular format in figure 1 that is striking. it all depends on the kinds of problems we pose in order to seek possible alignment with policy statements on mathematical skills and learning outcome 2 and/or learning outcome 1. separately listed multiplication tables, albeit well-organised, do not enable us to notice significant mathematics, as we have seen. it is the extended and deepened experience with tabular multiplication tables beyond “knowing one’s tables” that is going to forge continuities in the highly fragmented mathematics curriculum. thus a general observation is that tabulation needs to be reinstated into the multiplication tables in mathematics teacher education practice, which is one potential area where teachers can encounter tabulation. this means that teachers should be exposed to multiplication table charts that exemplify tabulation where they can be provided with opportunities to search for structure and pattern, and in turn, compression and decompression. the mathematics that emerges within and across the rows and columns has a great deal to do with the practice of table making, which robson and her colleagues have written so eloquently about. we pay a dear price if we ignore the technological and conceptual advances afforded by tabulation. almost every time i have shown prospective and in-service teachers figure 1, and pointed out that there is an astonishing amount of mathematics in it, i have received puzzled looks. unpacking student work second, teachers should have opportunities to unpack student work on multiplication tables such as that of the third grade student discussed earlier. the third grade students represented the multiplication tables in ways that departed from the well-known repeated addition model by focusing on rectangular area representations, which is intimately connected to the two-dimensional format in figure 1. in their shading, they decompressed or represented multiplicative relationships in figure 1 as rectangular areas. in doing so, they had to coordinate the shading of rows and columns in line with the two-dimensional tabular format common to tables per se. the work conceptually connects multiplicative thinking and rectangular area. the third grade work, however, shows an intriguing mix of additive and multiplicative thinking despite my clear instructions. the correct rectangular area shading has deliberately not been shown. even if it were, it is the finer details of their emergent multiplicative thinking that would be of interest and that teachers need to become familiar with as steffe and izsák alert us to. we are reminded by steffe (1988, p.136) that children “must not be forced to do things that they cannot do, such as learning their multiplication facts and algorithms for computation.” for children multiplication is a compression of mathematics and also about composites, which they need to learn to represent in various ways, as suggested in learning outcome 1. how and where would teachers have opportunities for learning to unpack such student work and the representations arising from multiplication tables? what would come in the way of teachers making sense of the rectangular areas or blocks that the third grade students shaded? i frequently hear teachers say something to the effect, but the children don’t know their tables. the amount of curricular time spent on rote memorising the multiplication tables through having children fill in random lists of multiplication facts with its restricted focus on compressed, arithmetical representations underscores this point. these multiplication facts do not provide any fresh combination with the rest of the mathematics curriculum. the gains of compressing third, there are irrefutable gains in reification or compressing, where the arithmetical processes at one level become mathematical objects at another level, leading to figure 1 as generalised arithmetic, as advocated in the policy statements. these can be used to open teachers’ eyes and ears to algebra, as faaiz gierdien 29 blanton and kaput (2003) advocate. these are specific instances of “knowing mathematics for teaching.” my prospective teachers have frequently been at a loss when they were asked to investigate or simply find the algebra in figure 1. in particular, they experienced difficulty in following how equivalent numerical expressions lead to generalisation. these expressions play an important role in perceiving arithmetic as becoming algebra. single multiplication lists or facts arranged like newspaper columns will enable teachers and students to notice the compression and decompression of arithmetical processes, such as we see thus far emerging from figure 1, in a limited way. sfard and linchevski (1994) remind us that bridging the “gap” between arithmetic and algebra is a difficult one. it is the tabulation imbedded in figure 1, accompanied by the use of mathematical skills that can make the transition from arithmetic to algebra more gradual not that there will be no cognitive difficulties along the way. compressing and decompressing the numbers in figure 1 depend on problem posing one of the mathematical skills in the policy documents. teachers and children encounter isolated multiplication tables very early in their schooling, implying that making connections to other branches of mathematics becomes difficult. the one prospective teacher’s reaction, “if these are the multiplication tables, why are we doing addition,” comes to mind. this student was baffled by the idea of wanting to add up the numbers in figure 1. in summary, how and where will teachers be afforded with opportunities to pose problems on the numbers in figure 1 in various ways, as written in the policy document in learning outcome 1 which can lead to learning outcome 2? teachers and people in general, need to know the reductive status of the multiplication tables in the mathematics curriculum and hence our inability to foster epistemic openness and interpretive possibilities with respect to number. this status makes us unable to notice mathematics beyond multiplication in figure 1. one of children’s first rote memorisation experiences in school is when they have to know their multiplication tables in sequence, as was a common practice in babylonia according to robson’s research. if we wish to understand multiplication we must first understand multiplicative thinking as an instance of epistemic openness and how surprisingly slowly it develops in children (clark & kamii, 1996). like the prospective teachers, i have been drilled in a rote memorisation of multiplication lists, probably in similar ways to mesopotamian scholastic mathematics but definitely not in the same volume. robson (2003, p. 42) notes that at that time there was a preference for arithmetical and metrological equivalences as lists. most children in school are surrounded by wall charts in the form of lists of multiplication facts. she cautions us that we pay a heavy price as a result of our collective blindness to the power of tabular formatting such as we find in figure 1. it is this formatting that i wanted my third grade students and currently, my prospective teachers to notice. observing the structure in figure 1 and the problems we pose with respect to the structure, is what we need to accomplish. modal multiplication tables practice in schools is not going to get us there. conclusion there is no doubt that the mathematics that comes about in this paper is dependent on the teaching practices associated with the two-dimensional format of the multiplication table in figure 1. any severance between the mathematics and teaching practices will not enable us to perceive the depth and richness of the problems we posed and investigated, starting from the case of the third grade students to the emergence of algebra at higher grade levels. it must be noted that this paper is doubtless flawed and may even be misguided when it questions standardised tests and teaching practices that emphasise rote recall of multiplication facts and exercises where children’s ability to perform multi-digit computation problems by relying on a rote memorisation of the so-called multiplication tables. however, such teaching practices rarely arrive at the underlying structure in the tabular multiplication table we started out with. also, they are not aligned with policy statements on the mathematical skills illustrated in this paper that are crucial to knowing mathematics for teaching. it is hoped that something worthwhile can be learned from the close probing of the history of tables and in turn, a truly tabular multiplication table such as figure 1. knowing multiplication tables for the purpose of computations has an important but limiting role. it is, however, the table making and tabulation processes that we inherited from the babylonians and later civilisations that allow us to revisit a simple musings on multiplication tables 30 multiplication table such as figure 1 to search for the algebra in it and see other mathematics such as those of the third grade student. the narrow floodlight of knowing one’s table is mostly confined to computation. echoing robson, today we can bring into the open sunlight our understanding of multiplication tables to a point where we can uncover structure, abstraction, and algebra through compression and decompression, whether it is in the elementary (primary) or high school. we are in need of these and other fresh combinations. references adler j. 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can be opened with acrobat and adobe reader 5.0 and later.) >> /namespace [ (adobe) (common) (1.0) ] /othernamespaces [ << /asreaderspreads false /cropimagestoframes true /errorcontrol /warnandcontinue /flattenerignorespreadoverrides false /includeguidesgrids false /includenonprinting false /includeslug false /namespace [ (adobe) (indesign) (4.0) ] /omitplacedbitmaps false /omitplacedeps false /omitplacedpdf false /simulateoverprint /legacy >> << /addbleedmarks false /addcolorbars false /addcropmarks false /addpageinfo false /addregmarks false /convertcolors /converttocmyk /destinationprofilename () /destinationprofileselector /documentcmyk /downsample16bitimages true /flattenerpreset << /presetselector /mediumresolution >> /formelements false /generatestructure false /includebookmarks false /includehyperlinks false /includeinteractive false /includelayers false /includeprofiles false /multimediahandling /useobjectsettings /namespace [ (adobe) (creativesuite) (2.0) ] /pdfxoutputintentprofileselector /documentcmyk /preserveediting true /untaggedcmykhandling /leaveuntagged /untaggedrgbhandling /usedocumentprofile /usedocumentbleed false >> ] >> setdistillerparams << /hwresolution [2400 2400] /pagesize [612.000 792.000] >> setpagedevice article information authors: sarah bansilal1 thokozani w. mkhwanazi1 affiliations: 1department of mathematics education, school of education, university of kwazulu-natal, south africa correspondence to: sarah bansilal email: bansilals@ukzn.ac.za postal address: 8 zeeman place, malvern 4093, south africa dates: received: 20 sep. 2014 accepted: 17 nov. 2014 published: 12 dec. 2014 how to cite this article: bansilal, s., & mkhwanazi, t.w. (2014). mathematical literacy teachers’ engagement with contextualised income tax calculations. pythagoras, 35(2), art. #246, 10 pages. http://dx.doi.org/10.4102/pythagoras.v35i2.246 copyright notice: © 2014. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. mathematical literacy teachers’ engagement with contextualised income tax calculations in this original research... open access • abstract • introduction • literature review • theoretical framework • methodology • research items • analytic framework used to distinguish between levels of use of the income tax rule • results • zero level • novice level • competent level • advanced level • discussion • concluding remarks • acknowledgements    • competing interests    • authors’ contributions • references abstract top ↑ this study focuses on teachers’ engagement with tasks based on the income tax tables issued by the south african tax authorities. the participants in the study are a group of 37 teachers who were enrolled in an in-service programme for mathematical literacy teachers. the purpose of the study is to explore the teachers’ interpretation and use of the rule used to calculate income tax. data were generated from written responses of the teachers to three tasks, as well as follow-up interviews with eight of the participants. the findings indicate that some teachers (8%) did not recognise any of the demands inherent in the income tax rule that they teach to their learners. most teachers (54%) were in the novice category, showing that they met some of the demands but need some help in carrying out the rule fluently. a further 32% were able to use the rule to work out the tax given various input incomes, but could not use the rule to find the input income when given the tax output, because they did not have the necessary algebraic skill. introduction top ↑ the introduction of mathematical literacy (ml) in south africa in 2006 led to exciting opportunities for mathematics literacy educators in terms of developing new classroom materials, assessment activities, as well as research studies that focus on real-life contexts. ml as a subject includes ‘elementary mathematical concepts and skills’ (commonly understood as those mathematics domains that are studied in the get phase) (department of basic education, 2011, p. 8). it also comprises a number of contextual domains that are deemed to be useful in attaining the life-preparation goals of ml, which seek to develop self-managing persons, contributing workers and participating citizens (department of basic education, 2011, p. 8). the importance of ml lies in this life-preparation orientation, a fact that seems to be forgotten by some vocal critics, such as jansen (2011), who view ml as a watered-down version of mathematics. the subject ml aims to develop skills that will enable learners to participate in (and not be excluded from) situations that use numerically based arguments. this study set around the theme of income tax fits in very well with the life-preparation orientation of ml and can therefore illustrate some of the value offered by the subject. the purpose of this study is to explore the varying levels of engagement with a contextual rule located in the income tax domain. by looking at what we call a contextual rule (somewhat different from the usual procedures and concepts encountered in school mathematics), we hope to contribute to a more nuanced understanding of what conceptual understanding in ml could be, and how this could relate with conceptual understanding in mathematics. an improved understanding of varying levels of understanding of contextual rules will also help teachers and teacher educators understand ‘what makes the learning of specific topics easy or difficult’ (shulman, 1986, p. 9), thereby contributing to the development of our (ml educators’) pedagogical content knowledge (pck) of ml. in this study we analyse teachers’ written responses to three ml tasks based on income tax calculations. the research question that underpins this study is: how do ml teachers engage with the income tax contextual rule? literature review top ↑ the introduction of the ml in south african schools has opened up spaces for research in many areas, including studies about learners’ perceptions of and attitudes towards the subject, teachers’ perceptions of the subject, teaching practices of ml teachers, professional development for ml teachers, conceptual understanding of mathematics concepts embedded in ml as well as conceptual understanding of ml concepts. in this study, the teachers can be considered as learners in an in-service programme so we first survey some literature about learners’ understanding of ml, before moving to studies conducted on teachers. venkat and graven (2008) examined learners’ perceptions of ml in grade 10 in an inner-city johannesburg school and were able to show that highly negative experiences of learning mathematics in grade 9 had been transformed into highly positive perceptions of learning ml in grade 10. through information gathered from questionnaires and interviews with selected learners, venkat and graven were able to attribute this positive perception of ml to the opening up of learning spaces within the classroom in terms of what learners described as shifts in the nature of classroom task and in the nature of interaction in ml. in terms of the nature of classroom tasks, learners viewed contextual tasks as more accessible, practical, ‘visualise-able’ and providing openings for communication, participation and sense making inside and outside school. venkat (2010) also looked at how learners’ mathematical proficiency improved whilst engaging in ml activities. she wrote that some strands that are usually under-represented in mainstream mathematics classrooms such as ‘strategic competence, adaptive reasoning and the development of a productive disposition feature strongly in ml lessons’ (venkat, 2010, p. 66). this statement highlights some differences between a mathematics and an ml classroom. another key difference between a mathematics and an ml classroom is the role of the context. with the former, contexts are often used to illuminate the mathematics or to illustrate how the mathematics can emerge (for example drawing upon a temperature gauge to motivate for addition and subtraction rules of integers). with the latter the focus is on the interpretation and exploration of the context. studies based on the intricacies of engaging with ml contextualised tasks can help us understand some of the complexities involved. bansilal and debba (2012) carried out a study with a class of 73 grade 12 ml learners, which explored learners’ recognition, interpretation and use of various contextual attributes. the context in the study (fifa world cup) was understood in terms of attributes that provide the resources for the interpretation of the event. in general, learners found it easier to work with calculations using the contextual rules than to engage in reasoning about the rules. a common tendency of relying on their everyday reasoning instead of contextual reasoning was identified. it was also found that some learners did not understand some of the contextual language, which limited their responses to the assessment items. vale (2012) carried out a study with 43 ml learners that was also based on an assessment designed around the fifa world cup. although the context was the same in the two studies (bansilal & debba, 2012; vale, 2012), the focal events (the focus of the tasks) differed. whereas the setting for the former study (bansilal & debba, 2012) was the soccer scoring formats, in the case of vale's (2012) study, the focal event was the pricing structure of the tickets for the matches. vale found that all the participants lost marks due to a failure to decode the contextual language. there have been many studies about the professional development of ml teachers and their classroom practice (botha, 2011; hechter, 2011; nel, 2012; pillay, 2006), and fewer about ml teachers’ knowledge and engagement with specific ml concepts. bansilal (2011) carried out a study on teachers’ engagement with the concept of inflation, which used a process-object framework to understand the teachers’ varying levels of engagement with the inflation rate signifier. another study by bansilal, mkhwanazi and mahlabela (2012) focused on 108 teachers’ interpretation and use of the transfer duty rule, which is used to calculate the transfer duty payable when somebody buys a house. it was found that whilst most of the group (89%) was able to carry out the routine calculations, only half the group were able to carry out more complex calculations. theoretical framework top ↑ greeno (1991) describes a conceptual domain in mathematics as an environment with resources at various places in the domain. knowing the mathematics domain lies in the ability to recognise, find and use those resources productively. our perspective is that ml is a subject that entails the use of mathematical tools and resources together with those from the contextual domain, in order to solve problems. we first clarify our meaning of context. gilbert (2006), p. 960) explains that the latin form contextus expresses ‘coherence, connection and/or relationship’. gilbert states that the function of the context is ‘to describe such circumstances that give meaning to words, phrases, and sentences’. duranti and goodwin (1992, p. 3) use the term ‘focal event’ to identify the phenomenon being contextualised: when the issue of context is raised it is typically argued that the focal event cannot be properly understood, interpreted appropriately, or described in a relevant fashion, unless one looks beyond the event itself to other phenomena (for example cultural setting, speech situation, shared background assumptions) within which the event is embedded, or alternatively that features of the talk itself invoke particular background assumptions relevant to the organisation of subsequent interaction. (duranti & goodwin, 1992, p. 3) the context thus involves two entities: a focal event and a field of action within which the event is being embedded. in this study the focal event is the calculation of the income tax. duranti and goodwin (1992, pp. 6–8) identify four ‘attributes’ of educational contexts, which are elaborated below in terms of how they relate to this study: contextual setting: this refers to the social and spatial setting within which the interactions take place. in this study the contextual setting is that of the income tax payable by south african taxpayers. behavioural environment: this refers to the framing that establishes ‘the preconditions for coordinated social action by enabling participants’ (p. 7) to predict what is about to happen. in this study the behavioural environment is the formative assessment within which the contextual task was presented. use of language: this refers to the ways ‘in which talk itself invokes context and provides context for other talk’ (p. 7). in this study we use the phrase contextual language to refer to words or phrases that hold a particular meaning within the context. for example, ‘200 free kilometres per day’ in car hire scenarios may refer to the situation in which the contract allows you to drive up to 200 km a day without incurring additional charges and ‘win by a margin of 2 or more’ in the context of soccer matches refers to the situation where the difference between the goals scored by the winning and the losing teams is 2 or more than 2. extra-situational background knowledge: this refers to the background knowledge that extends beyond the immediate setting, which is necessary for an appropriate understanding of the focal event. in this study we define two aspects of this contextual background knowledge that are pertinent to this study. (see bansilal, 2013, as well as bansilal & debba, 2012, for other examples of contextual attributes.) firstly contextual signifiers are the signifiers used in the context to convey specific information and which have a meaning that is bounded by the parameters of the context. for example, if the reported infant mortality rate in south africa in 2012 was 42.67, this figure represents the fact that ‘the number of infant deaths during the first year of life per thousand live births’ (kwazulu-natal department of education, 2009) was 42.67. secondly contextual rules are bound to the context and need to be interpreted by the learner. in this study the rule used to calculate the income tax payable on a certain income is an example of a contextual rule. to summarise, the focal event is the calculation of the income tax and the contextual attributes are the tools and resources used to throw light upon this focal event. methodology top ↑ this study is qualitative in nature, taking on an interpretative, naturalistic approach in line with denzin and lincoln's (2008) description that ‘qualitative researchers study things in their natural settings, attempting to make sense of, or interpret, phenomena in terms of the meaning people bring to them’ (p. 4). the purpose of this study was to explore the levels of engagement of ml teachers with the income tax contextual rule. the instruments for data collection were a set of three tasks that were specially designed for both assessment and research purposes and were administered as part of a routine class assessment to the participants who were enrolled for the advanced certificate in education: mathematical literacy (ace ml). the ace programme under discussion in this article was made up of eight modules consisting of six ml-specific modules and two generic education modules. the research study was located in one of the ml-specific modules which focused on numbers and operations in real-life contexts. there were 40 teachers who completed the assessment task, but 3 did not respond to the tasks on the income tax rule; hence, there were 37 participants whose responses were analysed. from the sample of 37 teachers, 8 participants consented to our request for an interview. the purpose of the semi-structured interviews was to find out more about their reasoning and reflections that influenced their written responses. thereafter, the process of analysis was undertaken by the two researchers (authors). the analytic framework was drawn up based on the contextual attributes theory and this informed the coding of the teachers’ responses. we now present a discussion of the three research items; this is then followed by details of the categories comprising the analytic framework. research items top ↑ in this study we explore teachers’ engagement with the contextual rule used to calculate the income tax. the calcula-tion of income tax is an important ml topic that can contribute to the attainment of the life preparation and citizenship goals of ml, by contributing to an understanding of how income tax is calculated. the three tasks under scrutiny involve the income tax rule used to calculate the amount payable by any employed person. the costs that are payable, are described in different levels and can be modelled by a piecewise function, where each piece is defined by a separate rule or formula over a specified domain. luthuli (2000) wrote an account of real-life applications of such piecewise functions by describing how one could use integer-valued functions to derive formulae to describe the real-life context or problem. an example of a piecewise function that models the income tax rule (for people under 65) for the first task (as described in figure 1) can be represented symbolically by f(x): figure 1: details of task 1, with suggested solution. the contextual rule presented as the income tax tables expresses the same information using different conventions (by drawing on contextual language and signifiers). in this study, three tasks were designed to investigate the varying levels of engagement by the teachers with this contextual rule. the first task (figure 1) required them to calculate the income tax payable for an income of r150 000 in the tax year 2008/2009. the second task (figure 2) asked the teachers to calculate the income tax using the 2009/2010 tax tables and to then calculate the monthly savings with the new (2009/2010) rates. the third task (figure 3) asked the teachers what the original income would have been, given an amount of tax that was payable. figure 2: details of task 2, with suggested solution. figure 3: details of task 3, with suggested solutions. analytic framework used to distinguish between levels of use of the income tax rule top ↑ in studying the contextual attributes of the situation, we identified three demands that need to be attended to in order to use the income tax rule successfully: identifying the appropriate subset (or interval) of the domain within which the income fits, carrying out the routine corresponding to that interval and subtracting the rebate. the first demand involves being able to decode the contextual signifiers 1–132 000 and 132 001–210 000 as denoting the set of values (or interval) within which the input of the rule (income amount) fits. once the interval is chosen, the routine linked to the interval is also identified. the second challenge is to carry out the appropriate routine to arrive at a numeric answer. this requires the decoding the contextual language used to describe the particular routine; in this case it involves understanding what ‘amount above r132 000’ meant. finally, once the routine is completed, it is necessary to recognise that the contextual rule requires a further operation of subtracting the specified primary rebate from the output of the appropriate routine. linked to this demand is the understanding that for amounts less than r46 000, there will be no tax payable because 18% of r46 000 is the value of the rebate. based on the elaboration of the preceding three demands, we propose four levels of engagement, which we will use as the categories of our analytic framework. these are: no (zero) engagement with the contextual rule: this describes the attempts of a person who does not recognise the role of any of the contextual resources and does not meet any of the demands. novice engagement with the contextual rule: this describes the attempts of someone who could be meeting some but not all of the demands embedded in carrying out the contextual rule. hence, a person operating at the novice level may recognise some but not all the different roles of the contextual resources in completing the routine or rule. competent engagement with the contextual rule: competent engagement is demonstrated when a person is able to understand and identify the roles of all the contextual resources and is able to consistently carry out the income tax rule fluently and reliably, in the form in which it is presented. such a person can be described as operating at a competent level. advanced engagement with the contextual rule: engagement at the advanced level is demonstrated when a competent user demonstrates a higher or more sophisticated use of the contextual rule. this could be in a situation when the rule may have to be carried out in a different manner than the way it is presented, when the input is an unknown, when the direction of the rule changes, when one is asked questions about properties of the contextual resources involved or when one is asked to find something other than the output of the rule. it may involve manipulating the rule or performing further transformations on the rule so that it is used in a different manner than just computing the result for a given input. it may also involve comparing two rules to make an insightful judgement. at this level, the person should be able to recognise the two representations (symbolic and the table) of the contextual rule as equivalent and to derive similar algebraic representation for related rules results top ↑ the results are reported here under the categories zero, novice, competent and advanced levels. we present an example of a response that was categorised under a particular level. where possible, we also include excerpts from interviews to clarify, or to provide further insight into, the teachers’ reasoning. the teachers’ verbal and written responses are reported verbatim without any language editing. the teachers are coded by using a combination of the first three letters of their surnames together with the last three digits of their student number. this was done to preserve anonymity whilst at the same time allowing us to go back to the original scripts, if necessary. zero level top ↑ there were three teachers who were placed at this level because they were not able to meet any of the demands enunciated in using the contextual rule across task 1 and task 2. these teachers did not seem to make sense of the different steps in the calculation. they carried out calculations but these were not based on the rule. one student (mab 914) wrote: he worked out the tax for the first part of task 2 in a similar manner: note that mab 914 followed the same steps for both the tasks: he first found a percentage of the whole amount and then subtracted the first value given in the description of the particular routine (r21 960 in 2008/2009 and r43 260 in 2009/2010). he did realise that he did not understand what the phrase 25% of the amount above r122 000 meant when trying to explain why he calculated 25% of 150 000. an excerpt from his interview appears below, where r stands for researcher and t for teacher (mab 914): t: ok, i thought 150, … 150 is more than that amount r: is more than which amount? t: eh 122 r: yah, ok, if 150 000 is more than 122 000? t: yes, i must just used 25% this exchange shows that the teacher interpreted the phrase 25% of the amount above r122 000 as describing that one should calculate 25% of any amount over 122 000. he did not recognise the speciality of the rule meaning take 25% of the difference between the income amount and r122 000. this is similar to the misunderstanding of learners about the transfer duty rule (bansilal et al., 2012). however, in addition to his misconception about the phrase, he also did not understand what to do with the fixed amount of r21 960, which he subtracted from the result of his first step. a further problem was that for task 2, mab 914 used the third option instead of the second one. in fact mab 914 said ‘i even raise my hands to make clear for me only to find that the time was not enough’, revealing that he recognised that he did not understand the instruction and raised his hands for help from his tutor but the time was up for the test. mab 914 attributed some of his difficulties to the fact that he was a specialist physical science teacher (who was retraining to become an ml teacher) and when he came across contexts that were related to science, he had no difficulties. however, he felt that the specialist language used in the tax task was more suited to ‘business study be’ or ‘ economics’ people, and was different from the physical science he was used to. he expressed this by saying: some of the terminology it was the first time to come across, sometime i too lost somewhere there. … it is just my first time to come across it at tertiary in fact it contradicts the physics, the physical science. … i am a science teacher, eh, those who are doing the financials is something that belong to the other subject. his comments show that he viewed these subjects as being in different domains. novice level top ↑ we classified the responses of 20 teachers as being indicative of novice engagement. these teachers demonstrate greater understanding of the stepwise rule because they have successfully met some, but not all, of the demands, unlike the three teachers from the first category who were unable to meet any of the demands embedded in the contextual rule. seven teachers chose an irrelevant option, of whom four mimicked the worked example provided at the beginning of the question, showing that they did not meet the first demand, arising from the piecewise nature of the rule. in mathematics a function is described as piecewise when the domain (values of the independent variable) is broken down into ‘pieces’ or subsets and each set is governed by a different formula (luthuli, 2000), as in this case. these participants chose the wrong option; however, they met the demand of subtracting the rebate from the output of the routine associated with the incorrectly chosen option. the four teachers responded similarly to kon 486, whose response appears in figure 4. figure 4: response of kon 486 to task 1. there were 17 teachers who chose the correct option but erred in other ways. for example, mhl 630 chose the correct option, but did not complete the routine in figure 5 we can see that the teacher identified the second option in the rule correctly, but he did not complete the routine correctly, because he did not add in the amount of r21 960. his interview comments suggest that his approach could have been cued by the worked example, which did not require the addition of a fixed amount. furthermore, mhl 630 subtracted r8280 from the smaller amount of r7000 to get r1280. perhaps this inconsistent result led to him only doing the first step in task 2 of subtracting r132 000 from r150 000 and then stopping. figure 5: response of mhl 630 to task 1. mhl 630 explained that although he had encountered tasks based on income tax calculations in class, he did not spend enough time working on the examples. when asked why he responded as shown, he explained in a disinterested manner: ‘i primarily did, yah, i'm not sure, maybe i was following this eh this pattern’, referring to the example preceding question 1. however, he felt that the reason for ‘messing things up’ was because he did not have enough time to study: i should get enough time, eh? put myself to eliminate some commitment, other commitment, family, as you said when we beginning this course all those stuff because they've cause disturbances. in this last statement he was referring to the orientation talk in which the module coordinator pointed out that this 16-credit module required a commitment of 160 notional hours of study. mhl 630 admitted he did not put in the recommended time to study the content. eight of the teachers who were categorised in this level had problems understanding the context-specific phrase ‘25% of the amount above r122 000’. these eight teachers produced responses similar to the following: here these teachers carried out all the steps in the routine. their only problem was the misconception of the meaning of ‘25% of the amount above r122 000’. they misinterpreted it as just a description of when the rule was to be applied (on amounts above r122 000) instead of recognising the speciality of the use in the context. this misinterpretation was also identified in the response of mab 914 (on the zero level) who explained his misconception in his interview. another participant who misinterpreted the rule but carried out all the other steps correctly was ndl 627 who wrote 25% of r122 000 instead of 25% of (r150 000 – r122 000). a further two participants on this level carried out all the steps correctly except that they did not subtract the rebate amount at the end. the teachers operating on this level may need some more practice in recognising the role of the various contextual attributes. perhaps if the speciality of the contextual language is emphasised, they may improve their competence in performing the contextual rule. competent level top ↑ we classified the responses of 12 participants as demonstrating competence in executing the contextual rule accurately. these were the teachers who were able to correctly calculate the tax that was due in the two years for the r150 000, thus demonstrating fluency in the use of the rule, but they were unable to produce a solution to task 3, which required a more sophisticated use of the rule. of these 12 teachers, six worked out the answers to task 1 and task 2 completely correctly, but did not obtain the correct answer to task 3. there were two teachers who calculated the two tax amounts due for the years 2008/2009 and 2009/2010 correctly, went on to find the difference but did not calculate the monthly savings due. in this study, what is under the microscope is their use of the income tax rule, which they have been able to use correctly in two situations and which we took as evidence of their competence. there were a further two teachers who did not find the difference or the projected monthly savings. again for the same reason cited above, their responses were categorised as indicative of operating on a competent level. there were two other participants who made minor errors in one question and not the other question who were classified in this category as well. for example, mad 282 found 25% of the difference between r180 000 and r122 000 instead of the difference between r150 000 and r122 000, but her calculation for the year 2009/2010 was completely correct. a second person, mad 301, calculated the sum of r21 960 + r7000 as r28 000 instead of r28 960. she did not make a similar error for the second calculation of the income tax due for 2009/2010, so her response was also taken as indicative of operating on a competent level. advanced level top ↑ for task 3, the output of the rule (income tax amount) was given and the teachers were required to calculate the input or the income that could have resulted in the given output. there were two teachers who were able to answer task 3 correctly, demonstrating their advanced skills in operating the contextual rule. these teachers were able to successfully recognise the inverse nature of the task and to set up an appropriate equation, which took the reverse of each of the steps into consideration, and finally to solved the equation correctly. there were 19 teachers who did not recognise the inverse nature of the question, which required a reversal strategy. zon 090 was one of these teachers and her response appears in figure 6. figure 6: response of zon 090 to task 3. fortunately, she was one of the people who was interviewed, which helped provide insight into why she chose the strategy that she did. when asked by the researcher why she wrote 18% of r125 000, she replied: ‘[b]ecause the income between 0 and r132 000 is 18% of that and i used that … to calculate.’ here she was trying to explain that 125 000 belonged to the interval 0–132 000, which was why she applied the routine pertaining to the first option. the researcher then read the question again to clarify that the r125 000 was the tax and the question asked for the annual income. zon 090 then said ‘i could not calculate the annual income because we were not taught how to calculate it.’ her response suggests that she did not have the algebraic skill to transform the rule to facilitate the calculation of the income (input) given the tax (output) that was payable. the rule when used in the current form takes the input of income and calculates the output of tax payable. zon 090 has made it clear that she expected a different formula that could be used to do the calculation for task 3 and was unable to see the rule in the different way required. however, she was able to use the rule in a direct manner in order to arrive at a value for task 1 and task 2. in the case of zon 090, using the rule to find something other than the output is not feasible, because of her restricted view of the contextual rule. there were 10 teachers who recognised that the question required a reversal strategy, but they set up an incorrect equation to go about solving it. mad 301 was one of those who tried a few approaches. he first wrote: getting nowhere with this approach, (note his algebraic error of dividing 125 000 by 23 760 instead of subtracting), he then tried again: however, he did not make further progress. in his interview he said task 3 was a problem for him. when asked why, he replied: to pay r125 000 tax, it goes back to that, it's a reversal thing. … looking at the tax he or she paid r125 000 i think i was just try … here i was not sure i definitely was not sure so for me it still a problem to reverse back. his comments show (unlike zon 090) that he recognised the different nature of the question, but he found it difficult to solve. whereas zon 090 did not recognise that the solution required a reversal strategy of the same rule, mad 301 did. there were also three teachers who did not take the rebate into account before setting up the equation. an example of this is mad 282 whose response appears in figure 7. figure 7: response of participant mad 282 to task 3. there were six teachers who recognised the inverse nature of the question and tried to solve the problem by systematically reversing or undoing each step. however, none was able to reach the correct amount. an example illustrating this by mew 713 appears in figure 8. figure 8: response of mew 713 to task 3. the response by mew 713 shows that she erred by not reversing the rebate step. she first subtracted r109 260 from r125 000 (top right). she then divided by 38%, obtaining the amount r41 421, which she added to r410 000 (although her use of the equal sign was not correct). she was able to mentally undo each of the steps in the contextual rule, but she struggled to express it in correct mathematical terms. for example, she took the equal sign as if it meant ‘results in’ instead of taking it as a mathematical symbol that expresses equality between two sides of an equation. thus it seems as if she wrote (15740/38) × 100 ‘results in’ 41 421, then adding r410 000, ‘results in’ r451 421.05. this is a common misconception of the equal sign that is often displayed by beginner algebra students (kieran, 1992). however, the participant's (mew 713) steps in undoing the operation at each stage were almost correct except at the beginning where she should have initially added back the rebate, before proceeding further. discussion top ↑ the results show that for this small sample of teachers, there is a range of levels of engagement with the income tax rule. the results can be demonstrated graphically, as shown in figure 9. figure 9: frequency of responses in each category. figure 9 illustrates that most teachers (20) were in the novice category, showing that they need assistance in meeting all the demands embedded in the income tax rule. they have demonstrated that they can take account of some, but have ignored others. the most common issue was the misunderstanding of the phrase ‘25% of the amount above r122 000’; thus, interventions must help them identify the speciality of the contextual language in this situation. the phrase has two purposes: firstly, it signals when the routine should be applied (on amounts above r122 000) and secondly it details how the routine should be carried out (take 25% of [the income amount – r122 000]). most errors in this category originated from taking account of the first purpose only, and not the second. the other obstacle faced by seven teachers was not being able to identify the correct option. in this case, it may help to present these teachers with various income amounts and ask them to identify the relevant option in each case. this practice would bring to their notice the role played by the signifiers ‘1–132 000’, ‘132 001–210 000’, et cetera, as alerting them to the set of values (or interval) within which the income amount falls. the teachers with zero engagement skills require much more assistance. perhaps they would benefit from instruction revolving around a simplified or cleaned context, in which the number of intervals or subsets of the domain is reduced, so that they can first pay attention to carrying out the contextual rule in a simpler context. it is evident that they cannot deal simultaneously with all the demands of the income tax rule in its ‘uncleaned’ form, so they need help in getting rid of some of the noise so that they can understand the rule in simpler situations first. the 12 teachers performing at the competent level need to be convinced that a rule can be used for other functions besides getting an answer for a given input. the calculation of the income tax can be described by an algorithm and levels of engagement with the algorithm can be understood by drawing upon usiskin's (2012) elaboration of the five dimensions of understanding, two of which are pertinent in this study. usiskin emphasises that there are different aspects or dimensions of understanding, which are brought into play in different situations. those teachers who are on a competent level can be described as demonstrating a skillalgorithmic understanding, which involves application of an algorithm. usiskin elaborates that a person exhibits a skill-algorithm understanding of a concept when they can complete the algorithm and get the right answer. a person exhibits a higher form of this same type of understanding when they know many ways of getting the right answer (that is, they know different algorithms) and choose a particular algorithm because it is more efficient than others (usiskin, 2012). teachers operating at the advanced level of engagement in our framework can be seen as displaying an understanding similar to what usiskin (2012) describes as property-proof understanding of a concept, which is understanding that involves justification of mathematical properties that underpin mathematical procedures and relationships: for many people, understanding has a completely different meaning than obtaining the correct answer in an efficient manner. you don't really understand something unless you can identify the mathematical properties that underlie why your way of obtaining the answer worked. ‘understanding’ is contrasted with ‘doing’. (usiskin, 2012, p. 508) teachers who were able to work at the advanced level exhibited an understanding of the algebra behind the rule and how it was possible to manipulate the given form in order to work out the input amount, showing that they understood the algebraic properties of the income tax rule. those who recognised the inverse nature of the question in the third task but were unable to model the situation using an equation may benefit from working with similar situations using other rules. it is worthwhile to note that the teachers could have solved task 3 by using a systematic reversal strategy which involves ‘undoing’ each step of the rule, without having to set up an algebraic equation. there were six teachers who attempted this strategy; however, none was successful. perhaps they may have coped better with such a strategy if the contextual rule was simpler. concluding remarks top ↑ in this article we took a real-life scenario (income tax) and designed questions around it for teachers of ml who were enrolled at an in-service course. one of the aims of ml is to promote informed decision-making in real-life situations by using mathematical skills to recognise, interpret and use the contextual resources appropriately. in this case we considered the case of the income tax rule and we investigated four levels of engagement with the rule, which were the zero, novice, competent and advanced levels. we found that most of the teachers were not competent at carrying out the rule because they did not attend to all the demands embedded in the contextual rule. less than half the group were able to use the contextual rule to compute the income tax payable for particular incomes. although the third task is not one that is commonly encountered in everyday situations, it has value in a situation when a person may want to estimate the income of somebody who claims to pay such tax. in such a situation, algebra offers additional tools to reach a vantage point. however, all except two ml teachers provided evidence of the algebraic skills required to set up a correct equation to model the situation. although task 3 could have been solved by a reversal strategy, few teachers chose that route. this illustrates that the interpretation and use of some contextual rules may require sophisticated mathematical skills. ml teachers require more than just a skill-algorithmic understanding of a rule that they will be teaching their learners. in order to deal with complex applications, a property-proof understanding of the rule may serve them better as teachers (usiskin, 2012). however, in this group, there were only two teachers whose algebraic skills displayed in their responses could be indicative of such an understanding. another issue that emerges in questions set around real-life contexts is that of working within and across domains. in this study it emerged that the contextual language posed a barrier to many teachers. by not being able to decode the contextual language, they did not access the specific information directing them to when and how the routine could be executed. the implications are thus that there are two domains of engagement that intersect and one has to learn the rules of engagement of each in order to progress and to consider the more challenging questions. a limitation of the study is that the study is set within a classroom setting. we acknowledge that understanding how people engage with contextual rules in their everyday situation is a complex task and that there is a substantial difference between a real-life situation as actually experienced and one that is recontextualised into a textual representation used in a classroom teaching or assessment activity. however, this study contributes to knowledge about the use of ml assessments based on real-life contexts. we hope that by analysing the responses of teachers to these tasks set around the particular contextual rule for income tax, we have elaborated on some of the demands associated with ml tasks. lessons learnt from this particular example of a contextual rule based on a piecewise function could be extended to other similar contextual rules such as municipal tariffs, transfer duty taxes, etc. we hope that other researchers are motivated to study engagement with similar contextual rules, in a classroom or in a real-life situation, in order to extend our findings or to refute them. acknowledgements top ↑ the authors would like to thank mr e. masondo for his help in conducting the interviews. competing interests the authors declare that they have no financial or personal relationship(s) that may have inappropriately influenced them in writing this article. authors’ contributions s.b. (university of kwazulu-natal) was the project leader. the data collection was supervised by both s.b. and t.w.m. (university of kwazulu-natal). the data analysis and write-up was done by both authors. references top ↑ bansilal, s. (2011). unpacking mathematical literacy teachers’ understanding of the concept of inflation. african journal for research in science, mathematics and technology education, 15(2), 179–190. available from http://hdl.handle.net/10520/ejc92749 bansilal, s. 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(2010). exploring the nature and coherence of mathematical work in south african mathematical literacy classrooms. research in mathematics education, 12(1), 53–68. http://dx.doi.org/10.1080/14794800903569865 venkat, h., & graven, m. (2008). opening up spaces for learning: learners’ perceptions of mathematical literacy in grade 10. education as change, 12(1), 29–44. http://dx.doi.org/10.1080/16823200809487193 abstract introduction methodology philosophical orientation discussion conclusion acknowledgements references appendix 1: key terms used in this article appendix 2: representations for addition strategies about the author(s) nicky roberts centre for education practice research, university of johannesburg, soweto, south africa citation roberts, n. (2019). the standard written algorithm for addition: whether, when and how to teach it. pythagoras, 40(1), a487. https://doi.org/10.4102/pythagoras.v40i1.487 original research the standard written algorithm for addition: whether, when and how to teach it nicky roberts received: 19 apr. 2019; accepted: 12 nov. 2019; published: 06 dec. 2019 copyright: © 2019. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract this article reflects critically on the guidance offered to south african teachers in two canonical texts: the curriculum and assessment policy statements (caps) and mathematics teaching and learning framework for south africa: teaching mathematics for understanding (tmu). i make explicit my philosophical orientation, and how ‘teaching mathematics for (relational) understanding’ is evident in both documents. distinctions are drawn between strategy, representation and procedure, and the progression towards efficient calculation strategies is emphasised (neither of which is clear in the caps). the suggestion made in the tmu framework is that teachers can shift from bundling concrete manipulatives for multi-digit numbers to the standard written algorithm in grades r–3, which contradicts both the caps and insights gleaned from mathematics education literature and two learning programmes that have shown positive results at large scale. in particular, the justifications for delaying teacher introduction of ‘break up both numbers’ strategies are discussed. further, when the ‘break up both numbers’ strategy is used, alternatives to the standard written algorithm which have been found to be more accessible to learners – expanded notation, write all totals, and new groups below – are offered. ways of using empty number lines and ‘drawing numbers’ to show the 5-wise and 10-wise structure are suggested. these alternative representations for breaking up both numbers are expected to be accessible to teachers for whom the standard written algorithm is a familiar calculation strategy. keywords: multi-digit; column addition; standard written algorithm; foundation phase; south africa; primary school; mathematics; curriculum. introduction any mathematics education community (researchers, teacher educators, government officials, subject advisors, materials developers, instructional leaders, teachers, trade union officials, etc.) ought to seek out, and provide, theoretically sound guidance on how to approach the teaching of particular topics in mathematics. ernest (2014) refers to policy in mathematics education as the ‘teaching sequence for the subject as planned and experienced by the learner’ and distinguishes four aspects which are commonly the focus of mathematics education policy debates: the aims, goals, and overall philosophy of the curriculum; the planned mathematical content and its sequencing, as in a syllabus; the pedagogy employed by teachers, and the assessment system. (p. 480) the four aspects are clearly related to each other, and ought to cohere. within such mathematics communities it is expected and appropriate that there are debates, discussion and contestation over what mathematics is, why it has value, and how it ought to be taught. absence of disagreement – with various positions and their underlying rationales being openly and hotly debated – ought to be cause for concern. despite the inevitable contestation, there is simultaneously a need for ‘sufficient consensus’ to steer the way mathematics is approached in schools. in south africa the legislated policy framework provides this ‘sufficient consensus’ and is articulated through a national curriculum policy. such curriculum policy is expected to be subject to revisions over time while maintaining sufficient stability to avoid disruptions to a large, yet fragile, public schooling system. currently, south africans are guided by a national curriculum and assessment policy statement (caps) for mathematics which gives specific learning outcomes for grade r – grade 12 (department of basic education [dbe], 2012a, 2012b). more recently further guidance has been offered with the publication of a mathematics teaching and learning framework for south africa: teaching mathematics for understanding (dbe, 2018), hereafter referred to as the ‘tmu framework’. importantly, the tmu framework does not replace the caps (dbe, 2018, pp. 2, 3, 10, 12). its purpose is articulated as follows: this (tmu) framework is not a new curriculum and does not replace the existing curriculum. instead it supports the implementation of the current curriculum through introducing a model to help teachers to change the way in which they teach. the framework model and the supporting exemplars are provided to offer guidance to teachers that will enable them to transform their teaching. this transformation should lead to teaching for understanding, so that learning for understanding will take place in all mathematics classrooms in south africa. (dbe, 2018, p. 10, emphasis in original) as such at the heart of the tmu framework is an intention to support a transformation in how mathematics is taught in south african schools. from the minister of education’s perspective, the tmu framework is a contribution to ‘the urgent need to pay particular attention to the development of a new curriculum for initial teacher education, induction and continuing professional development’ (dbe, 2018, p. 3). as such it may be viewed as an attempt to forge greater coherence between the remit of the department of higher education and training (responsible for initial teacher education), and that of the various branches within the dbe (responsible for national policy relating to assessment, curriculum and teacher development) and the various provincial departments of education (responsible for the implementation of national policy at provincial level). it is worthwhile reflecting on the nature of the transformation envisaged for ‘teaching mathematics with understanding’. in this regard i draw on hiebert (1999), who summarises traditional teaching of primary arithmetic as follows: most characteristic of traditional mathematics teaching is the emphasis on teaching procedures, especially computation procedures. little attention is given to helping students develop conceptual ideas, or even to connecting the procedures they are learning with the concepts that show why they work. (p. 12) in contrast, drawing across numerous successful studies in the teaching of primary arithmetic, hiebert (1999) identifies their common characteristics, asserting that the hoped for and envisaged teaching of mathematics should build directly on learners’ entry knowledge and skills, provide opportunities for both invention and practice, focus on the analysis of (multiple) methods, and ask students to provide explanations. notwithstanding its articulated focus on ernest’s (2014) ‘aspect 3: pedagogy’, the tmu framework identifies implications that go beyond pedagogy. it proposes potential changes to curriculum, assessment, learning and teaching support materials and information and communication technology (dbe, 2018, pp. 76–79). by way of concrete example, the dbe in collaboration with the national education collaborative trust has developed a detailed learning programme (comprising learner workbooks and detailed lessons plans) as one interpretation of what is advocated for by the tmu framework. this learning programme is currently being field tested, and its outcomes are expected to inform a planned process for strengthening the caps (meeting with dbe curriculum branch, june 4, 2019). given the potential influence of the tmu framework, it is therefore imperative that both it and the caps are reflected upon critically. their similarities and differences should be noted and motivated for. this article focuses on one particular aspect of guidance offered to teacher educators and mathematics teachers in the tmu framework which is a clear departure from that which is offered in caps: the ‘standard written algorithm (swa)’ for addition and subtraction, as illustrated in figure 1. figure 1: an illustration of the standard written algorithm (swa) for addition. following fischer et al. (2019) i use the phrase ‘standard written algorithm (swa)’ which may also be referred to as the ‘traditional vertical algorithm’ or ‘condensed column method’, among others. the caps refer to ‘adding and subtracting in columns’ while the tmu framework refers to a ‘column method for recording numeric work’. in south africa, this particular aspect of the mathematics education policy debate relates to how to teach ‘context free calculations for addition and subtraction’ in the foundation phase (dbe, 2012a). the clarification notes in caps delay the teaching of the swa to grade 5 in the intermediate phase. in contrast, the tmu framework advocates for inclusion of the swa for addition much earlier – in the foundation phase (which spans grades r to 3). methodology research questions the research questions being reflected on for this article are theoretical. i therefore first make clear my philosophical orientation towards mathematics which underlies my response to two research questions: first, what does research offer in relation to whether, when and how to approach teaching vertical algorithms? and second, how do the teacher guidelines on addition calculation strategies in the caps accord with, and differ from, those offered in the tmu framework? these questions are answered in order to reflect critically on the caps and tmu framework in order to inform the expected process of strengthening caps. methods and analysis in pondering these questions i drew on mathematics education literature pertaining to learning-teaching trajectories into number and multi-digit addition to frame the article and inform my document analysis. i conducted a detailed content analysis of the two dbe documents: the caps and the tmu framework. i ‘identified visual-quantitative learning supports and written-numeric aspects’ (fuson & li, 2009) of addition procedures given in the two canonical south african texts. for both documents my primary focus was on the foundation phase (grades r–3) and in the case of caps this extended to the intermediate phase (grades 4–6). i limited my attention to ‘methods, techniques or strategies’ relating to formal written methods for addition calculations. i examined every page and took notes of multi-digit addition examples. where the same method was repeated in a document i did not include it again, as my intention was not to examine frequency of occurrence. i then compared the examples offered in the two south african documents to each other, and to the research evidence presented from the literature. on receiving feedback from the peer review process for this article, i revisited both of the canonical texts again and engaged further with the literature. in addition i met with the government officials responsible for mathematics curriculum. this meeting provided further details about the purpose of the tmu framework and its status in relation to existing policy. following this further research and personal engagements i realised the need to, firstly, clarify the conceptual distinctions i made with regard to my use of the terms ‘procedure’, ‘strategy’ and ‘representation’, and, secondly, to include a learning programme (considering textbooks, learner books and teacher lessons plans) perspective. although the theoretical findings from academic research were of value, i felt i had been remiss in not considering how a couple of large-scale mathematics improvement interventions approached multi-digit addition. in order to include some engagement with guidance relating to multi-digit addition as offered to teachers (rather than researchers), i purposively selected two well-known early grade mathematics learning programmes: everyday mathematics from the usa (see bell et al., 2015) and mathematics recovery / count me in too which originated in australia and is now used in several countries (wright, martland, stafford, & stanger, 2006). i selected these two programmes as they are well documented, have been developed by universities over decades, were widely used, and have been empirically researched to show positive results at a large scale in primary schools mathematics. see ward (2009) and riordan and noyce (2001) for everyday mathematics, and wright et al. (2006) for mathematics recovery. these are also programmes which accord with the orientation to mathematics as teaching for relational, rather than instrumental, understanding (skemp, 1987), adopted and made explicit in this article. clarifying the use of the terms ‘strategy’, ‘procedure’ and ‘representation’ there is some conceptual fuzziness with regard to distinguishing a range of terms: procedure, method, model, strategy, algorithm, representation and technique. the various interpretations of, and contestations relating to, each of these terms is beyond the scope of this article. as such, i simply make explicit the ways in which these terms are used in the south african caps and tmu framework, and then how i use ‘strategy’, ‘procedure’ and ‘representation’ to analyse the documents, in this article. the caps explain that ‘in the early grades children should be exposed to mathematical experiences that give them many opportunities ‘to do, talk and record their mathematical thinking’ (dbe, 2012a, p. 10). this distinguishes the act of doing mathematics, from orally communicating about mathematics (to self and others), and from recording mathematics. the caps uses the term ‘techniques’ which it presents as a synonym for ‘methods or strategies’ offering a list of techniques appropriate for each grade in the foundation phase (dbe, 2012a, p. 21). to me, the caps does not adequately distinguish a strategy (way of thinking), from a representation (how such thinking is recorded), from a procedure (a generalised step-by-step rule or process on how to create a particular representation to depict a particular strategy). drawing on kilpatrick, swafford and findell’s (2001) definition of mathematical proficiency, the tmu framework deliberately distinguishes strategies from procedures: procedures: the processes through which mathematics is done. much of school mathematics involves procedural working which learners need to be able to perform fluently. strategies: the approaches used to do mathematical procedures and perform mathematical calculations. learners should be able to use a variety of strategies and to devise their own strategies when they solve mathematical problems and do mathematical calculations. (dbe, 2018, p. 7) i view a procedure as a specific step-by-step process which can always be followed to implement a particular strategy for a calculation (and hence, over time, procedures can be performed fluently). procedures may be followed mentally (as internal representations) or communicated using words (mathematics talk to self or others) or by drawings, number symbols, operations, gestures and so on (and so making use of external representations). the notion of a ‘representation’ as used in mathematics education becomes important here as this draws attention to the particular way in which a procedure is recorded: as most commonly interpreted in education, mathematical representations are visible or tangible productions – such as diagrams, number lines, graphs, arrangements of concrete objects or manipulatives, physical models, mathematical expressions, formulas and equations, or depictions on the screen of a computer or calculator – that encode, stand for, or embody mathematical ideas or relationships… to call something a representation thus includes reference to some meaning or signification it is taken to have. (goldin, 2014, p. 409) thomas (2014) argues that algorithms are a type of mathematical procedure: ‘the idea of an algorithm is closely related to what, in mathematics education terms, are often called procedures, since these may be accomplished using algorithms’ (p. 37). i view a strategy as a mental process – a particular way of thinking or approach to a calculation. as such strategies are internally represented to self, and then a strategy may be externally represented to others in numerous ways. importantly, there is not a one-to-one mapping between a strategy and a representation. a particular strategy may be represented in multiple ways. for example to calculate 34 + 7 = … a child may use a ‘count on in ones strategy’, and represent this in many ways: orally, using mathematics drawings or on a number line, as shown in figure 2. how to make use of a particular representation to communicate a strategy can be broken down into a generalised step-by-step process which is referred to as a procedure. figure 2: exemplar representations and related procedures for a ‘count on in ones’ strategy for 34+7. the standard written algorithm for addition, as shown in figure 1, records the steps of a procedure, with a representation using digits arranged in base-ten place-value columns, where a strategy of ‘break up both numbers using place value’ is adopted for addition calculations. the break-up-both-numbers strategy can be represented in other ways too, and the swa is not the only representation to support this strategy. philosophical orientation when critiquing canonical texts or policy documents, the critical commentary ought to make clear its philosophical positioning. there are underlying philosophies of what mathematics is, which mathematics is worth knowing, and how one might expect children to learn. these approaches are based on values and beliefs and necessarily differ by country, by national curriculum and by individual. it is therefore worthwhile to make explicit my philosophical orientation to mathematics learning. i take mathematics learning to involve learning ways of thinking. this approach has been succinctly defined by carpenter, franke and levi (2003) as follows: learning mathematics involves learning ways of thinking. it involves learning powerful mathematical ideas rather than a collection of disconnected procedures for carrying out calculations. but it also entails learning to generate those ideas, how to express them using words and symbols, and how to justify to oneself and others that those ideas are true. (carpenter et al., 2003, p. 1) my view on mathematics learning is compatible with how mathematics is defined in the south african caps: mathematics is a language that makes use of symbols and notations for describing numerical, geometric and graphical relationships. it is a human activity that involves observing, representing and investigating patterns and qualitative relationships in physical and social phenomena and between mathematical objects themselves. it helps to develop mental processes that enhance logical and critical thinking, accuracy and problem-solving that will contribute to decision-making. (dbe, 2012a, p. 8) notice the slight shift in emphasis in these definitions. while carpenter et al. (2003) attend to mathematics learning – which is described as a way of thinking – the caps attends to the mathematics itself, acknowledging it as part of human activity which helps to develop mental processes (ways of thinking). it is worth emphasising that i take mathematics learning to involve learning powerful ideas rather than a collection of disconnected procedures for carrying out calculations (carpenter et al., 2003). my position is informed by the fact that the nature of mathematics (and what is worth learning) has changed as a result of near ubiquitous and cheap access to calculating devices (such as calculators, computers and mobile phones). this change in access to technology for calculations has implications for mathematics pedagogy: in society today, there is general acceptance that ‘drill and practice’ of taught routines will not prepare children for life in technological society and that teaching approaches need to focus on the links that demonstrate the logical structure underlying numbers and number operations. rather than being shown how to do written calculations, children are to be encouraged to work mentally to observe patterns to predict results and to talk about the connections that can be made. (anghileri, 2000, p. 2) this view has driven much of the mathematics reform agenda, as evident in the nctm standards in the usa (see, for example, kilpatrick, martin, & schifter, 2003). skemp (1987) made an important contribution to this debate by distinguishing instrumental understanding (‘rules without reasons’) from relational understanding (‘knowing what to do and why’). the former involves ‘memorising which problems a method works for and which not, and also learning a different method for each new class of problems’ (skemp, 1987, p. 159), and is about developing ‘proficiency in a number of mathematical techniques’ (p. 156). while methods (without understanding) may be potentially useful in the short term, in the longer term this is quite detrimental, and generally involves conceiving of mathematics as a set of isolated, unrelated set of techniques which should be memorised (star, 2014, p. 305). the tmu framework emphasises teaching mathematics with ‘understanding’, which could refer to either relational or instrumental understanding. however, the tmu framework refers to a purposeful move to conceptual understanding rather than memorisation of procedures in this extract: although the four key dimensions (conceptual understanding, mathematics procedures, strategic competence, and reasoning) are interdependent and should be properly linked to optimise effective teaching and learning of mathematics, it could be argued that more emphasis should be placed on conceptual understanding since this is the metaphorical foundation on which all other dimensions build. the emphasis on conceptual understanding is a purposeful move to address the common teaching and learning practice which is characterised by memorisation of mathematical procedures with little understanding of how they were derived, why they work and when they are relevant. (dbe, 2018, p. 13) further evidence of support for relational rather than instrumental understanding appears in other parts of the tmu framework too: modern societies and economies are in a constant state of flux. it is no longer sufficient for learners only to learn how to reproduce the steps in the calculations that they are shown by teachers. (p. 15) if children learn procedures without understanding, their knowledge may be limited to meaningless routines. (p. 16) mathematics is not simply a collection of isolated procedures and facts; it consists of a web of interconnected concepts and relationships. if learners are taught mathematics as a series of disconnected procedures that need to be learnt off by heart, they are likely to experience mathematics as meaningless. it will also mean that they have more to memorise which deprives them of the opportunity to develop higher order thinking skills. (p. 18) these quotes reveal that the tmu framework hopes to contribute to a shift away from reproduction of steps to calculations shown by teachers and meaningless routines, towards mathematics as a web of interconnected concepts and relationships. the attempt to shift away from learning mathematics as instrumental understanding is not new in south africa. spending significant time on mastering taught written routines for quick and accurate calculation in school mathematics has been questioned for over three decades. olivier (1992) argued that ‘when the emphasis is on promoting understanding of number and algorithmic thinking, and not on speed and accuracy, the standard contracted and refined algorithms should be abandoned’ (p. 217). this forms part of global debates on whether to include written algorithms at all in mathematics curricula. kamii and dominick (1997) made a strong appeal ‘that the time has come to stop teaching the algorithms and, instead, encourage children to make the mental relationships necessary to build number sense’ (p. 60). i do not share kamii and dominick’s (1997) view that teaching written algorithms is irrelevant and should not be part of primary curricula. in this regard i follow kilpatrick et al.’s (2001) conception of having five strands of mathematical proficiency: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning and a productive disposition. this framing recognises that procedural fluency is a valuable strand of mathematical proficiency. my reasons for adopting this orientation are varied. firstly, i concur with olivier (1992) that promoting procedural fluency in the use of the standard contracted and refined algorithms (swa) should be abandoned. rather, olivier advocates that the learning of calculation should promote understanding of number, algorithmic thinking and cultural transmission. i also think there is intrinsic value in studying a variety of algorithms and socio-cultural mathematical artefacts developed in particular contexts for particular periods of time. this value has been strengthened with the future-looking focus on coding and the ‘algorithmics’ which underlies artificial intelligence. secondly, having a reliable ‘go-to’ algorithm for enacting a particular calculation strategy (strategic competence) frees up time and attention for other important mathematical processes such as adaptive reasoning, proof and explanation. i emphasise that i do not consider the ‘go-to’ algorithm to imply that the same strategy and representation must be used by all children in a class. i expect that children have their own go-to algorithm which is appropriately efficient for their grade level. thirdly, if we consider teachers to be products of their own mathematical learning experiences (see roberts, 2017), then the mathematical procedures that teachers have already automated is a resource which they bring with them into their own (re-)learning of mathematics. offering expanded written methods using columns (such as the ‘write all totals’ algorithm discussed below) would build directly on south african teachers’ existing knowledge and skills about column algorithms. discussion what does research offer in relation to when, and how, to approach teaching swa? i identified what, in my view, are fundamental insights about the developmental progression of young children’s sense of numbers from counting to efficient calculation strategies. each key insight is briefly discussed in turn, drawing on the literature informing it. i first consider the various strong justifications to delay the introduction of the swa as a taught way of recording the ‘break-up both numbers’ strategy. i then discuss the important considerations on how to use columns methods (of which swa is a very condensed form) when a ‘break up both numbers’ strategy is used. insight 1: the strategies that children use for single-digit addition and subtraction are well documented. researching young children’s thinking in a project referred to as cognitively guided instruction, carpenter and fennema (1999) identified three calculation strategies commonly used for solving additive relation problems and also considered these in a hierarchy of mathematical sophistication: direct modelling, counting, and calculating. direct modelling refers to the use of concrete apparatus such as manipulatives (like counters, or fingers) to enact a situation that closely resembles the problem situation. by counting, carpenter and fennema refer to strategies that make use of unit counting to calculate. for ‘counting in ones’, calculation strategies vary in levels of sophistication: count all. count on (from the first number, and then from the larger number). count up to reach a target. in this case counting strategies refer to unit counting, rather than counting in groups (such as twos, fives or tens). calculating refers to more sophisticated strategies which do not use unit counting. calculating strategies may use: counting in groups or counting on in groups (using the medium and large number sequences of counting in tens and hundreds, from any number). building on known facts (often knowledge of bonds of five and ten to ‘fill up or make tens’ and doubling or halving). the relationship between the numbers in the calculation for solving. all of these single-digit calculation strategies are also used for multi-digit calculations. at first learners need to ‘fill up the ten’ for calculations resulting in a solution that is more than ten. later the same strategies are used to fill up any multiple of ten. a ‘make a ten’ (or multiple of ten) strategy is included in ‘building on known facts’. knowledge of the bonds of ten (the whole number pairs which sum to 10: 1 and 9, 2 and 8, 3 and 7, etc.) is therefore central, as is knowledge of breaking down single digit numbers. using ‘the relationship between numbers’ includes solving a subtraction calculation using an unknown addend. so 9 – 7 = …, is solved as 7 + … = 9. here any addition or subtraction fact is seen as belonging to a family of equivalent number sentences. in line with this trajectory, fuson and li (2009) provide levels for the counting pathway into number making use of: count all, count on, ‘make a ten’ and doubles as key methods. it is important to notice that these trajectories do not expand on the shift into formal written calculations. their focus is primarily on the early grades (r–3). insight 2: children’s shift from counting in ones to efficient calculation strategies takes years efficient calculation strategies – which include use of formal written algorithms – take more than 3 or 4 years of formal schooling. from the netherlands, van den heuvel-panhuizen (2008) provides a ‘learning-teaching trajectory’ of young children’s likely progression for whole number calculation, but this trajectory makes specific reference to the expected progression towards more structured representations of number and additive relations. structured representations make use of quick-5, quick-10 and quick-100 arrangements, which refer to drawings or arrangements in which the objects or markings are spatially arranged so that it is quick and easy to identify groups of 5 or 10 or 100. importantly such a learning-teaching trajectory is not seen as strictly linear but includes discontinuities, individual differences in learning processes and so each level has a ‘certain bandwidth’ (van den heuvel-panhuizen, 2008, p. 13). working within this dutch tradition, treffers and buys (2008) provide attainment targets which are viewed as ‘benchmarks’, or reference points against which the development of children can be assessed. the teaching-learning trajectory in table 1 outlines 10 broad levels of development from counting to calculating, adapted from treffers (2008a), treffers (2008b) and treffers and buys (2008) in van den heuvel-panhuizen (2008). table 1: treffers and buys’s trajectory from counting to calculating. the treffers and buys (2008) framework (table 1) is a coarse-grained delineation of development levels which spans 7 years. when focusing on two-digit addition, levels 6–10 in the treffers and and buys (2008) framework are relevant. building on this tradition, kühne, lombard and moodley (2013) developed a ‘learning pathway for number’ for the south african context which reduced the 10 levels to 4 stages spanning from prior to schooling to grade 5. in its summary form, the ‘learning pathway for number’ is primarily organised by an increasing number range. however, in its detail it refers back to the treffers and buys framework and its reference to structuring (kühne et al., 2013, p. 83). this is not the first, nor the only, learning-teaching trajectory for number concept development. for example, murray and olivier (1989) formulated a theoretical model describing four increasingly abstract levels of types of computational strategies with two-digit numbers in a given range: count all, counting on, decade wise counting (the medium number sequence), but with a multi-digit conception, and condensed column methods (pp. 5–7). in addition to supporting the global evidence that the shift from counting to calculating takes time (as evident in their south african empirical data), murray and olivier make an important observation: ‘level 3 understanding provides sufficiently powerful computational strategies, so that the introduction of the standard written algorithms may be delayed, if they should be taught at all’ (p. 9). insight 3: as south african children learn base-ten place value in a multilingual context for number names, deep conceptual understanding may take additional time the south african context is important here: most foundation phase children learn mathematics in african languages and are introduced to english as an additional language. starting in grade 2 children are expected to count orally with meaning up to 100, and to write the number symbols up to 100. counting orally with meaning up to 100 means that children have to create an association between five aspects of each number: the visual stimulus, the chain of sounds for the spoken number word, the ten-wise place value structure of the number words, the written number words, and the number symbols (see fuson & li, 2009; fuson et al., 1997). they then change to learning mathematics in english in grade 4. figure 3 illustrates the five aspects of the number 23, in the south african language context, where an african language (such as isixhosa or sesotho) as well as english proficiency with multi-digit numbers is required. figure 3: five aspects of oral counting multi-digit numbers with understanding. using multi-digit number names with meaning requires children to notice word parts and associate these with decades and ones (fuson & li, 2009). the number words for a multi-digit number are linked in a chain of sounds. in english, the number symbol ‘34’ is said and heard as a continuous chain of sounds: ‘thir-ty-four’. this often leads to children linking (concatenating) the number symbols when writing multi-digit numbers. so thirty-four is written incorrectly as ‘304’. place value is part of the how numbers are named in some african languages (as with mandarin). this is not the case for english. when teaching in english, children must first learn that ‘eleven’ means ‘1 ten and 1’, and twelve means ‘1 ten and 2’ etc. in english the base-ten place value structure of numbers is not explicit in the english number names. the words ‘three-tens-four’ would be more regular and transparent than ‘thirty-four’. similarly, ‘one-ten-three’ would be more regular and transparent than ‘thirteen’. becoming secure with these five associations between the visual stimuli (of the real or imagined aggregate, the written number word, the written number symbol) and oral stimuli (of the chain of sounds for the spoken number name) takes time. when reading and writing is not yet fluent this is more difficult. when two languages are at play, additional time is required. all of these associations – together with fluency in reading and writing number symbols and operators – are prerequisites for formal written calculation methods. as a result, the introduction of formal written methods ought to be delayed until use of written number symbols and spoken number names is secure. insight 4: the standard written algorithm has been found to be error prone and to support an incorrect ‘single-digit-conception’ of place value there is much research that coheres on the finding that using a column method for addition and subtraction encourages a digit-wise conception rather than a multi-digit conception of large numbers. fuson et al. (1997) refer to a ‘concatenated single-digit conception’. the term ‘concatenate’ derives from the latin ‘con-’ (link) and ‘catenare’ (chain). in this article i use the simple phrase ‘digit-wise’ to refer to ‘concatenated single-digit’. with a digit-wise conception 234 is conceptualised as ‘two-three-four’ where each digit does not have its actual value. with a multi-digit conception, the magnitude of the number is explicit and 234 is conceptualised as ‘2 hundreds 3 tens and 4 ones’. various studies have shown that children may have an adequate multi-digit conception which they use for addition and subtraction calculations that are presented horizontally or in word problems. however, when presented with the same calculation vertically, they use a digit-wise conception and make errors (fuson et al., 1997, citing cobb and wheatley, 1988, davis, 1984, and unpublished south african data from murray). these findings cohere with the low success rates in using the swa evident among south african children (murray & olivier, 1989). common errors in using the column algorithm are also documented by kamii and dominik (1997), including the misconception of digit-wise conceptions of number for calculating highlighted by fuson et al. (1997). to avoid the digit-wise misconception, calculation methods for arithmetic emphasise a holistic concept of the numbers, where ‘numbers are kept as wholes in any partition’ (p. 41). as a result, in the early grades teachers are advised to avoid procedures that treat individual digits within a number independently. this caution applies to the swa where individual digits are treated independently. olivier (1992) argued that the swa is prone to error ‘because the techniques are difficult to understand relationally’, explaining that regrouping (at times referred to carrying or borrowing) is probably the source of most errors with swa (p. 217). in their small-scale study of 21 grade 4 students, flanders, torbeyns and verschaffel (2013) found greater speed and accuracy when the written algorithm was used in comparison to mental computation. significantly, this study was conducted with fourth graders (beyond foundation phase), and within an instructional culture ‘wherein routine mastery of written algorithms was emphasised’ (p. 139). in contrast, a recent study in france used a single subtraction word problem but had a larger sample of 2 619 grade 2 learners (fischer et al., 2019). it found that the column method was the seventh most accurate out of the 11 strategies and associated representations used by the children. the authors found that ‘a great number of students applied blindly the column procedure’ with the most successful students using multiple procedures. as a result, fischer et al. (2019) argued for ‘delayed teaching of the written algorithm’ (p. 13). i therefore note that the swa in the early grades (grades r–3) has been found to encourage the incorrect digit-wise conception of the place value of multi-digit numbers and has been found to be error prone. retaining numbers as whole numbers (such as 37 being 30 and 7) and not as digits (3 tens and 7 ones) in any partition is encouraged to avoid the digit-wise conception of multi-digit numbers. insight 5: cognitive load is an important consideration when reflecting on ways of recording the ‘break up both numbers’ strategy. given the above insights, it is not surprising that if it is to be used, the introduction and development of swa needs to be slow and incremental to ensure that a robust conceptual understanding of place value is first established. several alternative representations to record a ‘break up both numbers strategy’ appear in the education literature. some of these make reference to manipulatives or concrete materials. shifts towards arranging discrete objects or using structured materials in 5-wise and 10-wise groups are encouraged (roberts, 2015; saka & roberts, 2018). in my experience, particularly with an urban small class setting, disruptive learner behaviour was a significant feature of the mathematics classroom (see roberts & venkat, 2016). using manipulatives in very large classes (of 45 or more) would also be unfeasible. the same was evident in the usa, where in a foundation phase intervention, fuson and li (2009) found that the use of manipulatives (bottle tops, dienes blocks, counters) created classroom management issues and were difficult to show to a whole class to explain multi-digit addition and subtraction. they therefore turned to using concrete materials only initially (to make 10-sticks or strips and 100-squares on centimetre dot grids). this initial introduction soon moved into making sketches of quick-10s and quick-100s using 5-wise groups so that a viewer could see quickly (at a glance) how many there were in a drawing, as shown in figure 4. figure 4: mathematics drawings as a representation of place value. futon and li’s (2009) argument and suggested structured drawings draw on a review of asian texts (chinese, korean and japanese) as well as their empirical research with children using the math expressions texts in the usa. fuson and li (2009) offer various written methods for formal written calculations, but argue that the ‘new groups below’ and ‘write all totals’ (with its horizontal variation of ‘expanded notation form’) representations shown in figure 5 are the most mathematically desirable and accessible addition methods. figure 5: mathematics drawing and variations on swa. rather than have children overload their working memory, teachers should encourage children to: record intermediate steps (rather than expecting these to be held mentally) work flexibly in terms of either from ‘right to left’, or ‘left to right’ use structured drawings, together with expanded written methods. methods that record intermediate steps (rather than expecting these to be held mentally), and that allow for flexible working in terms of from right to left, or left to right, are advocated for. insight 6: mathematics recovery and everyday mathematics both introduce multi-digit addition with invented strategies, then using a ‘break up the second number’ strategy (and only everyday mathematics offers expanded written methods for communicating the ‘break up both numbers’ strategy). the two learning programmes reviewed, which have shown positive effects at a large scale, both prioritise learner invented strategies when operations are first introduced. they then guide learners towards multi-digit addition using a ‘break up the second number’ strategy. in the mathematics recovery learning programme, wright et al. (2006) make clear the prerequisite knowledge required for a child to be able to add and subtract multi-digit numbers, including: counting by tens off the decades (counting using the medium number sequence from any number), counting by hundreds off the hundreds (counting using the large number sequence from any number), and adding tens to any two-digit number. their teaching activities make use of a ‘break up the second number strategy’ which can be represented on an empty number line or using bundling sticks (manipulatives in ones, tens and hundreds). wright offers no teaching activities for the ‘break up both numbers strategy’. the authors of everyday mathematics are explicit in describing the first introduction to algorithms as an ‘invented procedures phase’: ‘when they are first learning an operation, everyday mathematics students are asked to solve problems involving the operation before they have developed or learned systematic procedures for solving such problems’ (bell et al., no date, p. 4). in the everyday mathematics learning programme, the introduction of written algorithms for addition and subtraction are introduced in grade 2, but proficiency in the use of at least one algorithm for each operation is only expected in grade 4. the swa is not taught (although it may be introduced by children, via their parents). interestingly, after its first few iterations, the authors of everyday mathematics introduced a common, widely accessible algorithm for each operation and considered this to be an important feature for classroom discussion. while children should invent and use a wide range of strategies, they chose to ensure that all children demonstrated familiarity with the ‘write all totals’ algorithm as its focal algorithm for addition (see figure 6). figure 6: example of ‘write all totals’ or partial sums algorithm in everyday mathematics. notice that ‘write all totals’ is a four-step procedure (add the 100s, add the 10s, add the 1s, add partial sums) which is recorded using columns. a digit-wise conception is avoided, as the 2 hundreds + 4 hundreds is recorded as 600 (and not as 6 in the hundreds column). similarly 60 + 80 is recorded as 140 and not as 1 hundred, 4 tens and 0 ones. this representation in columns also allows learners to work either from left to right (starting with the 100s), or from right to left (starting with the 1s). bell et al. (2015) note that children are inclined to work from left to right (when not directed by teachers to work from right to left), and that allows children to ‘begin the problem-solving process with a reasonable estimate of what the final answer should be’ (p. 8). this point is identified by kamii and dominick (1997) when they strongly asserted their rejection of algorithms: algorithms are harmful to children’s development of numerical reasoning for two reasons: (a) they ‘unteach’ place value and discourage children from developing number sense, and (b) they force children to give up their own thinking. children’s natural way is to think about numbers from left to right. however, algorithms require them to give up this thinking and to proceed from right to left and to treat each column as ones. (p. 58) everyday mathematics also offers another variation on the swa (but not as its focal algorithm) but all addition of partial sums precedes any exchanges. see figure 7. figure 7: everyday mathematics variation of the swa. the procedure here also allows flexible working from left to right, or from right to left. notice here that there are two important reductions of cognitive load (compared to the swa). firstly, all addition is done before any exchanges take place. so rather than alternating processes (add then exchange as needed) the child can focus first on adding, and only later on exchanging. secondly, the way of recording exchanges is documented in the relevant column to try and avoid a digit-wise conception of place value (so 16 ones is recorded in the ones column, before it is exchanged to be 1 ten and 6 ones). how do the teacher guidelines on addition calculation strategies in the caps accord with, and differ from, those offered in the tmu framework? as a result of the above distinction between strategy and representation, when analysing the caps and tmu framework, strategies for addition were distinguished from the various representations that could be used as means of communication, as shown in table 2. table 2: strategies and representation for addition in caps. the addition strategies (and their related grade progression) offered in the caps are shown in figure 8. figure 8: grade progression for addition calculation strategies in caps. the strategies, and the related progression, in caps are recognisable from the mathematics education literature. there are three important specifications relating to the expected progression for addition and subtraction techniques which are clearly stated in the caps and then exemplified in the clarification notes offered to teachers: adding and subtracting are considered together – with emphasis placed on using these operations as inverses. as such the structural relationship between a number triple (like 5-3-2) is made explicit. teachers are expected to emphasise that subtraction can be checked using addition (5 – 3 = …, can be considered as 3 + … = 5). there is therefore an emphasis on families of related number sentences for each addition or subtraction fact. in foundation phase ‘break down the second number’ is a strong focus as this can be utilised for both addition and subtraction without the need for regrouping. adding and subtracting in columns is delayed to grade 5. in term 1 a ‘write all totals’ representation for the ‘break up both numbers strategy’ is presented. from term 2 onwards this is complemented by a swa. more detail about the caps suggested progression for addition and subtraction strategies is evident in the clarification notes for teachers. figure 9 documents the addition strategies presented in the foundation phase (dbe, 2012a) and intermediate phase (dbe, 2012b) caps clarification notes by grade. to facilitate comparison, i have included the page numbers in the tmu framework that depict the same strategies (dbe, 2018). figure 9: addition strategies presented in caps clarification notes and the tmu framework. it should be noted, however, that progression in relation to the strategies for addition and subtraction (and the flexible use of various representations to record these strategies) is not made explicit in the caps document. it only becomes apparent when the entire document is analysed in relation to the expectations at each grade level. when comparing caps to the tmu framework it should be noted that the latter only offers a limited number of exemplar tasks. the examples chosen for foundation phase relate to addition and subtraction of whole numbers, whereas the examples chosen for intermediate phase relate to rational numbers (common fractions in particular). it is therefore to be expected that the tmu framework is less comprehensive than the caps. it is nevertheless instructive to note which example types were omitted from the tmu framework exemplars. the five main strategies in the caps each feature in the framework. the particular strategies not referred to in the tmu framework are made explicit in table 3. table 3: appearance of foundation phase strategies in caps and tmu framework. the tmu framework has much in common with the caps. by way of example, both the caps and the tmu framework recognise the need for learners to use any strategy, but that teachers should encourage them away from unit counting, towards more efficient strategies, and ways of recording calculations. the tmu framework offers an example of 29 + 15 = … where learners are expected to ‘use any method’. it also includes examples of ‘break up both numbers’ strategy using manipulatives (p. 26 and p. 27), ‘break up the second number in expanded notation’ strategy recording this using number sentences and on a number line (p. 27), and ‘break up the second number’ to fill up the tens, recording this on a number line (p. 28). a major distinction between the tmu framework and caps is the suggestion in the tmu framework for teachers to introduce the swa in the foundation phase (grades r–3). in the tmu framework the swa is offered as a way to record 34 + 37 = … using a ‘break up both numbers’ strategy (shown in figure 10). figure 10: an example of swa to solve 34 + 27 = … in the foundation phase exemplars of the tmu framework. this differs from the guidance offered to teachers in the caps, where any use of written methods in columns is delayed to grade 5 (dbe, 2012b, p. 157). it is only in term 3 of grade 5 that the swa appears in the caps guidelines for the first time, as shown in figure 11. figure 11: first example of swa in caps for intermediate phase. in addition, in the tmu framework the swa is the only representation for the ‘break up both numbers’ strategy, and no examples of expanded written methods are offered. as such the tmu makes a major departure from caps when it suggests that teachers should make use of concrete apparatus to work with ‘breaking down both numbers’ into expanded notation in order to introduce the swa in foundation phase. conclusion i have shown that both the caps and the tmu framework emphasise the need to teach mathematics for understanding. i have argued that the desired understanding is relational understanding (‘knowing what to do and why’) not instrumental understanding (‘rules without reasons’). i have argued that this is in line with the mathematics reform agenda where the desired transformation of mathematics teaching is away from teaching meaningless procedures towards teaching mathematics that builds directly on learners’ entry knowledge and skills, provides opportunities for inventions and practice, focuses on the analysis of (multiple) methods, and expects learners to provide explanations. i point out that the caps does not clarify the distinction between strategies, representations and procedures. i suggest, drawing on the tmu framework, that distinctions between a strategy (way of thinking), a representation (how such thinking is recorded), and a procedure (a generalised step-by-step rule or process on how to create a particular representation to depict a particular strategy) ought to be made explicit. i also note that the expected progression in relation to the strategies for addition and subtraction (and the flexible use of various representations to record these strategies) is not made overt in the caps document and is only evident when a grade by grade level analysis of strategies is conducted. i make explicit my belief in the value of inculcating both conceptual understanding and procedural fluency. i emphasise that this does not mean i think algorithms should be the starting points of mathematical learning. nor does my belief imply that the elegant and formal swa ought to be a desired destination for all children. in fact, we know that the highly condensed swa was used in a time when accuracy and efficient were most prized. in our current context – with ubiquitous access to calculators (in computers and mobile phones) – efficiency and accuracy are not as highly valued. the process of inventing, refining and reflecting on algorithms coupled with the ability to communicate with others about multiple strategies, and various ways of representing these, is of far greater importance and value. it is for this reason that i attend carefully to how i think formal written algorithms ought to be introduced to young children, and why i attend to various options for recording these (likely to suit a wider range of children), which i present in appendix 1. we know from both local and international literature that early introduction of the swa for addition and subtraction supports the incorrect digit-wise conception of place value for multi-digit numbers. we also know that deep conceptual understanding of place value is predictive of future mathematics attainment, that there are five correct conceptions of place value, each of which takes time to develop, and that the development from rote counting to efficient calculation takes about 7 years. the south african curriculum (caps) seems to build on this literature as it encourages a slow progression in relation to the development of the place value concept and progression with regard to efficiency of calculation strategies. in contrast, the tmu deviates from both the caps and the international literature. it proposes that the swa for addition can be modelled using manipulatives, and then used as a written method with digit-wise tens and ones columns in foundation phase. in addition, the tmu does not include the expanded column methods (such as ‘write all totals’ – either horizontally or vertically, which appear in caps) as possible alternatives to the swa. the expectation created in the tmu framework is that teachers can shift from bundling concrete manipulatives for multi-digit numbers to the swa in grades r–3. this contradicts both the caps and insights reported on above. the tmu framework is also discordant with the two learning programmes – everyday mathematics and mathematics recovery – where addition is introduced using a ‘break up the second number’ (which can be applied to subtraction without the need for regrouping or exchanges) before the ‘break up both numbers’ strategy is introduced. ‘break up both numbers’ is considered in everyday mathematics. when ‘break up both numbers’ is introduced, the swa is avoided and the expanded method of ‘write all totals’ is the focal algorithm (to be understood but not necessarily used by all learners). given how much conceptual knowledge for place value is to be constructed – and its critical importance in mathematical progression – the move from concrete manipulatives to the swa suggested in the tmu framework is too rapid. the introduction of the swa in foundation phase, when it is only in grade 2 that notions of place value are beginning to be constructed by learners, is too soon. this creates a lack of clarity in our policy landscape, which is only partially clarified by the repeated assertion that the tmu framework does not replace caps. so, if asked whether, when and how to teach the swa, i would respond as follows. while some mathematics educators reject the teaching of swa in its totality, i think exposure to algorithms in their historic context is worthwhile. as the swa is familiar to south african teachers, i think it is worthwhile for children to understand it, without emphasising speed and accuracy. however, i do not think it should be taught in foundation phase. i concur with the caps that the swa should only be introduced in grade 5 when place value in two languages is secure. in foundation phase i would teach multi-digit addition by first encouraging children’s invented strategies, then offering structured mathematics drawings and expanded methods. i would use ‘write all totals’ as a ‘go-to’ algorithm for the whole class. only once ‘write all totals’ is secure would i introduce the swa as an optional condensed method, which was used at a particular time in history, when speed and accuracy were valued. acknowledgements this article results from a symposium at the national science and technology forum in august 2018. i wish to thank all the mathematics education colleagues who took part in the robust debates at, and following, this symposium. in addition, i received invaluable critical feedback, which substantially changed this article, from colleagues within the curriculum unit of the dbe, and academic colleagues who reviewed the first version of this article. discussion with international colleagues emanating from an oral communication on this issue at the psychology of mathematics education (pme-43) conference was also very helpful. all of these role players share a common concern for improving mathematics in south africa which was evident in their critical engagement and generosity in sharing ideas. competing interests i declare that i have no financial or personal relationships that may have inappropriately influenced me in writing this article. authors’ contributions i declare that i am the sole author of this article. ethical consideration no children were involved in collection of empirical data for this article; therefore no ethical clearance was applied for. funding information this research received no specific grant from any funding agency in the public, commercial of not-for-profit sectors. data availability statement data sharing is not applicable in this article as no new data were created or analysed in this study. disclaimer the views and opinions expressed in this article are those of the authors and do not necessarily reflect the official policy or position of any affiliate agency of the authors. references anghileri, j. 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(2014). instrumental and relational understanding in mathematics education. in s. lerman (ed.), encyclopedia of mathematics education (pp. 304–307). dordrecht: springer. https://doi.org/10.1007/978-94-007-4978-8_79 thomas, m. (2014). algorithms. in s. lerman (ed.), encyclopedia of mathematics education (pp. 36–38). dordrecht: springer. https://doi.org/10.1007/978-94-007-4978-8_8 torbeyns, j., & verschaffel, l. (2013). efficient and flexible strategy use on multi-digit sums: a choice/no-choice study. research in mathematics education, 15(2), 129–140. https://doi.org/10.1080/14794802.2013.797745 treffers, a. (2008a). kindergarten 1 and 2 – growing number sense. in m. van den heuvel-panhuizen (ed.), children learn mathematics: a learning teaching trajectory with intermediate attainment targets for calculation with whole numbers in primary school (pp. 31–42). rotterdam: sense publishers. treffers, a. (2008b). grade 1 (and 2) – calculation up to 20. in m. van den heuvel-panhuizen (ed.), children learn mathematics: a learning teaching trajectory with intermediate attainment targets for calculation with whole numbers in primary school (pp. 43–50). rotterdam: sense publishers. treffers, a., & buys, k (2008). grade 2 (and 3) – calculation up to 100. in m. van den heuvel-panhuizen (ed.), children learn mathematics: a learning teaching trajectory with intermediate attainment targets for calculation with whole numbers in primary school (pp. 61–88). rotterdam: sense publishers. van den heuvel-panhuizen, m. (2008). children learn mathematics: a learning-teaching trajectory with intermediate attainment targets for calculation with whole numbers in primary school. rotterdam: sense publishers. https://doi.org/10.1163/9789087903954_004 ward, w. (2009). the influence of a reform-based mathematic program on third, fourth, and fifth grade student achievement. unpublished doctoral dissertation, seton hall university, south orange, nj. retrieved from https://scholarship.shu.edu/dissertations/1614 wright, r., martland, j., stafford, a., & stanger, g. (2006). teaching number: advancing children’s skills and strategies (2nd ed.). london: paul chapman publishing. appendix 1: key terms used in this article digit: the number symbols 0 to 9 which are used to represent all numbers. digit-wise: naming and thinking about a multi-digit number as comprising of its digits. 356 is ‘three five six’. unit-wise: depicting a number as comprising units (or ones). five-wise: depicting a number as comprising groups of 5. ten-wise: depicting a number as comprising groups of 10. this may include showing each ten as comprising 2 fives. quick 5: a drawing or arrangement in which the objects or markings are spatially arranged so that it is quick and easy to identify groups of 5. quick 10: a drawing or arrangement in which the objects or markings are spatially arranged so that it is quick and easy to identify groups of 10. quick 100: a drawing or arrangement in which the objects or markings are spatially arranged so that it is quick and easy to identify groups of 100. structured mathematics drawing: a drawing that clearly shows the five-wise and ten-wise structure of numbers. structured apparatus: apparatus which clearly shows the five-wise and ten-wise structure of a number such as an abacus or a 5-5-5-5 bead string. small number sequence: counting in ones from any number. 15, 16, 17, 18, … medium number sequence: counting in tens from any number. 15, 25, 35, 45, … large number sequence: counting in hundreds from any number. 15; 115, 215, 315, 415, … triad: a set of 3 numbers that form an additive relationship. this is also termed a ‘number triple’. whole-part-part diagram: a diagram depicting a whole and two parts of an additive relationship. this is a triad or number triple, where a context of length is applied. structured number line: a number line marked in ones semi-structured number line: a number line marked in fives or tens empty number line: a number line with no structured markings place value: the value of a digit according to its position in a multi-digit number. so the 3 in in 356, has a place value of 3 hundred. expanded notation: break up a number using the place value of each digit. so 356 = 300 + 50 + 6 appendix 2: representations for addition strategies the swa, which is familiar to many adults, is a very condensed version of a ‘break up both numbers’ strategy. but the mathematics education literature, and the example learning programmes, encourage invented strategies, and a ‘break up the second number’ strategy, before children are exposed to the ‘break up both numbers’ strategy. in this annexure, i show how each strategy may be represented. representations for ‘break up the second number’ strategy if being introduced by a teacher (and not brought or invented by a learner), this strategy should precede the ‘break up both numbers’ strategy. this strategy can be used for addition and subtraction and no regrouping (exchanging, borrowing or carrying is needed). this strategy can also be used flexibly for missing addend and missing subtrahend problems (such as 189 + … = 346; or 346 … = 189). notice that the ‘break up the second number’ strategy means you don’t break up the first number as well. in these examples you always start with 189, which is the first number in 189 + 157 = … expanded notation on an empty number line expanded notation and number sentences fill up the tens on an empty number line representations for ‘break up both numbers’ strategy if being introduced by a teacher (and not brought or invented by a learner), this strategy should be used after learners are secure with the ‘break up both numbers’ strategy for both addition and subtraction. depending on the numbers, this strategy may require regrouping (carrying or borrowing). also, this strategy can be difficult to use for missing addend and missing subtrahend calculations, such as 189 + … = 346; or 346 … = 189). notice that both 189 and 157 are broken up for 189 + 157 = … expanded notation, and mathematics drawings or manipulatives (horizontal) a ‘break up both numbers’ strategy, rearranging manipulatives or drawing quick 100s with squares, quick 10s tens with lines, and quick 1s with dots, then regrouping. rearranging mathematics drawings or manipulatives (vertical) a ‘break up both numbers’ strategy, rearranging manipulatives or drawing quick 100s with squares, quick 10s tens with lines, and quick 1s with dots, then regrouping. expanded notation in rows a ‘break up both numbers’ strategy, represented using rows and writing all totals (as whole numbers), then adding in columns. write all total or partial sums in columns a ‘break up both numbers’ strategy, represented using columns and writing all totals (as whole numbers) below. expanded column: add first then exchange a ‘break up both numbers’ strategy, represented using columns, writing all totals first (in their place value column), and then exchanging one place at a time. standard written algorithm (swa, condensed, traditional column, vertical) a ‘break up both numbers’ strategy, represented using columns, exchanging using digits at the top of in place value columns and making use of digit-wise place-value (1 hundred 8 tens and 9 ones). article information authors: debbie stott1 mellony graven1 affiliations: 1south african numeracy chair project, education department, rhodes university, south africa correspondence to: debbie stott postal address: po box 94, grahamstown 6140, south africa dates: received: 19 apr. 2012 accepted: 07 apr. 2013 published: 14 june 2013 how to cite this article: stott, d., & graven, m. (2013). the dialectical relationship between theory and practice in the design of an after-school mathematics club. pythagoras, 34(1), art. #174, 10 pages. http://dx.doi.org/10.4102/ pythagoras.v34i1.174 copyright notice: © 2013. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. the dialectical relationship between theory and practice in the design of an after-school mathematics club in this original research... open access • abstract • introduction • the context • research paradigm and empirical field • conceptualisation of the clubs • methods of data collection    • ethical considerations • club sessions overview • lessons from the pilot study    • initial club design using zone theory    • an overview of zone theory       • the zone of proximal development in zone theory       • the zone of free movement       • the zone of promoted action    • experiences from the pilot study • the zone of proximal development in the clubs    • contrasting the zone of proximal development in our initial design and what emerged from our observations    • emerging zone of proximal development constructs from the pilot club • conclusion • acknowledgements    • competing interests    • authors’ contributions • references abstract top ↑ in this article we describe the design process for an after-school grade 3 mathematics club, based on our experiences running a pilot club in a 2011 research and development project. working from a sociocultural perspective, we show the progression from an initial multifaceted design to a much simpler, more learner-centred design that speaks directly to our research foci and one which is based on empirical evidence. our experiences have brought to light the entwined and dialectical nature of the data collection and design processes and the significance of the post-club reflection sessions as a powerful data collection instrument for planning the club sessions. furthermore, we identify and shape the zone of proximal development for the purposes of our club as the critical design concept for each club session for each learner. introduction top ↑ graven (2011) has previously argued that after-school mathematics clubs hold the potential to address some of the challenges that young numeracy learners face. we argue that numeracy clubs can be conceptualised as communities where sense making, active mathematical engagement and participation, and mathematical confidence building are foregrounded. we have elsewhere addressed the challenges of designing the clubs so as to maximise learning in relation to each of these features (graven & stott, 2012). the initial design of these clubs drew largely on valsiner’s (1997) zone theory, which goos, dole and makar (2007a) further developed. this particular article focuses on how research data and our experiences piloting a grade 3 mathematics club led to the simplification of our pre-pilot design for the clubs and for the foregrounding of the zone of proximal development (zpd) (vygotsky, 1978) as the central organising feature for future club activities. this dialectical praxis illuminates the relationships between theory and practice and how the dialogue between the two elements informs each. the context top ↑ given the many negative perceptions and poor performance of learners in numeracy in south africa (bloch, 2009; fleisch, 2008; taylor, fleisch & shinder, 2008), national research funding organisations have begun to invest in research projects that focus on these specific issues. our work within one such project is focused on both development and research in this field. in development terms we aim to improve the quality of teaching of in-service teachers at primary level and to improve learner performance in primary schools as a result of quality teaching and learning. our research remit is to grow an area of research that looks towards finding sustainable solutions to the many numeracy education challenges faced in our area. as part of our developmental work, we work with 15 schools in the greater grahamstown area in south africa. our teacher development program involves working with 57 numeracy teachers (ranging from grade r to grade 6) who participate in fortnightly workshops focused on issues and challenges in numeracy teaching. learner activities are a key part of our developmental activities and are an area in which we are responsible for working directly with learners rather than via the teachers. as such, we facilitate learner numeracy proficiency by running learner-directed and learner-oriented mathematics activities as well as creating an ethos of ‘mathematics is fun’ in schools. some examples of these activities include mathematics bonanzas and mathematics relays. however, many teachers face the challenge that most of their learners do not have the necessary mathematical foundations to be learning at the grade level in which they are placed. in light of this, we decided to implement after-school mathematics clubs as a more focused and regular learner intervention as a possible way of addressing some of this challenge. these clubs were conceptualised by graven (2011) as being informal places where learning can take place in out-of-school time and have been elaborated on in previous work by graven (2011) and graven and stott (2012). the clubs are an opportunity for us to influence what happens with learners and they provide an empirical research field in which we can fully interact with the learners and thus be insiders to the learning process. research paradigm and empirical field top ↑ the research of our clubs and the nature of learning within it is informed by a sociocultural theory of learning and is largely interpretive and qualitative, drawing on a range of data collection methods. this article draws on our research experiences from a pilot club that we set up in the second half of 2011 for grade 3 learners aged between 8 and 10 years in one primary school in grahamstown. the language of learning and teaching in the school is english but many of the learners speak english as a second or third language. the school in which our pilot club ran is one of the poorer performing primary schools in grahamstown with regard to numeracy results, as indicated by the annual national assessments and data collected by our larger research project. the school was participating in the professional development activities offered by our project. the club ran once a week for an hour after school for 12 sessions. the club consisted of ten learners and two mentors (the authors of this article). the participating learners invited by the grade 3 class teacher included learners with a range of mathematical proficiencies. participating learners were those whose parents signed consent forms and for whom after-school transport arrangements could be made. in this sense the learners were an opportunity sample. we specifically chose to work with a small group of ten learners so that both the needs of the whole group and those of the individual clubs learners could be taken into account in the design process and in the sessions themselves. conceptualisation of the clubs top ↑ we define our mathematics clubs as informal, after-school clubs focused on developing a supportive learning community where learners can develop their mathematical proficiency, make sense of their mathematics and where they can engage and participate actively in mathematical activities. individual, pair and small group interactions with mentors were the dominant practices with few interactions with the whole club. we will briefly unpack some of these ideas here. firstly, we explain what we mean by ‘informal’. from experience with the associated teacher development project in 2011, we intentionally designed the clubs to contrast some of the more formal aspects observed in the classrooms of the participating schools. some of these contrasts are described in table 1. table 1: contrasted classroom and club environments. based on this initial conceptualisation for the clubs, our specific aim was for the clubs to promote mathematical proficiency as well as active participation. sfard (1998) describes the differences between two metaphors for learning. ‘learning as acquisition’ implies that learning is the acquisition of something which is then stored in the individual. learning as acquisition theories can be regarded broadly as mentalist in their orientation, with the emphasis on the individual building up cognitive structures. in contrast ‘learning as participation’ conceives learning as a process of becoming a member of a certain community, which entails the ‘ability to communicate in the language of this community and act according to its particular norms’ (sfard, 1998, p. 6). whilst some educators argue for the need for a paradigm shift away from (or even rejecting) acquisition perspectives in favour of participation, sfard suggests that these metaphors are not alternatives, but that each provides different insights into the nature of learning. she argues:an adequate combination of the acquisition and participation metaphors would bring to the fore the advantages of each of them, while keeping their respective drawbacks at bay. conversely, giving full exclusivity to one conceptual framework would be hazardous. (sfard, 1998, p. 11) we have purposely worked with the dialectical nature of these notions of acquisition and participation, drawing on sfard’s ‘metaphorical mappings’ (1998, p. 7). working within the sociocultural paradigm, the tensions between these two approaches are seen as normal. our design process, based on zone theory (discussed later), allowed us to plan for the types of interventions, activities, socio-mathematical norms and environment we had conceptualised. zone theory accommodates this dialectical relationship. from our experiences, different aspects of the clubs brought elements of either the participation or acquisition metaphors into focus at different times during the club sessions. this will become apparent when we discuss our data collection instruments in the following section. this dialectical approach to working with the two metaphors is by no means unique to our clubs and various studies provide some mathematical examples where this has taken place (askew, 2004; goos, galbraith & renshaw, 2002; jaworski & potari, 2009). the acquisition and participation metaphors are useful ways of accounting for evolving mathematical proficiency and participation as they occur in the social context of the club. figure 1 shows our interpretation of sfard’s (1998) metaphorical mappings in our club context. for us the line boundary between the two is intentionally blurred as we needed to work seamlessly with dialectical nature of the two notions in our clubs. figure 1: club metaphorical mappings. figure 2: zone theory design. methods of data collection top ↑ during the pilot club, we took the opportunity to pilot the data collection instruments that we planned to use for our research. sfard (2001, p. 24) draws our attention to the fact that research done under the participationist metaphor umbrella will focus on the growth of mutual understanding and coordination between the learner and the rest of the community (in this case our club) and the focus will turn to the activity itself and to its changing, interactional aspects of learning. this shift of focus implies that attention will need to be given to many factors that may be deemed irrelevant when working with a purely acquisitionist metaphor. since we work with both notions in the club, in order to capture both the participationist and acquisitionist aspects in our work, we designed, planned for and used two diverse data collection instruments. the first explored learners’ dispositions and participation of the club learners and the second allowed us to assess possible evolving mathematical proficiency in the club learners. for the first instrument, we designed a short interview that allowed us to gain insight into learners’ mathematical dispositions and to address the fifth strand of mathematical proficiency described below (kilpatrick, swafford & findell, 2001). the two club mentors administered this instrument as a one-to-one interview with each club learner in november 2011 and learner responses were written on the interview script (see graven [2012] for further information about the evolution of this instrument.) the second diagnostic instrument focused on collecting data on learners’ progress with regard to mathematical proficiency. our notion of mathematical proficiency draws on kilpatrick et al.’s (2001) definition of mathematical proficiency. this definition comprises five intertwined and interrelated strands: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning and productive disposition. this diagnostic instrument for assessing mathematical proficiency progress was derived from work done by askew and his team during their effective teachers of numeracy study (askew, brown, rhodes, johnson & william, 1997); we were given permission to use it for our research work. we administered an adapted version of it as a whole-club oral interview at two different points in time – once in september 2011 and again in november 2011. by administering this diagnostic instrument in the first few sessions of the pilot club, we had initial data that we could compare later and that allowed us to see if learners’ mathematical proficiency was evolving over time. the adapted instrument assessed: • understanding of the number system, including place value and fractions • methods of computation, including both known number facts and efficient and accurate methods of calculating • solving numerical problems, including complex contextualised word problems and abstract mathematical problems concerning the relationships between operations (askew et al., 1997, pp. 15–16). this instrument yielded largely qualitative data and allowed us to gauge progress in all but one of the five strands of mathematical proficiency (that of productive disposition), making it an ideal instrument for use in this pilot club. as part of the first author’s doctoral study we also collected less formal data from each club session in the form of observations, journal writings, learner workings, photographs and sometimes video. these were used to record explanations given by learners in the sessions as well as to document any interactions that took place between the learners and form secondary data to support data collected from the two key instruments described above. whilst we had not anticipated the significance of our post-club reflections, these became the key drivers for planning each subsequent session and re-planning and refining the data collection. these in-depth discussions took place between the mentors (the authors) every week directly following each club session and lasted on average half an hour. detailed notes and sometimes recordings were made of what was discussed. the story of how this became critical in the design of our clubs and for our research of the clubs is elaborated in the following sections. in this way we collected both quantitative and qualitative data for the pilot club, taking a broad, concurrent mixed method approach. ethical considerations ethical permission from the university and from the eastern cape department of education was obtained through the usual procedures. working as we were with young learners in the mathematics clubs, we obtained explicit informed parental consent in the form of signed letters, written in the appropriate home language, which explained the research and the anticipated learner involvement. in addition we received teacher, principal and district permission for each school and class that allowed the learners to participate in the mathematics clubs. as the learners were recorded via field and journal notes, video and occasional voice recordings, pseudonyms are used in subsequent publication of data. to this end, learner and where appropriate, school names have been changed. club participation was voluntary. learners were able to leave the club at any time if they, their teacher or parent wished them to do so. they were also free to remain in the club but withdraw from providing data for the research, in which case they are not included in any video or voice recordings. club sessions overview top ↑ the intention in the pilot club was to assess a number of aspects but we specifically wanted to evaluate the design process, the data collection instruments, the club size and promoted activities. space constraints prohibit an in-depth description of the club activities. the sessions were approximately an hour long and activities typically included playing mathematical games, using manipulatives such as place value cards and problem solving. the learners variously worked individually, in pairs or in groups. further examples are illustrated below. lessons from the pilot study top ↑ initial club design using zone theory before starting the pilot club and guided by our key conceptualisations, we devised a comprehensive design for the structure and planning of the clubs. the design we chose was based on an extended version of zone theory used by goos et al. (2007a). this design was multifaceted and gave us space to accommodate the dialectical tensions between the acquisitionist and participationist aspects of the club. on a more practical level it enabled us to conceptualise and plan how we thought the clubs might work in practice by providing us with a process for setting goals and planning and evaluating the ongoing learning programme in the mathematics clubs. as the process is iterative and cyclical we could use it to evaluate what was working and what was not and use this to plan and implement subsequent actions and activities in the clubs. furthermore, the decision to use this model highlighted the learning environment of the clubs as a dynamic and growing entity. goos et al.’s (2007a) version of zone theory uses zones of promoted action to describe and plan promoted learning activities, zones of free movement to describe and plan the learning environment and the zone of proximal development to describe and consider the learner’s existing understanding of mathematics. it is worth noting two other key features of their approach which they incorporated from work done by loucks-horsley, love, stiles, mundry, and hewson (2003). the first is the iterative nature of the design, which allows for ongoing evaluation of the planning and implementation of the environment, the activities and the possible learning that might take place. the second is that data analysis is the precursor to any planning or implementation (see figure 3). the significance of these features will become clearer as our story unfolds. figure 3: iterative nature of the design. an overview of zone theory valsiner (1997) expanded vygotsky’s (1978) notion of the zone of proximal development to include two additional zones of interaction: the zone of free movement (zfm) and the zone of promoted action (zpa). these describe the structure of a child’s development in terms of the environment and relationships between the child and other people in the environment (goos et al., 2010).the zfm, zpa and zpd can be seen as structures through which an adult or more knowledgeable other constrains or promotes a learner’s thinking and acting and as such the zfm/zpa combination interactively generates the environment in which that learner develops. blanton, westbrook and carter (2005) draw attention to a view that ‘the zfm and zpa are dynamic, interdependent constructs that are continually being reorganized in the learning process’ (p. 7). galligan (2008) also uses the word interdependent to describe the three zones and summarises the theory skilfully: ‘valsiner’s three zones constitute an interdependent system between the constraints put on the environment of the learner and the actions being promoted for the learner’ (p. 2). whilst the zpd may be a well-known concept to most in educational contexts and is discussed in detail later in this article, the zfm and zpa are less well known. for this reason we provide a brief overview of the zones as conceptualised in zone theory. the zone of proximal development in zone theory valsiner (1997) regards the zpd as a set of possibilities for development that are in the process of becoming actualised as individuals negotiate their relationship with the learning environment and the people in it. the zone of free movement hussain, monaghan and threlfall (2011) explain that the zfm represents the learner–environment relationship at a particular time in a certain environmental context. they point out that it is a dynamic and social construct and is created through the social and cultural interactions of all the people in the environment. it shapes the norms and values of the environment. galligan (2008) points out that the zfm itself can be set up either by the adult participants, by the learners themselves or through the joint action of all participants. in essence, the zfm is a function of what is allowed for the learner by the adult. on the one hand, the way an adult organises the zfm anticipates the nature of the child’s thinking about the concept being taught, at that moment and in the future. in this sense, the zfm ultimately channels the direction of development for the child, providing a framework for cognitive activity (blanton et al., 2005; galbraith & goos, 2003; galligan, 2008; goos et al., 2010; goos, dole & makar, 2007b). the zone of promoted action this is the set of activities, objects or areas in the environment through which an adult or more knowledgeable other attempts to persuade a learner to act in a certain way. the zpa describes what the adult is promoting. however the learner is under no obligation to accept what is being promoted, as in the case where learners may not wish to actively participate (blanton et al., 2005).the zpa should also be in a learner’s zpd. for example, having poor mathematics skills in a class which assumes basic mathematics skills may result in the learner’s inability or reluctance to participate or learn. on the other hand those learners who believe they already have the necessary skills may not participate either (blanton et al., 2005; galbraith & goos, 2003; galligan, 2008; goos et al., 2010; goos et al., 2007a). an overview of both zone theory and the iterative nature of the process are shown in diagrammatic form in figure 2 and figure 3. for a more detailed discussion of this initial design, see graven and stott (2012). figure 2 shows how the zones overlap to form a design for a learning community. figure 3 highlights the key features of the design process incorporated from loucks-horsley et al.’s (2003) work as highlighted in the discussion above. the three zones form the basis of the design, and the process is initially linear (analyse, set goals, plan, implement and evaluate). as a result of evaluation, the entire process starts again and, as dynamic constructs, the three zones are adjusted accordingly. experiences from the pilot study during the early part of the pilot, the design of the sessions became much simpler and a more focused design emerged. two things drove this change. the first was the data we collected using the diagnostic instrument in september and the second was the pivotal role of post-club reflection. timely analysis of the diagnostic data collected allowed us to use a data-driven approach to determine where individual learners were struggling. whilst post-club reflections were an unanticipated part of our data collection, they rapidly became a fundamental part of the ongoing design, thus entwining both the data collection and design processes. made possible by the small numbers of learners and by having two mentors, these reflections became focused on two aspects: the club activities, participation and progress as a whole and on individual members’ participation progress and mathematical proficiency progress. these reflections and the other data enabled us to plan interventions or activities for each subsequent club session that would promote each learner’s mathematical proficiency and would encourage their participation in the club. reflection was an integral part of our original iterative design process but it was more in the form of formal evaluation at the end of a particular implementation cycle. during the pilot we realised that our weekly, post-session reflections provided one of our key sources of rich learner data. they thus became an integral part of the data collection process from there on and provided significant input for each subsequent session design. we recognised that without these reflective events, we would not have had the insights into the learners, nor would we have arrived at a simpler, more learner-centred design that speaks directly to our research foci. we illustrate how this reflection influenced subsequent session design with an example. in session 6, we noted that the majority of the learners were very weak on number bonds. the bonds to 5 and 10 were not recalled automatically and learners used counting on by ones or used their fingers to work them out, for example, to work out the answers to 2 + 8 or 2 + 98. their grasp of basic number sense and basic bonds was severely limited so we noted that we needed to work on increasing their proficiency in these areas over a period of time. we realised that one session was not going to be sufficient to do this. after discussion, we decided to introduce some games that would allow learners to practise their bonds in an ongoing and fun way whilst also promoting interaction and communication amongst themselves and with others. in the next session we introduced a variation of a card game called pyramid which worked with the basic bonds to 10 and which they could play individually or in pairs. in subsequent sessions we used the same game to work with bonds to 11, 12 and 13. we gave each learner a pack of playing cards and asked them to teach the game to someone at home, so that they could practise it outside the club, with the intention of creating a third site of learning. to provide reinforcement in basic number sense and bond proficiency and to supplement the learning, we introduced the learners to a progressive series of workbooks (brombacher & associates, 2012), which we found to be accessible, to involve limited reading and allowed them to work through activities at home, at their own pace and without adult supervision. post-session reflections had another advantage that was unexpected. reflecting on each individual learner’s participation and mathematical proficiency in each session enabled us to document and record individual learner stories in a detailed manner. using the reflections, personal journal entries and other data, we were able to compile what we called ‘individual learner story cards’. here we recorded information about their mathematical proficiency, their disposition and attitude towards mathematics as well as their participation in the club. we could see from this how the learner was progressing (or not) in the club in terms of mathematical proficiency as well as the nature and level of their participation and engagement. through this process of reflecting on each individual learner’s progress and proficiency, we became increasingly aware of the emergence of each learner’s zone of proximal development in a way that we had not considered before. we saw distinctive and fluid zpds arising during sessions. we realised that by noting and discussing these we could use them to design subsequent activities, tailored for each learner. we also saw that the zpds arose both when we as mentors were mediating and increasingly in peer interactions, even though the learners had initially resisted this. in this way, over the course of the pilot, the zpd became the critical design concept for subsequent club sessions that supported individual tailoring of activities. whilst our initial design took into account the zpd, it was not foregrounded as a critical element; rather it was considered equally as one of the three elements (see figure 4). our post-club reflections allowed us to zoom into more detail about the individual learners and to focus on the zpd aspect. this is not to say that the two other zones became irrelevant; they were simply no longer considered with the same emphasis as the zpd and were less foregrounded. however, they were still a necessary aspect of the club design and allowed for zooming out to see and plan for the bigger picture of the club as a whole. figure 4: change of emphasis in the zones pre-pilot and post-pilot. the zone of proximal development in the clubs top ↑ a review of the literature shows the zpd to be conceptualised in many ways. chaiklin (2003) points out that ‘vygotsky’s concept of zone of proximal development is more precise and elaborated than its common reception or interpretation’ (p. 39). we briefly discuss these different interpretations as a way of making sense of this diversity.primarily conceptualised as ‘scaffolding’ of learning by adults and teachers, the notion of the zpd was developed by wood, bruner and ross (1976). scaffolds may take the form of more knowledgeable people or cultural resources external to the learner which support their learning. this enables them to build on their own existing knowledge and internalise new information. this interaction with a more knowledgeable other has also been called an expert-novice zpd. this view, as roth and radford (2010) point out tends to result in unequal status learning: the notion of zone of proximal development has come to be used widely to theorize learning and learning opportunities. unfortunately, following a simplified reading of its original definition and primary sense … the concept tends to be thought of in terms of the opposition of individuals. one of these individuals, a teacher or peer, is more capable than another individual, the learner. (p. 299, [emphasis original]) vygotsky (1978) talked a great deal about how children learn when playing with peers. some researchers believe that there is learning potential in peer groups and the interactions are seen as working in or creating a bi-directional or collaborative zpd (e.g. forman, 1989; goos et al., 2002). in this egalitarian conceptualisation of the zpd each partner possesses some knowledge and skill but requires the others’ contribution in order to make progress. as mentioned in the brief discussion on zone theory above, valsiner (1997) regards the zpd as a set of possibilities for development that are in the process of becoming actualised as individuals negotiate their relationship with the learning environment and the people in it. to explain how these possibilities actually emerge from the possible, valsiner proposed two additional zones: the zone of free movement (zfm) and the zone of promoted action (zpa), which have been discussed previously. for valsiner, these two additional zones create a better understanding of how the zpd operates in a specific learning context. meira and lerman (2001) argue that the zpd would be better conceptualised not as a physical space, in the sense of the individual’s equipment (either cognitive or communicative), but as a symbolic space involving individuals, their practices and the circumstances of their activity. this brief view of the different conceptualisations of the zpd affords us an overview of this complex notion and indicates that it is one that is still very current in mathematics education research. below, we talk about the concept of the zpd that emerged for us as most appropriate in our pilot club and thus position ourselves in this discourse. contrasting the zone of proximal development in our initial design and what emerged from our observations the zpd (as part of zone theory) was one of three equal elements of the initial design approach, which also included the zpa and the zfm as discussed earlier. following goos’s (2006) example, we characterised the zpd as ‘a set of possibilities for development that are in the process of becoming actualised as individuals negotiate their relationship with the learning environment and the people in it’ (p. 103). it was also important that the zpd construct was a way of recognising the status of learners’ existing understanding, which would enable us to plan activities and mediation approaches. this initial conception of the zpd was more in line with it being interpreted as a ‘physical space’ that existed in the learners themselves prior to club learning activities that took place. the assumption here is that learners would bring their potential zpd with them into the club session. this zpd would be enabled through mediated participation in the club activities but could also be constrained and enabled by what had been acquired in previous learning episodes. this conception also assumes that the learner would then take their zpd away from the club sessions too. however, what we saw during the pilot was a zpd that was much more fluid and less of a fixed set of pre-determined possibilities. it seemed to be largely determined by how the learners interacted with each activity. additionally, what they brought to each activity was clearly dependent on a whole range of social, emotional, health and other interactional influences. thus, what we mean by fluid is that the zpd was not durable from one session to another. again, we illustrate this with a couple of examples. we noticed that one of the club learner’s energy levels and health affected how she interacted and what kind of zpd emerged for her in any given session. when her energy levels were high, she was focused, engaged, participated in the talk and made great strides in her developing proficiency. at these times an expanded zpd emerged for her. on another occasion, by her own admission she was feeling unwell and we could see that she had reverted to her trusted finger counting strategy for working out the answers to problems posed in the activities. understandably, she engaged with the activities in a far more limited way than she had previously and thus she didn’t involve herself in any mathematical talk. a more limited and constrained zpd emerged in the club for her that day. another learner, a boy called reg, was usually quiet during sessions, did not contribute to ideas or explanations and did not seem to be fully engaged with the mathematics in a meaningful way. one could say that the zpds for reg in most sessions were hampered by his stifled participation. however, one day a boy he always sat with was absent and the difference in reg was visible: his eyes sparkled; he contributed to discussions and talked confidently to his partner during activities. for him that day, a large zpd emerged in his interactions with us and his peers that he partnered with for various activities. interestingly, he managed to continue this more active participation in subsequent sessions, almost as if his confidence had increased in that one session. what we observe from these examples is that the level of mathematical participation in the club was influenced not only by the activities promoted in the club but also by energy levels (e.g. tiredness), emotional and physical factors, interactional aspects and group dynamics. these observations resonate with the work of meira and lerman (2001, 2009) who have used the zpd as a tool to both analyse and design learning environments. they believe that when working from a sociocultural perspective, the zpd may be conceptualised as a ‘symbolic space involving individuals, their practices and the circumstances of their activity’ and that learners can be ‘pulled into their zpds by a combination of the activity, the actors, and appropriate communication’ (lerman, 2001, p. 103). this idea of pulling a learner into a zpd by a combination of factors (activity, actors and communication) indeed surfaced from our own observations. this reasoning led us to focus on meira and lerman’s work for exploring learning in the club. furthermore, levykh’s (2008) work echoed our experiences with learners: he draws attention to the affective aspects of the zpd. whilst aligning with meira and lerman’s work he further highlights the zpd as reflecting ‘constant changes in the emotional connections’ amongst the participants (p. 91). he states that vygotsky was ‘always a strong opponent of treating intellectual and affective aspects of human life as separate’ (p. 91) and that vygotsky combined affective and intellectual features in his notion of the zpd. meira and lerman (2001) work with a notion of the zpd where the zones emerge or not in the activity and as a result of different ways of participating and communicating in the classroom. they also point out that the zpd is not a generalised, context-free notion (meira & lerman, 2009, p. 203). meira and lerman are not the only researchers to conceptualise the zpd as a space for interaction and communication. radford (2010) claims that the zpd is a relational concept rather than an absolute one, one that is forged out of the interaction between students, and between the students and their teacher. ... the zpd is not a static thing that belongs to one particular student but rather a social, complex system in motion. (p. 116) roth and radford (2010) reveal another interesting way of conceptualising the zpd as a space for interaction and communication. they propose that asymmetries (a lack of symmetry) in the zpd are possible because of the ‘existing intercomprehension of interacting participants’. the participants can become each other’s teachers and learners regardless of their actual formal or institutional positions (p. 300). if the conversation is taken as the unit of analysis, in which each word spoken has two sides or possible meanings, any asymmetry within the unit, can be thought of differently. they summarise their conceptualisation thus: ‘the zone of proximal development is an interactional achievement that allows all participants to become teachers and learners’ (p. 307). intercomprehension can be described as a dialogue between people who use two different languages and in which each one makes efforts to understand the other. in the context of mathematics, we can see this intercomprehension occurring between a teacher and learner where the teacher is talking in accepted mathematical language and the learner is using their own everyday language. roth and radford are saying that because the participants are effectively speaking slightly ‘different’ languages, their interactions can cause them to swap roles to teach and learn from each other. in this conceptualisation, the teacher is not seen as the ‘more knowledgeable other’ and both can enter the zpd as equal partners. this idea of a zpd of equal interaction partnerships is echoed in work by goos et al. (2002). they use the term ‘collaborative zpd’ in their research on small group learning to emphasise the equal status interactions that occur in the zpd as opposed to expert–novice notions of the zpd, such as a more knowledgeable other. allal and pelgrims ducrey (2000) describe a learning setting where the type of interactive, formative assessment is similar to the way we use our diagnostic instrument described earlier. they point out that in this type of setting, there is a potential for multiple zpds to be created and for these to vary from one learner to another. the social interactions, dialogue and the appropriateness of the mediations with respect to the learners’ present level of mathematical proficiency cause these zones to come into being. in this type of context, the dialogues between the different participants allows exploration within the zone of proximal development created for a given child, at a given moment, by on-going social interaction. … in a certain sense, the zpd has no real existence outside the interactions that mediate the teaching-learning process. (allal & pelgrims ducrey, 2000, p. 146) this reflects the idea that the zpd is fragile and that it doesn’t exist outside of the social interactions that take place in the learning setting. it also reinforces the idea that the combination of interactions, dialogue and mediations all affect the emergence of a zpd in learners. the key ideas that surface from looking at this selection of conceptualisations are that the zpd is formed through interaction and communication during learning activities, that the participants in the zpd have equal status and that the zpd is a relational and complex system. in the next section we elaborate on the zpd as a construct as re-conceptualised based on our experiences during the course of the club pilot in 2011. emerging zone of proximal development constructs from the pilot club a combination of the concepts from this pool of ideas, with a bias towards those of meira and lerman (2009) and allal and pelgrims ducrey (2000), provides us with a set of four characteristics for our zpd construct that have become essential for the ongoing club design. firstly, the zpd does not exist prior to the activity and it is created, or not, by the social and dialogical interactions during the club activities, as part of the micro-culture of the club. these zpds do not ‘live’ in the learner and are therefore not necessarily carried forward to the next session of the club by the learners or into the classroom environment. in other words, they are fragile spaces. secondly, the learner’s current level of mathematical proficiency and confidence, along with their emotional and physical state, determine the type of interaction in which they can become involved and from which they can benefit. thirdly, the participants in the learning are, in most cases, of equal status. finally, learning may be initiated by learners as well as by mentors and is possible that there is learning for everyone involved in a specific interaction. conclusion top ↑ in this article we described a journey of a mathematics club design process based on our experiences with a pilot club. we highlighted the nature of the dialectic between the design process and the empirical field. we have shown how we used empirical evidence to move from an initial multifaceted design to a much simpler, more learner-centred design. our experiences have brought to light the entwined nature of the data collection process and the design, the significance of the post-club reflection sessions as a primary form of data collection for planning the club sessions and how the combination of features described formed a zpd construct tailored for our clubs which became the critical design concept for each session for each learner. these findings will continue to inform the ongoing design of learner clubs in our work. the data collection process and the club sessions will be designed and planned using the same data-driven and reflective process. this article reported on our learning journey from our pilot mathematics club in 2011 to our current clubs. in 2012 we have set up nine more clubs. we plan to explore the notion of the zpd in more depth in subsequent work. acknowledgements top ↑ the work of the sa numeracy chair project is supported by the firstrand foundation (with the rmb), anglo american chairman’s fund, the department of science and technology and the national research foundation. any opinions expressed in this article are solely those of the authors. competing interests the authors declare that they have no financial or personal relationship(s) that might have inappropriately influenced them when they wrote this article. authors’ contributions d.s. 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(1976). the role of tutoring in problem solving. journal of child psychology and psychiatry, 17, 89–100. http://dx.doi.org/10.1111/j.1469-7610.1976.tb00381.x article information author: craig pournara1 affiliation: 1wits school of education, university of witwatersrand, south africa correspondence to: craig pournara postal address: box 1531, pinegowrie 2013, south africa dates: received: 02 aug. 2013 accepted: 07 nov. 2013 published: 29 nov. 2013 how to cite this article: pournara, c. (2013). teachers’ knowledge for teaching compound interest. pythagoras, 34(2), art. #238, 10 pages. http://dx.doi.org/10.4102/ pythagoras.v34i2.238 copyright notice: © 2013. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. teachers’ knowledge for teaching compound interest in this original research... open access • abstract • introduction • compound interest in the south african school curriculum • research on conceptions of compound interest • mathematical aspects of teachers' knowledge    • a hierarchy of interest concepts    • a network of concepts relating to growth factor    • deriving a formula for compound interest • financial aspects of teachers' knowledge    • interest in the world of banking    • the compound interest formula as a model of compound growth    • knowledge of socio-economic issues and financial literacy • pedagogical aspects of teachers' knowledge    • student error – multiple compounding periods and linear thinking    • student error – percentage change and reversible operations    • implications for teaching compound interest • conclusion • acknowledgements    • competing interests • references • footnotes abstract top ↑ there is increasing acknowledgement that teachers’ knowledge for teaching mathematics is multifaceted and topic specific. given the paucity of research on the teaching and learning of financial mathematics in general, little can be known about teachers’ knowledge for teaching compound interest. however, since financial mathematics is a component of the school curriculum in south africa, and an important element of financial literacy more broadly, attention needs to be given to knowledge for teaching financial mathematics, and compound interest in particular. drawing from a larger study in which the author taught a financial mathematics course to pre-service secondary mathematics teachers, a theoretical elaboration is provided of the underlying mathematics of compound interest, and connections with the world of banking. based on findings from the study, two key student errors are identified: the over-generalisation of linear thinking in multiplicative scenarios, and the over-generalisation of reversible operations in percentage-change scenarios. taken together, teachers’ knowledge of relevant mathematics, of the banking context and of learners’ conceptions will contribute to building a knowledge-base for teachers’ knowledge for teaching compound interest. introduction top ↑ the issue of teachers’ mathematical knowledge for teaching has been on the agenda in mathematics education research since shulman’s seminal work in the eighties (shulman, 1986, 1987). there is now agreement that teachers’ knowledge is both multifaceted and topic specific (adler & ball, 2009). in areas of secondary mathematics that have been widely researched, such as introductory algebra, functions and calculus, there is a body of knowledge that elaborates the mathematical concepts at school level, that provides insights into learners’ conceptions in relation to the topics, and that proposes a range of topic-specific pedagogical approaches to support learning of the topic (e.g. even, 1990, 1993; thompson, 1994; thompson & thompson, 1996; zandieh, 2000). however, there are many concepts and topics in the school curriculum that have received little attention. financial mathematics is one of these. we know very little, for example, about learners’ conceptions of compound interest or about the most effective ways of teaching compound interest, and what we might claim to ‘know’ is largely anecdotal. therefore, before considering what knowledge teachers need for teaching compound interest, it is necessary to consider more deeply the notion of compound interest in the context of school mathematics. in this article i provide a theoretical elaboration of compound interest against which to explore learners’ conceptions, teachers’ knowledge and appropriate pedagogies. i focus on mathematical and financial aspects of compound interest that are pertinent to teaching at secondary school level. this article is drawn from a larger study of pre-service secondary mathematics teachers learning financial mathematics in a course specifically designed for teachers (pournara, 2013). the data drawn on in this article come from video records of contact sessions during the course and during small group tutorial sessions, as well as students’ written work from tutorials, journals and course assessments. whilst shulman proposed the distinction between subject matter knowledge (smk) and pedagogical content knowledge (shulman, 1986, 1987), this distinction is not so easy to operationalise (ball, thames & phelps, 2008). and so following others in the field (e.g. adler, 2005; adler & davis, 2006; huillet, 2009) i refer to mathematics for teaching as an amalgam of subject matter knowledge and pedagogical content knowledge, which consists of aspects of knowledge that are mainly mathematical and others that are mainly pedagogical. in addition, given my focus on financial mathematics, i add knowledge of financial aspects. in order to provide a backdrop for the article, i begin by locating compound interest in the south african school mathematics curriculum and use examples from local textbooks to illustrate typical tasks. i also briefly summarise the small body of research findings on learning of compound interest. thereafter i focus in turn on mathematical, financial and pedagogical aspects of teachers’ knowledge in relation to compound interest. compound interest in the south african school curriculum top ↑ the notion of interest on money is introduced in the south african mathematics1 curriculum in grade 7 (department of education, 2002), starting with simple interest, followed in later grades by compound interest with annual compounding, and then multiple compounding periods per year. in grade 11 the notions of nominal and effective interest rates are introduced, together with problems involving changes in interest rates and depreciation. here are three typical examples of textbook questions on compound interest: question 1: brett has r7500, which he wants to invest for 5 years. which savings plan will yield more interest: simple interest at 14% p.a. or compound interest at 12% p.a.? (cross et al., 2005, p. 60) question 2: a sum of money was invested at a nominal annual interest rate of 4.25% per annum compounded quarterly. after five years this investment was worth r4500. • what sum of money was invested? • give the effective annual interest rate for this example. (bennie, blake & fitton, 2006, p. 140) question 3: r28 000 is invested for 10 years. the interest is calculated at 9.3% p.a. compounded monthly for the first four years. after four years the interest rate is increased to 11.8% p.a. compounded quarterly. calculate the value of the investment at the end of 10 years. (laridon et al., 2006, p. 38) question 1 is typical in both local and international texts, and requires students2 to distinguish between calculating simple and compound interest. students will either do the necessary iterative calculations for five years, or they will use the simple and compound interest formulae. it is assumed here that compounding is annual since it is not stated otherwise. by contrast, in the world of banking, compounding is assumed to be monthly unless specified otherwise. question 2 is a typical grade 11 question dealing with quarterly compounding, and with nominal and effective interest rates. it requires students to identify the rate per period and the associated number of compounding periods. question 3 deals with a change in interest rates and in compounding periods. a key aspect here is the different rate per period for two time segments of the question. this question is typical of textbook questions that make use of timelines. from the perspective of the banking world, all three questions reflect very simplified scenarios. for example, it is assumed that deposits remain in the bank for full periods and, by implication that they are made at the beginning of the period. no attention is given to the number of days in the year or the number of days in a month. in question 3 the big increase in interest rate and the shift from monthly to quarterly compounding are not likely in the banking world. it is important to note the curriculum constraints with regard to financial mathematics. financial mathematics is allocated two weeks of teaching time, and 5% of the total marks in final assessments in grades 10–12 (department of basic education, 2011). given these curriculum constraints, one needs to be realistic about how deeply teachers can deal with compound interest. one also needs to consider the typical questions that learners will encounter in their text books, and in assessments. all this constrains what it is possible to teach and learn in schools. this, in turn, frames the nature and extent of the teachers’ knowledge for teaching financial mathematics in schools. whilst acknowledging the curriculum constraints, in this article i elaborate a range and depth of knowledge of compound interest that might enable teachers to teach the content with mathematical insight and increased awareness of the realities of the banking world. i do not propose that depth and breadth of knowledge is a requirement to teach compound interest well, but rather that it is a level to which teachers may aspire. research on conceptions of compound interest top ↑ whilst there is very little research on financial mathematics in general, the following findings can be gleaned from the limited research on students’ knowledge of compound interest and their ability to work with it. whilst it is difficult to determine the extent to which the findings may be generalisable beyond the original studies, the findings all resonate with my experiences of students’ learning of compound interest:• university students as well as the broader population have difficulty in executing compound interest calculations (dempsey, 2003; organisation for economic cooperation and development, 2005). • university students cannot easily distinguish the impact of simple interest from that of compound interest on the growth of a principal amount (beal & delpachitra, 2003). • university students, including ‘business majors’, lack knowledge of the impact of increasing the frequency of compounding (chen & volpe, 1998). • university students have difficulty in identifying which formula or procedure to use for time value calculations involving single amounts or multiple payments (jalbert, jalbert & chan, 2004). • high school learners may not convert the nominal annual rate to an appropriate rate per period, and may not easily distinguish whether a formula calculates the accumulated interest or the cumulative balance (geiger & goos, 1996). the workplace-based research by hoyles and noss and others (e.g. hoyles, noss, kent & bakker, 2010; noss & hoyles, 1996a, 1996b) provides the greatest insight into the learning of compound interest. they found that percentage, compound growth and graphs were pervasive in daily financial work, and that employees’ grasp of percentage and interest was deeply intertwined with their working contexts. however, they were not aware of the underpinning mathematics, such as the multiplicative structures connecting percentages and interest rates. hoyles et al. (2010) also emphasised the importance of being able to communicate with the customer to explain printouts, to answer queries, and to provide meaningful explanations about the benefits of one product over another. there are similarities between the demands of this work and teaching since both require an ability to distil the essence of the issue and to communicate this to another who is less knowledgeable. one of the key insights from bakker, kent, noss, hoyles and bhinder (2006) relates to interest and percentage increase or decrease, and is therefore relevant to school mathematics. based on interventions with employees, they argue that an approach of multiplying factors is preferable to adding or subtracting percentages. for example, in the case of adding interest of 6% p.a. or calculating a discount of 25% on some amount, p, they suggest that learners (or students or employees) should be encouraged to work with the multiplicative forms of and rather than . the multiplicative form has several advantages, including its efficiency and its similarity to the compound interest formula. mathematical aspects of teachers’ knowledge top ↑ in this section i distinguish the key features of interest in general and of compound interest in particular. i discuss links between compound interest and related mathematical concepts, and i show that two different derivations of the compound interest formula require different conceptions by learners. whilst mathematics is always in the foreground in this section, teaching issues are continually present, albeit in the background. i draw on ma’s (1999) notions of depth and breadth with regard to teacher’s mathematical knowledge. for her, depth of understanding concerns the ability to connect a concept or a topic ‘with more conceptually powerful ideas of the subject’ (p. 121) so that the power of a mathematical idea is related to its proximity to the structure of the discipline. a mathematical idea that is closer to the structure of the discipline underpins more topics and hence has more ‘mathematical influence’ (my term) and thus mathematical power. by contrast, breadth of understanding is related to the ability to connect a concept or topic with concepts or topics of similar or less conceptual power. inspired by ma’s work and drawing loosely on her idea of knowledge packages for primary mathematics, i propose the hierarchies shown in figure 1 and figure 2. concepts that are higher in the diagram build on those that are lower down. i use italics when referring to the nodes (text boxes) in the diagrams. figure 1: a hierarchy of interest concepts. figure 2: a network of concepts relating to growth factor. in network diagrams such as these it is often possible to argue that all components are linked in some way to each other, and thus to insert links between all nodes in the diagram. this is seldom productive because it obscures the key relationships in the mass of connecting lines. i have therefore chosen to represent only the key links between components. a hierarchy of interest concepts in figure 1 i present a hierarchical network of links between key mathematical concepts in simple and compound growth. the central (purple) cluster indicates mathematical concepts, with percentage at the lowest level, and extends to progressions. following ma (1999), this arrangement reflects that, for example, linear growth is a mathematically more powerful concept than geometric progressions because it underpins a larger number of mathematical topics (although other topics that it underpins, such as rates of change, are not indicated here). the right (green) cluster and left (light blue) clusters indicate concepts dealing with increasing and decreasing growth respectively. at the lowest level of the diagram, links are shown between percentage, percentage change and their application in financial contexts in the form of percentage points and basis points. building on percentage, and based on the recommendations of bakker et al. (2006), is the concept of growth factor. i use this term for the factor that is multiplied by the ‘starting value’, which could be the principal amount, original price, and so on. in the case of simple increase or decrease it is and in the case of compound increase or decrease it is . i shall use the term unit growth factor for in both simple and compound increase and decrease. this term is not indicated in the diagram. moving vertically, the diagram contains three horizontal clusters: linear growth, exponential growth and progressions, all of which build on growth factor. i place linear growth below exponential growth since constant additive change is a simpler concept than constant multiplicative change (brown, küchemann & hodgen, 2010). at the top level, progressions builds on the relevant growth from the levels below. moving horizontally within each cluster, linear growth is applied in the contexts of simple interest and straight line depreciation whilst exponential growth is applied in compound interest and reducing balance depreciation. geometric progressions are applied in annuities. the bold lines connecting growth factor to simple interest, compound interest and annuities indicate the importance of the notion of growth factor in calculations involving each of these concepts. the outer (dark blue) boxes on either side of the diagram indicate connections between the aspects of financial mathematics and other key mathematical concepts such as nth roots, logs and horizontal asymptotes. this reflects ma’s (1999) notion of breadth. these mathematical concepts are necessary when dealing with various aspects of compound increase and decrease at school level. for example, logs are required when determining the length (or number of compounding periods) of an investment. the connection with e applies to continuous compounding. a network of concepts relating to growth factor in figure 2, i expand the growth factor component of figure 1 by focusing on the processes and concepts involved in calculating interest with particular reference to the school curriculum. i therefore refer to interest at the base of the network. i distinguish three main components: the calculation method, the interest rate and the compounding frequency. each of these has several interrelated sub-components. different colours (or shades of grey) have been used to distinguish the three main components, their sub-components and the links. as in figure 1, i have represented only the key links between components. i begin with the calculation method: interest calculated on the principal amount is simple interest, whereas interest calculated on the latest balance is compound interest. daily interest calculations, as done in banks, are simple interest calculations that produce the balance on which interest is compounded at the end of the month, hence the links to simple interest and compound interest. with regard to interest rate, the key distinction is between nominal rates and effective rates. in addition, the rate per period is an essential component and might even be considered the central idea in the entire network since it connects interest rate, compounding frequency and calculation method. i include the interest rate of an annuity for completeness since it is not required in the school curriculum (because it requires numerical methods). consequently i have not inserted links to other sub-components such as rate per period and compound interest since this would complicate the diagram. the compounding frequency determines the amount of interest that will accumulate. i distinguish annual compounding from multiple compoundings per year, and continuous compounding. once again this is for completeness since continuous compounding is not part of the school curriculum. nominal interest rate is connected to effective interest rate (by means of a broken line) since the two notions are dependent on each other. nominal rate is linked to annual compounding, rate per period and simple interest because it is the rate used in the respective procedures. it is also linked to rate per period since it is one of the inputs that determine the rate per period. rate per period also has links from compound interest and multiple compoundings per year. the nominal rate, together with number of compounding periods, determines the interest rate for the smaller periods. the link from compound interest indicates the need to determine a rate per period when there are multiple compoundings per year. there is also has a link to effective interest rate since the rate per period is used in calculating the effective rate. effective interest rate has links from compound interest, multiple compounding periods, nominal rate and rate per period. these sub-components work together to produce ‘interest on interest’ and the effective rate is the annual interest rate, compounded once, that generates the same yield as interest on interest. effective rate also has a link from annual compounding and simple interest. the annual compounding link exists because the most important effective rate is the annual effective rate which assumes annual compounding. the simple interest link is important because it supports an explanation of the effective rate in the following manner: if accumulated interest is removed from the account after each interest period, the balance returns to the principal amount. this becomes a simple interest situation because interest is calculated on the principal amount each time. by contrast, if the interest remains in the account, then applying the same rate per period and appropriate compounding frequency will produce a higher yield. this yield, expressed as a percentage of the principal, is the effective annual rate. since effective rate can be seen as percentage increase, the link to percentage change in figure 1 has been included. continuous compounding has links from compound interest and rate per period. it assumes the number of compoundings per year tends to infinity, thus reaching the limiting situation which can be shown to equal peit where p = principal, i = nominal annual rate, and t = time in years. it is possible to make a link between rate per period and simple interest. this would be relevant if simple interest is determined more frequently than annually. a hypothetical example is a notice deposit where interest is paid out quarterly and thus a quarterly rate would be used to determine the accumulated interest per quarter. based on my teaching of financial mathematics to pre-service teachers, and conversations with teachers in schools, it appears that the notion of rate per period may be a key element of learning compound interest. it brings together compound interest, multiple compounding periods and nominal rate. this requires further research. deriving a formula for compound interest the compound interest formula is deceptively elegant and its subtleties take time to grasp. in this section i discuss two derivations of the formula from both a mathematical and a pedagogical perspective, and show what it might take to appreciate the formula as an algebraic representation of exponential growth. the first derivation uses an additive approach and requires an operational view (sfard, 1991). it builds directly on the process of adding interest to the principal amount. in figure 3, i illustrate the steps in this process. figure 3: an additive approach to deriving the compound interest formula. in line 1 the principal amount gains interest for one period, so the accumulated amount is the sum of p and the interest on p. this is then factorised to simplify the algebraic expression. the new expression, p(1 + i), needs to be seen as a single entity rather than the product of a quantity and scalar multiple – it is the closing balance at the end of period 1. the same process repeats at the end of each compounding period, but in choosing to work each time with the principal, p, rather than an iterative formula (that would substitute an), the algebraic representation starts to get clumsy. for example, in line 2, the student must recognise that p(1 + i) is the new amount on which interest will be gained, and be able to separate p(1 + i) from the adjacent i in order to recognise the similarity with the structure in line 1. as the process continues, it leads to the general formula for the accumulated amount at the end of period n, p(1 + i)n. in this context, it is obvious that n represents the number of times that interest is compounded.the intermediate steps shown in the ‘process’ column enable one to distinguish between the opening balance for each period and the amount of interest that accumulates in that period. from the point of view of algebraic manipulation, it is fortunate that the expression simplifies elegantly each time, reducing to an exponential form. in the ‘process’ column i have deliberately used square brackets to distinguish between the accumulated amount at the beginning of the period and the new factor of [1 + i] that emerges from factorisation. this step of factorising is key in establishing the formula. if one chooses to expand rather than factorise, one ends up with a very messy expression such as a3 = pi3 + 3pi2 + 3pi + p at the end of period 3. this expression does not point obviously towards the elegant formula. whilst those who have done more advanced mathematics may recognise in this expression the binomial expansion of (1 + i)3, those who are deriving the formula at school level will not have encountered this expansion and so their manipulation takes them away from the desired formula rather than closer to it. this may not be obvious to them – they have expanded and then collected like terms, so they may initially feel that they are making progress. however, in my experience, they soon realise that this formula gets increasingly complex and, although there may be a recognisable pattern and a repeatable process, the output for each period cannot be succinctly reduced to a ‘simple’ formula. students may not easily recognise the multiplicative structure in the process until they are able to produce the simplified expression. they may then accept that the growth is exponential only because of the form of the expression, not because of the process that has led to the formula. this makes it even more important that they understand the derivation of the formula – to recognise that adding to the whole a scalar multiple of the whole produces an exponential relationship. however, a disadvantage of the algebraic compression in the last column is that it may obscure the interest-gaining process until one recognises that multiplying repeatedly by a factor of (1 + i) models the process of growing by a proportion of the original quantity, which, in financial terms, is the gaining of interest on interest. this is not a simple transition. it requires that students operate with the object (1 + i). drawing on sfard’s (1991, 1992; sfard & linchevski, 1994) operational-structural distinction, initially students may see p(1 + i) only as the outcome of the process of algebraic simplification and not as an object to be operated with. once students accept p(1 + i)k and (1 + i) both as objects, they may be satisfied to work as in figure 4. figure 4: a multiplicative approach to deriving the compound interest formula. in figure 4, the student must recognise that for any month k, the expression p(1 + i)k‒1 represents the accumulated amount from the previous month, and that the compounding process can be modelled by multiplying by a factor of (1 + i). there is no explicit separation of the accumulated amount from the interest. each factor of (1 + i) should be seen as a unit growth factor over a particular time period, but it is not possible to determine the interest accumulated for any time period without further (albeit simple) calculations. as bakker et al. (2006) argue, becoming familiar with this multiplicative approach is an essential step in working with percentage. once students accept the exponential form, they are likely to recognise that the increase is always the same multiple of the previous amount. if the unit growth factor should change in the future after a certain number of periods m (where m < n), it may be possible for them to appreciate that the new formula will be an = p(1 + i1)m (1 + i2)n‒m where i1 and i2 are the different interest rates. financial aspects of teachers’ knowledge top ↑ in order to unpack (ball, bass & hill, 2004) the mathematics of compound interest for learners, teachers will likely draw on percentage calculations and exponential growth. in addition, unpacking may involve some reference to the world of banking and the ways in which interest works in banks. in this section i discuss two elements of the contextual knowledge of finance that will enable teachers to make appropriate links between the mathematics of compound interest and banking practice. i begin with daily interest calculations, followed by a discussion of the efficiency and accuracy of the compound interest formula as a model of compound growth. i conclude this section with some suggestions of the broader financial and socio-economic knowledge that would enable teachers to give learners access to the everyday world of finance. interest in the world of banking earlier i noted that the selected textbook examples did not reflect the complexity of the banking world where interest is calculated daily and (in most instances) compounded monthly. this means that interest calculations done each day use a daily interest rate, and interest is calculated on the balance in the account at midnight3. however, the interest is not added to the account until midnight on the last day of the month. thus we need to distinguish between calculating interest and compounding interest. each day banks use a percentage calculation to determine the amount of interest for the day. they do not make use of the simple or compound interest formula. below i summarise the details of daily interest calculations with monthly compounding. elsewhere i have used a ‘bucket analogy’ to illustrate this process of daily interest calculations and monthly compounding. (see pournara, 2012, for a detailed discussion of this analogy.) consider a scenario where r500 is invested for a year at 10% p.a. with daily calculation of interest and monthly compounding, as shown in the spreadsheet extract in table 1. whilst all calculations were done with maximum accuracy, only some columns show six or more decimal places. the daily interest rate is based on a day count convention of 365 days, where the daily rate is calculated by dividing the nominal annual rate by 365 irrespective of the actual number of days in the year. thus the cumulative interest for january is calculated as follows: 10% ÷ 365 × 31 × 500 = r4.25 (to two decimal places). note that more interest accumulates in january than in february despite the opening balance for february being higher. this is because there are fewer days in february. thus we see that in simple interest scenarios, the interest is not constant for each month. table 1: cumulative interest based on daily interest calculations and monthly compounding. if the spreadsheet is continued for a full year, it will give an accumulated amount of r552.356352 (to six decimal places) whilst the compound interest formula gives an answer of r552.356354 (to six decimal places). the difference is less than 0.0003%. the compound interest formula as a model of compound growth the compound interest formula provides a very accurate and efficient model of daily interest calculations with monthly compounding. i discuss this briefly, but see pournara (2012) for a more detailed illustration.in order to compare the accuracy of the compound interest formula with actual calculations done in the bank, one needs to consider its implicit assumptions as well as the different day count conventions. the formula assumes that all months have the same number of days and that there are an equal number of days in each year. there is a range of different day count conventions in use across the world. rand-based products use a 365-day convention whereas the typical standard for us dollar-based markets is 360 days. there are other markets that use an actual-day convention where the nominal annual rate is determined by the actual number of days in the year. in table 2 i summarise the accuracy of the compound interest formula as a model of these three different scenarios. i compare the answer from daily interest calculations (as shown in table 1) with the answer from the formula. i refer to the latter as the predicted answer. by choosing the most extreme case for each scenario, i indicate the maximum possible error. as can be seen, the maximum error is less than 1.72%. table 2: comparison of daily interest calculations versus predicted amounts. in this section i have shown that a gap exists between the notion of compound interest as presented in textbooks and the compounding of interest in the world of banking. if teachers are to bridge this gap for themselves and ultimately for their learners, they require a wide range of knowledge that is generally not found in school texts or university-level introductory financial mathematics texts. for example, they require contextual knowledge such as daily interest calculations and international day count conventions. they also require knowledge of basic modelling practices, including the notion of error in mathematical models, and appropriate metaphors and analogies to explain the daily banking process. knowledge of socio-economic issues and financial literacy ideally, mathematics teachers should give learners access to the everyday world of finance, in addition to mathematical basics of compound interest and annuities. to do this, they require knowledge of a range of financial concepts and conventions in addition to those mentioned above. this might include the notion of the time value of money, which is a fundamental construct underpinning all financial mathematics. knowledge of interest rates such as the repo rate and prime rate is important since these ultimately impact the banks’ rates. changes in the repo and prime rates provide opportunity to gain an appreciation of the substantial impact over time of small changes in interest rates, particularly on loans. the distinction between nominal and real interest rates is important because the latter takes inflation into account; thus, some knowledge of inflation is also beneficial for teachers. teachers also require knowledge of general socio-economic issues and financial literacy. it is well-known that levels of financial literacy in south africa are low (eighty20, 2008), that levels of national and personal debt are extremely high, and that the general public does not appreciate the negative impact of compound interest on borrowed money (south african reserve bank, 2012). therefore, mathematics teachers have a moral imperative to help learners to appreciate the power of compound interest on both investments and loans, thus increasing learners’ levels of financial literacy. whilst some might argue that this is not the responsibility of the mathematics teacher but rather of teachers of commercial subjects, i argue that an appreciation of the power of compounding comes from knowledge of exponential growth and it is the mathematics teacher who will open up learners’ access to this knowledge. inevitably this raises concern about the breadth (ma, 1999) of knowledge required of mathematics teachers. it could be argued that the knowledge of financial and economic issues identified above forms part of the knowledge base for general financial literacy of economically active citizens, and so a mathematics teacher who is financially literate will possess sufficient contextual knowledge. however, the detailed knowledge of daily compounding and day count conventions is not part of general financial literacy. this suggests that mathematics teachers’ knowledge of financial and economic issues is specialised and extends beyond general levels of financial literacy. pedagogical aspects of teachers’ knowledge top ↑ this section is concerned with aspects of a teacher’s knowledge base for compound interest that are primarily pedagogical. however, as will be seen, mathematical aspects are always present in the background, and at times are brought into sharp focus. i begin with two instances of student error that emerged in the study, and conjecture that these errors may be a consequence of overgeneralisation of previous learning (olivier, 1989). the first error relates to multiple compounding periods and the overgeneralisation of linear thinking. the second relates to depreciation and the overgeneralisation of reversible operations. thereafter i make suggestions for the teaching of compound interest with particular attention to key tasks and exemplifying mathematical practices. student error – multiple compounding periods and linear thinking results from the study (pournara, 2013) suggest that students have difficulty in distinguishing equal percentages of interest from equal amounts of interest, and may assume that equal interest rates per period imply equal amounts of interest per period. thus, they do not distinguish adequately between relative quantities and absolute quantities. this error may not be exposed unless one explicitly focuses on the amount of interest per period, because the compound interest formula provides the final amount, which obscures how interest accumulates each period. the error may be evidence of overgeneralising of linear thinking (de bock, van dooren, verschaffel & janssens, 2002; esteley, villareal & alagia, 2010), but it may also be a consequence of distortions from rounding and of the way we talk about accumulating interest. when dealing with multiple compoundings, the nominal annual rate is sub-divided into equal portions, depending on the number of compounding periods in the year, producing an effective interest rate per period. for example, a payment of r300 growing at a nominal annual rate of 9% p.a. compounded monthly gains 0.75% interest each month, since 9 is divided into 12 equal portions of 0.75. however, the amount of interest associated with this monthly rate is different each month because of compounding on the latest balance. in table 3 it can be seen that interest for the first month is r2.25 but interest for the second month is r2.27 (rounded to two decimal places). table 3: different interest amounts per month based on increasing balance. adding the twelve interest amounts gives total interest of r28.14, which is confirmed by the compound interest formula . but, although the formula is efficient, it hides the calculation of interest each month and thus the fact that the amount of monthly interest is not constant. the ratios of the interest from month to month are clearly constant (at 1.0075) but the arithmetic difference is not constant. however, this may be obscured or distorted by the chosen numbers and by rounding. for example, the differences in interest between months 9 and 10, months 10 and 11, and months 11 and 12 are 2c, 2c and 1c respectively. this appears to contradict the idea that amount of interest increases each month, and that the amount by which it increases also increases. this apparent contradiction is a consequence of rounding. table 4 gives the amount of interest rounded to five decimal places and shows that the interest amounts are indeed growing each month, and that the rate of increase is also increasing. however, when these figures are rounded to two decimal places, they produce the incorrect picture shown above. table 4: amount of interest rounded to five decimal places. another contributing factor is the choice of numbers. the choice of r300 and 9% in the above example yields very small amounts of monthly interest and even smaller second order differences. in order to illustrate the non-constant arithmetic difference more clearly, it would be wiser to choose values such as r5000 and 20%. however, it should then be acknowledged that an interest rate of 20% p.a. is not a realistic figure in the current financial context. a third contributing factor is the way we talk of interest. we speak of adding interest to the balance at the end of each month, and so it is not surprising that our tendency is to reason additively when comparing the interest amounts in the table. whilst we are indeed adding interest, the amount we add is generated by a multiplicative process. student error – percentage change and reversible operations as discussed above, percentage is a key concept underpinning financial mathematics, but it is a difficult concept to learn. one of the biggest sources of difficulty stems from the asymmetric nature of percentage increase and decrease; this difficulty extends into adulthood (parker & leinhardt, 1995). for example, increasing some amount a by 10% produces a new amount a', but decreasing a' by 10% does not fully undo the increase and so the answer is not a. it seems reasonable to conjecture that the source of this error is an over-generalisation of the symmetric nature of the familiar operations of addition-subtraction and multiplication-division. in financial mathematics this appears to lead to errors in distinguishing depreciation from discounting. in order to appreciate students’ difficulties, it is useful to compare and contrast appreciation, depreciation, compounding and discounting. in figure 5 i define each term and provide a formula for each process. figure 5: definitions of terms. the formulae show that the notion of unit growth factor applies to all four processes, that appreciation and compounding make use of the same formula, and that compounding and discounting are reverse operations (and hence symmetrical), but appreciation and depreciation are not. in my experience, students may view depreciation as the reverse of appreciation or compounding. consider question 2 from the textbook examples discussed above. using the compound interest formula, we can determine that . however, students may approach such problems by depreciating the future value for 20 periods as follows: . this error suggests that students are treating compounding and depreciation (with reducing balance) as reverse operations. this is not surprising on one level because the compounding process ‘adds on’ a percentage of the latest balance at the end of each compounding period, whilst the depreciation process removes a percentage of the balance. the error lies in the referent to which the percentage applies, but this may be less obvious when using the formulae than it is when doing a single percentage change calculation. whilst the discussion above may not be appropriately calibrated for school level, it highlights some of the knowledge required by teachers. firstly, mathematical knowledge that the ‘multiplicative differences’ (i.e. ratios) are constant but the ‘additive differences’ are not. secondly, knowledge of how learners may reason; for example, (1) learners may notice the different gaps between amounts and question the apparent contradiction that the rate of increase of the interest is not always increasing, (2) that a comparison based on additive reasoning is intuitive and sensible, (3) that the ‘additive problem’ does not lie in rounding of monetary amounts, but (4) that a comparison based on ratios is constant (and is based on the unit growth factor (1 + i). furthermore, teachers require knowledge of how to explain that the ratios are equal, and why the differences are not. this is indeed a challenging task. implications for teaching compound interest as noted earlier, the curriculum constraints place substantial limitations on what might be achievable in terms of teaching and learning compound interest. nevertheless, in this section i make some suggestions for opportunities to capitalise (pun intended!) on compound growth to expose learners to more general mathematical practices. i also identify four key tasks that lever up key mathematical ideas that extend beyond the realm of compound interest. although compound interest tasks at school level tend to focus on numeric work, there are various opportunities to attend to other mathematical practices, such as: • working to different levels of accuracy and the impact of rounding on financial calculations. • working inductively, as exemplified in the derivation of the simple and compound interest formulae. • working with inverse and reversible operations; see above for a discussion of appreciation, depreciation, compounding and discounting. • working with proof on two cases of counter-intuitive phenomena related to compound growth: students can use school-level algebra to prove that the time for an amount to double is independent of the amount of money, and graphical illustrations can be used to show powerfully that compound growth is slower than linear growth within the first compounding period. related to these practices i suggest three tasks that foreground the key ideas of compound interest whilst simultaneously exemplifying key mathematical ideas: • strategic choices for interest rates that do and do not lead to a recurring decimal value in relation to the compounding period. such choices are determined by the goals of the task: it may not be desirable to include the complexity of a recurring decimal when focusing on the impact of the number of compounding periods, but it is essential to work with a recurring decimal value to emphasise the impact of rounding errors in financial calculations; for example, 6.5% p.a. compounded monthly gives a nominal monthly rate of 0.541666…%. • when compounding, the growth of any amount tends to a limit. based on the limited range of examples of multiple compounding periods that learners are exposed to, they may assume that the amount of interest increases without bound as the number of compounding periods increases. however, it can be shown both numerically and analytically that the accumulated interest tends to a limit as the number of compounding periods tends to infinity. (see samson & pournara, 2013, for a detailed discussion.) • simple interest grows faster than compound interest in the first compounding period. whilst learners soon appreciate that compound interest has a higher yield than simple interest at the same interest rate over the same time period, it is not intuitive that this relationship is reversed during the first compounding period. for example, if r300 is invested at 12% p.a. with simple interest (1%) added at the end of each month, by the end of month 6 it will have grown to r318. by contrast if the same amount is compounded annually at the same rate, it will grow to only 300(1.12)0.5 = r317.49 by the end of six months. conclusion top ↑ in this article i have illustrated the specificity of teacher’s mathematical knowledge for teaching compound interest, and have argued that it has mathematical and pedagogical components, as well as requiring contextual knowledge of finance. i have proposed a theoretical elaboration of the notion of compound interest and discussed two errors made by students in the broader study. in discussing key practices of working with interest in the banking sector, i have shown how these practices are removed from the typical examples of compound interest questions found in textbooks. much work remains to be done to research the teaching and learning of compound interest, and the extent to which teachers’ knowledge enables learners not only to answer textbook and assessment questions, but also to develop a useable knowledge of finance beyond the classroom. acknowledgements top ↑ this work was supported financially by the thuthuka programme of the national research foundation (nrf, grant no. ttk2007050800004). any opinions, findings and conclusions or recommendations expressed are those of the author and the nrf does not accept any liability. competing interests i declare that i have no financial or personal relationships that may have inappropriately influenced me in writing this article. references top ↑ adler, j. 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(2000). a theoretical framework for analyzing student understanding of the concept of derivative. cbms issues in mathematics education, 8, 103–127. footnotes top ↑ 1. in this article i focus only on the secondary mathematics curriculum although financial mathematics is also a component of the mathematical literacy curriculum. 2. i refer to both students and learners in this article. in some instances the distinction is obvious, as in references to research with university students. in other cases the ambiguity is deliberate because i make the assumption that errors made by pre-service teachers are also likely to be made by learners in schools. 3. in reality banks may not carry out this process exactly at midnight. it will take place at some point after close of business, and forms part of a broader daily reconciliation process that may take several hours to complete. i therefore refer to ‘midnight’ in a metaphorical sense. the key issue is that interest is calculated (or compounded) at discrete points in time and not continuously. abstract introduction for the study literature review theoretical background methodology results and discussions conclusion acknowledgements references about the author(s) ifunanya ubah department of science and technology education, university of johannesburg, johannesburg, south africa sarah bansilal department of science, technology and mathematics education, university of kwazulu-natal, durban, south africa citation ubah, i., & bansilal, s. (2019). the use of semiotic representations in reasoning about similar triangles in euclidean geometry. pythagoras, 40(1), a480. https://doi.org/10.4102/pythagoras.v40i1.480 original research the use of semiotic representations in reasoning about similar triangles in euclidean geometry ifunanya ubah, sarah bansilal received: 27 mar. 2019; accepted: 08 nov. 2019; published: 13 dec. 2019 copyright: © 2019. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract many pre-service mathematics teachers in south africa are apprehensive about the content of euclidean geometry, because they did not study euclidean geometry in high school but will be expected to teach the content when they start their teaching career. this article reports on a study that explored the role of semiotic representations in pre-service teachers’ reasoning about the similarity relationship between triangles. data were generated from the written responses of 65 pre-service mathematics teachers as well as three semi-structured interviews. duval’s notions of conversions and treatments were used as a framework to understand the pre-service teachers’ struggles with negotiating movements between the visual and symbolic registers of representation. the findings revealed that many pre-service teachers struggled with identifying the similarity relationship between triangles appearing in various configurations of geometric objects. while some participants were easily able to draw upon the two registers to express the relationships, one student who initially made many errors was only able to discern the necessary relationships with the help of a concrete representation that could be physically manipulated. the study therefore provided an example of how a student’s errors could be used as a learning resource to lead to meaningful learning. keywords: duval; euclidean geometry; conversion; representations; preservice mathematics teachers; similarity; similar triangles. introduction for the study mathematics outcomes in south africa are very low and many researchers and stakeholders have expressed concerns about the poor performance in mathematics at school level, especially in geometry (luneta, 2015; mthembu, 2007; singh, 2006; van putten, stols & howie, 2010). euclidean geometry forms part of the core curriculum of mathematics in the further education and training (fet) band in south africa; however, many learners seem to have the opinion that the study of geometry is irrelevant to their daily lives (patkin & levenberg, 2012). in south africa, curriculum developers did not seem to be convinced about the important role of geometry in the mathematics curriculum. in 2006, as part of the many changes brought in by curriculum 2005, for instance, the euclidean geometry strand was made optional for those learners who opted to study mathematics in the fet band (department of education, 2006). one of the reasons for making it optional was the perception that teachers did not know the content well enough (bowie, 2009). when geometry was made optional, many learners chose not to study the strand. it was found that less than 4% of the grade 12 mathematics learners in 2008 wrote the third examination paper in mathematics where geometry was included (van putten et al., 2010). however, the education authorities changed their minds about the importance of geometry and in 2011 geometry was made compulsory again for these fet grades as part of the curriculum and assessment policy statements (department of basic education, 2011). when it was brought back into the core mathematics curriculum, teachers did not feel as confident about the strand since it had not been taught for such a long time. some researchers note that teachers avoided the teaching of geometry in school because of poor mastery of euclidean geometry (atebe & shaefer, 2009; ndlovu, 2011). some teachers find the euclidean geometry section difficult, even if they had studied it in high school and at tertiary level, let alone those who did not study euclidean geometry in high school or at tertiary level. many of our pre-service teachers were understandably anxious because they would be expected to teach the content when they start their teaching career. in order to help these students overcome their fears, the mathematics education department at a kwazulu-natal university designed a series of 3-hour workshops based on euclidean geometry that were run over six weeks. one of the key euclidean geometry concepts covered in the workshops was that of similarity of triangles, and it is this concept that this study focuses on. there is very little literature on students’ understanding of similar triangles using a euclidean geometry approach, and we hope that this study can add to knowledge in this field. in this study we delve into the area of semiotics which is the study of signs and sign symbols, how these signs are used to signify actions or objects, and the interpretation of these signs (moore-russo & viglietti, 2012) as noted by ernest (2006), a semiotic system is characterised by a set of elementary signs, a set of rules for the production and transformation of signs and an underlying meaning structure deriving from the relationship between the signs within the system (ernest, 2006). duval (2006) noted that mathematical objects cannot be perceived directly, but accessing them is bound to the use of representations. hence the role of semiotic systems of representation goes beyond just a means of labelling mathematical objects but they allow a person to work on mathematical objects and with them. the purpose of the study was to explore how semiotic representations influence students in their reasoning about the similarity relationship between triangles. to achieve this goal, the following research question was addressed: what role do semiotic representations play in pre-service mathematics teachers’ reasoning about the concept of similar triangles? literature review geometry is an essential part of mathematics and provides unique opportunities for mathematical modelling by drawing upon real-life examples (usiskin, 2002). the study of geometry provides opportunities for learners to visualise concepts that may be related to other areas of mathematics, including trigonometry, patterns and measurement. it has links with culture, history, art and design and it is the interaction with these vital human activities that provides opportunities to make geometry lessons interesting and stimulating (chambers, 2008). however, learners often do not view geometry as being relevant to their lives which may be because the logical and structured approach used in the study of geometry is so different from their previous experiences (patkin & levenberg, 2012). many learners find the study of euclidean geometry challenging (ngirishi & bansilal, 2019; singh, 2006; van putten et al., 2010). nationally in 2008 only 3.8% of the grade 12 mathematics learners wrote the optional paper 3 with almost half of them achieving less than 30% in that national examination paper (van putten et al., 2010). furthermore, in the latest national senior certificate mathematics examinations in 2018, based on a random sample of grade 12 learners’ responses, it was found that, in the second mathematics paper, learners performed worst in the euclidean geometry question based on similarity of triangles (department of basic education, 2019, p. 143). this result supports patkin and lavenberg’s (2012) contention that geometry is seen as the most complicated strand of the mathematics curriculum. it seems that the emphasis on the learning of deductive proofs of theorems, the correct use of symbolic notation and the structured requirements of providing appropriate reasons for statements that are made make the subject seem complicated (department of basic education, 2019). furthermore, the complexity of disentangling the various figures that make up more complex figural arrangements whose properties need to be discerned render the strand even more challenging for learners (department of basic education, 2019). many students are daunted by the learning of the formal logic and deductive reasoning that are necessary elements of euclidean geometry. one of the reasons why students find geometry difficult is the emphasis on the deductive aspect without a corresponding focus on the underlying spatial abilities (del grande, 1986). the use of deductive reasoning is an integral part of the study of geometry and is used to develop proofs about properties and relationships within and among figures. a proof is a set of deductive steps that are used to create a narrative starting from a known fact. it then proceeds in a step-by-step manner where each step is deduced from the result of the previous one until the unknown fact is justified (ngirishi & bansilal, 2019). it is generally expected that the proof, made up of the sequential statements, should be supported by valid reasons (serra, 1997). mudaly and de villiers (2004) highlight that the study of formal proof as part of the study of geometry is a useful means of developing deductive reasoning skills. another characteristic feature of the study of geometry is the necessary intertwining of the visual and symbolic or analytic representations where the one representation supports and underpins the others. as pointed out by del grande (1986), the development of spatial abilities is an important part of studying geometry. for some students it is this necessary dependence on the use of more than one representation that is experienced as a challenge. duval (2006) argues that in geometry it is necessary to combine the use of at least two representation systems, one for verbal expressions of properties and the second one for visualisation. duval (2006, p. 107) maintains that the ‘ability to change from one representation system to another is very often the critical threshold for progress in learning’. visualisation is the ability to ‘represent transform, generalise, communicates, document, and reflect on visual information’ (hershkowitz et al., 1990, p. 75). presmeg (1997, p. 304) focused on visualisation as a process ‘involved in constructing and transforming visual images’ which is the view that is taken in this study. bansilal and naidoo’s (2012) study focused on the use of visualisation and analytic strategies by 40 grade 12 learners when working with problems based on transformation geometry. the findings revealed that most learners easily carried out processes and calculations in the analytic mode when responding to the tasks, showing a limited movement across the two modes which are essential for a deepening of understanding of geometry. the authors argued that on the one hand, as the learners’ understanding improves, the learners find it easier to move across different representations. on the other hand, as learners work with different representations of an object, the different aspects emphasised by each representation contribute to a deeper understanding of the properties and relationships associated with the object (bansilal & naidoo, 2012). another study (bansilal, 2012), which drew upon duval’s theory about transformations within and between semiotic representation systems, focused on teachers’ success rates in solving problems based on the normal distribution curve. the findings of the study showed that the teachers achieved a higher success rate in processes that were based in one representational system compared to those processes that required coordination between two different representational systems. sinclair et al. (2016), in their review of research on geometry education, noted that there has been increased attention in recent years on visuospatial reasoning mainly in the teaching and learning of geometry, although researchers refer to this focus by different terms such as visualisation and visualising, spatial reasoning, visuospatial thinking as well as visual reasoning. sinclair et al. (2016, p. 696) note that they have in common ‘the activity of imagining static or dynamic objects and acting on them (mentally rotating, stretching etc.)’. rivera (2011) drew attention to the complexity of reasoning about visual representations. duval (2005), cited in sinclair et al. (2016), identified three cognitive processes as being central to learning. these three processes are visualisation, construction (using instruments) and a discursive process for communicating argumentation and proofs (sinclair et al., 2016). duval argued that ‘attending to the properties of a geometric figure involves dimensional deconstruction’; for example, in order to ‘see’ the properties of a rectangle, a student needs to concentrate on the sides and angles (cited in sinclair et al., 2016, p. 693). however, a study by gal and linchevski (2010) found that students had difficulty with deconstruction of figures when asked to point out shared parts (lines or angles) of triangles that have a common side. the students commonly identified intersecting lines as a shared side or the combination of two angles, with one falling in the one triangle and the other in the second triangle, as being shared or common. rivera (2011) noted that depending on what activity is being done, different kinds of visual representations can be generated which could be influenced by personal intuition, related to the development of a concept or process or produced as a means of solving a problem. in their study of a brazilian classroom, david and tomaz (2012) argued that visual representations should form a central part of structured learning activities. however, in their analysis, they found that learners held a dominant view that geometry required the learning of rules and norms and this interfered in their learning about calculating areas using drawings. it was also found that talking about and establishing common understandings about visual representations helps learners develop a conceptual understanding of geometric concepts (steenpass & steinbring, 2014). gal and linchevski (2010) identified that students experienced difficulties when trying to distinguish between the various configurations present in geometric diagrams. it was a challenge for students to identify those visual characteristics that were relevant to the solution of the problems and they could be side-tracked by those characteristics that were not relevant. the authors also noted that students try to deal with mental objects as if they were physical objects by trying to transform or manipulate the objects mentally. consequently, the difficulty of problems in geometry increases as the cognitive demand of the mental transformation increases. for example, the authors found that students found it easier to identify similar triangles in figures such as shape 1 in figure 1 (δab′c′ and δabc, where ∠ab′c′ and ∠abc are right angles) than that of shape 2(δklm and δknl, where ∠klm and are ∠knl right angles) because the second shape requires a greater cognitive effort for the mental transformation. figure 1: shapes requiring different mental transformations. identifying and understanding the errors that students make during the process of constructing their knowledge has occupied the attention of many researchers. however, such an enterprise is also valuable for teachers since knowledge of these errors can be used as a learning resource in their classrooms. chauraya and brodie (2018) argued that teachers need learning opportunities that can allow them to develop the skills of eliciting learners’ thinking about the errors they made. they found that as teachers focused on understanding learner errors, the teachers also improved their own mathematics knowledge (chauraya & brodie, 2018). going beyond just an acknowledgement of correct or incorrect answers, towards actively using the incorrect answers productively as a means to enhance understanding, may need a mind shift on the part of teachers as well as researchers. in this study we offer an example of how one student’s incorrect answers were used as a resource to develop meaningful learning. theoretical background according to duval (2006), there is no mathematical processing that can be performed without using a semiotic system of representation. this is because mathematical processing always involves substituting some semiotic representation for another. duval’s focus is not on single representations but on systems of representations which have rules for performing transformations of representations within the system without changing the mathematical object that is used. duval referred to these systems of representations as registers and specified two different types of transformations of semiotic representations that can take place during any mathematical activity, namely treatments and conversions. treatments involve transformations from one semiotic representation to another within the same system or register, while conversions involve changing the system while retaining the reference to the same objects (bansilal & naidoo, 2012). an example of a treatment could be carrying out a calculation while remaining in the same notation system. transforming the representation into the representation is a treatment because the notation system has remained the same. however the transformation from to 0.5 + 0.25 is an example of a conversion, because the algorithms for the addition of the decimal numbers are different from that of the numbers written in fractional form (duval, 2006). duval (2006) notes that a conversion comes in ‘for the purpose of choosing the register in which the necessary treatments can be carried out most economically or most powerfully’ (p. 106). for example, when asked to show that the opposite angles of a cyclic quadrilateral are supplementary, it would be very cumbersome to write out a verbal argument, whereas it would make sense to provide a visual representation of the quadrilateral which can be used to support and clarify the argument. a further purpose of conversions, according to duval (2006), could be to provide ‘a second register to serve as a support or guide for the treatments being carried out in another register’ (p. 106). an example of this function is when one is asked to solve a trigonometric equation such as: sin2x + cos 2x − cos x = 0. the algebraic register can serve as a support to expand this trigonometric expression using the rules of algebra (once the expansion for cos 2x is identified). methodology this interpretative study was located within a six-week intervention that was designed to help pre-service mathematics teachers improve their understanding of fet euclidean geometry. the purpose of the study was to explore how students used semiotic representations in reasoning about the similarity relationship between triangles. the participants of the study were 65 students who enrolled for the intervention. the data for the study were generated by the written responses of the participants to one task based on similar triangles, as well as semi-structured interviews that were conducted with 13 participants who volunteered to be interviewed about their understanding of the concept. the interviews were video-recorded and then transcribed verbatim by the first author. in order to ensure reliability, the transcripts were checked by the second author against the original recordings. for the purpose of this study, we draw upon interviews with three participants: sabelo, celo and vince, to highlight particular ways in which the semiotic representations were used to reason about the similarity of triangles. sabelo and celo had not studied geometry in the fet phase of their schooling while vince had written the third mathematics paper in grade 12, which was optional, and which included the study of euclidean geometry. the purpose of the interviews was to probe their reasoning about the concepts in the written tasks. however, the interviews were also used to help improve the interviewees’ understanding, hence they were interspersed with explanations of key concepts where necessary to clarify the thinking and strategies used in responding to the questions. ethical considerations ethical clearance for this research was obtained from the research ethics committee of the education faculty at the relevant university (ethical clearance number hss/0425/018a). after obtaining approval, each participant gave their written informed consent to participate in the research, allowing the use of their responses to the written task and interview extract for research purposes and assured anonymity in the use of these data. results and discussions we discuss the participants’ responses to the task which was based on identifying and naming the similar triangles that emerged from various geometric figures. the identification of the equal angles within the triangles to confirm the similarity of pairs of triangles, required knowledge of the properties of these figures which are typically studied in fet mathematics. the item analysis for the task based on the pre-service teachers’ written responses is presented in table 1. table 1: frequency of correct, wrong and no responses for items in the task. from the results in table 1, 39 (60.0%) out of 65 participants answered question 2.2 correctly, while 32 (49.2%) participants answered question 2.3 correctly. thirty (46.2%) participants answered question 2.1 correctly. question 2.4, question 2.5 and question 2.6 recorded poor performance while question 2.7 had the fewest correct responses (21.5%). the difference in success rates for question 2.2 (60%) and question 2.7 (21.5%) is quite striking, considering that the underlying figure in both diagrams is that of a crossed quadrilateral. the increased difficulty of question 2.7 supports the assertion by gal and linchevski (2010) that the difficulty of problems in geometry increases as the cognitive demand of the mental transformation increases. the mental transformation required in question 2.7 is more difficult to achieve than that of question 2.2 because δpqt needs to be mentally reflected across a vertical line so that the new configuration of the triangle enables the identification of the matching angles in the two triangles concerned. for question 2.2, the configuration of the triangles is such that the matching angles are easily identified. a second reason for the large number of correct responses in question 2.2 may be because participants were very familiar with the theorem that ‘angles on the same segment’ of a circle are equal (see interview extract for sabelo below for confirmation). the students also struggled with question 2.4, question 2.5 and question 2.6 whose complex figures may have limited the participants’ correct response. for questions in the task to identify the matching pairs of angles and hence the order of the triangle representation (such as δprs ||| δptq in question 2.2), it is helpful to move to the visual representation so that the properties of the figures could be discerned. figure 3 presents the response of sabelo who had problems with naming the triangles in the correct order. in figure 3, sabelo has been able to identify the triangles, but he was unable to represent most of the triangles in the correct naming order. he did not have a problem with question 2.2 however as he explained in the extract from the interview below. this shows that sabelo was comfortable with working with the visual representation in question 2.2 which drew upon the ‘angles of the same segment are equal’ theorem, and he was consequently able to represent this using the symbolic representation δprs ||| δptq: r: which of the six figures do you find very easy to solve? sabelo: question 2.2. r: why? sabelo: because the theorem involved is very easy; angles of the same segment are equal. r: then which angle is equal to what? sabelo: [gesturing at the points q and s in the figure] ∠q = ∠s [same segment], ∠t = ∠r, then p is common. hence δprs ||| δptq. figure 2: details of the task. figure 3: a representation of sabelo’s response. as seen above, sabelo had no problems identifying the angles that were equal in the two triangles using the ‘angles in the same segment are equal’ result and thereafter representing the symbolic relationship between the similar triangles correctly. the interview continued, where sabelo was probed about other questions: r: what are we required to do in question 2.1? sabelo: we are required to show that δbqp is similar to unknown triangle. r: how do we find the unknown triangle? sabelo: the diagram is confusing. r: let’s start by identifying the equal angles. sabelo: we start by identifying the equal angles; like ∠b is equal to ∠q (tangent and chord theorem), ∠q in δbqp is equal to ∠p in another triangle ∠p, is equal to ∠a then δbqp is equal to δqpa r: why is ∠q = ∠p? sabelo: because ∠p is exterior to ∠q. r: do you mean that ∠p is also exterior to ∠a? sabelo: yes, then δbqp ||| δqpa r: [moving on to the other question] now in questions 2.4 and 2.5, how do we get the unknown triangles similar to δpqr? sabelo: that question is confusing because of that 90°. i wanted to say that ∠s should be common, but s is not in δpqr. i need to study the question. in the above extract, sabelo’s misconception that ‘∠p is exterior to ∠q’ which made them equal was revealed. based on this incorrect deduction, sabelo has named the triangles in the order δbqp and δqpa respectively. sabelo’s steps can be viewed as a treatment that was carried out within the symbolic register. he has used the pairs of angles he identified as equal from the two triangles to set out the order of naming the two triangles and hence represented the similarity relationship incorrectly as δbqp ||| δqpa. with respect to question 2.4 and question 2.5, he was at a loss and did not know how to figure out the triangles that were similar or identify the equal angles. further problems with his reasoning in the visual representations are revealed in the following extract: r: what of question 2.7, how do we get the unknown triangle? sabelo: [he first draws the two triangles rts and pqt separately.] pq is parallel to rs, so it means that ∠q = ∠s [alternate angles], also ∠p = ∠r, and ∠t is common, hence δrts ||| δptq. the above interview extract shows that sabelo has once again made an incorrect deduction when working with the visual representation and identified angles incorrectly as being equal (). furthermore, his understanding of ‘common angle’ is similar to the misconception identified in the study by gal and linchevski (2010). as he did for question 2.1, sabelo applied the incorrect deduction to carry out a treatment within the symbolic register to represent the similarity relationship incorrectly as δrts ||| δptq. vince was one of the students who correctly represented the similarity relationships for all the questions in the task and agreed to be interviewed. during the interview he was asked to explain how he arrived at the correct answer for some of the items. r: looking at the given task [pointing to the script of the participant] what do you observe? vince: there are six different shapes included in different triangles and we are required to name triangles similar to the given triangle in correct order. r: how do we find the unknown triangles? let’s take 2.3. vince: in 2.3; δabe, is similar to? er let’s take δdce. so, we have ∠a = ∠d1 since line dc is parallel to ab, these are corresponding, and ∠b is equal to ∠c1 also corresponding angles. therefore, δabe is similar to δdce … r: now in questions 2.4, 2.5, 2.6 and 2.7, how do we get the unknown triangles similar to given triangles? vince: [he first marked off the equal angles in the diagram.] for question 2.4 and 2.5; δpqr is given; ∠p [pointing to δpqr] is equal to ∠s [pointing to δspr]; ∠q [pointing to δpqr] is equal to ∠p from δspr and ∠r = ∠r, hence δpqr ||| δspr [addressing the first part of question which is 2.4. question 2.5 is the second part which required one to show that δpqr ||| δsqp or alternately δspr ||| δsqp] from the interview extracts above, vince is comfortable with working with the visual representation and effortlessly moves to the symbolic representation. for question 2.2 he was able to connect the visual and symbolic representations without any hesitation, although for question 2.4 he first spent some time working within the visual register so that he could identify the equal angles using the visual representation before expressing the relationship symbolically. celo’s responses in figure 4 show that he was only able to identify the pair of similar triangles, and name them in the correct order for question 2.3, while his answers for the other six questions were wrong. celo was interviewed about his responses. r: if we look at your answer for question 2.3, it is δabe ||| δdce, how did you arrive at that answer? celo: this was easy because you have two parallel lines and dc is cutting the big triangle and parallel to ab. since the lines are parallel to each other, they divide the biggest triangle into two triangles, so the bigger triangle is similar to the smaller triangle. figure 4: a representation of celo’s response. his response is focused on how he identified the two triangles that are similar, but he has not mentioned the order of naming the triangles. he was then probed about this. r: but why would you say δdce, why not δcde? if you got δabe similar to δcde how did you work, it out so that you wrote δabe ||| δdce? celo: i just wrote it. hence, it was clear that celo did not assign any significance to the symbolic representation δabe ||| δdce and assumed it was sufficient to just identify the pairs of triangles that were in a similarity relationship. he seemed to have used the same reasoning for question 2.6 – he was pleased to just identify the triangles that were similar. the researcher spent much time explaining the significance of the order when using the ||| notation. the symbol ||| is a specialised notation, for example δabe ||| δdce shows that the two triangles are similar. however, the symbol ||| goes further than that and specifies the order of the corresponding angles (and sides). that is, angles a, b and e of δabe are respectively equal to angles d, c and e of δdce, while the ratios of the lengths of the lines ab:dc = be:ce = ae:de. in continuing, the researcher then asked him to try and work out the correct representation for the similarity of the triangles in question 2.1. in considering δbqp: r: so, let’s try to get the angles matched for δbqp, then we can get the right order. celo: okay er b er = q1 because of tangent … p is the same, and q = a2 [mumbling] r: right so δbqp is similar to triangle? celo: qap, but i was right [referring to his written answer that δbqp ||| δpaq] i identified the triangles, it is the right triangle. r: but the order was not correct. celo then went on to correctly represent the similarity relationship for question 2.2. he was then probed about question 2.4 and question 2.5 which he found difficult to work with. the researcher explained how the order could be found by taking ∠q = x, then working out ∠p1 to be 90º − x, then ∠p2 = x and ∠r = 90º − x. celo understood the calculations but did not seem convinced. he said he had a different method and tried to explain it: c: my method is different from yours – i use logic. i have δpqr here … now we got three triangles here. … now my method is like logic … i will take this triangle [referring to δspr] and put it over here [he gestured with his hand showing that he was moving the triangle to coincide with δpqr] r: are you trying to move it in your head, like inverting it? c: [continues] yes so p will come to this point [gesturing to point r on δpqr] the s will move over to like this [gesturing to ‘p’ on δpqr and ‘s’ on δspr]. r: can you draw it for me? can you tell me your order? c: [celo tried to draw the triangles that were manipulated to show the corresponding pairs of equal angles but did not quite succeed. he wrote δpqr ||| δsrp]. it was evident that celo was trying to visualise how the triangle could be moved around so the two triangles were oriented so that the similarity of the shapes could be easily discerned. however, it was difficult to mentally transform the image in his mind to match the symbolic representation that was needed. at this stage, the time was up so it was arranged that celo would meet up again and demonstrate his method. at the next meeting, celo came prepared with three different coloured triangles that were arranged as shown in figure 5. figure 5: celo’s cardboard models of the triangles. in figure 5 on the left-hand side, celo placed the model of δpqr using white cardboard on the desk with the angles marked as given. the right-hand side of figure 5 shows two other triangles overlaid on δpqr. the triangles are in two other colours with δpqs being dark grey while δpsr is made up of light grey cardboard. thereafter, celo then reflected the light grey triangle across the line sr and then moved the triangle so that the angle s in the light grey triangle coincided with angle p in the original white triangle δpqr as shown in figure 6. it could now be seen clearly that the light grey triangle was a dilation of the white triangle. figure 6: reorientation of the light grey and white triangles. by physically manipulating the light grey triangle, he was therefore able to easily match the pairs of equal angles corresponding to one another in the two triangles. he then proclaimed, ‘now you can see that δpqr is similar to δspr’. the reorientation using the physical representation helped him to identify the matching vertices which was needed for the symbolic representation. once the order of the corresponding vertices of the triangles has been established, it becomes easier to work within the symbolic register and express the ratios of the corresponding sides that were in proportion: pq/sp = qr/pr = pr/sr. at this stage the student did not need the visualisation of the two triangles but can make the necessary deductions using just the symbolic expression, δpqr ||| δspr. figure 7 shows how the three triangles in the original figure are rearranged to illustrate their similarity property, making it easy to recognise that δsqp ||| δspr ||| δpqr. figure 7: rearrangement of the three triangles. this demonstration provided insight into celo’s reasoning because it showed that celo was dependent on the cardboard cut-outs as a physical representation that could be manipulated. he needed to ‘see’ the orientation of the triangles so that he could draw out the symbolic representation. this means that he could not work within the symbolic register only – that is, to carry out a treatment, using the equality relationship between the angles of the triangles to express it using the similarity notation. however, when he had the physical triangle models that could be manipulated then he was able to use the similarity notation to express the relationship between the triangles. his problem is that he needed the visualisation processes to be merged with the symbolic representation. working within the symbolic register requires one to be convinced that if angles s, q and p of δsqp are respectively equal to angles s, p and r of δspr, then this means that δspr ||| δsqp. celo’s experiences show how important it is for students to be able to make connections between different representational registers. duval highlighted that it is sometimes necessary to move to another register, that is, to carry out a conversion, so certain properties of an object can be discerned. although an object in one register is the same as the object in another register, each register conveys certain properties that may not be so easily discernable in the other register. this function of conversions was illustrated by celo’s use of the concrete manipulatives which enabled him to manipulate the triangle models so that he could ‘see’ that one was an enlargement of the other. he was not able to discern the equal angles based on the visual representation only and needed to perform the transformations on the physical representation so that the matching angles could be identified. the interview with celo revealed that he found the mental transformation required in the visual representation too difficult. by drawing upon his cardboard cut-outs (physical representation) he was able to carry out a physical manipulation on the triangle models. furthermore, he was not able to engage properly with the symbolic notation of ||| because he used it as a sign indicating that two shapes are similar without giving any consideration to the order of the naming. the symbolic register did not give celo access to the objects and he needed to use the manipulatives comprising a physical or concrete representation so that he could work out the properties of the objects. it was clear that celo needs more opportunities for working across the three registers of representation for the concept of similarity. however, it was to celo’s credit that he recognised that the physical representation would help him access the properties of the objects unlike sabelo, for example, who was stuck when faced with question 2.4 and question 2.5 and did not have the means to move beyond this barrier. sabelo’s problem was that he did not understand the properties of the geometric figures well enough and needs more opportunities to improve his skills in this area. clearly the understanding of geometry requires fluency in moving between the visual representation using geometric figures and the symbolic representations which make use of symbolic notations for congruency, similarity, etc. sabelo was able to work with the symbolic register and carried out the treatments within the symbolic register but expressed the similarity relationship incorrectly because of his incorrect deductions. conclusion in this article, we studied the responses of 65 pre-service mathematics teachers to a euclidean geometry task based on similar triangles and focused on the role played by semiotic representations in identifying and naming similar triangles which arose in various configurations of geometric objects. it was found that most students struggled with the symbolic specialised similarity notation (|||). the symbol ||| is a specialised notation that denotes which two triangles are in a similarity relationship, for example δabe ||| δdce shows that the two triangles are similar. however, the symbol ||| goes further than the identification and specifies the order that the vertices must be arranged when the triangles are configured so that the matching of the corresponding angles and sides is evident. some students handled this challenge by carrying out mental transformations on the objects (gal & linchevski, 2010). when the cognitive demand of the mental transformation was higher, students found it more difficult to identify the similarity relationship. for example, it was found that for a pair of items (question 2.2 and question 2.7) that were similar in configuration except that the mental transformation required for question 2.2 was simpler than that required for question 2.7, students performed much better on the first item. sometimes students can carry out treatments that are based on incorrect deductions which leads to incorrect results as in the case of sabelo. his use of treatments in the symbolic register led to incorrect results because he incorrectly identified angles as being equal. his knowledge of the relationships and properties of the underlying geometry concepts was weak, so his incorrect deductions led him to incorrect formulations of the similarity relationships between triangles. duval (2006, p. 107) asserted that conversion-type activities play a crucial role in developing understanding of a concept and it is often the case that ‘the ability to change from one representation system to another is very often the critical threshold for progress in learning’ and the activity of conversion can lead to the mechanisms underlying understanding. this statement is true especially for the study of euclidean geometry where the artefacts are visual representations of configurations of objects such as angles, lines, polygons, circles, etc. in this study we found that skilful conversion movements between the visual and symbolic registers led to success in solving euclidean geometry problems as illustrated by vince’s approach. duval further notes that a key purpose of conversions in the learning process is to provide ‘a second register to serve as a support or guide for the treatments being carried out in another register’ (duval, 2006, p. 106). vince was able to use the symbolic register skilfully while the visual representation served as a support from which he derived the necessary symbolic representations. vince used the visual representation minimally to identify the matching angles using properties that were discerned from the visual representation and moved easily to express the similarity relationships based on these properties. for one student, the symbolic and the visual representation did not provide him enough access to the objects to allow him to discern the relationship and it was only after carrying out the rigid transformations using concrete representations of the triangles that he was he convinced about the similarity relationship and the order of the naming. celo needed the comfort of the physical representation of the objects that can be manipulated or rigidly transformed in order to facilitate the visual representation showing that the one triangle is an enlargement or dilation of the other as shown in figure 7, figure 8 and figure 9. this rearrangement allowed celo to ‘see’ the relationships between the objects in the visual representation and he could then express these relationships using the symbolic representation. a salient point relates to the fact that vince had elected to study euclidean geometry in school although it was not compulsory while sabelo and celo did not have that opportunity. the geometry workshop was designed to help students such as celo and sabelo. the results of this study showed that the students need much more help in navigating these concepts forming part of the euclidean geometry curriculum. celo’s initiative in making the cardboard cut-outs helped him to concretise some of the relationships embodied in the similarity of triangles concept. perhaps such teaching aids may be useful for other students such as sabelo who did not seem to see the connections between equiangular triangles and enlargements or reductions of the triangles, which are key to the concept of similarity. much of the earlier work in school in the earlier grades that focus on rigid transformations and enlargements or reductions of figures are meant to form the basis for this later work on similarity. hence, the use of these concrete manipulations is necessary for students to develop a more robust understanding of the concept. this suggestion resonates with the advice given by zazkis et al. (1996, p. 455) that ‘moving across [considering equivalent representations] in order to move up [increasing abstractions], at a rate appropriate for [the learners’ needs], may help them to make the connections necessary’. although moving across by deliberately drawing upon other representations may not be easy, zazkis et al. recommend such actions so that the students are able to develop a ‘a richer and more useful understanding of complex ideas’. in conclusion, we hope that this contributes some new knowledge in the field in terms of how learners’ misconceptions or errors could be turned into a resource to promote meaningful learning (chauraya & brodie, 2018), as was done in the case of celo. we consider that such a mind shift among researchers is important so that participants derive direct benefit from the research process, instead of the situation where participants’ errors are identified, elaborated and explained in a report, but very little feedback is given to the learners about their wrong answers. acknowledgements we acknowledge mdutshekelwa ndlovu of the university of johannesburg for proofreading the manuscript before submission, as well as moses mogambery who designed the activities and conducted the workshops. competing interests the authors declare that they have no financial or personal relationships that might have inappropriately influenced the writing this article. authors’ contributions i.u. was responsible for the initial analysis and conducting of interviews as well as drawing up the first draft. s.b. refined the analysis as well as the article. funding information this research was funded as part of the postdoctoral research support provided by university of kwazulu-natal. data availability statement data sharing is not applicable to this article. disclaimer the views and opinions expressed in this article are those of the authors and do not necessarily reflect the official policy or position of any affiliated agency of the authors. references atebe, h.u., & schäfer, m. 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(1996). coordinating visual and analytical strategies. a study of students’ understanding of the group d4. journal for research in mathematics education, 27, 435–457. 14-26 miranda & adler.doc 14 pythagoras, 72, 14-26 (december 2010) re‐sourcing mathematics teaching  through professional development    helena miranda & jill adler  marang centre for mathematics and science education  school of education  university of the witwatersrand  hmiranda@unam.na & jill.adler@wits.ac.za    in any education system, curriculum alone cannot fulfill reform demands imposed onto  teachers to help learners make meaning of subject contents they learn. this paper revisits  the notion of resources as explored in a mathematics teacher professional development  project that focused on algebra learning. as a way to promote the use of manipulatives in  teaching and learning algebra, the project introduced algebra tiles to participating teachers  and  investigated  how  the  tiles  facilitated  the  enaction  of  algebraic  meanings.  the  participating  teachers  learned  different  ways  of  helping  learners  interpret  and  solve  algebraic problems, with the use of algebraic tiles.       the need to make adequate interpretations of algebraic problems before we go into the discussion of the use of material resources, which is the core of this paper, we will first attend to the need for encouraging the enaction of adequate mathematical interpretations of mathematical problems by adopting a vignette from miranda (2004) which represents an algebraic task (figure 1) that was assigned to a group of grade 11 learners. the learners were working in small groups and this vignette represents only what happened in one of the small groups. figure 1: area problem involving a rectangle and a square the group members all expressed that this was an easy problem and that they could solve it quickly. they together constructed the (incorrect) equation (x  1)(x + 2) = x2  2 and tried to solve it. the group was later perturbed by the solution obtained which showed that x = 0. one learner suggested “that can’t be” and they all reasoned that x could not be zero because x is supposed to be a length. find x, when the area of the rectangle shown exceeds the area of the square by 2 cm2. x  1 x + 2 x x helena miranda & jill adler 15 when the teacher passed by, checking on this particular group’s solution, she told them that their solution was wrong. the teacher read out the problem to the group and told them to “think harder” about it and left them. the discussion of the above problem in miranda (2004) revealed how the learners were not able to come up with the solution to this seemingly easy problem. one would expect learners not to have much difficulty in solving problems of this kind especially with the provision of the geometric visual of the objects involved in the problem. yet one wonders what these learners were attending to in such a way that their interpretations of the problem itself did not mediate a successful solution method. miranda’s (2004) deeper analysis of the problem above suggests that one of the factors that could have hindered learners’ adequate interpretation of the problem is the word “exceed”. however, she notes that there could be other factors and that what is important to ask is: what pedagogic approaches teachers could take to help learners make more meaningful interpretations of mathematical problems and their contexts? how differently, for example, would learners approach this problem if they had access to other concepts and materials that allow for mathematical connections between algebra and geometry? in this paper we argue for the adoption of manipulatives, along with many other useful approaches, as a way to enrich learners’ learning experiences as they think mathematically about algebraic problems. the vignette above, indicates how the teacher’s response to the learners’ attempts to solve the problem does not go beyond “try-harder” suggestions and how learners fail to succeed in solving routine algebraic problems. this reflects dominant practice in namibian mathematics classrooms in which there is very little teaching for understanding (amoonga & kasanda, 2010) and discussions of learners’ problem solving methods and solutions rarely occur. it also opens space for working with teachers on expanding their pedagogical repertoires, through, for example, the use of concrete materials and multiple representations. this paper reports on a project that explored this pedagogical challenge with mathematics teachers in namibia. firstly, the idea of teaching resources is introduced. secondly, a re-conceptualization of the term resources is adopted with a focus on the notion of transparency among educational material resources. thirdly and finally material resources known as algebra tiles are discussed as explored in an action research project and how these may mediate the construction of adequate interpretations of algebraic problems. background in any education system, curriculum alone as a document cannot fulfill the current reform demands imposed onto teachers in order to help learners make meaning of the subject content they are supposed to learn. for example, in namibia, a mathematics teacher is expected to be creative and innovative enough to ensure that the teaching of mathematics is as much learner-centered as possible. however, many teachers of mathematics are unable to imagine what a learner-centred lesson could look like, let alone practice it (van graan, 1999). one of the main tenets argued for by those who call for learner-centred teaching is that learners should be actively engaged with the subject content as they try to relate it to their life experiences (norman & sphorer, 1996). in many namibian classrooms, engaging learners with the mathematical content is only done through textbook exercises in which the learners are expected to complete lists of exercises and have them checked right or wrong by the teachers. there is little focus on helping the learners to make sense of the mathematics content either through debate and discussion (miranda, 2004) or through the use of concrete learning aids (miranda, 2007). this paper revisits the notion of teaching and learning resources as explored in a mathematics teacher professional development project1 that focused on the teaching and learning of algebra. the project took place in northern namibia involving 10 secondary mathematics teachers from different schools in weekly in-service training sessions. the initial purpose of the larger study was to explore, together with the participant teachers, learner meaning-making of algebra as they solve mathematical problems. but it 1 this teacher professional development project was part of a larger research study (miranda, 2009a), which explored innovative means of professional development for mathematics teachers with a particular focus on how to help learners realise the need to make adequate interpretations of algebra. re-sourcing mathematics teaching through professional development 16 became apparent that the teachers encountered algebra in particular ways, and that they could only interpret learners’ solution methods to problems as either right or wrong. so, as part of a re-plan2 it became apparent that the teachers needed to first experience different ways of making meaning of algebra before they investigated and tried to interpret learners’ meaning-making of algebra. the implications this had on the study was that, even though learners also participated in the study, their involvement was monitored in parallel to, but not through, the teachers’ involvement, as was initially planned. one way of getting the teachers to experience the notion of meaning-making of algebra, was to engage them with both interpretive and hands-on algebraic problems, in which they explored how such activities open up possibilities for a diverse pedagogic approaches to algebra. for example, the purpose of the sessions, from which this paper emerges, was to investigate how the use of material resources may allow for the meaning-making of algebraic concepts and the explanation of some of the algorithms related to algebraic problem solving methods. in namibian mathematics classrooms, there is little awareness of the need to use material resources for enhancing mathematics learning. in this professional development project, mathematics teachers were engaged in activities of developing and adopting material resources, for mathematics teaching. the paper only discusses how some of those resources facilitated the development of various ways of helping learners construct adequate interpretations of algebraic situations. overall, the participants were also able to notice the differences and similarities between what they knew and what they came to know about the pedagogy of algebra while engaging with the resources provided by the project. re-conceptualising the notion of resources adler (2000) calls for a re-conceptualisation of resources as an important theme in mathematics education. she suggests that the term resource could be thought of “as the verb re-source, to source again or differently” (p. 207). ‘resource’, she adds, may also be considered “as both a noun and a verb, as both object and action” (p. 207). adler further classifies educational resources in three main categories, namely human resources, cultural resources and material resources. firstly, human resources include the teachers themselves and the pedagogic content knowledge that they embody. secondly, cultural resources include resources such as language, time, and other culturally available tools or concepts. thirdly, material resources are, for example, technologies, curricular documents, textbooks, and other tangible objects that may be incorporated into the teaching and learning process. this paper will focus on the third type of resources, material resources, which appear to be lacking or underutilised in many african mathematics classrooms. citing black and atkin (1996), adler (2000) argues that even though, human resources serve as a critical means to the successful implementation of curricular innovation and reform, such a success is equally dependent on the “availability of supportive material resources” (p. 205). at the same time adler cautions that while bearing in mind that limited resources may have a negative impact on learners’ mathematical experiences and performance, it should not be assumed that an increase of material resources will amount to better pedagogic practices. even if it leads to significantly better practices, this will not happen in unproblematic and linear ways (adler, reed, lelliot, & setati, 2002). also, one must realise that “resources are not self explanatory objects with mathematics shining clearly through them” (adler, 2000, p. 207). adler further argues that mathematics education needs to “shift from broadening a view of what such resources are to how resources function as an extension of the mathematics teacher in the teaching [and] learning process” (p. 207). alongside this, and drawing on the work of lave and wenger, adler introduces the term transparency that is needed in the use of any material resources for mathematical teaching and learning. defining transparency in terms of how the resources are contextualised and used, adler contends that resources need to be visible and invisible at the same time. on one hand, they need to be seen (visible) “so that they can be used [touched, felt, manipulated] and so extend the practices” (p. 214). on the other hand, resources need to be seen through (invisible) “so that they allow smooth entry into the practice” (p. 214). 2 this was an action research guided study which followed a spiral process of plan-observe-reflect-re-plan. for a description of this process, see miranda (2009a). helena miranda & jill adler 17 in other words, both teachers and learners need to move beyond the manipulation of material resources and see the mathematics through them, but not to be stuck with the materiality of the resources. constructing mathematical meaning through manipulatives this paper reports on how material resources known as algebra tiles (kitt & leitz, 2000; miranda, 2010) may serve as useful teaching and learning resources for making meaning of algebra. the term manipulative will be used in this paper in order to distinguish algebra tiles from other material resources such as chalkboards that are used in the teaching and learning of mathematics. as the word implies, manipulatives are either virtual or tangible materials that can be manipulated in shape or size as we use them to make meaning of our learning environment. for instance a computer is not a manipulative resource, but it may contain programs with virtual manipulatives within it, for example materials within a geometer sketchpad activity. one way of making mathematics meaningful to learners is to make explicit the close relationship among all its domains and how each can be used in making sense of the other. for example, the use of geometric diagrams and concrete materials may serve as a good source for bringing some geometric aspects into algebra. this also stresses the use of multiple representations in teaching and learning mathematics which has been, for a long time now, one of the foci in mathematics education. it has been argued that using multiple representations allows learners to understand mathematics concepts from different perspectives (see e.g., arcavi, 1999; duval, 1999, 2006; national council of teachers of mathematics, 2000). the need to use concrete materials has a long history in the field of mathematics education (szendrei, 1996). exploring the role of manipulatives in a mathematics classroom, szendrei (1996) argues that such materials “help pupils develop and understand the concepts, procedures, and other aspects of mathematics” (p. 427). she further alludes to the argument that adopting manipulatives in the teaching and learning process may also give equal opportunities to all learners to develop mathematical thinking in areas that are not readily supported by the abstract manipulation of mathematical structures. however, cautions szendrei, both teachers and teacher educators should be vigilant enough not to regard concrete materials as some ‘miracle drugs’ that will fix all learning problems that learners experience with mathematics. the role of such materials must be clearly explained in order to avoid confusion and frustrations as the learners handle the materials. namibia is one of the many african countries, in which the use of manipulatives in mathematics classrooms is not a common practice. unlike in some developed countries, the use of manipulatives is not explicitly required in most of the african school curricula (namukasa, 2005). therefore there are no rigorous standardised measures taken in order to ensure that possibilities for learning and teaching school mathematics through the use of manipulatives are taken advantage of. in a study that partly promoted the learning of mathematics through the use of semiotic artifacts, namukasa (2005) observes that african, specifically ugandan, learners appear less inclined than their north american (canadian) counterparts to use manipulatives when learning mathematics. this, she supposes, may be due to the fact that using manipulatives is not prescribed by the curricular documents; rather it is based on individual teacher’s decisions on whether to or not to make manipulatives part of their teaching. not making the use of manipulatives part of the school mathematics curriculum can therefore contribute to the teachers’ reluctance to go out of their way and try out different ways of adopting manipulatives in their teaching. a particular research interest in the use of manipulatives in mathematics classrooms of developing countries has recently emerged among african mathematics educators. there has been a realisation of how little literature there is that explores specific cases concerned with this phenomenon. for example, in a research study that involved ugandan pre-service mathematics teachers in the teaching of mathematics with manipulatives, namukasa and kaahwa (2007) introduced and investigated “the use of concrete, virtual and imagined materials/teaching aids in ugandan mathematics teaching” (p. 3). their findings suggest that there is need for mathematics education research to investigate the use of manipulatives in developing countries, especially those on the african continent. miranda (2007) also explored the need for the use of manipulatives as felt and expressed by mathematics teachers, and yet teachers do not have enough experience to draw from in order to develop any kind of materials that may be pedagogically useful. re-sourcing mathematics teaching through professional development 18 taking further the discussion on the use of additional material resources, adler et al. (2002) consider the educational implications that the acknowledgement of the need to use additional material resources has for teaching mathematics. for example, questions of provision, sufficiency and sustainability must always be accounted for, in order to make sure that all schools have access to the materials. also, since the impact of the power of any material resources lies in the way that they are used, there is a need for the teachers to teach learners how to use the resources, and hence the need for focused teacher development arises. this, adler et al. argue, will enable the teachers to realise that the resources do not have an automatic educational meaning but rather, the meaning emerges “through their use in the context of classroom practices and the subject [matter] being learned” (p. 69). in other words, the teachers should not just put the resources in front of the learners and expect them to be able to read the mathematics through them without proper guidance. as a response to the need to incorporate tangible materials in the teaching and learning of mathematics, a teacher professional development project was established and monitored in northern namibia (miranda, 2009a, 2009b). in addition to other activities of the teacher professional development project, the participating teachers were exposed to the manipulatives known as algebra tiles, for the first time. the purpose of introducing the manipulatives was partly to attend to the teachers’ expressed need of help with the development of teaching and learning materials and to investigate how the participants make algebraic meanings through their interactions with these materials. introducing algebra tiles as a way of promoting the use of manipulatives in the teaching and learning of algebra, the project introduced the algebra tiles to the participants and investigated how the tiles facilitated the enaction of elaborated algebraic meanings (miranda, 2009a). algebra tiles are pieces of paper or cards that appear in geometric forms, usually rectangles and squares, of different sizes. the materials are commercially available but one can also make them by cutting out soft or poster paper. this is the most affordable way of constructing such tools especially in teaching environments such as the namibian one. in the first in-service session on the algebra tiles the teachers spent time learning how to use the tiles and getting more comfortable with using them to represent algebraic expressions. it was in the second and subsequent sessions that the manipulation of algebraic expressions through the use of algebra tiles was explored to a larger extent. before the tiles were handed out, they were introduced by drawing three rectangles of different dimensions on the board. each of these rectangles represents one type of algebra tile in the kit. the dimensions of these rectangles are pre-given and so the participants had to determine the area of each rectangle. this is illustrated in figure 2. each kit of the algebra tiles contains a combination of many tiles; however, the group only explored three types of tiles, represented in figure 2 by (a) a square measuring x units by x units; (b) a rectangle measuring x units by 1 unit and a square measuring 1 by 1. as the group worked with the tiles trying to figure out their dimensions and total areas, they as a community adopted a language to give each tile a name. these names shaped the discussions that emerged for the rest of the time the group worked with the tiles. for example, (a) was referred to as the big square  blue in colour, (b) the blue strip  also blue in colour and (c) the small square  white in colour. big square blue strip small square (a) area = 2xxx  (b) area = xx 1 (c) area = 111  figure 2: the representation of some of the algebra tiles x 1 x 1 1 helena miranda & jill adler 19 from the chalkboard drawing of the tiles, a transition was made to the tiles by holding them, one by one, and asking the participants to state which rectangle represented each kind of tile in the kit. some conventional rules were also laid out for using these tiles in order to construct different geometric shapes. for instance, two or more tiles could only be placed next to one another if, and only if, the lengths of the adjacent sides were equal. for example, in figure 3, (a) and (b) would be sensible constructions, but (c) would not be, since x is not always going to have the value of 3, because it is a variable. (a) (b) (c) figure 3: sensible [(a) and (b)] and non-sensible [(c)] ways of aligning algebra tiles expanding algebraic expressions after the group revised the rules for using the algebra tiles and their measurements as explored in the first session of the project, the teachers were invited to use the tiles to construct geometric shapes of given dimensions. the objective of this brief task was to use the tiles to determine the area of the rectangle that would have been formed. a handout of different expressions was given for the teachers to choose the one they would like to start with. however, the teachers were also encouraged to think of other different expressions they might want to explore. the expression (x + 2)(x + 3) was the first to be explored. it was explained to the teachers that what we are trying to do is explore the algebra tiles and question: in what ways can we use them to, for example, introduce algebra to the learners or help learners make algebraic meaning as they solve problems? if we look at the two factors (x + 2) and (x + 3), for example, we can also view them as the dimensions of the sides of the geometric shape [rectangle] that we want to form with the tiles. so, if the width is (x + 2) and the length is (x + 3), what is the area of that rectangle going to be? the teachers took some time to renegotiate the dimensions of the sides of each tile before they started off building the required rectangles. this was done individually and then they all shared with one another what they had formed. figure 4 is a representation of what three of the teachers constructed. joe’s rectangle3 heita’s rectangle mia’s rectangle figure 4: a representation of teachers’ construction with the tiles 3 the names used here are not the real names of the teachers, but the pseudonyms as assigned in miranda (2009a). x + 1 x x + 1 x + 1 x + 2 x 3 re-sourcing mathematics teaching through professional development 20 it was interesting that each teacher tried to build a shape that looked different from what the others had constructed, perhaps to have multiple interpretations of the same problem. for example, mia could be heard remarking to heita that “mine is different from yours”. the next suggestion was to determine if they all had the same area, despite the different shapes formed by the tiles. it did not take mia and heita long to figure out the area of their rectangles, but joe was finding it difficult to make sense of what he built: miranda4: did we get the area? mia: yes. miranda: what is the area? mia: it is x squared, plus one, two, three, four, five x, plus six. miranda: mmhhh, x squared, plus five x plus six. did you get the same joe? [as he looked puzzled, frowning] joe: i am lost. mia: the area? miranda: yes, i mean the total area of the beautiful rectangle you constructed there. heita: what is this side? [pointing at one of the sides of joe’s rectangle] joe: x plus two? heita: and this one? joe: x plus three. heita: so, what is the whole area? [joe still looked puzzled; the group tried another strategy to help him] it was taking joe some time here to figure out the total area of the rectangle that he constructed. it appeared joe was attending to this construction in a way different from how the rest of the group could see it. heita, in trying to help him, interrogated joe on the dimensions of his rectangle. even though joe could answer these as “x plus two” and “x plus three” he could not immediately respond to heita’s question: “so, what is the whole area?” one amazing thing that was worth noticing was how much patience these teachers had with one another. after joe admitted that he really did not understand what everybody meant, mia and heita paused, looking at him while giving him ample time to make sense of his construction. then joe looked at miranda and asked: “what area now?” this time mia responded with a question similar to heita’s but phrased differently: mia: what is the area of this ‘big square’[pointing to the big blue square]? joe: it’s x squared. mia: what about the blue strips? what is the area of each one? joe: x? mia: and how many do you have of those? joe: five? miranda: and how many units [small squares] do you have? joe: i have six of them. heita & mia: now what is their total area, altogether? joe: x squared, five x, and six, ooohhh! [tapping his own head with the right hand] at this point, joe was laughing at himself when he finally realised that what he had in front of him was a rectangle with specific dimensions and hence a specific area. this, as he admitted, was a huge learning experience for him. what was even more interesting to the teachers is the connection between the area of 4 the first author as participant observer helena miranda & jill adler 21 this rectangle and what one would have obtained if they were to abstractly manipulate the given algebraic expressions. mia commented: “this is actually what we would get if we were to expand”. heita and joe went ahead to symbolically expand (x + 2)(x + 3) and ended up obtaining x2 + 5x + 6. the group tried a couple of more examples while comparing the areas constructed with the physical objects  algebra tiles  always matched with the product of the symbolic expansions. at the conclusion of the session, mia realised that the way we typically teach learners to expand algebraic expressions makes little sense as compared to how the tiles clarify this action. what we usually tell learners to do is “multiply first terms together, then the outside terms, then inside terms and then the last terms together” (the first-outside-inside-last, i.e. foil algorithm). for example in the case of (x + 2)(x + 3), learners would be told (or expected) to first multiply x by x (giving x2), then x by 3 plus 2 by x (giving 3x + 2x) and then 2 by 3 (giving 6), which is exactly the same as what is obtained with the tiles. in this way, the algebra tiles allow us to visualise what we are unable to see when manipulating algebraic expressions. they also help in visualising the geometric shapes  rectangles  and their relationship to algebra that we seem to take for granted in most cases. factorising algebraic expressions using algebra tiles the next time the professional learning group explored the algebra tiles, the session took a turn different from what was initially planned for. after the group reflected on what happened in the previous session of working with algebra tiles, one of the teachers (mia) had a very interesting question concerning the applicability of algebra tiles to other aspects of algebra. this is how the conversation proceeded: mia: maybe before you go aheadsomebody asked me a question whether these tiles can be used, for example, in algebraic expressions that um…for example, quadratic expressions which cannot be factorized. how do the tiles help here? we are unable to use them to do other things. miranda: to do what, like to factorize? mia: yes, to factorize. let’s say i want to…for example, if i have x squared plus three x plus one. i could not factorize it with the tiles. miranda: okay, that is a very good question. let’s see…what would we normally do to factorize that before we came to the tiles? are you saying it would not factor at all? mia: it would factor but not in the way that we factorize other expressions. miranda: how would we factor it? mia: we can complete the square. miranda: okay, oh, completing the square? i am sure we can do that with the tiles. why don’t we try it and see what we get? we can do almost anything with these things. mia: oh, really? mia insisted that the group specifically look at the expression she mentioned above (x2 + 3x + 1). everybody was then requested to take out the algebraic tiles whose total area is equivalent to this particular expression (see figure 5). figure 5: tiles with area totalling x2 + 3x + 1 x2 3x 1 + + re-sourcing mathematics teaching through professional development 22 after it was agreed that the tiles in figure 5 had an area of (x2 + 3x + 1), the teachers tried to organise them into a rectangle which did not work because they obtained a rectangle measuring (x)(x + 3) and then one unit remaining (see figure 6). figure 6: a representation of expressions that do not factor the group was then reminded of mia’s suggestion that since the expression (x2 + 3x + 1) would not factor right away they needed to complete a square. then miranda invited the group to consider the meaning of the phrase “completing the square,” especially in geometric terms. after some discussion, the group adopted an understanding that what this means is that the pieces x squared, three x and one should be used to form a rectangle that is a perfect square  in other words with all four sides equal in length. “how do we do that?” miranda probed. another teacher, seth, suggested that “we should add more pieces until all sides are equal”. later it was realised that this would change the original expression. and this is exactly what happens when one symbolically completes the square as long as we keep track of how much more we add. another teacher suggested: “first divide all that we have among the two sides [length and width], so that each side gets the same share of what we have”. this was done and is illustrated in the steps in figure 7. the piece x squared works out nicely because its length and width are already equal. in step one, a blue strip (x) is placed on each side  length and width. this leaves one blue strip unused, which must be further cut longitudinally into two equal parts with length x and width 2 1 (step 2). in the third step, the remaining tile, with an area of 1 is added, leaving more open space to be filled. figure 7: completing a square geometrically x x 1 step 1 step 2 step 4 step 3   + helena miranda & jill adler 23 the next challenge was to figure out how much area needs to be added in order to fill the open space, hence completing the square. to answer this question, the group concentrated on the pieces that had been added (the shaded pieces in figure 7). these measured 2 1 by 1, 2 1 by 2 1 , and 1 by 2 1 . these gave areas of 2 1 , 4 1 and 2 1 respectively. in total these three have an area of 4 5 4 1 1  . interpreting this means that the three shaded pieces that were added in order to complete the rectangle into a square, had an area of 4 5 . next the group revised the well-known algorithm used to complete the square through symbolic abstraction. this is further represented in figure 8. figure 8: completing a square symbolically transparency with the use of algebra tiles as noted earlier, transparency does not automatically come attached with any material resources. rather, it becomes evident in the way the resources are being adapted and used in context. for example, one cannot claim that the tiles used in the activities discussed above were transparent in themselves. there was a need for explicit instructions on how the teachers could use the tiles to represent algebra. hence, the transparency of the tiles lies in the way teachers were able to make connections between the geometric shapes of the tiles and their algebraic representations. the visibility of their geometric shapes, made the tiles visible and tangible, hence allowing for the facilitation of algebraic manipulations. this was however, not an advantage for all the teachers. for example, the tiles appeared too visible for joe, because he could only see the geometric shapes of the tiles but could not read the algebra through them. his aha moment of realising the link between the tiles and algebra surfaced when his colleagues intervened. in addition, the transparency with the tiles enabled the teachers to move beyond the technicality of the tiles themselves by bringing up the ‘what if questions’ and questioning what more the tiles could help them do. for example, mia suggested how the algebra tiles can be used to explain the foil strategy that is used when multiplying out factors of any algebraic expressions by opening their brackets. she further wondered if the tiles would help in factoring algebraic expressions that do not straightforwardly get factored by the foil strategy. further manipulations of the tiles allowed for the realisation of the power and new understandings that teachers possess and that could now help them in interpreting learners’ checking with symbolic abstraction methods x2 + 3x + 1 using the well-known algorithm for completing the square, we would go through the following steps: step 1: half the coefficient of x: 2 3 step 2: square what you got in step 1: 4 9 step 3: subtract 1 from the answer in step 2: 4 5 1 4 9  this would be the number we should add to our expression to complete a square and it is the same as the area of the tiles that we added to our rectangle to make it into a square. re-sourcing mathematics teaching through professional development 24 possible meaning-making of algebra as opposed to the right vs. wrong interpretations that were displayed at the start of the project. apart from enabling the participants to use multiple representations of algebra, the algebraic tiles were helpful in the participants’ learning of algebra in many different ways. it is not that these teachers did not, at the moment presented here, know how to expand or factorise algebraic expressions or to complete the square. they surely did, since they had been teaching this for quite a long time. however, in this study, the teachers were able to relearn those aspects of algebra again. they were learning how to expand and factorise expressions and complete the square differently. we say “differently” because even though they had learned these before, this was the first time they did this kind of learning with the algebra tiles where they paid attention to learners’ ways of interpreting mathematical problems. so, in the middle of doing what was familiar to them through the actions of manipulating algebra tiles and within the inter-actions among themselves, their capabilities were stretched to a certain extent. having said that then, it becomes evident that on one hand the algebra tiles served as a way of re-sourcing the teachers with different perspectives and approaches of teaching and learning algebra. on the other hand the project provided (resourced) the teachers with materials and activities that could be used in exploring algebra, in addition to other resources they might already have. there was a point where the teachers were able to set aside the concrete algebra tiles and start to make drawings in their books. as they became more fluent with the diagrammatic representation of the tiles, they could solve more algebraic problems much quicker. we now return to the vignette presented at the beginning of this paper’s discussion of learners’ attempt to find x if the area of a rectangle measuring (x + 2) by (x  1) exceeds the area of a square of side x by 2 cm2. the point here is not to claim that algebra tiles would have helped the learners trying to solve this problem. instead, we are trying to point to the range of possibilities of mathematical behaviours that could have emerged among the learners, should they have been exposed to algebra tiles prior to the area problem. figure 9 presents one possible way the algebra tiles could have helped in the representation of the problem and hence the meaning making and comparison of the areas of the two shapes (rectangle and square). of course as mentioned earlier, there are many factors, such as language barriers, that should be considered when analysing learners’ difficulties with problems of the kind in figure 1. yet one cannot ignore the range of possibilities that familiarity with algebra tiles could have opened up, and hence reducing the difficulties of mathematical interpretations. figure 9: possible representation and interpretation of problem in figure 1 conclusion this paper is about learning algebra through the use of manipulatives called algebra tiles through teacher professional development. the teachers first learned about the tiles as a new means of teaching materials before they were able to utilise them. in their learning they were able to help each other make sense of the tiles and the different geometric constructions they did with the tiles. this can be seen for instance in the case where mia and heita assisted joe to make sense of his own constructed tiles of the expression (x + 2)(x + 3). x  1 x + 2 x x helena miranda & jill adler 25 apart from the geometric shape that they encompass the algebra tiles also have some sense of measurement in them. such a measurement allows for flexibility because one can choose any parameters or variables to assign to the dimensions of each tile, depending on the nature of the problem being investigated and the meaning being construed. the paper also discussed the importance of ensuring that there is transparency with the use of any material resources by guiding the teachers how to use the materials. this will enable the teachers to realise that teaching and learning materials do not come with automatic transparency attached to them. rather, there is need to teach learners and help them to make sense of the materials and use them to make mathematical meaning through them. the participating teachers learned different ways of helping learners interpret and solve algebraic problems with the use of algebraic tiles. however, there is still need for curriculum to make specific stipulations with regard to the use of manipulatives as part of mathematics teaching and assessment. acknowledgements this paper was developed during a post doctoral fellowship by the first author in the school of education at the university of the witwatersrand. we would like to acknowledge the fund for support of international development activities (fsida) for assisting in the collection of the data for the research study from which this paper arises and the department of secondary education at the university of alberta, canada, where the original study took place. we also acknowledge the quantum research project, which supported the postdoctoral research activities under which this paper was developed. references adler, j. 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(pp. 52-65). okahandja, namibia: national institute for educational development. abstract introduction literature review theoretical framework methodology and research design results discussion conclusion acknowledgements references about the author(s) agnes d. qhibi department of mathematics science and technology education, faculty of humanities, school of education, university of limpopo, polokwane, south africa zwelithini b. dhlamini department of mathematics science and technology education, faculty of humanities, school of education, university of limpopo, polokwane, south africa kabelo chuene department of mathematics science and technology education, faculty of humanities, school of education, university of limpopo, polokwane, south africa citation qhibi, a.d., dhlamini, z.b., & chuene, k. (2020). investigating the strength of alignment between senior phase mathematics content standards and workbook activities on number patterns. pythagoras, 41(1), a569. https://doi.org/10.4102/pythagoras.v41i1.569 research project registration: project number: fhdc/2017/2520 original research investigating the strength of alignment between senior phase mathematics content standards and workbook activities on number patterns agnes d. qhibi, zwelithini b. dhlamini, kabelo chuene received: 03 aug. 2020; accepted: 20 oct. 2020; published: 17 dec. 2020 copyright: © 2020. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract improving the strength of alignment between educational components is essential for quality assurance and to achieve learning goals. the purpose of the study was to investigate the strength of alignment between senior phase mathematics content standards and workbook activities on numeric and geometric patterns. the study contributes to strengthening the intended curriculum and the workbook activities, since workbooks are widely used in south african public schools. the study employed the concurrent triangulation research design. the theoretical framework comprised webb’s alignment model and the surveys of enacted curriculum. the senior phase mathematics content standards and department of basic education workbooks were selected for content analysis. the findings showed that the alignment between senior phase mathematics content standards and workbook activities on numeric and geometric patterns was significantly acceptable with misaligned content and representations that require urgent attention. we recommend reconfiguration of both the workbook activities on numeric and geometric patterns and senior phase mathematics content standards to align content, cognitive levels, representations and assessment. further studies on teaching and learning that are aided by the workbook activities should be mandated. keywords: alignment index; cognitive levels; content standards; scale of agreement; number patterns. introduction the investigation of the strength of alignment ensures synergy between curriculum components’ main content standards, classroom instruction and assessment (polikoff & porter, 2014; porter, 2002). the extent of agreement between these curriculum components is referred to as alignment (roach, niebling, & kurz, 2008). the conceptualisation of alignment begins with common understanding of the educational components used in this discourse, content standards, classroom instruction and assessment. kurtz, elliott, wehby and smithson (2010) refer to these as follows: (1) the intended curriculum is reflective of the content standards as specified in the curriculum and assessment policy statement (caps) (department of basic education [dbe], 2011); (2) the enacted curriculum refers to the content of instruction taught by teachers in classrooms; (3) the assessed curriculum is depicted by the content measured by the various forms of assessment or tests during the academic year. hence, the conceptualisation between these three aspects of the curriculum in the alignment discourse is: the intended curriculum specifies content for instruction; the content taught by teachers during instruction portrays the enacted curriculum; the assessed curriculum depicts the assessed content that gauges levels of students’ achievement. the investigation of the strength of alignment normally begins with the determination of the content, the cognitive levels and representations of each of the documents (porter, 2002; webb, 1997). frequent studies on alignment are necessary to improve the agreement of curricula expectations, classroom instruction and assessment (russell & moncaleano, 2020). alignment is both horizontal and vertical. horizontal is between curricula (intended and assessed) and assessments while vertical is between learning materials, classroom instruction, professional development and learner outcomes (enacted curriculum) (webb, 1997). hence, alignment has the potential to strengthen the connections between what is taught, what is tested and what is intended by the curriculum (martone & sireci, 2009). the importance of measuring the strength of alignment between educational components remains a prerequisite for any education system (porter, 2002; webb, 1997). such importance hinges on the quality of content standards and how they filter to other educational components, such as assessment (roach et al., 2008). the current study focuses on the intended curriculum, caps, which is referred to as the senior phase mathematics content standards (spmcs) (dbe, 2011), and the assessed curriculum, the numeric and geometric patterns (ngp) workbook activities as an indication of what will be assessed in tests and exams. roach et al. (2008) observe that high expectations are attached to educational materials used to support curriculum delivery. similar expectations should be held of workbooks as they also play a role in supporting curriculum delivery. the influence of workbook activities on the quality of mathematics cannot be undermined when they serve as tools for practice, assessment and monitoring in schools (hoadley & galant, 2016). more specifically, ngp content promotes the possibility of learners’ generalisations from arithmetic to algebra (kieran, 2004). carraher, schliemann, brizuela and earnest (2006) suggest that algebra and arithmetic are interconnected and should be integrated in elementary mathematics. in 2012, the dbe in south africa rolled out an initiative to provide workbooks to grades 1–9 learners in public schools (hoadley & galant, 2016). this initiative, as shown in other education systems, was aimed at providing practice and monitoring tools in the form of worksheets (dbe, 2013; fleisch, taylor, herholdt, & sapire, 2011). practice tools provide extended opportunities for learners to revisit content and skills that ought to have been introduced in an earlier learning experience (hoadley & galant, 2016). in contrast, monitoring tools gauge the coverage of content as intended by the curriculum (fleisch et al., 2011). a review of the literature indicated that a few studies were conducted on workbooks in south africa (fleisch et al., 2011; hoadley & galant, 2016; mathews, mdluli, & ramsingh, 2014; pausigere, 2017); however, there is a dearth of similar studies in grades 7–9. textbooks are learning materials that place the content of the intended curriculum in the context of teaching and learning (roach et al., 2008). in contrast, workbooks supplement textbooks with worksheets for practice in preparation for tests and exams (hoadley & galant, 2016). the unavailability of textbooks resulted in the use of workbooks as teaching tools by some teachers in grade 3 and grade 6 (fleisch et al., 2011; mathews et al., 2014). similar practices may be expected with grades 7–9 workbooks. it is, therefore, critical to frequently compare educational components, as pointed out by hoadley and galant (2016), in textbooks, workbooks, assessments and curricular content. the aim of the study was to investigate the strength of alignment between spmcs (dbe, 2011) and workbook activities on ngp (dbe, 2017a, 2017b, 2017c). one research question was posed: how are the workbook activities on numeric and geometric patterns aligned to the senior phase mathematics content standards as stated in caps (dbe, 2011)? this investigation of alignment has the potential to provide crucial information to policymakers that is essential for developing improved content structure (mathews et al., 2014). this study contributes to the literature of quality educational support materials. it advances the literature in the methods of evaluating alignment of educational components, using the context of ngp. in addition, the simultaneous use of porter’s alignment model (quantitative) and webb’s (1997) alignment model (qualitative) signals the uniqueness of this study. most studies on alignment use a single method, hence the current study has a methodological significance. literature review alignment studies alignment studies on educational components have been conducted internationally in various disciplines (fitzpatrick, hawboldt, doyle, & genge, 2015; higgins, 2013). in south africa, ndlovu and mji’s (2012) alignment study was in mathematics, while edwards’s (2010) study was in physical sciences. some of these studies used webb’s alignment model while others used the surveys of enacted curriculum (sec). the current study uses the following as tools for alignment: (1) the grades 7–9 specification of mathematics content in numeric and geometric patterns, which is referred to as spmcs, for uniformity of alignment language (dbe, 2011) and (2) workbook assessment activities on numeric ngp (assessed) (dbe, 2017a, 2017b, 2017c). the review of literature on ngp in the sections to follow provides the rationale for using this content area. a handful of studies employed webb’s alignment with focus on investigating the alignment between content standards and assessment. fitzpatrick et al. (2015) conducted a study on the alignment between learning objectives and assessments used in therapeutics courses in canada. findings from their study revealed misalignment between assessment and learning objectives, where some of the learning objectives, such as communicator, collaborator, manager and advocate, had no matches in the course examination. as a consequence, to learning, these outcomes might be regarded as less important by learners since they do not form part of the examination, which might conflate the course. another alignment study was conducted between common core state standards (ccss) for mathematics grade 8 and three assessments: the national assessment of educational progress (naep), the american college testing (act) explore and the jefferson county public school (jcps) interim assessment (higgins, 2013). the findings highlighted the fact that the ccss for the grade 8 mathematics were significantly aligned with the jcps interim assessment. however, the other two assessments (the naep and the act explore) were found to be misaligned with the grade 8 mathematics ccss because of the implementation of new frameworks that neglected the ccss framework. there is a likelihood that the two assessments (naep and act explore) might not meet the acceptable level of the ccss. an alignment study that employed sec, and subsequently outlined the computed alignment index, was conducted by ndlovu and mji (2012). they explored alignment between the trends in international mathematics and science study (timss) for the 2003 grade 8 mathematics framework and the revised national curriculum statement’s (rncs) (dbe, 2002) assessment standards. the findings of this study revealed that the alignment index of 0.751 had misalignment that was significantly low, in accordance with the fulmer critical values. also, the rncs was found to be stronger in relation to knowledge and procedures under the three content categories (number, geometry and data) and weaker in terms of reasoning. edwards (2010) also employed sec to calculate the degree of alignment between grade 12 physical sciences exemplar papers and the final examination dealing with the south african curriculum content. the findings exposed alignment indices ranging from 0.76 to 0.79 for the physics component, which is higher than the chemistry component which had alignment indices ranging from 0.52 to 0.69. a huge discrepancy was obtained in the cognitive level ‘remember’ for both exemplar papers and the final examination. it is critical to cover the cognitive levels on the assessment as outlined by the curriculum to avoid discrepancies which result in misalignment. the alignment studies reviewed above employed different alignment models which provided various dimensions, such as the content structure and the alignment indices. to justify the significance of the alignment indices, studies used fulmer’s indices (ndlovu & mji, 2012; polikoff & porter, 2014). however, this review indicates that the use of quantitative means falls short of the depth and specifics of the misaligned content and cognitive levels. in contrast, the use of qualitative means to investigate alignment (webb, 1997) allows the determination of misaligned content, cognitive levels, content structure and ease of use of content (qhibi, 2019). there is a dearth of literature that uses both quantitative and qualitative means to determine the strength of alignment, where commonly used alignment models are employed, such as webb’s model, sec and the achieve model. hence the current study concurrently used porter’s and webb’s alignment models to corroborate alignment instead of testing significance quantitatively. studies on department of basic education workbooks studies (fleisch et al., 2011; hoadley & galant, 2016; mathews et al., 2014) were conducted on south african dbe workbooks whose primary purpose was to supplement textbooks and to provide learners with worksheets (dbe, 2013). as claimed by hoadley and galant (2016), the use of workbooks was not limited to monitoring implementation of the curriculum, but potentially they can inform quantitative coverage of the curriculum. mathews et al. (2014) conducted a study that focused on the utilisation of dbe workbooks in a grade 3 mathematics classroom. their findings reveal that some teachers were using dbe workbooks for teaching and learning, and for assessment activities, while others were using them interchangeably with textbooks. meanwhile, their primary purpose is to supplement textbooks and provide worksheets to the learners. this shows that some teachers are not using the workbooks as planned by the dbe. another study focused on evaluating the grade 6 mathematics dbe workbooks against the conventional textbooks in south africa using quasi-experimental design (fleisch et al., 2011). their findings highlighted the fact that learners who used conventional textbooks and those who used dbe workbooks exhibited the same improvement in mathematics performance. an alignment study was conducted by hoadley and galant between grade 3 literacy and numeracy dbe workbooks and the caps, focusing on content coverage, weighting of content areas and cognitive levels. the findings revealed that the content for grade 3 literacy workbooks comprehensively matched the caps content; however, skills such as handwriting and listening were not matched with the skills in the workbooks. the alignment of the caps grade 3 mathematics was strong in two content areas: (1) numbers, operations and relationships, and (2) patterns, functions and algebra. the alignment was moderate in another two content areas: (1) measurement and (2) data handling, while it was weak in one content area, namely space and shape. these findings suggest that the workbooks need to be strengthened in the future in areas where alignment was either moderate or weak. numeric and geometric patterns the generalised focus of ngp systematically integrates concepts such as arithmetic, algebraic thinking and reasoning (carraher et al., 2006). usiskin (1988) conceptualises algebra instruction into the following tenets: ‘1) algebra as generalised arithmetic; 2) algebra as a study of procedures for solving certain kinds of problems; 3) algebra as a study of relationships among quantities; 4) algebra as the study of structures’ (pp. 11–15). these tenets form the basis of ascertaining the linking of the concepts that constitute ngp. on one hand, the structural conception, symbols are referred to as structures that generalise numeracy (linchevski & livneh, 1999), and on the other hand, the procedural, computations with numbers lead to the need for some form of generality (sfard, 1991, 1995). algebra is used as a generaliser of patterns through the formation of generic structures (kieran, 2004). for example, if one is required to find the 100th even number, first a protocol dealing with numbers multiplied by two is formed: ‘2n’. ‘2’ is the constant difference and ‘n’ is the position in the term, where ‘2n’ constitutes the general rule of the sequence (dbe, 2011). substituting into the protocol, the 100th even number is computed to be constant difference × position of the term: 2n = 2(100) = 200. the generalisation of patterns in elementary mathematics uses the difference and the position in a term of a mathematical expression to develop a generalised generic protocol which constitutes algebraic reasoning (blanton & kaput, 2005). in contrast, with the learners’ conceptions that involve arithmetic computation of a constant difference, substitution in the protocol to determine any term in the sequence is referred to as algebraic thinking (kieran, 2004; pitta-pantazi, chimoni, & christou, 2020). in other instances, pictures from a context in the real world assist in learners’ problem solving and manipulation of number patterns while making sense of relationship of concepts; this constitutes geometric reasoning (pittalis & christou, 2010; spangenberg & pithmajor, 2020). du plessis (2018) conceptualises geometric thinking processes in number patterns for grades r–3 as follows: (1) sequenced items, the use of a variety of geometric shapes to denote a pattern, (2) the core, full repeat of geometric items, and (3) the form, a representation of how the geometric items are arranged in the pattern. the findings of the study (du plessis, 2018) revealed the following discrepancies in the content and cognitive levels of the caps: (1) the absence of an indication of growth in cognitive complexity in numerical and geometric patterns; (2) the core and form in the geometric patterns were limited to sequencing and could not pose complex thinking and problem solving. this structure of the curriculum could result in fragmented algebraic thinking and reasoning that is associated with ngp. results from other studies (bishop, 2000; girit & akyüz, 2016; pitta-pantazi et al., 2020) posed challenges with decreasing patterns, incorrectly assigned numbers where there were variables and generalising patterns where numeracy was not linked to algebra. these results show the persistent existence of algebraic thinking and reasoning in relation to numeric and geometric patterns (bishop, 2000). cognitive levels and standards in mathematics cognitive levels distinguish the level of thinking and the appropriate depth of understanding (zhuge, 2016). the most commonly used bloom’s taxonomy measures the cognitive domain in assessments ranging from low to higher order thinking (irvine, 2017). in content alignment, the sec (porter, 2002) uses cognitive levels and webb’s alignment employs depth of knowledge to determine standards in the content used for instruction. the caps adapted the timss cognitive levels to measure levels of thinking in mathematics assessments (dbe, 2011) comprising the following proportions: knowledge (25%), routine procedures (45%), complex procedures (20%) and problem solving (10%) (dbe, 2011). the timss cognitive levels were considered to be coherent in cognitive component, content component and problem solving as key elements of mathematics thinking (long & dunne, 2014). dbe (2011) describes ‘knowledge’ questions as those that use mathematical facts, straight recall and appropriate mathematical vocabulary. in addition, ‘routine procedures’ are described as questions that need simple applications and steps (dbe, 2011). kalobo and du toit (2015) highlight that ‘complex procedures’ involve unfamiliar problems or abstract problems that do not have a direct route to the solution, while ‘problem solving’ refers to unseen problems that require a higher level of cognitive skills and reasoning to solve problems. cognitive levels were used in this study to code spmcs and workbook activities on ngp in terms of their depth of understanding. one study used verbs to code rncs assessment standards (ndlovu & mji, 2012). similarly, verbs on the content standards were used to code spmcs while caps guidelines on cognitive levels were used to code workbook activities on ngp. theoretical framework the theoretical framework for this study comprised two alignment models: webb (russell & moncaleano, 2020; webb, 1997) and the sec (porter, 2002). webb’s alignment model uses content focus to judge alignment between content standards and assessment (webb, 1997). content standards in this case are spmcs on ngp while assessment comprises dbe workbook activities on ngp. it consists of six content focuses, three of which look at content, cognitive levels and content representation (categorical concurrence, depth of knowledge consistency and range of knowledge correspondence). the focus of this study was on the qualitative part of webb’s model and adapted these first three content focuses. the other three content focuses, the structure of knowledge comparability, the balance of representation and the dispositional consonance, fell outside the scope of the study. hence, the three webb’s content focuses employed in this study were as follows: (1) categorical concurrence, which verifies whether grades 7–9 content standards and assessment cover the same content on both algebraic and geometric number patterns. in this study, consistency in content was verified between the senior phase mathematics content standards and the workbook activities on ngp; (2) depth of knowledge consistency checks whether cognitive levels of content standards and assessments are in agreement. this content focus was verified if the cognitive levels between the workbook activities on ngp and spmcs were in agreement; and (3) range of knowledge correspondence checks how the representations of content standards are consistent in assessment. the range of knowledge correspondence covered on the spmcs and the workbook activities on ngp were compared in this study to explore their status of alignment. in other studies, the model was employed to investigate alignment of content standards and assessments (duke escobar, 2016; fitzpatrick et al., 2015; higgins, 2013; smith, 2012). the sec was employed to compute alignment indices between spmcs and workbook activities on ngp, where content proportions, assessment proportions and cognitive levels were used. content proportions are fractions or percentages used to compare how much content is covered by the cognitive levels, while assessment proportions are the fractions of assessment covered by the cognitive levels. the cognitive levels distinguish and classify the ability to think, understand and solve problems (zhuge, 2016). mathematical thinking in ngp is both algebraic and geometric, the arithmetic computations of common difference and terms of a sequence using algebra and pictures (pitta-pantazi et al., 2020). in contrast, mathematical understanding refers to the problem solving strategies and complex thinking of ngp which is constituted in the process of generalising number patterns using algebraic and geometric reasoning (blanton & kaput, 2005). various studies have employed the sec to investigate alignment between content standards, assessment and learning materials in terms of the alignment indices (ndlovu & mji, 2012; polikoff, 2015). ndlovu and mji (2012) describe the sec as the most effective and simplest procedure for evaluating alignment. the sec supplemented webb’s alignment model by outlining the degree of alignment in terms of the alignment indices. hence webb’s alignment model and the sec were adapted and deemed appropriate lenses for investigating the strength of alignment. methodology and research design we employed mixed methods to investigate alignment between workbook activities on ngp and spmcs. qualitative and quantitative data were generated, analysed and corroborated through the concurrent triangulation design (creswell, 2015). this design strengthened the findings of the study since limited knowledge exists about the alignment between workbook activities on ngp and spmcs. qualitative data using webb’s alignment model and quantitative data using porter’s alignment model were collected and analysed simultaneously to corroborate findings from the two data sets (onwuegbuzie & combs, 2011). figure 1 displays the procedure followed in implementing the design. figure 1: concurrent triangulation design. in this study, webb’s alignment guides the qualitative method. qualitative content analysis explains patterns of content, cognitive levels and representations of the mathematics content standards and the workbook activities on ngp. quantitative data were generated through porter’s alignment model. correlational prediction design was used in this study. the criterion variable is the alignment index which provided the forecast of the outcome using content and cognitive levels, which were the predictor variables (creswell, 2015). document selection the following documents were purposively selected in order to investigate the alignment between spmcs and dbe workbook activities on ngp: (1) dbe caps grades 7–9 mathematics (dbe, 2011), (2) dbe workbook, grade 7 mathematics (english) book 2 (dbe, 2017a), (3) dbe workbook, grade 8 mathematics (english) book 1 (dbe, 2017b) and (4) dbe workbook, grade 9 mathematics (english) book 1 (dbe, 2017c). the workbook activities on ngp were selected as this topic is crucial in the development of reasoning in algebra (bryman, 2016; etikan, musa, & alkassin, 2016). again, the selection of senior phase mathematics workbook activities on ngp was done to close the gap, since studies conducted on workbooks focused on foundation phase and intermediate phase (fleisch et al., 2011; mathews et al., 2014). data collection prior to data collection, dbe granted permission for the selection of the senior phase mathematics caps document and dbe workbook activities on ngp. the university to which the authors are affiliated granted ethical clearance. the instruments for collecting quantitative data were adapted from the sec (porter, 2002); they were matrices of content with cognitive levels and assessment with cognitive levels. the adaptations were spmcs on ngp with cognitive levels and workbook activities on ngp with cognitive levels (table 4 and table 5). quantitative data were generated through mapping spmcs and workbook activities on ngp with cognitive levels, and were represented by a score of 1 to represent a hit. a hit was used to show that content on spmcs matched with content in workbook activities. in cases where content on spmcs and workbook activities matched more than one cognitive level, the score was divided evenly using decimal fractions and sum to 1. the content proportions and assessment proportions in the matrices for grades 7–9 were used to compute the alignment indices. parallel to computing alignment indices, qualitative data were generated by mapping the spmcs and workbook activities on ngp for categorical concurrence, depth of knowledge consistency and range of knowledge correspondence, which were identified as themes (webb, 1997). content on spmcs and workbook activities on ngp was mapped on categorical concurrence, cognitive levels mapped on depth of knowledge consistency and ranges of patterns mapped on range of knowledge correspondence. data analysis in line with how qualitative data were generated, and to evaluate the degree of alignment between spmcs and workbook activities on ngp, data analysis was also based on categorical concurrence, depth of knowledge consistency and range of knowledge correspondence. three scales of agreement were adapted from webb’s content focus and were used to categorise the degree of alignment between spmcs and workbook activities on ngp. they were: (1) full alignment, which depicted equal corresponding matches for content standards, cognitive complexity and knowledge comparisons, (2) acceptable alignment, which is sufficient matches in terms of content standards, cognitive levels and knowledge comparisons, and (3) insufficient alignment, in terms of exclusion from the workbook activities on ngp when compared to the requirements of the spmcs (webb, 1997). these scales of agreement were adapted from webb’s content focus and were used to categorise the degree of alignment between spmcs and workbook activities on ngp. three mathematics subject advisors were appointed as content analysts and trained on how to match the spmcs and workbook activities on ngp. the krippendorff alpha was employed to measure the agreements and disagreements between the content analysts (krippendorff, 2011). the data analysis of the qualitative and quantitative data was done separately and corroborated afterwards. the quantitative data were analysed following porter’s (2002) alignment model, which uses proportions of the content and assessment to compute the alignment indices. the proportions for content (spmcs) and proportions for assessment (workbook activities) were calculated by dividing the proportions by the number of content standards and assessment activities. the alignment indices were calculated by adding the absolute differences of the content proportions and assessment proportions for all the content analysts on each grade. the absolute differences were then divided by 2 and the quotient was subtracted from 1 to obtain the alignment indices. alignment indices were then calculated using the formula: alignment index = , where ‘x’ represents the cell proportions in the content matrix and ‘y’ represents the cell proportions in the assessment matrix. porter’s rating scale ranges from 0 to 1, where ‘0–0.5’ means no to moderate alignment and ‘0.51–1’ means moderate to perfect alignment. quality criteria to ensure that the quantitative results are trusted, reliability and validity were assessed consistently throughout the study (ivankova, 2014). the matrices used to compute porter’s alignment index were adapted by mathematics subject advisors (content analysts) from porter’s alignment model to ensure their content validity (porter, 2002). triangulation of data from the spmcs and workbook activities on ngp resulted in the corroboration of the units of comparison in the three webb’s content focuses, namely categorical concurrence, depth of knowledge consistency and range of knowledge correspondence (ivankova, 2014; webb, 1997). to ensure interrater reliability of scales of agreements from the units of comparisons that the content analysts matched, their agreements and disagreements were measured using the krippendorff alpha (krippendorff, 2011; zapf, castell, morawietz, & karch, 2016). the computed krippendorff alpha of 0.999 was considered extremely reliable (krippendorff, 2011). finally, the simultaneous interpretation of results was ensured through the methodological triangulation of qualitative and quantitative data (creswell & clark, 2017). ethical consideration the permission for conducting this study was sought from, and granted by, the department of basic education. ethical clearance for the study was granted by the university to which the authors are affiliated. ethical clearance number: trec/10/2018:pg. results the overall qualitative results for the alignment between the workbook activities on ngp and the spmcs were on the scale ‘acceptable alignment’ using the three webb’s content focuses. on the other hand, the overall quantitative results indicated that the porter’s alignment index was in the range ‘moderate to perfect’ (0.73), for grades 7–9. this signifies that 27% of content was either not covered by the workbook activities on ngp or not specified by the spmcs, an indication of some misaligned content. webb’s alignment of spmcs and workbook activities on ngp we outline the degree of alignment in terms of the three webb’s content focuses, namely categorical concurrence, depth of knowledge consistency and range of knowledge correspondence. categorical concurrence this webb’s content focus was limited to investigating whether the content found in the workbook activities on ngp corresponded with the content required by the spmcs. the subtopics were used as units of comparison for content standards required by the spmcs that were compared to the content of the workbook activities on ngp (table 1). the content analysts adapted webb’s scale of agreement as follows: (1) full alignment, equal corresponding matches of content in spmcs and workbook activities on ngp, (2) acceptable alignment, sufficient matches of content in spmcs and workbook activities on ngp, with a few missing concepts in the workbook activities, and (3) insufficient alignment, exclusion of content in the workbook activities on ngp that is required by the spmcs. ironically, some parts of content were included in the workbook activities on ngp and were not a requirement of the spmcs in grade 7 and grade 8. the content analysts categorised this as ‘out of scope’. webb’s scale of agreement lacks a category that would capture this content. actually, webb’s scale of agreement matches from content standards (spmcs) to assessments (workbook activities on ngp) and not vice versa, which is linear. hence the ‘out of scope’ content was captured without matching but affected the final scale of agreement. table 1 shows the comparison between content identified on spmcs and workbook activities on ngp. table 1: grades 7–9 categorical concurrence and scale of agreement. table 1 illustrates the comparison between content identified by the content analysts on spmcs and workbook activities on ngp as well as the scale of agreement between the two components. most of the content identified on workbook activities matched with the content identified on spmcs, where scales of agreement ranged from acceptable alignment to full alignment (table 1). however, ‘out of scope’ content was also identified from the workbook activities on ngp. the grade 7 workbook activities on ngp were restricted to the following content: describe the rule for the pattern, 6, 14, 22, 30; describe the pattern, 2, 8, 32, 128, 512, … describe the pattern and draw a number line to show each, 8, 10, 14, 20, 28, … describe the rule in your own words, 6, 9, 12, 15, … calculate the 20th term using a number sequence, 2, 5, 10, 17. (dbe, 2017a, p. 3–10) these examples were limited to the description of number patterns, rules and drawing on number lines to show the patterns. the descriptions and drawings limited the extent of learners’ investigation and extension of the ngp and did not allow justifications, as outlined in the spmcs. the ‘out of scope’ content that appeared in the workbook activities were not matched since they were not required by the grade 7 content standards. in grade 7, the ngp are limited to a description in words and not in either drawings, algebraically or in context. however, such content was found in the grade 7 workbook activities and deemed ‘out of scope’. an example of content on workbook activities that was deemed ‘out of scope’ in grade 7 was extracted from dbe workbook: thabelo is building a model house from matches. if he uses 400 matches in the first section, 550 in the second and 700 in the third section, how many matches will he need to complete the fourth section, if the pattern continues? (dbe, 2017a, p. 9) patterns in context were found to be ‘out of scope’, because the content did not form part of the content standards requirement for grade 7. furthermore, ‘out of scope’ content was also identified in grade 8 workbook activities on ngp. the following example was extracted from the grade 8 workbook: calculate the number of matchsticks used, 4th hexagon has 4 matchsticks per side (dbe, 2017b, p. 58) this was considered out of scope since the skill of calculation was not outlined in the grade 8 content standards. however, all content covered in the grade 9 workbook matched the content with grade 9 content standards. the scale of agreement between spmcs and workbook activities on ngp under categorical concurrence was as follows: acceptable in grade 7 and grade 8, and full in grade 9. the acceptable alignment was obtained where content of the workbook activities on ngp sufficiently matched the content on the content standards, while full alignment was obtained where content of the workbook activities on ngp fully matched the content on the content standards (table 1). an example of grade 9 workbook activities that fully matched the content on the content standards has been extracted from the grade 9 workbook: describe the pattern by giving the rule and then extend it with three more terms, 2, 4, 6, 8, 10, … describe the pattern by giving the rule and then extend it by three terms, 2, 4, 8, 16, 32, 64, … describe the pattern by giving the rule and then extend it by three terms, 2, 4, 12, 48, 240, … (dbe, 2017c, p. 68) the content of these activities fully matched the content on the grade 9 content standards, since extension and description of rules of patterns are requirements of grade 9 content standards. depth of knowledge consistency this webb’s content focus was employed to verify whether the workbook activities on ngp measured the same cognitive levels as the spmcs. the cognitive levels emanated from the verbs of the ngp content standards in the spmcs that determined the cognitive complexity. the cognitive levels were knowledge, routine procedures, complex procedures and problem solving, which were sourced from the spmcs (dbe, 2011). the unit of comparison emanated from the matches of the cognitive levels of the spmcs and those of the workbook activities on ngp (table 2). the content analysts adapted webb’s scale of agreement as follows: (1) full alignment, equal corresponding matches of cognitive levels in spmcs and workbook activities on ngp, (2) acceptable alignment, sufficient matches of cognitive levels in spmcs and workbook activities on ngp, with a few missing cognitive levels in the workbook activities, and (3) insufficient alignment, exclusion of cognitive levels in the workbook activities on ngp that were required in the spmcs. table 2 illustrates the comparison of cognitive levels between spmcs and workbook activities on ngp. table 2: grades 7–9 depth of knowledge consistency and scale of agreement. the data in table 2 illustrate the comparison between cognitive levels identified in spmcs and workbook activities on ngp as well as the scale of agreement of the two components. the cognitive levels of the workbook activities on ngp matched with the cognitive levels of the spmcs, hence the scale of agreement fell on ‘full alignment’. the only cognitive levels identified between the spmcs and the workbook activities on ngp were knowledge and routine procedures. this was an indication that alignment between spmcs and the workbook activities on ngp in terms of the cognitive levels was full. however, two cognitive levels, complex procedures and problem solving, were not covered by both spmcs and workbook activities, an area for concern, since these cognitive levels should also be assessed as per caps requirements. the workbook activities on ngp in grades 7–9 were limited to describe, calculate and draw. an example from the grade 9 workbook activities on ngp is as follows: describe the pattern by giving the rule and then extend it by three terms, 2, 4, 8, 16, 32, 64, … 2. 25, 5, 1, 0.2, 0.04, … (dbe, 2017c, p. 69) the first pattern above required either knowledge of basic multiplication (2; 2 × 2; 2 × 2 × 2; 2 × 2 × 2 × 2; 2 × 2 × 2 × 2 × 2; 2 × 2 × 2 × 2 ×2 × 2) or knowledge of exponents (21; 22; 23; 24; 25; 26) to extend the pattern and determine the rule, while the second pattern required application of the simple procedure of dividing successive terms to get the difference, then deducing the rule. these workbook activities were matched with cognitive levels knowledge and routine procedures. the cognitive levels of these workbook activities matched with the cognitive levels of the grade 9 content standards, as description and extension of patterns fell under cognitive levels knowledge and routine procedures on the content standards. hence, the alignment between the workbook activities and grade 9 content standards was full. these activities were limited to two cognitive levels, knowledge and routine procedures. the same applies to grade 7 and grade 8: only knowledge and routine procedures were covered in ngp workbook activities. an example extracted from the grade 8 workbook is given below: what is the constant difference or ratio between the consecutive terms? 6, 24, 96, 384 8, 2, –4, –10 (dbe, 2017b, p. 56) the first pattern above requires knowledge and simple procedure of multiplication and division () to be able to determine the constant ratio. the second activity requires knowledge of integers and simple procedures of subtracting the previous term from the next term to be able to determine the constant difference. hence, these activities fell under knowledge and routine procedures, which also matched the same cognitive levels of the grade 8 content standards, where learners are expected to extend patterns with constant ratio. the scale of agreement between the grade 8 content standards and workbook activities on ngp was fully aligned. the following activities were extracted from the grade 7 dbe workbook: describe the pattern 2, 8, 32, 128, 512 describe the pattern and draw a number line to show each 10, 9, 7, 4, 0 (dbe, 2017a, p. 5–6) the first activity above requires knowledge of multiplication and the simple procedure of dividing the next term by the previous term to be able to describe how the pattern grows, whereas the second activity requires knowledge of integers, number lines and the simple procedure to subtract the previous term from the next term. hence, these workbook activities fell under knowledge and routine procedures. these cognitive levels matched with the cognitive levels on the grade 7 content standards, as description of patterns fell under knowledge and routine procedures. moreover, the patterns in context that were found in the workbook for grade 7 were labelled as problem solving by dbe whereas they, in fact, fell under knowledge and routine procedures. this is highlighted since problem solving require high levels of cognitive skills and reasoning to solve the problem (dbe, 2011; kalobo & toit, 2015). an example of such a pattern in context is extracted from the grade 7 workbook: lisa read 56 pages on sunday, 66 pages on monday, 76 pages on tuesday, and 86 pages on wednesday. if this pattern continued, how many pages would lisa read on thursday? (dbe, 2017a, p. 111) the activity can be solved by adding 10 pages for the next day as the pattern is growing by 10 without engaging high level of cognitive reasoning. the workbook activities on ngp were configured using cognitive levels stipulated in the spmcs. the overall scale of agreement on the depth of knowledge consistency was fully aligned. range of knowledge correspondence the spmcs and the workbook activities on ngp were based on the range of content. the unit of comparison included the ranges of content standards’ representations as required by the spmcs and tested by the workbook activities on ngp (table 3). the content analysts adapted webb’s scale of agreement as follows: (1) full alignment, equal corresponding matches of all forms of concept representations in the spmcs and the workbook activities on ngp, (2) acceptable alignment, nearly all forms of concept representations in the spmcs and workbook activities on ngp, and (3) insufficient alignment, exclusion of other forms of concept representations in the workbook activities on ngp that were required in the spmcs. surprisingly, there were certain forms of concept representations that were posed by the workbook activities on ngp which were not outlined on the spmcs in certain grades, compelling the content analysts to categorise them as ‘out of scope’ (table 3). table 3: grades 7–9 range of knowledge correspondence and scale of agreement. the data in table 3 illustrate the comparison of ranges of patterns identified on the spmcs and the workbook activities on ngp, as well as the scale of agreement between the two components. the scale of agreement between spmcs and workbook activities on ngp for grades 7–9 was ‘acceptable’. all the ranges of pattern representations required by the spmcs were covered by the workbook activities on ngp (full alignment), and out of scope ranges of patterns identified from workbook activities were also outlined (table 3). ironically, the workbook activities included ranges of patterns that were not outlined in the content standards, but were mentioned in the clarification notes. it would be good for ranges of patterns to be embedded in the content standards so as to ensure that these are not missed by users and also to clarify progression between the grades. hence ranges of patterns are labelled ‘out of scope’ since the focus of the study was on content standards. the ‘out of scope’ ranges of pattern representations were as follows: (1) grade 7: patterns represented algebraically, patterns represented on number lines, patterns in context, patterns with integers, patterns with whole numbers, patterns on drawing; (2) grade 8: patterns with integers, patterns with whole numbers; (3) grade 9: patterns with common fractions, patterns with decimal fractions. examples that show ‘out of scope’ content on ngp are extracted from workbooks: grade 9: describe the pattern by giving the rule and then extend it by three terms, (dbe, 2017c, p. 68) grade 7: describe the pattern and draw a number line to show each, 8, 10, 14, 20, 28 (dbe, 2017a, p. 6) the ranges of patterns for these workbook activities could not match the ranges of patterns on spmcs, hence were deemed ‘out of scope’. figure 2 shows ‘out of scope’ content identified in grade 7, which indicates the value of the term using drawing. figure 2: number patterns represented on drawing. this workbook activity was deemed ‘out of scope’ since description of patterns using drawing was not a requirement of the grade 7 content standards. the scale of agreement between spmcs and workbook activities on ngp under range of knowledge correspondence was as follows: acceptable alignment in grades 7–9. this resulted in the overall scale of agreement on range of knowledge correspondence between spmcs and workbook activities on ngp being ‘acceptable alignment’. the computed porter’s alignment indices the computed porter’s alignment indices for grades 7–9 are outlined in this section. the data in table 4 outline a matrix of content standards (ngp) and cognitive levels for the initial process of computing porter’s alignment index. the content analysts recorded hits of the matches between spmcs and cognitive levels. their averages were divided by the total number of content standards (table 4). table 4: grades 7–9 spmcs matrix. table 4 shows the content matrix, the average content proportions generated by the content analysts by mapping content standards (spmcs) with cognitive levels. the number of content standards in grades 7–9 were two in each grade, hence the average content proportions matched by the content analysts were divided by two. these content proportions were then used in conjunction with the assessment proportions (table 5) to calculate alignment indices. table 5: grades 7–9 workbook activities on ngp matrix. the data in table 5 are a synopsis of porter’s alignment of the assessment matrix for the workbook activities on ngp. the data were generated by content analysts by mapping workbook activities onto ngp with cognitive levels. the workbook activities on ngp were: 27 for grade 7, 12 for grade 8 and 7 for grade 9. the said activities were matched to cognitive levels of mathematics (dbe, 2011). the content analysts recorded hits of the matches, and their averages were divided by the total number of workbook activities on ngp (table 5). the two matrices (table 4 and table 5) were then used to calculate alignment indices between spmcs and workbook activities on ngp. the quotients in table 4 and table 5 were substituted in the formula used to calculate the porter’s alignment index (). for example, the index for grade 7 is 0.89, which was computed as follows: similarly, porter’s alignment indices for grade 8 and grade 9 were computed to be 0.60 and 0.71. these indices indicate that the alignment between the spmcs and the workbook activities on ngp for grades 7–9 are in the range ‘moderate to perfect’ alignment because they are in the range 0.51–1. the graphical representation of the alignment indices is shown in figure 3. figure 3: grades 7–9 alignment indices between senior phase mathematics content standards and numeric and geometric patterns workbook activities. the computed porter’s alignment indices were as follows: 0.89 (89%) for grade 7, 0.60 (60%) for grade 8 and 0.71 (71%) for grade 9. the information in table 4 and table 5 was also used to calculate discrepancies in cognitive levels between spmcs and workbook activities on ngp. these discrepancies are illustrated in figure 4. figure 4: grades 7–9 discrepancies between senior phase mathematics content standards and numeric and geometric patterns workbook activities. both weak and strong discrepancies were obtained between spmcs and workbook activities on ngp in grade 7, grade 8 and grade 9. a positive value depicts strong discrepancy, while a negative value portrays a weak discrepancy on ngp workbook activities for those cognitive levels. there was a strong discrepancy for knowledge in grade 7, while the discrepancies for grade 8 and grade 9 are weak. while there was a weak discrepancy in routine procedures in grade 7, there was a strong discrepancy for grade 8 and grade 9. there were no discrepancies obtained for the three grades on complex procedures and problem solving because the discrepancy value was zero. this study evaluated the strength of alignment between the spmcs and workbook activities on ngp in terms of the content structure and the alignment indices. this evaluation is brought together in table 6, which shows the results of the qualitative data analysis using webb’s alignment and quantitative data analysis using porter’s alignment. table 6: summary of the research findings. the degree of alignment on the alignment indices ranges from ‘moderate to perfect’ which signifies a scenario that parts of the spmcs and the workbook activities on ngp were misaligned. discussion the aim of the current study was to investigate the strength of alignment between the spmcs and workbook activities on ngp. this study elucidates two remarkable results for the determination of misalignment. firstly, webb’s alignment shows that certain parts of the spmcs and the workbook activities on ngp were acceptable in content and representations, and fully aligned on cognitive levels. secondly, the overall porter’s alignment index was in the range ‘moderate to perfect’ (0.73), positing some degree of bulk misalignment (27%) of content and representations between spmcs and workbook activities on ngp. in addition, complementarity between the two methods (creswell & clark, 2017), webb’s and porter’s alignment models, clarified, enhanced and supplemented the findings by exposing alignment and misalignment of content, representations and the degree of alignment between the spmcs and the workbook activities on ngp. corroborating webb’s and porters’ alignment in this study, the corroboration of the quantitative and qualitative results in a concurrent mixed method replaces the calculation of the traditional statistical significance of the alignment index (creswell & clark, 2017). the sec involves computing the statistical significance using the ranges provided in fulmer’s critical values (porter, 2002). to justify the significance of the results, we used the convergence model, that is, first compared and related porter’s alignment (73%) and the overall webb’s alignment (acceptable) as shown in table 6. subsequently, both results were further scrutinised and interpretations made to justify the misalignment (figure 5). the study by ndlovu and mji (2012) that aligned the rncs and timss concluded a computed porter’s alignment index of 0.751 which shows that the misalignment was significantly statistically low. the computed porter’s alignment index of 0.73 of the current study signals that 27% of the content and cognitive levels were misaligned. porter (2002) made assertions that the alignment indices, as tools of measuring alignment, are subject to extension and improvement. a handful of studies (ndlovu & mji, 2012; polikoff, 2015; polikoff & porter, 2014) have used the alignment index to evaluate alignment between content and assessment. noticeably, these studies made use of fulmer’s critical values to justify the significance of misalignment. however, in literature there is a dearth of extension that justifies the significance of the alignment in triangulation with qualitative methods. hence, we compare and contrast porter’s alignment index with webb’s alignment procedure. figure 5: the convergence of the results. webb’s alignment posits alignment that is ‘acceptable’ for content and representations on the scales of agreement, categorical concurrence and range of knowledge correspondence, while depth of knowledge consistency was found to be fully aligned. the ‘acceptable’ finding posits sufficient matches in terms of content and representations, which is short of ‘full’. this alignment is closely associated with the overall computed porter’s alignment index of 0,73, which is in the range ‘moderate to perfect’. comparatively, the two alignment models, porter’s and webb’s, posit the same picture, that is, there are missing content and representations for spmcs and workbook activities on ngp. according to porter (2002), when the alignment index increases, it seems that there is better alignment, but the determination of good alignment based on the index is still unclear. hence, making a judgement that 73% alignment is good could raise questions from policymakers who could claim the missing 27% content and representations would have an adverse effect on the content of ngp. in contrast, webb’s alignment was ‘acceptable’, with some content and representations that were coded ‘out of scope’, which also indicates some chunks of content that are either missing or misplaced in the wrong grade. russell and moncaleano (2020) advise that content standards that are not in agreement with assessment are more likely to fragment the enacted curriculum. hence, the convergence of both porter’s and webb’s alignment results exposes weaker links between the intended and assessed curricula (martone & sireci, 2009). interpreting the convergence of porter’s and webb’s alignment the workbook activities are formative assessment and form part of the assessed curriculum (hoadley & galant, 2016). their role as practice tools hinges on closing the gaps between the intended and the enacted curricula in preparation for summative assessment, another level of the assessed curriculum (kurtz et al., 2010; mathews et al., 2014). ideally, the summative assessment should be configured using the intended curriculum, the spmcs (dbe, 2011). in webb’s alignment, there is ‘missing’ and ‘out of scope’ content and representations in the workbook activities when compared to the spmcs; surely this reveals disagreement between these educational components. porter’s alignment shows discrepancies on how the spmcs and ngp workbook activities favoured knowledge and routine procedures in grades 7–9 (figure 4). also, that is coupled with the absence of complex procedures and problem solving. in addition, the alignment index of 73% infers that 27% of content was not in agreement between the spmcs and the ngp workbook activities. to interpret this convergence of the porter’s and webb’s results, we make reference to principles of the research design used in this study, the triangulation concurrent design, complementarity and integration (onwuegbuzie & combs, 2011). combining both the porter’s and webb’s results complements the existence of disagreements in content, cognitive levels and representations. this complementation of two data sources (onwuegbuzie & combs, 2011) serves as the significance of the disagreements between the spmcs and the ngp workbook activities. the corroborated results that jointly revealed disagreements between the spmcs and workbook activities pose concerns about the quality of the caps in ngp. some content on ngp is hidden in the learning outcomes (dbe, 2011), while alignment in both the webb’s and porter’s procedures matches only the outcomes and the assessments (russell & moncaleano, 2020). in fact, alignment begins with the content standards and verifies the extent that content standard is available in the assessment (porter, 2002). the methodological significance of the current study is the revelation of the missing (out of scope) content standards. there exists a dichotomy in the enacted curriculum: the workbooks activities supplement the intended curriculum. surprisingly, there is a claim that workbooks were configured using the content standards (dbe, 2017a, 2017b, 2017c). then, how can workbooks pose additional content than the intended curriculum (caps)? this implies a dilemma in the key aspects of the ngp content in these areas: (1) sequenced items, the algebra and geometric items were routine and could not extend to non-routine, (2) the core, the repeat of the algebra and geometric components of the patterns could not demand problem-solving strategies and complex procedures, (3) the form, the formulation of the generic protocol lacked rigour due to the absence of non-routine problem solving and complex procedures. the disagreements between the spmcs and the ngp workbook activities poses mismatch in content progression in the following areas: (1) algebra and geometry as generalised arithmetic in numeric and geometric patterns. the absence of complex procedures in the workbook activities is an indication that the process of generalising number patterns using algebra and geometry is fragmented (pittalis & christou, 2010; usiskin, 1988). the rigour applied in the generalisation of algebraic and geometric patterns in the ngp workbook activities was only based on obvious common difference and position of a term and missing the abstract nature of complex procedures. kalobo and du toit (2015) point out that complex procedures use abstract and unfamiliar problems. (2) algebraic thinking should pose a range of simple to complex computations involving numeric and geometric patterns (kieran, 2004; pitta-pantazi et al., 2020). the computations were limited to substitutions and simplifications in the generic protocols which lacked rigorous problem solving. in fact, in the workbooks they were referred to as problem solving, but during the analysis it was discovered that they lacked qualities of problem solving and were coded as routine procedures. kalobo and du toit advise that problem solving refers to finding solutions for problems that require higher level of cognitive skills and reasoning. (3) algebraic and geometric reasoning: the formulation of a generic protocol of ngp involved the difference in consecutive terms, the position in the sequence and pictorial patterns. some of the learning outcomes that posed the investigation of number patterns lacked key elements of algebraic and geometric reasoning. they were also reduced to the use of the difference and position in the term. long and dunne (2014) advise that assessment activities should be configured using verbs such as describe, investigate and justify which should be found in the content standards to promote algebraic and geometric reasoning in ngp. there was an obvious lack of these verbs in the matching of the spmcs and the ngp workbook activities. conclusion this study contributes to the existing literature on teaching and learning support materials by investigating the strength of alignment between the spmcs and workbook activities on ngp. the simultaneous use of webb’s alignment and porter’s alignment afforded the opportunity to study both the depth and quantity of the strength of alignment. the linearity of webb’s alignment requires reconfiguration to cater for ‘out of scope’ components during the matching. the dearth of studies that mix alignment methods afforded this study a contribution to existing literature which needs further research. also, the reconfiguration of the webb’s alignment should result in the consideration of content that is misplaced in the content standards. this study investigated solutions to the following initial research question: how are the workbook activities on numeric and geometric patterns aligned to the senior phase mathematics content standards? this investigation detected that, when configuring the workbook activities on ngp, reasonable attempts were made to conform to the spmcs. however, some chunks of content and representations of the workbook activities on ngp were either missing in the workbooks or out of scope when compared to the spmcs. this mismatch increases the possibility of negative effects on learners’ ability to generalise algebra using arithmetic, algebraic thinking, and algebraic and geometric reasoning relevant for ngp. the complementation of the two approaches employed in this study leads to the conclusion that certain parts of the workbook activities on ngp and the spmcs are misaligned with respect to content and representations. some content and ranges of patterns were found in the workbook activities whereas they are not requirements of spmcs. in addition, cognitive levels ‘complex procedures’ and ‘problem solving’ were not covered in spmcs and ngp’s workbook activities. although the misalignment was low, its effects may be devastating to the algebraic and geometric cognitive development of learners. the provision of problem solving and complex procedures relevant to specific grades in the senior phase workbook activities on ngp requires urgent attention. there is dire need to reconfigure the workbooks to conform to the content requirements of the spmcs for grades in the phase, which could avoid conceptual meddling. these findings require further research on a larger scale in order to address other content areas of the workbooks. also, further research is required on the pedagogical aspects of the workbook activities on the ngp which could inform the reconfiguration of the workbooks. limitations this study was limited to numeric and geometric patterns, while the findings leave a dilemma for further studies that may cover other content areas and topics in the workbooks as learning support materials. also, the alignment in the senior phase paves the way for an opportunity for studies on conceptual progression that results from the observed misalignment. implications for the teaching and learning of mathematics the review of literature indicated that most teachers in public schools use workbooks against the dbe’s intentions of supplementing the enacted curriculum. against this backdrop, the current study indicates misaligned and ‘out of scope’ content between the caps and ngp. the limitation of numeric and geometric patterns to routine problems deprives learners’ abilities for problem solving and complex procedures. where the need arises, where the workbooks supplement content standards that are missing in the caps, other complementary materials should be used to close that gap. similarly, this argument can be extended for inclusion of non-routine complex procedures and problem solving to allow higher order algebraic and geometric thinking. the disagreements between the caps and the workbook activities creates opportunities for the fragmentation of conceptual progression of the ngp in the senior phase. if not addressed, the fragmentation of algebra problem-solving strategies is most likely to filter to higher grades and post school and cause difficulty in the learning of advanced algebra concepts. policymakers and subject advisors should prioritise dissemination to teachers the supplementary content on problem solving and complex procedures to augment the discrepancies between caps and the ngp workbook activities. acknowledgements this research emanates from the dissertation of the first author at the university of affiliation. we thank the content analysts for their professional work of coding the qualitative data and quantitative data. competing interests there is no conflict of interest that links the authors to this article. authors’ contributions a.d.q. was the student under the supervision of z.b.d and k.c. z.b.d. conceptualised the article, k.c. worked on the logical presentation of the ideas and the methodology, and a.d.q. provided the first draft of the article. funding information the research received no specific grant from any funding agency in the public. data availability statement data sharing is not applicable to this article as no new data were created or analysed in this study. disclaimer the views expressed in this article are those of the authors which are neither from policy nor those of the university of the authors’ affiliation. references bishop, j. 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(2016). multi-dimensional summarisation in cyber-physical society. amsterdam: morgan kaufmann. https://doi.org/10.1016/b978-0-12-803455-2.00010-x abstract introduction theoretical framework and related literature methodology findings discussion conclusions acknowledgements references appendix 1 appendix 2 appendix 3 about the author(s) wajeeh daher mathematics education department, al-qasemi academic college of education, baqa, israel educational sciences faculty, an-najah national university, nablus, palestine amal tabaja-kidan al-qasemi academic college of education, baqa, israel faaiz gierdien faculty of education, stellenbosch university, south africa citation daher, w., tabaja-kidan, a., & gierdien, f. (2017). educating grade 6 students for higher-order thinking and its influence on creativity. pythagoras, 38(1), a350. https://doi.org/10.4102/pythagoras.v38i1.350 original research educating grade 6 students for higher-order thinking and its influence on creativity wajeeh daher, amal tabaja-kidan, faaiz gierdien received: 19 aug. 2016; accepted: 31 may 2017; published: 28 aug. 2017 copyright: © 2017. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract educating students for higher-order thinking provides them with tools that turn them into more critical thinkers. this supports them in overcoming life problems that they encounter, as well as becoming an integral part of the society. this students’ education is attended to by educational organisations that emphasise the positive consequences of educating students for higher-order thinking, including creative thinking. one way to do that is through educational programmes that educate for higher-order thinking. one such programme is the cognitive research trust (cort) thinking programme. the present research intended to examine the effect of the participation of grade 6 students in a cort programme on their creative thinking. fifty-three students participated in the research; 27 participated in a cort programme, while 26 did not participate in such programme. the ancova test showed that the students who participated in the cort programme outperformed significantly, in creative thinking, the students who did not. moreover, the students in the cort programme whose achievement scores were between 86 and 100 outperformed significantly the other achievement groups of students. furthermore, students with reported high ability outperformed significantly the other ability groups of students. the results did not show statistically significant differences in students’ creativity attributed to gender. introduction educating students towards higher-order thinking prepares them to be an integral part of the society, for this education strengthens their ability to confront life problems and find solutions to these problems (de bono, 1992b; papastephanou & angeli, 2007). this role of educating for thinking is behind educational organisations’ call for teaching for higher-order thinking. this preparation can also be an impetus for reforming future education in mathematics (kilpatrick, swafford & findell, 2001; national council of teachers of mathematics, 2000). higher-order thinking is associated with different thinking types. king, goodson and rohani (1998) say that higher-order thinking includes critical, logical, reflective, metacognitive, and creative thinking. in the present research, we are interested in students’ creative thinking that is being made the focus of curriculum (wilson, 2005), and being made an agenda for improving teaching and learning in the classroom (burnard, 2006). this centeredness of creativity in current educational practices in the classroom has gone a long way since the call of guilford (1950) for researchers’ attention to it. this call resulted in great interest and attention of the educational researchers in students’ creativity (archambault, 1970). torrance’s (1966) tests of creative thinking could be regarded as a milestone in the area of creativity measurement. we use three components of creativity suggested by torrance to examine grade 6 students’ mathematical creativity, as well as how this creativity is influenced by educating students in creative thinking and skills. some of the participating students participated in a cognitive research trust (cort) programme developed by de bono (1992a) as an educating programme that aims to optimise and develop learners’ thinking. we examine the differences in students’ overall creativity and its components as a result of educating them through the cort programme. research rationale and goals piggott (2011) argues that creativity in the mathematics classroom is not only related to what pupils do but also to what teachers do, where the mathematical experiences that teachers offer their students can open up opportunities for them to be creative. torrance (1972) describes several ways to teach students to think creatively. we adopted in the present research one of these ways, namely complex programmes involving packages of materials. as mentioned above, here we used the cort programme developed by de bono (1992a) to teach grade 6 students to think creatively. we examined how the participation in such programme affects students’ creativity in problem solving, where this creativity is represented by its three components: fluency, flexibility and originality, as well as overall creativity. our reported experiment follows different previous attempts (e.g. craft, cremin, burnard & chappell, 2007), but ours is concerned with mathematical creativity. in our case, we engage upper primary students with a programme that aims to cultivate their thinking. in the mathematics classroom, studies suggest open-ended tasks (e.g. mihajlović & dejić, 2015) or multiple solution tasks (e.g. levav-waynberg & leikin, 2012) for cultivating students’ mathematical creativity. here, we followed a different way of cultivating students’ creativity, namely the cort programme. the different studies, including the present, point to different possible ways to develop students’ mathematical creativity. this development would provide the student with powerful tools as a learner of mathematics in particular and of school disciplines in general. research questions question 1 how would participation in the cort programme influence grade 6 students’ mathematical creativity, including the three components of creativity (fluency, flexibility and originality) and including overall creativity? question 2 how would gender and mathematical achievement influence the creativity of grade 6 students in general and those who participated in cort programme in particular? theoretical framework and related literature recognising and nurturing students’ talents and thinking present an important challenge to educators (marin & halpern, 2011). this suggests the need for tools for such nurturing. the cort programme is such tool, concerned with developing thinking, especially critical and creative thinking (grissom, 2004). de bono (1992a, 1992b) developed cognitive tools for promoting thinking skills known as the cort with the first cort thinking lessons published in 1974. the cort programme (de bono, 1992a) assumes that teachers can educate for thinking, thus educating for thinking can be part of the curriculum. the cort programme provides teachers with tools to teach thinking skills. it consists of over 60 lessons focusing on developing thinking skills. these 60 lessons range over six sections: breadth, organisation, interaction, creativity, information and feeling and action. the cort section that was applied in the present study is cort (4) ‘creativity’. this section consists of 10 lessons: (1) yes, no and po (‘po’, a device for showing that an idea is being used creatively without any judgment or immediate evaluation), (2) stepping stone (the use of ideas not for their own sake but because of other ideas they may lead to), (3) random input (the input of unrelated spurious ideas into a situation may change the situation), (4) concept challenge (the testing of the ‘uniqueness’ of concepts may lead to other ways of doing things), (5) dominant idea (in most situations there is a dominant idea; in order to be creative one must find and escape from it), (6) define the problem (an effort to define a problem exactly may make it easier to solve), (7) remove faults (the assessment of faults and their removal from an idea), (8) combination (by examining the attributes of seemingly unrelated items, new items may be created either by fusion or by combination), (9) requirements (an awareness of requirements may influence the creation of ideas) and (10) evaluation (does an idea fulfil the requirements; what are its advantages and disadvantages?). in the present research, we used the creativity section of the cort programme to encourage grade 6 students’ creative thinking in mathematics. see appendix 1 for more detail on the activities given to the students in the frame of the programme. our interest in encouraging grade 6 students’ creative thinking in mathematics meets the call of educational institutions and researchers to encourage students to use high-order thinking skills, including creative thinking skills, because doing that prepares them to be 21st century citizens by possessing the appropriate skills (e.g. national council of teachers of mathematics, 2000). this interest in nourishing students’ creative thinking is due to its ability to support their problem solving through encouraging diverse solutions for a problem (imai, 2000). the prominence of creative thinking in education has been a response to the ongoing technological and scientific development (leikin, 2013; yazgan-sağ & emre-akdoğan, 2016). there is no single perspective or definition of creativity (leikin & kloss, 2011; mann, 2006; sriraman, 2005). mann (2006) claims that there are more than 100 definitions of creativity in the literature. ervynck (1991) defines mathematical creativity as the ability to solve problems or develop structured thinking, as well as make connections in the mathematical content. he emphasises that creative activity is not related to algorithms, but to a novel concept, definition, argument or proof. in the present study, we draw on a definition of creativity as including three components: fluency, flexibility and originality (guilford, 1950, 1975; torrance, 1966). fluency is associated with the number of correct answers that a student provides to a problem or correct questions the student poses with respect to a problem. flexibility is associated with the number of answer or question types suggested for a problem, or with the number of problem-solving or problem-posing strategies that have been implemented. originality is associated with the number of solutions offered or problems posed that very few or no other persons proposed (torrance, 1966). this is also true for the present study. specifically, when we evaluated the creativity components’ scores, we followed leikin (2009). recent studies have attempted to examine the effect of different variables on mathematical creativity. leikin and kloss (2011) studied the problem-solving performance of 8th and 10th graders in terms of the correctness of their solutions and the three components of creativity (fluency, flexibility and originality). they found that grade 10 students were significantly more successful and fluent when solving all the tasks, but nevertheless the differences between grade 10’s and grade 8’s flexibility, originality and creativity were task dependent. moreover, their results showed that originality determines creativity stronger than fluency and flexibility. achievement and gender are two variables that have been studied for their effect on students’ creativity, where different studies found different results concerning the effect of these two variables on students’ creativity. creativity and achievement haylock (1997) points out that mathematical attainment can limit students in overcoming fixation and in working with divergent problems, although it does not determine their mathematical work. this statement agrees with research on the relation between creativity and achievement, which has been long researched (baird, 1985; torrance, 1962). we argue that this agreement means that mathematical attainment could influence one’s creativity, but this influence could be mediated or moderated by other individual or social variables, such as motivation and teacher support. this conception of the relation between attainment and creativity can explain the literature that does not agree regarding the effect of achievement in a discipline on students’ creativity in that discipline. while some studies have reported high correlations between academic achievement and creativity (e.g. torrance, 1962), some have found low correlations between these two variables (e.g. baird, 1985). other studies have not found significant relations between achievement and creativity (e.g. nori, 2002). one of the studies that found significant differences in mathematical creativity and which is related to achievement is that of mann (2005) who explored the relationship between mathematical creativity and mathematical achievement, attitude towards mathematics, self-perception of creative ability, gender and teacher perception of mathematical talent and creative ability. the research results indicated that 35% of the variance in mathematical creativity scores could be predicted by the studied variables. mathematical achievement was the strongest predictor accounting for 23% of the variance. students’ attitudes towards mathematics, self-perception of their own creative ability and gender contributed the remaining 12% of variance. creativity and gender baer and kaufman (2008) argue that no simple conclusions can be drawn from the empirical evidence on gender differences in creativity test scores (general creativity tests, not specific in a specific discipline). they enumerate studies that report women scoring higher than men in creativity (e.g. misra, 2003), studies that report the opposite (e.g. cox, 2003) and studies that report no difference (e.g. kaufman, baer & gentile, 2004). in addition, some studies found significant and insignificant differences at the same time regarding the effect of gender on mathematical creativity. for example, evans (1964) reported significant differences in grade 7 and 8 students’ scores of some creativity measures that could be attributed to gender, where girls outperformed boys. at the same time, evans (1964) reported no significant differences between students’ creativity scores in grades 6 and 7. in the present research, our main interest is with the interaction of the two variables, achievement and gender, with implementing the cort programme and how these interactions affected grade 6 students’ creativity. to evaluate students’ creativity, we used multiple solution tasks. multiple solution tasks for evaluating creativity a multiple solution task is a task that requires the student to solve a mathematical problem in different ways (leikin, 2009). a multiple solution task has three solution spaces (levav-waynberg & leikin, 2012): expert solution space, individual solution space and collective solution space. an expert solution space includes the set of solutions to a problem known to an expert at a particular time. an individual solution space includes all the solutions produced by an individual. a collective solution space includes all the solutions produced by a group of students. these spaces are used for exploring students’ mathematical creativity. in the present study, we used multiple solution tasks to evaluate grade 6 students’ creativity in mathematics and to compare this creativity before and after participating in a cort programme. cognitive research trust programme for encouraging higher-order thinking a number of studies have shown that the cort programme affects significantly higher-order thinking, including creative and critical thinking (e.g. birdi, 2005). al-edwan (2011) explored the effectiveness of a training programme based on cort programme to develop grade 7 students’ critical thinking in a history course. the results showed statistical differences in the participating students’ critical thinking in history as a result of the cort training programme. moreover, melhem and isa (2013) explored the effect of using the cort programme on critical thinking skills among grade 6 student with learning difficulties in mathematics. they found that the training programme positively and significantly affected the participants’ critical thinking. in the present study, we intend to study the effect of the cort training programme on grade 6 students’ mathematical creativity. researchers have reported that the cort programme was effective for encouraging communication skills (e.g. alshurman, 2017). alshurman (2017) found that educating university students using the first section of the cort programme (breadth) resulted in statistically significant differences between the pre and post communication skills scores in the experimental group in favour of the post measurement. moreover, no statistically significant differences were found based on gender in the post scores. concerning educating for creative thinking skills, researchers found that the cort programme helped in encouraging these skills (e.g. al-jallad, 2006). al-jallad (2006) reported that using the cort programme, as an educating programme, was effective for developing creative thinking skills among the female university students of the arabic language and the islamic studies. little research has been done on the influence of the cort programme on mathematical creativity, which is the aim of the present research. research hypotheses hypothesis 1 participation in the cort programme results in significant differences between the experimental group and the control group in grade 6 students’ mathematical creativity, including fluency, flexibility, originality and overall creativity. hypothesis 2 participation in the cort programme results in significant differences between the scores of grade 6 students’ mathematical creativity, before and after the participation. hypothesis 3 creativity scores of students who participate in the cort programme will not differ significantly according to the interaction of their gender and their participation in the programme. hypothesis 4 creativity scores of students who participate in the cort programme will not differ significantly according to the interaction of their achievement and their participation in the programme. in the first and second hypotheses, we followed other researchers who pointed to the positive influence of preparing students to use higher-order thinking, critical thinking or creative thinking on their actual use of this thinking. as mentioned above, researchers found that using the cort programme resulted in significant differences in critical thinking scores (al-edwan 2011; melhem & isa, 2013). in the third and fourth hypotheses, we followed researchers who found that gender and achievement did not result in significant differences in creativity scores when students participate in an education programme that encourages thinking creatively. methodology research context and sample the present research intended to explore how a cort programme could affect grade 6 students’ mathematical creativity. the research was conducted in an arab public elementary school in a small town (with population of approximately 30 000) in the haifa district in israel. the school students come from middle socioeconomic backgrounds. the research sample included two groups, with overall 53 grade 6 students. the first group, the experimental group, included 27 students who participated in learning an arithmetic unit based on the fourth section of the cort programme, that is, the creativity section, which included 10 lessons that encourage students’ creativity. the second group, the control group, included 26 students who did not participate in the cort programme. the present study is a quasi-experimental study. a quasi-experimental is an empirical study used to find the causal impact of an intervention on its target population without a random assignment. according to the definition of quasi-experimental research (see, for example, kosslyn, 2017), this research shares similarities with the randomised controlled trial, but it specifically lacks the element of random assignment to experimental or control. instead, the quasi-experimental design allows the researcher to control the assignment of subjects to the treatment condition, using some criterion other than random assignment (in our case scores in creativity and its components). in addition to the above, the research design could be characterised as two-group design, with non-random selection and pre-test and post-test. this design is represented in table 1. table 1: research design. the experimental group learnt the cort creativity section in groups of 3–5 students, in which student discussion was encouraged. the students were encouraged not to be afraid of having different answers, even if they were strange or controversial, and to discuss these answers with the rest of the group members. moreover, the students had homework assignments to practise the new concepts and skills they learnt. data collection tools the data were collected through preand post-tests on creativity. the two tests were similar but not identical. an example of a question in the pre-test is: the sum of three natural numbers is 16. the sum of two of them is equal to the third. what are the three natural numbers? an example of a question in the post-test is: the sum of four natural numbers is 20. the sum of two of them is equal to the sum of the other two. what are the four natural numbers? data analysis tools to evaluate the components of creativity and overall creativity, we depended on the work of leikin and kloss (2011). we now describe our computations. fluency (fl) was evaluated by the number of solutions in the individual solution space. flexibility (flx) was evaluated after building groups of solutions for the multiple solution tasks. two solutions belonged to different groups if they employed solution strategies based on different representations, properties (theorems, definitions, or auxiliary constructions) or branches of mathematics. with respect to the corresponding solution spaces, we evaluated flexibility as follows: flx1 = 10 for the first appropriate solution. for each following solution, flxi = 10 if it belonged to a group of solutions different from the solutions performed previously; flxi = 1 if the solution belonged to one of the previously used groups but had a clear minor distinction; flxi = 0.1 if the solution was almost identical to a previous solution. a student’s total flexibility score on a problem was the sum of the student’s flexibility on the solutions in the student’s individual solution space. originality (or) was evaluated as follows: if p is the percentage of students in the group that produces a particular solution, then ori = 10, when p < 15% and for an insight-based or unconventional solution; ori = 1, when 15% ≤ p < 40% or for a model-based or partly unconventional solution; ori = 0.1 when p ≥ 40%. a student’s total originality score on a problem was the sum of the student’s originality of the solutions in the student’s individual solution space. the creativity (cr) of a particular solution is the product of the solution’s originality and flexibility: cri = flxi × ori. the total creativity score for a multiple solution task is the sum of the creativity scores for each solution in the individual solution space of a problem: the scores of the components of creativity and overall creativity were coded in spss 21 to test the research hypotheses. the following computations were performed: means and standard deviations of the different creativity components and the overall creativity, one-way analysis of co-variance (ancova) to verify the effect of the cort programme on students’ creativity, analysis of variance (anova) to verify the effect of the reported ability and the overall achievement on students’ creativity and eta squared (η2) as estimate of effect sized of the variables. as for the verification of the first hypothesis, we ran ancova and post-hoc tests to test whether there are significant differences between the creativity scores in the two research groups after the experiment, and ran paired t-test to test whether there were significant differences between the creativity scores in the cort group before and after implementing the cort programme. as to the distribution of the participants regarding the independent variables, table 2 describes the number and percentage of participants in terms of gender, reported ability and overall achievement. table 2: number and percentage of participants in terms of gender, reported ability in mathematics and overall achievement. the students in the two groups did not have significant differences in the components of creativity in the pre-test of creativity that the two groups took, as can be seen in table 3. table 3: means, standard deviations and t-test for creativity scores before the cognitive research trust program (n = 27 for the experimental group and n = 26 for the control group). ethical consideration before collecting the data, the second researcher received the permission of the ministry of education and the school headmaster. she also sought the permission of the participating students’ parents. findings the effect of participation in cognitive research trust programme on grade 6 students’ mathematical creativity to verify hypothesis 1, we examined the effect of participation in cort programme on grade 6 students’ mathematical creativity using two methods: running ancova and post-hoc tests to test whether there are significant differences between the creativity scores in the two research groups. in addition, to verify hypothesis 2, we ran paired t-test to test whether there are significant differences between the creativity scores in the cort group before and after the implementing the cort programme. differences between the scores of the two research groups’ creativity table 4 shows the means and standard deviations of the scores of creativity and its components in the two research groups after the experiment. table 4: means and standard deviations for creativity scores after the cognitive research trust programme (n = 27 for the experimental group and n = 26 for the control group). we see from table 4 that the scores of the experimental group are higher than those of the control group in creativity and its components. to examine the significance of the differences in the means of creativity and its components of the two research groups, we ran ancova. according to montgomery (2013), running ancova should be done after ensuring normality and equality of variance of the residuals of scores of the groups participating in the research. testing for normality of the residuals of creativity scores in the two research groups, using the shapiro-wilk test (shapiro & wilk, 1965), we found that the residuals of the creativity scores, including the components’ scores, were distributed normally. we also ran levene’s test (e.g. carroll & schneider, 1985) to examine the equality of variances of the residuals of creativity scores in the two research groups, which gave equality of variances of the residuals. we ran ancova to examine whether there was significant difference between the creativity scores of the two research groups that could be related to the intervention, that is, participating in the cort programme. though ancova takes into consideration the creativity scores before the experiment, we ran independent t-tests to examine the difference in creativity scores of the two groups before the experiment. the results showed, as described above, that the means of the creativity scores of the two research groups before the experiment were not statistically significant at the 0.05 level. the results of the ancova test showed significant differences in creativity scores after the experiment between the cort programme group and the control group, with f(1, 50) = 16.46, p = 0.000 for fluency; f(1, 50) = 18.177, p = 0.000 for flexibility; f(1, 50) = 17.13, p = 0.000 for originality; f(1, 50) = 13.95, p = 0.000 for creativity. a look at r-squared (r2) showed that r2 = 0.520 for fluency, r2 = 0.561 for flexibility, r2 = 0.441 for originality and r2 = 0.402 for creativity. the previous results show that the group of students who participated in the cort programme scored significantly higher in mathematical creativity and its components than the group of students who did not participate in the programme. these results indicate the acceptance of hypothesis 1. at the same time, they imply that the cort programme accounted for 40% of the total variance in creativity as a consequence of the cort programme. this percentage is the r2 value above for creativity and it indicates the contribution of the cort programme to the variance in creativity scores between the experimental and control group. this accounting for the variance is more in the components of creativity, where the r2 values ranged between 0.441 and 0.561. differences in the creativity scores of the cognitive research trust group before and after the experiment to verify hypothesis 2, a paired-samples t-test was conducted to compare the creativity scores of the cort group before and after the experiment. the results showed that the creativity scores in the post-exam were significantly higher than those in the pre-exam: t(52) = 4.677, p = 0.000, d = 0.783 for fluency; t(52) = 4.955, p = 0.000, d = 0.793 for flexibility; t(52) = 3.396, p = 0.001, d = 0.585 for originality; t(52) = 3.304, p = 0.002, d = 0.612 for creativity. the effect sizes indicated that the difference between the creativity scores after and before the cort programme were medium to large (dunst, hamby & trivette, 2004). these results indicate the acceptance of hypothesis 2. effect of gender on the creativity of grade 6 students who participated in the research in general and in cognitive research trust programme in particular the results of the independent sample t-test showed no significant differences before the cort programme in creativity scores of grade 6 students that could be attributed to gender. moreover, to verify hypothesis 3, we ran anova test to examine the effect of gender on the creativity of students who participated in the cort programme (the effect of the interaction between the programme and gender). the results gave: f(1, 49) = 3.162, p = 0.061 for fluency; f(1, 49) = 2.332, p = 0.133 for flexibility; f(1, 49) = 1.197, p = 0.279 for originality; f(1, 49) = 0.962, p = 0.332 for creativity. all the significance values of fs indicate that the interaction of gender with the intervention did not yield significant differences in the creativity of grade 6 students who participated in the cort programme. these results indicate the acceptance of hypothesis 3. effect of mathematical achievement on the creativity of grade 6 students who participated in cognitive research trust programme the results of anova test on creativity and its components’ scores before the cort programme showed significant differences in fluency scores of grade 6 students before the cort programme that could be attributed to achievement, f(3, 49) = 7.869, p = 0.000. anova also showed significant differences in flexibility scores before the experiment and that could be attributed to achievement, f(3, 49) = 8.011, p = 0.000. moreover, anova showed no significant differences in originality or overall creativity scores that could be attributed to achievement. a look at r2 showed that r2 = 0.325 for fluency and r2 = 0.329 for flexibility. at the same time, the post-hoc analysis, using bonferroni’s post-hoc test (e.g. garcia & herrera, 2008), showed that fluency scores before the experiment in the 86–100 achievement group (m = 0.97, sd = 0.23) was significantly higher than in the 0–55 achievement group (m = 0.64, sd = 0.33) and significantly higher than in the 56–70 achievement group (m = 0.62, sd = 0.26). furthermore, the post-hoc analysis, using bonferroni’s post-hoc test, showed that flexibility scores before the experiment in the 86–100 achievement group (m = 9.60, sd = 2.07) was significantly higher than in the 0–55 achievement group (m = 6.43, sd = 3.25) and significantly higher than in the 56–70 achievement group (m = 6.15, sd = 2.58). in addition, to verify hypothesis 4, we ran the ancova test to examine the effect of the interaction between the cort programme and mathematical achievement (the effect of mathematical achievement on the creativity of students who participated in the cort programme). the results gave: f(3, 45) = 2.598, p = 0.064 for fluency; f(3, 45) = 2.986, p = 0.041 for flexibility; f(3, 45) = 1.418, p = 0.250 for originality; f(3, 45) = 1.831, p = 0.155 for creativity. the significance values of fs indicate that the interaction of achievement with the programme yielded significant differences only in flexibility scores of grade 6 students who participated in the cort programme. these results indicate partial acceptance of hypothesis 4. post-hoc analysis was conducted using bonferroni’s post-hoc test. the post-hoc analysis showed significant differences in students’ fluency scores, as a result of the interaction between the programme and achievement. the fluency of students in the 56–70 achievement group (m = 0.70, sd = 0.30) was significantly lower than in the 86–100 achievement group (m = 1.74, sd = 0.73). at the same time, the post-hoc analysis showed significant differences in students’ flexibility scores, as a result of the interaction between the programme and achievement. the flexibility of students in the 86–100 achievement group (m = 16.84, sd = 6.48) was significantly higher than in the 0–55 achievement group (m = 8.67, sd = 3.21) and significantly higher than in the 56–70 achievement group (m = 7.00, sd = 2.98). the post-hoc analysis also showed significant differences in students’ originality and overall creativity scores, as a result of the interaction between the programme and achievement. the originality of students in the 86–100 achievement group (m = 7.75, sd = 6.16) was significantly higher than in the 56–70 achievement group (m = 0.07, sd = 0.03). moreover, the creativity of students in the 86–100 achievement group (m = 215.98, sd = 198.91) was significantly higher than in the 56–70 achievement group (m = 0.70, sd = 0.30). discussion the research results showed that the group of students who participated in the cort programme scored significantly higher in mathematical creativity and its components than the group of students who did not participate in the programme. at the same time, the research results showed that the scores of creativity and its components in the experimental group increased significantly as a result of the cort programme, that is, after the students were educated for creativity. both results are due to the variety of creativity tasks in the cort creativity unit. this variety characterised the cort creativity unit through the 10 lessons that differ in the meanings they associate with creative thinking (for example, the sixth lesson educates for defining the problem in order to facilitate solving it in various ways, while the fifth lesson educates for finding the dominant idea in the problem in order to escape this idea and thus be creative). this variety of the lessons of the programme supported the students in arriving at a systematic approach to creative thinking, and thus in internalising the meanings of mathematical creativity represented in fluency, flexibility and originality. this internalisation is a result of the multiple representations of creativity presented in the 10 lessons. furthermore, this internalisation made the students perform better in creativity and its components. moreover, the mathematical questions included in the cort creativity unit encouraged multiple solutions of a problem (see appendix 2), which supported students’ internalisation of the meanings associated with creativity. in addition to the explanation above, students’ discussions in the group (see appendix 3) motivated their thinking (williams & williams, 2011) and, as a result, encouraged their creativity in solving mathematical problems. this process also contributed to the participating students’ internalisation of the meanings of creativity, and thus motivated and increased the expressions of creativity and its components. furthermore, the homework assignments helped the students in the experimental group practise the creativity skills associated with the three components of creativity, which motivated their thinking creatively. the research results agree with previous studies which showed that cort creativity programme positively affected students’ performance in creativity (e.g. al-edwan, 2011; park & kwon, 2006). the research results also agree with leiken (2009) who found that an educating programme that emphasised multiple solutions of a mathematical problem increased students’ creativity. researchers do not agree on the effect of gender on creativity (baer & kaufman, 2008). the results of the present study agree with studies that found no significant differences in grade 6 students’ creativity that could be related to gender (e.g. kaufman et al., 2004). furthermore, the present study showed no significant differences in creativity scores that are related to the interaction of education based on the cort programme and gender. thus, the cort programme affected male and female students in the same way. this is also the case of the results reported by alshurman (2017) who found that educating university students based on the cort programme showed no statistically significant differences in the post scores that could be related to gender. the research results showed significant differences in fluency and flexibility scores of grade 6 students before the cort programme that could be attributed to achievement, but they showed no significant differences in originality or overall creativity scores that could attributed to achievement. researchers do not agree about the effect of achievement on students’ creativity (mann, 2005; nori, 2002). here too, the results were not unified. achievement affects fluency and flexibility more than originality and overall creativity. we argue that fluency and flexibility are related to the students’ formal learning of mathematics, which is not the case with originality, which is related to the production of novel ideas (leikin & kloss, 2011), apparently not taken care of in a satisfactory manner in the students’ formal learning of mathematics. in addition to the previous results, the post-hoc analysis showed that fluency and flexibility scores before the experiment in the 86–100 achievement group were significantly higher than in the 0–55 and the 56–70 achievement groups. these results indicate that achievement needs to be very high in order to produce significantly higher scores in fluency and flexibility. examining the interaction of achievement with the cort programme, the post-hoc analysis showed that, in the experimental group, the 86–100 achievement group significantly outperformed the 0–55 achievement group also in originality and overall creativity. these results show that the cort programme benefited the group with the highest achievement more than the one with the lowest achievement, but this was not the case with the other achievement groups. these results could be due to the more sophisticated learning means and learning strategies that the high achieving students have because of their richer learning history. here, we assume that high achieving students could afford richer learning strategies and problem-solving skills than others (stepanek, 1999). this explanation of the more sophisticated means and learning strategies agrees with castejón, gilar, veas and miñano (2016) who found that, in relation to learning strategies, underachieving students reported a lower use of strategies than the average and overachieving groups. this happened because, when learning, underachieving students processed less information and recovered it with more difficulty; in addition to that, they also transferred and applied less of what they learnt. furthermore, castejón et al. (2016) found that when underachieving students plan, they evaluate and control the learning pace and advancement to a lesser extent, which means that they apply less metacognitive strategies in learning. this application of less metacognitive strategies resulted in significantly lower scores of creativity (gutierrez-braojos, salmeron-vilchez, martin-romera & pérez, 2013). conclusions leikin (2013) points to the need for mathematics teachers to provide students with appropriate opportunities for developing their creative thinking. this is in line with the assumption that creativity is a public domain, so we should attend to developing it through special programmes (joussemet & koestner, 1999). we examined, in the present research, the effect of a cort programme that targeted creativity in a direct way on developing grade 6 students’ mathematical creativity. the research results implied that the programme affected students’ creativity positively, including fluency, flexibility, originality and overall creativity. the question whether educating programmes like the cort programme improve students’ creativity was raised by leikin (2009) who reviewed studies that found that it is possible to educate for fluency and flexibility but not for originality. in the present research, we found that educating for creativity can improve originality too. this could be due to the nature of the educating programme that targeted creativity in a direct way, so it resulted in improving not only students’ fluency and flexibility, but originality too. more research is needed to verify the issue of the effect of educating programmes on mathematical creativity, especially mathematical originality. one of the main questions that need special attention and further research is: what are the achievement levels of students who could be educated for creativity in general and specifically in originality? acknowledgements competing interests the authors declare that they have no financial or personal relationships that may have inappropriately influenced them in writing this article. authors’ contributions w.d. contributed mainly to the writing but was also engaged in the project design. a.t.-k. contributed mainly to the project design but was also engaged in the writing. f.g. contributed mainly to the writing process. references al-edwan, z.s.m. 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(2016). creativity from two perspectives: prospective mathematics teachers and mathematician. australian journal of teacher education, 41(12), 25–40. https://doi.org/10.14221/ajte.2016v41n12.3 appendix 1 example of a lesson work in groups to answer the following questions and reflect. question 1: which of the following numbers is the exceptional one: 21, 25, 26, 33? reflection 1: reflect on the exceptional number question. question 2: write the following term in the sequence: 2, 3, reflection 2: reflect on the following term question. question 3: what is the drawing and what is the background? reflection 3: reflect on the drawing and background question. question 4: search for a classic painting on the internet. discuss what the painting is about without searching for information on the painting. reflection 4: reflect on the painting question. reflection 5: reflect on the previous four questions. appendix 2 multiple solutions to a problem some solutions for the problem ‘which of the following numbers is the exceptional one: 21, 25, 26, 33?’: 21 is exceptional for it has 7 as a factor, 21 is exceptional for it has 21 as a factor, 25 is exceptional for it has 5 as a factor, 25 is exceptional for it has 25 as a factor, 26 is exceptional for it has 2 as a factor, 26 is exceptional for it has 13 as a factor, 26 is exceptional for it has 26 as a factor, 33 is exceptional for it has 11 as a factor, 33 is exceptional for it has 33 as a factor, 25 is exceptional for it is a square number, 26 is exceptional for it is an even number, 33 is exceptional for it is greater than 30. appendix 3 example of students’ discussion and reflection on the questions in the cort creativity unit wasim: at the beginning i thought only number 33 is exceptional because both of its digits are similar. ruba: i also thought that 25 is the only exceptional number because it is a square number. salma: this is the first time i have a question like this. it’s a beautiful question. i find this question interesting. wasim: yes, we never had a question like this in the class. they give more freedom for the students to answer questions. salma: i found difficulty in solving the question. i think we should be involved with more questions like this. they open our minds. ruba: i am sure i will be more open-minded in other questions and try to be more creative in answering mathematical questions. article information authors: renuka vithal1 ole skovsmose2,3 affiliations: 1school of education, university of kwazulu-natal, south africa2department of learning and philosophy, aalborg university, denmark 3graduate programme in mathematics education, universidade estadual paulista, brazil correspondence to: renuka vithal postal address: howard college campus, 2nd floor francis stock building, mazisi kunene road, glenwood, durban, south africa how to cite this article: vithal, r., & skovsmose, o. (2012). mathematics education, democracy and development: a view of the landscape. pythagoras, 33(2), art. #207, 3 pages. http://dx.doi.org/10.4102/ pythagoras.v33i2.207 copyright notice: © 2012. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. mathematics education, democracy and development: a view of the landscape in this editorial... open access • mathematics in action • uncertainties and contingencies • a view of the landscape • contested notions • references the links between mathematics education and democracy have been explored in a growing amount of literature. one might even claim a resurgence of the importance of this aspect in the current climate of financial crises and the ‘arab spring.’ the theme, for instance, of the 2012 conference of the international commission for the study and improvement of mathematics education was ‘mathematics education and democracy: learning and teaching practices’. the conference raised questions about democracy in the mathematics curriculum, in the mathematics classroom, in teacher education and in research. much of the writing in this area is, however, contextualised with reference to the so-called ‘developed world’: to the risks posed to long established democracies experiencing, rapid advances in science and technology and related societal changes. given the foundational place and role of mathematics in key areas of science, technology and the economy, questions emerge more forcefully about the kinds of mathematics and mathematics education needed, for whom, and how best to deliver these in diverse but also rapidly and unpredictably changing societies.the late eighties and early nineties are important to recognise for the theme of this special issue. in particular, scandinavian scholars were exploring theory and practices in education to strengthen their democracy, which coincided with the dawn of democracy in south africa and a period of rethinking the post-apartheid education system. it was also a period of ferment in mathematics education as, for example, the first conference on the political dimension in mathematics education conference (noss et al., 1990) was convened and the cultural dimensions of mathematics education were being put forward through research and practices in areas such as ethnomathematics. mathematics in action top ↑ mathematics operates in a variety of cultural and sociopolitical practices. it is part of everyday situations, professional contexts, technological enterprises and research procedures. it operates as part of a worldwide distributed technical rationality, which can be analysed in terms of mathematics in action (skovsmose, 2005; christensen, skovsmose & yasukawa, 2008): we send emails, we use credit cards, we contract loans, we get insured. workplaces include mathematics practices: part of production is automated, quality control takes place, cost-benefit analyses are conducted, goods are brought to the market, prices are set and advertised. all these practices are mathematics heavy. so is political decision-making where, for instance, implications of alternative economic policies are investigated through simulation models. mathematics is operating in technologies of surveillance, in health care, in weather forecasting and in ecological forecasting concerning the state of world. mathematics is a part of any form of technological enterprise as well as research processes in technology and science. globalised networking, with respect to communication, economy, production, distribution of welfare and poverty and social inclusion and exclusion, includes mathematics put into action. if we consider the scope of mathematics in action, it is not surprising that there is a huge concern for managing, in an efficient and proficient way, mathematics curricula as part of the educational system – a concern which recently has been expressed through international comparisons of students’ performances in mathematics. mathematics education signifies a worldwide means of developing and distributing a set of competencies and of labelling people through a finely graded exam system. at all levels of the general educational system, mathematics is a crucial component. it also forms part of a broad range of further education within science, technology, engineering, medicine, economy, management, et cetera. it would appear that mathematics education is responding to the fundamental demands of the modern labour market, which now takes the form of a knowledge market. mathematics and mathematics education are enacted in and are implicated in an increasingly uncertain world, a world in which technology has become pervasive and seeped into almost all facets of people’s lives: we may refer to the rapid spread of mobile phone technology in africa, for example. technological determinism holds that technological development sets the course of social development in general. an optimistic form of this determinism finds that technological development – due to its intrinsic laws and to the fact that it is science based – will eventually ensure social welfare on a grand scale. contrary to this, a pessimistic technological determinism depicts technology as a highway to a dehumanised world, due to the very rationality of it. we find moving beyond any form of determinism means moving beyond both optimism and pessimism. uncertainties and contingencies top ↑ mathematics-based technology brings us deep into a terrain of uncertainties and contingencies. situations are produced which can take society in very different directions. predictability of what technological innovations will emerge, how technology will be applied and with what consequences is minimal. different forms of development are leaping forth in a hazardous way, due to a technologically produced density of contingencies (skovsmose, 2005). mathematics in action makes up part of this terrain of contingencies. it is an integral part of an uncertain world. to illustrate this point: on the one hand, exploitation of natural resources is driven by technology, operating through a mathematically expressed rationality; on the other hand, mathematics enters into the models through which we seek to provide forecasts concerning the impact of those same technological enterprises. any long-term ecological implications cannot be identified without using mathematics-based models of simulation and forecasting. naturally, any such forecasting might be wrong – but whatever it is, it depends on mathematics. mathematics plays an important role in a huge variety of practices, the nature of which may differ greatly. these roles are not qualified in any particular way due to some assumed nature of mathematics. instead, mathematics forms part of technologically produced uncertainties.mathematics education forms part of open-ended social processes. on the one hand, one might assume that mathematics education blindly adapts to the demands for competencies expressed in dominant economic and technological structures. thus, one may interpret mathematics education as a way of developing a competency in following manuals, as a way of developing a prescription readiness, which is important in a multiplicity of work practices (skovsmose, 2008). on the other hand, mathematics education operates on market conditions in a globalised economy, where processes of inclusion and exclusion operate, not least through the educational system. there is, however, no transparent relationship between the competencies a mathematics education might provide and those competencies that mathematics-dense practices might presuppose. thus, one might find that mathematics education establishes citizenship and reflective insight in some situations for some students. mathematics education might even ensure new opportunities in life for groups of young people. it may be part of a string of processes of ‘social justice,’ and very many suggestions for what this could mean have been put forward. so, mathematics education is a crucial part of an unpredictable world. a view of the landscape top ↑ the extent to which links between mathematics education and democracy have been explored in a range of studies, in different parts of the world, is documented in this special issue in the literature review provided by aguilar and zavaleta. two articles make different proposals for mathematics education and this uncertainty. swanson and appelbaum put forward ‘refusal and disobedience’ as democratic action in mathematics education. they argue that ’globalisation and development discourses, via citizenship and nationalism, construct relationships with learners and mathematics education in very specific ways that delimit possibilities for egalitarianism.’ refusal ‘as a position of radical equality’ is a ‘refusal to participate in mathematics education’s colonising and/or globalising neo-liberal gaze.’ valero, garcia, camelo, mancera and romero propose that democracy be understood ‘in terms of the possibility of constructing a social subjectivity for the dignity of being’ and illustrate this possibility in mathematics education through ‘reassembling’ geometrical space in the columbian secondary school mathematics curriculum. drawing on notions of space from critical geography, the problem of territorialisation and latin american social epistemology, they show how ‘mathematical spaciality’ can be transcended ‘to social space and intimate space.’ they argue that ’decentring of the school mathematics curriculum may open the possibility for an educational project in mathematics that allows for different subjectivities.’ several articles show a concern with pedagogy and practice in mathematics classrooms. teacher practices such as ’listening’ and being able to ’promote dialogue and negotiation’ are engaged in three research articles based in the south african context: khuzwayo and bansilal; mhlolo and schafer; and brijall, bansilal and moore-russo. related to this, authors also emphasise the issue of ‘student voice.’ daher, writing from the palestinian context about student teachers’ perceptions of democracy in their mathematics class, describes how students want opportunities to express themselves in mathematics classrooms and to be allowed to be in ‘control of their actions.’ a conceptual development related to students with respect to democracy and mathematics education is the notion of ’foregrounds,’ developed by skovsmose. it refers to the future possibilities that a context reveals and provides for a person. in this special issue article skovsmose consolidates the concept of ‘students’ foreground’ by elaborating its educational meaning with respect to activities in the mathematics classroom. he shows how a ‘foreground might be ruined’ and turned into ‘a learning obstacle.’ using this same notion of foreground, but with reference to teachers, amin demonstrates, through memory work, ‘how exposure to mathematics teaching and learning when they were learners is implicated in shaping the foregrounds of teachers.’ in this way, amin provides an important new dimension to the discussions of foregrounds. it is clear that bringing democracy into mathematics education has many and varied implications for pedagogy. in her article, vithal extends a pedagogy of conflict and dialogue to integrate a pedagogy of forgiveness for a post-conflict society like south africa. this extension enables values of equity, social justice and reconciliation to become part of a mathematics curriculum. allied to the call for acknowledging social subjectivities and the dignity of being in the content of a mathematics curriculum by valero et al., vithal utilises, as a metaphor, south africa’s truth and reconciliation framework of multiple truths for bringing into dialogue and negotiation ‘multiple truths in mathematics.’ in this way, she attempts to connect a broad range of competing and cooperating developments in mathematics and mathematics education. contested notions top ↑ there is no doubt that in the triad of mathematics education, democracy and development, the concepts of democracy and development are deeply contested notions. the challenges faced by western or wealthy nations in this relation between mathematics education and democracy are, of course, also present in the context of societies variously described as ‘developing,’ ‘periphery,’ ‘south,’ or ‘third world’ in today’s networked and globalised world. but countries like south africa must engage simultaneously with, on the one hand, mathematics education and its role and function to deepen and strengthen democracy, and, on the other hand, enable and sustain key areas of development in order to overcome deep divisions, high levels of poverty and inequality.the notion of ‘development’ as it features in ‘development studies’ or in discussions of a ‘developmental state’ has not found any substantive voice in mathematics education literature. this, despite the fact that a vast majority of mathematics teaching and learning takes place in developing contexts with a lack of different human and physical resources. notwithstanding progress made in areas such as ethnomathematics, gender and equity, concerns about ‘development’ are still not profoundly researched and theorised in mathematics education. poverty and its related issues (e.g. youth unemployment), which have many implicit and explicit connections to mathematics education, do not seem to feature strongly in mainstream mathematics education research, literature, conferences and in theorising mathematics teaching and learning, even though they have major policy implications. however, some small movements in this area are emerging as can be seen in this special issue, in a survey team presentation on ‘socio-economic influences on students’ achievement’ at the most recent international congress on mathematical education (icme-12), as well as in the successive mathematics education and society conferences. these are likely to grow and become much more important for mathematics education in the future. references top ↑ christensen, o.r., skovsmose, o., & yasukawa, k. (2008). the mathematical state of the world: explorations into the characteristics of mathematical descriptions. alexandria journal of science and technology education 1(1), 77–90.noss, r., brown, a., dowling, p., drake, p., harris, m., hoyles, c., et al. (eds.). (1990). political dimensions of mathematics education: action and critique. london: institute of education, university of london. pmid:2378711 skovsmose, o. (2005). travelling through education: uncertainty, mathematics, responsibility. rotterdam: sense publishers. skovsmose, o. (2008). mathematics education in a knowledge market. in e. de freitas, & k. nolan (eds.), opening the research text: critical insights and in(ter)ventions into mathematics education (pp. 159–174). new york, ny: springer. 6205 de villiers pythagoras 62, december, 2005, pp. 31-35 31 a generalisation of the nine-point circle and euler line michael de villiers university of kwazulu-natal email: profmd@mweb.co.za "now i will have less distraction." leonhard euler, referring to losing the sight of one eye, quoted in mathematical circles (howard eves, 1969) introduction to most people, including some mathematics teachers, geometry is synonymous with ancient greek geometry, especially as epitomised in euclid's elements of 300 bc. sadly, many are not even aware of the significant extensions and investigations of apollonius, ptolemy, pappus, and many others until about 320 ad. even more people are completely unaware of the major developments that took place in synthetic euclidean plane geometry from about 1750-1940, and more recently again from about 1990 onwards (stimulated in no small way by the current availability of dynamic geometry software). the purpose of this article is therefore to give a brief historical background to the discovery of the nine-point circle and the euler line, and a simple, but possibly new generalisation and proof of the latter, that may be of interest to teachers and students. high school background the reader is reminded of the following three classic concurrency results from euclid's elements that are fortunately still mentioned (though seldom with proof) in a few south african high school textbooks. however, since these results are no longer required "theorems" for the final matriculation examination from about the late 1980s, it is likely that most teachers have simply ignored teaching them, thus producing a generation of children unacquainted with these remarkable results from our geometric heritage. (1) the medians (lines from the vertices to the midpoints of the opposite sides) of a triangle are concurrent at the centroid (centre of gravity) of a triangle (see figure 1). (2) the altitudes (perpendicular lines from the vertices to the opposite sides) of a triangle are concurrent at the orthocentre (see figure 2). (3) the perpendicular bisectors of the sides (lines through the midpoints of the sides and perpendicular to them) of a triangle are concurrent at the circumcentre, which is the centre of the circle through the three vertices (see figure 3). g figure 1. centroid h figure 2. orthocentre o figure 3. circumcentre a generalization of the nine-point circle and euler line 32 ceva's theorem in 1678 an italian mathematician named giovanni ceva discovered a surprising generalisation of the altitude and median (and angle bisector) concurrencies, namely, that if in any triangle, line segments ad, bf and ce are concurrent (with d, f and e respectively on sides bc, ac and ab), then af fc × c d db × be ea = 1 . conversely, if af fc × c d db × be ea = 1 , then line segments ad, bf and ce are concurrent (see figure 4). in ceva's honour, the line segments ad, bf and ce joining the vertices of a triangle to any given points on the opposite sides, are called cevians. ceva's theorem is a very important and useful theorem that has to form part of the standard armoury of any high school learner aspiring to be competitive at the third round level of the south african mathematics olympiad. learners who participate in the workshops and summer school of the mathematical talent search organised under the auspices of the south african mathematical society (sams) are well acquainted with this result, as are all the south african team members of the international mathematics olympiad (imo). homothetic polygons another valuable result that is usually also well known to successful mathematics olympiad contestants is the following theorem: if two polygons are homothetic (that is similar and their corresponding sides are parallel), then the lines connecting corresponding vertices are concurrent at their centre of similarity (see figure 5). the nine-point circle and euler line although leonhard euler was apparently the first person in 1765 to show that the midpoints of the sides of a triangle and the feet of the altitudes determine a unique circle, it was not until 1820 that brianchon and poncelet showed that the three midpoints of the segments from the orthocentre to the vertices also lie on the same circle, hence its name, the nine-point circle (see figure 6). the nine-point circle is often also referred to as the euler circle in honour of euler. it is also sometimes called the feuerbach circle in honour of karl feuerbach who in 1822 proved the stunning theorem that the nine-point circle is tangent to the incircles and excircles of the triangle! a result closely associated with the nine-point circle is that of the euler line (which euler presumably discovered more or less simultaneously), namely that the orthocentre (h), centroid (g), circumcentre (o) and the centre of nine-point circle (n) are collinear. moreover, hg = 2go and hn = 3ng. the historical background referred to above is widely available in standard historical resource books like boyer (1968), kramer (1970), etc. books such as these ought to be regularly consulted by teachers and lecturers in order to bring a much-needed historical perspective to mathematics in the classroom. for classic synthetic proofs of the results mentioned above, readers can for example consult any advanced geometry textbook like coxeter and greitzer (1967) or posamentier (2002). a b cd f e figure 4. ceva's theorem figure 5. homothetic polygons michael de villiers 33 experimental discovery in october 2002, i was wondering how one might generalise the nine-point circle and started off by considering what happens if instead of the concurrent altitudes, one took any three concurrent cevians (lines from the vertices to the opposite sides). next i constructed the midpoints of the segments from the cevian point h to the vertices as shown in figure 7, wondering whether there was any significance in them. dynamically dragging and manipulating the triangle with sketchpad for a while, it suddenly visually seemed to suggest that the feet of the cevians d, e and f, and the midpoints j, k and l all lie on an ellipse. this was immediately confirmed when i used a sketchpad tool to draw an ellipse through any five of these points, the ellipse passing through the remaining sixth point. much to my surprise, and delight, i next noticed with further dragging that this ellipse always passed through the midpoints of the sides of triangle abc (and turned into a hyperbola if d, e and f was dragged onto the extensions of the sides of the triangle). in other words, nine points in total lie on this uniquely determined conic! after labouring through long analytic geometry proofs (with the aid of the symbolic computer algebra of the ti-92), i later found out, much to my dismay, that the discovery was not novel at all, and was already known in the 1890s, appearing in some projective geometry texts (russell, 1893: 212). it also appears in standard projective geometry texts such as baker (1922: 41-42), which were required study material for entrance examinations into the mathematical doctoral programmes of oxford and cambridge in the 1920s and 1930s. it seems quite sad that such a beautiful projective geometry result has become forgotten and neglected. this nine-point conic result, however, contains a generalisation of the euler line as a corollary, which does not appear in any of the three references mentioned, and an internet search has also provided no explicit mention of it in the mathematical literature. furthermore, my initial analytic proofs of the nine-point conic and the euler generalisation (see de villiers, in press), although confirming the results, do not provide any satisfactory insight into why the results are true. in contrast, the proof given further down for the euler generalisation is not only synthetic, but also explanatory in terms of similarity. for the sake of completeness, the nine-point conic theorem and euler line generalisation are now formally stated, and a proof of the latter given. the nine-point conic given any triangle abc, and three cevians concurrent in h, then the feet of the cevians (d, e and f), the midpoints of the sides of the triangle (x, y and z), and the respective midpoints l, j and k of the segments ha, hb and hc, lie on a conic (figure 7). (note that an ellipse is obtained when the feet of all the cevians are on the sides of the triangle, but when some of the feet of the cevians lie on the extensions of the sides, the conic becomes a hyperbola. silvester (2001: 214-215) also explains how a six-point parabola can (theoretically) be obtained as a limiting case as the cevian point h is dragged off towards infinity.) figure 6. nine-point circle & euler line figure 7. nine-point conic and euler line generalisation a generalization of the nine-point circle and euler line 34 euler line generalisation given the above configuration for any triangle abc, then the centre n of the conic, the centroid g of abc and the point of cevian concurrency h, are collinear, and hn = 3ng. (note that the circumcentre o of triangle abc does not necessarily lie on this general euler line. however, note that just as with the nine-point circle, the centre n of the nine-point conic, is the common midpoint of segments xl, yj and zk – see proof below). proof construct the point m as the image of g in n under a half-turn. since g is the centroid of xyz (and abc), and n is the common centroid (midpoint and centre of gravity) of xl, yj and zk (as they are the respective diagonals of parallelograms xklz, jxyl and zjky), it follows that m is the centroid of jkl. and since bca is the image of the enlargement of jkl from h with scale factor 2, it follows that h, m, n and g are all collinear, and that hg = 2hm, mn = ng which implies that hn = 3ng. further euler line generalisation the above euler line generalisation generalises even further as follows. given any triangle abc with midpoints of the sides x, y and z and three cevians concurrent in h as shown in figure 8. with h as centre of similarity and scale factor 1 k , construct triangle ljk similar to abc. let n be the centre of similarity between ljk and the median triangle xyz. then h, n and g are collinear, and hn = 3 k − 1 ng . proof construct the centroid g' of triangle ljk. since abc maps onto ljk under the similarity situated at h, it follows that k gh hg = ′ and h, g' and g are collinear. since the median triangle is also similar to abc under a half-turn around g with a scale factor of 1 2 , it follows that ljk is similar to xyz with a scale factor 2 k . moreover, ljk is homothetic to xyz. therefore lines lx, jy and kz are concurrent at n (with 2 xn = knl , etc.). then since the centroid of the median triangle coincides with the centroid g of abc, and xyz maps onto ljk under the half-turn and similarity situated at n, it follows that ng ng' = k 2 and g', n and g are collinear. since the straight line through g and g' is unique, it follows that h, g', n and g are collinear. let hg' = x, then from the ratios into which hg is divided as shown in figure 9, it follows that hn ng = 3xk k + 2 × k + 2 kx( k − 1) = 3 k − 1 looking back instead of respectively using parallelograms and homothetic polygons to prove xl, yj and zk concurrent at n for the euler line generalisation and further euler line generalisation, ceva's theorem could be used. further note that both euler generalisations can be viewed as theorems about quadrilateral x(k 1) kx k•••• x(k-1) k + 2 2 •••• x(k 1) k + 2 x h gg' n figure 9. ratios between segments figure 8. further euler line generalisation michael de villiers 35 abch. an interesting physical interpretation of both generalisations is to consider finding the centroid n of point masses at a, b, c and h. for example, for the first euler line generalisation, consider unit masses at a, b, c and h. then n is the centroid of 2 masses at x and l, etc., and also the centroid of 3 masses at g and 1 at h; hence hn = 3ng. similarly, for the further euler generalisation, consider unit masses at a, b and c and mass k 1 at h. since the further euler generalisation no longer involves a conic, it is perhaps less interesting than the euler generalisation, which it generalises. indeed, this is often the case with generalisation, since the general case frequently involves fewer properties than the special case. concluding comment as mathematicians we have an educational obligation to share new developments in our discipline with our learners to combat the pervasive misconception that mathematics is a sterile and dead subject. euclidean plane geometry is particularly suited as learners can easily be led to some visual appreciation even without formal proof. moreover, it may just stimulate their own creativity and inspire them to engage in some mathematical research themselves. the availability of dynamic geometry software also encourages a kind of experimental approach in which it is easy to make and check conjectures that lie well within the means of average learners, and not just a select few. acknowledgements this article is dedicated to the recently deceased martyn cundy, co-author of the famous geometry classic mathematical models (published in 1951), who kindly suggested to me the elegant proof of the euler line generalisation, and which led to the further euler line generalisation. this research was partially funded by a national research foundation (nrf) grant from the spatial orientation & spatial insight (sosi) project, pretoria, gun no. 2050502. note: a dynamic geometry (sketchpad 4) sketch in zipped format (winzip) of the results discussed here can be downloaded directly from: http://my site.mweb.co.za/residents/profmd/9pointeuler.zip (this sketch can also be viewed with a free demo version of sketchpad 4 that can be downloaded from: http://www.keypress.com/sketchpad/sketch demo.html) references baker, h.f. (1922). principles of geometry, volume ii. cambridge: cambridge university press. boyer, c.b. (1968). a history of mathematics. new york: john wiley & sons. coxeter, h.s.m. & greitzer, s.l. (1967). geometry revisited. washington, dc: the mathematical association of america. de villiers, m.d. (in press). the nine-point ellipse: a rediscovery and proof by computer, the international journal for mathematical education in science & technology. eves, h. (ed.) (1969). mathematical circles. boston: prindle, weber & schmidt. kramer, e.e. (1970). the nature and growth of modern mathematics. princeton: princeton university press. posamentier, a. (2002). advanced euclidean geometry. emeryville: key college publishing. russell, j. w. (1893). pure geometry. oxford: oxford university press. silvester, j.r. (2001). geometry: ancient & modern. oxford: oxford university press. "life without geometry is pointless …” (author unknown) abstract introduction and background purpose of the study peer tutoring as an intervention strategy an overview on the arcs model of motivation in and adapted sense research question research methodology results discussion conclusion acknowledgements references about the author(s) abigail k. roberts department of science and technology education, university of johannesburg, johannesburg, south africa erica d. spangenberg department of science and technology education, university of johannesburg, johannesburg, south africa citation roberts, a.k., & spangenberg, e.d. (2020). peer tutors’ views on their role in motivating learners to learn mathematics. pythagoras, 41(1), a520. https://doi.org/10.4102/pythagoras.v41i1.520 original research peer tutors’ views on their role in motivating learners to learn mathematics abigail k. roberts, erica d. spangenberg received: 31 oct. 2019; accepted: 14 aug. 2020; published: 28 sept. 2020 copyright: © 2020. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract many learners are unmotivated to learn mathematics due to a lack of attention, irrelevance of mathematics, low self-confidence and dissatisfaction. however, peer tutoring can afford learners opportunities to engage with other individuals who have overcome similar challenges in the learning of mathematics and who can motivate them to become increasingly mindful of the task at hand at their own pace in a one-on-one relationship. this article reports on grade 12 peer tutors’ views on their role in motivating grade 8 and grade 9 learners to learn mathematics in relation to the four categories of learner motivation, namely attention, relevance, confidence and satisfaction (arcs). the four categories adapted from the arcs model of motivation were used as a lens to view and analyse the data using theoretical thematic analysis. this qualitative article utilised preand post-interviews as data collection instruments. ten of the best-performing grade 12 learners at an ex-model c school in gauteng province in south africa were purposively selected to participate in the research. the findings revealed that peer tutors view their role to motivate learners to learn mathematics peculiar to seven positions, which can inform future research on intervention strategies to improve mathematics performance. this article introduces research on an adapted use of the arcs model of motivation in motivating learners to learn mathematics, which is a novel way of bringing new perspectives to research on motivation in mathematics at secondary school level. keywords: mathematics; motivation; peer tutor; learning; secondary school. introduction and background in south africa, mathematics performance for grade 5 and grade 9 evident from the trends in international mathematics and science study (timss) reports is consistently poor in comparison with other developing countries such as botswana, indonesia, turkey and malaysia (mullis, martin, foy, & hooper, 2016). according to the 2015–2016’s world economic forum global competitiveness report, south africa’s mathematics performance is among the lowest of 140 countries (schwab, 2015). mathematics performance is a proxy of a learner’s ability to achieve academically (huang, craig, xie, graesser, & hu, 2016) and affects how the world views a country’s ability to perform internationally (bartelet, ghysels, groot, haelermans, & maassen van den brink, 2016). a key factor influencing learners’ mathematics performance is motivation (karakis, karamete, & okcu, 2016). ample research has shown that motivation is a necessary aspect of mathematics performance (cleary, velardi, & schnaidman, 2017; wilkins & ma, 2003; winberg, hellgren, & palm, 2014). learners who perform well often have high levels of motivation to achieve and therefore they put much effort into their learning (kim, park, cozart, & lee, 2015). when individuals are motivated their anxiety level decreases and their confidence increases (arroyo et al., 2014). motivated learners are also more focused and perform better in mathematics since they are more likely to persevere even when mathematics concepts are challenging (arroyo et al., 2014). therefore, it is essential to ensure that learners are motivated to learn mathematics and that they stay motivated (schukajlow, rakoczy, & pekrun, 2017). the concept of motivation is complex and various disciplines have taken many years to attempt to understand what motivation entails (güvendir, 2016). the oxford dictionary of english grammar (2010) (https://www.oxfordlearnersdictionaries.com/definition/english/) defines motivation as ‘a reason, desire, willingness or enthusiasm … for acting or behaving in a particular way’ (p. 963). rheinberg and engeser (2018) stated that motivation could be defined as the moment when individuals direct their life’s actions towards a positive goal. however, heckhausen (2018) claimed that motivation is more than the desire that controls an individual’s actions. specifically, motivation is learners’ natural yearnings and inclinations that influence how they act (heckhausen, 2018). ryan and deci (2000) also acknowledged that motivation is involved in causing learners to act and that learners’ motivation is often related to the communities that they find themselves in. for this article, the authors argue that motivation is the desire that learners have to learn mathematics, which can either be intrinsic, realised by themselves, or extrinsic, reached by teachers with the support of peer tutors who understand their role in motivating learners to learn mathematics. in the context of this study, the mathematics department at the school, the site of the current article, was also concerned about decline in mathematics performance of grade 8 and grade 9 learners over the past few years. this low mathematics performance could be linked to learners’ dependency on the teacher (heyd-metzuyanim & graven, 2016) and, specifically, their levels of motivation (karakis et al. 2016). although the mathematics teachers provided extra mathematics lessons twice a week after school hours, learners’ mathematics performance did not improve noticeably. in addition, many learners were unmotivated to attend the extra mathematics lessons due to inconvenient times, intimidation from other learners and aversion to the structured environment. peer tutoring as an intervention was suggested to address these challenges as learners might experience less intimidation when approaching fellow learners, and the daily contact in an unstructured environment could motivate them to focus on their mathematics learning. heyd-metzuyanim and graven (2016) acknowledged that a strategy to address poor mathematics performance is peer tutoring, which has the potential to increase learners’ motivation. one of the authors, in her capacity as a teacher in a high school, has observed that many learners are unmotivated to learn mathematics due to various reasons. some learners find it challenging to pay attention in the mathematics classroom, or view mathematics as not useful. others do not have confidence in their abilities to engage in mathematics, or could be unsatisfied with the quality of feedback received from teachers. peer tutors could assist teachers and learners in addressing these aspects if they comprehend their role in motivating learners to learn mathematics. several reasons could contribute to learners finding it challenging to pay attention in the mathematics classroom. many mathematics classrooms in south africa are crowded. according to ntow and adler (2019), such classrooms have a propensity for teachers to dominate discussions, while learners have limited chances to communicate openly or with other learners. however, peer tutors, either in the same grade level, or in the case of this study in a higher grade level, can provide learners who are struggling with mathematics with one-on-one attention that is not always possible in a crowded classroom. in addition, when learners are involved in a task, they have greater awareness of the content being taught. by engaging with peer tutors, learners could be encouraged to become increasingly mindful of the task and in return are often more motivated to continue with the task (kim et al., 2015). according to hernandez‑martinez and vos (2018), the usefulness of mathematics provides relevance to mathematics. however, from the authors’ perceptions, many learners are demotivated and view mathematics as irrelevant and not applicable in real life or for their future endeavours. however, when peer tutors demonstrate the future relevance of what the learner could achieve, they may enable the learner to envision what the future may hold, which could motivate learners to learn mathematics. according to keller’s (1987) arcs model of motivation, relevance ensures that learning fulfils the requirements for future study (izmirli & izmirli, 2015; milman & wessmiller, 2016). the authors argue that learners’ sense of themselves directly affects their motivation to achieve in mathematics. learners with a positive mathematics identity view mathematics as enjoyable and worthwhile and are confident in their ability to learn the subject and strive to become possible selves. according to oyserman, destin and novin (2015), possible selves refer to people’s aspirations and challenges for the future to form their identity. thus, positive selves in this study signify learners’ motivation and identity; thus, what learners envision themselves to become, what they desire to become, as well as their anxieties of who they do not want to become play into incentive and motive. focusing on the elements of identity, namely ideal self, ought self and learning experiences, learners are motivated to become the possible selves they have envisioned. oyserman et al. (2015) claimed that both the ideal self and ought self are desired identities a person wishes or feels compelled to become. the ideal self refers to what an individual would like to become, and the ought self refers to the characteristics that individuals believe they should have. learning experiences refer to the context where learning takes place such as the mathematics classroom. according to oyserman et al., learning contexts affect which identities transpire, the consequences of these identities for learning, and how learning difficulties are probably accounted for. perceptions of mathematics being difficult or a virtue one is born with could have a negative influence on learners’ self-confidence to engage in mathematics. peer tutors could be able to address learners’ beliefs about themselves and the mathematics to be acquired and assist learners with the mathematics content to be learnt. according to kim et al. (2015), learners tend to be more motivated if they perceive the importance of a task. therefore, the authors argue that when learners believe that they are prepared for mathematics tasks, learners’ motivation increases. however, mathematics tasks should not be too challenging for learners. when a task is perceived as too difficult, learners can be led to believe that they are failures, and this negatively influences their motivation. therefore, schukajlow et al. (2017) argued for more attention to the content in mathematics tasks as motivation is content-specific. satisfaction is linked to feedback (milman & wessmiller, 2016) and forms part of extrinsic motivation (güvendir, 2016). extrinsic motivation refers to the achievement of a goal to accomplish another goal such as praise or the avoidance of punishment for example negative feedback (güvendir, 2016). intrinsic motivation, conversely, is the innate longing of learners to achieve a desired outcome or simply complete a task for pure pleasure without seeking for a reward (winberg et al., 2014). during contact time, teachers often need to continue with new mathematics content to complete the mathematics syllabus and cannot immediately provide feedback to learners’ queries. neglecting learning difficulties in mathematics could negatively influence learners’ extrinsic motivation to learn the subject. assistance by peer tutors, however, may enhance the response time for feedback and enable learners to gain satisfaction for and from their work (izmirli & izmirli, 2015). learners would not have to wait for a formal lesson by their teachers, but could be assisted by peer tutors to build understanding of the subject content (karakis et al., 2016) at their own pace and in their own time (kroeger & kouche, 2006). the involvement of peer tutors in the learning process can enhance learners’ motivation and eventually lead to better mathematics performance (kim et al., 2015). despite the benefits peer tutors can add to the learning of mathematics, kim et al. (2015) acknowledged that there is a need for an inquiry on how peer tutors perceive their role in motivating other learners to participate in the mathematics classroom. the authors believe that peer tutors’ views regarding their role in motivating learners matter, because their views will determine how they will motivate other learners to participate in the mathematics classroom. many international studies on motivation in mathematics have been conducted (chue & nie, 2017; goldin et al., 2016; novak, 2014; schukajlow et al., 2017; winberg et al., 2014; yu & singh, 2018). among them, novak (2014) discussed the need for learners to have learning that learners believe to be applicable to their real-life experiences and that will be appealing to them. schukajlow et al. (2017) focused their research on affective constructs such as emotions and motivation, and how these constructs influence learners’ views and interest in mathematics. winberg et al. (2014) found that learners’ emotional experience of mathematics affects their motivation. if learners perceive the content to be interesting or useful, they are more likely to engage with it. ample research has also been dedicated to motivation in mathematics at school level in south africa (george & adu, 2018; grobler, moloi, & thakhordas, 2017; oswald & rabie, 2016; tsanwani, harding, & engelbrect, 2014). in particular, grobler et al. (2017) discussed the need to increase mathematics teachers’ self-motivation. oswald and rabie (2016) found that gifted learners who are motivated and persistent are more likely to achieve in mathematics. however, none of the aforementioned studies relates the role that motivation plays in the learning of mathematics to peer tutoring as an intervention strategy. purpose of the study this article reports on a study (roberts, 2019) with the purpose of establishing the views peer tutors hold with regard to their role in motivating grade 8 and grade 9 learners to learn mathematics in relation to the four categories of learner motivation, namely attention, relevance, confidence and satisfaction, adapted from keller (1987). peer tutoring as an intervention strategy motivation to achieve is often driven by a desire to be included and to be a part of something (kim et al., 2015) – for example, being included or accepted in a learning community consisting of participants who contribute to meaningful mathematics discussions and support in the teaching and learning of mathematics such as peers, tutors and teachers. the learning community and the views of learners as part of that community play an integral role in learners’ motivation (gardner, 1985), particularly in mathematics (lutovac & kaasila, 2014). peer tutoring is a strategy to create such a community that could address learners’ beliefs about themselves and the mathematics they need to acquire (hoops, yu, wang, & hollyer 2016). a community of learning could also motivate peer tutors in their perceptions of their role to learn new mathematics content and to accept their limitations pertaining to their own subject content (galbraith & winterbottom, 2011). with class sizes increasing at colleges and universities globally, tutoring and more specifically peer tutoring has grown in popularity (alegre, moliner, maroto, & lorenzo-valentin, 2020; nawaz & rehman, 2017). in south africa, tutoring has increased significantly since the 1990s (clarence, 2018). tutoring in south africa is an important intervention strategy that is utilised mainly at university level as an essential mediation that affects learners’ ability to retain the knowledge that they have acquired (layton & mckenna, 2016). the tutorial system enables students an opportunity to gain the necessary knowledge to pass (clarence, 2018; layton & mckenna, 2016). however, peer tutoring is not a common formalised intervention strategy used at the school level. the meta-analysis on peer tutoring in secondary education conducted by alegre, moliner, maroto and lorenzo-valentin (2019) also revealed that implications for practice at the secondary education level have not been investigated sufficiently despite ample existing literature on peer tutoring in mathematics at higher education level. peer tutoring in mathematics can be linked to increased academic performance. by affording learners the opportunity to engage with other individuals, learners can become increasingly mindful of the mathematics task and are often more motivated to continue with the task at hand (gardner, 1985; kim et al., 2015). peer tutors either can address motivation explicitly or, depending on the availability and access to individualised assistance, can implicitly inspire learners’ motivation to achieve in mathematics (grills, 2017). research on peer tutoring has indicated several benefits. peer tutoring could possibly shorten the response time for feedback from the teacher or allow learners the opportunity to linger over sections of work at their own pace, in their own time, after the teacher has moved on to new work (kroeger & kouche, 2006). this feedback enables learners to build their own understanding of the content (karakis et al. 2016). due to the encouragement of discourse and communication between the learner and the peer tutor, peer tutoring can address the need for peer approval (kim et al., 2015). peer tutoring focuses on a one-on-one relationship between the learner and peer tutor. this relationship has the potential to have a greater impact on the learner since the peer tutor and learner often have more in common (kroeger & kouche, 2006). despite the many benefits attached to peer tutoring, a disadvantage might be the perceived status difference between peer tutors and learners, which can result in negative relationships outside the classroom (de backer, van keer, & valcke, 2015). although both the peer tutors and learners have a joint responsibility in regulating learning, peer tutors play a more instructive role to guide learners in their learning of mathematics, while learners have to take ownership for their own learning. this difference in roles may result in some peer tutors becoming too directive in regulating learning, while learners may view peer tutors as the ones responsible for managing their learning. despite research showing predominately positive social interactions between peer tutors and learners (greene, mc tiernan, & holloway, 2018), de backer, van keer and valcke (2015) suggested further research on negative socio-emotional peer interactions since there is the opportunity that they might have a negative influence on the learners. regrettably, much research related to affective constructs in peer tutoring focuses on learners and not on peer tutors. kroeger and kouche (2006) revealed that peer tutoring at an inclusive middle school in the midwest (united states) affects learners’ attitudes towards mathematics positively. in addition, topping, cambell, douglas and smith (2003) discovered at a rural primary school in the united kingdom that mathematics games in peer tutoring have a positive impact on the attitudes of learners towards mathematics and encourage learners to continue with mathematics even when they find the content difficult. therefore, kim et al. (2015) argued for the need of an inquiry on how peer tutors perceive their role in motivating learners to engage in the mathematics classroom. an overview on the arcs model of motivation in and adapted sense keller (1987) developed the arcs model of motivation in response to a need for an ordered and more logical way of recognising and solving problems in learners’ motivation. this model endeavours to stimulate and encourage learners’ desire to achieve (karakis et al., 2016), and an adaptive arcs model provides thus an effective framework for understanding peer tutors’ views on their role in motivating learners to learn mathematics. keller’s arcs model of motivation focuses on four aspects of a learner’s motivation, namely attention, relevance, confidence and satisfaction (keller, 1987). this model is designed to create healthy learning environments while ensuring that learners’ motivation is addressed (izmirli & izmirli, 2015; karakis et al., 2016). the first category of keller’s (1987) arcs model of motivation, attention, aims to gain the learners’ attention by introducing different teaching methods and presentation (izmirli & izmirli, 2015). attention involves (1) capturing learners’ interest, (2) stimulating their inquiry and (3) maintaining their attention (keller, 2000). a strategy to capture interest includes the introduction of seemingly contradicting facts or experiences that the learner previously thought were true, known as the incongruity or the conflict aspect. another strategy is to provide different and varied ways of presenting mathematics content, such as using computer programs, videos or group activities to keep learners motivated (novak, 2014). demonstrating mathematics to the learners through visual examples or models – concrete representations – also aids in keeping learners interested in the subject. likewise, variation or humour may change the way the content is presented and can stimulate learners’ interest (keller, 1987). relevance is the second category of keller’s (1987) arcs model of motivation and one that ensures learning fulfils the requirements for future study or a chosen career (izmirli & izmirli, 2015; milman & wessmiller, 2016). the instructor, whether a teacher or a tutor, links content to familiar events or current happenings (milman & wessmiller, 2016) by (1) relating it to goals, (2) matching it with learners’ interests or (3) connecting it to learners’ experiences. it is useful for the learner to be able to link learning, where possible, to multiple situations. teachers should also link mathematics lessons to their future usefulness (izmirli & izmirli, 2015). some strategies could include (1) relating the mathematics curriculum to real-world issues, (2) addressing learners’ educational needs, (3) linking present content to future needs, (4) clearly laying out objectives, (5) encouraging group work and (6) addressing learners one on one (milman & wessmiller, 2016). showing the relevance of mathematics increases motivation because learners are able to see how mathematics content links to other subjects and it develops a greater understanding of why mathematics learning is necessary for them. thirdly, keller’s (1987) arcs model addresses the category of confidence. learners need to believe that they are capable of completing tasks. confidence includes how secure the learners feels in their learning environment (keller, 1987) and includes their success expectations. when learners believe that they can achieve, their chances of successfully completing a task are increased (keller, 1987); thus their success opportunities increase. learners’ confidence plays a vital role in their perseverance and achievement (keller, 1987). when learners feel confident to begin – and possibly complete – a task, thus taking personal responsibility, their self-efficacy is addressed (hodges & kim, 2013). focusing on what learners are able to do enhances confidence in learners. allowing learners to apply work and display what they have done affords them the chance of feeling good about their work (milman & wessmiller, 2016). satisfaction focuses on how learners feel about tasks they have accomplished. personal satisfaction refers to the pride a learner feels when a task is completed (izmirli & izmirli, 2015; keller, 2000). although intrinsic motivation is important, it can be difficult to impact and develop. intrinsic satisfaction occurs (1) when learners receive credit for their achievements (keller, 2000) that matches their intrinsic feeling about their accomplishments and (2) when learners feel that others have acted fairly towards them (keller, 2000). it is important that learners feel that they have been treated fairly, that the workload was reasonable and consistent and that there was no discrimination (keller, 2000). external motivation, through awards or praise, is easier to influence. extrinsic rewards can be either functional or representative like the receiving of recognition in practical and tangible methods such as marks, honours and other visible recognitions of achievement (keller, 2000). deriving from research studies, the authors do not view intrinsic or extrinsic motivation as better than the other. if learners are positively motivated and mathematics learning takes place, the type of motivation teachers provide or what learners prefer is of less concern. research question the research question is: what are grade 12 peer tutors’ views on their role in motivating grade 8 and grade 9 learners in learning mathematics in relation to the four categories of learner motivation adapted from keller (1987)? research methodology a predominantly qualitative research method was used to establish peer tutors’ views on their role as motivators in the learning of mathematics. this method was suitable as it was used for an exploratory purpose to seek deeper understanding of a real-life event in a context-specific setting (creswell, 2013). the four categories adapted from keller’s (1987) arcs model of motivation were used as a lens in designing the interview schedule, as well as in analysing the data using theoretical thematic analysis (braun & clarke, 2006). although qualitative data were collected, the data were also quantified to generate a deeper insight into the relevance of the codes in terms of frequency, and to minimise subjective inferences (keller, 2017). the study was conducted in an ex-model c school in the north of johannesburg. ex-model c schools are semi-private institutions in south africa receiving government funding for some staff salaries, but parents are also charged fees. these schools used to be for white children only during the apartheid era before 1991, but are now multicultural and highly diverse. the school had approximately 300 learners in grade 8 and 300 learners in grade 9. from a population of 15 grade 12 peer tutors at the school, 10 peer tutors were purposively selected based on the following criteria: (1) voluntarily participation, (2) attending a school with an existing peer-tutoring programme, (3) in grade 12, (4) selected mathematics as a subject and (5) obtained an average greater than 80% in grade 11. there were six female and four male participants. the peer-tutoring programme ran for nine months (february to october) from monday to thursday for an hour and a half after school. learners in need of additional support in mathematics could only be identified by the end of january after availability of the first cycle test results. the programme was concluded by the end of october before the final grade 12 examinations. the programme was conducted after school hours to ensure that it did not affect contact time during normal school hours, but also to allow sufficient time to support learners. there were two to three peer tutors assisting the learners on each day. learners who had received low marks for a test or a previous examination were encouraged by their teachers and the head of the mathematics department to attend in order for them to gain extra practice and assistance. one-on-one, semi-structured interviews with the 10 peer tutors were conducted at the beginning and the end of the nine months that the peer-tutoring programme ran. similar interview questions were utilised for both periods. the interview questions were conceptualised in relation to the research question (what are grade 12 peer tutors’ views on their role in motivating grade 8 and grade 9 learners in learning mathematics in relation to the four categories of learner motivation (attention, relevance, confidence, satisfaction) adapted from keller (1987)?) and the problem statement (many learners are unmotivated to learn mathematics because they find it challenging to pay attention in the mathematics classroom, view mathematics as not useful, do not have confidence in their abilities to engage in mathematics, and could be unsatisfied with the quality of feedback received from teachers). the interview questions were: what do you think your role as a peer tutor in the learning of mathematics is? elaborate. how would you as a peer tutor ensure that learners’ attention is maintained in the learning of mathematics? how would you as a peer tutor ensure that learners are able to see the relevance of mathematics? how would you as a peer tutor ensure that learners’ confidence in mathematics improves? how would you as a peer tutor ensure that learners feel satisfied about the mathematics they learn? the interviews were transcribed word for word and captured, using the software package atlas.ti, to engage with the details of the interviews. data were analysed using theoretical thematic analysis (braun & clarke, 2006). predetermined categories were used that aligned closely with the research question adapted from keller’s (1987) arcs model of motivation, namely attention, relevance, confidence and satisfaction, as well as synonyms or similar terms, for example ‘responsive’, ‘importance’, ‘comfortable’ and ‘pleased’. retrieving meaning from the categories was done by sorting the categories into subcategories and further breaking the subcategories into codes. to ensure trustworthiness, the research question guided the progress of the research and determined the choice of instrument. all data were compared against the adapted arcs model of motivation for consistency in results. furthermore, previous research done by keller (1987) was used as a guide to develop the interview questions and to strengthen the dependability of this study. a dense account of the context and a detailed description of the data collection and analyses procedures contributed to this study (rule & john, 2011), which also set the conditions for other researchers to decide on the credibility and transferability of the study. however, the contextual nature of this study only allows for transferability if adapted to a similar context. nevertheless, the authors supplied a thorough literature review from previous studies, conducted in and outside of south africa, for further reference. to ensure confirmability, member checking was done with the participants by verifying, and amending where needed, the correctness of the interpretation of data collected from the interviews (rule & john, 2011). all claims and findings were supported by data and controlled against previous findings from literature. faculty members at the university of johannesburg, who are experts in the field of mathematics education, checked all interview questions to ensure that they relate to answering the main research question and to establish that the interview questions are suitable to the purpose of the study. ethical consideration finally, because one of the authors was a teacher at the school, full ethical considerations (ethical clearance number 2018–021) were adhered to in order to address researcher bias. ethical considerations were implemented to protect the participants and the researchers. for this research study, it was important that the researchers valued the participants’ privacy. university of johannesburg faculty of education research ethics committee nhrec registration number: rec-110613-036. results table 1 illustrates the frequency of comments made in the interviews on what peer tutors’ views are on their role in motivating grade 8 and grade 9 learners in learning mathematics in relation to the four categories of learner motivation adapted from keller’s (1987) model, namely attention, relevance, confidence and satisfaction, as well as the number of participants who responded per code. the frequency (f) and percentage of responses from the peer tutors that were relevant to the four categories were calculated. the number (n) of peer tutors who responded to the codes was also determined. table 1: categories, subcategories and codes. all 10 peer tutors addressed certain aspects under the categories of attention and confidence. most of the comments were made under the category on attention (57 out of 107; 53.3%), while only 17 out of 107 responses (15.9%) referred to the category on confidence. seven peer tutors (18 out of 107 utterances; 16.8%) deliberated on some aspects under the category on relevance. although nine peer tutors mentioned a few aspects under the category on satisfaction, only 15 out of 107 (14.0%) remarks were made in this regard. attention three subcategories are classified under the category on attention, namely capturing interest, stimulating inquiry and maintaining attention (keller, 2000), which address learners’ motivation to participate in the mathematics classroom. first, nine peer tutors, thus most of the participants, referred to capturing of the learners’ interest as a strategy to get their attention. although none of the peer tutors stipulated that they used incongruity or humour as a way to get learners’ attention, eight peer tutors indicated that they captured learners’ interest by utilising visual representations. participant 1 commented that ‘using the white board [and using] different colours helped learners visualise [what was being asked]’ (female, pre-interview, march). five peer tutors discussed how they allowed learners to use different methods from the ones taught in class by the teacher to complete a task. according to participant 6: ‘i gave them more modern examples … [i also] gave them some tricks to make it easier for them’ (male, pre-interview, february). the finding is similar to both milman and wessmiller (2016) and novak (2014) who claimed that relating mathematics in a tangible and specific way is important in getting learners’ attention. from the finding, it can be deduced that the peer tutors believe that for the learners to increase their mathematics motivation it is important that they understood the mathematics in terms of their lived realities. three peer tutors discussed how they preferred working with worksheets with specific questions and methods to motivate learners to do mathematics. participant 1 said that worksheets worked well because it meant that the learners ‘actually had something to ask about’ (female, post-interview, august). this finding concurs with milman and wessmiller (2016) suggesting that to get and keep learners’ attention, teachers can use activities that involve numerous forms of collaboration. nine peer tutors addressed variation as the central way of capturing learners’ attention. variation is achieved by changing the way that the content is presented (keller, 1987). participant 8 found that it was important to try to ‘make it more interesting, not just [your] standard teaching way of doing [the lesson]’ (male, post-interview, october). five peer tutors mentioned using a variety of teaching methods and styles to keep the learners’ attention in mathematics. participant 4 spoke about how he would see what the learners wanted to do: ‘some [learners] preferred just to see it written down for them, or some [learners] preferred to do examples. they didn’t just all learn in the same way’ (male, post-interview, october). this finding corresponds with several researchers’ findings that varying methods of engagement are important to improve learners’ attention, which in turn affects their motivation (izmirli & izmirli, 2015; milman & wessmiller, 2016). this finding could indicate that the peer tutors were aware of the learners’ needs and were conscious of getting and keeping their attention. these strategies provided varied and stimulating activities, which may have been different to what the learners had experienced in the past and could contribute to learners being more motivated to engage in the mathematics classroom. secondly, six peer tutors, thus more than half of participants, addressed stimulating inquiry, under the category of attention. these peer tutors stressed the importance of asking questions and engaging with the learners. five of them specifically asked the learners questions. participant 8 stated, ‘i would ask them questions, so they can see for themselves where they are going wrong, so i [was continually] interacting with [the learners]’ (male, post-interview, october). three peer tutors addressing inquiry spoke about how they encouraged the learners to come prepared to ask questions, especially questions that the learners did not want to ask the teacher during class. this finding aligned with hodges and kim (2013) stating that mathematics teachers should encourage a sense of inquiry in order for learners to be more attentive. from this finding it is clear that the peer tutors believed they had the learners’ attention and were engaged with them in such a way that the learners felt comfortable enough to engage openly and candidly with the peer tutors. lastly, maintaining learners’ attention through participation under the category on attention was also addressed by six peer tutors, thus more than half of them. most of these peer tutors indicated that they worked alongside the learners. participant 3 interacted with the learners one on one until they had completed the question and remarked: ‘i never just did something on my own and made them watch. i kept communicating with them to see if they were following’ (male, post-interview, october). encouraging learners to participate with the content is important in keeping the learners’ attention. winberg et al. (2014) also found that the learners’ attitudes towards mathematics influence how they participate with the content, which can have a direct influence on their motivation to learn. by encouraging learners to participate not only with other learners, but also with the peer tutor and the content being presented, the peer tutors gave the impression that they were able to keep learners’ attention on the mathematics being completed, which might have motivated learners to learn mathematics. relevance the second category adapted from keller’s (1987) model of motivation is relevance. keller (1987) noted that relevance is not simply the content being taught but can also be the way the content is taught. under the category on relevance, three peer tutors felt that they did not explicitly address the relevance of mathematics with the learners, which could have led to some learners being unmotivated to learn mathematics. this acknowledgement may be interpreted that the rest of the peer tutors believed that the learners, who attended the tutorial sessions voluntarily, already knew that mathematics was relevant to their lives. participant 8 said, ‘i definitely think that they showed the initiative to come so that already shows that it is important to get good marks’ (male, post-interview, october). this finding is similar to chue and nie (2017) asserting that learners need to be encouraged to make an effort, so that different learning approaches, such as tutorials, can affect achievement and motivation. by attending the tutorials, learners might have felt that they were able to receive help that would benefit them in the future. this finding could be interpreted that the learners who attended the tutorials were demonstrating a degree of self-regulation by acknowledging that attending the tutorials could help them achieve their goals, which might have addressed the problem of some learners being demotivated to learn mathematics. the category on relevance comprises three aspects: relating to goals, matching interests and tying to experiences. none of the peer tutors discussed how they related the learners’ mathematics goals to show the relevance of mathematics. instead, they focused on how mathematics would affect the learners’ future. as learners were not exposed to the usefulness of mathematics in real life, it could lead them to be unmotivated to participate in the mathematics classroom. however, four peer tutors, thus less than half of the participants, addressed the aspect of matching interest of mathematics by referring to the future usefulness of mathematics. participant 7 said, ‘[i tried to] make them understand how much they should understand maths right now, like the basics of it. [i] make them see how important it is in life’ (female, post-interview, october). this finding is aligned with oyserman et al. (2015) arguing that when people are motivated to complete a task (or in this case, attend a tutorial) they are envisioning its usefulness to their future, they are creating an image of who they would like to be one day, or what they are hoping to achieve. by introducing the relevance of mathematics in real life, the challenge of learners being unmotivated to participate in the mathematics classroom was addressed. despite the previous, possibly negative, learning experiences a learner may have had with mathematics, they may still choose to engage in tutorials. when the peer tutors addressed and demonstrated the future relevance of what the learner could achieve, they could further enable the learner to envision what the future may hold, which may motivate learners to learn mathematics. none of the peer tutors referred to linking mathematics to learners’ experiences in order to demonstrate the relevance of mathematics and influence the motivation to pursue mathematics during the interviews. however, four peer tutors, fewer than half of the participants, addressed modelling. participant 1 commented on how she believed that the learners could understand the importance of mathematics since there were peer tutors who apparently enjoyed the subject: ‘[tutoring has] got a lot of people who like maths in one room’ (female, post-interview, august). learners need mentors to show them the possibility to achieve (moliner & alegre, 2020). peer tutors are able to guide learners in becoming increasingly self-regulated (heyd-metzuyanim & graven, 2016; hoops et al., 2016). this finding confirms what moliner and alegre (2020) stressed, namely that peer tutors are role models to learners. they can display what learners are able to achieve and they can assist learners with mathematics content. the peer tutors represent what they were able to achieve mathematically as they went through similar circumstances to the learners at an earlier stage. confidence the category on confidence consists of three aspects, namely success expectations, success opportunities and personal responsibility. none of the peer tutors indicated that they explicitly laid out learning requirements to the learners. however, three peer tutors, thus few of the participants, spoke about organising questions in increasing levels of difficultly, thus creating success opportunities. participant 10 said that he let the learners start with simple questions. ‘[i gave] them sums that would be easier for them to solve, so [the learners] don’t [have as much to] write [otherwise the sums] look confusing and that is intimidating’ (male, pre-interview, february). this finding corresponds with linnenbrink-garcia, patall and pekrun (2016) who contended that when a task is perceived as too difficult, learners could be led to believe that they are failures, and this negatively influences their motivation. however, according to pintrich (2003), when learners are presented with strategies that enable them to persevere with challenging tasks, and when they believe that they are prepared for the tasks, learners’ motivation increases. thus, this finding can be interpreted that when the peer tutors address learners’ confidence to complete difficult tasks, the learners are more likely to persevere because of an increase in their level of motivation. only one peer tutor mentioned the relevance of expectations. participant 1 claimed she felt that it was important that learners knew what was expected of them and that there was not only one way to get to the answer. this finding corresponds with lazarides, buchholz and rubach (2018) stating that it is important that the teacher show interest in the subject and the work being completed to have the most impact on learners’ interest. however, it could be deduced that most peer tutors did not address learners’ expectations and, therefore, did not boost their self-confidence to learn mathematics, which led them to be still demotivated to participate in the mathematics classroom. seven peer tutors, thus many of the participants, focused on personal responsibility, which involves attribution and self-confidence. these seven peer tutors stated that they primarily assisted in improving learners’ confidence by verbally attributing the learners’ success to the effort that the learners were putting in. participant 2 spoke about how important it was that she was patient when learners got an answer wrong. it was important for her that the learners knew they could do the work for themselves. she reflected, ‘i think it’s a big thing [for us] just being patient … letting them know they can do it and letting them know that we are here for them’ (female, pre-interview, march). although not explicitly addressed, four peer tutors, thus less than half of the participants, implied that they helped learners’ mathematics self-confidence by allowing learners to work more independently (keller, 1987). participant 7 said that she ensured that the learners knew she was there to assist but ‘eventually they would just do it themselves’ (female, post-interview, october). these peer tutors allowed learners to practise and repeat under low-risk conditions (keller, 1987). participant 8 stated that he would remain close to the learner but allow them to figure the sum out for themselves (male, post-interview, october). two peer tutors, thus very few of the participants, believed that they helped learners’ self-confidence by creating a safe space where the learners knew they did not have to be perfect. participant 9 said she encouraged the learners even when they made mistakes: ‘it’s okay, try again’ (female, post-interview, october). similar to the above-mentioned findings, but with reference to teachers, ryan and deci (2000) noted that competency, autonomy and relatedness are vitally important in a learners’ development with their community of learning. when learners are highly motivated it is often due to the teacher giving clear goals addressing what keller (2000) referred to as ‘success expectations’. when the teacher has provided tasks that are difficult, the teacher is providing the learner with what keller referred to as success opportunities. when teachers give learners a sense of autonomy, it enhances their personal responsibility. from the findings, the same applies to peer tutors. some peer tutors addressed the needs regarding learners’ competency, autonomy and relatedness by motivating learners through boosting their confidence. satisfaction although the category on satisfaction involves intrinsic satisfaction, rewarding outcomes and fair treatment, the peer tutors mainly remarked on rewarding outcomes when discussing learners’ satisfaction. only two peer tutors, thus very few participants, mentioned intrinsic satisfaction. natural consequences, in the arcs model of motivation, is allowing learners to use the skills that they have acquired and to reinforce their intrinsic pride (keller, 1987). participant 9 explained that she tried to show the learners that they are able to get better marks if they focus on their mistakes more closely before attempting to correct them. she indicated: ‘i told them, if you are getting like a 65% you can push for a 70%, so let’s try look at the small little things you are doing wrong’ (female, post-interview, june). this finding concurs with ryan and deci (2000) alluding to intrinsic motivation being important, as it builds on the learners’ inherent longing to learn. in mathematics, intrinsic motivation can often indicate how learners will achieve and what learners will be able to achieve in the long term (singh, granville, & dika, 2002). some of the peer tutors recognised learners’ inherent desire to learn and attempted to give them some personal strategies to enable the learners to take charge of their own learning and, by implication, futures. according to keller (1987), addressing the learners’ intrinsic satisfaction should help increase the learners’ motivation. five peer tutors, thus half of the participants, referred to the aspect of rewarding outcomes. two of these peer tutors specifically alluded to unexpected rewards (keller, 2000). unexpected rewards are compensations that encourage intrinsic motivation by rewarding boring tasks with unanticipated rewards (keller, 1987). participant 1 spoke about how she allowed the learners to complete the task on the teacher’s whiteboard, which really encouraged the learners to do the exercises (female, post-interview, august). participant 1 continued saying that she would have a competition (female, post-interview, august). often learners’ motivation to learn mathematics is linked to a reward or punishment. this finding aligns with güvendir (2016) claiming that a reward is a form of extrinsic motivation, which is more easily addressed, if perceived as worthwhile, than intrinsic motivation. thus, it can be deduced that the two peer tutors who responded on the aspect of rewarding outcomes are aware that creating a competition and using rewards motivate learners to work and they were satisfied with the work that they had executed. this finding demonstrates that small, seemingly insignificant, rewards do motivate learners to learn mathematics. lastly, five peer tutors mentioned positive outcomes, which can be given through verbal praise, giving personal attention to learners, informative feedback, as well as motivating feedback (keller, 1987). these peer tutors spoke about how they addressed satisfaction by verbally praising the learners when they completed work. participant 5 said he would inform learners of the difficulty of the work they had done. he would also praise them for doing it well. ‘[i] just told them to keep going: “that was really difficult you actually got it, well done”’ (male, post-interview, october). according to pintrich (2003), if learners believe that in the future there is a possibility that they will be able to overcome past failures, then they are more likely to persevere. from this finding, it could be deduced that peer tutors who encourage learners’ satisfaction in their learning of mathematics may promote learners’ motivation to persevere in mathematics, notwithstanding the challenges. the peer tutors empowered the learners to feel accomplished about the mathematics that they had done or were doing at that stage. none of the peer tutors spoke about any negative influences, such as avoiding threats and comparisons. it is likely that the peer tutors understood that creating a negative experience for the learners would disadvantage the learners in learning mathematics. however, it is also likely that the peer tutors were themselves positively motivated to learn mathematics and therefore mirrored their motivation to their learners. in addition, none of the peer tutors mentioned frequent reinforcing of what learners had learnt. the peer tutors could possibly have addressed it indirectly or subconsciously during the tutorials, but it was not evident from the collected data. discussion from the above results, some similarities were found pertaining to the peer tutors’ views about their role in motivating grade 8 and grade 9 learners to learn mathematics in relation to the four categories of learner motivation, namely attention, relevance, confidence and satisfaction, adapted from keller (1987). all 10 peer tutors understood their role to be a supporter to learners, which could be achieved through positively reinforcing what the learners had been taught in their mathematics lessons. the peer tutors also conceived their role to be flexible relative to how the learners learnt mathematics. the peer tutors discussed how they would attempt to improve learners’ confidence by explaining mathematics work to the learners in ways that they could understand best. furthermore, the peer tutors were open to explaining and giving learners’ work that matched with the learners’ interests. the peer tutors did not believe that they should simply assign work to the learners but were conscious of ensuring that the learners were able to complete the work that they were given without overwhelming the learners and causing them to give up. the main differences were about how the peer tutors believed that they would address the relevance of mathematics. many of the peer tutors discussed exploring future careers in mathematics and doing research with the learners so that the learners could apply mathematics to real-world problems that interested them. however, they did not discuss that they related the relevance of mathematics to learners’ personal, future plans. under the category of attention, it is apparent that the peer tutors were able to capture learners’ attention mainly through giving concrete examples and through using methods dissimilar to those taught in the classroom. capturing learners’ attention was the predominant method that was noted to get learners’ attention. the majority of the peer tutors addressed maintaining attention through participation. getting learners’ attention through using different methods of engagement was the most common method utilised by peer tutors. the findings indicate that peer tutors were able to get learners’ attention through concreteness and variation. peer tutors also believed learners’ attention could be maintained by means of stimulating inquiry, as well as through participation. this finding is in agreement with ryan and deci (2000) claiming that when learners have a feeling of connectedness to the task that they are completing, their intrinsic motivation is developed. winberg et al. (2014) also noted that when learners have a positive experience of learning and believe that they have connected to the community of learning, in this case mathematics, their intrinsic motivation is strengthened. this finding could indicate not only that the peer tutors prioritised participation in a community of learning, but also that learners’ intrinsic mathematics motivation could have been developed or strengthened during the tutorials. under the category of relevance, it is evident that most of the peer tutors believed that their role was to demonstrate to learners the relevance of mathematics by matching learners’ interests through demonstrating the future usefulness of mathematics and matching learners’ needs. this finding is similar to oyserman et al. (2015) discussing the need for motivation that encourages a person to achieve because of future prospects. weiner (2010) found that when learners believe that there are future incentives and successes in the offing, they are motivated to complete work immediately to enable them to reach their future goals. from this finding, it is evident that peer tutors usually have a good understanding of working towards a goal that could make mathematics relevant. they also realise the importance of linking the learners’ future interests with what they are currently working on in mathematics. under the category of confidence, the findings show that peer tutors understood their role to encourage learners to take personal responsibility for their work. they attributed their work to the learners’ effort and to encouragement of learners to become increasingly self-confident in their mathematics abilities. this finding coincides with cleary et al. (2017) arguing that when learners believe they can complete a task, often through preparedness, their motivation to continue in the future and to interact with the community of learning is positively impacted. finally, under the category of satisfaction, the peer tutors demonstrated that their role was predominantly to encourage learners to participate in the content, as well as with another person. this finding corresponds with hodges and kim (2013) stating that when learners are encouraged to participate with the content, it helps maintain their attention. however, none of the 10 peer tutors addressed fair treatment. according to keller (2000, p. 4), fair treatment can be done by supporting learners in securing a positive attitude about their successes. conclusion low mathematics performance not only influences learners’ progression in the subject but also affects how the world views a country’s ability to perform in the international domain. a strategy to enhance learners’ performance in mathematics is to motivate them to learn mathematics. when learners are motivated to learn mathematics, they are more likely to participate in the mathematics classroom. peer tutoring is one strategy that can assist in motivating learners. the authors of this article argue that if teachers are cognisant of and act on the role peer tutors play in motivating learners to learn mathematics, performance in the subject can improve. thus, this article reported on grade 12 peer tutors’ views on their role in motivating grade 8 and grade 9 learners to learn mathematics in relation to the four categories of learner motivation, namely attention, relevance, confidence and satisfaction, adapted from keller (1987). the findings in this article showed that peer tutors indeed understood their role to motivate grade 8 and grade 9 learners to learn mathematics. they knew that they needed to, and were able to, collaborate with the grade 8 and grade 9 learners to provide support in the learning of mathematics that was relevant and engaging. the peer tutors’ views on their role to motivate grade 8 and grade 9 learners to learn mathematics was to be (1) capturers of learners’ attention, (2) stimulators of learners’ mathematics inquiry, (3) maintainers of learners’ attention, (4) matchers of learners’ interests, (5) models of good mathematicians, (6) promoters of learners’ self-confidence and (7) encouragers addressing learners’ satisfaction in learning mathematics. these findings support the argument of the authors, namely that now that teachers are cognisant about the role peer tutors play in motivating learners to learn mathematics, they can develop peer tutors further in their roles and utilise them in this regard, which in turn can improve performance in mathematics. this article contributes to the understanding of peer tutors’ role as motivators regarding support to grade 8 and grade 9 learners in their learning of mathematics, which can be valuable for future intervention programmes utilising peer tutors at school level. this article also adds to the research on an adapted use of keller’s (1987) arcs model of motivation as a framework, specifically in motivating learners to learn mathematics, which is a novel way of bringing new perspectives on motivation in mathematics at secondary school level. an implication in developing peer tutors in their role to motivate learners to learn mathematics is to strengthen their skills regarding capturing learners’ attention, showing learners the relevance of mathematics, increasing learners’ confidence and ensuring that learners are satisfied by encouraging perseverance. a limitation of this article was that data were collected and analysed in a short period of only nine months. more data collection instruments could have ensured triangulation of the data. for example, observations on how peer tutors enact their role while assisting learners could have ensured the accuracy of their interpretation of their role. weekly reflection reports between the preand post-interviews could have cross-checked the evolution of peer tutors’ views on their role in motivating learners to learn mathematics. quantitative data could have established peer tutors’ levels of executing their role as motivators, but might also have examined the attitudes of both peer tutors and learners. the sample of 10 peer tutors at one particular school was not representative of all peer tutors in south africa. thus, the findings on the role peer tutors play in motivating learners is of importance to similar contexts utilising a peer-tutoring programme specifically to address learning in mathematics. however, the findings are not necessarily transferable to other school subjects and cannot be generalised to different contexts. although an adapted version of keller’s (1987) arcs model of motivation was applicable in the south african context to analyse peer tutors’ role as motivators in the learning of mathematics, it narrowed the role of peer tutors to only one aspect, namely motivation. more research is needed on other roles of peer tutors, such as being a critical friend, modelling appropriate behaviours, focusing on learners’ needs, supporting learners to complete tasks, preparing learning materials for tutoring sessions as directed by the teacher and monitoring progress toward academic goals. although peer tutors cannot substitute the content and teaching expertise of the mathematics teacher, their roles are important because they facilitate a learning climate that is friendly, but conversely rigorous. they extend in a supporting capacity on the learning that occurs in the mathematics classroom, which may assist learners in their progress in the subject. for future research in the area of motivation in mathematics, this research may be extended to a longitudinal study by focusing not only on peer tutors’ views on their role to motivate learners in their learning of mathematics, but also on how to develop peer tutors in various roles. a larger, more diverse sample emphasising different school demographics is also suggested. a further recommendation is a study investigating the affordances of a peer-tutoring intervention for learners in their learning of mathematics, because explaining a mathematics concept to another extends a learner’s own understanding of the subject. such an intervention could also deepen learners’ knowledge of mathematics by allowing them to practise challenging concepts and, in turn, develop their problem-solving skills. in conclusion, peer tutors’ understanding of their role as motivators aligns with research on an adapted use of keller’s (1987) arcs model of motivation and this model can be a useful lens to help address learner motivation. from the findings, the peer tutors were able to motivate learners and believed that learners’ motivation increased through the tutorials. for this reason, peer tutoring can be seen as a useful intervention strategy to help address learner motivation. going forward, the findings from this research can be expanded to further address motivation in mathematics to help improve the pass rates of south african learners. acknowledgements gratitude is expressed to the participants at the school in which this study took place. also, many thanks to colleagues, friends and family for their encouragement and advice. competing interests the authors have declared that no competing interest exists. authors’ contributions since this article is based on a.k.r.’s master study and e.d.s. was the supervisor of the study in mathematics education, a.k.r. collected the data, while e.d.s. provided academic inputs and technical editing to the manuscript. both authors contributed in terms of the conceptualising and writing the manuscript. funding information this research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors. data availability statement data sharing is not 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(2016). teacher support, instructional practices, student motivation, and mathematics achievement in high school. the journal of educational research, 111(1), 1–14. https://doi.org/10.1080/00220671.2016.1204260 article information authors: michael k. mhlolo1 hamsa venkat2 marc schäfer1 affiliations: 1mathematics education department, rhodes university, south africa 2marang centre for mathematics and science education, university of the witwatersrand, south africa correspondence to: mike mhlolo postal address: po box 94, grahamstown 6140, south africa dates: received: 21 july 2011 accepted: 14 mar. 2012 published: 16 may 2012 how to cite this article: mhlolo, m.k., venkat, h., & schäfer, m. (2012). the nature and quality of the mathematical connections teachers make. pythagoras, 33(1), art. #22, 9 pages. http://dx.doi.org/10.4102/ pythagoras.v33i1.22 copyright notice: © 2012. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. the nature and quality of the mathematical connections teachers make in this original research... open access • abstract • introduction    • the problem    • conceptualising a mathematical connection    • a framework for thinking about mathematical connections in practice    • our focus: different representations    • teacher’s representations of connections    • our conceptualisation of quality in teachers’ representations    • our analytical model    • methodology       • sample       • procedure       • validity and reliability       • ethical considerations       • data analysis       • excerpt 1 teacher b       • excerpt 2 teacher r       • excerpt 3 teacher r       • excerpt 4 teacher t       • excerpt 5 teacher b       • excerpt 6 teacher m • summary of findings    • implications • conclusion • acknowledgements    • competing interests    • authors’ contributions • references abstract top ↑ current reforms in mathematics education emphasise the need for pedagogy because it offers learners opportunities to develop their proficiency with complex high-level cognitive processes. one has always associated the ability to make mathematical connections, together with the teacher’s role in teaching them, with deep mathematical understanding. this article examines the nature and quality of the mathematical connections that the teachers’ representations of those connections enabled or constrained. the researchers made video recordings of four grade 11 teachers as they taught a series of five lessons on algebra-related topics. the results showed that the teachers’ representations of mathematical connections were either faulty or superficial in most cases. it compromised the learners’ opportunities for making meaningful mathematical connections. the researchers concluded by suggesting that helping teachers to build their representation repertoires could increase the effectiveness of their instructional practices. introduction top ↑ developed and developing countries, including south africa, have revised their mathematics curricula in recent years to take account of what they regard as the knowledge and skills learners require to participate in a globalising twenty-first century world. there seems to be some agreement that higher order cognitive skills and processes are necessary for more equitable educational outcomes and economic productivity (muller & subotzky, 2001). however, the major challenge has been how to convert this noble vision from the written into the taught curriculum. the problem in south africa, there has been general public discontent about learners’ actual gains in knowledge and skills despite the steady increase in pass rates since the advent of democracy. muller (2005) questioned the credibility of senior certificate pass rates given the opinion that standards have actually dropped: the largely invisible outcome, invisible to school educators, that is though not invisible to employers or university admission officers, was that the schooling system was emitting a cohort or two which had reduced opportunities to demonstrate higher-level cognitive skills, had possibly not even been taught them and, in far too many cases, therefore did not have them. (p. 43) this quotation raises two critical concerns: • that standards have actually dropped despite the upward trend in pass rates • that low-level cognitive skills were intentional or unintentional threats to the social and economic health of the nation. lolwana (2005) and edwards (2010) made similar observations about the low cognitive demand of mathematical activities and recommended that researchers do more to understand the specific cognitive levels in the intended, the tested and the implemented curriculum. it is from these observations and recommendations that the researchers saw a potential gap in knowledge. with specific reference to classroom practices (the implemented curriculum), this article raises the question of the extent to which high school mathematics teachers are creating opportunities for learners to acquire cognitively demanding mathematical connections to use in problem solving situations. the researchers’ entry point to this is by conceptualising mathematical understanding. whilst acknowledging that there is no consensus about the meaning of ‘understanding’, barmby, harries, higgins and suggate (2009) suggest three perspectives that the researchers found relevant for taking their ideas forward. barmby, harries, higgins and suggate (2009) argue that one shows a deep understanding of mathematics through: • connections made between different mathematical ideas • different representations of mathematical ideas • reasoning between different mathematical ideas. conceptualising a mathematical connection however, what exactly is a mathematical connection?blum, galbraith, henn, and niss (2007) suggest that the literature has identified two major types of mathematical connections. the first is recognising and applying mathematics to contexts outside of mathematics (the links between mathematics, other disciplines or the real world). the second concerns the interconnections between ideas in mathematics. most articles in the literature focus on connections in the real world and do not explore the area of mathematical connections or interconnectedness within mathematics (businskas, 2008). however, learners’ ability to make connections in mathematics itself is crucial for conceptual understanding (anthony & walshaw, 2009b) as well as for application outside the discipline. this study aims to contribute to the area of interconnectedness within mathematics. one can define a mathematical connection broadly as: • a relationship between ideas or processes that one can use to link topics in mathematics • a process of making or recognising links between mathematical ideas • an association a person might make between two or more mathematical ideas • a causal or logical relationship or interdependence between two mathematical entities (businskas, 2008). the literature has often captured these broad views into three ways of considering mathematical interconnectedness: • as a feature of mathematics (a priori − implying that it exists independently of the learner) • as a relationship that the learner constructs • as a process that is part of the activity of doing mathematics. this article concentrates on the first and last views. the researchers note the importance of the ability of learners to recognise mathematical connections (hodgson, 1995). however, we concur with weinberg (2001), who argued that, whilst learners might make connections spontaneously, one cannot assume that they will make them without some intervention.one should make learners aware of different possible mathematical connections. teachers play a crucial role in this intervention because they must teach lessons in ways that will enable learners to recognise and make sense of these mathematical connections. therefore, the view of the researchers is that considering mathematical connections as features (a priori) that exist independently of the learner would enable them to judge the process (whether or not the teacher was creating opportunities for learners to recognise them). therefore, the researchers’ next challenge was to develop a workable framework for classroom observations to enable them to examine how teachers structured learners’ opportunities for making appropriate mathematical connections. a framework for thinking about mathematical connections in practice the researchers borrowed from businskas (2008). in her study, she posed the question: what are the characteristics of the explicit mathematical connections that teachers are able to articulate from their practice? following the analyses she made of teachers’ responses to her study, she proposed a framework for identifying mathematical connections in practice. her model has the five categories that follow:• different representation (dr) as a form of mathematical connection • part-whole relationships (pwr) • connections where a implies b (im) • connections that show that a is a procedure for doing b (p) • instructional orientated connections (ioc) that show how certain concepts are pre-requisites for understanding related concepts. these five types of mathematical connections became the reference point for trying to understand the nature and quality of observed instructional strategies in selected south african grade 11 classrooms. whilst businskas (2008) only identified possible mathematical connections in practice, this study extended this focus because the interest of the researchers was also to identify and analyse the quality of these connections in practice (i.e. to link connections to their cognitive strengths or cognitive demands). our focus: different representations whilst the five categories provided the framework for the researchers’ broader study, the focus in this article is on different representations as types of connections. barmby et al.’s (2009) second view of mathematical understanding guided this decision to some extent.the ability to present a concept in several ways shows a deep understanding of that concept. there is a view in the literature that recognising and producing alternate representations is a particularly fruitful way of conceptualising what a mathematical connection is (gagatsis & elia, 2004) and that the transformation between these representations is a useful way of assessing whether learners are making connections (reead & jazo, 2002). the pedagogical implication is that teachers should use several modes of representation to improve learners’ understanding of mathematics and that these representations should be accurate and appropriate (national council of teachers of mathematics, 2000). therefore, the researchers decided to focus on different representations. an important finding in a broader study (mhlolo, 2011) also guided them. it showed that different representations were more prevalent than the other types in all 20 lessons that the researchers recorded (see figure 1). figure 1: summary of data counts for all four teachers in different categories. these results seem consistent with earlier classroom observations elsewhere. they showed that teachers constantly engage in a process of defining and constructing a mental image of some mathematical object and use instructional representations in the process (businskas, 2008; mcdiarmid, ball & anderson, 1989; stylianou, 2010). teacher’s representations of connections interpreting a mathematical connection as a recognition of two or more alternate representations lies at the intersection of research into mathematical connections and research into representations (businskas, 2008). in this intersection, the term ‘different representation’ could mean a mathematical connection (object) or the ways teachers present the mathematical idea (activity or process). the researchers use both of these interpretations in this article and felt that they had to clarify their position about using these terms. in this article, the researchers use the term ‘different representations’ (dr) in accordance with businskas’ (2008) definition to refer to an inherent feature of mathematics (object) − links, relationships or mathematical connections. these relationships exist inherently in mathematics, regardless of whether teachers or learners recognise them. businskas posits that this (dr) category comprises the two subcategories that follow. a is an alternate representation of b, where the two representations (a and b) of a mathematical concept or idea come from any two of the following modes − symbolic (algebraic), graphic (geometric), pictorial (diagrammatic), manipulative (physical object), verbal description (spoken), or written descriptions. for example, the graph of a parabola (graphic or geometric) is an alternate representation of f(x) = ax2 + bx + c (symbolic or algebraic). a is an equivalent representation of b, where ‘equivalent’ distinguishes between alternate representations and refers to concepts that are connected by representing them in different ways using the same mode of representation. one example is 13 + 12 equals 25 and f(x) = ax2 + bx + c equals f(x) = a(x – p)2 + q. another example is defining concepts by rephrasing verbal representations or written representations using different descriptors. on the other hand, one uses the term ‘representation’ to refer to an activity or process (the teacher’s way of presenting, or the teacher’s didactic strategy of converting, inherently linked or connected mathematical ideas). in this sense, a teacher can ‘represent’ an alternate link or an equivalent link, a part-whole relationship or any other similar mathematical connection. empirical evidence suggests that this activity of representing is a core activity of teaching mathematics (ball, 2001) because the ways teachers represent mathematical ideas are fundamental to how people understand and use those ideas. the focus of this article is the quality of teachers’ representations of mathematical connections. therefore, a question captures its next challenge: how can we begin to judge the quality of the alternate or equivalent representations (links) the teacher presents in a classroom situation? our conceptualisation of quality in teachers’ representations the third view of barmby et al. (2009) on mathematical understanding guided the researchers as they tried to determine the strengths or quality of teachers’ didactic strategies. it links connections and the reasoning between them. according to sierpisnka (1996), acts of understanding link what one must understand with the ‘basis’ or ‘reasoning’ for that understanding. ‘basis’ and ‘reasoning’ suggest answering the deeper questions of ‘why or how we know’. andrews (2009) combined these notions when he argued that teachers’ representations of mathematical connections, which (1) articulation, justification and argumentation from the teachers and/or (2) pressing for this reasoning from the learners accompanied, could lead learners into acquiring a deeper understanding of knowledge and skills. anthony and walshaw (2009a) had a similar view as they posited that effective teachers encourage their learners to explain and justify their solutions. they ask learners to take and defend their positions against the mathematical claims of other learners. this causes their attention to shift from procedural rules to making sense of mathematics because learners try less to find the answers and more to discover the thinking that leads to the answers. according to squires (2009), the trends in mathematics and science studies (timss) reports also show that teachers, in the countries that performed well, placed greater cognitive demands on learners by encouraging them to focus on concepts and their connections. the teachers in these countries maintained this cognitive demand when they insisted on justifications, explanations and meanings through questioning or other feedback (zurawsky, 2006). the findings from these studies also showed that, in classrooms in which teachers set instructional tasks and learners executed them at high levels of cognitive demand, the learners achieved better on measures of reasoning and problem-solving than did learners in classrooms where these tasks became merely following the rules, usually with little understanding (squires, op cit). our analytical model barmby et al. (2009) provided the researchers with a model. the researchers thought that this model incorporated all the notions of connections and their strengths through reasoning and justification. the model used the notion that learners should be fluent users of representations. however, the researchers thought that the model would be equally useful for analysing teachers’ representations of different mathematical ideas in class. the researchers felt that conceptualising mathematical representations, as barmby et al. suggested, left an important gap because, whilst it captured weak and strong mathematical reasoning, it did not seem to capture faulty connections or reasoning. the researchers felt that this was important, especially in the south african context where researchers have identified teachers’ understanding of the subject as a problem (adler, 2009; harley & wedekind, 2004). the researchers hypothesised that one could expect faulty representations of connections. therefore, they proposed adding a broken line (see figure 2) to show a faulty representation of a mathematical connection. figure 2: representational-reasoning model, adapted from barmby et al. (2009). consequently, they adapted the model slightly so that the mathematical concept or idea (algebraic equation) becomes the alternate representation (symbolic) and the lines show the strength with which the teacher connects or links it to another alternate representation (the graphical representation). using this adapted model of representation, broken lines show a faulty representation of a connection, thinner lines show a superficial or weak representation of a connection and thicker lines show a strong or deep representation of a connection. the researchers built on the models of barmby et al. and of businskas so that their coding of teacher representations: • level 0 (dr0) denotes a representation of a mathematical connection in the dr category that was faulty or incorrect • level 1 (dr1) denotes a representation of a mathematical connection in the dr category that was mathematically correct but superficial or routinely algorithmic, with no further explanation or justification • level 2 (dr2) denotes a representation of a mathematical connection in the dr category that was more than just mathematically correct; justification and/or further explanation follow. the researchers give examples of levels of actual representations in the data analysis section of this article. methodology sample this was a case study involving four grade 11 mathematics teachers. the researchers made video recordings of each as they taught a series of five algebra-related lessons. these teachers were part of a larger group of teachers who had agreed to take part in a large-scale research project called ‘implementing curriculum change in previously disadvantaged communities’. whilst some of the participating teachers in the broader study were not qualified to teach mathematics at further education and training (fet) level, the four who provided data for this article were. there were two women and two men teachers. their teaching experience ranged from 7 to 10 years. the four teachers were from four different schools in previously disadvantaged communities in a gauteng urban area. procedure the first author conducted the classroom visits. he observed one teacher teaching number patterns and the other three teaching functions and algebra.the decision to observe lessons on these two areas of mathematical content is consistent with the emphasis in the curriculum. it specifically requires teachers to structure learning experiences and situations to develop these key concepts and enable learners to ‘experience the power of algebra as a tool to solve problems’ (department of education, 2003, p. 13). star and rittle-johnson (2009) also observed that competence in algebra is increasingly being recognised as a critical milestone in the journey from primary to high school. international assessments have recorded learners’ difficulties in algebra and there is empirical evidence that the transition from arithmetic to algebra is a notoriously difficult one and presents teachers with several challenges (blum, galbraith, henn & niss, 2007). after considering all these factors, the researchers decided to find out what it would mean for teachers to make high quality mathematical connections in an area that the south african national curriculum statement for mathematics (department of education, 2003) emphasises but which is so notoriously difficult for both learners and teachers (koedinger &nathan, 2004). with regard to number patterns, warren and cooper (2008) posit that the links between patterns and algebra have wide acceptance. their argument is that learners begin their study of functions in the primary grades as they observe and study patterns in nature and create patterns using concrete models. learners in high school then expand their knowledge of algebra as they analyse a variety of different types of number patterns. validity and reliability the analytical tool for this study measured the quality of the mathematical connections that teachers used as they taught certain mathematical concepts in class. the researchers initially collected pilot data and then tested the tool on the pilot data. during this piloting period, two other mathematics experts (one who sets grade 12 examinations and one who is a team leader for marking them) validated the tool. the researchers presented the tool at several professional conferences and doctor of philosophy (phd) meetings, where it underwent rigorous peer reviewing before taking its final form. the researchers addressed the issue of reliability through replication because they observed four different teachers presenting different content. they obtained similar results for the dr0 category as it produced the highest frequencies. other research results from south africa (davis & johnson, 2007) and abroad (ball, 2001) corroborated this. this confirms the prevalence of dr0 representations in practice. because the researchers observed only four teachers, one cannot generalise the findings beyond the cases they studied. this is the nature of case studies. however, consistent with the objective of the study, the findings could lay the principles for making high quality mathematical connections in practice. ethical considerations the department of education granted approval to proceed with this study under permit t-728 p01/02 u-848. at institutional level, the university ethics committee granted approval under protocol 2007ec007. at school level, the first author received informed consent from the principals, teachers and parents of the learners who would participate in the study. at both school and individual levels, the researchers maintained the participants’ anonymity and confidentiality by using pseudonyms (teacher r, m, t and b) and the video recordings did not focus on either the teachers or the learners. data analysis the researchers provided six excerpts from the classroom interactions. they coded the teachers’ representations of mathematical connections at the three different levels of cognitive demand the article presented earlier.the researchers coded excerpts 1 and 2 as dr0 (faulty), excerpts 3 and 4 as dr1 (weak, superficial or rote) and excerpts 5 and 6 as dr2 (strong or higher order). in their analysis of these six excerpts, the researchers showed how they identified the different representations as equivalent or alternate (mathematical connections) in each case then justified why they placed the excerpts at each level of quality. in the broader study of businskas (2008), from which this article draws, she coded some of these excerpts more than once (i.e. in the same episode it was possible to see a part-whole connection, a generalisation, or an if-then connection). whilst one expects this in a typical classroom situation, this article does not discuss these other types of connections because it focuses only on alternate and equivalent representations. excerpt 1 −teacher b this lesson focused on quadratic functions and on how one determines the gradient of a curve at different points on it. the teacher began by defining the word ‘calculus’ to the learners: teacher: say for instance i mean it’s calculus. it has the word calculate within it. ok. so we will be calculating something but there are rules that we need to follow. commenton excerpt 1: the researchers regarded this as an example of an equivalent representation where the teacher attempted to define calculus by rephrasing it to ‘we will be calculating’. she represented the idea of calculus in a different way but in the same verbal mode using different descriptors. this intended to give the learners an understanding of the mathematical idea (of calculus). this was the only definition the teacher offered as an introduction to the series of lessons that followed. the researchers contend that, in the context of differential calculus and in the absence of further explanation, this representation is problematic because learners might not differentiate between calculating as any other mathematical operation applies it. it does not offer learners a clear idea of what calculus is as a mathematical concept in the context of differential calculus. therefore, the researchers coded it dr0. excerpt 2 −teacher r this lesson was on multiplying polynomials. the teacher had given learners different tasks to complete on the board and was summarising the observations.teacher: we were finding the products of binomials and trinomials, but look at the first terms in the second expressions [these were quadratic, so the teacher circled 2a2, a2, 3x2, 4a2 and b2]. the first term is squared. and if the first term is squared and we multiply it by a binomial, the answer is going to be cubed. the two terms in your answer is going to be cubed. and this introduces us to something else now. the difference between … how do we conclude this? you have been saying it right now. the difference between two cubes. let’s go to question number 3. did you do it in your books? let us look at this question. we are given: (a + b)(a2 – ab + b2). should we work it out or should we go straight for the answer? class: noooo, let us work it out. teacher (after the learners had completed it): what are you saying [names a learner]? learner: the answer is a3 + b3. teacher: what can you say about the answer? what can you say about the expression? what are we going to call this product [underlines a3 + b3]? we touched this yesterday. it’s the difference between… class: cubes, squares, terms, trinomials, exponents. teacher: this term cubed [circles a3] and this term cubed [circles b3]. so it’s a difference between two cubes. you got it? alright. now we agreed here [underlining a3 + b3] that this is the difference between two cubes.comment on excerpt 2: this is an example of alternate representation because the teacher is representing an algebraic expression (a3 + b3) verbally as the difference of two cubes. the researchers saw defining a3 + b3 as the difference of two cubes as mathematically problematic. therefore, they coded this as dr0. they contend that learners were unlikely to acquire a mathematically acceptable conceptualisation of the difference of two squares or two cubes with a representation like this. the researchers did not regard this as ‘a slip of the tongue’ because it occurred often throughout the series of lessons. excerpt 3 − teacher r in the same series of lessons, on multiplying polynomials, one learner completed her task as follows:learner: (3x2 + xy – 2y2)(x + 2y) 3x3 + 6x2y + x2y + 2xy2 – 2xy2 –4y3 3x3 + 7x2 y – 4y3 teacher: what are you saying about her approach? how did she approach this? she was finding the product of binomials and trinomials using the distributive law. did she apply the distributive law? [the teacher was concerned that the learner ‘did not’ apply the distributive law correctly because she did not re-arrange the two polynomials with the binomial on the left: (x + 2y)(3x2 + xy – 2y2). examples on the board, where the binomial was always on the left hand side and the trinomial on the right hand side, followed. however, in this case the textbook had given the task − with the trinomial on the left hand side].class: noo [meaning the learner had not applied the distributive law when dealing with this task]. comment on excerpt 3: the researchers contend that (3x2 + xy – 2y2)(x + 2y) equals (x + 2y)(3x2 + xy – 2y2) in terms of the commutative law of multiplication. however, the teacher did not recognise this here. the teacher insisted that the binomial must always be on the left hand side, suggesting that the learners should have rearranged (3x2 + xy – 2y2)(x + 2y) first before multiplying the polynomials. starting with the binomial on the left would achieve the same result (as one would have expected and the lesson proved). the researchers contend that the representation the teacher used here gives a limited conceptual understanding of the distributive law because the teacher associated it with a specific arrangement of the polynomials (binomial to the left and trinomial to the right). therefore, the researchers regarded it as algorithmic, superficial or rote and coded it dr1. excerpt 4 − teacher t this lesson focused on factorising binomials and trinomials. teacher: [writes on the board -x2 + 7x – 10]. so today, what we are going to do is factorisation where the first part has a negative coefficient. are we together? remember we said that the first number before x is the coefficient. what is the coefficient of x here [points to -x2]? class: 1, -1 [there is a debate about whether it was 1 or -1, but the teacher finally decided that it was 1]. teacher: let’s take that coefficient as what 1. we are saying 1 × -10 [the teacher now multiplies this coefficient 1 by the last term -10 in the trinomial and writes -10 on the board]. remember here we must find the common factors of what, -10. which are the common factors of -10? class: 5 and -2 [learners want to give other factors like -5 and 2, but the teacher discourages them]. teacher: let’s say they are the same because we just swap the signs. then, remember when we add those two factors they must give us the 7x [term in the middle of the trinomial]. which of these two factors will give us 7x? class: none of them. comment on excerpt 4: this is one of the excerpts where the researchers coded other forms of connections (like procedural connections) when dealing with the factors of -10. however, consistent with their focus in this article, they took the view that, at the point where the teacher says ‘let’s say they are the same’ with reference factors of -10, he is representing an equivalence (i.e. 5 × -2 equals -5 × 2) after swapping the signs. the product would indeed be -10 in each case. however, in the context of factorising trinomials, the researchers argue that it was necessary to think about the objective of breaking -10 into its factors. the interest is in factors of the last term -10 that will add to the middle term +7. swapping had worked very well before this task because the learners were dealing with tasks where the three terms of the trinomial were positive. therefore, the factors were also positive. for example, one knows from applying the commutative law of multiplication that 2 × 5 equals 5 × 2. similarly, 2 + 5 equals 5 + 2. therefore, suggesting that swapping signs between factors would apply equally in the case of 5 × -2 and -5 × 2 is typical of rote learning or applying well-rehearsed procedures without thinking about them. the researchers coded it dr1. they contend that swapping signs affects their sums, which relates to the middle term when factorising in this context. this task became more confusing because the learners had erroneously interpreted the coefficient -1 of the first term in this particular trinomial as 1. excerpt 5 − teacher b this lesson focused on functions and the different ways of representing them. teacher: so now i have written a function: f(x) = x2 + 1. this is what liner or quadratic? learner 1: it’s quadratic. teacher: you are saying it’s quadratic how do i identify that this is a linear function and this is a quadratic function? why are you saying this is quadratic? what if somebody says it is linear? class: it’s determined by the exponent of x. if x is to the exponent 1 we are talking of a linear function. if it’s a square it’s quadratic. teacher: okay. it’s quadratic. so in order for you to draw this graph what are you going to do? learner 2: you can do it in table form. teacher: how else can you do it if you did not want to do it in table form? learner 3: for some of us with a graphic scientific calculator i can just punch in the function and the calculator can draw the graph for me. teacher: [goes back to the table form]. so we have the function: f(x) = x2 + 1. we are going to substitute the x values into the function and then we will get what, the y values. so we will have [the teacher draws and completes the table with the rest of the class.] teacher: will this help you to draw the graph and if so how?learner 4: yes, it will, by drawing the xand yaxes then plotting the points. teacher: [the teacher asks one child to come to the board]. so show us how to draw the graph. [the teacher then says] we have drawn the graph of y = x2 + 1. comment on excerpt 5: the researchers interpreted this as an example of connecting a mathematical idea (quadratic function) using alternate representations (verbal, algebraic, tabular and graphic). in this excerpt, the teacher did not just accept correct answers without explanation or justification. she kept pressing for justification or reasoning from the learners. examples are ‘why quadratic’, ‘why not linear’, ‘how else can you do it’ and ‘will this help’. therefore, it is consistent with how the researchers have defined cognitively demanding activities in this article. the researchers coded it dr2. excerpt 6 − teacher m this lesson focused on arithmetic or linear sequences within the topic number patterns.teacher: right, suppose you are given a list of numbers starting with [writes 3, 6, 9… on the board]. somebody, tell me the next number. learner 1: 12. teacher: somebody else. learner 2: 12. teacher: 12. they all say the next number will be 12. anybody who does not agree? [the teacher raises his hand and pauses. after no other responses, the teacher folds his arms]. but, how do we know it’s 12? suppose somebody comes in from a distance and says it’s 13? why 12, why not 13? [says learner’s name]. learner 3: they are multiples of three. teacher: okay, somebody tell us what the tenth term will be. learner 4: term 10 will be 30. teacher: why? how do we know it is going to be 30? learner 5: there is a relationship between the term number and the value in this sequence and each time the term number is times 3 to get the term value. teacher: so what shall term n be? learner 6: 3n. comment on excerpt 6: here, the researchers contend that the teacher gave an equivalent representation or connection between a term number 3n and term value 30 in this particular sequence. recognising this equivalence is important for determining the general term for a sequence. it is also useful for forecasting or predicting unknown term values in the sequence. again, the teacher did not just accept correct answers (like 12, 30 or 3n) without explanation or justification. he insisted on justification or reasoning from the learners (‘why 12’, ‘why not 13’, ‘why 30’, ‘how do we know’ and ‘so what will be term n’). therefore, it is consistent with how the researchers have defined cognitively demanding representations in this article. the researchers coded it dr2. summary of findings top ↑ the researchers transcribed 20 lessons and coded 377 teachers’ representations in accordance with their coding system. a summary of their observations of the dr category of mathematical connections follows. figure 3 gives a comparison of the level of the quality of the representations of mathematical connections of each teacher. figure 3: comparison of the levels of quality of the representations of each teacher. note that the bars in figure 3 give data counts and not percentages. for example, in the case of teacher m, the researchers recorded faulty connections (dr0) 11 times. they then converted them to percentages for the discussion. the different representations were faulty in as high as 60% of the statements (as for teacher r), while 30% of them were rote or routinely algorithmic. this left only 10% of the representations with the potential to develop a deep understanding of concepts and procedures. however, teacher m’s representations of mathematical connections fell into level 2 (higher order) in 61% of the cases and into level 0 (faulty) in only 0.5% of the cases. this suggests that teacher m might have created more opportunities for learners to develop higher quality mathematical connections than did the other teachers. the researchers did not intend to delve into causal relationships or infer why this was the case. they were specifically interested in describing the nature and quality of teachers’ mathematical connections. figure 4 gives the summaries for the four teachers. figure 4: comparison of the quality of the representations. this graph shows that most of the four teachers’ representations of mathematical connections were either faulty (level 0) or superficial (level 1). the differences in the heights of the three bars in figure 4 might not reveal the magnitude of the problem.however, when one considers that level 2 of cognitive demand should be the target of classroom practice, then combining the levels 0 and 1 bars reveals a cumulative 70% off-target in the teachers’ representations. implications given the findings of the study, the researchers argue that most learners probably lost opportunities to develop a deep understanding of mathematical connections.the researchers acknowledge the limitations of the study, in terms of its generalisability, because of the small sample. however, the researchers note that davis and johnson (2007) made similar observations: that teachers spent most of classroom time in south africa on explaining mathematical ideas, principles and definitions. most teachers ‘briefly referred to definitions but without discussing or explicating the mathematical reasons for the productions of the definitions’ (p. 123). given this consistency in results, the researchers believe that helping teachers to build their representational repertoires, which consists of metaphors, analogies, illustrations, examples, explanations and demonstrations, and with further justification and explanation, might improve the effectiveness of their instructional practices and create opportunities for learners to learn higher order cognitive skills and processes. conclusion top ↑ there are developmental lessons that one could learn from this study. the researchers argue that their results confirm that different representations prevail in different categories or forms of mathematical connections in practice. this is consistent with the literature. therefore, they see potential in their results that researchers might want to test further, especially in south african classrooms. so far, researchers seem not to have identified what it means for teachers to enable learners to make strong mathematical connections. this could be a possible entry point into teacher support and enrichment programmes. acknowledgements top ↑ the researchers thank the department for international development for funding the phd study from which this article draws. the views this article expresses are not necessarily those of the funders. competing interests the authors declare that they have no financial or personal relationship(s) that might have inappropriately influenced them when they wrote this article. authors’ contributions m.k.m. 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(2006). do the math: cognitive demand makes a difference. research points, 4(2), 1−4. available from http://www.aera.net/uploadedfiles/journals_and_publications/research_points/rp_fall06.pdf article information authors: yael sarfaty1 dorit patkin1 affiliations: 1kibbutzim college of education, tel aviv, israel correspondence to: yael sarfaty postal address: kibbutzim college of education, faculty of mathematics, 149 namir road, tel aviv, israel dates: received: 17 jan. 2013 accepted: 04 may 2013 published: 28 june 2013 how to cite this article: sarfaty, y., & patkin, d. (2013). the ability of second graders to identify solids in different positions and to justify their answer. pythagoras, 34(1), art. #212, 10 pages. http://dx.doi.org/10.4102/ pythagoras.v34i1.212 copyright notice: © 2013. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. the ability of second graders to identify solids in different positions and to justify their answer in this original research... open access • abstract • theoretical background    • the van hiele theory       • critical attributes and non-critical attributes       • examples and non-examples       • prototype and non-prototype       • typical position and non-typical position • methodology    • research objectives    • research population    • research tools       • the physical solids (appendix 1)       • the structured interview    • research procedure       • research method and analysis method       • ethical considerations • findings    • the first research objective       • category 1: arguments based on perception of the solid in general       • category 2: arguments based on the attributes of the solid    • the second research objective • discussion and conclusions • acknowledgements    • competing interests    • authors' contribution • references • appendix 1    • photos of the physical solids abstract top ↑ from a young age children feel the need to identify two-dimensional geometric figures (shapes) and three-dimensional geometric figures (solids). the national council of teachers of mathematics indicates the importance of being able to identify and name various geometric figures by kindergarten age. one of the objectives of this study was to learn the ability of second graders to identify examples and non-examples of three generally known solids: cylinder, cone and pyramid, and to justify their identification based on the attributes (critical and non-critical) of those solids. another objective was to find out whether changing the position of the solids would result in those children maintaining their decisions regarding the name or changing their identification of the solids, giving arguments accordingly. findings of this study illustrate that children can identify and characterise solids presented to them in a typical position. however, they find it difficult to correctly identify the same solids in another position. an interesting finding was that most of the arguments given to justify their identification were based on the specific attributes of the solid rather than on the perception of the solids in general. findings of the present study suggest that it is highly important for learners to be acquainted with a variety of both non-examples and examples of solids. moreover, it is recommended that solids are presented to learners not only in the typical position, in order to improve their ability to identify them and understand that the name of the figure does not change when its position changes. theoretical background top ↑ identification and naming of geometric figures are performed from an early age. children are exposed to geometric figures long before they begin their formal schooling. the national council of teachers of mathematics (nctm) (2000) explicitly specifies that teaching programmes of this subject should begin at kindergarten age, thus enabling children ‘to analyse characteristics and properties of two-dimensional and three-dimensional geometric figures and develop mathematical arguments about geometric relation’ (p. 96). during their first years at school, young learners are expected to:• ‘recognise, name, build, draw, compare and sort twoand three-dimensional shapes; • describe attributes and parts of twoand three-dimensional shapes’. (p. 96) the mathematics curriculum for kindergarten in israel (ministry of education, 2010) includes the subject of three-dimensional figures or, as it is referred to in the curriculum, solids. pursuant to the curriculum children will learn: • ‘to identify and name solids. solids to be named are: cube, cylinder, ball, pyramid, cone, box, prism. • to identify shapes (identifying the faces) of which the solids are built’. (p. 44) later on, in the mathematics curriculum for primary school, the subject of solids appears in the second grade (ministry of education, 2006). according to the curriculum, children will learn about: • ‘solids: cube, box, cylinder, pyramid, cone, ball – initial acquaintance; • identification and naming of solids (in simple cases); • observing solids and describing them, including counting of faces, edges and vertices’. (p. 48) the van hiele theory according to the geometric thinking model conceived by van hiele (1997, 1999), children’s geometric thinking is developed in a hierarchical way, from visualisation (or recognition), to analysis (or description), to informal deduction (or ordering), and finally to formal deduction and rigour (or axiomatic approach). according to van hiele’s theory, partial mastery of a certain level is required though insufficient for mastering a higher level.the van hiele theory originally related to plane geometry only. some studies have recently applied the theory of plane geometry to other branches of mathematics, such as solid geometry (gutiérrez, 1992; patkin, 2010; patkin & sarfaty, 2012) and arithmetic (crowley, 1987; guberman, 2008). as this study engages in the geometric knowledge of young children, we focus on only the first two levels, that is, visualisation and analysis. on the first level – visualisation – learners become acquainted with different geometric figures, distinguish between them but are unable to identify and specify the components and attributes of these figures. they name a figure based on its appearance and can describe it by its similarity to a figure that is familiar to them. there is reference to the geometric figure in its entirety. koester (2003) describes an example of a child who determines that a certain shape is a rectangle because ‘it looks like a box’ (p. 436). the child’s explanation is wrong: he confuses a two-dimensional concept (rectangle) with a three-dimensional concept (box). on the second level of the van hiele model – analysis – learners can describe or analyse a geometric figure based on its features and characteristics. they recognise the figure by its attributes. clements and sarama (2000) present the case of a child who identifies a certain shape as a rectangle because ‘it has two pairs of equal sides and all right angles’ (p. 482). on the other two levels (which are not relevant to this study) learners develop higher levels of geometric thinking. critical attributes and non-critical attributes in geometry, arguments for determining names of geometric figures can be divided into two types. the first type relates to the critical attributes of the concept and the second relates to the non-critical attributes of the concept. tall and vinner (1981) distinguish between the terms ‘concept image’ and ‘concept definition’. the term concept image is used in order to describe the total cognitive structure relevant to the concept, including all the mental pictures, associated properties and processes. it is built up over the years through experiences of all kinds, changing as the individual encounters new stimuli and matures. concept definition is a formulation of words used to specify a concept. it may be learnt by an individual by memorisation or more meaningfully learnt and related to a greater or lesser degree to the concept as a whole. critical attributes must be present in every example of the concept and are derived from the concept definition (hershkowitz, 1989, 1990). an example of a critical attribute of a solid: ‘this is a triangular pyramid because all its faces are shaped like a triangle.’ non-critical attributes are found only in a subset of the concept examples. an example of a non-critical attribute: ‘this is a pyramid because it has a square-shaped basis.’ this is not a critical attribute because the pyramid basis can be any polygon and this attribute is not critical for determining that a solid is indeed a pyramid. burger and shaughnessy (1986) argue that non-critical attributes embody an element of visual argument. one of the objectives of mathematics education is to induce children to use only critical attributes as reasoning when identifying examples and building mathematical concepts. those who base their arguments on critical attributes function on van hiele’s second level. examples and non-examples when building knowledge and concepts it is extremely important to present children with examples and, at the same time, non-examples in order to facilitate concept formation in a swifter and fuller way (klausmeier & feldman, 1975; mckinney, larkins, ford & davis, 1983). in mathematics education, the use of examples and non-examples has been intensively studied in the context of geometric concept acquisition (cohen & carpenter, 1980; petty & jansson, 1987; vinner, 1991; wilson, 1986). one of the conclusions is that the use of non-examples constitutes part of the formation and creation of a concept (tsamir, tirosh & levenson, 2008). according to clements and sarama (2000), good and relevant non-examples, which can enhance the ability to identify and name geometric figures, encompass only part of the critical attributes but not all of the required critical attributes. the use of these can help learners to correctly consolidate the learnt concept. figure 1 presents an example and a non-example of a cylinder. figure 1: an example (a) and a non-example (b) of a cylinder. prototype and non-prototype the fact that there is an example or several examples that constitute a prototype is another important component of which we have to be aware when presenting examples. hershkowitz (1989) claims that all the examples that have common specific visual characteristics become prototypes. according to tsamir et al. (2008), a prototypical example is intuitively accepted as representative of the concept. that is, it is accepted immediately with certainty and without feeling that any kind of justification is required. figure 2 presents an example of a prototype cylinder, in which the altitude is longer than the diameter of the cylinder base, and a non-prototype cylinder, in which the diameter of the cylinder base is longer than the altitude. figure 2: a prototype cylinder (a) and a non-prototype cylinder (b). every geometric concept has at least one prototype example, which is the first to be acquired intuitively when learning a concept. the prototype plays an important role in the concept formation. hershkowitz (1989) maintains that, in addition to the required and sufficient attributes that are common to all the examples of that concept, examples that are a prototype of a shape have also special (non-critical) attributes ‘dominating and attracting our attention’ (p. 73). an example becomes a prototype due to its strong visual features. conversely, the other examples are rejected because they lack those special visual attributes of the prototype example (hershkowitz, 1992).when distinguishing between examples and non-examples of shapes, one can see that the prototype is the basis of prototypical judgement. that is, the prototype example is a reference framework against which learners judge the other examples instead of using the concept definition, that is, its attributes (hershkowitz, 1992). in such cases, learners stick to the prototype example which entails applying non-critical attributes to the rest of concept examples. as a result, learners who fail to identify a certain concept example make wrong decisions and do not consider it as one of the concept examples (hershkowitz & vinner, 1983). watson and mason (2005) argue that learners are familiar with a small variety of examples only. typical position and non-typical position when presenting geometric figures, one should pay attention also to the position of the figure. there are two kind of positions: a ‘typical position’ and an ‘atypical position’. in a typical position, the solid ‘stands’ on its basis. in an atypical position, the solid is ‘lying-on-its side’, for example, on the lateral surface of a cone or cylinder or on one of the faces of a pyramid’s lateral surface.tirosh, tsamir, levenson, tabach and barkai (2011) conducted a study that investigated pre-service and in-service teachers’ knowledge of two-dimensional and three-dimensional figures. the researchers presented examples of geometric figures in typical and atypical positions. burger and shaughnessy (1986) describe the direction or position of a geometric figure as a non-critical and irrelevant attribute. according to tirosh, tsamir and levenson (2010), studies show that children tend to treat non-critical attributes, such as the shape’s position (horizontal basis) or size (wide or narrow), as critical attributes. figure 3 presents an example of a cylinder in a typical and an atypical position. figure 3: an example of a cylinder in a typical position (a) and an atypical position (b). prototype figures are always depicted in typical position. in order to avoid adhering to one prototype example in a typical position, it is recommended that children be exposed to a large number of examples that represent a specific concept, as well as to many non-examples of the same concept in different positions. this will allow the learners to achieve a consolidated concept (nctm, 2000).this study focuses on the ability of second grade children to identify solids in different positions and to justify their answers. methodology top ↑ research objectives the first objective of this study was to learn second graders’ ability to:• identify examples and non-examples of three types of solids, cylinder, cone and pyramid, in a typical position • justify their identification based on the attributes (critical and non-critical) of those solids. the second objective of this study was to find out whether the children maintain or change their identification of the solids, giving arguments accordingly, when the position of the examples is changed. research population the research population consisted of 35 children from the second grade at a school located at the centre of israel. until the study, the children had not formally studied the subject of solids at school. it is important to note that these children were supposed to have learnt to identify and name solids such as cones, cylinders and pyramids in kindergarten, according to israel’s kindergarten mathematics curriculum (ministry of education, 2010). research tools the research tools included 14 physical solids (appendix 1) and a structured interview. the physical solids (appendix 1) fourteen physical solids were shown to the children: eight solids with mathematical names, namely two cylinders (solid 1, a ‘tall’ cylinder, and solid 2, a ‘flat’ cylinder); two cones (solid 6, a ‘clown hat’ cone, and solid 7, a ‘chinese hat’ cone); four pyramids (solid 9, a triangular pyramid, solid 10, a square pyramid, solid 11, a quadrilateral concave pyramid, and solid 14, a hexagonal pyramid). of these eight solids, five are clear prototypes (solids 1, 6, 7, 8 and 9) and three are perceived to a lesser extent to be prototypes (solids 2, 11 and 14).the remaining six solids are non-examples of those solids: solid 3 and solid 4 are non-examples of a cylinder, solid 5 is a non-example of a cone and solids 10, 12 and 13 are non-examples of a pyramid. the structured interview the structured interview comprised 28 questions, two questions for each of the 14 physical solids. the first question related to the typical position of each solid (position a) and the second question related to the solid’s position after it had been changed, usually to a ‘lying’ position (position b). in every question children were asked to justify their answer.for example, the researcher showed the learner solid 1 in position a, and asked the learner, ‘is this a cylinder? please explain why.’ then, the researcher changed the position of solid 1 to position b and asked the learner, ‘is this a cylinder? please explain why.’ research procedure during an individual meeting with every child, the 14 physical solids were presented by the researcher in two positions: a typical position (position a) and an atypical position (position b). the order of presentation of the solids was identical for every child. as mentioned previously, each child was asked to confirm or refute the name given by the researcher and then explain and justify their answer. the reasoning behind the identification of the solids was examined through the explanations. each interview lasted 30 min – 40 min. research method and analysis method the outcomes of the interview questions relevant to the identification of the solids were quantitatively analysed (frequencies of correct and incorrect responses). a qualitative content analysis was used for the arguments that were given by the learners as justification for their answers. all the arguments were classified into categories by the two researchers. no differences were found between their judgements of the arguments. ethical considerations permission to conduct the study was granted by the principal of the school, the teachers and the parents of the participating children. the methods, aims and objectives of the study were explained and discussed with all of them. the aims were explained to the participating children and they agreed to take part in the study. the name of the college and those of the children will not be divulged in this article. findings top ↑ the first research objective the first research objective was to learn the children’s ability to identify examples and non-examples of a cylinder, a cone and a pyramid in typical positions, giving reasons. table 1 indicates the frequency of correct and incorrect answers and presents the percentage of children who did not answer the questions. table 1: frequency (%) of identifying examples and non-examples of solids (in position a). table 1 illustrates that the percentage of children correctly identifying solids with a known mathematical name (cylinder, cone and pyramid) in typical position (position a) ranged between 42.9% and 100%. over 80% of the children correctly identified the solids known to them, that is, to which they are exposed in their day-to-day life: the prototypes (solids 1, 6, 7, 8 and 9). all of the children (100%) correctly identified solid 1 (the cylinder). the percentage of correct answers to the two cones (solid 6 and solid 7) was also high: 91.4% and 85.7%, respectively. as for the pyramids (solid 8 and solid 9), the correct answers ranged between 88.6% and 80% respectively.the lowest percentage of correct answers related to solids that have a known mathematical name (cylinder, cone and pyramid), but have a lower exposure in everyday life. for example, solid 2, the ‘flattened’ cylinder whose altitude is shorter than its diameter (65.7%), solid 14, the hexagonal pyramid (42.9%), and solid 11, the quadrilateral concave pyramid (11.4%). as for the non-example solids, the correct answers ranged between 42.9% and 82.9%. the percentage of correct identification of non-examples of a cylinder (solid 3) was the lowest (42.9%) and that of non-examples of a pyramid (solid 12) was the highest (82.9%). the answers were accompanied by arguments, 306 in total. as already mentioned, all of the children were asked to justify their answers but not all of them did so. nevertheless, it is noteworthy that none of the children gave more than one argument per answer and none gave no answer at all. all of the arguments for determining the answer were classified according to two categories: arguments based on perception of the solid in general (visualisation, first level according to van hiele theory) (category 1); and arguments based on the attributes of the solid (analysis, second level according to van hiele theory) with reference to and distinction between critical and non-critical attributes (category 2). category 1: arguments based on perception of the solid in general only 28 arguments (about 9%) were based on general perception of the solid. examples of these are:• ‘because it has the shape of a toilet paper roll’. • ‘because it has the shape of a cone’. • ‘because it looks like an ice-cream cone’. • ‘because it’s like a boat with a prow’. • ‘because it looks like a clown’s hat’. • ‘because it is like the pyramids in egypt – it is pretty both on the inside and the outside’. • ‘because it's like a circus tent’. category 2: arguments based on the attributes of the solid the other 276 arguments (about 91%) were based on the attributes of the solids. many arguments were given whilst touching and demonstrating the existence of that attribute in the solid. for example, they described the attribute of the curved surface of the cylinder while rolling the solid.as mentioned above in the theoretical background, the arguments based on attributes can be divided into two types: those based on critical attributes (195 out of 276) and those based on non-critical attributes (81 out of 276). examples of arguments based on critical attributes of the solid: • ‘this is a cylinder because it is round and has no vertices’. • ‘this is a cone because it is rounded at the bottom and pointed at the top’. • ‘this is a pyramid because it has pointed parts and some triangular parts’. examples of arguments based on non-critical attributes of the solid: • ‘this is a cylinder because it is long’ (solid 1) … ‘and this is not a cylinder because it is not tall and it resembles a drum’ (solid 2) – this non-critical attribute relates to the cylinder altitude. • ‘because there are seven vertices so it cannot be a pyramid’ (argument given for failing to identity solid 14 as a hexagonal pyramid) – a non-critical attribute relating to the number of vertices of the polygon which forms the pyramid basis. • ‘because there is a square here and if it was a pyramid we would have a triangle’ (while indicating the basis of the square pyramid, solid 9) – a non-critical attribute relating to the polygon that forms the pyramid basis. • ‘because at the bottom there is no triangle’ (whilst indicating the basis of the hexagonal pyramid) – a non-critical attribute relating to the polygon that forms the pyramid basis. the second research objective the second research objective was to find out whether a change in the position of the examples of the three types of solids (cylinder, cone and pyramid – solids 1, 2, 6, 7, 8, 9, 11 and 14) would result in a change in the children’s decisions regarding the name (transition from position a to position b), with relevant justification.table 2 illustrates that the number of correct identifications of solids cylinders, cones and pyramids in position b ranges between 11.4% and 85.7%. table 2: frequency (%) of identifying solids in position a versus position b. of the five more familiar solids (solids 1, 6, 7, 8 and 9), 80% of the children correctly identified two of them in position b (solid 1, the prototype cylinder, and solid 8, the triangular pyramid). on the other hand, in position a, over 80% of the children correctly identified all five. the greatest difference between position a and position b was demonstrated in solid 6, the cone (91.4% in position a and 71.4% in position b), and in solid 9, the square pyramid (80% in position a and 40% in position b). in the case of the three less familiar solids: solid 2, the ‘flattened’ cylinder showed no difference between the two positions (65.7%); only 8.6% of the children identified solid 11, the quadrilateral concave pyramid, in position b; and only 11.4% of the children identified solid 14, the hexagonal pyramid, in position b. the answers given to solids in position b were also accompanied by arguments, 108 in total. most of the arguments (73 out of 108) were given in connection with those solids whose name, according to the children’s identification, did not change, namely the solids that kept their name after changing position. below are some examples of justifications for keeping the name of the solid after changing the position. thirty-four arguments related to cylinders. for example: • ‘the same but reversed’. • ‘they have a fixed shape’. • ‘still a cylinder’. • ‘because it does not matter how we position it’. • ‘it remains a cylinder any way we turn it’. twelve arguments related to cones. for example: • ‘the same but lying on its side’. • ‘because it rolls like a cone’. • ‘it's still a cone regardless of how it looks’. twenty-seven arguments related to pyramids. for example: • ‘this is always a pyramid because no matter what, it always has a pointed tip’ (regarding the triangular pyramid). • ‘regardless of how we rotate it’. • ‘because it is still the same shape; we have only turned it’. • ‘one can put it in any way; it still remains a pyramid’. • ‘because it still has pointed tips at the end of every line’. below are some examples of justifications for not keeping the name of the solid after changing the position. three arguments related to cylinders. for example: • ‘a cylinder should stand upright, not lying on its side’. • ‘since it is lying on its side then it is not a cylinder’. twelve arguments related to cones. for example: • ‘because only when it is upright it is a cone’. • ‘because a cone is upright and needs a pointed top’. • ‘because it is lying on its side and it is short’. twenty arguments related to pyramids. for example: • ‘because now it has two pointed tips and a pyramid needs only one’ (when we laid the square pyramid on its side). • ‘because it is (indicating the basis, the square) too flat’ (the square pyramid). • ‘because in a pyramid we have to see the triangles and here we see a square’ (the square pyramid). • ‘because now it does not have the shape of a pyramid. there is a line above’ (the square pyramid). • ‘because it is lying on the side (indicating the hexagonal) and this is not a triangle’ (the hexagonal pyramid). to sum up, the children who identified the solid in position b as the same solid in position a specified again in their justifications that the solid did not change and gave the same attributes of the solid for position b as well. the reasons children gave when identifying the solid in position b as not the same solid stemmed from the change from the typical position to an atypical position. discussion and conclusions top ↑ the ability to identify, characterise and name geometric shapes is one of the skills to be developed in young learners in order to promote their mastery of the first and second geometric thinking levels. in this study we investigate second graders’ ability to identify three-dimensional figures (solids) in different positions and to justify their answers.findings show that more than 80% of the children correctly identify prototype cylinders, cones and pyramids in typical positions. a lower percentage was obtained in the identification of non-prototype cylinders, cones and pyramids. this was especially noticeable for those identifying the two cylinders (solid 1 and solid 2). the difference in the percentage of children who correctly named these two cylinder types in position a (100% vs. 65.7%) is due to the misconception that a cylinder altitude should be longer than the diameter of the cylinder base. it is to be assumed that the children who made an incorrect identification, turned a non-critical attribute, altitude, into a critical attribute following their exposure to the cylinder prototype, which is tall and ‘narrow’. this brought about a prototypical judgement followed by an incorrect decision (34.3%) not to include solid 2 as an example of cylinder. this is corroborated by the findings of vinner and hershkowitz (1983). over 50% of the children failed to identify the hexagonal pyramid (solid 14) in the typical position. we maintain that the reason for the low percentage stems from turning a non-critical attribute, the basis shape, into a critical attribute. only 11.4% identified the quadrilateral concave pyramid (solid 11) in the typical position. here, too, we believe that the low percentage stems from turning a non-critical attribute, the concave shape of the basis, into a critical attribute. in both cases, the same assumption led the children to decide that the solid is not a pyramid. we believe that the reason for relying on the prototype might be due to the children’s daily acquaintance with real-world objects that resemble the solids. for example, a toilet paper roll is a familiar cylinder (solid 1), a dreidel (a toy played with by children during hanukkah in israel) is a triangular pyramid (solid 7) and photos of the pyramids in passover haggadah (a book read on passover eve) show square pyramids (solid 8). conversely, children are not exposed on a daily basis to non-prototype solids. all of the arguments given by children who correctly identified the solids were correct. most of the arguments related to the critical attributes of those solids. only a few related to the figure as a whole. similar to the findings of burger and shaughnessy (1986), a large number of children in this study gave a visual description of the solid, for example, ‘it is round’, ‘it looks like a toilet paper roll’, ‘it’s like a wigwam’. the situation is different for those children who wrongly identified the solids, as they based their arguments on non-critical attributes of the solid. for instance, a mistake was made in identifying a hexagonal pyramid because the child did not know that a pyramid’s basis can be any polygon and not necessarily a triangle: ‘because in a pyramid we have to see the triangles and here we see something else.’ arguments based on prototypes frequently lead to a limited perception of a geometric figure. several researchers (hershkowitz, 1989; schwarz & hershkowitz, 1999) show that children tend to view only prototypes as examples of a specific concept and hence they frequently perceive other examples that are not prototypes as non-examples. wilson (1986, 1990) attributes great importance to the use of non-examples. she claims that exposing children to non-examples of a concept that demonstrate non-critical attributes develops children’s ability to distinguish between critical and non-critical attributes of concepts. regarding identification and naming of non-examples of solids with familiar names, the percentage of those correctly identifying the solids as non-examples is lower. this study suggests that this stems from insufficient exposure to non-examples of this type. this is in line with the arguments relating to one, non-critical attribute of the solid, for example, ‘it is somewhat like a round cylinder’ (an argument referring to non-example solid 3). when identifying solids in the different positions, the findings illustrate that many children find it difficult to identify known solids in atypical positions. a higher level of mastery was manifested with regard to familiar solids to which the children are exposed almost daily in israel: a cylinder (solid 1) and a triangular pyramid (solid 6). this finding is supported by the claim made by watson and mason (2005) concerning the small number of examples presented to learners, since the learners who participated in this study were supposed to have been acquainted with those solids in kindergarten (ministry of education, 2010). similarly, tsamir et al. (2008) and tirosh et al. (2010) stipulate that geometry teaching should include exposure to different and diverse types of examples and non-examples. to sum up, exposure to examples of solids in different positions and non-examples of the same concepts are an important stage in building children’s concept comprehension. the more learners are exposed from early childhood, starting from kindergarten, to varied examples and non-examples of solids in different positions, the more they can enhance their understanding of critical and non-critical attributes of those solids. consequently, at every stage of teaching this discipline, it is essential and recommended that children encounter as wide a variety as possible of solids, presented in different positions. it is recommended that further study be conducted involving a larger learner population. that study should focus on the effect of formal schooling in the subject of solids on children’s enhanced ability to identify the learnt solids in different positions and on the level of justification of their arguments. acknowledgements top ↑ competing interests the authors declare that they have no financial or personal relationships which may have inappropriately influenced them in writing this article. authors’ contribution y.s. (kibbutzim college of education) was the project leader as well as conducted the interviews and collected the data. both y.s. and d.p. (kibbutzim college of education) contributed to the conception and design of the study and both wrote and edited the final version of the manuscript. references top ↑ burger, w.f., & shaughnessy, j.m. (1986). characterizing the van hiele levels of development in geometry. journal for research in mathematics education, 17(1), 31–48. http://dx.doi.org/10.2307/749317clements, d.h., & sarama, j. (2000). young children’s ideas about geometric shapes. teaching children mathematics, 6(8), 482–488. available from http://www.jstor.org/stable/41197461 cohen, m., & carpenter, j. (1980). the effects of non-examples in geometrical concept acquisition. international journal of mathematical education in science and technology, 11(2), 259–263. http://dx.doi.org/10.1080/0020739800110218 crowley, m.l. 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(1987). sequencing examples and non-examples to facilitate concept attainment. journal for research in mathematics education, 18(2), 112–125. http://dx.doi.org/10.2307/749246 schwarz, b., & hershkowitz, r. (1999). prototypes: brakes or levers in learning the function concept? journal for research in mathematics education, 30, 362–389. http://dx.doi.org/10.2307/749706 tall, d., & vinner, s. (1981). concept images and concept definition in mathematics with particular reference to limits and continuity. educational studies in mathematics, 12, 151–169. http://dx.doi.org/10.1007/bf00305619 tirosh, d., tsamir, p., & levenson, e. (2010). triangle – yes or no? intuitions and triangles in kindergartens. literacy and language, 3, 161–175. [hebrew] tirosh, d., tsamir, p., levenson, e., tabach, m., & barkai, r. (2011). prospective and practicing knowledge preschool teachers’ mathematics knowledge and self-efficacy: identifying two and three dimensional figures. in b. roesken, & m. casper (eds.), current state of research on mathematical beliefs xvii. proceedings of the 17th mavi (mathematical views) conference (pp. 221–230). bochum, germany: ruhr-universität. available from http://www.ruhr-uni-bochum.de/imperia/md/content/mathematik/roesken/mavi17_final_content.pdf tsamir, p., tirosh, d., & levenson, e. (2008). intuitive nonexamples: the case of triangles. educational studies in mathematics, 69(2), 81–95. http://dx.doi.org/10.1007/s10649-008-9133-5 van hiele, p.m. (1997). structuure. zutphen, netherlands: uitgeverrj thieme. van hiele, p.m. (1999). developing geometric thinking through activities that begin with play. teaching children mathematics, 5(6), 310–316. available from http://www.jstor.org/stable/41198852 vinner, s. (1991). the role of definitions in the teaching and learning of mathematics. in d. tall (ed.), advanced mathematical thinking (pp. 65–81). dordrecht: kluwer. vinner, s., & hershkowitz, r. (1983). on concept formation in geometry. zdm: the international journal on mathematics education, 15, 20–25. watson, a., & mason, j. (2005). mathematics as a constructive activity: learners generating examples. mahwah, nj: lawrence erlbaum associates. wilson, s. (1986). feature frequency and the use of negative instances in a geometric task. journal for research in mathematics education, 17(2), 130–139. http://dx.doi.org/10.2307/749258 wilson, s. (1990). inconsistent ideas related to definitions and examples. focus on learning problems in mathematics, 12, 31–47. appendix 1 top ↑ photos of the physical solids solid 1 solid 2 solid 3 solid 4 solid 5 solid 6 solid 7 solid 8 solid 9 solid 10 solid 11 solid 12 solid 13 solid 14 article information author: margot berger1 affiliation: 1school of education, university of the witwatersrand, south africa correspondence to: margot berger postal address: division of mathematics education, school of education, university of the witwatersrand, private bag 3, wits 2050, south africa dates: received: 17 sept. 2012 accepted: 27 mar. 2013 published: 19 apr. 2013 how to cite this article: berger, m. (2013). examining mathematical discourse to understand in-service teachers’ mathematical activities. pythagoras, 34(1), art. #197, 10 pages. http://dx.doi.org/10.4102/ pythagoras.v34i1.197 copyright notice: © 2013. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. examining mathematical discourse to understand in-service teachers’ mathematical activities in this original research... open access • abstract • introduction • research goal • background    • the structure of the course • commognitive framework    • overview    • elements of mathematical discourse       • words       • visual mediators       • narratives       • routines • vignette and its background       • task       • pedagogical purpose of task    • description and analysis       • saming the unsame and endorsing the meta-rule of hand sketching       • representing an undefined point       • endorsing the meta-rule of hand sketching       • hand sketching y2 and y3       • saming two different mathematical objects       • unsaming two different mathematical objects       • saming two equivalent objects       • creating an endorsed narrative    • extending the analysis       • endorsing narratives in a technological environment       • visual mediation       • words       • routines • conclusion • acknowledgements    • competing interests • references • footnotes abstract top ↑ in this article i use sfard’s theory of commognition to examine the surprising activities of a pair of in-service mathematics teachers in south africa as they engaged in a particular mathematical task which allowed for, but did not prescribe, the use of geogebra. the (pre-calculus) task required students to examine a function at an undefined point and to decide whether a vertical asymptote is associated with this point or not. using the different characteristics of mathematical discourse, i argue that the words that students use really matter and show how a change in one participant’s use of the term ‘vertical asymptote’ constituted and reflected her learning. i also show how the other participant used imitation in a ritualised routine to get through the task. furthermore i demonstrate how digital immigrants may resist the use of technology as the generator of legitimate mathematical objects. introduction top ↑ in this article i examine the activities of a pair of in-service mathematics teachers (the students) as they engage in a particular mathematical task which allows for, but does not prescribe, the use of geogebra. the task was part of a mathematics course in which the mathematical notion of a function was revisited. the course itself was part of a larger programme offered by a south african university to practising high-school mathematics teachers who wished to improve their mathematics teaching knowledge and their academic qualifications. as will be seen, the students (i.e. the teachers) engage in the task in a surprising way: their use of geogebra is very limited and they use various mathematical terms such as ‘asymptote’ and ‘undefined’ very loosely. in order to understand what is happening and why, i situate my analysis of the students’ activities within sfard’s (2007, 2008) theory of commognition. the term ‘commognition’ (a mixture of the terms ‘cognition’ and ‘communication’) was introduced by sfard (2008) to refer to a theory, developed by her, in which cognition and communication are regarded as different expressions of the same phenomenon. specifically, cognition is an intrapersonal expression whereas communication is an interpersonal expression of a phenomenon. the theory draws both on vygotsky and wittgenstein and assumes a view of mathematics and of mathematics learning which resonates strongly with my own experiences of teaching and learning mathematics. moreover it provides tight analytic constructs within which to examine and interpret mathematical activity. i use these constructs to show how ‘words matter’ and how a change in the discourse (in particular the use of the term ‘vertical asymptote’) affects the mathematical activity of the students. i also use this theory to show how the use of computers as a tool in mathematical learning may require an explicit rewriting of the rules of what counts as mathematical activity. research goal top ↑ the research i present here is a case study drawn from a larger project. a primary aim of the larger project is to examine ways in which in-service teachers are able to deepen their knowledge of fundamental mathematical concepts, such as ‘function’. in this article, i focus on: • the ways in which two teachers engage in a task whose purposes is to enhance their knowledge of the distinction between removable discontinuities and vertical asymptotes. • how the teachers deal with the affordances and limitations of graphing software, in this case, geogebra. this has important implications for their use of technology as a teaching tool. these foci need to be contextualised within two constraints. the first is that all the students in the course were digital immigrants1 and had not used any educational software (such as geogebra) prior to this course. secondly, the intended purpose of geogebra in the particular task was as a generator of graphs. that is, it was intended as a tool of amplification rather than of cognitive reorganisation (pea, 1993). background top ↑ although the focus of this article is on learning mathematics, rather than on what constitutes mathematics for teaching, it is important to situate the article within the broader debate around the type of mathematics that pre-service or in-service teachers need to know in order to be effective mathematics teachers (ball, thames & phelps, 2008; davis & simmt, 2006; stacey, 2008). ball and her colleagues focus on the particular ways in which mathematics teachers need to engage with various mathematics procedures, representations and mathematical definitions in order to teach effectively. more generally, silverman and thompson (2008) argue that ‘the work of teaching for understanding is predicated on coherent and generative understandings of the big mathematical ideas that make up the curriculum’ (p. 501, my italics).in the southern african context, pournara (2009) grapples with issues about what constitutes appropriate knowledge for pre-service mathematical teachers; adler and davis (2011) explore the constitution of mathematics for teaching in various south african education institutions. they show how different courses are differently constituted with different foci and they ask how the different types of courses they describe ‘relate to teachers’ learning from and experiences of mathematics for teaching and ultimately, the quality of their teaching’ (p. 20). the article i present here goes a little way to addressing the first part of this question. the structure of the course in line with silverman and thompson (2008), i take it as axiomatic that mathematics teachers need to have a very good understanding of the fundamental ideas in mathematics, including mathematical concepts, definitions, procedures and connections to other fundamental ideas. in the south african situation (and probably in many other countries), this type of knowledge among practising high-school teachers cannot be assumed. for reasons born out of south african history, many high-school mathematics teachers in south africa have a degree or diploma in education rather than in mathematics and these teachers’ content knowledge is sometimes weak. within the south african context, fleisch (2008, p. 123) argues that small-scale studies point to very poor levels of conceptual knowledge and a large number of errors made by teachers in their own lessons. more specifically, venkat and adler (2012) point to problematic connections that several practising teachers in south african classrooms make between stated mathematics problems and their representations and the transformations of these representations. cognisant of these weaknesses in teachers’ knowledge and practice, our aim in this course was to revisit functions, extending and deepening teachers’ understanding of this fundamental concept (ma, 2010). allied to this, the focus in the course was on the teachers as learners of mathematics (rather than as teachers of mathematics).the course was structured as a part reading, part activity course. the class met once a week for a three-hour session over 11 weeks. students were expected to study a specific chapter from the prescribed pre-calculus textbook (sullivan, 2008) prior to their weekly session (this included doing examples at home). during the three-hour session, the class and lecturer discussed the topic and examples they had studied at home for the first hour. they were then presented with tasks around the topic which they did on their own or in pairs. some of the tasks required the use of geogebra, others did not. i designed the structure of the course; my colleague, lynn bowie, taught most of the course and designed, selected or adapted most tasks. commognitive framework top ↑ overview sfard’s (2008) theory of commognition is based on the premise that individual development is individualisation of ‘patterned collective activity’ (p. 93), and that thinking is individualised communication. in mathematics, patterned collective activity takes the form of mathematical discourse where a discourse is a special type of communication characterised by a range of permissible actions and reactions. mathematical discourse is characterised by its words, visual mediators, routines and narratives (ben-zvi & sfard, 2007; sfard, 2007). commognition is primarily a participationist theory. indeed learning only takes place through the individual’s thoughtful participation in mathematical discourse. accordingly mathematics learning is ‘tantamount to modifying and extending one’s mathematical discourse’ (sfard, 2007, p. 565).according to sfard, mathematics is an autopoietic system; that is, it creates its own objects. thus mathematical objects are necessarily discursive objects. sfard’s notion of mathematical discourse derives directly from this ontological view of mathematics and the power of her framework lies in the connections it makes between this ontological perspective and the ways learners engage in the discourse. the foundational basis of sfard’s mathematical discourse theory contrasts with other discourse theories which have developed from non-mathematical domains. indeed i find the inner consistency and coherence of sfard’s framework (sfard, 2008) most appealing for understanding mathematical activity. (this is not to deny that other discourse theories have been successfully used in the mathematics domain, for example, morgan’s, 2005, use of fairclough’s critical discourse analytic approach; shreyar, zolkower & perez’s, 2010, use of halliday’s systemic functional linguistic analysis.) sfard (2008) and ben-yehuda, lavy, linchevski and sfard (2005) have defined and illustrated the different components of mathematical discourse in the context of everyday and school mathematical activity. viirman (2011), in sweden, has used commognitive theory to examine the discourse of university mathematics lecturers teaching an undergraduate course on functions. however, the components of mathematical discourse have, as far as i know, not been elaborated to sites of teacher education. since such contexts have their own peculiarities, i extend sfard’s descriptions to include such contexts where necessary. elements of mathematical discourse words the use, in specific ways, of certain words or expressions such as ‘equal’, ‘function’ and ‘vertical asymptote’ indicates that we have mathematical discourse. although many of these words also appear in everyday discourse, their use in mathematical discourse (even school mathematical discourse) is well defined, albeit often implicitly. for example, in colloquial discourse we say: ‘i ate half an apple’ to indicate that we ate approximately half an apple. in mathematical discourse, we use the word ‘half’ to mean exactly half. for example, half of eight is four. word use is very important in that the use of the word constitutes its meaning (wittgenstein, cited in sfard, 2007, p. 571). in the task below, we see how the pair of students use the term ‘vertical asymptote’ to refer to any point whose value is undefined. i show that this loose and incorrect use of the term both reflects and reinforces an unrefined notion of the meaning of the term by the students and that a change in its use is necessary for distinguishing between a vertical asymptote and a hole (i.e. a removable discontinuity). sfard (2008) calls the process of associating one name with several seemingly different mathematical objects ‘saming’. saming, if applied to discursive objects that are all realisations of the same signifier, is part of the process of the learner’s construction of a new mathematical object. in the vignette below, students ‘same’ two objects which are not the same. these objects subsequently need to be ‘unsamed’. visual mediators visual mediators are visible objects such as symbols, graphs and diagrams which participants in a mathematical discourse use to identify the objects of their thinking or communication and to bring these objects into focus. sfard (2008) distinguishes between iconic mediators (such as graphs and pictures), symbolic mediators and concrete mediators (such as beads of an abacus). the use of these visual mediators constitutes mathematical thinking or communication and visual mediators are used as objects with which to think or communicate. in a geogebra context, it is fairly easy to generate graphs of various functions which can then serve as visual mediators. however, graphs generated by geogebra (and most other graphical software) may not reveal key aspects, such as removable discontinuities, of the function. in the task below, the students use hand-drawn graphs in which removable discontinuities may be represented by a hole rather than computer-generated graphs in which the discontinuities are hidden. narratives narrative is any text, spoken or written, that is ‘framed as a description of objects, of relations between objects, or processes with or by objects’ (sfard, 2008, p. 300). narratives are subject to endorsement and may be labelled ‘true’ or ‘false’. within formal mathematical discourse, the narratives that are approved by the academic mathematical community (according to specific, well-regulated rules) are called mathematical theories. these theories consist of various discursive objects such as axioms, theorems and definitions. the rules of endorsement for school mathematics are often different from the rules of endorsement for more formal mathematics. for example, the statement ‘the sum of angles in a triangle is always 180°’ is true within school academic discourse (in which the axioms of euclidean geometry form the backdrop). within a more formal mathematical discourse, where hyperbolic geometry and spherical geometry are admitted, the statement is not necessarily true. within the context of computer-based mathematical learning, mathematical narratives may be positively endorsed only if they conform to the traditional mathematical narratives (be it school or more formal academic mathematics). routines a routine is a repetitive and well-defined discursive pattern. a routine may be a procedure; it may also be a practice (rand mathematics study panel, 2003) such as generalising, justifying or endorsing (or rejecting) mathematical narratives. routines are regulated by certain rules; these may be rules about the objects in the discourse (object-level rules) or rules about the discourse itself (meta-rules), for example rules for what constitutes an acceptable proof. sfard (2008) also distinguishes between the how of a routine, that is, the set of meta-rules that constrain the course of action, and the when (applicability conditions) of a routine, that is, the set of meta-rules that constrain or determine when it is appropriate to use a particular routine. in the vignette below we see the difficulty that students have with the ‘when’ of using geogebra for their mathematical routines. computer-generated routines themselves require further scrutiny. that is, certain algorithmic routines are easily executed by the software; however, the computer’s routines do not always obey the rules of mathematical discourse. for example, geogebra (and most other graphical software) generates a continuous graph even if there are removable discontinuities in that function. consequently the user needs to be aware of how to interpret the computer output so that it is compatible with endorsed mathematical narratives. sfard (2008) usefully distinguishes between three different types of routines: explorations, rituals and deeds, each of which is distinguished by its own set of goals. explorations are routines whose purpose is to produce or verify an endorsable or endorsed narrative. for example, routines of solving equations, of proving a mathematical result, or of generating and investigating a mathematical conjecture are explorations. in the task below, students were expected to embark on a set of explorations in order to substantiate the definition of a ‘vertical asymptote’ and to determine the existence or not of a vertical asymptote at a particular point. a ritual is a routine whose goal is social approval. this is usually through the participant aligning her mathematical activity with other people’s routines. rituals involve imitation of others’ routines; as such they may be a very important part of mathematical learning. echoing vygotsky (1978), sfard (2008) argues that a ritual is the form that routines take in the zone of proximal development (p. 253). arguably we can identify ritualised routines in the episode below. finally, deeds are routines whose purpose is a change in objects, not just in narratives (as in the case of explorations). so for example, a child may be able to divide an actual set of six cookies amongst 12 children (a deed) although she may not be able to formulate or execute 12 ÷ 6. in the geogebra environment, a task which involves the use of a slider to change the value of a parameter and hence a graph may constitute a deed. vignette and its background top ↑ in this vignette, a pair of students, eva and tom, work on a mathematical task in which they are expected to recognise the presence or not of a vertical asymptote at a point of a particular function. as mentioned previously, the course was designed as a reading and self-study course and prior to this particular session, students were assigned self-study readings and activities from the textbook relating to rational functions. vertical asymptotes (definition plus symbolic and graphical exemplars of these) were part of this text as were removable discontinuities (which are called ‘holes’ in the textbook). vertical asymptotes are also part of the school syllabus and so students (who were teachers) should have been familiar with this notion. non-removable discontinuities are not part of the school curriculum but, as indicated, these should have been studied in the readings prior to this session. in addition, the students discussed examples of functions with vertical asymptotes in the first part of the session (prior to the task) although in this discussion, these vertical asymptotes were not directly related to their definition. eva has a bsc degree with a postgraduate diploma in education; tom has a degree in education. in an earlier survey eva claims to use the computer often and is very confident in its use; in contrast, tom claims that he does not use a computer often, but he is confident, as opposed to ‘very confident’, in its use. neither tom nor eva has used a computer in the learning or teaching of mathematics. eva does very well in the course; she obtains 87% in mid-semester test. tom is one of the weakest students in the class and he obtains 34% in the mid-semester mathematics test. in this vignette, the students were audio taped and their work was screen recorded as they worked on a task that was given to the entire class to work on in pairs. the researcher sat just outside the room in which the students were working, walking in periodically to observe what was happening. the pairs of students were told that they must treat this as a normal classroom session and that they could ask the researcher any questions as they would in a classroom setting. task the task was given as an instruction to do a particular question in the prescribed textbook (see figure 1). students were told that they could use geogebra if they wanted to. in addition, the warning: ‘be careful. geogebra isn’t perfect here’ was offered. the vignette took place during the sixth week of the class. figure 1: the task. pedagogical purpose of task this task can be done analytically. however, given explicit instructions to graph the functions it was expected that students would generate graphs of each function with geogebra (before or after algebraic simplification) or by hand (after algebraic simplification). for example, with respect to y1, we expected students to use geogebra to sketch y1 directly and to observe from the geogebra sketch that the graph of y1 the straight line y1 = x + 1.consequent to this, we expected them to write something like: alternatively, we expected students to first simplify y1 to get after this they could draw y1 = x + 1 by hand, leaving a hole at x = 1.we expected them to work similarly for y2 (a second degree equation). given the cumbersome nature of drawing a cubic or quartic equation by hand, we expected students to use geogebra to sketch y3 and y4 . whether the graphs were hand drawn or computer generated, it was hoped that students would notice that each function was not defined at x = 1 but that by cancelling the factor (x − 1) in the numerator and denominator, the problematic point, x = 1, could be removed. graphically this would be depicted by a hole at x = 1 (they could draw this on a printout of the graph). alternatively, they could use a theorem (the endorsed narrative) about the location of a vertical asymptote as given in their pre-calculus textbook (sullivan, 2008, p. 188) to decide that none of the given functions had a vertical asymptote at x = 1, or elsewhere. the theorem reads: a rational function , in lowest terms, will have a vertical asymptote x = r if r is a real zero of the denominator q(x). that is, if x – r is a factor of the denominator q(x) of a rational function , in lowest terms, r(x) will have a vertical asymptote x = r. note that that since this was a pre-calculus course we did not expect students to know the language of discontinuities. rather we expected them to speak of holes or functions not existing at a point. description and analysis in the boxes and descriptions below i show how the tools of discourse, that is, words and visual mediators (sfard, 2008), evolve as the activity proceeds. i also include spoken mediators since not all mathematical activity is visually mediated. i also highlight evolving aspects of the forms and substance of discourse, that is, routines and narratives (sfard, 2008), in the column ‘routines and narratives’. in the following section, extending the analysis, i organise the analysis in terms of the four characteristics of mathematical discourse. this leads to further insights into the students’ activities. saming the unsame and endorsing the meta-rule of hand sketching at the start of the task, eva (e) and tom (t) indicate that they will hand draw the graphs of the four functions and that they will use geogebra to check their graphs (line 3, 6). eva and tom talk about having a vertical asymptote at x = 1 (lines 10–12). box 1 contains the relevant transcript. box 1: setting the scene: saming and hand drawing. representing an undefined point after a little discussion eva writes the following: she does not write x ≠ 1. tom then hand draws line y = x + 1. on eva’s suggestion, but with full support from tom, tom draws a hole (a circle) at x = 1 (lines 36−38). that is, tom and eva’s sketch of y1 is in line with the pedagogical expectations of the task. tom, with implicit prior approval from eva (lines 9−12), also draws a vertical line at x = 1 to indicate a vertical asymptote (figure 2). see box 2 for the transcript relating to the representation of the discontinuity. figure 2: hand plots of y1, y2, y3 and y4 by eva and tom. box 2: representing the removable discontinuity. endorsing the meta-rule of hand sketching eva and tom then generate a plot of y1 in geogebra, presumably to confirm their hand plot. eva again notes that geogebra does not generate a discontinuous graph (line 39). eva and tom also note that geogebra does not draw an asymptote at x = 1 (lines 40–42). box 3 contains the pertinent transcript. box 3: geogebra does not draw asymptotes. hand sketching y2 and y3 when it comes to y2, eva simplifies the function algebraically; again she does not note that x ≠ 1. she writes: eva and tom then spend much time and energy hand drawing the resultant parabolas; they use calculus to find turning points. tom hand draws the graph correctly with hole at x = 1 (and also with vertical line at x = 1). see figure 2. eva and tom confirm their plot with geogebra. eva and tom now use geogebra to generate a graph of y3; however they ignore this geogebra-generated graph and draw the graph by hand. eva factorises the expression to get x3 + x2 + x + 1 and she and tom expend much effort hand plotting the resultant cubic using calculus. as before, tom hand draws the graph correctly with a gap at x = 1 (and also with vertical line at x = 1; see figure 2). saming two different mathematical objects eva and tom persist in talking and writing about x = 1 as a vertical asymptote (e.g. line 310) whilst at the same time recognising that there is a gap at x = 1 (line 312). box 4 contains the relevant transcript. box 4: saming 'point at which function is not defined', 'asymptote' and 'circle'. unsaming two different mathematical objects finally, near the end of the activity the researcher interrupts to ask if the students have any questions. eva states her concern that geogebra is not perfect because the asymptote is ‘not reflected when x is equal to 1’ (line 323). the researcher uses this opportunity to explain that for a vertical asymptote, one must have an expression in the form , a ≠ 0, in lowest terms. if the expression is in the form , one has a hole. (in this pre-calculus course, ‘hole’ is the word used both in the textbook and by the researcher to refer to a removable discontinuity.) that is, the researcher (r) ‘unsames’ vertical asymptote and a point at which the function is not defined. although it seems that tom does not entirely accept this explanation (see box 5, line 345, 348), eva endorses it in a later episode (box 6, line 406). the transcript for this episode is in box 5. box 5: researcher unsames 'vertical asymptote' and 'hole' box 6: eva implicitly sames 'hole' and 'removable discontinuity'. saming two equivalent objects importantly, the pair no longer speak of an ‘asymptote’ at x = 1 when they hand draw y4. rather, eva speaks of a ‘hole’ at x = 1; tom continues to speak of an ‘open circle’ (lines 406, 407 respectively). as before, they use geogebra to generate the graph of y4, which they then ignore. and, as before, they hand plot the graph (figure 2) using much effort and time. box 6 contains the relevant transcript. creating an endorsed narrative soon after this, eva and tom discuss their response to the question: ‘is x = 1 a vertical asymptote? why not? what is happening for x = 1?’ see box 7 for the transcript of their discussion. box 7: eva creates endorsed narrative. unlike eva, tom never articulates the difference between a removable discontinuity (a ‘hole’) and an asymptote, and it is not clear whether he ever makes this distinction. indeed, he only talks about the function not being defined at x = 1 (lines 434, 436, 440). for eva, the notion of ‘undefined at a point’ has different possible meanings: if we can cancel out terms which are zero at that point (line 444), we do not have a vertical asymptote. however, if we cannot perform this cancellation, we may have an asymptote. (i suggest that this latter statement is implicit in line 444.) interestingly, eva now talks of a ‘hole’ at x = 1, whilst tom continues to use the term ‘circle’. i discuss this further in the section headed ‘extending the analysis’. extending the analysis i use the four sfardian characteristics of discourse to organise and thereby extend the above analysis. endorsing narratives in a technological environment in this task, eva and tom do not exploit the affordances of geogebra as a graph sketching tool. as shown by the predominance of the category ‘iconic: paper and pencil graph’ in the transcripts, they spend much time and effort in hand drawing each of the graphs and only use geogebra for verification. this is despite the fact that they are easily able to use geogebra to generate the graphs (evidenced by the ease with which they generate geogebra graphs for verification purposes). in this particular task, the graph that geogebra generates is not consistent with the officially endorsed narrative (wherein a removable discontinuity is represented by a hole). however, it was expected that students (forewarned to ‘be careful. geogebra isn’t perfect here’) would recognise the limitations of the geogebra-generated graphs, but still use these computer-generated graphs as their primary visual mediator (largely to save time). but this is not what happened. one possible reason for eva and tom’s limited use of geogebra is an entrenched cultural attitude: mathematics is done by hand. technology is there as a tool for confirmation of hand-done mathematics, but not for doing mathematics. ontologically speaking, mathematical outputs produced by a computer are not part of the official mathematical narrative. indeed eva, who took the leading role (sfard, 2007) in the discourse, declared right at the beginning of the task: ‘we will do it sketching first and then we’ll check on geogebra’ (line 3). this attitude surprised me. this was the sixth week of this course and my colleague and i had stressed the value of using technology in doing mathematics: its narratives were endorsable, although some care had to be taken when interpreting its outputs. another possible reason for the privileging of hand-drawn graphs is in the students’ reading of the caveat (‘be careful. geogebra isn’t perfect here’). this statement was intended to alert the students to the fact that geogebra did not reveal removable discontinuities in its graphs. but the statement may have reinforced the belief (discussed previously) that narratives of computer-generated mathematics are not consistent with the official mathematical narrative. indeed eva implicitly justifies the hand sketching of all graphs by invoking this cautionary statement at the beginning of the task (line 3). it is important to note that the existence of the cautionary statement cannot be taken as the sole or even main reason why students privilege hand-drawn over computer-generated graphs. after all, the students use geogebra graphs to confirm their hand sketches and they are aware that the geogebra graph looks like their hand sketches, other than the lack of visible discontinuities: ‘but you see geogebra doesn’t do that. can you see that it’s like a continuous graph, né?’ (line 39). indeed i strongly propose that more technologically enculturated students would have printed geogebra graphs and then hand drawn holes or asymptotes. visual mediation as mentioned above, eva and tom do not exploit the visual mediation that the geogebra graphs afford. although geogebra generates a visual picture of the graph as if it were a continuous graph rather than a graph with a removable discontinuity at x = 1, it was expected that students would use this graph together with the algebraic reasoning, to recognise that x = 1 is a point at which the function is not defined. the students are clearly aware that the geogebra graph of y1 deviates from the endorsed narrative (see e.g. lines 39–42), although they erroneously assume an asymptote at x = 1.the question is: why do the students prefer to use hand-drawn graphs rather than computer-generated graphs as visual mediators? several reasons, some of which are discussed above, are feasible. for example, it may be that they do not accept geogebra graphs as compatible with the endorsed mathematical narrative (as previously discussed), or it may be that they need to represent the value at which the function was not defined iconically. in this task, computer-generated graphs were not iconic and students lacked the experience of how to turn them into iconic mediators (print the geogebra graphs and draw a hole on the graph). so they preferred to use hand-drawn graphs in which a ‘hole’ could be directly and iconically represented. with regard to the symbolic manipulation (an activity which is visually mediated), eva is able to simplify the expressions y1, y2, y3 and y4 into polynomials which are easier to hand sketch (although this hand drawing is still very time-consuming and effortful). further discussion is provided under the heading ‘routines.’ words in the vignette we see how the students initially use the term ‘vertical asymptote at a point’ to describe a point at which a function is not defined. see, for example, lines 7–12, 39–42, 310 and 323, where the students assume that if a function is not defined at a point, it necessarily has a vertical asymptote at that point. furthermore they consistently refer to the graphical representation of such a point as an (open) ‘circle’ (lines 38, 312). in sfard’s terms they ‘same’ vertical asymptote with a point at which the function is not defined.however, the researcher’s modest intervention (lines 323–351)2, in which the difference between a hole (usually represented by an open circle) and a vertical asymptote is explained, triggers a distinct change in eva’s discourse. indeed, post intervention eva ‘unsames’ the terms ‘not defined at a point’ and ‘vertical asymptote’ (lines 423–426) and she unsames the term ‘open circle’ (used to represent a point through which there is a vertical asymptote) with ‘hole’ (which she uses to represent a removable discontinuity) (lines 406, 435, 439). she moves on to characterise the conditions which lead to a ‘hole’ (numerator and denominator cancel at zero, line 444). mathematically speaking, neither ‘hole’ nor ‘open circle’ is a well-defined mathematical term. nonetheless, by no longer speaking of an ‘open circle’ but rather speaking of a ‘hole’ eva shifts her discourse from one in which ‘not defined at a point’, ‘open circle’ and ‘asymptote’ are samed to one in which these are unsamed; indeed, in the later discourse (lines 423–444) eva produces an endorsed narrative in which she reserves the word ’hole’ to refer to a removable discontinuity (line 444). in contrast there is little change in tom’s discourse; he still refers to the discontinuity as an ‘open circle’ (lines 407, 441) and he is uncertain as to whether he should erase the vertical asymptote at x = 1 or not (lines 345, 348). just to reiterate: although the term ‘open circle’ is as acceptable as the term ‘hole’ in pre-calculus discourse, ‘open circle’ was used, before the researcher’s intervention, to signify a point at which there was an asymptote. tom does not change his usage of this term and he does not sever this connection. furthermore, even though tom and eva no longer talk of an asymptote, tom argues that: ‘for x equals to 1 it’s undefined because of division by zero’ (line 434). this is not correct: at x = 1 the function is not defined because we have division of zero by zero. so unlike eva, tom does not change his discourse; nor does he produce an endorsed narrative around the notion of a removable discontinuity. implicit in the post-intervention discourse is a commognitive conflict. a commognitive conflict is a struggle, often implicit, which is the result of the concurrent use of two incompatible discourses. eva uses the term ‘undefined’ to refer to a point at which the function is not defined (because of a zero in the numerator and denominator) and at which there is a removable discontinuity − a ‘hole’ (lines 406, 435, 439). in contrast, tom uses the term ‘undefined’ to refer to a point at which a function is undefined because of division by zero (line 434). he calls such a point an ‘open circle’. for learning to take place, the commognitive conflict needs to be acknowledged and resolved. arguably it is the role of the teacher to support the resolution of such a conflict, rather than to bypass it, as in this vignette. routines eva and tom are involved in several routines, each of which they repeat for y1, y2, y3 and y4. specifically, they use hand graph sketching techniques to draw the four functions. these techniques include the routines of simplifying the original expression and routines involving point-by-point plotting and calculus. as has been discussed, in a context in which the students have access to a tool that they can use to sketch functions (albeit with some imperfections), executing routines to hand draw the graphs is inappropriate. in this case, i suggest that applicability conditions, that is, the rules that demarcate when a particular routine should be applied (sfard, 2008, p. 209), are unclear to the students. possible reasons for this non-appreciation of applicability conditions are given above (see ‘narratives’ and ‘visual mediators’).also, when simplifying the expressions for y1, y2, y3 and y4, the students do not explicitly state where the function is indeterminate. for this reason, the routines that they execute with regard to simplification contradict endorsed mathematical narratives. for example, eva writes: she does not indicate that x ≠ 1 although she and tom acknowledge that x ≠ 1 several times, for example, lines 7−8, 36, 310, 407. a further (non-endorsable) routine, extensively discussed above, involves the students’ sketch of the vertical line x = 1 to indicate a vertical asymptote at x = 1 (see figure 2). this drawing of an asymptote at x = 1 reflects and reinforces the saming of the notion ‘not defined at a point’ and ‘asymptote’ as discussed above. arguably, tom’s discourse after the researcher’s intervention (lines 421 onwards) has the quality of a ritual. that is, he is prepared to go along with eva’s explanation of what is happening at x = 1 (lines 422–444), even though he does not seem convinced that there is no vertical asymptote at x = 1 (evidenced by his checking whether he can erase the vertical asymptote from the sketch in lines 345, 348). sfard (2008) argues that ritual is a necessary stage in routine development. through the use of thoughtful imitation in routines the learner gains knowledge of the how of a routine. ‘imitation … is the obvious, indeed, the only imaginable way to enter new discourse’ (p. 250). in contrast, eva’s routines are primarily explorations. her goal is to produce an endorsed narrative (which she does). conclusion top ↑ in this article i have used commognition (sfard, 2008) to examine a pair of students’ activities as they engage in a mathematical task in an in-service course for mathematics teachers in south africa. i have found the framework and its characterisation of mathematical discourse very helpful in understanding certain surprising phenomena. in particular, i have used the analytic constructs to explain the reluctance of students to rely on computer-generated mathematical objects (e.g. graphs) as visual mediators. suffice it to say that in courses involving technology and digital immigrants, it is essential for the lecturer or teacher to make explicit the new rules of mathematical discourse, in particular rules around when it is appropriate to use a computer in mathematical learning.another very important aspect of the mathematical activities, central to sfard’s theoretical framework and evidenced through empirical data here, is the crucial role that words play in mathematical discourse. that is, it really does matter how we use words when talking about mathematical phenomena. this is starkly shown by the students’ unsaming of ‘vertical asymptote’ and ‘point at which the function is undefined’. it is further demonstrated by eva’s embracing of a new word, ‘hole’, rather than ‘open circle’ (which was previously samed with ‘vertical asymptote’) to describe a removable discontinuity. likewise, tom’s continued use of the term ‘open circle’ to describe a point at which the function is undefined both reflects and is constitutive of his non-distinction between different types of undefined points. this points to the importance of a teacher or researcher carefully listening to what students actually say and using their discourse as a way into their understandings. finally, considering learning as a change or extension of discourse, we can say that some learning has taken place: eva has changed her discourse and eva is able to distinguish between a vertical asymptote and a point at which a function is undefined. for tom, the learning is less evident. indeed i suggest that, for tom, learning in this vignette is in the form of ritual and imitation. acknowledgements top ↑ thank you to the national research foundation who supported the research underlying this article. thank you to my colleague, lynn bowie, who taught this course and who developed or adapted or selected many tasks for this course. competing interests i declare that i have no financial or personal relationship(s) which might have inappropriately influenced me in writing of this article. references top ↑ adler, j., & davis, z. (2011). modelling teaching in mathematics teacher education and the constitution of mathematics for teaching. in k. ruthven, & t. rowland (eds.), mathematical knowledge in teaching (pp. 139−160). dordrecht: springer. http://dx.doi.org/10.1007/978-90-481-9766-8_9ball, d., thames, m.h., & phelps, g. (2008). content knowledge for teaching: what makes it special? journal of teacher education, 59, 389−407. http://dx.doi.org/10.1177/0022487108324554 ben-yehuda, m., lavy, i., linchevski, l., & sfard, a. (2005). doing wrong with words: what bars students’ access to arithmetical discourses. journal for research in mathematics education, 36(3), 176−247. available from http://www.jstor.org/stable/30034835 ben-zvi, d., & sfard, a. (2007). ariadne’s thread, daedalus’ wings, and the learner’s autonomy. education & didactique, 1(3), 123–142. available from http://educationdidactique.revues.org/241 davis, b., & simmt, e. (2006). mathematics-for-teaching: an ongoing investigation of the mathematics that teachers (need to) know. educational studies in mathematics, 61(3), 293−319. http://dx.doi.org/10.1007/s10649-006-2372-4 fleisch, b. (2008). primary education in crisis. cape town: juta. ma, l. (2010). knowing and teaching elementary mathematics. teachers’ understanding of fundamental mathematics in china and the united states. (anniversary edn.). new york, ny: rouledge. morgan, c. (2005). words, definitions and concepts in discourses of mathematics, teaching and learning. language and education, 19(2), 103−117. http://dx.doi.org/10.1080/09500780508668666 pea, r. (1993). practices of distributed intelligence and designs for education. in g. salomon (ed.), distributed cognitions: psychological and educational considerations (pp. 47−87). cambridge: cambridge university press. pournara, c. 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(2012). coherence and connections in teachers’ mathematical discourses in instruction. pythagoras, 33(3), art. #188, 8 pages. http://dx.doi.org/10.4102/pythagoras.v33i3.188 viirman, o. (2011). discursive practices of two university teachers on the concept of ‘linear transformation’. in b. ubuz (ed.), proceedings of the 35th conference of the international group for the psychology of mathematics education, vol. 4 (pp. 313−320). ankara: pme. vygotsky, l.s. (ed.). (1978). mind in society. cambridge, ma: harvard university press. footnotes top ↑ 1. ‘digital immigrant’ is the term coined by prensky (2001) to refer to someone who was born before the widespread existence of digital technology.2. most of this transcript is not reproduced here. article information author: alwyn olivier1,2 affiliations: 1editor-in-chief, pythagoras 2research unit for mathematics education, university of stellenbosch, south africa correspondence to: alwyn olivier email: aio@sun.ac.za postal address: private bag x1, matieland 7602, south africa how to cite this editorial: olivier, a. (2011). pythagoras goes open access at www.pythagoras.org.za. pythagoras, 32(1), art. #48, 2 pages. http://dx.doi.org/10.4102/ pythagoras.v32i1.48 copyright notice: © 2011. the authors. licensee: aosis openjournals. this work is licensed under the creative commons attribution license. issn: 1012-2346 (print) issn: 2223-7895 (online) pythagoras goes open access at www.pythagoras.org.za in this editorial... open access • the golden route to open access • open access is not free • forward • references the publication of this issue of pythagoras finally realises the goals i envisaged for pythagoras in my 2008 editorial, titled “pythagoras is going places …” (olivier,2008): pythagoras is now formally an open access journal, freely accessible online for anyone to read, download and distribute. what i anticipated then was a do-it-yourself job where we would upload pythagoras articles on our own do-it-yourself website (the so-called ‘green route’), becausethe increased costs of outsourcing the production of pythagoras to a professional publishing house (the so called ‘golden route’) (harnard et al., 2008) would make it impossible for amesa (association for mathematics education of south africa) to continue offering print-issues of pythagoras as a ‘free’ benefit to members. however, we soon realised that real open access was only really possible with a professional publisher. so we went out and found the money and the publisher – pythagoras is now formally published by african online scientific information systems (aosis) openjournals publishing, and the print edition is still free to members. if you are reading the print-issue, i hope that you will agree that it looks much more professional than before! if you are reading this online, i trust that you appreciate the many other benefits of the website (searching, indexing, etc.). and if you are an author, i trust that you will soon reap the benefits of being read more widely and cited more often. the golden route to open access (back to top) the primary purpose of pythagoras as an amesa research journal is as a service to the scholarly community to support them to get their research published andavailable to the wide community. merely posting an article on the internet does not mean that many people are going to find it and read it. however, by publishing through the aosis openjournalsplatform, articles in pythagoras become much more discoverable, because the publisher has links to google scholar, crossref, directory of open access journals, et cetera. that means that when somebody searches for a key word, the probability that relevant pythagoras articles are listed and accessed is now much higher. the golden route open access model therefore first and foremost means better support for our authors, making their research universally accessible and discoverable, so their work is available to a larger audience, more people can read their work, so they can make a bigger impact in the research community, africa and internationally, and they may be cited more often. another benefit of the golden route is that only publishers may assign a so-called digital object identifier (doi) to each article. the doi for a document is permanent, whilst a uniform research locator (url) may not be, therefore referring to an online document by its doi provides more stable linking than simply referring to it byits url. using the ‘rolling publication’ model, an article is published online the moment it is finished, before it is later compiled with other articles into an issue and printed.it means that one’s research is made available more quickly. furthermore, the whole submission-review-edit-publish cycle is handled online through a manuscript management system and is supported by professional aosisadministrative staff. this frees the editor to concentrate on his academic role of engaging with authors and reviewers about the substance of submitted manuscripts. this is the way it should be! open access is not free (back to top) open access means that it is free to the reader. authors retain copyright through a creative commons attribution license, so there are no legal barriers. however, golden route open access is not free to publish, free to maintain or free to print! this change in publishing model has serious financial implications for amesa.amesa is carrying the full costs in 2011, but will introduce a publishing fee in 2012, where the authors (meaning their institutions) will pay 50% of the publishing costs. we believe such a business model will make our open access publishing economically feasible and robust. forward (back to top) it has been a long and arduous road from deciding to go open access, to researching different options, negotiating a contract, securing funding, developing the website, getting to know the manuscript management system, supporting authors and reviewers to handle the manuscript management system, et cetera. but it hasbeen worthwhile! i would like to thank the amesa council and the pythagoras editorial board for their support in following this route. now that we have established a sound infra-structure for pythagoras, we can address other means of support, that is, to support our authors in the writing process.our editorial team and reviewers are already playing an important role in helping authors to improve their manuscripts to get it publishable. we have now started a project in terms of supporting authors and reviewers through writing workshops at amesa congresses, as well as helping authors and reviewers with different aspects of the writing and reviewing process. i believe that through its support for authors and reviewers, pythagoras can play a role in the professional development of our mathematics education community, and i look forward to developing this role in the next few years. references (back to top) harnad, s., brody, t., vallieres, f., carr, l., hitchcock, s., gingras, y., et al. (2008). the access/impact problem and the green and gold roads to open access: an update. serials review, 34(1), 36−40. http://dx.doi.org/10.1016/j.serrev.2007.12.005 olivier, a.i. (2008). pythagoras is going places … pythagoras, 68, 2. breen 2 pythagoras 60, december, 2004, pp. 2-12 perturbating the assessment of individuals and groups: listening for challenges to mathematics teacher educators chris breen school of education, university of cape town, south africa email: cb@humanities.uct.ac.za in this article i begin by focusing on different ways in which the term assessment can be understood and practised. having done this, i turn my gaze onto one particular teacher education situation and explore student teacher assessment as they are prepared for a career in teaching. in describing some of the particular ways in which i try to heighten the awareness of this particular group of student teachers regarding assessment and evaluation, i reflect on the experience and pose questions for teacher educators in general to consider about their own practice. introducing assessment assessment is regarded by many in south africa as the backbone of outcomes based education and considerable time, effort and money is being spent on ensuring that educators are properly trained in assessment procedures (ieb asset, 2003). in the following sections i take a brief look at some different aspects of assessment in an attempt to broaden the field of vision. having done this, i will use the rest of the article to focus on one particular class of university students as a means of obtaining data that will allow a further exploration of some of the ideas and issues which have been raised in the introduction. assessment as evaluation the dominant form of assessment is as evaluation where schools and teachers are evaluated on the basis of marks obtained by individuals. one of the main purposes for this form of assessment is as a means of satisfying others that the promised skills have been delivered to the satisfaction of national standards. the assessment is usually framed by checklists, which are based on explicit criteria and a form of quantification. this approach is concerned with tallying, which shares the same etymological heritage of teaching as telling (davis 1996: 230) formative assessment black and wiliam (1998) provide a broader understanding of assessment in that they consider it to refer to all activities undertaken by teachers and by students in assessing themselves, which provide information, to be used as feedback to modify the teaching and learning activities in which they are engaged. such assessment becomes formative assessment when the evidence is actually used to adapt the teaching work to meet the needs of the students. their research survey shows that innovations, which include strengthening the practice of formative assessment, produce significant and often substantial learning gains. in seeking ways in which to improve formative assessment they include the necessity for feedback to any learner being based on the particular qualities of his or her work, as well as advice on what he or she can do to improve, avoiding comparisons with other learners. they state that the dialogue between learners and a teacher should be thoughtful, reflective and focused to evoke and explore understanding, and conducted in such a way that all pupils have an opportunity to think and express their ideas. this approach to assessment has as its main aim the improvement of the teaching-learning interaction, and it cannot be prescribed by outsiders or by pre-set schemas. it is an interactive process for which an extended view of the classroom situation is needed. assessment as sitting beside davis and sumara (1997) invoke learnings from complexity theory to draw a distinction between the terms ‘complicated’ and ‘complex’. they use ‘complicated’ to describe the modernist tendency to use machine-based metaphors to characterise and analyse most phenomena. “complex systems such as human beings or human communities – in contrast to complicated systems – are more dynamic, more unpredictable, and more alive” (davis and sumara, 1997: 117). taking a complex view of life means that the focus is on the interrelationship of things and the manner in which subsystems come together to form larger, more complex systems. the theory of enactivism (see chris breen 3 for example davis, 1996; maturana and varela, 1986; and varela, thompson and rosch, 1991) is concerned more broadly with the construction of a collaborative world. it involves becoming part of an ongoing existing world and the shaping of a new one, and acknowledges the role of the individual in affecting the world’s form, and this pushes enactivism into the realm of the moral. the theory of enactivism looks at each learning situation as a complex system consisting of teacher, learner and context – all of which frame and co-create the learning situation. davis (1996) explores the possible role of assessment within an enactivist position. he begins by noting that the root of the word assessment comes from the latin word assidere (to sit beside). he then goes on to argue that this means that assessment should be better understood as a focus on those teaching actions which are directed towards a fuller understanding of both a learner’s subjectivity and the learners’ collectivity. such an understanding will allow teachers to adapt their teaching approaches appropriately. in this enactive sense, assessment is participatory, and inseparable from instructing. a complicated view of assessment sees errors as symptoms of an underlying disease that can be located, isolated from other understandings, and removed. in this way, what should be understood as complex knowing is instead reduced to partitioned competencies by a complicated interrogation. in contrast, an enactivist position regards errors as important and essential focal points of any mathematical inquiry. errors signal moments where both teacher and learners have an opportunity to bring unformulated concepts to conscious awareness. such errors are not located inside particular individuals since they have been arrived at as a result of the interactions between teacher and learners in a particular context. on the contrary, they exist in the constellation of classroom events, and as such are to be welcomed as an opportunity for all to become engaged in a collaborative task of working on the unformulated concept. they are instances that call for negotiation as they prompt awarenesses of inconsistencies between subjective conceptions and general consensus – thus potentially presenting both for revision. the purpose is not to remediate them but to learn from them – that is to reform collective action every bit as much as to reform subjective action. introducing listening another way of thinking about different concepts of assessment is to focus on three types of listening (levin, 1989) that are required of the person doing the assessment. the most common form of listening found in the classroom is evaluative listening where the listener judges what the other is saying against the template of his/her own certainties. teachers typically judge whether the information which is offered is right or wrong, and, whatever they decide, deviate little from their plans. the familiar cycles of teacher question, learner short response, teacher evaluation that are associated with traditional teaching are typical of evaluative listening and consistent with the assessment as evaluation model. in interpretive listening, the listener tries to hear what the other is saying in order to interpret where they are and how they are feeling. it is ‘a sort of reaching out rather than a taking in’ (davis, 1996: 53). in this mode, the teacher’s focus is on accessing rather than assessing the learners’ ideas, and an emphasis is placed on the development of a skill of questioning which encourages the learner to think and explain their thinking. this form of listening seems to be the type called on in the model of formative assessment outlined in the previous section. in both these forms of listening, there is a split between the teacher and the learner where the teachers direct the learning from their own understandings. the third form of listening is called hermeneutic, and this describes an approach where listeners open themselves to others without holding on to their own assumptions. in this form of listening, both parties enter into a shared project of coming to a joint understanding of each other’s position. davis (1996: 234) comments that in this way, “the practice of assessment refers to testing one’s own hearing, and the word testing shares roots with text and texture”. the teacher becomes an important interactive and co-emergent part of the learning context. in a book based on his research, davis (1996) explores these ideas in more detail, and gives classroom examples of teachers operating in different lessons in each of these three ways of listening. assessment in a mathematics teacher education course having introduced the reader to some aspects of assessment and listening, i want to explore these perturbating the assessment of individuals and groups: listening for challenges to mathematics teacher educators 4 concepts further by focusing on one particular university class that i have taught. the data that follows was drawn from a postgraduate certificate in education (pgce) (secondary) mathematics method class of 16 student teachers. these student teachers came from a variety of backgrounds and their mathematics qualifications ranged from maths 1 (8 students) to maths 3 (5 students). at the time when i started teaching them, they had already completed a series of eight once-a-week three-hour sessions at the start of the year followed by their first period of five weeks of teaching practice in a local school. the data focused on four particular three-hour sessions, which are described in the next section. the four class sessions at the start of my first session, i set them the new task of writing at least an a4 page after each session “describing (an) aspect/s from the session in which you feel you gained an insight into yourself as teacher, learner or mathematician”. the journal entry had to reach me before the next session. the student responses to the four weekly three-hour sessions that i spent with them, which are described in brief below, forms the major part of the data that inform the rest of this article. in the first session, i described some of my work with adults who fear mathematics. i described some of the major shifts i had introduced into my pedagogy as i attempted to focus on cooperative work with an emphasis on listening to the diverse responses and processes employed by learners (see breen, 2001 for a detailed description of these features). i then introduced a mathematical activity, which asked them to build shapes with matches, and then visualise and generalise the emerging patterns, in an attempt to demonstrate some of these pedagogic shifts. at the start of the second session, i split the class randomly into five groups, each containing three members in it (one student was absent). after introducing the three different levels of listening (evaluative, interpretive and hermeneutic as outlined in the previous section), i emphasised the importance of listening as a tool for working with others. the student teachers were given the task to work on a given problem (painted cubes1) as a learning community where the focus would be on the process they developed rather than the solution 1 in the painted cubes problem, students are asked to imagine a large 3x3x3 cube which has been painted in red on the outside and then broken up into its constituent small blocks. how many of these blocks will have 3 sides painted? 2 sides? 1 side? 0 sides? generalise for an nxnxn cube. obtained. as an assignment they were asked “to describe the way in which their community came together and the contributions that the various members made to the experience”, and submit this to me in writing at least two days before the next session. in the third session, i started off by asking each group to talk about their experience of the previous week and we looked for similarities and differences in the group experiences. later on in the session, they were asked to get into pairs and assign one person the role of teacher and the other person the role of learner. a role play situation was set in which the teacher had asked the learner to stay after school to re-visit an incorrect answer that the learner had given to a question in a test. the teacher was given the task of following kierkegaard (1939: 30) by trying to “understand what the learner understands and in the way that he understands it”. the final session from which the data are taken involved the student teachers in writing a content examination. this will be discussed in more detail later. introducing perturbation an aspect of the enactivist position that has appealed to me revolves around the role of the teacher as perturbator. the idea is that learners each have their own construct, which is based on their biological constitution and historical and contextual experience in the world. learners will take up whatever aspects of the lesson that their constructs predispose them to accept at that moment. this is something which lies outside the control of the teacher. this means that the teacher’s role becomes one of maximising the possibility for take-up of key concepts and this can best be done by focusing on perturbating the learning environment. thus we argue that such notions as controlling learners and achieving pre-set outcomes must be set aside in favour of more holistic, all-at-once co-emergent curricula that are as much defined by circumstance, serendipity, and happenstance as they are by predetermined learning objectives. (davis and sumara , 1997: 122) one of my main aims in these pgce sessions was to try to provoke the student teachers into thinking differently about the potential in mathematics classrooms for embracing diversity of thought as a means of working on mathematical concepts. different answers or ways of working were chris breen 5 welcomed and appreciated as an opportunity to engage and work with others to optimise the potential for learning. acknowledging existing marks mathematics teacher educators who work with both primary as well as secondary student teachers often assume that this group of student teachers who have chosen to teach mathematics at secondary school level will have fewer issues remaining from their schooldays than their primary school colleagues, as a result of their superior track record of achievement in mathematics. previous work (breen, 1991) has shown that this is seldom the case, and this group proved to be no exception. several of the student teachers in this particular group were not at all complimentary about their school experiences. tony2 was the first to comment on this topic in his opening journal entry. i am reluctant to become a maths teacher for a number of reasons. whenever i think of maths being taught at schools, the only picture i have is a teacher drilling the learners to solve a problem in a particular way. most often maths classrooms i have been in at school were boring and the teacher doing the talking. the only thing keeping the learners awake is the fear of failing a test or exam. the only thing i remember about my maths teacher is her sarcastic comments. i think my greatest fear is that i would become like her.3 my work with those who struggle with mathematics (breen, 2001) has led me to try to assert (and practise!) a pedagogy that tries to break the normal power relationships. kathy soon conveyed her appreciation that my different approach had started to break patterns, which she felt had already been established within the group during the previous lecturer’s sessions in the early part of the year. i didn’t know if i’d be feeling how i had been feeling for every other tuesday before this: frustrated, angry, down and quite unintelligent (a nice way of saying stupid). i think it was more the company in the class than the teacher. there are a few people that act like the maths experts, better than the rest of us. they were always praised and loved it and we 2 names have been changed in all cases except where specifically requested to do otherwise by those concerned. 3 all extracts in italics are taken with permission from the journals of the student teachers. were always silent. so that is why i am still very quiet in class. and please don’t call on me to try and make me participate, i choke under pressure. cotton (2002) draws on lave and wenger (1991) and wenger (1998) to examine schools and classrooms as communities of practice. in this way, students in mathematics classrooms engage with each other in practice and develop a sense of self in relation to that community of practice. for some students there is a greater synergy and sense of belonging as they fit in with the group and the teacher’s expectations of the class, whereas for others, there is a sense of rejection and little sense of identity within the communities of practice. for those students for whom there is little sense of belonging and a lack of sense of identity, there is greater danger of exclusion from that community of practice. “they set themselves up in an alternative community of practice, which consists of a group of failure – those for whom mathematics is seen as difficult, complex and the learning of it unattainable – recognizable by ‘i was never any good at maths at school’.” (cotton, 2002: 1124). kathy’s comments strongly echo this description as she has clearly positioned herself as one of the ‘stupid’ ones. ross, on the other hand, later comments on how it was the actions of the teacher educator, which had positioned him in the opposing group of achievers. a couple of people verbalised that they were intimidated by me. this is definitely not due to my personality but probably because i probably got all the work this year correct and did it faster than the class. i definitely was never trying to impress the class but the teacher, mr. y, would always make me explain the work on the board because he noticed we weren’t being constructive when we had finished the work. it is interesting to note that ross links intimidation to a personality trait rather than to his achievement in mathematics. he also makes it the teacher’s responsibility for calling on him and does not offer any alternative action that might have been taken. community of practice the aim of the second session was to explore the ways in which the groups of three tackled the problem of setting up a community of practice to tackle the painted cubes investigation. the pgce course as a whole is one in which the benefits of group work are generally praised, but very little work is done in trying to tease out the difficulties perturbating the assessment of individuals and groups: listening for challenges to mathematics teacher educators 6 and stresses in getting individuals to work in groups. in this class, kathy had already commented on her dissatisfaction with the quality of group work in the previous teacher’s sessions. the maths method course was something i dreaded because we would be put in groups and those that knew would tell the rest of us. of the five groups, only one worked in a cooperative and collaborative way to solve the problem. michelle teamed up with kathy and tony in what (from earlier references) might have been expected to be a dysfunctional group. to the contrary, it was clear from observing the different groups that their interaction was positive and harmonious. michelle reported: i really enjoyed maths today. i felt that at long last there was no “i did 3 years of maths at university (so therefore i am better than you)” as opposed to us mere mortals who did less. the whole thing really stressed to me how personality compatibility contributes to the success of the group. i know that all three of us feel intimidated in class but between the three of us we all just got on and did the work while having fun. the comments of both tony and kathy support this view. tony wrote: i appreciated the exercise on learning communities. each member of the learning community brought something to the group. in trying to probe them at the start of the third session, it became clear that one important move that they had made was to start slowly and make sure that they all had the same understanding of the problem. kathy extended this by saying: i had such a great time today. it was because of each other that we discovered the patterns of numbers in the cubes. it was because of tony’s method of counting the squares on one side with only 1 side painted that i saw the pattern of (n-2). michelle, tony and i all had respect for each other and maintained that throughout the task. we really tried to create a learning community. telling in the previous section, i reported kathy’s dread of working in groups where the ‘clever’ ones would just tell the others the answer. clarke and lobato (2002) point out that ‘telling’ has had bad press since it has been linked to the form (i.e. whether or not the teacher is making a declarative statement or other type of assertion) rather than to the function of the teacher’s action. they reformulate ‘telling’ in terms of the functions of ‘initiating’ (if the idea originates with the teacher) and ‘eliciting’ (if it originates with the student/s). one of the groups appeared to be working reasonably well when i visited them although i was aware that one of the student teachers, michael, appeared to be paying more attention to solving the problem than to setting up the community. when prince submitted his report on the group’s interaction, a whole new picture emerged as to what had happened for him. i attempted to visualise it in my mind at first but i struggled with that so i decided that i would build just the one face of the cube and then see if i can work anything out from that. i took the blocks and put them together and just before i finished putting up one face, michael already had the answer. he gave the answer to us… he went on with the other blocks – the blocks that would have only two sides painted, one side painted and no sides painted. he just gave us like the answers to all that. i tried to ignore him because i was still looking at my face trying to figure out things but it was difficult for me to ignore the answers that he gave to us. so i don’t know if that put me under pressure or what. i don’t know if i should call it pressure but basically i just couldn’t figure out things any more. i couldn’t think. i had the face in my hand. i tried to move my fingers across it, trying to visualise things and make a few calculations. i couldn’t, because he had already given us the answer and what he was doing now as i was aware or trying to figure what was going on, he was sort of moving on to a 4 x 4 x4 cube and that put me under even more pressure. so i found myself asking him “how did you figure that out?” can you just explain that to me how you got the answers? and then he went on like this, ok listen prince, on the corners you have 4 and between them we have one blah blah blah and that makes... it didn’t make any sense to me. this extended extract gives a clear and powerful insight into the way in which being told an answer can freeze students’ thinking processes to such an chris breen 7 extent that they might end up colluding with the ‘teller’ in their own mis-education. fortunately, in this particular example, prince is a capable mathematician with a persistent streak who had the support of the third member of the group, joyce, in re-grouping and returning to the problem. ‘telling’ tensions the painted cubes task specifically asked them to work towards becoming a functional community of practice. nevertheless in two of the groups, one of the student teachers was sidelined. in the first of these groups, ntosh seems to have explicitly sidelined herself from the group at an early stage. according to karen, in our group ntosh identified herself as being not as able as arthur and i, and therefore sidelined herself before we had even started the task. arthur and i did not challenge her assumption but assumed the role of tellers. arthur made a similar observation in his report: me and karen seemed to think about the problem in a very similar way. ntosh was nervous and began by confessing to us that she was slow and took long to understand maths. i felt that she immediately restricted and boxed herself as an inferior mathematician. mason (2002) makes a distinction between notnoticing, noticing and marking.4 at the end of a particular shared incident, mason claims that there will be some who will select a specific aspect of the experience for writing down (marking); there will be some who, once reminded of the incident by someone else will be able to confirm that they remember that it did take place (notice); and there will be some who will not be aware of the incident despite the fact that others claim that it took place (not-notice). ntosh does not mark the moment where she raises the issue of her ability directly with the group, but she does write about her struggle with the problem. i did not understand this problem. arthur had a very good way of explaining the concept of how to get to it. but i had some difficulties in understanding this concept. they were trying to explain but i was kind of lost... both karen and arthur had different ways of solving the problem. but i was the only one left out of the group. 4 it is noticeable here that the word mark is used in a different sense from its assessment understanding. with ntosh now marginalised as active solver of the problem and positioned as the one who needs an explanation, the role of the other two members of the group becomes constituted in a different way. as members of the achieving group (‘the haves’), they take on the task of sharing with a member of the non-achieving group (‘the havenot’) – but they are both well aware of the tension that this brings as the following writing shows: i am very much a teller. i found myself spending most of the time explaining my understanding of the solution than listening to how arthur and ntosh had seen it. ... even after discussing hermeneutic listening and telling myself to listen i found my desire to tell was so overwhelming that i totally forgot about the listening part. when i realized that ntosh didn’t understand my explanation i merely repeated it more slowly – incorrectly assuming that the pace of my explanation had confused her and not the explanation itself. i really need to work on my listening skills. (karen) this activity really highlighted my tendency to do all the things chris has been warning us about. in this sense it was rather annoying as i found myself telling a lot and not really getting anywhere in terms of really explaining anything. this exercise highlighted how a problem can be approached in so many different ways. the options are in a sense endless. (arthur) more telling lessons the enactivist position believes that while the teacher can act as a disturbing agent, what is actually learnt is determined by the structure of the learner. the submission of both journals and assignment reports between sessions generally allowed me to come to a better understanding of the thoughts and lessons that had remained with individuals at the end of each session. this allowed me an increased opportunity to think about the appropriateness of content and method for the next session. this submitted material also helped me identify hooks that might be useful for further probing or perturbation. however, despite the fact that lessons can be prepared and plans made in this way, the direction of the session is inevitably influenced by what comes up during the session. the painted cubes community of practice assignment took place on a tuesday, and all writeups were handed in by the following friday. this perturbating the assessment of individuals and groups: listening for challenges to mathematics teacher educators 8 left me with a period of three days to read the reports and then to consider how to work with the insights that i had gained into their work. at the start of the exercise i had told them that their comments about each other would be kept confidential unless they gave me permission to quote them. informed by what they had each written, i decided that i would try to introduce those aspects, which their reports had suggested were significant to them, in a general and nondirected way. having done this i invited individuals within each group to comment on their own group’s process. in the session, issues of telling and exclusion were raised and individuals were generally willing to talk about their experiences with honesty and directness. prince had recorded his reflections on audiotape and both he and michael agreed to let the class listen to his story of how prince’s thought processes had been interrupted by michael’s desire to tell the answer. this session seems to have been a particularly successful sensitiser to the issues involved. the feedback from each student about the previous week’s group activities was absolutely fascinating. the main ideas that i learnt were that learning does not take place when you just attempt to tell or impose your understanding on someone else. this is what karen and i tried to do and it failed dismally. (arthur) joyce had been the third member of michael and prince’s group, and the debriefing session impacted on her as she considered what had happened. i was struck by the negative effect of interrupting a person’s thought process and telling them the answer. for me, this was quite an eye-opening discovery because i think that sometimes i do tend to give clues or answers to someone i am trying to help, thinking that i am actually helping them. in fact, i am just impatiently steering them towards my way of thinking about the problem. i suddenly put myself in the position of the person being interrupted and realised how frustrated i would feel if someone interrupted my thoughts just as i was coming to my own answer. (joyce) listening all three sessions had focused on ways in which the teacher could assist in co-creating a learning environment where each student felt confident enough to engage with the mathematical material being offered both on their own and within a group. the dimensions of the challenge that this would present seemed to have been seriously considered and pondered as shown in the following journal entry by joyce. quite often throughout this course we have been encouraged to listen to the learners and to build on their existing knowledge, rather than simply approaching a subject from our own perspective. however, today highlighted for me that this process goes a step further. it is not enough just to listen; as a teacher you have to literally put yourself in the position of the learner and try to get to grips with their way of understanding something. taking this process one-step further means sharing this way of understanding (whether it is right or wrong) with the rest of the learners so that their own understanding is improved. learning then takes place in a community. it all sounded fairly simple until we actually put these skills into practice today! it is quite difficult sometimes to work in groups in which each person’s way of understanding is very different from the next. i think the exercise also highlighted the fact that for a teacher to really listen takes effort and a lot of patience. sometimes we are so blinkered that we can’t even begin to consider someone else’s perspective. to adopt this approach in a classroom would be a huge challenge, but one that would hopefully have longer-lasting benefits for learning. reviewing assessment this section of the article will briefly re-examine the different types of assessment introduced in the first two sections. formative assessment the pgce class was able to experience a formative assessment approach from the lecturer where the contents and methodology of each session was based on their work and reflections from the previous session. in this way, each session was designed to meet the needs of the student teachers as argued earlier in the article. chris breen 9 sitting beside assessment an exclusive concern with the components of teaching has always been and continues to be inadequate for preparing teachers for the complex situations within which they will be working. we cannot teach everything that must be known for what is known and the circumstances of that knowledge are always shifting, evolving, unfolding. (davis and sumara, 1997: 121) during the teaching sessions described above, no student teacher was praised for getting the correct answer to a problem. instead, in keeping with the enactivist view, the class worked on the errors made by members of the class as a means of generating different understandings of the concept under discussion. in this way, feedback from student teachers was used to plan the next move. the sessions were co-created in the moment according to what the class offered. the teacher’s role was to engage with each student in a hermeneutic space so that he was able to enter into the student’s learning space and ask appropriate questions. the class was encouraged to mirror this example, both in pairs in the activity in the third session (described previously), and in the painted cubes community of practice activity in the second session. joyce’s final comment above gives an indication of the deep insights into this methodology that can be gained from the work done by the class. her insight that working in this way would be a huge challenge is encouraging, and, with this in mind, a full module at masters level has been developed where the enactivist theories espoused above are matched with mason’s discipline of noticing (mason, 2002). the module runs over twelve weekly sessions, builds on the ideas of varela, and interweaves readings on enactivism and practical work on critical incidents using the discipline of noticing, which provides an initial exploration of this topic (see breen, 2000; 2002). the further insights and skills gained by teachers completing this module are promising and provide a way forward for joyce (and other teachers) to continue with their studies in the future. the examination the pgce mathematics course as a whole required that student teachers prove that they have a sufficient command of the content of the schoolleaving syllabus, by obtaining a sub-minimum5 of 50% in a 3-hour examination set at the end of the first semester. the paper covered the work from both the school leaving mathematics papers (algebra, geometry, trigonometry and calculus). those failing to gain this sub-minimum at their first attempt have the opportunity to write a different paper at the end of the year. although the paper was intended to take 2.5 hours, student teachers were allowed to take up to four hours to complete it. they could also eat during the examination and get up and take a break when needed. despite these moves to soften the pressure of a return to assessment as evaluation, the contrasting effects on the different student teachers were clearly evident. the average mark for the class was 62%. five of the class failed to get the required 50%, while eight student teachers scored marks in excess of 80%. bronwen, one of the confident student teachers with a successful track record in mathematics, enjoyed herself (she obtained a mark of 95%). thank you for creating such a comfortable and non-threatening atmosphere for the exam today. it made the whole process quite enjoyable instead of daunting. when you stopped us to reflect on the experience i was actually very content. i was comfortable, working slowly and methodically and pleased with how i was progressing. thank you for a fair paper. bronwen’s only previous disquiet in these sessions had been in the painted cubes group exercise when she had come up against ross (he who had been accused of intimidating others). she felt on that occasion that he had become angry with her when she had stopped following his lead and instead changed direction. most of the class expected ross to finish in the top three in this examination – in fact his mark for this test placed him in the bottom half of the class. i discovered i did not remember as much as i had anticipated. at least i was not alone. someone else discovered the same thing and we were able to pat each other on the shoulder. during the test chris said we should pause and feel what it’s 5 the significance of this being a sub-minimum requirement is that all student teachers have to obtain this 50% pass in order to pass the course. in other words, outstanding marks obtained for other assignments cannot be included as an averaging opportunity to compensate for failing to obtain the required pass mark for the content test. perturbating the assessment of individuals and groups: listening for challenges to mathematics teacher educators 10 like to be writing a maths test again. unfortunately i did not feel anything in particular. the happy or unhappy bit will arrive with the results. the writing is just writing – it is just work. (ross) at the other end of the spectrum, the picture is bleak. tony experienced the whole examination as a return to the reality of his school experiences. he knew he was going to fail so he decided it was not worth spending time in preparation. his positive contributions in class and his role in creating a vibrant learning community with kathy and michelle are forgotten. the test comes – his selffulfilling prediction comes true and he’s back in his fixed position that he is not going to teach maths and that he hates it. he has been marked again. i have a phobia for maths. i am anxiously waiting for the year to come to an end. teaching maths is not in my dreams. (tony) kathy too is devastated by the experience of the examination and is so upset by her results and the way that it damages her that she writes a long reflection of over two pages in her journal after the final session. i know that i don’t want to teach matric6 maths. i like maths and numbers and the exercises we were doing in class… i have never done so poorly on an exam in my whole life. (kathy) the article could end at this point closing the circle and pointing out the way in which summative assessment has the potential to destroy the selfconfidence of individuals. the case for prioritising other forms of assessment for individuals seems to have been made. assessing confessions of a mathematics teacher educator evaluating assessment however, roth (1996) ends his article with a comment about the role of tertiary educators: the sad thing about all of this is that most of us who teach undergraduate and graduate courses use grades and thereby contribute to the system in this way. we contribute to the cultural reproduction of a system that many of us scathe. (p. 819) 6 this refers to the final school leaving matriculation examination. those in their last two years of school (grades 11 and 12) are prepared for this examination and thus, in kathy’s words, are being taught ‘matric maths’. roth has personalised the role that we play in reproducing this system, and in doing this, he challenges me to take personal responsibility for what happens in my classes. this echoes with some earlier writing of mine, in which i posed some probing questions to mathematics teacher educators, “what do we do with our own personal voices of discomfort? to what extent have we engaged in systemic change in our own institutions?” (breen, 1999: 117). it is time to turn the spotlight up a bit and look and listen a bit more closely to the people and their voices, and as i do so, the safe depersonalised style of writing immediately falls away and i am forced to engage with the individuals in my class. my heart becomes heavy as i think back to the way in which my initial sessions raised interest and enthusiasm for tony and karen (amongst others). i created the possibility for them to redefine their relationship with mathematics. they interacted with each other and the rest of the class and began to discover their ability to solve mathematics problems. they became animated in the mathematics class. and then i watched them crumble as i set and administered a test for which they did not have a chance of succeeding. roth’s challenge makes it too easy for me to pass all the responsibility for failure on to the system, or on tony and karen (the familiar ‘blame the victim’ syndrome?), especially if i am serious about our co-creating the learning environment. in the masters module mentioned earlier, i set a written assignment for the group of teachers taking the course in the same year. i gave them a flexible marking matrix and invited them to choose the individual matrix that they wanted me to use in assessing their work. student teachers weighted aspects such as ‘theoretical insights’, ‘personal insights’, ‘use of additional literature’, and ‘creativity’ differently according to how they believed that they had tackled the assignment. the challenge now is for me to explore ways in which i can break my unthinking contribution to the destructive evaluative assessment system currently embedded in this pgce course. my starting point will be to meet with those who have to re-write the examination at the end of the second semester to seek their advice and ideas. sitting beside or running away? acknowledging the role of the individual in affecting the world’s form effectively pushes enactivist thought into the realm of the moral. (davis, 1996: 190) chris breen 11 one of the ways that an enactivist position differs from constructivism is that the enactivist is forced into taking ethical stances because of the view of the interrelatedness of all components of the learning situation (see also varela, 1999). to what extent does this force the teacher to take uncomfortable stances? the issues raised above regarding evaluative assessment are easy to spot (although painful to raise). the victims are easily identified. a more difficult issue to pin down arises out of the ‘community of practice’ exercise. there is agreement in the group that ntosh was excluded (or excluded herself) from the mathematical contributions to the activity. arthur and karen agonised about this and acknowledged their predisposition to tell. in the other group where lyn, simon and fred worked on the problem, there was common agreement that fred had been excluded. fred is an older, quiet student. fred reported that: i was mostly constructing the figures and found out that my contribution in terms of visualisation was always not fully taken into consideration by the group. lyn had commented at the start: i was apprehensive about working with fred and simon as i’ve noticed in previous classes that they are not very interactive in classroom discussions. when the class met after the examination, i asked them to write down the names of those they thought would take the two top positions (bronwen and ross were by far the top choices). i also asked the class to estimate the mark each person thought they had obtained (again bronwen and ross came out on top). fred estimated that he had a mark of 55%, yet was one of the top three with a mark close to 90%! only one member of the class had positioned him in the top three. the class had made a decision on fred’s ability based on his lack of interaction in the class, and somehow fred seems to have internalised a lack of confidence or ability to judge his own performance. my dilemma is that both fred and ntosh have english as their second (or third?) language and their education was disadvantaged by the apartheid regime. in the feedback session, i make the comment that, for a group to maximise the possibilities of effective operation, all members of the group need to take responsibility for inclusion. in particular, i said that this meant that it was incumbent on all parties to both listen and to contribute. arthur picked up on this in his journal: however what was really interesting as well was that the responsibility also lies with the learner/ or the person who does not understand to be proactive and not allow themselves to be left behind or denied the chance to think. have i introduced an escape clause, which encourages the perpetuation of blaming the victim? where does my role as a mathematics teacher educator begin and where does it end? if i set a task and want to sit beside the learners as they work together, am i not morally obliged to tackle the issues as they arise? when can i safely say that i have fulfilled my obligations to the group? at the moment it feels as if i scratched a surface but beat a hasty retreat when i glimpsed what i was uncovering! in conclusion this article has taken as given the existence of assessment as evaluation as a means of judging the delivery of institutions and organisations. instead it has focused more closely on the possibilities of assessing the work of individuals and groups, and, in particular, on the challenges that this poses to the teacher – and the teacher educator! the issue of the teacher’s role as perturbator also presents some serious problems since this cannot be seen as a licence for the teacher to act irresponsibly. how much can one perturbate and what are the consequent responsibilities? how can teachers develop the appropriate skills and sensitivity for this type of role? clearly, with the class described in this article, i was able to obtain regular feedback from classroom discussion and the journal entries, which were submitted in sufficient time to impact on the next lesson. these were a great help. acknowledgements my thanks go to judy mousley, helen forgasz and helen doerr who co-ordinated and organised the residential scholar’s workbench in geelong, australia in july 2003 which provided the supportive environment for the writing of this article. this project was funded through a quality learning research priority initiative of deakin university, australia. references black, p. & wiliam, d., 1998, inside the black box: raising standards through classroom achievement. retrieved june 29, 2003 from url: http://www.kcl.ac.uk/education/publications/bl ack%20box.pdf perturbating the assessment of individuals and groups: listening for challenges to mathematics teacher educators 12 breen, c., 1991, “concerning mr. smith and his (very brief?) reign of terror”, pythagoras 25, pp 31-37 breen, c., 1999, “circling the square: issues and dilemmas concerning teacher transformation”, in jaworski, b., wood, t. & dawson, a. j., eds, mathematics teacher education: critical international perspectives, pp. 113122, london: falmer press breen, c., 2000, “re-searching teaching: changing paradigms to improve practice”, in clements, m.a., tairab, h. & yoong, w. k., eds, science, mathematics and technical education in the 20th and 21st centuries, pp. 94-103, department of science and mathematics education: universiti brunei darassalam breen, c., 2001, “coping with fear of mathematics in a group of pre-service school teachers”, pythagoras 54, pp 42-50 breen, c., 2002, “researching teaching: moving from gut feeling to disciplined conversation”, south african journal of higher education, 16(2), pp 25-31 clarke, d. & lobato, j., 2002, “to ‘tell’ or not to ‘tell’: a reformulation of ‘telling’ and the development of an initiating/eliciting model of teaching”, in malcolm, c. & lubisi, c., eds, proceedings of the 10th annual saarmste conference, pp. 1115-1122, durban, january 22-26 cotton, a., 2002, “joining the club: identity and inclusion in mathematics classrooms and in mathematics education research”, in malcolm, c. & lubisi, c., eds, proceedings of the 10th annual saarmste conference, pp. 11231130, durban, january 22-26 davis, b., 1996, teaching mathematics: towards a sound alternative, new york: garland publishing davis, b., & sumara, d., 1997, “cognition, complexity and teacher education”, harvard educational review 67(1), pp 105-125 independent examinations board assessment education and training, 2003, “become a registered assessor through ieb asset”, brochure distributed to professional development organisations kierkegaard, s., 1939, the point of view, london: oxford university press lave, j. & wenger, e., 1991, situated learning: legitimate peripheral participation, cambridge: cambridge university press levin, d., 1989, the listening self: personal growth, social change and the closure of metaphysics, london: routledge mason, j., 2002, researching your own practice: the discipline of noticing, london: routledgefalmer maturana, h. & varela, f., 1986, the tree of knowledge, new york: shambhala roth, w-m., 1996, “tests, representations, and power”, journal of research in science teaching 33(8), pp 817-819 varela, f.j., 1999, ethical know-how, stanford: stanford university press varela, f. j., thompson, e. & rosch, e., 1991, the embodied mind: cognitive science and human experience, cambridge, ma: the press mit wenger, e., 1998, communities of practice: learning, meaning and identity, new york: cambridge university press “in the middle of every difficulty lies opportunity.” albert einstein acknowledgment.doc 2 acknowledgment to reviewers  the quality of the papers in pythagoras crucially depends on the expertise and commitment of our peer reviewers. reviewing is an important part of scholarly work, making a substantial contribution to the field. reviewers’ comments serve two purposes, guided by two inter-dependent objectives:  pythagoras wishes to publish only original papers of the highest possible quality, making a meaningful contribution to the field. reviewers advise the editor on the scholarly merits of the manuscript to help him evaluate the manuscript and to decide whether or not to publish it. reviewers are encouraged to reject a manuscript if it is scientifically flawed, merely sets out observations with no analysis, provides no new insights, or is of insufficient interest to warrant publication.  pythagoras is committed to support authors in the mathematics education community. reviewers help the author to improve the quality of the manuscript. reviewers are encouraged to write their comments in a constructive and supportive manner and to be sufficiently detailed to enable the author to improve the paper and make the kind of changes that can eventually lead to acceptance. the following summary of outcomes of the reviewing process in 2010 shows that our reviewers do well in achieving both objectives: no. manuscripts processed 26 accept without changes 0 (0,0%) accept with minor changes (to the satisfaction of the editor) 5 (19,2%) reconsider after major revisions (re-submit, then re-review) 7 (26,9%) reject  the paper is not acceptable to be published in pythagoras 14 (53,8%) we would like to sincerely thank the following people who have recently acted as reviewers for pythagoras. we very much appreciate their time, expertise and support of pythagoras amidst pressures of work. benadette ainemani humphrey atebe margot berger piera biccard hennie boshoff lynn bowie aarnout brombacher iben christiansen michael de villiers clement dlamini tim dunne stephan du toit tony essien washiela fish vera frith rajendran govender vasuthavan govender piet human belinda huntley prince jaca mark jacobs shaheeda jaffer zonia jooste cyril julie luckson kaino erna lampen annapaula lombard caroline long sizwe mabizela elmarie meyer johan meyer andile mji henri moolman hanlie murray willy mwakapenda marc north craig pournara cas prinsloo mark schäfer gerrit stols nick taylor lindiwe tshabalala vijay reddy ingrid sapire manare setati jessica sherman marthie van der walt nelis vermeulen renuka vithal john webb dirk wessels helena wessels article information authors: helena wessels1 hercules nieuwoudt2 affiliations: 1research unit for mathematics education, stellenbosch university, south africa 2school of education, north-west university, south africa correspondence to: helena wessels email: hwessels@sun.ac.za postal address: private bag x1, matieland 7602, south africa dates: received: 10 sept. 2010 accepted: 17 feb. 2011 published: 19 july 2011 how to cite this article: wessels, h., & nieuwoudt, h. (2011). teachers’ professional development needs in data handling and probability. pythagoras, 32(1), art. #10, 9 pages. doi:10.4102/pythagoras.v32i1.10 copyright notice: © 2011. the authors. licensee: aosis openjournals. this work is licensed under the creative commons attribution license. issn: 1012-2346 (print) issn: 2223-7895 (online) teachers’ professional development needs in data handling and probability in this original research... open access • abstract • introduction    • classroom practices, teacher knowledge, beliefs and confidence       • profiling teachers • research design    • research approach    • research method       • research participants       • measuring instrument       • statistical analysis       • reliability       • validity       • ethical considerations • results    • section 1: preparation for teaching a unit in data handling and probability    • section 2: teaching practices, including materials or resources used by teachers and learners    • section 3: the topics sample and average in data handling and probability    • section 4: teacher confidence in the teaching of data handling and probability    • section 5: beliefs about statistics in everyday life    • section 6: teacher comments on learner survey items to determine content knowledge and pedagogical content knowledge    • sections 7 and 8: teacher background and professional development • discussion • conclusions and recommendations • acknowledgements    • competing interests    • authors’ contribution • references abstract (back to top) poor trends in international mathematics and science study (timms) results and widespread disappointing mathematics results in south africa necessitate research-based and more efficient professional development for in-service mathematics teachers. this article reports on the profiling of mathematics teachers’ statistical knowledge, beliefs and confidence in order to inform the development of in-service teacher education programmes in statistics for grade 8 and grade 9 teachers. ninety mathematics teachers from schools with culturally diverse learner populations in an urban region in south africa were profiled using an adapted profiling instrument (watson, 2001). although statistics formed part of quite a number of these teachers’ initial teacher education and about half of them were involved in professional development in statistics education, they still teach traditionally, rather than using a more data driven approach. teachers indicated high levels of confidence in teaching most statistics topics but showed low levels of statistical thinking when they had to apply their knowledge of concepts, such as sample and average in social contexts including newspaper articles and research reports. introduction (back to top) in south africa, as in many other countries, many mathematics teachers lack a sound grounding in statistics (north & scheiber, 2008; north & zewotir, 2006; wessels, 2008, 2009). this lack of grounding limits their confidence and competence in teaching data handling and probability and subsequently they are not able to teach their learners to become statistically literate. a series of professional development initiatives in statistics education started in south africa when the sixth international conference on teaching statistics (icots6) was held in the country in 2002. these initiatives were strengthened when the national statistics office (statistics sa) provided financial assistance for professional development in statistics education by launching the maths4stats campaign in 2006 (north & scheiber, 2008). despite this drive to improve statistics education in the country, many high school teachers still did not have any exposure to professional development in statistics by 2008 (wessels, 2009). in many cases professional development in statistics education was focused on grade 10–12 teachers, whilst statistics teaching in the lower grades did not receive enough attention. from discussions with teachers it was clear that because of minimal exposure to statistics in their initial training and later in their career, many of the grade 8 and grade 9 teachers lacked statistics content knowledge and pedagogical content knowledge. this lack of knowledge had an effect on the preparation of learners in grade 8 and grade 9 and consequently on the readiness of learners for the statistics done in grade 10–12. the need for professional development in statistics education, specifically for grade 8 and grade 9 statistics teachers, is therefore all too clear.the design of professional development initiatives in statistics education for teachers should be informed by research (borko, 2004; corcoran, 1995; ingvarson, meiers & beavis, 2005; shaughnessy, 2007; sowder, 2007). corcoran (1995) comments on the fact that in the past not enough attention was given to the planning of professional development and that the urgent need for effective professional development necessitates a focused approach: if today’s teachers are to be adequately prepared to meet the new challenges they are facing, this laissez-faire approach to professional development must come to an end. the needs are too urgent and resources too scarce to simply continue or expand today’s inefficient and ineffectual arrangements. (corcoran, 1995, p. 1) very little research has been documented about the statistical content knowledge and pedagogical content knowledge of south african teachers, their beliefs about statistics, their confidence to teach different statistics concepts and their levels of statistical thinking (wessels, 2009). classroom practices, teacher knowledge, beliefs and confidence ingvarson, beavis, bishop, peck and elsworth (2004) identified a number of crucial factors impacting on teacher effectiveness, including knowledge of the subject, pedagogical content knowledge, the organisation and application of this knowledge, teacher beliefs about teaching and mathematics, knowledge of the development of higher order thinking skills, and to a lesser degree, qualifications. it is interesting to note that experience was not found to have a significant impact on effective teaching but that classroom practices directly impacted on student outcomes. according to ingvarson et al. (2004) these practices: are reciprocally related to the teachers’ knowledge, beliefs and understandings. these in turn are shaped by the school and their educational and professional development experiences, as well as school system factors. (ingvarson et al., 2004, p. 23, 71) the crucial role that teachers’ knowledge as well as beliefs and confidence play in their practice necessitates further discussion of these aspects. teacher knowledge is a multifaceted concept. schulman (1987) identified seven different kinds of teacher knowledge necessary for effective teaching, including content knowledge, pedagogical content knowledge, curriculum knowledge, knowledge of students and their characteristics, knowledge of educational contexts, and knowledge of education ends, purposes and values. ball, thames and phelps (2008) have identified two essential categories of knowledge: subject matter knowledge and pedagogical content knowledge. they divide subject matter knowledge into common content knowledge, specialised content knowledge and knowledge at the mathematical horizon, whilst pedagogical content knowledge includes knowledge of content and students, knowledge of content and teaching, and knowledge of curriculum. burgess (2010) emphasises that these different knowledge variables are closely connected and do not act in isolation from each other. beliefs and confidence are important facets of teacher capacity. teacher beliefs about mathematics, teaching, students’ mathematical thinking, mathematics curricula, technology and gender play an important role in shaping their classroom practices and have to be taken into account when considering teacher practices (ambrose, 2004; philipp, 2007; remillard & bryans, 2004; staub & stern, 2002). graven (2004) argues that the development of confidence in mathematics ‘enables and supports mathematical learning necessary for ongoing professional development within mathematics education’ (p. 181). school enabling conditions, such as school leadership, the professional learning community at the school, the availability of technical resources, and programme coherence also impact on teacher efficacy (ingvarson et al., 2004). profiling teachers in order to plan meaningful professional development in statistics education for teachers, it is necessary to build a picture of their strengths and needs in the teaching of statistics. this picture should include information about their knowledge, beliefs, confidence, education and experience. the assessment of teacher knowledge is a controversial issue, but it is possible and necessary: measuring teacher knowledge, even using standardised modes of assessment, can be done in ways that honor and define the work of teaching, ratify teachers’ expertise, and help to ensure that every child has a qualified teacher. (hill, sleep, lewis & ball, 2007, p. 150) obtaining information to create a comprehensive picture of all facets that contribute to effective teaching requires multiple approaches. written assessments, including open-ended questions, interviews and classroom observation by trained interviewers and observers, discourse analysis, and teacher reflections are some of the approaches that could form part of the generation of a complete picture of teacher knowledge, beliefs and competencies (hill et al., 2007; schulman, 1987; watson 2001). watson (2001) argues that although it is desirable to use all the different approaches of gathering information about teacher knowledge and practice discussed in the literature for the sake of triangulation, it is not feasible to use such extended data gathering processes in most educational systems. in most instances an instrument is needed that can be administered to a large number of teachers in a limited, practical time frame, incorporating as many features of triangulation as possible. research design (back to top) the study adopted a qualitative–quantitative multimethod design using a questionnaire to profile teachers. research approach the project was undertaken to design a professional development sequence for mathematics teachers to improve the effectiveness of their teaching of data handling and probability. the project comprised two parts: firstly, to profile grade 8–12 mathematics teachers to determine their professional development needs in statistics education, and secondly, to use this information to develop a professional development course for grade 8–12 mathematics teachers. this article describes the first part of this research project, namely the profiling of teachers to gather information about their knowledge, practice and beliefs in the field of statistics and statistics education in order to determine their professional development needs. research method the profiling questionnaire was piloted with grade 4–9 teachers after their completion of a series of three in-service training workshops of four and a half hours each on data handling and probability (wessels, 2009). apart from improving the internal validity of the questionnaire, piloting the profiling instrument was important for two reasons: firstly, to evaluate its use in the south african context, and secondly, to assess further professional developmental needs of the group of teachers it was administered to. most of the teachers took more than two hours to complete the questionnaire; for the most part, their answers to the content-related questions were disconcerting. after completing the questionnaire, teachers in a group interview reflected on the questionnaire and expressed their need for further training in statistics. teachers’ comments confirmed the observation that the questionnaire was too long. two whole sections were subsequently eliminated from the questionnaire to shorten it. the preliminary brainstorming activity about factors influencing the teaching of data handling and probability and factors that should be looked for in employing a teacher to teach data handling and probability were removed. the second section in the original questionnaire covered the preparation for teaching a unit in data handling and probability of the teacher’s choice. the questions were similar to questions in section 3 on preparation for teaching the topics of sample and average, and this section was therefore omitted. the researchers were of the opinion that omitting the data that would have been generated by these two sections would not have a significant influence on the teacher profiles. research participants the sample consisted of 90 teachers from 23 diverse socio-economic schools in a large city in the north of the country, all with culturally diverse learner populations. twenty government schools and three independent secondary schools were included. only 90 out of 183 questionnaires that were distributed were returned: 39 questionnaires were received from schools with afrikaans as language of instruction and 51 from schools with english as language of instruction. in only 24 questionnaires all sections were completed. the gender distribution of the returned questionnaires was 22 male participants and 56 female participants, with 12 questionnaires not indicating gender. measuring instrument the teacher profiling instrument developed by watson (2001) has been adapted for use in several studies (fitzallen, 2004; watson, beswick & brown, 2006) and was also adapted for this study to profile grade 8–12 mathematics teachers (wessels & nieuwoudt, 2010). the adapted instrument used in this study consisted of a questionnaire divided into eight sections. the questionnaire included closed and open-ended questions about teachers’ preparation, practices, content and pedagogical content knowledge about specific statistics topics and their application, teacher confidence and beliefs, as well as their training and professional development in statistics education. questions in the eight sections can be cross-referenced to the different knowledge variables of mathematical knowledge for teaching, providing internal triangulation. the eight sections will be discussed under the results section. an instrument making judgements about teacher knowledge must be nonthreatening and encourage authentic teacher reflection on beliefs, knowledge and practices to pass ethical requirements (i. gal, personal communication, september 18, 2008; watson, 2006). shaughnessy (2007) describes watson’s profiling instrument as nonthreatening because it obtains information about teachers’ content knowledge and pedagogical content knowledge of statistics by asking them to suggest appropriate as well as an inappropriate answers which they think learners would give to a number of statistical tasks. the profiling instrument had to be shortened and some items had to be adapted for use in the south african context, for example, the currency in task 1 had to be changed from american dollar to south african rand. the questionnaire was designed to be used as either a survey or as a one hour to two hours semi-structured group or individual interview. due to teachers’ busy schedules, group interviews were generally not feasible. only one group of teachers at a specific school completed the questionnaire in a group interview; all the others responded individually. statistical analysis analysis of the profiling questionnaire data was mainly qualitative but was supported by a limited quantitative analysis. an adapted structure of observed learning outcomes (solo) framework was used to categorise teachers’ understanding of specific statistics topics in section 3 and of questions about the learner items in section 6. the solo model (biggs & collis, 1982, 1991) categorises observed outcomes on four levels of progression. on the prestructural level the individual is not engaging in the task and is focusing on irrelevant aspects. on the unistructural level responses focus on the problem but use only one relevant element of the task. on the multistructural level, two or more elements are used, although no relationships between them are observed and processing of several disjoint aspects of the elements are usually done in sequence. relational responses are characterised by the integration of multiple elements of the task. this adapted framework was used to categorise the developmental progression of increasing complexity that could be found in teacher descriptions (watson, 2006). reliability reliability of the questionnaire was improved after the pilot study by discussing questionnaire sections and questions with participants of the pilot study as well as with colleagues (peer examination). coding reliability for statistical thinking levels was established through double coding by two independent researchers. more than 80% agreement on coding was reached without discussion. the remaining differences were discussed and consensus reached. a complicating factor of analysing observed outcomes in survey questions like those in section 6 is that a teacher’s response might not have been the optimal response. an interview is needed to further probe a response in order to determine whether it was a functional or an optimal response (watson, 2006). validity the validity of the findings of this study is dependent on the accuracy of the self-reported information. although conflicting findings about accuracy are found in the literature, self-reported data are used in much of educational research because it is easier to collect, especially in larger studies (fletcher & barufaldi, 2002; lambert, 2003). internal validity of the questionnaire was improved through pilot study procedures such as administering the questionnaire exactly as it was planned to be administered in the main study; obtaining feedback from pilot study participants on questions; assessing the range of responses on each question; establishing whether replies can be interpreted according to the required information; revising questions and shortening the questionnaire (van teijlingen & hundley, 2001). the challenge in external validity is to describe the results in such a way that other researchers can successfully ‘transfer’ the lessons learnt in this research to other studies (bakker, 2004). transferability of the instrument was already proven through adaptation and use in a number of other studies (fitzallen, 2004; watson, beswick & brown, 2006). results generated from the use of a solo framework for determining levels of statistical thinking are similar to other studies conducted (watson, 2006; wessels, 2006). ethical considerations information about the research and anonymity of participants was shared with the mathematics head of departments of all participating schools and accompanying letters with this information were distributed to all participants together with the questionnaires; participation was voluntary. results (back to top) results are reported by section and relative values (percentages) for all responses on questions are given, not only for the 24 fully completed questionnaires. section 1: preparation for teaching a unit in data handling and probability less confident teachers spent more time on preparation than confident teachers and the teachers who spent little time on preparation, were less inclined to complete the questionnaire and to indicate confidence levels. during preparation of a unit on data handling, 78% of teachers consult someone, preferably a colleague, someone they meet at a course or with the head of department at their school. forty-one per cent of teachers spend one hour to six hours to prepare the overview of a unit plan in data handling and probability, whilst 26% spend more than six hours. section 2: teaching practices, including materials or resources used by teachers and learners two thirds (67%) of the respondents enjoy teaching data handling and probability. teachers regard data representation (24%) and probability (16%) as the topics their learners enjoy most, whilst probability (24%), data representation (15%), and measures of dispersion (15%) are regarded as topics learners struggle with the most. forty per cent of teachers prefer to address these difficulties in a traditional way by re-teaching the material, or by giving more explanations or activities, including remedial work or extra lessons. strategies, such as using different approaches, changing teaching methods, using group work and discussion, and giving projects or examples from real life scenarios, are used by only 18% of the respondents. using a calculator in teaching is common (96%), whilst 27% of teachers use computer software programs and 54% of the teachers who responded use concrete materials, including dice, coins and playing cards. twenty-two per cent of the teachers do not use any data sources, whilst most of those that do, choose to use data from school textbooks and other books. section 3: the topics sample and average in data handling and probability the adapted solo framework was used to determine levels of statistical thinking in teachers’ understanding of the two concepts sample and average (watson, 2006). almost 60% of teacher responses indicate higher levels of statistical understanding (multistructural and relational levels). this result is in striking contrast to teachers’ thinking levels when their understanding of the concept had to be transferred to a social context in a newspaper article (see ‘handguns’ article in section 6). teachers’ confidence levels for teaching the topic of sampling are, however, on average 4.1 on a scale of 5 with 77% of them rating their own confidence levels a four or a five (see section 4). these results are indicative of high confidence levels about theoretical knowledge despite an inability to apply this knowledge in context. teachers’ explanations of the meaning of average was also categorised using the adapted solo framework (watson, 2006). seventy seven per cent of responses were on the multistructural and relational levels of the solo framework, which is consistent with teachers’ high confidence levels on the teaching of the topic (see section 4). teachers seem to be more familiar with the topic of average than with sampling. it is therefore not surprising that teachers also expressed higher levels of confidence in the teaching of average than in the teaching of sampling (see figure 1). figure 1: teacher confidence levels. section 4: teacher confidence in the teaching of data handling and probability confidence levels of 1 and 2 on a scale of 5 were regarded as low confidence levels whilst levels of 4 and 5 were grouped together as high confidence levels (figure 1). responses indicating a confidence level of 3 on the scale of 5 as well as ‘no responses’ were not included in figure 1. areas where more teachers indicated somewhat lower levels of confidence are sampling and probability topics (figure 1). a clear trend that emerged was that teachers who attend professional development workshops or courses feel more confident than those who do not: 81% of teachers indicating high confidence levels did participate in professional development. the very high levels of confidence in the teaching of data handling topics are not consistent with the levels of statistical thinking identified in teachers’ understanding of applications of the concepts in social contexts (also see section 3 and 6). section 5: beliefs about statistics in everyday life ten statements were made that ranged from beliefs in their own abilities to understand and interpret statistical terms and representations, risks, opinion polls, probability of winning a lottery, to the importance of statistics in everyday life. examples of belief statements are ‘when buying a new car, it’s better to ask a few friends about the problems with their cars than to read a car satisfaction survey in a consumer magazine’ and ‘you need to know something about statistics to be an intelligent consumer’. south african teachers in general took an extreme position by strongly agreeing with almost all statements. the only issue where most teachers did not indicate strong agreement is the statement that weather reports are wrong so often that it cannot be taken seriously whilst a neutral viewpoint was expressed by 35% of the teachers differences in gender and mother tongue of teachers did not play a significant role in their beliefs about statistics and the role of statistics in everyday life. section 6: teacher comments on learner survey items to determine content knowledge and pedagogical content knowledge teachers were given five learner survey items and asked to give an example of both an appropriate and inappropriate response that learners could have given to each task. the first and third tasks included the application of sampling concepts; the second included graph interpretation, the fourth was about graphing and informal inferential reasoning whilst the last task comprised probability concepts. responses to this section were disappointing. many teachers did not respond to the questions at all; others only answered some of the questions. virtually no feedback from teachers indicated appropriate as well as inappropriate responses to any of the learner items. teacher responses were analysed with an adapted solo framework (watson, 2006). results of items number 2– number 5 will be given. because both item number 1 and number 3 cover the application of sampling concepts in social contexts, only item number 3 will be discussed.in survey item number 2, learners were given a pie chart on grocery market shares and asked to explain what the chart was about and if they noticed anything unusual about it (see figure 2). teachers’ responses on the first question requiring an appreciation of the context of the message in the chart show different ranges of structural complexity. prestructural responses resulted where teachers did not engage with the item, for example, ‘who is it?’ and ‘not relevant for teenagers’. fourteen per cent of responses only referred to technical aspects (unistructural), for example, ‘visually good ... students can see which is most/least’. however, many struggled with linking the percentage with its visual representation in the graph. comments indicating greater appreciation of the context in varying degrees of detail (multistructural level) emerged in 38% of teacher responses with answers, such as ‘is a visual picture of agents that sell groceries’ and ‘25.8% of the nationwide market is owned by woolworths’. remarks giving relative information about the different groups represented in the graph (relational), included statements, such as ‘it shows the relationship between the shares of different shops in the country’ and ‘it gives us information about the distribution of companies’ shares’ and represented 16% of the responses. for the second question, namely ‘is there anything unusual about the graph?’, 12% of teachers gave answers that did not focus on the statistical nature of the message but on other details, for example, ‘colour/dark or light may influence the response and order may influence perceptions’. responses such as ‘they know ... woolworths’ share prices have decreased in comparison with spar’ focused on a statistical aspect in the message but missed the point. more than half of the teachers (57%) responded on a higher level, noticing that the visual appearance of the chart is inappropriate and that the percentages did not add up to 100%. learner item number 3 cited a media report about a survey on handguns in schools and involved the interpretation of the concept of sampling: handguns task: about six in 10 united states high school students say they could get a handgun if they wanted one, a third of them within the hour, a survey shows. the poll of 2 508 junior and senior high school students in chicago also found 15 percent had actually carried a handgun within the past 30 days, with 4 percent taking one to school. would you make any criticisms of the claims in this article? (watson, 2001, p. 333) more than 50% of the teachers responded on an emotional level to the handguns task (item number 3) giving statistically inappropriate answers (prestructural level) when asked to criticise the claims in the report (see table 1). an example of a statistically inappropriate answer is ‘weapons must be banned; schools will be extremely dangerous with this number of weapons available’. high crime levels in south africa and the exposure of citizens to newspaper reports about violence in schools might have contributed to these emotional responses. only 21% of responses were on a statistically appropriate level (relational level). an example of a statistically appropriate response is: ‘chicago is only one area in the us and does not represent the whole us.’ learner item number 4 indicates a newspaper report stating claims about the detrimental effect of family cars on health of citizens: family car is killing us, says researcher: twenty years of research has convinced mr robinson that motoring is a health hazard. mr robinson has graphs which show quite dramatically an almost perfect relationship between the increase in heart deaths and the increase in the use of motor vehicles. similar relationships are shown to exist between lung cancer, leukemia, stroke and diabetes. (a) draw and label a sketch of what one of mr robinson’s graphs might look like. (b) what questions would you ask mr robinson about his research? (watson, 2001, p. 334) graphs presented for the first part of the task included bar, line and pie graphs. sixty-one per cent of the teachers represented mr robinson’s ideas with appropriate graphs, showing a relationship between time and at least one other variable. more than half of the teachers asked statistically inappropriate questions about mr robinson’s claims (see table 2). examples of such responses on the prestructural and unistructural levels are: ‘what must be done to reduce that high risks? what material can be used to avoid the accidents on the road?’ and ‘what about hiv aids?’ statistically more appropriate answers include: ‘how did he conduct the research? what is the logical connection between the different illnesses and car use?’ and ‘is the sample representative? did he investigate other causes of the illnesses?’ in these multistructural and relational responses scepticism of the claims emerge and it is suggested that different variables could be connected with the described health hazards. the context of learner item number 5 entailed independent coin tosses. a media report about coin tosses at the start of international cricket matches stated that a certain captain had lost 8 out of 9 tosses in his previous 9 matches as captain. issues included questions about the chance of one side of the coin coming up 4 out of 4 times, the chance of one side of the coin coming up after coming up 4 times in a row, et cetera. only two teachers responded to the question ‘suppose border decides to choose heads from now on. for the next 4 tosses of the coin, what is the chance of the coin coming up tails (and him losing the tosses) 4 times out of 4?’ neither one of these responses considered the multiplicative nature of independent events and therefore gave incorrect answers. answers to other questions in this item were equally few and disappointing; some teachers wrote ‘i don’t know’ or made a question mark as answer to some of the questions. the fact that only one of the 90 teachers responded to all the questions on learner item number 5 and that so few responded correctly or at all suggests a lack of knowledge and experience with probability concepts and is also consistent with the lower confidence levels indicated for the teaching of probability topics. figure 2: survey item number 2 – nationwide retail grocery market shares. table 1: teachers’ statistical thinking levels for item number 3 (handguns task). table 2: teachers statistical thinking levels for item number 4 (family car task). sections 7 and 8: teacher background and professional development gender distribution of respondents is 61% female, 24% male with 15% not indicating gender. teaching experience of participants range from a few months to 30 years with 55% teaching less than 14 years and 25% of teachers falling in the 5−9 years experience interval (figure 3). forty-eight per cent of grade 10–12 teachers had statistics training of 6 months or more during their initial teacher education, whilst only 33% of grade 8 and grade 9 teachers received such training. no trend regarding teaching experience and completion of the questionnaire could be found. as in the case of teaching experience, professional development experiences of teachers ranged widely from no professional development experience to a postgraduate diploma in statistics education. fifty-seven per cent of respondents did not participate in any professional development experiences. of those who participated in professional development, 52% participated in organised initiatives with colleagues whilst 38% did so on own initiative, mostly enrolling for a distant tuition course at a university. forty-nine per cent of the teachers indicated that they prefer the professional development training to be presented by an outside expert as opposed to a colleague (17%) or regional curriculum facilitator (14%). feedback to open-ended questions about professional development included: ‘development (professional) should be an ongoing process’; ‘teacher mentorship programmes are needed’; and ’professional development must be practical and practice orientated’. these teacher comments concur with garet et al.’s (2001) claim that ‘professional development is likely to be of higher quality if it is both sustained over time and involves a substantial number of hours’ (p. 933). one comment refers to teachers’ own responsibility to add to their knowledge: ‘reading up is essential’. a number of teachers emphasise that professional development courses should be presented by specialists – specifically specialists with classroom experience in teaching statistics. figure 3: teaching experience of responding teachers. discussion (back to top) reasons for including statistics in school curricula have been repeatedly emphasised over the past years (franklin et al., 2005; shaughnessy, 2007; wild & pfannkuch, 1999), specifically the usefulness of statistics and probability for daily life, its instrumental role in other disciplines, the need for a basic knowledge of statistics and probability in many professions, and the important role of statistics in developing critical reasoning. in the south african mathematics curriculum, the learning outcome ‘data handling’ (statistics) is part of the mathematics curriculum and states that learners should be able ‘to collect, summarise, display and critically analyse data in order to draw conclusions and make predictions, and to interpret and determine chance variation’. the kinds of skills learners need to develop in statistics are not explicitly stated in the curriculum document, but are evident in the verbal terms of ‘drawing conclusions and predicting’ (department of education, 2002, p. 88). these terms indicate inference skills, which is the essence of statistics (cobb, 2007; i. gal, personal communication, september 18, 2008; moore, 1990; paparistodemou & meletiou-mavrotheris, 2008). teachers need to move beyond the point where they are teaching only basic statistical knowledge, for example, graphs, mean, median, and mode, into the area of informal inference, teaching learners to ‘look beyond the data’ (franklin et al., 2005; makar & rubin, 2009). to be able to do this, teachers need to integrate and reason with multiple constructs (such as sampling, distribution, measures of shape, centre, and spread) when comparing different samples of data, estimating and drawing graphs of populations based on samples, and reasoning about the likelihood of competing statements being true (pfannkuch, 2006; zieffler, garfield, delmas & reading, 2008). teachers have to understand and feel comfortable with these constructs at least at an intuitive level to be able to use them as part of their reasoning and to develop learners’ statistical reasoning skills. results of this study clearly show that mathematics teachers in many cases lack specialised content knowledge of statistics or do not understand different constructs in statistics sufficiently to apply their knowledge and reason with these constructs in social contexts, such as newspaper articles and research reports.positive changes in teacher knowledge, classroom practice, beliefs and attitudes, and consequent changes in student learning outcomes would require ongoing professional development to empower teachers through contact sessions, classroom and online support, encouragement of reflection and engagement in professional communities of practice (burgess, 2010; shaughnessy, 2007; wassong & biehler, 2010). the length of the questionnaire contributed to the fact that so few of the teachers completed all sections. several teachers felt that it took too long and that they had to put in too much effort to answer some of the questions (e.g. learner items in section 6). one school returned all questionnaires without completing any questions. the mathematics head teacher explained that they did not have the confidence to complete it because of their lack of knowledge about data handling and probability. this comment is in line with comments of other teachers that they did not have enough content knowledge to answer the questions. seventy-three per cent of the teachers did not complete section 6, but 80% of teachers not completing this section did complete section 7 and/or section 8, suggesting that teachers found the questions either too difficult to interpret or answer, or too time consuming and skipped it altogether. with the exception of one school where teachers completed the questionnaire in a group, teachers completed the questionnaire in their own time and may not have felt obliged to complete all sections as may have happened when completing it in a group interview. analysis of the data was limited by the fact that many teachers did not complete the last few sections which included questions on teacher background, such as experience, initial teacher training and participation in professional development initiatives. the fact that only 24 teachers completed all sections of the questionnaire limited quantitative analysis of the profiling instrument. it is, however, useful to consider some trends in the responses of this group of 24 teachers. twenty-one teachers in this group indicated that they enjoy teaching data handling and probability; two said that they sometimes enjoy it whilst another one said that she enjoys it but sometimes finds it boring to teach. only 3 of the 24 teachers who completed all sections did not undergo statistics training during their teacher education, with 7 teachers having spent a semester studying statistics and 10 of them having undergone training in statistics for a year or more. the implication is that these teachers felt confident enough to complete all questions and might be more positive about teaching statistics, therefore putting more effort into the completion of the questionnaire. teaching experience (i.e. 2–29 years), and gender (10 male and 12 female respondents) did not play any significant role in this group. as reported in other studies, teacher responses on the different learner items show the need for rich class discussions and the necessity to link statistical concepts to social contexts and media examples (shaugnessy, 2007; watson, 2006; zawojewski & shaugnessy, 2000). the lack and inaccuracy of responses to the questions about probability in section 6 is disconcerting especially when taking into account the lack of emphasis in the mathematics curriculum on the topic of probability. probability is examined in an optional third paper in the senior certificate external grade 12 assessment (department of education, 2008, p. 12). anecdotal evidence suggests that probability is not taught in many schools during school hours and that some grade 8 and grade 9 teachers therefore regard probability, although part of the curriculum in these grades, as not important and do not teach it, pleading a full curriculum and not having enough time to cover all aspects. it is, however, crucial to introduce chance and probability concepts to students even much earlier than grade 8 and grade 9 (watson, 2006). it is imperative that professional development of teachers should include probability activities. conclusions and recommendations (back to top) the profiling instrument in this study provides a broad but incomplete picture of mathematics teachers’ statistical knowledge, practices and beliefs, supplying information about the professional development needs of the teachers profiled. this profiling instrument can be useful for institutions researching professional development needs of large populations of teachers, but can be equally useful to profile individual teachers or smaller groups of teachers. because the reasons for specific answers or the lack of answers to questions are not always clear from the responses in the questionnaire, semi-structured ethnographic teacher interviews, portfolios and classroom observations can be used to provide richer data and to add to the picture compiled by the profiling instrument. more focused studies with larger numbers of teachers are needed to enable a more thorough quantitative analysis to tease out relationships between and significance of different topics in the questionnaire. developers of in-service teacher education programmes in statistics, whether part of mathematics education or on its own, should take cognisance of the usefulness of a profiling instrument supplemented by data from interviews, portfolios and classroom observations for informing the design of such programmes. measuring and interpreting teacher knowledge and beliefs is not a clear-cut venture. explicit criteria to guide measure development are needed (hill, ball & schilling, 2008). the fact that teacher knowledge, beliefs and attitudes are interwoven implies that programmes developed for professional development must ‘ensure that all categories are targeted in a cohesive and connected way, in order to ensure that teacher knowledge develops effectively’ (burgess, 2010, p. 5). acknowledgements (back to top) the financial assistance of the national research foundation (nrf #sfp2007012300001) and the north-west university is hereby acknowledged. any opinions, findings and conclusions or recommendations are those of the authors and do not necessarily reflect the views of the supporting organisations. competing interests the authors declare that they have no financial or personal relationship(s) which may have inappropriately influenced them in writing this article. authors’ contribution h.w. conducted all the research and wrote the manuscript, whilst h.n. was project leader and made conceptual contributions. references (back to top) ambrose, r. 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(1999). statistical thinking in empirical enquiry. international statistical review, 67(3), 223−265. doi:10.2307/1403699, doi:10.1111/j.1751-5823.1999.tb00442.x, doi:10.2307/1403705 zieffler, a., garfield, j., delmas, r., & reading, c. (2008). a framework to support research on informal inferential reasoning. statistics education research journal, 7(2), 40−58. abstract introduction literature review theoretical framework methodology analysis and discussion of data conclusion acknowledgements references about the author(s) conilius j. chagwiza department of science and mathematics, faculty of science education, bindura university, bindura, zimbabwe aneshkumar maharaj department of mathematics, faculty of education, university of kwazulu-natal, durban, south africa deonarain brijlall department of mathematics, faculty of education, durban university of technology, durban, south africa citation chagwiza, c.j., maharaj, a., & brijlall, d. (2020). university students’ mental construction when learning the convergence of a series concept. pythagoras, 41(1), a567. https://doi.org/10.4102/pythagoras.v41i1.567 research project registration: project number: 567 original research university students’ mental construction when learning the convergence of a series concept conilius j. chagwiza, aneshkumar maharaj, deonarain brijlall received: 10 july 2020; accepted: 15 oct. 2020; published: 15 dec. 2020 copyright: © 2020. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract this article uses action-process-object-schema theory to study the mental constructions about the limit of series during a calculus 1 course at a university. the researchers also used the theory’s teaching methodology to teach the topic. a plethora of research on the limit concept is available and suggests that the concept is on record as being difficult for students to learn and comprehend. the study proposed a genetic decomposition on how undergraduate students might demonstrate their mental constructions in learning the limit of a series. students were taken through the activities-classroom discussions-exercises cycle. thirty students participated in answering questions based on the convergence of a series. the students’ written responses together with the interviews were analysed and based on the findings a revision of the preliminary genetic decomposition was done. we found that there were students who did not display the predicted mental construction indicated by the preliminary decomposition in the application of the definition for the convergence of a series, but displayed the predicted mental construction for the application of the series convergence tests. we also found that certain schema were necessary for the achievement of a complete understanding of the convergence of a series concept. the mental constructions within the missing schema were included in the modified genetic decomposition. this empirically enriched model is now expected to inform pedagogy on the convergence of a series concept. keywords: action; process; object; schema; limit; series. introduction many mathematical concepts in calculus and other courses depend heavily on the limit concept, like the definite integral as the limit of riemann sums, taylor series and the differential in multivariate calculus. convergent partial sums of a sequence may be used to define the limit of an infinite series. the limit of an infinite series can be defined as the limit (as n → ∞) of the sequence of partial sums. infinite series development was motivated by the approximation of unknown areas and for the approximation of the value of π (hartman, 2008). in about 1350, suiseth indicated that (stillwell, 1989). madhava (1340–1425) used to estimate π and for π (joseph, 2000). our experiences suggest that infinite series is one of those topics that many students do not appreciate. they are unable to connect infinite series application outside their calculus class which makes them believe that the infinite series application is not useful in their lives. however, infinite series play a vital part in the field of ordinary differential equations and in the field of partial differential equations. infinite series is an important aspect of calculus, riemann sums, and sequences and series. the infinite sums and series can be ‘well’ or ‘poorly’ behaved. for example, if one is required to find the sum of 1 + 2 + 3 + 4 + 5 + …, which is a ‘well’-behaved sum, one can add the numbers and get an ever-increasing sum. the sum of 1 − 1 + 1 − 1 + 1 − 1 + … represents an example of a ‘poorly’ behaved sum, since for two consecutive terms, the sum is zero. furthermore, technology can be used to discover convergence of series and their generation. technology can also be used to create symbolic, numerical and graphical representations, and to change among these different illustration methods. the purpose of this study was to explore the understanding of the limit of series using action-process-object-schema (apos) theory, for 30 students who had registered for a calculus course. this study aimed to add to the body of research on limits of series. furthermore, it suggested an alternative method for the learning of limits in relation to series in universities by proposing a genetic decomposition (gd) for the concept of a limit of an infinite series. we also use this proposed gd to explore the mental construction displayed by undergraduate students when dealing with the concept of a limit of an infinite series. the purpose of such an analysis was to observe whether our proposed gd was an accurate predictor of undergraduate students’ mental constructions and, if not, then what modifications were required in the proposed gd. in order to achieve this purpose, we formulated the following research questions: what mental constructions do undergraduate mathematics students reveal when solving problems involving the convergence of a series concept? how can these identified mental constructions of participating undergraduate mathematics students be used to refine the proposed gd? it was hoped that the answers to these two research questions would provide a model that will inform mathematics educators on the various mental constructions students possess when dealing with the concept of a limit of an infinite series. also, mathematics educators could use the modified gd, which was empirically developed in this study, to design questions for instruction, consolidation and assessment. this contribution to the field of the learning of undergraduate mathematics is one of the strong traits of apos theory. similar studies effectively formulated the modified gds: brijlall & ndlazi (2017) explored preservice teachers’ mental constructions of the concepts of injections, surjections and bijections and in another study (brijlall & ndlazi, 2019) presented a modified gd for integration techniques. literature review researchers have noted that there is a lack of research on the learning and teaching of infinite series (earls, 2017; earls & demeke, 2016). it is known from the researchers’ experiences that a source of cognitive difficulties for students is the limit concept (denbel, 2014). many students struggle with limits (cappetta & zollman, 2013; mccombs, 2014; patel, mccombs & zollman, 2014). those researchers identified specific student difficulties with limits. cappetta and zollman (2013) found that students have difficulties with understanding limits, involving the infinite processes of limits and the value of limits. additionally, metaphorical reasoning, a way of understanding a situation to resolve disequilibrium that students encounter when faced with a new problem situation (patel et al., 2014), develops without the knowledge of the lecturer and impedes students’ understanding of mathematical concepts. a student who understands a mathematical concept can move between numerical, algebraic, graphical and application representations. however, most students compartmentalise their thinking resulting in them staying with procedures even if such procedures result in illogical or contradicting results (mccombs, 2014). gulcer (2012) argues that even if lecturers move smoothly between limit as a number (end state) and as a process, students seem to focus only on limits as a process. further, strong students can view limits as both a value (a static end state) and as a dynamic process (never-ending). thus, the documented literature testifies to the lack of research on the limit concept in relation to series. misconceptions could be deeply ingrained in the mental map of an individual. some students hold the misconception that the limit of the sequence of partial sums and the sum of an infinite series are not the same. some students tried to determine the sum of the series by first adding all the terms (or as many as one could to determine a pattern) and then took the limit of the partial sums (martínez-planell, gonzalez, dicristina, & acevedo, 2012). martínez-planell et al. (2012) noticed in general that students relied on properties regarding finite sums, rather than looking at the limit of the sequence of partial sums or limit of the nth partial sum. furthermore, nardi and iannone (2001) discovered that students had difficulty accepting that the convergence tests can be inconclusive. earls (2017) carried out a study on students’ misconceptions on sequences and series, for second semester calculus. the main findings and some implications from that study were: when using the ratio test, some students got a value of 1, and concluded that the series converged to 1. such students do not know that the meaning of a value of 1 in the ratio test is that the test is inconclusive with respect to possible convergence of the series. when determining which series convergence test was to be used for a given problem, some students faced difficulties in selecting the appropriate test. the difficulties of such students include the identifying of an infinite geometric series, which results in them using a root test to determine the convergence of a given infinite (geometric) series. some students had trouble identifying the contrapositive of the nth term test and the logical equivalence of a statement. the implication here is that students require a greater exposure to these in the context of focused problems. distinguishing between the limit of a sequence and the sum of a series was also a challenge for some students. in particular students had difficulty with regard to the difference between a sequence of numbers and a sequence of partial sums. some students had difficulties with the limit and series notations, and their interpretations. when applying tests for convergence some students failed to check that the assumptions were satisfied. in particular this was noted when applying the integral and direct comparison tests. students had misconceptions about what could be concluded from the results of the convergence tests. there were several examples of students who thought that a series test would give the sum of the series, rather than just whether the series converged. each of the above findings has teaching and learning implications. these were taken into consideration by the researchers of the current study when they designed the questions and problems for the participating students. the study reported on in this article contributes to the limited literature on research related to the limit concept in the context of series. it also provides a proposed gd and modified gd, based on apos theory, which mathematics educators could use to design questions for instruction, consolidation and assessment. further, it gives an insight into the types of mental constructions that need to be focused on during the teaching and learning process. theoretical framework the apos framework describes the hierarchical developmental growth in understanding of mathematics concepts through mental constructions, namely action, process, objects and schema. apos theory was developed from the work of piaget and constructivist ideas (arnon et al., 2014). the theory is centred on the models of what might be taking place in the mind of a university student as they engage with mathematical concepts such as limit of a series. it comprises general descriptions of the mental structures and mental mechanisms (arnon et al., 2014). as actions are repeated and reflected on, students move from relying on external cues to having internal control over them. this is categorised by an ability to imagine carrying out the steps without necessarily having to perform explicitly each one, thus showing the ability to skip steps, as well as reversing them. this mental shift is made possible by the interiorisation mechanism. encapsulation occurs when the individual applies an action to a process; that is, the individual student sees a process as a static structure to which actions can be applied. when a process has been encapsulated into a mental object, it can be de-encapsulated back to its underlying process when the need arises. furthermore, the mechanism of coordination is indispensable in the construction of some objects. dubinsky and mcdonald (2002) define the components of apos theory as follows: actions: an action is transformation of objects perceived by the individual as essentially external and as requiring, either explicating or from memory, step-by-step instructions on how to perform the operation. an example of this is when students are required to determine whether the series is a convergent one. if the individual works s1 by substitution and then s2, s3, etc. then at each partial sum the individual demonstrates they are operating at an action conception. process: this is when an action is repeated and the individual reflects upon it so that they can think of performing the same kind of action, but no longer with the need of external stimuli. suppose the student wants to determine the convergence of a series say, and calculates the nth partial sum and avoids calculating individual partial sums. such a student has interiorised the procedure for finding out whether a series is convergent or not. an object: this is described as what is formed from a process. an individual becomes aware of the process as a totality and realises that transformations can act on it. an illustration of this is when a student can perform an action on the process conception of the monotonic increasing or continuity notions when applying convergence tests. a schema: this is an individual’s collections of actions, processes, objects and other schemas, which are linked by some general principles to form a framework in the individual’s mind that may be brought to bear upon problem situations involving related concepts. in our case we can speak of an infinite series schema. the major tool used in apos-based research is a gd, which is a hypothetical model of mental constructions that a student may need to make in order to learn a mathematical concept (arnon et al., 2014). a gd is a hypothesis that needs to be tested experimentally and is referred to as preliminary genetic decomposition (pgd; arnon et al., 2014). preliminary genetic decomposition for limit of a series the specific mental constructions relating to concepts of limit of series are detailed below. we drew upon the discussion by arnon et al. (2014, p. 51) on examples of what a gd is not, to refine our gd. this we did to avoid the common errors that can confound a sound description of a gd with description of teaching sequence or mathematical description of a concept. the expected apos level for the limit of a series are shown in the pgd in table 1. table 1: preliminary genetic decomposition of the limit of a series. methodology this section focuses on research design, participants, ethical issues, implementation of instruction and tasks. research design in this study, we used the interpretive research paradigm as it recognises that individuals with varied backgrounds and experiences contribute to the ongoing construction of reality in their context (wahyuni, 2012). the interpretive paradigm assumes that for researchers to understand some phenomena they should use the participants’ understanding of the limit of a series concepts. this study focused on gaining a deeper understanding of the university students’ understanding of limit of a series. participants the participants were students from a zimbabwean university who were first introduced to series in high school mathematics. however, in high school, formal definition of a limit of a series is not covered. in this study, all the 30 first-year university students volunteered to participate. there were 4 mathematics major and 26 statistics and financial mathematics major students who took part in this study. all 30 students attempted the tasks and their written responses were analysed and summarised in table 2. since this qualitative study could not present 30 × 7 written responses we decided to choose written responses representing the students’ display of a particular mental construction. we counted the number of responses that displayed the mental construction and those who did not. those who did not were categorised as n. for each of the seven tasks we adopted purposive sampling and chose an attempt we thought demonstrated the particular mental construction (as predicted by the pgd) and one from the n category. in this way we identified eight students who were then interviewed, to clarify our initial judgements. note that some students were interviewed more than once: for task 1a (st15 and st20), for task 1b (i) (st28 and st17), for task 1b (ii) (st20 and st13), for task 1b (iii) (st12 and st28), for task 1c (i) (st28 and st17, for task 1c (ii) (st28 and st17) and for task 1c (iii) (st1 and st14). table 2: complete summary of the categorisation of students’ mental construction, according to apos, on each of the test items on limit of series. ethical issues and criteria for evaluation of qualitative research for the sake of unrecognisability, the students were coded using tags ‘st1’, ‘st2’, up to ‘st30’. the order did not carry any implication. while enabling the organisation of data, the codes ensured that the responses could not be linked in any way in the publication of results to the original participant. to ensure that the data accurately reflected students’ thinking, a number of measures were taken. informed consent forms were given to all participants, and the researcher read and clarified its contents. participants were assured that their responses would firmly be preserved confidentially and that the data so gathered were for the use of the study only. it was communicated to the students that participation was completely voluntary, and that one could withdraw their services at any stage if they wished to do so. the researcher also outlined the nature, purpose and procedure of the study to the participants. further, the researcher clarified the participants’ concerns during the course of the study, whenever they arose. in addition to research ethics we considered the criteria for scientific rigour in so far as qualitative research is concerned. bitsch (2005) provided techniques to ensure credibility. for our study we resorted to persistent observation. through the activities-classroom discussions-exercises (ace) cycle students were first afforded the instruction on the convergence of an infinite series concept, then responded to seven tasks that addressed the pgd and later a selected group of students were interviewed to verify the researchers’ judgement of the mental constructions the students displayed. in this way triangulation was addressed. in order to satisfy persistent observation (bitsch, 2005) we carried out in-depth analysis to gain detail of the mental constructions demonstrated by the students. for transferability the modified gd will contribute to apos theory and pedagogy on the convergence of infinite series. in order to remove subjectivity of the researchers we carried out interviews with students with written responses that depicted the category of mental construction achievement. in this way we sought conformability by the removal of bias and prejudices. implementation of instruction the ace learning cycles were used in this research. the activities were done through the use of maple, a computer software or computer algebra system (cas) which is able to solve the problems on limits of series in numerical and symbolic form. it was used during the learning phase in this study. maple provides users with tools that are very easy to operate and are contained in palettes. palettes are used to simplify writing in worksheets. symbol palettes are used to write mathematical symbols, expression palettes are used to facilitate writing of mathematical expressions such as integrals, sigma series and root forms. the program allows students of different computer skills to work fluently (samkova, 2012). teaching calculus using a variety of computer facilities equipped with maple software facilitated lecturers to deliver material quickly, and students took lessons directly with practice. the learning phase included the use of computers, using maple. tasks the tasks (questions) were designed in line with the pgd. for the action conception we asked questions on recall of definition of convergence of an infinite series and the determination of the convergence of a particular infinite series by calculating individual successive terms of the series. to retrieve understanding on the process conception of the students’ mental constructions we asked students to prove convergence using the definition of convergence of an infinite series. this required students to apply the definition of convergence. an individual will be required to form the sequence of the partial sums. thereafter we asked the students to test specific series by using convergence or divergence tests. we thought that the student firstly would be expected to have a process conception of the features of the terms of the series in order to apply a specific convergence test; the student needed to: (1) make the correct choice of test, and (2) have a deep understanding of the behaviour of the given series, as the series must satisfy the hypotheses of the test for it to be used. by acting on a process conception of a mental construction via the application of a suitable convergence test, the student will display an object conception of the notion of the convergence of an infinite series. the question items given before the analysis of each response covered limits of series, with the first item checking the formal limit definition of an infinite series. this was followed by three items aimed at checking the use of methods to evaluate infinite series. the last three items checked the (series test) methods students employ to determine convergence and divergence of infinite series. analysis and discussion of data the participants exhibited diverse stages of understanding on the written limit questions. the students who gave correct responses to most of the test items were st1, st2, st12, st15 and st28, with st2 opting not to participate in the interviews. students st13, st14, st17, st20 and st21 failed to give correct responses to most of the test items and st21 opted not to participate in the interviews. through the interviews, we developed a deeper understanding of some of the ways of thinking that underpinned students’ responses to the questions. some of the interview questions were shared with all the students and others probed students’ experience with the topic. the rest of the questions were follow-ups to issues that arose in the students’ written solutions or to what they indicated during the interview. data analysis coding the test questions provided us with opportunities to analyse students’ written responses, which gave initial clues about their understanding of limits of sequences and series. these produced qualitative data and we used a coding modified from the work of asiala, cottrill, dubinsky and schingendorf (1997) to evaluate students’ responses as follows: a 1 was awarded for a response showing a step-by-step procedure for the solution (action level). a 2 was awarded for a response showing the performing of transformations mentally and prediction of outcomes (process level). a 3 was awarded for a response that showed encapsulation of the process into a total entity, or an object, ability to perform actions on that object, and displaying the capability to decompose an object to its underlying process when the need arises (object level). if the responses did not meet the postulates of the pgd the attempt was coded as n level with a zero (0) assigned. results emanating from data analysis the designed tasks granted a case analysis of each test question supported by authentic written responses and interview extracts from selected students. this was done to provide confirmation of the apos level at which the students operated, in terms of understanding the convergence of a series concept. the data presentation and analysis were done in line with the gd provided in table 1. task 1a tested students’ attainment of the action level conception. the results revealed that 63% of the students attained the action conception, with 37% failing to attain the action conception. tasks 1b (i) to 1b (iii) tested students’ attainment of the process level and the results revealed at least 63% of the students attained the process level conception. furthermore, responses to tasks 1c (i) to 1c (iii) showed that 67% of students managed to attain the object level conception. the relevant information displayed in table 2 was extracted for tables 3–9 to make it easier for the reader to follow the focused discussion on the different sub-questions. table 3: frequency of students’ responses for task 1a according to apos level. table 4: frequency of students’ responses for task 1b (i) according to apos level. table 5: frequency of students’ responses for task 1b (ii) according to apos level. table 6: frequency of students’ responses for task 1b (iii) according to apos level. table 7: frequency of students’ responses for task 1c (i) according to apos level. table 8: frequency of students’ responses for task 1c (ii) according to apos level. table 9: frequency of students’ responses for task 1c (iii) according to apos level. discussion of data this being a qualitative study allowed us to analyse the 30 students’ written responses and indicate the results in table 2. for an in-depth analysis eight students were interviewed and a discussion on an exemplar for each task was presented. each case discussed highlights the group of students falling in the category of the specific mental construction. in this subsection we provide the task given to students, a tabular summary of the results from the 30 students’ written responses to questions, a written attempt depicting the case of students not displaying the particular mental construction followed by one sample of those students demonstrating an attempt in line with the dictates of the pgd. after a discussion on each written attempt, we provide an interview excerpt followed by a discussion on the interview. for each task we verified whether the findings emanating from the data analysis concurred with the pgd or not. where there was non-alignment with our pgd we kept this in mind for the formulation of the modified gd. task 1a: define the limit of an infinite series responses of students according to apos levels: nineteen (63%) of the students gave the appropriate response to task 1a, thus operated at the action level of the apos theory. the responses to task 1a indicated that 11 (37%) of the students could not display an action level conception according to the apos theory, referred to as n level. of the 11 students, 5 did not attempt the question at all, while 6 gave incorrect and confusing responses. response at n level (incorrect response): the written response of st20 (see figure 1) does not illustrate the definition of a limit of a series. rather he seems to be referring to what a divergent series is. this response could be due to the individual learning mathematical concepts without a clear understanding. to check how he understood these concepts, we interviewed him. r: can you explain the definition of the limit of an infinite series you gave? st20: by the time i wrote the response, i mixed up issues. i thought of the existence of the limit of a function, yet the question required the definition of an infinite series. i cannot provide the definition accurately at the moment. figure 1: written response of st20 to task 1a. an example of a response of a student who operated at n level. this verbal response revealed that st20 operated at n level as he failed to appropriately state the definition of the limit of an infinite series. furthermore, his failure to state the definition from memory is an indication that he has not attained the action level. this question was based on our pgd to initiate the action conception of an individual’s mental construction of the limit of a series concept leading to the understanding of the convergence of a series concept. however, from the response of st20, we learn that we should provide opportunity for students to relate to the differences of convergent series and divergent ones. we kept this in mind when modifying our pgd. responses at an action level conception: st15 (see figure 2) managed to give a correct response to task 1a. her ability to recall the definition correctly is an indication that she operated at the action level of the apos theory. she exhibited a strong tendency to recall verbatim the definition, which is an indicator of the action level. we analysed the student’s written (figure 2) and interview responses and found that she displayed an action conception as this recall was an external manipulation. we realised that st15 used different symbols for the limit of the partial sums and the series itself. during the interview with the student she did indicate that it should have been the same. figure 2: written response of st15 to task 1a. task 1b (i): determination of convergence of series of responses of students according to apos levels: twenty-five (83%) of the students gave responses that showed they operated at the process level of the apos theory. their responses indicated they understood how to evaluate convergent series problems. five (17%) of the students’ responses showed that those students did not attain the process level of the apos theory. four students out of those five substituted n with infinity and concluded that the series diverges. responses at n level (totally incorrect response): the response provided by student st17 displayed that she did not know the definition of the convergence of a series. it could be that this student used the nth term test (the test for divergence). the student also used the = sign incorrectly in the second line. this therefore made us place her mental constructions at the n level conception. one of us interviewed her to check on her understanding of what she did. r: how did you determine that the limit of the series 1b (i) exists? st17: i tried to find the last term. then discovered that the last term goes to infinity. i am not able to determine that at the moment. it requires reading more about summation of series, maybe i have to start from advanced level work on series. it is difficult to find a formula for the general term in the sequence of partial sums. since st17 did not indicate, in her responses, the definition of a convergence of a series, it would seem that she did not have an action conception of the convergence of a series. cases of trying to find the last term of the series and the inability to find the relationship between infinity and limit diverges all pointed to failure by st17 (see figure 3) to attain a process conception of the convergence of a series. figure 3: written response of st17 to task 1b (i). an example of a response where the student operated at n level. response at a process level: figure 4 shows an example of a response of a student who operated at the process level. such students managed to give an expression of the general term for the partial sums and proceeded to show that the series diverges. figure 4: written response of st28 to task 1b (i). in the written response for st28 (see figure 4) we observe that he is applying the definition of the convergence of a series without stating it but uses the definition for this particular nth partial sum. this means that he has interiorised the action (recall of definition) into a process. st28 was able to apply the procedure as a whole without the need to plug in specific values. to further authenticate our assertion that he operated at the process level, an interview was held to investigate strategies used. r: how did you find the response to task 1b (i)? st28: if one adds up numbers that go on and on, then one would never come to a single value. so when i checked on the given question, i saw that was the case so i had to find a formula for the general term in the sequence of partial sums. then i borrowed the idea of finding the general expression of such a series from advanced level work. that is and the limit of the sequence terms is . hence the sequence of partial sums diverges to ∞ and it can be concluded that the given series also diverges. here the student did not work out each partial sum to form the sequence of the partial sums meaning he had interiorised the action of recall of definition of the convergence of an infinite series. hence the analysis of the interview responses confirmed that st28 possessed a process conception of the convergence of the specific series in the question. task 1b (ii): using the definition in the determination of convergence of a series responses of students according to apos levels: twenty (67%) of the students gave the correct responses, which were indicative of the process level of the apos theory. those students managed to make use of partial fractions, and the partial sums of the given series, in their responses to this question. furthermore, 10 (33%) of the students’ responses showed that they operated below the action level of the apos theory. all were confused as to what they should have done to solve the problem. responses at n level (mathematically incorrect response): figure 5 illustrates a response of a student who operated at n level. such students tried to add the resulting terms of the series, without success. figure 5: written response of st13 to task 1b (ii). the response of st13 in figure 5 showed that the convergence of a series concept was not addressed, since the preliminary notions for the limit of a series were absent. the response provided by st13 concurred with earls (2017) who found that students used inappropriate tests to determine the convergence of series. an analysis by the researchers of st13’s written response showed that he operated at not even the action level of the apos theory. a verification of our claim was done through an interview with the student. r: can you give the difference between a sequence and a series? st13: a sequence is a list of numbers, usually in an increasing order; and a series is when these numbers are added up and their sum may or may not be found. r: so how did you determine the convergence of a series given in task 1b (ii)? st13: i substituted the values 2, 3, 4 … into the given series expression then added the few. it was not possible to add up all the numbers to infinity as the numbers do not come to an end. r: that is okay. can you find the general term in the sequence of partial sums of question 1b (ii)? st13: um, i cannot. this topic on series was difficult for me at advanced level and still it is difficult for me. we placed him at the n level (not at a process conception) after taking into consideration the responses he gave to the interview questions. also, from the interview we observed that there was a need to consolidate differences between sequences and series. this was ignored in the pgd, so we then kept that in mind for our modified gd. responses at a process level: st12 managed to give the correct response to task 1b (ii) and an analysis of her response showed that she operated at the process level of the apos theory. st12’s written response showed evidence of omission of some steps in her solution method. she also showed the application of the definition of the convergence of an infinite series which meant that she had interiorised the convergence of an infinite series concept into a process. to verify our claim an interview was conducted. r: you gave the correct response to task 1b (ii), can you take me through your response? st12: i thought of advanced level work where we dealt with partial fraction and then general formula for series summation. so, i found the general formula for the given series. i discovered that this results in telescoping series which resulted in after some terms had canceled out. then taking limit as n → ∞ gives as the other two terms goes to zero. thus, the sequence of partial sums converges, so the series does converge to a value of . the dialogue with st12 verified our process mental construction placement of st12 from the written response. task 1b (iii): determination of convergence of a series by use of definition responses of students according to apos levels: nineteen (63%) of the students gave responses that indicted that they operated at the process level of the apos theory. eleven (37%) of the students’ responses showed that they were operating at the n level. among the 11, one did not attempt the question, while some of the students used sequence evaluation methods. responses at n level (totally incorrect response): figure 6 shows an example of a student who operated at the n level. such a student showed inability to determine the convergence or divergence of an infinite series. figure 6: written response of st20 to task 1b (iii). st20 provided an incorrect response, and the method used to answer task 1b (iii) was inappropriate and did not address the question. this student failed to attain a process conception as it seemed that st20 determined the limit of the nth term of the given infinite series instead of the partial sums. we interviewed him to check on how he understood these concepts and the associated difficulties he faced. r: you responded to task 1b (iii) as though you were answering a sequence problem. why is this the case? st20: i thought since a series is an addition of numbers of a sequence, then by applying the method of evaluating sequences, i would determine the convergence of the series. r: during the learning sessions, were the convergence of a series and those of sequences determined using the same methods? st20: no, we did not use such methods. r: so why did you use such a method? st20: i got confused during the time i was writing the test so i ended up using the wrong method. an analysis of st20’s written response and the interview excerpts gave strong indicators that the student operated at n level. responses at a process level: figure 7 illustrates a response of a student who operated at the process level. figure 7: written response of st28 to task 1b (iii). again, like the written responses of process conception attainment in the previous questions, the demonstration of the series convergence definition was pivotal in highlighting its interiorisation into a process. st28 showed the ability to form and work with telescoping series that led him to successfully come up with the correct response. to verify st28’s understanding of these procedures and confirm whether he operated at the process level, we interviewed him. r: you responded very well to task 1b (iii), can you explain how you arrived at your final response? st28: yaa, i had to think of partial fractions first, that is, expressing the given series as a sum of partial fractions. i wrote out the terms of the general partial sum for this series using partial fraction form. this resulted in a telescoping series, then i took the limit of the sequence of partial sums whose result is one as n → ∞. r: is it always the case that successive terms will cancel out? st28: this is not always the case; some terms cancel with some terms way down the list. in some case instead of successive terms cancelling, one term will cancel with another term that is farther down the generated list. the end result this time is that the initial two and the last two terms are left. r: what are the conditions that a series must satisfy in order for telescoping series method to be applicable to find the value of convergent series? st28: if we can express the series as partial fractions, have the difference of successive terms of the series, and get terms that cancel (some terms must be negative and others positive). if these three conditions are satisfied, then the telescoping series apply. his explanations during the interview verified our placement of his mental construction at a process conception. task 1c (i): using a convergence/divergence test to determine convergence for responses of students according to apos levels: the responses to task1c (i) illustrated that 20 (67%) students’ responses showed that they operated at the object level of the apos theory. ten (33%) students’ responses showed that they could not attain even the action level of the apos theory. two students attempted the question but gave totally confused responses. responses at n level (no response): st13 did not attempt question 1c (i) which is an indication that he operated at the n level. the student could have faced trouble with limit and series notation (brijlall & ndlazi, 2019; earls, 2017). specifically, the student did not appear to think of the limit of a series as a limit of partial sums. we interviewed him to check his understanding of the concepts under discussion and the probable difficulties he faced. r: you did not attempt task 1c (i). what challenges did you face in dealing with this question? st13: there are many methods of solving series limit questions for example, the integral test, ratio test, comparison series test, root test, alternating series test etc. i got confused as to which method was applicable to this particular question. r: do you understand the criteria for determining which method to use and when to use it? st13: exactly this where i have a challenges. i need more time to work on such problems. the student failed to make the necessary mental constructions needed to determine convergence or divergence of a series, because he failed to make the correct choice of the convergence test to be applied. he operated at the n level. responses at an object level conception: the written response in figure 8 supported our ideas in the formulation of the pgd for the object conception of the notion of the convergence of an infinite series. st12 was one of the students whose response showed the ability to apply the integral test to determine whether the series diverges. st12 checked on the assumptions of a series test, and found that they were satisfied before using this test. to further authenticate our claim that she operated at the object level, we carried out an interview with her on her response and strategies to answer the question. r: given , how do you determine whether the series diverge or converge? st12: in this case f(n) can be replaced by f(x) and the result is a non-increasing continuous positive function on a given interval. if x is made large, the denominator gets larger and so the function also decreases. thus, we apply the integral test. r: is it possible for one to determine the value of a convergent series using the integral test? st12: my understanding of this series test is that it only determines whether the series converges or diverges, but does not give the value of the series because we will have replaced a sequence by a function. figure 8: written response of st12 to task 1c (i). st12 demonstrated that she understood the behaviour of the given series to check whether the function is decreasing and continuous nature in the specified domain. we observe that other schemas played a role in this solution as mentioned by st12. reflecting on her verbal response we realised that to solve this task the student also required schemas for the concepts of increasing and decreasing functions and the notion of continuity of functions. those we omitted in our pgd, and they aided us in the modification of our pgd. in hindsight, one shortcoming in our interview question design was that we did not gather information about the domain of integration and the reason why n commenced at 2 rather than 1. task 1c (ii): use of a suitable test to determine the convergence of responses of students according to apos levels: table 8 indicates that 20 students (67%) gave responses that showed they operated at the object level of the apos theory. they managed to coordinate the conditions of the comparison and p-series test and came up with the required solution. ten (33%) students did not operate even at an object conception and so were coded as operating at the n level. responses at n level (totally incorrect response): figure 9 shows the response of a student who operated at the n level. such responses of students showed that they had not developed an object level conception for the determination of convergence of a series. figure 9: written response of st17 to task 1c (ii). the response given by st17 to task 1c (ii) is a case where the student tried without success to apply the comparison test. the student displayed an inability to successfully carry out the comparison test. the student failed to check if the assumptions of the series were satisfied before using the test, which supports the finding of earls (2017). the student failed to test if the assumptions of the direct comparison test were satisfied. st17 operated at n level; this was revealed after we had analysed the student’s written response. to confirm the accuracy of our analysis and the student’s understanding of these concepts, we interviewed her. r: can you tell me anything you know about the comparison test for the determination of convergence of a series? st17: i understand that given two series, say and with an ≤ bn then if the bigger is convergent then the smaller is also convergent; and if the smaller is divergent, then also the bigger one is divergent. r: that is true. can you explain to me how you came up with your response? st17: i tried to come up with two series and failed to do so. hence the response i provided. r: in your written response, why did you write ? st17: i wanted to, um i cannot give the reason for my response. r: then how did you come up with the conclusion that the series diverges? (the student did not respond) an analysis of the interview with st17 revealed that she had an action conception of the convergence of an infinite series. she managed to make the correct choice in the convergence test and dealt with the nth term of the given series, but incorrectly inserted the sigma signs. in so far as an object conception is concerned, we were correct to place her at having an n level understanding of the convergence of an infinite series, based on this task. responses at an object level conception: st28 (see figure 10) managed to give the correct response to question 1c (ii) and showed the ability to correctly select the most appropriate test for convergence, which he successfully carried out. those abilities signal the possession of the object conception. the individual was able to treat that concept as a cognitive object, which incorporates the ability to perform actions on that object and give interpretation or reason about its properties (suryadi, 2012). in order to endorse the student’s knowledge of such concepts, and our claim that she operated at the object level, the interview was conducted. r: what comment, if any, can you give about the partial sums of the infinite series? st28: with the series terms getting to zero in the limit, the limit might converge; but if series terms fail go to zero in the limit, then the series diverges. r: can you explain how you came up with the correct response to task 1c (ii)? st28: i made use of the terms in the infinite series, infinite process, the sum of infinite series and the links between them. these two series converge by the p-series test. therefore, their sum also converges. figure 10: written response of st28 to task 1c (ii). this student was able to display the conceptual structures indicated by our pgd in accordance with an object conception. task 1c (iii): use an appropriate convergence/divergence series test to determine the convergence of responses of students according to apos levels: table 9 indicates that 10 (33%) of the students failed to attain an object conception when applying the alternating series test. three of the 10 students who could not attain the object conception did provide an attempt, whereas the others did not provide any attempts. twenty (67%) gave responses indicative of operating at the object level, in the context of the application of the alternating series test. responses at n level (no response): st14 did not attempt task 1c (iii) which was an indicator that the student operated at n level. we interviewed her to ascertain the difficulties she faced and how she understood these concepts. r: you did not attempt question 1c (iii). what challenges did you face? st14: i found the question too difficult to answer. i need more time to work on such type of questions on the determination of convergence of series. the excerpt above indicated that st14 was not prepared to answer such questions. our analysis of her responses to task 1c (iii) was also supported by her interview response. she had not made the mental constructions necessary for answering such questions and operated at n level. responses at an object level conception: figure 11 gives an example of a student who operated at an object level. such students showed that they had developed the necessary mental constructions to determine series convergence. figure 11: written response of st1 to task 1c (iii). the student showed the ability to choose the appropriate convergence criteria based on the mathematical structure of task 1c (iii). a student is said have achieved object conception of a mathematical concept if the ability to treat that concept as a cognitive object is indicated, as well as the ability to carry out actions on that object, and give interpretation or reason about its properties (suryadi, 2012). to verify his understanding of these procedures and confirm his operational level, we interviewed st1. r: you gave the correct response to task 1c (iii). take me through your response. st1: this is an alternating series, so we have to carry out an alternative series test. the test is applicable if is an alternating series with an > an + 1 > 0 for all n and limn → ∞ = 0, then this series converges. r: can you explain how an infinite series (sequence of partial sums) has its limit as a real number? st1: if the infinite series converges, then its limit is a real number. one has to view an infinite process as one (a total and complete process) which we can arrive at through the use of a formula or convergence tests. the interview confirmed that st1 operated at the object level conception as indicated in the pgd. conclusion this article is based on research in which we studied 30 participants’ written responses to questions. for the purposes of triangulation interviews were conducted with eight of those participants involving tasks leading to them demonstrating their mental constructions both in written and verbal responses. in the discussion an exemplar for each task was presented. each case discussed highlighted the group of students falling in the category of the specific mental construction, supported by the relevant interview extracts. those discussions are relevant only for the eight students that were interviewed and cannot be generalised to the entire group of 30 participants. also, in this study we explored whether the students for a particular task either demonstrated a particular mental construction or not. this approach showed shortcomings so for future studies it will be necessary that for a full understanding of a mathematical concept, a student needs to exhibit all ‘levels’ or rather categories simultaneously. for the limits of series there was one task aimed at the action level. students’ responses matched some past research studies. the following conclusions are made from the tables in the discussion of data section. the information summarised in those tables was arrived at from an analysis of only the 30 participants’ written responses to the questions. in answering the first research question on what mental constructions undergraduate mathematics students reveal when solving problems involving the convergence of a series concept, we found that 19 out of 30 participants managed to attain an action conception of defining the limit of an infinite series. furthermore, the tasks on the definition of convergence of an infinite series 1b (i–iii) on the limit of the sequence of partial sums revealed that 25, 20 and 19 students displayed a process conception. the students’ response to tasks 1c (i–iii) revealed that 20 students operated at the object level conception, in each case. note that many of those 20 students overlapped with the students who displayed a process conception for tasks 1 b(i–iii). we also noted that a participant could be at the object level for one example of an infinite series, but at the n level for another example of an infinite series. for such observations from the data we believe there is a need for another study with this as a focus, which was outside the scope of the present study. during the analysis of the data and findings therefrom we observed that our pgd did not take into account: (1) at least an action conception of sequences and series with a purpose to display mental constructions which will accommodate for their differences, (2) at least an action conception of convergence and divergence of series with a purpose to display mental constructions which will accommodate for their differences, (3) an action or process conception of resolving rational functions into partial fractions, (4) at least a process conception of the increasing and decreasing function notions, and (5) a process conception of the notion of continuity of functions. based on these shortcomings we made revisions and present a modified gd (which addresses research question two) in table 10. this modified gd could be used by mathematics educators to design questions for instruction, consolidation and assessment. the modified dg gives an insight into the types of mental constructions that need to be focused on during the teaching and learning process. table 10: preliminary and modified genetic decompositions for series. acknowledgements competing interests the authors have declared that no competing interest exists. authors’ contributions all authors contributed equally to this work. ethical consideration ethical clearance was received by the university of kwazulu-natal (number hss/0953/016d). approval was also received by bindura university. funding information this research received no specific grant from any funding agency in the public, commercial or not-for-profit sectors. data availability statement the data are from the corresponding author’s phd thesis. disclaimer the views and opinions expressed in this article are those of the authors and do not necessarily reflect the official policy or position of any affiliated agency of the authors. references arnon, i., cottrill, j., dubinsky, e., oktac, a., trigueros, s., & weller, k. 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(2012). the research design maze: understanding paradigms, cases, methods and methodologies. journal of applied management accounting research, 10(1), 69–80. hockman pythagoras 61, june, 2005, pp. 31-41 31 dynamic geometry: an agent for the reunification of algebra and geometry meira hockman school of mathematics, university of the witwatersrand email: meira@maths.wits.ac.za this paper investigates the degree of separation or unity between the algebraic and geometric modes of thought of students in tertiary education. case studies indicate that as a student is inducted into the use of algebra the insightful and visual components of geometrical and graphical modes of thought are sidelined. based on vygotsky’s taxonomy of the psychogenesis of cultural forms of behaviour, i suggest that this separation occurs because the algebraic methods remain fixed at a naïve or algorithmic stage. the algebraic concepts may fail to be internalised because the stage of instrumental functioning of algebra as a ‘tool’ or ‘method’ of geometry is not successfully transitioned. i suggest that this stage of instrumental functioning may be stimulated by using dynamic geometry programs to promote the formation of images in conjunction with algebraic representations in problem solving. in this way the modes of thought in algebra and geometry in mathematics may be reunified. introduction algebra plays a fundamental instrumental role as a language in the practice of mathematics. it is used to reduce the complexity of problems and hence promote a method of solution through the application of an algebraic algorithm (grabiner, 1995; katz, 2004). while students may effectively use algebraic algorithms and routines, the purpose and/or usefulness of these ‘tools’ is often lost or misinterpreted by novice users. to these novice users, the algorithms are seen as an end in themselves. the signs and symbols, however, may not be used effectively as a means of reasoning. the descriptive and analytic powers of the algorithms in the initial problem or in other mathematical disciplines are often lost. furthermore, even if the learner has assimilated the algebraic tool, the geometrical or graphical insights relevant to algebra are often sidelined and neglected. thus the dynamic relationship between algebra and geometry (including graphs and visualisation) is broken. i believe that this break in the natural connection between these mathematical disciplines (tools for reasoning) prevents the novice learner from gaining a mature understanding of either. the functional unity of the component processes of geometry (through visualisation) and algebra of mathematics is the focus of the paper. this research follows experiences of others in the field, in particular, visualisation in algebra and analysis (yerushalmy et al., 1999; kawski, 2002; katz, 2004); extracting and giving meaning in formal theories through concept images (pinto & tall, 1999); cognitive units and algebra (crowley & tall, 1999) and relating to multi-representation (sierpinska et al., 1999). in order to understand the dynamics of change in algebra from its beginnings as a ‘method’ in geometry to its development as a legitimate branch of mathematics that sidelines geometric thinking, i examine a vygotskian perspective on the education of cultural forms of behaviour (vygotsky, 1994). vygotsky proposed that a tool (such as an algebraic algorithm) and the object it acts on (such as a complex mathematical problem) have a dynamic relationship that benefits both. while it may be understood that algebra is a ‘method’ of solving complex problems in geometry, it should also be understood that geometry is a ‘method’ of understanding the formal concepts in algebra. vygotsky mapped the functional use of tools and signs from their primitive instrumental beginnings to final sophisticated mental processes through four stages. these stages are detailed as: natural or primitive psychological; naïve psychology; instrumental function; ingrown or internalised. in this paper i propose that tools such as algebra develop and mature in a way similar to that of language, signs and symbols as described by vygotsky. i distinguish between the tools’ instrumental effects of cultural amplification (pea, 1987; bruner, 1966) that speed up or accelerate processes and cognitive functioning (pea, 1987) that illuminate or give insight into the processes. i suggest that as the use of the tool becomes more sophisticated its attributes are internalised and the dynamic geometry: an agent for the reunification of algebra and geometry 32 tool’s relationship to its objects of action fades. not only do the tools’ roles of amplification and cognitive functioning disappear but the dynamic feedback of subject and object may be severed. in what follows, i consider the pedagogical implications of reuniting geometry with algebra using dynamic geometry programs as the agent of reunification. i explore this separation/unification of algebra and geometry in two case studies. the first study shows how the separation of algebra from geometry has a detrimental result on problem solving within a class of university students in their second year of study. these students, majoring in mathematics, were participants in a first course in abstract algebra. i highlight the separation that occurs between existing graphical knowledge and newly acquired algebraic routines. the second study involves a mathematics education honours class in geometry. the class consisted of in-service senior school mathematics teachers. here i highlight the changed perspective of the students after they incorporate geometry into an algebraic problem solving activity. i review the status of the students in both case studies with respect to the described vygotskian taxonomy of the psychogenesis of cultural forms of behaviour. based on this evidence i suggest that dynamic geometry programs such as geometer’s sketchpad can be used to reunite the mathematical disciplines of algebra and geometry at school and at undergraduate level. such programs help produce a multi-perspective understanding of the mathematical content and highlight the illumination that the processes of algebra and geometry bring to each other. with such understanding the students may become aware of the instrumental function of algebra with respect to other branches of mathematics. algebra as a tool in mathematics tools used to accelerate and to illuminate algebra as a ‘method’, a ‘tool’, a ‘technology’ or ‘language’ in mathematics is widely accepted (applebaum, 1999; noss & hoyles, 1996). since algebra helps the user mediate or transcend the limitations of geometric thinking, learning and problem solving it may be classified as a cognitive technology (pea, 1987: 91). algebra as a tool may be used in mathematics in strategically different ways. that is as a tool of cultural amplification and of cognitive functioning (pea, 1987; berger, 1998: 15). in the case of cultural amplification, the tool provides the techniques for calculations, approximations or constructions that are otherwise laborious, complicated or simply tedious. the mathematics student/teacher/researcher benefits from using the tool by gaining insight into the nature of solutions to problems, through saved time and through maintaining interest that might otherwise dwindle. this attribute of the tool has been referred to by pea as a means of empowering human cognitive capacities. in the case of cognitive functioning, the tool provides a functional role of revealing changes and/or invariance that may be present. in particular, using the method of algebraic representation, geometric concepts are seen from an algebraic perspective. this approach adds rigour and depth of understanding. pea (1987: 96) refers to such technological attributes as the reorganisation of mental or cognitive functioning. he suggests that this technological feedback externalises thought processes, keeping a record of results and allowing patterns to be observed (1987: 97). these records are available for inspection, correction and reflection. this reflection leads to reformed action or to hypotheses that can be tested. in the case of algebra, i propose that the distinct attributes of cultural amplification and cognitive functioning emerge as algebra becomes part of the culture of mathematical practice of students of mathematics and mathematicians alike. as the use of algebraic techniques mature there is a shift in the dynamic relationship between algebra (the tool) and the geometric problem (the object or subject matter it was designed to serve). the induction of the learner into the use of algebra as a mathematical language is similar to the education of cultural forms and behaviours (signs, symbols and language) as explained by vygotsky (1994). vygotsky indicates that a tool develops from a naïve (external) to a sophisticated (internal) means of support to learning. i believe that the algebraic tool develops similarly. in what follows, i will make this path explicit. the education of cultural forms of behaviour vygotsky (1978), bruner (1986) and pea (1987) suggest that there is a dynamic link between the tool and the object it acts on. vygotsky notes that if one changes the available tools of thinking, the mind will develop a radically different structure (1978: 126). bruner (1986: 72) suggests that tools and technologies provide a means for turning around one’s thoughts and seeing the technologies in a new light. this changing dynamic is what vygotsky calls psychogenesis of cultural forms of behaviour. a person may master his/her external behaviour by a meira hockman 33 culturally acquired technique of signs. this method of using signs is “not only a key to the understanding of higher forms of a child’s behaviour which originate in the process of cultural development, but also a means to the practical mastering of them in the matter of education and school instruction” (vygotsky, 1994: 70). the method consists of natural psychological processes, and yet unites these processes in a complicated functional and structural way (1994: 61). vygotsky notes that remembering with signs and tools, such as maps and plans, may be an example of the cultural development of memory. the process of ‘remembering’ will be determined by the character of the signs or tools that are selected as an aid. that is, the cultural development consists of mastery methods of behaviour based on the use of signs. culture thus transforms nature to suit the ends of the child. i suggest that vygotsky’s mapping of the changing roles of the signs, symbols and language in learning throws light on how algebra, as a systems of signs, symbols and language, may become integrated into mathematical practice. algebra may be learned in a mechanical way, but may not be used as a means of reasoning in much the same way as a child may be able to speak but not reason in his/her native language. visualisation of the algebraic symbols, through graphs or geometry, reunites the tool with the object of action, to enhance the cultural development of algebra. this unification may thus determine the character of the system of signs, symbols and images that comprise algebra and geometry. the algebra and geometry should fuse into a functional unit of processes. vygotsky proposes that the cultural development of the child passes through four phases or stages that follow consecutively one after another (1994: 64). these stages form a complete cycle of cultural development of any one psychological function. following vygotsky, i list the four stages and elaborate to the mathematical domain: stage 1. in this stage natural or primitive psychological means are used to resolve a problem. that is, the task is not above the natural abilities of the child and he/she will master it using present memory and intellectual development. for example, the child may operate with quantities even though he cannot count. in the case of language, a student may be able to communicate in the language but does not understand how to reason or use words to draw conclusions. referring to algebra, this stage is epitomised by learners representing unknown quantities as “x” but not making use of this representation in the solution to a problem. for example, representing a parabola by 2ax bx c+ + with no understanding of the significance of the constants a, b, and c and variable x. stage 2. naïve psychology is the stage in which the sign or tools are adequately used in an algorithmic way. the task is above the natural capabilities of the child and the child is initiated into the use of new tools or technology. in this stage vygotsky explains that children may learn to use a mnemonic aid (the tool) to enhance memory. the solution of the problem requires the application of this mnemonic aid. the child solves an inner problem by means of the exterior object (mnemonic aid). the external object (say fingers or counters) takes on the functional importance of a sign (replacing the object). however, the child is unaware of how the object helped him solve the problem. the external connection between the method and the problem is not forged. for example, a child may imitate counting and repeat sequences of words but does not know for what purpose counting is used. in a similar way mathematics students may learn to build equations and solve problems using variables and fixed constants. for example, students may use the quadratic formula (the tool) to solve for the roots of the equation, but have no concept of what a root or quadratic equation actually signify. in this case the formula (the tool) plays a definite and functional role. however, while the student may mechanically repeat the methods on the same type of problem with success, he may not be aware of the significance or deeper meanings of the method. this is the stage that pea (1987) refers to as cultural amplification. stage 3. in this stage the instrumental function of the tool is established and the tool is used appropriately as an intellectual tool. the child learns how the method works and how to make proper use of techniques. the processes forming part of the method form a complicated functional and structural unity. the unity is effected by both the task that must be completed and by the means by which the method can be followed. the structure of the problem solving activity is moulded by available means (e.g. using fingers as a tool to help in the counting process by placing them in a one-to-one correspondence with the numbers). this is the stage of cognitive functioning (pea, 1987). in algebra, for example, the student may dynamic geometry: an agent for the reunification of algebra and geometry 34 recognise that curves bear a definite relationship to points on a straight line (the x-axis) and that this relationship may be expressed by a single equation. here, there is recognition that the quadratic formula (tool) relates to the shape and position of a parabola with respect to coordinate axes and that the roots are intersections with the x-axis. in this role the respective tools, used repeatedly and in diverse ways, illuminate and extend knowledge. stage 4. in this stage the use of the tool becomes ingrown or internalised. the external activity of using an object as a means of finding solutions passes on into internal activity. this assimilation indicates a maturation of the tools as a working strategy. a problem once solved leads to a correct solution in all analogous situations even when external conditions have changed radically (e.g. counting in the mind is an illustration of ‘complete in-growing’ of the technique of placing objects in one-to-one correspondence with the natural numbers). in the case of algebra, polynomial equations replace the graphs of the polynomials and their properties are now established algebraically with no reference to their physical reality. the algebraic representations replace the geometric objects and a geometric problem is solved algebraically. mental images or rough sketches of the changes produce insights. the algebraic method and the visual changes have been internalised. this internalisation promotes generalisation, where different examples are represented by the same or similar algebraic relationships. this stage marks the maturation of the role of the tool from technology to science. instrumental function while vygotsky points out that language as a tool of speech and a tool of reasoning have entirely different roots, he stresses that at a certain moment the two lines of development cross each other. at that moment a child discovers the ‘instrumental function’ of a word (1994: 68). prior to this stage the intellectual behaviour of the child indicates an independence of intellect from speech. following the stage of discovery of the functional importance of a word is the stage of transition from external to internal speech. vygotsky observes that these three main stages in the development of speech and reasoning correspond to the three main stages of cultural development of tools and signs as discussed above. the pre-speech reasoning corresponds to primitive and naïve behaviour (stages 1 and 2), while the instrumental function of a word, which vygotsky describes as the “greatest discovery of the life of a child” (1994: 69), corresponds to the third stage of the scheme. in this stage the tool is used appropriately as an intellectual resource. finally, the transition from external speech to internal speech corresponds to the transformation of external activity to internal activity as described in stage 4. similarly, i argue that the movement from algebra as a tool of amplification in stage 2 to algebra as a tool of cognition in stage 3 is reached when the student understands the instrumental function of algebra in solving geometric problems. this progress in roles from amplification to cognition provokes a change in the algebra itself and effects the relationship between algebra and geometry. in this case, the conscious reflections on the tool (algebraic method) causes the associated skills to mature into a branch of knowledge or a science in their own right (stage 4). that is the ‘method of algebra’ changes into the ‘science of algebra’ through a process of conscious reflection. the historical development of algebra (boyer, 1968; eves, 1953) mirrors this argument. while this fact is not used as an argument of the present thesis, it adds depth to understanding the way algebra matures as a discipline in a mathematical mind. historically we see the initial role of algebra as a ‘method’ in the service of giving rigor to geometry metamorphosise into its present role as a ‘central subject area or discipline’ in mathematics as its nature and attributes are internalised by mathematicians. as a discipline today modern algebra rarely shows its geometrical beginnings explicitly or draws on the visual images or intuition that geometry provides (atiyah, 1982; grabiner, 1995; hilton, 1990). i suggest that it is in vygotsky’s stage 3, when the method of algebra is used externally to support geometry, that the student may discover the true usefulness of algebra. conversely as algebra develops and matures into a discipline in its own right, geometry or geometrical intuition can be recalled to support new algebraic ideas. pedagogically it is in this stage that the links between algebra and geometry can be forged to produce mutual feedback and illumination. case studies of separated and unified practice the theory presented here is examined in two different situations. the first case study is undertaken in a regular second year abstract algebra class at university. the second case study focuses on a class of mathematics education honours students participating in a course in geometry. the two case studies were carried out meira hockman 35 for different purposes. in the first case study the aim was to support or contradict a hypothesis that students of abstract algebra do not integrate their new algebraic skills with existing techniques from other mathematical perspectives (graphical or geometrical). the second case study aimed at examining a change in attitude and approach by a cohort of students to a problem that appears to be purely algebraic, when a geometric approach is encouraged. in this case the students first attempted the problem with no interventions, and then redid the problem after visualisation was actively promoted. the two cohorts of students were substantially different. the first was a group of students who were studying full time at a south african university and majoring in mathematics. the second was a group of in-service mathematics senior school teachers who were taking a mathematics education honours degree part-time at the same university. in both cases the author was the course designer and lecturer. it must be noted that in both these studies, formal consent was received from the students to use their work, their responses and questionnaires as research data. case study one: an example of the separation in modes of thought math 204 abstract algebra is a first abstract algebra course offered to selected students in their second year of study at university. the students are selected on the grounds of their good achievement in an analysis course. the cohort was made up of 53 students, who completed three problems about bijective (one-to-one and onto) mappings that were selected from tutorial exercises. question 1(a) and (b) appeared to be algebraic but had geometric or graphical solutions. question 2 was theoretical with a solution that makes use of the formal definitions of one-to-one and onto mappings. the students had been introduced to algebraic methods of establishing whether given mappings were oneto-one and onto. these new algebraic techniques were introduced to support and complement the geometric methods that students had used at school. it was emphasised in the lectures preceding the case study that the geometric methods learned at school were still useful when the mapping could be represented geometrically or by a graph. the students were encouraged to use any method of solution to the problems. the exercise given to the students comprised the following (see equation 1): the exercise was chosen in order to examine the methods of solution, the accuracy of answers and to indicate whether or not the students, when in an algebraic environment, felt comfortable or inclined to use geometric representations to support their arguments or findings. the mapping in q 1(a) is not one-to-one and using the graphical representation a counter example can be found (figure 1). the range of the mapping in q1(b) can be drawn to produce support to the fact that the mapping is not a bijection (figure 2) since it is not onto ρρρρ ×××× ρρρρ. the solution of question 2 uses the algebraic technique for proving one-to-oneness both as a given property and a required property. it also calls for the use of the definition of onto mappings. as a result the algebraic manipulations are quite sophisticated. the problem is abstract and needs the application of theory (see equation 2). analysis of data in q1(a) out of the 53 students only six students used graphical representation to solve the problem. the remaining students all used a combination of lengthy algebraic techniques and counter examples to establish (correctly and incorrectly) the results. in q1(b) none of the students used any form of graphical representation to support their arguments. only two students who used graphical representation in part 1(a) gave correct answers in q1(b) (without using a graphical representation). it q1: state whether the given mapping is onto, one-to-one, bijective. (a) α : ν→→→→ν defined by ( )α n n n = +     1 2 2 if is odd if is even n n (b) α: ρ→→→→ρρρρ ×××× ρρρρ defined by α(x) = (x+1, x − 1) q 2: if βα is one-to-one and α is onto, show that β is one-to-one. equation 1: exercise for case 1 study. dynamic geometry: an agent for the reunification of algebra and geometry 36 is noted that all 53 students correctly proved the mapping in q1(b) is one-to-one, 48 correctly proved the mapping in q1(a) to be onto, and 38 of the 53 managed to answer q2. yet in all, only 11 students answered the three questions correctly. this study supports my contention that while graphical representation is available to students, this means of dealing with a problem is largely ignored in an ‘algebra’ course, once an algorithmic routine is learned and assimilated by the students. this study also highlights the fact that while algebraic techniques may be mastered they are not always the best method of solving a problem. in q2(b) the students were successful in showing that the mapping was one-to-one but half the class also, erroneously, ‘proved’ it was onto. similarly many students, following the technique for proving a mapping one-to-one ‘proved’ that the mapping in q1(a) was one-to-one. the algebraic routines and not the mappings appeared to be the focus of their attention. this case study indicates that when a mapping satisfies the bijective property the students are able to solve the problems using the algebraic techniques they have mastered. the problems involved the concepts of one-to-one and onto for which they had learned an algorithm. the mapping in q1(a) was onto and the mapping in question 1(b) was one-to-one and so both could be correctly solved using the algorithm. the majority of the students performed this task. in terms of vygotsky’s taxonomy they were at stage 2, indicating that they could use the algorithms. however, the responses to the mapping in q1(a) being one-to-one and the mapping in q2(b) being onto show that most of the students are not at stage 6 4 2 5 2 -2 -4 5 figure 1: the graph of α in q1(a) figure 2: the range of α in q1(b) 1 2 1 2 1 1 2 2, , with ( ) and ( )b b a a a a b a bα α∈∈∈∈ ⇒⇒⇒⇒ ∃ ∈ = =∃ ∈ = =∃ ∈ = =∃ ∈ = =b , since α is onto b (((( )))) (((( )))) (((( ))))(((( )))) (((( ))))(((( )))) (((( )))) (((( ))))1 2 1 2 1 2 1 2b b a a a a a aβ β β α β α βα βα==== ⇒⇒⇒⇒ ==== ⇒⇒⇒⇒ ==== ⇒⇒⇒⇒ ==== , since βα is one-to-one. therefore β is a one-to-one mapping. equation 2: solution to q2. q 1(a) “proved” map 1-1 q 1(a) proved map onto q 1(b) proved map 1-1 q 1(b) “proved” map onto q 2 proved all proved correctly no. of correct answers 17 48 53 24 38 11 table 1: the distribution of answers in the class. meira hockman 37 3. they ‘forced’ the algorithm to ‘work’ in order to ‘prove’ the results. i would suggest that the 11 students who achieved correct answers to all the questions were at stage 3. they recognised that the algorithm failed to give results in these cases and used alternative methods to find a solution. it is my suggestion that with the help of the geometric representation many of the students would have successfully answered all the questions. the graphical representations would help the students to focus on the actual mappings and to use the algorithms or set routines where appropriate. this support would help the students achieve the functional cognition of stage 3. evidence from case 2 indicates that as students broaden their perspectives to incorporate a graphical or geometric representation, their understanding of the algebraic concepts is enhanced. case study two: an example of the reunification of modes of thought sced 400 is a two-year, part-time, postgraduate course in mathematics education. 22 students (two students did not respond to questionnaire), all of whom are practicing teachers, participated in a geometry module as part of the course. this particular case study took place at the first lecture of this course. the study was broken into two parts: an initial investigation where a problem was solved by the students in groups in a classroom situation, and a follow up investigation where the problem was explored in a computer laboratory by students individually. the aim of the study was to investigate: • how students solved the problem. • how they categorised the problem. • whether students understood the nature of the problem and its solution. • changes in student responses resulting from joint geometric and algebraic exploration of the problem. the students’ responses to the questionnaire, their written solutions and comments were examined. this questionnaire was first completed after the initial work period, and then again after the computer laboratory session. in addition the students (working in groups) compiled a document recording their solutions, impressions and rough work of the two sessions and their reflections on the problem solving experience. the problem can be looked at in various ways. the problem: for which values of k will 2 2 1 1 x x k x x − + = + + have real roots? what are the maximum and minimum values of k for x real? equation 3: problem for case 2 study. ( ) 2 2 2 2 2 1 1 ( 1) 1 1 ( 1 ) 1 0 x x k x x k x x x x x k x k k − + = ⇒ − + = + + + + ⇒ − + − − + − = ( )2 2 2 2 ( 1 ) 4 1 (1 ) 2 1 4( 2 1) 3 10 3 ( 3 1)( 3) k k k k k k k k k k k ∆ = − − − − − = + + − − + = − + − = − + − 0 ( 3 1)( 3) 0 ( 3 1) 0 and ( 3) 0 or ( 3 1) 0 and ( 3) 0 1 1 and 3 or and 3 3 3 1 3 3 k k k k k k k k k k k ∆ ≥ ⇒ − + − ≥ ⇒ − + ≥ − ≥ − + ≤ − ≤ ⇒ ≤ ≥ ≥ ≤ ⇒ ≤ ≤ equation 4: algebraic solution to problem in case 2 study. dynamic geometry: an agent for the reunification of algebra and geometry 38 the algebraic approach is to find a quadratic equation in x and solve 0∆ ≥ (equation 4). the method does not require any geometric representation and does not refer to the polynomials comprising the rational function in any way. geometrically the problem could be looked at in three different ways. the dynamic geometry system can be used to animate the various positions of the parameter k. case 1. sketch intersections of the curves 2 2 1 ( ) 1 x x f x x x − + = + + and g(x) = k, allowing k to vary from { ( ) } to { ( ) }min f x x max f x x∈ ∈¡ ¡ as in figure 3. case 2. sketch the quadratic function 2 (1 ) ( 1 ) 1y x k x k k= − + − − + − , varying k through all real numbers, showing its limiting positions for real solutions (figure 3). case 3. sketch the discriminant ∆ of 2 (1 ) ( 1 ) 1y x k x k k= − + − − + − , and indicate where ∆ ≥0 as in figure 5. in figure 3 and figure 4 the continuously varying k can be used to corroborate the solution 1 3 3 k≤ ≤ , obtained either algebraically or from case 3 (figure 5), and that these bounds are the minimum and maximum values of k respectively. in this example the geometer’s sketchpad introduces different perspectives of the same problem. the confusing appearance of the quadratic equation ( ) 23 10 3k k k∆ = − + − that needs to be solved for 0∆ ≥ may be avoided or in turn explained, as the inequality 1 3 3 k≤ ≤ keeps reoccurring in each approach. analysis of data the 22 students completed the problem in four groups, comprising two groups of five (groups 1 and 4) and two groups of six (groups 2 and 3). the problems were discussed on the board and the solutions submitted. it was evident that all the students understood what was expected of them and completed the problem routinely. they expressed the fact that the problem was one that they were familiar with and was a standard problem at grade 11 and 12 at school. when asked about the geometrical significance of the problem they all drew the real line and indicated the interval where the possible solutions lay. the class remained unresponsive when pressed for a further geometrical interpretation of the problem: “initially … we believed the question to be only 2 -2 5 5 figure 3: graph in case 1. 4 2 -2 -4 -5 5 10 4 2 -2 -4 -5 5 10 4 2 -2 -4 -5 5 10 4 2 -2 -4 -5 5 10 4 2 -2 -4 -5 5 10 • position of curve for minimum and maximum values of 1 and 3 3 k k= = . • position of curve for 1 3 < k < 3 figure 4: graph in case 2. 4 2 5 figure 5: graph as in case 3. meira hockman 39 algebraic in nature. we could all reach the correct answer but were not sure of the meaning of this result” (group 4 reaction) and “initially we only relied on algebraic manipulation of the equation, with no idea that the equation (problem) could be dealt with geometrically” (group 1 reaction). the students were surprised to be asked about a geometrical significance. they felt the problem was an algebraic problem. the class then set up the curves with geometer’s sketchpad, using the various forms (see figures 3, 4, and 5) of the equation (intersecting curves) as indicated. in each case the solution 1 3 3 k≤ ≤ reappeared, although the equation occurred in equivalent forms. the class’ responses to the problem presumably changed as a result of the intervention. the results of the questionnaire indicate a swing from thinking strictly algebraically to mixing algebra and geometry. before the intervention 15 of the students classified the problem as being strictly algebraic and five classified it as being both algebraic and geometric (with two of the five indicating the geometric aspects of the problem lay in the solution on a number line). the work handed in concurred with this result. the only ‘geometry’ in their work was the representation of the solution to the problem on a number line. the questionnaire also indicated that all the students believed that you needed knowledge of quadratic formula, equations and roots to understand the problem, while knowledge of graphs, curves and their intersections was not relevant (only one student saw this aspect as being highly significant to the problem) for understanding. after the intervention with dynamic geometry all the students acknowledged the geometric or graphical significance of the problem and of its solution. the reflections of the students in the documents recording their impressions of the intervention are also illuminating. these expressions give evidence of the effectiveness in combining a geometrical and algebraic approach to this type of problem solving activity. “we also realise that the geometry aspect of the problem is very important as it attempts to show how these values are true for the given equation, it also gives insight in terms of explaining why the values [0.3;3]x ∈ will produce real roots for the equation”(group 1 reaction) “we were very impressed with sketchpad’s capability to show how the graph changed from parabola concave up, to straight line, to parabola concave down depending on k…the graphical approach using sketchpad did not provide exact answers but allowed us to view the problem from different graphical perspectives. we felt that this approach to teaching mathematics would act as a catalyst to develop conceptual understanding rather than procedural understanding. finally, for a complete understanding of the problem, we felt that we should consider the problem from as many angles as possible” (group 4 reaction). “the solution we got from this parabola confirms the solution we got from the algebraic method where 1 3 3 k≤ ≤ . we discovered that both the algebraic method and the graphical method algebraic algebraic and geometric before after before after classification of problem 15 2 5 18 understanding the problem 16 3 4 18 solving the problem 18 13 2 7 table 2: analysis of rational function questionnaire. dynamic geometry: an agent for the reunification of algebra and geometry 40 (sketchpad) complement each other but the graphical one gives insight into the solutions though sometimes the sketchpad does not give answers but estimations… the algebraic solution of inequalities in most cases gives learners problems especially where the inequality sign has to change. using both methods can be more helpful to learners because they will have more insight into the problems” (group 3 reactions). referring to the vygotsky taxonomy, i believe the students had all achieved the naïve understandings of stage 2 before the intervention. i suggest that introducing the graphical representations using sketchpad broadened the perspectives of the students. the intervention stimulated the students to ask questions about the meaning and interpretation of the results. the algebra could then be seen as bringing rigor to the geometric insights. in the case of group 3 and 4 there was an indication that the third stage of vygotsky’s taxonomy was engaged. this is evidenced by their clear and insightful reflections on the sketchpad intervention. conclusions the two case studies examined above serve to support the hypothesis that for most students in the groups there is a separation in the thinking modes of algebra from geometry. the first study underscored the fact that even among a mathematically talented group of students, the compartmentalising of the different disciplines was almost complete. the students were familiar with the process of checking one-toone and onto graphically from school, but the new algorithmic procedure took precedence in their solutions, even when it proved insufficient to the task. in the second study, no more than the algebraic algorithm was needed to solve the problem. however, insight and understanding into what the algorithm was doing was missing. the introduction of the geometric approach to the problem broadened the perspective of the students allowing them to see the purpose of algebra as a tool in geometry. here dynamic geometry played an important role. hence in both cases the separation robbed the mathematical endeavour of its depth and relevance. as i have noted previously, algebraic representation is used to reduce the complexity of a problem in mathematics. once the problem has an algebraic formulation a solution may be found through an appropriate algorithm. yet it is the resolution of the problem and not the application of the algorithm that is the centre of mathematical attention. if the attention of the student is drawn back to the given problem then the instrumental function of the algebraic tool will be established. the unity between the task that must be completed and the method used to complete it, indicates that stage 3 of the cultural development has been engaged. it is here that visualisation and the technologies of dynamic geometry can play a vital role. the graphical representations allow reflection on the purpose and process of the algorithm. an algebraic problem stated in a decontextualised form promotes algorithmic solution as the primary activity. thus the activity belongs at stage 2 of naïve amplification in vygotsky’s taxonomy. if the same algebraic polynomials are related to the curves they represent then through visualisation, with dynamic geometry or through rough sketches, a gateway to deeper understanding may be opened. this activity may encourage students to proceed into stage 3 of cognitive functioning. vygotsky suggests that it is in the important third stage of ‘instrumental functioning’ that the child masters his external nature by means of techniques or technical means. in this stage the tool has not yet been internalised and still serves as a technology in solving problems. i suggest that this instrumental function of algebra in solving problems in geometry lies at the gateway of the internalised science of algebra. here the geometry brings deeper understanding of the algebraic concepts. when viewing algebra as a tool we have access to the very process of formation of the higher forms of behaviour as manifest in abstract algebra. i believe that in order to understand the extent of concept development in abstract algebra we need to accentuate and observe the external process of algebra as a method of solution to problems in geometry and use geometry to support and extend algebraic manipulations. in this way we will hold the outer threads of the abstract, internal processes in our hands. references applebaum, p., 1999, “the stench of perception and the cacophony of mediation”, for the learning of mathematics 19, 2, pp 1118 atiyah, m., 1982, “what is geometry?” the mathematical gazette 66, 437, pp 179-184 berger, m., 1998, “graphic calculators: an interpretive framework”, for the learning of mathematics 18, pp 13-20 meira hockman 41 boyer, c. b., 1968, a history of mathematics, new york: wiley bruner, j., 1986. actual minds and possible worlds. cambridge: harvard university press. crowley, l. & tall, d., 1999, “the roles of cognitive units, connections and procedures in achieving goals in college algebra”, in zaslavsky, o., ed., proceedings of 23rd conference of pme, israel, 2, pp 225-232 eves, h., 1953, an introduction to the history of mathematics, philadelphia: saunders college publishing grabiner, j., 1995, “descartes and problemsolving”, the mathematical magazine 68, 2, pp. 83-97 hilton, p., 1990, “the role of geometry in the mathematics curriculum”, pythagoras 23, pp 15-20 katz, v. j., 2004, “stages in the history of algebra with implications for teaching”. plenary address, icme 10, copenhagen, denmark kawski, m., 2002, “interactive visualisation in complex analysis”, in proceedings of the 2nd international conference on the teaching of mathematics, crete, john wiley, p. 286 noss, r. & hoyles, c., 1996, windows on mathematical meanings: learning, cultures and computers, dordrecht: kluwer pea, r. d., 1987, “cognitive technologies for mathematics education”, in schoenfeld, a. h., ed., cognitive science and mathematics education, pp 89-123. hillsdale n. j.: lawrence erlbaum pinto, m. & tall, d., 1999, “student constructions of formal theory: giving meaning and extracting meaning”, in zaslavsky, o., ed., proceedings of 23rd conference of pme, israel, 4, pp 64-72 sierpinska, a., trgalová, j., hillel, j. & dreyfus, t., 1999, “ teaching and learning linear algebra with cabri”, in zaslavsky, o., ed., proceedings of 23rd conference of pme, israel, 1, pp 119-134 vygotsky, l., 1978, mind in society, london: harvard university press vygotsky, l., 1994, “the problem of the cultural development of the child”, in van der veer, r. & valsiner, j., eds., the vygotsky reader, pp. 57-72. oxford: blackwell yerushalmy, m., shternberg, b. & gilead, s., 1999, “visualisation as a vehicle for meaningful problem solving in algebra”, in zaslavsky, o., ed., proceedings of 23rd conference of pme, israel, 1, pp. 197 – 211 “if the human brain was so simple that we could understand it, we would be so simple that we couldn’t.” willem hendrik gispen microsoft word 50-56 mtetwa-mudehwe-munyira.docx 50 pythagoras, 72, 50-56 (december 2010) learning mathematics for personal understanding  and productions: a viewpoint    david mtetwa, lazarus mudehwe & sheunesu munyira  department of science and mathematics education  university of zimbabwe  dmtetwa@education.uz.ac.zw, lazarus@maths.uz.ac.zw & munyiras@maths.uz.ac.zw    in this paper we reflect on what makes mathematics more meaningful and more easily  understood and thus enabling the learner to apply it to everyday situations in his/her life  world.  we identify personal – in relation to ‘collective’ or ‘public’ – mathematising as one  key component  towards real understanding of mathematics.   we observe  that  today’s  mathematics learner is often typified by such orientations as approaching the subject with  timidity and  in a cookbook fashion, adopting a re‐productive rather than a productive  mode, and showing lack of intrinsic interest in the subject.  debilitating effects of some of  these  characteristics  in  relation  to  learning  mathematics  for  personal  development,  include  learner’s  failure  to  exploit  the  subject’s  natural  features  for  developing  own  mental  orientations such  as algorithmic,  stochastic,  reflective,  and  creative  thinking  so  essential  in coping with modern  life environments.   we propose  that, for  inspirational  effects,  learners should have closer contact with and appreciation for the activities and  practices of the professional mathematician. the mathematics teacher could enhance the  learner’s mathematical learning experience by orienting instructional designs in ways that  make the learning processes and outcomes more personal to the learner.  introduction naturally, i set myself the task of constructing all these functions. i conducted a systematic siege and one after another, carried all the outworks; there was however one which still held out and whose fall would bring about that of the whole position. but all my efforts served only to make me better acquainted with the difficulty, which in itself was something. all this work was perfectly conscious. at this point i left for mont-valerin, where i was to discharge my military service. i had therefore very different preoccupations. one day, while crossing the boulevard, the solution of the difficulty which had stopped me appeared to me all of a sudden. i did not seek to go into it immediately, and it was only after my service that i resumed the question. i had all the elements, and had only to assemble and order them. so i wrote out my definitive memoir at one stroke and with no difficulty.[henri poincaré] (bell, 1965, p. 551). this paper takes a reflective look at the problem of personal mathematising and problem-solving applications in everyday lives of people in general, and the school learner in particular. each person needs to develop some mathematical concepts and methods for understanding and better managing of everyday activities in our modern world (davis & hersh, 1980). for example, plain every language has some mathematical ideas involved. therefore for one to communicate effectively, one needs some basic mathematical concepts. fuller understanding of mathematical concepts and methods is achieved through personal mathematising by the learner her/himself. by personal mathematising we mean that each individual learner is involved in mathematical activities associated with the formation of a mathematical concept or method. david mtetwa, lazarus mudehwe & sheunesu munyira 51 some theoretical considerations the notion of mathematising has been popularised in the last few decades as‘pedagogical scaffolding’ aimed at achieving learning mathematics with understanding (freudenthal, 1973; 1983; 1991; gellert & jablonka, 2007). notable proponents of this theory are the dutch mathematician-cum-educator, hans freudenthal, and his followers in the realistic mathematics movement, whose work has contributed to an increased focus in learning processes in mathematics education (de lange, 1996; gravemeijer, 1999; gravemeijer & cobb, 2002; treffers, 1987; 1993). according to this school of thought, mathematising is the crucial skill or knowledge required in order for one to learn mathematics with understanding, and at the heart of mathematisation lies the idea of what they call the ‘reinvention principle’ that is partially captured in the following statement: children should repeat the learning process of mankind, not as factually took place but rather as it would have done if people in the past had known a bit more of what we know now. (freudenthal, 1991, p. 48) this notion of mathematisation has continued to be elaborated by other authors to include dimensions such as the epistemological in conjunction with the closely related notion of mathematical modelling, and the socio-cultural through the associated notion of demathematisation (gellert & jablonka, 2007). mathematical activities involve, at elementary level, investigations of quantity, that is number, measurement and the relationships that occur among them and of shape and its properties (davis & hersh, 1980). however, true mathematical activities involve the search for patterns and trends and deduction of valid conclusions from given premises or hypotheses. thus the study of mathematics should foster an analytic mind so that given a problem in any area of human endeavor one is able to look for trends and patterns and so go a long way towards a solution of the problem. for example, very intricate problems of chance and logic have been conquered through direct application of mathematical processes (hartsfield & ringel, 1994; zawaira & hitchcock, 2009). mathematical processes is a collective term that embraces many sub-processes involving specific and distinctive types of actions and objects – almost entirely mental – and interrelations among them (gray & tall, 2007). one of the key characteristics of mathematics is that of abstraction, described by damerow (2007, p. 22) as a ‘meta-cognitive construct’ that is necessary for the production of mathematical concepts, or as a deliberate effort requiring coordination of a variety of, often parallel, (mental) actions – a description that builds on piagetian conceptualisations (dubinsky, 1991; von glasersfeld, 1999). damerow further places the idea of representation or symbolic operation at the centre of abstraction whilst at the same time making linkages of these mental processes and objects to the socio-cultural life worlds of the learner. more recently gray and tall (2007, p. 38) have postulated that abstraction is driven by a phenomenon they have called ‘compression,’ saying: “we believe that the natural process of abstraction through compression of knowledge into more sophisticated thinkable concepts is the key to increasingly powerful thinking” (p. 38). abstraction plays a crucial role when, for instance, from a few concrete examples of an idea or method one picks out the essential points of the concept or points that makes the method work and then use these as conditions obeyed by more general systems (courant & hilbert, 1953). thus movement from specific examples to more general systems or generalisation is an important characteristic of mathematics. let us look at two examples of generalisation. example 1: addition of fractions: find the sum to work at, say, grade 4 level, one needs to realise that we have two fractions of different types, that is, in concrete terms, two pieces of fifths (two pieces of a whole divided into five equal parts) and one piece of thirds (one piece of a whole divided into three equal parts). how can these be combined so that we come up with something that tells us what part of the whole we end up with? one of the simplest methods is to divide each of the two fifths into a further three equal parts (thus getting six pieces of a whole divided into fifteen equal parts), and dividing the one third into five equal parts (thus getting five pieces of a whole divided into fifteen equal parts). combining them, we then have eleven equal parts of fifteenths. problem solved. learning mathematics for personal understanding and productions 52 the first generalisation is to say that this method works for all additions of fractions. further generalisation of the method is to say the method works for the addition of algebraic fractions. example 2: indices from the fact that we can write m for … (m factors), deduce the following rules for the manipulation of indices of positive whole numbers m and n: , (m > n), ( we can only show that these rules hold for positive whole numbers. we then generalise by assuming that the rules hold for indices of all positive whole numbers, negative whole numbers, fractions and zero. assuming that these rules hold for these indices, we then deduce the meanings of each of the following: , , / , / (skemp, 1971) we are saying here that the generalising and abstracting thought processes must happen deliberately and consciously (though these could happen subconsciously later). in other words it must be sort of a live self-conversation, something related to the notion of meta-cognition (gray & tall, 2007). our claim here is that these processes, which constitute what we have called personal and authentic mathematising, are not well developed in today’s learner. think, for example, of how the mathematician henri poincare had internalised the problem, and while performing different ordinary tasks, his subconscious was busy working on the problem, and hence obtained a solution in the end (refer to the quotation at the beginning of the introduction). if learners can activate the personal dimension and engage in authentic mathematising, they can more easily apply mathematical ideas and skills in solving everyday problems in their own life worlds. as most concepts in mathematics arose naturally from the need to solve everyday problems, learning these concepts has to involve use of similar everyday problems that inspired the development of those concepts. for example, the natural numbers and counting arose most probably from the need to record one’s possessions. the study of geometry might have arisen to serve the needs of builders, agriculturalists, and others (zawaira & hitchcock, 2009). personal mathematising thus leads to greater understanding of mathematical concepts and methods by each learner as the learner is involved in concept or method formation – in our view a more productive rather than re-productive learning outcome. clearly the learning of mathematics is more effective if concepts are introduced in as many different ways as possible. the learner is able to pick out the essential elements of a concept and discard the irrelevant elements (davis & hersh, 1980). consider the concept of twoness or the number two. the number occurs in two oranges, two pigs, two vegetables, two people, two things, etcetera. the learner will realise that the twoness does not involve color, quality of objects, type of objects, etcetera, but has to do with quantity (see damerow (2007) for a very good discussion of the notion of quantity and space). while the above discussion has centred on attempts to explicate the meaning of and mechanisms for acquiring mathematical knowledge or for doing mathematics, our interest in this paper is not so much to extend the discussion in that same vein. the main concepts, however, provide us with a theoretical context that in turn provides a language for talking about and lens for viewing personalisation of mathematics learning, which is the primary focus of the paper. some characteristics of a modern learner of mathematics we believe that without robust personal mathematising, the learner shows some or all of the following characteristics that indicate lack of full understanding: o learns in a cookbook fashion o reproduces for the teacher or for the public and not a producer for self or the public o is intimidated by mathematical language o shows little interest in the subject o is afraid of or avoids taking risks with own ideas or concepts. david mtetwa, lazarus mudehwe & sheunesu munyira 53 when one learns a concept in a cookbook fashion, one is often unable to apply the concept to problems that appear slightly different from the ones met before. in other words, there is no transfer of learning. take, for example, this problem which are often encountered in undergraduate mathematics: show that , 1, 2, 3, … is bounded above and below. this problem is basically a grade 4 problem where we show that if and only if , for example, because 5 7. this follows clearly from understanding our notation of common fractions: means that we have three pieces of a whole (unity) which has been divided into seven equal parts, that is, the seven relates to the number of equal parts into which the whole or unity has been divided. thus if the denominator is small, the size of the equal parts into which the whole or unity is cut is large. now applying this grade 4 idea to our undergraduate problem, we have:  and    hence    1 therefore the expression is bounded below by and above 1. the teaching of mathematics should significantly contribute towards producing a person who has a productive and satisfying life in the modern society. it should produce learners who can exploit the facility and power of mathematics as a tool or medium in their everyday lives to enhance their capacities as o consumers: sorting and valuing products, buying/selling/trading/exchanging, decision making (algorithmic thinking) o citizens: negotiating rights and privileges, assessing dangers, and balancing acts (statistical thinking) o family persons: planning, predicting, managing others, minimum/maximum problems (stochastic thinking) o social/cultured beings: communicating, the connected world (the information processing society: internet, cellphones), that is, interacting with others and the material world (reflective thinking). o individual beings (as opposed to an animal): intellectual satisfaction, that is, exercising imagination, beauty of form, elements of proof, logic (imaginative thinking) o professionals (any profession including mathematician): enjoy profession, producing and creating products, technical/ engineering aspects of mathematics (creative thinking) we suggest that it is best to think of this personalisation dimension that is being highlighted here as a general mental and affective orientation towards engagement with mathematical material. that orientation, or ‘habit of the mind and heart,’ is something we are saying needs to be attended to explicitly and developed effectively in learners. when well developed and functioning optimally and in concert with other complementary ‘orientation-cum-skills’ such as heuristic thinking (polya, 1973; shoenfeld, 1985a), commonsensical thinking (kilpatrick, 2007), symbolic thinking (damerow, 2007; gray & tall, 2007), visual thinking (tall & vinner, 1981; vinner & hershkowitz, 1980) and meta-cognitive thinking (shoenfeld, 1985b; silver, 1987), the learner should be able to gain better understanding of mathematics and use relevant aspects of the knowledge and skills productively in his/her daily life. we are not saying that all people should be molded like professional mathematicians; rather we are saying it would be useful for every individual to have some capacity to behave like a sort of amateur mathematician at the very least, if you like. it is needless to remind that mathematicians are ordinary human beings too. we suspect that there may be a general problem of under-valuing mathematics as learning mathematics for personal understanding and productions 54 something that may contribute significantly to the overall composite well-being of individual life of every person. and so we are saying mathematics assists and contributes to the realisation of the full potential of the person. teaching and learning the question of how we as educators can assist in the transformation of the current learner product (as described above) to the desirable product becomes an imperative. how can we intervene through learning and teaching? we offer some suggestions to the learner, teacher and to the public. mathematics is commonly regarded as something to be feared, something very difficult, hence often considered the exclusive province for geniuses. to help overcome this perception, we must make the learning environment non-threatening. furthermore we must encourage cooperation and collaboration in tackling problems in mathematics. adequate and relevant resources for teaching and learning should be made available in as many forms of presentation as possible and through as many different types of sources as possible, for example experiential, imaginative, simulations (as in many internet sources), written text, verbal renditions, physical representations, etcetera. meaningful learning of mathematics should help one to develop a number of mental abilities. if there is fruitful learning of mathematics, a learner is provided with opportunities to search for relationships between things, to look for patterns and trends among given data, observe unusual phenomena, etcetera. thus in general, learning in mathematics should develop heuristic thinking, a mental skill which is needed by all citizens. the ability to abstract is one that is developed very effectively in the learning of mathematics. as noted earlier, from a few concrete examples one isolates the crucial elements that makes them work and then applies them to any system in which these elements are present (courant & hilbert, 1953). thus, we obtain an abstract system, that is, a system that does not apply to one special kind but to many systems in which the conditions apply. the examples above of adding arithmetical fractions and of indices illustrate the point sufficiently. another important ability is that of visualising, that is, the ability to form mental pictures of real or imagined objects. in mathematics we constantly use symbols to represent numbers, statements, etcetera, and operate with these symbols. thus the ability to symbolise and to operate with symbols is developed considerably in the study of mathematics. indeed there is a calculus of logic where logic itself is operated symbolically. this is the whole province of the area of specialisation called mathematical logic. learners can appreciate the finer aspects of mathematics learning from studying the lives of creative mathematicians, in particular, about their passion for the subject and about their perseverance. we shall briefly describe here two anecdotes involving two great mathematicians: archimedes and abel. archimedes lived in the city of syracuse about 200 bc during the time of the punic wars between rome and carthage. the romans wanted to take syracuse but were prevented from doing so the first time by artillery designed by archimedes using levers and pulleys. frustrated, the romans retreated but returned to capture the city anyway. they found archimedes so involved in his mathematics that he did not notice that the city had been captured by the romans. he was busy drawing mathematical diagrams on dust. he only noticed that something was amiss when the shadow of a roman soldier fell over his diagram, to which he exclaimed “don’t disturb my circles!” (bell, 1965, p. 34). then, with abel  of the abelian groups fame: there is a charming picture of abel after his mathematical genius seized him by the fireside with the others chattering and laughing in the room while he researched with one eye on his mathematics and the other on his brothers and sisters. the noise never distracted him and he joined in the badinage as he wrote. (bell, 1965, p. 308) such anecdotes, whether true or exaggerated, serve to make us believe some kind of passion is a necessary condition or, at least, ingredient, for any meaningful personal production of mathematics. david mtetwa, lazarus mudehwe & sheunesu munyira 55 concluding remarks considering the changing times and the pervasiveness of modern technology in our daily lives, we see the notion of personal mathematising as both compelling and promising as a strategy to explore and exploit in the learning of mathematics. in personal mathematising the power and authenticity of mathematical procedures derives from the mathematics itself, rather than from authorities such as the teacher or the textbook, neither of which are readily available to consult with in these times we live in. an important task for mathematics teachers becomes one of trying to link mathematical concepts and procedures with activities that are meaningful in the learner’s lifeworlds, while at the same time cultivating in learners – and in themselves – a passion for doing mathematics for personal knowledge and use. it would be valuable to reflect on what has been discussed above, consider all suggestions made, and see how practicable they could be for the reader’s own personal real life situation. in doing so, the crucial question remains: “how can we get some sort of passion for learning and doing mathematics for personal use and knowledge into our learners?” we leave the question open for the reader to engage with. references bell, e. t. 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(1985b). meta-cognitive and epistemological issues in mathematical understanding. in e. a. silver (ed.), teaching and learning mathematical problem solving: multiple research perspectives (pp. 361-380). hillsdale, nj: lawrence erlbaum associates. learning mathematics for personal understanding and productions 56 silver, e. a. (1987). foundations of cognitive theory and research for mathematics problem-solving instruction. in a. h. shoenfeld (ed.), cognitive science and mathematics education (pp. 33-60). hillsdale, nj: lawrence erlbaum associates. skemp, r. r. (1971). the psychology of learning mathematics. london: penguin. tall, d. o., & vinner, s. (1981). concept image and concept definition in mathematics, with special reference to limits and continuity. educational studies in mathematics, 12(2), 151-169. doi: 10.1007/bf00305619 treffers, a. (1987). three dimensions: a model of goal and theory description in mathematics instruction. dordrecht, the netherlands: kluwer academic publishers. treffers, a. (1993). wiskobas and freudenthal: realistic mathematics education, educational studies in education, 25(1), 89-108. doi: 10.1007/bf01274104 vinner, s., & hershkowitz, r. (1980). concept images and some common cognitive paths in the development of simple geometric concepts. in r. karplus (ed.), proceedings of the 4th conference of the international group for the psychology of mathematics education (pp. 177-184). berkeley, ca: lawrence hall of science. (eric document reproduction service no. ed 250 186). von glasersfeld, e. (1991). abstraction, re-presentation, and reflection: an interpretation of experience and piaget’s approach. in l. p. steffe (ed.), epistemological foundations of mathematical experience (pp. 4565). new york, ny: springer verlag. zawaira, a., & hitchcock, g. (2009). a primer for mathematics competitions. oxford: oxford university press. graven 2 pythagoras 61, june, 2005, pp. 2-10 mathematics teacher retention and the role of identity: sam’s story mellony graven school of education, university of the witwatersrand email: gravenm@educ.wits.ac.za this paper provides a vignette of one teacher’s identity transformation from a temporary teacher of mathematics to a professional mathematics teacher (with a long-term trajectory within the profession) during the course of his participation in an in-service education and training (inset) program. thus the paper elaborates, from wenger’s (1998) social practice perspective, the role of identity (one of wenger’s four learning components) in teacher learning and teacher retention within the profession. the vignette provides textured insight into the way in which sam’s identity transformation was pivotal in his learning through participation within inset and other overlapping mathematics education communities. furthermore, i argue that identity (learning as becoming) was a key component of learning that kept sam in the profession of mathematics teaching. from sam’s vignette i argue that positive identity transformation should be one of the most important intended learning outcomes of inset and as such needs deliberate and focused attention in the design and implementation of teacher education programs. i therefore conclude the paper with some implications of sam’s vignette for the design of mathematics inset programs. introduction sam’s vignette is taken from a broader longitudinal qualitative ethnographic study that focused on understanding the nature of teacher learning within inset, at a time of major curriculum change (graven, 2002). the sample for the study included ten senior phase teachers of mathematics, from eldorado park and soweto (urban township) schools who volunteered to participate in a twoyear inset program called plesme (program for leader educators in senior phase mathematics education). the sample can therefore be described as a small, purposive, opportunity sample in the sense that the plesme teachers were not randomly selected but were volunteers from schools suggested by the districts. the teachers therefore are not ‘typical’ of the general population of mathematics teachers. the plesme program is set in context below. wenger’s (1998: 5) four components of learning provide the analytical framework for the study and for this paper. these components are: meaning-learning as experience; practice-learning as doing; community-learning as belonging and identity-learning as becoming. i supplement wenger’s four components of learning with a fifth component, namely confidence-learning as ‘mastery’ (see graven, 2004). these components are ‘deeply interconnected and mutually defining’ (wenger, 1998: 5). the focus on the component of identity, for this paper, should not imply that the other components are not also important for teacher learning and teacher retention. indeed these components provide the basis for the development of sam’s strengthened mathematics teacher identity and his longer-term trajectory within the profession. the relationship between the components of meaning, practice, community and confidence to identity will be illustrated in sam’s story. the story will highlight the way in which learning (as changing identity) in relation to these components is enabled through sam’s ongoing participation within the plesme community of practice and its overlapping communities. the context of sam’s story sam’s story occurs within the context of participating in the in-service program for leader educators in senior phase mathematics education (plesme), aimed at supporting teachers to make sense of the radical educational reforms they are expected to implement. south africa’s new curriculum (called curriculum 2005), premised on a learner-centred, outcomes-based approach to education, was launched in 1997. key principles include integration, relevance, learner-orientation, flexibility and critical creative thinking (national department of education, 1997). while the new curriculum has been revised (and ongoing mellony graven 3 revisions continue) these key principles have remained central to curriculum revisions. these curriculum changes provide the broader context within which sam’s learning occurs. the initial design of plesme included: weekly workshops; individual and group reflection sessions; classroom visits accompanied by the use of video to facilitate reflection and discussion; individual and collaborative practical activities to be done in school; and some written activities that accompanied practical activities. with time, other key practices, not part of the original design, evolved from within plesme. for example, teachers sharing frustrations and resources, commenting on each other’s videos of lessons and the inclusion of activities in which teachers were networked into the broader professional community of mathematics educators. plesme included fieldtrips to various teacher centres, district offices and the offices of mathematics inset organisations, mathematics education associations and curriculum development organisations. plesme teachers worked collaboratively to provide input into various mathematics curriculum documents and curriculum review documents. in addition plesme teachers attended mathematics education conferences such as the national conferences of the association of mathematics education of south africa (amesa). at these conferences the teachers presented papers relating to mathematics teaching ideas and/or challenges confronted in implementing new mathematics curriculum ideas. thus, in agreement with grossman, wineburg and woolworth (2001: 953) plesme worked with the assumption that ‘successful forms of professional development must offer multiple corridors for participation’. participation in broader professional networks was furthermore seen as a means of sustainability of teacher learning and participation within the field of mathematics education beyond plesme. while the participating teachers of plesme came into the program as teachers of mathematics the majority of teachers had not studied or intended to become mathematics teachers. in interviews and informal discussions with plesme teachers it became evident to me that the distinction between one’s identification as a teacher of mathematics or a mathematics teacher was a substantive issue which demanded focused attention (see graven, 2004). thus, while all plesme teachers were teachers of mathematics most were not mathematics teachers by training, identification or by choice. the challenge for plesme was therefore to help teachers to ‘become’ mathematics teachers in terms of mathematical competence and confident identification with mathematics teaching as their profession. plesme responded to this by strengthening teachers’ mathematical identities in two key ways. firstly it focused workshops on mathematical activities aimed at enabling teachers to: explore familiar mathematics topics in more depth; explore new mathematics topics introduced by the curriculum; explore mathematics problems and topics in relation to south africa’s socio, political and economic context; and to explore what it means to teach all of these in a learner-centred way. workshops thus developed mathematics content knowledge for teaching (shulman, 1986) and always contextualised discussions of ‘new’ pedagogies within mathematical contexts. the second was to maximise teacher participation in broader mathematics education communities through participation in conferences and field visits to overlapping communities. how is identity interpreted for this paper? according to lave & wenger (1991), learning is located in the process of co-participation and not in the heads of individuals; not located in the acquisition of structure but in the increased access of learners to participation, and it is an interactive process in which learners perform various roles. lave and wenger (1991) prioritise the importance of participation in the practices of a community and identity as primary features of learning. for them learning ‘implies becoming a full participant, a member, a kind of person…’ (p. 53) and is inseparable from a sense of identity. since participation in the practices of a community is essential for the development of identity (and therefore of learning) they refine the notion of community for the purposes of learning. they emphasise that to become a full member of a community of practice ‘requires access to a wide range of ongoing activity, old-timers, and other members of the community; and to information, resources, and opportunities for participation’ (lave and wenger, 1991: 101). from this perspective tools and techniques for learning are replaced with 'ways of becoming a participant' and 'ways of participating'. wenger (1998) characterises ‘identity’ as negotiated experience, community membership, learning trajectory, multimembership and as a relationship between our local and global ways of belonging to various communities. thus, he argues that we define who we are through: our participation and by the way we, and others, reify mathematics teacher retention and the role of identity: sam’s story 4 ourselves; the way we belong to various communities; where we have been and where we are going, and by combining various forms of belonging in various communities. wenger’s (1998) notions of learning trajectories, and ways of belonging and participating in various communities, are particularly key to the analysis of sam’s learning as changing identity. according to wenger (1998: 155), ‘as trajectories, our identities incorporate the past and the future in the very process of negotiating the present… learning events and forms of participation are thus defined by the current engagement they afford, as well as by their location on a trajectory.’ sam’s story will illustrate the way in which participation in plesme opens up new possibilities in relation to professional trajectories and ways of being. why focus the analysis of sam’s learning on the transformation of identity? identity is seldom the focus of in-service education and training (inset) or the focus of research on inset. i hypothesise that many inset providers view identity transformation as possible by-products of inset rather than part of the core business of inset. while designers and providers of inset might hope that through inset teachers will form positive ways of participating in and belonging to various related communities and hope that teachers will change who they are, their focus is usually restricted to changing what teachers know and do. this is possibly because designers and providers of inset view teacher identities (and their related identification and alignment to various communities) as being less controllable and less measurable than say teacher meanings and teacher practices. wenger (1998) himself notes that we can design teacher roles but we cannot design the identities that will form in uneven ways. but wenger (1998: 9) also notes that we need to reflect on all aspects of learning ‘when we choose to meddle with it on the scale which we do today’. inset, as we will see in sam’s story, ‘meddles’ with teacher learning in relation to teachers’ identities and their forms of participation in various communities. thus whether we view identity as a central component that should be a focus of the design and provision of inset or whether we see it as a by-product of a focus on practice and meaning, reflection on how this component interacts with teacher learning is essential. methods of data collection as explained above, the inset program plesme provided the empirical field for the study and the sample included ten teachers who participated in the program. in plesme i wore two hats. firstly, i was the co-ordinator of plesme. this was my full time vocation and i was accountable to my organisation, the university, the steering committee, donors, teachers and schools on the value and ‘success’ of the project. at the same time, i was a researcher in the process of conducting research on the nature of mathematics teacher learning in relation to inset within the context of rapid curriculum change. this duality of roles meant that i was both an ‘observer participant’ and a ‘participant observer’. working closely with teachers in plesme helped give form to the research and the research process and enabled more sensitivity and reactivity by myself in plesme. this study adopted a relatively longitudinal, qualitative ethnographic approach and the data collected by myself as the researcher included: teacher interviews, teacher questionnaires and classroom observations for each of the participating teachers. these took place approximately every six months throughout the two-year period of the inset. this data was supplemented with ongoing journal entries. while i collected data on all 14 participating plesme teachers, data analysis and the write up of the broader research study focused on ten teachers (see graven, 2002). these teachers were chosen because full data sets over the two-year period were available for them. i did not have full data sets for the other four teachers since they had either started late or did not teach mathematics at the senior phase (grades 7-9) at the time of the study. while the data analysis for all ten teachers revealed learning in relation to changing identity (graven, 2002), sam’s story was chosen for this paper because of his focus on his evolving mathematical identity throughout his description and explanation of his learning. the two primary sources of data that i use in this paper are interviews and questionnaires. there were, over the two-year period, four sets of interviews that all related broadly to teachers’ understanding and practice of the new curriculum and teachers’ understanding of their process of learning as related to plesme. interviews were semi-structured in the sense that they were conversations stimulated by a set of questions and probes. all interviews were conducted with teachers individually in an unoccupied classroom mellony graven 5 or office and were recorded for transcription. questionnaires asked similar questions to those in interviews and were useful in the sense that they provided another context for teachers to reflect on their practice sam’s story at the start of plesme in january 1999 sam taught primarily accountancy and business economics at a high school in eldorado park. sam had taught mathematics to grade 8 and 9 students in previous years but due to redeployments of teachers at his school he was forced to ‘fill the gaps’ and only taught one mathematics class. sam expressed that his intention was to remain in teaching for approximately five years and then move into a career in computers. the strengthening of sam’s identity as a mathematical being at the start of plesme in january 1999 sam struggled to make mathematical sense of various new mathematics specific outcomes as outlined in the new curriculum. for example: with (specific outcome) number 41, ‘critically analyse how maths relationships are used’ – that would be important for one of the outcomes, for instance specifics will come in here, where you can analyse specifics in certain diagrams and so on… as regards cultural products and so on you need mathematical things, shape, space and time…space and time? nothing comes to mind. it is too broad (sam interview, january 1999) by the end of plesme sam indicated both an enthusiasm and a strong ability to explain, with the use of examples from his classroom teaching, the mathematical meaning he made of the new mathematics specific outcomes. especially interesting in sam’s response to these outcomes two years later is the evidence of the development of a strong mathematical gaze on the world. this mathematical gaze emerged both in terms of 1 curriculum 2005 outlined 10 mathematical specific outcomes (sos). these included outcomes which contextualised mathematics in social, political, economic, ‘cultural’ and historical contexts. since these outcomes were new to teachers (i.e. they were largely absent in the previous curriculum), understanding these outcomes in relation to what they meant for teaching mathematics was a focus of the first six months of plesme workshops. in the excerpts included here sam is responding to his reading of: so4: “critically analyse how mathematical relationships are used in social, political and economic relations.” so8: “analyse natural forms, cultural products and processes as representations of shape, space and time.” finding ideas for mathematics teaching and as a critical mathematics gaze for approaching various aspects of his life such as checking the tax on payslips and comparing prices of different size packages in a supermarket. the following abbreviated extract indicates the development of sam’s mathematical gaze (with a critical stance) and of his strengthened identity as a person with mathematical expertise: okay this is one (referring to specific outcome number 4) i use quite a lot. it’s where for example in our nue (national union of educators) book one of the teachers came to me and said ‘listen they paid me wrong’… so she asked me now how much money. i said ‘okay this is the formula from the nue book so let’s see how much she has been paid’ and we went through that… and we sent a letter through… and the same with our increases they came to me and asked me how so so and so… and so when i get my pay-slip and i work out my money, the taxes, everything is fine. so i have the skill and the knowledge that they paid me correct… cricket as well if you look at your cricket scores… your bleach is normally 750 ml, now it gets to 1 litre and that changes your value… mg (myself as interviewer): ok so you think about that when you are buying the stuff? sam: ya mg: okay, in your teaching, what does this outcome mean for your learners? sam: like the cell phone project we went through it and they’ve seen what happened when you collect your data in order to make an informed judgement… with my surface area and volume when they wrapped their presents… even some of the taxi drivers i have seen them use a graph pasted on dashboard (sam interview, november 2000) this excerpt shows that sam clearly became a more confident ‘mathematical thinker’ who observed the world through a mathematical lens and used this lens to find ideas for mathematics teaching. furthermore the excerpt describes the way in which sam uses mathematical skills to critically analyse aspects of his environment: in order for me to question and change something i need to know it. if i don’t know it on a maths level i can’t change it. maybe that is why banks have lowered mathematics teacher retention and the role of identity: sam’s story 6 their interest rates now because someone questioned the interest… (sam interview, june 1999) sam’s changing identity in relation to his changing practice sam’s strengthened mathematical gaze and understanding of the various mathematical outcomes of the new curriculum influenced (and of course was influenced by) sam’s changing practice within teaching. in july 1999 sam applied for a mathematics post at a school outside of eldorado park (a well respected school in a predominantly white area). here he taught grade 8 and 9 mathematics classes and implemented various new curriculum ideas, contents and pedagogies which he had engaged with in plesme. at the end of the first year of plesme (november 1999) sam explained his changed practice as follows: i have experimented with groups’ sizes etc. pupils talk and explain to their peers. it is not so “quiet” in my classroom. i make better notes after and during the lessons. it is not so teacher-centred anymore. i will listen to different ways of getting to an answer. i know my pupils’ strengths and weaknesses and i can plan lessons accordingly. i know in what direction to move with my pupils. i know where they are – where to go and how i am going to get there (sam questionnaire, november 1999). in the second year of participating in plesme sam’s confidence in his teaching continues to grow. to illustrate sam’s increasing confidence over time and the development of his new ‘way of being’ in his classroom, i compare sam’s descriptions of his classroom practices in july 2000 to his description above. note that by july 2000 sam had been participating actively in plesme for 18 months. sam had thus spent 18 months debating various contents and methodologies espoused in the new curriculum and had experimented with them extensively in his teaching practice. by this time sam had also been teaching at his new school (as a mathematics teacher) for a year and his confidences within his school were clearly emerging. sam’s november description above clearly indicates changes in what sam knows and what sam does in the classroom. his increasing confidence in adopting more learner centred practices is implicit in his last three sentences which all begin with “i know …’. thus sam’s explanation of his learning is largely explained in relation to the components of meaning and practice. however, in july 2000, eight moths later, sam’s description of his changing practices moves beyond what he knows and does to explicitly include a range of affective terms, such as relaxed, better, confident and enjoy, which describe his changing way of being in the classroom. as a teacher i am much more relaxed… link maths to children’s everyday lives. i reflect on my lessons afterwards. try new and different things all the time. i am ten times better and confident than what i was two years ago. i enjoy my “maths” teaching so much “i will probably do it for a long time to come” (sam questionnaire, july 2000). the time that elapsed between the two interviews enabled sam to develop more experience both in terms of engaging in discussion about the new curriculum as well as in teaching it. sam’s descriptions of his changing classroom practice above indicate changing forms of participation and practice on both his part and on the part of his learners within the classroom community. for example, sam’s learners are encouraged now to explain their own methods resulting in increased engagement and more central participation2 in class. sam is more confident and relaxed in class and he now sees teaching as a learning process where he reflects on his lessons in order to adapt them and try new things in the future. sam’s changing identification by others in various communities in which he participates sam’s changing practices and new ‘way of being’ in the classroom results in (and is also the result of) a new identification within the school community. for example, sam explains: i must say you get your status as well in your community, it’s one of the good things, even with your children, your children see you as for example the other classes i don’t teach, the children say mr (sam) will you teach us next year for maths please? (sam interview, june 2000) this strengthened identity as a competent mathematics teacher who is able to implement new curriculum ideas and practices impacts on sam’s practice as a mathematics educator beyond his 2 central participation as opposed to peripheral participation which might involve a more passive or ‘quieter’ class where learners follow teacher methods rather than share their own ideas. this move to more central participation goes hand in hand with the movement from teacher-centred to more learner-centred practices. mellony graven 7 classroom and hence on his identification within the broader education community. for example, sam explained in interviews how he wished to work with other teachers to develop a mathematics project based on the 1999 elections, and added ‘i can do this because i know the outcomes from your (plesme) course’ (sam interview, june 1999). he explained how he had been asked to develop and co-ordinate learning materials and projects (related to various new curriculum outcomes) for all grade 8 and 9 classes in his school. sam further explained how his confidence influenced his interaction with others about mathematics education: like when you have your syllabus and your hod (head of department) will tell me i have to cover everything and then just to please him i must cover everything but now i am at the point where i can say listen there’s no need for me to cover everything. i’ve got the confidence and i say listen this is the time what i have, this is what’s important…(sam interview, november 1999) i can share my ideas with other teachers from other schools. i can join in with the obe (outcomes-based education) thing and express my opinions…yah, even when i am talking to parents now, its now where, how can i say, i can actually come up with examples and explain them’ (sam interview, november 2000) sam attributed his increased confidence to his belonging and alignment to his plesme group which allowed him to answer questions from others ‘and give them a good argument based on my pool of people i work with’ (sam questionnaire, july 2000). the quotes above are indicative not only of sam’s increased participation in the practice of sharing ideas with others but also illustrative of shifting relations between sam and members of the communities in which his participation increases. sam is now able to challenge others’ views and takes more ownership of his classes. sam increasingly identifies himself as a professional with the authority to make decisions in his classroom and to share information and ideas with others. at the same time others at his school, noting his competence and confidence, ask sam to take on more responsibility within the mathematics department. thus in a dialectical relationship sam’s strengthened identity as a more competent and knowledgeable professional leads to changing forms of identification within his classroom community and his school community. this in turn reinforces and strengthens the development of his identity as a more confident teacher with a greater investment in the profession of mathematics teaching. sam’s identity transformation and his multimembership in various communities as described earlier, alignment to the professional mathematics teachers association amesa was seen as an important part of plesme since it provided the opportunity for inducting teachers into the broader profession of mathematics education and extended the opportunities for teacher learning beyond the lifespan of plesme through membership in other more permanent communities. sam’s engagement and alignment to the amesa community was stated as a very important and enjoyable part of his learning. this is enthusiastically captured by sam as follows: i am not going “nuts.” i have met people at amesa who think like i do. what a relief…! i will present at this conference every year and will attend it every year. after five years of teaching this is the best thing i have experienced (sam questionnaire, july 2000) this quote provides insight into the way in which participation in and belonging to a professional association, such as amesa, reduces teacher isolation and provides a resource where teachers can engage with others about their ideas and their profession. it captures the importance of being able to engage with like minded people, to not feel so isolated within one’s practice and to find a shared purpose with others. the quote also indicates sam’s trajectory into the future when he says that he will attend the conference every year. furthermore participation in amesa leads to sam’s changing identification by others in various communities. thus, in the same way as sam’s identity and identification in terms of his participation within his school community changed, people in the broader community of mathematics education began identifying sam as someone with desirable expertise and enthusiasm. for example, at the end of 2000 sam was invited to apply for a job as a mathematics inset provider, he was asked by various mathematics educators to participate in their research projects, he was asked to give a presentation to mathematics teachers on his plesme experiences at a regional mini mathematics teacher retention and the role of identity: sam’s story 8 conference, and so on. the following year this recognition by others in the community of mathematics education continued. sam was offered the position of mathematics head of department at another school, he was asked to write pace setters for mathematics in the further education and training band (grades 10-12) and asked to make a teacher training video. all these examples are indicative of the changing status/identification awarded to sam by the broader community of mathematics educators. thus sam’s changing recognition and identification also resulted in (and was the result of) an increase in sam’s participation in professional activities relating to the broader profession of mathematics education. this recognition led sam to input into curriculum developments, trial curriculum materials, attend conferences, present at conferences, participate in discussions on current topical issues in mathematics education, develop learning materials and register for further studies in mathematics education. reflecting on sam’s identity transformation from a temporary teacher of mathematics to a professional ‘leader’ mathematics teacher all of the above parts of sam’s story come together to illustrate a profound transformation in sam’s identity from someone who temporarily taught mathematics while on a trajectory out of teaching to someone who identified himself strongly as a fully participating member of the professional community of mathematics educators with a great deal of expertise and enthusiasm to offer this community. thus sam explained that his five-year plan to move out of teaching was replaced by his ‘immersion’ in the profession of mathematics teaching. yeah because my main ideas when i started teaching five years ago i said i’m just going to teach for five years and that is it. and then i am going to go into my computers, because i’m coming to the end of my five years and i still want to do it. the computers are a hobby now (sam interview, november 2000) this immersion and sam’s new ways of belonging, ways of being and ways of participating in various communities kept sam invested in the profession and was accompanied by a long term trajectory. sam’s strengthened identity and longer-term trajectory is accompanied by confidence, enjoyment and enthusiasm for the profession as is clearly captured by sam in the following: i’m a maths teacher. i can work at a fast pace. instead of falling asleep, i’m busy and active, i calculate, i do what i do best, it’s maths…it (participation in plesme) makes me feel more secure in my job, where i can how can i say, it’s equipping me for obe (outcomes based education) and so on and redeployments. i won’t be the first on the (redeployment) list. because if i am first on the list i can always go up to the gde (gauteng department of education) and try and train some teachers and, yaah, i get satisfaction out of this. to me i’m at the forefront of the change. it’s one of my goals to be the best teacher, not just to get a salary (sam interview, june 1999). furthermore, as the above quote indicates, sam identifies himself as a leader in the field of curriculum implementation and change. this is clear when sam says, ‘i can always go up to the gde and try and train some teachers… i’m at the forefront of the change.’ similarly in a questionnaire of july 2000 sam explains his strengthened identity as a confident and competent mathematics teacher who is ready to lead others in curriculum change: i am ten times better and more confident than what i was two years ago. i enjoy my ‘maths’ teaching so much i will probably do it for a long time to come. i want to study and get my degree in maths education…i want to stay in the classroom…because of plesme i have options and i come to school with an even bigger smile… my mathematical sense has deepened. i can do lectures. i can conduct workshops. i think i am ready to work in eldo’s to help my fellow teachers to see what i have seen in maths education and maybe experience what i have experienced. i will present at this conference every year and will attend it every year (sam questionnaire, july 2000). these excerpts above indicate the projection of sam’s identity as a mathematics teacher/educator and leader in this field that is beyond present time. sam’s utterances indicate a clear mathematical trajectory that prioritises mathematics learning in the present and the future. sam’s trajectory involves continuing to increasingly establish himself as a leader in the field through further studies, ongoing participation, presentations at mellony graven 9 mathematics education conferences and supporting fellow teachers. in this way we see that sam’s identity, as a trajectory, ‘incorporated the past and the future in the very process of negotiating the present’ (wenger, 1998: 155). thus, sam’s story concurs with the research of grossman, wineburg and woolworth (2001: 996) who argue that communities provide opportunity for cultivating leadership where leadership is considered as ‘not a personality trait but an attribute of selfdevelopment in social relationships’. at the end of 2000, when sam was invited to apply to work in the field of mathematics teacher education, sam had to weigh up the extent to which his identity was more aligned to teaching mathematics to learners or working as a ‘leader educator’ with mathematics teachers. in the end, sam chose to remain in teaching, to register for the bachelor of science honours in mathematics teaching degree and to postpone working as a ‘mathematics teacher educator’ for later. sam qualified with his bsc hons degree two years later and is currently the mathematics hod at a school in eldorado park. he continues to be involved in mathematics education activities which extend beyond his school. notice that in relation to sam’s imagination3 of both his immediate and his longer-term future, mathematics is explicitly foregrounded. i have not argued that sam’s story is ‘typical’ of all the teachers in plesme. teacher identities indeed formed in uneven ways as teachers adopted new roles in relation to the new curriculum and their participation in plesme. the different trajectories of teachers were influenced by their mathematical and other histories. sam’s story was chosen because of his focus on his evolving mathematical identity throughout his description and explanation of his learning. grossman, wineburg and woolworth (2001: 942) refer to the essential tension between ‘professional development geared to learning new pedagogical practices and that devoted to deepening teachers’ subject matter knowledge in the disciplines of instruction’. indeed for some of the plesme teachers the subject specificity of mathematics tended to be backgrounded in their explanations of their learning through participation in plesme. however, all plesme teachers, like sam, explained their learning in plesme in terms of 3 according to wenger (1998: 173), identity involves ‘belonging’ to a community of practice. a central mode of belonging is ‘imagination’. imagination involves ‘creating images of the world and seeing connections through time and space by extrapolating from one’s own experience’. changing ways of being, new ways of participating in various education activities, new forms of identification and forms of belonging within various communities. in addition several teachers adopted similar trajectories to sam’s in the sense that they abandoned their past plans to exit the profession and replaced these with trajectories which increased their investment in the profession through participating in further mathematics studies. sam, like other teachers in plesme, linked his evolving identity to his participation and belonging in the plesme community. for example, he noted that participation in plesme made him feel more secure in his job and opened up new avenues, as captured by ‘because of plesme i have options’. thus belonging to the plesme community (and the overlapping communities which he participated in through plesme) provides the supportive context within which sam’s trajectory and identity evolve. this leads us to ask: what key aspects in the design of plesme supported sam’s (and other plesme teachers’) identity transformation? conclusions in concluding this paper i reflect briefly on two4 key aspects of plesme which sam and other teachers highlighted as important in the formation of their strengthened identities as confident mathematics teachers with longer-term trajectories in the profession. the first of these was a participatory approach to learning in which teacher professionalism was considered a key resource. the approach to learning that emerged within the plesme community of practice was one that assumed that the participating teachers were professionals with a great deal of experience to input into discussions about the new curriculum. the new curriculum was new to all and plesme would provide a community of practice in which teachers and presenters would jointly make sense of how the curriculum could and should be implemented. each would bring a different type of expertise into the community. while the discourse of such an approach might sound obvious and hard to reject, it was clearly not an approach which teachers had encountered in other inset contexts. thus the majority of plesme teachers, in describing their learning, highlighted the participatory approach and the professional identification they were 4 while there are many other aspects that could be elaborated on, for conciseness, i have focused only on those which were most prolific in teacher utterances. mathematics teacher retention and the role of identity: sam’s story 10 granted in plesme in contrast to approaches which ‘dictated’ what they should do. for example, sam noted, ‘what i pick up is how different the gde district treats us to workshops. here we are seen as a professional… (sam interview, june 1999). in wenger’s terms the participatory approach of the plesme practice would allow for teachers to move from peripheral participation (as in the case of teachers’ experiences of other workshops) to fuller participation in the plesme community and other overlapping communities. recall that for lave and wenger (1991: 53) learning ‘implies becoming a full participant, a member, a kind of person …’. the second key aspect on which teachers focused was the support they obtained through belonging to the plesme community and through their induction into other overlapping mathematics education communities. both of these aspects were enabled by the long-term and small-scale design of plesme. this design enabled teachers to become full participants and to develop a strong sense of belonging to the plesme community. the small scale of plesme allowed for strong relationships to develop between the members of plesme and allowed for the inclusion of ‘fieldtrips’ and financial support for teachers’ conference attendance. for all plesme teachers, belonging to plesme and to amesa were noted as key enablers of their learning and their changing ways of being. this belonging enabled teachers to draw on the community for support and also provided credibility5 outside of the community for their thoughts and views on mathematics education. sam’s story clearly illustrates that: ‘a community of practice is a living context that can give newcomers access to competence and also invite a personal experience of engagement by which to incorporate that competence into an identity of participation’ (wenger, 1998: 214) and for cultivating leadership. sam’s vignette has illustrated the mutual definition and interconnectedness of wenger’s learning components: identity (learning as becoming), community (learning as belonging), practice (learning as doing), meaning (learning as 5 recall sam explained that he could justify his arguments based on the pool of people he worked with (i.e. the plesme community) experience), and has also highlighted the centrality of confidence in this learning. the vignette furthermore reveals that sam’s strengthened mathematical identity increases his investment in the profession of mathematics education. this investment replaces his initial trajectory which involved exiting teaching and replaces it with a trajectory of ongoing learning and central participation within the field of mathematics education. as such his story highlights the importance of designing and implementing inset in ways that enable teachers to positively transform their identities through providing them with ‘access to a wide range of ongoing activity, oldtimers, and other members of the community; and to information, resources, and opportunities for participation’ (wenger, 1998: 101). while sam’s story is located within a mathematics inset context i believe that similarly inset in other learning areas should enable the investment and strengthening of teachers’ identities in relation to the learning area that they teach. references national department of education (nde), 1997, curriculum 2005: lifelong learning for the 21st century, nde, pretoria graven, m., 2002, mathematics teacher learning, communities of practice and the centrality of confidence, doctoral dissertation, faculty of science, university of the witwatersrand, south africa graven, m., 2004, “investigating mathematics teacher learning within an in-service community of practice: the centrality of confidence”, educational studies in mathematics 57(2), pp 177-211 grossman, p., wineburg, s. & woolworth, s., 2001, “toward a theory of teacher community”, teachers college record, 103(6), pp 942-1012 lave, j. & wenger, e., 1991, situated learning: legitimate peripheral participation, new york: cambridge university press shulman, l. (1986) “those who understand knowledge growth in teaching”, educational researcher, 15(2), pp 4-14 wenger, e., 1998, communities of practice: learning, meaning, and identity, new york: cambridge university press article information authors: caroline long1 sarah bansilal2 rajan debba2 affiliations: 1faculty of education, university of pretoria, south africa2department of mathematics education, university of kwazulu-natal, south africa correspondence to: sarah bansilal postal address: private bag x03, ashwood 3605, south africa dates: received: 27 june 2013 accepted: 23 apr. 2014 published: 26 aug. 2014 how to cite this article: long, c., bansilal, s., & debba, r. (2014). an investigation of mathematical literacy assessment supported by an application of rasch measurement.pythagoras, 35(1), art. #235, 17 pages. http://dx.doi.org/10.4102/ pythagoras.v35i1.235 copyright notice: © 2014. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. an investigation of mathematical literacy assessment supported by an application of rasch measurement in this original research... open access • abstract • background to the study • methodology • assessment and measurement • rasch measurement theory    • alignment of item difficulty and person proficiency on same scale    • dichotomous and polytomous item responses • requirements of the model    • item misfit • results from initial analysis    • individual item analysis       • item 3.2.3       • item 4.1.4c       • item 4.1.4       • item 1.3.2 • refinement of the instrument    • first round of rescoring    • results of first round of rescoring where the rescoring worked well    • results of first round of rescoring where the rescoring did not work well    • after resolving anomalies • implications for assessment in mathematical literacy    • allocation of marks for accuracy and method    • guessing    • item response dependency • concluding remarks • acknowledgements    • competing interests    • authors’ contributions • references • appendix 1 • appendix 2    • questions, marking guide and rescoring details • appendix 3    • category frequencies for each of the items • footnotes abstract top ↑ mathematical literacy (ml) is a relatively new school subject that learners study in the final 3 years of high school and is examined as a matric subject. an investigation of a 2009 provincial examination written by matric pupils was conducted on both the curriculum elements of the test and learner performance. in this study we supplement the prior qualitative investigation with an application of rasch measurement theory to review and revise the scoring procedures so as to better reflect scoring intentions. in an application of the rasch model, checks are made on the test as a whole, the items and the learner responses, to ensure coherence of the instrument for the particular reference group, in this case mathematical literacy learners in one high school. in this article, we focus on the scoring of polytomous items, that is, items that are scored 0, 1, 2 … m. we found in some instances indiscriminate mark allocations, which contravened assessment and measurement principles. through the investigation of each item, the associated scoring logic and the output of the rasch analysis, rescoring was explored. we report here on the analysis of the test prior to rescoring, the analysis and rescoring of individual items and the post rescore analysis. the purpose of the article is to address the question: how may detailed attention to the scoring of the items in a mathematical literacy test, through theoretical investigation and the application of the rasch model, contribute to a more informative and coherent outcome? background to the study top ↑ the subject mathematical literacy (ml), introduced in 2006 in south africa, is a compulsory subject for those grade 10–12 learners who do not study mathematics. the purpose in ml is not that learners learn more and higher mathematics: the emphasis in ml is on the use of mathematics to explore the meaning and implications of quantitative information presented in many real-life situations. the department of education (2003) defines ml as follows:mathematical literacy provides learners with an awareness and understanding of the role that mathematics plays in the modern world. mathematical literacy is a subject driven by life-related applications of mathematics. it enables learners to develop the ability and confidence to think numerically and spatially in order to interpret and critically analyse everyday solutions and to solve problems. (p. 3) ml differs from mathematics in purpose and in content. in mathematics, emphasis is placed on engaging with increasingly more complex and abstract mathematical concepts, the relations between them and some applications to problems. however, in ml the emphasis is specifically on the application of basic mathematics to understand situations in real life. there is some lack of clarity evident in the description of mathematical literacy noted above, as being mathematically literate requires a sound mathematical base of algebraic concepts and skills. the debate about the subject content of mathematics and mathematical literacy, though regarded as critical, is not the focus of this article. the juxtaposition of mathematics content with real life contexts has meant that many people are unclear about how competence or proficiency in ml may be demonstrated. it is clear that ml as a subject in its infancy requires much research in respect of teaching, learning and assessment, in order to generate debate and establish some consensus on the many contrasting perspectives within the ml field. in this study the focus is the grade 12 ml preparatory examination, which was set by a provincial department of education and is intended to prepare students for the final examination. we pay attention to one aspect, that is, assessment in ml, by identifying some issues arising from the analysis of the empirical data obtained from learners’ responses to this provincial preparatory assessment. the construct under scrutiny is the notion of proficiency in the subject ml. we apply rasch measurement theory (rmt) to investigate the validity and accuracy of the test in providing a measurement-like representation of ml proficiency in terms of person proficiency and item difficulty. a valid and reliable test would provide teachers with some indication of the levels of mastery of curricular elements and of developing proficiency in ml. it should also provide the department of education with an overview of the entire learner cohort taking ml. an application of the rasch model will help us to identify anomalies and inconsistencies amongst these assessment items and the accompanying scoring rubrics and working memoranda. in this article we consider the implications of considering the purpose of the test and the construction of rubrics so that they work coherently in the interests of valid measurement-like properties and consequently provide reliable information for teachers. some concepts underlying the rasch measurement theory are introduced to clarify the analytic process. the aim of the article is to investigate the domain of ml and offer some observations concerning the assessment of ml. methodology top ↑ this study, focused on the scoring of items, is part of a larger study on ml (debba, 2012). the instrument investigated here comprised 51 items, two of which were dichotomous items, marked either correct or incorrect. twenty items had a maximum of two marks, 14 items a maximum of three marks, 9 items had four marks and 6 items had five marks. the maximum possible score was 150. the participants in the study were 73 grade 12 ml learners.1the grade 12 kzn provincial preparatory ml examination paper is intended to assist grade 12 learners in their preparation for the final examination. it is set by a team of examiners selected by the education department and written under examination conditions. for the purposes of this study the ml 2009 preparatory test was re-marked by the third author to ensure that the final version of the marking was entirely consistent with the marking memorandum supplied by the kzn department of education. a rasch analysis supported the investigation of the test as a whole, the items and the ml learner responses. this analysis was conducted to identify factors that may have affected the coherence of the instrument for this sample of learners. the first requirement for this analysis was to capture the score obtained in each of the 51 items for each of the 73 students.2 the rasch model offers various statistics to help diagnose where the data differs from what is expected by the model. multiple means are applied to an analysis of this nature, to enable the subject expert to make an informed judgement. the output provides statistics on the test as a whole as well as the individual item statistics, in particular the fit residual statistic and the chi square probability statistic, which provide information on the fit of the items. in addition to these statistics we investigated the item characteristic curves (iccs) to identify which items were misfitting in the ways to be discussed. the research question directing the study is: how may detailed attention to the scoring of the items in a mathematical literacy test through theoretical investigation and the application of the rasch model contribute to a more consistent outcome? assessment and measurement top ↑ we note that mathematical literacy and its assessment have been introduced relatively recently into the south african high school curriculum. we agree with matters (2009) that assessment in the 21st century has a powerful influence, but this influence is only warranted if the assessment is of a sufficient quality to support the inferences, in this case the inferences about the mathematical literacy proficiency of learners that are drawn from the test results. the assessment process involves the theoretical exploration of the construct of mathematical literacy, the operationalisation of the construct in items designed to gauge proficiency, the compilation of a test instrument and the administration and marking.from the classical theory of measurement, and measurement in the physical sciences (wright, 1997), we note that the property of invariance of comparisons across the scale of measurement is a requirement. the application of the rasch model enables the calibration of item measures and the estimation of person locations on a common continuum that together fit the criteria of invariance for a particular frame of reference (rasch, 1960/1980; humphry, 2005; humphry & andrich, 2008). the comparative difficulty of any two items should be constant regardless of the abilities of the persons responding to the items. where the data do not conform to the measurement principles, the model will highlight the anomalies for further investigation. in the current study, the application of a rasch analysis highlighted anomalies and inconsistencies that constituted threats to the construction of measures in the sense understood in classical measurement theory. in particular, the allocation of marks was inconsistent with the grading of proficiency along a continuum. in this article, we investigate the outcomes of both the initial scoring memorandum and the revisions, utilising both rasch analysis and the educational considerations. rasch measurement theory top ↑ the fundamentals of rasch measurement theory (rmt) are covered in many publications (andrich, 1988; rasch, 1960/1980; wright & stone, 1979, 1999). here we note that with rmt there is an assumption that for the construct of interest there exists a latent trait in the learner that may be gauged through the operationalisation of the construct in various items. both learner ability, denoted by βn, and item difficulty, denoted by δi, may be represented on the same scale. this explanation is presented as follows by dunne, long, craig and venter (2012): each outcome of an interaction between a person and an item is uncertain but has a probability governed only by these two characteristics, that is by person ability (βn) and item difficulty (δi). the rasch model avers that the arrays of numbers βn and δi are on the same linear scale, so that all differences between arbitrary pairs of these numbers such as (βn − δi) and hence also (βn − βm) and (δi – δj) are meaningful. through these differences we may not only assign probabilities to item outcomes but also measure the contrasts between ability levels of items, and offer stochastic interpretations of these contrasts. (p. 7) alignment of item difficulty and person proficiency on same scale we have noted that a key feature of the rasch model is that the difficulty of items is located on the same scale as the ability of the persons attempting those items, precisely because the construct of interest underpins both the design of the items and the proficiency of learners. the focus of the model is on the interaction between a person and an item and is premised on the probability that a person v with an ability βv will answer correctly, or partially correctly, an item i of difficulty δi. the equation that relates the ability of learners and the difficulty of items is given by the logistic function: this function expresses the probability of a person v with ability βv responding successfully on a dichotomous item i with two ordered categories, designated as 0 and 1. here p is the probability, xvi is the item score variable allocated to a response of person v on dichotomous item i, x is the response, either 0 or 1, βv is the ability of person v and δi is the difficulty of item i (dunne et al., 2012). applying equation 1, we can see that if a person v is placed at the same location on the scale as an item i, then βv = δi, that is, βv − δi = 0, and the probability in equation 1 is thus equal to 0.5 or 50%. thus, any person will have a 50% chance of achieving a correct response to an item whose difficulty level is at the same location as the person’s ability level. similarly, if an item difficulty is above a person’s ability location, then the person has a less than 50% chance of obtaining a correct response on that item, whilst for an item whose difficulty level is below that of the person’s ability the person would have a greater than 50% chance of producing the correct response. in figure 1, the person location is represented on the horizontal axis, with the probability of a correct response located on the vertical axis. figure 1: category probability curves (item 2.2.3). dichotomous and polytomous item responses the rasch model was initially developed for the analysis of dichotomously scored test items.3 however, in many cases, tests require items that are scored at graded levels of performance. rasch (1960/1980) extended the model for dichotomously scored items to include a model for test items with more than two response categories, with possible scores of 0, 1, 2, … m4. these items are termed polytomous items. figure 1 models the conditional probability of a score of 0, 1 or 2 for a polytomous item (item 2.2.3a) with three categories. as the person ability increases, the conditional probability of a score of 0 decreases. by contrast, as ability increases the probability of obtaining a maximum score of 2 increases. also on this graph is the curve that shows the probability of a score of 1. in summary, this curve shows that when a person has very low ability relative to the item’s location, then the probability of a response score of 0 is most likely; when a person is of moderate ability relative to the item’s location, then the most likely score is 1 and when a person has an ability much greater than the item’s location, then the most likely response score is 2 (see also van wyke & andrich, 2006, p. 14). in figure 1, the thresholds,5 and the categories they define, are naturally ordered in the sense that the threshold defining the two higher categories of achievement is of greater difficulty than the threshold defining the two lower categories of achievement. the first threshold (τ1), which represents the point where a score of 1 becomes more likely than a score of 0, is about –1.10 logits. the second threshold, where a score of 2 becomes more likely than a score of 1, is approximately 1.25 logits. these thresholds show that progressively more ability is required to score a 0, 1 or 2 respectively on this item (van wyke & andrich, 2006, pp. 13–14). requirements of the model top ↑ we have noted that the central proposition for the rasch model6 is that the response of a learner to a dichotomous item is a function of both the item difficulty and the person ability and nothing else. the probability of a person achieving success on a particular item is entirely determined by the difference between the difficulty of the item and the learner’s ability.< the principle underlying the rasch model is: [a] person having a greater ability than another person should have the greater probability of solving any item of the type in question, and similarly, one item being more difficult than another means that for any person the probability of solving the second item is the greater one. (rasch, 1960/1980, p. 117) in rmt it is expected that the data will accord well with the model. the notion of ‘fit’, that is, accord with the model, is defined as ‘the correspondence between a data set and a statistical model’ (douglas, 1982, p. 32). the model provides indicators that alert researchers to where this principle of invariance of comparisons is not being met, which may result in item misfit. the fit residual is a measure of the difference between the observed response of each person to each item and that predicted by the model. the analysis process, whether showing a degree of conformity with the model or not, inevitably leads to greater understanding of the construct in question. item misfit as noted, it may be observed that the items are working well and are a good indicator of the learners’ proficiency. it may also be the case however that some items are highlighted as problematic. in subsequent sections we refer to particular examples where we focus on item functioning and the scoring rubrics. in some of the examples, the rasch model analysis confirms that the scoring rubric is working as required by the model. in other items, the analysis discloses that the scoring rubric is not working in an ordinal way.in a rasch analysis test of fit, the learners are placed into class intervals of approximately equal size. we have used four groups. the mean ability of the four groups becomes the horizontal coordinate of points in the diagrams, depicting the probability of answering correctly. where the data conform to the model, the theoretical curve (the expected frequencies) and the observed proportions (the empirically established average of the actual item scores in the four chosen groups) are in alignment. figure 2 shows the theoretical curve as expected by the model and the observed proportions, represented by black dots. figure 2: item characteristic curve for item 2.1.3, indicating fairly good fit. where the theoretical curve and the observed proportions are in alignment we assume fit to the model, but where the theoretical curve and the observed proportions deviate substantially we are alerted to some kind of misfit between the data and the model. there are four broad categorisations that describe how the observed proportions might relate to the theoretical expectation. in this section we describe a selection of items that fall into the categories of fairly good fit, under-discrimination, over-discrimination and haphazard misfit. firstly, the observed proportions may align with the theoretical curve, in which case there is a good fit to the theoretical requirement. figure 2 shows the item characteristic curve (icc) for item 2.1.3, in which the observed proportions are aligned fairly well with the theoretical curve. note that the fit residual, 0.601, is relatively small tending towards zero and within an acceptable range of good fit (–2.5 to +2.5). this relatively small residual means that the difference between the observed response of each person to each item and the expected response is small. a second phenomenon may be that the observed proportions are flatter than the theoretical curve, in which case the item does not discriminate enough. the pattern is labelled under-discrimination or underfit and is illustrated in figure 3. this unexpected pattern indicates that learners of lower proficiency appear to perform better than expected on this item and consequently, because of the interactive nature of item difficulty and learner ability, the high proficiency learners are falsely estimated to respond to the item as if the item was easier than it really was. the qualitative analysis suggests that a possible explanation lies with the marking rubric, which gives scores between 0 and 3. the scoring rubric allocates an arbitrary method mark and an additional mark for presenting information provided in the instruction. the allocation of marks appears to be more generous for the poorly performing learners and too constrained for the higher performing learners. figure 3: item characteristic curve for item 3.2.2, indicating under-discrimination. figure 3 presents the icc for item 3.2.2, which shows the observed proportions to be flatter than the expected theoretical curve. the fit residual indicating difference between observed response and that expected by the model is relatively high at 2.410.a third general category occurs when the observed proportions are steeper than the theoretical curves, in which case the discrimination is greater than expected, as shown in figure 4. over-discrimination in an item may unduly advantage high proficiency learners, whilst disadvantaging learners of lower proficiency. whilst traditional test theory asserts the greater the item discrimination the better, the case in rmt is that highly discriminating items provoke a concern that there is a marked dependence amongst responses in one form or another. an example of such a misfit is that of item 1.3.2, shown in figure 4. again we note that the fit residual is relatively high at –2.613. both a high negative fit residual and a high positive fit residual signal poor fit to the model. figure 4: item characteristic curve for item 1.3.2, indicating over-discrimination. the fourth general category occurs when the observed proportions are haphazardly but substantially different from the theoretical requirement, as in figure 5. this pattern demands specific investigation of the construct, an examination of the suitability of the item or the identification of another educational explanation. after analysis, item 4.1.1 was deleted from the test on the grounds of its misfit. this excision is discussed in the section refinement of the instrument. note that here the fit residual is 3.321, indicating a fairly large deviation from the model that should be investigated. figure 5: item characteristic curve for item 4.1.1, indicating haphazard misfit. results from initial analysis top ↑ from the initial rasch analysis, the summary statistics (table 1), person-item location distribution (figure 6) and person-item threshold distribution (figure 7) were generated. table 1 presents the initial summary statistics, which report the item mean as 0 (as set by the model) and the person mean as –0.2537. the standard deviation for the item location is 1.1378, whilst the standard deviation of the person location is just 0.3988. this contrast suggests that the spread of the items is high whilst the person locations are clustered together. cronbach’s alpha and the person separation index both indicate internal consistency reliability. table 1: summary statistics prior to rescoring. figure 6: the person-item location distribution prior to rescoring. figure 7: the person-item location distribution prior to rescoring. figure 6 illustrates the person-item location distribution (pild). the item location mean is set at zero; the person location mean is calibrated at –0.254. the item locations range from –2.2035 to 4.565 logits. the person locations are estimated between –1.414 and 0.441. the fact that the person location mean is lower than the item location mean suggests that the test was difficult for this particular learner cohort. reasons for the mismatch7 may be posited, for example the test questions might have been more easily answered if the cohort had been afforded more experience in basic algebra. discussion of explanatory conditions and factors may be found in debba (2012). the person-item threshold distribution (pitd) (figure 7) shows that the spread of the item threshold location ranges from –22 logits up to 22 logits, whilst the person location is from –1.4 to 0.4. the pitd representation indicates the distribution of the various categories in the items, for example an item that was weighted at 3 marks will have three thresholds. this wide distribution suggests that some of the 51 items may have been awarded too high a score. this is supported by the fact that for many items several of their possible score values between 0, 1, 2 … m, were not observed in this class of 73 learners or observed only once. there are at least 40 problematic thresholds. see the frequency chart (appendix 3).clearly this odd arrangement of thresholds and persons is out of alignment with what is expected of a balanced assessment. the items as originally conceptualised are not distinguishing the intended range of proficiency levels in this particular set of learners. the detailed discussion of thresholds, and the disordering of thresholds is not presented here. see andrich (2012) for a detailed discussion. both the summary statistics (see table 1) and the pitd (see figure 7) indicate some disjuncture between the items and the persons suggesting that there are some anomalies in the data. further investigation is required for both items and persons in terms of fit to the model8. in this article, and in the next section, we focus only on the possible anomalies that arise due to the scoring of items. individual item analysis in this section we present a short discussion about problems identified in particular items. we describe how the problems were highlighted and how the qualitative verification of measurement problems prompted rescoring. three items are discussed, firstly item 3.2.3, an example of an item that showed haphazard misfit (see figure 5), item 4.1.4, which shows how the allocation of two marks is not warranted, and item 1.3.2, which shows disordered thresholds. item 3.2.3 item 3.2.3a forms part of a question with a farming context (see appendix 2). the task is to determine whether doubling the dimensions of the bucket will double the volume of the bucket. the question requires a yes or no answer, and in requiring this response may not gauge the understanding of dimensions or of volume: the icc for item 3.2.3a (figure 8) shows a haphazard misfit, with learners of lowest proficiency on the test as a whole having an almost equal chance of obtaining a correct answer as learners of high proficiency on the test as a whole. information provided here and the qualitative investigation suggests a revision of wording of this question. it is possible that learners at the lower end of the proficiency scale could have randomly chosen yes or no. figure 8: item characteristic curve (item 3.2.3a). item 4.1.4c item 4.1.4c requires that the learner give two causes for the observed relative change in a child’s weight. the scoring rubric allocated two marks per reason. it was found that no learner obtained 1 mark without obtaining 2 marks and similarly no learner obtained 3 marks without obtaining 4 marks. the flat category curves in figure 9 show that the categories 1 and 3 are not functioning at all and category 2 is not functioning well. this outcome reflects the fact that the learners either got 0 marks (for providing no reason) or 4 marks (for providing two reasons). only rarely did a learner offer only one reason. figure 9: category probability curves for item 4.1.4.c. item 4.1.4 item 1.3.2 in item 1.3.2 the learner is required to calculate the deduction to his wages. this item was part of a broader problem context (item 1.3), which required calculating john’s wages, where the hourly pay was provided, the number of hours worked per day and the percentage deducted. the item itself is problematic as it depends on the learner answering the previous item (1.3.1) correctly. in addition, the scoring of part marks is odd, as once the learner identifies the 2,2% total deduction, they are likely with the help of a calculator to obtain a correct answer. the category probability curves for item 1.3.2 (figure 10) are used to illustrate disordered thresholds. figure 10 shows that the location of the first threshold (τ1) (the intersection of the curves for score 0 and score 1) has a difficulty of approximately 0.83 logits. however, the location of the second threshold (τ2) (the intersection of the curves for a score 1 and score 2) has a difficulty of approximately –0.15 logits. the location of the third threshold (τ3) (at the intersection of the curves of scores 2 and score 3) is approximately –0.68 logits. figure 10: initial category probably curves for item 1.3.2. the problem is that the location of the first threshold is greater than the location of the second threshold, whilst the location of the second threshold is also greater than the location of the third threshold. these reversed thresholds are due to the failure of the two middle categories, corresponding to scores of 1 and 2, to function as intended. at no point on the horizontal axis is a score of 1 most likely; neither is there an interval or point where a score of 2 is most likely. although persons with low ability relative to the item’s difficulty are still most likely to respond incorrectly and score 0 and persons with high ability relative to the same item’s difficulty are still most likely to respond correctly and score 3, persons with ability in the range –0.68 logits to 0.83 logits, where a score of 1 or 2 should be most likely, are more likely to score either 0 or 3. this disordering is evident where the high middle group is more likely to score 1 and where the low middle group is more likely to score 2.the disordering of the thresholds confounds the idea that thresholds between higher level categories are more difficult to attain than thresholds between lower order categories. these initial analyses help us to identify possible anomalies and inconsistencies in the scoring rubrics, which can alert us to possibilities that should be considered when devising scoring rubrics. there are strategies we can use post hoc to adjust the item scoring in order to exhibit ordered thresholds as shown in item 2.2.3a (figure 1). the verification that scoring rubrics are functioning as expected, and subsequent revision where necessary, contribute to the reliability of the ml examination paper in the context of this set of 73 learners. refinement of the instrument top ↑ in addition to looking at misfit statistics, we studied the associated category probability curves for each item, to further explore anomalies in the data. when we investigated the category probability curves for each item, we found that in most cases the thresholds were disordered. in the process of refining the instrument, iterative adjustments took place. the first step was to identify items that showed severe misfit according to the fit residual statistics and the chi square probability. in addition the iccs were investigated. where there were indications of anomalies we checked the item itself and the scoring rubric to identify any problems from both a mathematics education perspective and an assessment perspective. where the qualitative investigation confirmed the anomaly and it was deemed proper to adjust the scoring of the item, this step was taken. the item statistics were then reinvestigated by reanalysing and by rechecking the fit to the model with the revised scores. if the fit had not improved we would reverse the change. if there was no theoretical reason to support the rescoring of an item, then no rescoring took place. the details of the various items, together with the marking memorandum provided by the department of education, and details of the rescoring appear in appendix 2. first round of rescoring the scoring for the two dichotomous items, item 2.2.1b and item 3.2.3a, was retained. all other items were rescored according to the process identified above. one of the items, item 4.1.1 (figure 7), showed extreme misfit when investigating the icc. in addition the fit residual statistic (3.321) was outside the generally acceptable interval of between –2.5 and 2.5 and the chi-square probability (0.002) showed that the expected and the observed outcome were statistically significantly different. this item warranted further investigation to see whether the problem lay with the scoring and whether rescoring may resolve the misfit. after studying the evidence and finding that there were problems with the item which resulted in the item not contributing to the test, we deleted the item from the test. in item 4.1.1 learners were given four graphs from a growth chart used by parents and health workers to monitor the weight gain of infants. they were asked to find the normal weight of a baby boy at birth. two curves represented the weight of boys whilst two represented the weight of girls. the poor quality of the graphs and unclear titles contributed to the difficulties with this item. in addition, it was not clear how these graphs could be used to provide information about a ‘normal weight’. the assessment task did not make sense in the real-life context. this confusion of meanings illustrates a fundamental tension that exists between the intentions of assessment task designers and learner participants in the real-life context. the questions posed in the examination paper may not be the ones that are posed in the context of health workers who use growth charts to identify children whose health is at risk. results of first round of rescoring where the rescoring worked well after refining the scoring, we found that in several cases the rescoring improved the fit, whilst in few other cases it did not. the rescoring of item 1.1.3 (see box 1) resulted in the improved fit, the adjustment more closely approaching measurement principles. we provide the educational rationale for a change in scoring and also show the category curves both before rescoring and after rescoring to show how the rescoring helped improve the functioning of the categories. box 1: details for item 1.1.3. figure 11 shows that before the rescoring most of the categories were not working as they should, that is the allocation of scores 2 and 3 appeared somewhat redundant. furthermore, the icc for item 1.1.3 in figure 12 shows unduly high discrimination, labelled overfit. this problematic outcome may be explained as follows: learners obtaining the item’s answer correctly are unduly advantaged by scoring 4 points, whereas those answering incorrectly are unduly disadvantaged by ‘losing’ 4 possible points. figure 11: initial category probability curve for item 1.1.3. figure 12: initial item characteristic curve for item 1.1.3. the item, was rescored, moving from a five-category item (scoring 0, 1, 2, 3, 4) to a four-category item (scoring 0, 1, 2, 3). category 1 remained the same, category 2 and category 3 were recoded as 2, whilst category 4 was recoded as 3. a qualitative investigation in this case indicates that scoring 0, 1, 2, 3 can be justified. the details of the item and rescoring appear in box 1. in each case where the categories were not working according to the model, we applied multiple criteria before revising the scoring, keeping in mind principles of best test design (see van wyke & andrich, 2006; wright & stone, 1979). after this rescoring process, both the category curves (figure 5) and the icc (figure 6) indicated better fit according to the model. comparing figure 11 and figure 13, one can see that the categories are now working more appropriately. figure 14 shows the icc for item 1.1.3 after the rescore. when comparing figure 12 and figure 14, one can see that the fit between the observed and expected means for the persons in the four class intervals is much improved. a final check of the fit residual statistic and corresponding chi square values indicates that initially the fit residual statistic was –0.794 (chi square probability = 0.4317), whilst the final fit residual statistic is 0.082 (chi square probability = 0.8281), showing that the change affected the fit statistics positively. also note that with rmt, a p-value higher than 0.01 for the fit statistics indicates that the difference between the observed and expected is not statistically discernible and hence that the model is working well. figure 13: final category probability curve for item 1.1.3 figure 14: final item characteristic curve for item 1.1.3. results of first round of rescoring where the rescoring did not work well as explained earlier, item 4.1.1 was rejected as the fit was extreme and in addition it was judged to be a very poor indicator of mathematical proficiency. in addition, item 3.2.2, after rescoring, was now also identified as having extreme fit statistics outside of the acceptable interval. in a second process we investigated each icc again, and then the category probability curves, and found that the thresholds were disordered for items 1.2.1b, 1.2.2, 1.3.1, 1.3.2, 3.2.1, 3.2.3b, 4.1.4b, 4.1.4c, 5.1.4 and 5.3. (see appendix 2 for details of these items.) at this stage, we re-examined the questions to see whether we could justify rescoring these items again. we also examined the fit residual statistics to help us decide whether a rescoring was necessary or not. a summary of these processes and decisions appears in appendix 1. after resolving anomalies after the refinement of the scoring using information from both the qualitative investigation and the rasch analysis, we now examine the final summary statistics in table 2. table 2: final summary statistics. the figures in table 2 when compared to those of table 1 show improvement across most of the statistics. for example, cronbach’s alpha, a reliability index, has increased from 0.8845 to 0.8887. the standard deviation for the item location has increased to 1.615, showing that the items now have a larger relative spread. we note as well that the differences observed in the new person-item location distribution (pild) (figure 15) are an improvement on the initial pild (figure 2). figure 15: person-item location distribution after refinement to instrument. the distribution represented in figure 15 indicates a better balance than the original distribution represented in figure 2. we can see that the standard deviation for person locations has increased in the person-item location distribution from 0.399 to 0.808 (see figure 6 and figure 7). the person location spread was initially between –1.414 and 0.441. after the rescoring the spread of learner locations ranged between –2.460 and 1.176, thus providing greater discrimination of learner proficiencies. we judge the post-rescoring estimates of learner proficiencies to be more accurately reflected by the refined scoring.furthermore, one can see that the most difficult item, item 5.1.3, was far too difficult for this group of learners. its estimated difficulty level, calibrated at 5.448 logits, is much higher than the ability level of the most proficient learner in this test, who was estimated just above 1 logit. the items 4.1.2, 4.1.3, 4.2.1, 4.2.2, 5.1.4, 5.2, 2.1.2 and 3.2.3a were also beyond the ability level of the learners, but the difference δi – βv for this group of items was not as large as it was in the case of item 5.1.3. item 5.1.3 required learners to work out the price of an item before vat, given the final price including vat. the usual solution involves setting up an equation of the form a + 14% of a = final price, and they were required to find a. this item required the setup and solution of an equation and involves applying algebraic techniques. the ml curriculum document makes it clear that skill in such algebraic manipulation is not a focus of ml. the document states: ‘as a rule of thumb, if the required calculations cannot be performed using a basic four-function calculator, then the calculation is in all likelihood not appropriate for mathematical literacy’(department of basic education, 2011, p. 8). given such a stipulation, and noting that the item was far beyond the ability of this group of learners, it is recommended that a greater use of basic algebraic manipulation skills should be encouraged, to enable them to solve items such as item 5.1.3. we note that this item could also be solved by informal methods based on reasoning about percentage change or using proportion and perhaps opportunities for such reasoning should also be encouraged. on the other hand, three items, 4.2.3, 3.1.1 and 2.2.1a, were lower than the ability level of all the learners in the class, but the location of the items was less than 2 logits lower than the person with the lowest total score. this difference contrasts with the finding that in the higher ability ranking, one item was almost 5 logits higher than the location of the person with the highest total score, with two items being 2 logits or more higher and six other items being higher than the location of the person with the highest total score. we now present the person-item threshold distribution that was generated after the rescoring process. there is a marked difference between the initial pitd (figure 7) and the final pitd (figure 16). the distribution of the thresholds prior to rescoring ranged between –22 and +22 logits (see figure 7). now after rescoring the threshold distribution has been limited to between –4 and 8 logits. the number of thresholds has been reduced from 150 to 75 by the rescoring processes (see appendix 3). figure 16: person item threshold distribution post-rescoring. implications for assessment in mathematical literacy top ↑ in this study we examined the responses of 73 students to each of 51 items by using tools provided by the rumm2030 software (andrich, sheridan & luo, 2011). the analysis revealed some important issues, which are discussed below. allocation of marks for accuracy and method we have discussed the case of item 1.3.2, where marks were allocated for method and for consistent accuracy. this item was just one instance of many where the method and accuracy system did not work well. in fact, it unduly disadvantaged the person who did not address the question correctly. most learners who identified the method were able to obtain the correct answer because it involved just entering the numbers into the calculator and reading off the answer. the learners who answered correctly were thus unduly advantaged by getting additional marks. there were very few learners (too few as revealed by their respective category probability curves in figure 10) who achieved the method mark without achieving the accuracy mark. another related issue was in the case of item 4.1.4c, which asked for two causes of an observed change in the data. the scoring rubric allocated 2 marks per reason. the category curves in figure 9 revealed that the categories 1 and 3 were redundant. no learner attained the first mark without getting the second mark and similarly no learner attained 3 marks without getting the fourth mark. guessing questions that allocated a mark for answering yes or no to a question and then asked for a reason resulted in a haphazard response that could indicate guessing at the lower proficiency level. this pattern was revealed in some items by the icc where there was a haphazard misfit, such as in item 3.2.3 (shown in figure 10) and item 1.3.3 (not shown). we make a distinction here between the learners who know the answer to the question and the group who guess. it is the latter group who have a 50% chance of answering correctly if they guess. item response dependency in some cases, the answer to a previous item influenced the probability of success of the learner in a following item, as in the case of item 2.1.2a and item 2.1.2b as well as item 1.2.1a and item 1.2.1b. those learners who were able to get the first item correct were likely to get the second one correct. none of the learners who answered the first part of item 2.1.2a incorrectly achieved full marks in the second part of the item, item 2.1.2b, and none of the learners who answered item 1.2.1a incorrectly was able to obtain full marks in item 1.2.1b. the dependency of a subsequent item on an earlier item is regarded as unfavourable test practice. it is the independence of items that offers greater precision. concluding remarks top ↑ in this article we illustrated how the rasch model could be used in conjunction with professional judgment to check the validity of the assessment, by using the responses of 73 grade 12 ml learners to their preliminary examination. the process described in this article involved identifying items that did not fit the model. we described the items and the original scoring rubric. the rasch output was provided and the anomalies and inconsistencies were discussed. before initiating any changes in the scoring we sought educational reasons that warranted rescoring. thus, the rasch analysis and rescoring processes were guided at all times by the qualitative analysis that was conducted by an experienced ml teacher.an important advantage of the rasch analysis is that item difficulty and person proficiency are located on the same scale. without checking individual item validity, we cannot take the results at face value; instead, we need to verify the item fit. now that we have subjected the scoring of each item to rasch analysis and qualitatively investigated the structure of the scoring rubrics, we argue that the new scoring procedures for the instrument allow greater precision than the original instrument scoring. it must be emphasised that the scoring rubrics were not the only threats to validity in this test. the process that was conducted, which involved systematically rescoring each of the assessment items, has revealed the important role of scoring rubrics in contributing to the validity of assessments. examiners need to ensure that each mark that is allocated can be justified educationally. marks should not be allocated on the basis of time that is required to be spent during the examination. neither should marks be allocated for guessing. if these checks are not taken into account, then the total score has diminished meaning. if the total score is indicative of a position on a unidimensional scale, then differences between the score must reflect differences between proficiency levels of learners. a further process that could be followed is to study which items were experienced as more difficult and which items were experienced as easier in order to obtain a description of items on different levels of the scale. it may then be possible to identify different demands of the various questions. a study of a similar nature has been conducted by long (2011) who presented a comprehensive application of rmt to the multiplicative conceptual field. in her study, long used the rasch model to develop clusters of proficiency zones along the continuum representing the alignment of person proficiency and item difficulty. it may be the case that particular items are mathematically demanding because they require sophisticated use of algebraic tools (as in item 5.1.3). perhaps the increasing demand of the questions may be related to the contexts that were used. these analyses will be conducted and reported in a follow-up article. acknowledgements top ↑ we would like to acknowledge the valuable insights on the application of the rasch model to this study provided by professor tim dunne. competing interests we declare that we have no financial or personal relationship(s) that might have inappropriately influenced us in writing this article. authors’ contributions the data was collected by r.d. (university of kwazulu-natal) as part of his post-graduate studies. c.l. (university of pretoria) led the process of the rasch analysis, helped by s.b. (university of kwazulu-natal) and r.d. the write-up was done by all three authors, led by c.l. references top ↑ andrich, d. (1978). a rating formulation for ordered response categories. psychometrika, 43(4), 561–573. http://dx.doi.org/10.1007/bf02293814andrich, d. (1988). rasch models for measurement. newbury park, ca: sage publications. andrich, d. (2012). an expanded derivation of the threshold structure of the polytomous rasch model that dispels any “threshold disorder controversy”. educational and psychological measurement, 73(1), 78–124. http://dx.doi.org/10.1177/0013164412450877 andrich, d., sheridan, b., & luo, g. (2011). rumm2030 software and manuals. perth, australia: university of western australia. available from http://www.rummlab.com.au/ debba, r. (2012). an exploration of the strategies used by grade 12 mathematical literacy learners when answering mathematical literacy examination questions based on a variety of real-life contexts. unpublished master’s thesis. university of kwazulu-natal, durban, south africa. department of basic education (dbe). (2011). curriculum and assessment policy statement (caps). mathematics grades 10–12. pretoria: dbe. department of education (doe). (2003) national curriculum statements grades r–9 (schools). pretoria: doe. douglas, g. (1982). issues in the fit of data to psychometric models. education research and perspectives, 9(1), 32–43. dunne, t., long, c., craig, t., & venter, e. (2012). meeting the requirements of both classroom-based and systemic assessment of mathematics proficiency: the potential of rasch measurement theory. pythagoras, 33(3), art. #19, 16 pages. http://dx.doi.org/10.4102/pythagoras.v33i3.19 humphry, s.m. (2005). maintaining a common arbitrary unit in social measurement. unpublished doctoral dissertation. murdoch university, perth, australia. humphry, s.m., & andrich, d. (2008). understanding the unit in the rasch model. journal of applied measurement, 9(3), 249–264. long, c. (2011). mathematical, cognitive and didactic elements of the multiplicative conceptual field investigated within a rasch assessment and measurement framework. unpublished doctoral dissertation. university of cape town, cape town, south africa. available from http://web.up.ac.za/sitefiles/file/43/314/long,_m__c__(2011)__the_multiplicative_conceptual_field_investigated_within_a_rasch_measurement_framework_.pdf masters g.n. (1982). a rasch model for partial credit scoring. psychometrika, 47, 149–174. http://dx.doi.org/10.1007/bf02296272 matters, g. (2009). a problematic leap in the use of test data: from performance to inference. in c. wyatt-smith, & j.j. cumming (eds.), educational assessment in the 21st century: connecting theory and practice (pp. 209−225). dordrecht: springer. http://dx.doi.org/10.1007/978-1-4020-9964-9_11 rasch, g. (1980). probabilistic models for some intelligence and attainment tests (expanded edn. with foreword and afterword by b.d. wright). chicago, il: university of chicago press. (original work published 1960) van wyke, j., & andrich, d. (2006). a typology of polytomously scored items disclosed by the rasch model: implications for constructing a continuum of achievement. perth: murdoch university. wright, b.d. (1997). a history of social science measurement. educational measurement: issues and practice, winter, 33–45. wright, b.d., & stone, m.h. (1979). the measurement model. in b.d wright, & m.h stone (eds.), best test design (pp. 1–17). chicago, il: mesa press. wright, b.d., & stone, m.h. (1999). measurement essentials. wilmington, de: wide range, inc. appendix 1 top ↑ table 1–a: decisions after results of first round of rescoring. appendix 2 top ↑ questions, marking guide and rescoring details appendix 2 appendix 3 top ↑ category frequencies for each of the items appendix 3 footnotes top ↑ 1. the small sample size may in some senses present as a limitation, but should not detract from the study’s usefulness in alerting ml educators to the issues identified here. any teacher of a grade 12 ml class is likely to be concerned with fewer than 73 learners’ performance on any such test. larger counts of learners may occur in schools with several ml classes at grade 12 level. the general rule of thumb for the construction and development of test instruments is that the learner count is about ten times the maximum score count. in the case of this study, the information obtained from the small group is cross-referenced with substantive analysis and therefore generalisable in the sense that the same principles will apply. 2. missing responses were allocated zeros. 3. see dunne et al. (2012) for details of the analysis of dichotomous items. 4. in this study we use the rasch partial credit model, which is the default model in the rumm 2030 software. 5. the term threshold defines the transition point between two adjacent categories, for example scoring 0 and 1, or scoring 1 and 2.6. the discussion here will concern the dichotomous model. extensions of the model have been derived from this model for partial credit scoring by masters (1982) and rating scales by andrich (1978). 7. this mismatch is in itself not a problem; however, more information could be gleaned from a test situation that is better targeted. 8. in addition, the investigation of factors such as response dependency and differential item functioning is demanded in the interests of valid measurement. abstract introduction literature review theoretical perspective conceptual framework methodology ethical considerations findings discussion of findings, conclusions, and implications design principles implications of the findings for the next cycle conclusion acknowledgements references appendix 1 appendix 2 about the author(s) brantina chirinda division of mathematics education, wits school of education, university of witwatersrand, south africa patrick barmby division of mathematics education, wits school of education, university of witwatersrand, south africa citation chirinda, b., & barmby, p. (2017). the development of a professional development intervention for mathematical problem-solving pedagogy in a localised context. pythagoras, 38(1), a364. https://doi.org/10.4102/pythagoras.v38i1.364 original research the development of a professional development intervention for mathematical problem-solving pedagogy in a localised context brantina chirinda, patrick barmby received: 19 dec. 2016; accepted: 05 apr. 2017; published: 30 june 2017 copyright: © 2017. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract this article reports on the design and findings of the first iteration of a classroom-based design research project which endeavours to design a professional development intervention for teachers’ mathematical problem-solving pedagogy. the major outcome of this study is the generation of design principles that can be used by other researchers developing a professional development (pd) intervention for mathematical problem-solving pedagogy. this study contributes to the mathematical problem-solving pedagogy and pd body of knowledge by working with teachers in an under-researched environment (an informal settlement in gauteng, south africa). in this iteration, two experienced grade 9 mathematics teachers and their learners at a public secondary school in gauteng, south africa, participated in a 6-month intervention. findings from the data are discussed in light of their implications for the next cycle and other pd studies. introduction mathematics is an essential requirement for entry into south african universities and other tertiary institutions and is a ‘critical competency for the development of sorely-needed high-level skills’ (centre for development and enterprise, 2014, p. 1). however, south african learners’ performance in mathematics on local and international tests of educational achievement has been a major cause for concern. learner achievements in mathematics are still at an unacceptable level (department of basic education [dbe], 2014). after the release of the annual national assessments (ana) 2013 results, a diagnostic report was generated by the dbe showing that some of the numerous challenges learners experienced in certain mathematics topics were as a result of ‘ineffective teaching methods’ (dbe, 2014, p. 7). feza-piyose (2012) agrees with this finding and affirms that quality of instruction is one of the contributing factors to the poor performance of the majority of south african learners. a number of studies (adler & ronda, 2014; mji & makgato, 2006; mullis, martin, foy & arora, 2012; taylor & vinjevold, 1999) have also outlined numerous flaws in the teaching and learning of mathematics in south africa. many curricula (dbe, 2011; ministry of education singapore, 2013; national council of teachers of mathematics, 2000) consider problem solving an essential aspect of mathematics teaching and learning. the curriculum and assessment policy statement (caps) states that problem solving and cognitive development should be fundamental to all mathematics teaching. however, this is not what a number of researchers have found in south african mathematics classrooms. adler and ronda (2014) observe that south african teachers tend to implement traditional approaches in the classroom. they start the learning day by revising the previous day’s homework, demonstrating the new work with an example, and giving the learners some exercises to do. taylor and vinjevold (1999) found that some south african mathematics teachers spend a whole lesson doing problems that simply require the application of algorithms and hardly teach genuine problem solving. mji and makgato (2006) also perceive that outdated teaching practices have caused poor teaching standards in south africa. in response to these identified problems, we propose to design a professional development (pd) intervention to support mathematics teachers in teaching problem solving. traditionally, professional development is normally delivered in the form of workshops, college courses, seminars or conferences (villegas-reimers, 2003) but these approaches have been fiercely criticised for their ineffectiveness since they are not directly related to an individual teacher’s practice (hawley & valli, 1999). for this reason, we engaged the process of design-based research (dbr) to tailor a pd intervention that would be appropriate for teachers in a particular local context. we anticipated that dbr would permit us to refine and redesign our pd intervention based on multiple iterations in a naturalistic setting. the following were the research objectives of the larger project, from which the present study was taken: design an effective pd intervention for mathematical problem-solving pedagogy. explore the learning of participant teachers and learners from participating in the pd intervention. examine and evaluate the potential impact of the pd intervention. generate design principles that can be used to develop a pd intervention on mathematical problem-solving pedagogy for grade 9 teachers in a particular local context. this article reports on the first iteration of the pd intervention on mathematical problem-solving pedagogy. we formulated and sought to answer the following research questions: what is the impact of the pd intervention on learners’ learning processes? what is the impact of the pd intervention on teachers’ teaching of problem-solving? what factors facilitated learners’ learning and teachers’ development of pedagogy? what are the possible design principles required to generate a pd intervention on mathematical problem-solving pedagogy for grade 9 teachers in a particular local context? literature review problem solving traditional methods of teaching mathematics, prevalent in the south african context (adler & ronda, 2014), are very different from the teaching of problem solving that takes place while learners are ‘grappling’ with a problem (murray, olivier & human, 1998). traditional methods of teaching give emphasis to the teaching of algorithms, while problem solving is the whole process of dealing with a problem (wessels & kwari, 2003). problem solving is a fundamental characteristic of the teaching and learning of mathematics. the national council of teachers of mathematics (2000, p. 52) defines problem solving as ‘engaging in a task for which the solution method is not known in advance’. polya (1957) sees it as finding a path around a challenge or an obstacle and finding a solution to a problem that is not known. over the years as teachers, we have experienced that when a teacher genuinely teaches mathematical problem solving, learners are placed in the active role of problem-solvers by being confronted with unfamiliar tasks that have no readily known procedure or algorithm (murray et al., 1998). the teacher poses non-routine mathematical problems to learners for solving and they are expected to justify and explain their solutions. during problem-solving, the previously attained experience is applied to an unfamiliar situation that contains major obstacles in order to find a solution to the problem and learners are required to think deeply about what solution strategy to implement. dewey (1933), polya (1957), krulik and rudnick (1980) and barmby, bolden and thompson (2014) have identified steps in the problem-solving process. we chose polya’s steps to base our pd intervention on because they encapsulate the key aspects of mathematical problem-solving that comprises this study. polya (1957) proposed a four-phase problem-solving process, with identifiable strategies: understanding the problem. devising a plan or deciding on a strategy for attacking the problem. carrying out the plan; that is learners follow through with the strategy selected, carefully taking each step along the away. looking back at the problem, the answer and what one has done to get there. during the intervention, we worked with teachers on how to use polya’s steps in their teaching. we discussed how teachers could work with learners on understanding given problems. teachers were encouraged to ask learners questions like: do you understand what the problem is looking for? do you know all the words? can you repeat the problem in your own words? we discussed with teachers how they could help learners to create a plan to solve a given problem. teachers could ask learners questions like: what operation are you going to engage? a table? do you need to draw a picture? would you use an equation? teachers were encouraged to assist learners during carrying out the plan that is, doing calculations. we discussed ways teachers could help learners to persist with a chosen plan and if a plan does not work to discard it and choose another. after getting a solution to the given problem teachers were persuaded to always facilitate learners to review their answers by reflecting and looking back at what worked and what did not. effective professional development for teachers our objective for the main study is to design an effective pd intervention that can be used to support mathematics teachers in the teaching of mathematical problem solving. day (1999) defines pd as a process by which, either individually or with colleagues, teachers review, renew and broaden their commitment as change agents to the moral purposes of teaching, and by which they acquire and develop critically the knowledge, skills, planning and practice with children, young people and colleagues through each phase of their teaching lives. wei, darling-hammond, andree, richardson and orphanos (2009) see it as the processes and activities designed to improve teachers’ knowledge, the practice of instruction, and the learning outcomes of learners. in turn, quick, holtzman and chaney (2009) state that the main objective of pd is, through teachers, to increase learner achievement. effective pd should be based on constructivism (villegas-reimers, 2003); teachers should be treated as active learners who construct their own understanding (putnam & borko, 1997). gulamhussein (2013) concurs with this and stresses that during pd, teachers must be engaged through varied approaches so that they can participate actively in making sense of a new practice. reflection is one approach teachers can be involved in. it is what a teacher does when they look back at the teaching and learning that has occurred and reconstruct the events, emotions, and experiences of the situation (wilson, shulman & richert, 1987). the capability by an individual teacher to reflect on their actions is a crucial aspect of pd. schunk (2012) advocates that this process of reflection results in teachers incessantly changing and improving their views on effective teaching and this, in turn, results in their pd. effective pd must be perceived as a process that takes place within a particular context (villegas-reimers, 2003) and must be situated in classroom practice (putnam & borko, 1997). pd should focus on changing teaching practices positively while valuing particular teachers’ personal, social and professional needs (loucks-horsley & stiegelbauer, 1991). this is echoed by villegas-reimers who accentuates that pd is not similar in different settings, that is, what works for teachers in a certain area may not work for other teachers in a different area. consequently, we adopted classroom-based design research in order to design a pd intervention that would work specifically for teachers in our context of study. theoretical perspective social constructivism informed this study. the social constructivist perspective to learning mainly originates from the work of vygotsky (1986). it emphasises that culture and social contexts are important in understanding what occurs in a society and acquiring knowledge based on this understanding. learning is considered to be situation-specific and a context-bound activity (woolfolk, 2007), in line with the view of pd given above. this study took place in an under-resourced african context, where the teachers were working with large classes, with learners with english as a second language. the teachers had a wealth of experience of teaching mathematics based on their own cultural and social experiences. therefore, we began with the teachers’ experiences and how they were teaching problem solving, and we employed the dbr approach to develop the pd intervention within this context. social constructivism emphasises that knowledge is mutually built and constructed (vygotsky, 1986). by interacting with others, teachers or students share their views and automatically construct a common understanding connected to the concept under discussion. the notion of social constructivism is an appropriate perspective for this study because this study was structured in such a way that teacher social participation, reflections, and experiences evoked their learning of the teaching of problem solving in their particular contexts. participant teachers were required to share their experiences with other participant teachers and collaboratively learned from each other. in the pd workshops participant teachers spent considerable time interacting, investigating, explaining and discussing mathematical problem-solving pedagogy ideas. during lesson delivery we encouraged teachers to act as facilitators of learning and allow learners to actively construct knowledge individually from their prior experiences or collaboratively with their peers through discussions. conceptual framework the conceptual framework we drew on is that of problem-centred learning. the problem-centred teaching and learning approach is a learner-centred educational method that uses problem solving as the starting point for learning and as a ‘vehicle for learning’ (murray et al., 1998, p. 171). it means both the curriculum and instruction should begin with problems, dilemmas and questions for learners (wessels & kwari, 2003) and the subject should be allowed to be ‘problematic’ (hiebert et al., 1997, p. 12). learning occurs when learners actively grapple with problems for which they have no routine methods available (murray et al., 1998) and involves the learning of mathematics through real contexts, problem situations and models (van de walle, karp & bay-williams, 2013). for this study, teachers were encouraged to use problem-solving as a technique for helping learners to learn mathematics. learners constructed new mathematical knowledge and problem-solving skills after actively grappling with non-routine problems and reflecting on their problem-solving experiences and solution strategies. methodology research design design-based research (kelly, 2003), also known as educational design research (mckenney & reeves, 2012), design experiments (collins, 1992) and development research (van den akker, 1999), is a relatively new research approach in the field of educational inquiry. however, there is increasing recognition internationally of dbr as part of evidence-based educational research (design-based research collective, 2003). the design-based research collective further states that educational researchers and practitioners concur that educational research is frequently unconnected to the problems and issues of everyday practice. however, dbr aims to solve real-world problems by designing, enacting and sustaining interventions (van den akker, gravemeijer, mckenny & nieveen, 2006). for this study, dbr allowed us to be connected to the research problem in a specific setting. a number of researchers have attempted to give a definition of dbr and there is a discussion underway of what constitutes dbr (van den akker et al., 2006). wang and hannafin (2005)’s definition encapsulates the key aspects of dbr that comprise this study. wang and hannafin (2005) define dbr as a systematic but flexible methodology aimed to improve educational practices through iterative analysis, design, development, and implementation, based on collaboration among researchers and practitioners in real-world settings, and leading to contextually-sensitive design principles and theories. (p. 6) this definition implies that in dbr, researchers work as a team with practitioners to provide solutions to practical challenges that face a particular educational context. there is general agreement that dbr should generate valuable educational interventions and useful theory (van den akker et al., 2006). dbr uses the terminology of ‘intervention’ to refer to the object, output, activity or process that is designed as a possible solution to the identified problem. intervention is defined by mckenney and reeves (2012) as a comprehensive term used to encompass the various kinds of solutions that are designed. these various solutions can be educational products, processes, policies or programmes. this study developed a pd intervention for mathematical problem-solving pedagogy for grade 9 mathematics teachers in a certain district in gauteng. context and the design process the larger classroom-based design project focuses on designing a pd intervention on mathematical problem-solving pedagogy that can be further modified and used with schools in challenging contexts in south africa. the larger project has three iterative cycles. in march 2016 we conducted a baseline investigation with 31 teachers at 20 schools in the district of interest. the baseline investigation examined how grade 9 mathematics teachers in this district were using problem-solving in their teaching of mathematics. we ‘purposefully’ selected three schools a, b and c, out of the initial 20 schools. these schools were chosen because they could be conveniently accessed by the researchers and the grade 9 mathematics teachers reported that they were using traditional methods of teaching. this article reports on the first iteration of the larger project, which is the phase of the intervention in which we worked with two teachers in school a. teachers in school b and school c were investigated in cycle b and cycle c respectively. the goal was to work with at least two grade 9 mathematics teachers from each school; however, this depended on the number of teachers teaching grade 9 mathematics at a particular school. design of the professional development intervention the pd intervention is designed to take place within a period of 6 months. the goals we set for the pd intervention were to improve learners’ performance in mathematics and support teachers’ mathematical problem-solving pedagogy. we also aimed to explain and agree with participant teachers what mathematical problem-solving pedagogy is and what it is not. pd took place during the process of classroom instruction in order to link with classroom teaching (barber & mourshed, 2007). we conducted three pd workshops with teachers on the last wednesday of each month. the workshops were from 12:00 to 15:00, 3 h per workshop. this resulted in a total of 9 h of training for participant teachers. we carried out the training at a community centre in this district. we received funding from a non-governmental organisation and gave participant teachers a stipend to arrange work for their classes before leaving the school for the programme and to cover their lunch and transport costs. these workshops were different from the traditional ‘one-shot’ workshops in that the teachers attended the workshop three times, collaboratively learned from each other during the intervention, were actively engaged in meaningful discussion, planning and practice (loucks-horsley, hewson, love & stiles, 1998) and we observed and supported the teachers during the implementation. we selected pd activities that offered teachers the opportunity to become actively engaged in the meaningful analysis of teaching and learning. in the first workshop, we initially presented the workshop’s contents to the participant teachers who then watched two short videos on mathematical problem-solving pedagogy in action. the baseline investigation we conducted before implementing this intervention unearthed that participant teachers believed that teaching mathematical problem solving was about explaining to learners each and every concept, step-by-step, breaking down the topic, working out examples on the chalkboard and giving learners practice exercises to work on. therefore, these videos were to show the teaching of mathematical problem solving in action. we discussed the videos, focusing on what genuine mathematical problem-solving pedagogy entails and how to apply polya’s four steps of problem solving as a teaching process. teachers expounded on ways of introducing or posing the problems in such a way that learners understand the given problems. teachers collaboratively solved at least two ‘rich’ and open-ended mathematical tasks relating to the work they were teaching, and with our guidance discussed how to teach problem solving as a process. after attending the first pd workshop, teachers were encouraged to go and implement the new ideas in their lessons for a month. during this implementation stage, we observed, supported and guided the participant teachers as was necessary and audiotaped the lessons. it is imperative that teachers are supported during the implementation stage to address the specific challenges of changing classroom practice (gulamhussein, 2013), receive feedback and be given the opportunity to reflect critically (bell & gilbert, 1994). we encouraged teachers to reflect-in-action and reflect-on-action (schön, 1983) during the implementation. schön defines reflection-in-action as the teachers’ aptitude to reflect throughout the lesson rather than after the lesson. reflection-on-action encompasses teachers thinking about, commenting on, analysing and evaluating the lesson after it has been completed. throughout the implementation phase participant teachers were encouraged to continuously reflect on their experiences as learners of the pd programme and on their classroom practices. after the first implementation, we conducted the second workshop where the aim was for teachers to further collaboratively reflect on their teaching experiences and to review the audio tapes of the observed lessons. we selected crucial and relevant audio recordings that foregrounded participant teachers’ use of problem solving in their teaching. participant teachers analysed how they had taught mathematical problem-solving and they watched two further videos showing mathematical problem-solving pedagogy in action. once again teachers collaboratively solved mathematical tasks relating to what they were teaching and planned on how to teach similar tasks to their learners. after workshop 2 the teachers implemented new ideas for a month while being observed, supported, audiotaped and interviewed by the researchers. we also encouraged the teachers to continuously reflect on their experiences and classroom practices. the third workshop and the implementation process were similar to the second stage. data sources teacher data were collected through classroom observations and semi-structured reflective interviews. in order to get first-hand experiences when teachers implemented the problem-solving pedagogy, we observed them delivering lessons and recorded information on the spot as it occurred on the observation comment card. the observation comment card (see appendix 1) is a valid tool for the collection of data from teachers during the teaching of mathematical problem solving because it incorporates polya’s problem-solving processes. teachers need to assume certain roles in each problem-solving stage and these are all covered in the teacher observation comment card. classroom observations which were audiotaped were conducted before, during and after the pd intervention. we were at the school every wednesday and observed mrs x’s and mrs y’s lessons consecutively. in total, we had 13 observations with each teacher. audio recordings permitted us to record the observed lessons as they transpired and gave us the chance to re-experience the lessons at a later convenient time. semi-structured reflective interviews with the participant teachers were conducted with each teacher once a month and were conducted in their classrooms since it was during their free periods. in total, we had five interviews with each teacher. opie (2004) points out that interviews are extremely important as a data collection tool because they draw out data that participants may not display during observations. in terms of assessing the possible impact of the pd intervention on participant learners, we gave them mathematics attainment tests preand post-intervention and a self-reporting mathematical problem-solving skills inventory (mpssi) at the beginning and at the end of the intervention. we used teachers’ preand post-intervention mathematics attainment tests that covered the topics that the learners were doing (geometry and data handling). the questions on the pre-intervention test were different but similar in all respects to those of the post-intervention test. both tests had 15 questions and the marks were converted to a percentage. the attainment tests were useful in evaluating learners’ ability to use mathematical problem-solving skills because they required learners to supply the answers thereby avoiding guesses. the mpssi (see appendix 2) was developed by the researchers based on the mathematical problem-solving skills found in the literature (hiebert & wearne, 1993; kadel, 1992; polya, 1957; van de walle et al., 2013) and it was used to establish if grade 9 learners perceived themselves as having developed mathematical problem-solving skills after the intervention. learners were required to evaluate their own competencies on each item on a five-point scale. we chose the mpssi as a data collection tool because it provided learner data that supplemented the data from the preand post-intervention attainment tests and this allowed learners to fully contribute to the data collection process. data analysis our raw data included recorded notes on the observation comment card, audio tapes from the classroom observations and semi-structured interviews, test marks and responses from the mpssi. audio tapes from the classroom observations and semi-structured interviews were transcribed verbatim into written notes in order to be able to identify common patterns and experiences. we employed inductive data analysis to analyse the teacher qualitative data from the observations and the semi-structured interviews (hatch, 2002). according to mcmillan and schumacher (2014), qualitative data analysis is an inductive process of organising data into categories and identifying patterns among the categories. this concurs with creswell and plano clark (2011, p. 208) who affirm that qualitative data analysis involves ‘coding the data, dividing the text into small units, assigning a label to each unit and then grouping the codes into themes’. inductive data analysis permits the researcher to identify themes that emerge from the data (hatch, 2002) and not from predetermined categories. the statistical software package spss was used to analyse the learner quantitative data from the preand post-intervention mathematics tests and the mpssi. ethical considerations this was an ethical study; therefore, the researchers took precaution to protect the autonomy and anonymity of participant teachers. letters of permission were sent to and subsequently returned from the gauteng department of education, johannesburg north district of education, selected school principals, participant mathematics teachers, learners and their parents or guardians. participants were given detailed information about the proposed study and were clearly informed of the confidential nature of the research. we ensured that participation was voluntary; confidentiality was prioritised, and participants could freely withdraw from the study at any time without incurring any negative consequences, although none did. all responses were anonymised before analysis; neither the participant schools nor the teachers’ names were identified in any report of the results of the study. findings learners the mpssi results exhibited substantial gains in percentages for each question (see table 1). table 1 shows the preand post-intervention percentages for each of the 13 items on the mpssi, showing the percentage of learners that agreed or strongly agreed with each item. for example, on question 1 – ‘i always read the problem carefully to understand it’ – 34% of the learners agreed or strongly agreed with this statement pre intervention, and this figure rose to 86% post intervention. we concluded that the learners perceived an increase in their mathematical problem-solving skills after participating in the intervention. table 1: results from the mathematical problem-solving skills inventory. there was an increase in the learner attainment test scores and this indicated that there were gains in learner attainment. table 2 shows the average percentages on the preand post-intervention attainment tests with the standard deviation of each set of marks. we concluded that the pd intervention had a positive impact on learners’ performance. table 2: results from the preand post-intervention attainment tests. teachers findings from teacher observations the findings from the teacher classroom observations are presented under the three themes: understanding the problem, collaborative learning and encouraging metacognition. understanding the problem: the observations we conducted before implementation of the intervention unearthed that the participant teachers relied solely on the dbe prescribed textbook. they introduced the topic by demonstrating the examples in the textbook on the board and there was no clarification of the task at hand. the teachers did not verify if learners understood what was being demonstrated on the board. after demonstrating examples on the board, teachers would give learners homework from the prescribed textbook. after the first workshop teachers began to gradually implement what they had learned from the mathematical problem-solving pedagogy training. teachers began to encourage learners to read and understand the given problem before attempting it. teachers required learners to paraphrase the problem at hand in their own words as a way of demonstrating that they indeed understood. english language, the language of instruction, proved to be an obstruction to learners’ understanding. the researchers encouraged the teachers to code-switch as necessary. as the intervention progressed, teachers assisted learners to understand the given problems by discussing any unfamiliar terms in the problems and requiring them to underline key words in a given problem. in one instance during a data handling lesson, mrs y posed a problem on the total number of fatal accidents in each province on south african roads. in this problem learners were required to find measures of central tendency and dispersion. mrs y required the learners to firstly underline the key words in the problem and discuss in pairs the unfamiliar terms, that is, ‘measures of central tendency’ and ‘measures of dispersion’. learners were required to verbalise and restate the given problems in their own words to prove that they really understood the posed problem. in one lesson, we observed that mrs x encouraged learners to verbalise their thinking processes during an investigation to establish the minimum conditions for congruent triangles. collaborative learning: as the intervention progressed, teachers arranged learners in pairs or groups of three or four for discussions. learners were given the opportunity to brainstorm ideas and solution strategies, draw sketches or diagrams, look for patterns and generate solutions collaboratively. if learners’ chosen strategies failed, teachers encouraged them to try another one. we observed that mrs x put her learners into groups of three or four when they were investigating how doubling any or all of the dimensions of right prisms and cylinders affects their volume. encouraging metacognition: metacognition is the conscious monitoring of one’s own thinking and self-regulation of learning (van de walle et al., 2013). as the intervention progressed, teachers required that learners look back and reflect after getting solutions to check if the way they had solved the given problems was the best way, to clarify if there were other solutions other than what they had found and if their solutions were reasonable. findings from the teacher semi-structured interviews the findings from the teacher semi-structured interviews are presented under the four themes that emerged from the interview data. the four themes are: changes in perceptions about mathematical problem-solving pedagogy, appreciation of collaborative learning, increased awareness of learners’ needs and pd activities that had a positive impact on teachers’ professional development. changes in perceptions about mathematical problem-solving pedagogy: initially, teachers viewed the teaching of problem-solving as a step-by-step explanation, doing examples on the board and giving learners practice exercises. when we introduced mathematical problem-solving pedagogy, teachers perceived it as time-consuming and asserted that they had a lot of prescribed teaching material to cover. however, as the intervention progressed they began to understand and appreciate the teaching of mathematical problem solving as a process as illustrated by this quote from mrs y: initially, i was confused and showed learners how to work a problem using drilling methods but now i know how to teach mathematical problem solving as a process. (mrs y, female, teacher) appreciation of collaborative learning: both teachers valued collaborative learning as made explicit by mrs x: working with my colleague during the workshops and teaching was superb because we could support each other and we shared ideas and obstacles. collaboration made my learning effortless. i liked it. i became a better teacher by learning from my colleague. (mrs x, female, teacher) as the intervention progressed, participant teachers began to implement collaboration in their own teaching. learners were required to work in pairs or in groups and this kept them on-task. teachers were no longer the only source of knowledge and this helped with classroom management as learners had a responsibility of completing tasks in pairs or groups, as was highlighted by mrs y in one of the interviews: as i arranged learners to work in pairs, they were no longer moving in and out of the classroom. pairing them helped me with discipline and class control. (mrs y, female, teacher) increased awareness of learners’ needs: the two participant teachers revealed that they had become more aware that they had to clarify the task at hand and make sure the learners understood before solving the given problem. this is illustrated by these quotes from mrs y: this professional development intervention opened my eyes. i used to assume that learners understood the given questions but now i check if they really understand. if they don’t we read the problem together, paraphrase the questions and i sometimes require learners to verbally tell me what the question would be asking for. (mrs y, female, teacher) if my learners do not understand a given problem, i now re-read the problem for them or use their mother tongue to explain. this is to make sure that they understand the given problem. (mrs y, female, teacher) professional development activities that had a positive impact on teachers’ professional development: videos on mathematical problem-solving pedagogy that were shown during workshops were very positive for teachers, as articulated by mrs y: to be able to see mathematical problem-solving teaching in action was intriguing. i saw where i was getting it wrong. it helped me to realise areas that needed improvement in my own teaching. (mrs y, female, teacher) in addition, the teachers appreciated that we, the researchers who delivered the pd intervention were respectful to them and acknowledged their experiences, as expressed by mrs x: you treated me like an adult during the intervention. i have been to other workshops where i was treated like a child who knows nothing. you showed me respect. you were aware that i have fully trained to be a teacher and that i teach large classes. (mrs x, female, teacher) finally, the teachers valued solving problems and simulating how to teach problem-solving during the workshops, as explained by mrs y: to practically learn how to teach mathematical problem-solving during the workshops was helpful for me because it built my confidence before implementing the new ideas in the classroom. (mrs y, female, teacher) discussion of findings, conclusions, and implications factors that influenced change in learners the preand post-intervention tests and preand post-intervention mpssi demonstrated that the pd intervention had a positive impact on learners’ performance. this finding resonates with barber and mourshed’s (2007) results from their research of 25 national school systems: that there is a positive correlation between teachers’ pd and learners’ achievement. classroom observations confirmed that teachers were working in a different way with learners and emphasising problem-solving approaches. teachers were encouraging collaborative learning among learners. when learners grappled with the problems in pairs or groups, it allowed for richer and more worthwhile whole class discussion. after solving problems, teachers encouraged learners to look back on the answer and the solution process or method, and looking at the problem itself. this is the final and important step of mathematical problem solving. when learners were required to look back and justify their solutions, they were able to come up with solutions in many ways, were able to correct themselves and could easily discern if their solutions were appropriate. despite the apparent success of encouraging problem-solving approaches, we found that english was a major obstacle to learners’ grasping of the given problems. the finding on language as an obstacle to learning in south african mathematics classrooms is in agreement with what a number of researchers have disclosed in the past (adler, 2001; setati & barwell, 2006; webb & webb, 2008). factors that influenced change in teachers’ pedagogy the pre-intervention observations revealed that the lessons were teacher-led and teachers implemented traditional methods of teaching. this finding aligns with what other researchers have exposed (adler & ronda, 2014; chisholm et al., 2000; taylor & vinjevold, 1999). chisholm et al. (2000) lament that south african classrooms have remained teacher-centred. however, as the intervention progressed we noticed that the teachers became more facilitators rather than teachers, giving learners the opportunity to grapple with the given problems. there was more group work and discussion among learners, and learners began to work with and listen to each other. all these aspects that were being exhibited in the lessons after we implemented the intervention are highlighted as important in problem-based learning (murray et al., 1998; van de walle et al., 2013; wessels & kwari, 2003). teachers became more focused on working with learners to understand the given problems. understanding is the first important step in the teaching of mathematical problem solving. in situations where learners did not understand the given problems because of the language, teachers code-switched. this study fills a gap in the literature on professional development and problem-based learning as it unearthed that language and code-switching are important aspects to be considered when implementing a pd intervention in a multilingual context. we recommend that a pd programme should include a segment that supports teachers on how to appropriately conduct code-switching and to support learners with their language. initially, teachers were reluctant to participate in the intervention and implement the mathematical problem-solving pedagogy. they indicated that they were worried that they would fail to cover the caps syllabus within the prescribed time. this finding aligns with what slattery (2013) states: that the syllabus and task completion places pressure on teachers. however, as the intervention progressed teachers became comfortable with mathematical problem-solving pedagogy and started implementing the new ideas. we observed the participant teachers for a month before implementing the intervention and this created trust between us and the teachers, which resulted in teachers working comfortably with us. teachers appreciated that we respected them and acknowledged their experiences. it was important that we were responsive to respecting the participant teachers because respect is a key aspect in an african culture. when implementing the pd we took into consideration the lack of resources in our context. these are important aspects to add to the pd literature: that respect is an important aspect of the african culture and should be considered when supporting african teachers in a pd programme. we recommend creating a positive relationship with teachers before implementing a pd intervention and building on teachers’ experiences when training them. collaborative learning was beneficial to both participant teachers and learners. this outcome concurs with cordingley, bell, thomason and firth’s (2005) finding from systematically reviewing 17 studies of collaborative or sustained continuous pd in diverse contexts. they discovered that when teachers engage in collaborative pd, there are vast improvements in learners’ learning and behaviour, and in teachers’ practices. the semi-structured reflective interviews with participant teachers were imperative to the research process. as teachers looked back on classroom events during the interviews and made critical judgments about them, they modified their teaching behaviour and this resulted in them constructing knowledge about themselves, their teaching practices and their learners (schunk, 2012). as participant teachers knowingly and systematically reflected on their teaching experiences (farrell, 2007), we realised that they were consciously able to improve their own teaching. this procedure by teachers thinking about what they were doing and why they were doing it turned their experiences into meaningful learning. in this case, learning by teachers did not just happen but was derived from them constructing sense from their experiences and particular contexts. following up on teachers in the classrooms to check if they were correctly implementing the mathematical problem-solving pedagogy was advantageous as we were able to support teachers as necessary. this is different from the traditional ‘one-shot’ workshops and we recommend that pd practitioners should support the teachers in the classroom during the implementation stage. design principles design principles are one of the major outputs of dbr and mckenney and reeves (2012) term them theoretical insights into a phenomenon in question, which recommend how to address specific issues in a range of settings. while we acknowledge that we have gone through only one cycle of the dbr, the study generated a number of design principles relating to the designing of a pd intervention for mathematical problem-solving pedagogy. these principles will inform the next cycle of the study and are imperative to mathematics education and pd practitioners who are in the process of designing professional development interventions. the design principles include: a baseline investigation must be conducted to establish teachers’ perceptions and practices on the teaching of mathematical problem solving before implementing the pd intervention. facilitators of pd must create a positive relationship with participant teachers before implementing the intervention. pd should be built from teachers’ experiences and current knowledge of mathematical problem solving. respecting participants is important in an african context when implementing the pd intervention. facilitators of pd must observe teachers practically implementing the mathematical problem-solving pedagogy and support them as necessary. pd should be organised around collaborative problem solving. pd should support teachers on how to implement mathematical problem-solving pedagogy in a multilingual context. implications of the findings for the next cycle one of the most significant outcomes from this cycle is that language stands so much in the way of learners’ learning of mathematics. polya’s (1957) first step in mathematical problem solving is understanding the problem, and if learners struggle with language, it means that they do not even understand the given problems. language and the problem-solving process cannot be separated. therefore, in the next cycle, we will do a lot more with teachers on how we can support learners’ difficulties with problem solving, english language and code-switching. in this cycle we used teachers’ own tests to act as our preand post-intervention tests but in the next cycles we are going to design our own tests so that we can easily check on the reliability of these tests. in the next cycle we will also introduce learner task-based interviews to help us explain why learners were doing well in the post-intervention test and why they were giving better responses in the post-intervention mpssi. we anticipate that the learner task-based interviews will explain if learners’ improvements are real and if they would be doing problem solving differently from the way they would be doing it pre-intervention. in the next cycle participant teachers will also be observed delivering lessons by other participant teachers and given feedback. this development emerged from the usefulness of teachers observing practice (in the first cycle from videos) and the usefulness of working on and reflecting collaboratively on lessons. conclusion the findings of the study may prompt other researchers to develop pd interventions in local contexts. the study, having been done at one school, means that the transferability of the findings to larger contexts can be challenging. however, dbr does not make generalisability claims as ‘the effectiveness of a design is no guarantee of its effectiveness in other settings’ (collins, joseph & bielaczyc, 2004, p. 18). however, we believe that the above design principles can be used by other researchers as starting points for developing pd interventions for mathematical problem-solving 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(10th edn.). boston, ma: allyn & bacon. appendix 1 observation comment card observation comment card appendix 2 mathematical problem-solving skills inventory (mpssi) date: _______ name: __________________ grade: ______ dear learner please fill in the table below, where 1 = strongly disagree, 2 = disagree, 3 = neither agree nor disagree, 4 = agree and 5 = strongly agree mathematical problem-solving skills inventory (mpssi) reviewer acknowledgement pythagorashttp://www.pythagoras.org.za acknowledgement to reviewers the quality of the articles in pythagoras crucially depends on the expertise and commitment of our peer reviewers. reviewing is an important part of scholarly work, making a substantial contribution to the field. reviewers’ comments serve two purposes, guided by two inter-dependent objectives: • pythagoras wishes to publish only original papers of the highest possible quality, making a meaningful contribution to the field. reviewers advise the editor on the scholarly merits of the manuscript to help him evaluate the manuscript and to decide whether or not to publish it. reviewers are encouraged to reject a manuscript if it is scientifically flawed, merely sets out observations with no analysis, provides no new insights, or is of insufficient interest to warrant publication. • pythagoras is committed to support authors in the mathematics education community. reviewers help the author to improve the quality of their manuscript. reviewers are encouraged to write their comments in a constructive and supportive manner and to be sufficiently detailed to enable the author to improve the paper and make the changes that may eventually lead to acceptance. the following summary of outcomes of the reviewing process in 2012 shows that our reviewers do well in achieving both objectives: we sincerely thank the following people who have reviewed these manuscripts for pythagoras in 2012. we very much appreciate their time, expertise and support of pythagoras amidst pressures of work. aneshkumar maharaj anne watson ansie harding anthony essien antonia makina arthur powell belinda huntley benard okelo bill atweh brian greer busisiwe goba calvin jongsma caroline long clement dlamini craig pournara cyril julie danie strauss daniela coetzee-manning david mogari in an effort to facilitate the selection of appropriate peer reviewers for manuscripts for pythagoras, we ask that you take a moment to update your electronic portfolio on www. pythagoras.org.za, allowing us better access to your areas of interest and expertise, in order to match reviewers with submitted manuscripts. if you would like to become a reviewer, please visit the pythagoras website and register as a reviewer. to access your details on the website, follow these steps: 1. log into pythagoras online at http://www. pythagoras.org.za 2. in your ‘user home’ select ‘edit my profile’ under the heading ‘my account’ and insert all relevant details, bio statement and reviewing interest. it is good practice as a reviewer to update your personal details regularly to ensure contact with you throughout your professional term as reviewer to pythagoras. please do not hesitate to contact me if you require assistance in performing this task. rochelle flint submissions@pythagoras. org.za tel: +27 (0)21 975 2602 fax: +27 (0)21 975 4635 page 1 of 2 no. of manuscripts processed in 2012 (outcome complete) 47 accepted without changes 0 (0.0%) accepted with minor changes (to the satisfaction of the editor)1 11 (23.4%) accepted after major revisions (re-submit, then re-review)2 11 (23.4%) rejected after review – not acceptable to be published in pythagoras3 12 (25.5%) rejected without review – not acceptable to be published in pythagoras4 13 (27.7%) david mtetwa deborah moore-russo deonarain brijlall dirk wessels duncan samson ednei becher elizabeth pretorius eric gold erica spangenberg eugenia vomvoridi-ivanovic faaiz gierdien gawie du toit gelsa knijnik gilah leder hamsa venkatakrishnan hannatjie vorster hans niels jahnke hennie boshoff herbert khuzwayo 1.accepted after one round of review, with ‘minor’ changes as specified by reviewers and editor. 2.accepted after two or more rounds of review, with major changes specified by reviewers and editor. 3.includes three cases where authors did not resubmit after required to make major changes. 4.all submissions undergo a preliminary review by the editor (and associate editors) to ascertain if it falls within the aims and scope of pythagoras and is of an acceptable standard. includes six cases where authors did not resubmit after extensive feedback prior to reviewing. http://www.pythagoras.org.za http://www.pythagoras.org.za reviewer acknowledgement pythagorashttp://www.pythagoras.org.za page 2 of 2 if you would like to become a reviewer, please visit the pythagoras website and register as a reviewer. iben christiansen ida marais ingrid mostert jacob jaftha jacques du plessis jill adler johann engelbrecht jurie conradie kakoma luneta karen coe karin brodie kate le roux kosie smit leila goosen leonard mudau lindiwe tshabalala lorna holtman lovemore nyaumwe luckson kaino lyn webb lynn bowie marc schäfer marcus bizony margot berger marietjie potgieter marit johnsen-høines marthie van der walt mellony graven michael de villiers michael mhlolo michael samuel mogege mosimege murad jurdak neil eddy nelis vermeulen nick taylor norman webb nyna amin paola valero paula ensor percy sepeng peter gates piet human pragashni padayachee retha van niekerk richard barwell ronel paulsen sandra heldsinger sarah bansilal shaheeda jaffer sheena rughubar-reddy sibawu siyepu sizwe mabizela stefan haesen stephen lerman stuart rowlands sudan hansraj susan van rensburg tad watanabe temesgen zewotir thomas morman thulisile nkambule tim dunne tony cotton tracy craig umesh ramnarain vasuthavan govender vera frith washiela fish yusef waghid yusuf johnson zain davis reviewers (continued): article information authors: michael k. mhlolo1 marc schäfer1 affiliations: 1mathematics education department, rhodes university, south africa correspondence to: mike mhlolo postal address: po box 94, grahamstown 6140, south africa dates: received: 16 mar. 2012 accepted: 06 nov. 2012 published: 11 dec. 2012 how to cite this article: mhlolo, m.k., & schäfer, m. (2012). towards empowering learners in a democratic mathematics classroom: to what extent are teachers’ listening orientations conducive to and respectful of learners’ thinking? pythagoras, 33(2), art. #166, 9 pages. http://dx.doi.org/10.4102/ pythagoras.v33i2.166 copyright notice: © 2012. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. towards empowering learners in a democratic mathematics classroom: to what extent are teachers’ listening orientations conducive to and respectful of learners’ thinking? in this original research... open access • abstract • introduction    • theoretical underpinnings    • contextualising the problem of disempowerment • methodology    • participants    • procedure       • protocol 1 – evaluative event       • protocol 2 – relatability of learner contributions       • protocol 3 – listening orientations       • validity and reliability       • ethical considerations • analysis of results    • evaluative event 1 – multiplication of polynomials       • comment    • evaluative event 2 – general term of a linear sequence       • comment    • evaluative event 3 – graphical display of data       • comment    • evaluative event 4 – factorisation of trinomials       • comment • discussion    • practical implications    • limitations • conclusion • acknowledgement    • competing interests    • authors’ contributions • references abstract top ↑ in an effort to make education accessible, to ‘heal the divisions of the past and establish a society based on democratic values’, the south african department of education claims that a series of mathematics reforms that has so far been introduced is underpinned by the principles of ‘social justice, fundamental human rights and inclusivity’. critics however argue that the system has remained ‘undemocratic’ in that those groups of learners who were supposed to be ‘healed’ continue to underperform and hence be disempowered. in this study, we conceptualised a democratic and mathematically empowering classroom as one that is consistent with the principle of inclusivity and in which a hermeneutic listening orientation towards teaching promotes such a democratic and mathematically empowering learning environment. we then worked with three different orientations teachers might have towards listening in the mathematics classroom: evaluative, interpretive and hermeneutic. we then used these orientations to analyse 20 video-recorded lessons with a specific focus on learners’ unexpected contributions and how teachers listened and responded to such contributions. the results were consistent with the literature, which shows that teachers tend to dismiss learners’ ways of thinking by imposing their own formalised constructions. introduction top ↑ in south africa, despite many of the post-democracy education policies being acclaimed by curriculum experts, learners from previously disadvantaged schools continue to seriously underperform, prompting critics to argue that ‘there is little evidence that goals of transformation, including redress, equity and democracy have been achieved in practice’ (sayed & jansen, 2001, p. 2; see also fleisch, 2008; muller, 2005; volmink, 1994). learner underperformance has been shown to be acute especially in mathematics, a subject generally viewed as a critical filter (gatekeeper) for economic access, higher education, full employment and participation within a democratic society (stinson, 2004). in south africa, implicit within these ‘mathematics-as-a-gatekeeper’ debates is the argument that learners from previously disadvantaged schools are not being provided with the ‘key to the gate’ (stinson, 2004, p. 4) and are hence disempowered and excluded. in this article we argue that the concept of mathematics providing the key for passing through the gates to economic access, full citizenship and higher education is located at the core of learner–teacher relationships in the mathematics classroom. our proposition is that probably the most important factor to be problematised is the quality of those learner–teacher relationships. from a mathematical empowerment perspective and with reference to the learner–teacher relationship, volmink (1994) suggested that there is a kind of teacher guidance that has potential to be individually empowering whilst at the same time valuing the autonomy of the learner (democratic). the envisaged guidance must propelthose who have been marginalised, disinvited, and underrepresented … to come to see that they too can share in the creation of mathematics and the transformation of their world. (p. 52) key to this guidance is the shift in power relations so that the teacher listens to pupils in depth and allows them to make and express judgements whilst at the same time valuing their contributions. this article problematises this learner–teacher relationship and the shift (or not) in power relations by analysing how teachers who participated in this study responded to learner contributions. in doing so we worked with classroom transcripts and raised the following questions to guide our analyses: 1. to what extent do mathematics teachers give learners authority with respect to their novel comments and actions? 2. what are the possible implications for democracy and empowerment of the learner? theoretical underpinnings our concern in this article can be couched in the form of a question, borrowed from stinson (2004): if school mathematics is a gatekeeper, how might mathematics educators ensure that gate keeping mathematics becomes an inclusive instrument for empowerment rather than an exclusive instrument for stratification? (p. 9) the concept of empowerment has been dealt with from diverse perspectives including political, social and economic, but in essence empowerment provides the subject with the skills and knowledge to make socio-political critiques about their surroundings and to take action (or not) against oppressive elements of those surroundings (stinson, 2004). according to ernest (2002): empowerment is gaining of power in particular domains of activity by individuals or groups and the processes of giving power to them, or processes that foster and facilitate their taking power. (p. 1) ernest distinguished three different domains of empowerment concerning mathematics and its uses: mathematical, social and epistemological. none of these is either wholly discrete or unrelated in its modes of function. mathematical empowerment concerns gaining power over the language, skills and practices of using mathematical knowledge. social empowerment involves using mathematics as a tool for socio-political critique, gaining power over the social domains. ernest viewed epistemological empowerment as summative and the culmination of both mathematical and social empowerment in that the empowered learner will not only be able to pose and solve mathematical questions (mathematical empowerment), but also be able to understand and begin to answer important questions relating to a broad range of social uses and abuses of mathematics (social empowerment). in this summative sense, epistemological empowerment concerns the individual’s growth of confidence in not only using mathematics, but also a personal sense of power over the creation and validation of knowledge. we found that this epistemological view of empowerment enabled us to move our argument forward in two important ways. firstly, it allowed us to focus on the individual learners in a classroom situation. ernest (2002) suggested that whilst different models have been developed to analyse the three perspectives of empowerment, the most neglected in discussions of aims of teaching and learning are those that have direct focus on developing epistemological powers of the individual. yet lather (1991), for example, emphasised that empowerment is a learning process one undertakes for oneself: ‘it is not something done to or for someone’ (p. 4). epistemological empowerment addresses this issue in that it involves the development of learners’ personal confidence, their sense of mathematical self-efficacy, as well as their sense of personal ownership of and power over mathematics. the emphasis is on self-empowerment and this is critical to our argument in this article as we question power relations, that is, whether or not the learner is explicitly or implicitly treated by authority (the teacher) as a passive receiver and user of knowledge. secondly, epistemological empowerment concerns personal power over the creation and validation of knowledge. it has to do with the teacher not only encouraging learners to make contributions to the classroom discourse but also acknowledging and supporting such learners’ ways of understanding. it is this personal creation and validation of knowledge by the learners that we are putting under the spotlight in this article as we question the extent to which mathematics teachers acknowledge, support and build on learners’ personal routes or ways of knowing. according to boaler (2000): if the students’ social and cultural values are encouraged and supported in the mathematics classroom, through the use of contexts or through an acknowledgement of personal routes and direction, then their learning will have more meaning for them. (p. 17) within the multiplicity of perspectives on empowerment, different models have been suggested to analyse different scenarios. however, consistent with our concerns for self-empowerment of the learners, we felt compelled to turn to models of the development of the individual knower within a relationship of power, that is, the learner–teacher relationship. ernest suggested a model developed by belenky, clinchy, goldberger and tarule (1986), which we found valuable because of its direct focus on developing epistemological powers of the individual. whilst the belenky et al. model had earlier been tried in other studies which did not involve school-aged learners, becker (1996) cited in ernest (2002), found the model useful when she worked with 31 students of mathematics of both sexes. the model depicts stages of empowerment of the knower, in which students develop as epistemological agents from a position of complete passivity (passive receivers of knowledge) dominated by authority to one of epistemological autonomy and empowerment, as they progress through the stages as shown in figure 1. figure 1: model of epistemological empowerment. the model depicts shifts in power relations through the stages of epistemological empowerment, from the passive, silent reception of knowledge from authority, through to the active construction of knowledge coupled with the awareness of this power. the strength of this theory is that it accounts for the relative positions, including power relationships that hold between knowledge users and knowledge creators or authorities. the subject is explicitly treated in this theory as the receiver and user of knowledge and we equate this with the learner in our article. the ‘other’, implicit in this model, is the external source of knowledge and authority and in our case this is the teacher. in the first stage of this theory, silence and received knowledge, the learner or subject has no agency or power and simply receives passively. the transmission metaphor in its most authoritarian form reigns supreme, with the subject being the passive recipient of the absolute knowledge from the ‘other’. according to ernest (2002), the ultimate goal in this model is to achieve the stage of being a ‘constructing knower’, a learner who can combine intuition and the procedures and skills of mathematics to make sense of the world and confidently apply mathematical thinking. what this model suggests is that to achieve the epistemological empowerment of learners through mathematics, it is not enough for them to gain mastery over some mathematical knowledge and skills. there needs to be a personal engagement with mathematics so that it becomes an integral part of the learner’s personal identity. based on both a theoretical analysis and on personal experience, ernest (2002) identified seven different factors that are associated with a shift towards engagement, confidence and epistemological empowerment. however, this article specifically focuses on the following two that we found to be complementary and at the same time consistent with our objectives in this article: 1. a shift in power relations so that the teacher listens to pupils in depth, allows them to make and express judgements and values their contributions. 2. this in turn enables learners to have a sense of ‘ownership’ of their success – the sense that it results from their own powers and application. within this context of shifting power relations, stinson (2004) cites studies that show how teachers ‘teaching from a culturally relevant perspective’ could match their teaching styles to the culture and home backgrounds of their learners. in this article we view culture and home backgrounds as embracive terms to refer to all the ways of knowing that the learner brings to the classroom discourse. of particular relevance to our article is the observation that teachers working from a culturally relevant perspective build from students’ ethno or informal mathematics and orients the lesson toward their experiences, whilst developing the students’ critical thinking skills. the positive results of teaching from such a perspective are realised when students develop mathematics empowerment. these indicators were valuable in our analyses. contextualising the problem of disempowerment within the literature, researchers have shown how it is possible to relate the learning of mathematics to empowerment and democracy (ernest, 2002; muller, 2005; stinson, 2004; volmink, 1994). this relationship has been examined as a means to underpin a more equitable mathematics education system and to promote a more just society. however, mathematics education can also involve both overt and covert disempowerment, which can occur at all levels of the schooling system. skovsmose (2000) refers to this as an aporia in that on one hand mathematics education could mean inclusion and empowerment, yet on the other it could also mean suppression, exclusion and disempowerment. it is from this paradoxical observation that some researchers have used the metaphor of mathematics being either a pump or filter: a pump for some by propelling learners into educational opportunities and economic access, and a filter for others by limiting their access to careers and professions. in this context, volmink (1994) says:mathematics is not only an impenetrable mystery to many, but has also, more than any other subject, been cast in the role of an objective judge, in order to decide who in the society ‘can’ and who ‘cannot’. it therefore serves as the gatekeeper to participation in the decision-making processes of society. to deny some access to participation in mathematics is then also to determine, a priori, who will move ahead and who will stay behind. (pp. 51–52) south africa presents an ideal site to conceptualise how mathematics education can be overtly as well as covertly disempowering. an important step that was taken post-democracy was precisely to redevelop education as part of a democratic endeavour. however, a covert form of disempowerment could be seen in the manner in which the national curriculum has been crowned by some ‘nice-looking aims and objectives’ (skovsmose, 2004, p. 4) which seem not to translate into practice. for example, pass rates post-democracy have been on an upward trend, but jansen (2012) has described this as a ‘matric razzmatazz that conceals a sad reality’ as the matriculants who then enter tertiary education institutions find themselves hopelessly out of their depth, whilst those that leave for the job market do not seem to possess the basic numeracy skills they need to be of any use to potential employers. generally the allegation is that the cognitive demand levels of the examinations have been lowered to the detriment of learners from previously disadvantaged schools (fleisch, 2008; muller, 2005). a similar observation is that south africa was at the bottom of the log in all the trends in mathematics and science studies (timss) since 1995, giving a false impression that all learners were underperforming. however, further disaggregation of country marks revealed that learners from previously disadvantaged schools performed way below both the national and the international average whilst those from previously advantaged schools performed way above both (muller, 2005). such results exemplify covert disempowerment of such underperforming learners in terms of access to participation in mathematically related areas. methodology top ↑ participants this article works with archived data collected from four experienced (over seven years) grade 11 teachers, two men and two women, who took part in a broader study (mhlolo, 2011). these teachers were all from different previously disadvantaged high schools in south africa and, through their links with the university, had agreed to take part in the project. twenty lessons on number, algebra and data handling topics were video recorded and transcribed, generating a 300-page database from which we selected four evaluative events for this article. procedure protocol 1 – evaluative event in developing our analytical tool, we borrowed from adler (2005) who proposed the use of an evaluative event as a possible unit of analysis for what goes on in the mathematics classroom. adler defined an evaluative event as a teaching–learning sequence that can be recognised as focused on the acquisition or constitution of a particular mathematical object. andrews (2009) defined it as that part of a lesson where the teacher’s didactic intent remained constant. by describing observed pedagogic practices in terms of evaluative event series, it was possible to produce units for the analysis of the pedagogy. in this article we provide four such evaluative events, highlighting the unexpected learner productions. we note immediately a lack of consensus on what might constitute an ‘unexpected’ learner production in a particular lesson, leading us to the next protocol. protocol 2 – relatability of learner contributions we borrowed from fernandez, yoshida and stigler’s (1992) proposition of relatability of events, which builds on the presumption that not all relations between events must be presented for learning to occur because some events are not relevant to the content of the lesson. fernandez et al. suggest that following the teachers’ objectives (explicit or implied) for the lessons, and their responses to learner productions, one can judge whether or not learner productions were:1. relatable and relevant, hence expected and critical for learning to take place. 2. not relatable, hence unexpected and irrelevant for understanding the mathematical ideas. 3. relatable but, depending on the teachers objective, not immediately expected or relevant for critical understanding. in the absence of teachers’ explicit or implied objectives, fernandez et al. (1992) suggest that those characteristics of a lesson that make it easier to represent mathematical ideas coherently are closely related. these ultimately facilitate effective teaching and learning of content; hence, they are likely to enable learners to be authors and producers of mathematical knowledge. teachers’ responses to such learner productions are influenced by their listening orientations; hence, we were also interested in the extent to which such orientations were conducive to a democratic and empowering mathematics classroom. protocol 3 – listening orientations a listening orientation can contribute to mathematical empowerment in the sense that it is linked to a fundamental principle underlying the constructivist approach to learning mathematics: that one of the teacher’s responsibilities is to listen and determine or interpret the learners’ rationality and meaning (cobb & yackel, 1998). an important part of teaching mathematics is to support learners to voice their mathematical thinking and reasoning, nascent or flawed as it might be (brodie, 2010). when teachers do not listen to or do not understand their learners’ thinking, they are likely to be ineffective; hence, ball and forzani (2010) posit that:teaching without attention to learners’ perspectives and prior knowledge is like flying a plane in fog without instruments. this has big implications for equitable education because the greater the differences between learners and their teachers – in culture, language, and experience – the less precisely attuned the teaching is likely to be. (p. 41) in terms of promoting democratic values, when teachers do not listen to or do not understand their learners’ thinking they tend to dismiss such thinking by imposing their own formalised constructions onto the learners (cobb & yackel, 1998; davis, 1997), thereby instilling non-democratic values. according to the belenky et al. (1986) model this would be the stage dominated by authority and therefore less self-empowering for the learner. although there are various ways in which teachers can listen to their learners’ mathematical ideas, davis (1997) posits that not all listening orientations are conducive to and respectful of learners’ thinking. he discussed three different orientations teachers might have towards listening in the mathematics classroom: ‘evaluative’, ‘interpretive’ and ‘hermeneutic’. teachers with an evaluative orientation, according to davis, tend to listen to learners’ ideas in order to diagnose and correct their mathematical misunderstandings. with an evaluative listening orientation, the learners’ work is seen in light of how the teacher would approach the problem as well as their expectations for how the problem might be solved. teachers with an interpretive orientation, on the other hand, listen for something rather than listening to learners’ ideas. listening for something suggests that the teacher is not interested in what the learner is saying. according to brodie (2010), such teachers often ask questions that address particular aspects or points that they are looking for. when a learner produces an unexpected contribution, they usually do not entertain that response, but continue to look for a response that would be consistent with their thoughts. teachers with a hermeneutic orientation, on the other hand, interact with their learners, listening to their ideas and engaging with them in the messy process of negotiation of meaning and understanding. sherin (2002) referred to it as adaptation: teachers listen, interpret and respond to their learners’ mathematical ideas by modifying and building on both the learners’ and teacher’s existing subject matter knowledge and pedagogical content knowledge. thus, it is the hermeneutic orientation for teaching that is particularly needed if teachers are to take an adaptive or negotiating approach to the implementation of reform-based tasks (doerr & english, 2006). therefore, for us a hermeneutic listening orientation is at the heart of an empowering and democratic classroom. putting these three protocols together, our unit of analysis was an evaluative event. in it, we were interested in identifying the unexpected learner productions, and the manner in which the teacher listened to and dealt with such novel comments or actions. validity and reliability our measures for obtaining internal reliability included systematic selection of learner contributions from video clips, using a relatability tool to justify that such contributions were unexpected by the teacher. we then used a protocol to capture the teacher’s responses to such contributions to justify whether there was evaluative listening or hermeneutic listening on the part of the teacher. the protocol and analyses were shared with colleagues (peer examination) at conferences and local workshops. in order to enhance quality of our justification and conclusions, we submitted articles and conference contributions that were reviewed during the research period. ethical considerations approval to proceed with this study was granted by the department of education. at institutional level, approval was granted by the university ethics committee. at school level the first author obtained informed consent from the principals, teachers and parents of the learners who were to be involved in this study. at both school and individual levels the participants’ anonymity and confidentiality were maintained through the use of pseudonyms (e.g. teacher m, learner 2) and the fact that the video recording was not focused on either the teacher or the learner. analysis of results top ↑ whilst there are many learner contributions that we could have exemplified, we only present the four evaluative events (one from each teacher) that we chose for this article. the criterion for selecting these four was that, in our view, these were at the core of the mathematical object that was being focused on in each of the series of lessons and were therefore critical for learners to have a conceptual understanding of that particular object. the analyses we make thereof particularly focus on whether or not the teacher listened with the intention to reject or accept (evaluative), or to adopt and adapt (hermeneutic). evaluative event 1 – multiplication of polynomials in this event, the teacher was dealing with the multiplication of polynomials applying the distributive law. the examples that had been worked with so far were of the form (a + b)(c + d + e). the teacher wrote five tasks on the board; task e was (3x2 + xy − 2y2)(x + 2y). the learner who had been tasked to do this problem, worked on the board thus:learner 1: (3x2 + xy − 2y2)(x + 2y) 3x3 + 6x2y + x2y + 2xy2 − 2xy2 − 4y3 3x3 + 7x2y − 4y3 teacher: [to the class] what are you saying about her approach? how did she approach this? did she apply the distributive law? class: no. teacher: they emphasise here in brackets [pointing to the textbook] that apply the distributive law. what was she supposed to do first before she multiplied? [another learner is called] learner 2: [comes to the board works the task in accordance with the examples and gets the same result as learner 1] learner 1: there is the answer mum teacher: yes, what we said was the answer is the same but the approach was different. so next time you should read the question because the question says apply the distributive law ok. learner 3: [pointing to learner 1’s work] but mum i understand her approach better. teacher: we are following instructions. ok, ok if it was just ordinarily finding the product of binomials and trinomials really she was correct. but now in brackets there are those finer lines in a question that say we can get the same answer but if it was in an exam i was not going to credit her because she did not follow instructions from the question which is very important. do you understand me? comment in our view the teacher’s comment, ‘they emphasise here in brackets …’ showed that the teacher had a different expected image of the distributive law and learner 1’s contribution was unexpected. through the commutative law where a × b is equivalent to a × b, we argue that the learner’s contribution was relatable as we see it as a mathematically reasoned equivalent. nevertheless, because the learner did not rearrange the polynomials to start with the binomial on the left, the teacher raised the question for the class: ‘did she apply the distributive law’, leading to the contribution being rejected. evaluative event 2 – general term of a linear sequence in this lesson, the teacher introduced the general term of a linear sequence: teacher: [writes] 4; 7; 10; 13; … alright, i want us to observe a pattern here. term number 1 is 4. what has been done to this 1 to make 4; the same thing should be done to this 2 to make 7; the same thing should be done to this 3 to make 10; … class: multiply by 3 and add 1 teacher: so what shall the general term be? class: tn = 3n + 1 teacher: [writes] 3; 7; 11; 15; … alright you are given these first four terms of the sequence; the general term? ooo i’m seeing the same hands. why the same hands? eeeee, [name of learner]. learner 1: tn = 4n 1 learner 2: can i please ask a question. you see i just want to find out why isn’t that to find the tn = bla, bla, bla? [referring to tn = >4n 1]. why can’t you just add tose numbers i mean for example like three [t1] then you add one, two, three, then you get four [constant difference] then you put the four instead of getting all the other things for the tn. teacher: ok you can start afresh. what are you saying, what are you suggesting? learner 2: sir why can’t we just like find the differences? teacher: we find the differences fine, like in this case the difference is what, it’s four. learner 2: yaah it is four. teacher: it is four. learner 2: yaah. then why is it that you can’t write like tn = bla bla + 4? why do you have to write ‐1 that’s my question? teacher: right, the general term is some kind of a formula that will be used to generate all the terms of the sequence. it’s ok. learner 1: yes, yes. teacher: right. can you say tn = 4 is a formula? class: noooo. teacher: [to learner 2] ok, alright i thought you had made an observation. learner 2: sir i do have an observation! teacher: ok order, alright ok, let’s give somebody else a chance. [the learner is then ignored and the lesson continues] comment the teacher’s direct link between the tn and the values suggests the explicit rule as the object. however, in this rule the 3 and 4 which are constant differences in the first and second sequences respectively are masked, yet this is how the learner understood the number sequence. judging by the teacher’s response (‘i thought you had made an observation’, then ignoring the learner), learner 2’s talk was ‘unexpected’ but relatable as it suggests a recursive formula as an entry point. again we notice that the learner has no agency or power and simply receives the teacher’s version passively. evaluative event 3 – graphical display of data in this event, data had been collected in a survey about families. the top row in table 1 shows the number of children in each family whilst the bottom row shows the corresponding number of families. table 1: number of children in each family and their frequencies. the teacher began by asking the class how the data in the table could be displayed, and received ‘bar graph’ as one of the responses. a pupil was then asked to draw the first bar. the learner started by drawing x and y axes intersecting at 0 as shown in figure 2. figure 2: learner’s drawing of bar graph of survey data. the learner then drew a vertical bar, approximately a quarter of a unit wide and eight units high:teacher: is he correct? class: somehow, almost, maybe. learner: that bar shows a quarter and eight ma’am. teacher: ok so the zero was supposed to be where? here? the teacher then added another zero on the x-axis (what she was expecting) such that the pupil’s bar now sat between two zeros. she then asked whether the learner would have been correct if the zero was in this second position. although one pupil said, ‘maybe it’s incorrect’, the teacher ignored this, perhaps because the zero was now where she wanted it to be. she asked another pupil to add in the next bar for 14 and 1. this second bar was drawn across to the ‘one’ on the x-axis. another pupil drew the third bar for 20 and 2 going across to the ‘2’ on the x-axis and 20 high. on seeing that some learners were declaring the graph incorrect: teacher: ok fine, all right fine, our example ok, i chose it because i wanted you to see something. if you have zero, as a number included, don’t, this is the point of origin by the way [pointing to the intersection of the axes]. ok so don’t make zero your point of origin. so think this as a cartesian plane whereby the number before zero will be what, a negative one. comment judging by both the objective of the event (drawing a bar graph for categorical data) and the teacher’s responses, we argue that a two-dimensional chart with axes intersecting at 0 was ‘unexpected’ but relatable given the links that exist between such surfaces on which graphs are drawn. a graph is a representation of numerical values or functions by position of point, line or area on a two-dimensional surface or chart. each chart area contains an x-axis (horizontal) and a y-axis (vertical) which are the objects for its plot area. these axes can be named differently to correspond with the variables of interest. friel, curcio and bright (2001) warn that interpreting graphs that utilise two axes may present difficulties if the functions of the axes across these graphs are not explicitly recognised. it would appear that this error, in recognition of the functions of the two axes across graphs, is at the centre of confusion in this event. evaluative event 4 – factorisation of trinomials the teacher wrote the following trinomial on the board: a2 + 14a + 48. the lesson was on factorisation and the method involved finding factors of the last term (48) which would add to the middle term (14), hence:a2 + 14a + 48 = (a + 6)(a + 8) a few more examples were worked out including the following one with a negative middle term: n2 − 16mn + 15m2 = (n − m)(n − 15m) teacher: any question so far? learner 1: what if there is a division? teacher: division where? come and write it on the board. learner 1: [writes on the board] class: aaaaaaa! teacher: listen, is it possible that you find an expression like that? class: nooooo. teacher: let us proceed then. any other questions? comment from the teacher’s response, we argue that the learner’s contribution was unexpected. the relatability of the learner’s contribution, however, could be seen in terms of providing a counter-example, without which learners were likely to conjecture that all trinomials can be factorised this way. our view is that such factorisation is a critical tool when solving quadratic equations. discussion top ↑ we first examined the extent to which teachers gave authority to their learners’ novel comments and actions. in the first set of lessons, focusing on the multiplication of polynomials, a learner used her own personal method to deal with the expansion of brackets, which the class was also comfortable with. however, because she was not familiar with it, the teacher did not acknowledge it as an equivalent. the result is that, for the whole week, tasks on multiplication of polynomials were solved in the teacher’s way: binomial to the left, trinomial to the right. in the belenky et al. (1986) model, this depicts power relations at the first two levels: learners accept the pronouncements of the teacher’s authority passively and are able to repeat them. ernest (2002) sees this as epistemological disempowerment as it deprives the learners of their personal creation and validation of knowledge.in the second set of lessons that we analysed, the focus was on deriving formulae for linear number sequences. the teacher’s entry point was an explicit rule whilst one learner argued vehemently for a recursive rule. she was called to order by the teacher and the lesson continued in the teacher’s way. in this sense, we also see the transmission metaphor reigning supreme with the learners being passive recipients of the absolute knowledge from the teacher (ernest, 2002). with specific reference to the generation of a recursive or explicit rule for a sequence, blanton (2008) cautions that it is important to listen to how learners’ verbal statements imply that they are looking at the ways that quantities change (recursive), or that they are making a prediction based on the connection between the term number and its value (an explicit general term). recursive reasoning is seen as a building block for the eventual ability to use formulae that directly determine any unknown amount. such reasoning emerges naturally, as learners develop skip counting and the ability to add on. for learners in the early stages, instruction that encourages them to look for recursive patterns in functional situations is a recommended starting point for developing algebraic thinking (bezuszka & kenney, 2008). our view is that deriving these formulae through the recursive would have empowered the learners as they would have made sense of the relationships between the more familiar (recursive rule) and what was new to them (explicit rule). we argue that the teacher’s extraction of the general terms 3n + 1 and 4n – 1 in both cases was rote, limited and therefore disempowering both in terms of masking the idea of 3 or 4 being constantly added and in terms of its generalisability to other cases, such as 27, 30, 33, 36, … in the series of lessons focusing on drawing bar charts, we saw more of what brodie (2010) refers to as an interpretive orientation: the teacher did not build on the learner’s idea of a two-dimensional plane with axes intersecting at 0. the teacher’s shifting of zero to the position at which she was expecting it to be shows she was not listening with the intention to understand the learner’s mathematical reasoning. she was drawing the learner’s contribution close to her own established disposition. her directions thereof reflect a more procedural orientation than conceptual in that the learners were simply told that if zero is a number in their data set, then they should not put a zero at the origin. to us this was an indication of the teacher imposing her own formalised constructions onto the learners without saying why this is problematic. consequently, the teacher’s shifting of the zero to a new position does not resolve the issue of recognising the scaling of the x-axis in this context. the overall mathematical outcome here is a bar graph that does not represent the original data set particularly well, either in terms of mathematical structure and convention, or with reference to the real-world situation being represented, which is thus less empowering for the learners. in the last vignette focusing on factorisation of trinomials, we note the teacher soliciting questions from the learners: ‘any question so far?’ one learner raises a question that in our view would have provided a special case or counter-example to the type of tasks that were being dealt with. the teacher however did not attend (listen) to the mathematics within what the learner was saying and neither did he assess the mathematical validity of the learner’s ideas, and try to make sense of the learners’ mathematical thinking (ball & forzani, 2010). instead of determining the learner’s rationality (cobb & yackel, 1998) and teaching from the learner’s perspective (ball & forzani, 2010), he dismissed it with ridicule, presumably because it was not in line with his own formalised construction (davis, 1997). the literature alludes to such practices, such as gruenwald and klymchuk (2003) who noted that sometimes mathematics courses, especially at school level, are taught in such a way that special cases are avoided and learners are exposed only to ‘nice’ functions and ‘good’ examples. this approach can create many misconceptions, as explained in tall’s (1991) generic extension principle: if an individual works in a restricted context in which all the examples considered have a certain property, then, in the absence of counter-examples, the mind assumes the known properties to be implicit in other contexts. (p. 18) there are many areas of mathematics where opportunities abound for learners to investigate whether a conjecture is always, sometimes or never true (gruenwald & klymchuk, 2003). we note quickly that the quadratic formula and other algorithms were born out of a realisation that some quadratic expressions cannot be factorised using the algorithm that was used throughout the series of lessons. here was an opportunity for the teacher to negotiate and adapt the learner’s counter-example for other learners to conclude by themselves that sometimes a trinomial could not be factorised in this way. we therefore argue that the teacher’s action did not foster and facilitate learners to take power over the creation and validation of their mathematical knowledge (ernest, 2002). practical implications data analysed in this article suggest that the everyday practice of inviting students to contribute responses to a mathematical question or problem may do little more than promote cooperation. in order to be consistent with a democratic and empowering classroom, teachers who truly care about the development of their students’ mathematical proficiency need to show genuine interest in the ideas learners construct and express, no matter how unexpected or unorthodox. one way of achieving this is to take note of the learners’ contributions that might not have been resolved in class, consult peers and use such contributions as launching pads for further learning. limitations we note and acknowledge one reviewer’s observation that our vignettes do not seem to exemplify hermeneutic listening. perhaps this gap could best be explained in terms of the prevalence of ‘expected’ and ‘unexpected’ learner contributions. common classroom practice is that teachers go into class expecting certain questions to be raised by learners and for an experienced teacher such expected questions far outnumber the unexpected. the tendency is that learners’ questions are then shaped to be consistent with aspects that teachers are looking for – an interpretive orientation. it is for this reason that, in the context of our concept of a democratic classroom, we were specifically interested in identifying the ‘unexpected’ learner productions and the manner in which the teacher dealt with them. admittedly these are few, hence the reviewer’s concern, but the literature (brodie, 2010) confirms our findings that when learners produce unexpected contributions, teachers usually do not entertain such responses even though this is critical for learner empowerment in a democratic classroom. conclusion top ↑ all the learners’ contributions that we analysed had the potential to be exploited to enhance learners’ understanding of the related mathematical ideas. however, whenever they occurred in the lessons the teachers either explicitly or tactfully ignored them. such teacher behaviour is not consistent with giving epistemological empowerment to learners and suggests that the teachers are less democratic in their orientation. acknowledgement top ↑ we acknowledge the department for international development for funding the phd study from which this article draws. the views expressed in this article are not necessarily those of the funders. competing interests the authors declare that we have no financial or personal relationship(s) which might have inappropriately influenced our writing of this article. authors’ contributions m.k.m. 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(1994). mathematics by all. in s. lerman (ed.), cultural perspectives on the mathematics classroom (pp. 51–67). dordrecht: kluwer academic publishers. article information author: michael de villiers1 affiliation: 1mathematics education, university of kwazulu-natal, edgewood campus, south africa correspondence to: michael de villiers postal address: private bag x03, ashwood 3605, south africa dates: received: 15 aug. 2012 accepted: 27 sept. 2012 published: 30 nov. 2012 how to cite this article: de villiers, m. (2012). an illustration of the explanatory and discovery functions of proof. pythagoras, 33(3), art. #193, 8 pages. http://dx.doi.org/10.4102/ pythagoras.v33i3.193 note: this article is based on a presentation at the 12th international congress on mathematical education, july 2012, coex, seoul, korea. workshops on clough’s conjecture have also been conducted at the nctm annual meeting, april 2004, philadelphia, usa, as well as at the amesa congress, july 2004, north-west university, potchefstroom, south africa (de villiers, 2004). copyright notice: © 2012. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. an illustration of the explanatory and discovery functions of proof in this original research... open access • abstract • introduction    • extending the role of proof beyond verification    • computing technology and the changing role of proof    • proof as a means of discovery • clough’s conjecture    • geometric proof    • an alternative ‘algebraic’ proof    • revisiting the geometric proof    • generalising to equi-angled polygons • concluding comments • acknowledgements    • competing interests • references abstract top ↑ this article provides an illustration of the explanatory and discovery functions of proof with an original geometric conjecture made by a grade 11 student. after logically explaining (proving) the result geometrically and algebraically, the result is generalised to other polygons by further reflection on the proof(s). different proofs are given, each giving different insights that lead to further generalisations. the underlying heuristic reasoning is carefully described in order to provide an exemplar for designing learning trajectories to engage students with these functions of proof. introduction top ↑ it seems that the human brain is designed, or has evolved over time, not only to recognise patterns, but also often to impose them on things we observe. moreover, from a very young age, children naturally exhibit a need for an explanation of these patterns – a deep-seated curiosity about how or why things work the way they do. they ask questions about why the sky is blue, the sun rises in the east, or why more moss grows on the southern side of a tree (in the southern hemisphere).however, it sadly seems that once young children have entered the domain of mathematics in formal schooling, this natural inquisitiveness and quest for deeper understanding becomes severely repressed. largely to blame is probably the traditional approach of focusing primarily on the teaching, learning and practising of standard algorithms. these are still presented in many classrooms as mystical chants to be memorised, rather than focusing on understanding why they work, as well as on the meaning of the basic operations underlying them. lockhart (2002) laments this sorry state of affairs: by concentrating on what, and leaving out why, mathematics is reduced to an empty shell. the art is not in the ‘truth’ but in the explanation, the argument. it is the argument itself which gives the truth its context, and determines what is really being said and meant. mathematics is the art of explanation. if you deny students the opportunity to engage in this activity — to pose their own problems, make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs you deny them mathematics itself. (p. 5, [emphasis in the original]) extending the role of proof beyond verification traditionally, the verification (justification or conviction) of the validity of conjectures has been seen as virtually the only function or purpose of proof. most mathematics teachers probably see this as the main role of proof (knuth, 2002) and this view, to the exclusion of a broader perspective, also still dominates much of curriculum design in the form of textbooks, lessons and material on teaching proof (french & stripp, 2005). even the majority of research conducted in the area of proof has been done from this perspective (balacheff, 1988; ball & bass, 2003; harel & sowder, 2007; stylianides & ball, 2008). harel and sowder, for example, defines a ‘proof scheme’ as an argument that ‘eliminates doubt’, effectively restricting the role of reasoning and proof to only that of verification, although they acknowledge the explanatory role of proof in other places. in the past few decades, however, this narrow view of proof has been criticised by several authors (e.g. bell, 1976; de villiers, 1990, 1998; hanna, 2000; thurston, 1994; rav, 1999; reid, 2002, 2011). they have suggested that other functions of proof such as explanation, discovery, systematisation, intellectual challenge et cetera, have in some situations been of greater importance to mathematicians and can have important pedagogical value in the mathematics classroom as well. but these distinctions of other roles for proof are perhaps far older. for example, arnauld and nicole (1662) appear to be referring to the explanatory (illuminating or enlightening) role of proof by objecting to euclid because they felt euclid was ‘more concerned with convincing the mind than with enlightenment’ (cited by barbin, 2010, p. 237). computing technology and the changing role of proof mejía-ramos (2002, p. 6) argues that the search for deeper understanding is what makes many mathematicians reject ‘mechanically-checked formal proofs and computational experiments as mathematical proofs’, for example, the famous use of computers by appel and haken in 1976 to prove the four-colour conjecture (appel & haken, 1977). especially in the light of modern computing technology, such as dynamic geometry and symbolic algebraic processors, it is often the case that a very high level of conviction is already obtained before mathematicians embark on finding a proof. in fact, it can be argued that this ‘a priori’ conviction is more often a prerequisite and motivating factor (polya, 1954, pp. 83−84) for looking for a proof than the mythical view that ‘eliminating doubt’ is the driving force. on the other hand, although such computing tools enable us to gain conviction through visualisation or empirical measurement, these generally provide no satisfactory insight into why the conjecture may be true. it merely confirms that it is true, and although considering more and more examples may increase our confidence to a greater extent, it gives no psychologically satisfactory sense of illumination (bell, 1976) or enlightenment – for that, some form of proof is needed! in this regard, it is significant to note that young grade 9 children still display a need for some form of further explanation (deeper understanding) of a result, which they had already become fully convinced of after empirical exploration on sketchpad (mudaly & de villiers, 2000). within the context of algebra, healy and hoyles (2000) also found that students preferred arguments that both convinced and explained, strongly suggesting that the need for explanation is perhaps an untapped resource in lesson design and implementation. appreciation of the verification (justification) function of proof is most easily developed in fields such as number theory, algebra, calculus, et cetera. in these fields one can give spectacular counter-examples to conjectures with massive empirical support (e.g. as in stylianides, 2011). however, this is not quite the case with dynamic geometry. the difference is that one can transform geometric figures or graphs continuously (or at least closely) by dragging, as well as explore more deeply by zooming in to great levels of accuracy. with these facilities, one can usually find counter-examples to false conjectures fairly quickly and easily. it is possible to contrive didactical situations such as used in de villiers (2003, pp. 73, 85) where students are given sketches with measurements preset to one decimal accuracy, therefore deliberately misleading them to make a false conjecture. however, genuinely authentic examples in dynamic geometry that are accessible to high school students are few and far between. it seems more natural and meaningful that within a dynamic geometry (mathematics) environment, potential use may be made of this cognitive need for explanation and understanding to design and implement alternative learning activities. such learning activities could introduce novices for the first time to proof, not as a means of verification, but as a means of explanation and illumination (e.g. see de villiers, 1998, 2003), whilst the other functions of proof could be developed later or in other contexts. furthermore, by initially referring to a deductive argument as a ‘logical explanation’ instead of a ‘proof, it may help to focus attention on its role as a means of deeper understanding of a dynamically verified result rather than of conviction or verification. proof as a means of discovery quite often, logically explaining (proving) why a result is true gives one deeper insight into its premises. on further reflection, one may then realise that it can be generalised or applied in other circumstances. anderson (1996, p. 34) also clearly alludes to this aspect when writing, ‘proof can bring understanding of why methods work and, consequently, of how these methods might be adapted to cope with new or altered circumstances.’ rav (1999, p. 10) also describes this ‘productive’ role of proof when writing: ‘ ... logical inferences are definitely productive in extending knowledge by virtue of bringing to light otherwise unsuspected connections.’ more recently, byers (2007, p. 337) has made a similar observation: ‘a “good” proof, one that brings out clearly the reason why the result is valid, can often lead to a whole chain of subsequent mathematical exploration and generalization.’i have called this illuminating aspect of proof that often allows further generalisation, the discovery function (de villiers, 1990), and it appears also to be the first explicit distinction of this function (reid, 2011). for example, explaining (proving) viviani’s theorem for an equilateral triangle by determining the area of the three triangles it is divided up into, and noticing the ‘common factor’ of the equal sides of these triangles as bases, may allow one to immediately see that the result generalises to any equilateral polygon, because exactly the same ‘common factor’ will appear (de villiers, 2003, p. 26). nunokawa (2010, pp. 231−232) similarly claims that ‘explanations generate new objects of thought to be explored’. he gives an example of a problem involving two overlapping squares, and how explaining why the overlapping area remains constant as the one square remains fixed and the other is rotated, leads one to generalise to other regular polygons with the same feature. two other ‘discovery via proof’ examples are discussed and presented in de villiers (2007a, 2007b). of course, for novices and less experienced students such generalisation (or specialisation) from a proof is not likely to be as automatic and immediate as with an experienced mathematician. therefore, in didactically designing tasks to engage high school students, or even student teachers, with the discovery function of proof, sufficient scaffolding is often needed to provide adequate guidance for both the initial proof as well as for further reflection (hemmi & löfwall, 2011; miyazaki, 2000). jones and herbst (2012, p. 267) reporting on a study on the instructional practices of a sample of expert teachers of geometry at grade 8 level (pupils aged 13–14) in shanghai, china, identify two important factors in developing an understanding of the discovery function of proof, namely, variation of the mathematical problems as well as the questions asked by teachers to guide their students. it is important to also point out here that with the ‘discovery’ function of proof is not only meant a discovery made after reflecting on a recently constructed proof. as illustrated in de villiers (1990, p. 22, 2003, pp. 68−69), it also more broadly refers to situations where new results are discovered in a purely logical way by the application of known theorems or algorithms without resorting to any experimentation, construction or measurement. for example, using the tangents to a circle theorem, it is relatively easy to deduce logically (and proving at the same time) that the two sums of the opposite sides of a quadrilateral circumscribed around a circle are equal (and generalising to circumscribed 2n-gons). another illustrative example is given in de villiers (1999) involving the generalisation of a problem involving an area relationship between a square and a formed octagon. by dividing the sides into different ratios than the original, it was experimentally found with dynamic geometry that the area ratios remained constant. however, a purely inductive approach whereby the different ratios 0.1666; 0.3333; 0.4500 were looked at for the division of the sides into halves, thirds and quarters respectively, was not very helpful in finding a general formula and ultimately had to be derived logically. more generally, with the discovery function, it also means that a proof can reveal new, powerful methods of solving problems and creating new theories. logical reasoning and proof can show that certain problems are unsolvable such as, for example, representing 2 as a rational (fraction), squaring the circle or solving a quintic (or higher order) polynomial equation with radicals. hanna and barbeau (2010, pp. 90−93) suggest a nice example for classroom use, showing how the problem of finding the quadratic formula naturally leads to an introduction to students of the strategy of completing the square. grabiner (2012, p. 161) gives historical examples of how the distinction between pointwise and uniform convergence arose from counter-examples to cauchy’s supposed theorem regarding infinite series, and of how cantor’s theory of the infinite came about through trying to specify the structure of the sets of real numbers on which fourier series converge. similarly, the discovery (invention) of non-euclidean geometry came about from attempts to use indirect proof (reductio ad absurdum) to prove euclid’s 5th postulate. grabiner (2012, p. 162) describes this as ‘another triumph of human reason and logic over intuition and experience’. the main purpose of this article is to contribute further to the theoretical aspects of the role of proof by providing a heuristic description of some of my personal experiences of the explanatory and discovery functions of proof with a geometric conjecture made by a grade 11 student. after logically explaining (proving) the result geometrically and algebraically, the result is generalised to other polygons by further reflection on the proof(s). this conjecture and its generalisations could easily be turned into a set of guided learning activities that elicit ‘surprise’ amongst students (compare movshovitz-hadar, 1988); therefore creating a need for explanation, and provide an authentic mix of experimentation and proof of a possibly original result. clough’s conjecture top ↑ although it is a rare occurrence, nothing gives greater pleasure to a teacher than when one of their students produces a conjecture of their own. the conjecture need not be entirely original, but the excitement created in the classroom when something goes ‘outside’ or ‘beyond’ the textbook gives a much more ‘real’ sense of genuine mathematical discovery and invention. usually, students are also far more strongly motivated to want to solve such a problem because they perceive it as their own and not something old and boring from the textbook or the curriculum. by encouraging students, for example, to continually ask ‘what-if’ questions on their own until it becomes a regular occurrence, students are likely to more naturally start making more original conjectures of their own, providing an exciting injection to liven up the class. the availability of computing technology places at the disposal of students powerful new tools by which they can now easily make independent discoveries (arzarello, bartolini bussi, leung, mariotti & stevenson, 2012; borwein, 2012). a useful overview and analysis for the task-design of activities for promoting conjecturing is given by lin, yang, lee, tabach and stylianides (2012). moreover, speaking from my own experience also, honouring students by attaching their names to discovered results is a powerful motivator to continue further mathematical studies (compare leikin, 2011). during 2003, a grade 11 student from a high school in cape town was exploring viviani’s theorem using dynamic geometry. the theorem says that the sum of distances of a point to the sides of an equilateral triangle is constant (i.e. in figure 1 ppa + ppb + ppc is constant, irrespective of the position of point p inside triangle abc). the student’s further exploration led him to measure the distances apc, bpa and cpb, and then add them. to his surprise, he noticed that apc + bpa + cpb also remained constant no matter how much he dragged p inside the triangle. however, he could not prove it. figure 1: clough’s conjecture: apc + bpa + cpb is constant. his teacher eventually wrote to me to ask whether i could perhaps produce a simple geometric proof, as he himself could only prove it algebraically by means of co-ordinate geometry. below is the geometric proof i first produced, followed by further proofs, explorations and different generalisations of what has become known as clough’s conjecture (de villiers, 2004). geometric proof problem solving heuristics are valuable in that they often direct the problem solver towards a successful solution of a problem. george polya (1945) gives the following useful examples:have you seen it before? or have you seen the same problem in a slightly different form? do you know a related problem? do you know a theorem that could be useful? look at the unknown! and try to think of a familiar problem having the same or a similar unknown. here is a problem related to yours and solved before. could you use it? could you use its result? could you use its method? should you introduce some auxiliary element in order to make its use possible? (p. xvii) following polya’s heuristic, it seems natural to try and relate clough’s conjecture to viviani’s theorem and its proof. after several different attempts, i found by constructing perpendiculars to ab, bc and ca as ‘auxiliary elements’ respectively at a, b and c, that i obtained a triangle klm as shown in figure 2. figure 2: a geometric proof of clough’s conjecture. considering that ∠abk = 30°, it follows that ∠akb = 60°. in the same way the other angles of ∆klm can be shown to be equal to 60°; hence ∆klm is equilateral.next, drop perpendiculars from p to sides km, kl and lm respectively. it then follows that quadrilateral apcpa’ is a rectangle because all its angles are right angles. therefore, a´p = apc, and similarly, b´p = bpa and c´p = cpb. clearly the problem is now reduced to viviani’s theorem in relation to ∆klm. considering that a´p + b´p + c´p is constant, it follows that apc + bpa + cpb is also constant. qed. the preceding proof is quite explanatory (hanna, 1989) as one can almost immediately ‘visually see’ from the diagram in one ‘gestalt’, why the result is true and how it relates to viviani’s theorem. an alternative ‘algebraic’ proof in polya’s final step of problem-solving, namely, looking back, he asks amongst other things whether one can derive or prove the result differently. in doing so, not only is one developing a variety of problem-solving (proving) skills, but one may also gain additional insight into the result. recently, much has been written and researched about the value of posing such multi-proof tasks to students. dreyfus, nardi and leikin (2012) provide a comprehensive survey and review of this particular field.considering that there are several right triangles, it seems reasonable to try the theorem of pythagoras, and to apply it to each of these triangles and investigate where it leads. figure 3: an alternative, algebraic proof of clough’s conjecture. let ab = a, apc = x, et cetera, as shown in figure 3. we now need to show that x + y + z is constant. applying pythagoras to the right triangles adjacent to the hypotenuses ap, bp and cp, we obtain: it is often at this point, or even before reaching it, that a novice problem solver might lose hope of getting anywhere as it is not obvious from the start that this will lead somewhere useful. however, students should be encouraged to persist with such an exploration and not so easily give up and start asking for help. one might say that a distinctive characteristic of good mathematical problem solvers are that they are ‘stubborn’, and willing to spend a long time attacking a problem from different vantage points and not easily surrendering. in this regard, schoenfeld (1987, p. 190−191) also specifically refers to the importance of meta-cognition during problem-solving (i.e. maintaining a conscious awareness and control of a variety of possible approaches, and then monitoring how well things are going during the implementation of a possible approach). if we look at the set of three equations, however, an immediate observation is the cyclic fashion in which terms appear. this suggests that adding the left and right sides of the three equations, respectively, might lead to the quadratic terms cancelling out. indeed, doing so, after simplification, gives us the desired identity . considering that a is constant for a fixed equilateral triangle, it completes the proof. taking into account that is half the perimeter of the triangle, we also get the following bonus relationship: apc + bpa + cpb = pcb + pac + pba. although this algebraic proof appears less explanatory than the preceding geometric one, we have managed to find an additional property of the configuration that was not discovered experimentally, namely, that the sum of these distances is half the perimeter of the triangle. nor was this clearly evident from the geometric proof at all, although one could now go back armed with this hindsight and use basic trigonometric ratios in figure 2 to find that the side length of ∆klm is √3a; hence its height is (which is equal to its viviani sum). figure 4: an equilateral pentagon. however, more importantly, because of its cyclic nature, the algebraic proof suggests an immediate generalisation to equilateral polygons, giving a nice illustrative example of the discovery function of proof. it is not hard to see (at least for more experienced problem solvers) that from the structure of the proof, it will generalise as follows for an equilateral n-gon a1a2...an (refer to the notation in figure 4, showing an equilateral pentagon): by again adding the left and the right sides as before, we get a collapsing ‘telescopic effect’ with all the squares of ppn and xi cancelling out, and all that remains is 0 = na2 – 2a(x1 + x2 + ... + xn) which simplifies to , which as before, is also half the perimeter of the equilateral n-gon. revisiting the geometric proof let us now revisit our explanatory geometric proof. despite already knowing that clough’s result is true for a rhombus (as it has all its sides equal), let us nonetheless see if we can use the same geometric approach with it as for the triangle, and whether it provides any new insights. by constructing perpendiculars as before to ad, dc, cb and ba respectively at a, d, c and b as shown in figure 5, we find that the result is visually immediately obvious. for example, perpendiculars a and c are parallel to each other because they are respectively perpendicular to sides ad and bc. because it is easy to show that fph is a straight line, we see that ah + cf is simply equal to the constant distance between these two parallel lines. the same applies to the sum of the other two distances be and dg between the parallel perpendiculars b and d. therefore, ah + cf + be + dg is the sum of two constants; hence constant. qed. figure 5: a geometric proof for a rhombus. in many ways this proof is more explanatory than the preceding algebraic proof, which was more algorithmic, non-visual and required quite a bit of manipulation. moreover, following polya, and looking back critically and examining this geometric proof, one should notice that we did not use the equality of the sides of the rhombus at all! we only used its property of opposite sides being parallel − it depends only on the parallel-ness of opposite sides. this implies that the result will immediately not only generalise to a parallelogram, but also in general to any parallel 2n-gon (n > 1); in other words to any even sided polygon with opposite sides parallel, as the same argument will apply!so here we have another excellent example of the discovery function of proof, leading us to a further generalisation, without any additional experimentation. as shown in figure 6 for a hexagon with opposite sides parallel, exactly the same argument applies to the sum of the distances ah and dk respectively on opposite parallel sides, and lying between the two parallel perpendiculars a and d, et cetera. figure 6: a hexagon with opposite sides parallel. generalising to equi-angled polygons given that viviani’s theorem generalises not only to equilateral polygons and 2n-gons with opposite sides parallel, but also to equi-angled polygons, it seemed reasonable to investigate whether clough’s theorem is also true for polygons of this kind. considering that it is true for parallelograms, it is true for the quadrilateral case (a rectangle), but what about an equi-angled pentagon?a quick construction on sketchpad showed me that the result was indeed also true for an equi-angled pentagon. although i personally had no doubt about the equi-angled result from this experimental investigation, i was nonetheless motivated to look for a proof, because i wanted to know why it was true, as well as seeing it as an intellectual challenge (compare with hofstadter, 1997, p. 10). it was therefore not about the ‘removal of doubt’ for me at all! once again, one can try the same strategy used before by constructing perpendiculars at the vertices and attempt to relate it to something we already know, namely viviani’s generalisation to equi-angled polygons. given that abcde is a pentagon with equal angles as shown in figure 7, draw perpendiculars to each side at the vertices a to e, and label as k the intersection of the perpendicular from a with that of the perpendicular from e. similarly, as shown, label the other intersections of the perpendiculars as l, m, n and o. from q draw perpendiculars qj to ae and qx to ek (extended) to obtain rectangle ejqx. therefore, qx = ej. figure 7: an explanatory proof for an equi-angled pentagon. in the same way, construct rectangles to replace the other four segments af, bg, ch and di with the corresponding perpendiculars from q to the sides of klmno as shown. now note that ∠eab = 90° + ∠eak, but ∠okl = 90° + ∠eak, because ∠okl is the exterior angle of ∆eak. hence, ∠okl = ∠eab. similarly, it can be shown that the other angles of the inner pentagon are correspondingly equal to that of the outer one; hence that klmno is also an equi-angled pentagon. but we know that the sum of the distances from a point to the sides of any equi-angled polygon is constant, and because all these five distances are correspondingly equal to the distances ej, af, bg, ch and di by construction, the required result follows. qed. looking back at this proof, we can also see that we did not use the angle size (108º) specific to the equi-angled pentagon to show that ∠okl = ∠eab. this immediately implies that for any polygon with equal angles the same construction would produce another equi-angled polygon inside! hence, the result generalises, and we have here another lovely example of the discovery function of proof. another perhaps even easier way of logically explaining the theorem is shown in figure 8. by translating the segments bg, ch, di and ej as shown, and then constructing perpendiculars at a, b´, c´, d´ and e´, we produce another equi-angled pentagon (left to the reader to prove) and the result follows as before. figure 8: an alternative explanation (proof) for an equi-angled pentagon. concluding comments top ↑ although it is probably not feasible to attempt to introduce complete novices to the ‘looking back’ discovery function of proof with the specific examples illustrated here, i believe it is possible to design learning activities for younger students in the junior secondary school and even in the primary school. this could at least acquaint students with the idea that a deductive argument can provide additional insight and some form of novel discovery. for example, de villiers (1993) shows that to algebraically explain why the sum of a two-digit number and its reverse is always divisible by 11 can lead students to see that the other factor is the sum of the digits of the original number, which they may not have noticed from considering only a few cases. this activity has been done many times with both high school students as well as pre-service and in-service teachers. it has been very seldom that any of them noticed this additional property in the empirical phase, and they would express appreciative surprise at finding this out later from the proof when their attention was directed towards it. instead of defining proof in terms of its verification function (or any other function for that matter), it is suggested that proof should rather be defined simply as a deductive or logical argument that shows how a particular result can be derived from other proven or assumed results; nothing more, nothing less. it is not here suggested that fidelity to the verification function of proof is sacrificed at all, but that it should not be elevated to a defining characteristic of proof. moreover, the verification function ought to be supplemented with other important functions of proof using genuine mathematical activities as described above. it is also not suggested that the preceding examples be directly implemented in a classroom as their success will depend largely on the past experience, expertise and ability of the audience, the classroom culture, as well as the skill of the teacher as a facilitator of learning. for example, zack (1997, p. 1) contends that in her fifth grade classroom ‘for an argument to be considered a proof, the students need not only convince, but also to explain’. she then proceeds to give an example of how this broader ‘didactical contract’ with respect to proof motivated her students to actively engage in conjecturing, refuting and eventually developing a proof as a logical explanation through her continued insistence that they demonstrate why the pattern worked. leong, toh, tay, quek and dindyal (2012) similarly describe some success using a worksheet based on polya’s model to guide a high achieving student to ‘look back’ at his solution and push him to further extend, adapt and generalise his solution. one could speculate, and it might be an interesting longitudinal study, that students who’ve been exposed to several such activities are more likely to spontaneously start ‘looking back’ at their solutions to problems and start considering generalisations or pose new questions. problem posing and generalisation through the utilisation of the ‘discovery’ function of proof is as important and creative as problem-solving itself, and ways of encouraging this kind of thinking in students need to be further explored. johnston-wilder and mason (2005, p. 93) and mason, burton and stacey (1982, p. 9) have claimed that generalisation lies at the ‘heart’ of mathematics and is its ‘life-blood’, and give many instructive examples. it certainly is an important mathematical activity that students need to engage in far more than is perhaps currently the case in classroom practice. it is important to broadly distinguish between two kinds of generalisation, namely, inductive and deductive generalisation. with inductive generalisation is meant the generalisation from a number of specific cases by empirical induction or analogy, and is usually the meaning given to the word ‘generalisation’ in the literature. with deductive generalisation is meant the logical reflection (looking back on) and consequent generalisation of a critical idea to more general or different cases by means of deductive reasoning. in other words, generalising the essence of a deductive argument and applying it to more general or analogous cases. three examples of this deductive kind of generalisation have been illustrated in this article. schopenhauer (as quoted by polya, 1954) aptly describes the educational value of the process of further generalisation to assist in the integration and synthesis of students’ knowledge as follows: proper understanding is, finally, a grasping of relations (un saisir de rapports). but we understand a relation more distinctly and more purely when we recognize it as the same in widely different cases and between completely heterogeneous objects. (p. 30) in terms of learning theory, the process of generalisation corresponds to some extent to ‘superordinate learning’ as distinguished by ausubel, novak and hanesian (1978, p. 68), where an inclusive idea or concept is generalised or abstracted, under which already established ideas can be meaningfully subsumed. finally, it is hoped that this article will stimulate some more design experiments in problem solving as suggested by schoenfeld (2007), focussing not only on developing appreciation of the explanatory and discovery functions of proof, but also on other functions of proof such as systematisation, communication, intellectual challenge, et cetera. the aim is that ultimately, school curricula, textbooks and teachers can begin to present a more comprehensive, realistic and meaningful view of proof to students. acknowledgements top ↑ competing interests i declare that i have no financial or personal relationships, which may have inappropriately influenced me in writing this article. references top ↑ anderson, j. 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(1997). “you have to prove us wrong”: proof at the elementary school level. in e. pehkonen (ed.), proceedings of the 21st conference of the international group for the psychology of mathematics education, vol. 4 (pp. 291−298). lahti, finland: pme. available from http://meru-urem.ca/articles/zack1997(pme-finland).pdf article information authors: hamsa venkat1 jill adler1 affiliations: 1school of education, university of the witwatersrand, south africa correspondence to: hamsa venkat postal address: room 2, wmc corridor, wits school of education, st andrews road, parktown 2050, johannesburg, south africa dates: received: 13 aug. 2012 accepted: 21 oct. 2012 published: 11 dec. 2012 re-published: 21 dec. 2012 note: this article was re-published with a correction made on the last equation on this page. how to cite this article: venkat, h., & adler, j. (2012). coherence and connections in teachers’ mathematical discourses in instruction. pythagoras, 33 (3), art. #188, 8 pages. http://dx.doi.org/10.4102/ pythagoras.v33i3.188 copyright notice: © 2012. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. coherence and connections in teachers’ mathematical discourses in instruction in this original research... open access • abstract • introduction    • describing transformation activity – a literature review    • concepts and framing questions    • episodes       • episode 1       • episode 2       • episode 3       • episode 4 • discussion • conclusions • acknowledgements    • competing interests    • authors’ contributions • references • footnote abstract top ↑ in this article, we share our combination of analytical concepts drawn from the literature with a set of grounded framing questions for thinking about differences in the nature of coherence and connections in teachers’ mathematical discourses in instruction (mdi). the literature-based concepts that we use are drawn from writing focused on transformation activity as a fundamental feature of mathematical activity. within this writing, the need for connections between stated problems and the representations introduced and subsequently produced through transformation steps are highlighted. drawing from four empirical episodes located across primary and secondary mathematics teaching, we outline a set of framing questions that explore coherence and connections between these concepts, and the ways in which accompanying explanations work to establish these connections. this combination allows us to describe differences between the episodes in terms of the nature and degree of coherence and connection. introduction top ↑ teachers’ mathematical discourses in instruction (mdis), essentially the mathematical aspects of what teachers say, do and write as they interact with learners in mathematics classrooms, are a key feature of classroom practice. typically, these mdis include a problem, a selected representation that is subsequently transformed, and explanations and justifications for the representations selected and transformations performed. our interest in this article is in developing a language that can be used to describe a range of mdis. this interest is driven by our need to understand mdis that seem to us to disrupt coherence and connection in mathematical text in a range of ways, and thus impact on what mathematics is made available to learn.transformation of representations, through manipulations within and across different representation forms, is a central feature of mathematical activity (duval, 2006) and, therefore, of mdis. solving a problem in school mathematics often involves a set of steps through which one representation is transformed into another. for example, completing the square is comprised of a series of transformation steps that can act upon a quadratic function as input representation, if the stated problem is to find the turning point of the function. consider the problem: find the turning point of f (x) = x2 − 8x + 9 the first step to solving this stated problem could be to recognise that rewriting a quadratic expression as a perfect square, plus or minus some constant, allows us to ‘see’ vertical and horizontal shifts with respect to the parent function, and so the turning point, more easily. we would thus rewrite the function in the form f (x) = a(x − p)2 + q by completing the square: f (x) = x2 − 8x + 9 = x2 − 8x + 16 − 16 + 9 = (x − 4)2 − 7 what is important for this transformation activity1 is that within the mdi, the input representation introduced, the representations produced through transformation activity, and the accompanying explanations connect with each other and cohere with the stated problem. our observations, across primary and secondary classrooms within our respective projects, suggest that such coherence or connection is frequently, but varyingly, disrupted within mdis. in this article, we share our development of an empirically derived analytical language (elements of which are italicised above) that allows us to make visible a range of disruptions to connection and coherence that come into play across four contrasting teaching episodes. we focus here on input objects, transformational activity and accompanying explanations in order to ‘see into’ the micro-level production of mathematics in classrooms through describing differences in the nature and degree of coherence and connection and to consider the consequences for what is made available to learn. this micro-level focus on specific episodes within lessons follows our observation of the occurrence of disruptions at this level, rather than at the broader level of lessons or lesson sequences that have been taken up in prior research (e.g. sekiguchi, 2006). in order to present our thinking on making aspects of connection and coherence visible within transformation activity, we begin with a brief overview of the literature. we draw on writing focused on transformation activity and representations as these actions and objects are at the centre of all our episodes and, as noted already, at the heart of mathematical activity more generally. we also summarise evidence that points to the shortcomings that characterise practices in which transformation steps are emphasised at the expense of gaining understanding of the representations they act upon. from this review, we outline the key concepts that we found helpful in beginning to pull apart some of the range of procedural practices that we were working with. centrally, we home in on the stated problem, the selected input representation, subsequent sequences of transformation steps, and the interim and final representations produced in this sequence. these concepts are all covered in the literature we review.somewhat absent in this literature is a focus on the mdis that accompany transformation activity. teaching involves the giving of accompanying explanations alongside transformation and so, unlike the (often predictably) piecemeal learner discourses that are in focus in much of the literature on transformation-oriented activity, one expects mdis to be both coherent and connected, and to provide some of the rationales for the representations selected and transformation activity that is enacted. as noted above however, we see this expectation disrupted relatively frequently, and in a range of different ways. in order to consider the nature of these disruptions to coherence, we use a tentative set of framing questions, drawn from our grounded analysis of the episodes presented in this article, to analyse and differentiate the transformation activity in four selected teaching episodes. this could be criticised as somewhat circular: developing grounded framing questions from a dataset, and then using them to analyse the same dataset. our aim in doing this is to share this set of literature-drawn concepts and grounded framing questions in order to start conversations across the mathematics teaching and teacher education communities that can help to build a more robust language for thinking about what constitutes coherence and connection within mathematics teaching. we have already been through several iterations of concepts and framing questions, and have seen that our current formulation can be applied to a significantly broader group of episodes that we have encountered. describing transformation activity – a literature review duval (2006), from a semiotic perspective, describes mathematical activity as comprised by the transformation of one semiotic representation into another. for duval, mathematical transformations can happen within, or across, registers: encompassing natural language, numeric, symbolic and algebraic notations, graphical representations, geometric figures and tabular presentations. for our purposes, the focus is on the representations and transformations selected and produced within transformation activity, the turning of one representation into another, either within or across registers.paying attention to the representations selected and produced through transformation activity is described by haapasalo and kadijevich (2000), cited in haapasalo (2003), as important within transformation activity underlain by strong procedural knowledge. strong procedural knowledge, for them, involves: dynamic and successful utilization of particular rules, algorithms or procedures within relevant representation forms. this usually requires not only the knowledge of the object being utilized, but also the knowledge of format and syntax for the representational system(s) expressing them. (p. 98) this synchronous attention to both representational objects and transformation techniques is often described as lacking in school mathematics. an important second thread in the mathematics education literature is highly critical of the ways in which transformation activity has come to be configured within classrooms. artigue (2011), in her article for unesco on the challenges of extending basic mathematics and science education for all students, refers to international surveys to describe how schooling is very often unstimulating because the teaching of mathematics is framed by: • formal teaching, centred on learning techniques and memorizing rules whose rationale is not evident to the pupils • pupils [who] do not know which needs are met by the mathematics topics introduced or how they are linked to known concepts. (p. 21) implied within artigue’s formulation is a situation within which representations tend to be backgrounded, whilst transformation techniques are foregrounded. the need for a sense of the ‘problem’ that drives the selection of representations and the transformations enacted on them (the ‘raison d’être’) in coherent ways is highlighted. drawing from the analytical work of duval, and the critiques presented here, we see that the concepts of stated problem, input and subsequent representations produced through transformation activity are highlighted as fundamental to mathematical activity. the international literature base is replete with evidence of the consequences of pedagogies based on these kinds of practices for mathematical learning (thompson, philipp, thompson & boyd, 1994). de lima and tall (2008) provide evidence of learners enacting transformations of algebraic representations based on the ‘embodied actions they perform on the symbols, mentally picking them up and moving them around, with the added “magic” of rules’ (p. 3). such actions indicate inattention to the representations being operated on and the syntax of the registers these representations are located in; they thus frequently produce incorrect answers. learner performance in south africa across all phases attests to similar problems (department of basic education, 2011a, 2011b). more problematically, there is evidence in south africa of some of these actions occurring in the context of teaching rather than at the level of learner working (mhlolo, venkat & schafer, 2012). this leads to the need to focus on mdis. instructional explanation has been described as a ‘commonplace’ of mathematics teaching (leinhardt, 2001), and described in terms of the ‘orchestrations of demonstrations, analogical representations and examples’ (leinhardt, zaslavsky & stein, 1990). the word ‘orchestration’ points to the coherence and connection between problems and representations that we are focusing on, but does not, in itself, provide descriptors of what might constitute ‘good’ orchestration. similarly, whilst coherence was identified as a characteristic seen more frequently in some asian countries within the timms-video data (hiebert et al., 2003), what constitutes this coherence within teaching is not detailed. rowland (2012), in using the distinction made in leinhardt’s (2001) work between disciplinary and instructional explanations, notes that deductive reasoning characterises the former, whilst the need to ‘help students learn, understand and use knowledge’ through the use of ‘carefully devised analogy’ that render explanations ‘more accessible and more palatable’ (p. 59) is key to instructional explanations. in this more learning-focused category, there is a need to ‘establish’ rather than ‘state’ deductive connections, in order to support learner understanding of critical links. thus, we looked at the nature of both disciplinary and instructional explanations in mdis through framing questions that would allow us to analyse differences in connection and coherence between our episodes. in the teaching episodes we present in this article, all drawn from previously published work in the south african landscape, practices are exemplified at a range of problematic levels, beginning in artigue’s terrain (backgrounding of problems and representations and foregrounding of transformations), and moving to significantly deeper problems in relation to coherence and connection. our framing questions allowed us to disaggregate episodes in ways that provided some windows into understanding what constitutes coherence and connection. concepts and framing questions mathematical processing begins with a stated problem (sp). stated problems have to be solved through the introduction of an initial input representation, which could be a symbolic statement, visual representation, et cetera. in each of our episodes, we refer to this introductory offering to be operated upon as the input representation (ir). transformation activity (ta), constituted by the transformation steps enacted on this input representation, and the interim representations produced, then ensues. given our focus on teaching, these are associated with accompanying explanations from the teacher.we focus on the detail of transformation steps enacted and the interim representations produced through these transformations, noting, in the first instance, whether these representations retain connection to the input representation. it is worth noting here that algorithms in mathematics do sometimes break this connection at interim stages, and reinstate it at the final stage; the long division algorithm provides a well known example of this (see long [2005], for an elaboration on the differences between procedures and algorithms, and the deep mathematical structures underlying algorithms). we note this point to emphasise that these breaks in connection need not be innately problematic if the scope of application to representations and the mathematics underlying the transformation sequence are considered within the accompanying explanation. we therefore look at teacher explanations for whether or how they establish connections between representations and transformations, coherence with the stated problem, and reference to the scope of application and mathematical structure of the transformation being dealt with. the framing questions are presented below: • does the input representation cohere with the stated problem by providing an appropriate representation to transform? • does transformation activity produce representations that connect in mathematically defensible ways with the input representation? does this happen (1) across all interim representations or (2) at final representation? • does transformation activity, linked with the teacher’s accompanying explanation, serve to establish connections between its steps at each stage and the input representation? our questions reflect more basic notions about transformation activity than have been dealt with in the existing literature that deals with both cognition and semiotics (duval, 2008). in the literature we have summarised, the notion of coherence and connection in teachers’ selections and transformations of representations is largely assumed, and critique focuses on the absence of rationales for the transformations selected. our episodes suggest the need to suspend this assumption and focus on the detail of problem-representation-transformation connections as they play out in teaching. firstly, we look for whether the input representation presented to transform coheres with the stated problem. we then ask questions about how interim and final representations produced within transformation activity connect to the input representation. this often involves transformation activity that produces representations that maintain equivalence between the representation acted on and the representation produced. at a deeper level though, we can also ask whether this transformation activity, linked with the accompanying explanation, establishes connections with the input representation rather than simply assuming, or stating, the connection. in the next section, we present episodes drawn from previously published work. episodes episode 1 this episode was reported in adler and pillay (2007) and adler (2012) and is drawn from a study by pillay (2006). nash (pseudonym), a secondary school teacher, is described as well respected in his school, at which student performance in grade 12 national mathematics examinations was considered successful. the episode below is taken from the third in a unit of eight consecutive lessons on linear functions in a grade 10 class. in lesson 1 and lesson 2, nash had dealt with drawing the graph of a linear function first from a table of values, and then using the gradient and y-intercept method. all the linear function examples that were worked through in the first two lessons were in the standard form . in lesson 3, he moves on to demonstrate how to draw the graphs of functions that are not expressed in standard form. he begins with a few examples (e.g.), using the gradient-intercept method, and the manipulative work needed to get these into the standard form . this serves as motivation for the greater simplicity of the dual intercept method for drawing straight-line graphs. he returns to a function they had worked on,, and begins a discussion of ‘dual’ meaning ‘two’, eliciting from learners that the two intercepts are where the graph cuts the x and y axes. he demonstrates how to find the coordinates of the y-intercept by calculating the value of y when x = 0 and, similarly, the coordinates of the x-intercept. he writes (0; -3) and (2; 0) on the chalkboard and proceeds to sketch the axes, explaining how you can ‘estimate’ where the points are on each of the axes. he plots the two points and continues: nash: … all i have to do join these important points and i got [states and labels the line] – wasn’t that much easier? – there’s less mathematics to do [points to the calculations from the previous question] [than] when you come to write it in y form. simple, first make your x equal to zero – that gives me my y-intercept. then the y equal to zero gives me my x-intercept. put down the two points – we only need two points to draw the graph. then a learner asks a question: learner 1: you don’t need all the other parts? nash: … what’s important features of this graph? … we can work out … from here [points to the graph drawn] we can see what the gradient is … is this graph a positive or a negative? learners: [chorus] positive. nash: it’s a positive gradient … we can see there’s our y-intercept, there’s our x-intercept [points to the points (0; -3) and (2; 0) respectively]. after a brief discussion on the labelling of points on a graph, learner 2 and learner 3 ask nash: learner 2: sir, is this the simplest method, sir? learner 3: how do you identify which side must it go, whether it’s the right-hand side. [nash interrupts] nash: [response to learner 2] you just join the two dots. learner 2: that’s it? nash: yeah … the dots will automatically … if it was a positive gradient it will automatically … if this was [refers to the line just drawn] negative … that means this dot [points the x-intercept] will be on that side [points to the negative x axis] … because if the gradient was negative, how could it cut on that side? [points to the positive x axis] learner 2: is this the simplest method sir? nash: the simplest method and the most accurate learner 4: compared to which one? nash: compared to that one [points to the calculation of the previous question where the gradient and y-intercept method was used] because here if you make an error trying to write it in y form … that means it now affects your graph … whereas here [points to the calculations he has just done on the dual intercept method] you can go and check again … you can substitute … if i substitute for 2 in there [points to the x in ] i should end up with 0. our interest here is in how the teaching of the dual intercept procedure unfolds through transformation activity and its accompanying explanation. nash presents a function (an input representation) where the stated problem is to draw the graph of this line. this representation does cohere with the stated problem, and the preceding activity also shows that producing a line using the transformation sequences that have already been taught can be done, but is long-winded, making an alternative method useful. thus the need for an alternative transformation sequence is motivated in nash’s accompanying explanation. he then presents the steps to follow to obtain the coordinates of each of the intercepts. all the interim representations and the final representation do connect appropriately and cohere with the stated problem. whilst nash’s advocacy of the dual intercept method in all instances blocks some of the flexibility that a more selective match between stated problem and input representation would allow, and also does not provide ways of dealing with special cases (horizontal, vertical lines and lines through the origin), important features of the input and subsequent representations and their connections do remain in focus across nash’s lesson sequence. for example, the fact that the gradient can be derived from the application of the dual intercept transformation steps to a linear equation to produce the associated graph is explained in instructional terms, using the move between algebraic and graphical representations to emphasise how given information can be used to derive missing information. episode 2 in this episode, drawn from askew, venkat and mathews (2012), a grade 2 teacher is working on missing addend problems using a wheel representation with three concentric circles: 7 written on the inner circle, and the numbers 0–7 placed in random order around the outermost circle in separate sectors. askew et al. state: the task explained by the teacher was to fill in the intermediate circle with the numbers that needed to be added to the outer rim numbers of the wheel to make the number 7. (p. 29) the stated problem of the lesson, indexed by the title on the board hlanganisa (‘addition’ in zulu), is for the class to fill in the numbers that need to be added to the numbers on the edge to make 7. initial answers from some of the children indicate that they are interpreting the task in terms of addition of the numbers shown. in one episode, the teacher is focused on the problem: ‘what number is added to 3 to make 7?’ she shows the class three open fingers on her hand as she asks this, pointing to the 3 on the circle rim, and then shows seven fingers as she indicates the need to make 7, pointing to the 7 in the centre. some children are seen counting out seven on their fingers. when no correct answers are forthcoming from the class, the following exchange takes place: teacher: make 7 with your fingers. [shows seven fingers on her hands and several learners seen showing seven fingers] now hide three fingers. [teacher closes three fingers on her hand and asks class to do the same] which number can we add with this 3 to make 7? [teacher’s hand shows four fingers remaining open] now we made 7 and hide 3, and what is left? the number that when we will add with 3, we will get 7. [teacher goes over and helps a child to close the same three fingers and asks her to count what is left] learner: 4. [accepting 4, the teacher then counts out three toothpicks at the front as 1, 2, 3, then another 4 toothpicks as 1, 2, 3, 4. she then counts them altogether from 1–7] prior to and following this episode, we see instances of some learners able to give correct answers. however, we also see several learners who appear unaware of how many fingers to open, and what to do once they have one of the given numbers showing. here, a stated problem that is given in terms of missing addends comes to be ‘funnelled’ into a subtraction problem through a transformation step and associated explanation, and then verified by adding the two numbers as an addition problem. the teacher appears aware of the equivalence between missing addend problems and subtraction, but this equivalence is not established for learners; rather, the equivalence is simply assumed, and subsequently verified empirically. thus, a problem stated in terms of missing addends is worked out in terms of subtraction-based transformation activity, and checked through addition. essentially, the sum below is presented as the stated problem to be solved (though not in this form): whilst the transformation activity instead involves solving the following subtraction problem: it could be argued that in transformation terms nothing has gone wrong here, but given that we see several learners filling gaps in the follow-up missing addend individual activity, with 11 at the centre, simply by adding the two numbers seen, there is evidence that neither the missing addend nor the subtraction routine have actually taken ‘hold’ for broad swathes of the class. instead, the predominant interpretation of the problem involves ‘adding’ the two numbers that can be seen, a ‘putting together’ of the visual instruction to add with the numbers seen. here, we see transformation activity which, whilst connected in mathematically coherent terms to the input object, does not work to establish this connection. the switch to subtraction is followed as an embodied imitation in the teacher’s presence, but no explanation for establishing this switch as valid is provided; at the same time, no interim representational supports or associated explanations that link more directly to the stated problem are enacted. essentially, referring back to the literature, whilst the disciplinary explanation is coherent, an instructional explanation is lacking. episode 3 in this episode, drawn from davis (2010), a grade 10 teacher is working on integer addition sums, such as: -7 + 5. davis describes the teacher’s instructions to the class as follows (p. 384):teacher: so if the signs are the same ... what do you do? ... you take the common sign ... and then ... you add. … if the signs are not the same ... what do you do? you subtract. learners: [chorus] subtract. teacher: but first you take the sign of the what? the sign ... of the bigger number. you look at the bigger number between the two ... and then you take the sign ... of the bigger number. learners: [chorus] yes. teacher: this should always be the case. as was the case in episode 1, an ordered set of instructions is relayed to the class – ‘first you take ... and then you take ...’. some conditions for the application of transformation sequences are established at the outset: essentially ways to distinguish the input representation in order to recognise which transformation must be selected. davis (2010), discussing this episode in terms of operations (addition) and objects (integers), notes that: the regulative criteria required by the procedure indicate that the teacher and learners do not operate directly on the mathematical objects and relations being indexed (integer sums). they operate, instead on more familiar and intuitive objects and relations (‘whole number’ sums). (p. 385) in terms of our analytical concepts, -7 + 5 is the input representation that is transformed through a series of ta steps that provide an algorithm for solving the problem. in the interim stages, following the instructions would produce these representations: teacher: calculate the answer to the sum [sp]: -7 + 5 = [ir] if the signs are not the same ... what do you do? you subtract. [ta step 1] [following this ta step would result in the following interim representation]: 7 − 5 = 2 you look at the bigger number between the two ... and then you take the sign ... of the bigger number. [ta step 2] [following this ta step would result in the following final representation]: -2 in this episode, the backgrounding of the input representation is more significant than in the previous two episodes. within the transformation sequence that is presented, negative numbers simply do not figure. further, whilst the transformation sequence is driven by the need to solve the given problem, the algorithm presented enacts steps that produce an interim representation that is not equivalent to the input object (-7 + 5), even though equivalence with the input representation returns at the final stage. essentially, at the interim stage, transformation activity does not establish ‘reason-able’ connections between representations. as noted already, it could be argued that this is not problematic as a mnemonic device, given that correct answers across the range of input representations specified can be reliably produced. however, analyses of south african performance on timms items (howie, 2003) and our own data (adler, 2011) point to multiple basic errors in the realm of both integer calculations and algebraic manipulations requiring knowledge of integer sum transformations. further, thompson et al. (1994) note that the orientation to ‘answer-getting’ tends to work most reliably only for the learners ‘who understood the problem in the first place, and understood it in such a way that the proposed sequence of operations fits their conceptualisation of the problem’ (p. 9). in terms of our concepts, the input representation coheres with the stated problem, but the interim representation produced through the first transformation step does not retain equivalence with the input representation, even though this equivalence is recovered at the final stage. episode 4 venkat and mhlolo (2011) present an incident involving a grade 11 teacher working on a data-handling problem based on the data table shown in table 1, the input representation introduced in the lesson.in the process of asking generally about ways in which data can be presented, a student mentions the notion of a ‘tally table’. venkat and mhlolo note the subsequent return by the teacher, after several interim episodes focused on a range of other stated problems, to the notion of tallying. the teacher shifts attention from a focus on the meaning of the frequency values in table 1 with the following question: ok before we move on, somebody talked about tally ok. does anyone know how to tally the number 8? ... do you know or you want to try? having asked the question, she then adds a further column to her frequency table and gives it the title ‘tallies’. she then shows the class how to tally the number ‘8’, this being the first frequency value in her table. then, pausing to ask the class if they have seen this (pointing to her tally) before, she explains further and demonstrates: table 1: data table. ok so it’s one, it’s two, it’s three, it’s four and what happens to number five. [indicates the diagonal line]. and then it’s one, it’s two and it’s three. ok. your tally and your frequencies must be of the same number.in this episode we note that whilst transformation activity (producing the tally) does connect to the stated problem, the stated problem does not cohere with the frequency table as the input representation presented. thus, whilst equivalence is maintained, and indeed emphasised, in the teacher’s discourse between the input representation’s frequency values and the representation that is produced (the tallies), this equivalence in the absence of coherence between the stated problem and the input representation tends to nullify the equivalence that is produced. table 2 summarises the application of the analytical concepts applied to the four episodes. discussion top ↑ looking across table 2, we can see that in relation to the analytical framing of transformation activity developed, differences in the detail of access to the representations and transformations involved emerge. significantly, some episodes reflect more serious disruption of mathematical coherence and connection than others. we contend that having the discursive resources to see and talk about these differences is enabling for our work in teacher education.in episode 1, all the criteria are answerable in the affirmative, suggesting that basic connections and coherence are in place. nash’s practices, we believe, mirror the kind of teaching referred to in our earlier discussion of the literature. the fact that nash is viewed as successful, with learners under him viewed as performing successfully, backs up our claim of basic connections and coherence. thus, whilst concepts in the literature may well describe nash’s practice as ‘procedural’, the connections and coherence we have identified appear to open up access to appropriate transformation activity. in episode 2, the key issue is that a missing addend problem is assumed to be solvable through manipulating the sum into subtraction format. of course, this is mathematically correct, but the transformation from missing addend form to subtraction form needs to be established for learners through explanation and mediating representations that allow for the equivalence between the two forms to be appropriated. instead, an assertion or an assumption of equivalence is presented, rather than an establishment of the equivalence. an outcome of the assumed equivalence appears, in this lesson, to be ongoing difficulties with task completion. there is a notable absence of the analogical representations that rowland (2012) describes as important within instructional explanations: in this instance, representations that provide ‘direct models’ (carpenter, fennema, franke, levi & empson, 1999) of the stated missing addend problem. thus, the mdi fails to provide representations that scaffold the connection between the missing addend situation and the abstract understanding of number relationships needed to ‘jump’ to subtraction as an appropriate transformation step to enact. table 2: applying the analytical concepts. in episode 3, the problem seen in episode 2 is further compounded by the fact that the first transformation step indicated by the teacher’s instructions produces an interim representation that does not connect with the input representation. thus, the instruction that seeing one negative and one positive number means we have to subtract is arbitrary at this stage; it simply has to be remembered, and cannot be reasoned. whilst at the final stage, equivalence with the input representation is resurrected, one needs to take on trust that this will happen through the interim working. as in episode 2, the transformation activity does produce the correct answer with some efficiency. in this case though, connections between representations are broken at the interim stage. in episode 4, given that the stated problem is tallying, the presentation of a frequency table as the input representation is inappropriate as an object for the process of tallying to act on. thus, whilst equivalence between the tally graphic produced for each frequency is maintained, connections that could serve to establish the purpose of tallying processes in mathematics are not simply made invisible, but actively disrupted. conclusions top ↑ several comments emerge from our analysis. firstly, we note at the most basic level that if the input representation does not cohere with the stated problem, this appears to negate the possibilities for answering the other analytical questions in the affirmative. at the intermediate level, we suggest that two criteria allow for further disaggregation:• making transformation steps ‘reason-able’ by establishing connections between the representation and its transformation • producing transformation sequences that connect across representations. at the highest level, we have episodes that demonstrate coherence between the stated problem and the input representation, and connections between the representations produced through transformation activity where all three criteria are met. our sense is that the literature as it stands provides us with a discourse that can speak constructively to nash’s practice, but offers few insights into the kinds of limitations seen in our other episodes. the analytic concepts and questions that we have presented in this article were derived from analysing problematic episodes of the teaching of procedures involving the transformation of representations. whilst the framework still needs further testing, our application of these concepts and criteria to further episodes from our project data sets suggests that they may provide some general principles for basic coherence and connection within mathematics teaching. we therefore offer the concepts and questions developed and deployed in this article as a starting point that has some generality, and illuminating potential for the many classrooms in which the transformation activity that fundamentally underlies mathematical activity still appears to be problematic. acknowledgements top ↑ this work is based upon the research of the wits mathematics connect – primary and secondary projects at the university of the witwatersrand, supported, respectively by firstrand foundation, anglo american, rand merchant bank, the department of science and technology, and the firstrand foundation (frf) mathematics education chairs initiative of the firstrand foundation, the department of science and technology (dst). both are administered by the national research foundation (nrf). any opinion, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the institutions named above. competing interests we declare that we have no financial or personal relationship(s) which might have inappropriately influenced our writing of this article. authors’ contributions h.v. 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(2012). explaining explaining. in s. nieuwoudt, d. loubscher, & h. dreyer (eds.), proceedings of the 18th annual national congress of the association for mathematics education of south africa, vol. 1 (pp. 54−66). potchefstroom: amesa. available from http://www.amesa.org.za/amesa2012/volume1.pdf sekiguchi, y. (2006). coherence of mathematics lessons in japanese eighth-grade classrooms. in j. novotná, h. moraová, m. krátká, & n.e. stehlíková (eds.), proceedings of the 30th conference of the international group for the psychology of mathematics education, vol. 5 (pp. 81−88). prague: pme. thompson, a.g., philipp, r.a., thompson, p.w., & boyd, b.a. (1994). calculational and conceptual orientations in teaching mathematics. in d. aichele, & a. coxford (eds.), professional development for teachers of mathematics (pp. 79−92). reston, va: nctm. available from http:// www.patthompson.net/pdfversions/1994calc&concorientations.pdf venkat, h., & mhlolo, m.k. (2011). objects and operations in mathematics teaching – extending our understanding of breakdowns. in t. mamiala, & f. kwayisi (eds.), proceedings of the 19th annual meeting of the southern african association for research in mathematics, science and technology education (pp. 246−259). mafikeng: saarmste. footnote top ↑ 1.our use of the term ‘transformation activity’ coheres with duval’s (2006) use of the term, which refers to transformations within and across registers. it is more inclusive than kieran’s (2004) notion of transformation, which is restricted to transformation of algebraic expressions whilst maintaining equivalence across these manipulations. article information authors: pragashni padayachee1 hennie boshoff1 werner olivier1 ansie harding2 affiliations: 1department of mathematics and applied mathematics, nelson mandela metropolitan university, south africa 2department of mathematics, university of pretoria, south africa correspondence to: pragashni padayachee email: pragashni.padayachee@ nmmu.ac.za postal address: department of mathematics and applied mathematics, nelson mandela metropolitan university (south campus), port elizabeth 7000, south africa dates: received: 08 feb. 2010 accepted: 08 july 2011 published: 13 sept. 2011 how to cite this article: padayachee, p., boshoff, h., olivier, w., & harding, a. (2011). a blended learning grade 12 intervention using dvd technology to enhance the teaching and learning of mathematics. pythagoras, 32(1), art. #24, 8 pages. http://dx.doi.org/10.4102/ pythagoras.v32i1.24 copyright notice: © 2011. the authors. licensee: aosis openjournals. this work is licensed under the creative commons attribution license. issn: 1012-2346 (print) issn: 2223-7895 (online) a blended learning grade 12 intervention using dvd technology to enhance the teaching and learning of mathematics in this original research... open access • abstract • introduction • the context    • the blended learning dvd approach used in the 2007 isp    • instructional design of the dvds    • production of the dvds    • the saturday programme • research design    • research method       • participants       • action research       • validity and reliability       • ethical considerations • results and discussion    • themes which arose from the data       • new experience       • dvd as a resource       • the blended learning environment    • facilitators’ observations    • improvement in marks    • case study • conclusion • acknowledgements    • competing interests    • authors’ contributions • references abstract (back to top) this article describes the experiences and mathematics performance of grade 12 learners selected to participate in a mathematics intervention project using digital video disk (dvd) technology within a blended learning context. blended learning in the context of this study is defined as employing a variety of appropriate methods of delivery to enhance the teaching and learning process. dvd technology was used as an ingredient in this blended learning approach, since it is easily available and accessible to the majority of learners and the schools they attend. the study reported on here forms part of a larger study using action research methodology. this article reports on a single stage of the action research: implementing a change to improve the situation and observing the consequences of this action. mathematics incubator school project (isp) learners completed questionnaires with open-ended questions which pertained to their experiences of the blended learning approach. the observations of the facilitators were also recorded. a single school was used as a case study and the mathematics performance of learners who participated in the isp was compared with that of those who did not. the findings suggest that use of dvd technology in this blended learning approach impacted on mathematics learning and enhanced the mathematics performance of learners. introduction (back to top) the ongoing poor quality in mathematics teaching and learning in south africa is the ‘most important obstacle to african advancement’ (centre for development and enterprise, 2004, p. 239). at the heart of this concern is the fact that the present education system has disadvantaged learners by failing to meet their educational needs, especially regarding mathematics (evoh, 2009). it has been our observation in lecturing first-year mathematics students at a large metropolitan university in the eastern cape in south africa that many first-year students are under-prepared for mathematics. this under-preparation may be attributed to the shortage of adequately qualified teachers and lack of resources at schools attended by the majority of learners (adler, brombacher & human, 2000). many interventions in response to this crisis in the teaching and learning of mathematics are documented in the literature. there are interventions which specifically target the learner, such as the petrosa maths and science academy in mossel bay (government communication and information system, 2008), schools of excellence project (zenex foundation, 2007), and national dinaledi schools (department of education, 2004), whose aim it is to improve learners’ achievements in mathematics and science. there are interventions where the aim is to uplift the skills of the teacher, like the ergo programme (anglogold ashanti, 2009). however, none of these secondary school mathematics interventions make use of technology in the teaching and learning of mathematics within a blended learning approach. in this technologically advanced era it would seem natural to question whether ‘relatively accessible and affordable technologies’ can contribute towards addressing the poor quality of teaching and learning (nieuwoudt, nieuwoudt & monteith, 2007, p. 29). moreover, as technology becomes more advanced and easily accessible there is a shift from traditional, teacher-centred classroom instruction to a more learner-centred approach to teaching and learning (cohen, grady & springer, 2003). the traditional teacher-centred approach is inappropriate to outcomes-based education with its central focus on learner-centredness (badenhorst & de beer, 2004). however, technology alone will not create the ideal learning environment (luca, 2006). this idea is supported by singh and reed (2001), who cite research from stanford university and the university of tennessee when they suggest that blended learning is better than either traditional methods or the use of technology alone. blended learning incorporates different modes of delivery to enhance the learning experience and provide a more efficient and effective teaching experience. singh and reed (2001) suggest that in developing an efficient and effective blended learning model, consideration should be given to the achievement of learning outcomes when using the ‘right technology’ to match the learning styles of the learner. therefore blended learning involves use of an appropriate mix of delivery techniques and technologies to enrich the learning experience and achieve the outcomes of this learning (maguire & zhang, 2006). a need was identified for an intervention in the teaching and learning of mathematics in south africa that uses technology together with a variety of other delivery modes. the blended approach incorporating digital video disk (dvd) technology which is described in this article presents a financially feasible opportunity in the teaching and learning of mathematics to improve mathematics results of a grade 12 intervention group. the rationale behind this approach is that although the learners on this project did not have access to internet, they all had access to a dvd player. ideally the dvds should be used in a blended learning environment with the additional inputs of facilitators. however, the dvds are designed to be used as an independent learning resource as well. the fixed mathematics content of the dvd ensures that mathematical concepts are not watered down or misrepresented during presentations. this was particularly helpful to learners during the 2007 teachers’ strike, which impacted the mathematics learning of many of the learners. in this article a blended approach to teaching and learning mathematics is evaluated, with the focus on how learners experienced this approach and whether it impacted on their mathematical performance. the guiding research question for this research is: how can a blended learning approach that incorporates dvd technology contribute to improving the quality of teaching and learning in secondary school mathematics? the context (back to top) the mathematics incubator school project (isp) was initiated in 2004. the aim of the isp is to address the serious shortage of suitably qualified mathematics secondary school teachers and the shortage of students from eastern cape schools entering into higher education in the fields of science, mathematics and technology. the isp teaching and learning approach has undergone many changes over the past 4 years: • in 2004 traditional classroom teaching (no fixed lesson plans) was used • in 2005 traditional classroom teaching (fixed lesson plans) was used • in 2006 powerpoint slide presentations of topics were used during lessons • in 2007 a series of 20 dvds covering mathematics topics necessary for the then grade 12 mathematics curriculum was developed and used within the blended learning approach. the focus of the 2007 isp was to develop innovative, new, learner-centred learning programmes and strategies using appropriate technologies to facilitate the effective teaching and learning of mathematics. the blended learning dvd approach used in the 2007 isp a blended learning approach was adopted in these isp deliveries. graham (2004, p. 8) says that a ‘more effective pedagogy’ is one of the reasons for blended learning, since blended learning aims to create an environment that allows a shift to more learner-centredness. the challenge is to blend the right resources and activities to optimally enhance the learning experience. singh and reed (2001) highlight the following as key ingredients of a blended programme: • cognizance of the level of knowledge that the learners bring to the programme, different and shared learning styles of the learners, their motivation levels and ability to access technology • content analysis to determine selection of appropriate delivery modes • financial implications of delivery • infrastructure to support the use of technology. the aforementioned ingredients were considered in the design of the 2007 isp. learners’ levels of mathematics knowledge were based on their grade 11 mathematics results. however, learners attended different high schools and did not write a uniform mathematics examination. it became clear after the 2007 isp implementation that a pre-test should be administered in order to determine the mathematics knowledge of learners. the skills-driven learning model (valiathan, 2002) was used in the isp deliveries; this combines self-paced learning with instructor or facilitator support to develop specific knowledge and skills. valiathan (2002) expounds further that in this type of approach, interaction with the facilitator acts as a catalyst to achieve learning and keeps the learner from experiencing feelings of isolation. there are many possibilities which constitute a blended learning approach. in particular, live face-to-face: formal (facilitator-led classroom), live face-to-face: informal (mentoring by tutors and collaborative learning with peers) and self-paced learning (dvds) were used in all isp deliveries (rossett, douglis & frazee, 2003). in the blended learning environment reported on here, the use of easily accessible and affordable dvds was decided upon since the learners did not have internet access and broadband facilities to enable online learning or facilitate video streaming. also, many learners did not have computer skills, and were therefore not disadvantaged by this approach. the term blended learning here meant that each dvd covering a particular concept in the grade 12 mathematics syllabus was designed for viewing in a short session. this allowed for facilitator discussion, tutor interaction, peer interaction and also working on assigned tutorial problems during contact sessions, which were conducted on saturdays. once the learning outcomes were completed for a particular section, the dvd presentation was resumed, and so the cycle continued. learners were given hard copies of all the tutorials as well as solutions to the tutorial problems. each learner was also given a personal copy of the dvd to view at their own pace at home. learners were assessed on the concepts covered on a particular dvd upon their return the following saturday. table 1 represents a list of blends used in the isp deliveries. table 1: list of blends. a schematic model of the blended learning approach is shown in figure 1. figure 1: the 2007 blended learning approach. instructional design of the dvds each dvd covered a particular topic in the syllabus; the topic was then divided into micro lessons (parts). each part of a dvd topic started with an introduction and definitions of the concepts, and this was followed by fully explained examples pertaining to that particular concept. the intention was to give learners an understanding of the thought processes involved in solving a problem and to help learners who would have no idea where to begin (aminifar, porter, caladine & nelson, 2007). the part ended with tutorial problems to be attempted by the learners, intended to give them a way to assess their understanding of the micro lesson. a set of comprehensive solutions followed, that could be viewed after learners had attempted the tutorial problems. this approach ensured that immediate feedback was available to any learner who struggled with the tutorial problems. the entire dvd had voice narration integrated with animated powerpoint slides. table 2 shows the content topics covered in the 2007 mathematics dvd series. table 2: the 20 mathematics dvd topics. production of the dvds powerpoint slides with animation were developed for each mathematical concept and formed the basis of the recording of the dvds. recordings were done in a studio by a member of the isp team, normally a lecturer in the mathematics and applied mathematics department at the university in question. recordings entailed using a tablet pc and presenting the powerpoint slides with voice explanations. other software packages like autograph were used to illustrate sections requiring graphic drawing. whilst recordings were being done, a second academic staff member evaluated them to ensure that the dvds were free of mathematical errors. re-recording of an erroneous slide took place immediately if necessary. thereafter a technical media specialist edited the dvds to produce master dvds with a menu-driven system. the saturday programme learners attended a series of five-hour sessions held on saturdays, during which two dvds of one hour each were facilitated. after each session, learners engaged with tutorial problems. this was an interactive session, with discussions and assistance from facilitators as well as student tutors. the dvds were presented to learners using a dvd player connected to a data projector. research design (back to top) research method the research reported on in this article is part of a larger research study investigating the teaching and learning of mathematics using dvd technology within the context of blended learning. the focus of this research study was the 2007 isp which forms part of a larger project. the 2002–2006 isp were earlier cycles of this larger action research project. the evaluation of each cycle in this action research informed the next cycle in the project (e.g. boshoff & olivier, 2008). in 2007 the use of dvd technology was introduced in the isp, and this article focuses on use of dvd technology within a blended learning environment in the 2007 isp. participants a total of 184 grade 12 mathematics learners from local schools in the nelson mandela metropole were selected to participate in the 2007 isp. the best-performing learners were selected on the basis of their grade 11 mathematics marks and their interest in pursuing studies in science, engineering, technology and mathematics and science education. these learners were split into four groups, with on average 46 learners to one facilitator in each group. these learners and their facilitators were the participants in this research study. action research action research is the methodology used in the larger research study, and this study focuses on one aspect of the action research cycle. action research provides an important link between research and teaching, particularly when the research is actively conducted with the aim of informing teaching and learning and challenging ways of incorporating technology into the curriculum (manchester, ralph & shipova, 2005). action research was an appropriate design in the larger research study in order to reflect on the processes that were followed and to refine the isp before the next cycle of implementation. the following steps in the action research cycle were followed in the larger study: plan and design the intervention; implement the intervention; evaluate the intervention; reflect on the evaluation; and implement the changes – and so begins another cycle. this article focuses on the third phase in the action research cycle, namely the evaluation of the isp intervention, and in so doing concentrates on factors that contributed to a supportive and encouraging learning environment. in particular, the focus was on the experiences of learners and the impact of the blended learning approach on their mathematical ability. during the evaluation phase of the action research cycle a mixed-methods approach to collecting data was adopted, with both qualitative and quantitative methods employed. at the end of the 2007 programme 184 learners completed a questionnaire aimed at evaluating the teaching and learning approach they had experienced. the questionnaire contained open-ended questions and provided rich descriptive data with regard to the factors that provided a supportive and encouraging learning environment. further to this, qualitative data were also collected in the form of facilitator observation of the learning process using weekly reports. using quantitative methods, the learners’ final grade 11 mathematics results were compared with their final grade 12 mathematics results to determine whether an improvement in mathematical performance had occurred as a result of this blended learning intervention. the case study of a single school of 20 learners, 6 of whom were isp participants and 14 non-isp participants, is also described here. there was difficulty in obtaining the results of other learners at schools that participated in the 2007 isp. validity and reliability use of various methods of data collection and triangulation were vital in determination of an in-depth understanding of the programme under investigation. strategies for promoting validity and reliability (adapted from merriam, 2002, p. 31) used in this research study are outlined in the following section. • triangulation: the researchers used multiple sources of data and data collection methods to confirm emerging findings. triangulation was as follows: o data triangulation: questionnaires completed by learners and reports completed by facilitators. o methodological triangulation: use of qualitative and quantitative methods. • member checks: the researchers took the tentative interpretations of the data back to the learners and facilitators to see if they agreed with them. • peer review: the process of the research study, congruency of emerging findings with the raw data and tentative interpretations were discussed with colleagues. • researchers’ position: the researchers undertook critical self-reflection with regard to their worldview, assumptions and relationship to this study that could have influenced bias and affected this study. • adequate engagement in data collection: it was ensured that adequate data were collected such that the data became saturated. • rich, thick descriptions: the researchers tried to supply sufficient descriptions to enable readers to determine whether this situation matches their research context or whether the findings can be transferred. ethical considerations denscombe (2007, p. 142) outlines three core principles that should inform the ethical choices and guide the activities of the researcher. the principles as applied in this research study were as follows: principle 1: the interest of the participants should be protected. this research ensured the confidentiality of the participants. data were kept safe and no personal identities were revealed in disclosure of the results. principle 2: researchers should avoid deception or misrepresentation. the researchers were honest and made all the processes transparent. the researchers and participants developed a relationship of trust, and the researchers presented the data in a fair and unbiased way. principle 3: the participants should give informed consent. ethical clearance was given by the research committee of the university where the isp is based. the department of education gave its consent for the researchers to access participants from schools in the eastern cape. written consent was obtained from all the participants and in the case of minors consent was received from their parents. all participation was voluntary. results and discussion (back to top) analysis of the questionnaires, facilitator reports, comparisons of learners’ grade 11 and grade 12 mathematics marks and the case study are presented and discussed. in analysis of the data collected from learners’ questionnaires and facilitators’ weekly reports, the themes outlined below arose. themes which arose from the data new experience more than half of the learners felt that the approach was refreshing and a different way of learning mathematics. many learners said they found this way of learning enjoyable and exciting. a few learners said that the dvd was different from a textbook, since there was a ‘voice’ explaining the concepts. dvd as a resource learners felt that explanations of the mathematical concepts were good and the concepts were presented well. the dvd was particularly useful to them in sections that required visual representation (drawing graphs) and in calculus, since it was ‘not done well’ at school. many learners said they found the dvds helpful and that they found the mathematics easy to understand because of the many examples and illustrations on the dvds. a few learners said that they found it easier to concentrate using this method as opposed to having someone standing in front explaining what to do. on the other hand, a few other learners found it difficult to concentrate using the dvd approach, and one said it was easy to ‘move your concentration elsewhere’ unless you had a lecturer to intervene with discussion or explanations. many learners said the dvd approach gave more insight than school, and that they found it helpful to do a section on dvd in the saturday project before it was done at school, and that this helped their understanding of the section. learners also used the dvd to ‘test and check’ their understanding of a particular concept. they found that the dvd series was a good resource to consult when faced with homework or in preparation for a test or examinations. many learners said that the dvds allowed them to learn at their own pace and to watch a section over again until they understood the concepts. many found that they could watch the dvds at home and revise together with the resource material; this approach helped to enhance their understanding. however, some learners suggested that some of the dvds did not illustrate every step of a mathematical problem, which led to confusion. they asked for more detail and more examples on the dvds. the blended learning environment although most of the learners agreed that the dvd was a useful resource, many said they needed facilitators and tutors for further explanations and discussions. they needed the facilitators’ explanations to bridge gaps where the dvd skipped steps or where they had problems understanding. according to the learners, tutors, facilitators and discussions with other learners helped them see concepts from different viewpoints. learners felt that their english improved as a result of the blended learning approach, although one learner suggested that dvds be produced in other languages. learners said that they liked the way the isp presentations blended the tutorials, dvds and discussions, and that this made it easier for them to concentrate for longer periods. the issue that stands out and which was pointed out by almost all learners was the ease with which learners could view the dvd and replay and pause whenever they wanted to. however, some learners mentioned that whilst watching the dvd there were questions that needed explanation, and that the dvd should not be used alone. a few learners also said that their teacher used these dvds at their schools to teach certain topics, like calculus, for example. some learners formed study groups and watched the dvds in these groups over weekends. the blended learning approach using dvds presented a new way of learning mathematics to or for the learners, and most of them were positive that the method benefited their understanding of the subject. the approach allowed them the freedom to access a variety of different resources, and allowed them to work at their own pace and to revise at home. facilitators’ observations the facilitators’ observations of learners’ experiences of the blended learning environment were positive and encouraging. all four facilitators felt that the learners’ confidence improved as a result of this approach. in addition, one facilitator noted that as the project progressed, communication and engagement with the facilitators and learners, tutors and learners, and amongst the learners improved. a facilitator said that the level of mathematics questions posed by the learners improved with time, and the learners’ test performances improved steadily as the project progressed. all the facilitators agreed that the interaction in the blended learning approach was lively, although learners found it (i.e. dvd technology) strange at first. one facilitator said that the ‘dvd is a powerful resource allowing learners to work at their own pace and review solutions and procedures until they understood the concept’. all four facilitators said that they believed that dvds should be used in conjunction with other face-to-face methods of teaching and learning. they believed that used in isolation, the dvds would not prove such a successful resource in the teaching and learning of mathematics. evidence is provided which suggests that dvd technology impacted on the mathematics learning and enhanced the mathematics performance of the isp learners. in addition, the learners responded favourably to use of the dvd technology within the blended learning environment. however, some learners said that at first they needed to adjust to this new learning environment. they also indicated that once they had adjusted, they could see the advantages of being exposed to different modes of delivery. the majority of the learners agreed that this blended environment of teaching and learning mathematics fostered a deeper understanding of the subject for them. the most important point raised by learners was the fact that dvds alone were not sufficient to ensure success. they believed that the dvds together with facilitators’ and tutors’ explanations and discussion coupled with the hard copy resources were the best blended approach for their learning of mathematics. improvement in marks statistical analysis was undertaken to investigate whether differences between the final grade 11 mathematics marks and final grade 12 mathematics marks of the isp learners were significant. the descriptive statistics comparing the final grade 11 marks with the final grade 12 marks are shown in table 3. table 3: descriptive statistics for grade 11 and grade 12 mathematics marks (n = 184). the observed difference of 2.96 between the mean scores shows a statistical significance; however, no practical significance is noted. figure 2 illustrates that the distribution for grade 11 marks peaks with the majority of scores between 40 and 59, whereas the distribution for grade 12 is evenly spread, having moved from the middle to the other categories. the kolmogrov–smirnov test (k–s d = 0.245, p < 0.01) confirms a significant difference in distribution. figure 2: distribution of grade 11 and grade 12 mathematics marks. it is pleasing to note that in higher intervals from 60 upwards there is clearly an improvement of marks from grade 11 to grade 12. in the category of 70+ there is an increase from grade 11 to grade 12, from 4% to 22%. unfortunately, there also are more students who performed badly in grade 12 compared to grade 11: there is an increase in the under-40 intervals; in fact, there is an increase in the interval 0−39, from 14% in grade 11 to 26% in grade 12. figure 3 confirms that the distribution for grade 11 is flatter, with more results between 40 and 69, whereas the distribution for grade 12 is more evenly spread. figure 3: cumulative distribution of grade 11 and grade 12 mathematics marks. table 4 and figure 3 reflect the difference in distribution between grade 11 and grade 12. the relationship between the categories of marks for grade 11 and grade 12 is depicted in table 4. table 4: contingency table of grade 11 and grade 12 mathematics marks. it is interesting to note that no learners from the 0−39 category in grade 11 moved to the 70+ category in grade 12, and none of the learners in the category 70 in grade 11 moved to the 0−39 category in grade 12. it is also noteworthy that 25% of the learners in the 40−49 category in grade 11 moved to the 70+ category in grade 12. the quantitative data indicated statistical significance, and it seems that there was an improvement from the grade 11 to the grade 12 mathematics marks in many cases. however, it became clear from the quantitative data that not all learners responded positively to the blended learning approach. case study a school in the uitenhage district was used as a case study. there were 20 learners in this grade 12 mathematics class. six of these learners participated in the 2007 isp. for all 20 learners, their 2007 final grade 12 mathematics marks were compared with their 2006 grade 11 end-of-year mathematics marks. results of the learners who were on the isp improved, whilst those learners not on the isp generally presented lower grade 12 final results. to determine whether the difference was statistically significant, a mann-whitney u test (z = 3.38, p = 0.001, d = 3.20) was conducted. a cohen’s d test was conducted for practical significance. it was found that the difference in means was highly significant, since d > 0.8 reflects a large difference. the results are summarised in table 5 and table 6. table 5: case study – grade 11 and grade 12 end-of-year marks for mathematics. table 6: case study – difference in means of grade 11 and grade 12 end-of-year marks for mathematics. conclusion (back to top) the research question addressed in this article was ‘how can a blended learning approach that incorporates dvd technology contribute to improving the quality of teaching and learning in secondary school mathematics?’ from the study it is clear that this blended approach offers a workable teaching approach with definite advantages. one of the advantages of using dvd technology within a blended approach is accessibility of the subject content and presentation outside the classroom. qualitative results indicated that both learners’ and teachers’ experiences were largely positive, with concerns identified that offer opportunity for improvement. most learners attributed their better understanding of mathematical concepts taught to the dvd approach that was used in conjunction with other traditional modes of delivery. quantitative results indicated statistical significance when comparing the mean scores for grade 11 and grade 12 results, although not of practical value. according to calldo and du plooy (2008), the percentage of students passing mathematics on higher grade declined from 7.2% in 2006 to 6.9% in 2007. in the face of declining performance in mathematics in south africa in the 2007 matriculation examination, our results seem to suggest that the dvd approach of blended learning could in some way have contributed to the improvement in mathematics results that was noted amongst many of the isp learners, especially the better students. it is disappointing that a large group of the borderline students did not seem to benefit from this blended learning approach. this finding is reason for concern, since these are the students that should be targeted – and the reason for this disappointing deterioration has to be investigated. the fact that the dvd technology is easily accessible and affordable supports a case that the dvd approach could also help to address the shortage of adequately qualified teachers and lack of teaching resources at previously disadvantaged schools in south africa. harding, kaczynski and wood (2005) say that in order to be successful in using blended learning, one has to not only implement learning reforms but also, importantly, evaluate these reforms, and in so doing provide students with the best possible outcomes. the isp saw the implementation of dvd technology for the first time in 2007, and this research aimed to evaluate its value to the teaching and learning of mathematics. in order to provide learners with the best possible outcomes, ‘it is important that we continue to identify successful approaches of blended learning at institutional, programme, course and activity levels that can be adapted to work in contexts’ (graham, 2004, p. 19). our dvd blended learning approach still requires further development and refinement, especially with regard to the skills of facilitators within such a blended learning environment and the development of materials. these and other issues will be looked at in future research initiatives. acknowledgements (back to top) we are grateful to sasol for the funding that supported the 2007 isp and this research. competing interests we declare that we have no financial or personal relationships which may have inappropriately influenced us in writing this article. authors’ contributions h.b. was the project leader whilst h.b. and w.o. were responsible for the project design. h.b. collected the data. a.h. made conceptual contributions. p.p. performed the qualitative and statistical analysis, made conceptual contributions and wrote the manuscript. h.b., w.o. and a.h. reviewed the drafts of this research article. references (back to top) adler, 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(2007). the zenex foundation’s ten-year strategy 2006−2015. retrieved august 22, 2009, from http://www.zenexfoundation. org.za/strategy.php abstract introduction literature review mathematics word problems universal design for learning theoretical framework research methods and design sample selection of the participants mini-workshop training on the principles of universal design for learning teaching mathematics word problems data collection and analysis observations reflection sessions findings and discussion multiple means of representation conclusion acknowledgements references about the author(s) matshidiso m. moleko department of mathematics, natural sciences and technology education, faculty of education, university of the free state, bloemfontein, south africa mogege d. mosimege department of mathematics, natural sciences and technology education, faculty of education, university of the free state, bloemfontein, south africa citation moleko, m.m., & mosimege, m.d. (2021). flexible teaching of mathematics word problems through multiple means of representation. pythagoras, 42(1), a575. https://doi.org/10.4102/pythagoras.v42i1.575 original research flexible teaching of mathematics word problems through multiple means of representation matshidiso m. moleko, mogege d. mosimege received: 21 aug. 2020; accepted: 03 june 2021; published: 10 aug. 2021 copyright: © 2021. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract flexible teaching of mathematics word problems is essential to improve learning. flexible teaching is vital in terms of providing meaningful learning, creating inclusive learning spaces and making content accessible. as such, teachers need to strive to provide flexible teaching of mathematics word problems in order to optimise and maximise learning. in line with this notion, therefore, the qualitative case study reported in this article aimed to explore the implementation of one aspect of universal design for learning (udl), namely multiple means of representation (mmr), to guide flexible teaching of mathematics word problems. data were collected using focus group discussions, reflection and observation sessions in which five high school mathematics teachers and a head of department were involved. the teachers participated in a mini-workshop on the application of the udl principles which was organised to introduce and induct them to the approach. the study showed that mmr can be used to help guide flexible teaching of mathematics word problems by providing varied options for comprehension: options for language, mathematical expressions and symbols, as well as options for perception. the findings of the study recommend the need for teachers to adapt their teaching by considering the application of the mmr principle to guide and promote flexible teaching of mathematics word problems. keywords: flexible teaching; mathematics word problems; teaching strategy; universal design for learning; multiple means of representation. introduction learners who face challenges in terms of english proficiency find mathematics word problems (mwps) difficult to solve (vula & kurshimla, 2015). this is because mwps require learners not only to know how to work with numbers, but also to possess other skills such as to identify, understand, interpret, create, communicate and compute numbers (unesco, 2005, p. 21) which are skills they often lack. furthermore, learners have limited knowledge of mathematical vocabulary, which makes it difficult for them to understand and solve these problems. according to seifi, haghverdi and azizmohamadi (2012), many teachers find mwps challenging to understand and teach. as a result, they resort to teaching them in a ‘mechanical’ manner, which does not cultivate and deepen understanding (sepeng & madzorera, 2014). a mechanistic way of solving mathematical problems further causes learners to fail to develop personal connections and understanding between the mathematical concepts (goldberg & bush, 2003). according to liljedah, trigo, malaspina and bruder (2016), mechanical problem-solving refers to a method of solving mathematical problems by merely applying previously learned formulae to new similar situations. this way, true solutions can be reached by solving mathematical operations in a certain specific outlined order (bal, 2015). although this seems to be the preferred way of solving mathematical problems, the challenge it poses is that it limits learners’ critical thinking skills and through this method learners are unable to demonstrate the acquisition and ability to apply mathematical skills in new contexts as envisioned in the curriculum assessment policy statement (caps) document (dbe, 2011). solving mathematical problems mechanically promotes a high level of engagement with routine problems by merely applying formulae or certain defined steps without necessarily understanding the underlying concepts behind the applied procedures. this form of problem-solving limits the learners’ ability to solve and master the non-routine problems that are based on daily life and that guide learners in developing unique ways and strategies for problem-solving (anderson, 2009; elia, van den heuvel-panhuizen, & kolovou 2009; sahid 2011). learners who are exposed to this form of teaching practice often discount reality in their solution processes, thus generating conclusions that are mathematically correct but situationally incorrect or inaccurate (webb & sepeng, 2012). drawing from the above discussion, a deduction could be made that teachers who prefer mechanical ways of problem-solving do not make substantial efforts in terms of making their teaching flexible to ensure that all learners productively learn mwps. on the basis of this, therefore, in this study we argue that mwps could be taught productively through the application of the aspects of universal design for learning (udl), which promotes the idea of flexible teaching through inclusive practices. a consideration of inclusive practices is vital in terms of creating a learning environment where teachers could develop supportive relationships with learners and also increasing learner participation and engagement (guðjónsdóttir & óskarsdóttir, 2015). teachers who apply inclusive teaching practices show flexibility in their teaching in that they apply different strategies that accommodate a broad range of learners in their classrooms (engelbrecht, nel, nel, & tlale, 2015; ojageer, 2019). such flexibility is essential and beneficial because it allows teachers to respond to different learner abilities, needs and interests (murawski & hughes, 2009). teachers whose teaching is flexible make it easy for learners to follow precisely what they are trying to teach them. furthermore, teachers who apply flexible approaches are able to increase learner participation and engagement (hennessy, deaney, ruthven, & winterbottom, 2007) and this practice is important in ensuring that no learner is left behind (hill, 2007). other ways in which teachers could demonstrate flexibility and responsiveness include differentiating the instruction to address individual learner misunderstandings, building on learners’ interests, etc. (rock, gregg, ellis, & gable, 2008). our belief based on all this is that teachers who are flexible in their teaching carry the idea that their plans can change very swiftly, sometimes with notice and sometimes without. such an idea thus necessitates teachers to be proactive in terms of planning their teaching by considering various teaching strategies to accommodate the different learning styles. it also cultivates the culture of anticipating elements of diversity, which may necessitate adaptation of teaching strategies. in line with this, literature indicates that udl can be used as a front-loader (cooper-martin & wolanin, 2014). this means that the teacher has to incorporate the udl strategies during the creation of instruction and assessments, instead of adjusting lessons or assessments afterwards. it is against this backdrop that this study aims to explore the implementation of udl to guide flexible teaching of mwps. literature review universal design for learning is ‘an approach to teaching that consists of the proactive design and use of inclusive instructional strategies that benefit a broad range of learners including students with disabilities’ (scott, mcguire & embry, 2002, p. 2). this framework for teaching was coined to promote multiple teaching practices and integration of the current best approaches to engage learners to address their different learning styles. therefore, udl makes it possible for teachers to meet the learning needs through a fusion of teaching approaches, designs and technologies. dalton, mckenzie and kahonde (2012) note that udl is a teaching strategy used to accomplish the broader goal of inclusive education. on the other hand, van jaarsveldt and ndeya-ndereya (2015) state that udl is the most appropriate teaching strategy to address diversity within the classrooms and to promote flexible teaching that is all-encompassing. according to burgstahler (2008), udl is intended to maximise learning and to inculcate the culture of flexible and inclusive teaching practices (scott et al., 2002). universal design for learning inspires the teachers to anticipate, embrace and acknowledge diversity within the classroom. as such, teachers are expected to plan their lessons to address the diverse needs of learners from the inception of their teaching rather than to wait until teaching has taken place (israel, ribuffo, & smith, 2014). this therefore means that udl requires teachers to be proactive rather than reactive, in terms of addressing the needs of the learners, thus calling for teaching practice adaptation. mathematics word problems a mwp denotes text that describes a situation assumed to be familiar to the reader and poses a quantitative question that subsequently requires an answer to be derived through mathematical operations performed on the data provided in the text form, or otherwise inferred (greer, verscheffel & de corte, 2002). according to kasule and mapolelo (2013) mwps’ content is presented in the form of stories. palm (2009) refers to mwps as representations of real-life situations. texts of mwps are stretches of amalgamated forms (e.g. clauses or sentences), written in a particular dialect code (e.g. the english language) and register (e.g. mathematical terminology), and with a distinctive internal organisation (i.e. a textual structure) that can be logically and rationally understood by readers who bring with them expectations, interests, viewpoints, interpretations and prior reading experiences (oliveira et al., 2015). mathematics word problems are an essential part of the mathematics curriculum because of their ability to promote realistic mathematical modelling and problem-solving (van den heuvel-panhuizen & drijvers, 2014). these problems empower learners to realise the connections between real-life situations and their classroom mathematical knowledge (sepeng & madzorera, 2014). although a significant role of the mwp is undisputable in terms of promoting realistic mathematical modelling and developing personal connections between real life and classroom mathematical knowledge, this mathematical genre has proven to be challenging for most learners and teachers alike (seifi et al., 2012). teachers usually teach mwps in a mechanistic manner because they often find the teaching of these problems difficult (sepeng & madzorera, 2014). mechanistic teaching hampers problem-solving and this is often seen through learners who discount reality in their solution processes, thus generating conclusions that are mathematically correct but situationally incorrect or inaccurate (sepeng 2010; webb & sepeng, 2012). the teaching of mwps is further complicated by necessitating the learners’ ability to recognise, comprehend, construe, construct, communicate and work out numbers, which are skills most learners seem not to have (unesco, 2005). the teaching of word problems is made more intricate by the learners’ inability to read (gooding, 2009), which is evidenced by being unable to determine missing information, generate number sentences and set up calculation problems (fuchs et al., 2008). huang and normandia (2008) state that most learners commit more mistakes when solving word problems than when solving equivalent number problems and this is because word problems demand strong mathematical calculations along with other types of knowledge such as linguistic knowledge and analysis which are the skills that most teachers do not cultivate in learners (sepeng, 2013). on the other hand, the teaching of word problems proves to be more challenging especially in teaching and learning settings wherein english is a medium of instruction and a second language of the learners (essien, 2013). according to barwell (2009), such teaching and learning settings present teachers with challenges and extra demands for them to pay attention to mathematics, pay attention to english (the language of learning and teaching) and also pay attention to mathematical language and register. adding to the challenge of teaching mwps is the fact that teachers themselves also find word problems difficult to solve (seifi et al., 2012). deducing from the above, it is reasonable to note that mwps are a difficult genre to teach to learners. it is due to their complex nature that webb, campbell, schwartz and sechrest (1966) label them as ‘demon’ problems. the fact that they are often taught in a mechanistic manner, which does not cultivate understanding, shows that substantial efforts in terms of applying flexible and inclusive practices to cater for different learning styles are not made. it is against this backdrop that this study is purposed to explore the implementation of udl to guide flexible teaching of mwps. universal design for learning universal design for learning principles (which make up the udl framework for teaching) were employed in this study to guide flexible teaching of mwps. the udl framework is a broader framework that comprises three principles, namely multiple means of representation (mmr), multiple means of action and expression (mmae) and multiple means of engagement (mme) (center for applied special technology [cast], 2011). these three principles are linked to the three brain networks, namely the recognition, strategic and affective networks (grabinger, aplin, & ponnappa-brenner 2008). the recognition network (mmr) addresses the ‘what of learning’, while the strategic network addresses the ‘how of learning’ and the affective network addresses the ‘why of learning’ (rose & meyer, 2006). according to neuroscientists, the recognition network (mmr) makes it possible for learners to receive and analyse information, the strategic network (mmae) makes it possible for learners to generate patterns and develop strategies for action and problem-solving, while the affective network (mme) helps fuel motivation and guide the ability to establish priorities, focus attention and choose action (cast, 2011). in line with this, the cast team formulated a comprehensive framework to serve as a guide towards teaching that is flexible and inclusive (see table 1). table 1: three principles of universal design for learning. this study thus considers some of the aspects of this framework in order to analyse the teaching of mwps and to provide some guidelines in terms of applying flexible teaching methods. in order to make sense of the data, the emerging themes were organised according to the udl principles (i.e. mmr, mmae and mme) and their respective sub-themes. theoretical framework critical emancipatory research (cer), which borrows its roots from social constructivism (nkoane, 2012), was adopted to guide this study. in the same way as social constructivism, cer promotes the notion that knowledge is a product of social interaction. this framework thus espouses the notion of people working together to construct knowledge (tlali, 2013). the framework also promotes the notion of space creation for people to share ideas and knowledge with an intent to explore an issue of interest from manifold angles (tsotetsi, 2013). according to campanella (2009), cer requires people involved in the research project to be viewed as ‘capable speaking beings’ and not just objects that cannot think or do anything for themselves. it also advocates for the inclusion of all the people including the marginalised to identify solutions to their own challenges (mahlomaholo, 2009). on the basis of this therefore we regard cer as a lens that recognises the ‘silent’ and ‘silenced’ voices and as such affords all people including the marginalised opportunities to engage in issues of their concern, deliberating them from their point of view as informed by their lived experiences. researchers who apply this theoretical framework believe that knowledge is a key tool that should be turned into practice that transforms the situation and empowers the people (al riyami, 2015). deducing from the goal of cer, it is reasonable to indicate that it (cer) advances the agenda of human empowerment, transformation and liberation for better living or functioning. research methods and design research design in order to gather data significant to the teaching of mwps, in particular how the teachers teach this mathematical genre in line with udl’s multiple means of representation, an exploratory descriptive design was adopted. the exploratory descriptive design is usually used when there is limited existing information available on a topic in order to gain new insights and to understand the phenomena (grove, burns & gray, 2013). in line with this therefore an exploratory descriptive design was adopted in this study because there is limited existing information regarding the issue under investigation (i.e. the implementation of mmr to guide flexible teaching of mwps). furthermore, exploratory descriptive design was deemed flexible and appropriate in terms of gathering data that would help address the research question of this study: how can the multiple means of representation be implemented to guide flexible teaching of mathematics word problems? sample selection of the participants purposive sampling, which is an informant selection tool that is widely used in qualitative research, was used in this study (tongco, 2007). the purposive sampling technique refers to the deliberate choice of participants due to the qualities they possesses (etikan, musa & alkassim, 2016). when this technique is used, the researcher usually decides what needs to be known and determines the people who can and are willing to provide the information by virtue of knowledge or experience (bernard, 2002). this technique was selected in order to help the researcher meet or fulfil a specific purpose (i.e. to explore the implementation of udl in order to guide flexible teaching of mwps) (naderifar, goli, & ghaljaie, 2017). as such, five high school mathematics teachers including the head of the mathematics department in one school in the thabo-mofutsanyana district in the free state were selected to participate in the study. these teachers had more than 10 years of experience in the teaching of mathematics. besides the fact that these teachers were selected because they were teaching mathematics, their selection was also motivated by the fact that they had the necessary background to understand and teach these types of problems, which is the impression they gave during the first meeting when the researcher explained the rationale for conducting the study and also highlighted what she deemed to be the problem. they all expressed that they are familiar with this type of problem and they further alluded that these problems (mwps) were not only difficult to teach but also that the learners found them difficult to understand and solve. mini-workshop training on the principles of universal design for learning the teachers who were selected to participate in the study took part in the mini-workshop training on the application of udl principles. these teachers participated in the activities that involved the application of the three udl principles, namely mmr, mmae and mme. they later planned their lessons on the teaching of word problems in line with these principles. during the observation sessions, the udl guideline 2.0 (see table 1) was used as a tool to evaluate how they implemented the principles, and the gathered data were later analysed to indicate how the principles (mmr specifically for the purpose of this article) were implemented. teaching mathematics word problems during the teaching of mwps, the researchers sat in class to observe how the teaching was carried out. the sessions were both audio and video recorded. focus was placed more on how the teaching was carried out in order to later recommend the appropriate and more flexible teaching strategies that could be applied to teach this mathematical genre. the udl guideline 2.0 was used as a guide or point of reference to assess the teaching (cast, 2011). data collection and analysis focus group discussion the focus group comprised five high school mathematics teachers. the teachers were provided with opportunities to clarify and discuss their teaching practices during the focus group discussions. observations qualitative observations are the types of observations in which the researcher records field notes on the behaviour and activities of individuals at the research site (creswell, 2009). according to gibson and brown (2009) observational research can be conducted for a number of reasons; however, it is usually a part of a general interest in understanding, for one reason or another, what people do and why. in this study, the structured observation schedule was administered in four classrooms for three consecutive days. the researchers observed the sessions following the udl guidelines, which were to provide a guide for interpretation of how the word problems were taught. the classroom observation schedule focused on the following: the mmr used the consideration of mmae the application of mme. the three aspects above were focused upon in order to provide guidance in terms of multiple ways in which the content could be represented, multiple ways in which learners could be provided with options to demonstrate their learning processes, as well as various ways in which learners could be taught how to use the available formats, tools and technology to learn word problems. reflection sessions reflection sessions were conducted in order for the teachers to reflect upon their teaching practices. according to magalhães and celani (2005) reflection is a form of practice that involves, among others, the discussion of the aspects that were previously ignored, rethinking the situation and attributing newly generated meanings to situations already discussed. reflections therefore enable the participants to think critically about the issue, and thus give meaning to the experiences. in this study, the reflective sessions served as platforms where teachers shared their experiences in terms of teaching word problems as well as highlighting the strategies they used. these sessions were conducted after the teachers had given the lessons. findings and discussion this section reports on the findings and discussion, and recommendations. for the purpose of this article we discuss the findings that are related to the mmr. the results for the mme and mmae are not reported in this article but are reported elsewhere. we therefore focused on the aspects of udl (see table 1) that make up this principle (mmr) in order to draw the findings and make sense of the generated data. the data reported in the subsequent sections emerged from the observations, reflection sessions and focus group discussions. teachers were observed while teaching the different topics involving mwps. the udl framework as shown in table 1 was used to analyse and comment on the teaching of mwps. teachers’ experiences in terms of teaching mwps were also narrated through the reflection sessions and focus group discussions. the data that emerged from the three instruments revealed some of the best practices in terms of teaching mwps in line with the mmr principle. multiple means of representation the following sections provide an analysis of data in line with the udl principle of mmr. the examples that are discussed in the subsequent sections were chosen because they are word problems and the explanations provided, in terms of how they were carried out, show the operationalisation of the mmr principle. explanation teacher 2: in order to give the next three terms you need to first check the relationship between the terms. for example, two multiply by four is equal to eight. eight multiply by four is equal to thirty-two. again eight divide by two is equal to four and thirty-two divide eight is equal to four. when you check this, you will realise that four is a common number, which you either have to multiply the current number with it in order to get the next term or divide the next term with it in order to get the previous term. remember when we dealt with the relationships between the operational signs, i showed you how multiplication and division relate. the word problem given in episode 1 is a geometric series and this can easily be identified when the series is represented numerically as ‘2; 8; 32; …’. in order to respond to the three questions posed, the teacher had to address a few aspects in her teaching. in her explanation above the teacher first determined the ‘common’ number, which could be multiplied with the current term in order to yield the next term. the teacher also showed the learners that the same ‘common’ number can be obtained if the next term is divided by the previous term. thus the teacher ‘highlighted patterns’ as one way of ‘providing options for comprehension’ according to the udl mmr principle, thus enabling learners to represent the series correctly and to make generalisations. the critical features of the series were also highlighted by the teacher in line with this principle in order to help the learners solve the problem and establish the general formula. the teacher also indicated the big ideas regarding the given problem, thus showing the relationships among the terms as well as the operational signs used. according to cast (2011) teachers who consider these forms of practices provide options for comprehension which is necessary to assist learners receive and analyse the word problems. box 1: episode 1. box 2: episode 2. box 3: episode 3. explanation teacher 3: learners usually find it difficult to represent the two expressions that are reflected in this problem. in order to simplify it and make it understandable to all of them, i separate the two expressions and work them out separately. five is less than – part 1, i.e. 5 < three less a number – part 2, i.e. x – 3 not 3 – x according to teacher 3, learners find it difficult to solve and make a representation of the given problem (episode 2) because they fail to realise that there are two expressions embedded in it. learners also found this problem difficult to solve because they did not know which signs to use in order to represent and solve it. in order to simplify this problem the teacher broke down the problem into two simple comprehensible expressions namely: five is less than – part 1, i.e. 5 <, and three less than a number – part 2, i.e. x – 3 (not 3 – x). this way the teacher guided information processing. according to the teacher, learners understand the smaller parts of a given complex word problem and are able to make connections that lead to understanding the problem holistically rather than holistically solve the problem from the beginning. according to mevarech and kramarski (1997) guiding information processing enables leaners to be aware of problem-solving and this practice induces learners to activate the four-phase problem-solving model suggested by pólya (1973). explanation teacher 5: i find the use of different colours to be useful especially when i teach about the shifting of the graphs. according to teacher 5, demonstration of the shifting of the graphs can better be facilitated through the use of different colours (see figure 1). consequently, the use of different colours serves as a visualisation enhancing tool that promotes the visualisation skill, which is one of the vital skills necessary for mastering word problems. the teacher’s strategy for using different colours is supported by shabiralyani, hasan, hamad and iqbal (2015) who further stress that the use of different colours makes visual aids perceptible to the entire class. the different colours thus assist in providing a clear contrast and also making the plotted graphs easily visible. figure 1: shifting of the graphs. through the use of diagrams and accurately measured graph sheets, learners were able to see the shift that has taken place between the two given equations and this improved their problem-solving (kashefi, alias, kahar, buhari, & othman, 2015). such instruments thus increased the visual impact, interactivity and spontaneity which according to williams (2004) provides high-quality learning experiences and also improves learner focus. the example provided by the teacher shows the importance of customising information in different formats to maximise learning. the teacher’s example also confirms that visualisation is a skill, which teachers can nurture in learners through the use of different representations (e.g. diagrams, number lines) (alex & mammen, 2018; gilbert & auber, 2010). courtad (2019) also supports the notion of multiple representation in that it improves learner perception, helps decode language and symbols, and reinforces understanding. the data reported in this section outline what the teachers considered to be important in terms of providing options for language, mathematical expression and symbols. the data further suggest the need for teachers to vary their teaching and use the appropriate resources to optimise teaching. in terms of addressing the mathematical vocabulary, the teachers commented as follows: ‘it helps really to address vocab in your teaching. sometimes these learners fail to solve mathematical problems because of some of the words they do not know.’ (teacher 1) ‘that is so true because some of these terms that they are using, they also come across them in other subjects and if not clarified, they bring about confusion. for example a term such as “function” can cause confusion if not thoroughly explained.’ (teacher 4) the teachers indicated the importance of clarifying the mathematical vocabulary since it carries meaning in the given problems. according to teacher 1, these mathematical key terms are important to teach (riccomini, smith, hughes & fries, 2015) and to clarify because they carry meaning and directive in terms of what should be solved. as for teacher 4, clarification of these terms is important especially when it is linked to application of such similar terms in other ‘subjects’ as opposed to their application in mathematics in order to draw a clear distinction (owens, 2006; widdows, 2003). for example, the term ‘function’ is used within the mathematical context to refer to an equation, which denotes the ‘input, process and output’ concept. however, in life sciences, the same term ‘function’ describes the work of a specific organ in the body. the teaching of mathematical vocabulary further helps teachers to expose learners to terms that can be used ‘interchangeably’ (godino, 1996) (e.g. yearly or annually, altogether or sum, remainder or difference, etc.). chitera (2009) also recommends the practice of explicitly teaching mathematical vocabulary to help learners gain control over mathematical language, which is important for them to comprehend and master the mwps. it can be deduced from the example provided that a word in english may not necessarily be interpreted, explained and applied the same way as it is interpreted, explained and applied in mathematics (kashgary, 2011), which is why the teachers have to address this in their teaching to help eliminate confusion, thus promoting understanding language across disciplines. explicit teaching of mathematical vocabulary is also supported by literature, which indicates a strong correlation between mathematical vocabulary and comprehension of mathematical content (monroe & orme, 2002): ‘if we look at the example that was provided earlier … eeehhh that one of: “five is less than three less than a number”. learners usually read it literally as it is and they translate it into algebraic expression written as 5 < 3 < x which is incorrect. so when teaching a problem like this to learners i, break it down into simple expressions and then translate it into algebraic expressions that can easily be solved.’ (teacher 1) ‘breaking this word problem into simple expressions helps, for example “three less than a number”; you are able to indicate that it should be represented as x – 3 and not as 3 – x.’ (teacher 5) the above extracts indicate the significance of teaching mathematical language and structure. in order to achieve that, teacher 5 highlighted the need for teachers to break the complex word problems into simple expressions in order to correctly represent them algebraically. as such, teachers should emphasise in their teaching the fact that learners should not ‘literally’ read the word problems and use left to right translations; rather they should read them carefully and try to understand them holistically. for example: ‘five is less than three less than a number’. this word problem can be represented algebraically as: 5 < x – 3. in this case, the expression ‘is less than’ denotes ‘<’ and the expression ‘less than’ denotes ‘–’. looking at the expression ‘three less than a number’, the correct representation of the expressions is x – 3 and not 3 – x as it reads. in order for this expression to be correctly represented, careful reading, holistic understanding and knowledge of the mathematical structure and syntax are necessary. this will enable learners to realise that three less than a number is not 3 – x but x – 3. the holistic understanding of the problem helps to ensure the correct ‘placement or location’ of the variable (x) and the number (3). deducing from the teachers’ explanations, it is notable that the teaching of mathematical structure and syntax helps to translate texts into correct variable representations, which subsequently enable learners to solve the word problems. in order to clarify the syntax and structure, the teachers need to engage learners in reading, illustrate the representations of the word problems and show the application of the correct signs or symbols in the given problems. this way, the learners will realise that word problems are not necessarily represented in writing the same way as they are read and that the signs or symbols to be used depend on what the problem denotes or insinuates: ‘breaking down word problems helps to show learners how to write text in mathematical form. if learners read the whole word problem, it may be difficult for them to even realise that three less than a number should be represented as x – 3 and not 3 – x.’ (teacher 2) ‘to justify or clarify why the expression “three less than a number” is written as x – 3 and not 3 – x one can use money concept … t.’ (teacher 4) according to teacher 2, breaking down the word problems is one strategy that teachers can use to help learners with decoding of text and to convert such text into correct algebraic expressions with mathematical notations and symbols. since left to right translation does not always yield the correct interpretation and representation of text into variables, it is important for teachers to use various forms of explaining such representations. for example, in order to explain the swapping around of the number and the variable (e.g. 3 – x to x – 3) teacher 4 suggested the use of the ‘money concept’. the use of the money concept thus serves as another form of representation or medium through which the teacher could reinforce the correct mathematical representation of the mathematical expression. breaking down of the complex word problem into smaller comprehensible expressions also assists in terms of distinguishing the expressions (e.g. x – 3 versus 3 – x), thus promoting the correct interpretation, representation and decoding of text into mathematical notation and symbols: ‘what does “at least” mean?’ (teacher 4) ‘e bolela bonyane [it means little].’ (learner 1) ‘i think it refers to something smaller.’ (learner 2) ‘how would you represent this expression symbolically?’ (teacher 4) ‘i will represent it as “≤”.’ (learner 3) through lesson observation, the significance of promoting understanding across different languages when teaching mathematical concepts was notable. the learners were given the expression ‘at least’. mathematically, the expression denotes ‘greater than or equal to’ represented as ‘≥’ symbolically. however, during the lesson observation, learners seemed to have difficulty working out the word problem that contained that expression. this is because the expression was interpreted and understood by learners differently, according to various contexts and thus they used their backgrounds to attach meaning to the expression. for example, ‘bonyane’ in sesotho, ‘buncinci’ in isixhosa, and ‘okungenani’ in isizulu. the teacher seemed to be aware of this challenge, which is why she firstly asked learners what their understanding of the expression ‘at least’ is, thus eliciting their pre-knowledge. based on the answers they provided, the teacher realised that the expression was not understood within the mathematical context and that, as a result, learners used the incorrect mathematical sign to express it. the teacher then explained to learners that the expression does not refer to ‘less than’ or ‘small’ even though that may have been insinuated in their languages. the teacher further indicated that mathematically, the expression ‘at least’ denotes ‘greater than or equal to’ and does not refer to minimum as in ordinary english. from the lesson presentation and how the expression was explained, it became clear that there is a need for teachers to promote understanding across different languages and to explain mathematical expressions such that the meanings carried or embedded within the home language contexts do not become barriers towards learning and solving mwps. as such, the expression ought to be defined and explained correctly in context, so that learners do not misinterpret it and thus apply the incorrect mathematical symbols. nkambule (2009) also supports the issue of promoting understanding across different languages in order to reinforce understanding of mathematical concepts. during the reflective session, the teachers indicated the need to customise the display of information in order to provide options for perceptionsm offer alternatives auditory and visual information: ‘it helps most of the time to give word problems and the picture alongside to enable learners understand the word problem.’ (teacher 3) ‘that is what i do when i teach them. i make some drawing representations in order to explain the concepts.’ (teacher 4) teacher 3 noted the significance of providing pictures alongside the word problems as another form of representation to aid learners’ comprehension of word problems. this form of customising the display of information helps learners to understand and conceptualise the word problem. although adding a picture may cater more for visual learners, other learners with different learning styles may also benefit from this practice. the expression ‘it helps most of the time’ stresses the significance of such practice that should be cultivated when word problems are administered. according to teacher 4 this practice is not only supposed to be adopted when the learners are given assessments; however, the teachers should also apply it also in their teaching. for example: learners may be given a mathematical problem in the form of text. a picture that highlights some of the features mentioned in the text may be provided alongside the word problem. this will promote visualisation of the problem and assist learners to realise what the problem is all about and what it requires to be solved. this converges learner thinking in the ‘right direction’. it can therefore be deduced from what the teachers noted that some learners swiftly grasp content if information is presented to them in multiple formats. thus, learning and transfer take place with ease when multiple representations are provided because such representations allow learners to make connections that are necessary for them to master the word problems. conclusion this article explored how mmr can be implemented to guide flexible teaching of mwps. the study demonstrated that the mmr principle provides a flexible and a comprehensive framework for the analysis of teachers’ practice. this principle further encourages the consideration and balancing of multiple processes and orientations in teaching, rather than a more restricted focus. the application of mmr in the teaching of word problems thus serves as a positive contribution to the field, by promoting, among others, work to synthesise different perspectives and develop a more holistic view, in both teaching and research. the mmr principle constitutes three themes, namely providing options for comprehension, providing options for language, mathematical expressions and symbols, and providing options for perception. the study thus indicated the correlation among these three themes in terms of formulating mmr. for instance, the findings of the study indicate that in order for teachers to make it possible for learners to receive and analyse information, they (teachers) have to provide learners with varied options for comprehension, options for language, mathematical expressions and symbols, and options for perception. providing varied options is deemed important because learners differ in terms of how they receive, analyse and assimilate information. the study thus makes a contribution through lifting out and balancing these different themes within the mmr formulation by demonstrating the implications and significance of reinforcing comprehension of mathematical concepts, which is facilitated by the correct and appropriate use of language, mathematical expression and symbols, to make it possible for the content to be perceptible. the application of mmr also contributes towards shaping learners who are knowledgeable and resourceful (cast, 2011), which is one of the educational goals that teachers should strive to achieve. in line with mmr principle of udl, findings indicate that the productive teaching of mwps could be achieved by providing alternatives for comprehension, language, mathematical expressions and symbols, as well as providing alternatives for perception. the need for this principle to be applied is spelled out in the south african caps and its application is supported in order to help learners receive and analyse information. the application of this principle contributes in developing learners who are resourceful and knowledgeable. although this principle plays such a pivotal role in guiding flexible teaching of mwps, the caps does not specify how this principle can be applied, thereby leaving this task to the discretion of the teachers. however, in terms of applying the mmr principle when teaching word problems in this article, the following were regarded as good practices: highlighting patterns in a given word problem, outlining critical features of the given word problem, as well as the big ideas regarding the concept that is dealt with. elucidating mathematical vocabulary and symbols, teaching and clarifying mathematical vocabulary, syntax and structure as well as teaching learners how to decode text, mathematical notation and symbols were also regarded as good practices in terms of guiding flexible teaching of word problems. the findings of this study thus indicate mmr as a promising strategy to help guide flexible teaching of mwps. the teachers thus need to make their teaching flexible by considering this principle when teaching mwps, thus providing choices for comprehension, choices for language, mathematical expressions and symbols, as well as providing options for perception. acknowledgements competing interests the authors declare that there are no competing interests in the production or publication of this article. authors’ contributions this article is based on m.m.m.’s doctoral study, so m.m.m. collected and analysed the data used in the study. m.d.m. worked with m.m.m. to do further analysis, refinement and reshaping of the work included in this article, and finalised and submitted the article on behalf of m.m.m. and m.d.m. ethical considerations the ethical clearance was obtained from the university of the free state and the clearance certificate was obtained on 14-mar-2017 (ufs-hsd2016/1194). funding information the study on which this article is based was funded by the national research foundation (nrf). data availability data sharing is not applicable in this article as no new data were created or analysed in this article. disclaimer the views and opinions 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isixhosa ingrid e. mostert received: 03 mar. 2020; accepted: 26 aug. 2020; published: 21 dec. 2020 copyright: © 2020. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract word problems form an important part of the early grade mathematics curriculum in south africa. studies have shown that the relative difficulty of word problems differ: learners are more likely to solve certain types of word problems than others, with compare type problems being the most difficult. in order to help early grade learners understand and solve compare problems, it is important to understand the relative difficulty of different types of compare type problems and the factors that contribute to their relative difficulty. while these factors have been studied in english, less research has attended to word problems in other languages, such as isixhosa. in this study a typology of isixhosa compare type (difference unknown) word problem was set up. the typology included two dimensions, namely the problem situation and the comparative question. the relative difficulties of specific word problems from this typology were compared by analysing the results from an early grade mathematics assessment administered to two cohorts of grade 1–3 isixhosa learners in five rural eastern cape schools. the analysis showed that in isixhosa, as in english, some compare type problems are easier to solve than others. problems with ‘matching’ situations are easier to solve than problems with ‘no matching’ situations. problems with alternatively formulated comparative questions, specifically those using -shota or kangakanani, are easier to solve than those using a more classic formulation. this study highlights the importance of understanding the ways in which african languages express mathematical ideas in order to identify and leverage affordances for teaching and learning mathematics. keywords: word problems; early grade mathematics; compare problems; isixhosa; mathematical language. introduction word problems are a central, yet hard-to-teach, aspect of early grade mathematics. for example, in south africa word problems have been identified as a recurring weakness in the south african annual national assessments (anas) (department of basic education, 2012, 2014, 2015). research has shown that the relative difficulty of word problems differs: learners are more likely to solve certain types of word problems than others. for additive relation word problems, in other words any word problems involving addition and subtraction, compare type problems have been shown to be the most difficult for learners to solve. compare type problems are of the form ‘sbu has eight bananas and sive has five bananas. how many more bananas does sbu have than sive?’ while there has been some research into early grade word problems in south africa (e.g. petersen, mcauliffe, & vermeulen, 2017), and some research into word problems and african languages in higher grades (e.g. sepeng, 2013), there has been little research into early grade word problems in african languages. this is problematic as more than 75% of learners are taught mathematics in an indigenous african language in the first four years of formal schooling (spaull, 2016). internationally, different types of additive relation word problems and their relative difficulty (as measured by the percentage of learners who correctly answered the problem out of the total number of learners who were asked the question) have been studied in relation to english since the late 1970s. this work was pioneered by two research groups: the first led by carpenter, hiebert and moser (1981) and the second by riley, greeno and heller (1983). such studies have shown that there are a number of factors that influence the relative difficulty of different word problems. these include general factors such as problem length, grammatical complexity and whether learners use concrete aids or not, as well as specific factors such as semantic structure and the position of the unknown (riley et al., 1983). many of these factors relate to language. this raises questions regarding the extent to which these factors influence the relative difficulty of word problems in languages other than english, especially languages with linguistic features significantly different to english, such isixhosa, one of south africa’s nine official african languages. this study contributes to the limited research on early grade word problems in languages other than english by examining compare type word problems in isixhosa. the study offers a sub-typology for compare type (difference unknown) problems, both in english and in isixhosa. the relative difficulty of the problems in the isixhosa sub-typology is then empirically tested using data collected from an adapted early grade mathematics assessment (egma). the following two research questions are answered in this study: are certain types of isixhosa compare type problem easier for learners to solve than others? if so, is the relative difficulty influenced by: the formulation of the comparative question the problem situation? theoretical and methodological perspectives this article is informed by theoretical perspectives and methodological tools from linguistics that have proved helpful for research into the way different languages express mathematical concepts, as proposed in a recent paper by edmonds-wathen (2019). this article also draws on theoretical perspectives from variation theory, a general theory of learning largely developed by marton and booth (1997) and later extended by watson and mason (2005) in relation to mathematics learning. edmonds-wathen (2019) proposes using a typological framing for research on the diversity of mathematical expression in different languages and using interlinear morphemic glossing to present examples in different languages. these perspectives and methodologies are particularly pertinent for studies done by a researcher not fluent in the language that is being studied. edmonds-wathen points out that linguists often work with languages that they are unfamiliar with, either by working with translated texts or by working closely with bilingual speakers. she argues that mathematics education researchers can, and do, work in similar ways, with this study being a case in point. this study was undertaken by an english speaker with an emergent understanding of isixhosa. the researcher worked very closely with a number of isixhosa speakers to deepen her understanding of isixhosa, particularly in relation to compare type problems. typological framing typology is an area of linguistics that describes and classifies languages according to their structural similarities and differences (edmonds-wathen, 2019). typology strives to compare languages through an analysis that is framework-neutral (nichols, 2007). edmonds-wathen (2019) argues that because of this neutrality, ‘a typological approach may be useful to investigate mathematical expression in different languages, without privileging one language over another’ (p. 121). this is particularly important when comparing a language with a well-developed mathematical register (see halliday, 1978), for example english, with a language without a formal mathematical register or with a mathematical register that is still being strengthened, for example isixhosa. while this study does not adopt a strict typological approach, the study does strive to ensure that english was not privileged over isixhosa, for example by ensuring that a range of ways of expressing comparative questions in isixhosa was studied and not only those that correspond to the way in which english expresses comparative questions. however, as the researcher is not an isixhosa speaker, english and the linguistic features used to express comparative questions in english provided a starting point for the study, therefore implicitly privileging english. interlinear morphemic glossing one of the challenges of researching how different languages express mathematical ideas is how to present examples from a language different to the language of publication. to overcome this challenge, edmonds-wathen (2019) suggests using a simplified interlinear morphemic gloss.1 interlinear morphemic glossing allows the structure of the example to be presented. often the structure is lost if only an idiomatic translation is provided (edmonds-wathen, 2019). interlinear morphemic glossing is particularly helpful when presenting data from languages where word order is not necessarily a determinant of the function of a word. an interlinear morphemic gloss consists of four levels (edmonds-wathen, 2019). for this article the top level gives the isixhosa in sentence form. the second level gives the isixhosa morphemes (the smallest unit of a language that has its own meaning), the third level gives the english morphemic gloss, and the final level gives a free translation in english. morphemes are separated by a hyphen. if a morpheme is translated by more than one word the words are separated by a full stop. for example: level 1: umama uneembiza ezisibhozo neziciko ezihlanu. level 2: umama u-nee-mbiza ezi-sibhozo ne-ziciko ezi-hlanu level 3: mother she.is-with-pots that.are-eight with-lids that.are-five level 4: mother has eight pots and five lids. the simplified interlinear morphemic gloss is used in order to make the examples accessible to a mathematics education audience. for this reason, in some instances in this article, not every morpheme is glossed separately.2 for example, nouns and their prefixes, which indicate the noun class and the number (singular or plural) of the noun, are not glossed separately. in cases where strict glossing would detract from the comparisons being made, other glossing rules have also not been strictly followed. language of variation variation theory is a general theory of learning that has been applied specifically to learning mathematics. a central notion within variation theory is that in order to discern one aspect of a phenomenon, that aspect needs to be varied while other aspects remain unchanged (al-murani, kilhamn, morgan, & watson, 2019). the aspect of the phenomenon that varies is called the ‘dimension of variation’. watson and mason (2005) extended these ideas by defining the variation that is possible within a ‘dimension of variation’ as the ‘range of change’. for example, in the expression x + 3, one of the dimensions of variation is the addend (others include: the letter representing the variable, the operator and the order in which the variable and constant appear in the expression). the values that the addend can take (i.e. natural numbers, negative numbers, rational numbers and so on) constitute the range of change of this particular dimension of variation (al-murani, 2006). the extent to which a learner can discern the dimensions of variation and the corresponding ‘range of change’ of the expression x + 3 is an indication of how well the learner understands the algebraic expressions. in this article the ideas of a ‘dimension of variation’ and of a ‘range of change’ are applied not to an object of learning, but to an object of study, namely compare type word problems. in order to explore the full range of possible compare type problems, different dimensions of variation were identified and varied to set up a typology of compare type problems for isixhosa and for english. additive relation word problems additive relation word problems and the factors that influence their relative difficulty have been studied for english word problems since the late 1970s. in the following two sections relevant studies are discussed. word problem typologies early researchers categorised word problems describing the same mathematical problem but using different semantic structures into typologies of word problems (e.g. carpenter & moser, 1983; riley et al., 1983). recently these typologies have been combined into one comprehensive typology (mostert, 2019). the categories and labels from this comprehensive typology will be used in this article. at the highest level this typology consists of four different types of word problems, differing in terms of the number of sets being compared and whether the problem is dynamic or static (figure 1). figure 1: four main categories of word problems. each of these four main categories of word problems can be separated into subcategories in two ways. firstly, each category can be separated into two subcategories based on a number of factors or dimensions: the ‘direction’ of the change or equalisation, whether attributes or ownership are different in collection problems and whether the comparison is ‘more than’ or ‘less than’. secondly, by changing the position of the unknown, each category can further be divided into two or three subcategories resulting in a total of 22 subcategories (see figure 2). figure 2: twenty-two subcategories in the comprehensive typology. the word problems studied in this article are ‘more than’ compare problems where the difference is unknown (marked with † in figure 2). factors influencing difficulty of word problems as mentioned previously, there are a number of different factors that have been identified as influencing the difficulty level of word problems. the factor that is relevant for this study is the clarity of the problem for learners, both in terms of the problem situation and in terms of the formulation of the comparative question. verschaffel and de corte (1993) report that most mistakes made by young children when solving additive relation word problems are more likely to be because they represent the problem situation incorrectly, not, as was formerly widely believed, because they choose the incorrect arithmetic operation. this is evident from a number of studies, referred to and validated by verschaffel and de corte (1993), which demonstrate that problems can be rephrased, without changing their semantic structure, in way that is easier for learners to correctly represent the problem situation. for collection problems there are two cases of rephrasing that have been shown to increase the likelihood of learners solving the problem correctly. firstly, carpenter et al. (1981) found that if the problem ‘there are six children on the playground. four are boys. how many are girls?’, was changed to ‘there are six children on the playground. four are boys and the rest are girls. how many are girls?’, a higher percentage of learners answered the question correctly. secondly, lindvall and ibarra (1980) found that when the problem ‘together tom and joe have eight apples. three apples belong to tom. how many belong to joe?’ was changed to ‘together tom and joe have eight apples. three of these apples belong to tom. how many of these belong to joe?’, the problem was significantly easier for kindergarten children to solve correctly. such rephrasing is also possible for compare type problems, with some empirical data showing that rephrasing can increase the percentage of learners who correctly solve the problem. this will be discussed in detail in the next section. compare type problems in english compare type problems have been identified as the most difficult type of additive relation word problem for young children to solve (fuson, carroll, & landis, 1996). at least part of the reason why learners struggle to solve compare type problems is because standard compare type problems (‘sbu has eight bananas and sive has five bananas. how many more bananas does sbu have than sive?’) include quantifiers such as ‘more than’ and ‘less than’. quantifiers form part of later-developing languages skills, skills that mother tongue speakers continue to develop up to approximately age 9 (berman, 2004). if children are still learning to understand and use words such as ‘more than’ and ‘less than’ it is not surprising that they struggle to represent the correct problem situation for compare type problems. another reason why compare problems are so difficult for young learners to solve is that learners confuse the ‘classic’ comparative question ‘how many more?’ with the question ‘how many?’ or with the question ‘who has more?’. for example, the problem ‘sbu has eight bananas and sive has five bananas. how many more bananas does sbu have than sive?’, is often answered with ‘eight’ (correctly answering the question ‘how many bananas does sbu have?’) or with ‘sbu’ (correctly answering the question ‘who has more bananas?’) (roberts, 2016). in response to this potential confusion, roberts (2016) suggests first asking ‘who has more bananas?’, then ‘how many bananas does sbu have?’, before asking ‘how many more bananas does sbu have than sive?’. in terms of comparison, both mathematics education and linguistic research (e.g. kennedy, 2009) has focused on ‘more than’ compare type problems, neglecting ‘less than’ compare type problems (see figure 2). this focus on ‘more than’ problems in the literature is reflected in assessments, such as the egma, which only includes ‘more than’ compare type problems. because this study analyses data from the egma, only ‘more than’ compare type problems are considered. this is a limitation of the study and of research in general as ‘more than’ compare type problems are not necessarily equivalent to ‘less than’ compare type problems, especially for non-indo-european languages. isixhosa is one such language where they are not equivalent. it is also important to note that while there are three subcategories of compare type (more than) word problems, as can be seen from figure 2 and as exemplified in table 1, this study only considers ‘difference unknown’ compare type problems. table 1: exemplification of subcategories of compare (more than) problems. different types of compare type problems in this section, three variations of the standard, ‘difference unknown’ compare type problem are discussed. in as early as 1980, hudson constructed and tested a variation of the standard compare type problem. hudson’s (1980) variation differed from the standard compare type problem in two ways. firstly, the problem situation was set up to invoke the idea of matching by choosing birds and worms as the subjects of the story, namely ‘there are five birds and four worms’. secondly, the ‘classic’ phrasing of the comparative question ‘how many more birds than worms are there?’, was rephrased as ‘suppose the birds all race over and each one tries to get a worm! will every bird get a worm? how many birds won’t get a worm?’. hudson (1980) then presented learners with the same problem situation about the five birds and four worms, but posed the comparative question in two different ways, using the classic ‘how many more’ formulation and using the ‘how many won’t get’ formulation. table 2 shows the striking difference in results with a much higher percentage of learners being able to solve the problem with the ‘how many won’t get’ formulation. table 2: percentage of children with consistent correct responses. a second variation was introduced by roberts (2016), in what she refers to as ‘compare (matching) problems’. these are compare type problems that draw attention to the absence of elements by asking ‘how many elements are missing?’. an example of roberts’s compare (matching) problem is: ‘there are 11 locks but only 9 keys. how many keys are missing?’. roberts explains that: the choice of locks and keys is deliberate, as in this problem context it is implicit that each key fits uniquely with a particular lock. this unique 1:1 matching of each element in one set to each element in another set is not explicitly implied in the compare problem ‘i have 11. you have 9. how many more do you have than me?’ (p. 68) the unique one-to-one matching of roberts’s compare (matching) problem is embedded in the problem situation. as roberts (2016) points out, this is not the case for ‘standard’ compare type problems. it is also not the case for hudson’s ‘won’t get’ problems, where, unlike with locks and keys, one bird can get two worms or two birds share one worm. in hudson’s (1980) variation, the one-to-one matching is imposed on the problem situation by adding the phrase ‘each [bird] tries to get a worm’. while roberts (2016) did not use the same problem situation, she did empirically determine the facility score of a compare (matching) problem and a standard compare type problem (which she refers to as a compare (disjoint set) problem). for both pretests in her study the facility score of the compare (matching) problem was much higher than that of the standard compare type problem (see table 3). table 3: percentage of children with consistent correct responses. a third variation of the standard compare type problem appears in the egma used in this study (figure 3). this variation is similar to hudson’s ‘won’t get’ variation. firstly, the problem situation is set up to invoke matching, this time between children and oranges: ‘a mother has seven children, and she has two oranges’. secondly, the need for one-to-one matching is imposed on the situation rather than embedded in the situation. this is done through the phrase ‘if the mother wants to give each child one orange’. however the egma variation differs from the hudson (1980) variation in that rather than asking a ‘won’t get’ question, a ‘still needed’ question is asked: ‘how many oranges are still needed?’. figure 3: different types of compare type problems in the literature. while the three variations of the standard compare problem, summarised in figure 3, are helpful in showing that rephrasing a question can influence how easy or difficult it is for learners to solve, the problems differ both in terms of the problem situation and in terms of the formulation of the comparative question. this means that it is not possible to isolate the effect of the different factors on the level of difficulty of the different problems. in the next section a typology of compare type problems is set up, taking into account the variation that is possible for both factors. typology of english compare type problems drawing on the language of variation theory, the ‘dimensions of variation’ for compare type problems are (1) the problem situation and (2) the formulation of comparative question. the problem situation can either be one that invokes matching by referring to two things that learners might expect to go together (e.g. locks and keys or children and oranges) or one that does not invoke matching by referring to things that do not necessarily go together (e.g. sweets belonging to a girl and sweets belonging to a boy). for matching problems, it possible to further differentiate between problems that have one-to-one matching embedded in the situation and those in which the one-to-one matching is not embedded. in english, the comparative question, which constitutes the second dimension, can either be formulated in the ‘classic’ form, ‘how many more?’ or, for matching situations, the question can be formulated in one of a number of alternative ways such as ‘how many are missing?’ or ‘how many are still needed?’. using these two dimensions of variation and the range of change that is permissible for each dimension, it is possible to set up a typology of compare type problems. figure 4 provides an overview of the typology as well as showing how each of the four compare type problems discussed previously (see figure 3) fits into the typology. appendix 1, figure 1-a1 exemplifies each of the categories in the typology. figure 4: typology of english compare problems (including examples from literature). there are few important things to note about the typology. firstly, while it is possible to ask a classically formulated comparative question with a matching problem situation (e.g. ‘a mother has eight pots and five lids. how many more pots are there than lids?’), it is not possible to use an alternatively phrased question with a ‘no matching’ problem situation – the problem ‘a girl has seven sweets. a boy has five sweets. how many sweets are missing?’ does not make sense. this is the reason for the n/a cell. secondly, when a problem has a matching problem situation where one-to-one matching is not embedded but an alternatively phrased comparative question is used, an additional phrase (such as ‘each bird tries to get a worm’) must be added in order to impose the one-to-one matching on the situation. for this reason, the typology differentiates between ‘1-to-1 matching not embedded’ and ‘1-to-1 matching imposed’. see appendix 1, figure 1-a1 for examples of word problems in the different categories. finally, it is important to remember that this typology is only for ‘difference unknown’ compare type problems. once a broader typology has been set up showing the dimensions along which the problems can vary, it is possible to compare problems that only vary in terms of one dimension (either the problem situation or the comparative question) in order to establish the extent to which each factor influences the relative difficulty of compare type problems. in this study the influence of these two dimensions is explored for isixhosa compare type problems. in order to do this a typology for isixhosa compare type problems is set up, drawing on examples from canonical texts. compare type problems in isixhosa in a previous study, mostert and roberts (2020) describe the linguistic features of comparative phrases in isixhosa. in order to do this they analysed the examples of comparative phrases appearing in four canonical texts,3 written in english and translated into isixhosa (mostert & roberts, 2020). this set of examples included both comparison phrases (e.g. ‘there are more dogs than cats’) as well as comparative questions (e.g. ‘how many more dogs are there than cats?’). while the previous study only focused on the comparison phrases, this study focused on the comparative questions in these canonical texts, while also including comparative questions from the egma. as in the previous study, the isixhosa texts provide a valuable source of examples of how to formulate comparative questions in isixhosa, but are not a sufficient source of examples. because of the small number of examples and because, as mentioned previously, the author is not fluent in isixhosa, mother tongue isixhosa speakers were consulted to clarify and exemplify the range of possible formulations of comparative questions in isixhosa. before setting up a typology of isixhosa compare type problems, aspects of isixhosa grammar that are relevant for the study are discussed. relevant isixhosa grammar isixhosa is a nguni language spoken by more than 8 million south africans (of a total of 57 million). the other three nguni languages spoken in south africa are isizulu, isindebele and siswati. as a bantu language, isixhosa has many linguistic features that differ substantially from the linguistic features of indo-european languages. two features that are relevant for this study are flexibility of word order and a system of concordial agreement. isixhosa word order is not as rigid as english word order. in isixhosa the most important word in a sentence is emphasised by putting it at the beginning of the sentence. table 4 shows how the english sentence ‘sigqibo gave mveli bread’ can be constructed in four different ways in isixhosa, each emphasising different words. this flexibility of word order accounts for some of the variation of isixhosa compare type problems. table 4: example of word order variation possible in isixhosa. isixhosa, like other bantu languages, has a noun class system. this means that all nouns belong to a particular class which is determined by the noun’s prefix. in a sentence, any word (verb, noun, pronoun or adjective) associated with a noun has to show ‘agreement’ with that noun. this is achieved by adding a concord (a prefix) to the word, which contains similar-sounding letters to the prefix of the noun. this is referred to as concordial agreement. for example: izinja zininzi kuneekati. ‘ there are more dogs than cats.’ abantwana baninzi kunoomama. ‘there are more children than mothers.’ in the first example, the noun is izinja ‘dogs’ with the prefix iziwhile in the second sentence the noun is abantwana ‘children’ with the prefix aba-. in each sentence the adjective -ninzi ‘many’ takes a different prefix, as determined by the noun it is describing. for this reason, when referring to words on their own (i.e. when they are not referred to as part of a sentence), the root of the word is used (e.g. --ninzi) rather than one particular form of the word (e.g. -zininzi or baninzi). also relevant for this study is the use of loanwords in isixhosa. loanwords are words that are embraced by the speakers of one language (in this case isixhosa) from another language (the source language) (o’grady, dobrovolsky, & aronoff, 1997). in most cases nouns are borrowed; however, there are some languages that occasionally borrow verbs and adjectives (brown, 2003). isixhosa has many loanwords from english and afrikaans, most of which are nouns (e.g. ikati ‘cat’). however, isixhosa also has a few verb stems that are loanwords from english or afrikaans. these are used where no isixhosa words are available and are used in a phonetically adapted form e.g. -sarha ‘saw’ (from afrikaans ‘saag’) and -bhaptiza ‘baptise’ (from english) (oosthuysen, 2016, p. 282). in the following sections different formulations of isixhosa comparative questions will be discussed, first in terms of ‘classic’ comparative questions and then in terms of ‘alternative’ comparative questions. finally, a typology of isixhosa compare type problems will be set up. ‘classic’ comparative questions unlike english which only has one way to phrase the ‘classic’ comparative question, in isixhosa there are a number of different ways in which the ‘how many more’ question can be phrased (see figure 5). in this article three commonly used variations are discussed. one reason why variations are possible in isixhosa is because there are two question words that can be used in combination with two words expressing ‘more’. the two question words are -ngaphi ‘how many’ and kangakanani ‘to what extent’. the two words used to express ‘more’ are the adjective -ninzi ‘many/numerous/lots’ and the adverb ngaphezu(lu) ‘more/above’ (see mostert & roberts, 2020, for a detailed discussion on the use of -ninzi and ngaphezu(lu) in comparison phrases). figure 5: different formulations of the classic ‘how many more’ comparative question in isixhosa. the first formulation of a ‘how many more’ question in isixhosa uses -ngaphi ‘how many’ and ngaphezu(lu) ‘more/above’. in terms of word order, this formulation is the closest to the word order in english. like in english, the question starts with ‘how many’ (-ngaphi). it is therefore possible that this formulation can result in a similar confusion as in english in that learners might answer the question zingaphi iimbiza? ‘how many pots?’ instead of zingaphi iimbiza ngaphezu kweziciko? ‘how many more pots than lids?’. the second formulation also uses -ngaphi ‘how many’ but uses -ninzi ‘many’ to express ‘more’. in this formulation, and in the third formulation, the adjective -ninzi ‘many’ is used before the question word -ngaphi ‘how many’. because the formulation does not start with -ngaphi ‘how many’, it is possible that this formulation is less likely to result in learners answering the question ‘how many?’ instead of the question ‘how many more?’. note that due to the flexibility of word order in isixhosa the second formulation can be expressed in a number of different ways. for this study, however, only one variation was considered (see footnote in figure 5). the third formulation uses -ninzi ‘many’ and a specialised question word kangakanani ‘to what extent’. kangakanani can be used when asking about the difference between two nouns or sets of nouns. because the use of kangakanani precludes the use of -ngaphi, it was speculated that this third formulation would be the least confusing for learners and therefore the easiest for them to solve. this was tested in this study by comparing the third formulation with one variation of the second formulation. it is important to note that the question word kangakanani ‘to what extent’ does not necessarily have to have a numerical answer. as in english, when asked, ‘how many more stars than squares?’, it is possible to answer ‘many more’ or ‘a few more’. it is therefore important that isixhosa learners, or at least teachers, are aware that the practice of answering a kangakanani question with a numerical value is classroom based and is not necessarily used outside of the mathematics classroom.4 the canonical texts contain examples of all three formulations of classic comparative questions. while it is not possible to know what informed the choice of formulation in each example, this study sets out, in part, to provide research to better inform such decisions in the future. alternative comparative questions as discussed previously, in english there are a number of alternative ways of asking comparative questions in conjunction with a matching problem situation. these include ‘how many won’t get’ (hudson, 1980), ‘how many still needed’ (egma) and ‘how many missing’ (roberts, 2016). similarly it is possible to construct alternative comparative questions in isixhosa, as exemplified in figure 6. figure 6: alternative comparative questions in isixhosa (for matching (one-to-one embedded) problem situations). while the kusafuneka ‘still need’ and the ngazukufumana ‘won’t get’ formulations have direct equivalents in english, the -shota formulation does not and therefore requires some additional comments. the adapted loanword -shota is a loanword from the english verb ‘be short of’. while in english ‘be short of’ is most commonly used to refer to money (e.g. ‘i am short (of) three rand’ to mean ‘i have three rand less than i need’), in isixhosa -shota is commonly used to refer to being short of a wide range of things. in isixhosa -‑shota is either used with the prefix uwhen a person is short of something or with the prefix kuwhen there is a shortage of things (not belonging to a specific person). as part of a larger study, isixhosa adults (both teachers and other caring adults) were observed engaging with isixhosa learners and formulating comparative questions about specific problem situations. from these observations it appeared that the formulations that learners most easily understood were ones that included the verb -shota. this observation was the impetus for this study which, among other things, tests the hypothesis that -shota comparative questions are the easiest for learners to solve. like the ‘how many missing?’ question introduced by roberts (2016), asking ‘how many are short?’ draws attention to the absence of elements. it is important to note that some isixhosa speakers argue that it is not appropriate to use a loanword such as ‑shota in a mathematics classroom. others argue that because it is a word learners are familiar with and understand, it should be used as a means to help learners make sense of compare type problems, at least in teaching, if not in formal testing. typology of isixhosa compare type problems drawing on the typology of english comparative questions, and on the discussions of classic and alternative comparative questions in isixhosa, figure 7 provides a typology of isixhosa compare (difference unknown) word problems. as with the english typology, a complete version of the typology with examples for each category is provided in appendix 1, figure 2-a1. the typology and the identification of the dimensions of variation (problem situation and comparative question) and the range of possible change for each dimension make it possible to study the influence of one dimension by varying that dimension and keeping the other dimension the same. figure 7: typology of isixhosa compare type problems (with early grade mathematics assessment questions located in relevant categories). research design in order to answer the two research questions, as set out in the introduction, results from the south african version of the egma, based on the core egma (platas, ketterlin-geller, brombacher, & sitabkhan, 2014) and adapted by brombacher and associates, were used. the core egma includes one compare type problem out of a total of four additive relation word problems (q5–q8 in table 5). for this study, four additional compare type word problems were also administered (q1–q4 in table 5). table 5: different word problems used in this study. the four original word problems were translated from english into isixhosa by an accredited translator. the additional four problems were formulated in isixhosa through consultation with a number of isixhosa speakers. see appendix 1, table 1-a1 for the isixhosa formulation and english translations of the eight word problems. table 5 sets out the eight different types of word problems used in this study, based on the typologies set up in previous sections. the research for this study was done in two stages, corresponding to the two research questions. in the first stage, two additional compare type problems (q1 and q2) were added to the egma assessment in order to answer the first research question, namely whether, in isixhosa, different formulations of compare type problems had different levels of difficulty. this was answered by comparing q1, q2 and q5. the results of this comparison (discussed below) confirmed that in isixhosa, different formulations of compare type problems have different levels of difficulty. however, at this point it became apparent that the three formulations tested in the first stage differed in terms of more than one dimension. this led to stage two of the study in which the second research question was answered. in stage two, in order to establish which of the different dimensions had an effect on the difficulty level, q1 and q2 were replaced with q3 and q4. the addition of q3 and q4 made it possible to isolate the effect of the comparative question (research question 2.1) by comparing two differently phrased questions with the same problem situation (q1 and q4 as well as q2 and q3). it also meant that it was possible to isolate the effect of the problem situation (research question 2.2) by comparing two problems with the comparative question formulated in the same way but with different problem situations (q1 and q3). the relationship between the four questions and where they are located in the typology of isixhosa compare type problems is set out in figure 8. figure 8: questions used to isolate effect of different dimensions of variation. methodology in this section the methodologies used for data collection and data analysis are discussed in detail. data collection the egma was administered to isixhosa-speaking children in grade 1, grade 2 and grade 3. the data were originally collected to evaluate an early grade mathematics intervention in five isixhosa-dominant public schools in the rural eastern cape. all learners who were present on the day that the assessment was administered were tested. the egma was administered twice in 2019, once in may (stage 1, n = 242) and once in november (stage 2, n = 260). q1 and q2 were added to the egma administered in may and q3 and q4 were added to the egma administered in november. table 5 also indicates which questions were administered during each stage. the egma was administered individually by isixhosa-speaking adults. each word problem was read to the learner in isixhosa, first using isixhosa number names, and then using english number names. results (correct or incorrect) were recorded on tablets and then extracted into a spreadsheet for analysis. the guidelines for administering the egma state that if a learner incorrectly answers four questions in a row, they should not be asked the remaining questions in that section, the assumption being that the learner would not be able to answer any of the remaining questions. for this article, in each stage, only the results of learners who were asked all six word problems were analysed. data analysis in order to compare the difficulty level of two word problems, the facility score of each problem was calculated. the facility score is the percentage of learners who correctly answered the question out of the total number of learners who were asked the question. questions with a higher facility score were considered to be easier than those with a lower facility score. because neither cohort answered all five compare type questions (see table 5), it was necessary to establish whether the results of the questions administered only in stage 1 could be compared with the questions administered only in stage 2, and with those administered during both stages. in order to do this a pearson’s chi-squared test for homogeneity was done on facility scores of the four questions that were administered during both stages (q5–q8 in table 5). the test returned a p-value of 0.96 indicating that the two groups of learners were very homogenous, in other words the learners performed similarly on the four matched questions. in light of this, it is possible to compare any two questions, even if they were not answered by the same group of learners. while early research on the relative difficulty of word problems only considered the facility scores of the problems, subsequent developments in data analysis techniques now allow for more sophisticated comparisons. in this study, for each research question the word problems were first compared in terms of their facility scores. if there was a difference in facility score, a pearson’s chi-squared test for independence was used to establish whether the difference in facility score was significant or not (see appendix 1, table 2-a1 for a summary of the p-values for each research question). ethical consideration this study forms part of a phd study which has received ethical approval from the university of johannesburg, ethical clearance number: 2017-060. the data were anonymised and used only as aggregated data, which were not linked to individual children. ethical clearance was received on 08 september 2017. results the results will be discussed in relation to each research question. for each question the relevant problems will first be compared in terms of their facility scores and then in terms of the results of the pearson’s chi-squared test for independence. rq1: relative difficulty of different isixhosa compare type problems in this section three different types of comparison problems are compared, namely a standard compare type problem (q1), a matching (one-to-one embedded) problem (q2), and a matching (one-to-one imposed) problem (q5): (q1) iiswiti zentombazana zininzi ngeeswiti ezingaphi kwezenkwenkwe?         iiswiti         ze-ntombazana         zi-ninzi         ngee-switi         ezi-ngaphi         sweets of-girl         they.are-many         in.terms.of-sweets         they.are-how.many                  kwe-ze-nkwenkwe         compared.to-of-boy         how many more sweets does the girl have than the boy? (q2) ushota ngeziciko ezingaphi?         u-shota         nge-ziciko         ezi-ngaphi         she.is-short         in.terms.of-lids         they.are-how.many         she is short by how many lids? (q5) kusafuneka iiorenji ezingaphi ukuze umama akwazi ukunika umntwana ngamnye iorenji enye?         ku-sa-funeka         iiorenji         ezi-ngaphi         ukuze         umama         a-kwazi         it.is-still-necessary         oranges         that.are-how.many         so.that         mother         she-is.able         uku-nika         umtwana         nga-mnye         iorenji         e-nye                  to-give         child         by-one         orange         that.is-one                  how many oranges are still needed so that the mother can give each child one orange? figure 9 shows that, like in english, a smaller proportion of learners (47%) were able to solve standard compare type problems (‘no matching’ problem situation with a classic phrasing of the compare question) than both of the ‘matching’ problems (q2 and q5). there was less of a difference in the relative difficulty of the two different matching problems with a bigger proportion of learners able to answer the matching (one-to-one imposed) problem correctly (81%) than learners who answered the matching (one-to-one embedded) problem correctly (74%). the difference in facility scores was significant (p < 0.05). figure 9: facility score of three different isixhosa compare type problems. rq2: effect of different factors on relative difficulty of word problems in this section the different factors or dimensions that constitute a compare type problem are isolated to establish which factors influence the relative difficulty of compare type problems. the two factors that are considered are the problem situation and the phrasing of the comparative question. this is done by comparing different combinations of q1–q4, as outlined in the research design section. figure 10 shows that, for matching (one-to-one embedded) problems, a -shota -formulation of the comparative question is easier for learners than a -ninzi + --ngaphi formulation. it also shows that, for no matching problems, a -kangakanani formulation is easier than a -ninzi + --ngaphi formulation. finally, when a -ninzi + -ngaphi formulation is used, problems with a matching (one-to-one embedded) problem situation are easier than those with a ‘no matching’ problem situation. these results are discussed in detail in the following two sections. figure 10: facility scores of problems used to isolate factors influencing relative difficulty. rq2.1: effect of formulation of comparative question on relative difficulty to investigate whether the formulation of the comparative question influences the relative difficulty of a word problem, two different problem types were considered: matching (one-to-one embedded) problems and standard compare type problems. for each problem type the same problem situation was used but two differently phrased questions were asked. for matching (one-to-one embedded) problems, one question used the classic ‘-ninzi + -ngaphi’ phrasing (q3) and the other the alternative ‘-shota’ phrasing (q2): (q3)   iimbiza zininzi ngezingaphi kuneziciko?         iimbiza zi-ninzi nge-zi-ngaphi kune-ziciko         pots      they.are-many      by-they.are-how.many      compared.with-pots         how many more pots than lids? (q2) ushota ngeziciko ezingaphi?         u-shota nge-ziciko e-zi-ngaphi         she.is-short in.terms.of-lids are-they-how.many         how many lids are short (missing)? figure 10 shows that fewer learners answered correctly when the ‘-ninzi + -ngaphi’ phrasing was used (59%) and more learners answered correctly when the ‘-shota’ phrasing was used (74%). the difference in facility scores is significant (p < 0.05). this comparison is similar to the comparison hudson (1980) tested. like with hudson’s comparison, it is possible that the alternative ‘-shota’ phrasing is easier for learners to understand because it does not use quantifiers such as ‘more’ and ‘less’. for the ‘no matching’ compare problems, it was possible to compare two different formulations of the classic comparative questions, namely the ‘-ninzi + -zingaphi’ formulation (q1) and the ‘kangakanani’ formulation (q4). as can be seen from the english translations, this comparison is not possible in english where there is only one formulation of the ‘classic’ comparative question. (q1) iiswiti      zentombazana      zininzi      ngeeswiti      ezingaphi      kwezenkwenkwe?     iiswiti ze-ntombazana zi-ninzi ngee-switi ezi-ngaphi     sweets of-girl they.are-many in.terms.of-sweets they.are-how.many     kwe-ze-nkwenkwe     compared.to-of-boy     how many more sweets does the girl have than the boy? (q4) zininzi      kangakanani      iiswiti      zentombazana      kunezenkwenkwe?     zi-ninzi kangakanani iiswiti ze-ntombazana kune-ze-nkwenkwe?     they.are-many by.what.extent sweets of-girl compared.to-of-boy?     how many more sweets does the girl have than the boy? figure 10 shows that fewer learners answered correctly when the ‘-ninzi + -ngaphi’ formulation was used (47%) and more answered correctly when the ‘kangakanani’ formulation was used (57%). this supports the hypothesis that learners would find the ‘kangakanani’ formulation less confusing than the ‘-ninzi + -ngaphi’ formulation and begins to answer the question about which formulation of the ‘classic’ comparative question is most accessible to learners. even though facility scores for these two problems are not that different, the difference is still significant (p = 0.003 < 0.05). rq2.2: effect of problem situation on relative difficulty of word problem in order to establish whether reframing the problem situation without changing the question also has an effect on the relative difficulty level of compare problems, the facility scores for a standard compare problem (q1) and a matching (one-to-one embedded) problem (q3), both with the same formulation of the ‘classic’ comparison question (-ninzi + -ngaphi), were compared: (q1) iiswiti      zentombazana      zininzi      ngeeswiti      ezingaphi      kwezenkwenkwe?     iiswiti ze-ntombazana zi-ninzi ngee-switi ezi-ngaphi     sweets of-girl they.are-many in.terms.of-sweets they.are-how.many     kwe-ze-nkwenkwe     compared.to-of-boy     how many more sweets does the girl have than the boy? (q3) iimbiza zininzi ngezingaphi kuneziciko?     iimbiza zi-ninzi nge-zi-ngaphi kune-ziciko     pots      they.are-many      by-they.are-how.many      compared.with-pots     how many more pots are there than lids? figure 10 shows that even when a more difficult ‘classic’ comparative question is used, a higher percentage of learners correctly answered the matching (one-to-one embedded) question (59%) than the percentage of learners who correctly answered the no matching problem (47%). this suggests that changing the problem situation and not the question can, on its own, make it easier for learners to understand the problem. again, even though there is not a big difference between the facility score of these two questions, the chi-squared test (p = 0.006 < 0.05) confirms that the difference in facility score is significant. discussion these results raise a number of points regarding the relative difficulty of isixhosa compare type problems in early grade mathematics, some of which are also relevant for english. the study confirms that in isixhosa, as in english, while standard compare type problems (no matching + classic comparative question) are the most difficult to solve, when a standard compare type problem is modified, either by changing the problem situation or the formulation of the comparative question, the problem can become significantly easier for learners to solve (see figure 11). figure 11: facility score of different compare type word problems. the next two points are only relevant for isixhosa as they relate to specialised words that do not have an english equivalent. classroom observations suggested that comparative questions using the loanword -shota (e.g. q2) would be easier for learners to understand those using -ninzi -and -ngaphi (e.g. q3). this was confirmed by the results from this study. it is possible that this difference in difficulty level is because the ‘-shota’ formulation does not use the quantifier ‘more’ while the ‘-ninzi + -ngaphi’ formulation does. this difference in facility score suggests that teachers can use the ‘-shota’ formulation to introduce learners to compare type problems. it was speculated that the specialised question word kangakanani ‘to what extent’ would be less confusing for learners than questions using -ninzi and -ngaphi as these could be confused with --ngaphi questions. the results confirm this speculation: a higher percentage of learners correctly answered the -kangakanani question (q4) than those that correctly answered the ‘-ninzi + -ngaphi’ question (q1) when the problem situation was kept the same. the fact that isixhosa has a specialised question word that can be used when asking about the difference between two nouns or sets of nouns is an affordance that could be leveraged to mitigate the possible confusion between ‘how many?’ questions and ‘how many more?’ questions. the final two points relate to issues that are also applicable in english. while both matching (one-to-one embedded) and matching (one-to-one imposed) problem situations provide a useful teaching tool, in terms of the relative difficulty of comparison problems, matching (one-to-one imposed) problems are slightly easier to solve (see figure 11). one possible explanation for this is that in matching (one-to-one imposed) problems (e.g. q5) an additional phrase such as ‘each child can get one orange’ is required. this additional phrase makes the matching action explicit while in the matching (one-to-one embedded) problem (e.g. q2), the matching action is implicit. finally, the influence of matching problem situations is not only observed when used together with alternative formulations of the comparative question. when a more difficult classic comparative question is used with both a ‘no matching’ problem situation (e.g. q1) and with ‘matching’ problem situation (e.g. q3), the problem with the ‘matching’ situation is still easier for learners to solve than the problem with the ‘no matching’ situation. it is possible that the reason for this is because the matching problem situation invokes the action of matching which can be used to solve the problem. there are a number of limitations to be taken into account when interpreting the results. in order to fully establish whether there is a difference between matching (one-to-one embedded) and matching (one-to-one imposed) problems, it would be necessary to compare the two problem situations using all six different types of question. similarly, in order to establish whether there is a difference between a ‘-zingaphi’ question and a ‘kangakanani’ question, it would be necessary to compare all three different compare type problems (no matching, matching (one-to-one embedded), matching (one-to-one imposed)). finally, because the study did not compare all three classic comparative problems that can be formulated in isixhosa, it is not possible, at this stage, to establish which formulation is easiest for learners to understand. conclusion compare type word problems are notoriously difficult for learners but are also an important opportunity for learners to engage with the notion of comparison and of ‘subtraction as difference’. while the ‘standard’ formulation of compare problems (no matching problem situation with classic comparative question) is difficult, this and other studies have shown that certain formulations of compare type problems are easier for learners to understand and to solve, both in english and in isixhosa. these easier formulations provide a means of accessing the ‘standard’ compare type problems, allowing learners to make meaning of the problem situation without having to navigate complex language. this article has contributed to understanding the different factors that constitute a compare type (difference unknown) word problem. the typologies of english and isixhosa compare type problems provide a resource that can be used by materials developers and in further research looking in more detail at the influence of the different factors. this article also highlights the importance of studying the ways in which african languages express mathematical ideas in order to identify and leverage affordances for teaching and learning mathematics and, where different formulations are possible, to establish which formulation is most accessible for learners. while this article lays a foundation for studying compare type problems in other nguni languages, ultimately such research needs to be led by home language speakers in order for the linguistics features of african languages to be explored and described on their own terms, and not primarily in relation to english. acknowledgements this study would not have been possible without the insight and guidance of nicky roberts and the generous contribution of many isixhosa-speaking colleagues and friends, in particular: yolisa madolo, nobuntu mazeka, zinyaswa zuma, bambelihle nkwentsha, zola wababa, tholisa matheza, hlumela mkabile, zikhona gqibani, nolutsha parafini, ezile dalibango, pumza mfundisi and thulelah takane. competing interests the author has declared that no competing interest exists. author’s contributions i declare that i am the sole author for this article. funding information the study was funded by the south african national research foundation and the department of science and technology through professor henning, the south african research chair: integrated studies of learning language, mathematics and science in the primary school, grant number 98573. data availability statement the data that support the findings of this study are available on request from the corresponding author. disclaimer the views and opinions expressed in this article are those of the author and do not necessarily reflect the official policy or position of any affiliated agency of the author. references al-murani, t. 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(2005). mathematics as a constructive activity. mahwah, nj: lawrence erlbaum associates. appendix 1 figure 1-a1: typology of english compare word problems. figure 2-a1: typology of isixhosa compare word problems. table 1-a1: compare problems used in study. table 2-a1: p-values of problems that were compared for each research question. footnotes 1. see comrie, haspelmath and bickel (2008) for the leipzig glossing rules used widely in linguistics. 2. isixhosa is an agglutinative language and words are made up of many morphemes. 3. the curriculum and assessment policy statement (department of basic education 2011) and three sets of learner workbooks produced by the national department of basic education (2018), the national education collaboration trust and the nelson mandela institute. 4. thank you to bambelihle nkwentsha for pointing this out. article information author: suela kacerja1,2 affiliations: 1department of mathematical sciences, university of agder, norway 2department of mathematics, university luigj gurakuqi, albania correspondence to: suela kacerja email: skacerja@yahoo.com postal address: st olavs vei, 41-43, 4631 kristiansand, norway dates: received: 12 sept. 2011 accepted: 07 nov. 2011 published: 30 nov. 2011 how to cite this article: kacerja, s. (2011). albanian students’ motives for preferring certain real-life situations for learning mathematics. pythagoras, 32(2), art. #50, 9 pages. http://dx.doi.org/10.4102/pythagoras.v32i2.50 copyright notice: © 2011. the authors. licensee: aosis openjournals. this work is licensed under the creative commons attribution license. issn: 1012-2346 (print) issn: 2223-7895 (online) albanian students’ motives for preferring certain real-life situations for learning mathematics in this original research... open access • abstract • introduction    • background of the study and research questions • theoretical foundations • research design    • research approach       • data collection and participants       • participants and procedures       • ethical considerations    • research method       • realistic interviews       • reliability and validity       • initial mechanisms and contexts • findings    • new knowledge    • everyday use    • usefulness and relevance    • personal interest in a context    • perceived entertainment value    • community matters    • distance from gambling    • mathematics    • context-mechanism-outcome structure of students’ answers • discussion and conclusion • acknowledgements    • competing interests • references abstract (back to top) this article reports on a qualitative investigation of albanian students’ motives for preferring certain real-life situations to be used in school mathematics, and possible connections of these motives to characteristics and issues within albanian society. it is based on realistic interviews with students from grades 8 and 9 in a school in the district of shkodra in northern albania. interviews were conducted after students experienced five teaching units which dealt with mathematics embedded in real-life situations. results are expressed in terms of context-mechanism-outcome configurations as proposed by critical realist theory. most of the results are supported by those from a previous quantitative study which found that students prefer real-life situations which deal with computer games, being productive in a job, community development, and recreation, physical exercise and sport. an important mechanism uncovered in this study was the role of the mathematics that the topic introduced; degree of difficulty of the mathematics influenced the choice of preferred contexts by students. preferences expressed by students are therefore to be considered with care. introduction (back to top) inclusion of real-life situations in school mathematics textbooks and other teaching materials is discussed the world over (e.g. department of basic education, 2010; encyclopaedia britannica, 2011; julie, holtman & mbekwa, 2011; national mathematics advisory panel, 2008; organisation for economic co-operation and development, 2006; qualifications and curriculum authority, 2007). similar trends are emerging in the albanian school mathematics curriculum, where importance is placed on the need for students to recognise and use mathematical concepts in everyday life and in other school subjects (institute of curricula and training, 2008). the concepts of numeracy is introduced as ‘the group of mathematical concepts and skills that serve an individual in everyday life, at home, in his workplace, in community’ (institute of curricula and standards, 2006, p. 2). teachers are encouraged to bring examples from other school subjects into their mathematics teaching (institute of curricula and training, 2007). these latest changes in the albanian school mathematics curriculum should be reflected in mathematics textbooks and other teaching materials, and by the teachers who are expected to introduce these changes to students in the classroom. however, during selection of the real-life situations to use in teaching mathematics, students’ opinions are not usually taken into account. it is important that students’ voices are heard, yet there are virtually no research studies on students’ preferences in terms of the real-life situations to be used in mathematics (julie & mbekwa, 2005). there are indeed a few studies that include results about students’ preferences for real-life situations as secondary ones. one study is presented by kaiser-messmer (1993) on gender differences in school mathematics in germany, and one part of it deal with students’ preferences for different contexts. she not only found differences in preferences according to gender but also according to grade and level of course taken in upper secondary schools. in general girls were more inclined toward social topics and boys toward sports and technology. other studies on contexts in school mathematics deal with matters such as the effects of realistic contexts on learning (boaler, 1994; cooper, 1998; meaney, 2007) and assessment (van den heuvel-panhuizen, 2005), and how teachers use or choose contexts in teaching (chapman, 2006; pierce & stacey, 2006). background of the study and research questions the study presented is part of a multi-country project, relevance of school mathematics education (rosme), and is focused on grade 8–10 albanian students’ preferences of real-life situations to be used in mathematics. the target students in albania are 13–15 years old. the rosme project (julie, 2007; julie & holtman, 2008; julie et al., 2011; julie & mbekwa, 2005) aims to ascertain which real-life situations students in grades 8–10 are interested in dealing with in mathematics. a pragmatic outcome of the project will be research-based information on real-life situations that students find attractive for learning resources, which curriculum developers and other interested parties dealing with development of meaningful mathematical activities for students will be able to use. however, it does not hold that mathematical activities should only be driven by students’ interests.quantitative results of the rosme project in albania were reported in kacerja, julie and hadjerrouit (2010), where a hierarchy of students’ preferences for real-life situations in mathematics was obtained. in that earlier stage 24 interviews were held with students, referred to here as the ‘first interviews’, to ascertain their motives for preferences. results discussed in this article are from a second stage of the study that was informed by earlier results. in this stage the students participated in lessons with various real-life contexts before being interviewed. students’ motives for preferences are explained using the realist research paradigm. the current study sought to answer the following research questions: 1. what are students’ motives for preferring certain real-life situations for learning mathematics? 2. how, if at all, are these motives connected to contemporary issues in albanian society? theoretical foundations (back to top) this research is situated within a non-naïve, realist paradigm. this is different from naïve realism, in which a one-to-one relationship of reality and our description of it are assumed (pring, 2004). the main reason for choosing this paradigm was the belief that reality, that is, that students’ preferences for contexts exist, and thus an objectivist ontology is accepted. this reality is separated from our knowledge of it and is obtained through human cognition, so is subjective (au, 2007). in a realist paradigm a theory is given by the equation outcomes = mechanism + contexts, which mean that: ‘outcomes’ (are realised) only in so far as they introduce the appropriate ideas and opportunities (‘mechanisms’) to groups in the appropriate social and cultural conditions (‘contexts’). (pawson & tilley, 1997, p. 57) in this study students were taught five mathematics units dealing with real-life situations. the outcomes were students’ preferences as assessed through interviews, and the underlying mechanisms were students’ motives for their preferences. referring to pawson and tilley (1997, p. 66), to identify mechanisms in our study is to develop propositions about what it is in the mathematics lessons with real-life situations that drives students to have a positive, negative or neutral opinion about them. finding the mechanism, that is itself ‘a theory which spells out the potential of human resources and reasoning’ (pawson & tilley, 1997, p. 68), is to put together individual reasoning (choices) and collective resources (capacity). the starting point in finding a configuration of context-mechanism-outcome and testing and refining it is to conjecture about what might work, for whom and in which contexts, which is tested during data collection. students were asked directly about reasons for their preferences, but we cannot pretend that they are totally aware of the bigger picture; it is required of the researcher to conjecture about students’ reasoning in a wider and more complete context. students included in the study were members of albanian society, members of various families, studying at a certain school, belonging to certain groups and therefore having their own habitus (bourdieu, 1977) or history that shapes how they see the present and the future, which makes them perceive opportunities in specific ways. hence one should also take into account the context in which the units are introduced, not just the physical context but also the social norms, values and rules embraced by inhabitants of that physical location, since a mechanism can be activated or deactivated depending to a high degree on the context (pawson & tilley, 1997). in considering students’ motives for their preferences it was therefore important to refer to the albanian social environment where their habitus were shaped. research design (back to top) research approach the study presented used realistic interviews to gather data in order to answer the research questions. these interviews followed the experiences the students had with five teaching units of mathematics in real-life situations. data collection and participants the teaching units: the students participated in five units of mathematics embedded in real-life situations. in each of the units students learned about one real-life situation whilst using mathematics; this varied from application of known mathematics to relatively new mathematical knowledge for the students. five contexts were selected for the units: development indexes, lotteries and gambling, sport tournaments, secret codes and sending smss. these were chosen from different positions in the hierarchy of preferences that arose from the quantitative data analysis in 2008, from highly preferred to moderately preferred and least preferred items (kacerja et al., 2010). the most preferred item of the questionnaire was ‘determining the level of development regarding employment, education and poverty of my community’, and this was the realistic situation for one of the lessons. in this lesson information from the human development index (hdi) used by the united nations development programme (2009) to measure the degree of development of countries of the world was introduced to students. they learned how the hdi for each country is constructed, calculated albania’s hdi, and their homework was to collect data in their neighbourhood to calculate a simpler development index for the area they live in and to discuss it. the main mathematics concept here was the construction of a social index, which is normally embedded in econometrics and was in a sense new mathematics to them. one of the lesser preferred situations, ordered as 21st amongst 23 items in the 2010 questionnaire (kacerja et al., 2010), was ‘lotteries and gambling’. during the first interviews (kacerja, 2009) students referred to the negative material and social consequences of gambling. therefore one part of the lesson included a discussion of the following quote by einstein: ‘you cannot beat a roulette table unless you steal money from it’, after doing the mathematics and calculating different probabilities. one of the moderately highly preferred topics contextualised was about sport tournaments as part of the item ‘recreation, physical exercise, sport activities and competitions’, ranked 9th. the lesson brought to the fore elements that had to do with the number of teams and matches in a single-elimination tournament in sports such as basketball or football. the mathematics dealt with powers of a number and multiples, which they already knew from school mathematics, and geometric sequences, which were new to them. one of the medium-preferred items was ‘secret codes such as pin numbers’, ranked 14th on the questionnare. a lesson on secret codes and algebra was adapted from a chapter of the consortium for mathematics and its applications 1998 book modeling our world: course 1. students learned how to code a phrase by shifting letters or numbers, and how to decode a message coded by someone else. elements of cryptography were employed. students used different linear functions that were not totally new to them. one of the highly preferred items, ‘sending and receiving electronic messages’ (ranked 6th) was used for one topic about cell phone messages. students learned the route traversed by an sms as it moves from one mobile phone to another, the time taken for the call to be serviced, and the waiting time in a queue. the information students received from this lesson was new to most of them. the mathematics included some simple elements of queuing theory. participants and procedures a lower secondary city school was selected for data collection on the basis of the school principal’s willingness for collaboration after being contacted and having the study, its aims, the way the data would be collected and materials to be used for this purpose explained. the school was selected for convenience, and does not have a reputation for being the highest or lowest quality school in the city of shkodra. the mathematics teacher who collaborated with the researcher was asked to form a class with 22 students from grades 8 and 9, from amongst those who volunteered to participate. students were selected to ensure a gender balance and a mixed-abilities group in terms of their mathematical skills, so that the group would be as diverse as possible. in order not to interfere with normal school lesson plans, the five teaching units were conducted after normal school hours as extra lessons for students. each of the five teaching units lasted 45 minutes. the researcher was also the teacher of the lessons. the routine followed in every lesson was similar to that which the students were used to at school: the topic and some information about it were introduced, then the different sections or exercises were completed. some of the exercises were solved jointly on the blackboard, some individually in the classroom, and others were given as homework. after the five teaching sessions students were asked to volunteer to be interviewed. eighteen students volunteered and eight of these were selected to be interviewed. the gender balance was again a factor for selection, together with different degrees of students’ engagement during the lessons. students were interviewed individually at their school by the researcher for not more than 30 minutes. the procedure employed in each interview was the same: students were asked to place the five lessons in decreasing order of preference and to state the reasons for liking or disliking each of them. ethical considerations since no ethical guidelines exist for research in albania, both the school principal and the teacher were present when the project was explained to students and they were invited to volunteer to participate. all students who volunteered to participate signed a declaration that they understood the project and what was required of them, and they also informed their parents. in the excerpts from interviews below the initials used are pseudonyms to ensure students’ anonymity. research method realistic interviews realistic interviews seek to answer as to what works for whom in which circumstances (pawson, 1996). realistic interviews are in line with the realist paradigm that guides the study. the researcher already had an initial theory or conjecture about the matter at hand, from previous stages of the study, in terms of initial mechanisms and contexts. the strategy followed was therefore theory-driven: ‘the researcher’s theory is the subject matter of the interview, and the subject (stakeholder) is there to confirm, to falsify and, above all, to refine that theory’ (pawson & tilley, 1997, p. 155). the aim was to assess albanian students’ opinions of the five mathematics units which they were taught, and to establish the motives that guided their choices. interviews were conducted in albanian. a realistic interview starts with the researcher’s theory and goes through four stages that can be repeated in a cyclical way. the first two stages have to do with a ‘teacher-learner function’. here the interviewer’s active role is to inform or teach the interviewee the conceptual structure of the interview, its purposes and meanings, in order to have a common understanding. by doing so the interviewee will understand the nature of the information being sought, and why it is sought (pawson, 1996). in the introduction the researcher declared the object of the interview and explained its general aim as follows: i would like to know about the activities we did together, what are your interests, how did you like those, why do you like something and not something else. the idea in this interview is that i want to know from you why you like some … i.e. what interests do you have to learn the mathematics that is used in some fields, that is applied in these fields, and why do you like it, or what are the fields you like more, why do you like those, what do you connect those with. the two main concepts of the study, students’ interest in real-life contexts in mathematics lessons and the reasons for their preferences, were introduced to the interviewee. it was emphasised that it was the students’ opinions that were being requested. using the five units as a common basis for interviewing contributed towards mutual understanding, gave students some examples to think about, and made it possible to move from specific to more general discussions. teaching of the conceptual structure during the first two stages continued, by stressing the significance of students’ openness in declaring their liking or disliking of any or all of the lessons, offering them a conceptual structure in answering, and clarifying the interview structure: i want to know what you really think about these, so if you don’t like something you are free to say ‘i don’t like this because…’ and you tell your reasons. if you like something you normally say it with the reasons too. after the students ordered the five lessons from the most to the least preferred, the researcher went through each of the topics. in all of the following interview excerpts the notation r is used for the researcher, and a, b and so on for the individual students. r: good, let’s start with ‘sending smss’, which you ordered first. why is that? the students then answered questions about the reasons for their preference, keeping in mind the concepts mentioned earlier by the researcher. in this instance student b responded as follows: b: this is the topic i liked more because we use mobile phones every day and we send smss, but without knowing the time which we wait or the time when the sms arrives, how it arrives. i liked it very much because … i have learned something new about sending smss and i have some more knowledge than i knew [before] about this topic … that’s all about it. (girl, grade 9) in this way the student goes from the second stage, where a conceptual structure is learned, to the third stage, where she applies this structure to her answers. the two last stages are included under a ‘concept refinement function’, where the students express mechanisms, that is, motives that drive their preferences for specific contexts, to inform the researcher’s theory. the researcher applies these mechanisms to the initial conjectures and presents the finding to the students, who can then agree, disagree, or refine the theory’s conceptual basis. at this stage explanations, checking and repeated questions are important elements. the following is an example of the researcher applying the student’s reasoning to the theory, extracted from her previous answers, and offering it to the student to comment upon: r: so in order for a topic to be attractive, interesting for you, should it be something you do? b: hmm, there are also some things that one doesn’t do and which one is interested to know, but things one does in everyday life are more interesting to know compared to others. here the researcher was trying to clarify the role of the ‘everyday use’ as a mechanism in the student’s preferences. the student’s answer refined the researcher’s understanding about the connection between everyday life situations and interest in learning the mathematics used in these situations. the interviews continued like this in a cyclical manner until the researcher had clear answers to the questions and conjectures she had. reliability and validity to ensure reliability, care was taken to formulate questions as clearly as possible for every student. considerable time was taken during the whole study to emphasise the need for the students’ input on their reasons for preferences. anonymity of the interviews was ensured and students were informed in advance about the aims and objectives of the study. the interviewer was careful to avoid as much as possible any judgement of students’ answers, and to create a feeling of confidence during the interviews. reliability in interviews is related to ability to replicate the findings at other times, by other researchers (kvale & brinkmann, 2009). one thing that might put reliability at risk is that preferences can be time-bound but, as this study tries to show, preferences are influenced to a great degree by the societies in which individuals live. thus huge changes are not to be expected in short periods such as the time needed for this study.referring to pawson and tilley (1997), in a realistic interview the question of validity is not whether the data reproduce exactly the students’ beliefs, but whether they capture the relevant concepts for the researcher’s theory. during interviews the students’ thoughts and the researcher’s interpretations were continuously validated by the researcher, whilst applying students’ answers to the initial conjectures and asking repeated questions to check meaning. initial mechanisms and contexts in realistic interviews the researcher already has some initial conjectures about possible mechanisms, that is, motives, which drive students’ preferences and the context in which the teaching takes place, but also the broader context of the community. the following mechanisms are based on data collected from the first interviews with students and on the rosme study in south africa (julie, 2009), and are accompanied by some contexts within albanian society. ‘lotteries and gambling’ are deemed an inappropriate real-life context to be used in mathematics because of their undesirable effects (both social and material) on people’s lives. this is supported by a general negative opinion about the topic in albanian society, which is hypothesised as a possible mechanism for low preference for use of this real-life context in mathematics. albanian students participating in the first interviews expressed great interest in a secure future economic situation and upward mobility in life; thus connectedness of real-life situations to their future interests is a possible mechanism for preference. most parents want their children to achieve at school in order to have better possibilities for the future, and high school students themselves want to achieve in order to find a good job and attain a higher standard of living by studying further at university (world bank, 2005). students’ affiliation to and use of modern technologies for communication, such as ’sending and receiving smss’ or for safety in transactions such as ‘secret codes such as pin numbers’, is a mechanism that can make a topic interesting for them. in the south african rosme study a mechanism that came across was ‘community matters’ (julie, 2009); that is also important in albania, a country with a developing economy. regarding the general context of albanian society, there are three important aspects: transition from a 45-year-old communist totalitarian system to a market economy in 1990, with all the changes and challenges it carried; an economy where high poverty and unemployment rates predominate, especially in rural areas; and internal and external migrations (sultana, 2006) which have changed the structure of cities and brought phenomena such as brain and skills drains. findings (back to top) interviews were audio-taped, transcribed and analysed to identify categories which would characterise mechanisms driving the students’ preferences for real-life situations to be used in mathematics. these mechanisms, defined as students’ motives for preferences, are now described and illustrated. new knowledge in many cases students mentioned that much of the information they received whilst participating in the lessons was new to them. for example, a grade 8 girl (g) said: ‘i have learned things i didn’t think i would learn [in mathematics]’, referring to the unit about secret codes. students valued this as something which helped them to enrich their background, as demonstrated by another grade 9 girl (b), who asserted that: ‘this [development indexes] is a topic that makes you have a broader horizon [background].’ d, a grade 8 girl, said: ‘we get more information and we have a larger background.’ this mechanism was mostly mentioned in connection to other, more relevant mechanisms in order to add to the positive effect of them. its influence will therefore be discussed below. everyday use this category includes comments that students made about familiarity with a context, such as: ‘i chose sport tournaments first because i like football, but also because i know how they are organised’ (l, boy, grade 8). everyday use of a context was also directly mentioned by b (girl, grade 9): ‘this is the topic i liked more because we use mobile phones every day and we send smss.’ at the same time as the everyday use of a context influenced its choice, distance from it resulted in lack of interest from the students. this was the case, for example, for a boy (l, grade 8) who expressed his disinterest in learning mathematics used in agricultural matters since ‘i don’t think i will ever deal with agriculture’. usefulness and relevance terms used by students such as ‘useful’, ‘usable’, ‘valuable’, ‘needed’, ‘important’, ‘relevant’ and so on are included in this category, which was mentioned most often during interviews. in most cases it was a good reason for guiding choice of a context: … because sending smss is less valuable than indexes, much less valuable. (m, boy, grade 9) when i chose indexes before, it was … it was more … more necessary than this, lotteries. (a, boy, grade 8) because it is useful information in general, but also on mathematics part. (d, girl, grade 8) the usefulness of the contexts for actual or future life is emphasised and characterised as part of this category: yes, i liked it [indexes], because they are useful things in life. (g, girl, grade 8) this topic [lotteries] is … it gives you a … it teaches you a lesson, so in the future one should have in mind that these are to do with fate and it’s not that one can win with these. (a, boy, grade 9) the same student would be highly interested ‘about things that have to do with my future’. lack of perceived personal relevance of a context is a factor for according it lesser preference. this is supported by a grade 8 girl (g), who said ‘i don’t like mathematics used in agriculture because it is not relevant for me, since i will not do agriculture in the future’. personal interest in a context this category showed more interest when students were asked about the topic of sport tournaments. most of the boys expressed their personal interest in sports as an activity they personally do or follow closely:… because sport is one of my favourites. (m, grade 9) i liked this [sport tournaments topic] because i like sports very much and i liked this lesson better. (a, grade 8) i chose first sport tournaments because i like football, but also because i know how they work (l, grade 8). another grade 9 boy (a) chose sport tournaments first and reasoned as follows: i liked it, for example, because it gave me more knowledge about teams, how the champions league is organised, how many matches there could be, things i didn’t know … [i liked] the way this topic is organised, since it is also an entertaining topic and sports are something i like, for these reasons …yes, [i have] an interest in sports. a response from a girl, g (grade 8) points towards a lack of personal interest in sports as a reason for not preferring the topic sport tournaments: ‘i am not a sports fan, i don’t like them either.’ perceived entertainment value entertainment was often mentioned during interviews, when lack or presence of entertaining elements was perceived and used to describe lessons: entertaining, and the lesson i enjoyed the most. (g, girl, grade 8) i would like [to have a topic such as sport tournaments in my mathematics textbook], because it would be an entertaining lesson. (a, boy, grade 9) … it’s not just mathematics, it is entertaining also. (d, girl, grade 8) for some students the degree of entertainment overrides other aspects when choosing a preferred lesson. community matters under this category are incorporated mainly aspects from the lesson about development indexes, where relevance of information about one’s country and community is at the centre. students’ interest is clearly accompanied by their feelings of responsibility to know about their country ‘because it is in everyone’s interest to know more about our country and many other countries’ (a, boy, grade 8). as one girl put it, ‘since we live in this country, we should know our economic percentages [meaning economic figures and indexes], how developed is our education, to know the development of the country we live in’ (b, grade 9). however, even with its relevance this topic was not one of the most preferred, because of mathematics aspects which will be discussed further. distance from gambling gambling was one of the most discussed topics during the first interviews (kacerja, 2009) because of its negative effects. in the current realistic interviews students expressed distance from it as an activity that they never take part in. they gave reasons such as: i wouldn’t do it myself, and i would advise my friends not to do it. because i have learned in this lotteries topic that the probability [to win] is very low if we gamble. (b, girl, grade 9) gambling is seen as an activity that only boys can do, whilst ‘[girls] during their life they don’t have possibilities to deal with lotteries’ (a, boy, grade 9). mathematics the effect of mathematics has to be considered, since it emerged in the interviews that it was one of the mechanisms often mentioned as influencing students’ choice of contexts. the students mostly talked about the degree of difficulty and ‘trickiness’ of the mathematics:secret codes, i liked it, it was not so tiring topic … i liked it also because mathematics was of a lower level [means easier] than the everyday [school] mathematics. (b, girl, grade 9) [indexes topic] has too many calculations, and one can get lost, it’s tricky and it looked to me more like a difficult mathematics lesson than entertaining like others. (a, boy, grade 9) i liked it a little bit [indexes] but the problem was that it was tricky [mathematics] and i put it lower than others. (l, boy, grade 9) as can be seen, the mathematics is the overriding issue; this relationship is a matter requiring further research. context-mechanism-outcome structure of students’ answers the characteristic of all of the mechanisms presented above is that when combined with each other, their effects change: the same mechanism can produce different outcomes. for example, the everyday use of a context, modern technologies such as sending smss, accompanied by new information about it can give rise to disinterest (outcome o1) when it is not perceived as relevant or valuable to the students. however, everyday use of modern technologies for communication (sending smss), including security matters (secret codes), accompanied by new information about it when ‘it is like a game’ (a, girl, grade 8), highlight its utility value and intrinsic quality. for example: ‘[secret codes could be used] when i don’t want to say something out loud, i can talk with codes with my friends’ can result in increased interest (outcome o2), enthusiasm and students’ engagement in mathematical activities (outcome o3). for all these mechanisms, the contexts that make it possible for them to induce the respective outcomes are related to students’ age (13–15 years), and their knowledge of and the possibility of them using modern technologies. even though albania is one of the poorest countries in europe, its figures in terms of using mobile phones are higher than averages in the european union. pin codes are also used extensively in mobile phones, atm machines and so on. when accompanied with the mathematics involved, another important mechanism, community matters, brings about two different outcomes. community matters bringing new information are perceived as useful in education, occupations and life in general, and can lead to elevated interest (o2) in following and learning the lesson. however, when the unit is identified by students as using difficult mathematics and too many tricky calculations requiring more attention, the same mechanisms and contexts can end up at a lowered or lost level of interest (o4). this is one of the cases of mathematics overriding context choice, as happened for this girl: ‘[indexes] would have been first on my interest, but because it has many calculations i put it third’ (b, grade 9). the greater context of albanian reality is also an indicator and predictor of the relevance perceived by students. issues about community matters, such as health, education, poverty and unemployment, are sensitive and perceived by students as such. therefore following higher studies that will lead to a good and stable job in order to have a safe future is in itself a desire and objective for albanian youngsters, as l (boy, grade 8) states: [in the future] i wanted to be a lawyer, but there are too many of them now, i should find another field … if you want to find a job when you grow up, there are many [lawyers] and you don’t find it. the everyday use of sports topics associated with boys’ personal interest in sports as a preferred activity in which to participate or watch, conveying new information, makes boys recognise it as a relevant and entertaining topic, which can therefore result in a higher level of interest (o2) in learning mathematics. this is not the case with girls; since sports are not of everyday use to them, their personal interest in sports is also not that developed. in the absence of these two mechanisms, girls’ interest in the unit is low (o1). at the basis of students’ answers about lotteries and gambling was their positioning of themselves at a certain distance from it. albanian students participating in both stages of the rosme study admitted that they have either used lotteries very seldom or not at all, but that friends around them use them. this has to be understood within the larger reality of albania, where gambling was considered a negative, illegal behaviour during the communist era (1945–1990). nowadays lotteries and gambling are a growing industry regulated by law, the phenomenon remaining mainly a male one. even though it is illegal for people under 21 years to gamble, school children still enter casinos or gambling establishments (gambling pushes minors toward crime, gazeta shqip, 30 january 2009).   discussion and conclusion (back to top) the current study aimed to uncover students’ motives for choosing their preferred real-life situations for use when learning mathematics. these motives are presented here under the label of ‘mechanisms’ as part of a realist paradigm. the same research question (the first one) drove an earlier, qualitative part of this study based on interviews with students in grades 8-10 (kacerja, 2009). in the research presented in this article, the difference is that students were interviewed after they had experienced mathematics embedded in some real-life situations that they were then asked about. the purpose of introducing this difference was to ascertain earlier found mechanisms, and to determine whether the teaching units would add further mechanisms. the second research question was to detect and explain any connection between students’ motives for preferences and societal matters in albania. by formulating the context-mechanism-outcome structure from students’ answers, an answer is provided to this question. albanian societal matters are introduced above as contexts for the realistic theory, and their influence on students is disclosed as part of the realist equation outcomes = mechanism + contexts. some of the views that students expressed during realistic interviews reflected some tendencies of albanian life. this can be seen, for example, in students’ distance from gambling as an activity which in the albanian community is considered problematic and not desirable. another tendency was expressed in students’ need for a good education and work position as desired outcomes for a satisfying life. a study by the world bank (2005) revealed that most albanian parents want their children to achieve at school in order to have better possibilities for the future. one can therefore see the parents’ ideas reflected in their children’s objectives. further examples can be connected with students’ interest in community matters as important aspects in a developing country. during interviews one student discussed the economic crisis, using the example of remittances from albanian emigrants for their families in albania to explain the relevance of being knowledgeable about one’s country. these issues are objects of everyday discussion everywhere in albania. the realist approach embraced in this study made it possible to consider not only students’ motives for their expressed preferences, but also to relate these motives to the context of the students’ everyday life in their community. this helps to find the roots of their motives in the influences that society exercises on students. the key matter in the realistic interviews is the flow of information which, as explained earlier, allows the understanding of concepts and conjectures to be assessed by both participants in the interview (pawson & tilley, 1997). this openness allowed continuous checking of the meanings obtained from the conversations. one of the confirmed motives was connected to lotteries and gambling, which, as argued above, is connected to recent developments in the albanian context. gambling was considered as a male activity by students, as is the case in the greater albanian society. it was, however, noticed that during interviews students saw a possible use for the information they received from the lesson, to advise friends who deal with gambling. this result is to some degree different from those of previous questionnaires, where the topic was amongst the least preferred, and from the first interviews where only the negative aspects of gambling were highlighted. a similar finding was presented in south africa, where the mechanism was defined as ’personal regression via irresponsibility and/or addiction’ (julie, 2009). we can speculate here that putting the teaching unit’s emphasis on being able to calculate gambling’s negative consequences in order to become aware of its dangers helped in making students’ attitudes towards learning the mathematics related to the topic more positive than in previous results. this issue of least preferred real-life situations, where an appropriate mathematical treatment of the context can contribute to learners engaging with it in a more informed manner, has been discussed previously (kacerja et al., 2010). students’ affiliation with modern technologies, a motive revealed by the first interviews, manifested itself in their everyday use of technologies and was related to the topics ‘sending and receiving smss’ and ‘secret codes’, ordered 6th and 14th respectively in the list of preferences (kacerja et al., 2010). in the realistic interviews this difference is inverted. the same mechanism in the two topics generates two different results: the first topic is mentioned for its lack of relevance, whilst the second one is perceived as entertaining. this is an indicator of the relevance of integration of mechanisms and the influence of contexts. personal interest in an activity and perceived entertainment value are two other motives occurring in most responses during the interviews, seen as desirable qualities for a topic to be preferred. this was the case with the item ‘sport tournaments’, that was clearly more preferred by the boys; for girls the lack of the two motives made it uninteresting to them. however, no gender differences were found on the item ‘recreation, physical exercise, sport activities and competitions’, which means that the choice of a topic belonging to the same item can also make a difference. in terms of social life in albania, for boys (especially those playing football or basketball) following national or international matches, wanting to be football players in the future and discussing sports are very usual, but it is not the same for girls. the influence of the albanian reality in students’ preferences is also highlighted in connection with the ‘community matters’ mechanism, and is emphasised in the ‘development indexes’ topic. the relevance of the topic for their future educational and professional life, as a motive, was closely connected to this as well as to other topics. the two mechanisms find their reflection in the ‘need for a safe economic future and upward movement’ as a motive disclosed during the first interviews. findings from the qualitative data are consistent with those from the quantitative data. one mechanism displayed for the first time during this stage of the research was the issue of the mathematics introduced in the lessons. in some cases this mechanism overrode the choice of the topic, as happened with the ‘development indexes’. the difficulty of mathematics, its trickiness, and the amount of calculations steered students’ interest away, even with the topic’s perceived relevance. however, it is impossible to find a connection between students’ characteristics and lowered interest because of difficult mathematics, nor between the kind of mathematics and the degree of interest, from the data obtained. further research is needed to explain this relationship. related to this latter topic, care should be taken to consider the quantity and difficulty level of new concepts (such as life expectancy, gross enrolment ratio, adult literacy index and gross domestic product [gdp]) when introducing these to students. apparently these notions require more time to be grasped; thus a unit like this could be extended over two or more lessons to facilitate understanding. other units could also be extended to two lessons in order to create some space for discussions about each topic, concepts and knowledge used or gained, and generalisations. some discussion was included in each topic as part of the unit, but it is seen that discussion is crucial to enhance critical thinking which, from personal impressions seems to be missing or at least not so developed in albanian schools. a study by sahlberg and boce (2008) supports this impression, and points toward development of productive learning of the skills needed within a knowledge society. these issues and education in general are especially vital in albania, for their contribution ‘to furthering democracy, to promoting an active citizenry, as well as to creating a vibrant, skilled workforce essential to the country’s competitiveness, especially in a context marked by resource scarcity’ (sultana, 2006, p. 10). amongst the contexts that influence the level of preference of a teaching unit described in this article, the context of the classroom environment is equally important. students confirmed that they enjoyed the topic ‘secret codes’ the most, not only because it was entertaining but also because they could work together and everyone could participate. this suggests that such topics can be even more interesting if there is a classroom climate where participation, discussion and critical thinking are appreciated and fostered. other factors of influence could be the design of the lesson, the way it is taught and the relationship created between teacher and students, as in every other lesson. it must be emphasised that the focus of this study was the real-life situations and students’ perception of these. questions of generalisation relate to discussion of results as being time-bound, as mentioned in the section on reliability. kvale and brinkmann (2009, p. 265) argue that ‘analytical generalisation may be drawn from an interview investigation regardless of sampling and mode of analysis’. it requires detailed descriptions and arguments about the transferability of results to other subjects and situations. in this study the findings can be generalised at a local level, namely in terms of albanian students. this can be said since, as discussed above, some issues, values and norms in albanian community are reflected in the students’ words when giving their reasons for preferring or not preferring to learn mathematics in specific contexts. acknowledgements (back to top) my special thanks go to my supervisors cyril julie and said hadjerrouit for reading and commenting on several drafts of the manuscript. competing interests the author declares that she has no financial or personal relationships which may have inappropriately influenced her in writing this article. references (back to top) au, w. 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(2007). mathematics. programme of study for key stage 3 and attainment target. available from http://curriculum.qcda.gov.uk/uploads/qca-07-3336-p_ict_ks3_tcm8-401.pdf sahlberg, p., & boce, e. (2008, march). are we teaching for a knowledge society? evidence from albanian upper secondary schools. paper presented at the annual meeting of the american educational research association, new york. sultana, r.g. (2006). facing the hidden drop-out challenge in albania. evaluation report of hidden drop-out project. piloted in basic education in 6 prefectures of albania. tirana, albania: unicef. united nations development programme (undp). (2009). human development report 2010. the real wealth of nations: pathways to human development. available from http://hdr.undp.org/en/reports/global/hdr2010/chapters/en/ van den heuvel-panhuizen, m. (2005). the role of contexts in assessment problems in mathematics. for the learning of mathematics, 25(2), 2–9. available from http://www.jstor.org/stable/40248489 world bank. (2005). albania: poverty and education in albania – who benefits from public spending? annex ii. to albania long term development issues and options. washington, dc: world bank. article information authors: paola valero1 gloria garcía2 francisco camelo3 gabriel mancera3 julio romero3 affiliations: 1department of learning and philosophy, aalborg university, denmark2department of mathematics, universidad pedagógica nacional, colombia 3faculty of science and education, universidad distrital francisco josé de caldas, colombia correspondence to: paola valero postal address: sohngaardsholmsvej 2, 9000 aalborg, denmark dates: received: 14 apr. 2012 accepted: 18 sept. 2012 published: 15 nov. 2012 how to cite this article: valero, p., garcía, g., camelo, f., mancera, g., & romero, j. (2012). mathematics education and the dignity of being. pythagoras, 33(2), art. #171, 9 pages. http://dx.doi.org/10.4102/ pythagoras.v33i2.171 copyright notice: © 2012. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. mathematics education and the dignity of being in this original research... open access • abstract • introduction • from democracy to the dignity of being • from mathematics to social subjectivity with mathematics • from mathematical spatiality to social space and intimate space • social subjectivity for the dignity of being with mathematics • acknowledgements    • competing interests    • authors’ contributions • references abstract top ↑ on the grounds of our work as researchers, teacher educators and teachers engaging with a socio-political approach in mathematics education in colombia, we propose to understand democracy in terms of the possibility of constructing a social subjectivity for the dignity of being. we address the dilemma of how the historical insertion of school mathematics in relation to the colonial project of assimilation of latin american indigenous peoples into the episteme of the enlightenment and modernity is in conflict with the possibility of the promotion of a social subjectivity in mathematics classrooms. we illustrate a pedagogical possibility to move towards a mathematics education for social subjectivity with our work in reassembling the notion of geometrical space in the colombian secondary school mathematics curriculum with notions of space from critical geography and the problem of territorialisation, and latin american epistemology with the notion of intimate space as an important element of social subjectivity. introduction top ↑ we do not need to start this paper with a vivid depiction of the conditions of poverty, violence and disrespect that many students in mathematics classrooms experience in their daily lives. such situations and the implication of mathematics education in (re)producing them have been documented in research (e.g. valero & pais, 2012). such situations still exist and justify revisiting the discussion about the connection between mathematics education and democracy in a special issue of a research journal such as pythagoras. the connection between mathematics education and democracy has been the topic of international journal issues (e.g. zdm: the international journal on mathematics education 30[6], zdm 31[1]) and of scattered papers (e.g. de mattos & batarce, 2010; skovsmose & valero, 2008). in the last decade it seems as if topics such as ‘equity’ and ‘social justice’ have displaced the discussion about democracy in the mathematics education research literature. still, the editors of this special issue call researchers to report on the advances in the topic and to explore through theoretical and empirical discussions the significance of the connection between mathematics education and democracy in relation to development.we have been working in a collaborative team of teachers, teacher educators and researchers grappling with recontextualising the ideas of critical mathematics education as proposed by skovsmose (1994) and vithal (2003) to study and transform mathematics education practices in classrooms and schools in so-called ‘at risk’ communities in bogotá, colombia (camelo, mancera, romero, garcía & valero, 2010; garcía et al., 2009). attending to the specificities of the context and listening to students and teachers have led us on an intellectual search for a resignification of the possible links between mathematics education and democracy. in the process we drew on diverse sources such as contemporary latin american philosophy, political and pedagogical literature, critical educational research grounded on the work of michel foucault, and critical and political research in mathematics education. our intention is to bring to international research in mathematics education the discussion of a theoretical framing to rethink the role of school mathematics in constructing historical subjects who strive for living in and with dignity. we build on our research work with teachers and children in communities whose life possibilities are far from reaching the promises of the globalised discourses of social and economic democracy. we start our article formulating an understanding of democracy in terms of the possibility of a social subjectivity for the dignity of being. we then address the issue of how the historical insertion of school mathematics in relation to the colonial project of assimilation of latin american indigenous peoples into the episteme of the enlightenment and modernity is in conflict with the possibility of a social subjectivity in mathematics classrooms. we then present the theoretical grid on which we constructed a curricular proposal for working with students in a colombian classroom on the notion of space. the curricular proposal is an example of an attempt to decentre the core of the mathematics curriculum by opening its possible meanings with other related discursive fields in which notions of space are found. the curricular proposal brings together euclidean geometry, critical geography and the problem of territorialisation, and contemporary latin american philosophy with the notion of intimate space as an important element of social subjectivity. we argue that such a decentring opens the possibility for forms of subjectivity that are denied in the mainstream mathematics curriculum. we conclude with some remarks about the significance of the types of studies that we have undertaken in contributing to a resignification of the connections between mathematics education and democracy in different historical and geographical contexts. from democracy to the dignity of being top ↑ connecting mathematics education and democracy as an idea became possible in mathematics education research in the decade of the 1980s. it is a very recent thought in mathematics education that was made intelligible in the confluence of at least three trends. firstly, during the 1980s mathematicians and mathematics educators questioned the outcome of the new maths movement in strengthening the construction of a school mathematics curriculum for an elite of selected students who would continue to study mathematics at a higher level. concerns for a ‘mathematics for all’ (damerow, dunkley, nebres & werry, 1984) entered the discussion of mathematics educators in a time of consolidation and expansion of educational systems around the world. secondly, in mathematics education research there has been a growing adoption of sociological and philosophical theoretical frameworks in the study of persistent student failure in mathematics (lerman, 2000). the sociocultural-political trend in mathematics education research has made it possible to open up the object of research of the field from narrow problems of teaching and learning to understanding them as social practices (valero, 2010). thirdly, there was a global trend of making education the pillar of democratisation as expressed by the united nations educational, scientific and cultural organisation (unesco) in, for example, the global agenda of ‘education for all’ (unesco, 1992). contrary to previous curricular trends in different countries, this particular international agreement was an attempt for the universalisation of basic education for all children in the world, within a discourse of democratisation and expansion of access to education. it is in the intermesh of these different trends that mathematics education started to be related with power in society, and with the generation and maintenance of mechanisms of in/exclusion. linking mathematics education to democracy is so recent a formulation that it still may cause many people to wonder. yet, such an idea has rapidly been adopted in certain research and even policy circles as a new, powerful justification for the needed improvement of educational practices in mathematics classrooms (e.g. gutierrez, 2010).the result is the construction of a discourse that posits mathematics education, in the eyes of mathematics educators, as the school subject that can save excluded children from their lack of a future (lundin, 2012). from policy to research documents, assertions such as ‘mathematical (and scientific) competence is the key to the welfare of our nation in a global economy’ or ‘children who are equipped with better mathematics will have a better future’ contribute to selling the myth that mathematics learning can be a way of saving the world, the nation, and the individual. concomitantly, mathematics education research is positioned to be the scientific discipline with the knowledge, evidence and techniques for achieving such a noble goal (popkewitz, 2004). we want to start our rethinking of mathematics education and democracy with the assumption that mathematics education research and practices, if implicated, are neither the cause of nor the solution to the harsh stratification and reduction of millions of people in the world to misery and violent conditions of life. pais and valero (2011, 2012) have argued that a political − with capital p − reading of mathematics education treats the different practices that form part of it as economic, social, cultural and historical forms of reasoning and acting. we are compelled to modestly recognise that the redemptive narrative of mathematics education research is no more than an exemplar of a discursive practice that brings forward the privileged function that mathematics as a school subject has performed in the construction of modern, capitalist societies and subjectivities. failing to do so would allow researchers to keep on disavowing the intricate network of historical, social, political and economic relationships in which mathematics teaching and learning are constantly formed. democracy as the striving for a chance of a dignified life is therefore a task to which mathematics educators can contribute a grain of sand only. nevertheless, even such a minute contribution is worth the effort of thinking it through. it provides an entry into a way of thinking about how mathematics education is one of the areas of the school curriculum that, as we will argue, contributes strongly to the governing of children’s conduct. in particular, in the case of what a striving for ‘democracy’ may mean in colombia, we have been confronted with the conflict that emerges between the universalising discourse of ‘mathematics for all’ as configured in the 1980s and now circulating internationally, and the particular conditions for the creation of a social subjectivity in colombia. in our research (garcía et al., 2009) we concluded that any serious attempt to the develop curricular ideas in mathematics for students positioned as excluded had to be woven around their significance for the construction of a social subjectivity. what does this mean? in contrast to other views of curriculum organised around central mathematical ideas or competencies, we proposed the displacement of the core, traditional mathematical concepts as the centre of the curriculum. such a decentring opens the space for subjectivity to become the articulating axis around which mathematical forms of reasoning and acting could be organised. in making subjectivity an articulating point for the curriculum, it is possible to open up to other forms of being than the ones that are historically and culturally embedded in the traditional mathematics curriculum. this displacement emphasises the idea that knowing is not separated from being, an idea that has been recently expressed by socio-cultural theories of mathematics learning (e.g. radford, 2008), but also by foucaultian studies of science and education: forms of knowledge are effected and effect power as they bring together knowing and being as two sides of a coin. forms of knowledge do not only bear the rules of how one knows and what it is to be known, but also impose ways of being on the knowers. knowing fabricates particular subjectivities through its technologies (e.g. daston & galison, 2007; popkewitz, 2009). if knowing and being are inseparable, the question emerges of what the forms of knowing and being are that the mathematics curriculum effects in children, and whether those forms of subjectivities are desirable. this becomes a central question of mathematics education seen as a technology of the self (foucault, 1982), confronted with a desire for ‘democracy’. zemelman (1997) argues that in the context of latin american societies and their history, the concept of social subjectivity is an epistemological category that not only points to the necessity of thinking about human beings as necessarily collective beings, but also to the imperative of rethinking social processes on the grounds of the recognition of the multifaceted, complex latin american reality. in contrast to european notions of the subject as a monadic individual, zemelman’s notion of social subjectivity emphasises the social constitution of concrete historical subjects who articulate time and space for the construction of new, possible projects of a collective future. against a historical context where colonisation and a colonised education has taught latin americans ‘how to be a subject that is constantly thinking in being what one is not’ (rivas, 2005, p. 117, our translation), the construction of a social subjectivity, in particular relation to education, is an attempt to make subjects aware of their historical position, for them to know and think the world with others, with the intention of possibly generating new common visions of future conditions of living. the particular colonial history of latin america, as varied as it is in the different countries of the continent − as well as distinct from the colonial history of africa and many asian nations − not only occupied the minds of the colonised indigenous population. it created historical forms of subjectivities with two main characteristics. the civilising rule of the colonial powers installed successful technologies of the self that generated ideas of the white european as the norm to strive for − thereby abjecting all who are not like the coloniser. learning to desire assimilation or to be ‘what one is not’ became a central characteristic of the colonial subjects (guillén, 1996; quintar in rivas, 2005). the other characteristic is that all forms of subjectivity that do not assimilate to the norm and that dare to challenge it need to be silenced and exterminated (guerra, 1997). guerra argues that the notion of democracy and citizenship that emerged in latin america during the consolidation of the nation states at the beginning of the 20th century is rooted in an idea of sameness and correspondence. it opposes the ideas of difference, diversity and heterogeneity. democracy as a form of governmentality − the combination of particular techniques of government and their rationality working at both the level of the individuals and their selves and also at the level of the population (foucault, 1988; lemke, 2002) − to reach sameness was clearly a pillar of the flourishing totalitarian regimes in the 1970s. these regimes positioned themselves as ‘democratic’ because they strove to guarantee a high degree of civil rights for a unified nation. in this way the regimes succeeded effectively in restricting the expression of social subjectivities. in colombia, díaz (2010) claims that the notion of sameness as a central characteristic in the formation of a nation state with a democratic political regime got inserted in the political constitution of 1886 through the declaration of colombia as a nation unified by the catholic, apostolic and roman religion. furthermore, such a declaration posited the organisation and direction of public education in the hands of the catholic church and its various representative orders, in particular the society of jesus. such a configuration guaranteed a laic education for all colombians. it simultaneously effected the abjection of people with different political, cultural and sexual expressions and orientations. if these have been the power effects of colonisation, then the challenge of an education that allows reconstructing a social subjectivity is to build a new interpretation of democracy. rather than keeping on chasing the european and north american ghosts of freedom, equality, and fraternity, democracy can be thought of as striving for respect for difference and multiple possible senses of future. democracy is about reclaiming the right of being in dignity what one socially is and could potentially be on the grounds of one’s reality. from mathematics to social subjectivity with mathematics top ↑ the expansion of school mathematics education and its particular unfolding in countries such as colombia cannot be separated from the history of spanish colonisation (from the 16th to beginning of the 19th century) and the formation of an independent national state (from the 19th to mid–20th century) (meyer, ramirez & soysal, 1992). as part of the colonial power, the teaching of mathematics in the territory now identified as colombia was made possible by the royal botanical expedition. the expedition was a large ‘scientific’ enterprise in the americas that, side by side with the economic exploitation of the new continent, intended to document for the european naturalists − headed by the swede karl linnaeus − the botanical wonders of the new world. josé celestino mutis, spanish medical doctor, mathematician and jesuit priest, came to the americas as the personal physician of the viceroy and was commanded to lead the royal botanical expedition. he is recognised as having started the first course in mathematics at the ‘colegio mayor de nuestra señora del rosario’ in 1761 (sánchez & albis, 2012). for a naturalist like josé celestino mutis, mathematics was important as a reasoning method and a practical tool for all people: ‘peasants, citizens, plebeian, courtiers, soldiers, artificers. wise, secular, ecclesiastic, all in a word no matter condition and status should devote themselves to such a useful study’ (josé celestino mutis cited in sánchez & albis, 2012, p. 110). the tools of mathematics were essential in the effort to create generalised, ideal typologies of natural species that reflected the epistemological virtue of ‘scientists’ at that time (daston & galison, 2007). the entry of mathematics as part of the commitment of the spanish catholic colonisers to evangelise the indigenous populations − and the economic exploitation of natural resources − went hand in hand with the insertion of the colonised in the european, classical episteme (foucault, 1971) within which scientific rationalities and discourse were configuring in the second half of the 18th century.in a historiography of mathematics teaching in colombia, sánchez and albis (2012) show how, from its very beginning, mathematics teaching mainly at universities was strongly associated with how the prominent figures of colombian society − a creole elite descending from spanish ancestors − strove to bring the country in pace with the thinking of the european and later north american powers. from the middle of the 19th century to the beginning of the 20th century, the vision of the construction of a nation state that had to promote economic development through the advancement of engineering for the taming of colombia’s tropical nature was associated with the placing and growth of mathematics in the military school and, later on, in the new national university of colombia. in that context, the discussions on the role of mathematics for the development of the country can be interpreted as that element through which rational and scientific thinking was to be brought to the population. furthermore, the expansion of mathematics from universities to schools also represents the move towards the insertion of as many people as possible into the modern episteme. the teaching and learning of mathematics as a school subject for the masses is a recent invention that is related to the universalisation of education at the beginning of the 20th century (radford, 2004). popkewitz (2008) has studied the effects of education and educational sciences into the constitution of the cosmopolitan citizen of modernity in the usa during the 20th century. cosmopolitanism refers to the ‘enlightenment’s hope of the world citizen whose commitments transcended provincial and local concerns with ideal values about humanity’ (p. 1). the school mathematics curriculum is a powerful technology of the self. the technologies of the self are the techniques that human beings have historically developed in practice to understand themselves as human. these technologies: permit individuals to effect by their own means, or with the help of others, a certain number of operations on their own bodies and souls, thoughts, conduct, and way of being, so as to transform themselves in order to attain a certain state of happiness, purity, wisdom, perfection, or immortality. (foucault, 1997, p. 225) thinking with foucault, whilst the school mathematics curriculum seems to teach children a valued and useful knowledge, it primarily teaches all of the people involved particular ways of being a subject. it embodies and makes available the cosmopolitan forms of reason, which build on the belief of science-based human reason having a universal, emancipatory capacity for changing the world and people. human agency, the hope for progress, science as a means to direct and achieve progress, and planning of time result in a thesis about who is the human subject: the being with a ‘homeless mind’ (popkewitz, 2008, p. 29). the homeless mind is a type of ‘individuality that is both an object and a subject of reflection’ and that places ‘individuals in a relation to transcendental categories that seem to have no particular historical location or author to establish a home’ (popkewitz, 2008, p. 30). this thesis is made possible, amongst others, when quantification operates the displacement from qualities of knowing to quantities that can be operated and modelled as facts, as well as when science − both the natural and the human sciences at that time under configuration in their current divisions − makes of the world of things and humans an object of reflection and planning. the mathematics curriculum as a school subject at that time − and still today − became one of those areas of schooling that most effectively could ‘enlighten’ all population into this type of being. from the turn of the 19th century to current times the mathematics curriculum is an important technology of the self that inserts subjects into the forms of thinking and acting needed for people to become the ideal cosmopolitan citizen. if we look at the particular history of colombia, the ideal of the cosmopolitan subject of europe and the usa travel to and become reinscribed in the particularities of colombian history. here it is important to point out that in the history of the usa, the reformist education agenda was attached to the lutheran narrative of redemption for the growing urban population. on the other hand colombian education was entrusted to the society of jesus in its mission of catholicising and evangelising the native indigenous population (ahern, 1991). the political alliance between colonial power and the catholic church was a doubly effective strategy for not only subordinating the colonised to a new rule, but above all, for europeanising them through making them loyal to god and the spanish crown (herrán, 1998). in the second place, the process of consolidation of a nation state in the 1960s was closely connected to the advance of the agendas of international cooperation for development and modernisation from international agencies such as the world bank. in her study of the mathematics curriculum in colombia, garcía (2003) argues that in the 1960s the national government responded to the crucial challenges of school drop-outs caused by the enlargement of coverage given the changes in the distribution of the population from rural to new urban masses. the response to both expand and strengthen education followed development policies of international agencies that pushed the adoption and implementation of technocratic planning strategies in education. good education for development meant not only the political and administrative steering of education for the needs of economic and social development in the country, but also the introduction of educational technology to make the processes of education more effective, flexible and constantly educational. a few years later, this formulation became clearer when educational effectiveness − qualitatively and quantitatively − was also connected to the optimisation of investment in education. in this way the whole logic of education for development as part of the steering of the state was accompanied by concrete educational technologies governing children’s conduct. these two trends were epitomised some years later in the formulation and implementation of the curricular renovation in the 1980s. the first unified school mathematics curriculum was formed under the leadership of carlos eduardo vasco. being a philosopher and mathematician, having pursued graduate studies in the usa and germany, and belonging to the society of jesus, carlos vasco was consultant for the colombian ministry of education from 1978 to 1993. he was in charge of the first systematic attempt to spread psychology-based ideas about mathematics education amongst teachers (molano, 2011). the new technology of school mathematics was called the ‘system approach’, and it was meant to be differentiated from other attempts to include a modern mathematics approach to school mathematics in colombia. the system approach defined a system (s) as a set of objects and their relations and operations. every mathematical system may be defined in terms of a subset of objects (a), a subset of operations (o), and a subset of binary relations between the objects of a, (r). in general terms, a system may be defined as s = (a, o, r). the curriculum for school mathematics proposed to work with eight kinds of systems: number systems, geometric systems, metric systems, data systems, set systems, operations and relations systems, and analytical systems (men, 1991, pp. 9−17). as a complement of this ontology of school mathematics, its epistemology was founded on jean piaget’s ontogenetic epistemology as a theory of child development that could be operationalised to think about children’s cognitive development in mathematics. the curricular reform of the 1980s in colombia was possible in the confluence of different theses about the hope for a new, mathematically competent, modernised colombian child who could become the cosmopolitan citizen needed for the progress of the country. mathematics education technology entered the administration of schoolchildren with the double authority of mathematics, cognitive psychology and educational research, under the leadership of a jesuit priest. since the 1980s the official colombian school mathematics curriculum has incorporated the notions that travel around the international discussions on mathematics education research, as well as the global trends of national education. the epistemologies available for the curriculum remain rooted in constructivist theories of learning derived from piagetian ontogenetic epistemology. new curricular reforms have brought the language of ‘outcomes-based education’ in the 1990s, associated to the era of unesco’s plan education for all (valero, 2007). more recently, the language of competences and standards inserted in the logic of the organisation for economic co-operation and development (1989, 2001) have found their way into curricular documents. as new discourses are reconfigured and some elements apparently displaced, there is continuity in the fundamental assumption that mathematics education is about the fabrication of the rational, effective, global cosmopolitan child of the 21st century. so far we have argued that, from its very introduction in colombia, mathematics education has contributed to fabricating particular historical subjects as part of its insertion in the colonial and national project of assimilating the culture, the economy, the political regime and also citizens into the developed, scientific western, modern world. at this stage many people would think: but what is the problem? that is an aim to reach and it is only desirable that mathematics education fulfils such social role. as león and zemelman (1997) point out, the problem is that the faithful allegiance of latin american elites to the order historically established by the white european rationality has reduced any other form of being to that of an undignified existence, prevented from doing and enacting its own history. the system of reason of modernity and its current manifestations render it almost impossible to construct a social subjectivity based on the dignity of being latin american, or being colombian. when played in the terrain of school mathematics practices, the educational technologies that insert pupils in the grid of the modern, cosmopolitan and now global subject, effect the abjection of all those who do not conform to the norm by singling out who and how those who are different need to be brought into redemption. popkewitz (2008) argues that any cultural thesis about the subjects of schooling produce abjections. abjection is the way that exclusion is generated as the effect of defining the norm for inclusion and its hope for those who are not part of that norm. when the curricular guidelines in mathematics declare the hope for the future rational, cosmopolitan citizens, they are at the same time declaring who is not seen as being part of those who comply with the norm. the mathematics curriculum, as a technology of the self, brings about compliance with the norm in children’s minds, bodies and conduct, and thus operates inclusions and exclusions. from mathematical spatiality to social space and intimate space top ↑ in the case of the schools, children and teachers that we have been working with, the existence of a deficit discourse on the students who live in the misery belt of bogotá is an expression of the effective use of the tools of mathematics educational technologies not only to teach children mathematics, but also to create a clear position of exclusion for them. teachers collectively constructed the class 703 − grade 7, group 3 − as those who ‘have low values’, ‘have little interest in their learning, especially in the learning of mathematics’ and: … do not have a defined centre of attention and their dispersion generates a complicated dynamic in the classroom. they run over those students who have the desire of getting involved in the activities proposed. (garcía et al., 2009, p. 18) such discourse embeds in itself the thesis that the ideal school child is one with ‘high values’, who is interested in learning mathematics, who respects other children, and who gets involved in the proposals of the teachers. in other words, the normal, desired child is a child who has learned to be and behave according to the norms of some kind of middle class, ‘culturised’ people. in this way the children in class 703, their families, and their experiences are positioned as deviant and in need of remediation and salvation. in this case, most teachers found the children so ‘deviant’ that they had given up on them. it was far more than evident that the exclusion of these students was already operated in the constant contrast between teachers’ expectations of the children being rational, cognitive children against the students’ engagement in the world. the issue at stake then became how we could possibly engage as teachers and as researchers in going beyond our understanding of that situation, and how we could engage students recovering one thing: their dignity of being the social, historical human beings that they were, with a possibility of imagining a future. in what follows we do not want to play the saviour of the children of class 703. in fact, one year after our work at the school took place, some of the teachers involved in our team left the school for other jobs, the school principal was replaced, and very probably, the whole situation returned to what it was. elsewhere (e.g. camelo et al., 2010; garcía et al., 2009) we have reported the design of teaching sequences that built on skovsmose’s (2001) landscapes of investigation as an important tool for realising into practice some of the concerns of critical mathematics education. in this article, we depart from that literature to reinterpret the design and activities from the point of view of the conflict between the subjectivities that the mathematics curriculum offers, and the possibility of a mathematics education for a social subjectivity. as mentioned previously, class 703 was the ‘problem class’ of the school: the headache of all teachers. it was a group of 39 grade 7s, whose age varied between 11 and 15 years. the school was located in an area at the outskirts of bogotá, where some years ago there were fields which developed into shantytowns of people displaced to the big city due to the many sources of violence in the colombian countryside. for conducting this participatory research all the ethical considerations related to the involvement of teachers and students were followed. teachers and students participated voluntarily. the teachers involved were also part of the collaborative research team. children and their parents were informed and asked for permission to participate. in the following account we keep the names of the teachers, since they are the same researchers, but we keep anonymous the names of the children. when we started our work at the school in 2008, francisco camelo was a mathematics teacher there. together with other teachers in mathematics, science, biology and physical education, francisco has been challenging the statement that ‘there was nothing to do with these kiddos’. the concept of students’ foregrounds (alrø, skovsmose & valero, 2008; skovsmose, 2005) allowed us at that time to move away from a deficit explanation of children’s ‘disengagement’ with school mathematics and education on the grounds of the lack in their background. instead, the possibility of thinking about the relationship between students’ engagement in their (mathematics) education and their interpreted possibilities of future was an alternative. we faced the challenge of conceptualising and performing mathematics teaching or learning units that built on the students’ foregrounds and that introduced them into a landscape of investigation. how to do that? where to start? ‘listen to the students, ask them about their lives and don’t imagine what may be interesting for them’, paola valero reminded all in the research team. ‘the concept of space is important in mathematics. why don’t we start there?’ gabriel mancera mooted an idea. the idea resonated with gloria garcía. she was part of a research group with critical geographers and was participating in a discussion on representations of time of space and the construction of territorial identities. gonzalo peñaloza thought this idea could be connected with his experience of working with social cartography with teachers in bogotá, as a way for teachers and students to inquire about the school community, the children and the problems that could generate interdisciplinary learning and social action (peñaloza et al., 2006). it seemed that the idea of space could be fruitful as a basis for creating a scenario for mathematics learning. reaching this choice was not a straightforward decision. there were many discussions, readings and interpretations amongst the research team that made possible the configuration of that idea. according to the colombian curricular guidelines (national ministry of education of colombia [men], 1991), the mathematics curriculum in secondary schools should promote notions of space in euclidean geometry and to a lesser extent in projective geometry. the curricular contents tend to be reduced to the establishment of geometric figures and their properties. in euclidean geometry, space is constructed on the grounds of the reflection on the properties of geometrical shapes, made evident through the use of ruler and compasses. combined with the cartesian coordinate system, it allows one to think about space as a system of positions that can be described in a precise and uniform way (gálvez, 1985). projective geometry invites an active exploration of tridimensional space in an external or imagined reality, and through the representation of solid objects in space. the guidelines describe the cognitive processes that children need to attain as a result of the teaching of central notions of euclidean and projective geometry in defining space: it is expected that students move from an intuitive or sensorimotor understanding of space (related to the practical capacity to act in space, to manipulate objects, and to locate them in an environment) towards a conceptual and abstract space related to the capacity of representing internally the space. (men, 1991, p. 56) the concept of space to be reconstructed in the students’ understanding is that of a rational, referential space with fixed points in two or three dimensions. it is assumed that the conceptual development of the child will lead to an internal and abstract representation which will contribute to making a decontextualised child, freed from the practical capacities of acting with objects in space, particularly of those spaces where everyday life occurs. it is evident that the curricular view of space and spatiality was in sharp contrast with the experience of many of the students in the class 703. their personal history in the neighbourhood was that of a social space in a geographical location in transformation by the practices of living and surviving, filled with the bad smells of one of the largest city landfills in the vicinity. feelings of detachment and attachment to new geographical locations due to the forced or voluntary displacement from the hometowns or homelands to the big city due to political violence were part of children’s lives. the intimate spaces where some children had learnt to be were far more filled with dirt, bodies, practices, conflicts and feelings than the clean spaces invoked by the euclidean space promoted by school mathematics. in our work the issue of how to articulate a teaching or learning experience in mathematics classrooms that could allow students to go beyond the modern subjectivity towards a social subjectivity took form in a new conceptual grid, enabling us to think of space as incorporating the children’s social experience of space. skosvmose’s suggestion of building semantic fields for organising open, landscapes of investigation (skovsmose, 1994) was a starting point that allowed us to bring together two additional perspectives on space. the euclidean and projective geometrical space was reconfigured with the notion of territorialisation of critical geography, and the notion of intimate space in latin american social epistemology. seen retrospectively, the reconfiguration of the notion of space with these two other semantic spaces on ‘space’ can be reformulated as an attempt on our side to decentre the notion of euclidean space as the core, fundamental way of thinking about space that is part of a traditional, mathematics-centred curriculum. we concur with deleuze and guattari (1987) on the strategy of decentring the mathematical core notions of the curriculum by displacing their meaning into related though non-mathematical semantic fields. the strategy of decentring is a conscious attempt from our side to articulate the curriculum around the construction of subjectivity. in the displacement of a unified and unique notion of space, students were invited to fill with their bodies, experiences and practices the clean, empty void of school euclidean space, thus allowing the possibility of being and knowing who they are, and of imagining their future. contemporary social theory has reclaimed the thinking about space from the regime of mathematisation to the field of political, economic and historical thinking (e.g. lefèbvre, 1991). such displacement has affected traditional geographical discourses, making possible the emergence of critical geography and the concern for the inseparability of physical and geographical space from the practices and processes of social and cultural identity formation and subjectification (e.g. crang & thrift, 2000). in latin america, critical geography contributes a geopolitical analysis of the relationship between space and power in processes of the organisation of territories locally, regionally and globally. it also addresses the issue of territorial appropriation and representation of different peoples and communities, and the claims to the right to the territory through performing critical, social cartographies based on participatory information systems (delgado, 2006). the notion of intimate space (tapia, 1997), rooted in contemporary latin american philosophy, allowed us to connect the relevance of thinking about space, spatiality and the construction of a social subjectivity. following the recovery of space as an important category for thinking about society and practice, tapia asserts that a social space − as a possibility of constituting a material and cultural world − unfolds from the relationship between the self and the others when different positions are shaped and delimited. thus, analysing the social space is a matter of ‘ordering correlative positions, that is, ordering coexistences’ (p. 159). becoming subject means a double move of recognition of otherness and recognition of the self. the possibility of a social subjectivity therefore also requires an intimate space of action but in full awareness of the other. this is different from, say, an individual space which could tend to be closed in itself. the intimate space is close to the subject but in coexistence with the other. the thinking about space in the connections between these three related semantic fields −geometrical space, territorialisation, and intimate space − led us to propose to students a quite different learning landscape than the one made available by the traditional mathematics curriculum. we established three entry points into the learning landscape: who am i? who are we as members of class 703 in our school? who are we as inhabitants of this locality in the capital city? each one of these entry points, besides being thought of as a field for learning, was also thought of as a ‘nodal point’ of social subjectivity which, according to zemelman (1997, p. 30), allows connecting intimate spaces of being to collective spaces of action for the search for a different viable future. in each nodal point a series of activities combined different mathematical notions related to space as well as many other topics of the curriculum. however, the mathematical activity was always carried out allowing connections to the students’ experiences in the family, in the school and in the locality. for example, in the first nodal point ‘who am i?’ students were asked to write a story about themselves in their family. jeimy, one of the students, wrote: before there were huts made of hay with no public services before there were no paved streets and before there were less people. now there are public services the houses have one or two floors there are more people 128 parcels before there were fewer the neighbourhood has public lighting we still miss the pavement in the streets and to have a park were we children can have fun and improving the community room to have a community dining room and some other necessities that we all our hope for. (jeimy’s letter in the activity ‘who am i’, our translation). the personal and family histories were connected to the growth of the locality and the territorialisation of that space in the communities where students live. google earth was used as a tool for locating important sites of practice. the maps were used for tracing the paths and movements that students usually perform in their locality. the maps also brought together the practices of the communities around the school with the practices of the children and their families. many students could locate their houses and make sense of the relationship between them, their families, the school, the economic activities in the locality, and even sites and practices that threatened them − such as drug distribution, the activities around the huge sanitary filling in the locality, et cetera. in each nodal point, moving between mathematisations, socially constituted geographical spaces and the closeness with the experience of children was a strategy for decentring the sense of space as a clean, abstract mathematical object. social subjectivity for the dignity of being with mathematics top ↑ there could be many ways of thinking about the relationship between mathematic education and democracy. we have argued that any attempt at theorising about such an idea and attempting to realise it in educational practice needs to consider the effects of power of the school mathematics curriculum in promoting the rational, objective, homeless, cosmopolitan subject of modernity. the thesis of the mathematically competent child, however, is inscribed in particular times and spaces. thus, there is no universal analysis of how the school mathematics curriculum is fabricating subjects. we need to pay attention to the study of the cultural histories of the constitution of what counts for democracy and in which conditions of possibility mathematical subjectivities are inscribed in national histories. if retaking the issue of mathematics education and democracy intends to be a move beyond redemptive discourses of empowerment with and through mathematics that effect a clear abjection of all those children whose forms of life and experience do not align with those of the cosmopolitan child, then we need to consider seriously how we wish to understand democracy and mathematics education at each historical point and society.in colombia it was evident to us that the forms of knowing and being made available by the national curricular guidelines in mathematics were implicated in the systematic exclusion of children who did not fit the norm established by the thesis of the child as an abstract, piagetian cognitive agent on which the curriculum was intended to operate. we decentred the key concepts of the mathematics curriculum such as that of space in an attempt to facilitate the emergence of social subjectivities. the decentring of the school mathematics curriculum may open the possibility for an educational project in mathematics that allows for different subjectivities. such a possibility is precisely an alternative for a democracy that reclaims the dignity of being. acknowledgements top ↑ this article is part of the study of learning environments in mathematics and exclusion processes in the classroom project funded by the colombian institute for the advancement of science and technology (colciencias). the project is supported by the universidad pedagógica nacional de colombia, universidad distrital de bogotá, and aalborg university in denmark.the article is the result of many years of collaborative research, together with other teachers and teacher educators such as gonzalo peñaloza, sandra samacá, claudia salazar and maria rosa gonzález. we also thank the undergraduate and graduate students who have participated with us in the work in the schools that make part of the project. we thank thomas popkewitz and the wednesday group at the university of wisconsin, usa, gelsa knijnik from unisinos, brazil, and the members of the smerg group at aalborg university, denmark for their comments on previous drafts of this article. competing interests the authors declare that they have no financial or personal relationship(s) which may have inappropriately influenced them in writing this article. authors’ contributions g.g. 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(1997). sujetos y subjetividad en la construcción metodológica. in e. león, & h. zemelman (eds.), subjetividad: umbrales del pensamiento social (pp. 21−35). barcelona: anthropos. article information authors: mdutshekelwa ndlovu1,2 andile mji2 affiliations: 1institute for mathematics and science teaching (imstus), department of curriculum studies, university of stellenbosch, south africa 2faculty of humanities, tshwane university of technology, south africa correspondence to: mdutshekelwa ndlovu postal address: faculty of education (imstus), university of stellenbosch, private bag x1, matieland 7602, south africa dates: received: 28 june 2012 accepted: 24 oct. 2012 published: 10 dec. 2012 how to cite this article: ndlovu, m., & mji, a. (2012). alignment between south african mathematics assessment standards and the timss assessment frameworks. pythagoras, 33(3), art. #182, 9 pages. http://dx.doi.org/10.4102/ pythagoras.v33i3.182 copyright notice: © 2012. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. alignment between south african mathematics assessment standards and the timss assessment frameworks in this original research... open access • absract • introduction • theoretical framework of alignment studies    • definition of alignment and scope of alignment studies    • porter’s model for evaluating alignment • purpose of the study • methodology    • the document analysis methodology    • computation of the porter index    • critical values for the strength of the alignment index • results and discussion    • the structure of the content and cognitive domain matrices for the components of the 2003 grade 8 timss mathematics assessment frameworks    • the structure of the content and cognitive domain matrix for the grade 8 rncs for mathematics    • computed porter indices of alignment for this study    • the structure of discrepancies between timss 2003 framework components and rncs assessment standards    • a comparison of the rncs-timss test discrepancies with south african students’ performance in timss 2003 • conclusions • acknowledgements    • authors’ contributions    • competing interests • references absract top ↑ south africa’s performance in international benchmark tests is a major cause for concern amongst educators and policymakers, raising questions about the effectiveness of the curriculum reform efforts of the democratic era. the purpose of the study reported in this article was to investigate the degree of alignment between the timss 2003 grade 8 mathematics assessment frameworks and the revised national curriculum statements (rncs) assessment standards for grade 8 mathematics, later revised to become the curriculum and assessment policy statements (caps). such an investigation could help to partly shed light on why south african learners do not perform well and point out discrepancies that need to be attended to. the methodology of document analysis was adopted for the study, with the rncs and the timss 2003 grade 8 mathematics frameworks forming the principal documents. porter’s moderately complex index of alignment was adopted for its simplicity. the computed index of 0.751 for the alignment between the rncs assessment standards and the timss assessment objectives was found to be significantly statistically low, at the alpha level of 0.05, according to fulmer’s critical values for 20 cells and 90 or 120 standard points. the study suggests that inadequate attention has been paid to the alignment of the south african mathematics curriculum to the successive timss assessment frameworks in terms of the cognitive level descriptions. the study recommends that participation in timss should rigorously and critically inform ongoing curriculum reform efforts. introduction top ↑ hencke, rutkowski, neuschmidt and gonzalez (2009) make the important remark that the trends in international mathematics and science study (timss) examines the effectiveness of curriculum and instruction in relation to student achievement. there is increasing global interest in and attention paid to the resultant rankings of participating countries, making the very participation in timss a high-stake local decision. as a consequence of the heightened (political and educational) stakes, the relevance of the tests to local curricula has come under sharp scrutiny, which makes the issue of alignment of the south african (sa) curriculum with timss important for educators, curriculum workers, test developers and policymakers. hencke et al. (2009) concede upfront that whilst timss assessments were developed to represent an agreed-upon framework with as much in common across countries as possible, it was inevitable that the match between test and curriculum would not be identical in all countries. however, the more aligned a national curriculum is to what is common across countries the greater the chance of that country’s students performing well. in other words, rather than reject the common core assessments as irrelevant it might be beneficial to investigate in depth what discrepancies exist between sa’s curricula and timss, with special focus on the overlapping content. mullis, martin, ruddock, o’sullivan and preuschoff (2009) refer to the timss curriculum model as consisting of an intended curriculum, an implemented curriculum and an attained curriculum, all of which are familiar terms in curriculum theory. for instance, porter (2004, p. 1) suggests that a curriculum can be divided into four aspects: the intended, enacted, assessed and learned curriculum. the enacted curriculum refers to instructional events in the classroom whereas the assessed curriculum refers to student achievement tests. mullis et al.’s (2009) attained curriculum refers to student achievement in those tests. for cross-national tests such as timss to be valid, it is critical that their assessed curricula correspond with the intended national curricula. moreover, assessments aligned with the assessment standards can guide instruction and raise achievement (martone & sireci, 2009; polikoff, porter & smithson, 2011). in view of the foregoing it is expected that, in order to be relevant, cross-national studies or tests should provide curriculum information that can help countries to improve the quality of their education systems on the basis of benchmarking performance (reddy, 2006). this makes curriculum matching analysis a logical starting point. in bemoaning the absence of extensive use of alignment research in the classroom, martone and sireci (2009) point out lost opportunities to help policymakers, assessment developers and educators to make refinements so curriculum, assessment and instruction support each other in achieving what is expected of students. in an attempt to bridge this gap the aim of the study was to analyse the alignment between sa’s grade 8 mathematics curriculum and timss by means of the porter (2002) procedure. to achieve this goal the remainder of this article gives the theoretical background to alignment studies in general and shows why the porter index was chosen. thereafter we spell out the research questions guiding the study and outline the procedure for determining the index before presenting and discussing the results. the article concludes with summary observations and recommendations. theoretical framework of alignment studies top ↑ definition of alignment and scope of alignment studies for purposes of comparing the grade 8 revised national curriculum statements (rncs) for mathematics and the timss assessment we analyse measures of curricula and assessment alignment based on research that has developed methods for judging the extent and nature of alignment (e.g. porter, 2002; porter & smithson, 2001; webb, 2005). alignment can be defined as the degree of agreement, match or measure of consistency between curriculum content (content standards) for a specific subject area and the assessment(s) used to measure student achievement of these standards (bhola, impara & buckendahl, 2003; näsström, 2008; näsström & henricksson, 2008). a major feature of alignment studies is the development of common languages of topics and categories of cognitive demand for describing content in different subject areas such mathematics, reading and science (berends, stein & smithson, 2009, p. 4). the underlying logic is that if standards specify what and how well students should be learning and tests measure what they know and can do, then the two ought to be synchronised (herman & webb, 2007, p. 1). in other words, the language of the assessment items must match the language of the outcomes stated in the rncs or its successor, curriculum and assessment policy statements (caps) (department of basic education, 2011a). similarly, the content and cognitive domain language of the caps should match that of the timss assessment frameworks as closely as possible. alignment, thus, has both content and consequential validity in terms of the knowledge and skills prescribed and tested (bhola et al., 2003, p. 21) although the alignment between standards and assessment has been most commonly studied (e.g. bhola et al., 2003; herman & webb, 2007), the alignments between standards and instruction as well as between instruction and assessment have also been studied (e.g. porter, 2002). in curriculum theory and practice, standards have lately come to refer to ‘descriptions of what students are expected to know and be able to do’ (näsström, 2008, p. 16), which makes them synonymous with the intended relationship between educational objectives and subject matter content. in sa, the term ‘outcomes’ has been used widely to frame statements about both subject matter content and anticipated learning behaviours. porter’s model for evaluating alignment from the three commonly used primary models of evaluating alignment, that is webb’s (1997, 2005) depth of knowledge procedure, rothman, slattery, vranek, and resnick’s (2002) achieve procedure model and porter’s (2002) surveys of enacted curriculum index, we opted for the last one. unlike the other two approaches, the surveys of enacted curriculum index does not rely on direct comparison of assessments or assessment items with objectives or standards. instead, content analysts first code the standards and assessments onto a common framework, a content taxonomy, developed by subject matter experts. the taxonomy defines content in terms of two variables: topics or sub-topics and levels of cognitive demand. the two variables compare favourably with webb’s categorical concurrence and performance centrality. analysts place assessment items and objectives from standards documents into the taxonomy and the documents are then represented as matrices of proportions, where the proportion in each cell (topic and cognitive demand) indicates the proportion of total content in the document that emphasises that particular combination of topic and cognitive demand. the matrices for standards and assessments are then compared, cell by cell, and an alignment index is calculated. we believe that the porter procedure achieves in two dimensions what the webb and achieve procedures do in four measures. more importantly, the porter alignment model ‘can be applied to analyse the match between any two of curriculum, instruction and assessment’ (liu et al., 2009, p. 795). it was therefore appropriate for our purpose since we wanted to compare two curriculum documents: the timss frameworks and rncs.the calculated porter alignment index ranges from 0 to 1 with 0.5 as its centre since it uses absolute differences, a characteristic that has to be taken into account when interpreting the computed values. fulmer (2011) has recently provided critical values for the strength of the porter index of alignment based on the number of cells and number of standard points used. furthermore, the porter procedure agrees with bloom’s taxonomy of educational objectives which has also been used by timss and the rncs. purpose of the study top ↑ the purpose of this study was to determine the degree of alignment between the grade 8 rncs for mathematics and the 2003 timss assessment frameworks by means of the porter index. timss regularly assesses learners at the grade 4 and 8 levels and sa has previously participated (in 1995, 1999, 2003 and 2011). the grade 12 level has not been consistently assessed. we chose the grade 8 curriculum because it is a transitional grade between primary and secondary phases. the 2003 results were the latest available of south africa’s participation in timss because the 2011 results were still pending at the time of this article. the following research questions guided the study:• what is the structure of the content and cognitive domain matrices for the components of the 2003 timss grade 8 mathematics assessment frameworks? • what is the structure of the content and cognitive domain matrix for the rncs grade 8 mathematics assessment standards? • what are the computed porter indices of alignment within and between the components of the 2003 timss assessment frameworks and the rncs assessment standards for grade 8 mathematics? • what is the structure of discrepancies in emphasis between the rncs assessment standards and the 2003 timss assessment objectives? • how do the discrepancies assessment objectives compare with sa’s performance in timss 2003? to help answer these questions we adopted the document analysis methodology in this study. methodology top ↑ the document analysis methodology the methodology of document analysis was adopted for this study as it entails systematic and critical examination rather than mere description of instructional or curriculum documents (center for teaching and learning, 2007). document analysis is also referred to as qualitative content analysis (daymon & holloway, 2011), an analytical method used in qualitative research to gain an understanding of trends and patterns that emerge from data. the aim of qualitative document analysis is to discover new or emergent patterns, including overlooked categories (daymon & holloway, 2011, p. 321). statistical reports within a qualitative study should reveal ways in which the data and statistics have been organised and presented to convey the key messages and meanings intended. the qualitative document analysis in this study is organised and presented statistically by means of the porter alignment procedure to convey messages and meanings about the strength of the alignment between the timss (2003) mathematics assessment frameworks and the rncs for grade 8 mathematics. table 1: content and cognitive domains for mathematics used in timss 2003. table 2: coding of the rncs assessment standards according to timss 2003 cognitive levels. for the empirical work the first author worked with two experienced mathematics in-service facilitators for the senior phase and the further education and training phase. the grade 8 rncs mathematics assessment standards were compared with the timss assessment objectives and a common template consisting of 110 standard points or fragments as follows: number: 32 (8 whole number points, 5 integer points, 10 fractions and decimals points, 2 irrational number and financial mathematics points, and 7 ratio, proportion and percentage points); algebra: 26 (3 patterns, 7 algebraic expressions, 7 equations and formulas, 9 relationships/functions); measurement: 16 (3 attributes and units, 13 tools, techniques and formulae); geometry: 18 (3 lines and angles, 6 two-dimensional and three-dimensional shapes, 3 congruence and similarity, 4 location and spatial relationships, 3 symmetry and transformations); and data: 17 (4 data collection and organisation, 4 data representation, 5 data interpretation, 4 uncertainty and probability). in the common template, 88% of the rncs assessment standards were covered whilst 89% of the timss assessment objectives were covered. the facilitators were introduced to the mathematics cognitive domain categories used in timss 2003 (mullis et al., 2003, pp. 27−33) and asked to code the cognitive domain levels elicited by each standard point in the template according to the verbs used (see table 1). following the level descriptors, the two facilitators independently coded the standards and the author allocated marks according to their coding, totalling 1 score point per standard point. table 1 shows the content and cognitive domain categories, the weightings and the verbs or descriptors that characterise the cognitive levels. table 2 shows an example of five selected rncs standard points coded following airasian and miranda’s (2002, pp. 251−253) procedure of coding objectives according to the revised bloom’s taxonomy of educational objectives. assessment standard point 1.1, for example, uses three verbs, the first of which is in the ‘knowing’ category and the other two are both in the ‘using concepts’ category. standard point 1.2 uses the same three verbs but for a different content sub-topic (fractions instead of integers); standard point 2.1 solves problems and standard point 4.1 investigates and extends at the reasoning level. reliability was assured by the independent coding of the experts who were given copies of the relevant pages for the classification of timss assessment objectives as they appear in mullis et al. (2003, pp. 27−33) and implored to adhere to these as closely as possible. (the inter-rater kappa reliability index could not be computed because it applies to items falling in mutually exclusive categories.) computation of the porter index as already noted, the porter procedure analyses the extent of alignment between two matrices or matrices of frequencies (fulmer, 2011, p. 384). it produces a single alignment index, ranging from 0 to 1 to indicate how closely the distribution of points in the first matrix (of standards) aligns with the second matrix (of assessment). the alignment index p is arrived at in four steps as shown in figure 1: 1. create matrices of frequencies for the two documents being compared and label these as x and y. 2. for each cell in matrices x and y, compute the ratio of points in the cell with the total number of points in the respective matrix. label the matrices of ratios as x and y. 3. for every row j and column k in matrices x and y (the matrices of ratios), calculate the absolute value of the discrepancy between the ratios in cells xjk and yjk. 4. compute the alignment index using the formula , where j is the number of rows, k is the number of columns in each of matrices x and y, and xjk and yjk are ratios of points in the cells at row j and column k for each of ratio matrices x and y respectively. critical values for the strength of the alignment index a greater number of cells in the matrices will yield a range of likely values that is lower than for matrices with fewer cells. hence the total number of cells in the a and b matrices can have an effect on the significance of the alignment index. when we also consider that the centre of the distribution of indices is not zero, as noted earlier, we need to assess how far an observed alignment index is from 0.5. fulmer (2011) generated a matrix of means and critical values for alignment indices with results also demonstrating the expected (mean) distribution of pattern of alignment indices (see sample entries in table 3).in addition to matrix-size dependence, the alignment index also depends on the number of curriculum or standards statements or test items being coded. if the total number of cells in the matrix is n (= j × k) then for matrices a and b, j = 2 and k = 2 yields n = 4. in this study we used fulmer’s (2011) estimates of the critical values as determined by the number of cells and standard points. table 4 shows sample reference (or critical) value estimates from the corresponding number of cells and standards points. figure 1: porter alignment index example calculation for 2 × 2 matrices. table 3: sample mean alignment indices by number of cells and standard points. from results presented by porter (2002), for instance, the alignment between the standards of four us states (and the nctm) and their own assessments ranged from 0.30 to 0.47 for 30 standards points. six content areas and five cognitive levels were used, which meant 30 squares made up matrices a and b. table 4 gives a critical value of 0.7372 for the lower quantile ( ) if a two-tailed test is used at the alpha level of 0.05 (i.e. lower than might be expected by chance). therefore one can conclude that alignment amongst assessment and standards was very low. liu et al. (2009) used a coding structure with five content categories and six cognitive levels (hence 30 squares again) to compare the alignment of physics curriculum and assessments for china, singapore and new york state, china and singapore had alignments of 0.67, which were significantly lower than the mean ( ) at the 0.05 level (below the critical value of 0.7372); new york’s alignment index of 0.80 was equivalent to the mean. results and discussion top ↑ the structure of the content and cognitive domain matrices for the components of the 2003 grade 8 timss mathematics assessment frameworks three matrices were derived in respect of the three components of the timss assessment frameworks: the timss 2003 assessment objectives, the timss 2003 target percentages and the released timss 2003 test items. the content and cognitive domain matrix for the timss 2003 assessment objectives was derived from the list of objectives given in the timss assessment frameworks document (mullis et al., 2003, pp. 27−33). table 5 shows the results of the coding of the 98 (fine-grained) timss objectives. the numerical values form the required matrix. the 98 fine-grained objectives were accorded equal weight guided by the estimated time to be devoted to each of them. of the 110 fine-grained standard points (assessment objectives or standards) in the template, 12 were not amongst the timss assessment objectives.the content and cognitive domain matrix for the timss 2003 target percentages (table 6) was computed by extrapolation from exhibit 2 (mullis et al., 2003, p. 9) showing the target percentages of timss 2003 mathematics assessment time devoted to content and cognitive domain for the grade 8 level. time devoted was assumed to be equivalent to the importance attached to the respective categories in the frameworks as underpinned by the respective objectives. the content and cognitive domain matrix (table 7) for the released timss 2003 test items was derived from tallying the coding of all the released timss 2003 test items. the assumption was that the items were accurately coded and accurately reported on. in the coding process multiple-choice items were allocated one score point each whilst constructed response items were allocated marks, depending on the amount of work to be done, so that at least one third of the assessment came from constructed response items (martin, mullis & chrostowski, 2004, p. 35). the results of the tallying (table 7) agreed with exhibit 2.24 in martin et al. (2004, p. 60) and were converted to proportions. table 4: sample reference values for indices of alignment by number of cells and standard points. table 5: results of coding the timss 2003 assessment objectives by content and cognitive domain. table 6: derived target percentages of timss 2003 mathematics assessment devoted to content and cognitive domain by grade level. table 7: timss 2003 grade 8 mathematics content and cognitive domain matrix for test items. the structure of the content and cognitive domain matrix for the grade 8 rncs for mathematics table 8 shows the content and cognitive domain matrix obtained for sa’s grade 8 mathematics curriculum. all of the 98 rncs assessment standards whose content was covered by timss were coded. the resultant score points were totalled for each content and cognitive domain category and converted to the proportions shown. computed porter indices of alignment for this study table 9 shows the calculated raw cell-by-cell differences between rncs assessment standards and timss assessment objectives. these raw differences were converted to absolute differences, from which the porter index of alignment was computed using the formula . the computed index was 0.735. the mean-simulated alignment index for a 5 × 4 comparison with 20 cells is 0.9635 (see table 3). using a two-tailed test, at the 0.05 alpha level, we looked to the 0.025 and 0.975 quantiles in table 4. close to 100 standards points and matrices of 20 squares each were used, so the critical values for 90 standard points are 0.9302 and 0.9758 respectively whilst those for 120 standard points are 0.9333 and 0.9843 respectively. the computed alignment value is well below 0.9302 and 0.9333 in the 0.025 quantiles ( ). the alignment was therefore significantly lower than would be expected by chance at the 0.05 level. further iterations were conducted to determine the pair-wise alignment amongst the rncs assessment standards and the three components of the timss 2003 assessment frameworks. the indices obtained are as shown in table 10. surprisingly the pair-wise alignment is significantly lower in all instances, without exception, suggesting a low internal consistency even amongst the timss components themselves. the structure of discrepancies between timss 2003 framework components and rncs assessment standards the first structure of discrepancies investigated was between rncs assessments standards and the timss assessment objectives. figure 2 shows the structure of (mis)alignment by content and cognitive domain.from the graph it is evident that the rncs assessment standards were stronger than timss assessment objectives in all cases where the bars extend upwards above zero but weaker in those cases where the bars extend downwards below zero. whilst there is a common criticism of teachers concentrating on knowledge of facts and procedures, the tables shows that the rncs was weaker than timss objectives in this cognitive level in four of the content domains, namely number, algebra, geometry and data. the rncs was stronger with respect to routine problem solving in number and data but weaker in measurement and geometry. a similarly mixed picture emerged in respect of reasoning. the second structure to be investigated was between the rncs objectives and the timss target percentages. table 8: grade 8 rncs mathematics content and cognitive domain matrix. table 9: raw cell-by-cell differences between the rncs assessment standards and the timss 2003 assessment objectives. table 10: porter indices of alignment amongst the timss components and the rncs. figure 2: discrepancies between rncs assessment standards and the 2003 timss assessment objectives. figure 3: discrepancies between rncs assessment standards and timss target percentages. figure 3 summarises the structure. surprisingly, the rncs was stronger on knowledge of facts and procedures in number, measurement and geometry but weaker in algebra and data. taken together, these discrepancies were significant. a third and final structure to be investigated was that between the rncs and timss test items. figure 4 summarises the discrepancies. a marked shift in this comparison is that the rncs was stronger with respect to knowledge of facts and procedures in number, geometry and data. the rncs was, however, weaker in respect of reasoning in all content categories. routine problem solving was almost evenly split, above in algebra, measurement and data, but below in number and geometry. figure 4: discrepancies between rncs and assessment standards and timss 2003 test items. figure 5: south african students’ performance by content and cognitive domains in timss 2003 compared to the international average. table 11: comparison of rncs-timss discrepancy with sa performance in timss 2003. the next logical question is whether there was any relationship between the structure of discrepancies and south african students’ performance in timss 2003. given the discrepancies within the timss components themselves, the ultimate question is whether the discrepancies between the rncs and the timss 2003 test items had any correlation with student performance as they were partially in force at the time of the 2003 timss assessments. a comparison of the rncs-timss test discrepancies with south african students’ performance in timss 2003 figure 5 was compiled after extracting south african students’ performance in each of the released test categories by content and cognitive domain relative to the international average. it is already well known that south african students performed below the international average across the board (e.g. reddy, 2006). beyond that, however, the intention in this study was to additionally investigate if the pattern of discrepancies between rncs assessment standards and timss assessment objectives was in any way related to south african students’ performance (i.e. the achieved timss curriculum). table 11 and figure 6 attempt to answer that question. it is evident that student performance correlated negatively with discrepancies in number but positively with discrepancies in algebra, measurement, geometry and data. that is, the narrower (or positive) the discrepancy was, the closer the performance was to the international average in all content domains except number. in number, sa students performed worst in items on using concepts even though this was not the weakest cognitive domain representation in the rncs assessment standards. in algebra, sa students performed worst in the knowledge of facts and procedures and this was the weakest category. in measurement they performed the worst in routine problem solving which was the weakest category of rncs. in geometry they performed worst in the routine problem solving category which was the second weakest in the rncs curriculum. in data handling they performed worst in using concepts, which was also the weakest point in the rncs. overall, sa learners performed worst in using concepts, suggesting little conceptual understanding being achieved by the curriculum. routine problem solving was second worst. this pattern has implications for the intended curriculum which determines what curriculum materials should emphasise and ultimately what teachers should teach in the classroom. brijlall (2008) notes that the lack of problem-solving skills in sa may be a result of the way it has been taught in schools: individual solution by learners, presentation of abstract problems foreign to learners. there is little doubt that the ultimate answer lies in the implemented curriculum but what feeds into the implemented curriculum is the intended curriculum. figure 6: comparison of the rncs-timss discrepancy in assessment objectives and sa students’ performance in timss 2003: (a) number, (b) algebra, (c) measurement, (d) geometry and (e) data. conclusions top ↑ the study reported in this article set out to investigate the alignment of south africa’s rncs for grade 8 mathematics with the timss 2003 grade 8 mathematics assessment frameworks. from the results we conclude that the computed porter index of 0.751 suggests that the misalignment was low enough to warrant urgent attention, from curriculum designers, assessment practitioners, educators, teacher educators and policymakers alike, in order to enhance prospects of improved performance in future participations. in particular there is need to pay attention to the observed discrepancies between the content and cognitive domain emphases. the fact that, even where the rncs curriculum was stronger than timss, performance was still generally poor suggests the likelihood of a gap between the intended curriculum and the implemented curriculum. such a gap further suggests a possible mismatch in emphasis between the intended curriculum and the curriculum support materials that actualise it. however, this conjecture requires further investigation. the study also points to the likelihood of a consequential gap between the implemented (sa) curriculum and the attained (timss) curriculum reported by reddy (2006, p. xiv). from a developing country perspective, what is even more disconcerting is that the three components of timss do not appear to be aligned. that the misalignment is statistically significant calls into question the value-neutrality of timss which currently appears to be a constantly shifting target that only well-resourced, developed countries can cope with. finally, participation in timss should not be another bureaucratic ritual. rather, it should rigorously and reflexively inform curriculum reform and innovation. in an increasingly globalised knowledge economy the school system needs to be globally competitive in the gateway fields of mathematics and science education. a simple illustration of the current disconnect is that, despite sa’s participation in previous timss studies, the recently published curriculum and assessment policy statements (caps), which are largely a refinement of the rncs, proclaim to have been influenced by the cognitive domain levels used in timss 1999 (department of basic education, 2011a, p. 55, 2011b, p. 59). this is so in spite of changes in 2003 (when the country participated in timss for the third time) and further changes in 2007 (when the country did not participate), which were carried over to 2011. accordingly, the influence of the revised bloom’s taxonomy on timss as evident in the 2007 and 2011 frameworks was apparently not taken into account in the latest curriculum revisions. this suggests that even the newly introduced annual national assessments (ana) for grades 1−6 and 9, together with the senior national certificate examinations, will continue to be guided by out-of-sync domain categories. by implication, curriculum and assessment will continue to be out of step with international trends resulting in mixed messages for teaching and learning. bansilal (2011), for example, calls for a closer alignment of curriculum implementation plans with classroom realities. it is ironic that although sa teachers and educationists have complained of rapid curricula changes, the national curriculum statement has not changed at the same pace as timss. accordingly, educators, teacher educators and education researchers should be engaged more constructively in the curriculum and assessment reform processes for sustainable curricula coherence to be achieved. reddy (2006, p. xiv) reports that during the period of timss 2003, sa teachers consulted disparate curricula documents to determine what and how they taught. as affirmed by airasian and miranda (2002, p. 253), severe misalignment of assessment, standards and instruction can cause numerous difficulties. given the extent to which the misalignment of the sa curriculum has gone relatively unchecked, the school system will continue to buckle for some time to come when subjected to international scrutiny. the latest of such scrutiny is the global competitiveness report (schwab, 2012) and the southern and east african consortium for monitoring educational quality report (spaull, 2011), in which sa ranks very unfavourably. acknowledgements top ↑ this research was funded by the tshwane university of technology through its postdoctoral fellowship placement of the first author. however, the opinions expressed do not necessarily reflect the views of the university. the authors are also grateful to jeram ramesh and cerenus pfeiffer for their time and expertise in the coding of the rncs grade 8 mathematics assessment standards and the timss assessment objectives. we are further grateful to iben christiansen’s comments on an earlier draft of this manuscript. authors’ contributions m.n. (university of stellenbosch) was the researcher responsible for the empirical study as well as the writing of the manuscript. a.m. (tshwane university of technology), as postdoctoral supervisor, shared his expertise during the main study and the development of the manuscript. competing interests we declare that we have no financial or personal relationship(s) which might have inappropriately influenced our writing of this article. references top ↑ airasian, p.w., & miranda, h. (2002). the role of assessment in the revised taxonomy. theory into practice, 41(4), 249−254. http://dx.doi.org/10.1207/s15430421tip4104_8bansilal, s. (2011). assessment reform in south africa: opening up or closing spaces for teachers? educational studies in mathematics, 78, 91−107. http://dx.doi.org/10.1007/s10649-011-9311-8 berends, m., stein, m., & smithson, j. 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(2005). alignment analysis of mathematics standards and assessments, michigan high schools. madison, wi: author. abstract introduction acknowledgements references about the author(s) levi ndlovu stellenbosch university centre for pedagogy, department of curriculum studies, faculty of education, stellenbosch university, cape town, south africa mdutshekelwa ndlovu department of science and technology education, faculty of education, university of johannesburg, johannesburg, south africa citation ndlovu, l., & ndlovu, m. (2020). the effect of graphing calculator use on learners’ achievement and strategies in quadratic inequality problem solving. pythagoras, 41(1), a552. https://doi.org/10.4102/pythagoras.v41i1.552 original research the effect of graphing calculator use on learners’ achievement and strategies in quadratic inequality problem solving levi ndlovu, mdutshekelwa ndlovu received: 30 apr. 2020; accepted: 08 oct. 2020; published: 18 nov. 2020 copyright: © 2020. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract the purpose of this mixed methods study was to investigate the effect of a graphing calculator (gc) intervention on grade 11 learners’ achievement in quadratic inequality problem solving. the quantitative aspects of the study involved an experimental and control group design in which the experimental group received instruction using the gc activities and the control group was taught without using the gc. the qualitative aspects of the study involved script analysis and task-based interviews. we used three data collection instruments: a quadratic inequality problem solving test used both as a preand a post-test administered to both the experimental and the control group learners, a written task analysis protocol and a task-based interview schedule. the results of the dependent samples t-test confirmed a statistically significant improvement in the quadratic inequality problem solving achievement of the experimental group with a cohen’s d effect size of 1.3. the dependent t-test results for the control group were also a statistically significant improvement but with a smaller cohen’s d of 1.2. the results of the independent t-test indicated that the experimental group achievement was significantly higher than that of the control group with a cohen’s d effect size of 0.79. script analysis of selected learners’ post-test solutions also showed that learners in the experimental group employed more problem-solving strategies (at least three – symbolic, numeric and graphical). interview results of purposively selected learners also affirmed that experimental group participants perceived the gc intervention to have prepared them more effectively for multiple solution strategies of the quadratic inequality problem tasks. the researchers recommend the integration of gcs in the teaching and learning of mathematics in general and quadratic inequalities in particular. however, more research is needed in the integration of the gc in high-stakes assessment. keywords: quadratic inequality problems; problem-solving strategies; graphing calculator; realistic mathematics learning; learner achievement and strategies. introduction the mathematical topic of quadratic inequalities plays a significant role in the solution of some real-life optimisation problems as given in the pre-post-test examples of this study. this topic, however, requires prior knowledge of other mathematical topics such as algebra, linear inequalities, quadratic equations, quadratic functions and geometry (bicer, capraro, & capraro, 2014; el-khateeb, 2016; halmaghi, 2011). el-khateeb (2016) adds that solving inequalities demands basic knowledge of properties and applications of functions that include domain, range and intervals (increasing or decreasing). this implies that the teaching and learning of quadratic inequalities should be underpinned by a strong mathematical background of foundational concepts, algebraic manipulation skills, related contexts of application and geometric visualisation. it is important for learners to understand the relationship of comparison and to develop the competencies of explaining the inequality relations, using terms such as ‘greater than’, ‘less than’, ‘greater than or equal to’, ‘less than or equal to’, the associated symbolisations (>, <, ≥, ≤) and how all these differ from the relationship of equality (=). yin (2005) attests that a sound knowledge of inequalities is critical in the study of engineering, astronomy and science. civil engineers may, for example, want to design artefacts such as bridges or apply optimisation methods to improve the efficiency or quality of a product. this suggests that proficiency in quadratic inequalities may increase learners’ confidence in application and transfer of knowledge to real-life situations (tsamir & bazzini, 2004). apart from applications in engineering and science, such real-life contexts may also include business, economics and agriculture within the experiential world of learners. examples may be to maximise profit taking into account multiple factors and therefore solutions of systems of inequalities (e.g. linear programming). learners should thus develop the appropriate mathematical skills of solving inequality problems and interpreting their solutions accurately. the general study of inequalities is also important for the understanding of epsilon-delta proofs in later real analysis studies for those learners who might pursue advanced studies in mathematics. despite this elaborate rationale for the study of inequalities, high school learners frequently view quadratic inequalities to be one of the most difficult topics in the mathematics curriculum (bicer et al., 2014; halmaghi, 2011). these difficulties extend to the solution of quadratic inequalities by learners. as an important topic, the south african curriculum and assessment policy statement (caps) for mathematics in the further education and training (fet) phase requires all the learners (grades 10–12) to study quadratic inequalities and their usefulness in solving real-life problems (department of basic education [dbe], 2011). in this regard, mathematics teachers are at liberty to use digital tools such as graphing calculators (gcs) to enhance the conceptual understanding of quadratic inequalities and reduce conceptual obstacles. the international society for technology in education (iste, 2016) recommends the use of digital technologies, including gcs, to improve the teaching and learning of mathematics. research also shows that integrating these information and communication technologies (icts) also has the potential to develop learners’ critical thinking skills and provides them with the opportunity to interpret solutions of inequalities sensibly (abramovich & ehrlich, 2007; tsamir & almog, 2001). although the use of technology can provide learners with a rich learning environment as advocated by the dbe’s (2011) caps (intended curriculum), there has not been a widespread adoption of gcs either in the implemented curriculum or in the assessed mathematics curricula of south african high schools. problem statement despite the caps expressing the desirability for all fet learners to understand quadratic inequalities, the national senior certificate (nsc) results contrastingly show that many learners repeatedly have trouble solving inequalities correctly (dbe, 2016, 2018, 2019). from the first authors’ experiences grade 11 learners struggle to understand this topic when introduced to it and seem to rely mainly on rote memorisation of procedures. learners also have difficulty making connections between quadratic inequalities and their pre-concepts such as linear inequalities and quadratic equations. in addition, learners have difficulty providing meaningful solutions in interval notation, leading to errors regarding the domain and range. they also interpret wrongly the connectives ‘and’ and ‘or’ in various problem contexts. as already noted earlier, literature also confirms that many learners experience difficulties when solving quadratic inequality problems in particular (bicer et al., 2014; halmaghi, 2011; verikios & farmaki, 2010). from the nsc diagnostic reports (dbe, 2016, 2018, 2019) many learners appear to procedurally treat the inequality as an equation rather than an optimisation problem. this misconception results in inaccurate solutions and consistently limits learners’ ability to provide sensible answers (yin, 2005). providing learners with relevant or realistic (optimisation) contexts as advocated in realistic mathematics education (rme) might help them make better sense of their answers. connecting algebra with functions using a gc is a missing aspect of classroom practice that may help learners to construct more accurate visual representations of quadratic inequalities and their solution sets. the gc’s representational affordances appear to have the underexplored potential to meaningfully support, through activity and intertwinement principles of rme, the integration of algebra and functions. for example, learners with difficulties in describing the domain, range, maxima and minima, as well as restricted values, on graphs, including the representation of solution sets by means of the interval notation, might find participation in the gc-enhanced activities more illustrative and therefore more meaningful. in support of this potential, bicer et al. (2014) argue that working with graphs, algebraic representations and tables of values simultaneously can help learners gain experience that makes their quadratic inequality problem solving more comprehensible. the function-based approach advocated by verikios and farmaki (2010) is a missing link that can be harnessed within the gc environment to more effectively develop learners’ contextual quadratic problem-solving strategies and visual thinking skills. functions of real-life contexts represent horizontal mathematisation instances in rme of which a typical example is the function for the height of an object in free fall: f(h) = –16t2 +h0, where t is time in seconds and h0 is the initial height in metres. this is one of the gaps that this study sought to fill since the gc is capable of concurrent algebraic, numeric (table of values) and graphic representations. in this study we expected the use of the gc to promote the development of learners’ visualisation skills, and meaningful representation and interpretation of quadratic inequality results (abramovich & ehrlich, 2007; tsamir & almog, 2001). we thus reasonably expected the use of the gc to simultaneously mediate the effective teaching and learning of quadratic inequalities embedded in real-life contexts in which the real-world context would serve as a pivot for meaningfulness and relevance of mathematics to real(istic) situations envisaged in rme’s characterisation of mathematics as a human activity as articulated in the theoretical framework. purpose and research questions the purpose of this study was to explore the effectiveness of a gc intervention in the teaching and learning of context-based quadratic inequalities to grade 11 learners. to achieve this goal, the study addresses the following overarching research question: to what extent can the use of graphing calculators influence grade 11 learners’ ability to solve quadratic inequality problems embedded in realistic or real-life contexts? the following sub-research questions guided this study: what is the effect of the pedagogical use of gcs on learners’ achievement in solving (contextual) quadratic inequality problems? what strategies do learners in a gc-mediated classroom use to solve contextual quadratic inequality problems? what are the students’ perceptions of (contextual) quadratic inequality problem solving in a gc-mediated classroom? literature review research on learners’ achievement in solving quadratic inequalities although inequalities are an important mathematics topic, there is limited literature on their teaching and learning in general, particularly quadratic inequalities. this paucity of the literature signifies the persistence of a limited understanding of why learners continue to experience challenges with the topic. verikios and farmaki (2010), for example, affirm that not enough research has been conducted with respect to learner’s understanding of quadratic inequalities. a better understanding of learners’ misconceptions (sources of misunderstanding) can lead to a better understanding of how to teach quadratic inequalities more effectively. grewal’s (1994) study of 311 grade 12 learners taking higher grade mathematics in south africa found that they did not have sufficient prerequisite knowledge, and did not display a satisfactory level of mastery in solving quadratic inequalities. as seen earlier, the development of sufficient prerequisite knowledge may lead to a stronger mastery of quadratic inequality problem solving ability among learners. as seen earlier, bicer et al. (2014) found that many middle and high school learners in the united states have misconceptions and difficulties regarding quadratic inequality problem solving. they also found that college preservice teachers experienced the same difficulties with solving and interpreting quadratic inequalities accurately. understanding the difficulties encountered by students can help teachers teach their learners to become better quadratic inequality problem solvers, and more so in a gc-mediated environment. yin’s (2005) single case study found that the learner solved quadratic inequalities procedurally without really coming to grips with concepts like the domain, nor was she able to logically check her answers. this is a problem that a gc-mediated environment can help solve. also, limiting quadratic inequality problem solving to algorithmic procedures without introducing (optimisation) contexts that are experientially real to students as envisaged in rme might obscure their conception of critical prior knowledge of the domain and range concepts and hinder their ability to check and make proper sense of their answers. in their study, tsamir and reshef (2006) investigated whether to present grade 10 students with a single method or with several methods to solve quadratic inequalities. they introduced the students to three differently sequenced approaches – the sign chart, the graphic and the logical connectives approach – and found that almost all students correctly solved the quadratic inequalities and liked being introduced to different approaches. however, most preferred the graphic approach to the other two methods. thus, we considered that presenting learners with multiple strategies of solving quadratic inequalities, within a gc intervention, might constitute more effective teaching and learning of the topic. aided by the use of the gc, the function-based approach is likely to be more effective. following balomenou, komis and zacharos’s (2017) recommendation on the use of digital tools, this study investigated the efficacy of gc use to fill the research gap in the teaching and learning of quadratic inequalities embedded in realistic contexts. below we survey literature on the use of the gc in school mathematics education in search of a solution for effective ict-intensive teaching and learning of quadratic inequalities. research on the effect of graphing calculator use in the mathematics classroom early research reveals that gcs significantly affected the teaching and learning of mathematics, particularly functions and graphs (dunham & dick, 1994) and was later expanded to statistics, geometry, trigonometry, algebra, modelling and calculus concepts (muhundan, 2005). several researchers report that the use of gcs in mathematics education improves learners’ achievement in solving algebra problems in applied contexts, interpreting graphs and general cognitive understanding (e.g. chen & lai, 2015; karadeniz, 2015; parrot & leong, 2018; wareham, 2016). this suggests that learners who use gcs can have relative advantage in mathematical problem solving. additionally, learners using gcs can consistently display more innovation, speed and accuracy in their problem-solving strategies as well as better reasoning in their answers and better visualisation of graphs and abstract concepts (hunter, 2011; thomas, 2016). the use of a gc can also potentially improve learners’ organisation of written work, and the correct use of notation and symbols (shahriari, 2019). however, for the successful use of gcs there must be a shift in classroom culture (parrot & leong, 2018), where learners become the constructors of their own mathematical knowledge. this is in line with rme’s activity and guided reinvention principles of allowing learners the freedom to become active participants and to reinvent mathematics with the teacher’s guidance (freudenthal, 1991). ndlovu (2019), for example, states that the created opportunities enabled learners to use visual models, diagrams and symbols for exploring and figuring out the solutions. rich (1991) also found that gc use changed the classroom dynamics in that learners became more willing to interact, discuss and share their strategies of solving mathematical problems, resulting in improved retention of content. however, this study examined the effect of a gc intervention on the learners’ achievement and strategies in quadratic inequality problem solving. regarding problem-solving strategies, kenney (2014) proposed five that are applicable to a gc environment, namely: (1) linking verbal and symbolic representations, (2) recognising conventions and their properties, (3) making connections to symbols and their graphs, (4) linking symbols with their numeric representation (e.g. inequalities and tables) and (5) recognising the meaning of symbols within the context of the problem. kutzler (2000), on the other hand, proposed three strategic steps for solving more complex and real-world problems using the graphing calculator, namely: (1) choosing the model and translating words into the mathematical expressions, (2) applying the available algorithms to solve the model problem and (3) translating the model solution into a real-world solution. these strategies became a core of the activity principle in the rme approach to provide guidance for solving real-world quadratic inequality problems in a gc environment. rme as a theoretical framework we adopted rme as a theoretical framework for this study. rme, developed in the netherlands for the teaching and learning of mathematics, views mathematics as a human activity in which contextual or experientially real problems are used as the starting points for learning (drijvers, 2019). it takes a cue from freudenthal’s (1991) advocacy that the learning of mathematics must be experienced as meaningful or authentic sense-making by the learners. in the rme approach, learners are also afforded the opportunity to use their own informal strategies and models to solve problems and to discuss with the teacher and fellow learners (theodora & hidayat, 2018). to this end, rme was relevant as a lens to understand quadratic inequality problem solving in a gc-mediated learning environment. we highlight four key constructs in this framework that are applicable to this study, namely the meaning of ‘realistic’, activity principle, didactic phenomenology and mathematisation (drijvers, 2019; ndlovu, 2014; van den heuvel-panhuizen & drijvers, 2014). realistic in the context of digital tool use in rme means that learners should experience activities with digital technology as meaningful (drijvers, 2019). that is, the tool must be fit for purpose by helping learners to solve the problems they want or are tasked to solve and produce intelligible representations and results. the activity principle views mathematics as learning by doing, and thus digital tool use ought to give learners opportunities to actively explore, reinvent mathematics and (re)construct their own mathematical understandings as the chief actors (drijvers, 2019), not as passive observers of the teacher’s use. these activities ought to provide learners with experientially real situations that will make them appreciate the purpose and functionality of the digital tools. in this study, learners experienced the use of a gc through the meaningful and purposeful activity of graphing quadratic functions and related quadratic inequalities to produce meaningful solutions (symbolic, numeric and graphic) and to make connections among these. a didactic phenomenology view of digital tool use means that as the tool gets integrated into the mathematics classroom it can become part of the shared technology-rich classroom discourse that becomes a phenomenon worth studying (drijvers, 2019). in other words, when digital tools such as gcs and dynamic geometry software become ubiquitous in the classroom their exploratory use for different topics begins to emerge spontaneously. didactical phenomenology also concerns the analysis of how mathematical thought objects can help to organise and structure phenomena by means of specific activities or concepts (van den heuvel-panhuizen & drijvers, 2014). in this case, the teacher may organise the quadratic inequalities (being the phenomena or mathematical objects) into possible instructional activities (or tasks) that support meaningful learning. the same principle provides opportunity for arranging the instructional activities into social groups for learners to collaboratively and interactively discuss their strategies and (re)inventions (van den heuvel-panhuizen & drijvers, 2014). mathematisation, on the other hand, refers to the activity of doing mathematics (van den heuvel-panhuizen & drijvers, 2014). in the horizontal dimension, mathematical procedures are used to organise and solve problems embedded in real-life or realistic contexts. such procedures include learners being able to schematise, formulate, transform and visualise the realistic or contextual problems mathematically and to transfer knowledge between different domains (van den heuvel-panhuizen & drijvers, 2014). this is often referred to as moving from the world of life to the world of symbols which is comparable to mathematical modelling and problem-solving approaches. the vertical dimension entails the solution of a problem, the generalisation of the solution and the further formalisation thereof (van den heuvel-panhuizen & drijvers, 2014). this approach uses models, schemes, symbols and diagrams as conceptual tools for developing the mathematical relations. this is often referred to as a transition within the world of symbols from simple to more complex and better organised mental structures or schemes (van den heuvel-panhuizen & drijvers, 2014). the digital tool becomes the medium of mathematical expression. research methodology research design this study used a sequential mixed methods research design to investigate the effects of using a gc on grade 11 learners’ achievement and strategies in quadratic inequality problem solving. the quantitative aspects of the study focused on the classical (teaching) experiment that employed a pre-test-post-test control group design (creswell, 2011; laurens, batlolona, batlolona, & leasa, 2018) to address the first research sub-question. the time allocated for teaching and learning for the two groups was four weeks and the same teacher-researcher taught both groups. in particular, the experimental group was exposed to gc-mediated instruction in quadratic inequalities while the control group was taught using the traditional pen and paper method. the dependent variable in the teaching experiment was learners’ achievement in quadratic inequality problem solving measured by means of learning gains between the pre-test and the post-test. in answering the post-test, the experimental group was not allowed to use gcs to determine the solutions because in the nsc examinations candidates are not allowed to bring programmable calculators but they write the same test. we expected that if they had experienced greater learning gains they would demonstrate greater conceptual understanding and perform better than the control group in the post-test. this was also in compliance with the caps document, which requires the use of technology to enable learners to create tabular, symbolic and graphical representations (dbe, 2011). experimental group learners were given adequate time (i.e. a week) to familiarise themselves with the use of the digital tool in the designed activities. both the experimental and control groups were exposed to kenney’s (2014) and kutzler’s (2000) problem-solving strategies. the control group used more than one pen and paper visualisation method of solving quadratic inequalities (graphical, number line and table). these methods allowed learners to link verbal and symbolic representations to their graphs and solutions. both groups were given the same learning tasks that required the use of these strategic procedures. the strategies suggested by kenney and kutzler assisted in the development of the rubric for assessing the problem-solving abilities of learners in a gc-enhanced classroom. the qualitative aspects of the study involved script analysis to address the second research sub-question, and task-based interviews to answer the third and final research sub-question. participants the population for the study consisted of grade 11 learners in one of the public secondary schools in gauteng, south africa. a combination of purposive and simple random sampling strategies was used to select the learner participants for the study. purposive sampling was used to identify the public high school in ekurhuleni north district that fell in the underperforming category (as measured by 2016 nsc results of the schools in mathematics). purposive sampling is a non-random method of sampling where the researcher selects information-rich cases for in-depth study (patton, 2002) and from which the most can be learned (merriam, 2009). this sampling helps to identify any common patterns of particular interest and value when recording the key experience and shared dimensions of a setting or phenomenon (patton, 2002). this sampling strategy befits the qualitative aspect as this approach seeks to understand the behaviour of the two groups of learners when solving quadratic inequalities in different classroom environments but in similar school settings. we drew a random sample of 40 learners from a population of grade 11 learners doing mathematics at the selected school. the 40 learners were randomly assigned to two groups of 20 learners each. one of the groups served as the experimental group and received gc-mediated instruction as the intervention. the other group served as a control group and received non-gc instruction to develop quadratic inequality problem-solving skills. at the beginning of this study, the two groups wrote the same pre-test at the same time. approximately four weeks after the administration of the pre-test and completion of the series of intervention lessons, participants of both groups wrote the post-test on contextual word problems of quadratic inequalities to help us answer the first research question. in addition, we drew another random sample of two learners (a boy and girl) from the gc group after the post-test. these learners were interviewed to explore their perceptions of a gc-facilitated learning environment. research instruments we used three instruments to collect data, namely self-developed quadratic inequality problem solving tests (qipsts) to assess problem-solving proficiency, a self-developed assessment rubric for script analysis informed by kenney’s (2014) and kutzler’s (2000) strategies, and a self-developed semi-structured interview schedule to ascertain students’ perceptions of the gc-mediated learning environment. the tests were validated by experts prior to the research. learning gains in quadratic inequality problem-solving proficiency would indicate the effects of the gc intervention within and between the experimental and the control groups. all the items for the learning activities and the tests were designed using the previous examination question papers and grade 11 mathematics textbooks about quadratic inequalities. identical rme-aligned quadratic inequality problems were set for the pre-test and post-test as described below. learners were given two quadratic inequality word problems to solve in both the pre-test and post-test as provided below: 1. a small-scale taxi owner’s daily profit is given by f(x) = −2x2 + 70x, in which x is the number of trips made per day. find the number of trips that must be made per day if the profit is to be greater than or equal to r600. this item used the commercial context of a taxi owner to test whether learners could formulate the corresponding quadratic inequality (ability to distinguish between an equation and an inequality) with a negative coefficient of x2 (handling of signs when multiplying by negative values), formulating the inequality correctly (as a maxima problem) from the given word problem, graphing the inequality correctly (as a concave down graphic), as well as interpreting the domain correctly (as a solution set and not a single value). many working-class students use public transport in south africa and therefore the context would easily resonate with their daily experiences of commuting to and from school or to and from shopping centres. this formed the realistic context that was experientially real to south african learners (see figure 1). 2. a small-scale farmer wanted to fence off her chickens within a rectangular area on her farm. the breadth and length are x metres and 2x – 3 metres. determine the values of x that can give an area of her chicken run larger or equal to 20 metres2. figure 1: visualisation of solution set for q1 generated using a graphing calculator. this small-scale chicken farming context item also tested whether learners could formulate the requisite quadratic inequality (using prior knowledge of the rectangle area formula and ability to distinguish between an equation and an inequality) with a positive coefficient of x2 (handling of signs when multiplying by positive values), formulating the inequality correctly (as a minima problem) from the given word problem, graphing the inequality correctly (as a concave up graphic), as well as interpreting the domain correctly (as a solution set and not a single value). figure 2 is a visualisation of the solution set from the area formula f(x) = x(2x – 3) = 2x2 – 3x, which leads to the inequality 2x2 – 3x ≥ 20. many working-class or township school learners eat chicken as one of the cheapest meat types available in south africa and the context would easily resonate with their daily experiences of not only eating chicken but also how they are raised. this formed the realistic context that was experientially real to south african learners (see figure 2). figure 2: visualisation of the solution set for q2 generated using a graphing calculator. we used at least two markers (the teacher-researcher and a moderator) who obtained similar results using the same rubric. the issue of reliability of the pre-test was also addressed by conducting a pilot study with grade 11 learners from a different school and both tests were marked by two teachers to compare their scores. learners’ scripts were scored using the memorandum in the appendix, a modified version of the analytic problem solving rubric developed by charles, lester and o’daffer (1987) which has been widely used in other problem solving research (yeo, 2011). the paired samples t-test was used to determine the statistical significance of within-group quadratic inequality problem solving learning gains. we used the independent samples t-test to determine the statistical significance of the between-group problem solving learning gains attributable to the gc intervention when compared to those of the control group. we calculated cohen’s d using the 2016 statistical package for social sciences (spss) to ascertain the practical significance or effect size of the gc intervention on the experimental group, the non-gc intervention on the control group and to compare both groups. the assessment rubric was used to describe qualitative changes in learners’ strategies manifested in the their scripts. we used in-depth interviews to ascertain students’ perceptions of the gc-mediated learning environment. ethical consideration the study was approved by stellenbosch university (ethical clearance/project number: 2023). results of the research effect of the gc intervention on learners’ achievement in quadratic inequality problem solving this section answers the first research question of this study which we broke down into four two-tailed null sub-hypotheses: h01: there is no difference in the pre-test mean scores of the experimental and control groups, i.e. µ1 = µ3. h02: there is no difference between the preand post-test mean scores of the experimental group, i.e. µ1 = µ2. h03: there is no difference between the preand post-test mean scores of the control group, i.e. µ3 = µ4. h04: there is no difference in the post-test mean scores of the experimental and control groups, i.e. µ2 = µ4. µ1 is the experimental group pre-test mean score; µ2 is the experimental group post-test mean score; µ3 is the control group pre-test mean score, and µ4 is the control group post-test mean score. to test the first hypothesis, we conducted a between-group samples t-test using spss and the formula: the results show that there was no statistically significant difference between the experimental group (µ1 = 24.40; sd = 14.34) and the control group (µ3 = 20.05; sd = 13.98) in the pre-test since t = 0.97156 and p = 0.337416, which is greater than 0.05. this meant that both groups had similar (homogeneous) problem-solving abilities before the gc intervention to the experimental group. it is important to establish the pre-intervention knowledge levels of the experimental and control groups in order to control for pre-existing knowledge. this confirmed that random assignment established equivalence between the two groups. to test the second hypothesis, we conducted a paired samples (within-group) t-test using spss and the formula: the results showed a statistically significant improvement in learners’ problem-solving abilities of quadratic inequalities from the pre-test (µ1 = 24.40, sde = 14.34) to post-test (µ2 = 45.25, sde = 20.65), with t = 5.62 and p = 0.00002, which is less than 0.05, in the experimental group. this meant that there were significant problem solving learning gains in the experimental group. to ascertain the size of the effect we first used excel to obtain r = 0.602 and then calculated cohen’s d using g*power and the formula to determine whether the difference was practically significant (or big enough to worry about). we obtained a d-value of 1.256 (standard deviations), which indicated a large effect size since it is greater than cohen’s (1988) threshold of 0.8. this meant a practically significantly large improvement of approximately 1.3 standard deviations in the experimental group’s (within-group) quadratic inequality problem solving achievement. however, this alone does not tell us whether gc use was more effective than the traditional approach used for the control group. to test the third hypothesis, we conducted a paired sample (within-group) t-test using spss and the relevant t-test formula for paired samples. the results showed a statistically significant improvement in learners’ quadratic inequality problem solving learning gains from the pre-test (µ3 = 20.05; sd = 13.98) to the post-test (µ4 = 30.6; sd = 16.17), with a t-value of 5.30 and p = 0.00004, which is less than 0.05, in the control group. to ascertain the size of the effect we first calculated the correlation between the pre-test and the post-test and obtained r = 0.83576 which we used to calculate cohen’s d of 1.187 which indicated a large effect size since it is greater than cohen’s (1988) threshold of 0.8. this indicated a large improvement of approximately 1.2 standard deviations in the control group’s quadratic problem solving achievement. hence, not only were learning gains significant in both the experimental and control groups between the pre-test and post-test, but also both groups’ learning gains were large. what remains is to establish, by testing the fourth hypothesis, is whether the experimental group’s achievement was significantly higher or practically significantly higher than that of the control group. the establishment of this fact is important to show whether the gc intervention in the experimental group was superior to the control group’s traditional teaching approach or not since so far both were statistically significant and large. this leads us to the fourth hypothesis. to test the fourth hypothesis, we conducted an independent samples (between-groups) t-test using spss and the relevant t-test formula for independent samples. we found that the experimental group’s post-test mean score (µ2 = 45.25; sd = 20.65) was significantly higher than the control group’s mean score (µ4 = 30.6; sd = 16.17) since t = 2.49775 and p = 0.016945, which is less than 0.05. to ascertain the size of the difference in effect we computed cohen’s d to determine the magnitude of the differences (effect size) in the means and obtained d = 0.789937. this value was more than the threshold of 0.50 for moderate effect size and marginally short of 0.8 for large effect size (cohen, 1988). this indicated that the learning gains of the experimental group were more than those of the control group (by a moderate margin of approximately 0.79 standard deviations). that is, learners who used gcs in class became better quadratic inequality problem solvers than those in the control group who used traditional pen and paper methods, albeit by a moderate margin. this means that the experimental group’s learning gains were not only statistically significantly higher than those of the control group but also practically significantly superior. problem-solving strategies used by learners in the experimental and control groups this section answers the second research question: what strategies do the experimental and control groups learners display when solving contextual quadratic inequality problems? an analysis of learners’ written responses, in conjunction with the assessment rubric given in table 1, revealed strategies they used to solve the quadratic inequality problems. learners’ strategies are summarised in table 2. table 1: rubric for quadratic inequality problem solving test. table 2: frequencies of learners using quadratic inequality problem-solving strategies. table 2 shows that there was a considerable difference between the two groups in the use of problem-solving strategies. seventeen (85%) learners in the experimental group used graphical strategies (i.e. making a model or diagram) while only six (30%) from the control group did so. the other most frequently used problem-solving strategies by learners from the experimental group were making tables (10 learners, 50%), looking for patterns (13 learners, 65%) and checking for accuracy (i.e. logical reasoning; 13 learners, 65%). this indicates that learners from the experimental group more often made attempts as evidenced by the way the strategies were distributed. learners’ perceptions of the gc-mediated quadratic inequality problem solving environment this section answers the third research sub-question: what are the learners’ perceptions of contextual quadratic inequality problem solving in a gc-mediated environment? in-depth semi-structured interviews were conducted with two purposefully selected learners to explore their perceptions of gc use in solving the context-based quadratic inequality problems. to improve readability, we corrected grammatical errors and omitted filler words, but maintained the integrity of the meaning attributable to the interviewees (oregon department of transportation research section, 2010). below are the transcripts involving a female learner, mk (pseudonym), and a male learner, gh (pseudonym): teacher-researcher (tr): does the use of the graphing calculator help you to understand the contextual quadratic inequality problems? mk yes, it did help me to understand the problems of quadratic inequalities. tr please explain how it did help you. mk a graphing calculator uses the graphs or number lines to solve the inequalities, which made it easy for me to see where solutions lie. tr does the graphing calculator provide you with opportunities to use more than one method [e.g. table of values, graphs, factors or quadratic formula] in solving quadratic inequality problems? mk yes, it is possible to use more than one method. tr can you explain a little bit. mk i used the graphing calculator to calculate the critical values, sketch the graph and then check the solution from the table of values. tr did the graphing calculator use affect your choice of strategies to be applied in solving quadratic inequality problems? mk yes, it does. i never liked using graphs to solve quadratic inequalities but the graphing calculator made me use and enjoy them. tr which of the following strategies were most helpful in solving quadratic inequality problems: graphs, tables, checking, drawings, etc.? mk the most helpful strategies for solving quadratic inequality problems are drawing graphs and making table of values. tr using a graphing calculator in quadratic inequality problems does not assist to master the concepts of critical value, interval notation, behaviour of function. mk actually it does because i was able to figure out the behaviour of the function and to use critical values for writing the interval notation. tr does the use of the graphing calculator help you to understand the contextual quadratic inequality problems? gh absolutely yes. tr please explain how it did help you. gh because the graphing calculator shows the quadratic graphs, i was able to figure out the x-intercepts and the shape of the graph, and then predict the region of the solution. tr does the graphing calculator provide you with opportunities to use more than one method [e.g. table of values, graphs, factors or quadratic formula] in solving quadratic inequality problems? gh definitely, it allows me to use both graphs, table of values and number lines. tr can you elaborate a little bit. gh the graphing calculator use afforded me a chance to work with different methods such as calculating the critical values and deciding the inequality solutions from the sketched graph. tr using graphing calculator does not affect your choice of strategies to be applied in solving quadratic inequality problems. mk i think it can affect because i now solve the quadratic inequalities using graphs and number lines that are usually displayed on the graphing calculator. the use of the graphing calculator made me feel more comfortable with graphs. tr which of the following strategies were most helpful in solving quadratic inequality problems: graphs, tables, checking, drawings, etc.? gh the most helpful strategies were drawing graphs, number lines and making table of values. tr using a graphing calculator in quadratic inequality problems does not assist to understand the concepts of critical value, domain, interval notation, behaviour of function. gh oh yes, it does assist. i am now able to predict the behaviour of the function and to write domain using correct interval notations. learners’ responses to the interview were positive and almost similar as they affirmed gc use assisted them to understand quadratic inequality problems better. they responded by saying that gc use ‘definitely’ and ‘absolutely’ helped them to ‘understand the problem’, ‘predict the behaviour of function’ and ‘figure out the region of the solution’ in quadratic inequalities. this showed that the learners got familiar with the gc environment as envisaged by the activity principle and phenomenological views of ict tool use in the mathematics classroom. furthermore, learners indicated that the gc use provided them with opportunities of employing more than one approach to solve quadratic inequalities. this meant learners were able to switch between the various representations with ease and could then have more time to reflect on the reasonableness of their solutions and engage in additional random exploration of the topic tasks. admittedly, learners indicated that the gc use had an impact on their choices of problem-solving strategies. both learners mentioned that the availability of the gc inspired them to use graphical strategies and number lines as they are frequently displayed. through the use of the gc learners were able to master the concepts of critical values, domain, interval notation and graphs. these concepts were critical in determining the solutions of the quadratic inequalities. using the t1-83 plus gc, learners can see how the quadratic inequality is solved graphically as displayed on its screen in figure 3. figure 3: solution of a quadratic inequality displayed on the graphing calculator. in this case, the graph displays the solution set of the quadratic inequality x2 − 2x − 3 > 0, which is a disjunction of x = −1 and x = 3. the use of the gc, according to the learners, demonstrates the graph with ‘its x-intercepts which are the critical values’ for the inequality. this enabled the learners to visualise the graph with critical values which made it easy to determine the solution of inequalities. the visual images of the graphical representations helped learners to understand easily the connections between the graphs and solutions of quadratic inequalities. learners of the control group, in contrast, were exposed to the three methods of solving quadratic inequalities prescribed in the caps document (i.e. graphical, table and number line). while the methods have visual aspects, some of the learners mastered the quadratic inequalities through memorising the procedures. as such, the learners were able to determine the critical values (x-intercepts) of the quadratic expressions but could not write meaningful solutions of the inequalities. learners further struggled to express their solutions in set builder or interval notations. this is consistent with the findings of the dbe (2018), which indicated that learners treated the inequality as an equation and their solutions did not make sense. discussion of results in this section we summarise and discuss our findings focusing on the quantitative for researchers who plan to integrate the gc in their classrooms. effect of graphing calculator use on learners’ quadratic inequality problem solving achievement this study addresses the issue of whether gc use affected the learners’ achievement and strategies in solving quadratic inequality problems. in this study the gc appeared to be an appropriate instrument for improving learner’s achievements. the findings, using an independent t-test, showed that there was a statistically significant difference in the mean scores of the post-test for the experimental and control groups. such findings are compatible with the results obtained in the previous research, which showed that learners exposed to gc-mediated environment can achieve significantly higher scores in problem solving than those who were not (rich, 1991; tan et al., 2011). this study adds the case of quadratic inequalities in which rme principles were observed in terms of both the contexts of problems and the integration of the gc, where each learner had access to their own device. the results also support thomas’s (2016) findings that learners who persistently use a gc can achieve more learning gains and be capable of greater flexibility in their problem-solving strategies. incorporating kutzler’s (2000) and kenney’s (2014) problem-solving strategies into the gc meditated classroom could have led to more systematic experimental group students’ involvement in quadratic inequality problem solving activities and more insightful reflection on the effectiveness of their problem-solving strategies. however, not all learners of the experimental group were able to operate the gc in the intended way. some of them experienced technical and conceptual obstacles leading to 15 ‘no attempts’. the number of ‘no attempts’ was, nevertheless, higher (double) for the control group learners. learners experienced technical difficulties in setting appropriate viewing windows of gcs. they also made some syntactic errors in entering negative numbers with a different minus sign than used for subtraction. this is in line with the previous findings that revealed that the use of gc in teaching and learning often turns out to be more complex than expected (karadeniz, 2015). effect of graphing calculator use on learners’ problem-solving strategies in written tasks the findings from the learners’ written task solutions revealed that there was a difference between the experimental group and control group in the way they answered and spread their problem-solving strategies in quadratic inequalities. despite the fact that the control group was exposed to different visual methods (e.g. graph and number line), they had to memorise some of the necessary procedures to understand the concept. the use of a gc as a visual tool helped learners in the experimental group to intuitively master the properties of algebraic processes, which are missed during the traditional learning process. with the use of a gc, learners can repeat the algebraic processes supported by the visual images and representations. in that regard, the results did show that learners approached questions differently. specifically, the experimental group consistently used graphical and numerical representations to solve problems and performed better. this is consistent with the results of earlier research, which indicated that using gcs leads to a different distribution of solution strategies, particularly the increased use of graphical strategies (chen & lai, 2015; harskamp, suhre, & van streun, 2000). this further shows that gc use remarkably expanded the learners’ repertoire of problem-solving strategies to include more visual procedures. this was also supported by other researchers who reviewed that learners become better problem solvers when gcs are used in class (parrot & leong, 2018). this indicates that the gc-supported learning modelled by kutzler’s (2000) and kenney’s (2014) problem-solving strategies afforded learners of the experimental group opportunities to explore quadratic inequality problems by linking strategies such as making a model or diagram, using table of values, linking verbal and symbolic representations and using logical reasoning to infer the meaning within the context of the problem. learners’ perceptions of graphing calculator use in quadratic inequality problem solving the results of the in-depth interviews of the two learners from the experimental group showed that the gc availability had a positive effect on their learning of contextual quadratic inequality problems. in their responses, learners explained that graphing calculators enabled them to use more than one problem-solving strategy (e.g. graph, table of values, number line, etc.). the use of different strategies enhanced the learners’ understanding of quadratic inequality problems. this supports the findings obtained by prior researchers (montijo, 2017; parrot & leong, 2018) reporting that the multiple linked representations provided by a gc in the form of graphic, tabular and computation improved the learners’ global view of problem solving. according to kenney (2014), such problem-solving strategies may be beneficial for learners as they link the algebraic and geometrical affordances. the use of different strategies enabled learners to enjoy and feel comfortable with quadratic inequality problems as they were able to move back and forth among these forms. this is consistent with the results of parrot and leong (2018) who found that such representations give learners more time to think about the problem itself without focusing on long algebraic procedures. the results also align with ndlovu’s (2019) findings that the use of a gc increased the learners’ confidence in solving more challenging problems. in addition, the findings showed that learners perceived that gc use affected their choices of problem-solving strategies in quadratic inequalities. the interview results indicate that the consistent use of gc stimulated and increased the learners’ application of graphical strategies. this finding is supported by chen and lai (2015) who reported that gc use led to changes in the mathematics learners’ solution strategies that they employed. we also found that the use of gcs enabled learners to understand and feel comfortable with functional graphs. the indications are that learners mastered well the properties of functions which made it possible for learners to visualise the solutions of the quadratic inequality problems. these results are consistent with previous studies which indicated that the use of gcs in teaching and learning was helpful to learners’ cognitive understanding, visualisation and achievement in mathematics classrooms (karadeniz, 2015). this is in line with the recommendations of dbe (2018) that integrating algebra with functions can improve learners’ visual understanding of quadratic inequalities. this means learners can develop a balance between algebraic solutions and graphical strategies through the use of a gc. conclusion information obtained from the results of this research showed that exposing learners to a gc-supported learning environment helped them to improve their achievements and to develop formal problem-solving strategies in quadratic inequalities compared to those who used non-gc methods. the use of a gc created an exceptionally enabling learning environment that became more suitable for learners to be engaged with experimental activities. these experimental activities helped the learners to critically identify the main facts of the problem, to draw its model supported by charts, tables and visual images, and to reflect on the selected strategies of solving quadratic inequalities. the gc served as a tool for integrating algebra with functions so that learners can have a better visual understanding of quadratic inequalities. using a gc potentially raised learners’ cognitive achievement in quadratic inequalities, in particular as they were able to observe the different representations connected to the concept. the varied representations (algebraic, arithmetic, geometric, number pattern) of quadratic inequalities helped learners to gain insight into the big ideas in mathematics. as learners switched between different representations and contexts of the mathematical concept, they were able to realise that such differences in fact are interconnected and interrelated by a single mathematical idea. such perceptions are critical in raising learners’ procedural fluency and applying abstract concepts to concrete real-life mathematical experiences. the idea is that learners who are consistently exposed to gc-supported learning can see the close relationship between concepts and procedures, which is a strong foundation of this study to write meaningful interval notations of quadratic inequalities as solution sets. recommendations for teachers the findings of our research study led us to the following recommendations for teachers who desire to adopt gc-supported lessons in quadratic inequality problems: teachers should consider integrating gcs in teaching and learning of quadratic inequalities as their availability in class can improve learners’ academic achievement and solution strategies. this tool, gc, potentially integrates algebra with functions so that learners can have a better visual representation of quadratic inequalities. teachers should consider using gcs in their teaching and learning of quadratic inequalities as learners are afforded opportunities to employ different approaches (graphical, table of values, number line) for increasing the learners’ choices of problem-solving strategies. teachers should consider integrating gcs in teaching and learning quadratic inequalities as they increase the use of a graphical approach that enables learners to connect the x-intercepts with quadratic inequality solutions. this can help learners to distinguish equations from inequalities. teacher training institutions should assume the lead in developing preservice teachers on how to carefully create a rich learning environment supported by the use of gcs for learners to realise the multiple representations (graphical, symbolic and numeric) of quadratic inequalities. on the other hand, we recommend that learners should be given more time to use gcs in order to fully master all their key functions and to reduce syntactic errors in entering negative coefficients of quadratic inequalities. this can also help to minimise the technical difficulties experienced in setting appropriate viewing screens or windows of gcs. it is further recommended that educational policymakers, including universities, must support the drive for progressive changes in the educational approach, in both teachers’ and learners’ roles in classrooms, where technology is in use. if learners are allowed to use gcs in their assessment, they can have a clear impression of their possible effects on their solution strategies of quadratic inequality problems. acknowledgements gauteng department of education; mathematics teachers and 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(2005). understanding student’s quadratic inequality misconception through an in-depth interview. makalah disajikan dalam 3. international qualitative research convention 2005. qualitative research experience across discipline, pp. 31–45. universiti teknologi malaysia & qualitative research association of malaysia: sofitel palm resort, senai, johor, malaysia. 36-49 naidoo & huntley.docx 36 pythagoras, 72, 36-49 (december 2010) procepts and proceptual solutions  in undergraduate mathematics    inderasan naidoo  department of mathematical sciences  university of south africa  naidoi@unisa.ac.za  belinda huntley  st john’s college, johannesburg  huntley@stjohnscollege.co.za    this  paper  explores  and  promotes  the  notion  of  ‘procept’  in  an  undergraduate  mathematics course in linear algebra for first year pure and engineering students. on the  basis  of  students’  preference  for  procedural  to  conceptual  solutions  to  mathematical  problems,  this  paper  augments  the  role  of  certain  concepts  in  pure  and  applied  mathematics, particularly in the problem‐solving approaches at the undergraduate level  by providing novel solutions  to problems solved  in  the usual  traditional manner. the  development of the concept of ‘procept’ and its applicability to mathematics teaching and  learning  is  important to mathematics education research and tertiary pure and applied  mathematics  didactics  in  south  africa,  welcoming  the  amalgamation  of  the  theories  developed  at  pre‐tertiary  level  mathematics  with  theorems  and  proof  at  the  undergraduate level.  this paper emanated from a series of semester lectures that we presented to first year pure and engineering mathematics students in an undergraduate linear algebra course at a large university in gauteng in south africa. the novelty of this paper is twofold. first, it transcends the boundaries between elementary (and secondary) mathematics education research and tertiary pure and applied mathematics didactics, welcoming the amalgamation of the theories developed at pre-tertiary level mathematics education with an approach to teaching theorems and proof at the undergraduate level. the paper explores and promotes the notion of procept in an undergraduate mathematics context which emanated from the teaching unit of skew lines in 3-dimensional space, . secondly, it augments the role of certain concepts in pure and applied mathematics, particularly in the problem-solving approaches at the undergraduate level by providing novel solutions to problems solved in the usual traditional manner. the classical approach in determining whether two non-parallel lines in are skew is to equate their parametric equations and then solve the resultant system of equations by gaussian elimination. this procedural approach to solving such a problem has always been the traditional computational mode of instruction in an undergraduate level linear algebra course. the nature of the solution then determines the orientation of the lines, either they intersect or they are skew. we provide an immediate (vector) characterisation that determines when two non-parallel lines in are skew. we also propose a novel approach to determine the point of intersection of two lines in in contrast to the traditional, procedural approach using gaussian elimination. furthermore, we consider and propose a proceptual solution to the shortest vector between two skew lines in . inderasan naidoo & belinda huntley 37 process, object and symbol are integrated in a common entity defined by gray and tall (1994), as an elementary procept. this amalgam of a process that produces a mathematical object, and a symbol that either represents the process or the object are considered for simple arithmetic in gray and tall (1991). the notion of a procept emphasises the importance of having a single mathematical entity encompassing both a process and an object/concept. this duality between process and concept broadens to a wider audience in undergraduate mathematics. for instance, in differential calculus the partial derivative , lim , , dually represents both process (as gets smaller) and concept (the limit itself). in integral calculus, the definite integral procept of a continuous function on the interval , , lim ∆ is the limit of riemann sums with sample points in the subintervals , of , , represents the area under (for positive ) or the work done by a force in moving an object from to . the symbols ∂ ∂λ and invoke both a process and a concept involved and the expressions above may be viewed in a range of different contexts. the definition of continuity of a function at a point , lim as a mathematical equation is an amalgam of two elementary procepts. in undergraduate linear algebra, the symbol for two vectors in is a dichotomy of a process (a determinant calculation) and an outcome that produces a mathematical object (a vector cross product). the application of the cross product procept of two vectors in tends to be fairly procedural in most traditional undergraduate texts (e.g. anton & rorres, 2005; edwards & penny, 1988; salas & hille, 1995). the same can be said about the procedure of gaussian elimination (or the least-squares solution) and its applications. undergraduate students are expected to perform routine calculations involving the cross product of two vectors in , for example, finding the equation of a plane containing three points, finding the area of a parallelogram or triangle, or determining the volume of a parallelepiped or tetrahedron in applications of vectors in . most of the instruction at this level is aimed at equipping students with the necessary procedural knowledge in which the focus is on the process to compute or manipulate. the algorithm of gaussian elimination in finding the solution to a system of equations is another typical example of the application of procedural knowledge in solving particular undergraduate linear algebra problems, for example, determining the geometric orientation of three planes in or finding bases for the eigenspaces of a 3 3 matrix. piaget (1972) highlighted cognitive development as involving the duality of process and concept: ...mathematical entities move from one level to another; an operation on such “entities” becomes in its turn an object of the theory, and this process is repeated until we reach structures that are alternately structuring or being structured by “stronger” structures. (p. 70) skemp (1987) proposed a general varifocal theory in which a schema seen as a whole is a concept and a concept seen in detail is a schema. dubinsky (1991) speaks of encapsulation of processes as objects, sfard (1991) of reification of processes as objects, and gray and tall (1991, 1994) see the symbol as a pivot between process and concept  the notion of procept. gray and tall (1994) introduced the idea of a procept as the symbolisation of an object that arises from processes carried out on other objects. such procepts can be viewed in two distinct but related ways, as a process or as an object. a procept is considered as a cognitive construct in which the symbol can switch from a focus on a process to compute or manipulate, to a concept that may be thought of as an entity that can be manipulated. thus, a procept refers to the amalgam of concept and process represented by the same symbol. procepts and proceptual solutions in undergraduate mathematics 38 tall et al. (2001) believe that procepts are at the root of human ability to manipulate mathematical ideas in arithmetic, algebra and other theories involving manipulable symbols. they allow the biological brain to switch effortlessly from doing a process to thinking about a concept in a minimal way. (p. 81) a student possessing a strong proceptual understanding of some mathematical idea is able to effortlessly switch between viewing it as a process (and doing the process) and seeing it in the condensed form as an object. based on these ideas of a procept and proceptual thinking and understanding (a process of one kind and a concept of another) we propose the notion of a proceptual solution as the type of alternative solution described in this paper, a solution realised with the integration of a particular process and a particular concept. we highlight that this paper associates particularly the work of gray and tall (1991, 1994) to the mechanics of certain aspects of undergraduate mathematics. in the examples chosen in this paper, by doing the processes of gaussian elimination, a least-squares solution technique, and applying the procedure of cramer's rule together with relating the concept of the cross product of two vectors in , we develop an alternative solution to finding the point of intersection of two lines in , as well as a similar type of solution in determining the shortest vector between two skew lines. these examples serve to illustrate the differences between the more traditional, fairly procedural approaches to methods of solution with the alternative proceptual solutions encompassing both the process and the concept. in the next section we provide the necessary notation and preliminary material on vectors and lines in that we require. then in examples 1 and 2 we provide a traditional solution via gaussian elimination to find the point of intersection of two lines in , as well as the shortest distance between two lines that are skew. certain aspects of the solution presented in examples 1 and 2 will be subsequently required. we then consider linear independence and the necessary theory on the rank of a matrix. a criterion for the existence of the intersection of two lines in is given and an alternative solution to the point of intersection is presented, derived from using the rank of a matrix and cramer's rule. the point of intersection is considered as a least-squares problem as stated in theorems 5 and 6 where a vector solution is realised by integrating the procedure in solving a (unique) least-squares problem with the concepts of the dot and cross product. the section on proceptual solutions concludes with arriving at a vector solution to the shortest vector between two skew lines in and is similar to the approach of finding the point of intersection, save for the use of the process of solution by matrix inversion or that of cramer’s rule. preliminaries given the points , , , , , and , , in , we use the bold lowercase vector notation of their position vectors as , and respectively. we will also make use of the following notations and matrix representations of given vectors. we will form the matrix with vectors as row vectors given by or column vectors . the vectors , and are linearly independent provided that for scalars , the equation has only the trivial solution 0 for each 1, 2, 3. otherwise, the vectors , and are linearly dependent. alternatively, a viable result disseminated (and proved) to undergraduate students using a inderasan naidoo & belinda huntley 39 matrix determinant is that , and are linearly dependent if and only if 0. (edwards & penny, 1988, p. 134) given and the position vectors of the points , , and , , the joining vector from to in is , , . the projection of the vector in the direction of, or on is and the scalar component of on is where ∑ is the usual dot product and is the euclidean norm of . the cross product of with is the vector calculated as the determinant1 where , and are the standard basis vectors of . a line ℓ in is usually characterised by two points on the line. alternatively, ℓ is determined by a point , , on the line and a non-zero direction vector , , for the line. the vector equation for ℓ is then given by ℓ: , where , , is the position vector for each point , , on ℓ. varying the scalar through , traces out the line ℓ. using the algebra of vectors, one then realises the parametric equations for ℓ as , . the distinct lines ℓ : , and ℓ : , are parallel in the case when their direction vectors and are parallel, i.e. for some scalar or using the cross product . the following result will be used in subsequent sections of this paper. lemma 1: let , and be non-zero vectors in , then 1. if and only if 0. 2. (lagrange’s identity) 3. 4. 1 we all know that a 3 3 determinant is a mapping of the form  , which is certainly not an element of , whereas for all vectors and in . what we should see/read in is simply a little memory aid, i.e. a ‘recipe’ to help us to remember how to calculate the vector product. in other words, it is as if you calculate a determinant, and treat the vectors , and as numbers for the time being. it should be emphasised that this is not really a determinant. we regularly fall into similar traps when we deal with determinants, by saying, for example, “multiply the second row of the determinant by 2”, instead of “multiply the second row of the matrix of which the determinant is taken, by 2”. if one is not properly informed and aware of the difference between the ‘loose and informal’ and the ‘formal and strict’ way of referring to these processes, the whole point of distinguishing between process and concept could be missed. procepts and proceptual solutions in undergraduate mathematics 40 the expansion of the vector triple product (3) in the above lemma appears in craig (1951) and elementary results on vector and matrix algebra may be accessed in anton and rorres (2005), or edwards and penny (1988). the point of intersection: gaussian elimination the lines ℓ : , and ℓ : , intersect if and only if there are unique reals and for which (salas & hille, 1995). the two lines are skew lines if they are not parallel and do not intersect. the (traditional) approach of salas and hille to determine whether the distinct lines intersect (and then to find the point of intersection) is to consider the parametric equations for ℓ and ℓ . the two equations are equated, using the parametric form of , to realise a 3 2 linear system of equations. thereafter, the method of gaussian elimination is used to resolve the non-square system. if no solution arises in the resulting algorithm, the conclusion is that the lines do not meet. further, consulting the direction vectors it is noted that, if and are not parallel then the lines are skew. in addition, the shortest distance between lines ℓ and ℓ is calculated using a projection vector, … . . equation 1 if the system has a unique solution given by and then the point of intersection is determined using the value (or ) in the parametric or vector equation for ℓ (or ℓ ), i.e. finding in ℓ or in ℓ . we illustrate this application of procedural knowledge in the two concrete examples below. example 1 consider the lines with vector equations : 8, 14, 7 2, 1, 5 and : 41, 8, 7 1,4,1 . for the above lines determine if the lines intersect. if so, find the point of intersection. if not, then find the shortest distance between ℓ and ℓ . solution: we follow the classical process in salas and hille (1995) where ℓ and ℓ are parametrically represented as ℓ : 8 14 7 2 ℓ : and 5 41 8 7 4 . the above parametric equations are equated to give 8 2 14 7 5 41 8 4 7 the above then produces the 3 2 system of equations 2 4 5 49 22 0 the augmented matrix : : : 49 22 0 of this system is then solved by gaussian elimination. reduction produces the equivalent 1 2 : 0 7 : 0 0 : 49 218 169 from which the last row shows that the system is inderasan naidoo & belinda huntley 41 inconsistent. we can therefore conclude that the lines do not intersect. furthermore, since the direction vectors for ℓ and ℓ are not parallel, the lines are not parallel. consequently, the two lines are skew. to find the shortest distance between ℓ and ℓ , the point 8, 14, 7 that lies on ℓ is chosen and the point 41, 8, 7 that lies on ℓ is used. the procedure is continued by forming the joining vector . the shortest distance is then calculated using equation 1 to give √ , where the direction vector for ℓ is 2, 1, 5 and that of ℓ is 1, 4, 1 . an example where the lines do intersect is given below. the same procedure that is followed in example 1 is employed but with a different consequence. example 2 determine whether the two lines intersect, and if so, find the point of intersection where : 1, 1,5 2, 4, 1 and : 4, 1,4 1,6,1 if the lines do not intersect, then find the shortest distance between them. solution: proceeding as in the previous example would result in the augmented matrix : : : 3 2 1 1 2 : 6 4 : 1 1 : 3 2 1 performing the gaussian algorithm on the augmented matrix gives 1 0 : 0 1 : 0 0 : 1 2 0 the system has the unique solution 1 and 2. the conclusion is then that the two lines ℓ and ℓ intersect in the unique point 5, 7, 3 realised by using either 1 in the parametric equation for ℓ or 2 in the parametric equation for ℓ . the above calculations are the norm presented to students procedurally in solutions to problems of this type (e.g. anton & rorres, 2005; edwards & penny, 1988). we reflect on this technique and ascertain whether the solution methodology can be improved and the problem resolved by vector methods. in this procedural approach no existence criterion is presented to acknowledge that the lines do intersect or not and this formed the basis for the investigation in the next section. the point of intersection: cramer's rule consider the distinct lines ℓ : and ℓ : in with position vectors , , for the point , , on ℓ and , , for the point , , on ℓ . the vectors , , and , , are the direction vectors for ℓ and ℓ respectively. it is immediate that the shortest distance given in equation 1, between ℓ and ℓ provides a necessary and sufficient condition that determines an intersection. certainly, if 0 then ℓ and ℓ must intersect, otherwise ℓ and ℓ are parallel or skew. this translates to a dot product criterion to determine whether ℓ and ℓ intersect or not. since 0 if and only if 0, we may conclude that ℓ and ℓ intersect in a unique point provided that the scalar triple product 0. procepts and proceptual solutions in undergraduate mathematics 42 based on the given vector equations of the two lines, we immediately have the determinant criterion 0 which confirms that ℓ and ℓ intersect. if this determinant is non-zero and the direction vectors and are not parallel, then the lines are skew. this determinant criterion is equivalent to linear dependence. in our particular case of lines in , this criterion for linear dependence may be contextualised in the following result. theorem 1: the distinct non-parallel lines ℓ : and ℓ : in intersect in a unique point if and only if the vectors , and are linearly dependent. this also has a geometric interpretation. the scalar triple product gives the volume | | of the parallelepiped defined by the vectors , and . certainly, if the volume is zero then the three vectors must lie on the same plane. consequently, the two lines must intersect. another geometric interpretation is given by smith and henderson (1985), where it is shown that the skewness of two lines ℓ and ℓ necessitates the existence of a tetrahedron with volume | |. again, if the volume of the tetrahedron is zero, the vectors , and lie on the same plane so that the two lines intersect. once the fact that the two lines ℓ and ℓ do intersect is established, we can proceed with the traditional approach. however, on a closer look behind this traditional procedure, an interesting observation is made based on the existence of an intersection that has just been described. suppose that 0 , then and exist such that , , , , this then gives the linear system which as a matrix equation, becomes in the notation of the defining vectors, the latter matrix equation is … . . equation 2 we note that based on the solutions of example 1 and example 2 in the previous section, the deliberate bold digits in the matrices makes for a shorter route to the solution as described by the matrix equation 2. from the given equations of two lines in , ℓ : and ℓ : we may directly form the augmented matrix : or : and then pursue the gaussian algorithm in attaining a solution where . if ℓ and ℓ intersect (non-zero scalar triple product of the vectors , and ), the solution for and in equation 2 may be simplified with the use of cramer's rule as illustrated below. since equation 2 is consistent with a unique value for and , the rank of the coefficient matrix and that of the augmented matrix must be given by the existence and uniqueness theorem of a solution of a system of linear equations in theorem 2 below (lipschutz & lipson, 2001, p. 79). inderasan naidoo & belinda huntley 43 theorem 2: consider a system of linear equations in unknowns with augmented matrix : . then a) the system has a solution if and only if b) the solution is unique if and only if . the rank of a matrix , symbolically , is usually defined as the number of non-zero rows in an echelon form of . the symbol is an amalgam of a process (finding an echelon form of using gaussian elimination or theorem 3 below) to produce a mathematical object. indeed, the rank of a matrix is also a procept. now the rank of a matrix may be determined by the following determinant criterion. the proof may be found in beesack (1962). theorem 3: the rank of a matrix is if and only if has some submatrix with a non-zero determinant and all square submatrices of larger size have determinant zero. in examining the matrix equation 2, at least one of the three 2 2 submatrices of the coefficient matrix has to have non-zero determinant. selecting that submatrix realises a 2 2 square system with appropriate omission of the corresponding component of the column vector given by . applying cramer's rule to the resulting square system gives the unique solution to and without the use of gaussian elimination. we illustrate this next by providing an alternative solution to example 2 via cramer's rule using theorem 4: theorem 4: in equation 2, if the coefficient matrix has at least one 2 2 submatrix whose determinant is non-zero, then equation 2 has the same solution as that of the (sub)augmented matrix determined by if and only if 0 if and only if , and are linearly dependent. proof: suppose that the coefficient matrix of equation 2 has a submatrix such that 0 and that the solution to the subsystem determined by is the same as that of equation 2. since 0, the subsystem determined by has a unique solution, say and . these are then particular solutions to equation 2 so that by theorem 2, : 2. consequently by theorem 3, 0, showing that the three vectors , and are linearly dependent. for the sufficiency, suppose that , and are linearly dependent and that the coefficient matrix of equation 2 has a submatrix with non-zero determinant. since the three vectors are linearly dependent, the 3 3 matrix whose columns are that of , and has zero determinant. using determinant properties (column switching and factorisation) we see that 0 consequently, since 0 the rank of the coefficient matrix of equation 2 is 2 and the rank of the augmented matrix, : 2. thus equation 2 has a unique solution. suppose that . we show that the subsystem determined by , namely has the same solution as that of equation 2. the solution to this subsystem, say and , may be resolved by cramer's rule. thereafter, as can be verified, a routine substitution of and into satisfies equation 2. hence, since equation 2 has a unique solution it is the same as that of the subsystem determined by . the same conclusion is realised if is any of the other two submatrices with non-zero determinant. returning to example 2, we may use the method given by theorem 1 in determining whether the two lines intersect and finding the point of intersection (by theorem 4), if it exists. we have procepts and proceptual solutions in undergraduate mathematics 44 3, 2, 1 , 2, 4, 1 and 1, 6, 1 . using the scalar triple product (as a matrix determinant), 0. thus ℓ and ℓ intersect in a unique point that may be found by solving for and in the (2-rank) equation 2, i.e. 2 1 4 6 1 1 3 2 1 . the coefficient matrix has rank 2 and all its 2 2 submatrices have non-zero determinant (easily verified). the unique solution may be derived using cramer's rule by solving the subsystem, say 2 1 4 6 3 2 to get 2 and 1. using 2 in ℓ (or 1 in ℓ ) we get the position vector for the point of intersection 5, 7, 3 . selecting any of the other two subsystems will give this same solution. hence, it all culminates in identifying and selecting a submatrix (that has non-zero determinant) of the coefficient matrix of equation 2 in arriving at a solution without the use of the gaussian algorithm. proceptual solutions the point of intersection between two skew lines according to tall (1995), mathematical growth starts from perceptions of, and actions on, objects in the environment. successful actions on objects use symbolic representations flexibly as procepts – processes to do and concepts to think about – in arithmetic and algebra. the concept image of a procept uses the symbol to link to suitable processes and relationships in the cognitive structure. thus, although we may not have anything in our mind which is like a physical object, we have symbols that we can manipulate as if they were mental objects. tall et al. (2001) consider the word procedure as a specific sequence of steps carried out one step at a time, while the term process is used in a more general sense to include any number of procedures with the same effect: those who are procedurally oriented are limited to a particular procedure, with attention focused on the steps themselves, whilst those who see symbolism as process or concept have a more efficient use of cognitive processing. (tall et al., 2001, p. 90) in this section we will recall some basic facts on the least-squares problem. thereafter, we consider the least-squares procedure in solving an inconsistent system of equations together with the cross product concept in to arrive at a proceptual solution to the point of intersection of two distinct lines in . suppose that is an matrix with and . let be any given vector. the least-squares problem involves finding a vector such that the euclidean norm is a minimum, i.e for each . is called a least-squares solution of . it is well-known (e.g. osborne, 1961; edwards & penny, 1988) that a least-squares solution satisfies the normal equation . the algorithm to solve a least-squares problem would involve the process of finding the product . thereafter, computing and then performing gaussian elimination on the augmented matrix for the normal equation . naturally, there is only one least-squares solution if the matrix is invertible. a useful criterion for the unique least-squares solution is given by theorem 5 in lay (2005) which is particular to our situation. theorem 5: the matrix is invertible if and only if the columns of are linearly independent. in this case, the equation has only one least-squares solution , and it is given by . we also have the following well-known theorem (see e.g. edwards & penny, 1988): theorem 6: if the matrix has rank , then the matrix is invertible. inderasan naidoo & belinda huntley 45 returning to , the lines ℓ : and ℓ : and our equation 2 , we see that if the three vectors , and are linearly dependent then the coefficient matrix of equation 2 has rank 2 so that is invertible. furthermore, the direction vectors are not parallel so that the columns of are linearly independent. by the results above, equation 2 has a unique least-squares solution given by where . however, since we are in , is a 2 2 matrix and in this unique solution case, is easily found using its determinant and its adjoint. the process in finding the unique value of and is continued by finding and thereafter computing the final product . now, amalgamating into the above process the vector concepts of the cross product and the dot product, produces the following direct solution. in the context of the vector equations of the lines, we realise a vector solution to the point of intersection without inference to matrix algebra. we first remark that and by lagrange's identity in lemma 1, . then 1 1 1 1 1 … . lemma 1 3 1 1 which is the direct (proceptual) least-squares solution to equation 2. thus, two distinct non-parallel lines ℓ : and ℓ : in intersect in a unique point whose position vector is or , where and procepts and proceptual solutions in undergraduate mathematics 46 if and only if the vectors , and are linearly dependent. it is noteworthy that and in the above solution can be easily resolved using a determinant calculation together with that of the vector as 1 and 1 . the shortest vector between two skew lines we now consider the procedure of finding a solution to the shortest vector between two skew lines in by the process of matrix inversion (or cramer's rule) incorporating the concept of the cross product and thus realising a proceptual solution. to this end, consider the two skew lines ℓ : and ℓ : in . apart from directly finding the shortest distance between ℓ and ℓ , using the norm of the projection of in the direction of the vector , one also procedurally determines the shortest vector between ℓ and ℓ as follows. the general vector joining ℓ and ℓ is found algebraically as . the shortest vector between ℓ and ℓ is then the one which is perpendicular to both ℓ and ℓ . in effect this gives the two equations 0 and 0 realising a 2 2 system of linear equations in and . the approach is to solve for these two variables using cramer's rule. once and are found, the shortest vector is calculated. although tedious, this approach also provides a way to the shortest distance between the two lines. to develop a proceptual solution to the above problem, we now solve the dot product equations above for the arbitrary skew lines ℓ and ℓ . lemma 1 in the preliminary section will be required. now, 0 0 and 0 0 produces the matrix equation … equation 3 on closer inspection, the above equation 3 is precisely … . equation 4 where as before. since ℓ and ℓ are skew their direction vectors, and respectively, are not parallel. thus 0. if one considers the coefficient matrix in the above equation 4, we have by lagrange's identity 0. hence, we have the same unique solution 1 . cramer's rule could also be used with the use of lemma 1 to produce the same result. alternatively, the shortest vector between the skew lines ℓ and ℓ may be found using differential calculus by minimising the norm of the general joining vector . inderasan naidoo & belinda huntley 47 in finding just the shortest distance between the skew lines, srinivasan (1999) illustrates this procedure in an example, but presents no underlying theory. we now examine the underlying theory governing this procedure as follows: let , . expanding, , 2 2 2 . minimising would produce and that will give the shortest vector between ℓ and ℓ . to this end we generate two equations 0 and 0. then 0 2 2 2 0 and 0 produces the same matrix equation 3, the solution of which is determined above. looking at the solution for the point of intersection (of the two lines), it turns out that the values of 1 and 1 are exactly the same as that of the parameters for the shortest vector between the two skew lines. in all, the linear dependence of the three vectors , and is crucial in the orientation of the two nonparallel lines ℓ : and ℓ : in . if these three vectors are linearly dependent, then the two lines intersect in a unique point. otherwise the two lines are skew. we summarise these findings in our concluding theorem. theorem 7: let ℓ : and ℓ : be two non-parallel lines in . then a) ℓ and ℓ intersect in a unique point whose position vector is given by or if and only if 0 if and only if , and are linearly dependent b) ℓ and ℓ are skew and the shortest vector between ℓ and ℓ is given by if and only if 0 if and only if , and are linearly independent where 1 and 1 . returning to the least-squares problem, when a least-squares solution is found for , the euclidean norm is the least-squares error. in the inconsistent system equation 2 (two skew procepts and proceptual solutions in undergraduate mathematics 48 lines) the least-squares error is the norm of the shortest vector (the shortest distance between the two skew lines). indeed, if the least-squares error is zero then the lines intersect in a unique point. concluding remarks in south africa, the general perception is that secondary school teaching of mathematics tends to be fairly procedural and that students that enter university are better equipped to deal with procedural problems than with conceptual problems (engelbrecht, harding, & potgieter, 2005). teaching for procedural knowledge means teaching definitions, symbols and isolated skills in an expository manner without first focusing on building deep, connected meaning to support those concepts (skemp, 1987). teaching for conceptual knowledge, on the other hand, begins with posing problems that require students to reason flexibly. through the solution process, students make connections to what they already know, thus allowing them to extend their prior knowledge and transfer it to new situations (national council of teachers of mathematics, 2000). this paper introduces proceptual solutions as a type of alternative solution to problems at tertiary level mathematics, the particular case of solving for the point of intersection of two lines and the shortest vector between two skew lines. such proceptual solutions require a more efficient use of cognitive processing, as they require a cognitive switch between doing a process and thinking about a concept. we also propose and encourage a proceptual approach to mathematics instruction at tertiary level. such a proceptual view, which in our example amalgamates the gaussian elimination, cramer's rule and leastsquares solution processes and the dot product and cross product concepts, encourages a cognitive switch from a focus on the process of algorithmic manipulations of symbols to a deeper, conceptual understanding and thinking about the concepts. as proceptual thinking grows in conceptual richness, procepts can be manipulated as simple symbols at higher levels or scaffolded to perform computations, to be decomposed or recomposed at will. the unique features of the proceptual solution as described in this paper, suggest that it might be of benefit for mathematics educators to look for such solutions in the teaching and learning of mathematics. we also mention that the direct (vector) solutions presented allow greater access to such type of problems and deeper applications of vector concepts in the sciences and engineering undergraduate linear algebra courses. references anton, h., & rorres, c. 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(1995). cognitive growth in elementary and advanced mathematical thinking. in l. meira & d. carraher (eds.), proceedings of the 19th conference of the international group for the psychology of mathematics education (vol. 1, pp. 61-75). recife, brazil: universidade federal de pernambuco. tall, d., gray, e., bin ali, m., crowley, l., demarois, p., mcgowen, m., … yusof, y. (2001). symbols and the bifurcation between procedural and conceptual thinking. canadian journal of science, mathematics and technology education, 1(1), 81-104. doi: 10.1080/14926150109556452 article information authors: caroline long1 tim dunne2 hendrik de kock3 affiliations: 1centre for evaluation and assessment, faculty of education, university of pretoria, south africa 2department of statistical sciences, university of cape town, south africa 3independent consultant, south africa correspondence to: caroline long email: caroline.long@up.ac.za postal address: po box 2368, houghton 2041, south africa dates: received: 06 aug. 2013 accepted: 10 nov. 2014 published: 12 dec. 2014 how to cite this article: long, c., dunne, t., & de kock, h. (2014). mathematics, curriculum and assessment: the role of taxonomies in the quest for coherence. pythagoras, 35(2), art. #240, 14 pages. http://dx.doi.org/10.4102/pythagoras.v35i2.240 copyright notice: © 2014. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. mathematics, curriculum and assessment: the role of taxonomies in the quest for coherence in this original research... open access • abstract • introduction    • teaching and learning • proposed model • the purpose of bloom's taxonomy and associated challenges    • the difficulty of classifying items in terms of bloom's taxonomies • similar conceptual and taxonomic efforts in mathematics • trends in international mathematics and science study frameworks: a taxonomy • rncs and caps taxonomies in international perspective • challenges in 21st century assessment • selecting a taxonomy • conclusion • acknowledgements    • competing interests    • authors’ contribution • references • appendix 1    • worksheet 2: grade 9: patterns, functions and algebra    • worksheet 4: grade 9: measurement • appendix 2 • appendix 3 • footnotes abstract top ↑ a challenge encountered when monitoring mathematics teaching and learning at high school is that taxonomies such as bloom's, and variations of this work, are not entirely adequate for providing meaningful feedback to teachers beyond very general cognitive categories that are difficult to interpret. challenges of this nature are also encountered in the setting of examinations, where the requirement is to cover a range of skills and cognitive domains. the contestation as to the cognitive level is inevitable as it is necessary to analyse the relationship between the problem and the learners’ background experience. the challenge in the project described in this article was to find descriptive terms that would be meaningful to teachers. the first attempt at providing explicit feedback was to apply the assessment frameworks that include a content component and a cognitive component, namely knowledge, routine procedures, complex procedures and problem solving, currently used in the south african curriculum documents. the second attempt investigated various taxonomies, including those used in international assessments and in mathematics education research, for constructs that teachers of mathematics might find meaningful. the final outcome of this investigation was to apply the dimensions required to understand a mathematical concept proposed by usiskin (2012): the skills-algorithm, property-proof, use-application and representation-metaphor dimension. a feature of these dimensions is that they are not hierarchical; rather, within each of the dimensions, the mathematical task may demand recall but may also demand the highest level of creativity. for our purpose, we developed a two-way matrix using usiskin's dimensions on one axis and a variation of bloom's revised taxonomy on the second axis. our findings are that this two-way matrix provides an alternative to current taxonomies, is more directly applicable to mathematics and provides the necessary coherence required when reporting test results to classroom teachers. in conclusion we discuss the limitations associated with taxonomies for mathematics. introduction top ↑ in the current global educational climate, some degree of regulation is deemed necessary in both the curriculum document prescription and in systemic assessment (kuiper, nieveen & berkvens, 2013). if teachers are to be judged by the outcomes of systemic assessments then at least the components making up the curriculum and the assessment tasks should be made explicit, so that the classroom activities may be aligned and reasoned judgments may be made by teachers concerning their classroom focus. in the first part of this article, we propose a model for assessment that integrates both external and classroom-based educational functions. in order for this model to function optimally, there is a need for coherence in the description of educational objectives, classroom activities and assessment; we therefore need a common language across all three educational processes. in the second part of the article, we provide an overview of the main cognitive categories in bloom's taxonomies, both the original and revised versions, the various frameworks from the trends in international mathematics and science study (timss), as well as the recent south african curricula. bloom's taxonomy of educational objectives was initially conceptualised to assist curriculum planners to specify objectives, to enable the planning of educational experiences and to prepare evaluative devices (bloom, engelhart, furst, hill & krathwohl, 1956, p. 2). because the educational objectives are phrased as general cognitive processes, including activities such as remembering and recalling knowledge, thinking and problem solving, it is necessary to rephrase the particular statements in terms of the subject under consideration (andrich, 2002). in fact, the taxonomy may be ‘validated by demonstrating its consistency with the theoretical views’ that emerge in ‘the field it attempts to order’ (bloom et al., 1956, p. 17). the process of thinking about educational objectives, defining the objectives in terms of the mathematical tasks and relating these tasks to the teaching activities and assessment tasks is an important exercise for the policymakers, curriculum designers, test designers and teachers. an overview of the other taxonomies in use in timss and in the various south african curriculum documents provides the background to the planning, communication and feedback processes for the grade 9-11 monitoring and evaluation project with which the authors are currently engaged. the broad question arising from the project needs is: can the (three) essential elements, an externally designed monitoring component, a classroom-based formative assessment component and a professional development component, be logically and coherently aligned for the purpose of informing teaching and learning? the subquestion is: how may we best design assessment frameworks (the design tool specifying the purposes, structure and content of an assessment instrument) in such a way that there is coherence from the mathematical knowledge to be taught and learned, through the design of a set of assessment instruments, to providing diagnostic and practical feedback to teachers about learner performance and needs? the congruence of curriculum, in the sense of what subject knowledge is to be learned, pedagogy, in terms of how particular concepts and skills are to be learned, and assessment, how the two former elements of the educational experience are to be assessed, is the central concern of this article. teaching and learning good, and especially excellent, teachers cannot all teach to the same recipe. of course, the same mathematics canon underpins their teaching and student learning and the ultimate goal is attainment of the abstract and powerful predicative knowledge of mathematics. but, the route to this end goal along a developmental path is through the operationalisation of mathematics in terms that can be grasped by the young and aspirant mathematicians (see also vergnaud, 1994). also, because learners are able to draw from appropriate contexts the mathematical understanding underpinning the formal mathematics, the creative teacher draws on contexts pertinent to the learners and appropriate for generating mathematical understanding. when teachers plan assessments for their classes, and even for the clusters of classes in a school, the assessment is generally geared to what the learners have been taught. the contents of the test will not be unexpected. the language will be familiar. but in the case of external assessment for qualification purposes, or national systemic studies in which school performance or teacher performance is monitored, or large-scale assessments in which many different countries are involved, the attainment of coherence of language across countries, schools and individual teachers is more difficult to achieve. countervailing these limitations of an external systemic type of testing is the view that the outcomes of systemic-type assessment should not be the only, nor the primary, source of information for a school evaluation (andrich, 2009). the lynchpin of coherence here is attention to the validity of the test components, consistency across the collection sites, so as to generate thus generating reliable test data, and attention to the overall validity of the assessment programme, including the purpose for which the assessment outcome is to be used (messick, 1989). adherence to these requirements is not easy to attain. the attainment demands clear communication about the curriculum contents and about the expected responses from learners. for example, the expectation from the teacher may be that individual concepts, with associated procedures, are acquired; in contrast, the examiner may require the learner to apply problem solving skills to a mathematical task requiring multiple concepts. bloom et al. (1956) refer to the need to understand the educational context of the learner in order to correctly align educational assessment and correctly categorise the cognitive levels. some criticism of external systemic-type testing is noted here. schoenfeld (2007) warns that the type of assessment items generally given in tests of larger scale may often work against the type of problem solving process, extended and thorough in nature, advocated by polya (1957). others point to socio-economic factors that impact on the school culture, and therefore on learning and teaching, that warrant deeper consideration (nichols & berliner, 2005, 2008; usiskin, 2012; wolk, 2012). questions about teacher autonomy and professionalism, and about who has the professional authority to monitor professional teachers, are of paramount importance. these critiques are noted here but are not the concern of this article. webb (1992), in response to dissatisfaction with what he perceives as inadequate testing processes, proposes that mathematics education requires a specific assessment programme. he argues that the then-current assessment models had been based on outdated psychological models designed for purposes no longer relevant. aligned with this view, we explore later in the article a taxonomy proposed by usiskin (2012) that has been operationalised in the university of chicago schools mathematics project textbooks (ucsmp; see http://ucsmp.uchicago.edu/). see for example the algebra textbook, teacher's edition (mcconnell et al., 2002). proposed model top ↑ in answer to the critique of current assessment practices and problems experienced in practice, bennett and gitomer (2009) propose a model that provides articulation between three components: systemic assessment (monitoring), formative assessment (classroom-based diagnostics and classroom teaching) and professional development (see figure 1; see also bennett (2010, 2011). figure 1: model for system assessment. for the external monitoring component we propose along with bennett and gitomer that any mode of assessment should be aligned with cognitive models that are currently acknowledged as supporting learning. the implication is that when a test is designed for monitoring purposes that both the critical subject knowledge and the associated requirements from a cognitive development perspective are to be considered. here we note that modern scientific techniques for the generation and analysis of test data may be used to provide information about the individual student, and to ensure reflection on the test instruments themselves and their constituent items. suitably supported, these methods also permit the tracking of individual needs and performance in the classroom and evidence for the extent of change, progress and redress of performance for the specific child. these techniques, critical to the model, are explored elsewhere. see dunne, long, craig and venter (2012) and long, dunne and mokoena (2014) for discussion on techniques for analysis and reporting of assessment results. the classroom-based formative assessment component of the model requires that teachers be provided with information obtained through the monitoring component. this information should reflect both apparent learner proficiency and item performance characteristics. the feedback needs to be sufficiently specific to enable the teachers to reflect on how best to meet the emerging needs of the learners as detected within the assessment. we acknowledge that there will be circumstances in which this reflection will have to be accompanied by improvement of teacher mathematical skills, which is the intent of the professional development component. the professional development component of this system of interventions should be informed by a deeper insight into the nature of the knowledge domains. in essence, the professional development component is required to build with the teachers a model of mathematical development against which teachers may gauge the progress of their learners. the intended curriculum constitutes an essential but incomplete part of this professional development function. the component also involves identifying with the education role players and re-examining the various necessary factors involved in acquiring mathematical proficiency. teachers and decision-makers together explore the reasons for these critical factors being absent from the school classroom and address strategies to address that absence. in order to promote congruence at the three sites, an explicit model of conceptual development from the perspective of mathematics and of cognitive development from the perspective of learning is required (see vergnaud, 1988). these two components have both a hierarchical trajectory and horizontal breadth encompassing both related mathematics concepts and the required cognitive engagement. in order to make explicit at any one time the breadth and depth of knowledge and the responses required of individuals, an explicit description of the particular knowledge field is required. the purpose of bloom's taxonomy and associated challenges top ↑ when bloom gathered a group of assessment specialists together in the mid-20th century, his purpose was to provide the assessment community with a common language about learning goals which would facilitate communication across subject matter, persons and grade levels. in an attempt to ensure development of ‘higher mental processes’ bloom (1994, p. 2) proposed a common framework for the setting of examinations and for the assessment of these examinations (cited in andrich, 2002, p. 40). as noted previously, this framework was initially conceptualised as an assessment tool which could aid in the classification of items for item banking purposes. the educational objectives explicated in the taxonomy could then be translated into behaviours that would provide evidence that the objective had been achieved (andrich, 2002, p. 41). the aim of the common framework was to help curriculum designers ‘specify objectives so that it becomes easier to plan learning experiences and prepare evaluation devices’ (bloom et al., 1956, p. 2). this common language and vocabulary was to serve as a basis for determining the specific meaning of broad educational goals that informed both the local and the international community. it was also a means for ‘determining the congruence of educational objectives, activities and assessments’ (krathwohl, 2002). the establishment of a broad base of descriptions that could describe a range of educational experience was to guard against the limitations of any curricula that had been narrowly conceptualised. for the assessment community a bank of items covering a range of question types was to provide a solution to the increasing demand for the construction of assessment items. bloom's original taxonomy embraced cognitive, affective and psychomotor skills. the cognitive processes included six major components: knowledge, comprehension, application, analysis, synthesis and evaluation. the affective aspect included five major components: receiving, responding, valuing, organising and characterising. there was a third component named psychomotor skills (bloom et al., 1956). this conceptualisation of educational objectives, embracing a broader view of knowledge and the inferred cognitive responses, was groundbreaking at the time and the effect on education has been an exponential growth in taxonomy use. it is of interest here that though knowledge is specified as a component, defining this component does not prove that straightforward. whilst there is an element of memory involved, in that recalling facts, terms, basic concepts and answers forms part of this component, this component also embraces knowledge of specifics (terminology and specific facts), knowledge of ways and means of dealing with specifics and knowledge of the universal and abstractions in a field (principles and generalisations, theories and structures). the idea behind the taxonomy is that it does not only specify breadth but also unfolds a depth of engagement within a particular topic. in this respect the elements of the taxonomy have been regarded as hierarchical, moving from simple to complex, concrete to abstract, so creating a cumulative hierarchy of knowledge and skills (bloom et al., 1956; krathwohl, 2002). although this idea of hierarchy has been acknowledged as groundbreaking, there has been critique from a number of sources. one of these critiques is that the elements do not necessarily form a hierarchy (usiskin, 2012, and others). another view is that whilst the first three elements of the taxonomy are somewhat hierarchical, the last three, in contrast, can be conceptualised as distinct but parallel (anderson & krathwohl, 2001).1 another critique is that the act of cognition is so highly interrelated and connected across its features that any attempt to classify and confine the thinking process is bound to fail. here we note that bloom et al. (1956) were acutely conscious of the danger of the fragmentation arising from the use of particular focuses and advocated a degree of classification that did least violence to the construct under investigation. this critique is partly addressed by the revised bloom's taxonomy that arranges the existing elements into two dimensions, placing types of knowledge on the vertical dimension and the cognitive process dimensions on the horizontal dimension (see table 1) (krathwohl, 2002). three types and six cognitive processes permit a two-way 3 × 6 array of classifications. table 1: revised bloom's taxonomy manifesting two dimensions used to classify assessment items in the mathematics monitoring and evaluation project. a further observation that results from the use of the taxonomy rather than the original conceptualisation is that for each subject area, the observable behaviours and the different levels of thinking and performance will manifest differently. the abstract nature of the taxonomy requires that for each subject area, the six levels have to be recontextualised by curriculum developers, examiners and classroom teachers who know the subject discipline (andrich, 2002). the reconceptualisation of the taxonomy into two dimensions makes the adaptation for the different subject knowledge domains somewhat easier. the inclusion of metacognitive strategies as a separate category in the revised bloom's taxonomy is regarded by some as a major advance in that without metacognition, it is argued, learning cannot be claimed (anderson et al., 2001). the difficulty of classifying items in terms of bloom's taxonomies it is at this point that we reflect on two sets of three items, designed as formative assessment resources, for the formative assessment component of the project. the first set focuses on algebra (see worksheet 2 in appendix 1). our attempts to classify the items that were originally created to cover a range of cognitive processes proved difficult. the individual items have been given a temporary home populating the cells. but how does one distinguish ‘remember x conceptual knowledge’, ‘understand x procedural knowledge’, and ‘apply factual knowledge x’ (represented in table 1)? we note here that the selection of a cell or cells within which to locate the item depends not only on the understanding of the mathematics involved but also on the approach that the learner may take to solving the problem. an issue arises for a class within which a particular problem has been discussed: re-use of the problem will be classified as ‘remember and apply’, whereas if the particular topic has not been dealt with in class the learner may be required to analyse the problem and apply conceptual understanding. this difficulty confirms the statement that by bloom et al. (1956) that ‘it is necessary to know or assume the nature of the examinees prior educational experience’ (p. 20), in order to classify test questions. similar conceptual and taxonomic efforts in mathematics top ↑ similar work in mathematics education, in parallel or in conjunction with the work of bloom, krathwohl and colleagues, has been conducted in an attempt to achieve congruence from the curriculum, through the pedagogical domain, and into assessment. distinctions between types of mathematics knowledge, relational understanding and instrumental understanding by skemp (1976) describe the theoretically distinct though practically linked constructs. he describes relational understanding as the ability to deduce specific rules and procedures from more general mathematical relations. instrumental understanding describes the ability to apply a rule to the solution of a problem without understanding how it works. this contrast, however, refers to the learner's understanding and may be an objective for teaching, but cannot easily be distinguished in an assessment item. the somewhat different terms conceptual knowledge and procedural knowledge are identified by hiebert and lefevre (1986) following scheffler (1965). the distinction is made between conceptual knowledge in which relations are established between concepts and procedural knowledge elements which are sequential in character. conceptual knowledge is attained by ‘the construction of relationships between pieces of information’ or by the ‘creation of relationships between existing knowledge and new information that is just entering the system’ (hiebert & lefevre, 1986, p. 4). hiebert and lefevre make a secondary distinction between primary level relationships and the reflective level constructs. the primary level refers to elements of knowledge that are at the same level of abstraction, whilst the reflective level refers to a higher level of abstraction that occurs when two pieces of knowledge initially conceived as separate pieces of knowledge are abstracted to become a principle or concept that is generalisable to other situations. these levels of abstraction align with the purpose of mathematics education expressed by vergnaud (1988), which is to transform current operational thinking into more advanced concepts that are generalisable across varied situations. procedural knowledge, in hiebert and lefevre's (1986) definition is described as knowing the formal language, or the ‘symbol representation system’, knowing algorithms and rules for completing tasks and procedures and knowing strategies for solving problems. in practice, the two perhaps conceptually distinct knowledge types are intricately linked and cannot be distinguished (long 2005; usiskin, 2012; vergnaud, 1988). subsequently, kilpatrick, swafford, and findell (2001) included conceptual understanding and procedural fluency, similar in essence to the terms used by hiebert and lefevre (1986), as two of five strands necessary for mathematical proficiency. the other three strands are adaptive reasoning, strategic competence and a productive disposition (kilpatrick et al., 2001, p. 141). in essence, the kilpatrick strands focus on features of learner activity in the mathematics classroom to which a teacher may properly attend. whilst these strands are useful for the purpose of planning learner activity, they do not function as a taxonomy or typology for purposes of categorising curriculum knowledge or for guiding the design of a test instrument, nor as an instrument to judge teacher competence. in an attempt to make the design of curriculum, the stating of objectives, the educational activities and the assessment thereof coherent and iteratively cyclical, usiskin (2012) and colleagues at the ucsmp have conceptualised an elaborated view of what it means to understand mathematics, which comprises five dimensions: skills-algorithm understanding, property-proof understanding, use-application understanding, representation-metaphor and history-culture understanding. this elaborated view of understanding mathematics is conceived from the learner's perspective and as such should be useful in terms of teaching and learning. this taxonomy of understanding will be discussed in connection with the project to which it was applied in a later section. given the above alternative distinctions made, we reflect on the process of categorising and describing items for the purpose of communication and for providing feedback to the teachers in our project. we turn to the items on worksheet 2 and worksheet 4 (in appendix 1). are these items easily classifiable as conceptual knowledge or understanding, or procedural knowledge or procedural fluency? a partial answer from bloom et al. (1956) is that the classification depends on knowledge of or an assumption about the learners’ prior knowledge. nevertheless, large-scale studies and national systemic programmes require guiding assessment frameworks. trends in international mathematics and science study frameworks: a taxonomy top ↑ the timss frameworks have been used to inform curricula and provide a framework against which tests may be constructed and results reported. though with no direct evidence it appears that the third international mathematics and science study, as it was known in 1995, timss-repeat (1999) and the trends in international mathematics and science study 2003, 2007 and 2011 have all engaged with bloom's taxonomy, both the original and revised versions, and with the various categorisations made in mathematics education literature. in order for the international large-scale studies to make a claim for both reliability and validity, it is essential that they make explicit the frameworks informing the design of the assessment instrument, including both the content domains and the cognitive domain. the early timss studies, in 1995 and 1999, used the term performance expectations to provide the second dimension. these expectations were as follows: knowing, using routine procedures and problem solving at grade 4; representing situations mathematically, using more complex procedures, generalising and justifying at grade 8 (schmidt, mcknight, valverde, houang & wiley, 1996; see also table 2). table 2: conceptual and cognitive domains: bloom's (original and revised), timss 1995/1999, 2003, 2007/2011, rncs and caps. the categories used in timss 2003 for the cognitive domain were: knowing facts and procedures (recall, recognise or identify, compute, use tools), using concepts (know, classify, represent, formulate, distinguish), solving routine problems (select, model, interpret, apply, verify) and reasoning (logical, systematic thinking, including both inductive and deductive thinking) (mullis et al., 2003). in 2007 and 2011, the categories changed somewhat to knowing (recall, recognise, compute, retrieve, measure, classify or order), applying (select, represent, model, implement, solve routine problems) and reasoning (analyse, generalise, synthesise and integrate, justify, solve non-routine problems) (mullis et al., 2005, 2009) (table 2). without going into detail, one may observe broad similarities across the timss frameworks with the bloom's taxonomies, both original and revised. we note here that our items from worksheet 2 and worksheet 4 (in appendix 1) may be allocated to timss content domains fairly easily as the topics in the framework are elaborated to a fine level of detail. the difficulty still remains with assigning a cognitive domain to the items or, to phrase the challenge differently, to assign the expected response of the learner. both the timss frameworks and the bloom's taxonomies (both original and revised) have influenced curricula planning in many participating countries, including south africa, over recent decades. rncs and caps taxonomies in international perspective top ↑ in this section we comment, in relation to bloom's taxonomy and timss, on the south african curricula, the revised national curriculum statement (rncs) introduced in 2002 (department of education, 2002), though only fully implemented some 5 years later, and the curriculum and assessment policy statement (caps) (department of basic education, 2011), introduced in 2011 and implemented from 2012 to 2014. the categorisations – knowledge, routine procedures, complex procedures, and problem solving – in caps (dbe, 2011) are similar to the timss 1995 and 1999 categories. the earlier rncs curriculum used the same content categories, but had more elaborated cognitive dimensions, which were more akin to the bloom's categories and the timss 2007 categories. table 3 provides a summary with the rncs and caps categories somewhat roughly aligned. table 3: curriculum framework for test design purposes with caps -percentages for grade 9. applying the matrix of content domain categories (mathematical concepts) and the cognitive domain categories (the responses expected from individuals) allows test designers to cover the broad range of knowledge requirements expected by the curriculum. of course such a matrix of content and learner activity level inevitably sets up artificial distinctions between subject topics and between the responses expected.2 there are likely to be many occasions when the test designer will be in a quandary as to which category to assign a particular item. table 3 provides an example of a standard framework providing content and cognitive domains. the cells would then be populated according to the curriculum requirements, for example as laid out in caps. the requirement is for the designer to populate the tabular framework with suitable types and numbers of items for each cell in the matrix. in the case of our worksheet 2 and worksheet 4 items, they may all be allocated to the category application, which includes routine procedures and complex procedures (table 3), although some may argue that the items all belong in the knowledge category. the authors acknowledge that the depth of description in these taxonomies and frameworks has not been provided in this article. they are listed rather to show how there are similarities and differences across these frameworks which then point to the complexity of constructing such taxonomies. it is necessary for the taxonomy and the various frameworks to be transformed into subject-specific descriptions (andrich, 2002; van wyke & andrich, 2006). the timss descriptions have achieved this requirement (see mullis et al., 2003, 2005, 2009). an interesting divergence to be explored is that whilst bloom's original and revised taxonomies claim a hierarchy of cognitive processes, the timss framework claims only minimal hierarchy of cognitive domains with a range of difficulty within each cognitive domain (mullis et al., 2003, p. 32). challenges in 21st century assessment top ↑ the challenge to test developers in the 21st century is to achieve some congruence between tests used for monitoring or summative purposes, for the active classroom and classroom-based assessment. we also propose that in addition to the alignment required for these modes of assessment, there is also critical engagement in a professional development cycle. the congruence of educational objectives, teaching and learning activities and assessment envisaged by bloom (see krathwohl, 2002) is difficult to achieve. however, given the importance of aligning assessment practices with classroom practices, it is necessary to have a framework that is explicit and is in some respects common to both settings. the current monitoring and evaluation project under consideration in this article has an external monitoring component: there is also in the design a feedback component provided to teachers in the interest of improving teaching and learning. the model for this project based on the work of bennett and gitomer (2009), and bennett (2010, 2011), has been explained earlier in the article. the content of the assessment programme requires reviewing and making decisions about substantive mathematics knowledge.3 in addition, the fact that there is feedback to the teachers means that there should be a common conceptual language and some congruence of expectations across all three sites, the curriculum, classroom teaching and external assessment. in this project the problem emerged of making explicit the content of the curriculum framework for learners in grade 9. (the project also encompassed grades 8, 10 and 11; the focus in this article is grade 9.) the research team believed that making the framework explicit would serve three purposes: firstly, it would provide some direction to the constructors of the test items and provide an overview of the test; secondly, the explicit descriptions could provide feedback for teachers; thirdly, in the interests of democratic participation, the design of the test would be transparent (see appendix 2 and appendix 3). the broad question, as stated earlier, is: can the three essential elements, a monitoring component, a formative assessment component and a professional development component be logically and coherently aligned for the purpose of informing teaching and learning? the subquestion is: how may we best design assessment frameworks (the design tool specifying the purposes, structure and content of an assessment instrument) in such a way that there is full coherence from the mathematical knowledge to be taught and learned, through the set of assessment instruments to providing diagnostic and practical feedback to teachers about learner performance and needs? in the early phases of the project, we explored alternatives to the current practice which draws on bloom's taxonomy and variations as described in the national curriculum documents. in particular, we examine the function of taxonomies in guiding a monitoring and developmental process. selecting a taxonomy top ↑ there is the obvious difficulty of assigning mathematics items to one particular mathematics category, but when assigning an item to a cognitive category, the problem is one of presuming how the learner will respond to the item. whether an item is categorised as knowledge, routine procedures or complex procedures and problem solving (as in the caps, table 2) depends on the level of knowledge acquired by the learner; this level relates directly to what has been taught (bloom et al., 1956; usiskin, 2012). a possible solution to this predicament of interpretation is to limit the categories to mathematics components rather than attempting to second-guess how the generic learner will respond. this approach, focusing on the mathematical content of the question, may circumvent the difficulty test designers have in manipulating the mathematics to fit a cognitive category. a second criterion for the selection of a taxonomy, or criteria for categorising test items, is for the categories to align with teaching and learning. the question to consider here is whether or not feedback from a particular category will provide information to the teacher about needs and interventions. the first approach we took in this project was to describe in detail what we expected of the learner responding to the item. this approach is exemplified in the patterns, functions and algebra component of a grade 9 test illustrated in the table in appendix 2. the cognitive requirements form the horizontal headings across the table. the purpose of making this content aspect explicit was to inform teachers of the contents of the test so that they could make a reasoned judgement about the performance of their classes in relation to the test during this external monitoring programme. in other words if the teachers knows that item x covered probability, and she also knows that she made a judgement call to leave probability out of the grade 9 work plan with the view to having an intense focus in grade 10, she would understand her students lack of performance in this section. a second approach categorised the items in terms of the dimensions of understanding identified by usiskin (2012; see also appendix 3). using three criteria for a useful taxonomy, that is, firstly to stay true to the mathematics, secondly to guide a balanced assessment and thirdly to provide useful feedback to teachers, we thought to explore the potential of the five dimensions of understanding proposed by usiskin, which are also operationalised in the ucsmp high school textbooks. he proposes that for a full understanding of concepts, five dimensions are necessary: the skills-algorithm dimension of understanding deals with the procedures and algorithms required to achieve answers. this dimension includes the understanding of procedures and algorithms, which usiskin (2012) and others assert is much deeper than what has been called procedural understanding or procedural fluency (see kilpatrick et al., 2001; long, 2005). the understanding and ability to carry out a skill invariably involves at base the understanding of the associated concept and requires all sorts of skills. this dimension of understanding mathematics concepts is what is mostly addressed in school classrooms and found in systemic type tests. the property-proof understanding of concepts deals with the principles underlying, for example, the number system and operations. it may be argued that a procedure is only really understood when one can identify the mathematical properties that underlie the procedures. knowledge of the properties and being able to ‘prove’ that the procedure works enables one to more confidently generalise the procedure to other problems. here we may contrast conceptual understanding with procedural understanding, although as argued previously this distinction has to be qualified. the use-application understanding of mathematics deals with the applications of mathematics in real situations. a person may understand how to perform some procedure and may know why his method works, but they cannot fully understand unless they know when, why and how to use the skill and procedure in applications. applications are not necessarily higher order thinking, but rather a different type of thinking according to usiskin (2012). usiskin (2012) avers that the three types of understanding previously described do not give a complete picture: to fully understand a concept a person must be able to represent the concept in different ways. the representation-metaphor understanding refers to the pictures, graphs or objects that illustrate concepts and that can be used interchangeably with symbolic representation. such analogies may need to be in one or more of verbal, figural, graphical or tabular modes and may need to invoke more than a linear ordering or more than a single static dimension. they may also require a location in time. the fifth is the history-culture dimension. whilst this theme is an important dimension of understanding, it cannot easily be tested where responses require only short answers. it reflects a sense of the interrelatedness of mathematical content and its embedding in the social fabric of experience. some key consequences of this dimension of understanding include an appreciation of the utility and creativity associated with mathematical thinking and problem solving at the level of the learner and an insight into the proximity of mathematics. we suggest it has motivational consequences. in applying this revised taxonomy we faced two dilemmas. the first was that we had to conform in some degree to the status quo. the caps document, the legal framework guiding teachers in their everyday teaching and assessment, requires strict adherence. in that document the four levels applied are knowledge, routine procedures, complex procedures and problem solving (dbe, 2011). we generally use three categories, knowledge, applications and problem solving. the second dilemma was where to include problem solving. usiskin's (2012) focus is on understanding a concept. does problem solving form part of the dimensions of understanding, so, for example, could we place problem solving into the category use-application, or should it have a category of its own? the process ‘problem solving’ has many different interpretations. in some sectors problem solving means a ‘word sum’; to others the term means encountering a problem never seen before by the learner cohort. this salient but inherently unverifiable definition is very difficult in practice because a teacher or test designer may never know whether a learner has seen a particular problem type previously or not. the good teacher, enthusiastic parent or grandparent and the internet could all have a part to play in rendering a really good problem routine, in that it becomes something the child has seen and perhaps solved before. problem solving according to polya (1957) has distinct phases. the problem solver when confronted with a problem they have not seen previously needs to firstly understand the problem, then think about the strategy to use, then ‘generate a relevant and appropriate easier related problem’, then ‘solve the related problem’, and finally ‘figure out how to exploit the solution or method to solve the original problem’ (schoenfeld, 2007, p. 66). taking this process seriously means that problem solving is not possible in a standard testing situation. we have to acknowledge here that our tests are omitting a very significant part of mathematics. in fact, schoenfeld (2007) asserts that the types of questions and answers common in many mathematics classrooms work against the generation of good problem solvers in those classrooms. in the case of problem solving we have compromised and included the notion of problem solving as a separate category, although knowing that the items allocated to that category are only shadows of what polya would describe as a real problem. so, in table 4, we have assigned the items from worksheet 2 and worksheet 4 (see appendix 1) to one of the dimensions, knowing that the allocation to another single dimension may be argued and that a single item may well span two categories, true to the nature of mathematics applications. each of these four dimensions of understanding, skills-algorithms, property-proof, use-application and representation-metaphor, has aspects that can be memorised; they also have potential for the highest level of creative thinking, for example the invention of a new algorithm (usiskin, 2012). each of the dimensions is relatively independent of the others. each of the understandings has proponents who teach mathematics largely from that single perspective. usiskin (2012) claims however that the understanding of mathematics is multidimensional, with each of these dimensions contributing some elements of the notion of understanding. table 4: dimensions of understanding, levels of processing, and possible weightings. we argue that this taxonomy of dimensions provides, firstly, a necessarily mathematics-specific taxonomy and, secondly, that these dimensions support good teaching practice and that therefore feedback to teachers in terms of these dimensions may be helpful. an interesting observation is that by including an additional somewhat hierarchical dimension the taxonomy becomes three-dimensional: the mathematical knowledge as listed in the curriculum, the dimensions of understanding and the levels of complexity involved. note that the explicit weightings in table 4 are aligned to the south african curriculum documents and would differ depending on the content domain and on the constitution of the class and their aspirations. having a class of aspiring engineers may warrant more emphasis on problem solving and the creative application of mathematics, whilst also not neglecting routine algorithms that are an important component of the engineer's tool box. conclusion top ↑ at the heart of the matter for the curriculum designer, the teacher and the assessment specialist is an advanced understanding of mathematics that takes into account the interconnections between the current school mathematics topics, the connections to the earlier concepts and the progression in subsequent years to more advanced topics (usiskin, peressini, marchisotto & stanley, 2003). also required is the exploration of alternate definitions, the linking between concepts, knowledge of a wide range of applications and alternate ways of approaching problems (usiskin et al., 2003). this background knowledge informs the designer in any systemic testing programme. in a model such as the one envisaged by bennett & gitomer (2009), there is the potential that attention be paid to the critical areas of mathematics and that these areas are aligned with the classroom. the fact that we are constrained by existing test programmes, which serve some purpose in the current system, implies that we need to find way of progressively adapting the existing requirements. we must simultaneously bear in mind both that administrators and teachers are change weary and that any changes need to be thoroughly debated and explicit consensus reached about the role and forms of systemic tests. we note here that we have developed formative assessment resources (worksheet 2 and worksheet 4 in appendix 1) that are linked to the monitoring component, and are designed in sets with each of the items covering different dimensions and ranging in difficulty. the purpose of these products is for teacher use in the classroom so that the teacher does not have to rely only on external monitoring for feedback about teaching and learning, but will also have useful resources at his or her disposal. monitoring and accountability purposes can accommodate time lags that intervention strategies cannot afford. as has already been observed, professional development that does not relate to the classroom experience may not be useful. in addition, systemic assessment that gives no thought to its diagnostic relevance in the classroom must be questioned. the dilemma here, as with the levels advocated by bloom, is how to operationalise these levels or components of understanding, in such a way that they manifest evidence that the objectives of the curriculum have been met or that learner proficiency is being developed and exhibited. bloom's levels or timss cognitive domains convey very little in themselves unless they can be interpreted for a specific mathematical context (as they have been in the timss frameworks). for these systems of categories to be useful, they have to be further elaborated by the subject specialist. the usiskin taxonomy serves this purpose. devices such as table 4 guide the designer of an assessment programme towards appropriate balance and coverage of curriculum and attempts to cover different types of cognitive engagement for the context of a specific grade. however, further mathematical insight is required to populate such a multidimensional framework with appropriate items. these insights include the apparent difficulty level of items within each cell in the table. a norm-referenced instrument can emerge, suitable for diagnostic and intervention purposes, which will require some form of marking memo. such an instrument can be valuable in every classroom but perhaps at different times and stages to suit the progress of the learners in each context. this variability suggests the importance of collaborative projects that construct comparable assessment instruments using a common design framework across the targeted curriculum and then share access to the resulting variety of classroom-focused instruments. any additional criterion-referencing, as may be desired for adjudicating individual learner attainment in a classroom summative assessment or in a systemic testing programme, will require some external specification of explicit outcome criteria for various levels of performance quality. these criteria require judgments about the extent to which each of the constituent items are indicators of the required performance levels. these judgments should also be explicitly recorded and may influence memo mark allocations. the related matter of conditions for the legitimacy of addition of marks to establish a single overall performance total is a separate non-trivial issue, but is not discussed further in this article. the challenge presented to the mathematics education community by vergnaud (1994) is that the analysis of concepts and processes must be from a mathematical perspective. he asserts that no linguistic or logical system or natural language description, or levels of abstraction, such as bloom's taxonomy, can provide the ‘concepts sufficient to conceptualise the [mathematical] world and help us meet the situations and problems that we experience’ (vergnaud, 1994, p. 42). it is the precision of symbolic representation and well-defined concepts in mathematics that conveys both the essential aspects of the mathematical situation and the schemes used by the learner of mathematics. this somewhat radical stance challenges educational researchers and practitioners, whilst being pragmatic in the current policy environment, to keep in mind the essential mathematics. a related challenge is to maintain the distinction between a learning environment, which requires extensive investigation and engagement with meaningful contexts, and an external assessment programme, which inevitably focuses on the outcomes of a process. short-circuiting the learning process with obsessive testing may be counterproductive. here we are reminded of two modes of evaluation, that of the connoisseur, a genuine appreciation of the art of teaching, and that of the critic, an inspector that moves in with a checklist (eisner, 1998). bloom et al.'s (1956) aim in formulating the taxonomy of educational objective was to extend the repertoire of teaching through engagement with the taxonomy. we envisage that the ideas expressed in this article will provide the impetus for further discussion. acknowledgements top ↑ the ideas in this article have been generated whilst working on a project conducted by the centre for evaluation and assessment at the university of pretoria. the project was funded by the michael and susan dell foundation. the theoretical developments have been the work of the authors. competing interests the authors declare that they have no financial or personal relationship(s) that may have inappropriately influenced them in writing this article. authors’ contribution c.l. (university of pretoria) conceptualised the article, with major contributions from t.d. (university of cape town) and h.d.k. (independent consultant). most of the writing has been the responsibility of the first author, with critical review and insights into the curriculum provided by the co-authors. all three authors have been involved in the monitoring and evaluation project, and have contributed to the conceptualisation of the products, that is, the frameworks and tables. references top ↑ anderson, l., & krathwohl, d.a. (2001). taxonomy for learning, teaching and assessment. a revision of bloom's taxonomy of educational objectives. new york, ny: longman. anderson, l.w., kwathwohl, d.r., airasian, p.w., cruikshank, k.a., mayer, r.e., pintrich, p.r., et al. (2001). a taxonomy for learning, teaching and assessing: a revision of bloom's educational objectives (abridged edition). new york, ny: longman. andrich, d. 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(2012). common core vs. common sense. education week, 32(13), 35–40. appendix 1 top ↑ worksheet 2: grade 9: patterns, functions and algebra determine the general term for this pattern: factorise fully: 20x2 – 45y4 a, b and c show different representations of a linear function. which one of the following three representations does not represent the same linear function? worksheet 4: grade 9: measurement the area of a square is 4 m2. calculate the area of the shape if one side of the original square is doubled. calculate the volume of the cylinder. use 22/7 as an approximation for π. an airplane flies 300 km due north. however the pilot ignored the constant side wind which took him off course. the flight path is shown in the figure below. how far is he from his original destination? appendix 2 top ↑ table 1−a2: adapted cognitive domain categories (grade 9). appendix 3 top ↑ table 1−a3: items allocated to usiskin's dimensions (grade 9). footnotes top ↑ 1. interesting connection here with the van hiele levels. 2. see long (2011, p. 234) for a detailed discussion. 3. in the early stages of the project these tasks were performed by the research team. in later stages of the project this task became a joint function of both researchers and teachers. http://www.pythagoras.org.za open access page 1 of 1 reviewer acknowledgement acknowledgement to reviewers in an effort to facilitate the selection of appropriate peer reviewers for pythagoras, we ask that you take a moment to update your electronic portfolio on https:// pythagoras.org.za for our files, allowing us better access to your areas of interest and expertise, in order to match reviewers with submitted manuscripts. if you would like to become a reviewer, please visit the journal website and register as a reviewer. to access your details on the website, you will need to follow these steps: 1. log into the online journal at https:// pythagoras.org.za 2. in your ‘user home’ [https://pythagoras. org.za/index.php/ pythagoras/user] select ‘edit my profile’ under the heading ‘my account’ and insert all relevant details, bio statement and reviewing interest(s). 3. it is good practice as a reviewer to update your personal details regularly to ensure contact with you throughout your professional term as reviewer to pythagoras. please do not hesitate to contact us if you require assistance in performing this task. publisher: publishing@aosis.co.za tel: +27 21 975 2602 tel: 086 1000 381 the editorial team of pythagoras recognises the value and importance of the peer reviewer in the overall publication process – not only in shaping the individual manuscript, but also in shaping the credibility and reputation of our journal. we are committed to the timely publication of all original, innovative contributions submitted for publication. as such, the identification and selection of reviewers who have expertise and interest in the topics appropriate to each manuscript are essential elements in ensuring a timely, productive peer review process. we would like to take this opportunity to thank all reviewers who participated in shaping this volume of pythagoras. we appreciate the time taken to perform your review(s) successfully. andrew v. talmadge anita l. campbell bruce may bruce brown cerenus pfeiffer christiaan venter deonarain brijlall dirk c.j. wessels duncan mhakure ednei l. becher erica d. spangenberg faaiz gierdien ifunanya j.a. ubah kabelo chuene kgaladi maphutha marie joubert mdutshekelwa ndlovu mellony h. graven michael k. mhlolo michal tabach mogege d. mosimege mohammad f. gierdien monde mbekwa peter pausigere piera biccard rina durandt rolene liebenberg rose s. maoto shaheeda jaffer simon a. tachie stanley a. adendorff tšhegofatšo p. makgakga vasuthavan g. govender vimolan mudaly washiela fish willy mwakapenda yip-cheung chan zalman usiskin zwelithini b. dhlamini http://www.pythagoras.org.za� https://pythagoras.org.za� https://pythagoras.org.za� https://pythagoras.org.za� https://pythagoras.org.za� https://pythagoras.org.za/index.php/pythagoras/user https://pythagoras.org.za/index.php/pythagoras/user https://pythagoras.org.za/index.php/pythagoras/user mailto:publishing@aosis.co.za article information author: nyna amin1 affiliation: 1school of education, university of kwazulu-natal, south africa correspondence to: nyna amin postal address: 9 normandy crescent, westville 3630, south africa dates: received: 23 may 2012 accepted: 03 oct. 2012 published: 05 dec. 2012 how to cite this article: amin, n. (2012). excavating memories: a retrospective analysis of mathematics teachers’ foregrounds. pythagoras, 33(2), art. #178, 9 pages. http://dx.doi.org/10.4102/ pythagoras.v33i2.178 copyright notice: © 2012. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. excavating memories: a retrospective analysis of mathematics teachers’ foregrounds in this original research... open access • abstract • introduction    • foregrounds and the south african context    • narratives in mathematics education    • producing narratives       • ethical considerations • mathematics teachers’ narratives    • sindiswe    • aziz    • nisha • discussion    • mathematics teachers’ experiences       • foregrounds and experiences of learning mathematics       • mathematics and development       • images of mathematics teachers • mathematics, foregrounds and development • acknowledgements    • competing interests • references • footnotes abstract top ↑ this article offers a retrospective analysis of mathematics teachers’ experiences as learners in school to examine their foregrounds. based on skovsmose’s notion, foregrounds are perceptions of possibilities, impossibilities, hope and despair made by an individual whose life chances are compromised by socio-political adversities and economic deprivation. since foregrounds are imagined it is not possible to be certain about how the future is realised without engaging in a longitudinal study spanning decades. one way to attempt to do so is through memory work, that is, by excavating the memories of persons who are already living in their foreground. three mathematics teachers were recruited through snowball sampling to share their memories of teachers of mathematics, learning mathematics in schools and how they envisioned the future. the analysis revealed how exposure to mathematics teaching and learning when they were learners is implicated in shaping the foregrounds of teachers, and the ways in which teacher foregrounds are connected to personal and national development. the participants’ narratives confirmed that, although a foreground is shaped by prevailing conditions, a view to a viable future requires an optimism that goes beyond present situations. introduction top ↑ there is some evidence that the south african education system is in a crisis. bloch (2009), for example, sketches in stark detail the ingredients of a ‘toxic mix’ in education that threatens the good future of the next generation. for a country that inhabits and exhibits the full spectrum of positions simultaneously on a continuum from undeveloped to advanced development, bloch offers a systems approach to analysing the woes besetting the country: poor teaching, poor learning, dysfunctional schools, ill-prepared and under-prepared teachers, poverty, poor health and nutrition, lack of resources, poor management of schools, problematic education policies, militant professional teacher associations and lack of political will. the crisis bloch refers to exteriorises challenges that appear to be insurmountable. although a relativist argument may be made that it depends on which schools and which learners are referred to, skovsmose (2005) guards against relativist stances: ‘crises belong to reality. they are not merely conceptual constructions, meaning a different way of looking at reality may create a picture we can appreciate’ (p. 17). the crisis in education is most strongly articulated in mathematics as there is, ostensibly, a relationship between mathematics competency and levels of attainment and national development. it is mathematics performance in general that continues to support the notion of a failing education system. evidence of the importance of mathematics is tangible in the countless official reports (e.g. department of social development, 2010; united nations development programme, 2011) and reports in the popular press (e.g. macfarlane, 2011; parker, 2012) concerning educational performance and the lack of readiness of learners to study at tertiary level due to poor mathematics proficiencies. underpinned by current grand narratives of a crisis in education, this article offers the personal narratives of three teachers of mathematics to disrupt the domination of negative discourses. the narratives are recollections about the teachers who taught mathematics when they were learners at school. all three participants experienced poverty and deprivation and grew up during a period of political repression, economic hardships and uncertain futures. the challenging circumstances the participants faced are common to most children growing up in ghettoes, and sprawling low-income townships and settlements. surrounded by poverty, and deprived of cultural capital, how did they imagine the future? and what made it possible for each one to become a mathematics teacher? the notion of imagining the future, described as a foreground by skovsmose (2005), is explained in more detail in the next section. it is important to focus on the mathematics learning experiences of those who currently teach mathematics in schools to complement and destabilise the many systems and meta-analytical insights. the quest to improve mathematics performance in south africa requires exploration of all aspects that have a direct or indirect impact, and even of aspects that may appear to be only remotely connected to learning success. mathematics teachers, i argue, are not produced in isolation: past experiences and internal and external influences (see e.g. samuel, 1999), are implicated in the making of teachers and, similarly, of the national development landscape. indeed, both personal and national developments are connected in and through teachers. foregrounds and the south african context as in many developing contexts, children in south africa are expected to have better prospects than the preceding generations. however, the woes besetting south african education are unlikely to be resolved in the short term so there is an urgency to support skovsmose’s proposal that it is more important to know learners’ aspirations about ‘[h]ow the future might look from the perspective of a ghetto, from the fourth world, or for a group of disposable people’ (skovsmose, 2005, p. 191). skovsmose’s description of certain learners as ‘disposable people’ has specific reference to those whose lives are ravaged by a combination of socio-political adversities and economic deprivation. these are individuals on whom hardships are imposed by contextual complexities and for whom political systems have neither affection nor the will to improve their lives. they have, to state it differently, been relegated to the periphery, perhaps forgotten and ignored by those who wield political control. under these conditions skovsmose asks how ‘disposable persons’ envisage the future. are their prospects promising or dismal? skovsmose’s notion of foregrounds (future looking) differs markedly from psychological perspectives that a foreground (object) can be distinguished from its background area, or be reversed or merged. it is also dissimilar to casati’s (2003, p. 283) interpretation of ‘representational advantage’, that is, the figure (foreground) is more prominent and remembered for longer periods than the ground (background). skovsmose’s notion, by contrast, is allied to socio-political concerns to reveal the aspirations that ‘disposable persons’ have despite their uninspiring situations, which include a range of possibilities, impossibilities, hope and despair. interestingly, skovsmose demonstrates that the future is not necessarily perceived as hopeless and condemned. furthermore, foregrounds are complex, uncertain and unstable phenomena comprising multiple possibilities, barriers and propensities suggested by a context (background). foregrounds cannot be predicted and are vulnerable to change even when a context remains unchanged because foregrounds are imagined and mediated by experiences and interpretations of experiences. the imagined future possibilities undoubtedly change as foregrounds shift and change. how the future is perceived is closely related to development. two types of development are relevant: personal (as it applies to an individual) and national (as it applies to a nation state). the south african development indicators report (department of performance monitoring and evaluation, 2010) lists education and its connection to mathematics achievement as key to national development. the importance accorded to mathematics is not accidental. the development report highlights a drop of 11% in performance from the trends in mathematics and science study (timss) of 1999 to timss 2003. despite criticism raised about the timss studies (see e.g. bracey, 2000), the development report concludes that ‘no matter what index is used south africa performs poorly in mathematics and science’ (2010, p. 51). of course the development report and timss studies are grand statements about the state of learning mathematics; they do not provide insights about individual teachers and their perceptions of teaching and learning mathematics. against this backdrop, i wondered how mathematics played a role at a personal level of development. what kinds of mathematics learning experiences inspire a foreground filled with possibilities or make it improbable? and what are the ways in which to study foregrounds? studies about ‘disposable persons’’ views of the future could lead to multiple paths of uncertainty. it would be more useful, i presumed, to begin in individuals’ foregrounds and to work backwards. memory work (king, 2000; onyx & small, 2001) presented the means to do so as the inquiry required going back in time to recall how participants viewed the future and its relevance to their present status as mathematics teachers. conceptualising the study as an inquiry of past memories provided a methodological approach to study foregrounds because teachers, in a way, are already living in the foreground and their stories enable retrospective analyses. since mathematics is central to development, i chose to work with black mathematics teachers who were construed as ‘disposable children’ when they were young. they grew up in the shadows of apartheid, lived in townships ‘disfigured by poverty’ (desai, 2002, p. 8), attended poorly resourced schools and were offered a watered-down education, especially in key subjects like mathematics and science. additionally, access to higher education was restricted by political design to a few institutions that admitted black individuals to a few careers where employment possibilities were viable. despite the harsh combination of a repressive political system and an adverse socioeconomic context, the participants in this study became teachers of mathematics. they relate personal narratives of unexpected foregrounds that provide insight about the nature of foregrounds. in particular, their narratives enabled an exploration of the links between mathematics teacher foregrounds and teaching, their influence on learner foregrounds and connections to personal and national development. in essence, this article thus makes two important contributions: methodological and empirical. firstly, it reports on a study that makes use of memory work to produce narratives as a means to research foregrounds as conceptualised by skovsmose (2005) and, secondly, it provides empirical understanding of mathematics teacher foregrounds and their entanglement with education, development and learner foregrounds. narratives in mathematics education the idea of narratives in mathematics education is not a novel one (see e.g. doxiadis, 2003; solomon & o’neill, 1998). in practice it is evident in numerous ways: the lives of great mathematicians envelope and underpin the curriculum; narratives of learners have been generated to understand the complexities of acquiring mathematics knowledge (burton, 2001); problems have been situated in narratives (scieszka & smith, 1995); and teachers’ stories have been used for continuous professional development (jaworski, 1998). burton (1996) makes the argument that ‘a narrative approach to mathematics and its pedagogy is consistent with a view of mathematics as being socially derived and with the understanding of mathematics as being socially negotiable’ (p. 32).by contrast, solomon and o’neill (1998) argue that the narrative genre is one of the solutions to underachievement in mathematics. it is also a way to provide descriptions that relate to the personal and its impact on an individual’s life. in this section, a reading of great, grand, political and foreground genres is offered with personal narratives solicited methodologically for analysis in a subsequent section. great narratives refer to stories about outstanding individuals, their anguish and passion, brilliance and innocence, tragedy and triumph, commitment and sacrifice. these are the kinds of stories that supposedly move, inspire and awe as exemplified by early mathematicians like pythagoras (gorman, 1979) and euler (bell, 1937), the tragic geniuses abel (livio, 2006; turnbull, 1993), galois (livio, 2006; singh, 2002) and lobachevskii (gray, 1989), and latter-day giants like wiles (singh, 2002) and nash (nasar, 2001), whose trials and triumphs have been turned into movies. fictionalised narratives of great mathematicians (doxiadis, 2000; tahan, 1993) have also played a role in inspiring learning about mathematics. it is assumed that mathematics teaching that incorporates the lives of great mathematicians allows learners to make emotional connections to the discipline (doxiadis, 2003), consistent with the value of storytelling in education in general. it was the move, however, from absolutist to fallibilist philosophies of mathematics education that created more opportunities to present historical narratives of great lives. history constructed from biographical accounts of great mathematicians gave credence to the claim that mathematical knowledge is ‘a product of on-going social processes and a reflection of cultural, political, and practical norms and values’ (latterell & wilson, 2001, p. 1). consequently, great narratives can be viewed as social creations supporting particular interpretations of truth. grand narratives1 (lyotard, 1979) similarly propose truths about the usefulness (van oers, 1996) and values (wilson, 1986) of mathematics. the british qualification and curriculum authority (2008) asserts that: mathematics is fundamental to national prosperity in providing tools for understanding science, engineering, technology and economics. it is essential in public decision-making and for participation in the knowledge economy. … mathematics equips pupils with uniquely powerful ways to describe, analyse and change the world. (p. 1) the hegemonic2 notion of the importance of mathematics for ‘national prosperity’ ensures that access to careers in the medical, health and physical sciences, and the economic, technological and engineering fields is limited to those who succeed in the discipline. furthermore, international studies and national schooling agendas and development goals are shaped by performance in mathematics. for example, south africa’s position at the bottom of performance league tables in the timss studies of 1995 and 1999 (howie, 1999, 2001) has placed intense focus on mathematics and science as keys to national development and access to high status occupations. in recent years, politicised narratives of mathematics (e.g. adler, 2001; apple, 1995; keitel, 1998; skovsmose, 1994; vithal, 2003) questioned the ways in which social injustices operate and proposed possibilities for recourse. the focus is on philosophies and practices that bring a political dimension to teaching and learning and this has expanded the traditional realm of mathematics in classrooms beyond pure discipline matters to include socio-political and cultural aspects and notions. the education of girls (walkerdine, 1989), equity (vithal, 2002) and race and ethnicity (lim, 2008) are some issues central to critical examination of mathematics curricula and teaching practices in the discipline. the mathematics classroom the literature suggests is implicated in reproducing iniquities, and in marginalising those most vulnerable. the literature perused offers three uses in the analysis of personal narratives of the participants in this study. the literature on great narratives which lauds high intelligence and ability for mathematics learning will be interrogated for its role in mathematics teachers’ foregrounds. literature on grand narratives of mathematics as a discipline provides a template against which the experiences of participants will be probed for the effects of national agendas and inspirational stories on mathematics teachers’ foregrounds. finally, the ideas from politicised narratives will be deployed to scrutinise mathematics teaching influences on the barriers and limitations on ‘disposable’ individuals’ foregrounds. producing narratives for an inquiry purporting to offer an alternative to systems approaches and large-scale studies, a method commensurate with qualitative research was chosen. the intention was to produce narratives that enabled an analysis of teachers’ foregrounds. to produce narrative data meant deploying a social method that would encourage participants to talk about their past as well as their present experiences. i opted for three participants using a snowball technique. the usefulness of the snowball approach was twofold: it allowed me to carefully identify and search for a participant to fit the profile of a ‘disposable’ person during childhood and it invited participant assistance with identification and recruitment of two more individuals. the first participant was known to me as a fellow mathematics teacher. he referred me to the second participant and she referred me to the third individual. the participants were three black teachers. the selection of participants was deliberate as these individuals were schoolgoers during the apartheid era of the 1970s, a period of intense political oppression for black people in the country. during that time, the schools they attended were generally poorly resourced with unqualified or under-qualified teachers, and the curriculum was narrow and inadequate. future possibilities were compromised because there were limited opportunities for upward mobility and fewer career choices. the participants, marginalised by the politics of race, space and place could, in a sense, be conceptualised as ‘disposable’ individuals in the way skovsmose intended. the three participants, aziz, sindiswe and nisha (pseudonyms), a man and two women, have each been teaching mathematics in high schools for black learners for an average of 28 years. aziz teaches at a suburban school in kwazulu-natal, whilst nisha and sindiswe teach at township schools in gauteng. each came from a deprived socio-economic background. aziz’s father was a recovering alcoholic, nisha was an orphan and sindiswe’s father was unemployed for a long time. all three attended schools in kwazulu-natal during their childhood and now they all teach at schools they described as having poor and deprived contexts, serving children whose lives are comprised by socio-political and economic adversities. aziz is the most experienced (32 years) and sindiswe the least (26 years). they have known each other for about 15 years and meet each year at a national mathematics conference. the data was generated through unstructured interviews, which allowed for probing and some latitude to hone in on insightful experiences. two telephonic interviews were conducted with each participant. telephone interviews were convenient for two reasons: 1. nisha and sindiswe teach in a province approximately 600 km away, making face-to-face interviews a costly task. 2. not knowing them meant anonymity was assured and this, presumably, encouraged them to speak freely. aziz was also interviewed telephonically to ensure consistency of data production. the ideas explored in the interviews were related to understanding their past (as learners of mathematics) and present (as teachers of mathematics). their stories, reconstructed from interviews as first-person narratives to enhance the presence of the teller of the story, that is the research participant, are interpretations of their past experiences inside school and outside of school. the stories are not evaluated for their truth claims because memories of the past are not objective or untainted; they are reconstructions, interpretations and reinterpretations, reflections, and perhaps even fictionalised accounts. whether, objective, subjective or fictional, it is the expressed meanings constructed by participants as experiences that are valued and, to that extent, accepted as personal truths as opposed to forensic truths3 for the purpose of analysis. ethical considerations permission to conduct fieldwork and ethical clearance was applied for and granted4. participants were contacted, informed about the study and invited to volunteer their involvement in the study. before and during the interviewing process, care was taken to ensure that participants were not coerced or compelled to participate or to answer questions they did not want to. afterwards, participants were given access to transcripts of the recorded interviews and to the narratives constructed from the transcripts. none of the participants made changes to the narratives. mathematics teachers’ narratives top ↑ three narratives were constructed from transcripts of recordings of each interview. each narrative recalls a participant’s encounters with mathematics. the narratives are reconstructions with fragments that i deliberately selected to tell a story for the purposes of the inquiry. the narratives do not contain data not offered by participants. although the data were restructured by shifting the order of statements and edited for clarity (the removal of ‘um’ and repetitious use of words like ‘he never told, he never told us’), they were not augmented, although portions of the interviews, for example about the weather and discussions about colleagues which were not related to the themes discussed in this article, were omitted. sindiswe pythagoras was my mathematics teacher. i know everybody is surprised when i say that. in primary school i learnt a little bit about numbers and the four main operations. i was fascinated by numbers. but when i got to high school geometry was a new field. if i remember correctly, i hadn’t even heard the word geometry before. and there was this maths teacher who spoke a lot about pythagoras. he referred to him as the greatest mathematician. we were very interested. who was this great mathematician, and what made him great? but he never told us. he walked around like a professor and used big words that we could not understand like ‘integers’ and ‘complex numbers’. he said only he knows about mathematics and he could decide whether we passed or failed. almost all the children in the school failed mathematics. he said we were too stupid to do mathematics. i believed him, for a while. fortunately, my dad got a job with a big company and we had to move.’ in my new school i learnt geometry, algebra, trig and graphs. i passed matric with a b in maths. now i teach maths. i really want to visit old pythagoras. i wonder how he will react when i show him the maths results of my students – most of them pass maths. the kids i teach will change this country with their mathematical abilities. i am glad to play a part. i might never have been a teacher if i had remained in that school. aziz i have been teaching mathematics for close to thirty years. it is heaven and hell, especially in a poor school like ours. the kids find it so hard. of course there are those who enjoy it and do very well, but most of the time it is so frustrating. i think a lot about why they relate negatively to mathematics. a lot of their problems have to do with the home. when i was young, my father always told us that mathematics opens doors and careers. we could be anything we want to be as long as we did well in mathematics. whenever he drove us around town, he made us point out numbers, he explained how mathematics was used in the construction of roads and buildings. we identified the usefulness of numbers in shops and stores, and while we were shopping he made us calculate gst [general sales tax] on each item. those were the days before vat [value added tax]. i think i was made to feel so comfortable with numbers and calculation so that when we did mathematics in school it seemed so normal and so easy to understand. i am grateful to my dad for helping me to get to grips with maths. there was not a single teacher in primary school or high that was qualified to teach properly. some of them knew less than us. the funny thing is that i became a teacher. because of the realities, i could only become a teacher. you couldn’t just decide to go to varsity [university] and do law or anything. so, mathematics opens doors, if the doors are there to be opened.nowadays i don’t think the children in my class have quite the same opportunities to play with numbers. i think that is the key, playing with numbers, otherwise it is so vague and abstract. the children here believe that they will never master the subject. so this country will continue to flounder and fail. we will never measure against international standards and competencies. mathematics is so important and yet so unvalued. nisha you know, i was once a learner, a very, very good student, i might add, but a very weak maths learner. so why was i weak in maths? let me tell you. it was the maths teachers. they were unkind and very unsocial. they exuded and expected perfection. one of the teachers spent so much time just instructing us on how to write numbers. he was fascinated by the number 2 and after he wrote the number on the board he would stand back and admire it and say something like, ‘isn’t that just the most beautiful symbol? it’s curvy and stylish, delicate and a symmetrical piece of art’. isn’t it strange for a mathematician to describe a science as an art? stranger yet was the teacher who killed us with mental tests, both oral and written. we were never given enough time and were made to feel stupid when a test was not completed in the time limit. she stressed speed and her famous words were ‘quick, what’s the answer?’. another character, in high school, would stand in front of the board, mumble to himself and voila, solve the problem, but he could not explain the steps. often i tried to make sense, but the logic defied me. i was kept out of the inner world of mathematics. i came to grips with maths through my own initiatives. i literally taught myself. come to think about it, i’d forgotten these episodes and perhaps it explains the reasons i spend so much time explaining, or trying to make maths fun. maybe that’s why i chose to be a maths teacher. my subconscious memories want me to make it work for kids. would i have ever thought of becoming a teacher? i don’t think so. the possibility was as remote as climbing mount everest. but i believed in myself and here i am. my learners will never think of me as a bad maths teacher. discussion top ↑ mathematics teachers’ experiences each participant’s narrative recounts memories of days when they were learners. in other words, these are teachers’ memories of learning mathematics. their stories provided insights of the formation of subjects who resisted, despised and embraced mathematics. mathematics, it appears, was the site of personal struggles and achievements. the analyses of the three narratives are restricted for this paper to three themes: foregrounds and experiences of learning mathematics, mathematics and development, and, images of mathematics teachers. the final section hones in on the connection between mathematics, foregrounds and development. foregrounds and experiences of learning mathematics the data suggests that the three participants were inspired to learn mathematics in dissimilar ways. sindiswe’s experiences were mediated by two individuals who were radically unalike. teacher ‘pythagoras’ displayed arrogance, imbued with notions of elitism. in an era and an arena when education was highly valued in unprivileged apartheid spaces because black persons had limited access to education, he was able to strut about and act as a gatekeeper of sacred knowledge. sindiswe related how it was impossible to know whether he had content but not pedagogical knowledge or whether his cryptic musings about the greek pythagoras was a facade that veiled his ignorance of mathematics. a fortuitous move to another city activated the acquisition of crucial mathematical knowledge. a change of school reversed teacher pythagoras’ evaluation that she ‘was too stupid to do maths’. a change of teachers enabled personal development and growth. it is reasonable to conclude then that foreground too was changed, making teaching as a career possible. aziz’s memories of learning mathematics were located outside school. his father taught him mathematics. based on his experiences in school he recounted that the methods his father used were unusual when compared to those used in school. the lessons were not planned and occurred incidentally and opportunistically. his father used everyday, practical living to teach him. whereas sindiswe and her peers were explicitly told that they were ‘too stupid’, aziz thought the ‘mathematics teachers were stupid’ as they could not make learning fun. during the interviews he related how he ‘sailed through’ mathematics in high school because of his father’s tutoring. aziz’s narrative related learning experiences that were joyful and his foreground reflected possibilities to ‘be anything he wants to be’. nisha’s experiences of learning mathematics were not ideal as her teachers were idiosyncratic and eccentric and kept her ‘out of the inner world of mathematics’. she experienced teachers who were technical perfectionists, or interested in the aesthetics of numbers or could not teach problem solving. the consequence was that she became her own teacher. nisha’s negative experiences of learning in school led to positive outcomes, making her a self-reliant and independent learner. her foreground was of her own making, as she was not reliant on external structures and processes. the grand narrative about learning mathematics is that it requires teachers. but what the participants’ personal narratives reveal is that the mathematics teacher is not necessarily a schoolteacher; they can be a parent or a learner too. each teacher, whether a professional, a parent or a learner, does influence the way the future is ruined or rescued. learning mathematics can be inspired by school, home or self. all paths have the same effect, that is, learning does take place, but school learning is hegemonic in the gramscian (1977) sense of being valued above other learning spaces even though ‘mathematics education is everywhere’ (skovsmose, 2005, p. 2009). mathematics and development based on teachers’ memories, two themes recur in the narratives. the first relates to mathematical ability and personal development and the second to mathematics and national development. the memories of schooling were similar for both female participants. sindiswe and nisha were made to feel ‘too stupid to do mathematics’. the idea that girls do not achieve as well as boys in school because girls are wired for language and boys for abstract reasoning has been disproved (halpern, benbow, geary, hyde & gernsbacher, 2007). perhaps the answer lies in social processes: girls are marginalised in class because boys dominate teachers’ attention and gender norms ensure that girls learn to be girls and deny their ability (paechter, 2000). if so, we can deduce that sindiswe’s and nisha’s foregrounds, from these perspectives, are predetermined by a combination of social, cultural and biological forces. but sindiswe’s and nisha’s narratives demonstrate that the future can either be ruined by poor teaching or rescued through self-intent. they acquired mathematical competency through change of fortune or self-motivation, resilience and tenacity.aziz’s experiences of mathematics are also about poorly qualified teachers who could not make learning fun and it was the education provided by his father that compensated for this poor teaching in school. the narratives reveal that possibilities and impediments to national development were contingent. sindiswe and nisha were optimistic about the future based on their contribution to mathematics teaching. remembering their own experiences of feeling stupid yet now able to produce successful mathematics learners, they felt the future was better: that the learners they taught had far more competency than they did as learners. aziz, by contrast, who enjoyed ‘play[ing] with numbers’, and who now teaches ‘vague and abstract’ mathematics, does not believe that mathematics will be mastered. one can infer that he is reliant on parents to contribute to mathematics learning and his assumption that parents do not explains the poor performances on standardised tests. although the logic deployed by aziz to explain poor performance is questionable, his claim is closer to the reality of the present (howie, 2001) than the experiences of sindiswe and nisha. all three participants attested to a concern for development: sindiswe and aziz commented on mathematical performance and national development and nisha on mathematical performance for personal development. sindiswe voiced optimism for the country’s progress because she believed that her role as teacher was a crucial contribution to learning success. aziz, by contrast, feared that the outlook for the country was bleak because the children in his class were not supported by their parents. in his opinion, his role as teacher was not as vital as are parents’ roles. nisha, on the other hand, judged personal determination and self-reliance as key to personal growth and success. one explanation for the sweeping statements uttered by each participant may have to do with their awareness that they teach a high stakes discipline. those who teach subjects like mathematics or literacy are more scrutinised than teachers of, say, history or music. these are the subjects that are most mentioned in the press because mathemacy and literacy are important competencies for success in contemporary times. therefore, it is not surprising that participants made huge claims about learner foregrounds. foregrounds were interpreted as promising (sindiswe and nisha), or ruined (aziz). images of mathematics teachers discursive images of mathematics teachers emerged from the narratives. sindiswe’s image was a great individual, but one whose greatness was symbolised by a name, arrogance and power. ‘pythagoras’ did not teach mathematics; instead he explained how he wielded power: he tantalised learners with his knowledge of mathematical concepts and terms but did not follow through with teaching. he also controlled the pass rate. he was the ‘professor’ in his ivory tower or the ‘imitator’ who pretended to know mathematics. sindiswe came to this realisation only after exposure to another context. in this instance, the importance of having a comparative basis can be seen in the change of foreground – the perception of a future teaching mathematics was not possible until she experienced success in mathematics. consistent with research findings (skovsmose, alrø & valero in collaboration with silvério & scandiuzzi, 2007) mathematics teachers’ experiences reveal that foregrounds are not fixed: they change with changing conditions. aziz’s teacher was the parent-as-teacher, a loving guide and inspirer who turned mathematics into a game outside the classroom. aziz as a teacher-as-teacher conjured an image of frustration as he reflected on his own practice. aziz’s portrayal was self-generated: someone who was disappointed, frustrated and pessimistic. he was not able to appropriate the methods of teaching employed by his father in his own classroom. he is delpit’s (1988) version of the teacher who ‘teaches other people’s children’ – disinterested in their foregrounds. nisha provided a number of images of mathematics teachers. one was a ‘perfectionist’ who was more interested in the aesthetic proportions and forms of numbers. another image was that of a teacher who was unkind and unsocial, made excessive and unreasonable demands and constructed mathematics as an obscure and abstract subject, compromising and perhaps denying hope and possibility. a third was an image of a teacher who made mathematics a mystery, with secret processes taking place in the mind, as he did not teach algorithmic processes. he was the brilliant teacher, but could not communicate or make visible the mysteries of mathematics. these teachers, one can conclude, sabotaged nisha’s foreground. images of teachers were powerful sources that stirred or stifled the foregrounds imagined by aziz, nisha and sindiswe. when life conditions were severe, as they were during apartheid, nisha, sindiswe and aziz needed teachers who made mathematics inviting; their foregrounds, instead were shaped by chance or luck. mathematics, foregrounds and development top ↑ undertaking a retrospective analysis of teachers’ foregrounds has yielded a number of insights about mathematics teaching, mathematics teachers’ foregrounds and the links to development. tracking early experiences in school unveiled the ways the participants were inspired to embrace mathematics or to resist it. who taught mathematics and how it was taught opened or closed foregrounds. furthermore, mathematics teachers were not necessarily trained and credentialed professionals. of the three participants, one was taught by a parent and another taught herself. these tangential learning opportunities liberated their imaginations which could have been constrained by context and background.the importance of gazing at the lives of mathematics teachers lies in its connection to learner foregrounds. as explained by skovsmose (2005), teachers are ideally positioned to take on the responsibility of enabling learners to imagine a better future. what comes through is that more than thirty years ago, when sindiswe, aziz and nisha were learners in school, what were their foregrounds, that is, how did they see their future? for a period of time, sindiswe and nisha faced poor prospects; changes in context and personal belief changed the way they saw their futures. by contrast, aziz’s positive encounters with mathematics and messages to ‘be anything’ might have foretold a future that surpassed that envisioned by nisha and sindiswe. but the analysis reveals a reversal of sorts. despite advantage or disadvantage, positive or negative encounters, good or poor teaching, they all became teachers. that is, no matter what foreground each imagined, they ended up with remarkably similar prospects. the reversal is in their outlook or foregrounds for their learners and the country. sindiswe and nisha, who had more reason to be unsuccessful, embraced teaching with confidence, enjoyed their roles and believed that they teach to make a difference to both learners and the country. by contrast, aziz, the benefactor of a progressive approach to learning mathematics outside of school, was pessimistic about the future. it appears that his early experiences resulted in limited success. he resisted the role of inspiring, creative teacher as he believed those functions were the obligation of parents. the absence of parent involvement in the learning of mathematics, in his opinion, meant a bleak future for learners and the country. it also communicated his impotence to make a difference. undoubtedly, the conclusion one can draw is that teachers’ outlooks are implicated in the foregrounds of children. what explains aziz’s (and sindiswe’s) choice of career if the grand narrative about mathematics and success is to be relied on? why did they not aspire for careers such as scientists, medical doctors or economists? what explains their choice to pursue teaching which is, by contrast, less prestigious and provides a moderate income? one has to remember that when the career choice was made more than three decades ago by all three participants, south africa was under apartheid rule. in aziz’s words, all ‘doors were not open’ to black people. in spite of good results, black persons could not pursue any occupation they desired as there were severe restrictions on access to higher education and work prospects. as disposables of the apartheid era, aziz’s, sindiswe’s and nisha’s foregrounds were already constrained by political ideology. it required personal belief in their own ability for them to become teachers. what is significant about the achievement of these individuals is that they chose to become mathematics teachers despite experiences of poor teaching at school. a foreground can thus be seen as a combination of context, education and personal belief to perceive prospects. the context is both political and personal. ‘disposable’ persons do not choose their context; it is imposed. the context, in this case, had structural impediments, reflecting social, economic and political dimensions beyond their control. in a sense, a foreground is shaped by prevailing conditions and to have a view of a viable future requires an optimism that goes beyond present situations. such optimism is doubtful when the present is so consumed with socio-economic hardships that possibilities cannot be imagined. perhaps it requires the mediation of teachers for those marginalised by socio-economic, political and cultural impediments to envision a future with possibilities. based on the narratives it seems that teachers like sindiswe and nisha, who transcended the bleakness of their own situations to rescue their futures, are ideally placed to inspire those growing up in situations of adversity. if we accept that a foreground is not perceived in isolation, that it is a combination of self with others, context and structures, then we can assert the importance of education in general and mathematics in particular for personal and national development. national development does not occur in a vacuum with national policy and programme drivers. south africa’s attempts through curriculum change, regulating teachers’ work through the setting of norms and standards and progressive methods of assessment have not improved the educational outcomes for those most affected by socio-economic disadvantages. the answer, based on the experiences of nisha and sindiswe, is that it perhaps lies at the level of personal development efforts. individuals, i want to argue, be they teachers or learners, who take charge of their own development are more likely to be vital constituent parts of national development as attested to by the narratives of nisha, who overcame personal and political challenges, and sindiswe, who displaced bad memories of poor teaching with successful experiences. consequently, personal and national developments are in a symbiotic relationship, connected to each other; a change at either level has an impact on both levels. national and personal development cannot be isolated; their connections are intricate and non-dissociative and both play a role in an individual’s foreground. ultimately, the grand narrative, which is also a politicised narrative connecting mathematics achievement and national development, is dependent on personal narratives of optimism, resilience and beliefs. in terms of great and political narratives, teachers are more likely to be inspired by local and contemporary individuals, such as a parent or a teacher, as evidenced by aziz. sindiswe’s memories about teacher pythagoras are unusual. although it may seem curious, she is more likely to make the connection with a caricature of a teacher she once knew rather than with the greek mathematician. it appears that the lives of great mathematicians are far removed from the lives of children in poor and deprived contexts and unlikely to contribute to viewing a promising future. by contrast, political narratives are subtle, suggesting that a high-stakes discipline like mathematics colonises a grand discourse around the discipline rather than paying adequate attention to issues of poverty, marginalisation and social oppression. aziz, nisha and sindiswe were not beneficiaries of liberating and transformative discourses which are implicated in limiting or opening foreground possibilities. whilst some may argue that the teaching of mathematics is unrelated to political agendas, vithal’s (2003) study highlights the way a teacher is pivotal in bringing issues of a socio-political dimension into the mathematics classroom. to conclude, the production of personal narratives has been useful to map the experiences of teachers and to provide some insight into how they understand the learning of mathematics and to track the consequences of that learning in their adult lives. through the lived experiences of mathematics teachers, it is possible to extract the grand and political narratives that inhabit mathematics teaching and learning. teachers’ narratives may be a way to understand mathematics teaching and also its connection to foregrounds and development. the claims, however, are emergent, suggesting that mathematics performance, teaching and learning are complex and unravelling their many aspects, through teacher narratives, requires more research. acknowledgements top ↑ i am grateful to renuka vithal for inviting me to respond to ole skovsmose’s presentation at the symposium mathematics, education, democracy and development: challenges for the 21st century on 04 april 2008 at the university of kwazulu-natal, durban, which inspired the empirical work underpinning this article. competing interests i declare that i have no financial or personal relationship(s) which may have inappropriately influenced me in writing this article. references top ↑ adler, j. 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(2011). human development report 2011. sustainability and equity: a better future for all. new york, ny: united nations development programme. available from http://www.gm.undp.org/hdr_2011_en_complete.pdf van oers, b. (1996). mathematics education and the quest for meaning. in l.p. steffe, p. neshar, p. cobb, g.a. goldin, & b. greer (eds.), theories of mathematical learning (pp. 91–113). hillsdale, nj: lawrence erlbaum. vithal, r. (2002). differentiation, in contradiction and co-operation, with equity in mathematics education. african journal of research in mathematics, science and technology education, 6, 1–20. vithal, r. (2003). in search of a pedagogy of conflict and dialogue for mathematics education. dordrecht: kluwer academic publications. walkerdine, v. (1989). femininity as performance. oxford review of education, 15(3), 267–279. http://dx.doi.org/10.1080/0305498890150307 wilson, b.j. (1986). values in mathematics education. in p. tomlinson, & m. quinton (eds.), values across the curriculum (pp. 94–108). lewes: falmer press. footnotes top ↑ 1. the notion of grand narrative was coined by lyotard (1979) to describe the generalisations that supposedly apply universally. he was especially critical of the ways in which diversities, differences and peculiarities were ignored or dispensed with. 2. hegemony is an idea that dominates through a combination of power (ideological truth) and complicity (by those who do not recognise ideological truth). 3. posel (2004) provides a theorisation of the four kinds of truth (forensic or factual truth, personal or narrative truth, social or experiential truth, and healing truth), generated by south africa’s truth and reconciliation commission. 4. the application for ethical clearance was approved by the university of kwazulu-natal’s humanities and social sciences research ethics committee. article information authors: mdutshekelwa ndlovu1 dirk wessels2 michael de villiers3 affiliations: 1institute for mathematics and science teaching (imstus), department of curriculum studies, university of stellenbosch, south africa2department of curriculum studies, university of stellenbosch, south africa 3school of science, mathematics and technology education, university of kwazulu-natal, south africa correspondence to: mdutshekelwa ndlovu email: mcn@sun.ac.za postal address: university of stellenbosch, faculty of education (imstus), private bag x1, matieland 7602, south africa dates: received: 06 oct. 2010 accepted: 25 oct. 2011 published: 23 nov. 2011 how to cite this article: ndlovu, m., wessels, d., & de villiers, m. (2011). an instrumental approach to modelling the derivative in sketchpad. pythagoras, 32(2), art. #52, 15 pages. http://dx.doi.org/10.4102/pythagoras.v32i2.52 copyright notice: © 2011. the authors. licensee: aosis openjournals. this work is licensed under the creative commons attribution license. issn: 1012-2346 (print) issn: 2223-7895 (online) an instrumental approach to modelling the derivative in sketchpad in this original research... open access • abstract • introduction • theoretical framework    • the instrumental approach to technology use in the classroom       • instrumentation       • instrumentalisation    • the teacher’s instrumental orchestrations in a technology-rich classroom    • research questions • research approach    • methodology       • participants and ethical considerations       • teaching experiment methodology and post-task-based interviews       • modelling functions       • modelling the average rate of change of a function       • modelling the derivative as the instantaneous rate of change       • modelling the derivative as the rate of change function • findings and discussion    • modelling the function concept in sketchpad       • post-task interview protocol for student mt       • protocol analysis a    • modelling the average rate of change of a function       • post-task interview protocol for student tm       • protocol analysis b    • modelling the derivative as the instantaneous rate of change       • post-task interview protocols for students tm and dj       • protocol analyses c    • modelling the derivative as the rate of change function       • interview protocols for students tm and mn       • protocol analysis d • conclusion    • limitations of the study • acknowledgements    • competing interests    • authors’ contributions • references • footnote abstract (back to top) encouragement to integrate information and communication technologies into mathematics education curricula is an increasingly universal phenomenon. as a contribution to the discourse, this article discusses the potential use in the classroom of the geometer’s sketchpad® (key curriculum press, emeryville, ca, united states) mathematics software in modelling the derivative and related concepts in introductory calculus. in an empirical study involving first-year non-mathematics major undergraduate science students, a hypothetical learning trajectory (hlt) was conjectured and implemented for students to experience the visualisation and multiple representations of calculus concepts on the cartesian plane with a computer graphic interface. the utilisation scheme is interpreted through the lens of the instrumental1 approach proposed by trouche. the hlt was partly informed by the historical development of the derivative as synthesised from the literature on the history of calculus and partly by the affordances, enablements, constraints and potentialities of sketchpad itself. the findings of the study suggest that when exposed to the capabilities of this software, learners can experience geometer’s sketchpad® as an effective visualisation tool or instrument for the representation and learning of the derivative and related concepts in introductory calculus. however, the effectiveness of this tool is not a given or a foregone conclusion − it is a product of the teacher’s instrumental orchestration, gradual learner mastery of the software syntax and careful resolution of theoretical-computational conflicts that can arise during early use of the instrument. introduction (back to top) there is increasingly firm encouragement to integrate information and communication technologies into mathematics education curricula in many countries of the world today, aided and accentuated by an unprecedented global proliferation of digital technologies. south africa, or the southern african region for that matter, is no exception in this global euphoria, as shown by emphasis on the use of available technology in the curriculum and assessment policy statements (caps) (department of basic education, 2011, p. 12). the imperative to introduce computing technologies into the classroom not only brings with it many new opportunities for learning and teaching, but is also strongly encouraged from within the mathematical and scientific community itself. for example, borwein (in press) points out that the computer has become one of the main and indispensable tools for experimentation and research in many parts of pure and applied mathematics. today most problems in the ‘real world’, in the scientific community and society broadly, can only be tackled, modelled and investigated with the aid of computing technology. indeed, there has been a noticeable shift in the last 50 years towards favouring numerical methods and discrete modelling with difference equations and recursion in industry as opposed to the classic modelling techniques with continuous algebraic functions and traditional calculus. for example, the problems engineers and scientists frequently investigate today almost as a rule lead to differential equations which can only be solved numerically. computing technology is therefore also seriously challenging the very content and focus of aspects of traditional mathematics curricula at school and university, and gradually beginning to force curriculators to start grappling with the ‘uncomfortable’ question about what is still relevant and what is not, and which new skills are essential for a modern computing society. in order to integrate the new technologies into the classroom, the challenge for teachers is to master the new tools and simultaneously to find ways to enhance or empower learners’ mathematical learning. one difficulty cited by robert and rogalski (2005) has been that teachers’ practices are both complex and stable. some studies (e.g. jenson & williams, 1992) show that the integration of technology initially complicates rather than simplifies the teacher’s life. lagrange and monaghan (2009) additionally argue that the availability of technology amplifies the complexity and, as a consequence, challenges the stability of teaching practices: techniques used in ‘traditional’ settings can no longer be applied in a routine-like manner when technology is available. a new repertoire of teaching techniques (e.g. where each learner or small group of learners has access to a computer in the classroom) demands a renegotiation of the traditional didactic relationships between the teacher, mathematics and the learner. even if only the teacher has a laptop or computer connected to a data projector in the classroom, this affords the teacher a much wider range of teaching possibilities than a static blackboard or overhead projector. these new techniques are likely to be related to already existing ones as well as to teachers’ underlying views on mathematics education (pierce & ball, 2009). in spite of the existence of innovative techniques, integration of some of the technologies in mathematics classrooms remains marginal (e.g. haspekian, 2005) and, ironically, lags behind the abundant availability of technology in the public domain. in order to help teachers benefit from the new technology in mathematics teaching, it is pertinent to deliberate on the mediation possibilities that the new technological tools offer. to contribute towards such a discourse, this study sought to investigate the viability of using the geometer’s sketchpad® dynamic mathematics software as a pedagogical tool to enhance learners’ understanding of the concept of ‘derivative’ in introductory calculus. this initially poses challenges to many learners because of the underlying limit concept that is difficult to grasp at first encounter with calculus, in terms of ruptures (discontinuities or false continuities) that occur in transitions from the finite to the infinite and the infinitesimal. although it might be easy for learners to equate the ‘derivative’ with ‘the thing that changes xn into nxn–1’ (wu, 1999, p. 13), on its own this traditional symbolic manipulation approach evokes little conceptual understanding in learners (tall, 1997, p. 289). dynamic geometry software such as sketchpad, cabri and geogebra, and computer algebra systems (cas) such as derive, maple and mathematica allow for numeric, graphic and symbolic or algebraic representations to be generated simultaneously in both static and dynamic forms, in a manner that has potential to foster deeper conceptual understanding based on real-time visualisation in the graphic interface. although these new tools hold so much promise, they do not immediately translate into efficient mathematical instruments in the classroom or realise the epistemic value of the associated instrumented techniques; that is, the way in which they contribute to the understanding of the objects they involve may not be immediately accessible to learners (artigue, 2002, p. 245). that these software packages are relatively new tools makes their efficient integration into mathematics classrooms a pedagogical research imperative. based on ndlovu (2008), this article describes a possible theoretical framework for a sketchpad-mediated teaching sequence from an instrumental approach. the research questions that guided the study are then presented and the research methodology and procedure explained before presenting and discussing the results. finally, some conclusions and recommendations are drawn. theoretical framework (back to top) the instrumental approach to technology use in the classroom the theory of instrumentation inspired the instrumental approach to tool use in technology-rich environments, as elaborated by trouche (2004) to describe the context of human-machine interaction. the approach helps us to understand the influence of tools on the mathematical approach and on the building of learners’ conceptual understanding (knowledge) through a process referred to as ‘instrumental genesis’ wherein, in our case, the dynamic mathematics software sketchpad becomes the mathematical work tool to empower students’ learning of the derivative and related concepts. this instrumental genesis depends on the software constraints and potentialities (constraints, enablements, affordances, and potentialities), students’ knowledge in the form of utilisation schemes (usage schemes and instrumented action schemes), and the teacher’s instrumental orchestrations in the form of didactical configurations and exploitation modes, elaborated upon below.the term ‘tool’ is preferred to the term ‘machine’, and is used in the sense of something which is available for sustaining human activity (trouche, 2004, p. 282). before a tool can be used, it is a material artefact or given object (e.g. hammer or computer) or cultural artefact (e.g. language). the tool or artefact has potential to shape the environment in the sense that ‘tools wrap up some of the mathematical ontology of the environment and form part of the web of ideas and actions embedded in it’ (noss & hoyles, 1996, p. 227). however, when a tool or artefact is appropriated by the subject and integrated into his or her own activity, it becomes an instrument which is a psychological construct (verillon & rabardel, 1995) in cognitive ergonomics. more precisely then, an instrument can be considered to be an extension of the body or ‘a functional organ made up of an artefact component and a psychological component’ (trouche, 2004, p. 285). the organ construction procedure is referred to as instrumental genesis, a complex process needing time and linked to the artefact’s characteristics (its potentialities and constraints) and to the subject’s activity, knowledge and former method of working (trouche, 2003, 2004). this procedure is characterised as consisting of a combination of two sub-processes: an instrumentation process directed towards the subject, and an instrumentalisation process directed towards the artefact (trouche, 2003, 2004). instrumentation instrumentation is the process by which the computer user is mastered by his or her tools or by which the artefact prints its mark on the subject by allowing him or her to develop activity schemes within some boundaries or limits – the constraints (obliging the user in one way and impeaching in another way), enablements (effectively making the user able to do something), potentialities (virtually opening possibilities) and affordances (favouring particular gestures or movement sequences) of the artefact (noss & hoyles, 1996; trouche, 2004). guin and trouche (1999, 2002) distinguish between three types of constraints: internal constraints intrinsically linked to the hardware, command constraints linked to the syntax of the various commands, and organisation constraints linked to the interface between the artefact and the user (e.g. symbolic, numeric and/or graphic). instrumentalisation instrumentalisation can go through different stages that can include discovery and selection of relevant functions, personalisation and transformation of the artefact, sometimes in directions unplanned or not conceived of by the designer (trouche, 2004) but inadvertently enabled by the architecture and configuration (enablements and potentialities) of the artefact. for example, sketchpad was not meant ‘to generate’ but ‘to show’ the derivative of a function graphically and symbolically. drijvers (2002, p. 225) refers to this gap between the (symbolic) solution process and the final result as the top-down and ‘black box’ character or non-transparency of a computer algebra system (cas). however, by plotting the tangent slope of a function at any given point against changes in the independent variable, sketchpad can enable the user to dynamically or progressively generate the graph of the derivative of a function. similarly, one can numerically and graphically simulate the average rate of change and instantaneous rate of change of a function in real time on screen. it is this adaptability of the software that the teacher is at liberty to exploit to enhance a deeper understanding of mathematics concepts. thus the idea of instrumental genesis reflects the fact that using a tool is not a one-way process, but a dialectic relationship between the subject acting on his or her personal instrument and the instrument acting on the subject’s thinking (haspekian, 2005, p. 118). trouche (2003) points out that because of this dialectic it is not always possible to distinguish clearly between these two processes. the teacher’s instrumental orchestrations in a technology-rich classroom trouche (2004, p. 296) introduces the term ‘instrumental orchestration’ to point out the necessity for a given institution (e.g. a teacher) to provide external steering of learners’ instrumental genesis. he argues that this necessity is seldom taken into account when considering the environmental organisation of a technology-rich classroom in so far as the learners’ or teacher’s workspace and time are concerned. from an anthropological point of view, lagrange (2000) notes that the teacher’s task is likely to be complicated by the use of a new environment, since mathematical knowledge and conceptualisation are highly dependent on techniques. as in the paper-and-pencil tradition, the teacher has the didactic responsibility to organise the tasks so that the learners develop and master some techniques of achieving understanding of the mathematical concepts at stake. from an instrumental point of view, the notion of a ‘scheme’ is articulated to address this complexity and consequently to inform the hypothetical learning trajectory (hlt). the scheme is part of the psychological component of an artefact organisation in that it is a dynamic functional entity with goals and anticipations, rules of action, gathering of information, control-taking and some operative invariants (trouche, 2004). trouche consequently views a scheme as performing three main functions: a pragmatic function which allows the user of the instrument to do something; a heuristic function which allows the user to anticipate and plan actions; and an epistemic function which allows the user to understand what he is doing. the teacher’s instrumental orchestration should therefore help learners to actualise the various functions of their schemes, the most important of which is the epistemic – to develop mathematical understandings and knowledge using the artefact cum instrument. trouche (2004) also distinguishes between two kinds of artefact utilisation schemes: usage schemes oriented towards the management of the artefact (e.g. turning on a computer, adjusting the screen contrast, choosing a particular key, etc.); and instrumented action schemes oriented towards the performance of specific tasks (e.g. computing a function’s limit). the teacher’s instrumental orchestration has to take into account that a given artefact can lead to various instruments and utilisation schemes or a system of interdependent artefacts or instruments which in turn call for chains of usage and instrumented action schemes. for example, dynamic mathematics software (an artefact by itself) has to be installed onto a computer (which is another artefact), and a projector (another artefact) and/or a smart board (yet another artefact) can be used for whole-class discussions. learners working in sketchpad can copy and paste their mathematical productions into microsoft word (which is yet another artefact). for a given artefact then, the teacher must consider various geneses enrolled in complex systems of various interacting instruments (haspekian, 2005). in a sense, using a new tool implies a reconsideration of the teacher’s mathematical organisation (of sets of tasks, techniques and theories) to guide learners’ instrumental genesis – individually, in groups, or through whole-class activities. to clarify instrumental orchestration further, drijvers, doorman, boon, reed and gravemeijer (2010) distinguish between three elements within a teacher’s instructional activity: a didactic configuration, an exploitation mode and a didactic performance. a didactic configuration is metaphorically defined to be an arrangement of the artefacts in the classroom environment, and as such refers to the configuration of the teaching setting (comparable to the choice and layout of musical instruments for an orchestra) to induce a sound mathematical discourse. an exploitation mode entails the way in which the teacher decides to exploit a didactical setting or configuration for the benefit of his or her instructional intentions, including decisions about how a mathematical task will be introduced and worked through (e.g. whole-class, group work, individual work, etc.). a didactic performance entails the ad hoc or on-the-spot decisions taken during the teaching-learning session regarding what question to pose, how to respond to a particular learner’s response or input, and how to deal with the unexpected. research questions whereas ndlovu, wessels and de villiers (2010) focused mainly on the quantitative results of the pre-test, post-test quasi-experimental design aspects of the main study, this article reports on the qualitative aspects of the study. the learning trajectory hypothesised was the culmination of an analysis of the historical development of the derivative in the context of motion (e.g. zeno’s paradoxes, speed and time) and of drawing tangents and finding rates of change or instantaneous speed (see ndlovu, 2008; ndlovu et al., 2010). the main research questions which guide this report are: (1) how can sketchpad be used as a technological teaching tool to enhance students’ understanding of the derivative in introductory calculus? and (2) in what ways does such a teaching or learning trajectory improve the quality of domain-specific understanding of the derivative and related concepts? the following sub-questions guided the study: (1) how can sketchpad be used to model pre-concepts of the derivative? and (2) how can the derivative be modelled in sketchpad, to lead to its dual meaning as ‘the instantaneous rate of change’ and ‘rate of change function’? research approach (back to top) methodology participants and ethical considerations this research is based on a teaching experiment conducted by the first author involving 20 undergraduate non-mathematics science major students at a university in zimbabwe (ndlovu, 2008). the students volunteered to participate in the study, were informed about its purpose and consented to participation. they were also advised that they were free to withdraw from the study at any stage and were assured that their names would be kept confidential. for this reason the names used in this article are pseudonyms. participation in the experimental group was on a first come, first served basis. written permission to conduct the study was obtained from the host university department with the support of the academic registrar. permission was also granted by the departments whose computer laboratories were used. teaching experiment methodology and post-task-based interviews we use the term ‘teaching experiment methodology’ to refer to a context in which the teacher’s role, as described by steffe and thompson (2000, p. 277−279), is continually to postulate possible meanings that lie behind students’ language and actions, so that their actions guide the teacher to construct a frame of reference in which what they can do seems rational. the methodology thus had the general form of a developmental research design, elaborated upon by cobb, confrey, diseiza, lehrer and schauble (2003) and gravemeijer (1994), amongst others, in which the teacher progressively develops new hypotheses to explain students’ ways of operating and designs tasks to provoke creative activity. the distinguishing characteristic of the research design therefore rests in how it links the development of an hlt, which is then implemented and later adjusted in the light of the implementation experiences. essentially then an hlt involves the design of learning materials and activities that take into account the expected thinking or conceptual development that students might engage in with a given technological tool to achieve a particular learning goal (drijvers et al., 2010; simon, 1995). there is relatively widespread consensus amongst mathematics educators that deeper conceptual understanding, higher problem-solving processes, and internal constructions of mathematical meanings have to be developed in addition to traditional procedural and algorithmic learning (goldin, 1997, p. 40). the quest to understand students’ thinking has enabled the use of structured interviews for the dual purpose of: (1) observing the mathematical behaviour of learners engaged in problem solving and (2) drawing inferences from the observations. consequently, after each session of the implemented learning trajectory students were interviewed to ascertain their mathematical experiences of calculus concepts as mediated by sketchpad under the instrumental orchestration of the teacher-researcher. the didactic configuration was such that each student had access to his or her own computer in the computer laboratory, where the personal computers were arranged in rows. the exploitation modes entailed availing worksheets with tasks for students to execute individually or in pairs. students were encouraged to share experiences and observations or indeed to solicit the teacher-researcher’s assistance whenever desired. as part of the teacher-researcher’s didactic performance, the sessions were intermittently interrupted by whole-class discussions to clear up any common misunderstandings or unexpected behaviour of the software. a qualitative approach was adopted to enable both an interpretative and explanatory analysis of learner perceptions of their mathematical and software experiences. a brief description of the classroom activities which constituted the intervention follows. these activities will be discussed with reference to the sketchpad enablements and constraints for modelling functions, the average rate of change of a function, the derivative as the instantaneous rate of change, and the derivative as the rate of change function. modelling functions starting with models of informal real-life situations such as distance-time covariations, consistent with the historical context of discovery and invention of calculus, students were tasked to plot various functions in sketchpad, leading to graphic and symbolic representations. figure 1 shows the graph of y or f(x) = x2. an analysis of the figure shows that sketchpad enabled the user to ‘plot the function’ and represent it both graphically and symbolically on the cartesian plane. this is curve sketching made easy, and exacted at the click of a button. the software also enabled the user to plot a point on the graph. a further enablement is that such a plotted point (a) can be animated to trace or follow the path of the graphical representation. figure 1: function modelling and simulation in sketchpad. an organisation constraint of the animation is that it can only be seen within the visible range of values of the domain and co-domain or the viewing window. the scale of the viewing window can be varied by the user by means of a usage scheme (e.g. dragging point (1;0) on the x-axis) to perform a pragmatic function. in turn, by virtue of sketchpad enablements the animation can be paused, reversed or resumed by the user – an instrumented action scheme performing an epistemic function of understanding the locus of the point being animated. more importantly, the coordinates of the plotted point can be plotted and a sample table of values created as the animation progresses, thus enabling a numeric representation. an internal constraint of sketchpad regarding this numeric representation is that one cannot start with a table of values and then plot – the user is obliged in one way and impeached in the another way, the opposite direction. however, once such a sample table of values has been created, the points on it can be plotted on the graph (e.g. points a1 – a11 in figure 1 were plotted from the table on the left). again this appeared to be an internal constraint of sketchpad, in the sense that the user is obliged to sample points first on an existing graph plotted apriori rather than the other way around – a familiar pencil-and-paper sequence. overall, the enablements and affordances of sketchpad software enable one to create a static graphic, static symbolic (algebraic), and static numeric (table of values) representations of the function on the one hand, and a dynamic graphic (animated point), dynamic symbolic (for curves or lines which vary their position), and dynamic numeric (table of values) representation on the other. modelling the average rate of change of a function commencing with informal real-life situations, such as average velocity, corresponding to the motion context of discovery and invention of the calculus as motivation, students were assigned function-plotting tasks in sketchpad, leading to graphic and symbolic representations. figure 2 shows the graph of y or f(x) = x2 + 1 with a secant line through points a and b. sketchpad enables the user to construct a second point on a graph and to construct a line passing through the two points (a and b). furthermore, it allows the user to measure or calculate the vertical (y-) and horizontal (x-) displacements between two plotted points on the cartesian plane. this enables a step-by-step calculation of the slope of the straight line joining the two points; this enablement also appears to be a potentiality of sketchpad. alternatively, the slope of a straight line can be measured directly using an available command − both an affordance and a command constraint of sketchpad, since this is what the artefact favours and can by nature do. in both cases the result is the same: the average rate of change of a function between any two points of its domain. when animated, the line becomes a continuous sequence of secants at different positions of the graph and the results can be tabulated, as shown in figure 2. figure 2: graph of f(x) = x2 + 1 and the secant line ab (modelling the average rate of change). in other words, sketchpad potentially enables the user to create a static graphic, static symbolic (algebraic), and static numeric (table of values) representation of the secant on the one hand and a dynamic graphic (animated secant), dynamic symbolic (as the secant varies its position), and dynamic numeric (table of values) representation on the other – virtually opening up possibilities or potentialities for the user. these affordances and enablements of sketchpad highlight the epistemic function of the associated instrumented action schemes or gestures oriented towards carrying out specific tasks. modelling the derivative as the instantaneous rate of change using real-life situations informally as a point of departure, such as speed at an instant, or at time of collision of objects as background, students (guided by didactic configuration of worksheets) were required to construct the tangent as the limit of the secant line in sketchpad. this resulted in the graphic and symbolic representations shown in figure 3. figure 3: modelling instantaneous rate of change in sketchpad. in addition to the enablements and affordances for representation of the average rate of change of a function described above, through its syntax-linked command constraints sketchpad enables the user to move one point (b) to another point (a) along a graph (by dragging or via a self-created action button) or vice versa. the secant line joining the two points shifts accordingly and stops at the destination point as the tangent to the graph at that point (a potentiality). it does not matter whether b is dragged to a from above or from below – a convenient enablement. this opens up to the sketchpad user the possibility or potentiality to dynamically create the tangent as a limit of a secant at any point: that is, as the rate of change of a function at any instant, leading to a construction of the instantaneous rate of change concept image of the derivative. this exemplifies instrumented action schemes performing an epistemic function. in other words, sketchpad enables the user to create a static graphic, static symbolic (algebraic) and static numeric (table of values) representation of the tangent as a limit of the secant, on the one hand, and a dynamic graphic (animated secant as shown by path traced), dynamic symbolic (as the secant varies its position as shown by sample table readings), and dynamic numeric (sample table of values) representation, on the other. however, a command constraint or limitation of the representation capabilities is an apparent violation of the definition of a secant line when it becomes a tangent or when point b mathematically or precisely reaches point a. theoretical-computational conflicts emerge, and require mathematical resolution and explanation. giraldo and carvalho (2002, p. 1) define a theoretical-computational conflict to be any situation in which a computational representation for a mathematical concept is, at least potentially, contradictory to the associated theoretical formulation. for example, in figure 3, the conflicting differential quotient results were resolved by means of the teacher-researcher engaging in a didactic performance that initiated a whole-class discussion about the effect of increasing the precision of the decimal measure of the slope. this revealed that it was a rounded-off value to a limited number of decimal places, and the limit was then investigated from both above and below to arrive at an intuitive understanding of its existence – an epistemic function of the associated instrumented action scheme. modelling the derivative as the rate of change function starting with real-life situations, such as speed at any instant, as motivation, students worked on worksheets (didactic configuration) to represent the rate of change function as shown in figure 4. sketchpad enables the user to plot a point given its coordinates (enablement) and to trace that point as its coordinates change (enablement). if the x-values of point a are plotted against the slope values of tangent ab, point c is obtained (affordance). when the tangent is animated (enablement), point c follows a path that can be traced to represent the slope function of the original graph (potentiality). alternatively, sketchpad enables the user to find the symbolic derivative (an affordance or command constraint) and to plot its graph directly (an organisational constraint). figure 4: modelling the rate of change (slope) function in sketchpad. in other words, sketchpad enables the user to generate a static graphic, static symbolic (algebraic), and static numeric (table of successive coordinates of point c) representation of the tangent slope changes with respect to changes in x-values, on the one hand, and a dynamic graphic (animated tangent effect on point c as shown by the path traced) and dynamic numeric (last row of x-y table of values) representation, on the other. however, one organisation constraint of sketchpad is that the tangent may disappear when animated, causing theoretical-computational conflicts that need to be resolved to merge the mathematics with the technology. a second (organisation) constraint is that the equation of the derivative can only be represented statically and a dynamic symbolic (algebraic) representation makes no sense. a third (command) constraint is that the symbolic derivative is given as an answer that does not show the steps used to arrive at it (which also makes it an organisation constraint). findings and discussion (back to top) qualitative data in terms of some sample post-task interview protocols with individual students are now presented and then analysed. we use the notation tr for the teacher-researcher and students mt and dj and so on for the individual students. bold type, for example, plot new function, refers to commands in the sketchpad drop-down menus as illustrated in figure 5. figure 5: example of commands in the sketchpad drop-down menu. modelling the function concept in sketchpad post-task interview protocol for student mt a1. tr: can you describe to me how you would plot a function in sketchpad? choose a function of your choice. [instrumented action scheme]a2. mt: from the graph menu i would select plot new function [an affordance] and define using the calculator keypad that appears [instrumented action scheme performing an epistemic function]. [mt demonstrated that on the computer and obtained the graph in figure 6] figure 6: student mt’s function plot. a3. tr: how would you represent the function in dynamic graphic form? [enablement]a4. mt: i would first construct a point on the graph of by highlighting the plotted graph of [enablement] and choosing point on function plot [affordance] from the construct menu and animating or dragging the point. [enablement] a5. tr: do you mean construct point on function plot? a6. mt: yes construct, not plot. a7. tr: what happens when you animate the constructed point? [organisation constraint] a8. mt: the point follows the graph of the function. a9. tr: how do you create a table of values for in sketchpad? [enablement] a10. mt: i select the plotted point and label it a [enablement], then select abscissa (x) and ordinate (y) from the measure menu to display the coordinates. [demonstrating sketchpad affordance] a11. tr: how do you create the table after that? a12. mt: i select the values for xa and ya [enablement] and select tabulate from the graph menu to create a table of values for the two variables [organisation constraint]. then animate and double-click the table as the point moves along the graph. [demonstrating sketchpad enablement] a13. tr: are all the table data points visible on your graph? [organisation constraint] a14. mt: let me see, the point (-0.01; -130.77) appears to be out of sight. [internal constraint of sketchpad] a15. tr: what happened to the animated point when it disappeared at the bottom? a16. mt: it reappeared again at the top of the graph. [internal constraint] a17. tr: what does that tell us about the values of y when x = 0? a18. mt: there is some connection between the large negative y-value and the large positive y-value yet the y-axis is an asymptote and there should be no values of y. [theoretical computational conflict] a19. tr: can you give examples of real-world situations that can be represented by the function you have chosen? a20. mt: maybe the electrical resistance of a conductor which is inversely proportional to its size, or speed and time the faster you run, the less time you take, etc. protocol analysis a in the interview there was a satisfactory description of the sketchpad affordances or favourable movement sequences for the instrumented action schemes of defining and transforming a function from its static symbolic or algebraic form (a2). the static symbolic form, , entered in the calculator keypad (a built-in artefact of the software accessed by means of the usage scheme) was transformed into the static graphic visualization on the graphic interface (an organisation constraint of sketchpad). the static graphic representation was transformed into a dynamic representation by constructing a point (a4), by means of a sequence of enablements and affordances (a6), on the function plot. this was a process of the student mastering of the artefact instrumentalisation. the plotted or constructed point was successfully animated (a4), a potentiality enabled by sketchpad, and observed to follow the static graphical representation as its locus (a8) – an organisation constraint linked to the interface between the artefact and the user. this was a powerful conceptual development potential of the software in that it demonstrated continuity as an important property of the function. however, the continuity of the function was violated when the animated point disappeared at the bottom (a14) and reappeared at the top of the y-axis, the asymptote, as if there was no break in movement (a16) – an internal constraint of the artefact induced by the limited viewing window. a probing of the anomaly by the teacher (a didactical performance) provided an opportunity for the student to reconcile with the fact that the graph was undefined at x = 0 (a17). this suggested that there could be deep mathematical understandings created by a discussion and resolution of the apparent theoretical-computational conflicts. regarding the care needed when using software as a generator of mathematical concepts, noble, nemirovsky, wright and tierney (2001, p. 87) remind us that mathematical concepts reside not in physical materials but in what students do and experience. similarly, the ability to make connections between representations (instrumental genesis) clearly lies outside the materials themselves, but can be enhanced or inhibited by the teacher’s instrumental orchestration strategies − exploitation modes in particular. we can only gain access to the connections the learner makes by focusing his or her attention on the meanings attached to the mathematical relationships in the activity they are undertaking – the epistemic function of their instrumented action schemes. it is clear that whereas mt was able to make interconnections between the multiple representations and to link the inverse function informally to some real-life situations, there were still some command and organisation constraint problems to contend with. modelling the average rate of change of a function post-task interview protocol for student tm b1. tr: can you describe to me how you would use sketchpad to demonstrate, calculate or measure the average rate of change of a function between any two points of its domain? [potentiality of sketchpad]b2. tm: firstly select the function [enablement] and choose plot point on function from the construct menu, twice, to plot two points on the graph of the function. [affordance] b3. tr: is it plot point or construct point on function plot? [didactic performance] b4. tm: thank you for that correction, it should be construct, not plot. [command constraint] b5. tr: that’s ok, you can proceed. b6. tm: then select the points [affordance] and choose label points from the display menu to label the left point and the point on the right. [affordance] b7. tr: can you demonstrate to me how you would construct a secant line in sketchpad? [potentiality] b8. tm: choose line from the construct menu to construct a line passing through the highlighted points, and this is the secant line. [instrumented action scheme supported by an enablement – choice of menu, and an affordance – favoured menu command] b9. tr: how would you find the slope/gradient of the secant line in sketchpad? [didactic performance of prompting an instrumented action scheme in the student] b10. tm: to find the slope in sketchpad i will select the secant line [an enablement] and choose slope from the measure menu [an affordance]. [syntax] b11. tr: how would you find the equation of the secant line in sketchpad? [didactic performance] b12. tm: i would select equation from the measure menu, instead of slope, this time. [an affordance] b13. tr: how would you explain to a friend what this gradient represents? [mathematical concept] b14. tm: i can say change in y-values divided by change in x-values. b15. tr: how can you express the slope symbolically? b16. tm: to symbolically express the gradient, let there be two points a and b such that their coordinates are (x1;y1) and (x2;y2). then secant line gradient = potentiality] b17. tr: could this be the same as ? [didactic performance] b18. tm: yes, if we substitute y2, y1, x2, and x1 for f(x+h), f(x), (x+h) and x respectively. b19. tr: what happens when the secant line is animated? b20. tm: the coordinates for points a and b, the slope and equation of the secant line all change as the animation goes on. [organisation constraint] b21. tr: can you give real-world examples of the average rate of change? b22. tm: average speed of a bus from bulawayo to gwanda on its way to beit bridge or any two points of its journey. protocol analysis b tm started with the selection of the graph of the function, an enablement, and the construction of two points on the function plot (b2) – an affordance of sketchpad on the construct menu favoured by the sequence of software manipulations. there were, however, some semantic conflicts and difficulties – a command constraints problem. the ‘construct point’ command was referred to as ‘plot point’ (b4). there was some inconsistency between the traditional paper-and-pencil mathematical language, where a point is ‘plotted’ not ‘constructed’, and sketchpad mathematical language. in his narrative the student seemed to be rationalizing this terminological debacle. however, it was clear that the task was to construct two points (b8) on the graph (function plot) first an instrumented action scheme supported by a sketchpad enablement (choice of menu) and an affordance (favoured menu command). the labelling of points was optional and therefore an enablement making the user able to do something.through his didactic performance the teacher prompted the student to employ an instrumented action scheme for finding the slope of the secant in sketchpad (b9). the student was able to select the secant (enablement) and to choose slope (an affordance) from the measure menu (b10). similarly, the student was able to find the equation of the secant using the relevant sketchpad affordances (b12). in other words, the student started with a graphic representation of the secant (an organisation constraint) and proceeded to the numeric slope (an affordance), which he then expressed symbolically as a differential quotient (b16). the combination of representations amounted to a potentiality of sketchpad, in that it was not a given but a possibility. the results of the animation (enablement and affordance) gave rise to various average rates of change as the secant slid along the function plot (a command constraint). the activity thus gave students options of representing the average rate of change as a measure of slope or as a method of calculating, and as a variable measure dependent on any two points on the graph. modelling the derivative as the instantaneous rate of change post-task interview protocols for students tm and dj these two students were interviewed separately, but for convenience their responses are given together below.c1. tr: can you describe to me how you would use sketchpad to demonstrate/calculate the slope of a graph at a given point of its domain? [a potentiality supported by an instrumented action scheme] c2. tm: by drawing a tangent line through that point by choosing line from the construct menu [command constraint]. to calculate slope using sketchpad, i select the tangent line through given point [enablement], then i choose derivative from the graph menu. [command constraint error for this student] c20. dj: drawing a tangent through that point by choosing line from the construct menu [enablement]. to calculate slope using sketchpad, i select the tangent line through the given point [enablement], then i choose slope from the measure menu [affordance]. it will then be shown on the screen. [organisation constraint error] c3. tr: can you show me how to construct a tangent line when given a secant line in sketchpad? [teacher’s didactic performance] c4. tm: by selecting point b and dragging it towards point a it’s the rate of change of function at that point. [enablement] c22. dj: by creating an action button to move point b to point a to form a tangent at a. it’s the slope of the function at that point a. [see figure 7] figure 7: construction of the tangent (a) before animation (b) after animation. c5. tr: what is the value of the gradient of the tangent line at the point of contact?c6. tm: this refers to the increase in y divided by increase in x of a tangent. [instrumental genesis] c25. tr: is it always an increase? [didactic performance] c26. dj: no…. c7. tr: can you explain the meaning of the value of the gradient of the tangent at a point? [didactic performance] c8. tm: instantaneous rate of change at that point. c9. tr: is the value always positive, negative or both? [didactic performance] c10. tm: the value of the gradient on a graph can be positive, negative or non-negative. c30. dj: no. c11. tr: can you explain your answer? [didactic performance] c12. tm: it cannot be both positive and negative at the same time. c32. dj: it can be negative if xand yare changing in opposite directions but cannot be both positive and negative at the same time. c13. tr: can you give real-world examples of the instantaneous rate of change? [didactic performance] c14. tm: speed at an instant. c34. dj: speed at point of impact in a car accident. c15. tr: how would you express the gradient of the tangent symbolically as a limit of the secant line gradient? c16. tm: [mathematical understanding] c36. dj: limb → a gradient of secant = gradient of tangent c17. tr: is there any other way of expressing the limit? c18. tm: [note error] c38. dj: protocol analyses c the demonstration and calculation of the slope of a graph at a given point was a potentiality which sketchpad availed, requiring execution of the appropriate instrumented action schemes (c1). both tm and dj incorrectly explained that they would ‘draw a tangent line’ in sketchpad by choosing line from the construct menu (an indication of command constraints). their descriptions were more like paper-and-pencil than sketchpad methods. tm was unable to calculate the slope unaided, due to the use of an incorrect menu command (c2), a command constraint, and proceeded to find the ‘derivative of the tangent line’ in sketchpad (c2) – another command constraint. however, dj was able to measure the slope of a given line (tangent) by selecting or highlighting the line (enablement) and choosing an appropriate command in the measure menu to measure the slope (an affordance favoured or privileged by the menu) (c20). in both cases the difficulties encountered were technical, in that the students were not able to operate the software in the intended way (mehanovic, 2011), an indication that they were still in the process of mastering the software (instrumented action schemes in their instrumentalisation process) to accomplish their instrumental genesis. the second question of the interviews (c3 and c21) redirected the students towards using the secant line as the starting point. this redirection evidenced the teacher-researcher’s instrumental orchestration efforts through the exploitation mode of didactic performance. eventually both students constructed two points, a secant joining the two lines, and dragged one point (b) towards the other (point a) (c4) or used an action button to obtain the tangent by animating point b to point a (instrumented action schemes) (c22). this led to a dynamic graphic illustration of the tangent as the limiting position of the secant to represent the instantaneous rate of change, as in figure 7 (an organisation constraint linked to the interface/viewing window as well as a potentiality made possible by sketchpad). modelling the derivative as the rate of change function interview protocols for students tm and mn d1. tr: in sketchpad, how would you construct the graph of the function showing how the tangent gradient changes with respect to x? [potentiality of sketchpad to be supported by appropriate instrumental action schemes]d2. tm: i would start by selecting the gradient [enablement] and measuring its slope [affordance], and then isplaying its x-coordinates using the slope and abscissa commands. [an enablement within the command constraints] d14. mn: i would follow activity c, select f(x) = x2 and display the algebraic derivative using the derivative command [an affordance] and then plot it using the plot function command from the graph menu. [an affordance favoured by sketchpad] d15. tr: what does the plotted function represent? [didactic performance] d16. mn: the function plotted represents the gradient at any point. [epistemic function of the instrumented action schemes] d3. tr:how would you plot the slope of the tangent line against the x-values in sketchpad? d4. tm: select the slope and the abscissa and choose plot as (x,y) from the graph menu. point c is plotted. [potentiality achieved through appropriate enablements – selection, and affordances plotting] d18. mn: move point b to point a using the action button [instrumented action scheme]. plot x against slope as (x,y). that is, point c [see figure 8]. select point c and choose trace point from the display menu [an enablement followed by an affordance]. deselect everything and select line ab [tangent] and select animate line from the display menu [heavy demand on syntax – command constraint]. the path taken by point c represents the gradient of the function at any point. [potentiality] [attention can be easily drawn away from the mathematical to the syntactical processes – command constraints] figure 8: generation of the rate of change or slope function in sketchpad. d5. tr: if you trace point c and animate the tangent, what pattern is the path traced by the plotted point? [didactic performance]d6. tm: it’s a straight line. [organisation constraint] d20. mn: the path traced by c is a straight line. d7. tr: how would you represent this path numerically in sketchpad? [didactic performance] d8. tm: constructing a table of values and clicking repeatedly to enlarge it during animation. [enablements] d22. mn: select abscissa (x) and ordinate (as slope of ab). from the graph menu, select plot as (x, y). tabulate x and slope values to form a table of values. [enablements] d9. tr: how can you express the function (path) symbolically as a limit of the differential quotient for the general point on the graph of f(x)? [mathematical meaning of the instrumented action scheme] d10. tm: limit x everywhere: d24. mn: d25. tr: how can you verify your findings in the previous step using sketchpad? [instrumentation] d12. tm: choosing derivative from the graph menu, as we did in the first step. [affordance] d26. mn: select the function label f(x) = x2 [enablement] and the derivative command from the graph menu. [affordance] select plot function from the graph menu to plot the graph of the derivative. [affordance] protocol analysis d the teacher-researcher requested the performance of a potentiality of sketchpad task that needed to be supported by appropriate instrumented action schemes. tm started by selecting the tangent drawn in the previous activity (c) (a time-saving affordance of sketchpad), and made the appropriate selection of an abscissa (x) and tangent slope value for the point of tangency, point a (d2) – an enablement. he then plotted point c using the graph menu (an affordance of sketchpad when two points have been selected). mn also started with work in activity c and specified the function whose tangent was under investigation to be f(x) = x2. however, he selected the derivative command from the graph menu straight away (d14) (an affordance of sketchpad when a differentiable symbolic function is selected) and proceeded to plot the derivative (an affordance) employing appropriate instrumented action schemes. he was also clear that the function represented the derivative as the gradient at any point on the graph (d16). it was evident that mn took advantage of the recently introduced sketchpad shortcut to find and draw the derivative of a function, even though this was a less illuminating or ‘non-transparent’ characteristic of software, reminiscent of the ‘black box’ and flashcard metaphors of technology use (drijvers, 2000; cuoco & goldenberg, 1996). however, the use of a shortcut signified the extent to which mn had mastered the software – an instrumentalisation process – and how, in turn, he was being influenced by the instrument in his solution strategies – an instrumentation process. differences in the progress the individual students had made highlighted the idiosyncratic nature of student experiences of and adaptation to the software environment. the idiosyncratic experiences have implications for the didactic configurations – specific arrangements of the artefactual environment, and the exploitation modes of these configurations in the teacher’s instrumental orchestrations (guin & trouche, 2005, p. 1022). tm was able to plot the tangent slope against the x-coordinate (or abscissa) of the point of tangency (d4). he managed to trace point c (an enablement) and to animate the tangent (d6) (an enablement). mn, on the other hand, was more precise about the tracing and animation commands and appeared more conversant with what the trace represented (d18) − again a sign of growing fluency in his instrumental genesis processes. the uneven growth in fluency in instrumental genesis processes has implications for the teacher’s instrumental orchestration: students might take time to be familiar with the orchestration of new tools and teachers need to be patient with them, at least in the initial stages. however, both students managed to recognise the pattern of the trace to be a straight line graph (d6 and d 20). both were also able to form sample tables of values (d8 and d22) – enablements − but once again, mn was more elaborate in explaining the sketchpad command constraints or complex syntax involved (d22). expressing the path of c algebraically as a limit of the differential quotient was performed correctly (d10 and d24). however, tm’s solution was more accurate (d10). the verification of the derivative using sketchpad was not problematic to execute (an affordance). however, the absence of a solution procedure meant that sketchpad acted algebraically as a flashcard or black box, as alluded to above – a giver of answers (an internal constraint). because of this propensity, cuoco and goldenberg (1996, p. 15) observe that computers are often used badly, in a manner which does not creatively tap the capabilities of either the computer or the learner. conclusion (back to top) the purpose of this study was to investigate how, if at all, sketchpad could be used to enhance students’ mathematical understanding of the derivative in introductory calculus, and to obtain qualitative feedback from students in the experimental group regarding their experiences. the findings suggest that the introduction of technology into the classroom requires instrumental orchestration skills from the teacher, with which to facilitate the processes of instrumental genesis amongst learners. the qualitative results suggest that provided with individual computers and worksheets, learners can work collaboratively to achieve a reasonable understanding of the mathematical concepts at stake. however, the pace of learners’ appropriation proficiency of the artefact for mathematical understanding – instrumental genesis – can vary from learner to learner. the findings are consistent with earlier studies, such as those of roddick (2001), ellison (1994), queseda (1994), kendal and stacey (2001) for learning environments involving cas which have similar symbolic algebra capabilities to sketchpad. the additional interest in this study was on the extended potential of dynamic mathematics environments that combine both cas and dynamic graphic capabilities. there is a need to constantly reconcile the occasional tension experienced between paper-and-pencil methods and meanings on the one hand and the software meanings and methods for the same constructs and concepts on the other. technology cannot orchestrate itself to articulate mathematical understandings to learners; the human instructional agent remains indispensible in appropriating the artefact to accomplish specific didactic goals and purposes. this has implications for curriculum design and software programming. the study needs to be replicated with improved materials, larger class sizes, more implementation time available, and a wider scope of calculus concepts. we therefore cautiously conclude that a careful use of dynamic mathematics software promises some potential to improve the conceptual understanding of mathematics concepts in general and calculus in particular. that potential cannot be realised unless explored and documented carefully. limitations of the study the limitation of this study is that the results cannot be generalised since the number of students was limited. the fact that students were able to progress at different paces suggested that they had idiosyncratic experiences that would be impossible to replicate. to enhance the reliability of the feedback, the same questions were used in the structured post-task interviews, in as much as the same worksheets were used for all learners. to enhance both the internal and external validity of the study, the interview questions were based on the worksheets to ensure correspondence. the replicability of the study is also dependent on the teacher-researcher’s fluency with sketchpad use. acknowledgements (back to top) the initial research was partly funded by the university of south africa under a postgraduate bursary scheme. however, the opinions expressed do not necessarily reflect the views of the university. competing interests m.d.v. is a south african agent for key curriculum press, promoting and selling sketchpad software and curriculum materials. however, the software capabilities and possibilities discussed in this article are not limited to sketchpad but are shared by most dynamic geometry software programs, including some freeware. the authors declare that this interest did not in any way influence them in writing this article, and that their intension is not to promote sketchpad specifically, but to promote all dynamic geometry software and cas programs as vehicles for teaching and learning mathematics. authors’ contributions m.n. was the teacher-researcher responsible for experimental and project design as well as the writing of the manuscript. d.w and m.d.v. as first and second promoters respectively shared their expertise during the main study and during the development of the manuscript. references (back to top) artigue, m. 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(1999). on the education of mathematics majors. in e.a. gavasto, s.g. krantz, & w. mccallum (eds.), contemporary issues in mathematics education (pp. 9−23). new york, ny: cambridge university press. footnote (back to top) 1.the term ‘instrumental’ should not be confused with skemp’s (1976) use of the terms ‘instrumental’ and ‘relational’ understanding. article information authors: linda van laren1 busisiwe goba1 affiliation: 1department of mathematics education, university of kwazulu-natal, south africa correspondence to: busisiwe goba postal address: private bag x03, ashwood 3605, south africa dates: received: 14 feb. 2013 accepted: 12 may 2013 published: 12 june 2013 how to cite this article: van laren, l., & goba, b. (2013). ‘they say we are crèche teachers’: experiences of pre-service mathematics teachers taught through the medium of isizulu. pythagoras, 34(1), art. #216, 8 pages. http://dx.doi.org/10.4102/ pythagoras.v34i1.216 copyright notice: © 2013. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ‘they say we are crèche teachers’: experiences of pre-service mathematics teachers taught through the medium of isizulu in this original research... open access • abstract • introduction • theoretical framework • methodology • findings and discussion    • communication in isizulu during lectures    • academic concepts and isizulu vocabulary    • translation of teaching and learning materials    • status of the numeracy education module taught in isizulu • conclusions • acknowledgements    • competing interests    • authors' contributions • references • footnote abstract top ↑ language of instruction, both in schools and higher education institutions (heis), is a contentious issue in south africa. at our university in the kwazulu-natal province of south africa a language policy was approved in agreement with the language policy for higher education, recommending promotion of isizulu as a medium of instruction for teaching and learning. in complying with this policy the school of education of our university offered three education modules (numeracy, life skills and literacy) in the foundation phase (fp) postgraduate certificate in education (pgce) through the medium of isizulu. against this background we conducted a qualitative study with 16 fp pgce pre-service teachers selected purposefully from the 2011 cohort who completed the fp numeracy education module in isizulu. participants were generally positive about their experiences of being taught in isizulu, but highlighted challenges related to isizulu as a medium of instruction for academic concept explanations, translation of teaching and learning materials, and the status of isizulu as an instruction medium at a hei. introduction top ↑ in post-apartheid south africa the government actively promotes multilingualism. one vehicle through which multilingualism has been promoted is recognition of the 11 official languages in south africa. previously only english and afrikaans were afforded higher status as mediums of instruction at higher education institutions (heis). however, the languages in education policy (department of education [doe], 1997) notes that schooling is to maintain home language(s), but it also mandates that access to and acquisition of additional language(s) should be provided. in addition, learners in the foundation phase (fp) (grade r−3) are expected to learn all subjects in their mother tongue. despite the good intentions of the languages in education policy, there are complications with the issue of languages spoken in south africa, especially in urban areas where there is mixing of different ethnic groups (howie, 2002).higher education institutions are required to take cognisance of the languages in education policy and expected to provide professional development for teachers in the fp using mother tongue instruction. desai (2012) adds that: the state will have to develop resources in these [african] languages, resources such as teachers trained to teach through the african mother tongue, textbooks, reading material and terminology lists. african languages are likely to be seen as viable choices as languages of learning and teaching if they have some currency in domains other than the private. (p. 58) in keeping with the recommendations of the language policy for higher education (doe, 2002), our university in the kwazulu-natal province of south africa formulated a language policy which includes promotion of mother tongue instruction in modules within its colleges as well as the fostering of multilingualism (university of kwazulu-natal, 2006). the university’s strategic plan (2007−2016) provides strategies and goals for implementing its policies, vision and mission. the strategic plan indicates that english will remain the primary academic language but that isizulu should be developed as a medium of instruction. in addition, the strategic plan indicates that the promotion of isizulu will be encouraged through appropriate resources (university of kwazulu-natal, 2013). naturally, the school of education at the university realigned its programme offerings to accommodate the new university-wide policy on language. accordingly, since 2008 the fp postgraduate certificate in education (pgce) life skills and literacy education modules are offered in mother tongue (isizulu) as well as in english. the numeracy education module was only offered for the first time in isizulu in 2011 because of unavailability of materials in isizulu and staff able to lecture in isizulu. what does the literature say about mother tongue instruction in schools and higher education institutions? the literature points out that when young children learn in a second language, their academic progress is inhibited (alexander, 2000, 2009; cummins, 1981; skutnabb-kangas, 2009). in addition, the literature suggests that young children gain culturally, socially, linguistically and cognitively when they learn in their mother tongue (alexander, 2009; kajoro & halai, 2012; mashiya, 2010, 2011). there is a dearth of literature about mother tongue instruction and teaching numeracy (mathematics)1 education modules (green, parker, deacon & hall, 2011). instead, the literature focuses mainly on the language of instruction in teaching numeracy and mathematics to school learners in multicultural contexts (adler & setati, 2000; setati, 2008). furthermore, the literature usually considers proficiency in a language in relation to understanding of mathematics (chen & li, 2008; kajoro & halai, 2012). researchers also highlight performance of english second language learners in mathematics (essien, 2010), and the challenges these learners experience when learning mathematics in multilingual south african classrooms (adler, 1998; adler & setati, 2000; setati, 1998, 2008; vandeyar, 2010). globally, english is regarded as the dominant symbolic language necessary for access to and status in sociopolitical, economic and educational arenas (mufanechiya & mufanechiya, 2011). particularly in africa, english is still regarded as the language of education and provides opportunities for economic and social upliftment (kajoro & halai, 2012; mufanechiya & mufanechiya, 2011; setati, 2008). researchers in mother tongue instruction argue that if learners, students and teachers are made to feel that their mother tongue is inferior and ‘is not the language of education’ (mufanechiya & mufanechiya, 2011, p. 196) then a negative attitude develops towards mother tongue as a medium of instruction. in addition, ndamba (2008) provides an example of an english honours degree having higher status than an honours degree in mother tongue. further, ndamba points out that underperformance in mathematics is due not only to learners’ lack of mathematical knowledge but also to their incompetence in the language of instruction. given the lack of emphasis in the literature on issues of teaching and learning of mathematics in the fp, this study explored the experiences of fp pgce pre-service teachers who completed the numeracy education module that was taught through the medium of isizulu in 2011. the research question that guided this study was: what were fp pgce pre-service teachers’ experiences of being taught a numeracy education module through the medium of isizulu? the article first discusses the theoretical framework provided by lesh, post and behr (1987). secondly, the methodology used in the study is presented, and thirdly we discuss the findings and present our conclusions. theoretical framework top ↑ an important aspect of any lecture or school classroom setting is the social interaction that occurs between the students or learners and the lecturer or teacher (vygotsky, 1978). although the interactions are through speaking, hearing and gestures, the interchange of knowledge, skills, attitudes and values in lecture rooms is also through symbolic, socially defined language norms. mathematics is considered to be a language on its own (usiskin, 1996). the south african foundation phase (grade r–3) national curriculum statement lists specific skills that school learners need to develop in mathematics, including: ‘develop the correct use of the language of mathematics’; ‘learn to listen, communicate, think, reason logically and apply the mathematical knowledge gained’; ‘learn to pose and solve problems’; and ‘build an awareness of the important role mathematical plays in real life situations including the personal development of the learner’ (doe, 2011, pp. 8−9). thus, the national curriculum statement emphasises that mathematics is yet another language that learners need to contend with in the classroom. moreover, one of the core functions of mathematics is seen as facilitating problem solving, which requires additional knowledge, skills and attitudes on top of everyday spoken language knowledge. how are learners assisted to negotiate the language of mathematics and the language of instruction?lesh et al.’s work (1987) on language in the teaching and learning of mathematics is a relevant lens through which to explore our research question. the seminal work of lesh et al. offers transformations that enhance mathematics development and promote problem-solving ability. these authors distinguish five types of representation necessary in mathematics learning: (1) … knowledge is organized around ‘real world’ events that serve as general contexts for interpreting and solving other kinds of problem situations; (2) manipulatable models … in which the ‘elements’ in the system have little meaning per se, but the ‘built in’ relationships and operations fit many everyday situations; (3) pictures or diagrams – static figural models that, like manipulatable models can be internalized as ‘images’; (4) spoken languages – including specialized sublanguages related to domains like logic, etc.; (5) written symbols – which, like spoken languages, can involve specialized sentences and phrases … as well as normal english sentences and phrases. (p. 33) the connections between representations are presented diagrammatically in figure 1. figure 1: a diagrammatic representation of the translation model. lesh et al. (1987) argue that there are valuable ways of translating between and transforming within the five representations that are important in mathematical learning and developing problem-solving ability. in figure 1 the double-headed arrows within each of the five representations indicate transformations, whilst those outside the five representations denote translations. even though all of the representations are important in the teaching of mathematics − use of real life contexts, manipulatives, pictures, written symbols and verbal symbols − we concentrate on the latter because the four other representations are not relevant for analysing the data generated for this article. we pay attention to the fp pgce pre-service teachers’ experiences during the transformations within the verbal symbols representation. in particular, we look at the verbal symbols transformed through code-switching during lectures, translation of learning materials into different languages and communication with learners and teachers during professional practice at schools. methodology top ↑ the methodology employed is a qualitative case study of fp pgce pre-service teachers who completed a numeracy education module that was offered through the medium of isizulu in 2011. to explore the pre-service teachers’ experiences, we worked in an interpretive paradigm. the experiences presented in this article are our interpretations (mcniff & whitehead, 2006). when conducting this research, we had ‘insider and outsider’ (cohen, manion & morrison, 2011) perspectives. we were ‘insiders’ by conducting research at our own hei. at the same time, we were ‘outsiders’ since we were not the numeracy education module presenters. this allowed for confidence in our data as it was ‘buttressed’ (miles & huberman, 1994) by the fact that we collected data in locations where we teach and research. we acknowledge that this analysis of qualitative data is further influenced by the fact that we lecture in mathematics education modules through the medium of english whilst our home languages are not english. we have taught mathematics education for more than 20 years through the medium of english to students whose mother tongue is also not english. we are interested in issues related to teaching through a second language. the fp pgce pre-service teachers, from two different campuses at our university, were invited to take part in the focus group discussions. at the time of data collection there were 58 students registered for the numeracy education module at both campuses. twenty-four were registered at campus a and 34 at campus b. of the 24 registered in campus a, 8 attended the module in isizulu and the remainder in english. of the 34 registered at campus b, 17 attended the module in isizulu and the remainder in english. the participants were selected purposefully from the pgce fp pre-service teachers registered in 2011 who completed the numeracy education module in isizulu because we were interested in their experiences. all of the pre-service teachers who were registered for the module offered in isizulu were invited to take part in the focus group discussions but only 25 volunteered to take part in the discussions. table 1 shows the number of pgce fp pre-service teachers registered for the isizulu medium numeracy education module in 2011 at the two campuses and those who participated in the study. table 1: the number of pgce pre-service teachers registered for the isizulu numeracy education module and the number of participants in this study. the participants completed undergraduate degrees through the medium of english at a south african hei. thereafter they registered to study numeracy, life skills and literacy education in a fp pgce programme through the medium of isizulu because isizulu is their mother tongue.before the relevant data were collected, all of the university’s ethical clearance requirements were met. written consent was requested and all 16 participants willingly signed the consent form that clearly stated that their responses were to be used for research purposes. the information provided was not related to sensitive issues, but we decided not to link responses to the real names of particular participants. we conducted one focus group discussion at each campus with the pgce pre-service teachers registered at the respective campus. the discussion focus was on these pre-service teachers’ experiences after having attended the numeracy education module offered in isizulu. during the focus group discussions we used unstructured group discussions to encourage the pre-service teachers to share their experiences. the focus group discussions were transcribed verbatim and, where necessary, translated from isizulu into english. we were guided by the research question and the selected theoretical framework to inductively look for ‘emerging patterns, associations, concepts and explanations’ (nieuwenhuis, 2010, p. 107). the ‘describe, compare, relate’ method described by bazeley (2009) was used to elaborate on the analysis findings. after the data was inductively analysed, four areas emerged that relate to the presentation of the module in isizulu: communication in isizulu during lectures, academic concepts and isizulu vocabulary, translation of teaching and learning materials, and status of the numeracy education module taught in isizulu. findings and discussion top ↑ each of the four areas that emerged from the findings is discussed under a separate subheading. the discussions are based on the participants’ responses and are explained and illustrated using selected responses. in the excerpts, students are identified by campus, campus a or campus b, and then by a unique number in the particular campus group (e.g. student b3). communication in isizulu during lectures all of the participants indicated that the pgce numeracy education module was taught largely in isizulu, with minor switching from isizulu to english where there was no isizulu meaning for the mathematics education concepts being discussed. for them it was helpful that the module was taught in their mother tongue; they could follow the lecture easily and they enjoyed being taught in isizulu. two example responses are: student b1: it becomes easy when someone teaches you in isizulu; you do not have to think what the person is saying, sometimes they do not have to even finish the sentence, you just get them. student b2: even what the lecturer teaches, it was simplified and it stayed in you [sic]. i haven’t studied it but i still remember how a child in a certain stage would approach a sum. so it was very helpful – numeracy is enjoyable in isizulu. communication is at the centre of concept development (vygotsky, 1978). this means that the participants were probably more at ease doing mathematics education through the medium of isizulu which is their mother tongue. as students point out, they could anticipate and remember the concept development because it was conveyed in their mother tongue. development and supplementation of the participants’ existing knowledge in mathematics education was in a language medium in which the students felt confident. academic concepts and isizulu vocabulary although the participants appreciated that the module was taught in isizulu, they experienced some constraints in using mathematics education concepts in isizulu. they found that there was insufficient isizulu vocabulary for the required concepts explored in the module. isizulu was limiting because the emphasis was on describing rather than in-depth theorising of numeracy education in relation to teaching and learning. for example:student b2: i think the isizulu vocabulary needs a lot of additions. student a3: learning using only isizulu has challenges. for example, when explaining vygotsky’s theory we would say: ‘iqhaza likathisha ukuqinisekisa kulokhu abantwana abakwaziyo’ [the teacher’s responsibility is to build on what the learners already know], … in english you can explain more fully … zpd and scaffolding … in a test, but trying to explain these concepts in isizulu you would get confused and not sure whether you are repeating yourself. the above responses indicate that the pre-service teachers consider isizulu vocabulary to be limited when it comes to exploring the academic concepts of teaching numeracy. for example, concepts such as the zone of proximal development (vygotsky, 1978) are relevant for numeracy education. student a3 pointed out that he understood the difference between the zone of proximal development and scaffolding. however, when he tried to define these concepts in isizulu, he could not easily distinguish between them: he found the concepts difficult to explain in isizulu. similar feelings were expressed by other participants who experienced difficulty in formulating the meanings of theories and terms in isizulu and mentioned that academic terminology is difficult to translate into isizulu. in addition, the pre-service teachers pointed out that the correct number names used in isizulu are unfamiliar to school learners. furthermore, the participants themselves seldom use the isizulu number names for counting, as in everyday life they count in english. as a result, studying a numeracy module in isizulu was difficult because the isizulu spoken by participants is often mixed with english words. student b1 commented on her experience whilst completing her professional practice teaching at a fp school: student b1: what i noticed about teaching numeracy in isizulu and how you pronounce the numbers in isizulu, for example ‘isishiyagalombili’ [eight], you find that learners are not used to those words … even us as teachers. … even when you are teaching the learners, you find that the learners are more used to the english words when referring to numbers. mathematical language building and subsequent numeracy education concept development requires communication of concepts. however, the participants were required to work within both english and isizulu. here participants had to contend not only with the unfamiliar sound of terms in numeracy education, but also with the fact that their own understandings might be difficult to communicate because of the use of different languages. the meanings of the words the participants were required to use in developing appropriate mathematics education knowledge had to filter through thought, language and experience in two different languages, but only one language was allowed as a means of communication. translation of teaching and learning materials another challenge noted by the participants is the translation of materials, literature and assessment tasks. the course materials provided for the numeracy education module were mostly in english, with isizulu notes prepared by the lecturer and distributed during lectures. however, the pre-service teachers often needed to consult literature in the library, where all the articles and readings were in english. translation of this literature was time-consuming and difficult because of insufficient vocabulary in isizulu. for example: student b9: although we do get notes in isizulu but you find that some course packs are written in english, for example, when one has to do an assignment, you will have to start by translating the information. … sometimes in that process you lose the meaning and you find that you end up not understanding what you wanted to say. … when you have to study for the test, most of the materials are given to us in isizulu but because we also have to refer to books written in english and you have to translate even the title. … so that is the challenge we often come across. as mentioned before, the pcge pre-service teachers completed their undergraduate degrees in heis where the instruction medium is english. the participants commented on the fact that they found it challenging to complete assessments in isizulu as they have not studied in isizulu for a long time. the following comments further illustrate the difficulties experienced: student a3: we got used to study in english from our first degrees and now we are forced to change to isizulu and so the words are sometimes difficult. … when we are writing tests and assignments it is so difficult, translating english to isizulu. student a2: we find it hard to write in isizulu because it has been a very long time since we last wrote in isizulu. the above responses signal the difference between spoken and written language. cummins (1980) and lim and presmeg (2011) support this assertion, pointing out that there is a difference between basic interpersonal communicative skills and academic language proficiency. this means that the above participants’ language competency in isizulu was not sufficiently developed to ensure proficiency in academic language. as a result, they experienced difficult in writing tests and assignments in isizulu. the materials used in the numeracy education module were written in english and some were translated into isizulu as lecture notes. however, the participants were of the opinion that the translated notes lacked depth and were mere summaries. student b10 and student b4 also mentioned the disadvantage that isizulu materials were not provided timeously: student b10: we shouldn’t receive the english course packs before the isizulu course packs. in addition to the materials in english and isizulu not being provided simultaneously, the quality of the translations and the manner in which the translations were presented were also questioned by the participants: student b4: there should be more people translating materials from english to isizulu. … there should be more readings written in isizulu so that we can learn successfully in isizulu. … we should not take people who are not isizulu speakers and allow them to translate materials. student a2: i think the notes that we have in isizulu are summaries because they are in point form. however, not all participants noted negative experiences related to the learning materials. student a3 felt that the availability of module materials in both english and isizulu was an advantage to isizulu-speaking pre-service teachers. they could read both sets of module materials to check their understanding of the concepts: student a3: if the materials could be isizulu only we wouldn’t be able to understand because there are some words we do not know in the language. using only the english language also poses the same problems. having both languages helps, it’s just that there is a lot of work. assessment is an integral part of teaching and learning and has to match the mode of delivery of the module. the tests and examination papers were printed in both english and isizulu for lecturers’ convenience as the isizulu students sat in the same venue as other pgce pre-service teachers who were registered for the identical numeracy education module that was presented through the medium of english. at different times in the timetable, the same lecturer taught the numeracy education module in isizulu or in english. some participants found the use of both languages in the tests confusing, whilst others found this beneficial: student a3: writing becomes easier in english while you would need to give yourself lots of time analysing a test written in isizulu so that you can understand it. it’s just that it would be good if we would only have tests written in isizulu. interviewer: so, in other words, you are saying, using isizulu and english simultaneously in the tests, caused the confusion. student a5: i tend to differ; i … would read a paragraph in isizulu without getting any meaning. then i go back to english; it is only then i am able to understand what is going [on]. as well as the benefits or problems of writing tests and examination papers provided in both languages, participants pointed out that there were problems with completing tests within the specified time limits. the time constraints during tests and examinations meant that the participants did not have sufficient time to translate and write ‘proper’ isizulu. they tended to mix isizulu with english when they could not immediately think of the correct or appropriate isizulu concept. as a consequence, the participants were penalised for including english words in the tests or examination instead of using only isizulu: student b5: i am worried about the test because … i did not know that i had to write in isizulu but i wrote most of the things in isizulu which makes me worry about the exam. [in] the exam i would be mixing isizulu with english and i don’t know if [examiners] will mark it or not because we don’t have much material [provided in isizulu] which allow us to write in isizulu. another participant highlighted the fact that translation into isizulu is time consuming: student b2: also it takes us more time to get to the point as opposed to our counterparts writing in english, sometimes you would find that they have finished writing but we haven’t because it is harder to fully explain a concept in isizulu and also the people who are marking expect you to write in genuine isizulu. contrary to the tests or examinations written in both languages, the assignment was provided only in isizulu for the pre-service teachers who were registered for the numeracy education module offered in isizulu. however, the participants were required to read and understand the literature in english and translate the information before submitting the assignment. the process of writing the assignment thus took much longer for the pre-service teachers writing in isizulu. computer word processors do not usually have spell check available in isizulu so the participants checked for typographical errors manually. consequently, they took longer to complete assignments than the pre-service teachers who submitted their assignments in english as word processors have english spell-checking facilities. the students’ experiences expressed below reiterate cummins’s (1980) distinction between basic interpersonal communicative skills and academic language proficiency: student a2: when we are writing assignments we end up having to ask other people for translations of certain english words, not that we forget, we know it because it is a simple language, we know the basics but in terms of writing academically, it’s difficult. student a1: what i do when i am writing my assignments; when i start to realise that i’m starting to fail getting the message across in isizulu, i write an english word in brackets. student a7: it puts strain on me when i have to start reading the literature in english, translate to isizulu, you know, it becomes a lot of work. … [rather], read in english, understand in english, write in english and type in english. in the translation model developed by lesh et al. (1987), one of the five representations necessary for ensuring mathematical concept development in learners is ‘verbal symbols’. this model was designed by these authors for consideration in the development of mathematics and numeracy education. if more than one language is used in mathematics education then transformations within the verbal symbol representation become complicated. this occurs because verbal symbol translations, as described in lesh et al.’s model, are usually considered to occur within a single spoken language. however, when translation between two languages, such as isizulu and english, are necessary then the verbal symbol representation will require additional translations: within each of the two languages and between the two languages. the single verbal symbol representation presented in lesh et al.’s model requires adaptation to show two different sets of verbal symbols. figure 2 shows how this adaptation in verbal symbol transformation may be represented diagrammatically. the figure shows the two sets of verbal symbols that the pre-service teachers engaged in during the numeracy education module. the participants noted that the language used for numeracy education concepts in english is not sufficiently supported by equivalent academic language in isizulu. the participants thus found conversion between the two sets of verbal symbols challenging and time consuming. furthermore, the translations of materials were mainly from english into isizulu: the students were often required to read and understand learning materials related to numeracy education in english and then to translate their understandings into isizulu. the translations were problematic because of the technical and theoretical terminology in numeracy education. this means that the double-headed arrow between and within the two sets of verbal symbols in an ideal situation could not be achieved, and so broken-line arrows represent the existing, limited, incomplete translations of academic concepts. figure 2: diagrammatic representation of verbal symbols. the challenges experienced by the pre-service teachers are not only as a result of the difficulties experienced in translation, but also with the lower status ascribed to isizulu as a medium of instruction by their peers and practising teachers. status of the numeracy education module taught in isizulu according to the participants, not only does isizulu have insufficient academic vocabulary for concepts taught in the numeracy education module, but it also has lower status. those registered for this module noticed that their peers and practising fp teachers considered this module, delivered through the medium of isizulu, to be inferior to and of lower status than the equivalent fp numeracy education module offered in english:student a2: basically people look down upon us because we are learning in isizulu. they tend to think that our qualification is not the same as the one for those learning in english. interviewer: how do you take your peers’ comments? student a1: personally since i understand the reason behind us learning in isizulu … is because of the child’s interest. i am not offended. but to someone who does not understand … to him/her we are just playing games. student a2: they say we are crèche teachers. the difficulty faced by the pre-service teachers taking the numeracy module in isizulu was that their peers, who were studying the same module in english, considered them inferior and labelled them as ‘crèche teachers’. although the pre-service teachers were all studying identical numeracy education module content and fp pgce qualification, nonetheless they were considered to be teachers of ‘nursery school’, where no serious teaching and learning takes place. the implication is that fp pgce teachers will only be care givers, instead of teachers, of young learners. however, some participants studying in isizulu did not take offence: they understood the need to study in isizulu, as this would eventually benefit isizulu mother tongue learners. despite the benefits of mother tongue instruction, negative attitudes towards its use as the language of instruction persist. similar concerns were expressed in the research findings of ndamba (2008): african students considered a degree in english to be more prestigious than a degree in an african language. the lower status accorded to mother tongue instruction is often a result of english being the lingua franca in africa. according to policy documents, the official language of instruction at the majority of kwazulu-natal fp schools should be isizulu. however, the participants were pressured by school mentors during professional practice to use english rather than isizulu when teaching mathematics. the mentors are practising fp teachers who oversee the pre-service teachers’ teaching during their professional practice at schools. the mentors were of the opinion that their fp learners would experience language difficulties if the fp mathematics lessons were conducted through the medium of isizulu. in grade 4, the medium of instruction is usually english in kwazulu-natal, so the teachers prefer to teach fp mathematics also through the medium of english, despite this being against official national policy imperatives: student b7: in the school that i was in, the principal said ‘don’t teach in isizulu too much because they sometimes get lost’, so we ended up having to add some english. student b6: i found it to be a bit challenging. … my mentor teacher was speaking a lot of english in class, did everything in english. when i teach in isizulu the learners did not have a problem learning in isizulu, but the mentor teacher … [said] if it happens that learners go to other schools they need to be able to speak english, [so that they are not] labelled as unsuccessful learners. [also, i] explained to the mentor that the person coming to assess me expects me to teach in isizulu, so it is important that i teach in isizulu and learners get to understand better, … using their home language. the response of participant student b6 clearly indicates that there is a mismatch between reality and what the university’s language policy is attempting to achieve through taking cognisance of and initiating implementation of the languages in education policy (doe, 1997). despite peers and practising fp teachers’ poor perceptions of teaching of numeracy through the medium of isizulu and their reluctance in taking up the recommendations of language policy documents in teaching and learning numeracy, the participants were generally positive about being instructed through this medium at their hei. conclusions top ↑ obviously it would be naïve to think that merely changing the medium of instruction in a pre-service teacher education module is sufficient to change the poor mathematics achievements of south african learners. there are many political, social and economic factors that also need to be addressed in the south african teaching and learning context. however, the participants’ experiences described in this article give a clear indication that pre-service teachers are willing to teach young learners mathematics in their mother tongue. the challenges communicated by the participants are generally connected to technical delivery issues that were experienced during the isizulu numeracy education module. the resolution of these issues does not seem insurmountable. for example, it should be possible to develop isizulu technical and theoretical terminology that is acceptable, appropriate and correct for use in numeracy education teaching and learning. there are a number of names for counting in isizulu, but there are many other terms and concepts that need to be ‘created’ to enhance numeracy education development through the medium of isizulu. in other words, the verbal symbols in isizulu that relate to numeracy education require formalisation, acceptance and usage by isizulu academia, particularly in the area of mathematics education, as well as by the larger zulu-speaking community. in addition, if our university is sincere about achieving its strategic goal to promote multilingualism, then it should establish a centre dedicated to assisting academics in translating learning materials. however, one of the most challenging issues relates to the status of modules presented through the medium of isizulu. in south africa, english is considered to be important for economic, political, social and educational betterment. this leads to english being afforded a higher status than the other 10 south african official languages. this influences members of the teaching community, as well as of the community at large, to believe that teaching and learning through english is ‘better’. moreover, parents may consider that the benefits of learning to speak english fluently from a young age far outweigh the benefits of being taught in mother tongue. it will thus be difficult, but necessary, to convince learners, teachers and parent communities that providing opportunities for mother tongue instruction in mathematics is significant. furthermore, to initiate mother tongue instruction at higher education institutions, each institution needs to start by using a ‘bottom-up’ approach within their community to elevate the status of mother tongue instruction. for example, instead of translating teaching materials from english or afrikaans into a mother tongue, academics at the higher education institution could develop their own authentic mother tongue teaching material. this is particularly important for fp pre-service teachers who are being prepared to teach young learners in their mother tongue. improving the status of mother tongue instruction within a higher education institution is the first important step in dispelling the impression that teaching fp pgce students numeracy education in their mother tongue results in preparing merely ‘crèche teachers’. acknowledgements top ↑ we thank the department of higher education and training and the european union for funding the primary education sector policy support programme from which the data for this article comes. our views are expressed in this article. competing interests we declare that we have no financial or any other competing interests that might prevent us from executing and publishing unbiased research. authors’ contributions l.v.l. 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(1978). mind in society: the development of higher psychological processes. cambridge, ma: harvard university press. footnote top ↑ 1. in south africa, the school subject ‘mathematics’ was changed to ‘numeracy’ in the fp with the introduction of outcomes-based education in 1998. in2011, with the launch of the curriculum and assessment policy statement (doe, 2011), ‘numeracy’ was renamed as ‘mathematics’. article information authors: verona leendertz1 a. seugnet blignaut1 hercules d. nieuwoudt1 christo j. els2 suria m. ellis3 affiliations: 1school of natural science and technology for education, faculty of education sciences, north-west university, potchefstroom campus, south africa2school of human science, faculty of education sciences, north-west university, potchefstroom campus, south africa 3department of statistical consultation services, north-west university, potchefstroom campus, south africa correspondence to: seugnet blignaut postal address: private bag x6001, potchefstroom 2520, south africa dates: received: 16 may 2013 accepted: 27 nov. 2013 published: 11 dec. 2013 how to cite this article: leendertz, v., blignaut, a.s., nieuwoudt, h.d., els, c.j., & ellis, s.m. (2013). technological pedagogical content knowledge in south african mathematics classrooms: a secondary analysis of sites 2006 data. pythagoras, 34(2), art. #232, 9 pages. http://dx.doi.org/10.4102/ pythagoras.v34i2.232 copyright notice: © 2013. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. technological pedagogical content knowledge in south african mathematics classrooms: a secondary analysis of sites 2006 data in this original research... open access • abstract • orientation and research problem • theoretical framework • research design and methodology    • secondary data analysis    • sites 2006 south african mathematics dataset    • procedures • findings    • technological pedagogical content knowledge clusters       • impact of ict use       • insufficient ict knowledge       • increase in learners' knowledge, skills and affect       • teacher practices       • confidence       • barriers • conclusions • acknowledgements    • competing interests    • authors' contributions • references abstract top ↑ this article reports on a secondary data analysis conducted on the south african mathematics teachers’ dataset of the second information technology in education study (sites 2006). the sample consisted of a stratified sample of 640 mathematics teachers from 504 randomly selected computer-using and non–computer-using schools that completed the sites 2006 teachers’ questionnaire, which investigated their pedagogical use of information communication technology (ict). the purpose of the current investigation was to investigate the level of technological pedagogical content knowledge (tpack) of mathematics teachers, and how tpack attributes contribute towards more effective grade 8 mathematics teaching in south african schools, using the tpack conceptual framework. the findings are presented according to the three clusters identified through the association between the main variables of the tpack model and other variables on the sites 2006 teachers’ questionnaire: (1) impact of ict use, (2) teacher practices and (3) barriers. a cramér v of between 0.3 and 0.4 was considered to signal a medium effect that tended towards practically significant association, and a cramér v of 0.4 or larger was considered to signal a large effect with practically significant association. the results indicate that the tpack of mathematics teachers contributes towards more effective grade 8 mathematics teaching in south african schools. orientation and research problem top ↑ education in south africa is constantly transforming according to the department of education requirements, such as the development of 21st century learning outcomes which allow learners to use information in different contexts (department of education [doe], 2002; law & chow, 2008b). mathematics teaching is an indispensable part of the curriculum and fulfils an important role in the development of higher order thinking skills to accomplish specific tasks for the achievement of relevant pedagogical outcomes, conceptualisation, abstraction, generalisation, problem solving and information processing (nieuwoudt & golightly, 2006). mathematics teaching aims to develop learners to (1) have an acute awareness of how mathematical relationships are used in social, environmental, cultural and economic relations; (2) foster a love of mathematics; (2) recognise that mathematics is a creative part of human activity; (4) obtain profound theoretical understandings in order to make sense of mathematics; and (5) apply mathematics in physical, social and mathematical problems (doe, 2002). the use of information and communication technology (ict) provides scaffolding for the socio-economic development and building of much needed skills in a newly industrialised country (marais, 2009). ict has become the mode of choice of communication amongst people in all spheres and its use in south african education increases daily (doe, 2004). ict supplements, supports and facilitates curricula (ertmer, addison, lane, ross & woods, 1999). mathematics curriculum outcomes can be addressed through various ict-driven activities: to perform calculations efficiently and to the required degree of accuracy, from describing patterns and relationships in the foundation phase to following laws and meanings of exponents in the senior phases (doe, 2002; mcdonald, 1988). ict provides teachers with opportunities to assist in their teaching practices, to develop mathematical knowledge and skills, and to interact with learners, parents, peers, colleagues and the global society (haddad & draxler, 2002). although many mathematics teachers acknowledge the value of ict in teaching and learning, the pedagogical use of ict in south african schools remains dismally limited (law & chow, 2008a).the second information and technology in education study (sites) 2006, the most recent in the series of ict use in education studies conducted by the international association for the evaluation of educational achievement, investigated the pedagogical use of ict across the world. south africa performed poorly in the integration of ict into teaching and learning practices (law & chow, 2008a). even though ict infrastructure in schools for teaching and learning has more than doubled, from 12.3% in 1999, to 26.5% in 2002 (doe, 2004) and 38% in 2007 (doe, 2007), the situation is not conducive for integration at all levels of education. the e-education white paper stipulates a three-phase plan to have a fully integrated ict system in all spheres of education, administration, management and teaching and learning, by 2013 (doe, 2004). however, as the stance of mathematics teaching and learning is a much-debated issue and concern in south african schools (prew, 2013), role players in education have to take notice of the findings from sites 2006 if we want an ict-integrated and competitive education system. however, merely introducing ict into teaching and learning is not sufficient to ensure its effective use for teaching and learning. teachers should be competent with sufficient tpack in order to adequately integrate ict into their pedagogical practices (mishra & koehler, 2006). this article poses the question: to what extent does tpack contribute towards more effective grade 8 mathematics teaching in south african schools? theoretical framework top ↑ mathematics teachers in south africa have different philosophical and ontological views on what mathematics teaching should encompass. this is especially true of how mathematics should be used, facilitated or taught in specific situations. whilst many view mathematics teaching as conforming to a set of rules, others regard it as either a deductive or an inductive learning processes (huetinck & munshin, 2000). nieuwoudt (2006) recommends that the teaching and learning of mathematics should be viewed from an ontological-contextual perspective comprising six inter-related aspects: the teacher, learner, content, intention, live interaction and the context. these components collectively enable learners to perform learning tasks. teachers, who facilitate teaching and learning, should have clear goals about what they want to achieve during teaching and learning interventions. mathematics teachers ought to have the relevant mathematical knowledge, skills, attitudes and values that they want their learners to achieve in order to facilitate learners in a specific context (doe, 2002). with the introduction of ict into teaching and learning, mishra and koehler (2006) built upon shulman’s theory and constructed the technological pedagogical content knowledge conceptual framework (tpck), renamed by schmidt et al. (2009) as tpack, which describes the relationship between technology, pedagogy and content. in order for technology to add value to teaching and learning, it cannot be regarded as context-free, but must be linked to pedagogy. figure 1 presents the tpack framework with the seven components and shows the interface of the integration of content, pedagogy and technology. figure 1: the technological pedagogical content knowledge (tpack) conceptual framework. shulman (2004) discusses the categories of knowledge that facilitate teaching: content knowledge and pedagogical knowledge.content knowledge refers to the quality and organisation of knowledge in the thought processes of teachers. mathematics teachers should have appropriate content knowledge in order to be able to teach the subject fluently (ball, thames, & phelps, 2008; mishra & koehler, 2006). pedagogical knowledge refers to the expertise of teachers in selecting appropriate methods of teaching the particular content to learners. pedagogical content knowledge, the interface between subject and pedagogical knowledge, is referred to as specialised content knowledge (shulman, 2004). pedagogical content knowledge becomes evident when teachers have the ability to build on their learners’ prior knowledge and adapt their teaching strategies to best facilitate the new content to learners (mishra & koehler, 2006). content knowledge, pedagogical content knowledge and technological knowledge are important dimensions of effective teaching with ict. technological knowledge refers to the ability and skills to use the variety of technologies such as books, chalk and blackboard, as well as technologies such as computers, the internet and digital resources, to teach learners the required content. technological content knowledge refers to how content can be taught with the use of technology. technological content knowledge is the area where technology and pedagogy link. this knowledge base is where the teaching and learning occur due to the existence, components and capabilities of the various technologies. teachers with technological content knowledge select a specific teaching strategy, as well as the most appropriate ict, to teach curriculum content to learners (draper, 2010). the central part of figure 1 is the intersection between content knowledge, pedagogical knowledge and technological knowledge; this is known as technological pedagogical content knowledge (tpack), which is not merely knowledge of technology, pedagogy and content individually, but also the basis of effective teaching with ict (koehler & mishra, 2009). tpack aims to support skills development of teachers for acquiring and explaining how technology-related subject-specific knowledge is applied during teaching and learning activities (koehler & mishra, 2009). mathematics teachers are a diverse group with different teaching and learning styles; therefore, no single permutation of content, pedagogy and technology will be applicable to every teacher (koehler, 2012). however, overall tpack requires mathematics teachers to have: • an understanding of how to represent mathematical concepts with technology • pedagogical skills that utilise technologies constructively to transfer content • fundamental knowledge of what brands mathematical concepts as difficult or easy to grasp • knowledge of how technologies can assist learners in solving mathematical misconceptions • knowledge of learners’ existing mathematical knowledge and their learning styles • knowledge of how to utilise technology to construct new knowledge using learners’ existing knowledge (koehler & mishra, 2009). learners can engage in an abundance of activities in which they utilise ict; they can conduct research and communicate through ict with peers and their teachers to augment their learning. ict has the potential to enhance both the theoretical and practical aspects of teaching and learning: speeding up and enhancing work production, linking school curricula with reality, supporting exploration and experimentation, and providing immediate feedback. ict focuses attention on overarching issues, increasing the relevance of underlying abstract objects to improve motivation and engagement (osborne & hennessey, 2003). this, however, solely depends on how it is used by the teacher. more importantly, such activities will not be effective unless learners engage with technological devices. it is the teachers’ responsibility to simultaneously impart ict competency whilst focusing on mathematical pedagogical knowledge and developing learners’ attitude and values in a constructivist learning environment (draper, 2010; galloway, 2007). research design and methodology top ↑ secondary data analysis the study followed a methodology of secondary data analysis of the south african mathematics teachers’ dataset of sites 2006 (brese & castens, 2009). secondary data analysis refers to the empirical exercise that uses specific research questions and existing data for further analysis, using either the same or different statistical procedures (smith, 2008). for t-tests and analysis of variance (anova), the data consist of a set of scores, such as intelligence quotients, attitudes, time, errors and so on. each subject has one quantitative score. for chi-square (χ²) analysis, however, the data are frequency counts in categories. each subject is observed and placed into one category. the frequencies of observations in categories are counted and the chi-square test is calculated from the frequency counts. chi-square analysis compares the observed frequencies of a category to frequencies that will be expected if the null hypothesis is true (spatz, 2008). in this case, statistical significance tests (e.g. chi-square tests) are used to show that the results of the correlations between the items in the sites 2006 mathematics teachers’ questionnaire are practically statistically significant. chi-square can be considered as effect sizes: a cramér v value of less than 0.3 is considered to show a small effect with no or very little practically significant correlation; a cramér v of between 0.3 and 0.4 is considered to signal a medium effect that tends towards practically significant correlation; a cramér v value of 0.4 or larger is considered to show a large effect with practically significant correlation. the sites 2006 dataset provided opportunities to explore the associations between variables that were not calculated or reported during the main study (law et al., 2008). sites 2006 south african mathematics dataset in the 1990s, the international association for the evaluation of educational achievement initiated comparative research on the use of ict in education. sites 2006 aimed to determine the influence of ict on teaching and learning in schools (blignaut, els & howie, 2010). to date, three sites studies (modules) have contributed to the understanding of how ict affects the teaching and learning of mathematics and science in grade 8 in schools across the world. sites 2006 (module 3), an international longitudinal comparative large-scale survey, investigated how school and system level factors influence teachers’ pedagogical adoption of ict (law & chow, 2008a). in 2005, the 22 participating countries and education systems adapted, translated and piloted the survey instruments and trained their local project personnel and supplementary fieldworkers. data collection took place during 2006 (blignaut et al., 2010). sites 2006 administered three questionnaires: (1) a questionnaire for school principals, (2) a questionnaire for ict coordinators and (3) a questionnaire for mathematics and science teachers (brese & castens, 2009). researchers from the university of twente, hong kong university and the iea data processing centre in hamburg, germany, coordinated the main study. the national research coordinators of each country managed local data collection and data capturing (law et al., 2008). the international study population comprised schools with learners enrolled for mathematics and science in the target grade, that is, schools with students studying in the grade that represents eight years of schooling (blignaut et al., 2010). the sample size per education system was fixed at a minimum of 400 schools. in each school the teacher of one of the classes in the target grade (grade 8 mathematics or science) was randomly selected to participate (law et al., 2008). no official report was written of the sites 2006 study for south africa. this secondary data analysis focused on an aspect that was not analysed during the sites 2006 study. the dataset used for this secondary data analysis comprised a stratified sample of the 640 mathematics teachers from about 500 randomly selected computer-using and non–computer-using schools who completed the sites 2006 teachers’ questionnaire (blignaut et al., 2010). the second author of this study was the co-national research coordinator for sites 2006 in south africa and had access to the south african dataset. procedures in order to address the research question, that is, to explore to what extent tpack can contribute towards more effective grade 8 mathematics teaching and learning in south african schools, the tpack conceptual framework was applied. the sites 2006 teachers’ questionnaire comprised eight parts: • part i: information on the target group • part ii: curriculum • part iii: teacher practice • part iv: learner practice • part v: learning resources and technology infrastructure • part vi: impact of ict use • part vii: information about schools • part viii: specific pedagogical practices that use ict associations were calculated between item 21j (part viii) in the sites 2006 teachers’ questionnaire, i know which teaching or learning situations are suitable for ict use (tpack), and other variables from parts ii–viii (available from brese & castens, 2009) that focused on teaching and learning practices in grade 8 mathematics classrooms. tpack clusters (themes) were identified by the association study. statistical relationships (associations) between categorical variables were determined with chi-square tests and effect sizes to determine if the relationship was large enough to be significant (ellis & steyn, 2003). only the effect sizes of significance are reported. the statistical consultation services of our university assisted in the cross tabulation analysis using spss® 16.0 for windows (spss, 2007). findings top ↑ this section presents the findings that grade 8 mathematics teachers who have tpack 21j (i know which teaching or learning situations are suitable for ict use) are more effective teaching mathematics in south african schools. the findings grouped into four clusters: (1) impact of ict use, (2) teacher practices, (3) confidence and (4) barriers, according to the categories of the sites 2006 questionnaire. two categories are used to display the findings: percentages (tpack and no tpack or yes and no) and effect sizes (figure 2). figure 2: clusters (themes) identified through the associations between tpack and other variables on the sites 2006 teachers' questionnaire. technological pedagogical content knowledge clusters figure 2 shows the effect sizes of the four main tpack clusters. the following section will discuss these clusters. impact of ict use the main variable (21j: i know which teaching or learning situations are suitable for ict use) correlated with questions 19a–k from part vi and items 20a–n from part vii of the questionnaire; these items focused on the impact of ict on different teaching, learning and planning aspects. for the purpose of this secondary data analysis, there was little difference between the three categories and the researcher combined the categories ‘ict knowledge to a certain extent (a little and somewhat)’, ‘adequate ict knowledge (a lot)’ and ‘no knowledge to use ict (not at all)’ to create two analysis criteria: tpack (i know in which teaching and learning situations to use ict) and no tpack (i do not know in which teaching and learning situations to use ict). the impact of tpack clustered into two categories: (1) insufficient ict knowledge by the teachers and (2) increase in learners’ knowledge, skills and affect. these two categories correlated to a medium effect that tended towards a practically significant association and a large effect with practically significant association (figure 2). insufficient ict knowledge two hundred and twenty-two teachers responded affirmatively to the survey question ‘i know which teaching and learning situations are suitable for ict use’. the researchers posit that this portion of teachers represented the ideal population that had adequate and integrated technological, content and pedagogical knowledge (figure 1), henceforth referred to as tpack. teachers who responded negatively to this question (n = 282) did not have all the elements of tpack, but could relate to another category of the framework. in order to investigate which category of knowledge those non-tpack teachers should improve, the responses to questions 19a–k were correlated with their responses to question 21j (table 1). table 1: correlations between teachers' with tpack and teachers' knowledge and skills. amongst the teachers who knew which teaching and learning situations were suitable for ict use (tpack in table 1), 39% did not think they had sufficient knowledge and skills to incorporate new teaching methods. also, 61% reported that they did not know in which situations to incorporate ict, and also did not know when to incorporate new teaching methods (large effect, v = 0.47). another large practically significant association (v = 0.47) was evident between tpack and collaborating with peers and colleagues. not knowing in which situations to use ict, to a lesser extent though still with practical significance, correlated with absence of knowledge about searching for resources, incorporating new ways of learning and performing administrative tasks.a portion of non-tpack teachers found it difficult to provide individualised feedback; this correlation tended towards practical significance. this association could be due to insufficient ict, subject or pedagogical knowledge (thompson, 1988). a medium effect indicated the correlation between tpack teachers and increased work pressure. overcrowded classrooms mean teachers are overworked. new or additional teaching requirements, without adequate training, make teachers less likely to embrace technology at their disposal for fear of increased work demands (mofokeng & mji, 2009). these findings are in concurrence with the work of al-senaidi, lin and poirot (2009), who indicate that poor ict knowledge leads to hindrances in the performance of teachers with insufficient tpack. although ict was available to teachers in many cases, their tpack did not seem to improve as there were other contributing factors preventing ict integration. this suggests that although teachers may have access to computers, without adequate continuous professional teacher development, it is unlikely that ict will be utilised optimally (mofokeng & mji, 2009). increase in learners’ knowledge, skills and affect according to the tpack framework, an ideal teacher is able to integrate knowledge of technology, mathematics, content and pedagogy (figure 1), knows in which teaching and learning situations ict use is suitable and has a positive effect on their learners’ knowledge and skills. the questionnaire probed whether such knowledgable teachers reported whether such effects were observed in their learners. three indicators were used: change, no impact and decrease in learners’ knowledge, skills and effect. the correlations between the effect of the teachers’ tpack status and the change in their learners’ knowledge and skills are given in table 2. table 2: correlations between the effect of teachers' tpack and status and the change the learners' increase in knowledge and skills. the teachers relating to this part of the analysis had tpack. the analysis of this group of teachers yielded quite uniform results for all skills. between 70% and 76% of these teachers indicated a change in the mathematics knowledge and skills in their learners since they started to implement ict in teaching and learning, whilst 19% of those teachers indicated no impact. less that 10% indicated a decrease in their learners’ skills. it is not possible to pinpoint the reasons for the decrease of knowledge and skills amongst learners, as a number of factors can influence their learning. what is important is that all the mentioned skills in the learners increased in classes taught by tpack teachers, with a medium association effect.from these results it is evident that ict, when integrated effectively into the curriculum, will pave the way for learners to achieve the critical and learning outcomes in mathematics (doe, 2002). teacher practices mathematics teachers should embrace tpack to deepen their teaching and learning experiences across the curriculum in order to ensure that the outcomes for mathematics are met (mishra & koehler, 2006, p. 1026). tpack is essential for ict integration in mathematics classrooms and the use of ict should form an integral part of teaching and learning practices. in order to investigate which category of teacher practices with ict the non-tpack teachers needed to focus on, the responses to questions 9a–c, 9g, 9h, 9j, 9m and 14a–l from part iii were correlated with the responses to question 21j (table 3). teacher practices with ict of teachers with tpack (table 3) showed a medium associated effect. many teachers (63%) with no tpack did not employ ict during teaching, but some (37%) tried to use ict when teaching content to their learners. teachers who had tpack (30%) were able to present essential information, provide rich demonstrations and facilitate interactive classes using ict. tpack enables teachers to build on their own, as well as their learners’, existing knowledge in order to develop strong attitudes and positive beliefs towards mathematics (mishra & koehler, 2006; swan et al., 2002). table 3: correlations between teachers' tpack and teacher practices with ict. confidence multiple studies on ict integration have identified insufficient confidence as a major barrier to ict integration in mathematics (mofokeng & mji, 2009). with an increase in confidence, mathematics teachers become more enthusiastic, interested, motivated and committed to embracing ict in their teaching and learning environment (attwell & hughes, 2010; uyangor & gör, 2010). the main variable (21j) was correlated with question 21i from part vii, which focused on mathematics teachers’ confidence in the pedagogical use of ict; this correlation is shown in table 4. table 4: correlations between teachers' tpack and confidence to prepare lessons that involve the use of ict. a high practically significant association (v = 0.63) was indicated between tpack and mathematics teachers’ confidence in preparing lessons that involved learners’ use of ict. the 8% of teachers who were confident to prepare lessons that involved ict were those who had adequate tpack to select the appropriate strategies and resources best suited to the teaching and learning of mathematics. the absence of confidence of the non-tpack teachers prevented them from using ict for mathematics teaching and learning. if teachers are meaningfully exposed to continuous professional teacher development in tpack they will become confident to explore the vast array of ict resources (daly, pachler & pelletier, 2010). barriers the main variable (21j) was correlated with questions 23d, 23i and 30a from part vii, which focused on the barriers to the use of ict (table 5). the respondents had the option to select from two categories (yes or no). the non-tpack mathematics teachers were requested to indicate whether they experienced barriers when using ict during these activities. figure 2 illustrates the items relating to barriers in using ict. table 5: correlations between non-tpack teachers' and barriers in using ict. a practically significant association (v = 0.42) was evident between non-tpack and the use of ict for teaching-related activities. mathematics teachers’ inadequate use of ict for teaching-related activities was a major barrier to tpack. identifying useful ict tools (v = 0.37) and ict-related pedagogical skills (v = 0.32) showed a medium associated effect. non-tpack teachers experienced difficulties in selecting appropriate useful ict tools and they acknowledged their insufficient tpack.in order to address teachers’ limited tpack, it is essential for mathematics teachers to participate in training for the integration of ict. only 15% of the south african mathematics teachers participating in sites 2006 had access to ict training, 19%to technical training and 32%to introductory courses (law & chow, 2008a); even fewer had access to training on the integration of ict in the curriculum (schoolnet south africa, 2008). intel® teach (intel, 2012), a large teacher development program, identified one representative per school from 1000 schools across south africa to attend ict integration training. their programme evaluation indicated that south african teachers demonstrated meagre ict competencies, and that it would take at least five years to adequately train teachers to develop tpack (thomson & wilson-strydom, 2005). however, none of these programmes were focused on developing mathematics teachers’ ict competencies. the majority of the mathematics teachers in the sites 2006 study indicated that they wanted to prepare quality and interesting presentations and learning materials even though they had no ict competencies (law & chow, 2008b). the development of the teachers’ tpack would ensure ict integration required by the e-education white paper (doe, 2004). it becomes evident that the first and foremost responsibility of the department of basic education (dbe) is continuous professional teacher development. if teachers receive appropriate professional development relating to ict integration, ict can become embedded in their pedagogical practices (mishra & koehler, 2006). teachers would have access to support in their preparation and facilitation (howie & blignaut, 2009). they would better assist their learners to methods of problem solving, contribute towards the development of 21st century learning skills and achieve the outcomes for grade 8 mathematics (doe, 2002; nieuwoudt & golightly, 2006). mathematics teachers are willing to integrate ict despite inadequate tpack. however, insufficient resources at schools impact on teachers’ ability to create rich learning environments to stimulate learners’ curiosity and interest in mathematics. it is disheartening that only 10% of schools had tutorial software and 35% had general software for the teaching and learning of mathematics (law & chow, 2008b). in some provinces, only 1% of schools were connected to the internet for teaching and learning purposes. the numbers of computers in school have increased dramatically since the first sites study in 1999 (pelgrum, 2001): 51% of schools in 2007 throughout south africa had computers at their disposal for curriculum activities (doe, 2007). the dbe should address the social and economic disparities across the provinces and provide equal access to ict to all schools in terms of ict continuous professional development and ict infrastructure. even though the numbers have increased since sites 2006, not all schools have equal ict profiles, and not all achievement outcomes are on par with the aims of the white paper on e-education (doe, 2004). the white paper on e-education specifies that all managers and teachers should obtain access to personal computers to assist them during administrative tasks and during lesson preparation (doe, 2004). in 2008 the dbe initiated the laptop-for-all initiative (mahlong, 2012), aspiring to provide educators with laptops for the integration of teaching and learning, thereby improving the quality of education in, particularly, mathematics and science by 2011. the laptop-for-all initiative never got off the ground. the initial criterion was the number of years in service of the dbe and only 11.43% teachers received laptops. similarly to other dbe initiatives, this critical ict initiative is two years behind schedule, with little hope for the fulfilment of its aims. in a report, it was announced that the teacher laptop initiative is alive again and that broader plans for ict progression are in the making (mahlong, 2012). however, no strategic plan is in place to support the initiative, and inadequate planning and insufficient resources discourage teachers to integrate ict in their classrooms (thomson & wilson-strydom, 2005). in terms of the above findings, the professional development of teacher tpack is challenged. conclusions top ↑ the research question that underpinned this secondary data analysis of the south african data of the sites 2006 was to investigate the level of tpack and to what extent tpack of mathematics teachers contributes towards more effective grade 8 mathematics teaching in south african schools. this study indicates that increases in tpack of mathematics teachers, confidence in their ability to use ict and levels of mathematical content knowledge will contribute towards more effective grade 8 mathematics teaching in south african schools. the results indicate that mathematics teachers with adequate tpack acknowledged that ict had a positive impact: (1) they were able to access an abundance of resources for teaching and learning, (2) they could communicate with colleagues and peers regarding teaching and learning practices, (3) they became more innovative with their teaching and learning activities, (4) they could conduct all their administrative work, (5) they were able to facilitate interactive lessons and (6) they were confident using a variety of teaching and learning strategies best suited to achieve the outcomes of the curriculum. raising the standard of mathematics education is a priority for the dbe (dbe, 2012). the dbe advocates that ict interventions could address the shortcomings in the education system (parliamentary monitoring group, 2011). the authors therefore advocate addressing and developing the tpack and confidence of mathematics teachers. the dbe (2012) is looking for ict solutions to address teacher training, professional development and provide complementary teaching and learning resources so as to improve the quality of classroom teaching. it is imperative that role players within the education system strengthen and support the dbe and work collaboratively to address backlogs in the system. acknowledgements top ↑ this work is based on research supported by the national research foundation of south africa for the second international technology in education survey 2006 grant (icd2006071400008) and the north-west university. the researchers wish to thank these institutions for the funding. competing interests the authors declare that we have no financial or personal relationship(s) that might have inappropriately influenced us in writing this article. authors’ contributions v.l. 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(2022). research resilience in the covid era. pythagoras, 43(1), a714. https://doi.org/10.4102/pythagoras.v43i1.714 editorial research resilience in the covid era rajendran govender copyright: © 2022. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. in the previous two years, we had to grapple with an unforeseen global covid-19 pandemic and ensure our business continuity. the pandemic made us rethink a lot of what we thought we knew in the academic research field, while coping with the impact of the pandemic on our lives and health and on our families. since the cataclysmic effects of the pandemic experience are still with us, it would be remiss not to reflect on its impact on publication in pythagoras in 2020, 2021 and now in 2022. firstly, it has been refreshing and encouraging to have witnessed the growth in the number of submissions to pythagoras, from our beginner researchers in the field of mathematics education in collaboration with their supervisors to senior research collaborators, which after a rigorous review process have been published in pythagoras. congratulations to all the authors, who may rest assured that their publications will help in enhancing meaningful teaching and learning across mathematics classrooms at school and tertiary levels. secondly, some of the articles published in this issue provide new insights into how to respond to and engage in the new normal brought about by covid-19. for example, the article ‘exploring low-tech opportunities for higher education mathematics lecturers in an emergency techno-response pedagogy’ puts forth a conceptual framework for emergency remote teaching of mathematics in higher education environments that could enable academics and students to better prepare for future pandemics that may affect physical access to the classroom. the contestation of the role of home language in the learning of mathematics is ongoing, and this volume brings to the fore the benefits that accrue when learners’ home languages are taken into account in mathematics teaching and learning. in addition, the article ‘weathering the storm: learning strategies that promote mathematical resilience’ illuminates the importance of learners setting aside time to study mathematics successfully, despite the adversities and challenges they are exposed to in their daily lives. it is the distilling and articulation of findings of this kind that may enhance the quality of the teaching and learning of mathematics across schools and higher education institutions and even catapult it into a changing world. going forward, it will be interesting to explore the design and execution of online assessment mathematical tasks that probe higher-order thinking and stimulate creative and innovative ways of working. abstract introduction significance of the study literature review research methods and design research findings conclusion acknowledgement references about the author(s) lisnet mwadzaangati department of curriculum and teaching studies, chancellor college, university of malawi, zomba, malawi citation mwadzaangati, l. (2019). comparison of geometric proof development tasks as set up in the textbook and as implemented by teachers in the classroom. pythagoras, 40(1), a458. https://doi.org/10.4102/pythagoras.v40i1.458 original research comparison of geometric proof development tasks as set up in the textbook and as implemented by teachers in the classroom lisnet mwadzaangati received: 12 oct. 2018; accepted: 10 nov. 2019; published: 10 dec. 2019 copyright: © 2019. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract this qualitative case study examined similarities and differences between circle geometric proof development tasks set up in the malawian grade 11 mathematics textbook, and those that are set up and implemented by teachers in the classroom. data generation included analysing the content of circle geometry proof tasks from the mathematics textbook and video recordings of geometric proof development lessons taught by three teachers. the mathematics discourse in instructional framework for textbook analysis (mditx) by ronda and adler and the framework for analysing the level of cognitive demands of mathematical tasks by smith and stein were used to identify and analyse the mathematical tasks as set up by the textbook and as set up and implemented by three malawian secondary school mathematics teachers in the classroom. the findings revealed that the textbook presented the geometric proof development tasks at a high level as they included both empirical exploration tasks and formal proof tasks. despite this task setup in the textbook, only one teacher involved the learners in empirical exploration tasks and maintained the high cognitive level of the tasks during instruction. the other two teachers only presented the formal proof tasks. although the formal proof tasks that were set up by the two teachers were of a high cognitive level, the procedures that were used during task implementation resulted in reduction of the cognitive level of the proof tasks. i therefore conclude that teachers’ ability to set up and implement high cognitive level tasks that promote learners’ understanding and discovery of deductive geometric proofs depends not only on the availability of a good quality textbook, but also on the teacher’s conceptual ability to make effective use of textbook content. keywords: task setup; task implementation; geometric proof development; mathematics textbook; geometric proof tasks. introduction one of the aims of teaching secondary school mathematics in malawi is to promote learners’ logical reasoning, problem-solving and critical thinking skills (ministry of education, science and technology [mest], 2013). euclidean geometry is regarded as the main area of mathematics that is a key source for teaching mathematical argumentation and proof, developing learners’ deductive reasoning and critical thinking (kunimune, fujita, & jones, 2010). but the malawi national examinations board (maneb) chief examiners’ reports indicate that secondary school learners fail to develop geometric proofs at national examinations (maneb, 2013). poor teaching practices are highlighted as a major cause of learners’ inability to understand geometric proof development (maneb, 2013). the reports emphasise that due to lack of both content knowledge and pedagogical knowledge, the teachers are not creative in conducting effective lessons to support learners’ understanding of geometric proof development. studies conducted in different parts of the world also indicate that despite the importance of reasoning and proving in learners’ learning, many learners face serious challenges in proof development (kunimune et al., 2010; otten, males & gibertson, 2014; stylianides, 2014). these studies support maneb’s by arguing that learners’ challenges in proof development should be attributed more to classroom inappropriate practices that mainly emphasise rules of verification and devalue or omit exploration. as a result, the learners memorise the rules without understanding the process of proof development; hence, they are able to reproduce similar proofs but cannot apply the principles to develop a different proof (ding & jones, 2009). use of exploratory teaching strategies is suggested as one way of helping learners to understand geometric proof development (ding & jones, 2009; jones et al., 2009). this implies that the solution for improving classroom practices for enhancing learners’ understanding of geometric proof development lies in teacher professional development and teacher education. stylianides (2014) argues that apart from teacher professional development and teacher education, textbooks are the other important but less explored and insufficiently exploited solution for improving classroom practices. this is because mathematics textbooks of a particular country often mirror the national curriculum (fujita, jones, & kunimune, 2009). as ronda and adler (2016) claim, ‘textbooks have some similarity to classroom lessons – of course, without possibilities for actual (as opposed to imagined) contributions from learners in the development of the texts’ (p. 1097). as such, many mathematics teachers use textbooks to decide the type of tasks to implement in class and the way of implementing such tasks; hence, mathematics textbooks have an influence on learners’ opportunities in proof development (stylianides, 2014). this might imply that availability of high cognitive demanding tasks in mathematics textbooks will always result in instructions that promote learners’ understanding of geometric proof development. but henningsen and stein (1997) argue that, sometimes, a task could be set up in the textbook to require high-level cognitive demands from learners, but during classroom implementation phase it could be transformed in such a way that learners’ thinking focuses only on procedures, with no conceptual connections. this implies that teachers’ ways of setting up and implementing tasks might affect the cognitive level at which learners engage with the tasks. this study builds on the findings from my phd project (mwadzaangati, 2017a) which aimed at exploring mathematical knowledge for teaching geometric proof development. what i found is that most of the tasks that were used by the teachers were taken from secondary school mathematics textbooks and were presented in the form of either a statement or a diagram, or both statement and diagram. (mwadzaangati, 2017a, p. 143) one of the recommendations in my phd dissertation was that future research should focus on comparing the textbook content on geometric proof development and the teachers’ ways of implementing the textbook content. therefore, this study aimed at examining similarities and differences between geometric proof development tasks (referred to as proof tasks hereafter) set up in the textbook, and those that are set up and implemented by the teacher in the classroom. in this study, task setup means the way tasks are presented either in the textbook or in the classroom. specifically, this study aimed to answer the following two questions: what geometric proof development opportunities do proof tasks in malawi secondary school mathematics textbooks offer to learners? how do opportunities in the proof tasks in the textbooks compare with opportunities in tasks that are set up and implemented by the teacher in the classroom? significance of the study de villiers (1999) distinguished six functions of proof as follows: verification (concerned with the truth of a statement) explanation (providing insight into why it is true) systematisation (the organisation of various results into a deductive system of axioms, major concepts and theorems) discovery (the discovery or invention of new results) communication (the transmission of mathematical knowledge) intellectual challenge (the self-realisation or fulfilment derived from constructing a proof). (de villiers, 1999, p. 3) the functions of mathematical proof that are suggested by de villiers (1999) imply that teaching and learning of geometric proof focuses not only on learners’ ability to verify mathematical statements, but also on their ability to explain why a certain mathematical statement is either true or false, argue in a logical manner, construct new knowledge and communicate their arguments. this implies that the teaching and learning of geometric proof development promotes some of the aims of teaching secondary school mathematics in malawi such as logical reasoning, critical thinking and problem-solving. this means that teachers are expected to plan and implement proof tasks in a manner that would promote learners’ opportunities to achieve the five functions of mathematical proof that are suggested by de villiers. since textbooks are the only resource that malawian teachers use to plan and teach their lessons, it was necessary to compare the cognitive levels of proof tasks as set up in the textbook and as set up and implemented by the teacher during instruction. literature review geometric proof development geometric proof development is a process of constructing a sequence of arguments from x to y with supportive reasons, and hence is also called deductive reasoning and proving (cheng & lin, 2009). x is the given information (hypothesis) while y is the statement to prove (conclusion). reasoning and proving are central to learning of geometric proof development because they offer learners opportunities to make sense of mathematics through pattern identification, generation of conjectures and development of arguments (otten et al., 2014). apart from formal proof development, deductive reasoning and proving support learners’ capabilities in other mathematical elements including developing, outlining, or correcting an argument, deriving a formula, making or testing a conjecture (bergwall, 2017). it is, however, reported that many learners experience challenges in deductive geometric proof development (battista, 2007; kunimune et al., 2010; otten et al., 2014; stylianides, 2014). several reasons have been advanced for this problem. usiskin (1982) argued that learners do not succeed in secondary deductive geometric proof development because their prior knowledge in the geometry course is poor. he claimed that the learners come to learn secondary geometry before they have reached the level of formal deduction that was proposed by van hiele (1999). jones (2002) noted three reasons for learners’ difficulties in learning to develop geometric proofs. firstly, the learning of geometric proving is complex because it requires coordination of a range of competencies. secondly, the teaching approaches used during geometric proving lessons tend to concentrate on verification and devalue, or omit, exploration and explanation of how and why the proof works. thirdly, learning to prove involve learners’ ability in making the difficult transition from informal geometric reasoning to formal deductive geometric reasoning. jones (2002) argued that these reasons imply that teachers find it difficult to provide learners with meaningful experiences to enable them to understand geometric proof development. battista (2007) explained that geometric proof development continues to be challenging to learners because it involves geometric reasoning which mainly requires spatial reasoning. as such, learners face challenges in using spatial reasoning to see, inspect, and reflect on spatial objects, images and relationships (battista, 2007). heinze (2004) identified three reasons why learners fail to develop geometric proofs. these include insufficient knowledge of facts and concepts for developing the proof, deficits in methodological knowledge about mathematical proofs, and lack of knowledge with respect to developing and implementing a proof strategy. this brief review has shown that there are two major causes of learners’ challenges in geometric proof development. the first challenge lies in the nature of geometric proof development itself as being a complex domain. the second reason is use of inappropriate teaching and learning strategies which prolong learners’ challenges in understanding a domain that is already complex. this implies that the solution for this challenge lies in teachers’ ability to provide learners with meaningful experiences for understanding geometric proof development. this study argues that teachers’ ability to set up and implement high cognitive level tasks is an interplay of many factors including the type of curriculum material available to the teachers as well as the teachers’ ability to make effective use of these materials. design and use of mathematics textbooks in malawi malawian textbooks are expected to mediate both the intended and the implemented curriculum. as such the national curriculum determines the content of textbooks of a particular educational level in malawi. this is because textbooks are the main content resource used by teachers in malawi just like in other developing countries which are characterised by lack of teaching and learning resources (ronda & adler, 2016). this implies that textbook design is expected to determine the teaching and learning of a particular subject or concept to a greater extent. malawian public secondary school textbooks writers and publishers are commissioned by mest. purchasing and supplying textbooks into public secondary schools is also authorised by mest. as such, textbooks that are available in malawian public secondary schools are only those that are recommended and supplied by mest. for secondary school mathematics education, there are two types of textbooks that were recommended by mest for use in the secondary schools. the content of these mathematics textbooks combines all branches of mathematics at secondary school mathematics level in malawi including geometry, algebra and arithmetic. teachers are expected to decide whether to use either both textbooks or only one textbook depending on the content. as mellor, clark and essien (2018) noticed, two different textbooks can present the same topic in different ways, hence creating different affordances for learners to learn the topic. the two textbooks that are recommended by mest are written for both teachers and learners. the teachers are expected to be guided by the two recommended mathematics textbooks when planning and implementing their lessons. the van hiele levels of geometric thought according to van hiele (1999), learners progress through five levels when learning geometry. these are visualisation, analysis, informal deduction, formal deduction and rigour (van hiele, 1999). geometric proving starts from the informal deduction level. learners at informal deduction level can deduce properties of a shape, recognise classes of shapes and follow formal proofs, but they do not know how to construct a proof starting from different or unfamiliar premises (crowley, 1987). at formal deduction level, learners can develop a proof in more than one way, because they understand the interaction of necessary and sufficient conditions of a proof (crowley, 1987). van hiele explains that most of the secondary school geometry is at the formal deduction level. the van hiele levels of geometric thought have a sequential and advancement property. the sequential property is linear and hierarchical; as such, learning and attainment of higher levels always depends on attainment of the lower levels (crowley, 1987). the advancement property implies that progress or lack of progress from one level to the next level depends more on the content and methods of instruction received rather than the age of the student (crowley, 1987). if a method only helps a student to memorise geometric concepts without understanding, it reduces the learners’ ability to progress to the higher level (crowley, 1987). this means that learners’ challenges in geometric proof development which is at formal deduction level might be a result of the quality of content and methods that are used by the teachers. the advancement property of the levels also implies that learners’ challenges in geometric proof development might affect their advancement to the next level of geometric thought. as already indicated, the main source of content and methods for malawian teachers is the textbook; therefore, a study on textbooks and teachers was necessary. studies on geometric proof development textbooks well-designed mathematics textbooks are regarded as a good resource for supporting learners to understand reasoning and proving (thompson, senk, & johnson, 2012). textbooks mediate between the intended and the implemented curriculum; hence, they are widely used in classrooms around the world (fujita et al., 2009). mathematical tasks presented in the textbooks offer potential sources for opportunities to learn reasoning and proving (bergwall, 2017). thus, the learning of mathematics can be influenced by the textbook content (mellor et al., 2018). this influence might be greater in developing countries where textbooks remain the most readily available resource due to constraints of teaching and learning resources (ronda & adler, 2016). due to the widely agreed upon importance of textbooks to learners’ learning, many scholars have called for studies on how proofs are presented in textbooks (fujita et al., 2009; ronda & adler, 2016; stylianides, 2014). otten, gilbertson, males and clark (2011) examined the reasoning and proving activities, justifications given, as well as the nature of mathematical statements around which reasoning and proving take place in six us secondary textbooks. they found that the mathematical content of the exposition part of the textbook was general in nature; for example, they generalised how to prove properties of a particular shape like a rectangle, but the content of the exercise tasks was specific to a mathematical concept, for example finding values of angles. otten et al. (2011) concluded that although the textbooks contained many theorems and proofs, there were rare tasks that asked the learners to develop the proof on their own. most of the tasks required the learners either to provide a rationale or to determine the truth value of a mathematical claim but not to provide a deductive argument. fujita et al. (2009) analysed the content of the textbooks commonly used for teaching learners geometric proof development in lower secondary school in japan. they found that deductive reasoning is prominent in japanese textbooks as expected from the curriculum specification. however, the japanese textbooks only presented geometric proof development in a formal way without convincing illustrations that can help learners to understand the difference between formal proof and experimental proof. fujita et al. (2009) claim that this type of geometric proof presentation in the textbook might only help the learners to understand the process of geometric proof development but not to understand and appreciate its discovery function; hence, they are unable to understand the notion of generality as well. this agrees with de villiers (1999) who identified discovery or invention of a new mathematical result as one of the functions of mathematical proof. fujita et al. (2009), therefore, suggested that textbooks should design geometric proof development in a manner that provide learners with opportunities that can help them to understand and appreciate more fully the notion of generality of proof. bowie (2013) analysed a grade 10 south african textbook chapter on quadrilaterals to find out how it managed tensions inherent in transition between informal and formal geometry. she found that in some instances, the book used tightly prescribed investigations, generalisations and definitions to manage the transition. this implied that the textbooks struggled with the transition from informal to formal geometry. thus, this review has shown that analysis of textbooks in relation to geometric proof development has focused on different issues. these include illustration of differences between experimental verification and formal proof (fujita et al., 2009), proof presentation (otten et al., 2014), mediating of transition between informal and formal geometry (bowie, 2013; thompson et al., 2012), significance and methodological challenges of analysing reasoning and proving in textbooks (stylianides, 2014), proof and proving in general (stylianides, 2009). while the focus of previous studies was on the content and tasks set up in the textbooks and their affordances to reasoning and proving only, this study focuses on both geometric proof task setup in textbooks as well as task setup and implementation in the classroom. theoretical framework the study is guided by two analytical frameworks: mathematics discourse in instructional analytic framework for textbook analysis (mditx) developed by ronda and adler (2016) and mathematical task analysis framework developed by smith and stein (1998). the mditx framework comprises five key elements: object of learning, examples, tasks, naming and word use and legitimations. the object of learning is what the learners are expected to be able to do at the end of the lesson (ronda & adler, 2016). in this study, the object of learning was developing geometric proofs. examples are a particular case of a larger class used for drawing reasoning and generalisations (ronda & adler, 2016). learners’ textbooks are expected to contain an example space (set of examples) which would enable the learners to attend to a particular feature of the object of learning. a mathematics textbook can contain two types of example spaces: worked example spaces and example exercise spaces (ronda & adler). worked examples are those whose solutions are provided, while exercises examples are those whose solutions are not provided by the textbook or the teacher. tasks are what learners are asked to do with the examples like solving, proving, measuring or drawing (ronda & adler). naming and word use is the way of naming mathematical concepts. adler and ronda (2015) argue that the specific words that we use for naming mathematical concepts and the way we name the procedures or actions carried out on them affect learners’ attention in particular ways. legitimations are the mathematical and non-mathematical criteria that are communicated to substantiate the key steps in procedures or in statements about the object of learning. this study used three mditx elements: (1) object of learning, (2) examples and (3) tasks, to partition the content in a circle geometry chapter of a textbook. the mditx framework was used for analysing both textbook and lesson observation data because the ways of identifying these three elements (object of learning, examples and tasks) in the textbook and in the lesson is similar (adler & ronda, 2015; ronda & adler, 2016). however, although the mditx framework contains different levels for analysing cognitive level of tasks, it does not focus on how to analyse the implementation of the tasks. therefore, a framework by smith and stein (1998) was used for analysing the cognitive level at which the tasks were implemented by the teachers. the mathematical task analysis framework by smith and stein (1998) comprises four categories of cognitive demands of a task: (1) memorisation, (2) procedures without connections to concepts or meaning, (3) procedures with connections to concepts and meaning, and (4) doing mathematics. the categories of tasks depend on the level of learners’ thinking; hence they offer learners different opportunities of learning depending on the level of the thinking involved and the activities expected in each category. memorisation tasks involve exact reproduction of previously learnt facts without showing their algorithms, for example listing the properties of a rectangle. procedures without connections to concepts or meaning are tasks that require the use of an algorithm without showing understanding of how the algorithm works (smith & stein, 1998). an example of procedure without connections is using given information to come up with statements for proving that two triangles are congruent. memorisation and procedures without connections are classified under lower-level tasks because they place little demand on learners’ thinking and explanations. procedures with connections to concepts require some degree of cognitive effort because they involve thinking about how to apply a procedure to a task (smith & stein, 1998). an example of a procedure with connections task might involve developing a multi-step proof that requires learners to apply several geometric properties. doing mathematics tasks also demands considerable cognitive effort because the procedure is not known to the learners, so they are required to explore and understand the nature of mathematical concepts, processes, or relationships to be used in solving the task. an example of a doing mathematics tasks might involve exploring a geometric theorem and its proof both empirically and formally to understand how and why it works. this study used smith and stein’s mathematical task analysis framework as a guide for analysing cognitive changes as the geometric tasks passed through different phases from the textbook to the classroom. as stein, glover and henningsen (1996) point out, a task can be viewed as passing through three phases: firstly, as curricular or instructional materials (textbook), secondly, as set up by the teacher in the classroom and, thirdly, as implemented by learners during the lesson. as such, smith and stein’s framework of analysing tasks was used and acted as a lens for analysing the cognitive level of geometric proof development tasks as they passed through these three phases. research methods and design the study utilised a qualitative case study design with the aim of gaining in-depth understanding of the issue being studied as in line with yin (2009). data for the study were generated through qualitative content analysis of a secondary school mathematics textbook and lesson observations. one textbook, strides in mathematics 3, which was written by hau and saiti (2002), was analysed because it was the only textbook that was being used by all the teachers who participated in the study. deductive proof tasks and examples on nine circle geometry theorems were analysed to examine their cognitive level. to find out how teachers set up and implemented the geometric proof development tasks in the classroom, i analysed lessons by three teachers for form 3 mathematics. this is part of the data that i collected and analysed for my phd study. the three teachers taught the entire topic of circle geometry at secondary level as prescribed in the curriculum. in total, i observed and videorecorded 43 lessons from these teachers. the three teachers were from three different government funded secondary schools. the teachers, paul, john and kim (pseudonyms), were purposively selected on the condition that they were qualified teachers with a minimum of a bachelor of education degree and teaching experience of six years. the assumption for selecting such teachers was that by the end of six years, the teachers would be familiar with the content of the recommended mathematics textbooks and would have selected what they considered to be the best textbooks for their teaching. in malawi, secondary education is four years and beginning teachers (teachers with less than three years of teaching experience) are usually allocated junior secondary classes (grade 9 and grade 10) while experienced teachers are allocated senior classes (grade 11 and grade 12). as such, the teachers would have acquired experience in selecting and implementing deductive geometric proof tasks and examples by the end of six years. data analysis geometric task as set up in the textbook is the way the task is presented as well as how the learners are expected to engage with the task. geometric task as set up by the teacher is the way the task is presented by the teacher in the classroom, which can be either in elaborate form (including verbal directions and explanations) or short and simple form (such as telling the learners to begin work on a set of problems displayed on the chalkboard). task implementation is the manner in which learners actually work on the task (stein et al., 1996). analysis of lesson observation data involved transcribing the video recordings, identifying units of analysis (segments with deductive geometric proof development tasks and their implementation) and analysing the cognitive level of the task setup and task implementation. identification of the object of learning, the examples and the tasks as set up in the textbook, and as set up by the teacher in the classroom, was done by using the mditx analytical framework suggested by ronda and adler (2016). analysis of the cognitive level of the tasks as set up and as implemented by the teacher in the classroom was done by using the stein et al. (1996) framework. analysis of the textbook data involved several steps. as the circle geometry chapter is already partitioned into sections according to the theorems covered, the first step was to partition each circle geometry theorem section into object of learning, examples and tasks using the mditx analytical framework. the second step involved identification of deductive geometric proof development tasks which were the main focus of the study. the third step involved examining what each task required of the learners and relating the requirements to the four categories of cognitive demands to determine their cognitive level using stein et al.’s analytical framework. to ensure credibility of the findings, the transcribed lesson observation and textbook data were also analysed by another researcher and the differences and similarities were discussed. the issue of transferability was addressed by providing thick descriptions on how data was collected and analysed (bryman, 2008). to ensure dependability of the findings, the records of all phases of the research process were kept in accessible form (bryman, 2008). ethical considerations during data collection and analysis, i observed several school-based research ethical principles proposed by cohen, manion and morrison (2007) and berg (2001). these included following official protocol to gain access and acceptance into the school (cohen et al., 2007), informed consent, and the issue of confidentiality (berg, 2001). prior to going to the schools, i received approval from my supervisory committee permitting me to conduct a study with at least three secondary school teachers who were teaching circle geometry during the period of data collection. to gain access into the schools, i obtained an introductory letter from the department where i was studying to the education division manager. upon approving the request to conduct a study in the said secondary schools, the education division manager granted me a permission letter addressed to the head teachers and mathematics teachers of the schools. to seek informed consent from each teacher, i followed several steps (berg, 2001). firstly, i explained to each teacher the purpose of my study and how i was going to conduct the study. secondly, i assured the teacher that they would be treated anonymously during reporting, as pseudonyms would be used instead of real names. thirdly, i assured the teachers that the data collected from their classrooms would only be used for purposes of this study and be treated confidentially. consent to observe the teachers’ lessons was sought every day before going to the classroom. research findings the findings are presented in three subsections: tasks as set up by the textbook tasks as set up by the teacher in the classroom task implementation by the teachers. analysis of tasks as set up by the textbook there are nine circle geometry theorems in the textbook. two theorems are on chord properties of the circle, three theorems are on angle properties of chords in the circle, two theorems are on properties of cyclic quadrilaterals, and two theorems are on concyclic points of a circle. in summary, analysis of the tasks set up for the nine theorems show that learners were expected to do two types of tasks on each circle geometry theorem. firstly, they were expected to do an empirical exploration task where they were to draw a geometric diagram, do some measurements, and then deduce a theorem. secondly, they were expected to develop a formal proof for each theorem that was deduced from the empirical exploration task. at the end of these tasks, the formal proof is given, but the textbook emphasises that the learners should first attempt to develop the formal proofs on their own and then compare their proofs to those given it the textbook. there are two limitations in terms of the way empirical tasks are presented in the textbook. the first limitation is that there is one instance where the textbook begins with a formal proof task then moves to an empirical exploration task. this strategy is not supported by some researchers who argue that learners should first be engaged in an empirical exploration before learning a formal proof to enhance their understanding of how and why the formal proof works (ding & jones, 2009; jones et al., 2009; stylianides, 2005). the second limitation is that for all empirical tasks, the learners are asked to draw one diagram, measure its angles or lines, and deduce a theorem from the results of a single case. there is only one empirical exploration task where learners are expected to draw several diagrams or generalise from multiple cases. this shows that most of the empirical exploration tasks that are set up in the textbooks do not conform to some mathematical principles which discourage generalising from a single case (marton & tsui, 2004). despite the limitations, the tasks that are set up in the textbook can be regarded as of high cognitive level because they promote engagement of learners in both an empirical exploration task and formal proof task. some examples of the empirical exploration tasks set up in the textbook are shown in figure 1, figure 2 and figure 3. figure 1: empirical task on chord properties of the circle. figure 2: empirical task on angle properties of the circle. figure 3: empirical task on angle properties of a cyclic quadrilateral. empirical task on chord properties of the circle figure 1 presents the empirical task on chord properties of the circle. according to the mditx analytical framework, the section title in figure 1 shows that the object of learning is chord properties of the circle. although the activity is titled as ‘challenge’, i partitioned this segment under example exercise because it is a particular case from which learners are expected to deduce or generalise that a perpendicular line drawn from the centre of the circle to a chord bisects the chord (ronda & adler, 2016). the task is for the learners to first try to develop a formal proof to show that ax = xb and then afterwards verify empirically that ax = xb. it is noted that the textbook does not provide clear guidelines regarding the type of resources to be used for drawing the diagrams. this might be the case because the mathematics books that the teachers and learners use in form 1 (grade 9) provide clear guidelines on construction and drawing of geometric diagrams. therefore, at this level, the learners already know the materials to use when drawing a geometric diagram. in figure 1 the learners are required to follow the given procedure to draw a circle with a chord and to drop a perpendicular from the centre of the circle to a chord. using the diagram that they draw, the learners are required to think of ways of developing a proof to show that a perpendicular line drawn from the centre of the circle bisects the chord. although a hint is provided regarding the construction to be made (join oa and ob), the task is under the doing mathematics category because it requires learners to identify another hint (in this case a theorem) for developing the proof (stein, smith, henningsen, & silver, 2009). as cheng and lin (2009) argue, the most critical part of geometric proof development is hypothetical bridging which involves identifying or constructing a theorem that can link the hypothesis and the conclusion. the task in figure 1 shows that the textbook promotes engagement of learners in both formal and empirical understanding of the proof. two limitations of empirical exploration task setup are observed in figure 1. the first limitation is that the learners will be required to generalise from a single case since only a single diagram will be used to deduce the theorem. the second limitation is that the learners will begin by developing a formal proof then do the empirical exploration task. this means that the empirical task will be used for the purpose of verification but not for discovery of a proof, hence it might not promote learners’ understanding of geometric proof development. empirical exploration task on angle properties of the circle figure 2 presents empirical exploration tasks that learners are expected to carry out before learning the formal proof development on angle properties of chords in a circle. the object of learning for the empirical tasks in figure 2 is angle properties of chords in a circle. there are two empirical tasks that learners are required to do before developing a formal proof. i also partitioned these activities under example exercise because they aim at helping learners to come to two generalisations or conclusions (ronda & adler, 2016). the first generalisation as stated in the textbook is that ‘the angle, which an arc of a circle subtends at the centre of a circle, is twice that which it subtends at any point on the circumference’ (hau & saiti, 2002, p. 29). although the focus of this study was not on naming or word use, it is worth commenting that the way the textbook has stated the theorem implies that it does not matter whether the angles are in the same segment or not. for example, it might mean that the angle size of reflex ∠aob is equal to twice the size of either ∠acb or ∠adb or ∠aeb. but for the theorem to be true, the angles are supposed to be in alternate segments. for example, in the case of the diagram produced in figure 1, it is the size of the obtuse ∠aob which is equal to twice the size of either ∠acb or ∠adb or ∠aeb on the circumference. the correct way of stating the theorem would be: the angle that an arc of a circle subtends at the centre of a circle is twice the angle that it subtends at any point on the circumference in the alternate segment. this implies that the textbook has not stated the theorem in a correct manner; as such, if the teachers do not notice this mistake, they might not phrase the theorem correctly during the proving lesson. the second generalisation as stated in the textbook is that ‘angles in the same segment of a circle are equal’ (hau & saiti, 2002, p. 29). figure 2 shows that activity 1 requires learners to draw a diagram according to the given steps and recognise that ∠acb or ∠adb or ∠aeb are produced by a common chord ab, hence they are named as angles in the same segment of a circle. activity 2 requires learners to measure the angles (∠acb, ∠adb and ∠aeb), identify relationships between or among these angles, and deduce a theorem that angles in the same segment of a circle are equal. thus activity 1 and activity 2 require the learners to do several tasks, for example drawing, measuring, relating angles and deducing theorems. although the limitation of generalising from a single case is also observed on exploration tasks in figure 2, both tasks are under the doing mathematics category because they require learners to explore the diagram in relation to the values they get and to deduce theorems that they will be required to discuss with their teacher. this implies that the learners would be engaged in making connections between their findings and the lines drawn on the diagram and to justify their conclusions. after doing the measuring activity and drawing conclusions, the learners are expected to learn how to develop the formal proof. this way of sequencing geometric proof development tasks supports learners’ understanding of the difference between empirical proof and formal proof, and it also promotes learners’ understanding of formal proof (ding & jones, 2009; jones et al., 2009; stylianides, 2005). empirical task on angle properties of a cyclic quadrilateral figure 3 presents the task on angle properties of a cyclic quadrilateral. as shown by the section title, the object of learning for the activity in figure 3 is angle properties of a convex cyclic quadrilateral. i partitioned the activity in figure 3 under example exercise because it aimed at helping learners to deduce or generalise that opposite angles of a convex cyclic quadrilateral are supplementary. thus, the task for this example required learners to do an empirical exploration by following the three steps that have been provided in figure 3. this implies that the task in figure 3 is also under the doing mathematics category because it requires learners to make explorations and use the empirical results to deduce a theorem (smith & stein, 1998). the task in figure 3 is an example of an empirical task that takes into account the mathematical principle of generalising from several sets of examples (marton & tsui, 2004; ronda & adler, 2016). the study found that all the circle geometry theorems in the textbook that were analysed used a similar approach. there is an empirical activity that learners are expected to do before developing the formal proof. however, there is one instance where the learners are expected to develop a formal theorem before doing an empirical exploration. all empirical exploration tasks require learners to do different mathematical activities including drawing diagrams, doing measurements on the diagram, identifying geometric relationships between or among different parts of the diagram, and deducing theorems based on the findings. after doing each empirical exploration task, the learners are expected to reflect on the inductive proof to develop a deductive geometric proof without any given hints on theorems or geometric properties to be applied. as such, the learners are expected to devise their own strategies of developing the proof. in the course of devising the strategies, the learners might develop new problem-solving skills. this shows that the inductive proof that learners develop through empirical activities and the deductive proof that they later develop might enhance learners’ abilities not only to verify the theorem, but also to discover new knowledge and new ways of problem-solving (de villiers, 2012; ding et al., 2009). this also means that the deductive geometric proof development tasks set up in the textbook are in the doing mathematics category, hence they are of high cognitive demand (stein et al., 2009). as such, the tasks that are set up in the textbook have the potential to engage the learners in making connections among different features of geometric content (ronda & adler, 2016), to link formal and informal geometry (bowie, 2013), and to make logical and clear explanations (deductive proving), hence promoting other functions of proof like explaining and justifying mathematical concepts (de villiers, 2012). analysis of tasks as set up by the teachers in the classroom the findings showed that deductive geometric proof development task setup and implementation by teachers in classrooms was slightly different. it was observed that paul and john set up formal proof tasks only during all proving lessons for the nine theorems on deductive proof development. kim always started from an empirical exploration task to formal proof task in all proving lessons for the theorems. empirical tasks set up by kim in the classroom figure 4 and figure 5 present some of the empirical tasks set up by kim in the classroom before presenting the formal proof task. figure 4: empirical exploration tasks on chord properties of the circle. figure 5: empirical exploration task on angle properties of the circle. both figure 4 and figure 5 show example exercises that the learners were asked to do in class. the main object of learning for the examples and tasks in figure 4 was proving the theorems about the chord properties of a circle. the specific object of learning for activity 1 in figure 4 was discovering that a perpendicular line drawn from the centre of the circle bisects the chord of the circle, while the object of learning for activity 2 was proving that if a line drawn from the centre of a circle bisects the chord, then the line is perpendicular to the chord. the tasks for both examples required learners to measure either lines or angles and deduce theorems based on the results of empirical exploration tasks. figure 4 shows that the learners were expected to draw a circle with a perpendicular line from the centre of the circle to chord, measure lines ad and bd, compare lengths of the lines, and deduce a theorem. the task setup in figure 4 is similar to the task setup in the textbook (figure 1). however, in addition to the textbook example, kim set up example 2 where learners are required to draw another chord whose length is equal to the first chord, draw a perpendicular line from the centre of the circle to the chord, compare its length to the first perpendicular line, and then deduce a theorem. kim might have extended example 1 to ensure that both chord properties of the circle that were being proved during the lesson are deduced and tested using empirical explorations. the object of learning for the task in figure 5 is proving that angles in the same segment are equal. as such the learners are expected to do a similar empirical exploration task of drawing a circle with several angles subtended by the same arc, measuring the angles, and deducing a theorem from the results. however, the task is slightly different from the one in figure 2 as it is only concerned with angles in the same segment of the circle. this might also be the case because on this day the object of learning was only about angles in the same segment. although the limitation of generalising from a single case is also observed in kim’s exploration tasks, there is an attempt to use the exploration approach that is suggested in the textbook. in general, the tasks set up by kim can be categorised under the doing mathematics category as they required learners to make explorations and discoveries in reference to the results from empirical explorations. figure 4 and figure 5 also show that kim made some modifications to the textbook activities according to the theorem that was being explored on a particular day, as well as to ensure that learners discovered each theorem on their own using empirical evidence. formal deductive geometric proof development tasks as set up by the teachers in the classroom the findings showed that most of the formal proof tasks set up by the three teachers in the classroom were similar and they were taken from the same textbook (strides in mathematics 3). the similarity of the tasks might suggest that the teachers made either few or no alterations on the textbook tasks. figure 6(a) and figure 6(b) present examples and tasks set up by the teachers in the classroom for development of formal deductive geometric proofs. the tasks were set up for proving a third theorem on circle geometry and were from lesson number 3. figure 6: examples and tasks as set up by the teachers in the classroom. (a) john’s example for formal proof and (b) kim’s task for formal proof. figure 6(a) and figure 6(b) show that the tasks are similar despite differences in the way the diagrams are drawn. the tasks involve developing a formal deductive geometric proof to show that angles in the same segment of a circle are equal. as noticed in figure 6(a), the angles that are subtended by the same arc have already been identified and necessary construction has also been provided (lines ao and ob are already drawn). the learners are required to decide the theorem to be used for developing the proof and to construct proving statements. the example and task set up by john in figure 6(a) is similar to the one that was set up by paul under the same object of learning. the only difference was that paul added labels to the angles in the same segment. he labelled ∠adb as x and ∠acb as y and wrote that the task was to prove that x = y. although some of the decisions were already provided by paul and john, the tasks are still of high cognitive value because the critical part of hypothetical bridging (cheng & lin, 2009) is not disclosed. hence the task is under the doing mathematics category as it requires learners to identify the hypothetical bridge and make geometric connections to develop the proofs. to be able to do the task of proving that angles in the same segment are equal using the example in figure 6(b), the learners are required to identify angles subtended by the same arc at the circumference, to decide the construction to be added to the diagram, to decide the hypothetical bridge to use for developing the proof, and to construct proving statements. this implies that kim’s formal proof task is also under the doing mathematics category as there is no hint for deciding the construction to be added to the diagram and the hypothetical bridge to be used. this analysis has shown that the deductive geometric proof development tasks that were set up by the three teachers in their classrooms were similar to those that were set up in the textbook. the analysis has also shown that the formal proof tasks are of high cognitive level because they required learners to make several decisions on angles to be proved, construction to be made and the hypothetical bridge to be used. task implementation by paul the lesson observation data showed that during geometric proof development, paul explained everything that was supposed to be done to develop the proof. for example, when proving angles in the same segment of the circle are equal, he started by showing the learners that angles x and y were the ones that were being subtended by arc ab. then he told the students that they were going develop the proof by using the theorem developed in the previous lesson which states that ‘an angle that an arc of a circle subtends at the centre of a circle is twice the angle subtended by the same arc at any point on the circumference of the circle’. this is the way the theorem is stated in the textbook. this shows that paul did not realise that the theorem was not stated correctly in the textbook. later on, paul went ahead explaining and writing the proving statements and their reasons on the chalkboard until he arrived at the conclusion. this shows that despite the task being of a high cognitive level, paul implemented it at a lower level as learners were not involved in any form of explaining or justifying the truth of the theorem. although the textbook encouraged the learners to first attempt to develop the formal proof on their own before referring to the proof development approach provided in the textbook, paul did not give learners an opportunity to express their views on how the proof could be developed. this implies that despite being set up as a high cognitive level task both in the textbook and on the chalk board, the proof task was implemented at a low cognitive level as paul did not involve the learners in any form of thinking, explaining and exploring of how the proof could be developed. hence, the mode of task implementation reduced the cognitive level of the task from a high level to a low level (stein et al., 2009). task implementation by john john also used an approach of question and answer to teach the learners how to develop the deductive geometric proofs. in all lessons, john provided guidance on the theorems or geometric properties that were applied when developing the formal proofs. for example, when developing the proof using the task in figure 6(a), john started by telling the learners that the lesson was about proving that angles in the same segment of a circle are equal. he explained that ‘angles in the same segment are the angles that are subtended by the same chord or arc at the circumference’. john stated the theorem the way it is stated in the book that he was using (strides in mathematics 3). however, john’s description of angles in the same segment is partially not correct because it has left out an important point that is stated in the textbook. he does not specify that the angles should be in the same segment of the circle. this might imply that john’s wording or explanation of the theorem was not correct as it implied that the equal angles that are subtended by the same arc might lie in alternate segments as well. after describing angles in the same segment, john engaged the learners in the following conversation: 1. john: so, we start with the first step, what are we given? 2. student: circle with centre o. 3. john: what else? 4. student: chord ab. 5. john: yes, chord ab which is subtending angles c and d at the circumference, and angle y at the centre. this is what we are given [and he writes on the board]. so, what are we asked to prove? 6. student: angle c equals angle d. 7. john: yes, this is what we should prove. so, for us to show that these angles are equal, let us consider the theorem we proved yesterday. what theorem did we prove yesterday? 8. student: angle at the centre is two times the angle at the circumference. 9. john: yes, that is true, yesterday we proved that angle at the centre is two times angle at the circumference. so what statements can we come up with using that theorem to prove that angles c and d are equal? 10. student: angle y is two times angle c, reason is that angle at the centre is two times angle at the circumference. 11. john: correct [while writing the statement on the chalkboard], what is the other statement? 12. student: angle y is two times angle d, same reason, angle at the centre is two times angle at the circumference. the lesson conversation shows that john implemented the task using the question and answer method in a stepwise process. firstly, he asked the learners to identify the given information (utterances 3–6). secondly, he asked the learners to identify the statement to prove (utterance 6). lastly, he involved the learners in constructing the proving sentences (utterance 10–13). the lesson extract shows that learners did not struggle to construct the proving statements. this might be because john had already provided the hint for the hypothetical bridge to be used for developing the proving statements (utterance 10). since identification of the hypothetical bridge is the critical part of deductive geometric proof development (cheng & lin, 2009), then this shows that the learners were not involved in high cognitive level thinking. according to smith and stein (1998), provision of hints that are critical to solving of a task reduces the cognitive level of the task. this implies that the cognitive level of the task was reduced from high level to low level because of the mode of implementation (charalambous, 2010). some might argue that john provided the hint with an aim of trying to help the learners to gradually realise how to develop the proof. but polya (1945) discourages teachers from providing much guidance to support learners; instead he suggests that teachers should ask different types of questions that would support the learners to identify the hypothetical bridge on their own. task implementation by kim kim mainly used group work when teaching the learners to develop proofs for the theorems. the learners would discuss and develop the proofs in their groups and then present them to the whole class. for example, when implementing the task in figure 6(b), kim asked the learners to draw a diagram similar to the one that he drew on the chalkboard in their groups and discuss how to develop the proof for the theorem by stating that angles subtended by an arc in the same segment of a circle are equal. as the learners were developing the proof in groups, kim checked their work and provided some guidance to those learners who experienced difficulties in developing the proof. after about 15 minutes, kim asked the learners to report their work by giving an oral account regarding how they proved the theorem. the following is a conversation between kim and students after the group discussions: 13. student 1: [while pointing at the diagram drawn by kim] we joined ab at o, then let ∠aob be 2y. so ∠aeb equals y, ∠acb equals y, and ∠adb is also equal to y. therefore, angles aeb, acb and adb are equal. 14. kim: why do you say that angles aeb, acb and adb are equal to y? 15. student 1 because angle at the centre is two times angle at the circumference, so if we have 2y here [pointing at ∠aob] then this is y [pointing at ∠aeb], this is y [pointing at ∠acb], this is also y [pointing at ∠adb]. 16. kim: yes that is true, but you need to remember to justify your statements with reasons. did you all use this approach, or is there any group with a different approach? 17. student 2: yes, we joined ao and bo. then two times ∠aed equals ∠aob, two times ∠adb equals ∠aob, and two times ∠acb equals ∠aob. the reason is the same, angle at the centre is equal to two times angle at the circumference. 18. kim: yes, that is also a correct method. so, apart from the two ways that have been presented, any group with a different way? 19. student 3: yes, we joined ao and bo, ∠aeb equals half ∠aob, reason is angle at the circumference is equal to half angle at the centre. ∠acb equals half ∠aob same reason, ∠adb equals half ∠aob, same reason, therefore angles aeb, acb and adb are equal. the lesson conversation shows that kim involved the learners in exploring the proof of the theorem and the learners came up with their own ways of developing the proof. all learners made the same construction on their diagrams, and they also used the same property – which states that the angle subtended by an arc at the centre is twice the angle subtended by the same arc at the circumference of the circle – as their hypothetical bridge (utterances 15, 17 and 19). this was the theorem that the learners proved in the previous lesson. despite making the same construction on the diagram and using the same hypothetical bridge, the proofs were developed using slightly different approaches (utterances 15, 17 and 19). although the proof explained by student 1 (utterance 15) is similar to the one presented in the strides in mathematics 3 textbook, the letters that have been used are different. this implies that when the learners explored how to develop the proof, they made several connections with their previous knowledge. furthermore, the learners were able to explain their proofs clearly, and they provided good justifications for their proving statements. this shows that kim implemented the task in a manner that involved learners’ high cognitive levels of thinking, hence the task maintained its high cognitive level during implementation (stein et al., 2009). discussion of the findings the findings reveal that the textbook that the teachers used for teaching deductive geometric proof development contained both empirical exploration and formal proof tasks which were of a high cognitive level. the empirical exploration tasks promoted discovering of proofs as they required learners to deduce theorems from the results of an empirical activity. the formal proof tasks promoted verification, explanation and systemisation as they required learners to identify a hypothetical bridge for constructing logically sequenced proving statements (cheng & lin, 2009; de villiers, 1999). this shows that the textbook has included rich opportunities for learners to engage in doing mathematics (stein et al., 2009) and in reasoning and proving in a coherent manner (stylianides, 2014). despite the limitation that the empirical task set up in the textbook might lead to misconception that it is acceptable to generalise from a single case in mathematics, it is worth acknowledging that the textbook attempted to provide opportunities for empirical explorations. some might argue that the empirical exploration tasks set up in the textbook were poorly designed because they mainly require the use of pencil and paper and not computer software. it is argued that computer software can enhance learners’ ability in developing geometric proofs better than pencil and paper tasks because they organise and promote learners’ thinking (mariotti, 2000). it has also been observed that pencil and paper diagrams are difficult for learners to grasp because they might deviate learners’ focus from properties of the diagram to properties of construction (mariotti, 2000). despite this flaw, pencil and paper tasks fit well with the context in which the study was undertaken. as already explained, malawian secondary schools operate under resource constrained conditions where textbooks are the most available teaching and learning resource. the textbook tasks might have been designed in this way to enable teachers and learners to use the resources that are accessible to them, for example mathematical sets, pencils and paper. the instructions for doing the empirical exploration tasks might serve as attempts to address limitations of pencil and paper tasks. as such, a didactical design of the textbooks fits well with the malawian context. the instructional approach used in the textbook is called presenting a proof problem as an experimental problem and a formal proof (ding & jones, 2009). this approach helps learners to understand the formal proof and to appreciate the discovery function of proof in mathematics through the new problem-solving strategies that they devise and the reflections that they make on the developed proofs (de villiers, 2012; ding & jones, 2009; jones et al., 2009). the task setup approach used in the textbook is also supported by hanna (2000) who suggests that during mathematical proof development, learners should be provided with opportunities to explore different paths to the solution outcome by using a combination of inductive and deductive reasoning processes. kim provided learners with rich opportunities to explore and understand the proof development process and its advantages by setting up and implementing tasks similar to the textbook tasks. the findings from kim support the argument that textbooks are supposed to have similarity with classroom lessons (ronda & adler, 2016). the approach that was used by kim is also recommended by ding and jones (2009) who explain that helping learners to understand why a proof works, and then using that understanding to further generalise or specialise the result, is one way of encouraging learners to understand and appreciate the discovery function of mathematical proofs. however, although the textbook has presented geometric proof development as both an empirical and formal process, paul and john only presented the formal proofs in the classroom. this implies that paul and john did not utilise the textbook guidance regarding learners’ tasks for supporting understanding of geometric proof development. this finding agrees with stylianides (2014) who argues that equipping teachers with textbooks containing high quality proving tasks does not necessarily imply that the teachers will implement the tasks presented in the textbooks faithfully. this means that although high quality mathematics textbooks have a potential of improving learners’ understanding of geometric proof development (otten et al., 2014), the realisation of this potential depends on how teachers implement the tasks in the classroom. the findings also showed that the formal proof tasks that were set up by paul and john in the classroom were similar to those that were in the textbook, hence they were of a high cognitive level. however, the tasks decreased in their cognitive level because learners were not involved in activities that would enhance their reasoning. as charalambous (2010) argues, a high level task can decline in its cognitive level when learners are not involved in exploration and explanation of their procedures during task implementation. charalambous (2010) argues that the teacher’s way of explaining and representing mathematical tasks largely depends on the breadth and depth of their conceptual understanding of mathematics. this agrees with the findings in mwadzaangati (2017a) which showed that john and paul displayed some limitations in conceptual understanding of the deductive geometric proof development process while kim displayed competencies in deductive geometric proof development (mwadzaangati, 2017a, 2017b). this suggests that depending on their conceptual understanding of geometric proof development, teachers can either elevate or decrease a task’s cognitive demand during either task setup or task implementation. conclusion this study examined similarities and differences between deductive geometric proof development tasks as set up in the textbook, and as set up and implemented by teachers in the classroom. the findings showed that the geometric proof tasks in the textbook were set up at a high cognitive level and included explorations and explanations. the textbook contained two types of deductive geometric proof development which can be categorised as empirical exploration tasks and formal proof tasks. empirical exploration tasks were those that required the learners to draw diagrams, do some measurements on either lines or angles of the diagram and deduce a theorem. formal proof development tasks required the learners to explore how to develop the formal proof. despite the availability of both empirical exploration tasks and formal proof tasks, john and paul involved the learners in formal proof tasks only. kim followed the textbook procedure by involving learners in both empirical exploration tasks and formal proof tasks. in terms of the formal proof tasks, the findings showed that all three teachers set up high cognitive level tasks in the classroom but their modes of implementation were different. kim involved the learners in exploring and explaining how to develop the proofs, hence he maintained the high cognitive level of the tasks during implementation. paul and john implemented the formal proof tasks at low cognitive level as they provided the learners with hints for the most critical stages of deductive proof development. this study, therefore, concludes that ability to support learners’ understanding in deductive geometric proof development does not only depend on the availability of a well-designed textbook, but also depends on the teachers’ conceptual ability to use the textbook effectively. since the data analysed in this study were only from the teachers’ lessons and the textbook content, i suggest further studies to increase our understanding of the reasons behind the choices that teachers make during task setup and implementation in the classroom through other methods of data collection like interviewing the teachers. since naming and word use and legitimations were also observed to be problematic for the teachers, future research might also focus on examining how teachers explain geometric concepts and legitimise them during the lesson. acknowledgement competing interests i declare that i have no financial or personal relationships that may have inappropriately influenced me in writing this article. author’s contributions i declare that i am the sole author of this article. funding information this study was kindly funded by norwegian programme for capacity building in higher education and research for development (norhed) under improving quality and capacity of mathematics teacher education in malawi project. data availability statement data sharing is not applicable to this article as the author did not seek consent from the participants to share the data. disclaimer the views and opinions expressed in this article are those of the author and do not necessarily reflect the official policy or position of any affiliated agency of the author. references adler, j., & ronda, e. 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(2009). case study research design methods (4th ed.). los angeles, ca: sage. abstract introduction theoretical framework data analysis and research results discussion conclusion and implications acknowledgements references about the author(s) vimolan mudaly department of mathematics education, school of education, university of kwazulu-natal, durban, south africa citation mudaly, v. (2021). constructing mental diagrams during problem-solving in mathematics. pythagoras, 42(1), a633. https://doi.org/10.4102/pythagoras.v42i1.633 original research constructing mental diagrams during problem-solving in mathematics vimolan mudaly received: 09 june 2021; accepted: 31 aug. 2021; published: 29 nov. 2021 copyright: © 2021. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract in mathematics, problem-solving can be considered to be one of the most important skills students need to develop, because it allows them to deal with increasingly intricate mathematical and real-life issues. often, teachers attempt to try to link a problem with a drawn diagram or picture. despite these diagrams, whether given or constructed, the student still individually engages in a private discourse about the problem and its solution. these discourses are strongly influenced by their a priori knowledge and the given information in the problem itself. this article explores first-year pre-service teachers’ mental problem-solving skills. the emphasis was not on whether they solved the problems, but rather on their natural instincts during the problem-solving process. the research shows that some students were naturally drawn to construct mental images during the problem-solving process while others were content to simply leave the question blank. the data were collected from 35 first-year volunteer students attending a second semester geometry module. the data were collected using task sheets on google forms and interviews, which were based on responses to the questions. an interpretive qualitative analysis was conducted in order to produce deeper meaning (insight). the findings point to the fact that teachers could try to influence how students think during the problem-solving process by encouraging them to engage with mental images. keywords: mathematical images; mental manipulations; problem-solving; visualisation introduction solving mathematical problems is generally a useful way to develop a deeper understanding of difficult concepts. but mathematical problems are difficult for most students, as was affirmed by tambychika and meerah (2010, p. 142) when they cited students who stated that ‘mathematics problems are really difficult. i did not know how to do it. that’s why i did not finish it. i don’t like maths’. this is not new or unique. there is a general tendency to despise what cannot be done easily and problem-solving falls well within this category. students often struggle with various aspects of a problem and much has been written in this field. despite all the research ‘students’ problem-solving skills still require substantial improvement’ and the ‘one challenge that we have to grasp in our classroom practices is reversing this situation by helping students construct their mathematical knowledge with problem solving’ (vale & barbosa, 2018, p. 23). guzman (2018, p. 53) acknowledged that ‘problem-solving has a special importance in the study of mathematics’ and further explained that ‘it can also be used, as a teaching method, for a deeper understanding of concepts’. the latter is perhaps a significant function of problem-solving. to achieve understanding of concepts through problem-solving is not an easy process. there are often complex steps that need to be followed and these need to be practised through constant use. yin (2010, p. 2) also felt that the ability to solve problems ‘is at the heart of mathematics’ but was convinced that ‘visualization is at the heart of mathematical problem solving’. this is indeed significant and forms much of the argument around which data are presented in this article. walker, winner, hetland, simmons and goldsmith (2010) also mention that the principles and standards for school mathematics and the common core standards: explicitly describe visualization as a tool for problem-solving and also recognize the essential role of being able to represent and interpret mathematical ideas and problems in visual forms, including graphs, sketches, and diagrams. (p. 22) vale and barbosa (2018) also found that: recent research in the area of cognition in particular about problem solving processes, indicates that for certain types of tasks, the use of strategies that requires visual representations may have advantages over the use of other representations, facilitating problem solving. (p. 24) yin (2010) further cited two important passages from the singapore ministry of education (moe) website, which allude to the possibilities that visualisation offers during the problem-solving process: visualization is the ability to see and understand a problem situation. visualizing a situation or an object involves mentally manipulating various alternatives for solving a problem related to a situation or object without benefit of concrete manipulatives’. (moe, 2001, p. 51, cited in yin, 2010, p. 2) visualization can be a powerful cognitive tool in problem solving. (moe, 2007, p. 13, cited in yin, 2010, p. 2) these citations capture the essence of visualisation and convey, without any ambiguity, that visualisation involves more than just seeing. it is deeper and involves mental processes. whiteley (2004, p. 1) stated from his own experiences and collected anecdotes that ‘it is an illusion that mathematical reasoning is done in the brain with language’. so, if it is not done using language, how do we engage with mathematical reasoning in the brain? there is the allusion to the idea that much of our mental reasoning occurs using visual representations and manipulations. this is not easy. again, whiteley (2004, p. 3) postulated that ‘we create what we see. visual reasoning or “seeing to think” is learned. it can also be taught and it is important to teach it’. so what are these mental representations? chi (2008, p. 67) captured this deftly by describing a mental model as an internal representation of a concept (such as square) or an inter-related system of concepts (such as the quadrilaterals) that corresponds in some way to the external structure that it represents. ibrahim and rebello (2013, p. 1) also emphasised this idea of mental representations by claiming that ‘a key aspect of teaching and learning requires that the learner visualize, i.e., form images both internally (mentally) and externally (with paper and pencil or technology)’. they further surmised that the: importance of external representation lies in the fact that it may provide an associated “mental image” of the physical processes in terms of principles and underlying concepts being considered, hence supporting comprehension. (ibrahim & rebello, 2013, p. 1) walker et al. (2010, p. 22) also suggested that ‘the ability to visualize what cannot be seen directly is considered a critical skill in mathematics and science’. their claim that visualisation is a critical skill is, in a sense, supported by an argument drawn from the arts. they found that visualisation: seems to be a fundamental habit of the artistic mind. artists do not just magically ‘see’ in their mind’s eye, but deliberately and systematically analyze shape and space into familiar simple forms, construction lines, angles, and size ratios. (walker et al., 2010, p. 23) this is an analogous argument for the way visualisation is used in mathematics understanding as well. rösken and rolka (2006, p. 1) found in their study that ‘students that do not show visualization on their paper were able to solve problems correctly. this highlights once again the importance of pictures in the mind’. this is probably the heart of the ability to solve mathematical problems – the ability to manipulate images in the mind. gutiérrez, ramírez, benedicto, beltrán-meneu and jaime (2018, p. 4) noted important elements of visualisation. significant among them were mental images, external representations and the process of visualisation. they defined mental images as ‘any kind of cognitive representation of a mathematical concept or property by means of visual or spatial elements’ (gutiérrez et al., 2018, p. 4). external representations are ‘any kind of verbal or graphical representation of concepts or properties including pictures, drawings, diagrams, etc. that helps to create or transform mental images and to do visual reasoning’ (gutiérrez et al., 2018, p. 4). the process of visualisation is ‘a mental or physical action where mental images are involved’ (gutiérrez et al., 2018, p. 4). this would imply that every physical action is somehow related to a visualisation process because it would be difficult to conceive of a process where no mental action occurred without a mental image. this image may be in the form of a picture, diagram, icon, symbol, word and so on. presmeg (1986, p. 42) affirmed this idea when she stated that ‘after all, diagrams are a frequent accompaniment to mathematical thinking’. theoretical framework this article uses the iterative visualisation thinking cycle. the cycle begins with the learner engaging in a physical or visualisation activity (doing stage). the stage is characterised by visualisation of physical or mental activities. the physical activity may involve the use of a manipulative. but despite the physical engagement, all activities are related to some mental activity – they happen in the head – since the hands move as dictated to by the brain. the visual mediators (symbols, graphs, notations, images, and so on) also influence the visualisation activities that occur in the mind. this process of engaging physically (using manipulatives) and mentally is the meaning making stage. physical and mental objects are manipulated in order to create deeper understanding. the process is iterative because it allows for postulations and testing. if the hypothesis is correct then it is easy to move ahead, but if the hypothesis is shown to be incorrect then the process will start again but with added information (of what does not work and what new information should be employed). in this visualisation process the tendency is eventually to ‘see’ the concept. ‘seeing’ is translated to understanding. once the concept is understood well (‘i got it’), it needs to be converted into a mathematical, symbolic form. through this process new knowledge is formed or old knowledge is transformed. research focus the study focuses on the way students conduct self-discourses with themselves as they work through a problem in mathematics. this is an attempt at trying to ascertain what students think and how they think, as they work through a mathematics problem from the perspective of the visualisation process. figure 1: iterative visualisation thinking cycle. research aim and research questions the broad aim of this research was to explore how learners thought about problems and their solutions before attempting to actually write out an answer. in essence, the research question explored how first-year university students responded to simple mathematical problems by getting them to recall their mental activities and processes. research methodology general background and sample this is a qualitative, interpretivist study of first-year undergraduate mathematics students’ use of visuals during a problem-solving activity. the undergraduate class comprised 51 students, generally from poor socio-economic backgrounds. all of the students would have completed 12 years of formal schooling and one semester of university courses, including a module on geometry, which served to reinforce and build on their existing knowledge in order to create more knowledgeable classroom practitioners in the future. all 51 students would have passed a geometry course in the first semester in order to be accepted into the more advanced geometry course in the second semester. the intention of the module is to help the students to employ general principles of mathematics, more specifically geometric procedures for application. the purpose is to develop personal confidence and competence in geometry. this, it is hoped, will prepare teachers for the effective facilitation of learning (teaching) of geometry in secondary schools. all students were invited to participate in the geometry tasks provided. the students were fully aware that this was for research and did not count towards their module mark. only 35 students volunteered to participate. instrument and procedures the research was conducted during the coronavirus disease 2019 (covid-19) pandemic and university shutdowns. all participants were asked to complete tasks that appeared on google forms. students were informed that any diagrams drawn could be emailed separately. the interviews were conducted via zoom or telephone, based on the student responses (needs and availability). the instrument itself comprised 19 questions and each attempted to capture their responses in different ways. in some instances diagrams were provided and in others the students had to create the diagrams themselves. data analysis and research results the first question that the students had to complete was: what is the value of angles a and b in figure 2? these types of questions are based on the students’ tendency to simply accept properties of a diagram as they see them. figure 2: question 1 of the task sheet. all 35 students reacted to the question by simply looking at the diagram and claiming that the value of angles a and b are 90° each. this is exactly what was expected. this is obviously a problem with visualisation. learners sometimes respond quickly to what they see and do not interrogate the details of the given information that may influence the response to the question. their reason for claiming the angles to be 90° is that both of the figures were squares (‘these shapes are squares and all angles in a square are 90 degrees’). besides the diagram, no other details were given and therefore the assumption that the shapes were squares was based on their experience of what a square looked like. ‘seeing’ without insight is problematic. overconfidence with knowledge can be dangerous and this is a serious pitfall of visual methodologies. a similar question was repeated to ascertain the veracity of students ‘jumping to conclusions’ by simply looking at the diagram. students were asked to identify the value of a in figure 3. figure 3: question 2 of the task sheet. again, 34 of the students confidently stated that the value of angle a is 90°. their reasons were all similar: ‘angle in the semi-circle’, implying that the angle in the semi-circle is equal to 90°. one student said ‘i don’t know’ because, as he later stated in the interview, he ‘could not recall the theorem’. in figure 2 and figure 3 the answers seemed to be what they looked like but in the absence of more details, these conclusions were incorrect. seeing more in the diagram is a common problem and this has to be dealt with deliberately in a mathematics classroom. but, these were university students engaging in a geometry module in the second semester of their first year. after having completed all of the euclidean geometry in the first semester, these ‘errors’ should not occur. they would have worked through several examples and should have acquired the experience to draw conclusions from what is given and what is assumed. while it can be argued that incorrectly drawn diagrams may result in incorrect conclusions, these diagrams were correct but merely impersonated the students’ a priori knowledge of known theorems. this allusion to what students already know creates confidence in their minds. the responses from the first two questions represented by figure 2 and figure 3 illustrates the understanding that the process of visualisation cannot be viewed only as a visual act, something that can be seen with the eyes alone. visualisation involves sight and insight. visual mediators and imagination can be used to mentally manipulate mathematical figures, shapes and symbols. in both of these figures the students ought to have considered the properties of the figures by reflecting on what they actually represent. a shape should be defined mentally by its properties. if the properties are not inherent in a figure then the mind should not conclude that that figure is a particular shape. diagrams (whether mental or physical) cannot be considered metonyms for their properties. a study by samkoff, lai and weber (2012, p. 49) found in their research that ‘participants’ reasons for using diagrams included noticing mathematical properties’. if the properties did not exist in the diagram then those properties cannot be inserted because they ‘looked’ like they were there. participants were then asked an algebraic question that could be resolved using either mental or pencil and paper methods. the question was: ‘the sum of the supplement and the complement of an angle is 130°. find the measure of the angle. write out your solution.’ the solution to the problem was that the angle is 70°. of the 35 students, 9 produced no answer. their reasons were that they did not know what to do. one student explained that she ‘could not imagine what this was’. eight students were able to provide the correct answer. three of these students were able to draw diagrams to help solve the problem (see figure 4) but from a visualisation perspective the other five solutions were of greater significance. the students did produce the correct algebraic solution but when asked how they arrived at the equation 180° – x + 90° – x = 130o (the students used different variables) without using a diagram, one student stated that she could clearly ‘see the images in my head, and the expressions were easy thereafter’. another student also recalled ‘trying to imagine the supplement and complement in my head’. there was a particular ease with which these five students translated the question into a mental problem and they mentally manipulated the symbols to produce the equation. this is exactly the process of meaning making. none of the other 18 students could provide a suitable justification for their answers. they seemed to have guessed their solutions and many did not understand the terms supplement and complement. a deep reflection on the results reveals that the students who used a diagram or who claimed to have mentally manipulated the given data were able to solve the problem. samkoff et al. (2012, p. 50) further concluded that ‘diagrams can be used to provide novel and more accessible explanations for mathematical phenomena, or highlight aesthetics that are less accessible through symbols and logic’. figure 4: student’s drawings – reconstructed using sketchpad. the next question provided a diagram but still required the students to reflect on how they could find the solution. the question required them to determine whether it was possible to find the sum of a, b, c and d in figure 5 and, if so, what the solution was. the answer to the question was that yes it was possible and the sum was 540°. figure 5: the diagram that seems to present little information. every student felt that no solution was possible. there was no attempt made to use the different triangles available to seek a solution. when probed about their responses there was a general assumption that no relationships could be found among all four variables. the students were very adamant, when it was suggested that a solution was possible, that ‘i cannot see how these letters can be linked to find a solution’. one student stated that ‘it can’t be found because of some missing information’. insight requires the students to process the visual data, by using what is given, to create possible solutions. vision enables us to process the given data at a glance, provided the a priori knowledge can be linked immediately with what is given. a simple iterative process of seeing the different triangles and then noting the angles in terms of these triangles would have guided them towards a solution. the next question attempted to test their ability to iterate between internal and external processes. they were asked to determine which triangles could be divided into two isosceles triangles and then explain their answers. this required both internalisation and externalisation processes because students had to draw, reflect, redraw and reflect again. this process ought to recur until the students find the correct answer. it seems that there was limited utilisation of the iterative process. a number of students recognised that it could be a right-angled triangle and drew it. their reflections thereafter produced the hypothesis as stated by one student that ‘in the right triangle, a line segment will be drawn from the midpoint of the hypotenuse to the right-angled vertex’. or the hypothesis that another student arrived at that stated that in the ‘right-angled triangle, drawing a line perpendicular to the hypotenuse of the triangle to the vertex of the right-angle, divides the right triangle into two isosceles triangles’. they then proceeded no further. it did not occur to these students that they should reflect on their solution and determine how those new triangles would become isosceles. the iteration simply stopped without any further testing and reflection. but, there were students who first declared that ‘the triangle must be a right-angle one’. on further reflection the student stated that ‘it must be an isosceles right-triangle which can be divided into two isosceles triangles because two sides are of exactly equal lengths’. while this might not be the complete solution, it did show an added layer of reflection that moved the student’s thinking deeper and produced a little more understanding. even the student who represented his thoughts by saying that ‘it must be a right-angle isosceles triangle … a line must divide a triangle in such a way that it bisects the right-angle’ showed an added layer of reflection. clearly, the student was close to the solution but he simply did not articulate it well enough. to illustrate an eloquent response to the question one student stated: ‘this must be an isosceles right-angled triangle which can be divided into two isosceles triangles by constructing a perpendicular line at the midpoint of the hypotenuse thus bisecting the opposite angle, giving rise to two new isosceles right-angled triangles.’ when asked to explain why she presented more information than was required, the student declared that ‘this is exactly how i saw it’. it means that in her mind she could ‘see’ the segment emanating from the midpoint of the hypotenuse to the vertex of the opposite hypotenuse. she could ‘see’ that this segment would be perpendicular and it would bisect the 90° angle. all of this was happening in her mind because she did not draw a diagram. these mental manipulations were deliberate and based on her previously known knowledge. the proposition that can be drawn from this evidence is that if the students mentally deliberated on the problem, used an iterative process of reflecting on what is given, what is required and what is already known, then there is a possibility that they could find a solution. some came close in this example but many were very tentative in their responses (‘right-angled isosceles triangles, a line must divide a triangle in such a way that bisects the angle’). the responses received for the question related to areas that were mainly well thought of and presented. the question stated that four points are evenly spaced around a circle as shown in figure 6. connecting the points makes a square. the diameter of the circle is exactly 5 cm. they had to determine the area of the shaded region and explain how they arrived at this solution. figure 6: areas of circles and squares. there were a number of mental knowledge processes that had to occur silently for this question to be answered. students had to recognise that angles of a square are 90° (this would then mean that the diameter of the circle is the same as the diagonal of the square), the radius is half the length of the diameter or diagonal,(the side of the square can be calculated by using the theorem of pythagoras and the radius of the circle, and the formulae for the area of the circle and the square. there would be various other bits of information that the mind must recall, such as calculations, symbols, notations, procedures and concepts (like area, for example). this is a large amount of information to process at one time but through practice and engagement, this becomes easier. the students were able to explain their mental procedure by saying: ‘find radius which is 2.5. use the pythagoras theorem to find the length of the side of the square. the shaded region is the difference between the area of the circle and the square.’ while none of the subtle processes are described, they are inherent in the response. to get the radius, the student knew that the diameter was 5 cm. by calculating the side of the square, the student recognised that the diameter and the diagonal were the same. these become part of the known knowledge and can be ‘seen’ at a glance. the next question sought to understand the students’ thinking when they are faced with problems that require solutions that are not immediately discernible from the question itself. the data collected did not seek the solution itself but delved deeper into the way the students were thinking. the question itself was: ‘in a quadrilateral two angles are equal. the third angle is equal to the sum of the two equal angles. the fourth angle is 60° less than twice the sum of the other three angles. find the measures of the angles in the quadrilateral.’ before they could solve the problem, they were asked the following set of questions: when you read through this problem, describe exactly what happened in your head. give as much detail as possible. when you see a problem such as this one, list the sequence of activities you engage in, in order to solve it. what do you really do? what do you consider to be the most important part in this solution process? these questions were important in order to understand how students first react to problems and what they actually think and do during the problem-solving process. so, they were given the problem and were then asked to answer the questions below. when you read through this problem, describe exactly what happened in your head: give as much detail as possible there were various responses to this question but they generally captured the idea that the students created mental images and manipulated these images to firstly understand the problem and then find the solution. one student stated that in his head he tried ‘to construct the figure and to guess what could be the angles’. another student related her experience by saying that she found: ‘the first three angles easily but the fourth angle i found it difficult to understand mentally. it then came to my mind that i have to write down the sizes of all the angles so that i will arrive at the answer.’ she was alluding to an internal and external process. a number of students indicated that their first reaction was to visualise the diagram. these are examples of student responses: i instantly visualized the problem and tried to picture what kind of quadrilateral it could be. an image of a quadrilateral was visualized, then i started to add each bit of information to that quad. i drew a diagram in my mind and i talked to myself. automatically the structure appeared in my mind, all information was stored together there is sufficient evidence here to show that the students’ initial reaction was to create mental images and then try to fit the given information into that image. some students simply began by drawing a physical diagram: ‘i took the pen and drew a 4-sided figure and put the labels a, b, c and d on the vertices and named the 2 angles as unknown “x”’. in all of these cases an image played a crucial role in the solution. when you see a problem such as this one, list the sequence of activities you engage in, in order to solve it: what do you really do? the general practice was to draw a diagram from the information that the mind had processed and then to fill into the diagram the given information. other students would begin by underlining the key words in the problem so that they could focus on the key words and concepts directly when attempting the solution. one student provided clear details by saying: ‘after thinking and drawing my diagram, the first thing i did was to let the two equal angles equal to x. i used that to find the other angles in terms of x. i added the angles of this quadrilateral and equated them to 360°. then i solved for x. once i found the value for x, i solved for each angle.’ another student indicated that he would: ‘read each sentence individually. then draw a rough sketch. then start filling in information which tells me what i have and what i need. then proceed to get the information i need. then solve.’ the common theme in the process of solving was to draw a diagram and then enter the given information into the diagram (‘you consider how many angles/sides you are given and draw the figure such that everything you are told in the statement appears in your figure’). there is a reason for inserting all the information into the diagram. it seems that their mind processes the information together when they look at the diagram. a trained mind can pick up all the information at one glance like sherlock holmes who would look at a crime scene and have a fairly good idea of what transpired. this comes with practice. what do you consider to be the most important part in this solution process? still intent on probing the students’ thought processes, the next question asked what they considered to be the most important part in their solution process. this question also evoked a myriad of responses. many indicated that reading the question carefully was important; others felt that understanding the question was crucial. only seven students felt that drawing a diagram was most important. when other students were probed about not finding the diagram to be important some replied: but i already had the diagram in my head. this was easy to see. the image was already determined. thinking came with the diagram. i did not need to draw, it automatically appeared in my mind. all of that happened inside here [tapping the head]. all of these responses pointed to the fact that visualisation played a central role in the problem-solving process. the next question sought to establish the mental conversations they engaged in with themselves. the exact details were not conclusive but a few examples are provided. one student stated that his internal conversation contained questions such as: ‘what figure am i working with? will this figure give me correct answers? does this figure include all aspects of the statement given? can i use x as a variable?’ these are just some of the questions that he asked himself. another student indicated similar thoughts: ‘what information is given? what do i need? how do i draw this? what is required of me to solve this problem?’ a third student’s mental conversations reflect the following thoughts: ‘can i find two unknown angles? i can make one of them equal to x. sum of angles of a quadrilateral is equal to 360°. i can add all angles and those will equal to 360°’. these conversations seem to concentrate on what they have, what can they do and how they could manipulate the information given. much more needs to be done in this area. discussion the evidence shows that while visuals are important in mathematics, they must be used appropriately and in conjunction with previously acquired knowledge. overconfidence in drawing conclusions too quickly from illusory diagrams poses a problem. students must be taught to explore what is actually given in the context of the visual. the words in the problem must harmonise with the diagram given or created physically or mentally. the second finding shows that if students draw diagrams in order to solve problems then they are likely to solve the problem. there are two possible reasons for this. in drawing the diagram deeper meaning is extracted from the problem itself. if the problem is better understood then a solution is likely to be found. the second possibility is that diagrams provide a global picture of the problem which can be mentally manipulated by the student in order to explore different possibilities quickly. this is akin to using computer software to produce many examples in a short time unlike when using pencil and paper methods. the third finding relates to the way students attempt to link given information to previously learned mathematical knowledge. if the link is not explicit, students struggle to see the connection. they are able to manipulate images in their minds but often they do not extend themselves far enough to see new possibilities. this was evident when they could not determine which triangles could be divided into two isosceles triangles. they were getting close to the solution and with a little more reflection they probably would have found the solution, but they would just stop without exploring the problem further. conclusion and implications the final finding gave us a glimpse of the conversations that some of the students had with themselves. there was an indication that these students devised plans in their heads, organised their information in their heads and worked towards a solution in their heads. but, most importantly, they constructed and transformed diagrams in their heads. they were able to form pictures in their minds before they could draw them on a page. in some instances they were seeing the solution before writing the solution. despite this finding, it was evident that many students were still hesitant to draw physical diagrams or even formulate one in their minds. this is similar to samkoff et al.’s (2012, p. 50) declaration that ‘researchers have noted that students are often reluctant to use diagrams, even for problems where their use might be highly productive’. there is probably a lack of practice in the use of diagrams and this probably stems from the underusage of this heuristic tool when solving problems in school. cooper and alibali (2012, p. 281) argued that ‘many studies also suggest that the usefulness of visual representations depends on students’ ability levels’. while this may be the case, i believe that practice and prior knowledge are key factors. with the increasing use of technology, diagrams become an even more priceless tool. jones (2013) found that: diagrams are invaluable in aiding the teaching and learning processes; it is also clear that the processes involved in using diagrams are surprisingly complex this points to the need for more research into diagrams in the teaching and learning of geometry. (p. 41) perhaps, we need to acknowledge that diagrams are critical in the learning process because they offer great prospects for establishing understanding as the student engages in commognitive discourses with the self. in thinking and reflecting, meaning can be extracted from physical or mental images. more research must be conducted to explore the relationships between the students’ creation of mental and physical images, the way they communicate these ideas with themselves and how this leads to deeper understanding. acknowledgements competing interests the author declared that no competing interest exists. authors’ contributions i declare that i am the sole author of this research article. ethical considerations all the necessary ethical considerations were included in our application through the ethics office at ukzn. funding information this research received no specific grant from any funding agency in the public, commercial or not-for-profit sectors. data availability data sharing is not applicable to this article as no new data were created or analysed in this study. disclaimer the views and opinions expressed in this article are those of the author and do not necessarily reflect the official policy or position of any affiliated agency of the author. references chi, m.t.h. 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(2004). visualization in mathematics: claims and questions towards a research program. paper presented at the 10 international congress on mathematics education. copenhagen, cambridge: cambridge university press. retrieved from http://www.math.yorku.ca/~whiteley/visualization.pdf reviewer acknowledgement pythagorashttp://www.pythagoras.org.za acknowledgement to reviewers the pythagoras editorial team recognises the value and importance of peer reviewers in the overall publication process – not only in shaping individual manuscripts, but also in shaping the credibility and reputation of our journal. we are committed to the timely publication of all original, innovative contributions submitted for publication. as such, the identification and selection of reviewers who have expertise and interest in the topics appropriate to each manuscript are essential elements in ensuring a timely, productive peer review process. we would like to take this opportunity to thank the following reviewers who participated in shaping this issue of pythagoras: andile mji anthony essien belinda huntley caroline long dirk wessels erna lampen faaiz gierdien hamsa venkat we appreciate the time taken to perform your review successfully. hugh glover johan meyer john malone michael de villiers percy sepeng sharon mcauliffe willy mwakapenda in an effort to facilitate the selection of appropriate peer reviewers for pythagoras, we ask that you take a moment to update your electronic portfolio on www.pythagoras.org. za for our files, allowing us better access to your areas of interest and expertise, in order to match reviewers with submitted manuscripts. if you would like to become a reviewer, please visit the journal website and register as a reviewer. to access your details on the website, you will need to follow these steps: 1. log into the online journal at http://www. pythagoras.org.za 2. in your ‘user home’ [http://www.pythagoras. org.za/index.php/ pythagoras/user] select ‘edit my profile’ under the heading ‘my account’ and insert all relevant details, bio statement and reviewing interest. 3. it is good practice as a reviewer to update your personal details regularly to ensure contact with you throughout your professional term as reviewer to pythagoras. please do not hesitate to contact me if you require assistance in performing this task. rochelle flint submissions@pythagoras. org.za tel: +27 (0)21 975 2602 fax: +27 (0)21 975 4635 page 1 of 1 microsoft word 64 front cover final.doc pythagoras 64, december, 2006, pp. 37-44 37 developing mathematical literacy through project work: a teacher/teaching perspective1 renuka vithal faculty of education, university of kwazulu-natal vithalr@ukzn.ac.za the implementation of the new mathematical literacy curriculum in south africa is assuming several different conceptions of mathematics and therefore also being realised through a range of different pedagogies. in this paper i begin from a particular privileging of a critical perspective in mathematics education, which i argue is one (among others) of the forces shaping the new south african curriculum reforms, particularly the mathematical literacy curriculum. if so, then the case for a specific pedagogy, that of project work, can be shown to support the development of a mathematical literacy from a critical perspective. in this paper a particular set of conceptual tools, principles and practices associated with project work, as developed in the scandanavian context but researched in south africa, are elaborated from the perspective of teachers/teaching of mathematical literacy. 1 a version of this paper was presented as a keynote address: vithal r. (2006) developing mathematical literacy through project work, in harikirat s. et al. (eds) shaping the future of science, mathematics and technical education. proceedings of 11th annual conference of the sultan hassanal bolkiah institute of education, universiti brunei darussalam, 22-25 may, pp. 117-128. mathematical literacy from a critical perspective arguably, many of the intentions in the critical, developmental and learning outcomes enunciated in the new mathematical literacy south african curriculum policy document (department of education, 2003) resemble those embodied in the broad emerging literature that explores the social, cultural, political, historical and economic dimensions of mathematics education, both as a field of practice and as a field of study. that is to say that a particular interpretation being made in this paper of mathematical literacy is one that is more closely aligned to that from a critical perspective. the notion of a “critical mathematics literacy”, first used in the early ’80s, sought to link in a more direct way concerns in mathematics education with a critique of society (see for example frankenstein, 1987; 1990). but here too the concepts and terms are contested as is the very idea of combining mathematics with literacy. skovsmose (1994), for instance, has coined the term mathemacy, which he argues brings together both a democratic competence and a critical competence in mathematics education. mathemacy as an integrated competence “implies that the guiding principles for mathematics education are not any longer to be found in mathematics but in the social context of mathematics” (skovsmose, 1994: 117). one of the means for achieving this kind of mathematical literacy that has been particularly well demonstrated in the scandanavian context is through project work (see also mellin-olsen, 1987). it is this specific theorising and practices of project work, but taught and researched in the south african context in my own work with student teachers and learners, that inspires its exploration in this paper as a means for achieving a particular kind of mathematical literacy – one that allows for a critique of society in strengthening democracy. such a conception of mathematical literacy seeks to provide students with opportunities to begin to “read the world (understand complex issues involving justice and equity) using mathematics, to develop mathematical power, and to change their orientation toward mathematics” (gutstein, 2003: 37). whatever the terms – numeracy (in the uk see brown, 2003) or quantitative literacy (in the usa see steen, 2001) – an emerging concern is with linking knowledge, skills, attitudes and values produced through mathematics literacy to those needed for effective participation in democratic life in the 21st century. developing mathematical literacy through project work: a teacher/teaching perspective 38 project work project work is a well established pedagogy but it may be engaged from many different theoretical orientations and practised in different ways within mathematics education (see as an example boaler, 1997). according to the third international mathematics and science study (timss, 1999) a substantial number of south african teachers claim to engage project work (mullis et al., 2000). the mathematics teachers of approximately 40% of learners surveyed (the third highest of 38 countries), stated that they sometimes or always assigned mathematics homework based on small investigations and individual or small group investigations compared to the international average of 18%. in this conception of project work, it is primarily used as a continuous assessment practice rather than as a classroom pedagogy per se. that is, it usually involves a small self-study on a topic or problem selected by the teacher to be done outside class-time. however, in this paper quite a different conception of project work is being explored. the particular conception of project work as a substantial classroom pedagogy, specifically from a critical perspective, is perhaps most well developed both in theory and practice in the scandinavian context (see olesen & jensen, 1999). problem orientation, participant directed, interdisciplinarity, and exemplarity are central conceptual tools in this practice of project work. this more “progressive” and “open” project work pedagogy, in which learners take greater responsibility for their learning while working on problems that are jointly negotiated with their teachers, has been explored and researched with respect to all levels of school mathematics (skovsmose, 1994; christiansen, 1996; nielsen et al., 1999; vithal, 2003; 2004; 2005) and also university mathematics (vithal, christiansen & skovsmose, 1995; niss, 2001). in this paper this set of conceptual tools and practices for developing a mathematical literacy from a critical perspective is extended to include assessment and practical organisation. these are argued from a teacher/teaching perspective as being necessary (but by no means sufficient) to sustain and make project work viable in a mathematics classroom. while this practice of project work offers a means for a different kind of mathematical literacy to develop, it also produces particular tensions. each of these six conceptual tools and their associated practices are discussed in the rest of this paper with reference to a range of projects undertaken in diverse south african mathematics classrooms. problem orientation a key foundational principle in this conception of project work is that the curriculum experience is not organised around the structures of the discipline of mathematics but rather around some critical problem, question or theme related to the context of the school, community or broader society and that is deemed important by the learners. learner interest in the problem is therefore a critical feature of learning because it shapes how they would invest in dealing with the problem and take ownership (skovsmove & nielsen, 1996). in her work with young learners in grade 6, paras (1998) reports how learners chose a topic of building a fence around a section of the school ground to prevent their balls from rolling down an embankment during play time over a project on solving the problem of a traffic jam each morning at the entrance to the school grounds. they argued that the second was an adult problem and not really a concern for them. however, the kinds of problems that are chosen need not be rooted in reality. projects can also be selected for challenging learners’ imagination such as creating a dream playground which was implemented in one school with poor facilities for the learners (vithal et al., 1997). learning as action, which underpins project work, implies that students must own the reason for learning and cannot be forced to be critical. in practice, according to skovsmose (1994), this means that any topic, issue or problem that is selected has to be known by the broad range of learners; belong to their daily life; be interesting and important to learners; can be described in nonmathematical terms; allow them to develop a personal in-depth understanding of the multifaceted aspects of the chosen focus; and allow mathematical ideas, skills and concepts to emerge. however, inherent in this freedom to choose is the tension of the authority of the teacher to teach mathematics. for example, learners often chose topics in which it was difficult to see any mathematical potential, such as interviewing sports personalities. even when topics were chosen with possible mathematical potential these would be subverted by learners away from mathematics and the teacher’s intervention was required to keep the mathematics in view. so the freedom to choose exists in a kind of contradiction but also cooperates with the structure of the discipline and the renuka vithal 39 classroom and the obligation to teach mathematics, even if it is a contextualised mathematics. teacher interests are constantly in tension with learner interests when classrooms themselves are construed as democratic spaces. for example, a project that learners chose in order to question the school about inadequate and under-utilisation of sporting facilities was (re)directed by the teacher toward a survey. the survey, however, became a minor aspect as learners pursued other activities to make a case for their complaint. but in this the “failure” of a project can also be seen as its “success” if a broader view of learning mathematics is taken. what learners learn through constituting the class as a democratic space is how they can have voice, and how that voice could challenge authority to meet their educational needs. an important aspect that emerges here is in how freedom and democracy coexist with structure and authority in any classroom that enacts this kind of approach to project work (vithal, 2003). these play out in a productive tension that provides learners with opportunities to understand societal conditions and features that they are expected to become aware of in acquiring a mathematical literacy. participant directed the notion of being participant directed refers to the democratised relationship and responsibilities for the work in the project as being jointly undertaken by both teachers and learners. learners and teachers share ownership for the processes, activities and the direction the project takes. in this co-construction of the project is a recognition of different knowledge, skills, abilities and vested interest between learners and teachers, and among the learners themselves involved in any one project. typically project work is undertaken in groups and the diversity in the group often means that compromises are made in agreeing to do a particular project. the participant directed nature of project work within groups makes it important for the project to provide entry to and engagement with the problem or topic at a number of different levels and for different capacities and talents. in a project on “how much is spent on our education” learners in any one group all contributed different resources to create a chart representing their different graphs of parental expenditure. they debated over several days how to represent the lists of costs their parents incurred for their schooling in different types of bar graphs or pie graphs. here what was observed was that learners tended to choose what they deem to be easier and hence they chose to draw bar graphs. but what was also observed was a strong teacher intervention. graph drawing was first attempted by learners with reference to a mathematical text that they were given by the teacher. only later was a lesson conducted by the teacher on how to draw these graphs when other groups also chose this mathematical representation for their project – for example a group investigating the problem of too much homework who were listing how they spent their time after school. the participation of the teacher as a facilitator in this conception of project work includes not abdicating her responsibility and obligation to provide access to her knowledge and skills resource base to her learners. whether a single project is undertaken by a whole class with different activities delegated to different groups (as was the case in the fence building project by paras, 1998) or multiple projects are undertaken by different groups in the same class, projects allow for different capacities and talents to find expression in relation to the mathematics. for example, in the sports project investigating the problem of inadequate facilities, some learners interviewed the school principal and physical education teacher about this while others solicited information about sporting preferences among the learners in the class. in being activity or task based and multiple participant directed, the different abilities and interests of learners assisted them to investigate and understand a particular problem from multiple perspectives. but as in all team efforts there were debates in different groups about the differential participation of some members and learners had to deal with members not doing their work, marginalising and silencing. in south africa, given our apartheid history, differences along race, gender and class dimensions often get played out in much sharper tensions as learners from still deeply segregated residential areas with huge differences in access to basic resources in their home settings come together in a classroom to discuss societal problems. the tension here is in how difference is left unchecked to sow divisions and more conflicts or is actively worked with and valued as diversity which is recruited to deeper understanding and possible actions on the inequalities of societies to move toward equity. the contextualised nature of mathematical literacy within a project work pedagogy opens for engagement with this broader integrated mathematical competence. in real terms this means legitimately raising questions in a mathematics classroom, for instance, about the gendered nature of work. in an activity of developing mathematical literacy through project work: a teacher/teaching perspective 40 measuring the ground, observed in two different schools during project work, boys were found refusing to allow girls to use measuring instruments such as builders’ tapes, which required strong intervention from the teacher and a discussion about the south african constitution and the rights of women (paras, 1998; vithal, 2005). the point is that diversity is in fact required to bring to the surface issues of equity so that they can be engaged. mathematical literacy carries the potential for developing critical “pupil-citizens” who gain the experience of working with and in diversity, which can be carried to broader societal settings. learners learn through and with mathematics the “language of decision-making” (nielsen et al., 1999: 15) as a project unfolds and different participants together with the teachers push the project in different directions. to engage effectively in collective work within different kinds of group settings, to organise and manage themselves responsibly, is an important critical outcome identified in the new curriculum. interdisciplinarity definitions of mathematical literacy assume a contextualised mathematics. this contextualisation refers on the one hand to links with reality (whether past, present or future) and assumes on the other hand that any exploration of a topic or theme will require connections with other subject areas. if the starting point for a project is not some narrow mathematical content to be learnt, “interdisciplinarity implies drawing on different disciplines to the extent that they are useful for the treatment of a specific problem” (vithal et al., 1995). the problem oriented aspect of project work which argues for choice in problem selection and seeks societal relevance entrenches interdisciplinarity, which recognises the separation of disciplines but equally takes into account their relations and intersections toward a synthesis of knowledge and skills needed to better understand the same object of study from different perspectives and points of view (bastos & costa, 2000). in a project on redesigning the agricultural science garden and in the fence building project, the teachers saw the need to link with work being done in geography on scale drawings. project work opens real possibilities for linking mathematics in authentic ways to other subjects but this requires effort and facilitation on the part of the teacher. the english language teacher was asked to help the learners in the fence building project to write a letter to the department about assistance with funds. the question of what mathematics should be included in the letter and how it should be represented was discussed. the letter that was eventually chosen came from a group who were considered “weak” performers in mathematics but who had incorporated all the key information. this demonstrates how a broader conception of mathematics and what counts in being regarded as being good at it can be widened in mathematical literacy achieved through project work. integration and progression in the disciplines that takes account of learners’ knowledge and skills at their grade level is made possible in this conception of project work. in both the projects on redesigning the agricultural science garden and the fence building one, learners had to undertake the task of measuring the perimeter in appropriate units, using different measuring instruments in groups, agreeing in the group on discrepancies in readings and measures, recording this information, making conversions and operating on this in decimals and representing the data in scale drawings. different levels of competence and areas of background knowledge and skills were brought together as these tasks were undertaken. in this way project work gives learners with other interests and strengths a different entry into mathematics. teachers remarked on how learners who did poorly in mathematics or showed a fear or dislike for the subject in the traditionally taught mathematics class, were showing improved performance in mathematics in this approach. that is, valuing their other strengths seemed to have a positive impact on their mathematics both in achievement and attitude (vithal, 2004; 2005). the strength of any interdisciplinary approach rests on the strengths of the particular disciplines. once learners decided or were guided into a particular activity and were given a reasonable opportunity to demonstrate their capacity to undertake the task, teachers intervened with direct exposition teaching if necessary. for instance, lessons were taught on how to draw bar graphs when it became clear several learners were struggling with identifying the scales on the xand y-axis and representing the information from their respective projects. the difference in this kind of exposition teaching from traditional teaching is that the teacher is working from learners’ data and the reasons for learning belong much more strongly to learners than to the teachers. a particular challenge in the teaching and learning of a contextualised mathematics, is that the teacher has to ensure that neither learners’ renuka vithal 41 understanding of the mathematics or that of the context gets compromised. when the learners in the fence building project called the different companies on quotes for different types of fencing and forgot to ask for the sales tax (value added tax of 14%), some discussion ensued about whether they should call the companies back to get the information or work it out themselves. a lesson was conducted by the teacher on percentages using the given quotes which allowed them to check the costs and discover their budget was inadequate. this interdisciplinary aspect of project work brings to the fore a different power of knowing mathematics. it makes visible in a direct and explicit way where and how mathematics and its systematisation is used and can be used in relation to the problem or topic being investigated. much of mathematics is taught assuming that learners can and will be able to make these connections. in fact even when explicitly taught, they were often not made. most of the letters that were drafted to send to the department of education for support with funds to build the fence made no reference to the perimeter, metered costs, etc. until this was brought to learners’ attention. when making a case to the principal about the lack or provision and under-utilisation of the existing sporting facilities by the sports project group, the survey of learner preferences was not completed or presented as evidence to counter the school’s reasons of timetabling and space. being able to use mathematics accurately and effectively to improve their own life conditions both inside and outside school add and create much more powerful reasons for learners to learn and succeed in mathematics other than to gain good grades and for access to further educational opportunities. exemplarity the exemplarity principle is central to justifying an alternative project-based curriculum and has been interpreted for a critical mathematical literacy in the work of skovsmose (1994). it deals with a different approach to curriculum selection that argues for “exemplification” rather than “coverage” (bishop, 1988). the main idea is that some larger totality or complexity can be reflected in and comprehended by focusing on some smaller part of it, a particular problem or phenomenon. by working on a single problem, opportunities may be created for developing knowledge, skills, attitudes and values for understanding the topic in a broader context of society and of the disciplines themselves. construing learners as epistemic subjects interested to know their world and to act meaningfully in it to improve their situation, requires that the curriculum is organised so that its subject matter is relevant and meaningful from the perspective of individual experience and relevant to a conceptual understanding of society (rasmussen, 1991). by working on a single problem of “how much money is spent on my education” learners had the possibility to come to know multiple realities. they came to know their family reality of how much parents sacrificed to pay for their school needs, and the reality of the school that was trying to address multiple competing needs for a school library, hall and sporting facilities. however, they left unexplored the national reality of the broader education funding system inherited from apartheid with its still continuing deep inequalities. each project carries a range of potentialities that open for exploring different realities and complexities. in actuality a limited number are engaged by virtue of the directions the different participants pursue and their vested interests. the focus on learner interest may be linked to the notions of foregrounds and backgrounds of the participants in any project, which comes to shape what becomes part of actuality and what remains as potentiality. foregrounds, according to skovsmose (1994), refers to those possibilities that a social situation reveals to an individual. for learners in school the social situation seldom went beyond the totality of the school to national issues and imperatives. in seeking to address the problem of high schools fees, learners did question and come to know the broader system of education funding. however, focusing on potentialities and actualities we can come to understand not what projects fail to do but rather what they do for particular reasons. the extent to which the project is pushed into different realities such as into the broader community or even developing a national or global perspective depends on the dynamics of particular groups engaged in a project and the role and orientation of the teacher. understanding of a particular phenomenon within a broader societal complexity requires group or collective reflection to arrive at multiple vantage points from which to view and investigate the different aspects of a topic and at different levels. in this respect diversity of groups become important. the all girls race and class mixed group dealing with the project on time spent after school, through their joint vested interest to address the problem about being given too much homework, came to see how this issue had dimensions of race, class and gender found in developing mathematical literacy through project work: a teacher/teaching perspective 42 broader society as they interviewed different members of the community and the classroom and compared each other’s data on what they did after school. for instance, they found out that some boys from poor backgrounds did chores after school that were traditionally done by girls. the critical dimension of mathematical literacy carries an imperative to go beyond investigating and understand the injustices and inequalities in society, but incorporates the imperative for the curriculum to provide the tools for meaningful action to improve social conditions related to learners’ lives. learners’ intentions and interests to learn mathematics may be enhanced through an exemplary organising of the curriculum. in this they can come to understand how mathematics can be useful and be used in their interests but also how it can be problematic in dealing with real life problems that have multiple dimensions to them. assessment in south africa project work is currently more commonly used as an assessment strategy (mullis et al., 2000) where it is typically assigned as homework involving small investigations by individuals or small groups. this practice has emerged mainly as a result of the introduction of continuous assessment since the mid ’90s. the pedagogy of project work being presented here is somewhat different, regarded as more “progressive” in advocating a more “openedended”, “learner-centred” and “activity-based” approach. project work makes it possible for learners to see that a problem can be solved in many different ways and gives the possibility for these to be demonstrated with different explanations and justification. the participant directed aspect of project work creates space for negotiating assessments with learners. agreement can be sought on what aspects of the project, at what time and what will be taken as evidence for assessment. in this conception of project, which is typically undertaken over an extended time as a classroombased activity, assessment can be done on an ongoing basis. moreover, it allows multiple assessment formats to be integrated – individual, peer and group. the new south african mathematics curriculum allows for teachers to engage the new assessment policies which encourage many of these practices. if mathematical literacy is conceptualised as an integrated competence then it has to be assessed more broadly both in the different forms of knowing and formats. assessments often indicate to learners what is being valued in the classroom and by teachers. the different activities that make up project work means that many different activities can be valued in this broader connected conception of mathematics. these include for example: the planning done; the workings of the group as a collective; presentations; broad engagement of topic or problem at different levels; writing reports and other documents; content learned; etc. despite these positive possibilities, the issue of assessments in project work has been raised routinely by practitioners who must balance different approaches and assessments imperatives. for instance, the tension between group and individual assessments – the higher weighting and valuing of individual examination scores and the ways in which it is counterproductive for group work is one critique and a tension in project work. the question that came up in the project work is that of why should learners cooperate if what eventually gets most valued is individual performance. learners within school compete for marks, and for access to further educational opportunities. another difficulty that may emerge is that the interdisciplinarity aspect requires teachers to have a much broader knowledge of not only mathematics but also have some understanding of societal issues and a range of other disciplines to be able to guide and direct learners to the relevant sources. practical organisation in this much broader conception of project work involving a “research-like” process, learners are given greater independence in their learning to identify a problem, collect information, analyse and draw conclusions, write reports and present their findings and position on the matter under investigation. the teacher’s role becomes one of guiding the process in the role of a facilitator or supervisor. given that this is the mathematics classroom and learners are expected to learn this mathematics, even as they may go into other subject matter, the teacher cannot renege on the didactical contract between herself and learners to ensure that the necessary mathematics is learnt, albeit a contextualised mathematics. however, the teacher is also a student in listening and learning about what knowledge and skills and values each learner brings into the project by virtue of their having lived particular lives in particular contexts. in the education project, participants became aware of the huge costs in funds and time some parents incurred in travel to send their children to a renuka vithal 43 particular school and the burdens this created for learners who had to rise two hours earlier than other learners, while in the homework project girls from poorer families showed in the calculations done how much time they spend after school in family chores of childminding, cooking, etc., while others participate in extracurricular activities of music and dance lessons. project work processes and activities give teachers access to background knowledge as well as foreground information, about the dreams and aspirations of learners that can guide their intervention in particular projects. the interdisciplinary nature of project work creates opportunities and imperatives for groups of teachers to work together in small teams as collectives to deal with different content and skills needed. depending on the kind of project, the mathematics teachers interacted with language, science and geography teachers. implementation of this kind of project work could require reorganising some of the practical features of classroom life or in the functioning of schools to facilitate the activities of the project. for instance, a group of teachers could agree to pool their teaching periods to create block sessions for extended work on the project. further, the classroom can be transformed to simulate a particular environment such as a newspaper office if learners are involved in developing a newsletter to have the experience of being journalists (nielsen et al., 1999). since in this conception of project work, real outcomes in changing some aspect of learners’ lives are aimed for, it also means the teacher has to pave the way for learners to meet with some authorities such as the school principal or a local councillor. in the fence building project, the teacher sought to invite a land surveyor to show how grounds are measured and by whom; and in a project on electricity provision the plan included involving the electricity company to explain the different electricity systems being implemented and their different costs. such projects can also include excursions to different places to give learners first hand real life experiences and interaction with people who do the actual kinds of work they are investigating in the project. conclusion the presentation of what learners uncover through the activities of the project to people it most affects and involves, makes the mathematics powerful in a different way from that experienced in the traditional classroom only. even if the entire curriculum is not organised through project work, as is the case in some institutions such as aarlborg and roskilde universities in denmark (niss, 2001; olesen & jensen, 1999; vithal et al., 1995), the opportunity to engage in one project, chosen and undertaken in a way that realises the exemplarity principle, makes it possible for learners to experience and know both the power and limitations of mathematics. through project work they can learn how to work in diverse groups inside schools and in broader society; and come to see how schools in general but mathematics teaching and learning in particular can provide the curriculum tools to act in their own interests and those of their families and communities to address societal injustices and inequalities. the conceptual tools, principles and practice discussed here in this form of project work are offered in the spirit of inspiring teachers to experiment and explore in their mathematics literacy classrooms. to take some calculated risks to evolve new and different ways of teaching mathematics to achieve the policy intentions of the new mathematics literacy curriculum that invites for creative possibilities that could be shared and built on. references bastos, r., & costa, f. (2000). transdisciplinarity and curriculum organisation. in j.p. matos & m. santos (eds.), proceedings of the second international mathematics education and society conference, 26-31 march (pp.169-179). portugal: university of lisbon. bishop, a.j. (1988). mathematical enculturation: a cultural perspective in mathematics education. dordrecht: d. reidel publishing company. boaler, j. (1997). experiencing school mathematics: teaching styles, sex and setting. buckingham: open university press. brown, m. (2003). research and national policies in primary numeracy. in b. putsoa, m. dlamini, b. dlamini, & v. kelly (eds.), proceedings of the 11th annual conference of the southern african association for research in mathematics, science and technology education research, 11-15 jan. swaziland: waterford kamhlaba. christiansen, i.m. (1996). mathematical modelling in high school: from idea to practice. revised doctoral dissertation. denmark: aalborg university. department of education (doe). (2003). national curriculum statement grades 10-12 (general) developing mathematical literacy through project work: a teacher/teaching perspective 44 mathematical literacy. pretoria: department of education. frankenstein, m. (1987). critical mathematics education: an application of paulo freire’s epistemology. in i. shor. (ed.), freire for the classroom: a sourcebook for liberatory teaching (pp. 180-210). new hampshire: boyton and cook publishers. (first appeared in 1983 in the journal of education, 165(4), 315339.) frankenstein, m. (1990). critical mathematical literacy. in r. noss, a. brown, p. dowling, p. drake, m. harris, c. hoyles, and s. mellinolsen (eds.), political dimensions of mathematics education: action and critique. proceedings of the first international conference (pp.106-113). london: institute of education, university of london. gutstein, e. (2003). teaching and learning mathematics for social justice in an urban, latino school. journal for research in mathematics education, 34(1), 37-73. mellin-olsen, s. (1987). the politics of mathematics education. dordrecht: d reidel publishing company. mullis, i.v.s., martin, m.o., gonzalez, e.j., gregory, k.d., garden, r.a., o’connor, k.m., chrostowski s.j., & smith t.a. (2000). timss 1999 international mathematics report: findings from iea’s repeat of the third international mathematics and science study at the eighth grade. chestnut hill: international study center, boston college. nielsen, l., patronis, t., & skovsmose, o. (1999). connecting corners: a greek-danish project in mathematics education. aarhus: forlaget systime. niss, m. (2001). university mathematics based on problem-oriented student projects: 25 years of experience with the roskilde model. in d. holton (ed.), the teaching and leaning of mathematics at university level: an icmi study. dordrecht: kluwer academic publishers. olesen, h.s., & jensen, j.h. (eds.) (1999). project studies – a late modern university reform? copenhagen: roskilde university press. paras, j. (1998). improving the playground: a fence-building project in mathematics. pythagoras, 46/47, 57-62. rasmussen, p. (1991). the role of the first year in the educational structure and pedagogy in aalborg university. in a. lorentson & a. kolmos (eds.), quality by theory and practice of higher education: proceedings from the first european conference on the first year experience in higher education. tnp-series no. 6. denmark: aalborg university. skovsmose, o. (1994). toward a critical philosophy of mathematics education. dordrecht: kluwer academic publishers. skovsmose, o., & nielsen, l. (1996). critical mathematics education. in a.j. bishop, m.a. clements, c. keitel, j. kilpatrick & c. laborde (eds.), international handbook of mathematics education. dordrecht: kluwer academic publishers. steen, l.a. (2001). (ed.) mathematics and democracy: case for quantitative literacy. usa: national council on education and the disciplines, the woodrow wilson national fellowship foundation. vithal, r. (2003). in search of a pedagogy of conflict and dialogue for mathematics education. dordrecht: kluwer academic publishers. vithal, r. (2004). devan; mathematics; and project work. south african journal of education, 24 (3), 225-232. vithal, r. (2005). gender justice, human rights and measurement in the mathematics classroom. international journal of mathematical education in science and technology, 36(8), 827-841. vithal, r., paras, j., desai, s., zuma, z., samsukal, a., ramdass, r., & gcashbe, j. (1997). student teachers doing project work in primary mathematics classrooms. in p. kelsall & m. de villiers (eds.), proceedings of the third national congress of the association for mathematics educators of south africa. july 7 – 11. durban: university of natal. vithal, r., christiansen, i., & skovsmose, 0. (1995). project work in university mathematics education. educational studies in mathematics, special issue on advanced mathematical thinking. 29, 199-223. pyth_v42(1)_2021_contents.indd http://www.pythagoras.org.za open access table of contents original research early career teacher’s approach to fraction equivalence in grade 4: a dialogic teaching perspective benjamin shongwe pythagoras | vol 42, no 1 | a623 | 15 november 2021 original research insights into the reversal error from a study with south african and spanish prospective primary teachers calos soneira, sarah bansilal, reginald govender pythagoras | vol 42, no 1 | a613 | 25 november 2021 original research constructing mental diagrams during problem-solving in mathematics vimolan mudaly pythagoras | vol 42, no 1 | a633 | 29 november 2021 original research novice and expert grade 9 teachers’ responses to unexpected learner offers in the teaching of algebra julian moodliar, lawan abdulhamid pythagoras | vol 42, no 1 | a624 | 15 december 2021 original research reasoning under uncertainty within the context of probability education: a case study of preservice mathematics teachers samah g.a. elbehary pythagoras | vol 42, no 1 | a630 | 17 december 2021 57 71 82 90 103 original research the contribution of online mathematics games to algebra understanding in grade 8 tichaona marange, stanley a. adendorff pythagoras | vol 42, no 1 | a586 | 29 july 2021 original research flexible teaching of mathematics word problems through multiple means of representation matshidiso m. moleko, mogege d. mosimege pythagoras | vol 42, no 1 | a575 | 10 august 2021 original research professional development for teachers’ mathematical problem-solving pedagogy – what counts? brantina chirinda pythagoras | vol 42, no 1 | a532 | 25 august 2021 original research design principles to consider when student teachers are expected to learn mathematical modelling rina durandt pythagoras | vol 42, no 1 | a618 | 29 september 2021 original research grade 9 learners’ understanding of fraction concepts: equality of fractions, numerator and denominator methuseli moyo, france m. machaba pythagoras | vol 42, no 1 | a602 | 21 october 2021 1 9 19 31 44 page i of i table of contents vol 42, no 1 (2021) issn: 1012-2346 (print) | issn: 2223-7895 (online)pythagoras original research bridging powerful knowledge and lived experience: challenges in teaching mathematics through covid-19 karin brodie, deepa gopal, julian moodliar, takalani siala pythagoras | vol 42, no 1 | a593 | 13 august 2021 original research reflecting on dilemmas in digital resource design as a response to covid-19 for learners in under-resourced contexts pamela vale, mellony h. graven pythagoras | vol 42, no 1 | a599 | 09 november 2021 118 132 reviewer acknowledgement pythagoras | vol 42, no 1 | a670 | 21 december 2021 149 vol 42, no 1 (2021) focussed collection: teaching and learning mathematics during the covid-19 abstract introduction conceptual framework methodology findings and results discussion conclusion acknowledgements references about the author(s) rina durandt department of mathematics and applied mathematics, faculty of science, university of johannesburg, johannesburg, south africa citation durandt, r. (2021). design principles to consider when student teachers are expected to learn mathematical modelling. pythagoras, 42(1), a618. https://doi.org/10.4102/pythagoras.v42i1.618 original research design principles to consider when student teachers are expected to learn mathematical modelling rina durandt received: 09 apr. 2021; accepted: 30 july 2021; published: 29 sept. 2021 copyright: © 2021. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract this article sets out design principles to consider when student mathematics teachers are expected to learn mathematical modelling during their formal education. blum and leiß’s modelling cycle provided the theoretical framework to explain the modelling process. learning to teach mathematical modelling, and learning to solve modelling tasks, while simultaneously fostering positive attitudes, is not easy to achieve. the inclusion of real-life examples and applications is regarded as an essential component in mathematics curricula worldwide, but it largely depends on mathematics teachers who are well prepared to teach modelling. the cyclic process of design-based research was implemented to identify key elements that ought to be considered when mathematical modelling is incorporated in formal education. fifty-five third-year student teachers from a public university in south africa participated in the study. three phases were implemented, focusing firstly on relevance (guided by a needs analysis), secondly on consistency and practicality via the design and implementation of two iterations, and lastly on effectiveness by means of reflective analysis and evaluation. mixed data were collected via a selection of qualitative instruments, and the attitudes towards mathematical modelling inventory. through content analyses students’ progress was monitored. results analysed through spss showed significant positive changes in their enjoyment and motivation towards mathematical modelling. student teachers require sufficient resources and opportunities through their formal education to participate regularly in mathematical modelling activities, to develop competence in solving modelling tasks, and to augment positive attitudes. this study adds value to the global discussion related to teachers’ professional development regarding mathematical modelling. keywords: attitudes towards mathematical modelling; design-based research; design principles; formal education; learning mathematical modelling; mathematical modelling; professional development; student mathematics teachers introduction student mathematics teachers, who will also teach mathematical modelling in their future professional role as teachers, should purposefully and strategically be prepared for this task because both teachers and students find the topic cognitively demanding. a well-prepared teacher should include real-life examples and applications in mathematics teaching as an essential component in mathematics curricula to develop problem-solving and cognitive abilities in learners (department of basic education, 2011), but also to help learners to better understand the world, to support mathematics learning (e.g. concept formation), to develop various mathematical competencies and appropriate attitudes, and to contribute to an adequate picture of mathematics (see blum, 2015; wessels, 2017). usually with application questions the focus is to link a mathematical topic to reality, but with real-life questions the focus is to link reality to mathematical topics. the difference in focus is explained through posing the following questions (stillman, galbraith, brown, & edwards, 2007): where can i use this particular piece of mathematical knowledge, as opposed to, where can i find some mathematics to help me with this problem? it is possible for teachers to learn both modelling and the teaching of modelling. borromeo ferri (2018) has emphasised the necessity of trainee mathematics teachers to have vast opportunities to deal with modelling activities on a theoretical and a practical level. biccard (2019) confirmed that teachers should be given opportunities to experience tasks in the role both of learner and teacher. one such possibility is to follow a design-based research methodology (compare biccard, 2019). thus, two primary directives provided a stimulus for this study. firstly, student teachers require adequate preparation in respect of their ‘knowledge-in-action’ of mathematical modelling, while also fostering positive attitudes towards this topic. this ‘knowledge-in-action’ includes competencies both as modellers themselves and as teachers of modelling activities (see blum, 2015; durandt & lautenbach, 2020; ng, 2013), and experiences that provide a stimulus for growth in attitudinal aspects such as motivation, value, self-confidence and enjoyment of mathematical modelling (chamberlin, 2019; jacobs & durandt, 2017). secondly, mathematical modelling has a positive influence on the teaching and learning of mathematics and is included in the current south african mathematics curriculum (department of basic education, 2011). these two directives are not only limited to the south african context but are also relevant for other contexts where the professional development of future teachers is well established and mathematical modelling included in curriculum documents (anhalt & cortez, 2016; blum & borromeo ferri, 2016; tan & ang, 2013). the aim of the study reported in this article was to identify suitable design principles by exposing student teachers (both as modellers and as teachers) to a well-planned set of mathematical modelling activities, while monitoring their development in competencies and their change in attitudes over time. the research questions informing this study were: what design principles can be identified to prepare student teachers for facilitating mathematical modelling activities? what shortcomings and progression in student teachers’ mathematical modelling competencies can be identified during the intervention? what change in student teachers’ attitudes towards mathematical modelling activities can be identified during the intervention? awareness of the ways mathematical modelling can be learnt through incorporating a selection of key elements and fundamental principles in their formal education could contribute to decreasing the cognitive burden of student teachers and cultivating positive experiences. the article will report on relevant theoretical aspects, the design-based strategy (organised through different phases) that formed the situational context to provide an opportunity for student teachers to learn modelling, as well as the results and findings from mixed data explaining their shortcomings, progression and experiences, ending with concluding remarks. conceptual framework the intention is this study was to identify key elements and fundamental principles to guide the integration of mathematical modelling in the formal education of student mathematics teachers in a purposeful and strategic way. two underlining theoretical perspectives are relevant: (1) the mathematical modelling process and the competencies needed for doing mathematical modelling, and (2) key characteristics of a design-based study. learning mathematical modelling mathematical modelling is a cyclic process that describes the translation between reality and mathematics in both directions. this process consists of certain sub-processes as shown, in an ideal-typical form, in figure 1 (the modelling cycle from blum & leiß, 2007). figure 1: the modelling cycle according to blum and leiß (2007). the sub-processes are to construct a situation model from a real-world problem, to generate real-world facts, data and relations, to simplify and structure the data, to mathematise (represent the data mathematically), to work within mathematics, and finally to interpret and to validate the mathematical results with respect to the real-world situation. a variety of illustrations of the modelling process exist in the literature (e.g. stillman, kaiser, & lampen, 2020). the advantage of the illustration from blum and leiß (2007) is that individual steps (1–7) separate the phases of a typical mathematical modelling process. students usually do not follow those steps in linear order when solving modelling tasks, but often ‘bounce’ between them. each step can potentially be a cognitive barrier for students (blum, 2011, 2015; stillman, 2019; stillman, brown, & galbraith, 2010). it seems important to develop mathematical modelling competency and sub-competencies related to the sub-processes of the modelling cycle. mathematical modelling problems are more challenging than traditional word problems, or application problems. word problems (common in school textbooks) usually follow prior instruction for a specific theme in mathematics and relate only to a segment of the real world (comap-siam, 2016). some authors refer to word problems as level 1 modelling problems (tan & ang, 2012). when solving such problems, the modelling process is limited to mathematisation, mathematical procedures following on prior instruction, and direct interpretation. an application problem could be compared to a level 2 modelling problem (tan & ang, 2012) as context and meaning is added to the word problem and required in the solution. these problems are common in assessment tasks at the secondary level. however, students still do not have an opportunity to put their analysis back into a real-world situation (comap-siam, 2016). also, the attempt to contextualise the mathematics may make the mathematical work seem completely irrelevant. mathematical modelling problems, or level 3 modelling problems (tan & ang, 2012), are open ended and generated from a real-world situation that requires the complete modelling process. no clear pathways are suggested in such problems and the modeller will seek for any mathematics to solve the problem and do research about the context and assumptions about the context. these problems might not be common in assessment tasks at the secondary level, but the benefit of incorporating such problems in the mathematics classroom has been established. the following examples from the secondary environment, where students are learning how to write equations and draw graphs of linear functions given the slope and vertical intercept, can explain the difference between those three levels: example of a level 1 problem: emil works at a retail store that pays r100 per week plus r2 for each item he sells. last week he sold 85 items. how much did he earn last week? example of a level 2 problem: emil works at a retail store that pays r100 per week plus r2 for each item he sells. write a linear equation representing the relationship between emil’s weekly income and the number of items he sells. example of a level 3 problem: the holidays are approaching, and your best friend karen would like to make some money to purchase gifts. she found one job that will pay r2 per hour above the minimum wage. another job offers to pay half the minimum wage plus commission in the amount of r2 per item she sells. which job is better? help karen to make the decision. for various examples of modelling problems and an overview of the literature see niss and blum (2020). the teaching and learning of mathematical modelling are difficult mainly due to the cognitive demand of modelling activities. student teachers often have misconceptions about what mathematical modelling entails (anhalt & cortez, 2016), do not always understand the value of such activities, have mixed feelings about the topic and feel under-prepared to teach mathematical modelling (blum & borromeo ferri, 2016; ng, 2013). an example of an exploratory study with a sample of mathematics educators on difficulties in teaching mathematical modelling in france and spain is reported by cabassut and ferrando (2017). results confirmed that most educators were positive about mathematical modelling, although some educators lack self-confidence. most difficulties were experienced in relation to teaching the topic. these difficulties were specific to time (e.g. time to prepare for tasks, time on tasks), students’ involvement, and resources. borromeo ferri (2018) highlights four key competencies that should be developed in teacher education in order for teachers to teach mathematical modelling effectively and appropriately. the competencies are: (1) theoretical competency for practical work, (2) task competency for instructional flexibility, (3) instructional competency for effective and quality lessons, and (4) diagnostic competency for assessment and grading. developing these professional competencies in student teachers might narrow the gap between research and practice in teaching mathematical modelling. one way of narrowing the gap could be through design-based research (dbr) which is linked particularly well with the teaching and learning of mathematical modelling. characteristics of design-based research and design principles dbr is a flexible methodology aiming at improving practices through iterative cycles of analysis, design, development, and implementation via interventions, and it focuses on the collaboration between researchers and practitioners with the intention to extend and identify new design possibilities (abdallah, 2011). to identify key elements and to develop suitable design principles to guide the integration of mathematical modelling into the formal education of mathematics student teachers, abdallah’s view of dbr seemed suitable. seven key characteristics of dbr are widely agreed upon (anderson & shattuck, 2012; reeves & mckenney, 2013; wang & hannafin, 2004) and provided a suitable validation to structure the study: first, dbr’s authentic nature and naturalistic context suggests a grounding in real-world contexts where participants can interact socially with one another, similar to everyday life. second, dbr generates design principles that are contextually sensitive with the intention to improve practice. these principles are informed by theories, literature sources and the contextual variables, and are refined over multiple iterations. third, dbr’s rigorous methodology faces the difficulty of incorporating a variety of factors. thus, the outcome should be the culmination of the interaction between designed interventions, human psychology, personal histories or experiences, and local contexts. to improve the design, data collection and analysis are conducted simultaneously, while credible findings and meaningful solutions to the envisaged problem are produced by the rigour and reflection of the dbr process. fourth, dbr’s fragile, complex and ‘messy’ nature requires mindful consideration from the researcher and a close collaboration between the researcher and participants to continuously refine the flexible design in the context. this close collaboration is possible through an appropriate well-structured or disciplined approach. fifth, dbr functions via unique processes to transcend the local context, for example to refine the design continuously and iteratively (here the focus is on the processes). sixth, dbr transcends the local context and attempts to provide an answer for why the problem occurs, what must be done and what we can learn from this to inform the practice of others (here the focus is on the elements related to the local context). seventh, dbr generates credible evidence and useful knowledge that might also be used in another context. to ensure a purposeful and strategic intervention for student teachers to learn mathematical modelling, it is vital to consider these characteristics of dbr. the product of a dbr study is to contribute to local (analogue to the situation and sample in this study) instructional theory through identifying key elements and fundamental principles. nieveen and folmer (2013) emphasise that design principles, stemming from a dbr study, should provide insight into an educational intervention, and should communicate the purpose and context, key characteristics, design and procedural guidelines, as well as implementation conditions of the intervention. dbr has not often been used in studies to improve the professional development of teachers in mathematical modelling. however, one example of such a study to improve the didactical practices of primary school mathematics teachers through modelling was conducted by biccard (2013). the study reported on in this article provided a unique opportunity to integrate mathematical modelling in the formal education of student teachers. methodology by following a pragmatic approach (creswell, 2013), considering the key characteristics of dbr, the researcher was looking for ‘what works’ in preparing student teachers for mathematical modelling activities. research design with the suitability of dbr having been theoretically established earlier, the research was conducted via three main phases (view figure 2). the first phase (cycles 1–3) was in preparation for the experiment with a focus on needs analysis. this phase was informed by an in-depth literature review, the researcher’s personal experiences, ideas from other practitioners and specialists, and a pilot study. the second phase (cycles 4–6) focused on experimenting in the classroom keeping consistency and practicality in mind. this phase included all mathematical modelling activities structured over two iterations. the third phase (cycles 7 & 8) required a retrospective analysis of all qualitative and quantitative data with the intention to identify key elements and principles to prepare student teachers for mathematical modelling. although the design of the three phases is supported by literature (see reeves & mckenney, 2013), the phases reflected in figure 2 included eight cycles, uniquely crafted for this study, that describe the design step by step. the intention with the eight cycles was to distil contextually sensitive design principles that might be suitable in further studies or in other contexts. both qualitative and quantitative data were collected in cycles 2, 4 and 6. the qualitative data provided information on mainly the shortcomings and progression in student teachers’ mathematical modelling competencies and the quantitative data provided information on the change in student teachers’ attitudes towards mathematical modelling activities based on this exposure. figure 2: the cyclic process of design-based research (dbr) in this study structured over three phases involving eight cycles. a pilot study was conducted (see cycle 2 in figure 2) one year before the implementation of the main intervention. the sample and conditions in the pilot study were similar to the main study. on all occasions the context was a lecture room, in a real-life setting. the intention with the pilot study was to test the validity of the data collection instruments, to confirm the identified needs of participants and to improve the researcher’s experience with a dbr strategy. findings from the pilot study improved the preparation for the first iteration (cycle 4 in figure 2), and some of its results have been published (see durandt & jacobs, 2014; jacobs & durandt, 2017). similarly, findings from cycle 4 improved the preparation for the second iteration in cycle 6 and some results have been published (see durandt & lautenbach, 2020). each data collection cycle (2, 4 & 6) was followed by a cycle where the results were conceptualised and draft design principles were identified and refined (cycles 3, 5 & 7). through this cyclic process the findings from one cycle were incorporated in the design of the next cycle ensuring a continuous refinement of the key elements and principles relevant to the learning of mathematical modelling in this context. the pilot study (cycle 2) consisted of one session of 90 minutes and participants acted as modellers. iteration 1 (cycle 4) was similar to the pilot study. in iteration 2 (cycle 6) participants acted as both modellers and teachers of modelling. four sessions of 90 minutes each were planned, two sessions allocated to each role. rigour was maintained in all research phases by following all activities as planned and a methodological specialist attended the sessions. multiple data collection instruments were used throughout the research phases to identify key elements and fundamental principles relevant when student teachers learn mathematical modelling. these instruments are related to learn the cyclic process of mathematical modelling (worksheet and poster documents), to identify shortcomings and capture experiences (open-ended questionnaire), and to determine student attitudes towards mathematical modelling (atmmi). for the focus of this article, only data collected from cycles 4 and 6 will be reported on, which mainly informed the retrospective analysis. iterations in cycles 4 and 6 (see figure 2) participants were exposed to modelling activities that ranged from solving easier tasks (like an application problem or level 2 modelling example; see above for an explanation) to more challenging, open-ended and complex modelling tasks (like a level 3 modelling example; see above for an explanation and below for an example). the intention with the range in complexity of tasks was to develop mathematical modelling competencies in student teachers, for them to learn sub-competencies and develop more positive attitudes towards such activities over time. the tasks asked participants to be modellers themselves, but also teachers of modelling. while acting as modellers themselves, they were expected to work their way through the modelling cycle in groups and record their work on a predesigned worksheet. these activities were included so that student teachers might develop both theoretical and task competencies as described by borromeo ferri (2018). one such example is discussed in a later section and for other examples of modelling activities and tasks used in the study, see durandt (2018). in groups, they participated in discussions, designed their own modelling tasks, reflected on and evaluated their own and others’ work, prepared posters and presented proposed solutions. in these activities, participants acted in both roles, as modellers themselves and also as teachers, with the intention to develop instructional and diagnostic competencies as emphasised by borromeo ferri. participants a population of 55 third-year student mathematics teachers from a large public university in south africa participated in the main study that was completed in 2018. the participants did not share their mathematics class with other mathematics students (like engineers or computer science students) and this separation was automatically done by the university system. the participants were arranged in the same 10 comparable groups in all iteration phases by using purposive sampling procedures. relatively small groups of four to six members were formed, with each group containing at least a high, a moderate and a low achiever. the achievement level was determined by their formal course marks in mathematics. the group selection was mandatory (determined by the researcher) and purposeful in planning for multiple dependent variables such as the complexity of the modelling activities, student teachers’ initial or limited exposure to mathematical modelling tasks, and the envisaged and expected valuable collaboration among participants. instruments to learn mathematical modelling student teachers were exposed to mathematical modelling tasks to learn modelling themselves, to develop mathematical modelling competencies and sub-competencies and with the intention to develop positive attitudes towards the topic. one example is the ‘location problem’ (view figure 3) where participants (in groups) were asked to use the data and make a recommendation to the department of town and regional planning on the best location for a day-care centre for toddlers. figure 3: real-world data from the ‘location problem’ to inform a recommendation on the best location for a day-care centre for toddlers. participants were expected to work through the processes of the modelling cycle, record their work on a predesigned worksheet with a four-step plan (view table 1), similar to the idea of a plan used in other studies like the ‘solution plan’ used in the disum study (schukajlow, kolter, & blum, 2015) and aligned with the main processes stemming from the modelling cycle (view figure 1). table 1: elements of the predesigned worksheet: four-step plan. after solving the task, all groups participated in discussions, both in groups and as a whole class. then, each group prepared and presented a poster to demonstrate visually how they used the modelling cycle to find a possible solution to the problem. see figure 4 for an example from group 9 after solving the ‘location problem’ in iteration 2. figure 4: poster example from group 9 after solving the ‘location problem’ in iteration 2 and showing how they moved through the processes in the modelling cycle. they also evaluated their own work and the work of others by using an evaluation sheet (selecting one of three criteria: high, medium, low). later, at the end of each iteration, participants reported their experiences by answering an open-ended questionnaire individually. the questionnaire included a section on biographical information of participants as well as a section on their perceptions of the mathematical modelling experience and support they might require. one example of a question from the questionnaire is: how can the university further support you (during your teacher training) in becoming an even more effective teacher of mathematical modelling (make concrete suggestions)? the qualitative data collection instruments (worksheets, poster and open-ended questionnaire) were designed at the end of cycles 1, 3 and 5. content analysis methods were used to analyse the data (saldaña, 2016). students’ worksheet and poster documents were marked according to a framework deduced from the sub-processes of the modelling cycle (see figure 1) and compared through the cycles (see table 2). data collected through the open-ended questionnaire were analysed according to themes, and further separated in categories and sub-categories (see figure 5). this was done via pen and paper and using atlas.ti software. strategies to maintain the trustworthiness of the qualitative component of this study included a thick description of the methodology (in section 3) and an external subject specialist confirming coding categories and qualitative findings (as suggested by creswell, 2013). figure 5: reflective analysis and evaluation of all qualitative data: themes, categories, and sub-categories. table 2: overall mean scores as percentages related to developing competencies (theme 1) collected in cycles 4 and 6 via worksheets and poster presentations. instrument to determine attitudes the attitudes towards mathematical modelling inventory (atmmi) was used to gain information individually regarding student teachers’ attitudes towards mathematical modelling at the end of each iteration. the atmmi, adapted from schackow (2005) that followed the original atmi instrument from tapia and marsh (2004), is a locally tested instrument that consists of 40 likert-scale items arranged from ‘strongly disagree’ to ‘strongly agree’ (5 possible responses). the items were grouped in four sub-scales: enjoyment (10 items to determine whether mathematical problem-solving and modelling challenges were considered enjoyable for participants), value (10 items, to determine whether mathematical modelling knowledge and skills were considered worthwhile and necessary for participants), self-confidence (15 items, to determine the expectations about doing well in respect of mathematical modelling, and how easily modelling was mastered by participants) and motivation (5 items, to determine the desire of participants to learn more about mathematical modelling and to teach the topic). data were collected on paper at the end of iteration 1 (cycle 4) and at the end of iteration 2 (cycle 6). the statistical software package for the social sciences (spss, version 24) was used to analyse the data. internal consistency was confirmed by acceptable cronbach’s alpha coefficients (in all sub-scales > 0.8). the results were similar to previous recorded alpha values by schackow and tapia and marsh. content validity was confirmed by mathematical modelling specialists from other south african universities, construct validity by previously recorded factor analysis (see tapia & marsh, 2004) and sight validity through the pilot study. a non-parametric test (wilcoxon signed ranks test) was used to evaluate the change in attitudes of student teachers towards mathematical modelling after exposure to a series of mathematical modelling activities, and a parametric test (one-sample t-test) was used to determine the effectiveness of the methodological design. due to the sample size (30+) and the robustness of statistical approaches (pallant, 2010), the parametric test was a possibility. ethical considerations standard ethical measures were taken according to the literature (creswell, 2013; teddlie & tashakkori, 2009) and the procedures at the university (ethics clearance number 2015-024). at the beginning of the modelling activities in cycles 2, 4 and 6, participants were briefly informed about the planned activities. participation was voluntary and participants signed an informed consent form. the researcher had a dual role for the purpose of this dbr study, both designer and researcher. this is a limitation and was purposefully addressed by balancing the roles, and through the observations recorded by the methodological specialist during implementation. findings and results student teachers’ shortcomings and progression in modelling activities based on data collected from the worksheets, posters and open-ended questions (collected in cycles 4 & 6, view figure 2) themes were identified reflecting student teachers’ shortcomings and progression in mathematical modelling activities. all themes focused on elements and principles related to developing competencies (theme 1), building perceptions (theme 2) and providing resources and opportunities (theme 3). the categories and sub-categories included in these themes intend to sketch a picture of the challenges participants experienced throughout the intervention. figure 5 shows an overview of the themes, with categories and sub-categories. the first theme, developing competencies, groups the needs (to a lesser or greater extent) of student teachers to develop particular competencies as they proceeded through the intervention in this study. there were four sub-categories: modelling competencies, mathematical competencies, utilising mathematical models and facilitator competencies (see table 2). modelling competencies refer to how participants proceeded through the different phases of the modelling cycle and interpreted their findings in terms of real-world relevance. mathematical competencies refer to the mathematical representation of real-world problems, relevant mathematical content selection and the need for relatively accurate calculations. utilising mathematical models refers to an investigation of model efficiency (by criteria from meyer, 2012) and an exposure to real-life model applicability. teacher competencies describe the preparation work (e.g. designing of a modelling task) and the facilitation work (e.g. facilitating class discussions reflecting on the modelling process) of participants as they took on the role as teachers of modelling. for example, in the ‘location problem’ (see sub-section 3.4) as participants took on the role of modellers themselves, they structured the information and represented the one-way streets and number of cars travelling along specific roads visually (like in the display in figure 4), but they had trouble constructing a suitable mathematical model (a linear system of four equations with four variables) to represent the situation. even if they managed to construct a suitable mathematical model, some groups struggled to solve the system of equations. such difficulties caused blockages in the modelling cycle. one group made a calculation error (where the variables that represent the number of cars travelling along a specific road were equal to negative values) and they struggled to interpret their mathematical result within the context of the example. table 2 represents the overall mean scores as percentages of participants in groups who mastered the specific competency. the second theme, building perceptions, highlights the importance (based on former and current opinions) of developing a positive disposition in student teachers towards mathematical modelling activities (for example the ‘location problem’). this theme is divided into two sub-categories: ideas about mathematics and ideas about mathematical modelling (informed by data findings from the open-ended questionnaire). ideas about mathematics refer to participants’ thinking about mathematics as a subject, the teaching thereof and valuing the subject. ideas about mathematical modelling are related to participants’ disposition based on their exposure to modelling activities as modellers themselves and as facilitators of such activities throughout their formal education (for example, in this study their exposure to modelling activities in cycles 4 and 6; see figure 2). qualitative data collected from the open-ended questionnaire (in cycles 4 & 6) showed: approximately half of the participants commented on a positive and enjoyable school learning experience in mathematics. they used phrases like ‘interesting and a good challenge’, ‘my favourite subject’, and ‘key to success’. a fifth of the participants described their school learning in mathematics in a traditional and partially negative way. they experienced the class as uninteresting and could not understand the link between mathematics and real life. one participant commented ‘mathematics was theoretically based and failed to integrate it into a real-life problem’, while another wrote ‘bored in class’ and ‘confusing’. two-thirds of the participants commented positively on their involvement in the modelling activity. some of the participants found the experience extremely interesting, exciting and stimulating, and realised the meaningfulness of mathematics. for others, it began with some frustration, but it improved, and they were willing to learn. for example, participants commented ‘the topic is scary when still reading but once you get the way of using it, it is very interesting’. the majority of participants (41) described the modelling task as overwhelming and challenging, while a quarter of the participants felt that they had some idea to solve the problem. participants mentioned key aspects as challenges in the modelling cycle, such as finding a strategy and deciding on a point of departure, selecting relevant information and expressing the information mathematically. comments such as ‘we were confused on where to start and what to start with’, and ‘we could not understand the problem’ explained their point of view. they also mentioned the need to think ‘out of the box’. findings from the data revealed 49 participants indicated the modelling task was sufficiently real-world related and realistic. the third theme, providing resources and opportunities, indicates the required support highlighted by student teachers to develop in the teaching and learning of mathematical modelling throughout their professional education. this theme is divided into three sub-categories (informed by data findings from the open-ended questionnaire): pre-tertiary education, tertiary education and post-tertiary education. the categories pre-tertiary education and post-tertiary education are beyond the scope of this study although the researcher recognised from the findings that support during these phases could contribute towards the development of student teachers’ competencies in mathematical modelling and the further enhancement thereof. the category tertiary education refers first to the group work opportunities required by student teachers to gain modelling competencies (as modellers and facilitators) and to develop proficiency in collaborative actions, and second to particular curriculum considerations to ensure continuous contextualised exposure (to content and teaching methodologies) of mathematical modelling activities. for example, participants worked together in small groups to find a real-life solution for the ‘location problem’. qualitative data collected from the open-ended questionnaire (in cycles 4 & 6) showed: approximately half of the participants commented positively on the collaboration and highlighted the manner in which the group worked as a team, the manner in which the modelling problem stimulated interaction, and the manner in which all group members shared ideas. for example, a participant commented ‘our group worked well together and it seemed that we liked the challenge’. contrary to this, others experienced the group work as being rather negative. they mentioned challenges such as language and cultural barriers, group members’ contributions (or lack thereof) and no agreement on a strategy, inactive group members, confusion and, in some cases, group members being just concerned with getting an answer. approximately two-thirds of the participants (32) preferred a mandatory group allocation, while one-third (19) preferred a voluntary selection. most participants (48) confirmed their groups’ modelling abilities improved over the period of exposure to the modelling activities. the majority of participants (51) regarded themselves as semi-active or active (4 non-active) in their particular group. similarly, they (50) also viewed themselves as constructive contributors. they gave a number of reasons for their participation, such as the group member who came up with the solution, or completed the worksheets, presented and took on the role as group leader. a majority (54) indicated they would benefit from continuous exposure to material regarding the teaching of mathematical modelling and the approach to modelling tasks. for example, a participant wrote ‘i would benefit a lot for it will give me skills and knowledge’. then again, 46 participants (with 5 no responds) confirmed they would participate in discussions regarding the teaching and learning of mathematical modelling. most respondents approved the use of an electronic platform (such as social media or an electronic learning environment), but some preferred face-to-face contact. reflective analysis of quantitative data within each sub-scale, the researcher compared total scores and investigated cross-tabulation results for respective items. due to the comparative nature of the analyses in dbr phase 3, only cases displaying the necessary information in both iterations were considered (enjoyment 44 cases, value 43 cases, self-confidence 44 cases, and motivation 43 cases). hence, missing data were excluded pairwise. the wilcoxon signed ranks test as statistical technique is regarded as suitable to answer the research question related to change in attitudes over time (pallant, 2010). this test converts scores to negative or positive ranks and compares them at iteration 1 and 2. enjoyment: more positive ranks (26) than negative ranks (14) on the enjoyment scores of participants were detected, but no significant change in enjoyment mean scores following participation in the mathematical modelling intervention over two iterations. table 3 shows z = –1.86 and p = 0.06, with a small to medium effect size (r = 0.20) according to the criteria by cohen (1988). table 3: test statistics for the enjoyment sub-scale. findings from cross-tabulation data in the enjoyment sub-scale revealed at the end of iteration 2 that most participants (over 80%) mildly to strongly agreed on the following aspects: (1) mathematical modelling is a very worthwhile topic and they wanted to further develop their mathematical modelling skills, (2) the process taught them to think and they recognised the importance of the topic in everyday life, (3) they were not sure how the topic can be utilised in learning mathematics although it seems important for mathematics students for any grades of teaching, (4) the usefulness of studying mathematical modelling at a higher education level, and (5) their belief that the topic will support them with problem-solving in other areas and a strong background in modelling could help any mathematics teacher. value: more negative ranks (24) than positive ranks (17) on the value scores of participants were detected and no significant change in value mean scores following participation in the mathematical modelling intervention. table 4 shows z = –0.182 and p = 0.86, with a small to medium effect size (r = 0.20) according to the criteria by cohen (1988). table 4: test statistics for the value sub-scale. findings from cross-tabulation data in the value sub-scale revealed at the end of iteration 2 that participants mildly to strongly agreed that they: (1) got a great deal of satisfaction out of solving a mathematical modelling problem (70.2%), (2) enjoyed being involved in a mathematical modelling session (66.7%), (3) liked to solve real-world problems in mathematics (69.8%), (4) preferred a mathematical modelling task rather than writing an essay (73.2%), (5) thought they liked the topic (59.5%), (6) preferred real-world problems to other mathematical themes (29.3%), (7) found the topic very interesting (76.2%), (8) were comfortable expressing their own ideas on how to solve a modelling problem (69.8%), and (9) were comfortable suggesting possible solutions to a modelling problem (73.1%). furthermore, participants (86%) mildly to strongly disagreed with the statement that mathematical modelling is dull. self-confidence: more positive ranks (26) than negative ranks (15) on the self-confidence scores of participants were detected and no significant change in self-confidence mean scores following participation in the mathematical modelling intervention. table 5 shows z = –1.42 and p = 0.16, with a small effect size (r = 0.15) according to the criteria by cohen (1988). table 5: test statistics for the self-confidence sub-scale. findings from cross-tabulation data in the self-confidence sub-scale at the end of iteration 2 revealed that participants mildly to strongly disagreed with the following: (1) mathematical modelling is a feared topic (50%), (2) it created a feeling of dislike (69.8%), (3) their minds went blank when confronted with the topic (62.8%), (4) mathematical modelling made them nervous (51.2%), and uncomfortable (67.5%), (5) they experienced terrible strain in a mathematical modelling session (66.7%) and just thinking about the topic made them nervous (67.4%), (6) they were confused in a modelling session (55.8%), and (7) they felt insecure when attempting mathematical modelling (64.3%). however, participants mildly to strongly agreed with the statements: (1) mathematical modelling did not scare them (45.3%), (2) they had self-confidence with the topic (41.9%), (3) they would be able to solve such a problem without too much difficulty (26.6%), (4) their expectation would be to do fairly well in future modelling sessions (64.3%), (5) they will learn mathematical modelling easily (45.2%), and (6) they believed they were good at solving real-world problems (48.9%). d) motivation: more positive ranks (25) than negative ranks (14) on the motivation scores of participants were detected and no significant change in motivation mean scores following participation in the mathematical modelling intervention. table 6 shows z = –1.71 and p = 0.09, with a small to medium effect size (r = 0.20) according to the criteria by cohen (1988). table 6: test statistics for the motivation sub-scale. findings from cross-tabulation data in the motivation sub-scale revealed at the end of iteration 2 that participants mildly to strongly agreed with: (1) they could pass a course on mathematical modelling for mathematics teachers (62.8%), (2) they were willing to learn more about mathematical modelling in future (65.1%), (3) the challenge of mathematical modelling is appealing (54.7%), and (4) they would be keen to enrol for a course on mathematical modelling for mathematics teachers (65.2%). furthermore, participants mildly to strongly disagreed that they would like to avoid teaching mathematical modelling (67.5%). finally, the researcher investigated the percentage change in scores between iteration 1 and 2 for all sub-scales (by utilising the formula: the purpose of this test was to evaluate the effectiveness of the mathematical modelling intervention over two iterations on participants’ attitudinal scores and more broadly on the methodological design. table 7 shows positive percentage changes in all sub-scales, and significant percentage changes in both the enjoyment (t[43] = 2.12, p < 0.05 two-tailed) and motivation sub-scales (t[42] = 2.16, p < 0.05 two-tailed). table 7: one-sample t-test for percentage changes in each sub-scale. discussion at the beginning of the study, student teachers had misconceived ideas of the modelling process according to the data collected in cycle 4, and they struggled to find their way through the modelling cycle. they experienced challenges in each phase of the modelling cycle, such as mathematisation, working with mathematics, interpretation and reflection. these findings correlated with findings from other studies (compare zeytun, cetinkaya, & erbas, 2017). towards the end of the study, student teachers displayed an improved understanding of the modelling process which was evident in their poster presentations in cycle 6. anhalt and cortez (2016) presented similar results. thus, in order to design an intervention for student mathematics teachers to learn mathematical modelling during their formal education in the south african context, the designer is best advised to emphasise the development of student teachers’ mathematical and modelling competencies, capabilities in utilising mathematical models and facilitator competencies, all of which should focus on demonstrating practicality. the following procedures, which flow from the qualitative findings after student teachers were exposed to a series of modelling activities, might be considered: (1) providing an authentic modelling activity with a familiar mathematical content and real-world context, also reported by ikeda (2013), (2) providing opportunities in demonstrating the applicability of mathematical models in a real-world context (compare meyer, 2012), (3) promoting active participation and socially constructed knowledge by means of class discussions and collaboration (supported by anhalt & cortez, 2016), (4) encouraging individual development and building of self-confidence in mathematical and modelling competencies (also supported by the atmmi results; see sub-section 4.2), (5) providing support by scaffolding the processes in the mathematical modelling cycle (supported by other studies, for example blum 2015), and (6) creating an opportunity to practise aspects of teaching of modelling, develop competency in preparation work (e.g. the selection of mathematical modelling tasks), and facilitation work. furthermore, student teachers were able to build an appropriate belief about mathematical modelling and to develop a positive disposition towards it by participating in the series of modelling activities. at the beginning of the intervention, they started with a vague idea and even misconceptions about mathematical modelling, but towards the end, most participants had developed an understanding of the modelling process. they also had predetermined ideas about mathematics (compare jacobs & durandt, 2017). this correlated with beliefs about school mathematics explained in ӓrlebäck (2009) and the way students act in a typical manner based on their mathematics belief system (kaiser, 2017). they therefore required opportunities to build positive perceptions about mathematical modelling which is also evident from the atmmi findings. in the final atmmi results, a significant positive percentage change in student teachers’ enjoyment scores over the two iterations was indicated. positive enjoyment scores in mathematical modelling can be linked to positive feelings of satisfaction, a willingness to participate in activities, and the persistence to develop competencies. özdemir and üzel (2012) also reported positive enjoyment results. likewise, a significant positive percentage change in student teachers’ motivation scores over the two iterations was indicated in the atmmi results. an increase in motivation could encourage student teachers to become interested in mathematical modelling and to display the curiosity needed to continue studying mathematical modelling. thus, student teachers could grow in confidence when participating in modelling activities. these results contradict some findings from kreckler (2017) where students at the secondary level were exposed to a series of modelling activities and although a significant increase in the global modelling competence independent of grade and topic was reported no significant changes in motivation could be identified. the atmmi showed a positive although not significant percentage change in student teachers’ self-confidence scores over the two iterations. higher confidence equals lower anxiety and higher confidence scores relate to performance and ability and provoke enjoyment (compare maxwell, 2001). the atmmi value scores also reflected a positive but not significant percentage change, although at the end of the intervention, more than half of the student teachers could not really understand the value of mathematical modelling. the value of learning mathematical modelling is related to personal goals, and the person’s perceived usefulness of mathematical modelling. one reason might be that the specific modelling tasks used in the intervention did not link to the personal preferences and realities of some participants and, as a result, they could not see the ‘gain’ in these activities. özdemir, üzel and özsoy (2017) explained value is determined by teachers’ knowledge, thus a lack thereof could also be regarded as a reason why student teachers did not fully value the modelling activities in this study. overall, the increased scores indicated a positive change in aspects (enjoyment, self-confidence, value and motivation) determining the attitudes of student teachers towards mathematical modelling, and ultimately creating a more positive disposition towards the topic. thus, in order to design an intervention for student mathematics teachers to learn modelling during their formal education in the south african context, the designer is best advised to emphasise the building of student teachers’ perceptions about mathematics and mathematical modelling, to emphasise practicality via some procedures like: (1) establishing an inviting modelling climate to promote mathematical modelling practices among student teachers, (2) creating an authentic context, to which student teachers could relate, and that would reflect the real-world usability of the knowledge, and (3) promoting reflective activities, such as class discussions to replicate the authentic context, and to demonstrate the value of real-world application. student teachers required multiple opportunities to learn mathematical modelling over a period of time. available resources, like knowledge, and tools such as technology, as well as practical matters, such as time constraints, were other concerns. opportunities to learn mathematical modelling should involve activities with student teachers both as modellers themselves, and as facilitators of modelling. from the beginning, student teachers were grouped in mandatory groups, and they experienced a number of challenges in understanding the mathematical modelling tasks, continuous participation of all group members and collaboration among members (also indicated by answering the question in the questionnaire). thus, in order to design an intervention for student mathematics teachers to learn modelling during their formal education in the south african context, the designer is best advised to provide student teachers’ with opportunities and required resources (both characteristics to demonstrate practical emphasis), via some procedures like: (1) promoting continuous and frequent exposure to mathematical modelling activities throughout student teachers’ professional development, (2) grouping student teachers in comparable groups and rotating responsibilities among group members in order to ensure productive participation and the development of self-confidence in all members (compare goos, 2004), and (3) managing the classroom regarding practical matters (e.g. time management and logistical arrangements). the dbr approach, chosen for this study with a pragmatic view, resulted in rich data shedding light on student teachers’ shortcomings and progression in modelling competency and their positive attitudinal experiences, similar to the results of other studies (e.g. anhalt & cortez, 2016; biccard, 2013). both the approach and the findings of this study could support the international discussion. a limitation is that dbr guides theory development but usually takes place through several iterations and this study could thus be further developed through more iterations. additionally, a narrower lens on aspects of mathematical modelling (e.g. reflective activities) could enhance metacognitive development on an individual level. conclusion in this study, a cohort of third-year mathematics student teachers at a south african institute of higher education (grouped in 10 groups) were exposed to a series of well-planned mathematical modelling activities over two iterations – both as mathematical modellers themselves and as teachers of modelling. both qualitative and quantitative data were collected during the dbr phases. qualitative data were analysed by means of content analysis methods, and quantitative data were analysed with spss. both the t-test (parametric) and the wilcoxon signed rank test (non-parametric alternative) were utilised to determine participants’ mean change in attitudes towards mathematical modelling (concerning motivation, value, self-confidence, and enjoyment) over time. finally, a reflective analysis on all data contributed to desirable elements and underlying principles that might inform local instructional theory. the elements and principles ought to develop student teachers’ mathematical modelling competencies, building positive perceptions, and to provide them with resources and opportunities to learn modelling themselves and to learn how to teach modelling. towards the end of the study, student teachers displayed an improved understanding of the modelling processes and sub-processes and showed significant positive percentage changes in their enjoyment and motivation towards mathematical modelling over the course of the study. awareness of the student mathematics teachers’ experiences when mathematical modelling is learnt through incorporating a selection of key elements and fundamental principles in their formal education could contribute to decreasing the cognitive burden of student teachers and cultivating positive experiences. acknowledgements the author is grateful to prof. geoffrey lautenbach who provided valuable insight in designing the research project. competing interests the author has declared that no competing interest exists. authors’ contributions i declare that i am the sole author of this article. funding information this work is based on the research partially supported by the national research foundation (nrf) of south africa, unique grant no. 106978. data availability the data that support the findings of this study are available on request from the author. the data are not publicly available due to restrictions (i.e. information that could compromise the privacy of research participants). disclaimer the views and opinions expressed in this article are those of the author and do not necessarily reflect the official policy or position of any affiliated agency of the author. references 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(2017). understanding prospective teachers’ mathematical modeling processes in the context of a mathematical modeling course. eurasia journal of mathematics science and technology education, 13(3), 691–722. guest editorial making mathematics accessible  for multilingual learners   mamokgethi setati  richard barwell    university of south africa  university of ottawa, canada   setatrm@unisa.ac.za  richard.barwell@uottawa.ca  why do we need a special issue of pythagoras on teaching and learning mathematics in multilingual classrooms? while there is a reasonable amount of research done in this area of study in southern africa (e.g. adler, 2001; kazima, 2007; setati, 2005), much of this work is not published locally. in the past ten years pythagoras has not published a single article on learning and teaching mathematics in multilingual classrooms1. furthermore the work that is already published in this area of study is not easily accessible to practicing school mathematics teachers. the majority of learners in south africa learn mathematics in a language that they are not fluent in. this area of research is therefore important for equity and ensuring that all learners have access to mathematics. until these learners can have equal access to mathematical knowledge it will be impossible to produce the number of engineers, technologists and scientists that south africa so desperately needs. there is therefore a clear need for a collection that is of relevance to the south african context focused on teaching and learning mathematics in multilingual classrooms. a recent analysis of research on multilingualism in mathematics education in south africa for the period 2000 – 2007, shows that there is a paucity of published research in this area of study (setati, chitera, & essien, in press). as editors of this special issue, we believe that it is crucial that pythagoras, the only accredited south african mathematics education journal, focus on teaching and learning mathematics in multilingual classrooms now. an analysis in which kahn (2005) uses language as a proxy, shows that it is mainly learners who 1 tobias (2003) focuses generally on the relationship between language and mathematics rather than multilingual issues. learn in a language that is not their home language who do not succeed in grade 12 mathematics. this finding is consistent with what howie (2003, 2004) found in her analysis of the performance of south african learners in the third international mathematics and science study (timss). it is thus crucial that more research is undertaken to explore why it is that learners who learn mathematics in a language that is not their home language do not perform at the expected levels. that said, we hold the view that poor performance by multilingual learners cannot be solely attributed to their limited proficiency in english. we agree with setati et al. (in press) that learner performance (and by implication, mathematical achievement) is determined by a complex set of interrelated factors. poor performance by multilingual learners thus cannot be solely attributed to the learners’ limited proficiency in english (suggesting that fluency in the language of learning and teaching will solve all problems) in isolation from the pedagogic issues specific to mathematics as well as the wider social, cultural and political factors that infuse schooling. there is a need for research that can identify other factors that interact with the fact that these learners have limited proficiency in the language of learning and teaching to contribute to their poor performance. there is also a need for research that can point to initiatives that can be implemented to ensure success in multilingual mathematics classrooms. in this issue we publish six papers by authors all of who work in multilingual mathematics classrooms in southern africa. while these authors work in different countries, they all regard language as a resource. they critically engage with policy, past research and the demands of teaching and learning mathematics in multilingual classrooms to explore ways in which mathematics can be made more accessible to learners in these classrooms. rather pythagoras, 67, 2-4 (june 2008) 2 mamokgethi setati & richard barwell than fall prey to creating dichotomies, the authors present ways in which the different home languages of the learners can be drawn on to support their mathematics learning. in so doing the authors point to the fact that teaching and learning mathematics in multilingual classrooms is a complex cognitive and socio-political phenomenon, not to be simplistically analysed. in his paper, dlamini discusses the relationship between examination results in mathematics and in english language in swaziland. in that country, students are not admitted to university to study mathematics or sciences without a pass in english language – even if they have high scores in all other subjects. dlamini argues that this language policy is based on a false assumption – that proficiency in english is necessary for success in mathematics. he bases this argument on analysis of examination data for the country as a whole, as well as for students in one school. these analyses demonstrate that there is no connection whatsoever between students’ performance in mathematics and their performance in english. hence, denying university places in mathematics to students because they do not have a pass in english language is to deny places to potentially talented mathematicians. this policy affects a large number of students. dlamini argues that a country such as swaziland can ill afford to turn away such a large pool of potential mathematical talent. his article forcefully makes the point that in multilingual societies, mathematics educators cannot afford to ignore broader language policies. setati, molefe and langa report on their innovative study in which they explore a new approach to teaching mathematics in gauteng’s multilingual classrooms. this approach involves the teacher giving his grade 11 learners mathematics problems in both english and in the main african languages used by learners in his class. the research team then investigated the effects of this strategy. they found that access to multiple languages was beneficial for learners. of particular interest is the way the learners made use of the different language versions – they tended to regularly switch between them, sometimes referring to english, sometimes referring to the african language version they had in front of them. the different language versions were therefore mutually supportive. their analysis demonstrates that common assumptions about, for example, needing to choose between teaching and learning in english or teaching and learning in an african language are simplistic. what seemed to be effective in this study was using several languages at once! this paper therefore represents a significant challenge to received wisdom about teaching and learning mathematics in south africa’s multilingual context. webb and webb explored some similar issues to setati et al., but with a particular focus on the nature of classroom discourse. in their study, which was conducted in the eastern cape, they worked with teachers in an in-service program to develop the use of ‘exploratory talk’ in their mathematics lessons. exploratory talk is talk that involves careful reasoning, characterised, for example, by the sharing of different ideas and the use of words like ‘because’. research in the uk and mexico has shown that working with teachers to develop exploratory talk is effective in improving learners’ mathematical attainment. an interesting feature of webb and webb’s study is that exploratory talk intersects with codeswitching. they present evidence that at least some of the teachers in the program did start to see more exploratory talk in their mathematics lessons and that this entailed use of both isixhosa and english. like the two preceding papers, vorster’s study also examines the issue of how multiple languages may be used in teaching mathematics. in the study, which focused on a geometry topic in grade 8, two teachers were provided with notes, a glossary of relevant mathematical terms and an end of topic test in both setswana and english. the teachers used these materials as well as active codeswitching during their teaching. as in setati et al.’s study, vorster reports that learners had a positive attitude to this planned use of their main language alongside english. interestingly, while the teachers in the study were also enthusiastic, they were concerned about the use of setswana mathematical terminology. these concerns can be traced to the context of the study, in which mathematical terminology is usually only in english. hence the learners were not necessarily used to using setswana terminology. it may be that with more systematic use of terminology in learners’ main languages in addition to english, this issue would not arise. bohlmann and pretorius work in the field of reading research. as part of a recent study of grade 7 reading in two township schools, they also collected data on students’ attainment in mathematics. this attainment was indicated by their performance on examinations prepared by 3 making mathematics accessible for multilingual learners 4 their teachers. bohlmann and pretorius’s analysis produces an interesting and, for some, perhaps a surprising result: students’ mathematical attainment is closely linked to their reading proficiency. this finding should not, however, be so surprising. success in mathematics depends in part on reading and interpreting mathematical texts, particularly in the context of curriculum 2005. mathematics educators have not often considered broader factors in learning mathematics in multilingual contexts. this study alerts us to an important non-mathematical aspect of learners’ education that potentially has a big impact on mathematical attainment. finally, this special issue ends with kazima’s discussion of some of the different policy options available for teaching mathematics in multilingual classrooms. she considers three countries – tanzania, nigeria and malawi – who have adopted different approaches, including, in some cases, the development of mathematical terminology in african languages or, in other cases, the adoption of english terminology. kazima highlights some of the complexity involved in developing a language policy for mathematics. this complexity relates to many of the issues apparent in the other papers of this collection. a particular tension seems to arise around the issue of whether to follow current practice and hence stay with what teachers and learners are used to, or whether to attempt to change current practice, with the hope of leading to more successful learning. it is apparent from all the papers that these kinds of tensions are ever present. acknowledgements the collaboration that led to the production of this special issue was supported by the national research foundation (ttk2007051500040). any ideas expressed are, however, those of the authors and therefore the national research foundation does not accept any liability. references adler, j. (2001). teaching mathematics in multilingual classrooms. dordrecht: kluwer. howie, s. j. (2003). language and other background factors affecting secondary pupils’ performance in mathematics in south africa. african journal of research in mathematics, science and technology education, 7, 1-20. howie, s. j. (2004). a national assessment in mathematics within international comparative assessment. perspectives in education, 22(2), 149-162. kahn, m. j. (2005). a class act – mathematics as a filter of equity in south africa’s schools. perspectives in education, 23(3), 139-148. kazima, m. (2007). malawian students’ meanings for probability vocabulary. educational studies in mathematics, 64(2), 169-189. setati, m. (2005). teaching mathematics in a primary multilingual classroom. journal for research in mathematics education, 36(5), 447-446. setati, m., chitera, n., & essien, a. (in press). research on multilingualism in mathematics education in south africa: 2000 – 2007. african journal of research in mathematics, science and technology education. tobias, b. 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can be opened with acrobat and adobe reader 5.0 and later.) >> /namespace [ (adobe) (common) (1.0) ] /othernamespaces [ << /asreaderspreads false /cropimagestoframes true /errorcontrol /warnandcontinue /flattenerignorespreadoverrides false /includeguidesgrids false /includenonprinting false /includeslug false /namespace [ (adobe) (indesign) (4.0) ] /omitplacedbitmaps false /omitplacedeps false /omitplacedpdf false /simulateoverprint /legacy >> << /addbleedmarks false /addcolorbars false /addcropmarks false /addpageinfo false /addregmarks false /convertcolors /converttocmyk /destinationprofilename () /destinationprofileselector /documentcmyk /downsample16bitimages true /flattenerpreset << /presetselector /mediumresolution >> /formelements false /generatestructure false /includebookmarks false /includehyperlinks false /includeinteractive false /includelayers false /includeprofiles false /multimediahandling /useobjectsettings /namespace [ (adobe) (creativesuite) (2.0) ] /pdfxoutputintentprofileselector /documentcmyk /preserveediting true /untaggedcmykhandling /leaveuntagged /untaggedrgbhandling /usedocumentprofile /usedocumentbleed false >> ] >> setdistillerparams << /hwresolution [2400 2400] /pagesize [612.000 792.000] >> setpagedevice abstract introduction literature review research design and methodology findings discussion and conclusion recommendations acknowledgements references about the author(s) methuseli moyo department of mathematics education, faculty of education, university of south africa, pretoria, south africa france m. machaba department of mathematics education, faculty of education, university of south africa, pretoria, south africa citation moyo, m., & machaba, f.m. (2021). grade 9 learners’ understanding of fraction concepts: equality of fractions, numerator and denominator. pythagoras, 42(1), a602. https://doi.org/10.4102/pythagoras.v42i1.602 original research grade 9 learners’ understanding of fraction concepts: equality of fractions, numerator and denominator methuseli moyo, france m. machaba received: 31 jan. 2021; accepted: 04 aug. 2021; published: 21 oct. 2021 copyright: © 2021. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract our research with grade 9 learners at a school in soweto was conducted to explore learners’ understanding of fundamental fraction concepts used in applications required at that level of schooling. the study was based on the theory of constructivism in a bid to understand whether learners’ transition from whole numbers to rational numbers enabled them to deal with the more complex concept of fractions. a qualitative case study approach was followed. a test was administered to 40 learners. based on their written responses, eight learners were purposefully selected for an interview. the findings revealed that learners’ definitions of fraction were neither complete nor precise. particularly pertinent were challenges related to the concept of equivalent fractions that include fraction elements, namely the numerator and denominator in the phase of rational number. these gaps in understanding may have originated in the early stages of schooling when learners first conceptualised fractions during the late concrete learning phase. for this reason, we suggest a developmental intervention using physical manipulatives to promote understanding of fractions before inductively guiding learners to construct algorithms and transition to the more abstract applications of fractions required in grade 9. keywords: fraction; equality; numerator; denominator; misconceptions; manipulatives. introduction transitioning from whole number concepts to rational numbers traditionally poses a considerable challenge to the mathematically developing mind and may become a stumbling block in the way of maturation in number conceptualisation (durkin & ritle-johnson, 2014; simon, placa, avitzur, & kara, 2018). experience with whole numbers makes the transition to rational numbers somewhat abstract (simon et al., 2018). bruce, bennett and flynn (2014) explain that fractions are multiple digits (numerator and denominator) that represent one quantity, making different interpretations of fractions possible. bruce et al. argue that a fraction has several meanings, depending on the context in which it is used: for example, a part of a whole, a part of a set, ratio and rate. to fully understand all situations in which fractions express different meanings and to discern which meaning applies in a particular context, some conceptual understanding is required. conceptual understanding in the context of this article refers not to isolated facts, but to the ability of learners to make meaningful connections between fractional elements such as the numerator and the denominator. for example, in their studies, deringöl (2019) and simon et al. (2018) found that learners faced challenges in terms of viewing the numerator and the denominator as representing a whole together. learners tend to see a fraction as a pair of numbers representing quantities with no relationship implied between those quantities (stafylidou & vosniadou, 2004). furthermore, in his study, deringöl found that learners had difficulty comprehending the principle of identical pieces in the piece-whole relationship. they failed to understand that an object must be divided into identical pieces when defining and representing fractions. the grade 9 learner results of the annual national assessments (ana), from 2012 to its conclusion in 2018, present a bleak picture. this test was administered at phase exit levels: grade 3 marks the end of the foundations phase (fp), grade 6 the end of the intermediate phase (ip) and grade 9 the end of the senior phase (sp). over the years, a pattern of steep decline has emerged from one exit level to the next. in grade 9, the performance dropped to 5%. in 2014, only 3% of grade 9 learners countrywide achieved above 50% in the ana (mathematics), according to the report by the department of basic education (dbe, 2014, p. 63). grade 9 is a threshold grade, the successful completion of which gives access to the sp or further education and training (fet) phase. several fraction concepts, including rate and ratio, are included in the grade 9 curriculum, notably in formulae, algebraic expressions or equations. these also occur in various contexts in all mathematical content areas. thus, a conceptual understanding of rational numbers and their applications is essential in grade 9 mathematics. literature review fractions as a transition to rational numbers the troublesome nature of teaching and learning fractions has been widely reported (bruce et al., 2014; gabriel, 2016; stafylidou & vosniadou, 2004). the transition from whole numbers to rational numbers such as common fractions is gradually introduced during the first six years of schooling (braithwaite & siegler, 2018). braithwaite and siegler’s (2018) study found, resonating with our findings, that many learners’ knowledge of fractions is adversely affected by whole number bias: the tendency to focus on the separate whole number components (numerator and denominator) of a fraction rather than on the fraction’s magnitude (ratio of numerator to denominator). several researchers, including ni and zhou (2005) and brown (2019), have described initial rational number learning as an experience of conceptual change. the work of siegler, thompson and schneider (2011) challenges the view that these differences are fundamental to rational number learning, proposing instead that children should be encouraged to see rational numbers and whole numbers as one unitary formal system, united by the property of magnitude. this property can be represented symbolically on the number line; that is, this property and the use of the ordered number line should be the basis for teaching rational numbers. in their attempt to address the challenge of the transition from a whole number to a rational number, van de walle et al., (2016) illustrate in their textbook how children learn fractions, and then shows pre-service teachers the most effective methods of teaching fractions through hands-on, problem-based activities. the initial concrete mode of instruction is followed by pictorial and diagrammatical representations of fractions and by the end of the ip learners are expected to perform calculations with fractions using numerical symbols (van de walle, folk, & bay-williams, 2010). this transition is intended to prepare the ip learner for the formal abstract applications of rational numbers in algebra, geometry, probability, data handling and measurement applications in the sp. a critical component of the understanding of rational numbers and proportional reasoning aids the transition from informal (pre-algebra) to formal (algebra) mathematical thought; however, this is often handled as an add-on, after fraction and ratio concepts (doyle, dias, kennis, czarnocha, & baker, 2016), when a ratio should be expressed as a comparison of a part to a whole. for example, the ratio of the number of girls in a class (9) to the number of students in the class (16) can be written as the ratio of . the general difficulties that arise from learners’ transition from whole numbers to rational numbers, including their development of proportional reasoning, are compounded by their struggle to advance from informal concrete to formal abstract work with fractions. although the teaching protocol of moving from the concrete through representational work to symbolic work with fractions seems to follow a logical developmental sequence, we doubt whether it succeeds in equipping learners for the work with fractions that is required in the sp, specifically in grade 9. in this article, we conclude that learners face challenges in demonstrating a flexible ability (the application of fractions to other examples, related concepts and contexts). it appears that the most basic fractional concepts have not taken root. constructivism adapted from piaget piaget (1964a) believes that when the developing mind interacts with new situations or ideas, a continuous interplay of two cognitive processes occurs, namely: accommodation happens when learners reflect on the new experience and realise that it does not fit into their existing schema; they, therefore, modify their cognitive structure to accommodate the new idea. assimilation occurs when learners make meaning of the new idea by relating it to their existing knowledge, organising their cognitive structures to incorporate the new idea in their existing schema. piaget (1964a) argues that learners need information, concepts, ideas or a network of connected ideas to think, and they will think according to the knowledge they already have at their disposal (in their cognitive schema). the deadweight of facts learnt off by heart, by memory, without thought given to meaning (that is rote learning), robs the learner of the potential excitement of relating ideas or concepts to one another and the possibility of divergent and creative thinking (van de walle et al., 2016). what is inculcated in learners because of rote-memorised rules, in many cases, is the manipulation of symbols that have little or no attached meaning. this makes learning much more difficult because rules are much harder to remember than integrated conceptual structures that are made up of a network of connected ideas. such learners’ understanding of a whole divided into equal-sized portions is indicated in figure 1. in addition, careless errors are not picked up by learners themselves because the task has no meaning for them: they are unable to anticipate the kind of result that might occur. figure 1: example of a fraction as part of a whole. an integrated network of connected ideas brings about conceptual understanding. understanding can be thought of as the measure of the quality and quantity of connections of an idea to existing ideas. for example, learners should see the numerator and the denominator in the fraction not as separate values, but as numbers that are related. understanding depends on the existence of appropriate ideas and the creation of new connections. the greater the number of appropriate connections to a network of ideas, the better the understanding. skemp (1974) argues that learners’ understanding could exist along a continuum. at one pole, the ideas are loosely connected or isolated from each other. this is the pole of so-called instrumental understanding (van de walle et al., 2016). at the other pole, an idea is associated with many others in a rich network of related ideas (the relationship between numerator and denominator, for example). this is the pole of so-called relational understanding. knowledge learned in a rote way, for example without regarding the numerator and denominator as a unitary value, is almost always at the pole of instrumental understanding, where ideas are nearly always isolated and disconnected. noureen, arshad and bashir (2020) suggest that constructivism holds that learners are not blank slates or empty vessels waiting to be filled with knowledge. instead, from a rich array of previous experiences, knowledge and beliefs, learners construct new knowledge. gupta and gupta (2017) agree that just as in cell theory, where all cells are explained as arising from pre-existing cells, knowledge already resides in the human mind. what is needed is ways to explore this knowledge. piaget’s (1964a) cognitive constructivism implies that learners engage with ideas on their own, work through tasks, sift through the material at hand and either accommodate or assimilate the present experience into their existing mental schemata. they do not merely absorb the ideas as teachers present them; rather, they are creators of their knowledge (hatano, 1996). the purpose of this study was to explore learners’ understanding of fundamental fraction concepts, such as numerator and denominator, and of the fraction itself, used at grade 9 level. it was based on the ideas of constructivism in a bid to understand whether learners’ transition from the whole number to rational numbers would allow them to deal with more complex fraction concepts. the theory of constructivism was particularly pertinent in that when learners construct knowledge on the concept of a fraction by themselves, misconceptions inevitably arise. this means that misconceptions are part of learning; however, appropriate intervention is required to rectify such misconceptions. the concept of the fraction when establishing the foundations of the concept of the fraction, the following ideas should be applied. conceptualising a fraction as part of a whole fitri and prahmana (2019) argue that most learners still consider a fraction as a meaningless symbol and assume that the numerator and the denominator are separate numbers. the fraction could not for instance be understood to mean the quantity formed by one part when a whole is partitioned into eight equal parts. the key to understanding fractions as part-of-a-whole is to identify the whole and the equal fractional parts. as such, fractions have meaning only concerning the whole to which they apply. van de walle et al. (2016) explain that the conceptualisation of fractions concerns the relationship between the part and the whole. in an example using a geometric shape, a rectangle divided into six equal parts, each part equals one-sixth of the whole; therefore, five parts are five-sixths of the whole and six-sixths make one whole, as illustrated in figure 1. van de walle et al. (2016) emphasise the importance of the language used in learning fractions. to prevent misconceptions, they advise that in the acquisition of the concept, learners should be able to say, as in our example for instance, ‘my whole is a rectangle. it is divided into six equal parts. each part is a sixth of the whole’. conversely, given the whole (undivided) rectangle, they should be able to explain that ‘to find one-sixth of the whole, i divide it into six parts of equal size, and shade one part. the shaded part is one-sixth of the whole’. conceptualising fractions as numbers simon et al. (2018) argue that to explain fractions as part of geometric shapes is insufficient, as it excludes an understanding of some important aspects of fractions. in a ‘part-of-a-whole’ geometric shape, part shading clearly shows the fraction; however, it does not convey the idea of a fraction as a number. the notion of a fraction as a number means a fraction can be expressed in the form of a/b where a and b are integers but not equal to 0. similarly, deringöl (2019) believes that the ‘part-of-a-whole’ conceptualisation limits the understanding that a fraction can be greater than 1 and it narrows thinking to the idea of partitioning one whole. simon et al. (2018) advise that learners understand fractions as numbers that expand the number system beyond whole numbers, and recommend the use of number lines as a key representation tool to convey fraction concepts. we share the concern of these scholars that defining a fraction as a part of a whole is only a part of the understanding of the concept. nevertheless, in our study, we explored learners’ understanding of a whole divided into equal-sized portions (van de walle et al, 2016). representing a fraction as ‘part-of-a-whole’ forms part of many south african textbooks and as such has justifiable use in the fostering of this idea (laridon et al., 2005). conceptualising the fraction notation alghazo and alghazo (2017) believe that learners tend to see a fraction either as a division calculation on its own or as a set of two numbers written above and below a fraction line. in a study on equivalent fractions, jigyel and afamasaga-fuata’i (2007) concluded that some learners perceived the components of fractions, the numerator and denominator, as two unrelated whole numbers, which led to misconceptions. in a bid to investigate the influence of this misperception on learning, jigyel et al. (2007) conducted a test on learners to choose the most appropriate way of reading . options provided were: two-fifths, 2 over 5, 2 upon 5 and none of the above. jigyel et al. found that two-fifths was the commonest answer, although a significant number of learners chose two over five. a learner selected from the 2 over 5 group explained that ‘there is a two on a line above five, so it is two over five’. the separation of the 2 from the 5 may influence the perception that is not a number in its own right that can be positioned on the number line. jigyel et al. (2007) warn that if learners acquire the fraction concept by counting the shaded and total number of parts in a geometric diagram, they may perceive fractions as a relationship between two unrelated counts. seeing as shading two out of five parts may lead them to perceive as 2 ÷ 5. this may result in learners recognising not as a single quotient value with a specific position on the number line, but rather as one number on the number line, divided by another number on the number line. deringöl (2019) clarifies the meaning of the components of fraction notation, arguing that most learners do not realise that the denominator and the numerator represent the part-of-a-whole representation of a fraction. they do not understand that the bottom number is telling what is being counted and the top number is counting how many parts one has of the available whole. as a result, for , for example, they might call 5 the numerator and 6 the denominator, yet the terms are meaningless to them (van de walle et al., 2016). misconceptions various conceptualisations of misconceptions exist in the literature. some theorists emphasise the causal aspect, while others focus on the consequences of misconceptions. in our opinion, both perspectives hold value and should be explored, as we do in this study. perspectives on the nature of misconceptions the causal perspective of misconceptions: for smith, disessa and roschelle (1994), misconceptions are flawed ideas that are firmly held by learners and that interfere with learning. drawing on the notion that learners do not enter the learning situation as blank slates, smith et al. argue that as they interact with new material, learners interlink ideas they already have. therefore, when learners construct knowledge, the activity of reconstructing and reorganising what they already know and synthesising new knowledge may include the synthesis of misconceptions. olivier (1989) asserts that misconceptions are errors made repeatedly, each time learners deal with similar situations, with the result that the repetitive misappropriation of a concept eventually becomes routine for the learner. vamvakoussi and vosniadou (2010) explain misconceptions as ‘synthetic concepts’ as they are the by-product of a broadening of knowledge of the number system from natural numbers to rational numbers. ojose (2015) describes misconceptions as misunderstandings and misinterpretations based on incorrect meanings. in this regard, we cite the faulty understanding of the equality of fractional parts as an example. frequently, the qualification that fractional parts are equal is misunderstood. learners might, for example, say that the parts in the rectangles in figure 2 are sixths, not taking into account that in these rectangles the whole is not divided into equal parts. figure 2: rectangle divided into six parts. prediger (2006) describes the obstacles faced by learners as didactic stumbling blocks created by the method of teaching, or epistemological obstacles stemming from the structure of mathematical content. one way or the other, that which misleads is the source of obstacles in the way of conceptual understanding. learners seemingly construct their knowledge based on their own experiences. if these experiences provide them with only limited views of a particular concept, this may close their minds to other related aspects of the concept. these narrow experiences result in constructions that inhibit further understanding and are called limiting constructions. pitkethly and hunting (1996) refer to them as inhibitors – mechanisms that inhibit the development of new and enlarged rational number knowledge. based on findings of other research projects, d’ambrosio and mewborn (1994) anticipated that fourth-graders in their study would have constructed many fraction schemes that would limit their further understanding. taking a different view to the studies mentioned above, however, they avoided labelling the children’s constructions as misconceptions because they believe it is important to view children’s constructions as objects for a study that can provide insight into and a new understanding of their thinking. the consequential perspective of misconceptions: rather than delving into the possible causes of misconceptions, sarwadi and sharhill (2014) focus on their result or effect, namely systematic errors as consequences of misconceptions. both the causal and the consequential perspectives imply that misconceptions are an undisputed reality and an inevitable part of knowledge development that will always be there; however, just as they came about, so they can be corrected and appropriately dealt with, if and when they surface. in our study, we started with the consequence (the error manifested) and moved to the cause. confrey (1990, p. 18) asserts: ‘a misconception is a “conceptual stumbling” block, inconsistent semi-autonomous schemes, and cognitive process responsible for errors in problem-solving’. manifested misconceptions in fractions resnick et al. (1989) classified misconceptions in fractions into three categories, of which the whole number misconception is particularly pertinent. the whole number misconception is what ni and zhou (2005) refer to as the whole number bias, limiting one’s view of numbers to whole numbers. bruce et al. (2014) hold that this bias is the result of an emphasis on whole number counting at an early age, which tends to reinforce a stubborn conceptualisation of numbers as whole numbers. in keeping with the argument of jigyel et al. (2007), durkin and rittle-johnson (2014) believe that the transition from natural numbers to rational numbers opens the way for the development of misconceptions. van de walle et al. (2010) argue that learners build on their prior knowledge of whole numbers, which may both support and inhibit their understanding of fractions. one such example would be the common misconception where learners over-generalise their understanding of whole numbers, subsequently regarding both the numerator and denominator as whole numbers and not as an element of the new fraction concept. ashlock (2010) describes the nature of manifested misconceptions as learners trying to make sense of fractions by either over-generalising or over-specialising the rules they know already. this implies that as much as previous knowledge is essential, it contains elements that can prejudice conceptualisation in fractions. machaba (2016) asserts that misconceptions are likely to arise when learners construct knowledge by reconstructing and reorganising prior knowledge with new knowledge. inappropriate linking of prior knowledge to new knowledge may jeopardise conceptual development. this study was conducted to explore the understanding of fundamental fraction concepts used in more advanced applications that are required at the grade 9 level. in this article, we argue that introducing numerical fraction notation before fraction concepts are properly understood could be regarded as malpractice. for this reason, we suggest the developmental intervention of using physical manipulatives as a remedy for ill-conceived fraction concepts before guiding learners to inductively construct algorithms and transitions to the more abstract applications of fractions, as required in grade 9. work on fractions by grade 9 learners was explored by posing the following research questions: how do grade 9 learners define, describe and represent the concept of fractions? what misconceptions are evident when learners define, describe and represent problems involving fractions? what are the root causes of learners’ misconceptions in the learning of fractions? research design and methodology in this qualitative study, an instrumental case study research design was employed, using a test and interviews as data collection methods. in the larger study, a test was administered, consisting of four questions with 15 items in total. for this article, we have chosen four question items because they are particularly relevant to our argument. the purpose of the test was to assess learners’ conceptual understanding of the concept of fractions. as learners define fractions, they demonstrate what they know and understand about this concept. eight learners were purposively chosen for an interview based on how they answered the test items. a grade 9 class of 40 learners completed the test on the topic of fractions. from the fp onwards, these learners had been exposed to fraction terminology, notation, calculations using all four operations, related applications such as rate and ratio, and problem-solving involving fractions. the first author marked the tests and classified each learner’s responses to each question as correct, incorrect or partially correct (see table 2). codes were randomly assigned to each learner to serve as references whenever necessary, for instance l1 for learner 1. the interview was tape-recorded so that the researchers could transcribe the information correctly. ethical considerations ethical clearance was obtained from the university of south africa, college of education ethics review committee (2018/11/14/46511/024/43/mc). the first author went through the required process, applying for ethical clearance from the university. since the study was conducted at a government school, he had to follow protocol and obtain clearance from all structures that govern the school up to the district level. he sought permission from the gauteng department of education (gde) to research one of its institutions. he requested permission from the principal of the school to conduct the study with grade 9 learners and described the purpose and the rationale of the study. the school governing body (sgb) was informed of the study in a meeting with the principal. after permission was granted, the grade 9 learners were informed of the study in good time so that they could decide whether to participate voluntarily in the study. those learners who agreed to participate were guaranteed anonymity and confidentiality. pseudonyms were used in the study, and responses were kept confidential. researchers made it clear to participants that the study had nothing to do with their course work, and would not contribute towards promotion marks. the main instrument for data collection learners’ written responses to the test constituted the main body of data analysed in the study. the four (out of 15) test items selected for this article and under discussion here are shown in figure 3. figure 3: written test (selection of four items). the introductory section of the test was not at grade 9 level, where the key concepts and skills include, among others, calculations with common fractions and mixed numbers using all four operations. the rational numbers in these operations include exponents and roots of fractions and require prior knowledge of multiples and factors, equivalent forms of fractions (common and decimal fractions and percentage), reciprocals and problem-solving with fractions. fractions then occur in applications in other mathematics content areas, other than numbers, operations and relationships, such as algebra, geometry and data handling in measurement contexts. whereas grade 9 is the exit level of the sp, the concepts and skills tested in the items in this test fell midway in the curriculum for term 1 of grade 4, the first grade of the ip. at this stage of their schooling career, learners should have transitioned from the fp with sufficient prior knowledge to complete the items in questions 1.1 to 1.4 in figure 3. this set of items could appear as is, in a grade 4 textbook. the testing of these basic fraction concepts and skills in the present study was intended to regress across grades and phases to determine the roots of problems manifested in grade 9 learners’ dealings with the complex requirements of the four test items. at face value, this leap would seem rather optimistic, since the researchers’ work with grade 9 learners had alerted them to the possibility of gaps in their knowledge originating as far back as term 1 of grade 4, if not further. findings in the initial step of the analysis, the outcomes of learners’ written responses were categorised as correct (c), partially correct (pc) or incorrect (ic). the use of tables, percentages and frequencies is associated with quantitative research; in this case, the researchers wished to determine the extent of the prevalence of misconceptions to enable valid inferences and further analysis. table 1 presents the classification of all 40 learners’ responses. the eight learners who were selected for follow-up interviews are highlighted in bold. table 1a: learner responses to test items 1.1 to 1.4 in three categories. table 1b: learner responses to test items 1.1 to 1.4 in three categories. the number and percentage of correct answers in items 1.1–1.4 are indicated in table 2. table 2: correct learner solutions per test item. discernible patterns of misconceptions and faulty understanding showed up in learners’ written responses. the most common misconceptions demonstrated in learners’ written responses are explained in table 3. table 3: most common misconceptions and faulty understanding in items 1.1 to 1.4. particularly concerning was the fact that these basic constituents of the fraction concept were lacking or faulty in an average of 85% of the grade 9 learners tested. the researchers assumed that learners who lacked an understanding of these basic fraction concepts would be unable to cope with the complex requirements of grade 9 fraction applications. below are the verbatim verbal responses of four of the eight learners (l6, l11, l15 and l17) who were interviewed after the tests had been marked. the interview responses supported the patterns we observed in the written responses as explained in table 3 and provided more insight into the reasoning that preceded the written responses. in the following section, we report various learners’ verbal responses to each question. to avoid repetition, for each question we selected participants whose answers reflected those of most of the interviewees. question 1.1 what is a fraction? table 5 contains selected verbatim responses from l6, l11, l15 and l17 during the interview. interviewees responded to questions and prompts from the researcher (r) about question 1.1. table 4: verbal responses of selected participants about question 1.1. table 5: verbal responses of selected participants about question 1.2. the main concerns arising from these responses were: the idea of fractions as equal parts of a whole is distorted. the phenomenon fraction is muddled, probably by the (too early) introduction of fraction notation as numerical symbols, which were probably misunderstood at the time they were taught. the routine use of diagrams such as two-dimensional geometric shapes or the picture of a pizza seems to have had limiting effects on true conceptualisation. the terms are used as empty and meaningless words. question 1.2 write the fractions represented by the following diagrams table 6 contains verbatim responses of l6, l11 and l38 on the questions and prompts of the researcher (r) during the follow-up interview. the responses pertain to question 1.2. table 6: verbal responses of l6 on question 1.3(a) and (b). verbal explanations by l6, l11 and l38 fairly represent the opinions of all interviewees. although test item 1.2 had shortcomings in that the idea of fraction-as-equal-parts of a whole could not be fairly tested by this item with three different fraction diagrams, the misunderstanding of fractions as unequal parts of a whole became particularly evident in learners’ answers to this item and was confirmed by their responses in the interview. the main concern here was that learners failed to recognise that the parts in diagram a (circle) were unequal and the shaded part did not qualify as a third, unlike those in diagram b and diagram c, which were equal parts. instead, when a shape was divided into three parts, for example, they interpreted all parts as thirds, irrespective of their size. question 1.3 (a) shade of the rectangular mat below. (b) write down in words what you will do to a shade of the rectangular mat figures 4 to 7 and tables 6 to 9 reflect the written and verbal responses about question 1.3 (a) and (b) by four selected interviewees, l6, l17, l34 and l38. figure 4: l6’s demonstration of question 1.3(a). figure 5: l17’s illustration for question 1.3(a). figure 6: l34’s illustration for question 1.3(a). figure 7: l38’s illustration for question 1.3(a). table 7: verbal response of l17 on questions 1.3(a) and (b). table 8: verbal response of l34 on questions 1.3(a) and (b). table 9: verbal response of l38 on question 1.3(a) and (b). this question tested very simply the idea of a fraction of a set made up of several objects. in this case, there are 15 blocks on a mat, with three blocks constituting one-fifth of the set. most learners misunderstood this conceptualisation of a fraction. instead, they tried to represent the numerical symbol of the fraction in a visual form by shading three blocks and five blocks. most of these learners shaded three blocks on top of five blocks, in keeping with fraction notation. working with 15 squares was difficult for some, so they used five little squares – the number they thought they needed. question 1.4: in the fraction , what does the 5 represent? what does the 3 represent? the responses in table 10 from l6, l15, l17 and l38 were common to all respondents. table 10: verbal response of l6, l13, l15 and l38 on question 1.3(a) and (b). this question was focused on the interpretation of a fraction in terms of its components. instead of explaining the meaning of each component, though, most learners provided a rote answer; they felt it sufficient to label 5 as the denominator and 3 as the numerator. how teachers had tried to explain fraction notation and the function of the numerals became clear from these answers; however, whether the numerals had meaning for the learners in terms of the fraction of which they formed part could not be determined. common among learners’ responses were notions such as: the idea of something being cut into parts, and then counted. the top number is out of the bottom number (like a test score, 3 out of 5). the position of the numerals: the top number and the bottom number. the top number is divided by the bottom number. discussion and conclusion one objective of this study was to gain insight into grade 9 learners’ level of understanding of the fundamental concepts of fractions. in the data collection process, all participants wrote a test and the researchers conducted a follow-up interview with 20% of the sample to establish how learners had reasoned while completing the test. the results discussed in the previous section led us to conclude that the understanding of basic fraction concepts that is required at the grade 4 level was almost non-existent among the grade 9 participants in this study. learners have a partial understanding of a fraction as part-of-a-whole the equality of fraction parts most learners used words such as ‘sharing’, ‘dividing’ and ‘cutting’ when defining a fraction; 50% of the learners defined a fraction as a part of a whole, and l15 explained it as ‘a piece of something complete’. ‘piece-whole’ is the most basic concept in fractions. however, it appears that most learners experienced difficulties in mastering this concept. this may have been the only meaning learners had encountered in the classroom. learners also had difficulty with the principle of identical pieces in the piece-whole relationship. they failed to understand that an object must be divided into identical pieces when defining and representing fractions. the question of how many identical pieces the whole was divided into, and of the number of these pieces that students were to colour in or select, was not understood. evident in these responses was an absence of the qualification that fractional parts are equal shares or equal portions of a whole unit (van de walle et al., 2016). for example, in this study learners referred to three partitions in an object as ‘thirds’, irrespective of the size of the parts. this resonates with what altıparmak and özüdoğru (2015) found: the learners in their study had partitioning misconceptions, that is, using unequal parts of a whole while adding. they also misunderstood how to add the numerators and denominators of fractions. in a study of turkish learners, deringöl (2019) found that they had difficulty in dividing a whole into equal fraction pieces: they demonstrated misconceptions by failing to indicate in their drawing that each piece was identical. it was evident from their incorrect definitions that learners’ faulty prior knowledge of fractions and errors such as dividing an object into unequal parts had influenced their conceptual understanding of fractions. learners in this study could not add new knowledge to their existing knowledge because they could not make sense of what they had already been taught (machaba, 2016). they could not organise, structure or restructure their experience in the light of available schemes of thought (van de walle et al., 2016). if they are to overcome these difficulties, the piece-whole relationship should be taught first to learners when teaching fractions to develop prior knowledge of the concept of fractions. learners will then be able to build on this knowledge as they continue to learn about mathematics. regarding the numerator and the denominator of a fraction as separate learners could define a fraction as a number formed by two numbers. l17 said, for example, ‘it is a number formed by two numbers’. learners could understand the numerator and the denominator in the fraction as separate values, but they did not realise that these numbers were related. they could not conceive that the numerator and the denominator represented a whole (deringöl, 2019; simon et al., 2018). they could see that a fraction was a pair of numbers representing quantities, but with no relationship implied between those quantities (stafylidou & vosniadou, 2004). they may have understood that the first number was reflected in the number of shaded parts and that the second number corresponded to the number of total parts. the scope of fractions no learner included fractions larger than 1 as part of their fraction definition. this led us to conclude that the part-of-a-whole conceptualisation had limited their understanding of fractions to values between 0 and 1. l6, for example, said, ‘when you eat a slice of pizza it is a fraction’. learners perceived a fraction as deriving from a full set of parts making up the whole. a key effect of this conception was their inability to conceptualise an improper fraction, as having more parts than the number of parts in the source of those parts, the whole, did not make sense to them (deringöl, 2019; kieran, 1981; simon et al., 2018). when learners related the fraction concept to real life, they referred to solitary items. our observation was similar to that of deringöl (2019) who argues that the ‘part-of-a-whole’ definition limits the development of the idea that a fraction can be greater than 1. fractions of a set learners’ definitions excluded the notion that fractions can be formed by part of a collection of discrete items. jigyel et al. (2007) found that this limitation may be mostly the result of linking fractions to pictures of shaded parts of a model such as circles or rectangles, and less frequently to part of a group. in our study, we observed that even when the geometrical figure constituted a set of equal parts (question 1.3 [a] and [b]), this was not recognised as such and learners struggled to make sense of it in terms of their unitary fraction concept. we conclude that the partial understanding of fractions might be traced back to the definition of a fraction as ‘part of a whole’, encountered in several south african textbooks, such as laridon et al. (2005). it was clear from learners’ responses that they had a limited understanding of the term ‘fraction’. terms used when learning fractions are hollow words learners could label but not explain the meaning of the 3 and the 5 in the fraction almost all said that 3 was the numerator and 5 the denominator, without explaining what this meant. learners did not comprehend the piece-whole relationship between the numerator and the denominator: they failed to understand that pieces constitute the whole and that the numerator is a piece of that whole, nor did they realise that the numerator and the denominator are related. most were unaware that the bottom part of the fraction indicates how many parts the whole is divided into and is the name of a fractional part. if the number is 5, for instance, it means we are counting the fifths. nor did they understand that the top part of the fraction tells them how many of the parts are shaded or identified. the top number counts or enumerates. the numerator thus indicates the number of fifths they should consider or count (van de walle et al., 2016). moreover, fraction jargon appeared to be procedural rather than relational, without the true meaning being attached to expressions such as part of a whole, equal parts, two over three, three divided by five, a piece of something, a section cut from something, numerator on top and denominator at the bottom. this corresponds to findings by jigyel et al. (2007) and deringöl (2019); in their studies, learners could not read fractions correctly as they did not fully understand the concepts of numerator and denominator. they would read as 3 over 5, for example, which obscured the meaning of numerator and denominator. if they had understood it, they would have read it as three-fifths: 3 as a number that counts and 5 as a number that indicates what is being counted. rather than relational understanding, learners should have developed an instrumental understanding of fractions, what skemp (1976) describes as ‘rules without reasons’. indeed, learners wavered between unrelated fragments of fraction learning; they tried various perspectives, such as the quotient sub-construct and the whole number perspective, yet their knowledge remained devoid of meaning and, as such, completely dysfunctional as a tool with which to solve algebraic or context-based problems at grade 9 level. getting to the root of the problem while we do not have convincing data from learners’ solutions strategies to support the root of the problem, the interviews seemed to suggest that the root cause of misconceptions was learners’ prior knowledge, the teaching paradigm they had been exposed to, teaching practices and the over-generalisation of the whole number to the concept of fractions. this is discussed further below. prior knowledge our findings revealed that learners’ prior knowledge of fundamental fraction concepts was either lacking or flawed. the resulting misconceptions affected their application of fractions in algebra and other contexts in grade 9. from a cognitive constructivist perspective, when a learner interacts with an experience, situation or idea – in this instance a fraction concept – one of two things may happen: either the new experience is integrated with the person’s existing schema, a process called assimilation, or the existing schema is adapted to accommodate the new idea or experience, a process called accommodation or adaptation (piaget, 1964a). in this study, there was almost no evidence of assimilation of concepts into the learner’s cognitive schema. there was little sign of integration of the idea of fractions (new experience) in learners’ existing schema, which in this instance could have been knowledge of natural numbers. new knowledge of fractions was not assimilated by the appropriate schemata (cognitive ideas). the new concept of a fraction did not fit with concepts that were already part of learners’ cognitive schemata. there was thus disequilibrium. the existing schema of natural numbers did not give meaning to new experiences with and ideas about fractions. learners lacked the basic conceptual understanding of numerator and denominator that would have enabled them to understand what fractions are all about. in this study learners engaged in a process of adaptation, where they merely revised their existing knowledge to accommodate the new knowledge. the notion of the fraction did not fit into their existing schema, and instead of altering by assimilating the new fraction concepts, they ‘forced’ the new ideas into their unaltered existing schema. for example, the new knowledge of fractions should have been assimilated into their knowledge of natural and whole numbers, thus altering their number schema conceptually. teaching paradigm from a cognitive constructivist perspective, we assign the failed acquisition of basic fraction concepts to, at least in part, the behaviouristic teaching of instrumental knowledge, which is the traditional pedagogical approach in many south african schools. in this style of teaching, learners passively receive the knowledge poured into them without being given the opportunity to construct meaning for themselves. in the classrooms in which this study was conducted, learners were seated in orderly rows, and group discussion was kept to the minimum. in a contrasting approach, cobb (1994) advocates that learners should construct mathematical knowledge experientially. in this way, they will learn to understand the world of their personal experiences. ultanir (2012) agrees that real understanding is constructed through experience and not bestowed on the learner. the findings of this study led us to conclude that a shift from the traditional approach of direct teaching to a constructivist approach is needed. in such an approach, knowledge acquisition is facilitated rather than bestowed; ideas are negotiated rather than imposed, and learners are actively involved in ‘doing’ mathematics rather than simply passively executing mathematics. it appeared from this study that learners were not engaged in constructing knowledge on their own by working through set tasks, allowing their minds to sift through the materials they were working with, and integrating new ideas with existing ones. constructivism rejects the notion that children are ‘blank slates’ with no ideas, concepts and mental structures. they should be encouraged not merely to absorb ideas as teachers present them but, rather, to be creators of their understanding. teaching practices some pedagogies seem to have had a limiting effect on conceptual understanding: most learners defined fractions from the part-of-a-whole perspective and their incorrect definitions were aligned with this sub-construct. we believe that the practice in teaching fractions of routinely shading parts of one whole geometric figure or cutting a pizza or a cake into pieces to indicate a fraction has to do with this limited conceptualisation (deringöl, 2019; van de walle, 2016). the part-of-a-whole sub-construct addresses area model type situations and learners in this had not been exposed to the continuous model, as evidenced by their failure to comprehend equivalence questions. this corresponds to gabriel’s (2016) argument that learners always consider fractions as entities smaller than 1; thus, they have difficulties placing fractions on the number line. traditionally, there is an over-emphasis on whole number teaching and insufficient transitional bridging to rational numbers. it appears that prior knowledge of whole numbers encourages misunderstanding of the concept of fractions (machaba, 2016). rote teaching and learning of the elements of fractions, such as fraction notation (jigyel et al., 2007) and terminology (maharaj, brijlall, & molebale, 2007), hamper true understanding. in this study, learners displayed rigidity in their dealings with equality, fraction representations and whole number over-generalisations. these aspects should become part of learners’ active vocabulary, but because teachers do not serve as models by using fraction terminology in their everyday classroom discourse, this does not happen. overgeneralisations this study found a generalisation of knowledge from whole numbers to work on fractions. fractions were thus regarded either as two separate, unrelated whole numbers or a whole number divided by another whole number. we concur with bruce et al. (2014) that this is the result of regarding a fraction as simply two whole numbers that can be treated separately. the fact that learners thought of the numerator and the denominator as separate values may have been because they used their knowledge of natural numbers when they encountered fractions. olivier (1989), machaba (2016) and deringöl (2019) all found the source of these misconceptions to be mostly an overgeneralisation of previous knowledge (knowledge that was correct in an earlier domain) to an extended domain (where it was no longer valid). this meant that learners who had been taught whole or natural numbers in their early stages of learning, for instance, later overgeneralised this knowledge and extended it to other knowledge about fractions; while fractions are taught, their representation with symbols such as numerator and denominator is introduced without setting the conceptual basis for these symbols. recommendations analysis of the data revealed gaps in essential aspects of fraction knowledge that should have been dealt with at an earlier stage in these learners’ education. we concluded that particular concepts were faulty. these included: equality of fractional parts the numerical conceptualisation of fraction notation the idea that the whole can comprise several objects the scope of fractions beyond 1. according to the specific knowledge and skills described in the curriculum assessment policy statement (caps) (dbe, 2011) document, these concepts are learned in grade 3 and grade 4, leading us to conclude that the problems that surfaced in grade 9 originated with the transition from fp to ip. however, the mode of teaching is also developmentally appropriate for each phase. drawing on the classical piagetian cognitive-developmental theory as summarised by mcleod (2009), we regard the movement from concrete to abstract learning as a cognitively appropriate progression. in this conceptualisation, fraction learning should ideally follow a developmental sequence: the initial encounter with the notion of fractions would typically be taught experientially, using physical manipulatives, such as cutting an apple into equal parts or grouping a set of balls in equal quantity sub-sets. this would be followed by pictures of fractions, such as sketches or photographs of the apple pieces or the groups of balls. after this, diagrams, for example geometrical shapes such as circles and rectangles, would represent the fraction of a whole or sets. once the concepts have been firmly established, learners can work with fraction symbols only, without the need for physical, pictorial or diagrammatic representations. from what we have observed of the way learners responded and explained their understanding during interviews, we conclude that the traditional way of teaching fractions is to start with diagrams. unfortunately, we could link several instances of fraction misconceptions to misinterpretations of diagrams and pictures. this suggests that teaching fractions primarily with the use of diagrams or geometrical shapes does not adequately support the development of fundamental fraction concepts. we found evidence that these diagrams often formed an integral part of the misconceptions (for instance, the typical ‘shaded part’). in keeping with our approach of cognitive constructivism, we would therefore suggest a rethink of the building process of the fraction concept, starting with physical manipulatives, even if this is regarded as inappropriate at the grade 9 level. grade 9 learners cannot be introduced to algorithms without understanding the basics of fractions. this implies that learners must unlearn their misconceptions and relearn and solidify the basic concepts by constructing meaning from basic fractions concepts. to design cognitively appropriate remedies, we therefore recommend that teachers diagnose misconceptions in the light of their manifestation, to establish at what point conceptualisation becomes problematic. the purpose is to deconstruct misconceptions at their point of origin by reconstructing foundational fraction concepts from the roots up. we believe that this is the path to structural cognitive modification, where new concepts are assimilated into existing schemata through relational (as opposed to instrumental) understanding. acknowledgements competing interests the authors declare no competing interest exists. authors’ contributions the involvement of the first author in this study was part of his master’s research degree. the involvement of the second author, as a supervisor, was more on reconceptualising the article and writing it. funding information this research received no specific grant from any funding agency in the public, commercial or not-for-profit sectors. data availability data were collected as a part of master’s degree, which will be uploaded in the unisa repository. disclaimer the views and opinion expressed in this article are those of the authors and 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(2010). elementary and middle school mathematics: teaching developmentally (7th ed.). boston, ma: allyn & bacon. abstract introduction literature overview and conceptual framework research design and methods statement 1: do you think learning without online mathematics games is boring? statement 2: do you feel comfortable playing online mathematics games? statement 3: do online mathematics games contribute to an understanding of abstract concepts? statement 4: do you feel the use of online mathematics games helped you in understanding algebra better? statement 5: do online mathematics games increase concentration levels of learners during lessons? statement 6: do online mathematics games arouse your interest in class? statement 7: is algebra difficult without being taught online mathematics games? recommendations conclusion acknowledgements references about the author(s) tichaona marange department of research, faculty of education, cape peninsula university of technology, cape town, south africa stanley a. adendorff department of research, faculty of education, cape peninsula university of technology, cape town, south africa citation marange, t., & adendorff, s.a. (2021). the contribution of online mathematics games to algebra understanding in grade 8. pythagoras, 42(1), a586. https://doi.org/10.4102/pythagoras.v42i1.586 original research the contribution of online mathematics games to algebra understanding in grade 8 tichaona marange, stanley a. adendorff received: 07 oct. 2020; accepted: 03 june 2021; published: 29 july 2021 copyright: © 2021. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract this study explores how online mathematics games contribute to grade 8 learners’ understanding of basic principles and more sophisticated aspects of algebra. this project documents the trajectory of a purposive sample of 30 grade 8 learners doing mathematics and one mathematics educator. the study is premised on the argument that learners with the guidance of the teacher can grasp algebraic concepts better and learn to manipulate these imaginatively and independently, by integrating new online mathematics games into standard classroom teaching of mathematics. the study was located within the interpretive qualitative research paradigm and used a case study approach. data were collected by means of (1) lesson observations, (2) questionnaires and (3) semi-structured interviews. the data collected were analysed through the lens of the sociocultural theory, social constructivism and the activity theory. this study supports the view, set out in the literature reviewed, that the way in which resources are utilised can substantially improve the teaching and learning of algebraic concepts. teachers should encourage learners to venture into the world of online mathematics games to learn algebra because they help learners to be creative, look for patterns, make conjectures, collect data, express their own thoughts, accept the ideas of others and establish structured forms of cooperation. the teacher’s role is to show and guide the learners how to use online mathematics games to solve mathematics problems. this study’s main recommendation, among others, is a revision of the curriculum to integrate online mathematics games into all subjects in classrooms at all levels. keywords: online mathematics games; algebra; paradigm; pedagogy; curriculum; case study. introduction as the result of complex historical, socio-economic and political factors, post-liberation south africa presently is embroiled in a predicament in mathematics education, especially in algebra, which has resulted in the country positioned last in the third international mathematics and science study (smith & hardman, 2014, p. 22). the scenario points to a need for novel, more attractive and engaging methods for teaching and stimulating its learning. learners who experience difficulties in grasping basic mathematics concepts often have poor prior knowledge in the subject. learners who are introduced to online mathematics games often gain self-motivation and intense interest (mclaren, mayer, adams, & forlizzi, 2017, p. 49). although algebra provides access to forthcoming studies and mathematically significant ideas, it often acts as a wall that blocks the path of many learners. as drijvers (2003, p. 2) points out, the use of a formal algebraic system creates confusion to many learners who struggle to grasp the basic first steps and subsequently fail to later catch up. online mathematics games introduce learners to fundamental steps of learning where they easily attain knowledge without consciously sensing that they are being taught through gameplay guided by the teacher. an online mathematics game enables learners who would otherwise have given up to flourish in mathematics. the aim, therefore, is to assess how and why online mathematics games attract, educate, engage and stimulate learners to find out more about algebra. there is limited literature on how online mathematics games advance learners’ understanding of algebra. becker (2005, p. 2) corroborates that while the potential value of online mathematics games is high there is limited in-class research focusing on the use of online mathematics games, partly because of considerable resistance from more conservative educators. most teachers demand proof of the effectiveness of online mathematics games as learning tools before attempting or adopting them in the classroom. to convince more teachers, it would be necessary to try online mathematics games through pedagogy – connecting elements of existing game designs with accepted learning and instructional theories (becker, 2005, p. 2). south africa is currently having a crisis in mathematics education, especially in algebra, which has seen it amass a lower aggregate score (379) compared to asian countries in the third international mathematics and science study (2019). given the potential benefits of online mathematics games to mathematics education nationally, including stimulating learners’ interest in the subject, placing this new technology in schools could arrest the deepening crisis in an affordable, democratic and timeous manner. additionally, this will enable shifts in pedagogical practices and potentially benefit learners’ learning. this study, therefore, aims to investigate how online mathematics games help grade 8 mathematics learners understand algebra. many authors argue that algebra teaching in south africa could be improved by providing suitable and well-designed online mathematics games that address learners’ weaknesses in algebraic concepts (jupri & drijvers, 2016, p. 2). the research question guiding this study is: how do online mathematics games contribute to algebra understanding in grade 8? in order to answer this general research question, the study asked the following sub-questions: (1) what aspects of grade 8 algebra are improved through learners’ exposure to online mathematics games? (2) how does an online mathematics game contribute to learners’ understanding of algebra in grade 8? literature overview and conceptual framework there is substantial evidence in the literature on the advantages of online mathematics games for teaching mathematics (barkatsas, kasimatis, & gialamas, 2009; becta, 2001; bokhove & drijvers, 2012; drijvers, 2003; jupri & drijvers, 2016; van ameron, 2003). learners live in a world experiencing the 4th industrial revolution where online mathematics games expose them to electronic information at a more rapid pace than when educators teach them. in that regard, the use of online mathematics games is a shift from the traditional instruction model of knowledge to an autonomous, active and collaborative learning process through learners’ engagement (paraskeva, mysirlaki, & papagianni, 2010, p. 499). online mathematics games wiersum (2012, p. 24) defines online mathematics games as activities that are (1) governed by rules, (2) have a clear underlying structure and (3) show specific mathematics cognitive objectives. the proponents of technology use in mathematics teaching point to studies that indicate that computers and calculators can support and enhance problem-solving environments. computers can decrease the amount of time required to master skills, allowing for more time to be spent on developing conceptual understanding (hardman, 2005, p. 1). by using online mathematics games, teachers would create a rich concrete learning environment as it is believed that we understand better where visuals are included. the use of online mathematics games is effective only when the games are used to encourage learners to think and make connections between objects and an abstract mathematics idea (murray, 2001, p. 28). focus should be on ensuring that learners are fully engaged mentally rather than seeing it as just a game. previous research on online mathematics games zoltan dienes’s work (1969 cited in moyer, 2001, p. 177) convinced researchers that the use of ‘multiple embodiments’ was needed to support learners’ understanding. online mathematics games have been proposed as a potential learning tool by both educational researchers (barab, thomas, dodge, carteaux, & tuzun, 2005; gee, 2003; squire, 2003) and online mathematics games developers (adrich, 2004; prensky, 2001). these scholars have argued that using online mathematics games in education invoked intense engagement in learners. they also argue that active learning is encouraged among the learners (garris, ahlers, & driskell, 2002). for ricci, salas and cannon-bowers (1996), there is empirical evidence that online mathematics games may be effective tools for enhancing learning and understanding of complex subject matter. on the other hand, other scholars such as mahmoudi, koushafar, saribagloo and pashavi (2015, p. 423) argue that there is no clear causal relationship between academic performance and use of online mathematics games. their view is that there is a lack of an empirically grounded framework for integrating online mathematics games into the classroom (mahmoudi et al., 2015, p. 423). this study used two online games, namely dragon box algebra and algebra meltdown. with dragon box, algebra concepts are simplified into a simple game and it is an innovative educational game that through gameplay teaches learners concepts relating to solving algebraic equations. dragon box covers the following algebra concepts: addition, multiplication and division. by controlling cards and attempting to confine the dragon box on the other of the online game, the learner progressively learns the tasks required to confine x on the other side of the linear equation. when playing the online game, the cards transform into variables and mathematics symbols. as learners progress through the levels, they earn dragons. algebra meltdown is an online mathematics game that can be used to solve linear equations. in this online game, learners unravel linear conditions by controlling them through the reactor. the game teaches learners to understand linear equations by using the number machine concept. these online mathematics games were designed to reinforce algebraic concepts. they are both intuitive and engaging and contain fun elements. learners develop skills to solve equations in a playful and colourful game environment, where they are encouraged to experiment and be creative (pearce, 2013). theoretical framework in order to have in-depth information on whether online mathematics games contribute to learners’ understanding of algebra, the study used the following theories: activity theory, socio-constructivism theory and sociocultural theory. these theories are relevant and help us comprehend the development of learners’ understanding in the process of learning mathematics. in addition, the chosen theories agree on the notion that knowledge is inseparable from practice. learners understand better by doing and mediating through tools and signs. the activity theory views the human mind as the product of our interaction with other people, objects and artefacts in the context of everyday activity (kaptelinin & nardi, 2006, p. 345). moyer (2001, p. 176) postulates that the abstract thinking processes of learners are closely anchored in their concrete perceptions of the world. the point is that the active use of online mathematical games allows learners to develop a repertoire of images that can be used in the mental manipulation of abstract concepts. engestrom (1987) used the model shown in figure 1 to explain how a wide range of factors work together to create an activity. activity theory can form a solid basis for developing online mathematics games that create concerted learning conditions, for example networks of practices, where mathematics learners cooperate with different subjects, articles and instruments of the online mathematics game, under stated guidelines, and make networks through division of labour. when learning algebra, the subjects used artefacts or instruments such as the dragon box online mathematics game. in the process, the learners were required to follow the rules for algebra meltdown and dragon box to understand algebra. dragon box covers the following algebra concepts: addition, multiplication and division. according to pearce (2013) when playing dragon box, the levelled puzzles start simple and easy, with graphics only. then gradually some of the images are replaced with numbers, letters and mathematics operators as well as adding complexity until learners are solving algebra without the need of graphics. when playing algebra meltdown at the beginning of each level, the researchers line up at the generator’s outlets, with discourse over their heads to demonstrate the iotas (number) they want. the player is given an equation and the player needs to find the correct number of atoms to put in the reactor. if the player chooses the wrong answer or takes too much time, the player will lose points. figure 1: activity theory diagram. engestrom and sannino (2010, p. 4) state that learners acquire something new when they figure out the concepts by themselves. the basic relations in action hypothesis as deliberated in this research are sketched out beneath: subject: this is the focal point of the examination; for our motivations the subject is the learner. mediating artefacts: these are apparatus that the subject utilises to follow up on the item space. altogether, the apparatus intervenes thought during the cooperation between the subject and the setting within an activity. crucially, apparatus are not non-partisan. they have built-up history of utilisation and convey inside them social implications (kaptelinin, 2005, p. 10). it is critical to consider the object as both material and ideal, that is, the item contains inside it both the subject’s inspiration (for going about) just as genuine material issue space is followed up on (kaptelinin, 2005, p. 11). first, as the item embodies the rationale for the presence of the activity, and as it is the subject’s inspiration that drives this, the researcher used interviews because they can be helpful apparatus for unloading thought processes. second, what the instructor says and does in the classroom empowers us to build up an image of what it is that the person is dealing with in the exercise; subsequently, recognisable proof of the devices utilised by the educator empowers one to recognise the item in the framework. rules: these are standards, conventions and social collaborations of the schoolroom, which drive the subject’s activities (hardman, 2005, p. 259). community: the subject is an individual from a network who takes up interest in following up on the mutual item. there is division of work inside the network, with obligations, assignments and control unceasingly being negotiated (hardman, 2005, p. 260). in this research, the network contains the educator and the learners who cooperate on a mutual issue in the mathematics classroom. in a wider sense, the teacher and learners are individuals from the network of the school. division of labour alludes to the arrangement of duties, errands and force relations inside a mathematics classroom just as all through the school. the introduction of the online mathematics games can possibly compel a move in the job of the instructor and learners, with learners working more as instructors of other learners in the mathematics classroom. the sociocultural theory is rooted in the concept that learning takes place in cultural contexts and is mediated by language and other symbol systems (john-steiner & mahn, 1996, p. 191). teachers need to be aware that the way in which learners think is largely influenced by their discrete culture, which informs each learner’s learning approach. individuals and their daily life environment are intertwined. the focus should be on activity within socially assembled, recognised situations (john-steiner & mahn, 1996, p. 191). constructivism was used to examine the ways learners built on their cognitive structures through online mathematical games. the online mathematics games were selected to generate cognitive disequilibrium within the learners’ existing conceptual structures. learners were required to accommodate new conceptual understandings, potentially attaining cognitive equilibrium. according to hein (2007, p. 1), constructivism is a theory of learning based on the idea that knowledge is constructed by the knower, based on mental activity. research design and methods the study follows a qualitative approach using an interpretive paradigm, with a case study design. qualitative research encompasses the study of groups of people so the researcher can guide and support the creation of a hypothesis (frederick, 2013, p. 1). a key element of qualitative research is that it generally draws from inductive reasoning processes to interpret and structure the meanings that can be derived from data. in inductive reasoning, the researcher uses the collected data to generate ideas (thorne, 2000, p. 68). according to adendorff (2007, p. 50), research applying the interpretative approach involves socially meaningful action through detailed observation of people in their natural settings. the goal is to arrive at understanding, and interpretation of how people create and maintain their social worlds. the interpretative paradigm was used to gain deeper understanding of why learners fail to understand algebra (cohen, manion, & morrison, 2011, p. 116). a case study was used as it provides a chronological narrative of events relevant to the case. this approach makes it possible for a single researcher to undertake the study without the need for a team. additionally, the results are easily understood by a wide audience as they are frequently written in everyday, non-professional language, mix a portrayal of occasions with an investigation of them, give an ordered account of occasions, permit researchers to hold the comprehensive and important qualities of genuine occasions, and focus on individual participants looking to comprehend their view of occasions and the researcher is necessarily associated with the case (cohen et al., 2011, p. 182). the study was undertaken at a high school in the metro north education district in cape town, western cape. the school accommodates all learners from different social backgrounds. the school employs english as the language of teaching and learning. the researcher chose a grade 8 class because that was the foundational class for high school mathematics. the school climate was conducive to learning which made it easy to engage learners. the school has limited technological resources: mathematics learners are not readily exposed to technological aids during lessons. the site was chosen because it was easily accessible and logistically convenient. in line with wegner’s (2007, p. 214) argument, convenience sampling was chosen as it allowed the researcher to select convenient sampling units. the choice of a single school allowed for an in-depth case study and addressed time constraints and accessibility problems. the participants for this study comprised 30 grade 8 learners and the mathematics teacher. the class was large enough to facilitate in-depth comprehension of the contribution of online mathematics games on algebra understanding in grade 8. thus, the sample consisted of 31. the composition of the sample is described below: grade 8 mathematics learners n = 30 grade 8 mathematics teachers n = 1 total sample size: n = 31 data collection data collection lasted four weeks in the month of august 2018. the principal and the teacher were informed about the study, and permission was granted. all participant learners provided informed consent with their parents signing the parent consent forms. the online mathematics games were used as supplement to the class instructions. no additional time was provided for playing the online mathematics games. the games were played during the regular class time for 15 minutes per period to reinforce the concept taught during that lesson and to make abstract concepts clear to the learners. learners were directly observed as they played online mathematics games during the lesson. as noted by creswell (2014, p. 190), the process of qualitative observation entails the recording of observed behaviour and activities of individuals at the research site. lessons were observed to determine the extent of the contribution of online mathematics games to learners’ understanding of algebra. the interactions and the enthusiasm of the learners during the lessons were also observed. a questionnaire was also used to gather data from the participants. through the questionnaire, participants’ perspectives and perceptions on the use of online mathematics games when doing algebra were recorded. according to van vuuren and maree (2002, p. 281), the use of questionnaires is economical, ensures anonymity, gives participants enough time to think about the answers they want to give and provides room for uniform procedures. the questionnaires were completed by all 30 participants in the research. they became the lens through which to measure and understand how the participants viewed their own development after using online mathematics games (schmidt et al., 2009, p. 130). moreover, interviews were conducted to generate feedback on the contribution of online mathematics games to learners’ understanding of algebra. participant learners were interviewed individually, with the researcher audio recording the conversations. the interviews focused on their views on online mathematics games. the interview questions for the learners were developed based on their experiences during the lessons. random sampling was used to choose six participant learners from a sample of 30 learners to take part in the interviews (creswell, 2014). the study utilises the inductive data analysis approach by employing data triangulation as it combined field observation and interviews. themes were identified and analysed to capture the essence of online game-based learning through the voices of those who have participated directly in its implementation. the study attained ethical clearance from the cape peninsula university of technology’s ethical committee for the faculty of education and the western cape education department. all ethical measures were complied with. the study granted anonymity to research participants to ensure confidentiality. this contributed to the researcher gaining the trust of the participants thereby ensuring that the data are trustworthy. analysis of results table 1 shows the participants’ perceptions on the contribution of online mathematics games to understanding algebra in grade 8 mathematics. table 1: participants’ perceptions on the contribution of online mathematics games. the results in table 1 uncovered the accompanying discernments concerning the contribution of online mathematics games to better understanding algebra in grade 8 mathematics by participant learners. statement 1: do you think learning without online mathematics games is boring? the results in statement 1 revealed that the majority (97%) of participants agreed that learning without online mathematics games was boring while only (3%) disagreed with the statement. statement 2: do you feel comfortable playing online mathematics games? in statement 2 the results revealed that most of the participants (83%) agreed that they felt comfortable playing online mathematics games while (17%) disagreed with the statement. statement 3: do online mathematics games contribute to an understanding of abstract concepts? statement 3 uncovered that most of the members (67%) agreed that online mathematics games contributed in understanding abstract concepts while (17%) disagreed with the statement. statement 4: do you feel the use of online mathematics games helped you in understanding algebra better? in addition, the results in statement 4 revealed that most of the participants (67%) felt that the utilisation of online mathematics games helped them understand algebra better while (17%) disagreed with the statement. statement 5: do online mathematics games increase concentration levels of learners during lessons? statement 5 revealed that most of the members (80%) agreed that online mathematics games increased concentration levels of learners during lessons while (20%) disagreed with the statement. statement 6: do online mathematics games arouse your interest in class? in statement 6 the majority of participants (80%) agreed that online mathematics games aroused their interest during lessons while (20%) disagreed with the statement. statement 7: is algebra difficult without being taught online mathematics games? statement 7 showed that most of the participants (80%) agreed that algebra was difficult when taught without online mathematics games while (20%) disagreed with the statement. looking closely at the learners’ questionnaire responses, the following game effects were observed: the online mathematics games had an experiential nature. that allowed learners to interact with familiar environments in the games and construct their mathematics concepts through completing game missions. the online mathematics games changed learners’ perceptions on mathematics. the learners understood the relationship between mathematics and real life. as a result, their mathematics phobia was diminished (paraskeva et al., 2010, p. 499). six learners were interviewed after they had used online mathematics games to solve algebra. each interview took 20 minutes and was audio-recorded followed by transcription. various themes emerged that indicated the fluidity of the teacher’s and learners’ perspectives and understanding of online mathematics games. in-depth probing was employed to obtain additional relevant information. the main themes identified from the analysis of interviews related to learners’ beliefs on online mathematics games and learner motivation. the following are some of the participants’ comments during interviews. learner 2 said that: ‘i didn’t really understand what equivalence was. the dragon box really helped me a lot because it simplified the algebraic concepts into a game. i now understand that both sides of the equal sign have the same value.’ the gameplay simplified the difficult algebra concepts as echoed by learner 2. as argued by star and seifert (2006, p. 290), algebra is a source of considerable confusion and negative attitudes among learners. this is because learners seem to find algebra difficult to understand. when learners find something difficult to understand, they become confused and bored. learner 3 said that: ‘for an expression like 2(a + b), i did not know that a and b are used to generalise a pattern. i used to get 4ab as my answer. but now after gameplay, i now understand the context 2(a + b) = 2a + 2b.’ this narrative confirms van de walle’s (2013, p. 257) view that the variables used in algebra take on different meanings depending on context. for instance, in the equation 4 + x = 6, ‘x’ is the unknown and 2 is the solution to the equation. but in the statement a(x + y = ax + ay, ‘x’ and ‘y’ are being used to generalise the pattern. failure by learners to understand the contexts lead to difficulties in solving the algebraic expression and equations. the questions asked during the interviews were important as they allowed the researchers to understand whether learners were familiar with the online gameplay as well as understanding the reasons for using online mathematics games. interviewing the learners helped to cross-validate data from observations made during the lessons. the learners were of the view that the online mathematics games positively affected their algebraic understanding, focus skills and motivation. the narratives below support the views noted above. learner 2 stated that: ‘when our teacher introduced gameplay in our algebra lesson, i started looking forward to the next lesson. the lessons were now vibrant, accommodating and inspiring. i now understand mathematics concepts more than i did when we depended on textbooks only. gameplay during lessons is the right way to learn mathematics.’ the interviews confirm that the learners’ achievements and motivation were positively impacted by the online mathematics games. further, the comments express feelings of increased enthusiasm and eagerness towards learning through computerised games. learner 5 commented that: ‘the first week the gameplay made me feel embarrassed. most of the learners were advancing to the next levels whilst i remained at the first. though the teacher kept on guiding and explaining on how to play, as i kept on playing, i ended up being angry with myself.’ this isolated incident revealed by learner 5’s comment shows that although learners can face challenges, they may continue to play until they win or advance to the next levels of the game. this determination by the learners, coupled with the frustration, can cause low self-esteem or aggressive behaviour, especially if the child keeps losing the game (annetta, minogue, holmes, & cheng, 2009, p. 80). although a few learners expressed negative responses regarding the online mathematics games, the interviews showed that to a greater extent the online games positively influenced the learners because online mathematics games help learners to ‘be creative, look for patterns, make conjectures, collect data, express their own thoughts, accept the ideas of others and establish forms of cooperation’ (siemon et al., 2016, p. 6). the common challenges learners experienced such as failure to understand basic algebraic concepts, namely variables, algebraic expressions and equations, were perfected. the games stimulate cognition, motivate the learners to remain engaged and promote learning in an integrated manner. the responses were varied but reflected individual progression in terms of how each participant developed their skills, knowledge of game play and confidence in their abilities. recommendations this research provides a starting point for looking into the integration of online mathematics games into the teaching and learning environment for mathematics. this research provides a platform for integrating online mathematics games into the teaching and learning environment for all conventional mathematics topics. based upon the findings and caveats of this research, teachers may safely encourage learners to use online mathematics games in the learning of algebra because online mathematics games help learners to be creative, to look for patterns, make conjectures, collect data, express their own thoughts, accept the ideas of others and establish forms of cooperation (siemon et al., 2016, p. 11). there is need for further research to be conducted on a larger scale, with many schools and in different areas to test the findings of this study on the contribution of online mathematics games to algebra understanding in grade 8 mathematics. these findings indicate that learners spend a considerable amount of time playing digital games and tend to identify with the characters they adopt or take on in the course of playing the game. thus, we need to further examine what boys and girls prefer in games in order to develop educational online games that suit different personality types and, more importantly, exploit multiple skills and intelligences, in true reflection of real-life collaborative environments, in the context of learning algebra specifically and mathematics in general. according to paraskeva et al. (2010, p. 503) online mathematics games must negotiate the difficult area between engaging the initial interest of learners, without becoming addictive and detrimental to academic performance. i have suggested that the above concern can be averted by integrating elements that limit play sessions and oblige learners to engage actively in an external educational task before continuing play, ideally in collaboration with other learners. alternatively, the online mathematics game itself can function as the basis for an innovative and motivating homework assignment. conclusion the relation between online mathematics games and self-esteem cannot be clearly defined yet, since research so far has yielded conflicting or ambiguous results (paraskeva et al., 2010, p. 503). given the crucial role of self-esteem in academic performance and social adjustment, however, it is worthwhile to investigate the issue further. finally, the suggested potential of online mathematics games with regard to improving mathematical understanding and development of algebraic concepts is an encouraging finding and should be more fully exploited. acknowledgements from the first author: my sincere gratitude is conveyed to the following people who made it possible for my research to be successful: my supervisor, dr stanley a. adendorff for the support and guidance you provided me since i started this journey. you have given me so much of your time and expertise. thank you for your ideas, suggestions and valuable insights and inspiration. my editor, dr rusenga, thank you for your meticulous proofreading and editing, constructive suggestions, inspiration and technical expertise in formatting and the layout of this thesis. my wife evelyn for the love, care, support and motivation which directly contributed to my achievement. thank you for not feeling ignored when i dedicated much of the time to my studies. my two children, moreblessings and asher, i thank them for their patience when i could not attend to their needs immediately. the principal mr young for giving me permission to conduct research at the schools as well as guidance and advice. competing interests the authors have declared that no competing interests exist. authors’ contributions t.m. was responsible for initiating and formulating the research, identifying the sample and collecting data, and writing up the research. sa.a. advised and guided the conduct of the research, proofread, edited and gave recommendations on the write-up, supplied resource materials and finalised the article. ethical considerations education faculty ethics committee (efec), cput (certificate number efec 1-6/2018). the researchers made it clear that participating in the research was voluntary and that if for some reason participants wanted to withdraw from it, they had the right to do so at any time. participants were informed that they would be protected from physical or psychological harm, discomfort or danger that might arise due to any research procedures. participating learners were told that the information gathered would be confidential and no one would know their identities except for the researchers. the names of the subjects were removed from all data collection forms as pseudonyms were used. the researchers obtained permission to conduct the research from western cape education department, cape peninsula university of technology, and the principal of the participating school. funding information this research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors. data availability data lists and summaries, figures and tables can be obtained from t.m.’s med thesis. disclaimer the views and opinions in this article are those of the authors and do not necessarily reflect the official policy or position of any affiliated agency of the authors. references adendorff, s.a. 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(2013). elementary and middle school mathematics: teaching developmentally (7th edn.). new york, ny: longman. van vuuren, d., & maree, a. (2002). survey methods in market and media research. in m. terre blanche, & k. durheim (eds.), research in practice: applied methods for the social sciences (pp. 269–286). cape town: university of cape town press. wegner, t. (2007). applied business statistics. cape town: juta & co. ltd. wiersum, e.g. (2012). teaching and learning mathematics through games and activities. acta electrotechnica et informatica, 12(3), 23–26. https://doi.org/10.2478/v10198-012-0026-2 article information author: michael mhlolo1 affiliation: 1faculty of humanities postgraduate studies, central university of technology, south africa correspondence to: michael mhlolo email: mikemhlolo@yahoo.com postal address: private bag x20539, bloemfontein 9300, south africa dates: received: 22 feb. 2015 accepted: 09 june 2015 published: 30 june 2015 how to cite this article: mhlolo, m. (2015). investigating learners’ meta-representational competencies when constructing bar graphs. pythagoras, 36(1), art. #259, 10 pages. http://dx.doi.org/10.4102/pythagoras.v36i1.259 copyright notice: © 2015. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. investigating learners’ meta-representational competencies when constructing bar graphs in this original research... open access • abstract • introduction    • basic-level constituents of a graph    • the constituent parts of bar-like graphs    • value bar graph    • distribution bar graph    • the histogram • methodology    • participants    • units of analysis • data    • unit 1: construction of a table    • unit 2: drawing the axes    • unit 3: construction of the bars    • unit 4: the final bar-like representation • discussion    • unit 1: constructing the frequency table    • unit 2: drawing the axes    • unit 3: constructing the bars    • unit 4: the final representation • implications • acknowledgements    • competing interests    • ethical considerations • references abstract top ↑ current views in the teaching and learning of data handling suggest that learners should create graphs of data they collect themselves and not just use textbook data. it is presumed real-world data creates an ideal environment for learners to tap from their pool of stored knowledge and demonstrate their meta-representational competences. although prior knowledge is acknowledged as a critical resource out of which expertise is constructed, empirical evidence shows that new levels of mathematical thinking do not always build logically and consistently on previous experience. this suggests that researchers should analyse this resource in more detail in order to understand where prior knowledge could be supportive and where it could be problematic in the process of learning. this article analyses grade 11 learners’ meta-representational competences when constructing bar graphs. the basic premise was that by examining the process of graph construction and how learners respond to a variety of stages thereof, it was possible to create a description of a graphical frame or a knowledge representation structure that was stored in the learner's memory. errors could then be described and explained in terms of the inadequacies of the frame, that is: ‘is the learner making good use of the stored prior knowledge?’ a total of 43 learners were observed over a week in a classroom environment whilst they attempted to draw graphs for data they had collected for a mathematics project. four units of analysis are used to focus on how learners created a frequency table, axes, bars and the overall representativeness of the graph vis-à-vis the data. results show that learners had an inadequate graphical frame as they drew a graph that had elements of a value bar graph, distribution bar graph and a histogram all representing the same data set. this inability to distinguish between these graphs and the types of data they represent implies that learners were likely to face difficulties with measures of centre and variability which are interpreted differently across these three graphs but are foundational in all statistical thinking. introduction top ↑ traditionally instructional focus in the statistics classroom has been on learners’ construction of various graphs with the instruction being didactic in nature but with little attention being given to the analysis of reasons why the graphs were constructed that way in the first place (friel, curcio & bright, 2001). similar concerns have been expressed by disessa, hammer, sherin and kolpakowski (1991, p. 157), who have suggested: one of the difficulties with conventional instruction … is that students’ meta-knowledge is often not engaged, and so they come to know ‘how to graph’ without understanding what graphs are for or why the conventions make sense. watson and fitzallen (2010) suggest that little is likely to be achieved by providing a collection of data (found in the textbooks) and having children practise drawing graphs in isolation. a recommendation that is consistent with current views of ‘data handling’ that goes beyond ‘statistics’ is put forth by shah and hoeffner (2002), who suggest that research on learners’ abilities to construct graphs, and how this relates to their ability to comprehend graphs, was particularly relevant for project-based activities in which learners create graphs of data that they collect for themselves. due to the fact that collected data are grounded in real-world contexts, disessa (2004) argues that an ideal environment is usually created for learners to demonstrate their meta-representational competence. such competence includes learners’ abilities to invent or design a variety of new representations, explain their creations, understand the role they play and critique and compare the adequacy of such representations. learners’ meta-representational competence is the very resource out of which expertise is constructed (disessa & sherin, 2000) and a number of researchers have used other terms such as phenomenological primitives (p-prims) (disessa, 1993, 2004), cues (davis, 1984) or ‘met befores’ (tall, 2008) in support of the existence of such a pool of knowledge. although previously activated knowledge structures (disessa, 1993) are acknowledged as critical resources, tall (2008) cautions that it should not be taken for granted that new levels of mathematical thinking are necessarily built logically and consistently on previous experience. empirical evidence has shown that the existence of prior knowledge can also lead to negative outcomes in the form of ‘misconceptions’ (english, 2012). given this dichotomous nature of prior knowledge, disessa and sherin (2000) suggest that we should understand this resource in more detail for its theoretical and practical import in learning. we should raise questions about the nature and content of these intuitive ideas, where they come from and how they are involved, both productively and unproductively, in learning. these are the questions that steered this analysis of grade 11 learners’ instructional activities during the process of constructing bar graphs. the learners worked with data that they had collected for themselves for a mathematics project that was part of their curriculum requirement. the article aims more specifically to tease out evidence of the knowledge representation structures that were stored in the learners’ memory and the extent to which this pool of knowledge was (in)adequate as a resource for bar graph construction. basic-level constituents of a graph given this objective, it is doubtful whether one could discuss adequacy, productivity or effectiveness in graph construction without making references to conventions that guide us in validating our concept of adequate, truth, correctness and accuracy in such mathematical activities. with this in mind it seems appropriate to develop an understanding of the way graphs are structured to appreciate the way in which they communicate information. in doing so i acknowledge watson and fitzallen (2010), who point out that due to the more recent emergence of the field of statistics there is more flexibility on what the conventions should be, unlike algebra and other areas of mathematics where conventions are more fixed. despite this variability in nomenclature and conventions, especially in statistical graphs, researchers warn that writing realistic assessment items and resources to mark them would not be easy if there was no movement towards convergence on conventions (kosslyn, 1989; shah & hoeffner, 2002; watson & fitzallen, 2010). consistent with this need to move towards convergence on conventions, this article borrows from kosslyn (1989) who suggests a schema for the analysis of graphs that can be used to communicate information clearly and concisely. kosslyn argues that even though there are many types of graphs they are all made up of the same basic-level constituents. the elements include the ‘background’, the ‘framework’, the ‘specifier’ and the ‘labels’ (kosslyn, 1989, p. 188). figure 1 illustrates the basic-level constituents of a typical graph. figure 1: the basic-level constituent parts of a graph. the background is the pattern over which the other component parts of a graph are presented. in most instances the background is blank as it is not necessary to include a pattern or picture. the framework represents the kinds of entities being related, in this case weight on the x-axis and speed on the y-axis. the specifier conveys specific information about the entities represented by the framework by mapping parts of the framework (in this example weight) to other parts of the framework (speed). the specifier may be a point, line or bar and is often based on a pair of values (x and y values). the labels of a graph are an interpretation of a line or region. they may be letters, words or pictures that provide information about the framework or the specifier. to analyse graphs it is necessary to understand the interrelated connections amongst these constituents of a graph. so how do these basic-level constituents help us to distinguish between the different types of bar-like graphs? the constituent parts of bar-like graphs although there is variability in naming these bar-like graphs in this article i adopt terminology used by cooper and shore (2010) as well as watson and fitzallen (2010). the decision was guided by what i viewed as (1) the consistency with which their work builds on kosslyn's (1989) work, (2) their long-standing history of contribution to making sense with graphs, (3) clarity in the way they exemplified the links between these graphs and (4) the need to maintain consistency in the discussion. watson and fitzallen (2010) posit that bar-like representations are of three major types (value bar graphs, distribution bar graphs and histograms), which are presented as historically developing from one into the other in that order. this article does not intend to dwell much on the historical development of these graphs but suffice it to say that, especially at primary and secondary school level, these bar-like representations are often simply referred to as bar graphs, so that their distinction is unclear. this is despite the fact that the differences between these bar-like representations merit an entirely different interpretation of centre and spread. according to cooper and shore (2010), it is only recently that more attention has been given to distinguishing between these graphs. watson and fitzallen (2010) use the following example to show the links and differences between these bar-like representations: ‘in a class of 12 children a survey was taken to find out how many books each child read. the results of the survey then generated the … data [shown in table 1]’. table 1: the number of books read by 12 students. value bar graph cooper and shore (2010) argue that the simplest and perhaps the most popular way in media and research articles would be to represent such data as shown in figure 2. figure 2: number of books read by each of the 12 children. such a representation is often encountered by learners as early as preschool and is typical of the way in which data is represented in elementary and middle school curricula. without discrediting other terms that have been used elsewhere, in this article i will refer to it as a value bar graph consistent with cooper and shore's (2010) terminology. similarly, records of rainfall throughout the year are usually presented in such value bar graphs with the vertical axis showing the amount in centimetres or inches and the horizontal axis showing the months of the year from january right through to december as in figure 3. figure 3: rainfall for beijing and toronto the critical distinguishing features in both cases (figure 2 and figure 3) are that bars represent values of single cases (number of books read by each child or the amount of rainfall that fell in each month) and in both cases the mean can be interpreted as the height at which all bars would be level as shown with the superimposed horizontal line in figure 3. one might notice that even the most rudimentary measure of variability (the range) is also perceived on the vertical axis (difference between the highest and lowest bars). other measures of variability in the data are also perceived through the vertical axis and would then be judged by deviations from the mean – the superimposed horizontal line in figure 3. notice that this superimposed horizontal could also have been drawn in figure 2 to enable visualisation of variability from the mean number of books read. admittedly such a representation would only be useful when dealing with a small number of cases or data, hence such ‘value bar graphs’ are suitable in elementary and middle school work. cooper and shore (2010) warn of misconceptions that manifest when this correct perception in a value bar graph is juxtaposed onto other more complex bar-like representations, resulting in learners incorrectly interpreting such measures. in order to appreciate this difference in perceiving variability in data, let us look at how the distribution bar graph is developed from such a value bar graph. distribution bar graph let me point out here that, historically, bar-like representations are rooted in geographical analysis of population statistics where a large amount of information was gathered (cooper & shore, 2010). despite the fact that different data representation techniques have been developed over the years the goal in data handling remains focused on analysis of large multivariate data sets; hence, learners should develop the skills of dealing with summaries (not cases) of large amounts of information. the same example of the number of books read by 12 children is used to show the transition from a value bar graph to a more complex distribution bar graph which aggregates data. looking across the data in table 1, there are five possible values the data could take: 0, 1, 2, 3 and 4. it is important to note that just like we could write the children's names in any order so we could also write the values in any order because in this context these are mere labels. the frequencies for each value are determined by the counts of children who read that number of books, as in figure 4. figure 4: distribution bar graph for the number of books read by 12 children. the resultant graph is an aggregation of data (distribution bar graph) as opposed to single cases that characterise a value bar graph. we immediately notice how in the distribution bar graph, the individual cases are lost as we can no longer tell how many books were read by each of the children. according to cooper and shore (2010), these two types of graphs (value bar graph and distribution bar graph) may superficially look the same. both have qualitative values (categories or case names) usually on the horizontal axis and numerical scale on the vertical axis. in each case the height (or length) of the bars represents the value of the data counts. however, the difference between the two graphs is that each ‘bar’ for a value bar graph represents data associated with an individual (number of books read by each child) whereas a distribution bar graph collects together number of books read and reports their total frequency. they also differ in that, visually, the method to judge variability is exactly the opposite. for example, the highest bar in a value bar graph measures the maximum score (highest number of books read by a learner) whereas the highest bar in a distribution bar graph measures the mode (the number of books read by most learners). these are clearly different measures, the former being a measure of variability and the latter being a measure of centre. to elaborate further on this point, if we superimposed a horizontal line for the mean (the height at which all bars would be level) in the value bar graph (figure 2 and figure 3),variability in the data (how far above and below the mean) is perceived through variation in the bar heights. on the other hand the centre for a distribution bar graph implies a typical categorical value (modal) found on the horizontal axis. furthermore, in the case of the distribution bar graphs, bars of approximately equal height indicate great variability, whereas for value bar graphs, the same visual display of approximately equal bar heights indicates little variability. so in summary, we notice immediately that in distribution bar graphs, measures of centre and variability are no longer perceived from the vertical axis as in the case of the value bar graph. for data sets that have a typical value (mode), the greater the frequency of that modal category compared to frequencies of other categories, the more alike the data are and thus the less variable the data. the more the data differ from the modal category, to the extreme point that there is no longer a concentration of values, the more variable the data. the extent to which the modal category's frequency stands apart from the frequencies of other categories therefore determines the appropriateness to refer to the mode as a typical value (cooper & shore, 2010). the histogram within the group of bar-like representations, the histogram is an innovation developed from the distribution bar graph. according to cooper and shore (2010), its use of bars makes the histogram visually similar to the two other types of graphs (value bar graphs and distribution bar graphs) discussed earlier and thus it can potentially be confused with them. categorical scales come in three fundamental types: nominal, ordinal and interval. whilst value bar graphs and distribution bar graphs usually plot nominal and ordinal data respectively, in a histogram, each bar represents the frequency of intervals of continuous data. i will use an example to illustrate how histograms represent continuous data. let us say we want to count the number of people in a region who are aged 50 years and older. however, we might not want to report a separate count for every individual case of the 1000 people that fall within this age range (a value bar graph) and neither do we want to report on an individual age from 50 to 100 (a distribution bar graph). this age range (50–100) could then be converted into interval scale by subdividing the full range into smaller ranges, for example, ranges labelled 50–59, 60–69, 70–79, 80–89, and 90–99. according to few (2005), an interval scale starts out as a quantitative scale that is then converted into a categorical scale by subdividing the range of values into a sequential series of smaller ranges of equal size (intervals) and by giving each range a categorical label. age is a typical example of a continuous variable and in figure 5 we see how the histogram summarises the data. figure 5: a histogram showing the distribution of ages of people in a region. histograms are best used with data where non-integers are actually possible; hence the bars are drawn adjacent to each other as they represent intervals of continuous data. the numbers on the horizontal axis correspond to the midpoints of the intervals (e.g. 55 in the first interval of 50–60), which determine where a particular data point gets counted on the histogram. due to the use of the midpoint value the raw data values are no longer accessible in a histogram. the reader therefore is less likely to calculate a measure of variability and even when an attempt is made, accuracy is lost in measures of centre such as the mean as they become more estimates. in a histogram the counting of a particular data point at the midpoint of intervals is supported by cooper and shore (2010), who argue that at times we may want to read the trend of the distribution. we can achieve this by creating a histograph or frequency polygon from a histogram. a frequency polygon displays data by using line segments connecting points plotted for the frequencies at the midpoint of each class interval. a histograph is used only when depicting data from the continuous variables shown on a histogram. given these conventions, the analysis then focused on the extent to which learners’ representations were consistent with or in violation of these conventions. methodology top ↑ participants this article works with archived data collected from four experienced (over seven years on average) grade 11 teachers, two male and two female (mhlolo & schäfer, 2012). twenty lessons on number, algebra and data handling topics were video recorded and transcribed, generating a 300-page database. this article focuses on the lessons from one male grade 11 teacher who was observed teaching data handling to a class of 43 learners. prior to the lessons, the learners had been tasked by this teacher to collect data on the number of children in different households around the school. this was for a mathematics project which formed part of their curriculum requirements. the lessons from which this article draws data could be described as learner-centred in that the teacher took more of a back seat and wanted to see how the learners would handle the data they had collected. this presented an ideal environment for the researcher to understand how the learners assimilated their prior knowledge in a typical problem-solving situation. the lessons were demarcated into four units of analysis and the criteria for demarcation are briefly discussed. units of analysis there is general consensus on the view that learners’ meta-representational competence is the very resource out of which expertise is constructed (disessa & sherin, 2000) and a number of researchers have used other terms such as phenomenological primitives (p-prims) (disessa, 1993, 2004), cues (davis, 1984) or ‘met befores’ (tall, 2008) in support of the existence of such a pool of knowledge. kosslyn (1989) suggests that in order to analyse learners’ meta-representational competence for graphs it is necessary to examine their understanding of the interrelated connections amongst three broad constituents of a graph: a frequency distribution table, a framework and a specifier. consistent with this suggestion, in this article, analysis unit 1 focuses on how learners created the table for the graph, unit 2 on how they drew the axes and unit 3 on construction of the bars. unit 4 was added to focus on the final bar-like representation that was drawn by learners. whilst connections between these interrelated constituents of a graph are necessary, an observation made by few (2005) was that most people walk through these choices as if they were sleepwalking, with only a vague sense of what works or why one choice is better than another. data top ↑ we pick up the conversation after the learners had drawn a frequency table on the board showing the results of the survey of the number of children in different households. initially the table had been drawn without the tally column. in the extracts below, ‘t’ stands for teacher, ‘l’ for learner and ‘chorus’ indicates a group response. unit 1: construction of a table t: so what do we do next after you have drawn the frequency table? chorus: we make tallies. we make a pie chart. we make a graph. [after a while it is agreed that the table should have tallies.] l1: [comes to the board and makes a tally of the number 8 as requested by the teacher.] t: have you ever seen something like this? chorus: yes t: where? chorus: last year. last of last year. the previous maths teacher. t: so the previous maths teacher showed you how to tally? ok, can you complete the table then. [the table is then completed as shown in figure 6.] figure 6: frequency table for the number of children in different households. unit 2: drawing the axes t: now after this information, how can you display this information? what it is like here, the information has been collected and now it has been organised. ok now how are you going to display this information? l2: in a graph. t: graph, we have different types of graphs and also we have different types of data. it's grouped and ungrouped. the way you display grouped data is not the same way as you display ungrouped data. so what type of a graph? l3: bar graph. t: can somebody show us how to go about it? l4: [comes to the board and draws two axes labelled as in figure 7.] t: ok what do you call this line? [points to the horizontal axis.] l5: the x-axis. t: now on the horizontal or the vertical ok you need to have either the number of children in each family and on the other you need to have maybe type of frequency whatever. l6: [comes to the board and labels the horizontal axis as ‘number of children in different families’. the vertical axis is labelled as the frequency axis.] figure 7: axes drawn for the bar graph. unit 3: construction of the bars t: now how are you going to display your data? where, ok here it is number of children [pointing to the horizontal axis]. we start with what? now because it's a bar graph, how would you put it here? like this is the bar [teacher drawing examples of horizontal and vertical rectangular blocks]. now how are you going to display your 0 and 8? l7: [comes to the board and draws the first bar in between 0 and 1 on the horizontal axis in figure 8.] t: is he correct? chorus: somehow, almost, maybe. l8: that bar shows a quarter and eight, ma'am. t: ok so the zero was supposed to be where? here? l9: [goes to the board and places a second 0 at the point where the bar intersected with the horizontal axis making the first bar sit between the two zeros as shown in figure 7 and figure 8.] t: so if the 0 was here he would be correct. l10: maybe it's incorrect. l11: it's incorrect. t: let's see if you can put the bar for 1 and 14. let's see. let's try. l12: [comes to the board and places the second bar showing a frequency of 14 as shown in figure 8 and figure 9.] l13: [commenting after the second bar had been drawn] it's wrong. t: [asks yet another learner to draw the bar showing 2 and 20. an interesting observation made is that whilst the first two bars had been drawn adjacent to each other, this third bar was disjointed as shown in figure 8. the graph was re-drawn for clarity (figure 9).] [after some long discussions on whether or not the graph was representing the data accurately, it was erased.] l14: [comes to the board and draws another new set of axes. the zero which was at the intersection of the horizontal and vertical axes is then removed leaving the second zero and the other values as they were on the abandoned axes.] chorus: [learners take turns to draw the bars on this new set of axes as shown in figure 10.] figure 8: graph showing the number of children in different households. figure 9: learners’ bar-like graph of survey data. figure 10: bar-like graph of the number of children in different households. unit 4: the final bar-like representation t: if you were to display something like this to a person who doesn't know mathematics will that person be in a position to read? ok remember that you have organised your data and now you are displaying your data, can a person be able to read this? l15: i think maybe you have to label whether which side is talking about number of children and the households. [this comment came because the axes had not been labelled.] t: ok now turn to the notice board. look at the graph of inflation. this type of graph is called a bar graph. look at it and the one we have just drawn. what is the difference? [there was a graph in class showing inflation rates from 1999 to 2009.] chorus: the spaces, it's decorated, it's neatly displayed. [lesson ends] discussion top ↑ the questions steering this analysis were: what is the nature of learners’ prior knowledge for graphs? where do these ideas come from? how are they involved both productively and unproductively in the process of constructing bar graphs? each unit of analysis attempts to answer these three questions. unit 1: constructing the frequency table from the discussion that took place during the process of making a frequency table for the collected data, it is evident the learners brought the knowledge of tallying from the ‘previous teacher’. it can be argued that the knowledge of tallying was neither supportive nor problematic since with or without the tally column the students would still have been able to construct a correct bar graph. however, the agreement by learners that frequencies should be ‘tallied’ opened up a number of questions about their procedural and conceptual understanding of tallying. let us recall that a tally is a mark used in recording a number of acts or objects, most often consisting of four vertical lines cancelled diagonally or horizontally by a fifth line. tallying or counting is the act of finding the number of elements of a finite set of objects through a one-to-one correspondence. it is meant to avoid visiting the same element more than once. after tallying the value of the final object gives the desired number of elements (cardinality) in that set. so if the learners’ frequency table had a column of frequencies, by implication tallying had already been done. therefore from the learners’ wanting to tally the number 8 or 14 or 20 (frequencies) it can be concluded that the purpose of tallying and when it should be done were not clear to them. this suggests that learners had a superficial understanding of the concept. unit 2: drawing the axes when prompted to show the information on a bar graph, what is evident is that learners brought their prior knowledge of a framework of a graph with an x-axis and a y-axis intersecting at 0 and scaled on both axes as shown in figure 7. students meet this type of framework more often when solving equations graphically. was this prior knowledge supportive? to a certain extent this prior knowledge was supportive for, according to friel et al. (2001), graphs share similar structural components. the framework of a graph as discussed earlier gives information about the kinds of measurements being used and the data being measured. the simplest framework has this l-shape that learners drew, with one leg (x-axis) standing for the data being measured and the other leg (y-axis) providing information about the measurements that are being used. this was important for the learners to be able to represent their data on a bar graph. however, to a larger extent, it is evident that their prior knowledge of axes was not very productive as they later struggled to draw the bars for their data. when both the x-axis and the y-axis have numerical information, as was the case in this task, learners needed to have a deeper knowledge of numbers in order to figure out which numerical information goes onto which axis. curcio (1987) reports that the mathematical contents of a graph, that is, number concept, relationships and fundamental operations contained in it, were factors in which prior knowledge seemed necessary for graph comprehension. the recommendation was that the relationship between the subject matter of number and choice of graph form should be further investigated. it is evident that learners did not have a clear understanding of this relationship. by drawing a framework of a graph with an x-axis and a y-axis intersecting at 0 and scaled on both axes learners implied a functional relationship between the variables depicted on the axes. yet bar graphs by convention are not used to convey functional relationships (follettie, 1980) because such a graph of categorical data displays the relative magnitudes without implying a functional relationship. therefore, conventionally a bar graph of categorical data would have a scale only on its frequency axis. in a similar study on high school and college students, delmas, garfield, ooms and chance (2007) also speculate that learners do not actually understand what the axes represent. friel and bright (1995) caution that interpreting graphs that utilise two axes may present difficulties if the nature of data that they represent across different graphs is not explicitly recognised. when considering graphs with any of these frameworks as tools for data reduction, one should note the differences in the nature of data that are represented on these axes. in the case of a value bar graph, distribution bar graph or histograms, the major difference is in what is represented on the x-axis. for example, in a value bar graph drawn with vertical columns, the columns are positioned over a label on the x-axis that represents a nominal measure. a nominal measure refers to data that consist of names or categories so that the data cannot be arranged in any specific ordering scheme. the nominal level of measurement occurs when the observations do not have a meaningful numeric value, for example numbers assigned to soccer players. the values of nominal variables cannot be meaningfully compared to see if one is larger than another, cannot be added, subtracted, multiplied or divided nor can the mean be calculated (what most people call the average). so in this case, the x-axis does not have a low end or a high end, because the labels on the x-axis are categorical and not quantitative. learners get experience of such categorical bar graphs much earlier than functional graphs. they draw graphs of weather in a week where the horizontal axis is labelled with the days of the week as early as grade 1. so one can argue that learners’ pre-knowledge of symbolic functional graphs where the numbers on the x-axis represent a scale like on a number line was a stumbling block to understand how to represent categorical data as labels without a scale or order. unit 3: constructing the bars after drawing the axes, it was evident that learners did bring their prior knowledge of matching the height of bars with the frequencies (see figure 8 and figure 10). generally a bar graph plots the number of times a particular value or category occurs in a data set, with the height of the bar representing the number of observations of that score or that category. it is evident from figure 8 and figure 10 (see marks placed between 5 and 10 and 10 and 15 on the vertical axis) that this knowledge was productive in terms of matching precisely the height of bars for 8, 14 and 20 with the frequencies. the problem however surfaced in terms of where these bars sit. by placing the 0 at the origin the class struggled to draw the first bar showing 8 families with 0 children each and the subsequent bars were also problematic. this suggests that learners were unable to distinguish the data set that they were dealing with. distinguishing between sets of data as discrete cases, discrete categories or grouped numerical data along some scale is a critical factor for constructing appropriate representations of the data. in all the three representations of categorical data, that is, value bar graphs, distribution bar graphs and histograms, categories of the variable are typically marked at the midpoints of the category on that particular axis (horizontal if it is a column graph and vertical if it is a bar graph). from the way learners drew their bars, it is evident that this convention was not recognised as their bars were sitting on two different numbers at the same time. another evident failure to recognise a convention was that at times learners drew joint bars as in the histogram and at times disjoint bars as in a bar graph, yet conventionally histograms must have joint bars and bar graphs must have disjoint bars. a study on learners’ conceptual understanding of statistics by delmas et al. (2007) identified learners’ inability to recognise critical differences between histograms and other graph types that use bars. this would have been expected given that empirical evidence shows that at school level these graphs are usually referred to as bar graphs and only recently has more attention been given to distinguishing between these graphs (cooper & shore, 2010). unit 4: the final representation let us recall that the learners wanted to represent their own collected data on a bar graph. the question then is: to what extent did they achieve this objective? we notice from the basic-level constituents discussed earlier that the learners’ representation is neither a value bar graph, nor a distribution bar graph nor a histogram. whilst the heights of bars matched with the frequencies, they were joint bars and were sitting on two different values on the horizontal axis in violation of the midpoint convention that guides where bars should be located in value bar graphs, distribution bar graphs and histograms. the overall mathematical outcome here was something close to a histogram but did not represent the original data set particularly well, either in terms of mathematical structure and convention or with reference to the real-world situation being represented. this suggests that learners’ meta-representational competences were inadequate for bar graph construction. when numbers are used in bar graphs, the axis that assumes a categorical scale could represent three fundamental types: nominal, ordinal and interval data. these categorical contexts of number are problematic even with adults given that the majority of time spent on number and operations in the earlier grades focuses on numbers in their quantitative contexts, with learners usually encountering the categorical contexts of number only when dealing with data handling tasks. this suggests that to communicate effectively using graphs, one has to understand the nature of the data, graphing conventions and a bit about visual perception. without guiding principles rooted in a clear understanding of graph design, choices are arbitrary and the resulting communication fails to represent the information effectively, as was the case in this class. implications top ↑ this article has both theoretical and practical implications. in terms of theory, this article has shown that due to the more recent emergence of the field of statistics, there is much more flexibility in nomenclature and lack of convergence on what the conventions should be. watson and fitzallen (2010) show how for example at both primary and high school levels, these bar-like representations are often simply referred to as bar graphs so that their distinction is unclear. yet from this article it has been shown that the methods of judging both centre and variability are clearly different across such bar-like representations. cooper and shore (2010) show how an understanding of measures of centre and variability was the single most important foundational concept in all statistical thinking. so in order to teach these concepts effectively, curricula need to be constructed and implemented carefully; writing realistic assessment items plus having the resources to mark them is not easy if graphs continue to be referred to loosely as bar graphs. all this points to the need to converge on some specific naming of these bar-like representations and this article suggests that cooper and shore's way of distinguishing between value bar graphs, distribution bar graphs and histograms guides us towards such convergence in nomenclature. in terms of concept formation, as long as these bar-like representations are referred to loosely as bar graphs, learners will not make connections between the different graphical representations of quantitative data and their corresponding ways of conveying information on measures of centre and variability for that data. research indicates that learners entering college may have only a superficial understanding of centre and variability and are likely to have particular difficulty extracting information about those measures when data are presented in graphical form (cooper & shore, 2010). yet franklin et al. (2007) maintain that an understanding of variability in data is the single most important foundational concept in all of statistical thinking. a solution to this problem might be addressed by this convergence in conventions as suggested in this article. in terms of practice, this study argues that knowing the ways in which these types of bar-like graphs (value bar graphs, distribution bar graphs and histograms) represent certain types of data may help teachers make decisions about the level of complexity for instruction. whilst the so called ‘bar graph’ is often encountered by students as early as preschool, this article argues that the level of complexity of categorical data that is handled by learners at that early stage is low. this is the kind of data that is best represented in what has been defined in this article as the value bar graph. friel et al. (2001) show that the transition from these case value bar graphs to distribution bar graphs may be confusing if this transition is not carefully considered and explored because the axes must be redefined. this confusion is evident in this article: learners wanted to draw a bar graph but they ended up with something close to a histogram, suggesting that they could not distinguish between these types of bar-like graphs. the view is that teachers should create a gradual transition from drawing graphs with objects themselves (value bar graphs) to the more abstract distribution bar graph (rangecroft, 1994). a similar suggestion put forth by franklin et al. (2007) was that both primary and secondary learners engage in tasks that require them to integrate deep understanding of graphical representation along with measures of centre and spread through a steady progression from value bar graphs, through distribution bar graphs to histograms. acknowledgements top ↑ i acknowledge the department for international development for funding the phd study from which this article is drawn. the views expressed in this article are not necessarily those of the funders. competing interests the author declares that they have no financial or personal relationships that may have inappropriately influenced them in writing this article. ethical considerations the department of education granted approval to proceed with this study under permit t-728 p01/02 u-848. references top ↑ cooper, l.l., & shore, f.s. (2010). the effects of data and graph type on concepts and visualisation of variability. journal of statistics education, 18(2), 1–16. curcio, f.r. 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(1989). understanding charts and graphs. applied cognitive psychology, 3, 185–226. http://dx.doi.org/10.1002/acp.2350030302 mhlolo, m.k., & schäfer, m. (2012). towards empowering learners in a democratic mathematics classroom: to what extent are teachers’ listening orientations conducive to and respectful of learners’ thinking? pythagoras, 33(2), 79–87. http://dx.doi.org/10.4102/pythagoras.v33i2.166 rangecroft, m. (1994). graph work – developing a progression. in d. green (ed.), the best of teaching statistics (pp. 7–12). sheffield: the teaching statistics trust. shah, p., & hoeffner, j., (2002). review of graph comprehension research: implications for instruction. educational psychology review, 14(1), 47–69. http://dx.doi.org/10.1023/a:1013180410169 tall, d. (2008). the transition to formal thinking in mathematics. mathematics education research journal, 20(2), 5–24. http://dx.doi.org/10.1007/bf03217474 watson, j., & fitzallen, n. (2010). the development of graph understanding in the mathematics curriculum: report for the nsw department of education and training. sydney: nsw department of education and training. available from http://www.curriculumsupport.education.nsw.gov.au/primary/mathematics/assets/pdf/dev_graph_undstdmaths.pdf abstract introduction constituents of early number concept development methods results discussion acknowledgements references footnotes about the author(s) hanrie s. bezuidenhout department of childhood education, faculty of education, university of johannesburg, johannesburg, south africa elizabeth henning department of childhood education, faculty of education, university of johannesburg, johannesburg, south africa citation bezuidenhout, h.s., & henning, e. (2022). the intersect of early numeracy, vocabulary, executive functions and logical reasoning in grade r. pythagoras, 43(1), a646. https://doi.org/10.4102/pythagoras.v43i1.646 original research the intersect of early numeracy, vocabulary, executive functions and logical reasoning in grade r hanrie s. bezuidenhout, elizabeth henning received: 16 aug. 2021; accepted: 06 may 2022; published: 15 sept. 2022 copyright: © 2022. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract the current quantitative study, a naturalistic field experiment, was conducted in a public primary school in soweto, johannesburg, with the objective to examine how children’s achievement on four assessments at the beginning of grade r, namely their numeracy, their mathematics-specific vocabulary, their executive functions, and their logical reasoning capabilities, predicted their performance on a numeracy assessment at the beginning of grade 1. a purposive intact group of 59 participants was assessed at the beginning of their grade r year and again when they entered grade 1. the results of the study indicate that, apart from existing or prior numeracy knowledge at the beginning of grade r, mathematics-specific vocabulary was the strongest predictor for numeracy attainment at the beginning of grade 1. we suggest that early grade teachers consider young children’s number concept development as a cognitive, developmental psychology phenomenon and that they help learners build a lexicon of mathematics-specific qualifiers in their teaching with words that represent concepts of, among others, space, position, comparison, inclusion, sequence and magnitude. keywords: early numeracy; mathematics-specific vocabulary; number concept development; executive functions; logical reasoning; early grades pedagogy; grade r. introduction grade r children’s mathematics-specific vocabulary is a strong indicator of their achievement in grade 1 numeracy (purpura & logan, 2015; toll & van luit, 2014). grade r numeracy also predicts grade 1 achievement since mathematics concepts have been shown to develop hierarchically (bezuidenhout, 2018; clements & sarama, 2014; fritz, ehlert, & balzer, 2013). executive functions (i.e. working memory, inhibitory control and cognitive flexibility) (cragg, keeble, richardson, roome, & gilmore, 2017) and logical reasoning (morsanyi & szücs, 2015; nunes et al., 2007) have also been shown to significantly contribute to number concept development. after identifying these four factors as critical contributors for grade 1 numeracy from the literature, we designed an experiment to investigate predictive correlations between grade r numeracy, mathematics-specific vocabulary, executive functions and logical reasoning, and numeracy attainment at the beginning of grade 1. we were specifically interested in the sample we have chosen because the school from which the sample was purposefully selected follows a custom-designed, dual language of instruction model: the children’s home language (isizulu or sesotho) is their language of instruction in grade r but in grade 1 their tuition changes to a dual language modality when they learn mathematics through the medium of english, with some code-switching in class discussions (henning, 2012). the sample was selected from a public school in soweto, johannesburg. we propose the argument that the development of four constituents, namely numeracy, mathematics-specific vocabulary, executive functions and logical reasoning, during grade r collectively prepares children to develop numeracy concepts in grade 1. specifically, in a context where the language of instruction in grade 1 differs from that of grade r, we view mathematics-specific vocabulary as a particularly important enabler for learning mathematics in grade 1. our data confirmed that, apart from grade r numeracy, mathematics-specific vocabulary is the strongest predictor for grade 1 numeracy achievement. the study was undertaken because of the ongoing concern in south africa about children’s consistently weak performance in primary school mathematics (department of basic education, 2014, 2017; southernand eastern africa consortium for monitoring educational quality, 2007, 2017; trends in international mathematics and science study, 2015, 2019). gustafsson (2019) gives an overview of some of these results. recognising that young children’s poor attainment, and possible ensuing learning difficulties in early mathematics assessments, is a worldwide concern (chinn, 2015), we acknowledge that it is a specifically serious concern in south africa. one aspect of this is that in many schools in the country, children learn mathematics through the medium of english, which is not their home language. some children learn mathematics in a dual language mix in the early grades, which may further compromise their learning (henning, 2012). it is not clear how children map their early mathematics concepts onto linguistic representations. odic, le corre and halberda (2015) investigated such mapping and found that language is used to ‘point down’ to approximate numbers before approximation does the obverse. we regard this as one of the important findings in recent research, indicating that semantic mapping cues conceptual understanding. this view is also proposed by spelke (2017), in a proposal that natural language is the source of concepts more than cognitive evolution or culture. the argument that dowker and nuerk (2016) propose in this regard is that familiar vocabulary is crucial for forming linguistically named concepts, much as was argued by vygotsky (1986), who proposed that there is a constant pattern of interaction between the development of concepts (such as number concepts) and the development of language (kozulin, 1990). dowker and nuerk (2016) accentuate not only vocabulary, but also linguistic structures such as syntax, grammar and morphology. we propose that linguistic representation intersects with cognitive modelling. with this view in mind, we argue that young children rely on vocabulary that represents a concept as a ‘semiotic mediation’ tool (henning, 2013; vygotsky, 1986) for learning. as the data of our study suggest, many children struggle to link the english mathematics vocabulary that they encounter in grade 1, after they had been initiated into the vocabulary of the sesotho and isizulu languages of their grade r classrooms and their homes and communities. vygotsky (kozulin, 1990) proposed that children engage in ‘inner conversation’ or ‘inner speech’ to reason logically about concepts. in this sense, there is thus a relation between the development of number concepts, language (also in the form of inner speech) and logical reasoning. the skill of being able to reason logically about properties of numbers and the relations between numbers is required for the development of numeracy skills (nunes et al., 2007). we add to this argument by proposing that children may need to switch back and forth between their home language inner speech and their interpretive and expressive language in english when they encounter tuition in english. this may increase the cognitive load of the working memory and other executive functions. the research question of the study is: what are the predictive associations between children’s achievement on grade r1 assessments of numeracy, mathematics-specific vocabulary, executive functions and logical reasoning, and a numeracy assessment in grade 12? constituents of early number concept development early numeracy the importance of early development of number concepts has been shown in many studies (aunio & räsänen, 2016; balala, areepattamannil, & cairns, 2021; clements & sarama, 2014; desoete, 2015). research from these and other studies has shown that a variety of early numeracy skills predict overall later mathematics performance. in particular, counting (sarnecka & carey, 2008; wynn, 1990), calculation, number line and magnitude comparison (lefevre et al., 2010), ordinality (lefevre et al., 2013), cardinality (frye, braisby, lowe, maroudas, & nicholls, 1989; sarnecka & wright, 2013) and numeracy-related logic (aunio & niemivirta, 2010) have been shown to contribute to mathematical skills and concept development. fritz and her colleagues (fritz, ricken, balzer, willmes, & leutner, 2012; fritz et al., 2013) identified five levels of number concept development. while each level describes a specific level of development, comprising the concepts of, first, counting, and then ordinality, cardinality, part-part whole understanding and relationality of numbers, the conceptual levels collectively describe a continuum of number concept development. these levels informed the development of the assessment instrument we used to assess numeracy, namely the marko-d sa (mathematics and arithmetic competence diagnostic, as translated from german). this interview-based test was developed in germany and translated into four south african languages. it has also been validated in south africa (henning et al., 2021; bezuidenhout, henning, fitzpatrick, & ragpot, 2019). the test items were designed to assess the specific concepts assigned to each of the five levels of number concept development. around the age of 2, children begin to distinguish number nouns from other parts of speech and realise that these words refer to quantities, although they are not yet able to connect a specific word to the corresponding quantity (fritz et al. 2013). they learn the order of the ‘count list’ and use their fingers or objects to link number words to objects which they count in a one-to-one correspondence (level 1: counting). after learning the ordinal properties of numbers (level 2: ordinality), young children develop a sense that each number is not only part of a sequence, but also represents a specific quantity (level 3: cardinality). they learn that the quantity can be decomposed into a specific number of units, for example 5 = 1 + 1 + 1 + 1 + 1, and that the quantity, or the set, can be broken into smaller sets, such as 5 = 3 + 2 or 5 = 4 + 1 (level 4: part-part-whole). this understanding brings about the realisation that numbers have relationships with each other, an important one of which is that the next number on the number line is always one more and that the one before is one less (level 5: relationality). such combined understanding of the principles of counting, ordinality and cardinality makes it possible for children to determine the relationship between cardinal units. language for learning number concepts several studies have shown a relationship between early mathematics learning and language (davidson, eng, & barner, 2012; negen & sarnecka, 2012; purpura, hume, sims, & lonigan, 2011; yang, dulay, mcbride, & cheung, 2021). an explanation offered by dowker and nuerk (2016) for this relationship is that frequent exposure to mathematics-specific vocabulary increases the chance of a child developing an understanding of the conceptual properties of words. gopnik and meltzoff (1997) argue that: [a]aspects of linguistic input can have quite striking effects on conceptual development. children who hear language relevant to a particular conceptual problem are more likely to solve that problem than children who do not. (pp. 208–209) studies have also shown that both the quantity and quality of parents’ (gunderson & levine, 2011; levine & bailargeon, 2016) and teachers’ (klibanoff, levine, huttenlocher, vasilyeva, & hedges, 2006) ‘number talk’ influences the development of early number concepts. one of the most serious challenges in this regard in south africa is that even if children are exposed to mathematics-specific vocabulary during their early development years, the language of instruction in school may differ from the language in which they have mostly encountered initial mathematics-specific language. apart from different vocabulary use, variations in semantic structures may also contribute to young children’s confusion. differing semantic structures include the inversion of the order of tens and units in some languages, such as vyf en twintig [directly translated as five and twenty] in afrikaans. children may also rely on transparency of number nouns themselves, which adds to conceptual clarity in some languages. for instance, leshome le motso o mong [eleven] in sesotho, means ‘ten and one’ and leshome le metso e mmedi [twelve] means ‘ten and two’. although some words can become very long (like the isizulu word for eight, isishiyagolombili), children can no longer apply the rule on which they have relied to learn number words when learning in a different language, like english, where there is less conceptual clarity. in english, ‘eleven’ and ‘twelve’ have no real connection to 10. this type of semantic representation may confuse a child. susan carey (2009) introduced the notion of ‘input analysers’ as the ‘evolutionary constructed’ mental mechanisms that guarantee ‘the relevant (mental) representations [that] refer to aspects of the environment that are important to survival’ (p. 29). these metaphorical input analysers enable humans to represent sensory and perceptual entities in the world mentally. she proposes that for input to be mentally represented, various formats of input are ‘filtered’ by an innate mental input analyser. although perceptual (seeing, hearing, tasting, feeling and smelling) and symbolised input (for example, words and notations that represent reality) contribute to the process of conceptual development, input analysers also enable humans to reason about concepts, such as mathematical concepts, on a representational level (carey, 2009). in this view, if children’s linguistic input, such as the phonology of a language, its vocabulary and grammatical structure during early development years, is different to the linguistic input at school, it is likely that the input analyser will have difficulty filtering and thus making sense of linguistic input that does not match a child’s existing filtering system. in the sample of this study children were exposed to their home language (isizulu or sesotho) in the preschool years, while the language of instruction in grade 1 is english – which includes not only different vocabulary, but also different phonemes, morphemes, syntax and grammar as well as prosody. the tone, pauses, voice inflections and syllabic emphasis of languages differ. much as spelke (2017) and odic et al. (2015) have proposed, our data suggest that there is a link between the fine-grained aspects of language input, such as the vocabulary, and early numeracy. executive functions: manifested in classroom engagement increasingly, research findings suggest that the development of mathematics skills also relies on children’s executive functions (cragg & gilmore, 2014; prager, sera, & carlson, 2016). some studies show that early numeracy and executive functions are correlated, and others indicate that executive functions are predictors for mathematic conceptual development (blankson & blair, 2016; zaitchik, iqbal, & carey, 2014). these authors argue that the three components of executive functions, namely cognitive flexibility, inhibitory control and working memory, collectively influence conceptual development. tasks such as problem-solving, reasoning and planning make use of a combination of the three executive functions (diamond, 2013)3. working memory enables children to monitor and code relevant information and revise information by replacing old information with new, appropriate information (miyake et al., 2000). for instance, when the question “how many more is 5 than 3?” is presented orally or in written language, children must hold all the information of the question in their memory, while calculating the solution. cognitive flexibility – also referred to as ‘shifting’ of mental sets, attention switching or task switching – allows one to shift between multiple tasks (miyake et al., 2000) and to hold focus and refocus attention to relevant tasks (fitzpatrick, 2014). for example, when a second question, “what is 3 plus 7?” follows the question “how many more is 5 than 3?”, children must be able to switch from subtraction to addition. inhibitory control allows children to deliberately inhibit responses to certain stimuli and choose more appropriate responses (miyake et al., 2000). for instance, focusing one’s attention on a different task or switching between tasks requires inhibition of the automatic inclination to continue with a previous task. inhibitory control enables children to decide to change activities or to inhibit automated responses. fitzpatrick and pagani (2012) have established a link between children’s executive functions and their classroom engagement actions and argue that classroom engagement can be used as ‘placeholder measure’ for executive functions. classroom engagement includes tasks such as cooperation with other children, the ability to follow rules and instructions, listen attentively, work neatly and independently and complete work on time. well-developed cognitive flexibility allows children to disengage and re-engage in classroom tasks, which, in turn, increases participation in classroom activities. inhibitory skills, such as avoidance, effortful control of behaviour, emotional and social self-regulation, facilitate an increase in learning activities (fitzpatrick & pagani, 2012). by means of working memory children can briefly store, maintain and rehearse information, which can increase their engagement in cognitive activities. logical reasoning handley, capon, beveridge, dennis and evans (2004) and morsanyi and szüs (2015) have established a connection between logical reasoning and mathematical competence. both mathematical tasks and logical reasoning involve complex cognitive processes which depend on the retrieval and application of normative rules and rely on abstract processes and symbolic representations (morsanyi & szüs, 2015). for instance, children should know that if two sets consist of the same number of objects, then the objects in one set are in one-to-one correspondence with those in the other. in terms of counting (level 1 of numeracy assessment in this study), if set a is in one-to-one correspondence with set b, and c is in one-to-one correspondence with a, then sets b and c are equal. in terms of ordinality (level 2 of the numeracy assessment), children can reason about the relations between numbers to compare and quantify values. for instance, if 8 is larger than 7, then it should also be larger than the first six numbers. or, if 1 is smaller than 2, it should also be smaller than 3 and so forth. in terms of cardinality (level 3 of the numeracy assessment), children can rely on the inversion property of addition and subtraction to reason about quantities. for instance, 1 + 2 = 2 + 1. they may also use logical reasoning to decompose (level 4 of numeracy assessment) numbers when adding or subtracting. for instance, they can reason that 5 + 8 equals 5 + (5 + 3) and thereby know which decomposed values of 8 will assist them best in the addition task. in this study, we used the revised culture fair test (cft-r) to operationalise logical reasoning. in this test, children are required to reason about differences and similarities and be able to classify and identify patterns. methods study design in this naturalistic field experiment we studied how four competencies that were measured at the beginning of grade r (i.e. numeracy, mathematics-specific vocabulary, executive functions and logical reasoning) predict numeracy after one year when the children enter grade 1 (see figure 1). studying the naturalistic ‘intervention’ – namely learning during the grade r year – allowed the researchers to investigate how these competencies develop in the ‘real world’ as opposed to a controlled environment. figure 1: study design. the hypotheses of the study were: h1-0: there is no relationship between learners’ grade r numeracy (assessed in their home language) and their achievement in grade 1 numeracy (assessed in their home language and in english). h1: a relationship exists between learners’ grade r numeracy (assessed in their home language) and their achievement in grade 1 numeracy (assessed in their home language and in english). h2-0: there is no relationship between learners’ grade r mathematics-specific vocabulary (assessed in their home language) and their achievement in grade 1 numeracy (assessed in their home language and in english). h2: a relationship exists between learners’ grade r mathematics-specific vocabulary (assessed in their home language) and their achievement in grade 1 numeracy (assessed in their home language and in english). h3-0: there is no relationship between learners’ grade r logical reasoning (assessed in their home language) and their achievement in grade 1 numeracy (assessed in their home language and in english). h3: a relationship exists between learners’ grade r logical reasoning (assessed in their home language) and their achievement in grade 1 numeracy (assessed in their home language and in english). h4-0: there is no relationship between learners’ grade r executive functions (classroom engagement) (assessed in their home language) and their achievement in grade 1 numeracy (assessed in their home language and in english). h4: a relationship exists between learners’ grade r executive functions (classroom engagement) (assessed in their home language) and their achievement in grade 1 numeracy (assessed in their home language and in english). participants at the beginning of their grade r year, 65 sesotho and isizulu speaking children’s early numeracy, mathematics-specific vocabulary, executive functions and logical reasoning were assessed in either sesotho or isizulu. at the beginning of grade 1, 67 children’s early numeracy was assessed, in both english and in the children’s home language. of this sample, 59 had come from the previous year’s grade r group. the sample (n = 59) was thus a purposive intact group. the school was selected because of its dual language of instruction model. the results of children who were not assessed either in grade r or grade 1 (due to being retained in grade r or grade 1) were included when data were imputed for a multiple regression analysis. 56% of the sample were male, 48% were isizulu speaking and 52% sesotho speaking. the average age of participating children was 6 years and 2 months in grade r and 7 years and 2 months in grade 1. test instruments the south african version of the marko-d (bezuidenhout et al., 2019) was used to assess early numeracy. to test knowledge of mathematics-specific vocabulary, we administered the custom-designed meerkat maths language test (mmlt), which consists of 26 items, assessing numerical language qualifiers (such as more, many, few, just as many), comparative language (such as smaller, taller, same size) and spatial language (such as in between, first, in front of). the mmlt is a five-minute interview-based test in which the items correspond with the concepts assessed in the marko-d sa. the children only needed to point to the correct picture, showing their understanding of a mathematics-specific word. we did not wish to assess executive functions in an unnatural controlled environment (bezuidenhout, 2018), and rather used a teacher inventory of classroom engagement to describe behavioural manifestations of executive functions. fitzpatrick and pagani (2012) and aunio et al. (2019) argue that a behavioural scale of classroom engagement, administered by the teacher, reflects executive functioning aptly. for example, the successful completion of instructions relies on working memory, shifting between tasks and adapting to routines requires flexibility, and focusing attention on a particular task requires inhibition. the teacher inventory captures teacher ratings of self-regulated learning and productive work behaviour in the form of classroom engagement (gioia, isquith, guy, & kenworthy, 2012). cronbach’s alpha of 0.83 for the classroom engagement measurement of the current study was lower compared to 0.94 in a canadian study (fitzpatrick & pagani, 2013). the scores correlated with the other instruments used in this study. for the assessment of logical reasoning, subtests 3 (similarities), 4 (complete the rows) and 5 (classification) of the cft-r (coppard, 2018) were used. in terms of ethics, the school’s board and parents gave written consent for children to participate in the research. all results were treated as confidential information. participants could withdraw their consent any time during the project. ethical clearance was obtained from the faculty of education research ethics committee (ethical clearance no. 2017-053), university of johannesburg. ethical considerations the board of the school and ethics committee of the university of johannesburg approved this research (ethical clearance no. 2017-053). the authors certify that the study was performed in accordance with the ethical standards. all participants remain anonymous. results an independent samples t-test revealed no significant gender differences on the assessments of early numeracy, mathematics-specific vocabulary, executive functions or logical reasoning. isizulu children outperformed the sesotho children on early numeracy in grade r (means = 20.55 vs 17.11, p ˂ 0.01) and grade 1 (means = 25.71 vs 21.93, p ˂ 0.01), while there was no significant difference when tested in english. there were no significant tester effects. table 1 shows that early numeracy scores increased from grade r to grade 1 when children were tested in their home language in grade 1 (means = 18.93 vs 23.86) but decreased when they were tested in english in grade 1 (means = 18.93 vs 17.73). there was a statistically significant difference between english and home language early numeracy scores in grade 1 (means = 23.86 vs 17.73). table 1: descriptive statistics for independent, dependent and control variables. table 2 summarises bivariate correlations between predictors and outcome variables. except for early numeracy in grade r, mathematics-specific vocabulary had the strongest correlation with grade 1 early numeracy (tested in english) (0.35, p < 0.01). early numeracy in grade r (home language) and grade 1 (home language) showed a correlation of 0.4 (p < 0.01). however, grade r mathematics-specific vocabulary, executive functions and logical reasoning were not significantly correlated with grade 1 early numeracy scores when tested in their home language. finally, english and home language early numeracy scores (in grade 1) had a correlation of 0.28 (p < 0.05). table 2: correlations between predictor and outcome variables. to determine associations between each predictor in grade r and outcome variable in grade 1, multiple regression analyses were calculated, while controlling for language of instruction. because the grade r predictor variables correlated only with the grade 1 english early numeracy scores and not with the grade 1 home language early numeracy scores, a regression model was only estimated for the english assessment. six regressions, which examined associations between each of the predictors while controlling for language of instruction, were run in total on imputed data. in terms of model fit, all the regressions accounted for a significant proportion of the variance in the grade 1 early numeracy scores. to examine how each independent variable contributed to the variance in early numeracy, we ran two regression models. firstly, we examined each predictor separately, controlling for language of instruction (given that the predictors are statistically and theoretically closely related). of the individual predictors, early numeracy in grade r was the strongest predictor of grade 1 early numeracy (β = 0.42, p ˂ 0.05) and mathematics-specific vocabulary the second strongest predictor (β = 0.16, p ˂ 0.05). in the second regression analysis, while still controlling for home language, grade r numeracy was omitted as a predictor. in this model we examined both concurrent associations (between grade r mathematics-specific vocabulary, executive functions and logical reasoning, and grade r numeracy) and prospective associations (between grade r mathematics-specific vocabulary, executive functions and logical reasoning, and grade 1 numeracy). mathematics-specific vocabulary, executive functions and logical reasoning significantly contributed to the variance in this second model. table 3 shows concurrent and prospective associations between grade r cognitive skills (mathematics-specific vocabulary, executive functions and logical reasoning) and numeracy in grade r and grade 1. table 3: standardised regression coefficients depicting associations between cognitive skills and number concept development for grade r and grade 1. mathematics-specific vocabulary, executive functions and logical reasoning were all significant predictors for early numeracy in grade r and grade 1. in terms of concurrent variance, 30% to 47% of the variance in grade r numeracy (with logical reasoning explaining the most variance and executive functions the least) were explained by concurrent cognitive skills (mathematics-specific vocabulary, executive functions and logical reasoning). in terms of prospective associations, grade r predicting skills explained 11% to 14% of the variance in grade 1 numeracy with mathematics-specific vocabulary explaining the most variance and logical reasoning the least. in summary, although grade 1 children’s early numeracy scores increased from grade r to grade 1, when tested in their home language, the scores decreased significantly when early numeracy was tested in english in grade 1. there was also a statistically significant difference between the home language and english scores of early numeracy in grade 1. apart from grade r numeracy, mathematics-specific language was the strongest predictor for grade 1 numeracy, which indicates that children’s knowledge of mathematics-specific vocabulary and language of instruction play vital roles in the development of young children’s number concepts. based on the results presented, we conclude that h1 was confirmed, while the other three hypotheses were only partly confirmed: relationships between all four independent variables (grade r numeracy, mathematics-specific vocabulary, logical reasoning and executive functions) and the english assessment of numeracy in grade 1 was confirmed, but in the sample of this study there exists only a relationship between one of the independent variables, namely grade r numeracy, when numeracy was assessed in the children’s home language in grade 1. discussion the results of this study show that children find it hard to integrate number concepts that have developed prior to learning english with mathematics vocabulary in english. this finding resonates with vygotsky’s theory of the intersect between pre-linguistic concepts and pre-conceptual language as a starting point for promoting conceptual change and language ‘labels’ (kozulin, 1990; vygotsky, 1986). according to this theory, language and concepts develop concurrently while one supports the other. as children develop vocabulary, they learn the conceptual properties of words and phrases such as more, less, in front of, bigger, just as many and also the number nouns, in tandem. spelke (2017) has come to propose number nouns as the origin of numerical cognition. according to levine and baillargeon (2016) and spelke, children align words to constructs they intuitively know or recognise because of words used in their environment. in this sense, language not only supports the development of number concepts, but mathematics-specific vocabulary itself can be seen as input for the development of number concepts and is filtered comfortably by the innate input analyser (carey, 2009). the young children in classrooms such as those in our study sample enter a multilingual ‘maze’ (henning, 2012) in the first year of formal education, which may negatively influence their opportunity to develop number competency in the building-block years of mathematics progression (clements & sarama, 2014). also, because english is a language with strong local social and economic currency, to which only a few children who come from ‘township’ areas are exposed prior to formal education, the participants in this study had not yet stored english mathematics vocabulary for fast memory retrieval. in order to make sense of mathematics in a grade 1 class, children are required to not only align a new english label (word) to an already developed number concept, but also to its equivalent in their home language. this process could easily overload the working memory. for this reason, we propose that foundation phase teachers’ pedagogical content knowledge should include a precise understanding of the contribution of all levels of linguistic input (dowker & nuerk, 2016) for early numeracy. further research should focus on ways to include intentional development of mathematics-specific vocabulary in preparation for formal mathematics education. our findings also indicated that there is an association between number concepts and other contributors, namely logical reasoning skills and executive functions. for this reason, we also propose that teachers should understand individual differences in young children. these differences include variations in executive functioning, classroom engagement and logical reasoning abilities. dowker (2008) proposes that teachers need to know which specific skills contribute to number concept development and know how to strengthen each individual child by developing that specific skill. knowledge of contributing skills for the development of number concepts would, for instance, enable teachers to explain why the isizulu speakers in this study outperformed sesotho speakers on the grade r and grade 1 assessments of numeracy in the home language versions of the tests. by including an understanding of the levels of linguistic influences and knowledge of executive functions in their pedagogy, teachers will know that lexical and grammatical composition of number names (lefevre, 2018), or longer number names, possibly overload the working memory. in conclusion, the development of early numeracy, mathematics-specific language, executive functioning and logical reasoning during grade r prepares children for formal education of number concepts in grade 1. this study’s findings emphasise that apart from the development and learning of number concepts in grade r, mathematics-specific vocabulary is the most important enabler for learning mathematics in grade 1. specifically, in the context where the language of learning in grade 1 differs from the language of learning in grade r, the importance of developing mathematics-specific vocabulary in dual language mode during grade r is highlighted. further research should be conducted to find ways to include the intentional development of mathematics-specific vocabulary before children enter grade 1 and ensure that they learn the terms in the language of future instruction, with bilingual scaffolding engineered by the teacher and the curriculum. acknowledgements competing interests the authors declare that they have no financial or personal relationships that may have inappropriately influenced them in writing this article. authors’ contributions h.s.b. and e.h. authors contributed equally to the design and implementation of the research, to the analysis of the results and to the writing of the manuscript. funding information this project was funded by a global excellence stature 4.0 (ges) postdoctoral fellowship from the university of johannesburg. data availability data is available from the authors on reasonable request. disclaimer the authors declare that the work is their own and that it was written in their own words. all citations from literature are acknowledged in text and referenced. we agree that subject to the ownership of all intellectual property rights in this work, the approved version of this work may be published by the pythagoras journal. references aunio, p., korhonen, j., ragpot, l., törmänen, m., mononen, r., & henning, e. 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(2014). the effect of executive function on biological reasoning in young children: an individual differences study. child development, 85(1), 160–175. https://doi.org/10.1111/cdev.12145 footnotes 1. assessed in their home language: isizulu or sesotho. 2. assessed in both their home language and in english. 3. there are researchers who do not share this view, such as clements, sarama and germeroth (2016) who published a review about causal connections between executive functions and mathematics learning. article information authors: patisizwe t. mahlabela1 sarah bansilal1 affiliations: 1department of mathematics education, school of education, university of kwazulu-natal, south africa correspondence to: sarah bansilal email: bansilals@ukzn.ac.za postal address: 8 zeeman place, malvern 4093, south africa dates: received: 06 jan. 2014 accepted: 29 sept. 2015 published: 18 dec. 2015 how to cite this article: mahlabela p.t., & bansilal s. (2015). an exploration of learners’ theorems-in-action used in problems on ratio and proportion. pythagoras, 36(2), art. #252, 10 pages. http://dx.doi.org/10.4102/pythagoras.v36i2.252 copyright notice: © 2015. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. an exploration of learners’ theorems-in-action used in problems on ratio and proportion in this original research... open access • abstract • introduction • strategies used in solving missing value problems • a framework for understanding ratio and proportion • methodology • findings    • performance on question 1    • performance on question 2 • discussion    • the use of the doubling and halving strategies    • additive and subtractive strategies    • unitary strategy    • cross multiplication strategy • conclusion • acknowledgements    • competing interests    • authors’ contributions • references abstract top ↑ the purpose of this study is to explore grade 9 learners’ understanding of ratio and proportion. the sample consists of a group of 30 mathematics learners from a rural school in the province of kwazulu-natal, south africa. data were generated from their responses to two missing value items, adapted from the concepts in secondary mathematics and science test set in the united kingdom over 30 years ago. the study utilised vergnaud’s notion of theorems-in-action to describe the learners’ strategies. it was found that the most common strategy was the cross multiplication strategy. the data reveal that the strategy was reduced to identifying and placing (often arbitrarily) three given quantities and one unknown in four positions, allowing the learners to then carry out an operation of multiplication followed by the operation of division to produce an answer. the study recommends that the role of the function underlying the proportional relationship should be foregrounded during the teaching of ratio and proportion. introduction top ↑ there have been numerous studies focused on ratio and proportion which have looked at learners’ strategies and errors (ben-chaim, keret & ilany, 2012; chick & harris, 2007; jiang, 2008; lamon, 2007; long, 2011; md-nor, 1997; misailidou & williams, 2002; olivier, 1992). many researchers agree that understanding proportional relationships is a long-term developmental process (ben-chaim et al., 2012) and forms a fundamental building block for many other areas in mathematics as well as in the sciences. many phenomena studied in daily life, physics, chemistry, biology, geography, agriculture, woodwork and needlework and economics are defined using proportion (ben-chaim et al., 2012; olivier, 1992). it is evident that a poor understanding of proportion can limit a learner’s access to various concepts and subjects. this article reports on part of a bigger study which investigated grade 9 learners’ understanding of ratio and proportion (mahlabela, 2012). the bigger study used eight test items adapted from the original concepts in secondary mathematics and science study (hart, 1981). in this article we report on the strategies and errors exhibited by the learners in two of these items, which consist of missing value problems where three quantities are given and the task is to find the fourth one. strategies used in solving missing value problems top ↑ to understand learner errors, one has to look at the methods or strategies that the learners use to arrive at the incorrect solutions. errors could be the results of incorrect strategies or the results of incorrect use of correct strategies. some strategies identified in the literature are now briefly described. the for every strategy entails finding the simplest ratio first and then multiplying by a factor that yields the required result. the unit value strategy is similar except that the simplest ratio is reduced to a unitary ratio. this strategy has been identified in studies conducted in various countries (hart, 1988; jiang, 2008; md-nor, 1997; misailidou & williams, 2003). the multiplicative strategy (within measure space approach) entails determining a ratio of measures from the same space and using it as a factor (hart, 1988; md-nor, 1997; misailidou & williams, 2003). the cross multiplication strategy based on setting up a proportion is also described as the rule of three by some (md-nor, 1997; vergnaud, 1998) or algebraic method (jiang, 2008) or the use of the formula x/a = y/b (hart, 1984; olivier, 1992). in the constant difference or incorrect addition strategy or additive strategy, the relationship within the ratios is computed by subtracting one term from another and then applying the difference to the second ratio (hart, 1988; md-nor, 1997; misailidou & williams, 2003). according to misailidou and williams (2003) the additive strategy is the most commonly reported erroneous strategy in the research literature. long (2011) made a similar finding in the trends in international mathematics and science study (timss) with south african learners. the building-up strategy involves establishing a relationship within a ratio and then extending it to the second ratio by addition (md-nor, 1997, p. 34) and is also referred to as the addition and scaling strategy (hart, 1988), because it involves a multiplicative strategy combined with an additive one. the incomplete strategy (misailidou & williams, 2003) involves using the same number given for the measure space. long (2007) found that in timss some responses were incorrect because of ‘incomplete reasoning’. she says that learners’ ‘reasoning took them part way towards the answer’ (p. 16). an incorrect strategy is the incorrect doubling method. since there are problems involving ratio and proportion where doubling is a correct strategy, the word ‘incorrect’ signals that the use of doubling was wrongly implemented in the setting. a framework for understanding ratio and proportion top ↑ researchers (ben-chaim et al., 2012; olivier, 1992; vergnaud, 1998; wu, 2006) point out that proportional reasoning is just mathematical reasoning based on the concept of a linear function without a constant term. proportional reasoning requires a recognition that a situation is completely described by a function of the form shown in equation 1: knowing for all x implies that for any two non-zero values x1 and x2, equation 2 is true: we say the four numbers f(x1), x1, f(x2) and x2 form a proportion which can be expressed in terms of ratios: f(x1): x1 = f(x2): x2. the proportional relationship can also be written as equation 3: equivalently, it can be expressed as the ratio representation x2:x1 = f(x2):f(x1). the relationship expressed in equation 2 is such that the two quantities forming the fraction are from different measure spaces; this expression is referred to as a rate and the value is actually the constant coefficient (vergnaud, 1998) or the constant of proportionality, c, from equation 1. the term measure spaces is used by vergnaud to express the fact that certain quantities belong to one measure space, which may have particular measures of, for example, length or cost, and others belong to a second measure space, which may have dimension of weight, distance or length, for example. considering equation 3, we note that the fraction formed by the two quantities in each case has no dimension because they are from the same measure space. vergnaud (1998) refers to this as a scalar fraction. hence, the proportional relationship can be expressed as an equality between two rates (equation 2) or as an equality between two scalar fractions (equation 3). vergnaud (1998) uses the term theorems-in-action to describe the cognitive processes and yet-to-be concepts of a learner. a theorem-in-action is the set of mathematical relations considered by learners when they choose an operation or a sequence of operations to solve a problem. these theorems are implicitly held by learners and may not be articulated. sometimes these theorems-in-action are not correct, but they can help teachers and researchers to analyse learners’ intuitive strategies and give insight into their thinking. long (2011, p. 109) asserts that identifying ‘theorems-in-action provides a way to make a better diagnosis of what students know and do not know’. an important property of the linear function f(x) is described by equation 4: vergnaud (1998) draws on this property to illustrate learners’ implicit theorems-in-action when solving a problem based on a train travelling at constant speed, which took 16 minutes to travel 40 km from town a to town b. learners were asked for the distance between town b and town c, given that it took the train 36 minutes to travel from town b to town c. a learner wrote: 40 × 2 = 80; 80 + 10 = 90. vergnaud explains that 36 minutes can be decomposed into 2 × 16 minutes + 4 minutes, and 4 minutes = ¼ of 16 minutes. therefore, the corresponding distance is 2 × 40 km + ¼ of 40 km. note that one can manipulate equation 4, to derive a further property used by vergnaud (1998), that is f(m.1) = c(m.1) = c.m = f(1).m = m.f(1), thus leading to , and hence to equation 5: where m is any real number vergnaud describes another way of solving proportionality problems, which he calls the rule of three. equation 6 follows directly from equation 3: in equation 6, f(x1), x1 and x2 are known and f (x2) is required. methodology top ↑ this qualitative study was conducted with a group of 30 grade 9 learners from one school in kwazulu-natal, south africa. the school is situated in a rural area, with most learners coming from impoverished backgrounds. the school was selected because of its proximity to the authors, which can be described as convenience sampling. ethical requirements were fulfilled according to the ethical procedures stipulated by the local university. the original larger study (mahlabela, 2012) used eight questions that were adapted from the concepts in secondary mathematics and science study. however, in this article we only report on learners’ responses to two items, because of the variety of rich responses that emerged from these. after an initial marking and coding of the scripts, five learners were identified and selected for semi-structured interviews. learners are referenced as learner 1 to learner 30, according to the order in which the scripts were collected; the numbering does not denote any difference in ability between the learners. the research questions that underpin this study are: what strategies do learners use to solve the two test items? what are the underlying theorems-in-action associated with these strategies? what do the theorems-in-action suggest about the learners’ understanding of ratio and proportion? the learners’ written responses were analysed by studying their final answers and their working details to identify the strategy that was used. in some cases we were unable to identify a definitive strategy and this was described as ‘miscellaneous’. interview responses helped to support our strategy classification. findings top ↑ performance on question 1 learner responses are analysed separately: question 1a and question 1b first and the rest later. table 1 shows how learners responded to question 1a and question 1b; table 2 shows performance in the rest of the questions. table 1: learner responses to question 1a and question 1b (n = 30). figure 1: the onion soup recipe (question 1). figure 2: response of learner 13 to question 1a. one way of solving question 1a and question 1b would be to halve the ingredients in the given recipe since soup is being prepared for four people. more than 70% of the participants responded correctly to both questions. some learners used the incorrect strategy of incomplete halving, where they just indicated that they needed half of the ingredients given in the recipe, but did not work out what half the actual amounts were. other errors were related to incorrect cross multiplication as some learners obtained the solution of 4 in question 1a as shown in figure 2. learner 13 omitted an equal sign between the equivalent fractions. the ratio of people in the two recipes (given recipe for eight people and the recipe for four people) is 8:4, which the learner wrote as (8 people)/(4 people). the ratio of water amounts is 2:x, where x is the amount of water needed to make soup for four people. she correctly formulated the two scalar fractions described in equation 3 by comparing quantities from the same measure space (i.e. quantities of the same kind or elements of the same set). to find the amount of water needed (value of x), she multiplied 8 (the number of people given in the recipe) by 2 (the amount of water needed to make the soup for eight people) and then divided the product by 4 (the number of people that soup was needed for). we refer to this error as incorrect cross multiplication. the learner explained her method of cross multiplication and, interestingly, provided an explanation different from the written one appearing in figure 2: interviewer: i can see that your answer to this question is 4 pints. tell me how you got 4 pints? learner 13: i took 8, the number of people in the recipe and wrote it down. i then took 2, the number of pints needed to make soup. i wrote 8 over 4, the number of people that i want to make the soup for. because i do not know the number of pints of water needed, i wrote over x [the unknown]. i then cross multiplied. i got 8x and 8. i divided by x, although i am no longer sure this is how i did the calculation [she realises that this solution is different from the one which appears on the script]. i then wrote x and moved 8 to the other side [as if she has the equation 8x = 8]. learner 13 in her written response in figure 2 seemed to confuse cross multiplication with the rule for multiplication of fractions, that is, multiplying the numerators of the scalar fraction to get the numerator in the answer and multiplying denominators to get the denominator in the answer (however the confusion is not exhibited in her interview). a perusal of the learner scripts revealed that the error in figure 2 recurs in other solutions obtained using this strategy. one way of solving question 1d and question 1e would be to halve (to obtain ingredients for four people), halve again (obtain ingredients for two people) and then add the result to the first halving to find the ingredients required for 6 people – called the build-up strategy. for example, to determine the amount of water needed to make soup for six people, learners could halve 2 pints of water (water needed to make soup for eight people) to get 1 pint (amount of water needed to make soup for four people). they could halve 1 pint of water again to get ½ pint (amount of water needed to make soup for two people). the amount of water needed to make soup for six people is then 1 pint of water + ½ pint of water = 1½ pints of water. the problems could also be solved by the use of the for every strategy or the multiplicative strategy. the performance of learners is shown in table 2. over half (60%) of the learners answered question 1c correctly, more than 66% correctly answered question 1d, but only 10% answered question 1e correctly. this is not surprising as question 1e required halving and addition of fractions whilst question 1d required halving and addition of whole numbers and question 1c required halving of whole numbers and addition of a whole number to a fraction. the incomplete use of the build-up strategy (incomplete build-up strategy) was observed in all three questions. in question 1c, for example, 6% of the learners either halved once to get 1, or halved twice to get ½ learners who halved twice did not add the first answer to the second answer. many learners also used an incomplete strategy (using the same number given for the measure space). for example, in question 1e it is stated that ½ pint of cream was needed to make soup for eight people. about 13% of the learners said that ½ pint of cream was needed to make soup for six people too. some strategies were based on incorrect cross multiplication. other incorrect answers obtained by the learners, categorised as ‘other’, were diverse. the solutions emanated from the incorrect performance of either basic operations, conversions from one unit to the other or both. for example, in question 1d, learners divided 24 by 8 and obtained 2 as the quotient (computational error). in question 1e, learners divided (½ × 6) by 8 and obtained 0,75 (computational error). some learners found the product of 6 and ½ to be 7,2 (a computational error) and then some correctly divided 7,2 by 8 to obtain 0,9, whilst others divided 7,2 by 8 to obtain an incorrect answer of 9. in question 1d one learner incorrectly converted ½ to a decimal as 1,5. to arrive at ½ in question 1e, learner 15 incorrectly answered as shown in figure 3; this was categorised as an incomplete strategy. it however includes the same misconception as that displayed by learner 13 in figure 2, who multiplied fractions instead of ‘cross multiplying’ across an equal sign, because there was no equal sign between the fractions. learner 15’s response in figure 3 shows that he has modelled the relationship correctly by mapping 8 people to 6 people and ½ pint of cream to the unknown volume of cream. however, by not expressing the relationship as equivalent fractions, the learner has taken this as a product of two fractions (similar to the misconception learner 13 displayed in question 1a). however, learner 15 has furthermore ignored the unknown and carried out the computation with numbers only. a third error is that he has interpreted the quotient of (8 × 6) and ½ as 4 × 3, which represents two errors: taking divided by ½ as divided by 2 and also taking pq/2 as p/2 × q/2. thus, there are many algebraic misconceptions in these few lines. performance on question 2 the question in figure 4, a version of piaget’s popular eel question, was the second question in the research instrument. piaget’s eel question is concerned with the amount of food given to eels of different lengths, the amount being proportionate to the length of the eel (hart, 1981). table 3 shows the performance of learners for each sub-question. figure 3: response of learner 15 to question 1e. table 2: learner responses to question 1c–e (n = 30). more than 50% of the learners responded correctly to each sub-question, except in question2d. the performance of learners in question 2a was outstanding as 90% of the learners responded correctly to this sub-question. many errors in this question resulted from the incorrect use of cross multiplication strategy. figure 5 shows how learner 6 arrived at the incorrect answer of 25 sprats in question 2a(i). learner 6 used the equal sign incorrectly by expressing equality between quantities from two different measure spaces (2 sprat = 5 cm and x sprats = 10 cm). in order to apply the cross multiplication strategy, two equivalent rates or scalar fractions would be needed, which learner 6 did not identify correctly. in fact, the result was obtained by multiplying the length of eel a (5 cm) by the length of eel b (10 cm) and then dividing the product by 2 (the number of sprats that a is fed with). figure 4: the eel question (question 2). the responses of learner 17 (see figure 6) indicate the use of the cross multiplication rule. however, the choice of which of the available numbers should be the multipliers and which should be the divisor seems to systematically follow a set formula. in the first case it was that the number of sprats for eel ; this was similar in each case. learner 17 has thus consistently used the cross multiplication rule as , where x is the length of the eel and f(x) is the number of sprats fed to the eel. figure 5: response of learner 6 to question 2a(i). figure 6: response of learner 17 to question 2. table 3: learner responses to question 2 (n = 30). note that at no point in these or any other solutions which used the cross multiplication rule was there a point where any learner expressed equality between two scalar fractions or two rates. for example, consider the response of learner 14 to question 2 in figure 7. the learner has modelled the situation correctly by associating the quantities from one measure space to the next, in the correct order, for each of the questions. she then carried out a cross multiplication across the vertically opposite numbers and created an equation in each case which she then solved correctly. all of these steps were done four times, without once expressing an equation in the form of equation 2 or equation 3, without which none of these relationships holds true. the fact that none of the learners expressed such equations suggests that they were influenced by teaching methods. the incorrect addition strategy was also observed in learner responses. participants suggested that the eels should be given two more or fewer sprats or fish fingers, depending on the length of the eel. actually the addition strategy leads to the correct answer for question 2a. if eel a is fed two sprats, then eel b should be given two more sprats, because eel b is twice the length of eel a. however, the strategy does not work for other sub-questions. when used in question 2b, it yields an incorrect answer of 14 (obtained by 10% of the participants). there were also instances where learners added or subtracted 5 (the difference in eel lengths). for example, an incorrect result of 17 sprats in question 2b was obtained by adding 5 sprats to 12 sprats. the incorrect doubling or incorrect halving strategy was also identified in some responses. the strategy works correctly in question 2a(i), but immediately fails in question 2a(ii), yielding an incorrect result of 8. the reason for the incorrect answer of 4½ in question 2c is incorrect halving. figure 7: responses of learner 14 to question 2. figure 8 illustrates correct and incorrect solutions, based on the unitary strategy from the script of learner 16. learner 16 divided or multiplied the eel length by the number of sprats or mass of fish fingers it is fed. she divided where the number of sprats or mass of fish fingers was given for a longer eel in order to find how much should be given to the shorter eel. to determine how much the eel should be given, its length should be divided by the quotient obtained during the first division. in question 2c the longer eel is 15 cm. the eel is fed 9 sprats; hence, learner 16 divided 15 by 9 and got 1,67. the length of eel b (10 cm) was then divided by 1,67. she obtained an answer of 5,99 which she then incorrectly rounded off to 5,10. the correct answer is 6. her attempts at question 2d reveal her uncertainty. she first wrote 10 ÷ 2 = 5, followed by 25 ÷ 2 = 12,5 (instead of 25 ÷ 5, which is the quotient of the first answer). she then struck off the division signs, replaced them with multiplication signs and repeated the same method: 10 × 2 = 20, then 25 × 2 = 50. similarly for question 2e, she first wrote 15 ÷ 9 = 1,67 and 25 ÷ 1,67, which is correct. however, she then replaced this with 15 × 9 = 135 and 25 × 1,67 = 41,67. learner 16 seemed to have confused herself about the operations that she needed to carry out. a scrutiny of question 2f shows that she seems to have worked through her confusion and presented the correct operations and sequence of operations as she did for question 2c. however, she has made a slip and swopped the two solutions. that is, her response for question 2f(i) is the solution to 2f(ii) and vice versa. figure 8: response of learner 16 to question 2. additional information on learner errors was obtained through interviews. the interview with learner 8 is now presented in three excerpts, each of which details a different strategy. the dialogue below (excerpt 1) elaborates on how learner 8 arrived at his solution for question 2a: interviewer: you said that if eel a gets 2 sprats, then eel b must get 4 sprats. … how did you arrive at that? learner 8: eel a is 5 cm long, right. if eel a gets 2 sprats, then eel b gets 4 because 5 is half of 10. so if eel a gets 2, eel b gets 4. interviewer: if a gets 3? learner 8: then b gets 6. learner 8 seems to recognise proportion very well in this problem. he realised that eel b is double the length of eel a and therefore should get double the number of sprats given to eel a. the conversation (excerpt 2) continued by looking at question 2b as follows: interviewer: if eel b gets 12 sprats how many should eel a get? learner 8: if eel b gets 12 sprats, then eel a should get 6. interviewer: good. if eel b gets 12 sprats, you said in your script eel c gets 24 sprats. how did you get 24? learner 8: 24? i was in a rush then, i must have said 12 × 2 and got 24. i should have said if b gets 12, c gets 30. interviewer: 30? … how did you get 30? learner 8: i said 15 × 2. interviewer: why did you say that? learner 8: eel c eats more than eel b. interviewer: more? learner 8: yes. interviewer: why do you multiply by 2? learner 8: i was in a hurry and the bell was ringing. in excerpt 2, learner 8 multiplied 12 by 2 to get 24 and multiplied 15 by 2 to get 30, showing that he has now tried to extend the doubling strategy which yielded the correct answer in excerpt 1. however, doubling does not work in this case. he seems convinced that eel c should get twice what eel b gets. the continuation of the conversation (excerpt 3) now reveals a different strategy to question 2d: interviewer: let us look at this one [question 2d]. you said that if eel x gets 2 grams fish fingers, eel z must get 7 grams. how did you get 7? learner 8: i added 5. interviewer: ok. you said if eel x gets 2 grams, eel z gets 2 + 5 grams? learner 8: i said 2 + 5 and got 7. interviewer: where did you get 5 from? learner 8: i said 25/5 and got 5, then here i said 2 × 5 to get 10 [length of eel x] and if i multiply 5 by 5 i get 25 [length of eel z] over here. interviewer: here [pointing at the learner working] you multiplied 5 by 2 and here you multiplied 5 by 5? so when you get 2 you add 5? learner 8: yes, and i got 7. the conversation with learner 8 in excerpt 3 reveals that he has used an incorrect additive strategy: adding 5. however, his explanation is interesting. he correctly identified that the eel lengths are multiples of five with eel x being the second multiple of 5 (2 × 5) and eel z being the fifth multiple of 5 (i.e. 5 × 5). however, having identified that the length of eel z was the fifth multiple of 5, he then added the 5 to the 2 (from number of sprats that should be fed to eel x. he has not realised that he had obtained the required answer of 5. note that learner 8 has described three different strategies in the three excerpts of the interview, showing his uncertainty about the underlying relationships. a conversation with learner 27, who also used an additive strategy, went as follows: interviewer: let us look at this one [question 2d]. eel x is given 2 grams of fish fingers. how did you get the answer that eel z should get 8 grams of fish fingers? learner: if i give eel x 2 grams, then i must give eel y 4 grams, looking at how their lengths differ [observing a pattern of lengths]. i saw the lengths were 10, 15 and 20. i think i did not see that eel z is 25; i think eel z was supposed to get 8 grams if its length was 20 [seems to be trying to establish a pattern of 2, 4, 8 for eels x, y, z respectively]. i was not supposed to give eel z 8 grams. i was supposed to give eel z 12 grams. interviewer: now your pattern is 2, 4, 8 for 10, 15, 20. why is it not 2, 4, 6, 8? an eel of length 20 would get 6 grams and it would make sense to give eel z 8 grams. learner: actually i think that is exactly what i did. learner 27 suggests that he tried to establish a pattern based on repeated addition, that is forming an arithmetic sequence. for eel length, he saw patterns or addition by 5. for fish finger mass the learner saw patterns of addition by 2. discussion top ↑ the study found that learners used various strategies. there was also evidence that learners shifted between strategies, using different ones for the same question as learner 13 and learner 8 revealed in their interviews. this tendency demonstrates their uncertainty about the underlying relationships, which led them to adopt different ‘methods’ at different times because the methods seem arbitrary and are not grounded in the properties of the proportional relationship. learners often used incorrect mathematical notations such as 2 sprats = 5 cm (e.g. learner 6 in figure 5). this suggests little comprehension of what a proportional relationship entails. learners’ misconceptions about number and algebra added to their confusion as was the case with learner 15 in figure 3, who displayed multiple errors, such as multiplying the equivalent fractions as well as incorrectly simplifying to . the underlying theorems-in-actions associated with the strategies are now presented. the analysis of the underlying theorem-in-action is a tool that can be used to check the validity of a particular strategy and also helps us identify the scope and limits of application of a strategy. the use of the doubling and halving strategies it seems as if many learners recognised that some problems could be solved by halving or doubling or a combination of the operations. this implies that the operations were intuitive efforts in trying to obtain fitting answers. however, many of them performed incorrect operations or stopped short of completing all the steps. most learners were able to solve question 1a and question 1b, which required just one operation of halving correctly, whilst some learners recognised the need for halving but did not know which quantity to halve. doubling also led to the correct answer for question 2a(i), which is discussed under additive strategies. the underlying theorem-in-action based on equation 4 is that f(½ x) = ½ f(x) or f(2x) = 2f(x), which can yield the correct answer to the given proportionality problems. however, some learners may have doubled the elements across a measure space, such as learner 8 in interview excerpt 2, who doubled the length of the sprat y and expressed it as the number of fish fingers required for y. at other times learners doubled or halved quantities in the same measure space inappropriately such as those who used the halving strategy incorrectly in question 2c. additive and subtractive strategies some learners resorted to addition or subtraction, which was incorrect in many cases. however, addition using the build-up strategy can lead to the correct answer. for example, question 1e could be solved by first reducing the given ingredients by two factors and then adding the results. this is possible because (using equation 4, where x is the number of people and f(x) is the amount of cream needed for x people). another instance when addition led to a correct answer was in question 2a(i). the length of the eel increased from 5 cm to 10 cm. hence the number of sprats increased from 2 to 4; this increase could be seen as an addition of 2. here the function could be seen as where x is the length of the eel x and f(x) is the number of sprats fed to eel x. hence f(10) = 4 = 2f(5) = 2(2) which is also equal to 2 + 2. so this is true only because 2 + 2 = 2 × 2 in this instance. however in general, f(2x) ± 2 + f(x). for question 2d learner 8 used an incorrect additive strategy, as explained in excerpt 2. he correctly identified that the eel lengths are multiples of five, with length of eel x being the second multiple of 5 and the length of eel z being the fifth multiple of 5. hence the number of fish fingers for eel x would be 2 (i.e. then and the number of fish fingers for eel z would be 5 . however, having identified that the length of eel z was the fifth multiple of 5, he then added 5 to the 2. he did not realise that he had obtained the required answer of 5. his solution can be expressed as f(z) = f(x) + fz), which does not hold in the case of a linear function unless x = 0 and f(x) = 0. also in question 2d, learner 27 used an additive strategy in a different and also incorrect way. as the lengths of the lengths of the eels increased by 5 cm, in a corresponding manner he increased the number of fish fingers by 2. hence, his incorrect solution can be expressed as f(x) = f(10) = 2, f(15) = f(10) + 2, f(20) = f(10) + 4; f(25) = f(10) + 6, statements which are not true. unitary strategy in general there were few responses that indicated the use of the unitary method; however, learner 16’s responses to question 2c×2e are based on a version of the unitary method (equation 5). however, it seems that he was uncertain about which operation to use and what the result of the operation represented. in working out 15 ÷ 9 = 1,67 for question 2c the underlying theorem in action was (equation 5). that is, he worked out . that is, . he then went on to work out 10 ÷ 1,67 which is actually working out , also using equation 5, here where f(m) = 10, f(1) = 1,67 and m is unknown. so his method was correct; however, he entangled himself in trying to keep track of the operations and divisions and multiplications whilst also trying to identify which elements were from the different measure spaces. cross multiplication strategy vergnaud refers to this strategy as the rule of three strategy. suppose that the three quantities x1, x2 and f(x1) are given and f(x2) is required. then, using equation 2, we can follow five steps: however, as shown in this study, even those learners who correctly carried out the strategy moved directly from step 1 to step 5, leaving all the intervening steps or transformations out. it is those omitted steps that demonstrate why the strategy works. in particular, it was noticed that learners expressed step 1 as step 1*, without using any equal or ratio sign: also, none of the learners expressed step 2 in their working details. their strategy therefore was not based on knowledge of ratios and proportional relationships, but on meaningless operations of ‘cross multiply and divide’. even the work of learner 14 who arrived at the correct answers show that she moved directly from step 1* to step 5 without any of the intervening steps. this learner was able to consistently reproduce step 5 without including any of the transformations between step 1 and step 5, which is quite remarkable. however, many of the other learners’ lack of knowledge of the specific transformations that make the cross multiplication rule work led to mistakes such as those demonstrated by learner 13 in question 1a, who formulated the correct relationship in step 1*, but who got confused when carrying out the operations and multiplied the scalar fractions and , where f(x) was the amount of water needed to make soup for x people. learner 15, in figure 3, made a similar mistake, but his response suggested many other misconceptions besides the one held by learner 13. the errors displayed by learner 17 in question 2 can also be explained as incorrect application of the cross multiplication rule. learner 17 only produced step 5 in each case of the form , instead of , which is the result of the correct application of the cross multiplication rule. these results support olivier’s (1992) advice given more than 20 years ago that: any teaching strategy which merely supplies pupils with recipes such as … cross multiplication and the unitary method which can solve certain classes of stereotypes proportional problems … cannot be effective. (p. 301) vergnaud (1998) notes in his study that only 1% of the participants used the rule of three, explaining that this low rate was because most learners consider that there is no meaning in multiplying quantities from different measure spaces (f(xi) x2). hart (1981) reports that ‘of the 2257 children … only 20 [15 from the same school] wrote down an equation of the form and used it consistently and correctly’ (p. 89). ‘there was little evidence that the taught rule was remembered and used by children’ (p. 21). ‘teaching an algorithm such as is of little value unless the child understands the need for it and is capable of using it. children who are not at a level suitable to the understanding of will just forget bd the formula’ (p. 101). in this study the cross multiplication method was the most common method, even though it is not an intuitive strategy. of the 12 learners who showed some working details, all displayed evidence of using this strategy in their responses. it is therefore clear that the recipe for this method was taught to the learners, which many teachers under pressure may decide to do. however, not a single learner, even those who produced correct responses, were able to provide an explanation that showed reasoning beyond step 1*, suggesting that they did not know why the strategy worked. the cross multiplication strategy was clearly taught without ensuring an understanding of when and why it works; this effect was evident in the learners’ responses. it is acknowledged that the dynamics of the classroom are complex and often teachers face dilemmas about whether to teach for conceptual understanding or to focus on getting good results by focusing on procedures. however, teaching procedures without understanding, as alluded to by olivier (1992) above, is not effective. in fact, this study shows that learning procedures without understanding the background does not even lead to good results. conclusion top ↑ in this article we analysed learners’ strategies to tackle questions based on ratio and proportion by identifying the underlying theorems-in-action. the identification of the theorems-in-action provided insight into whether their strategies were correct. this also helped us understand whether or not some incorrect methods produced correct answers coincidently because of the numbers that were used. as was shown, some of these incorrect theorems–in-action revealed procedural ways of working that contradicted properties of the direct proportional relationship. it was also shown that many learners opted for the cross multiplication rule; however, many were confused about which quantities should be the multipliers and which should be the divisor in the rule of three. we argue that the confusion emerged because none of the learners’ responses indicated equation 2 or equation 3, which are the crux of the proportional relationship. they then carried out operations without understanding the algorithm and why it worked. the number of possible permutations of four numbers in four positions is 4p4 = 24. hence, there are 24 possible ways of arranging four numbers in terms of two equivalent fractions and only eight of them correctly represent the proportional relationship. in the absence of knowing the functional relationship that dictates exactly which fractions are equal, and why they are equal, learners seem to have guessed and placed the quantities (three given and one unknown) in arbitrary positions, allowing them to carry out the operations of ‘cross multiplication’ and division to arrive at an answer. the study also found that no learner mentioned the words ratio or proportion. we can thus infer that most of these learners are working out the problems (some doing it successfully) without realising the meaning of equivalent ratios, or knowing the conditions under which four quantities form a proportional relationship. hence they were carrying out the procedures without engaging with the linear function of the type y = kx, which defines the relationship between the quantities. the role of the function is not addressed, even in textbooks, and perhaps this omission is a large factor accounting for learners’ struggles with solving such missing value problems based on proportional relationships. without reference to the linear function y = kx learners can rely only on memorised strategies. we therefore suggest that shortcuts such as the cross multiplication strategy should be not be used with secondary school learners without a concurrent emphasis on the role of the function that sets the conditions for the proportional reasoning. acknowledgements top ↑ competing interests the authors declare that they have no financial or personal relationship(s) that may have inappropriately influenced them in writing this article. authors’ contributions the data collection and analysis was carried out by p.m. (university of kwazulu-natal). s.b. (university of kwazulunatal) led the write-up of the article. references top ↑ ben-chaim, d, keret, y, & ilany, b.-s. (2012). ratio and proportion. research and teaching in mathematics teachers’ education (preand in-service mathematics teachers of elementary and middle school classes). dordrecht: sense publishers. http://dx.doi.org/10.1007/978-94-6091-784-4 chick, h.l., & harris, k. (2007, december). pedagogical content knowledge and the use of examples for teaching ratio. paper presented at the annual conference of the australian association for research in education, fremantle, australia. available from http://www.aare.edu.au/data/publications/2007/chi07286.pdf hart, k.m. (1981). children’s understanding of mathematics: 11–16. london: john murray. hart, k.m. (1984). ratio: children’s strategies and errors. a report of the strategies and errors in secondary mathematics project. windsor, berkshire: nfer_nelson. hart, k. (1988). ratio and proportion. in j. hiebert, & m. behr (eds.), number concepts and operations in the middle grades (pp. 198–219). reston, va: national council of teachers of mathematics. jiang, c. (2008, august). strategies for solving word problems on speed: a comparative study between chinese and singapore students. paper presented at the 11th international congress on mathematical education, monterrey, mexico. available from http://tsg.icme11.org/document/get/773 lamon, s.j. (2007). rational numbers and proportional reasoning: toward a theoretical framework for research. in f.k. lester (ed.), second handbook of research on mathematics teaching and learning (vol. 1, pp. 629–668). charlotte, nc: information age publishing. long, c. (2007). what can we learn from timss 2003? in m. setati, n. chitera, & a. essien (eds.), proceedings of the 13th annual congress of the association for mathematics education of south africa (vol. 1, pp. 1–23). white river: amesa. long, c. (2011). mathematical, cognitive and didactic elements of the multiplicative conceptual field investigated within a rasch assessment and measurement framework. unpublished doctoral dissertation. university of cape town, cape town, south africa. available from http://hdl.handle.net/%2011180/1521 mahlabela, p.t. (2012). learner errors and misconceptions in ratio and proportion: a case study of grade 9 learners from a rural kwazulu-natal school. unpublished master’s thesis. university of kwazulu-natal, edgewood, south africa. available from http://researchspace.ukzn.ac.za/xmlui/handle/10413/6490/browse?value=mahlabela%2c+patisizwe+tennyson.&type=author md-nor, m. (1997). investigation of the teaching and learning ratio and proportion in malaysian secondary schools. in proceedings of the british society for research into learning mathematics, 17(3), 32–37. london: british society for research into learning mathematics. available http://www.bsrlm.org.uk/ips/ip17-3/bsrlm-ip-17-3-full.pdf misailidou, c., & williams, j. (2003). diagnostic assessment of children’s proportional reasoning. journal of mathematical behavior, 22, 335–368. http://dx.doi.org/10.1016/s0732-3123(03)00025-7 olivier, a.i. (1992). developing proportional reasoning. in m. moodley, r.a. njisane, & n.c. presmeg (eds.), mathematics education for in-service and pre-service teachers (pp. 297–313). pietermaritzburg: shuter & shooter. vergnaud, g. (1998). a comprehensive theory of representation for mathematics education. journal of mathematical behavior, 17(2), 167–181. http://dx.doi.org/10.1016/s0364-0213(99)80057-3 wu, h. (2006, october) professional development: the hard work of learning mathematics. paper presented at the fall southeastern section meeting of the american mathematical society, tennessee, usa. available from https://math.berkeley.edu/~wu/hardwork2.pdf abstract introduction and background to the study literature review research method and design findings and analysis discussion conclusion acknowledgements references appendix 1 opening conversation discussion conclusion opening conversation discussion conclusion footnotes about the author(s) julian moodliar school of education, faculty of humanities, university of the witwatersrand, johannesburg, south africa lawan abdulhamid school of education, faculty of humanities, university of the witwatersrand, johannesburg, south africa citation moodliar, j., & abdulhamid, l. (2021). novice and expert grade 9 teachers’ responses to unexpected learner offers in the teaching of algebra. pythagoras 42(1), a624. https://doi.org/10.4102/pythagoras.v42i1.624 original research novice and expert grade 9 teachers’ responses to unexpected learner offers in the teaching of algebra julian moodliar, lawan abdulhamid received: 18 apr. 2021; accepted: 04 oct. 2021; published: 15 dec. 2021 copyright: © 2021. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract in south africa, limited studies have been conducted investigating responsive teaching and little is known about how teachers respond to unexpected events ‘in the moment’ that did not form part of their planning. in this article, we report how a grade 9 novice and expert teacher responded to unexpected learner offers during the teaching of algebra using a qualitative case study approach. three consecutive lessons for each teacher were video recorded, transcribed and analysed. our units of analysis for episodes were teachers’ responses to unexpected learner offers and we coded the responses as ‘appropriate’ or ‘inappropriate’. indicators used to highlight the degree of quality of the response were ‘minimum’, ‘middle’ and ‘maximum’ if a response was coded as appropriate to a learner’s offer. once lessons were analysed, the first author conducted video-stimulated recall interviews with each participant to gain insight into the two teachers’ thoughts and decision-making when responding to unexpected learner offers. the findings from this study illustrated that the novice teacher failed to press learners when their thinking was unclear, chose to ignore or provided an incorrect answer when faced with an unexpected learner offer. conversely, the expert teacher continuously interrogated learner offers by pressing if a learner offer was unclear or if she wanted learners to explain their thinking. this suggests that the expert teacher’s responses were highly supportive of emergent mathematics learning in the collective classroom space. keywords: responsive teaching; unexpected events; contingency; secondary school; algebra; novice and expert teachers introduction and background to the study international literature on responsive teaching suggests that quality teaching is highly improvisational as teachers ask learners questions based on learners’ thinking ‘in the moment’ (sawyer, 2004). moreover, teachers need to listen interpretively and respond constructively to learner offers rather than ignoring or dismissing a learner’s offer (davis, 1997; jacobs & empson, 2016). borko and livingston (1989) have noted that responding to unexpected learner ideas is an important albeit challenging facet of effective teaching. abdulhamid and venkat (2018) have acknowledged that limited research has been conducted in the south african terrain concerning responsive teaching and the research base is at a distance from responsive teaching described in the international literature. evidence of responsive teaching at primary school level in south africa suggests that ‘in the moment’ responsiveness is rare due to lessons being highly scripted and that there is a lack of evaluation of learners’ offers accompanied by chorusing in the collective classroom space (abdulhamid & venkat, 2018; hoadley, 2012). however, the situation at secondary level is different since empirical evidence suggests that teachers often evaluate learners’ offers using an initiation-response-evaluation (ire) format such as a question-and-answer exchange but the nature of their engagement with learner offers is often superficial (brodie, 2007). currently, little is understood about how teachers respond to learners when faced with unexpected incidents that did not form part of their planning. this knowledge in action demonstrated when teachers deal with unexpected events forms part of the contingency dimension associated with the knowledge quartet formulated by rowland, huckstep and thwaites (2005). in recent years, rowland, thwaites and jared (2015) have proposed that there are three triggers of contingency associated with responding to unexpected events and these include (1) responding to unexpected learner offers, (2) teacher insight and (3) the sudden (un)availability of pedagogical tools such as technology during teaching. based on the paucity of research in south africa associated with contingency, we were interested to explore how senior phase expert and novice teachers (specifically at grade 9) responded to unexpected learner offers while teaching a fundamental topic such as algebra. our focus on algebra1 emanates from difficulties noted by examiners of high stakes examinations such as the matric examination: the department of basic education (dbe) notes the ‘algebraic skills of the candidates are poor’ and that the majority of learners ‘lacked fundamental and basic mathematical competencies which could have been acquired in the lower grades’ (dbe, 2019, p. 179) such as grade 8 and grade 9. in both international and local terrains, algebra has been continuously noted as being abstract for learners and the transitioning from arithmetic to algebra is difficult since learners erroneously link arithmetic principles with algebra (herscovics & linchevski, 1994; macgregor & stacey, 1997). in both south africa and internationally, attention has been predominantly focused on learner errors since errors and misconceptions are believed to provide a lens for understanding learners’ thinking (gardee & brodie, 2015; makonye & luneta, 2014; pournara, hodgen, sanders, & adler, 2016). however, tsui (2003) notes that it is imperative that researchers in different domains such as language and mathematics direct their efforts to understanding how the knowledge bases of novice and expert teachers differ and explore how these knowledge bases are developed. the purpose of this study was to investigate how a novice and expert grade 9 mathematics teacher responded to unexpected learner offers within their lessons and gain insight into their thoughts and decision-making when responding to learners via video-stimulated recall (vsr) interview. the following two research questions guided the investigation: how do novice and expert senior phase teachers respond to unexpected events in the teaching of algebra? what are the novice and expert senior phase teachers’ thoughts and decision-making in response to unexpected events during the teaching of algebra? we begin this article with an overview of responsive teaching, and how it is seen in the south african context. we then provide a discussion on contingency in mathematics teaching, as a vehicle for responsive teaching. a discussion of why a focus on novice and expert teachers is presented. the literature base on algebra acted as a vantage point for understanding and commenting on the nature of teachers’ responses to contingent moments. details of the observed lessons are presented and then analysed based on a modified form of weston’s (2013) coding protocol. we present an interview outline involving questions that interrogate teachers’ decision-making in response to contingent moments in the teaching of algebra. subsequently, analysis and findings from the study are presented. literature review overview of responsive teaching in the context of classroom discourse, responsive teaching refers to when a teacher follows up on learners’ questions or offers which open up possibilities for mathematical learning in the collective classroom space. over the last 40 years, ire interactions have been widely studied by researchers and there are two main bodies of literature associated with responsive teaching: ‘deficit’ approaches and ‘affordance’ approaches (abdulhamid & venkat, 2018; brodie, 2007; mehan, 1979; sinclair & coulthard, 1975). the ‘deficit’ approach is associated with teachers listening evaluatively to learner offers, asking questions that are predetermined and not contingent on learner offers (davis, 1997). moreover, ‘funnelling’ is a common phenomenon associated with ‘deficit’ approaches wherein teachers narrow the questions to a point that learners can answer them with minimal cognitive effort (bauersfeld, 1980; stein, grover, & henningsen, 1996). conversely, ‘affordance’ approaches refer to teachers asking ‘authentic’ questions that are not pre-planned and teachers let learner offers steer the trajectory of the lesson (borko & livingston, 1989; jacobs & empson, 2016; nystrand, gamoran, kachur, & prendergast, 1997; sawyer, 2004). the improvisational and spontaneous nature of responsive teaching associated with ‘affordance’ approaches is a concerted effort between the teacher and learners (sawyer, 2004). this is comparable to theatre actors working without a script as part of improvisation since the conversation taking place is unplanned and dependent on how interactions unfold between the actors (sawyer, 2004). the south african research base on responsive teaching is limited and stands at a distance from responsive teaching described in the international literature (abdulhamid & venkat, 2018). the nature of responsive teaching in south african primary mathematics classrooms has been recently examined qualitatively by researchers and it has been noted that evaluation of learner offers ‘in the moment’ are rare in the collective classroom space (abdulhamid, 2016; abdulhamid, 2017; abdulhamid & venkat, 2018; hoadley, 2012). a consequence of not evaluating learner offers is that ‘learners may well remain unaware of the extent to which their offers and narratives are “endorsable” from a mathematical perspective’ (abdulhamid, 2017, p. 200). in stark contrast to the south african research base, abdulhamid (2017) notes that the international literature does not focus on whether learner contributions are evaluated but rather the focus is on what opportunities for learning arise as a result of the teacher’s evaluation of the learners’ contribution. in secondary mathematics classrooms, brodie (2007) has illuminated that there is evidence of evaluation of learner offers which differs from the situation that currently exists at primary school level. however, the nature of teacher engagement with learners’ offers is often superficial (brodie, 2007). from our literature search, we found limited research conducted in south africa investigating responsive teaching. little is understood about how teachers respond to learners when faced with unexpected incidents that did not form part of their planning associated with contingency. moreover, little is known about teachers’ thoughts and decision-making when responding to unexpected events in the south african terrain. contingency in mathematics education in the 1970s, alan bishop initiated research into contingency where he was interested in how teachers deployed strategies to buy time when faced with unexpected events that caught them off guard (borko, roberts, & shavelson, 2008). in recent years, there has been a surge in contingency research internationally related to how teachers respond ‘in the moment’ to unexpected events that did not form part of their planning (coskun, bostan, & rowland, 2021; foster, 2015; mason & davis, 2013; mason & spence, 1999; rowland, turner, thwaites, & huckstep, 2009; rowland et al., 2015; rowland & zazkis, 2013). rowland et al. (2015, p. 76) note that it is impossible for a teacher to predict all the unexpected events that can occur within a lesson but ‘the teacher who anticipates such obstacles to learning is better placed to plan a less “bumpy”, more joined-up learning experience for his or her students’. as previously stated, rowland et al. (2015) have outlined that there are three triggers of contingency, namely (1) arising from learners’ offers, (2) arising from teacher’s insight and (3) arising from the (un)availability of a pedagogical tool. the first trigger of contingency takes into account how the teacher responds to learners’ contributions that are unanticipated and this is further subdivided into three classifications which include (1) the learner’s response to a question initiated by the teacher, (2) a learner’s spontaneous response to an activity or discussion and (3) the unexpected incorrect offer stated by a learner (rowland et al., 2015). when faced with an unexpected learner contribution, rowland et al. (2015) have proposed that teachers can decide to (1) ignore the learner’s offer, (2) acknowledge but continue with the lesson or (3) acknowledge, respond and incorporate the learner’s offer into the lesson. for instance, rowland et al. recount an incident where a teacher asked learners to find the area of a rectangle with dimensions of 22 cm and 28 cm and one learner unexpectedly stated that the area would be 25 cm multiplied by 25 cm. the teacher decided to ignore the incorrect offer and continued with the lesson. failing to acknowledge the unexpected incorrect offer denied the teacher a chance to interrogate the learner’s thinking ‘in the moment’ and a teachable moment was missed (boaler & humphreys, 2005; nesher, 1987). also, within this first trigger of contingency, the teacher has a choice of acknowledging a learner’s spontaneous response to a demonstration (rowland et al., 2015). for example, borko et al. (1992) present an illuminating example of a pre-service teacher attempting to construct a response ‘in the moment’ to a learner’s spontaneous question of why one needs to ‘invert-and-multiply’ the second fraction when dividing a fraction such as . the pre-service teacher attempted a diagrammatic representation and abandoned her attempt after she realised that her representation was incorrect (borko et al., 1992). the second trigger of contingency emanates from teacher insight when the teacher becomes aware that something is amiss by reflecting in action during the progression of the lesson (schön, 1987). rowland et al. (2015) recognise that this trigger is rare among pre-service and newly qualified teachers and predict that it is more common for expert teachers. corcoran (2008) describes a pre-service teacher in ireland making use of butter beans to illustrate division to a class of primary school learners. the teacher designed division questions requiring learners to use a division problem structure associated with grouping. however, by reflecting ‘in the moment’ she senses something is amiss and realises she unintentionally guided learners to the incorrect division structure of partition. as a result of teacher insight, the teacher re-directed learners to the appropriate division structure needed to answer the division problems. the third trigger of contingency accounts for the availability or sudden unavailability of a pedagogical tool such as technology which disrupts the trajectory of the lesson. for instance, when technology becomes unavailable, this will place demands on the teacher to deviate from their lesson plan. rowland et al. (2015) recount an incident in a pre-service teacher’s lesson where a computer package, autograph, useful for sketching functions, refused to open. the teacher needed to act ‘in the moment’ and decide how to respond to this sudden unavailability of a resource. ultimately, the teacher resorted to using the whiteboard to sketch the functions. however, it is not always the absence of a pedagogical tool that can perturb the intended trajectory. there are times that a resource can be incorporated into a lesson that did not initially form part of the teacher’s lesson image. for example, a teacher in a study by rowland et al. found a 1 to 100 square lying on a table which did not form part of her lesson plan and decided to incorporate it into the lesson based on subtraction. however, she did not realise that unexpectedly incorporating this pedagogical tool encouraged learners to respond spatially instead of symbolically which unwittingly undermined her envisaged lesson image of teaching subtraction symbolically. in this article, we focus on the first trigger arising from learners’ unexpected offers during classroom interactions. firstly, we focus on this trigger since it was the only trigger evident across the novice and expert teachers in our study. secondly, in the current curriculum and assessment policy statements (caps) in south africa there is a press for strong pacing and curriculum coverage (bertram, mthiyane, & naidoo, 2021) which has led to a situation of less attention being paid to learners’ contributions during the course of teaching. hence, more attention on this first trigger is needed to capture the subtle nuances for understanding how teachers deal with unexpected learner offers in their teaching. what is meant by a ‘novice’ and ‘expert’ teacher? there are differing conceptions in the literature as to what constitutes a ‘novice’ and an ‘expert’ teacher by mathematics education researchers. for instance, novice teachers are typically equated with pre-service teachers (borko & livingston, 1989; borko, livingston, & shavelson, 1990; borko et al., 1992; stockero & van zoest, 2013). for other researchers such as berliner (1988), the term encompasses pre-service teachers and teachers in their first year of teaching. moreover, berliner goes further to categorise teachers with two to three years’ experience as ‘advanced beginner’. we adopt huberman’s (1993) interpretation of a novice teacher to be an in-service teacher in their first year to third year of teaching as this interpretation is broader than that of berliner (1988) and excludes pre-service teachers. in terms of defining an expert teacher, there is a lack of a universal consensus for what constitutes an expert teacher (yang & leung, 2013). berliner (2001) notes that interpretations of expert teachers are strongly linked with the culture and contextual background in which the study is being conducted. researchers such as palmer, stough, burdenski and gonzales (2005) note that little research has been conducted to identify a common set of criteria among expert teachers in different subject areas and these researchers identified four common criteria across different studies used to identify expert teachers. the first criterion used to select expert teachers is the number of years the teacher taught the subject which was usually at least 10 years. the second common criterion was the teacher being recommended by the school principal or colleagues as an expert teacher. the third criterion was that teachers needed to be in possession of an appropriate degree and teacher qualification. lastly, performance-based criteria that were ‘normative’ and ‘criterion-based’ were used to select expert teachers. for our study, we found all four criteria delineated by palmer et al. (2005) useful when conceptualising an expert teacher. if we considered only one of the criteria, this would be insufficient to describe an expert teacher. for example, if one considered a teacher in terms of years of experience yet the teacher did not have an appropriate degree this would not capture the essence of an expert teacher. we consider a teacher to be an expert if they had at least 10 years’ teaching experience, were in possession of a relevant academic qualification such as having at least an honours degree and were selected to act as a mentor teacher to pre-service teachers or novices. several studies originating in the 1980s conducted by researchers interested in cognition and improvisation have examined novice and expert teachers on the basis of their planning, instruction and post-lesson reflection on classroom incidents (borko & livingston, 1989; borko et al., 1990; leinhardt & greeno, 1986). in an important and highly cited study conducted by borko and livingston (1989), they noted that the nature of novice teachers’ planning was rehearsed, scripted word for word, done at unit level and usually prepared the day before the lesson, whereas expert teachers engaged in flexible, unrehearsed, long-term planning which was done at chapter rather than unit level and expert teachers had mental plans for lessons with written planning kept to a minimum (borko & livingston, 1989). borko et al. (1990, p. 43) state that the efficient planning skills of expert teachers is attributed to rich, interconnected mental schema which permit them ‘to determine what information is relevant to their planning tasks and to plan more efficiently’. moreover, expert teachers anticipated unpredictable incidents in their planning and possible learner misconceptions whereas novice teachers did not anticipate such contingencies in their planning (borko & livingston, 1989). in terms of instruction, expert teachers delivered coherent lessons whereas novice teachers tended to deliver mostly disjointed lessons (borko & livingston, 1989). moreover, novices had difficulties constructing explanations and making in-flight decisions when faced with unexpected learner questions as a result of their inexperience in skilled improvisation (borko & livingston, 1989). however, expert teachers were successful at answering unexpected questions, providing demonstrations and examples in response to unexpected learner offers as a result of their skilled improvisation (borko & livingston, 1989). in terms of the post-lesson reflections, novice teachers focused more on their classroom management and effectiveness as a teacher rather than key classroom incidents (borko & livingston, 1989). however, post-lesson reflection of expert teachers focused primarily on their learners’ understanding of concepts (borko & livingston, 1989). during the 1990s, researchers in israel analysed the lessons of novice and expert mathematics teachers building upon the work of borko and livingston (even, tirosh, & robinson, 1993; tirosh, even, & robinson, 1998). for instance, even et al. (1993) found that novice teachers delivered lessons that lacked connectedness in linking topics and ideas from earlier lessons whereas experts delivered coherent lessons that linked with ideas and topics discussed in previous lessons. in recent years, researchers have started to examine novice and expert teachers in terms of their ability to notice classroom incidents. huang and li (2012) examined how 10 novice and 10 expert mathematics teachers in china noticed classroom events when watching videotaped lessons. it was evident that novice teachers found noticing to be more challenging in relation to expert teachers and novices’ noticing was less developed than experts (huang & li, 2012). huang and li (2012) found that: expert teachers paid significant and greater attention to developing mathematical thinking and ability, developing knowledge coherently, and developing high-order thinking; they also paid significant and greater attention to teachers’ enthusiasm and passion, and students’ participation. (p. 428) teaching and learning of algebra algebra is an important branch in mathematics which serves as a precursor to other important topics such as trigonometry and analytical geometry (watson, jones & pratt, 2013). however, many learners experience difficulties transitioning from arithmetic to algebra and this has been widely documented in many studies since the 1970s (collis, 1978; herscovics & linchevski, 1994; macgregor & stacey, 1997). common reported misconceptions include learners attempting to conjoin unlike terms as a result of the lack of finality that some expressions pose such as b + 3 (macgregor & stacey, 1997). difficulties can also arise when teachers encourage a ‘fruit salad approach’ to algebra encouraging learners to let letters represent initials or labels for variables such as b for banana rather than the number of bananas (boaler & humphreys, 2005; hallagan, 2006). we note that researchers in south africa and internationally have predominantly focused on learner errors since errors and misconceptions are believed to provide a lens for understanding learners’ thinking (gardee & brodie, 2015; makonye & luneta, 2014; pournara et al., 2016; watson et al. 2013). however, we agree with tsui (2003) that more attention should be directed to understanding how the knowledge bases of novice and expert teachers develop and differ rather than focus primarily on learner errors. theoretical framework mason and spence (1999) have theorised that there are two distinct forms of ‘knowing’, referred to as ‘knowing about’ and ‘knowing to’. planned aspects of teaching are related to ‘knowing about’ whereas ‘knowing to’ relates to the contingency dimension of the knowledge quartet. ‘knowing to’ refers to active knowledge that is needed when improvising and responding to unexpected events (mason & spence, 1999; rowland et al., 2015; sawyer, 2004). mason and spence (1999) have categorised ‘knowing about’ into three divisions referred to as ‘knowing that’, ‘knowing why’ and ‘knowing how’ as illustrated in figure 1. ‘knowing that’ links with the foundation dimension of the knowledge quartet and this refers to subject matter knowledge that the teacher possesses. mason and spence (1999, p. 145) posit that ‘knowing why’ suggests being able to provide a rationale for actions performed and ‘having stories i tell myself to account for something’. ‘knowing how’ refers to a teacher being able to perform a procedure in front of the class such as being able to solve an equation by means of completing the square. it is pertinent to note that ‘knowing about’ something does not ensure that a teacher would be able to respond to unexpected events since ‘knowing about’ ‘gives little indication of whether that knowledge can be used or called upon when required’ (mason & spence, 1999, p. 138). figure 1: beyond ‘knowing about’. our interest is associated with ‘knowing to’ since this is knowledge in action which requires improvisation. moreover, ‘knowing to’ provided a suitable lens to examine interrelated concepts dealing with the phenomenon of contingency associated with teachers’ responsiveness to unexpected learner offers (figure 2). figure 2: possible teacher response pattern to triggers of contingency research method and design a qualitative case study approach was used to allow for a detailed exploration of the novice and expert teachers’ responses to unexpected learner offers within their grade 9 mathematics classrooms (merriam, 1998). in february 2020, we observed and videorecorded three consecutive lessons of two teachers purposively selected teaching grade 9 algebra from the same school. both teachers were teaching simplification of algebraic expressions and distributive law when data collection took place. prior to videorecording, both teachers were asked to prepare lesson plans for the observed lessons. context of the study and participants the study is conducted in a well-resourced, private school for girls located in johannesburg east which caters for 600 learners. tara2 is a novice teacher in possession of a bachelor of science degree and was currently in her first year of teaching when the study was conducted. tara completed a teacher internship the previous year at the same school while registered part-time for a postgraduate certificate in education (pgce) and finished the pgce at the end of 2020. cassandra is an expert teacher with 20 years’ teaching experience, was recommended by the head of department as an expert teacher and is in possession of a master of education. cassandra has taught grade 9 mathematics every year since she started teaching. the criteria used to select cassandra met all four of the criteria outlined by palmer et al. (2005) in the literature review. cassandra has also served as a mentor to novice teachers over the years and is currently serving as a mentor to tara. overview of data analysis process once data were collected, we transcribed verbatim teacher talk and teacher-learner interactions for each lesson. our units of analysis for episodes were teachers’ responses to unexpected learner offers and we coded the responses as ‘appropriate’ or ‘inappropriate’ as per weston’s (2013) coding protocol. a response was coded as ‘appropriate’ when teachers acknowledged unexpected learner offers and these offers provided potential learning opportunities for the class. conversely, ‘inappropriate’ responses included teachers ignoring or dismissing learner offers that had potential for learning and this hindered learners’ access to mathematics ‘in the moment’. if a response were coded as ‘appropriate’, we coded the quality of the response as ‘minimum’, ‘middle’ or ‘maximum’ as per the weston coding protocol (figure 3). figure 3: schematic representation of coding protocol used to analyse teachers’ responses to unexpected learner offers if a teacher chose to acknowledge a learner offer that had potential for learning but provided an incorrect response, this was coded as ‘minimum’. however, if a response was correct, provided potential for learning but there was a lack of probing the learner’s offer ‘in the moment’ we coded this as ‘middle’. a response coded as ‘maximum’ was evident when a teacher gave a detailed explanation based on the unexpected learner offer and pressed the learner demonstrating that the full potential of the incident was exploited. researchers note that ‘press’ is a fundamental aspect of the questioning process and concerns the teacher’s decision to follow up on what a learner has said to understand their thinking ‘in the moment’ (boaler & humphreys, 2005; kazemi & stipek, 2001). once episodes were analysed, the first author conducted vsr interviews with each participant to help participants reflect on ‘their thoughts at the time’ (mackey, 2012, p. 27) and give us insight into their decision-making in response to the unexpected learner offers they encountered. contingent incidents were watched together with participants to help them recollect these events. firstly, general questions were asked such as ‘did you deviate in anyway whatsoever from the lesson plan you designed for this lesson?’. subsequently, more specific questions related to contingent incidents were asked such as ‘can you elaborate on how you responded?’ or ‘if you were to repeat this lesson, would you respond to the learners’ contributions in the same way or differently?’ the sample of semi-structured questions during the vsr interview is included in appendix 1. ethical considerations permission from the university (2019ece050m) was received prior to data collection. informed consent was obtained from teachers, learners and their guardians. names of all participants have been kept anonymous to preserve the confidentiality of participants. findings and analysis four contingent incidents emerged within tara’s lessons and three contingent incidents were noted in cassandra’s lessons. all contingent incidents were related to the first trigger of contingency associated with teachers’ responses to unexpected learner offers and we present an overview of the findings in table 1. as previously mentioned, if a response was present, we coded it as ‘appropriate’ or ‘inappropriate’ and we coded the response patterns that were ‘appropriate’ as ‘minimum’, ‘middle’ or ‘maximum’. tara chose to either ignore or acknowledge a learner offer and none of her responses was coded as ‘maximum’. conversely, cassandra’s responses were all coded as ‘appropriate’ and ‘maximum’ since she continuously interrogated learner offers and her responses provided ‘maximum’ potential for learning. we provide below a presentation of the dialogue between the teachers and learners and provide an analysis of each teacher’s response to the contingent incident. table 1: an overview of the response patterns of tara and cassandra to unexpected learner offers. tara’s response patterns to unexpected learner offers contingent incident 1: ignore or dismiss in tara’s first observed lesson, a learner expresses confusion as to why can be rewritten as 1x−1 when tara states that an expression with a negative exponent cannot be classified as a polynomial. the learner asks tara to explain the ‘exponentive one’ and tara chooses to dismiss the unexpected learner question. failing to acknowledge the learner’s important question hindered the learners’ access to mathematics ‘in the moment’ and we coded her response as ‘inappropriate’ as this question needed to be addressed. similarly to novices in borko and livingston’s (1989) study, tara postponed answering the question as she was caught off guard by the question. tara’s response is unsurprising as stockero and van zoest (2013) note that 18% of the response patterns of novices in their study tended to dismiss or ignore an unanticipated learner offer. moreover, in the vsr interview, tara acknowledged that she did not anticipate the question in her lesson plan as she thought that their previous grade 8 teacher had taught this concept. box 1: tara’s response to contingent incident 1. box 2: tara’s response to contingent incident 2. contingent incident 2: acknowledge but continue in tara’s second lesson, she spent the beginning of the lesson explaining to learners why can be rewritten as 1x−1 and then demonstrates negative exponents using a base of two. subsequently, she uses numerical bases of three and six and asks learners to use their calculators and see that the answer will be a fraction if the exponent is negative. a learner asks tara what the answer would be if it were –2−2 and tara chooses to acknowledge the learner’s question. tara writes that the answer to –2−2 is instead of and then erases the whiteboard. tara attempted to construct an explanation ‘in the moment’ to an unanticipated question and realises that her attempt is incorrect similar to the novice teachers’ unskilled improvisation in borko and livingston’s (1989) study. tara’s ability to reason ‘in the moment’ is momentarily frozen. she refrains from further answering the learner’s question by stating that she will come back to the question and continues with the lesson. tara’s response is reminiscent of a pre-service teacher in a study by borko et al. (1992) where the teacher attempted a diagrammatic representation not planned in advance to illustrate conceptually why one needs to ‘invert-and-multiply’ but abandons the attempt when she realises her answer is incorrect. we coded tara’s response as ‘minimum’ since she chose to acknowledge but provided an erroneous response to a learner’s question which did not support mathematical learning in the collective classroom space. stockero and van zoest (2013, p. 135) note that when the teacher provides an incorrect response, ‘the error is likely to affect what students take away from the lesson’. in the vsr interview she admitted that ‘i don’t want to now go through something else where they’re going to look at me so lost and confused again’ and was ‘scared to want to answer their questions’ as they were ‘not really in line with what i actually want them to know’. tara felt she was ‘going down a rabbit hole with this question’ and she thought it was best to postpone further answering the question. contingent incident 3: acknowledge but continue later in tara’s second lesson, a learner asks her ‘what if it says, so x to the negative fraction?’ tara acknowledges the ‘interesting question’ but decides to continue with the lesson as not answering the question was likely going to have a neutral impact for the class. we coded her response as ‘middle’ rather than ‘minimum’ since her response did not introduce any errors. box 3: tara’s response to contingent incident 3. box 4: tara’s response to contingent incident 4. we note the question asked by the learner needed to be interrogated to understand her mathematical thinking ‘in the moment’ as the question was unclear and needed clarity. tara chooses to superficially engage with the question by recognising that the learner asked a ‘interesting question’, gives a brief response and decides to continue with the lesson as if the question did not occur. this response pattern was noted by 26% of novices in stockero and van zoest’s (2013) study and the authors note that novice teachers typically did not capitalise on these ‘pivotal teaching moments’. during the vsr interview, she revealed that she ‘was just trying to keep things as simple as possible’ and was willing to ‘explore these questions but not right now’ as she felt she had a weak class. contingent incident 4: acknowledge but continue during tara’s third lesson, she is recapping distributive law and starts to simplify the algebraic expression: (x + 1) + (2x + 3). tara starts to distribute the invisible one into the first bracket and is interrupted by a learner who asks ‘madam, isn’t there an invisible one before … wouldn’t it be four?’ tara chooses to acknowledge the learner’s question but does not interrogate the learner’s thinking ‘in the moment’ to understand her mathematical thinking. tara is listening evaluatively rather than interpretively to the learner’s question (davis, 1997) and offers a procedural explanation of distributive law. we coded tara’s response as ‘middle’ since the response provided does support mathematical learning to a certain extent but tara could have probed the learner with a follow-up question to get the learner to clarify and explain her thinking (boaler & humphreys, 2005; kazemi & stipek, 2001). boaler and humphreys (2005, p. 37) note that probing or pressing learners’ thinking is important since it allows learners ‘to elaborate their thinking for their own benefit and for the class’. the lack of press or probing is not uncommon among novice teachers as a study conducted by moodliar (2020) noted that the novice teachers in his study failed to interrogate learner’s thinking ‘in the moment’. during the vsr interview, the first author probed tara’s response to the learner and she knew that the learner ‘wanted to jump in already and say one plus three is four’ but thought it best to procedurally explain how to obtain the answer as this specific learner skips steps and does not show all her working details. box 5: cassandra’s response to contingent incident 1. cassandra’s response patterns to unexpected learner offers contingent incident 1: acknowledge and incorporate a learner expresses confusion with the wording of the cassandra’s question which asks: by how much is the sum of 2x2 + 3x – 2 greater than the sum of (−x + 2) and (−5x2 – 8)? cassandra spends time trying to understand the source of the learner’s confusion and realises that the learner is struggling to translate the wording into mathematics. sawyer (2011, p. 1) contends that ‘what makes good teachers great’ is their ability to engage in skilled improvisation. cassandra is able to engage in such skilled improvisation by drawing upon her repertoire of pedagogic strategies and constructs a series of numerical questions ‘in the moment’ to help the learner interpret the initial question and make sense of the wording of the question. cassandra effectively dealt with the learner’s question without derailing the lesson and her successful improvisational performance is analogous to how expert teachers responded to unexpected learner offers in borko and livingston’s (1989) study. we coded her response as ‘maximum’ since her response provided maximum potential for learning. when probed as to how she responded ‘in the moment’, it was revealed that cassandra was able to pre-empt the question. during the vsr interview, cassandra revealed that when a learner expresses confusion with such questions, she has learnt to ‘replace all this algebra with just some integers’ to serve as a ‘tool that they can use when they sort of get confused with x’. box 6: cassandra’s response to contingent incident 2. box 7: cassandra’s response to contingent incident 3. contingent incident 2: acknowledge and incorporate in cassandra’s second lesson, she is teaching distributive law and works out how to simplify (x + 5)(+2). a learner then asks cassandra ‘if it’s x plus five and there is a two outside the bracket?’ cassandra chose to acknowledge and incorporate the learner’s example into the lesson as she sees the potential of the learner’s question. cassandra probes the learner with follow-up questions such as ‘what do you think?’ and ‘why?’ in order for the learner to clarify and elaborate her thinking. subsequently, cassandra asks the learner ‘what can you tell me about the previous example and this one?’ during the vsr interview, it was evident that cassandra thought that learners might think that the previous example she worked out with the class of (x + 5)(+2) might produce a different result from the learner’s question of (x + 5)2. cassandra wanted the class to make a connection and realise that the two questions yielded the same answer. thus, deciding to deftly incorporate the example ‘in the moment’ proved fruitful and maximised learning opportunities for the class. we coded her response as ‘maximum’. contingent incident 3: acknowledge and incorporate in cassandra’s third lesson, a learner expresses confusion as to whether the answer to 3x(x – 5) is 3x + 3x2 – 15x or 3x2 – 15x. cassandra decides to acknowledge and try to make sense of the learner’s question rather than dismissing it. cassandra probes the learner to understand why she wants ‘to add another three x’ as her reasoning is unclear to cassandra. cassandra realises ‘in the moment’ that the learner wants to carry over the 3x since ‘it’s already there’ and resorts to explaining procedurally that each term gets multiplied during distributive law meaning that 3x will not remain. subsequently, cassandra probes the learner as to whether she added in an extra term and cassandra’s response to the unexpected question provided maximum support for learning. during the vsr interview when cassandra was probed on her response to the learner, she stated that ‘i was thinking that she probably forgot to multiply’ and reflects that ‘because the last term is 15x’ the learner did not ‘realise that they also have to multiply by another x which makes it the 3x2’. we coded cassandra’s response as ‘maximum’. discussion cassandra displayed skilful improvisation in responding to unexpected events in the teaching of algebra, exhibited openness in listening interpretively to learner offers and allowed learner offers to steer the trajectory of the lesson (davis, 1997; sawyer, 2004). cassandra continuously interrogated learner offers by engaging in press if a learner offer was unclear or if she wanted learners to elaborate their thinking. this proved fruitful and provided maximum learning opportunities for the class. conversely, tara failed to press learners when their thinking was unclear, chose to ignore or provided an incorrect answer when faced with an unexpected learner offer. this can be possibly attributed to her lack of experience when faced with unexpected learner offers and her lack of rich, interconnected mental schema to quickly access pedagogical content knowledge ‘in the moment’ (borko & livingston, 1989). another interesting finding was that vsr served as a useful catalyst to help tara and cassandra reflect on their thoughts and decision-making in response to contingent events that emerged across their three lessons. this was evident as both teachers at certain points during the interview were able to articulate a rationale for their responses to learners’ questions ‘in the moment’. furthermore, the vsr interview allowed cassandra to note potential limitations in how she responded to certain contingent incidents and envisage alternatives to how she could have responded. cassandra recognised the potential of the vsr interview as it allowed her to consider aspects of her teaching that she did not previously consider and stated that ‘it’s always good to revisit and look at your teaching. we should all be recording our lessons’. however, tara was not able to envisage improvements that could be made to her lessons and her reflections were related to her own effectiveness such as her mannerisms and classroom management during teaching similar to that of novice teachers’ post-lesson reflections in veenman’s (1984) study. internationally, researchers have echoed concerns that teacher preparation programmes need to equip novice teachers for the intricate and challenging job of teaching through ‘intellectually ambitious instruction’ (lampert et al., 2013) or ‘high leverage practices’ (ball & forzani, 2009). barker and borko (2011, p. 292) note that ‘for the most part, novice teachers’ initial opportunities to sharpen the second domain of presence – the contingent, interactive aspects of practice – do not occur until they are actually in the classroom’. we agree that many novice teachers are unprepared to engage in the improvisational act of teaching and require opportunities in teacher preparation programmes to practise how to notice and respond to unexpected learner offers. this suggests that teacher education programmes in south africa need to consider how to prepare pre-service teachers for the demanding work of responsive teaching as novice teachers are generally underprepared in handling unexpected learner offers. this could be a potential avenue of research for mathematics education researchers. conclusion findings from the analysis indicated that the novice teacher did not press or probe learners to understand their mathematical thinking ‘in the moment’ when faced with an unexpected question whereas the expert teacher continuously pressed learners in all contingent incidents with follow-up questions to understand their thinking. findings from the vsr interview indicated that both teachers were able to articulate a rationale for their decisions when reflecting on contingent incidents. moreover, cassandra was able to note potential limitations in how she responded to certain contingent incidents and envisage alternatives to how she could have responded. preparing pre-service teachers for the demanding work of responsive teaching is needed and this is an avenue of research relatively understudied in south africa. acknowledgements the authors would like to thank all who consented and participated in this project. without the participation of the teachers and learners, this work would not have been possible. competing interests the authors declare that they have no financial or personal relationships that may have inappropriately influenced the writing of this article. authors’ contributions the first author wrote an initial draft of the manuscript. the second author provided critical feedback on the initial draft of the manuscript. both authors equally contributed to revising the manuscript in light of the reviewer comments. ethical considerations ethical clearance was obtained from the wits university human research ethics committee (non-medical) 2019ece050m. funding information j.m. thanks the university of the witwatersrand for the postgraduate merit award which provided financial support during the course of this research. the 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(2013). conception of expert mathematics teacher: a comparative study between hong kong and chongqing. zdm, 45(1), 121–132. https://doi.org/10.1007/s11858-013-0487-5 appendix 1 tara interview questions introduction i would thank tara for allowing me to have videorecorded her lesson and making time available for this interview to discuss the lesson. semi-structured questions opening conversation what was your intended goal for each lesson (lesson 1, 2 and 3)? in other words, what were you hoping that learners would be able to do, or understand, at the end of each lesson? did you feel that you achieved what you had intended for each lesson? why or why not? discussion responding to student ideas (selected episodes from the lesson) lesson 1: selected episode to discuss: considering lesson 1, did you deviate in anyway whatsoever from the lesson plan you designed for this lesson? in your teaching, please can you comment on how you handled a learner’s contribution where she asks why there is a x−1 when rewriting as 5(x−1) – 8x? what were your thoughts and decision making at that moment in time when responding to the learner’s question? if you were to repeat lesson 1, would you respond to the learners’ contributions in the same way or differently? if differently, please can you elaborate. lesson 2: selected episodes to discuss: episode 1: considering lesson 2, did you deviate in anyway whatsoever from the lesson plan you designed for this lesson? in the lesson, a learner asked a question as to simplify –2−2. what was your thinking in the moment as to how to respond to the learner and what was your reason for putting aside further answering that question? episode 2: a second contingent event arose in this lesson when a learner asked how you would simplify an expression with a base x that has a negative exponent that is a fraction. what was your thinking in the moment as to how to respond to the learner’s question and what was your reason for not further exploring the question? if you were to repeat lesson 2, would you respond to the learners’ contributions in the same way or differently? if differently, please can you elaborate. lesson 3: selected episode to discuss: episode 1: considering lesson 3, did you deviate in anyway whatsoever from the lesson plan you designed for this lesson? the class is working on simplifying (x + 1) + (2x + 3) and you were working out the solution with the class. you wrote x + 1 + 2x so far and a learner asks whether the answer would be four. what were you thinking in that specific moment? please can you tell me why you responded in this way. if you were to repeat lesson 3, would you respond to the learners’ contributions in the same way or differently? if differently, please can you elaborate. conclusion is there anything else that you would like to add? what has been your experience with the three triggers of contingency (i will explain what i mean by the three triggers of contingency to the teacher) beyond the lessons i observed? cassandra interview questions: introduction i would thank cassandra for allowing me to have videorecorded her lesson and making time available for this interview to discuss the lesson. semi-structured questions opening conversation what was your intended goal for each lesson (lesson 1, 2 and 3)? in other words, what were you hoping that learners would be able to do, or understand, at the end of each lesson? did you feel that you achieved what you had intended for each lesson? why or why not? discussion responding to student ideas (selected episodes from the lesson) lesson 1: selected episode to discuss: considering lesson 1, did you deviate in anyway whatsoever from the lesson plan you designed for this lesson? you started answering the following question: “by how much is the sum of 2x2 + 3x – 2 greater than the sum of (−x + 2) and (−5x2 – 8)?”. you wrote 2x2 + 3x – 2 – [(−x + 2) + (−5x2 – 8)] and a learner asks why they have to minus the big bracket and expresses confusion. can you elaborate how you responded and what were you thinking as to how to respond to the learner’s question? was it an expected question? if you were to repeat this lesson, would you respond to the learners’ contributions in the same way or differently? if differently, please can you elaborate. lesson 2: selected episode to discuss: considering lesson 2, did you deviate in anyway whatsoever from the lesson plan you designed for this lesson? cassandra, you asked the class a question regarding how to simplify (x + 5)(+2) and you finished answering the question with the class. subsequently, a learner asks you ‘if it’s x + 5 and there is a two outside the bracket?’ you stated it was a ‘good question’. was this an expected question for you? what were you thinking at that moment in time as to how to respond? if you were to repeat this lesson, would you respond to the learners’ contributions in the same way or differently? if differently, please can you elaborate. lesson 3: selected episode to discuss: considering lesson 3, did you deviate in anyway whatsoever from the lesson plan you designed for this lesson? a learner asks you after simplifying 3x(x – 5) if you just write 3x or 3x + 3x2 as your answer. what was your thinking as to how to respond to her? if you were to repeat this lesson, would you respond to the learners’ contributions in the same way or differently? if differently, please can you elaborate. conclusion is there anything else that you would like to add? what has been your experience with the three triggers of contingency (i will explain what i mean by the three triggers of contingency to the teacher) beyond the lessons i observed? footnotes 1. by focusing on algebra, this is not to suggest that algebra has more contingent events than other topics in the mathematics curriculum. rather, the motivation for pursuing contingency with this topic stems from our own experiences in teaching this topic and the unexpected events we encountered. 2. all the teachers’ names are pseudonyms intended to preserve the anonymity of all participants. no learner names are mentioned. abstract introduction research methods and design results discussion conclusion acknowledgements references footnotes about the author(s) jayaluxmi naidoo department of mathematics and computer science education, school of education, college of humanities, university of kwazulu-natal, durban, south africa citation naidoo, j. (2020). postgraduate mathematics education students’ experiences of using digital platforms for learning within the covid-19 pandemic era. pythagoras, 41(1), a568. https://doi.org/10.4102/pythagoras.v41i1.568 original research postgraduate mathematics education students’ experiences of using digital platforms for learning within the covid-19 pandemic era jayaluxmi naidoo received: 19 july 2020; accepted: 16 oct. 2020; published: 30 nov. 2020 copyright: © 2020. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract within the digital era, as global society embraces the fourth industrial revolution, technology is being integrated swiftly within teaching and learning. within the coronavirus disease (covid-19) pandemic era, education institutions are preparing robustly for digital pedagogy. this article reports on a study focusing on 31 postgraduate mathematics education students’ experiences of using digital platforms for learning during the covid-19 pandemic era. the study was located at one teacher education institution in kwazulu-natal, south africa. the research process encompassed three interactive online workshops and two online discussion forums, which were conducted via different digital platforms (zoom, moodle and whatsapp). the study was framed using the theory of communities of practice, which denotes a group of people who share an interest which is enhanced as group members support and interact with each other. qualitative data generated during the interactive online workshops and discussion forums were analysed thematically. the results exhibit challenges and strengths of using digital platforms as experienced by the participants. the results of this study suggest that before using digital platforms for mathematics learning, it is important for students to be encouraged to practise and engage collaboratively within digital platforms. the study adds to the developing knowledge in the field concerning using digital platforms for learning mathematics within the covid-19 pandemic era. keywords: covid-19; digital platform; fourth industrial revolution; mathematics; moodle; technology; whatsapp; zoom. introduction technology is increasingly used by society in education, business and general daily activities (bell, 2011; qurat-ul et al., 2019). also, as we embrace, the fourth industrial revolution (4ir), there are various debates on how existing educational contexts can be adapted to support and incorporate the use of technology-based tools. technology-based tools in this study refers to electronic, digital or any other teaching tools that are supported by technology and are used within educational contexts to facilitate learning (ertmer & ottenbreit-leftwich, 2012). within the context of this study, digital platforms provided the participants and the lecturer with a diverse array of technology-based tools for communicating and attaining new knowledge and skills to enhance and guide the online learning process. also, within this study, digital platforms and web-based resources were used with computers or mobile gadgets which supported pedagogy using text, audio and video (peachey, 2017). thus, digital platforms assisted the online community of practice (participating postgraduate mathematics education students) in using digital resources and engaging with learning materials online. these platforms are combined software solutions which supported online learning within this study. moreover, the coronavirus disease (covid-19) pandemic has transformed life resulting in social and educational lockdowns globally and within the era of the 4ir. considering the novel covid-19 pandemic conditions, higher education institutions are inclining towards using digital tools to support digital pedagogy. digital pedagogy, which is a strategy for teaching and learning using digital platforms, is seen as a technique to avoid the spread of the contagious covid-19 virus (murgatrotd, 2020). also, the contemporary 4ir education contexts support pedagogy that encourages critical thinking, hands-on learning, collaboration, problem-solving approaches, the use of digital platforms and technology-based pedagogy (goertz, 2015). thus, technology within the 4ir education context is significant to improve the digital education setting (boholano, 2017; klopfer, osterweil, groff, & haas, 2006). digital education settings support the effective integration of technology-based tools (for example, computers and mobile devices) for teaching and learning (buzzard, crittenden, crittenden, & mccarty, 2011). this study aimed to respond to the main research question: what are postgraduate mathematics education students’ experiences of using digital platforms for learning within the covid-19 pandemic era? fourth industrial revolution the 4ir comprises various methods of integrating technology within societies and human bodies (schwab, 2016). within the 4ir era, technology-based tools and digital platforms are transforming the way we conduct our lives. this transformation is unsettling society since the 4ir has changed the way nations subsist and is epitomised by blending the virtual and physical domains (schwab, 2016), shaping a comprehensively linked and progressive society. students may be at different levels concerning readiness for progressing within the 4ir: students may be digital natives or digital immigrants. digital natives are acquainted with the use of technology-based tools and digital platforms (for example, the internet, computers and other online tools or platforms). the term is generally used to characterise students who are familiar with technology-based tools (helsper & enyon, 2009). these students will probably succeed when using technology-based tools and digital platforms when learning. in contrast, digital immigrants are students who may acquire the knowledge of how to use technology-based tools, but rather than working online initially, they may examine printed information first before referring to the internet for support (helsper & enyon, 2009). these students may need added assistance when using technology-based tools and digital platforms. as is evident, lecturers need to be aware of their students’ abilities concerning their acquaintance of using technology-based tools and digital platforms when learning. this knowledge will support the lecturer when presenting the notions of the 4ir within the education environment. also, education institutions need to adjust to prepare students sufficiently to thrive within these circumstances (butler-adam, 2018; mensch, 2017; schieffer, 2016; thieman, 2008). the 4ir impacts the purpose that higher education institutions play in preparing students for succeeding within our technologically advanced society. to take advantage of 4ir opportunities, more so within the era of the covid-19 pandemic, we need to transform our pedagogy to include the successful use of technology-based tools and digital platforms. using digital platforms and technology in mathematics teaching and learning the digital world has changed education environments, and teaching and learning are being transformed through using technology-based teaching tools (grand-clement, devaux, belanger, & manville, 2017; jeffrey, milne, suddaby, & higgins, 2014; lazarus & roulet, 2013). integrating technology in mathematics pedagogy supports teaching and learning and has positive effects on student performance (cheung & slavin, 2013; mlotshwa & chigona, 2018). however, within south africa, mathematics teachers’ use of technology-based tools for teaching and learning is negatively affected by their lack of computer skills. thus, many mathematics teachers are hesitant to transform their pedagogy to incorporate the use of technology within their classrooms (stols et al., 2015). nevertheless, there are many online mathematics teaching and learning websites and several educational applications that can influence students’ learning and achievement (pope & mayorga, 2019). this online teaching and learning support is available for mathematics teachers to engage with since student performance in mathematics has been improved through using digital tools for teaching (cheung & slavin, 2013). in the study conducted by cheung and slavin (2013), the authors focused on researching the link between mathematics achievement and the use of technology. also, that study focused on exploring computer management learning and is therefore related to this study. computer management learning is similar to moodle, which is a learning management system (lms) being used at the participating university in the study under focus. the use of moodle in the teaching and learning of mathematics within higher education supports students’ learning (lopes, babo, & azevedo, 2008) by stimulating students’ interest which results in a positive effect on student performance (handayanto, supandi, & ariyanto, 2018). moreover, the use of digital platforms, for example moodle, results in an improvement in students’ performance in mathematics (jayashree & tiwari, 2016). also, successful mathematics students and those that have challenges may benefit from the use of digital tools (bruce, 2012) since the use of technology-based tools in mathematics has shown a positive link with student achievement (cheung & slavin, 2013). furthermore, the implementation of digital platforms for the learning of mathematics encourages supportive relationships to be developed within communities of practice (mlotshwa & chigona, 2018). however, it is also important to note that the success of using digital tools also depends on the design of the digital tools and platforms as well as the time allocated for completing specific content within the curriculum (drijvers, 2013; sahal & ozdemir, 2020). moreover, the use of digital tools in teaching and learning can lead to students becoming easily distracted, and as a result they may not complete their academic work timeously (mbukusa, 2018). therefore, it is dependent on the mathematics teacher to select the most helpful digital tool to achieve the full potential of digital pedagogy to support students’ learning. thus, it is the teacher’s responsibility to enable progression in education (montrieux, vanderlinde, schellens, & de marez, 2015). especially within the current covid-19 pandemic era, the use of digital platforms to support pedagogy within the 4ir is essential. some of these digital platforms used at the participating university include zoom1, whatsapp2 and moodle3. thus, the digital world has entered education spheres, with digital devices and platforms being used to deliver education (grand-clement et al., 2017). given the ubiquity of smartphone technology and the ease with which it is used in daily life, excluding smartphones from higher education courses confines prospects for these courses to be contemporary and realistic (schuck, 2016). to ensure that courses and pedagogy are current and realistic, many higher education institutions are advancing to integrate digital devices and digital platforms (for example tablet computers, laptops, smartphones, netbooks, social media, zoom, google meet, microsoft teams, google drive and google classroom) within pedagogy. theoretical framing: communities of practice this study was framed using wenger’s (1998) communities of practice (cop) theory, a theory of learning that has its own set of norms and emphasis in which the fundamental unit of analysis is the cop (graven & lerman, 2003). a cop revolves around events that are of interest to members of that community (wenger, 1998). a cop may create an environment of thinking, insight and responses that supports the connection between researchers and practitioners such that the information generated is more valued and substantial (hearn & white, 2009). wenger’s (1998, p. 4) cop theory is founded on four principles: individuals are social beings, information concerns ‘valued enterprises’, knowing is about participation and experiences in the world and meaning is what learning produces. this theory maintains that a cop is formed by people who partake in the activity of communal learning within a public space. thus, a cop comprises groups of people who share an interest in something, and through cooperation they learn how to advance what they adopt (wenger & wenger-trayner, 2015). furthermore, within the cop theory, learning is identified as being made up of four elements: practice, meaning, identity and community (wenger, 1998). the relationship between this study and the cop theory is clarified as follows: a cop shares an interest; within the scope of this research the shared area of interest was the use of digital platforms for the learning of mathematics. within the ambits of cop, members of the community partake in the activity of communal learning. since technology can surpass location and time, virtual cops, which rely primarily on technology to connect the online cop, are becoming increasingly popular (dubé, bourhis, & jacob, 2006). within the domains of this study, shared learning within the virtual cop took place via three interactive online workshops and two discussion forums. thus, within the ambits of a virtual cop, members of the cop in this study were the postgraduate mathematics education students and the mathematics lecturer. the postgraduate mathematics education students in this study were also practising mathematics teachers at school level. the researcher was the mathematics lecturer who taught these participants at the research site. a virtual cop may use traditional tools, for example a telephone, and more advanced technological tools, such as emails, cell phones, virtual meeting spaces, digital platforms and websites, to create a shared virtual collaborative space (dubé et al., 2006). the virtual cop under study supported each other as they interacted within the interactive workshops and online discussion forums; they shared resources online and discussed challenges with the mathematics content being discussed. the learning process was collaborative and interactive (osterholt & barratt, 2010), which involved all members of the virtual cop. thus, this theoretical framework provided the framing for this study which focused on one main research question: what are postgraduate mathematics education students’ experiences of using digital platforms for learning within the covid-19 pandemic era? research methods and design general background this qualitative study, which aimed to explore postgraduate mathematics education students’ experiences of using digital platforms for learning, was located within an interpretive paradigm. data were generated via three interactive online workshops and two discussion forums at one teacher education institution in kwazulu-natal, south africa. the participating university’s research committee approved gatekeeper access and ethical clearance. the study incorporated three interactive online workshops and two online discussion forums with participants. the participants were purposively selected for convenience since the researcher taught these participants. participants participants were emailed an informed consent sheet outlining the purpose and process of the study; the participants’ right to leave the study without prejudice was also noted on the informed consent sheet. pseudonyms were used to protect the anonymity and confidentiality of participants. forty-two postgraduate mathematics education students were invited to participate in the study, and 37 agreed to participate in the study (23 male and 14 female). six participants were selected at random to join the pilot study. hence, the remaining 31 postgraduate mathematics education students, who were also practising mathematics school teachers, participated in the main study. pilot study the pilot study was conducted with six participants who were part of the postgraduate mathematics education class. these participants were randomly selected and were not included in the sample for the main study. all interactive workshops were piloted; during the pilot study, internet connections were limited, sluggish and unsteady since the workshops were conducted at peak times when internet service providers were supplying numerous customers which led to slow network speed and unstable networks. hence, the workshops took longer than anticipated. also, during the pilot, individual participants were unclear about what was required of them for the discussion forum questions. to avoid similar issues during the main study, through negotiation with the participating postgraduate mathematics education students, the online workshops were held at off-peak times to ensure the speed and steadiness of internet networks. also, the online discussion forums were piloted to explore the learning of mathematics by using digital platforms as experienced by the participants of the pilot study. as a result of conducting the pilot study, the questions used during the workshops and discussion forums were revised to enhance the trustworthiness of the research instruments and process. thus, by conducting the pilot workshops and piloting the discussion forums, the consistency and dependability of the study were maintained. this was important for ensuring that the results were due to the study itself and not as a consequence of any other peripheral factors. main study thirty-one postgraduate mathematics education students agreed to participate in the main study. data were generated via three interactive online workshops and two online discussion forums. although 31 participants were involved in the workshops, due to various reasons, only 15 participants (eight male and seven female) were available to take part in the discussion forums. to assure the participants of their anonymity, pseudonyms were used. table 1 represents the pseudonyms used for the participants in the discussion forums. table 1: the 15 postgraduate mathematics education students who participated in the discussion forums. online workshops the purpose of the interactive workshops was to explore postgraduate mathematics education students’ experiences of using digital platforms within the covid-19 pandemic era. a sequence of online mathematics workshops (n = 3) via zoom (a digital platform used at the participating university) was held. these workshops were facilitated by the lecturer, who is also the researcher. these workshops were mandatory and were part of a postgraduate mathematics education module for which the participants were registered. all participants were provided with notes, powerpoint presentations of case studies, and examples of online assessments focusing on using digital tools and free online websites within a digital mathematics education environment. within this environment, the lecturer presented the module content and shared resources with the participants online via email, zoom chats and moodle. the content for the first workshop focused on academic writing, and the second workshop concentrated on assessing and providing feedback for algebra. the third workshop was dedicated to identifying school learners’ misconceptions in geometry. at the end of the third workshop, all participants were invited to participate in two discussion forums (using whatsapp and moodle digital platforms). the researcher presented content for all three workshops, and the postgraduate mathematics education students engaged with and discussed the content that was introduced with members of the virtual cop. the discussion forums were not compulsory. online discussion forums the purpose of the two discussion forums was to explore postgraduate mathematics education students’ experiences of using digital platforms for the learning of mathematics. the discussion forums were conducted online using digital platforms, for example moodle and whatsapp chats. questions were placed on these digital platforms, and the participants had one week to respond to questions posed. if further probing was required for clarification purposes, the researcher would ask specific participants follow-up questions in each discussion forum. each discussion forum began with a few general questions to place the participants at ease. the discussion forum then progressed to specific items focusing on the participating postgraduate mathematics education students’ experiences of using digital platforms for learning about mathematics academic writing, learning how to assess and provide learners with feedback for mathematics problems in algebra as well as how to identify learner misconceptions in geometry. during the explanation of the research process, participants were asked to respond at least once to each question posed during the discussion forums. the discussion forums focused on the following key questions: what were the postgraduate mathematics education students’ experiences of using digital platforms for learning? what were the strengths of using digital platforms for the learning of mathematics? what were the challenges of using digital platforms for the learning of mathematics? the discussion forums were recorded; transcriptions of the discussion forums were sent to each participant for perusal to ensure the correctness of the information captured within these forums. data analysis data analysis in the form of coding and categorising of themes was based on the conceptual framework of the study, that is, cop theory. the relationship between the data generated and the notions of practice, meaning, identity and community (wenger, 1998) were examined. data analysis included the following steps: firstly, open coding was used to ensure acquaintance with the data and to classify codes after inspecting the qualitative data; secondly, all data were perused, and codes were processed into themes. the data that were related to each other were grouped into themes. thirdly, the themes were scrutinised to ensure that all codes within each theme revealed a connection. finally, the similarities and difference between the participants’ responses were compared, and subthemes were identified. also, to confirm the accuracy of results and to provide participants with the chance to clarify their responses, member checking was undertaken. ethical consideration ethical clearance for this study was obtained from the ethics committee of the participating university. the participants were assured that pseudonyms would be used to ensure and protect their confidentiality and anonymity when using all generated data. participants provided informed consent to participate in the study, noting that they allowed the use of their responses to the online workshops and discussion forums for research purposes. ethical clearance number: hss/1562/016. results postgraduate mathematics education students had various experiences when using digital platforms for learning. these experiences are presented in themes and subthemes that follow. strengths of using digital platforms the postgraduate mathematics education students indicated that there were strengths in using digital platforms for the learning of mathematics. their sentiments are presented in the subsequent subthemes. experiences of using whatsapp discussion forums the majority of the participants had positive experiences of using whatsapp for the discussion forums as reflected in their discussion forum excerpts. anne: …was easy and quick…convenient to respond… kavir: …no problems with whatsapp…could get quick feedback…easy to go back and see what i missed…i work on my own first…then i can discuss my answers…also each to ask the group for maths resources…group gave ideas on how to give feedback in algebra… lungi: …easy to use whatsapp…use it all the time with my family and friends…also could chat quickly about other stuff…not only maths… nancy: …whatsapp is easier to use than moodle…could quickly look at what i missed and follow the conversation…the lecturer responded quickly to help us…also a good way to keep in touch with each other… based on the evidence, the participants agreed that whatsapp assisted in providing immediate feedback and offered a means of communicating instant social messages. these participants were participating interactively within their virtual cop using digital platforms. the sharing of mathematics worksheets and resources was easy and quick to distribute to the virtual cop using whatsapp. this engagement via digital platforms is important for enhancing academic success and development in mathematics within the era of the 4ir. digital platforms provide unlimited access to module material and resources the participants valued the notion of creating a mathematics repository, uploading videos, resources and recordings of the workshops and discussion forums as reflected in the excerpts that follow. nancy: …recordings and videos were easy to view at any time…this helped me…i could revisit the workshops at any time…clarify my answers…had time to think about what i was learning… lungi: …during the lockdown…difficult to concentrate…had to see to my family…at night i could look at the workshop videos…help me understand and catch up…could think…i even learnt about other misconceptions learners have in geometry… nomsa: …needed time to think about what was discussed…i could look at the recordings after the workshop…helped me with understanding assessment methods better…worked on my task…then shared with the group…we discussed and shared our views…the lecturer gave us help if we needed it…other students also shared resources online…we developed an online maths data bank… siya: …couldn’t concentrate totally during the workshop…too many distractions at home. i could look at the video when i had quiet time to focus…this was useful…i could look back on what i learnt and think carefully about the maths…i could look at the videos at any time…the online maths data bank with examples and resources also helped me… xolani: …the uploaded content helped…i could learn at any time…the data bank we developed also assisted with checking misconceptions and providing feedback in maths…i could study anytime and anywhere i even could learn in my car… as was evident, the majority of the participants appreciated the uploaded workshop videos and recordings. in addition, the virtual cop developed an online data bank with mathematics examples of assessing and providing feedback as well as resources on identifying misconceptions in geometry. these resources were valued by the participants and provided them with new learning experiences as they discovered, for example, other misconceptions that learners may acquire in geometry. they felt at ease that they could view the uploaded material at any time when it was suitable for them. it was evident that the participants were prepared to engage with online material. in addition, with the uploading of the online data bank developed by the virtual cop, the participants had access to mathematics support at any time. this is an important step for achieving academic success in mathematics within the era of the 4ir. support received within the digital community the virtual cop formed by using digital platforms created a support mechanism for the participants, as reflected in the subsequent excerpts. delani: …felt belonged…the class cared about me…my child was sick…i could talk to the class on whatsapp about what was going on at home… they helped me…gave me advice…told me about the drive-through covid-19 testing places around me… leon: …could use whatsapp to find out how everyone was…if anyone was going through a bad time…we offered advice and help. good to talk to people who knew what you were going through…better to use whatsapp than moodle…got replies almost instantly…could also check up on each other…the group helped me complete the algebra question… patrick: …when we had problems…helped each other…got support from the group…it was good…i could submit my maths work with success…we created our own maths resource bank for the group…all shared our worksheets and examples that we use in our own class… thandi: …the online group provided good help…not only about the maths work but also about family problems… based on the preceding excerpts, creating a supportive virtual cop was beneficial for the participants. they valued the online support they received concerning the mathematics being discussed as well as the online social support they received during the covid-19 pandemic. the participants’ willingness to embrace the online platforms and support other members of their cop while they learnt about mathematics academic writing, assessing and providing feedback in algebra and identifying learner misconceptions in geometry was encouraging. in addition, the development of the online repository with mathematics resources and mathematics examples supported active online interaction and engagement. this robust engagement and interaction via the internet using digital platforms is supported within contemporary 4ir lecture rooms. collaboration is encouraged within the digital community collaboration and the sharing of ideas were essential to the participants, as reflected in the excerpts that follow. delani: …worked well in the group…shared problem-solving and how to give feedback to learners…discussed answers…we placed examples in the data bank to help each other…the lecturer assisted and explained concepts further…it was good to belong to the online group…learnt new assessments methods from resources shared online… devi: …discussed how to teach learners who had misunderstandings in geometry…shared advice about how to help learners with understanding geometry better…the group placed their examples in the online data bank. we helped each other as a group…could complete the task through this help…realised we needed to do our work on our own first and then discuss and share our answers… john: …if someone needed help we gave support. we looked at each other’s answers first…then we gave advice about what worked for us in the class…we asked the lecturer to assist or we needed more explanation…could check on each other at any time using whatsapp…working in the group was helpful… nomsa: …no extra effort was required…convenient and was quick to respond to questions and share mathematics resources that work for us…got others’ feedback quickly…i could also look at the data bank…i could complete my work with this help… patrick: …some of us had the same problems…for example the geometry problems were confusing…we could share advice…shared resources online…we worked together…it was useful…helped each other… to take advantage of 4ir opportunities, more so within the era of the covid-19 pandemic, the lecturer in this study transformed pedagogy to include the use of technology-based tools and digital platforms. it was evident from the discussion forums that members of the virtual cop in this study engaged with each other online and shared resources and examples online using a data bank. for example, based on the evidence provided, if the participating postgraduate mathematics education students within the virtual cop needed further assistance when solving the mathematics problems under discussion, they sought help from the lecturer who was also a member of the virtual cop and shared resources with each online. these resources included examples of assessments in mathematics, how to provide feedback in mathematics as well as common misconceptions in geometry. these examples and resources were uploaded into a data bank created by the virtual cop. as was evident, while the participants mentioned that collaboration was of value when using digital platforms, they valued their interaction within their virtual cop. however, they also indicated that they experienced challenges when using digital platforms. these challenges are discussed in detail in the following section. challenges of using digital platforms the participants indicated that there were challenges in using digital platforms for the learning of mathematics. their views are presented in the subsequent subthemes. need training to use digital platforms effectively the participants did not generally use zoom and were only exposed to this digital platform during the covid-19 pandemic. they needed to practise and prepare for using this digital platform as is reflected in the following discussion forum excerpts. devi: …was difficult at first…got easier as i experimented and tried out zoom …i think we need enough training before using zoom… john: …zoom is something different…needed to learn how to work with zoom…then only was i ok in the workshop…still had problems raising my hand…the buttons on the zoom system changed…. siya: …the workshops were a bit difficult to follow…my zoom link was not stable…wish we could have face-face workshops…needed to learn quickly how to use zoom… xolani: …had to learn how to use zoom…i studied the guide…but still asked my friends and my child for help…i felt i was missing out…needed extra time to learn… as was evident, zoom workshops were a new form of learning for the participants. this new learning method required practice and collaborative engagement within the virtual cop for the participants to navigate this digital platform successfully. these participants exhibited characteristics of digital immigrants, and it was evident within this virtual cop that the 4ir impacts the purpose that higher education institutions play in preparing students for succeeding within a technologically advanced society. also, during the period of the workshops, the zoom platform and functions were revised to a minor extent; this resulted in the participants having to relearn and practise how to use specific revised techniques. this took place during the second workshop. as a result, considerable teaching and learning time was lost due to addressing technical issues during the second workshop. devices, data and resources for working within digital platforms are expensive to use digital platforms, the participants required access to digital devices and data. some participants experienced difficulties with this, as reflected in the following excerpts. kavir: …i relied on the university wi-fi and computers…with covid…was forced to work at home…this created lots of expenses…had to buy a second-hand laptop and data for internet… lungi: …the zoom workshops took a lot of data…was expensive to use. i had to download information from the internet…these free websites are not free…too costly…a difficult time to make ends meet at home…no one else is working at home because of covid…lucky for the free online data bank we created in the group…these resources helped me a lot… rani: …the resources that we need is expensive to download…takes time to download…costs me a lot of data…lots of money…the online data bank worked well for me… thulani: …i paid fees to study at the university…thought i would use the university library. with covid…i am working at home…i don’t have a computer…i am using my phone…also costs lots of money to work from home…did not think about this expense earlier on…did not budget for this… participants revealed that devices and data were expensive to purchase, especially as this was unforeseen at the beginning of the 2020 academic year. the participating university offers only the contact mode of learning under normal circumstances; however, due to the covid-19 pandemic, the university was required to provide lectures online or remotely via the use of digital platforms. as was evident, it was important for lecturers to be aware of their students’ challenges concerning the availability of technology-based tools to engage with digital platforms when learning. this knowledge supported the lecturer when presenting the notions of the 4ir within the education environment. through engaging within the virtual cop, all members were made aware of challenges experienced by members within the cop. the creation of the online data bank supported students with examples and resources for mathematics. since these were free mathematics resources, this was of benefit to the virtual cop during this challenging pandemic. based on the evidence provided, both the participating postgraduate mathematics education students and the lecturer needed to adapt quickly to this new pedagogical approach to achieve success within the contemporary 4ir lecture context. using digital platforms within the covid-19 pandemic era is socially complex the covid-19 pandemic era resulted in many people working from home with their families around them; this created difficulties for the participating postgraduate mathematics education students as captured in the following excerpts. lungi: …my children with me at home…difficult to attend workshops with them…they are young and need me…found it hard to concentrate and take care of my children at the same time… nancy: …live with my parents and my husband and small children…during covid we are all at home in a tiny house…my husband does not understand that i need to work…have other work at home that i need to do first…was very stressful for me…my husband was complaining all the time…i could not cope… thandi: …as a woman…lots of responsibilities at home…especially with covid. everyone is home…need to clean and cook and take care of my family…it was hard for me to attend online workshops and complete my other responsibilities…that is why i prefer to study from campus… thulani: …no privacy at home…we are all at home in one house during lockdown…difficult for me to respond and ask questions in the workshops…everyone at home is listening to my workshop… xolani: …my child is also studying…also has zoom lectures…have a small house…need to plan so that we can have space to work…distracting when we have lectures at the same time…sometimes i sit in my car during the lectures… working and studying from home, especially during the covid-19 lockdown conditions, created many challenges. often family responsibilities needed to be addressed first before the participating postgraduate mathematics education students could start with academic work. as was evident, members of the virtual cop within this study experienced challenges attributed to the covid-19 pandemic. fortunately, the digital world has transformed education contexts that embrace notions of the 4ir. teaching and learning were transformed using technology-based tools within the education context under study. the participating postgraduate mathematics education students and the lecturer did not have to engage within the virtual cop at the same time. teaching and learning could occur successfully at any time using digital platforms. while members of the virtual cop within this study may not have been able to attend the online workshops, they were provided with the opportunity to view the uploaded content, workshops and videos at any time. in addition, the members of the cop could consult with the resources in the online repository if they needed more assistance. after that, the members of the virtual cop within this study could engage with other members of the online community through whatsapp and the discussion forums on moodle to seek further guidance and assistance. using digital platforms for learning mathematics is time consuming and uncomfortable learning how to use digital platforms was difficult and often took much time, as reflected in the following excerpts. anne: …took long for me to log onto to moodle and follow the conversation…also when i posted a question…it took long for me to get an answer… nancy: …i used moodle in my undergrad days…but uncomfortable to use. it is hard to follow the conversations since there are so many and everything is sent to your email…i am looking at emails and moodle…need to catch up… nomsa: …waited long for the class to respond on moodle…i needed help immediately…so i had to use whatsapp to get help…this is when we started sharing maths resources with each other…this was useful… rani: …moodle is ok…but sometimes people take long to respond…need to wait until someone replies…it is limiting me… siya: …it was tricky using moodle…took time for me to find the conversation and then i did not know to reply…this wasted much time… xolani: …had a problem accessing moodle…forgot my password…needed to set up a new one…so i had to catch up with the discussion…was stressful…i felt i was left behind…lucky for the whatsapp chats…kept me informed… as was evident, the participants mentioned the challenges of using digital platforms. while the participants used moodle previously, they did experience a few issues. when they needed a quick response, moodle was tricky and uncomfortable to use and did not assist with providing rapid feedback. it was evident in this study that the participants needed to engage within their virtual cop for support and guidance with the mathematics problems under study. as a result, the participants used the digital platforms available to them to communicate and engage with other members of the virtual cop quickly and easily. in addition, the online repository with resources and examples that were developed by the cop was useful when the participants needed added support. lecturers who embrace the notions of the 4ir ensure that their students have access to various technology-based tools and digital platforms to enhance and support teaching and learning within the era of the 4ir. to further interrogate the results and to exhibit the significance of this study, a comprehensive discussion is included in the following section. discussion the qualitative results provide evidence of postgraduate mathematics education students’ experiences of using digital platforms during the covid-19 pandemic era for the learning of mathematics. firstly, the participants generally experienced positive experiences of using whatsapp chats; however, it was evident that they experienced difficulties when using zoom and moodle. still, it is important to note the whatsapp platform may create challenges for students when balancing online activities and academic planning, and may also distract students from finishing their assessments timeously (mbukusa, 2018). however, based on the evidence, it was apparent within the virtual cop under focus that through collaboration and shared interest (wenger, 1998), the participants managed to complete the mathematics academic writing, algebra and geometry tasks as discussed during workshops. thus, these participants embraced the notions of a cop by focusing on sharing ideas and providing support to other online community members. the participating postgraduate mathematics education students within the virtual cop also developed an online repository with resources and examples that provided information regarding identifying learner misconceptions in geometry and assessing and providing feedback in algebra. this repository was available for any member of the virtual cop to access at any time. thus, communication and the sharing and discussion of ideas is essential to mathematics pedagogy, and it is important for mathematics teachers to become actively involved in adding to the development of knowledge (lazarus & roulet, 2013). in this study, the use of digital platforms allowed participants to engage within the virtual cop actively, and the community members shared ideas through individual and collaborative attempts (mlotshwa & chigona, 2018; osterholt & barratt, 2010). this was evident during the academic writing, algebra and geometry workshops, where the participants showed evidence of practising the mathematics tasks independently first to promote meaning and understanding. only when the participants encountered challenges, did they seek assistance from the virtual cop via the discussion forums or whatsapp. the discussion forums on moodle and whatsapp created an added layer of mathematics support for members of the virtual cop. thus, whatsapp is a novel teaching method that can appeal to students and provide them with prospects for further learning (mbukusa, 2018). the actions of the participants in this study exhibit the fundamental elements of learning: practice, meaning, identity and community within a cop (wenger, 1998). the participants shared experiences of good practice and discussed their own meaning-making of the resources that were shared in the online data bank. this sharing of ideas and resources created an empowered online community which helped shape the identity of each member of the virtual cop. similarly, wenger and wenger-trayner (2015) proposed that members of a cop need to collaborate around ideas related to the content under study which was evident in this study. the participants showed evidence of networking with their virtual cop members to learn mathematics together. the participants also exhibited evidence of embracing notions of the 4ir by engaging with technology-based tools and digital platforms, in addition to developing an online repository to learn mathematics. similarly, ertmer and ottenbreit-leftwich (2012) suggested that the effective incorporation of technology-based tools within education contexts would support learning. in this study, mathematics learning was supported and enhanced using digital platforms. secondly, the participants indicated that the uploaded lectures, videos, recordings and online data bank were beneficial in that they could access information at any time. in this way, the participants were not compelled to engage with content, other members of the online cop and resources during a specific period and while they were in a certain location. this is important to consider since research (jeffrey et al., 2014) has maintained that the use of online pedagogy disregards the confines of time, place and obstacles while empowering active collaboration between members within the cop under focus. within the ambits of cop, collaboration is supported so that learning is enriched by sharing and discussion (wenger, 1998; wenger & wenger-trayner, 2015). as was evident, in this study, the participants embraced the notions of the 4ir by using digital platforms. the participants collaborated and discussed solutions for the algebra and geometry tasks within the virtual mathematics cop. they further collaborated to develop an online repository for learning mathematics. the lecturer formed part of the virtual cop by presenting course content and guiding the learning process. it was evident that the implementation of digital platforms for learning mathematics fostered and developed supportive relationships within the virtual cop under focus (mlotshwa & chigona, 2018). thirdly, while the participants mentioned that there were strengths in using digital platforms, they also indicated that they needed to practise and required training and time to prepare for using these digital platforms efficiently. they believed that practising using these platforms first before being expected to engage within these platforms formally (for example, the zoom digital platform) would assist them in feeling more comfortable with participating in the online workshops. moreover, the idea of practising using the online platforms would support and equip students to engage with each other and the content being discussed. similarly, research (thieman, 2008) maintains that for students to achieve success with digital pedagogy, they need to be prepared adequately since the use of technology within the 4ir education context is important (boholano, 2017). students need to know the nuances of the online class and what is expected of them within the online course. to assist with achieving preparedness in this respect, an orientation session needs to be included before engaging with online pedagogy (mensch, 2017). fourthly, in this study, it was evident that limited access to digital devices and data for the internet influenced the participants’ use of online learning and digital platforms (klopfer et al., 2006). the participants believed that they need to have the basic requirements for digital pedagogy (for example, digital devices and access to data and the internet) to achieve success with digital learning. furthermore, postgraduate mathematics education students indicated that the virtual cop fostered active interaction and supported the acquisition of new knowledge using various digital platforms. this result resonates with research by jeffrey et al. (2014), which maintained that students need to be encouraged to reflect on their learning so that they are provided with diverse new learning experiences and resources. as was evident in this research, using digital platforms, the participants were encouraged to become responsible for their mathematics learning. for example, members of the virtual cop within this study came up with the idea of sharing mathematics resources online to support their learning of mathematics. this idea escalated into the creation of an online mathematics data bank. this showed evidence of the participants independently taking responsibility for their own mathematics learning. this independence promoted their ability to learn mathematics at any time and place. similarly, linking students and online platforms does not necessarily take place in a physical education context; this is ubiquitous due to their access to the internet (bell, 2011). also, to promote student success within the 4ir, education institutions need to adjust traditional pedagogy to prepare students suitably (butler-adam, 2018). finally, the challenge of working from home during the covid-19 pandemic area was also a significant difficulty. it led to many social and academic issues for these participating postgraduate mathematics education students. the advantages of collaboration and communication offered by using digital platforms within the context of covid-19 was significant. lecturers need to note that students are sociable people and require feedback and support from peers and the lecturer during these extraordinary circumstances. apart from communication about the specific challenges concerning the learning of mathematics being discussed, during lockdown conditions within the covid-19 pandemic, students may not have other people who may relate to what they are experiencing within their contexts. a virtual cop builds feelings of trust and camaraderie, and supports the online community in curbing the perception of social isolation (schieffer, 2016). thus, for emotional, health and social benefits, students need to discuss and interact with their peers and lecturer during this unprecedented era. conclusion this study was conducted to respond to the question: what are postgraduate mathematics education students’ experiences of using digital platforms for learning within the covid-19 pandemic era? the article concludes with possible suggestions for lecturers who wish to use digital platforms for the learning of mathematics. these suggestions are based on the experiences of the participants as uncovered in this study. firstly, the difficulties of using digital platforms within the era of covid-19 are vital for lecturers to note. to alleviate this issue, it is important for the lecturer to ensure that students have access to digital devices and data for using the internet. lecturers need to establish that the resources being used are easy to use, readily available, inexpensive and data efficient. also, students need to be provided with orientation sessions on the use of digital platforms to better prepare them for using these platforms effectively and successfully. secondly, the challenges experienced by the participants in this study are important to note. the participants indicated that they needed space and time to practise and engage with the online mathematics examples first before interacting within the online community. lecturers need to consider providing students with the content and material for discussion in advance of the online lecture or workshop. this allows students to study the content independently first. also, the participants in this study valued the collaborative online discussions, which, for example, focused on identifying learner misconceptions in geometry. the participants reflected on and discussed their experiences with their virtual cop, and through this reflection on experiences, collaborative ideas revolving around teaching to avoid misconceptions in geometry were circulated online. also, the participants shared examples and resources for assessing algebra, providing feedback in algebra and identifying learner misconceptions in geometry with their virtual cop. through this sharing of ideas and resources, the participants developed an online mathematics data bank for identifying learner misconceptions in geometry and assessing and offering feedback in algebra to mathematics learners. this online mathematics data bank was available on moodle and was available for any participant within the virtual cop. thus, as we embrace the 4ir and as we engage within digital platforms, lecturers need to create a similar virtual cop to engage with each other to develop data banks and repositories to support mathematics teachers. these online repositories provide an added layer of support to address mathematics teaching and learning challenges, and due to the nature of digital platforms, these online repositories and data banks are accessible at any time. thirdly, when preparing for online pedagogy during covid-19 pandemic conditions, lecturers need to acknowledge that students have family responsibilities to deal with during these extraordinary circumstances. lecturers need to recognise that students are sociable people and need support, encouragement and feedback from other members of the online community. the advantages of the collaboration inspired by using digital platforms within the era of the pandemic are significant. students may need to use these digital platforms for social and emotional support during this unprecedented time to deal with family and academic issues. finally, this study is not without limitations. the study was located at one teacher education context during the covid-19 pandemic era. it was conducted to explore postgraduate mathematics education students’ experiences of using digital platforms for learning within the covid-19 pandemic era. due to the importance of the study, further systematic studies based on other teacher education institutions nationally and globally could complement and provide additional insights on the topic. new empirical research could also test the strength of the research process and instruments. new empirical research may also assist in determining further noteworthy experiences of postgraduate mathematics education students on the use of digital platforms for the learning of mathematics. the experiences of postgraduate mathematics education students, as discussed in this article, focuses on their use of digital platforms for learning within the covid-19 pandemic era. the implications, results and limitations as discussed in this article add new knowledge to the field. also, this new knowledge is of benefit to lecturers globally as we embrace the use of digital platforms for the learning of mathematics, especially within the context of the covid-19 pandemic era. acknowledgements competing interests the author declares that she has no conflicting interests that may have inappropriately influenced her in writing this article. authors’ contributions the author declares that she is the sole author of this article. funding information the author is thankful to the national research foundation (nrf), who partially funded this research (grant number: ttk170408226284, uid: 113952). data availability statement data sharing is available on request. disclaimer the views expressed in this submitted article are those of the author and do not reflect the official position of the institution or funder. references bell, f. 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(2015). communities of practice: a brief introduction. retrieved from https://wenger-trayner.com/introduction-to-communities-of-practice/ footnotes 1. zoom is a video conferencing software application that allows you to interact virtually with friends, colleagues, students and family when face-to-face communication is not possible. zoom is a digital platform used for teaching and learning at the participating university during the covid-19 pandemic. 2. whatsapp is a free messenger application that uses the internet to send and receive messages and calls. whatsapp was used unofficially at the participating university by lecturers and students as a digital platform to communicate via messages, images, audio and video files during the covid-19 pandemic. 3. moodle is a learning management system used at the participating university. it is an open-source e-learning digital platform. article information author: kakoma luneta1 affiliation: 1department of childhood education, university of johannesburg, south africa correspondence to: kakoma luneta email: kluneta@uj.ac.za postal address: private bag x09, bertsham 2013, johannesburg, south africa dates: received: 07 mar. 2014 accepted: 26 may 2015 published: 30 june 2015 how to cite this article: luneta, k. (2015). understanding students’ misconceptions: an analysis of final grade 12 examination questions in geometry. pythagoras, 36(1), art. #261, 11 pages. http://dx.doi.org/10.4102/pythagoras.v36i1.261 copyright notice: © 2015. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. understanding students’ misconceptions: an analysis of final grade 12 examination questions in geometry in this original research... open access • abstract • introduction • the theoretical framework • errors in geometry • methodology • synopsis of student responses to the questions    • question by question analysis of students’ errors       • question 3.1.1       • question 3.1.2       • question 3.2.1       • question 3.2.2       • question 3.2.3       • question 3.2.4       • question 3.2.5 • discussion • conclusion • implications • acknowledgements    • competing interests • references • appendix 1    • 2008 national examination grade 12 paper 2 question 3 (simplified) abstract top ↑ the role geometry plays in real life makes it a core component of mathematics that students must understand and master. conceptual knowledge of geometric concepts goes beyond the development of skills required to manipulate geometric shapes. this study is focused on errors students made when solving coordinate geometry problems in the final grade 12 examination in south africa. an analysis of 1000 scripts from the 2008 mathematics examination was conducted. this entailed a detailed analysis of one grade 12 geometry examination question. van hiele levels of geometrical thought were used as a lens to understand students’ knowledge of geometry. studies show that van hiele levels are a good descriptor of current and future performance in geometry. this study revealed that whilst students in grade 12 are expected to operate at level 3 and level 4, the majority were operating at level 2 of van hiele's hierarchy. the majority of students did not understand most of the basic concepts in euclidian transformation. most of the errors were conceptual and suggested that students did not understand the questions and did not know what to do as a result. it is also noted that when students lack conceptual knowledge the consequences are so severe that they hardly respond to the questions in the examination. introduction top ↑ geometry is the ‘study of shapes, their relationships, and their properties’ (bassarear, 2012, p. 463). it has a long history arising from the practical measurement of land in ancient egypt and the study of properties of shapes in greek geometry (cooke, 2007). geometry has been identified by research as the field of mathematics that offers ‘enormous potential of bringing the subject alive’ (chambers, 2008, p. 187). it is an exploratory subset of the discipline and has links with culture, history, art and design. it is the interactions with these vital human constructs that provide opportunities to make geometry lessons interesting and stimulating (chambers, 2008). according to gonzález and herbst (2006), geometry is the only high school subject in which students routinely deal with the necessary consequences of abstract properties and in which students are held accountable for reading, writing and understanding mathematical proofs. knowledge of geometry remains a prerequisite for study in fields such as physics, astronomy, art, mechanical drawing, chemistry, biology and geology. atebe and schäfer (2011) assert that students’ general mathematical competencies have been linked closely to their geometric understanding. research has also noted that geometry is difficult to teach as well as to learn. coordinate or analytical geometry, for instance, requires not only geometrical knowledge, but also a vast amount of knowledge in working with coordinates on a 2d (two-dimensional) or 3d (three-dimensional) set of axes. these additional concepts make geometry more complex and require an intricate manner of thinking. van der sandt (2007, p. 2) concedes that in south africa geometry is regarded as a ‘problematic topic’ at secondary school level. analysing transformation geometry involves many different types of knowledge as defined by shavelson, ruiz-primo and wiley (2005) and others (e.g. hiebert, 1986; rittle-johnson & alibali, 1999) such as procedural, conceptual, strategic and declarative knowledge. the theoretical framework top ↑ piaget (1971), supported by ding and jones (2006), writes that children's geometrical understanding develops with age and that for children to create ideas about shapes they need physical interaction with objects. clement, swaminathan, hannibal and sarama (1999, p. 193) assert that ‘children's representation of space is constructed through the progressive organisation of the child's motor and internalised actions’. van hiele (1986, 1999) on the other hand tried to analyse the various aspects involved in the learning of geometry and space. van hiele (1986, 1999) introduced the existence of five levels of geometrical thought (bahr, bahr & de garcia, 2010; musser, burger & peterson, 2011). for van hiele, students develop their knowledge of geometry in accordance with these levels. in the first level (visualisation and recognition), students can identify a shape, but are not able to provide its properties. the shape is judged only by its appearance. the second level (analysis) is predominantly descriptive: students are able to identify particular properties of shapes, but not in a logical order. the third level (abstraction and relationships) is informal and deductive: students can combine shapes and their properties to provide a precise definition as well as relate the shape to other shapes. there is a logical order to the properties and they are deduced from one another. the fourth level is more formally deductive: students apply formal deductive arguments such as in proofs. theorems with an axiomatic system are established. the fifth level (rigour and axiomatics), also known as the meta-mathematical level (van der sandt, 2007, p. 1), is characterised by ‘formal reasoning about mathematical systems by manipulating geometric statements such as axioms, definitions, and theorems’ and at this juncture students can ‘compare systems based on different axioms and can study various geometries in the absence of concrete models’ (burger & shaughnessy, 1986, p. 31). understanding these levels enables teachers to identify the general directions of students’ learning and the level at which they are operating (lim, 2011). the first three levels involve the development of procedural fluency in geometry, whilst the last two display the development of conceptual understanding (kilpatrick, swafford & findell, 2001). pegg (1985) explains that van hiele's theory is divided into two parts: the first part is the hierarchical sequence of the levels, which shows that each level must be fully developed by the student before proceeding to the next level. the second part is the development of intuition in students and the phases of learning that influence geometric learning. van hiele's levels provide teachers with a framework within which to conduct geometric activities by designing them with the assumptions of a particular level in mind and they are able to ask questions that are below or above a particular level (lim, 2011; van de walle, 2004). the levels are also a good predictor of students’ current and future performance in geometry. according to jones (2003, p. 128), ‘the van hiele model of mathematical reasoning has become a proved descriptor of the progress of students’ reasoning in geometry and is a valid framework for the design of teaching sequences in school geometry’. van hiele's (1986) theory of geometry with its focus on geometrical reasoning has been linked to piaget’ five stages of child development and the role they play in learning geometry (pusey, 2003). van hiele's levels of geometrical thought were the guiding principles for studying exiting grade 12 students’ knowledge of transformation geometry and for determining the level at which the average student in the sample operated. significantly, such information can inform tertiary and vocational institutions as to the instructional support in transformation geometry that students need, as they engage with fundamental mathematics and physics courses. the findings are also crucial to the grade 12 teachers as the study delineates common conceptual and procedural errors in transformation geometry that they look out for when teaching the topic. the research question is: what were the most common error that students in grade 12 displayed in the examination scripts on the geometry question? errors in geometry top ↑ school curricula worldwide cover four main learning outcomes in geometry (bahr et al., 2010; bassarear, 2012; department of education, 2006). by the time students complete school they should be able to: analyse the characteristics, properties and relationships of two-dimensional and three-dimensional geometrical shapes (euclidean geometry). specify locations and describe spatial relationships using coordinate geometry and other representation systems (coordinate geometry). apply transformation and use symmetry to analyse mathematical situations (transformation geometry). use visualisation, spatial reasoning and geometric modelling to solve problems. research has delineated that errors occur mainly because students have difficulties in understanding the instructional strategies adopted by the teacher (confrey, 1990). in geometry the communication of information at different levels of reasoning of the sender (the teachers) and the receiver (the student) become a major cause of misconception (lim, 2011). this is especially true in the case of geometry. when the teacher operates and communicates at different levels of geometric thought to those of the students, concepts are not understood or acquired fully. it is necessary for teachers to know their students’ level of geometrical thought and to operate at those levels. michael (2001, p. 11) defines misconceptions as ‘conceptual or reasoning difficulties that hinder students’ mastery of any discipline’. according to drews (2005, p. 18), a misconception could be the result of ‘a misapplication of a rule, an overor under-generalization, or an alternative conception of the situation’. swan (2001, p. 154) views misconceptions as ‘natural stage of conceptual development’. for students to be able to confront underlying conceptual difficulties, overcoming misconceptions is required (van der sandt & nieuwoudt, 2003). luneta (2008, p. 386) defines errors as ‘simple symptoms of the difficulties a student is encountering during a learning experience’. according to swan (2001, p. 150), an error could be the result of ‘carelessness or misinterpretation of symbols or text’. misconceptions manifest in students’ work as errors, which implies that errors are symptoms of misconceptions students possess. according to confrey (1990, p. 33) misconceptions emanate from ‘a line of thinking that causes a series of errors all resulting from incorrect underlying premises’. knowledge of students’ errors is essential and teachers should provide opportunities for students to display their errors as these will be essential stepping stones for effective instruction. it can be argued that knowledge of students’ levels of geometric reasoning is essential for effective teaching. the most common errors in transformation geometry are the result of students operating at levels that are different to their teachers’. it is evident that people reasoning at different levels may not understand each other and this is true for teachers and students. a student reasoning at level q will not understand a teacher reasoning at level q+1 (pusey, 2003). by establishing the levels at which grade 12 students are operating in transformation geometry, the study hopes to inform university mathematics lecturers on the level of interaction at which to anchor their discourses. methodology top ↑ this study conducted an analysis of 1000 grade 12 mathematics scripts. these were obtained from the department of education with this purpose in mind. the scripts were randomly selected from the entire 2008 batch of 108 000 scripts. the selection was not based on schools, but was merely an assortment of scripts. after being sampled the 1000 scripts were stratified into three groups according to student ability. group 3 was made up of students who attained between 0% and 32% (n = 333), group 2 obtained between 33% and 55% (n = 334) and group 1 between 56% and 100% (n = 333). the study focused its analysis on the 2008 grade 12 national examination paper 2 question 3 (see appendix). this question is made up of seven parts that assess student knowledge of different aspects of geometry. the study used a content analysis technique in which each question was analysed according to the content it contained (students errors). according to berelson (1952), content analysis is a technique used for objective, systematic and quantitative description of the manifest content of communication. kerlinger (1986) emphasises content analysis as a method for studying and analysing communication in a systematic, objective and quantitative manner for the purpose of measuring variables. in this instance the student answers are an indication of their ability to interact with geometry examination questions. the variable being measured is their responses (misconception and the associated errors) against the correct answers. the analysis made inferences to the communication (student's answers) by systematically and objectively identifying specific characteristics of the student's errors in the answers. this study was encapsulated in the epistemological framework of constructivism and the theoretical perspective of interpretivism. this implied that the study viewed learning as being informed by both the students and the teacher. this means that the interpretation of the students’ understanding of mathematics was pivotal to the theory that was developed. the unit of analysis was the errors students displayed on each of the seven parts of question 3 (appendix 1). the students’ errors were classified as conceptual errors – or those errors that were due to non-conceptual understanding of the concept – and procedural errors – these were errors that were related to the incorrect use of the procedure to solve the problem. this included the inappropriate use of formulae, application errors, which are errors that apply to the misuse of rules, and careless errors, which are those that students made unknowingly and which could be corrected by the student if they were so prompted (luneta & makonye, 2011). synopsis of student responses to the questions top ↑ table 1 provides a synopsis of how the three groups answered the seven parts of question 3. the first analysis focused on ‘correctly answered’ questions and classified the responses from the three groups of students into three categories: ‘partially answered’ questions, ‘incorrectly answered’ questions and ‘not answered’ questions. using van de walle's (2004, p. 346) classifications of the products that emanate from van hiele's thoughts the researcher was able to classify the questions accordingly. van de walle's classification defines the levels as: level 1, the products of thought are classes or grouping of shapes that seem alike; level 2, the products of thought are the properties of shapes; level 3, the products of thought are relationships amongst properties of geometric objects; level 4, the products of thought are deductive axiomatic systems of geometry; level 5, the products of thought are comparison and contrasts amongst different systems of geometry (van der sandt & nieuwoudt, 2003). table 1: students’ attempts to answer question 3. table 1 shows that the question on transformation geometry required students to operate at the knowledge level of bloom's taxonomy (evaluation, synthesis, analysis, application, comprehension and knowledge). the third column shows the van hiele levels products of thought for each question derived from van de walle (2004). students in group 3 had the highest number of ‘not answered’ questions, students in group 1 had the highest number of ‘correctly answered’ questions (85%) and those in group 2 had the highest number of ‘partially answered’ questions. it was also evident that group 1 students had the highest number of ‘common mistakes or careless errors’. this was due to the fact that they attempted most of the questions. the middle group had the highest number of procedural errors. most students in group 3 did not attempt the questions, so it was not easy to identify their errors. it can however be assumed that since group 3 could not provide answers to most of the questions, they lacked conceptual understanding of the concept of transformation geometry. table 1 also shows that the majority of the students were operating below level 2 of van hiele's levels of geometric thought and that only 44% of group 1 could operate at level 4. figure 1 is developed from column 3 and column 4 in table 1. the figure illustrates that only 493 (49,3%) students could respond correctly to question 3.1.1, pegged at level 1 of van hiele's levels of geometric thought. it further depicts that 539 (53,9%) students could respond correctly to questions 3.1.2, 3.2.1, 3.2.2 and 3.2.3, pegged at level 2, only 400 (40%) students could respond correctly to question 3.2.4 that was pegged at level 3 and only 183 (18%) students could respond to a question pegged at level 4 of van hiele's levels of geometric thought. figure 1: student groups’ average van hiele levels of geometric thought on the questions. it is worth noting that the sixth column (incorrect) of table 1 would have produced the converse of figure 1 where group 3 would have had the highest bars and group 1 the shortest bars depicting incorrect answers. question by question analysis of students’ errors question 3.1.1 students were required to give the coordinates of the image of a point after reflection in the line y = x. according to table 1 students operating at van hiele level 1 would be able to respond to this question. the majority of the group 1 students were able to answer this question, whereas only 13% of the group 3 learners managed. many learners gave coordinates suitable for a 180° rotation or a reflection in the y-axis. this implied that the majority of the students who completed the examination could not differentiate between the lines y = x, the x-axis or the y-axis. there were a number of students who demonstrated a total lack of understanding of what the question was about. the example (see figure 2) (e3.1.1a = error in question 3.1.1 part a) shows a student using a formula that was hardly related to the question. thirty percent of the group 3 students did not offer a feasible answer (i.e. they did not understand that the question was asking for the coordinates of a point). conceptually the majority of students did not know the line y = x and therefore could not correctly reflect the object on that line. figure 2: a formula not related to the question that was asked (e3.1.1a). question 3.1.2 table 1 shows that to respond to this question correctly students need to operate at van hiele level 2. this question required students to give the coordinates of the image of a point after undergoing a 180° rotation. once again, this question was answered correctly by most students in group 1. by contrast, 20% of students in group 3 did not attempt the question and a further 72% of these students got the answer wrong. an analysis of the students’ responses showed that most of them did not understand the question and therefore did not know what to do. many students wrote coordinates for p′ that did not resemble the original coordinates of point p. it seemed as if some students confused the transformation rule for a reflection in y = -x with the rule for a 180° rotation: they used the rule (x; y) → (-y; -x) instead of (x; y) → (-x; -y). this question entailed a gradual geometric move from analysis to abstract levels of geometrical thought. but the low performance on question 3.1.1 showed that most grade 12 students could not operate at van hiele level 2 or above. the major misconception in this question was that the students could not differentiate the operative rules required for reflection, rotation and translation – rules of rigid transformation in euclidian geometry. question 3.2.1 in this question, students were asked to give the coordinates of a point after a 90° rotation in a clockwise direction. this question also required students to operate at van hiele level 2. nine percent of the sample group rotated d in an anti-clockwise direction. twenty percent of the group reflected d about the x-axis and 13% reflected d about the y-axis. the common errors showed the students’ lack of knowledge of the difference between rotation and reflection. question 3.2.2 table 1 again shows that students were required to operate at van hiele level 2 in order to be able to answer this question. this routine question required students to draw a sketch of a polygon after rotating it 90° in a clockwise direction. only 25% of the students got this question correct and these were from group 1. the most common error made was a reflection about the x-axis. these students probably tried to rotate the polygon in a clockwise direction, but did not understand that the polygon's orientation in space needed to change as well. some students reflected the polygon about the y-axis (probably in an attempt to rotate it in an anti-clockwise direction). this shows that these students had the same misconception regarding the rotation of a rigid shape. a significant number of students were not able to transform the polygon without altering its shape or size and others rotated the polygon 180° or translated it. it showed that most of the students (75%) could not tell the difference between a rotation, a reflection and a translation. in fact, it meant that they could not tell the difference between a rigid and a non-rigid transformation. figure 3 shows an example of the conceptual errors the majority of the students made. figure 3: typical student conceptual error on 3.2.2 (a) and the memorandum (b). below are three more examples of the various errors that students made regarding this question; they all show lack of knowledge of transformation geometry. the student's work on the immediate left is a reflection on the y-axis but the coordinates are wrong. the x-coordinates are depicted as y coordinates and vice versa. this was very common amongst most students in both group 3 and group 2 despite the fact that basic coordinate geometry is learnt at primary school: ‘plotting, reading and writing coordinates, year 2’ (hansen, drews & dudgeon, 2014, p. 170). the example on top right is from a group 1 student who seems to have knowledge of rotation but does an anti-clockwise rotation of 90°, which was not what was asked. figure 4 further shows that a number of students did not understand the principle that rotation, like reflection and translation, is a rigid transformation. such an answer was common in many (72%) of the students’ scripts; this is despite the fact that translation, reflection and basic rotation are taught at foundation phase (hansen et al., 2014). figure 4: students lack of knowledge of rigid transformation (e3.2.2d). question 3.2.3 this question required students to enlarge a polygon by a given factor and find the image of enlargement. more than half of the students answered the question correctly (method marks were often awarded following on from their answers to question 3.2.1). students were asked to give the coordinates of d″ after enlarging the polygon by a factor of 3 through the origin. it was difficult to analyse the incorrect answers, as often the given values were not related to the coordinates of d or d′. this implies that students were not conceptually grounded on transformation that involved enlargement and whilst table 1 shows that the question required them to operate at level 2 of the van hiele hierarchy, they were predominantly operating at level 1 as can be seen from table 2. table 2:: explanations of students’ average errors on each of the seven parts of question 3 and the resultant van hiele classification. question 3.2.4 for this question, students had to create a transformation rule for the combination of the two transformations in 3.2.1 and 3.2.3. table 1 shows that this question was at level 3 of the van hiele levels of geometrical thought (abstraction and deduction). because of its high cognitive demand, less than 50% of students in group 1 and very few in group 2 and group 3 managed to answer this question correctly. it is not surprising to note that 13% of the sampled students did not attempt this question and 50% did not write an appropriate response. this question involved complex thought procedures, but is an examination question that grade 12 students are reasonably expected to tackle. the students who understood what the question required managed to write a transformation rule to represent the enlargement by a factor of 3, but few could represent the rotation of 90° algebraically. a typical partially correct answer was (x; y) → (3x; 3y). question 3.2.5 to respond to this question students needed to operate at level 4 of the van hiele hierarchy according to table 1. approximately 18.3% of the sample group got this question correct. thirteen percent of the students stated that the area abcde:area a″b″c″d″e″ is 1:3. a few of the students understood the implications of scaling the area, but did not write the ratio correctly (i.e. area abcde:area a″b″c″d″e″ = 1:9). half of the students were not able to use an appropriate method to address this question. in some instances students used incorrect formulae to determine the answer as shown in figure 5. a″b″c″d″e″ is the result of enlarging abcde by a factor of 3; therefore, the ratio of the image and the object is 3:1 and ratio of their area will be the square of these units. figure 1 shows that this was the most difficult question amongst the majority the students sampled, such that even in group 1 only 44% got it right. students’ answers ranged from an attempt to finding an area of a triangle such as the first answer in figure 5 to unknown mathematical manipulations that did not make sense at all as can be seen in the second answer in which the student was adding x and y coordinates. figure 5: students’ errors that were not related to the question (e3.2.5e). table 2 shows students’ common errors on each question and their identified van hiele levels. van der sandt and nieuwoudt (2003, p. 200) confirm that ‘students’ answers can be classified according to the van hiele levels of thinking they reflect by using description levels provided by the mathematical accuracy and how complete the solution to the activity is’. from the students’ errors the table also shows the researchers’ identified products of thought (description levels) and their resultant van hiele levels. discussion top ↑ table 2 shows that students in the sample were mostly operating at level 1 of van hiele's levels of geometric thought. by contrast, table 1 shows that in order for the students to respond correctly to all the questions except 3.1.1, they needed to operate at level 2 or above. furthermore, research (luneta, 2014, p. 80) suggests ‘that learners who successfully complete grade 12 should be operating at level 3 of the van hiele theory’. it is possible to suggest that most students in grade 12 operate at or below level 2 of the van hiele hierarchy. there are several reasons for this. for instance, according to van der lith (2007), there are two ways of finding the equation of a straight line: using the formula y – y1 = m(x – x1) or the general form y = mx + c. connecting the procedure of the gradient and the equation of a straight line is essential for deducing the equation (this is understanding), but the majority of students could not solve a straight line problem, despite the fact that graphs of a straight line are taught in grade 9 (bassarear, 2012, p. 545). according to the curricula, transformation geometry (translation, reflection and rotation) is introduced and taught at primary school (bassarear, 2012, p. 545). it is disconcerting that the majority of students in grade 12 could not solve tasks that are gauged to be at a grade 7 level of accomplishment and at level 2 on van hiele's hierarchy of geometrical understanding. question 3.2.4 on enlargement is equivalent to a grade 7 question, which requires students to find the image of a polygon by multiplying the coordinates of the original polygon by 3 (bassarear, 2012, p. 546). of the seven questions, only question 3.2.5 was at a level higher than level 2 of van hiele's levels of geometrical thought. the results show that most students did not have basic knowledge of geometry, such as the meaning and characteristics of the three rigid geometric transformations of reflection, translation and rotation. students swapped rules that governed reflection with those that dealt with rotation or translation. this means that students did not seem to know the difference between the three transformations. the findings of this study are consistent with other studies in south africa (atebe & schäfer, 2011; siyepu, 2005), which also found that the majority of learners were operating at the pre-recognition level and that a very small number of students operated at van hiele's second level. the study further showed that students enter university operating at a level of geometric thinking that is not appropriate for learning the university mathematics curriculum, which requires students to have reached level 4 and level 5 (lim, 2011). some of the explanations of this problem is that geometry is not taught very well from primary school onwards. de villiers (1998) asserts that ‘unless we embark on a major revision of the primary school geometry curriculum along the lines of van hiele's hierarchy, it seems clear that no amount of effort at the secondary school will be successful’. geometry at level 1 should be based on perception of the concepts involved, which at primary school entails the identification of polygons and their properties and non-standard orientations. for instance, a student fixated on the natural shape of a trapezoid will fail to notice that all three figures in figure 6 are in actual fact trapezoids. figure 6: different orientations of trapezoids. teaching geometry to learners in a standardised way leaves them incapacitated when a change in the natural orientation of a figure is affected. for instance, in figure 7a both figures are squares but many students would think the first figure is not a square (but a rhombus) because that is the conventional way that a rhombus has been presented to them. the figures in figure 7b are all pentagons, but most students are familiar only with the first orientation, because that is how most teachers and textbooks represent a pentagon. figure 7: different orientations of a square and a pentagon. the most common errors were procedural. students were not able to engage with simple geometric relationships, reflections about a particular line, rotation of shapes in standard angles of 90°, 180° and 360° about a specific point. every question answered had a higher occurrence of procedural errors than common mistakes or conceptual errors. whilst there is no evidence from this study to back up this point, other research (hansen et al., 2014; van der sandt, 2007) asserts that the main explanation why a number of students are fluent in the use of procedural methods to answer mathematics questions is because that is the most dominant way that many teachers teach mathematics. the majority of mathematics teachers hardly explain concepts in ways that enable students to acquire conceptual understanding that leads to conceptual knowledge. hansen et al. (2014, p. 156) assert that students were less likely to notice attributes of shapes in geometry ‘because [of] the conventional way geometry is being taught’. the study concurs with research (centre for development in education, 2010) on teachers that points to the fact that most mathematics teachers in south africa do not have the appropriate skills, content knowledge, as well as the pedagogical content knowledge, necessary to be effective in a mathematics classroom. most mathematics teachers do not seem to have the knowledge and instructional skills required to explain concepts, but rather their teaching consists of algorithms that students are instructed to follow. researchers (verschaffel, greer & de corte, 2000) assert that the root cause of the lack of mathematical skills and conceptual understanding, which results in a cognitive deficit amongst mathematics students, is not their failure to acquire these skills, but rather the tendency to revert to ‘rules of the game’: the teachers and the textbooks present mathematics in simplistic, solvable, uncontroversial operations or procedures where there is only one precise correct answer, which can be obtained by performing one mathematical operation. a number of textbooks and teachers explain the concept in a routinised and standardised way. for instance, most texts introduce the concepts of angles at the centre and on a circle by having the subtended angle always facing upwards and subtended by the minor arc as in figure 8b rather than figure 8a (hansen, et al., 2014). figure 8: the angle subtended by an arc at the centre is twice the angle subtended on the circle. lim (2011) concedes that transition from level 2 to level 3 is even harder for students, but it is a goal of all mathematics teachers for students to attain level 3 at the end of their secondary schooling. if they do not achieve this, they will only have a superficial understanding of geometry and will regard it as a bundle of unrelated concepts, rules and properties. the transition from level 2 to level 3 is acquired by students responding to ‘why’ questions and justifying their reasons, for instance ‘why this is a rotation and not a reflection?’, ‘why it is an enlargement and a translation?’ ‘why it is a polygon and not a polyhedron?’. conclusion top ↑ this study confirmed, on the one hand, previous findings of the literature (a confirmatory study); on the other hand, it explored thinking in geometrical patterns and revealed a number of errors that grade 12 south african students made in their final examinations. the analysis hinged on students’ answers to question 3 and this question was mainly on coordinate geometry (specify locations and describe spatial relationships using coordinate geometry and other representational systems) and transformation geometry (apply transformation and use symmetry to analyse mathematical situations). these sections of geometry require learners to mainly operate on levels 1 to 4 of the van hiele hierarchy. however, the results revealed that the majority of students in grade 12 operate at level 2 of van hiele's levels of geometrical thought. whilst the research did not establish the main reason behind this, literature validates that most mathematics teachers are not grounded in instructional strategies that enable students to learn mathematics effectively. hansen et al. (2014) indicate that teachers’ vocabulary of geometry terms is pivotal in ensuring that students acquire the knowledge of the subject and that lack of it has been a source of errors. it was also established that most of the errors were those related to procedures for solving questions on geometry and because most of the learners were conceptually weak, their procedures were flawed too. implications top ↑ examining the literature and the results derived from the study the first question that this study helped to answer was: how to help students understand high school geometry. the study of geometry, like the other sections of mathematics, starts from early childhood. the first geometrical concepts form the basis for the rest of geometry in school curricula. thus the best approach involves changing how mathematics and especially geometry is taught before high school. some points to consider are: improve geometry teaching in the foundation and intermediate phases so that students’ van hiele levels of geometrical thought are brought up to at least to the level of abstract or relational. include more justifications, informal proofs and ‘why’ questions in geometry teaching during foundation phase and intermediate phase. the van hiele levels explain the understanding of spatial ideas and how one thinks about them. the thinking process that one goes through when exposed to geometric contexts defines the levels of operation and they are not dependent on age (battista, 2007; van de walle, 2004). van de walle (2004) insists: while the levels are not age-dependent in the sense of the developmental stages of piaget and a third grader or a high school student could be at level 0 […] age is certainly related to the amount and types of geometric experience that we have. therefore, it is reasonable for all children in k-2 range to be at level 0. (p. 347) hence, one can expect children in the first grade to be in the first level of van hiele's hierarchy – the visual level – as they are interacting with formal classroom geometry for the first time. this means children recognise geometric figures based on their appearance and not based on their properties. at this level, children are mainly learning the names of some shapes, such as square, triangle, rectangle and circle. at elementary level children should investigate geometric shapes so that they will reach the second van hiele level (descriptive or analytic). that is when they can identify properties of figures and recognise them by their properties, instead of relying on appearance. but all this requires teachers both at primary and secondary schools who are grounded in the content of geometry and are able to teach in ways that equip learners with conceptual knowledge and not only procedural knowledge. teachers should from an early stage instruct children to use rulers, compasses and protractors to draw shapes. acknowledgements top ↑ i would like to acknowledge the gauteng department of education who funded the main study of which this article is a part. competing interests i declare that i have no financial or personal relationships that may have inappropriately influenced me in writing this article. references top ↑ atebe, h.u., & schäfer, m. 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(2000). making sense of word problems. lisse, the netherlands: swets & zeitlinger. appendix 1 top ↑ 2008 national examination grade 12 paper 2 question 3 (simplified) 3.1 given that p (-√2; √3) is on a cartesian plane. determine the coordinates of the image of p if: 3.1.1 p is reflected in the line y = x. 3.1.2 p is rotated about the origin through 180°. 3.2 polygon abcde on a grid has coordinates a (1; 1), b (1; 2), c (2; 3), d (3; 2) and d (2; 2). each of the points abcde on the grid is rotated 90° about the origin in a clockwise direction. 3.2.1 write down the coordinates of d′, the image of d. 3.2.2 sketch and label the vertices a′b′c′d′e′ on the image of abcde. 3.2.3 the polygon a′b′c′d′e′ is then enlarged through the origin by a factor 3 in order to give the polygon a″b″c″d″e″. write down the coordinates of d″, the image of d′. 3.2.4 write down the general transformation of a point (x; y) in abcde to (x″; y″) after abcde has undergone the above two transformations. that is, rotation in a clockwise direction through an angle of 90°, followed by an enlargement through the origin by a factor of 3. 3.2.5 calculate the ratio of area abcde:area a″b″c″d″e″. abstract introduction and problem statement the van hiele theory of geometrical thinking commognition conclusion acknowledgements references footnote about the author(s) sfiso c. mahlaba department of mathematics education, faculty of education, north west university, mafikeng, south africa vimolan mudaly department of mathematics education, faculty of education, university of kwazulu-natal, pinetown, south africa citation mahlaba, s.c., & mudaly, v. (2022). exploring the relationship between commognition and the van hiele theory for studying problem-solving discourse in euclidean geometry education. pythagoras, 43(1), a659. https://doi.org/10.4102/pythagoras.v43i1.659 review article exploring the relationship between commognition and the van hiele theory for studying problem-solving discourse in euclidean geometry education sfiso c. mahlaba, vimolan mudaly received: 15 oct. 2021; accepted: 11 may 2022; published: 29 july 2022 copyright: © 2022. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract this article is an advanced theoretical study as a result of a chapter from the first author’s phd study. the aim of the article is to discuss the relationship between commognition and the van hiele theory for studying discourse during euclidean geometry problem-solving. commognition is a theoretical framework that can be used in mathematics education to explain mathematical thinking through one’s discourse during problem-solving. commognition uses four elements that characterise mathematical discourse and the difference between ritualistic and explorative discourses to explain how one displays mastery of mathematical problem-solving. on the other hand, the van hiele theory characterises five levels of geometrical thinking during one’s geometry learning and development. these five levels are fixed and mastery of one level leads to the next, and there is no success in the next level without mastering the previous level. however, for the purpose of the curriculum and assessment policy statement (caps) we only focused on the first four van hiele levels. findings from this theoretical review revealed that progress in the van hiele levels of geometrical thinking depends mainly on the discourse participation of the preservice teachers when solving geometry problems. in particular, an explorative discourse is required for the development in these four levels of geometrical thinking as compared to a ritualistic discourse participation. keywords: commognition; van hiele theory; euclidean geometry; geometrical thinking; visual mediators. introduction and problem statement a review of the grade 12 national senior certificate examination diagnostic analysis from 2016 to 2020 reveals that the average pass percentage of grade 12 mathematics learners in south africa is below 60% (department of basic education, 2016, 2017, 2018, 2019). this comes after van putten, howie and stols (2010) felt that south african teachers are not well prepared to teach euclidean geometry. in alleviating the situation, machisi (2021) suggests the use of unconventional teaching approaches such as the van hiele theory-based teaching and learning approach which as machisi (2021, p. 1) concluded ‘meets learners’ needs better than conventional approaches in learning euclidean geometry’. furthermore, in alleviating the difficulties faced by mathematics learners, sfard (2008) proposed that teachers should aim to transform learners’ discourse participation from ritualistic to explorative discourse participation. in particular, sfard proposes this transformation of discourse participation because she believed that learners’ mathematical thinking can be encoded from the way they communicate about mathematics. we have seen sfard’s theory being applied in other mathematical domains in south african research such as functions (mpofu & mudaly, 2020), numeracy (heyd-metzuyanim & graven, 2016) and equations (roberts & le roux, 2019). while commognition has a potential for alleviating difficulties in all domains of mathematics, the van hiele theory is specifically dedicated to guide teachers on how to alleviate learning difficulties related to euclidean geometry. the current theoretical article locates the problem in the fact that these theories are currently operating in isolation yet they have a similar purpose of improving learning in mathematics. wang (2016) combined these two theories (commognition and van hiele) with a focus on the elements of mathematical discourses, but this article takes a different approach by focusing on the type of discourse participation. the aim here is to harness the power of commognition in improving geometrical knowledge through the van hiele levels when solving geometry problems. thus, we discuss the tenets of each theory, then discuss how the study viewed the amalgamation of commognition and the van hiele theory as means of enhancing geometry understanding during problem-solving. the findings reported in this theoretical article are from a larger phd study but this article focuses only on the relationship between commognition and the van hiele theory when studying problem-solving discourse in euclidean geometry. thus, no empirical data will be cited to in this article. the van hiele theory of geometrical thinking the van hiele theory of geometrical thinking was developed by van hiele-geldof (1957) and van hiele (1957) towards the completion of their doctor of philosophy degrees at the university of utrecht. the van hieles posited that children go through five levels of thought development in their geometrical learning. these levels include recognition (level 1), analysis (level 2), order (level 3), deduction (level 4) and rigour (level 5). this theory was developed within the contexts of learners who are still in their secondary school education but in this review article it is viewed from the lens of preservice teachers (psts). thus, any reference to psts is equivalent to secondary school learners in the context of this study. these four levels of geometrical thinking also have their own descriptions that determines the type of learning that occurs in that level. these characteristics per level are summarised below: level 1: psts recognise names and recognise figures as a whole (i.e. a square and a rectangle are different). level 2: psts begin not only to recognise objects by their global appearance but also to identify their properties with appropriate technical language (e.g. a triangle is a closed figure with three sides). level 3: psts begin to logically order these properties through short chains of deduction and understand the interrelationship between figures through their properties. level 4: psts begin to develop longer chains of deduction and understand the significance and roles of postulates, theorems and proofs. level 5: psts understand the role of rigour and can make abstract deductions that allow them to understand even non-euclidean geometries. the van hiele theory is characterised by the existence of four characteristics summarised by usiskin (1982, p. 4) and de villiers (2012) as follows: fixed sequence: psts progress through the levels invariantly which means that a pst cannot be at van hiele level n without having passed level n–1. adjacency: at each level of thought, the intrinsic knowledge from the previous knowledge is extrinsic in the current level. distinction: the linguistic symbols and network of relationships connecting these symbols are distinct in each level. separation: psts who are reasoning at different levels cannot understand each other. these characteristics describe the manner in which psts are to proceed through the levels and what is important to consider in each level. within the five levels of geometrical thinking, the most pertinent ones in the curriculum and assessment policy statement (caps) are the first four levels (levels 1–4) which we focused on in this study because they also apply to psts’ education. at level 1, geometrical figures are recognised by their visual appearance (form) only, without any reference to their properties and any relationship that might exist between them. at this level, psts are able to relate geometrical figures with objects they see in their everyday lives, for example a rectangle looks like a door. these activities are critical at this level as the foundation for the next level (yi, flores, & wang, 2020). in level 2, geometrical figures are identified based on their properties, without considering the relationship that exists between these properties. thus, secondary school psts see a geometrical figure in isolation, not related to other figures. a square can be recognised as having four equal sides and four right angles without relating the property of right angles to a triangle. as psts develop to level 3, they begin to see the relationships between the properties of geometry figures. they now can relate a square and a rectangle by ordering their properties and deducing one from the other. at level 4, psts’ thinking and reasoning are concerned with understanding the meaning of deduction and proof. they can understand the role of theorems, postulates and properties of geometrical figures when doing proofs. there are critical issues about these levels that apply to the development of thought in geometry, especially for psts to use in their instruction. the language and signs used at each level are distinct, such that a relationship that is true at one level might not be true at another (van hiele, 1959). the second issue to be aware of is that people who reason at different levels cannot understand each other. hence, teachers need to attempt to reason at the level of learners, understand their routines and narratives to scaffold them to the next level. teachers must continuously support learners to construct their deductive relational system in geometry (van hiele, 1959), without imposing the relational system of the teacher onto the learners. these levels are critical in the analysis of thinking and reasoning in geometry, because they reveal the characteristics of thinking for both learners and teachers. since this study had its main focus on psts’ thinking when solving geometry problems, it seemed useful to incorporate these levels, as they are indicators of geometrical thinking. even though these levels were not assessed directly in this study, they are pertinent in geometry problem-solving. the psts’ behaviour when participating in geometry problem-solving can be related to each van hiele level and teachers’ discourse participation when solving euclidean geometry problems (see table 3). commognition this section provides the details of sfard’s (2008) theory of commognition, with a particular focus on the aspects of the theory that relate to this study. we begin with a brief explanation of the theory in general and a few key tenets of the theory. thereafter, we explain how the theory relates to learning and thinking, and how mathematics learning is a discourse, as we locate the current study within the theory of commognition. this section aims to describe the keywords and the language of sfard’s commognition and how they are key in describing psts’ mathematical discourse when solving geometry problems. furthermore, this section aims to explain how commognition has enabled the study to explain psts’ discourses and what improvements can be made to the theory in the future. commognition in a nutshell in 2008, sfard published a book titled thinking as communicating: human development, the growth of discourses, and mathematizing, which explains a theory that can guide and be used to understand mathematical learning. sfard (2008) uses the amalgamation of ‘communication’ and ‘cognition’ to coin the term ‘commognition’, which she describes as communicating about thinking. she defines commognition by putting into perspective this amalgamation, stating that commognition ‘stresses that interpersonal communication and individual thinking are two facets of the same phenomenon’ (sfard, 2008, p. xvii). here, sfard asserts that thinking is correlated to communicating, stating that ‘thinking is defined as the individualized version of interpersonal communication’ (sfard, 2008, p. xvii), closely relating thinking to bakhtin’s (1986, p. 126) idea of the ‘superaddressee’. sfard emphasises that communication and thinking are inseparable. she describes the underpinnings of the theory from the significance of communication, objectification, and elements of mathematical discourse that are significant in mathematics classrooms. in her elucidation of commognition, sfard differentiates between colloquial and mathematical discourse, where the former is considered to be everyday, spontaneous discourses and the latter is specifically related to mathematics. she posits four characteristics of the latter kind of discourse. a discourse becomes mathematical because of its word usage, visual mediators, narratives and routines. commognition is driven by the processes of objectification which is characterised by a double elimination of using metaphors to generate new discourse. this double elimination is characterised by the processes of alienation and reification. according to sfard (2008, p. 44), reification ‘consist[s] of substituting talk about actions with talk about objects, [while alienation] consists in presenting phenomena in an interpersonal way’. a reified talk includes utterances like “thabo has developed the concept of geometrical proof and problem-solving” while an alienated talk includes utterances like “the sum of all interior angles of a triangle is 180°”. the results of objectification are mainly abstraction, which helps in differentiating discourses of mathematicians and that of, for example, street vendors about similar issues. it is this power of objectification that differentiates between colloquial and mathematical discourse. sfard asserts that mathematical discourse is characterised by word use, visual mediators, narratives and routines. these characteristics are used to differentiate between three routines: rituals, deeds and exploration; here we only focus on routines and explorations. thus, in this study we use the elements of mathematical discourse to differentiate between a ritualistic and an objectified discourse using each of their characteristics (see table 1). all these will be discussed in the following sections in more detail. table 1: the comparison between rituals and explorations in commognition. why commognition? commognition is a discursive theory that is utilised here for its theoretical potential to explain psts’ thinking during geometrical problem-solving. it is a theory that acknowledges that everyone thinks on a daily basis but that others do not have direct access to this thinking. hence, commognition regards thinking as ‘an individualised version of interpersonal communication’ (sfard, 2008, p. 81). according to this view, thinking cannot just be an isolated activity, but becomes ‘the act of communication in itself’ (sfard, 2008, p. 82). this means that whatever utterances are made through discourse by an individual are a consequence of that individual’s thinking, and the best way to study that individual’s thinking is to analyse their communication in discourse. furthermore, school learning, as it is for teacher training, should present an opportunity to extend the discourses of learners and psts (ben-zvi & sfard, 2007). commognition further recognises that, just like learning, thinking develops from a patterned collective activity. commognition recognises that thinking can be objectified or disobjectified, but rests mainly on the significance of explaining mathematical thinking through disobjectified discourses. therefore, thinking can be explained by analysing the discourses of psts. thinking, as a patterned collective activity, happens through communicating with others and ourselves. thinking is therefore dialogical (sfard, 2008), and thinking is modified and changed as we communicate with others. mathematics is considered a difficult subject in south african schooling. furthermore, geometry is seen as the topic where learners perform the poorest and where even teachers struggle to teach geometry effectively (naidoo & kapofu, 2020; tachie, 2020; van putten et al., 2010). most teachers who do manage to get learners to pass geometry use the drilling of theorems and how to prove them. some teachers rely mainly on the possibility of questions being repeated in the standardised tests (machisi, 2021). to improve the dire situation of teachers with insufficient geometry knowledge, which leads to learners performing poorly in geometry, we need to approach this problem as a collective, ensuring that teachers are properly trained to teach geometry in secondary schools. we need to tap into psts’ thinking when they solve geometry problems to see how they think and then design proper tasks and teaching strategies to enhance their level of geometry thinking to a suitable one, where they would be able to teach geometry effectively to ensure meaning making within learners. in this way, commognition offered a window to tap into psts’ thinking when they solve geometry problems, to understand their thinking. in commognition, thinking is voluntary, individuals engage in thinking through their continued participation in the mathematical discourse. hence, in this view, geometry learning for psts is a consequence of their continued participation in the community of mathematics, which mainly originates from the participationist theories of learning (lave & wenger, 1991). as a human activity, participation in the activity of communication has emotional implications, which need to be understood properly, especially if the communication is among competing peers (heyd-metzuyanim, 2013). preservice teachers need to move from being ritualistic participants in the discourse of geometry to being individual geometry problem-solvers who can teach geometry effectively. since vygotsky (1978), the activity of being a peripheral participant in geometry discourse for psts begins with the help of a more knowledgeable other. hence, the role of the lecturer (trainer) is important as a knowledgeable other. hence, if psts are learning, they become more and more independent of the lecturer and their thinking as they learn does not require the aid of the lecturer, as they become independent thinkers. objectification in discourse the use of metaphors is common in all discourses, where words are partitioned into an unfamiliar discourse because of their familiarity and their readiness to be used in that discourse (sfard, 2008). the building of mathematical knowledge from concrete objects has long been recognised (dienes, 1960). in the current article, it seemed significant to distinguish between psts’ discourse about objects and how they communicated about mathematical objects. the ‘process in which a noun begins to be used as if it signified an extra-discursive, self-sustained entity (object), independent of human agency’ (sfard, 2008, p. 300) is known as objectification. an example, in euclidean geometry there can be a statement such as “angle abc is equal to angle bca because of angles opposite equal sides” instead of “this angle is equal to that angle because this is an isosceles triangle”. in commognition, objectification is considered to encapsulate two inseparable discursive moves: reification and alienation. according to sfard (2008, p. 44), ‘reification is the act of replacing sentences about processes and actions with propositions about states and objects’. hence, in this article reification describes psts transforming their talk about the process of problem-solving into talk about objects. reification allows psts to be concise about what they are communicating, which makes it more flexible and applicable in mathematical discourse. a reified version of the statement “in the majority of the tests and tasks dealing with euclidean geometry proof in school, he regularly did well and achieved very good marks” is “he has acquired the concept of euclidean geometry proof”. alienation on the other hand, involves the removal of the reified discourse from the actor. alienation refers to ‘using discursive forms that present phenomena in an impersonal way, as if they were occurring of themselves, without the participation of human beings’ (sfard, 2008, p. 295). alienation includes the use of passive voice in a particular mathematical sentence, for example “the angle between the tangent and the chord is equal to the angle subtended by the chord in the alternate segment”, which removes any personal attachments to the statement (sfard, 2021). alienation allows psts to engage in the discourse of geometry problem-solving in an impersonal way. these alienated geometry discourses can be thought of as theorems, axioms or postulates, etc., since they are monological1 (bakhtin, 1986). in this view, a geometrical proof is the final stage of the process of objectification, where the human experiences and constructions are removed from the discourse; it is the stage of alienation itself. this is a hint that geometry teaching and learning should not begin with the process of proving, but that of investigation and exploration (de villiers & heideman, 2014). alienation is seen as contributing to the genesis of mathematical knowledge and understanding (morgan & tang, 2016). hence, once psts can alienate a certain mathematical discourse, they begin to understand and construct mathematical knowledge, specifically in mathematical discourse and not just colloquial discourse. objectification has been shown to have several advantages in the process of mathematical learning. ben-yehuda, lavy, linchevski and sfard (2005) show that objectification may lead to mathematical discourses that contribute to increased levels of mathematical performance. objectification further makes the way we communicate about mathematics more effective and provides a method of attaching objects into our mathematical discourses. once we objectify, we create an ‘object’ or a ‘thing’ that has permanence in our mathematical discourse, which can also be an abstract entity. through this objectified discourse, psts accumulate knowledge through participating in successive mathematical discourses that increase in complexity and applicability. the reification process relates directly to the mathematical objects objectified in discourse and it allows psts to endorse the discourse as a mathematical one. hence, objectification, in this case, underlies the patterned ways in which we think. however, objectification removes personal experiences of learning and thinking in the discourse. as sfard (2008, p. 56) puts it, objectified ‘descriptions deprive a person of the sense of agency, restrict her sense of responsibility, and, in effect, exclude and disable just as much as they enable and create’. this is possibly a consequence of objectification removing the psts’ thinking and learning experiences and the way in which they might communicate in their everyday lives in the discourse. the objectification of mathematical discourse means that colloquial discourse is reduced into a more specific mathematical discourse, consisting of specific word usage, routines, narratives, and some visual mediators. in geometry, this can be articulated by the colloquial utterance “this angle is equal to that angle” compared to the more objectified utterance “<abc = <bca”. these, as explained in commognition, are the main characteristics of a mathematical discourse (sfard, 2008). hence, mathematics from this point of view, can be identified as a discourse. mathematics as a discourse sfard (2008, p. 161) describes mathematical discourse as ‘an autopoietic system’, whose end products are exactly the constituents or objects of its present discourse (cf. sfard, 2021). with its salient features of lave and wegner’s (1991) idea of ‘legitimate peripheral participation’, commognition further describes mathematical learning as participation in the mathematical discourse (sfard, 2008). vygotsky (1978) highlights the significance of communication in learning, and lave and wenger (1991) highlight the significance of participation in the process of learning. furthermore, the idea of mathematics as a discourse has been strengthened by researchers such as nardi, ryve, stadler and viirman (2014) and sfard (2014), saying that to not consider mathematics as a discourse would be ludicrous. the objects produced in a mathematical discourse are defined as abstract discursive objects containing mathematical signifiers and they are the products of objectification (sfard, 2008). a simple geometrical example would be constructing an accurate diagram from a given statement. the notion of signifiers is important, also, in considering mathematics as a discourse. in commognition, ‘signifier’ refers to any primary object that encapsulates its realisation procedures (sfard, 2008). an example of a signifier can be ‘words or symbols that function as nouns in utterances of discourse participants’ (sfard, 2008, p. 154). specifically, mathematical objects are defined as abstract discursive objects with distinct mathematical signifiers (sfard, 2008). a geometrical diagram that is used to solve particular geometrical problems can be recognised as a signifier. human beings are able to take part in different types of discourse, and in some discourses humans fall short. for example, it might be difficult for a visual artist to participate in a specialised mathematical discourse, in the same way that it would be difficult for a mathematician to participate in a discourse about project management or geography. furthermore, sfard (2008) differentiates between two types of discourses. colloquial discourses are everyday-life discourses, visually mediated by pre-existing objects (sfard, 2008). colloquial discourses of mathematics use everyday language that is reified and colloquial narratives can be endorsed by psts through their engagement in discourse with the knowledgeable other or through repetition. mathematical discourses are, however, distinct from colloquial discourses. what makes mathematical discourses distinct is the fact that mathematics is characterised as a domain-specific discourse that can be identified by its word usage, visual mediators, routines and narratives (sfard, 2008). in south african mathematics education, the use of words and symbolic information is common, and it seems like an attempt to strengthen a formal mathematical discourse. this is also evident in the strong way in which symbolic language is evident in geometry proofs with very limited word usage. for example, most geometrical proofs are completed with statements in symbols such as “<abc = <dac” with a few justification statements in words such as “tan-chord theorem”. mathematics is presented in an objectified way in south african high schools. the discourse does not go through the process of reification, but is alienated. this form of mathematical learning is what freudenthal (1973) refers to as the anti-didactical view. teachers and learners do not engage in discourse for the objective of reification, but simply aim to memorise alienated mathematical discourses. hence, mathematics, and especially geometry, is challenging for many south african learners, as described in this article. let us take a closer look at the elements of any mathematical discourse as delineated in commognition. word usage the key to identifying the realm in which a discourse belongs is the keywords used by interlocutors in the dialogue. if you hear people dialogising and the keywords ‘tangent’, ‘chord’ or even ‘straight line’ occur, we know this dialogue belongs within the constraints of geometrical discourse in the high school curriculum. mathematical discourses in schools are similar. there is no way one can talk of ‘quadratic equations’ or ‘the tan-chord theorem’ in any subject other than mathematics. even though some word usages in mathematics appear in colloquial discourse, they form part of the formal mathematical discourse that helps in understanding mathematical concepts. the significance is highlighted by sfard (2008) clearly when she states ‘word use is an all-important matter because being tantamount to what others call “word meaning,” it is responsible for what the user can say about (and thus see in) the world’ (p. 133). an ‘integral’ in colloquial discourse is different from an ‘integral’ in mathematical discourse and to converse with a mathematician about an integral will not be the same as that with any other citizen who has no knowledge of mathematics. hence, in this study, the focus here was placed on psts’ use of mathematical keywords. using this commognition, one can observe psts regarding whether in their utterances they substituted a word from their colloquial discourse with a key mathematical word, and how this affected their understanding of their thinking in problem-solving. this can allow the researcher to make comments on the psts’ use of mathematical keywords in their practice and the use of words in mathematics learning in south african schooling. visual mediators these are the visible objects that can be used by interlocutors in communicating (sfard, 2008). these visual objects are significant, because they can act as identifiers of colloquial and mathematical discourse. in mathematical discourse, visual mediators are only developed to mediate a specific discourse that has developed or will develop at a specific time and in a specific space. these visual mediators can be used as thinking aids in mathematical discourse and they can enhance the discourse. shapes, notations and geometry diagrams included in this review provide clinical examples of visual mediators. preservice teachers engage individually with these visual mediators. especially in geometry, diagrams are important and, in most cases, they are key to solving geometry problems. if a geometry problem is posed in words, psts need to come up with their own diagram to solve the problem. this visual diagram can provide psts with discursive prompts, which can allow them to recall specific knowledge and ways of mathematical problem-solving. furthermore, visual mediators are cues for visual thinking and many prominent mathematicians have relied to a large degree on their visual thinking for success (clements, 1981; shepard, 1978). commognition holds that this visual thinking and imagery is enhanced by the ability of visual mediators to induce multiple ways of thinking about a certain problem. narratives in sfard’s (2008) words, narratives are: any sequence of utterances framed as a description of objects, of relations between objects, or of processes with or by objects, that is subject to endorsement or rejection with the help of discourse-specific substantiation procedures. (p. 134) narratives in discourse can be thought of as ideas that need to be discussed and endorsed mathematically, and once a certain narrative is endorsed, it is considered a theory (sfard, 2008). the goal of mathematics is to produce endorsable narratives through the process of alienation in objectification. this means that psts need to produce narratives that can be derived through the use of mathematical rules and methods. in geometry, postulates, axioms and theorems can be considered as narratives, since they can be derived using mathematical laws. these different narratives can be used in isolation or in conjunction with others to solve different mathematical problems. a geometrical example of a narrative used in this way is a theorem such as “the sum of angles of a triangle is equal to 180°” to support a particular geometrical statement. routines routines are significant and special in mathematical practice. routines, according to sfard (2008), ‘are repetitive patterns characteristic of the given discourse’ (p. 134). routines are repetitive and well-articulated discursive patterns, which may include the process of mathematical generalisation or completion of certain procedures (berger, 2013). routines are regulated by mathematical rules, such as what counts as a definition of a square, what constitutes a mathematical proof or even how to calculate the area of a triangle. once a narrative has been endorsed mathematically, it can be used as a routine or a reason to endorse other mathematical narratives. different patterned ways of communicating about geometry exist and psts can use any method. these can be observed in the way psts use words and visual mediators to derive new or substantiate existing mathematical narratives. for example, a geometry proof can be approached from different perspectives and if a proof is elucidated in an objectified manner, it stands to be true unless it is refuted by others. furthermore, the context, the teacher, the classroom environment, the learners and other factors may influence psts’ discourse about their thinking. differentiating between two types of discourse participation in commognition mathematics education has been characterised by different beliefs about teaching and learning. a recent development was commognition, a belief that equates learning to communicating about thinking, which becomes a ‘legitimate peripheral participant’ in mathematical discourse (lave & wenger, 1991; sfard, 2008). rooted within sociocultural theory, commognition provides an appropriate analytic framework for analysing psts’ discourses. participating in the mathematical discourse during learning can be described as based on the complex dyadic relationship between discursive routines in rituals and explorations. to describe psts’ thinking, their utterances were examined for the objectification of the object of geometry problem-solving, and the substantiation of their endorsed narratives and considered explorations. in this subsection, we explain the distinction between ritualistic and explorative discourse participation in commognition. rituals are socially oriented actions performed to conform to society. one engages in rituals to avoid punishment, please someone or for gain certain rewards (lavie, steiner, & sfard, 2019). for example, if grade 12 learners memorise and reproduce proofs for certain euclidean geometry theorems because these proofs are required in the examination, the participation of the learners in classroom discourse becomes purely ritualistic. the main driver of rituals is societal expectations. when one feels obliged to perform in a certain way to please someone, then one engages in rituals. if learners in mrs x’s classroom were directing all answers to her without giving reasons and the teacher was the one endorsing the answers as correct or incorrect through giving applause for positive evaluation, these learners’ participation in this discourse was solely motivated by getting applause or positive evaluation from the teacher (heyd-metzuyanim & graven, 2016). in a way similar to ritualistic participation in the discourse, psts continually sought affirmation and approval from the interviewer as they solved geometry problems. in ritualistic discourse participation, interlocutors obtain discursive cues by imitating their knowledgeable other peers or teachers or rely on previous experiences. these discursive actions are scaffolded by others; hence their level of applicability is highly restricted. here, ritualistic discussants aim to act in harmony with no degree of flexibility in their actions, merely to imitate what others are doing. as a consequence of their high reliance on performance instead of knowing, rituals tend to follow strict rules to ensure that they can be produced by others. in ritualistic discourse participation, all rituals that do not succeed are simply repeated instead of being corrected or modified, which is a further indication of the rigidity in the applicability of rituals. this form of discourse participation does not require special substantiation of the produced narrative, because it focuses mainly on the how of a routine (as explained earlier). hence, the steps for the process are listed clearly and the performer needs to follow these steps to reproduce the ritual. as opposed to rituals, explorations are not aimed at pleasing or conforming to societal expectations, but at advancing theory. particularly, routines can be characterised as explorations if they produce narratives contributing to a mathematical theory instead of tangible objects (sfard, 2008). explorative participation in mathematical discourse is characterised by objectification, where words denote realistic mathematical objects, and the participant views different realisations of the same thing (such as x2–4 and (x–2)(x+4)) to be interchangeable (heyd-metzuyanim & graven, 2016). rituals are mainly ‘process-oriented’, while explorations are ‘outcome-oriented’ (lavie et al., 2019). explorative activities aim at inventing, producing or discovering some truths or facts about mathematical objects. these routines are applicable in a variety of contexts because the resulting narratives are endorsed and form part of a mathematical theory. psts who exhibit this meta-level type of thinking can use these endorsed narratives in a wide range of contexts while providing multiple, but acceptable, means of substantiations for their routines. too often learners are considered ‘good’ at mathematics because they conform very well to society’s expectations, and those who do not conform are usually labelled as ‘outcasts’ and ‘weak at mathematics’ (heyd-metzuyanim & graven, 2016). this view may affect learners’ formations of their identities in the classroom community. participation in explorative discourse requires a good understanding of some already endorsed mathematical knowledge; hence rituals can be thought of as significant building blocks towards explorations (lavie et al., 2019; nachlieli & michal, 2019). sfard (2008, p. 223) claims that rituals and deeds are ‘developmental predecessors of explorations’. hence, meta-level learning of mathematics moves from ritualistic discourse participation towards explorative participation, and the ritual develops into exploration through the process of rationalisation. at the beginning of learning, when a routine constitutes objects or metarules unfamiliar to the learner, learning becomes highly unlikely. hence, at this stage, for beginners to be conversant with a certain mathematical discourse, they rely on imitating a knowledgeable other in that specific discourse. the beginner imitates the rules and procedures of the knowledgeable other, adapting them until they individualise the mathematical discourse, but at this early stage, their discourse participation is ritualised. at this stage, beginners know how to perform a certain routine, but have not struck the balance as to when to perform it. for learners to be able to transform their rituals into explorations, they need to reflect continually on their performance of the ritual, while examining the rationale for the expert performance as they participate in mathematical discourse. in vygotsky’s idea, ritualistic discourse participation occurs in the zone of proximal development (zpd), where the learner can participate in the collective patterned thinking or the performance of a routine, but is incapable of individual performance. by contrast, explorations look for the connectedness of different routines so that once learners have acquired explorative discourse participation, they hold a network of interconnected routines, instead of disconnected ones. the explorative form of objectifying particular routines allows learners to compress mathematical knowledge to allow them to solve a multitude of mathematical problems with few routines. furthermore, commognition recognises that metalevel learning occurs ‘when the learner is exposed to commognitive conflict’ (sfard, 2008, p. 260). sfard (2008, p. 296) describes the commognitive conflict as the conflict that ‘arises when communication occurs across incommensurable discourses’. interlocutors who are in commognitive conflict participate in discourses that differ in their utilisation of words, mediators and routines, which might allow them to endorse narratives that seem contradictory. commognition warns the commognitive researcher not to confuse commognitive conflict with cognitive conflict. the difference is tabulated in table 2. table 2: comparing cognitive conflict and commognitive conflict. the link between commognition and the van hiele theory a brief explanation of the relationship between the two theories as conceived in the study suffices. in particular, this relationship is described based on the type of discourse participation evident from psts and how this type of discourse participation was related to each level of the van hiele theory. firstly, the relationship is explained based on teachers’ failed behaviours during geometry problem-solving instead of their successes, because this will raise awareness of the severity of the need to develop competent geometry teachers in south africa. teachers’ behaviours, described in table 3, were constructed based on the properties of each van hiele level of geometry thinking. these behaviours can range from any example that characterises behaviour at a certain van hiele level of geometry thinking through the processes described by the van hieles. the relationship between teachers’ behaviours and the van hiele levels is a critical one but elementary, because the teachers’ behaviours were formulated based on the properties of the van hiele levels. nevertheless, what is not evident here is that, since competency at each level is dependent on competency at the preceding levels, teachers’ struggles at a particular level might not be because of struggling with geometry reasoning related to that level, but the preceding levels. for example, a failure to prove a geometry theorem (level 4) might be hindered by not only a deficient competency to order objects according to the relationship that exists between their properties (level 3), but also psts’ visualisation skills (level 1). table 3: relationship between preservice teachers’ discourse participation and the van hiele levels of geometrical thinking. a pmt who is operating at a ritualistic van hiele level 1 not only struggles with the visual identification of geometrical objects using their appearances but also fails to link descriptions with their visual appearances. this means that this pmt is not able to identify a theorem or property using the appearance of the diagram in a particular problem. this pmt is then not ready to proceed to the other levels of geometrical thinking and must stay in that level until they have mastered the explorative van hiele level 1. wang (2016) found that pmts uses phrases related to visuals such as ‘looks like’ even when substantiating their arguments. this explanation goes for all the levels and, as a critical finding of this study, this progression is better explained in figure 1. as seen in figure 1, progression to the next van hiele level requires that psts operate fully in the explorative discourse participation where they rely on their own thinking and experiences and use objectified discourse when talking about solving geometrical problems. table 3 shows how pmts’ discourse changes as their geometrical thinking moves to a higher van hiele level (wang, 2016) but this does not indicate that their discourse participation changes. changes in discourse participation from ritualistic to explorative (discourse of experts) requires that pmts apply some reification to their discourse allowing them to switch from colloquial to mathematical use of euclidean geometry discourse (sfard, 2021). these changes are observed in how pmts use the elements of mathematical discourse in their discourse participation during geometry problem-solving. that is why pmts who cannot differentiate between sufficient and necessary properties, conditions or even geometrical steps for a particular proof to be true are indeed still lacking not only van hiele level 2 but also level 1 (sfard, 2007). in peer discussions, such changes in discourse (and perhaps discourse participation) can be a consequence of the coalescence among the discourses of the peers (ben-dor & heyd-metzuyanim, 2021; kaur, 2020). figure 1: a commognitive analysis of pre-service teachers’ geometrical thinking development through van hiele levels of geometrical thinking. the transformation in discourse is a critical and necessary condition for mathematical learning (heyd-metzuyanim, 2018) and, in many instances, transforming from a ritualistic to an explorative discourse may be dependent on the scaffolding from knowledgeable others. as such, transforming to being able to do deductive geometrical proofs individually (from level 3 to level 4) requires that one participates in the discourse of proof ritualistically first and, through support and scaffolding from knowledgeable others, one gradually transforms to explorative discourse participation (ben-dor & heyd-metzuyanim, 2021). this is also supported by cooper and lavie (2021) who found that developments in mathematical thinking and specifically learning cannot avoid ritualistic discourse participation (cf. kaur, 2020). as such, table 3 as corroborated by figure 1 indicates that development in geometrical thinking is reliant on pmts transforming their discourse from ritual to exploration. thus, pmts participate in the euclidean geometry discourse with the aim of producing endorsable narratives about geometrical objects. however, the transition to either a higher van hiele level or from ritualistic to explorative discourse participation is complex and a deeper understanding of the discourses at each van hiele level and how transformation occurs from ritualistic to explorative discourse participation is warranted. for example, pmts can be operating at van hiele level 4 but their word usage, visual mediators, narratives and substantiation of euclidean geometry routines may differ depending on their discourse participation (wang & kinzel, 2014). even further, pmts may be operating at a ritualistic van hiele level 4 and still have differing discourse in their talk about geometrical objects. thus, the transformations in geometrical thinking represented in figure 1 are complex and sometimes may constitute more complex discourse relationships than the ones represented in figure 1. conclusion upon conducting this literature study, it was found that development to higher van hiele levels was dependent on the discourse participation of the psts according to figure 1. thus, in conclusion, we discuss the study’s main contribution to the theories of commognition and the van hiele theory of geometrical thinking. in particular, we discuss how the findings from the study amalgamated commognition and the van hiele theory of geometrical thinking to contribute to new knowledge. we use figure 1 to guide the discussion and this allows us to show how pmts’ discourse participation can be used to enhance their development of geometrical thinking through the van hiele levels of geometrical thinking. the below discussion is guided by the characteristics of the van hiele levels and how psts participated in discourse in this study. thus, we discuss this contribution based on the movement from level 1 to level 2, movement from level 2 to level 3 and so on ending at level 4 which is the level at which grade 12 psts should be operating. furthermore, the discussion is guided by the type of discourse participation teachers displayed in each level, from the findings. development from van hiele level 1 to level 2 the identification of geometrical figures by their appearances only is the critical determining skill in level 0 because only the recognition of the figure is required. the psts who could not identify geometric figures by their appearances during discourse and those who relied on scaffolding from the interviewer lack this critical geometrical skill. furthermore, in advanced geometry, theorems can also be identified through the appearance of the diagrams and psts who could not identify particular theorems that were available in the diagram lack visualisation. from the literature findings, the lack of visualising theorems can hinder psts’ problem-solving but once a scaffold was given by the interviewer, they were able to complete subroutines that involved the theorem as exploration. thus, psts who lack this visualisation skill should not be allowed to progress to level 1 because visualisation is a prerequisite in level 1. they must stay at level 0 as indicated by the red arrow in figure 1. however, psts who could identify geometrical objects based on their appearances individually and could compare these objects with objects they encounter daily could be allowed to move to level 1 as they have mastered the skill of visualisation as indicated by the green arrow in figure 1. that is because the former is still reliant on ritualistic discourse participation while the latter relies on explorative discourse participation. development from van hiele level 2 to level 3 the defining characteristic in van hiele level 1 is that psts should have moved from not only recognising objects by their appearances but also linking the objects to their properties. however, interrelating the properties of objects is still not developed at level 1. the theoretical findings indicate that psts who cannot name properties of geometrical objects without scaffolding and those who used colloquial discourse to name these properties must not move to van hiele level 3. furthermore, psts who mentioned incorrect properties to gain social acceptance, relying on the usage of ritualistic discourse and past experiences should also remain in ritualistic van hiele level 2. these psts have not mastered the ability to link a geometrical figure with its property and thus they should remain in ritualistic van hiele level 2 until this skill is mastered. those psts who are promoted to explorative van hiele level 3 are those who can individually link a particular property to a particular geometrical figure individually. these psts can use objectified discourse to mention and link geometrical properties with their geometrical figures which according to the theoretical findings relates to explorative discourse participation. this link can allow psts to produce endorsed narratives about the geometrical world and these psts can be promoted to explorative van hiele level 3 because they show mastery of the skill of linking geometrical figures with their properties using explorative discourse. development from van hiele level 3 to level 4 at level 3 psts do not just link properties with their geometrical figures but can logically order and interrelate properties to understand the relationship between geometrical figures. mastery of level 3 means that psts are getting ready for logical deduction required in proofs. at this level, psts should not be reliant on scaffolding to link properties and still be promoted to level 4 because that shows that they have not mastered van hiele level 3. furthermore, the formulation of meaningful definitions is critical in level 3 and if psts still rely on scaffolding they are not ready for level 4. the understanding of properties and theorems is critical in proofs and if one has not mastered the link between properties and the relationship between figures, then one is not ready to move to level 4. this is because psts will produce a whole proof without giving proper reasoning for their arguments or statements with correct reasons. only psts who show logical understanding of how properties of different figures link and are able support their narrative individually using objectified discourse should be allowed to proceed to level 4 which is the last van hiele level required in the caps. discourse participation at level 4 level 4 requires that psts be able to use experiences from the previous levels to understand the role of properties, theorems and the links therein when doing geometry proofs. at this level, psts now begin to develop longer arguments to perform geometrical proofs and can successfully substantiate each argument with an endorsed mathematical narrative. those psts who continually rely on the same procedure applied in exactly the same way instead of being flexible in their strategies are not ready for level 4 and they should be demoted to level 3. at this level, relying on procedures, subroutines and scaffolding from others to obtain a proof does not guarantee that one will be able to prove similar problems in the future; independence is required to master level 4. justifying statements during proofs using phrase-driven and colloquial discourse instead of objectified discourse also shows that one has not mastered level 3; thus, they must be demoted to level 3. a pst who performs geometrical proofs independently and uses explorative discourse when talking about geometrical proofs can be thought of as ready to teach geometry at grade 12 level in the caps. acknowledgements competing interests the authors declare that no competing interest exists. authors’ contributions s.c.m. completed the article individually and v.m. was the supervisor of the phd study and contributed by checking the accuracy of the information in the article. ethical considerations this article followed all ethical standards for research without direct contact with human or animal subjects. funding information this research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors. data availability data sharing is not applicable to this article, as no new 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(2010). making euclidean geometry compulsory: are we prepared? perspectives in education, 28(4), 22–31. vygotsky, l. (1978). mind in society: the development of higher psychological processes. cambridge: havard university press. wang, s. (2016). discourse perspective of geometric thoughts. heidelberg: springer spektrum. wang, s., & kinzel, m. (2014). how do they know it is a parallelogram? analysing geometric discourse at van hiele level 3. research in mathematics education, 16(3), 288–305. https://doi.org/10.1080/14794802.2014.933711 yi, m., flores, r., & wang, j. (2020). examining the influence of van hiele theory-based instructional activities on elementary preservice teachers’ geometry knowledge for teaching 2-d shapes. teaching and teacher education, 91, 103038. https://doi.org/10.1016/j.tate.2020.103038 footnote 1. a monologically understood world is objectified and corresponds to a single and unified authorial consciousness. http://www.pythagoras.org.za open access page 1 of 1 reviewer acknowledgement acknowledgement to reviewers in an effort to facilitate the selection of appropriate peer reviewers for pythagoras, we ask that you take a moment to update your electronic portfolio on https:// pythagoras.org.za for our files, allowing us better access to your areas of interest and expertise, in order to match reviewers with submitted manuscripts. if you would like to become a reviewer, please visit the journal website and register as a reviewer. to access your details on the website, you will need to follow these steps: 1. log into the online journal at https:// pythagoras.org.za 2. in your ‘user home’ [https://pythagoras. org.za/index.php/ pythagoras/user] select ‘edit my profile’ under the heading ‘my account’ and insert all relevant details, bio statement and reviewing interest(s). 3. it is good practice as a reviewer to update your personal details regularly to ensure contact with you throughout your professional term as reviewer to pythagoras. please do not hesitate to contact us if you require assistance in performing this task. publisher: publishing@aosis.co.za tel: +27 21 975 2602 tel: 086 1000 381 the editorial team of pythagoras recognises the value and importance of the peer reviewer in the overall publication process – not only in shaping the individual manuscript, but also in shaping the credibility and reputation of our journal. we are committed to the timely publication of all original, innovative contributions submitted for publication. as such, the identification and selection of reviewers who have expertise and interest in the topics appropriate to each manuscript are essential elements in ensuring a timely, productive peer review process. we would like to take this opportunity to thank all reviewers who participated in shaping this volume of pythagoras. we appreciate the time taken to perform your review(s) successfully. alwyn olivier andré du plessis aneshkumar maharaj anita l. campbell anthea roberts anthony essien belinda huntley benita p. nel carol macdonald caroline long cerenus pfeiffer deonarain brijlall duan van der westhuizen duncan mhakure eunice k. moru george ekol hamsa venkatakrishnan helen sidiropoulos iben m. christiansen ifunanya j.a. ubah ingrid e. mostert jacques du plessis joseph baidoo kabelo chuene kakoma luneta karen e. junqueira lyn webb marc north marie joubert mdutshekelwa c. ndlovu michael murray michael d. de villiers michael k. mhlolo million chauraya mohammad f. gierdien neil eddy nokulunga s. ndlovu patrick barmby paul mokilane piera biccard rina durandt rose s. maoto sarah bansilal shaheeda jaffer sheena rughubar-reddy tarun k. tyagi vanessa scherman vasuthavan govender yip-cheung chan zingiswa m.m. jojo zwelithini b. dhlamini http://www.pythagoras.org.za� https://pythagoras.org.za� https://pythagoras.org.za� https://pythagoras.org.za� https://pythagoras.org.za� https://pythagoras.org.za/index.php/pythagoras/user https://pythagoras.org.za/index.php/pythagoras/user https://pythagoras.org.za/index.php/pythagoras/user mailto:publishing@aosis.co.za editorial pythagoras, 68, 2 (december 2008) 2 editorial pythagoras is going places … alwyn olivier stellenbosch university pythagoras@amesa.org.za you will be very much aware that this issue of pythagoras is very late. i very much apologise! i will be franc and admit that i underestimated the task! i was overwhelmed and underprepared for the task! i underestimated the amount of time that needs to be invested to just manage the day-to-day editor’s task – acknowledging receipt of submissions, finding suitable reviewers, distributing the papers to reviewers, following up reviewers, chewing on different evaluations, giving feedback to authors. i found that i needed good organisational skills to manage – store files and e-mails and find them again! truth is that my little system basically crashed and my communication with eager authors and reluctant reviewers faltered. i underestimated the specialised knowledge needed for the task – i still had to learn how to edit a journal. it takes time to establish procedures and policies, e.g. about the so-called journal house style. so i had to learn much on my own, mostly by learning from others. suddenly i was reading the apa manual and downloaded the gaurdian style book! it has been suggested that editing pythagoras is not for an amateur working in his “free time”, but that amesa should hand over pythagoras to some publishing house so that it can be done professionally. however, it is clear that it will escalate costs and it will probably then not be possible to continue offering pythagoras as a “free” benefit to members. therefore it is not the route to go. however, it will be necessary for amesa to invest more in pythagoras to develop an infrastructure and involve more people. the amesa council has accepted my proposal to appoint three associate editors. they are michael de villiers (ukzn), anthony essien (wits) and dirk wessels (stellenbosch). i am sure that this larger team must be more effective. we will start by re-building a database of reviewers and streamlining our procedures and policies … council has voted that we establish a little office for pythagoras in my building, with a little infrastructure and administrative support. i also have the support of the dean of education for the pythagoras office. this should help in being more efficient! council has made the principled decision to go the open access route, where pythagoras will be freely available to everyone. pythagoras is already available online to members and buyers through sabinet. we will now also join african journals online. we are starting a new website where pythagoras will be freely available. to do this, we will be using an open source journal management program that will streamline the process. nearly everything should happen “automatically”: authors will submit their papers online, the program will acknowledge receipt, then suggest reviewers, send the paper with a covering letter to reviewers, reviewing will likewise be online, etc. there are also national initiatives to improve the quality of south african journals, e.g. the academy of science of south africa (assaf) has established a committee on scholarly publishing in sa and they have an annual national scholarly editors’ forum. i have therefore become a member of an active community of editors, and i find that very supportive. all in all, i am very confident that we are building a structure where pythagoras will go from strength to strength – improving its quality while simultaneously becoming more accessible to more authors and to its readers. abstract introduction description of the south african numeracy chair project covid-19 digital resource drive theoretical and conceptual framework research method and design results and discussion: a displaced ecology of learning results and discussion: digital affordances and constraints discussion: the process and principles of digital resource design feedback conclusion acknowledgements references footnote about the author(s) pamela vale department of education, faculty education, rhodes university, makhanda, south africa mellony h. graven department of education, faculty education, rhodes university, makhanda, south africa citation vale, p., & graven, m.h. (2021). reflecting on dilemmas in digital resource design as a response to covid-19 for learners in under-resourced contexts. pythagoras, 42(1), a599. https://doi.org/10.4102/pythagoras.v42i1.599 note: teaching and learning mathematics during the covid-19. original research reflecting on dilemmas in digital resource design as a response to covid-19 for learners in under-resourced contexts pamela vale, mellony h. graven received: 19 jan. 2021; accepted: 19 aug. 2021; published: 09 nov. 2021 copyright: © 2021. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract the coronavirus disease 2019 (covid-19) pandemic and the resulting school closures in south africa necessitated a major shift in how to support learners’ ongoing mathematics learning. for 10 weeks learners were strictly confined to their homes with restrictions that prohibited seeing any person outside of their household. the only means to access learners and parents in their homes was to reimagine our south african numeracy chair project work and transform it from predominantly face-to-face interventions to digital modalities. as a result, we initiated a project of digital resource development and distribution, particularly focused on our local community in the eastern cape. twenty-two existing resources and 36 purpose-designed resources were shared via facebook. through in-depth post hoc reflection of the rapid digitalisation of our materials and ways of working we address these questions: (1) in relation to learners’ new ‘ecology of learning’ during lockdown what digital access modality and platforms were most fit-for-purpose in sharing mathematics learning resources? (2) what principles informed resource design and adaptation for digital distribution and use? (3) what dilemmas were confronted in making decisions about resource design and distribution?. these questions are answered through a document review and post hoc reflections on the noted dilemmas. we share some feedback received and discuss implications of our work and the dilemmas confronted for the provision of quality digital resources for supporting mathematics learning in historically disadvantaged and under-resourced communities in a post pandemic world. keywords: south africa; covid-19; digital resources; task design; ecology of learning; under-resourced communities. introduction the coronavirus disease 2019 (covid-19) pandemic and the resulting school closures in south africa necessitated a major shift in how to imagine providing support for learners’ ongoing mathematics learning. schools were closed from 18 march 2020, initially for a period ending on 14 april 2020. the announcement of a 21-day ‘hard’ lockdown (stage 5) from midnight on 26 march 2020 represented the first extension of this closure. it was further extended to 30 april 2020, and again extended with slightly fewer restrictions (stage 4) to 31 may 2020. this meant that for 5 weeks all learners were strictly confined to their homes (republic of south africa [rsa], 2020a). schools remained closed for a further 5 weeks with restrictions remaining in place that prohibited learners from seeing any person outside of their household (spaull & van der berg, 2020). the strict regulations on movement of people during this time meant that no project or person could reach learners and parents in their homes to distribute much-needed numeracy and literacy learning resources. it was only on 08 june 2020 that a staggered re-opening of schools began; however, another school closure was imposed from 23 july 2020. the revised plan for phasing grades back into schools was published on 02 august (rsa, 2020b) and had grade 5 and grade 8 returning to school as late as 31 august 2020. the number of school days available for instruction was significantly lower than the number scheduled on the pre-covid-19 school calendar: grade 5 and grade 8 lost 42% and grade 4 and grade 9 39% of their official school days in the pre-covid-19 calendar (hoadley, 2020). there was further loss of instruction time due to the timetable models promoted by the department of basic education (dbe, 2020) in order to ensure social distancing. these included platooning (half of the school attending in the morning, and half in the afternoon); attending on alternate weeks, or days of the week; and a bi-weekly rotational attendance model (dbe, 2020). schools that were able to maintain the strict social distancing requirements with all learners attending were permitted to do so (rsa, 2020c), but for the vast majority of schools this was not possible and as a result the remaining contact instructional time was effectively halved (hoadley, 2020). hoadley (2020) reports that of 7162 schools surveyed in august 2020, only 8% were able to accommodate all learners on a regular daily timetable. this schedule endured for the remainder of the school year and attendance remained extremely poor. van der berg, van wyk and selkirk (2020) explain that the learning losses for learners are likely to be longer than the tally of instructional days lost. while the extent of the learning losses is as yet unknown, gustaffson and nuga (2020) predict that grade 12 outcomes could be below expectations for a period reaching as far forward as 2031, indicating the cumulative loss that young learners are likely to carry through their entire school journey. hoadley (2020) explains that the disparities among households with regard to educational resources being available in the home, and the effects of the lockdown on food security and health, mean that the severity of the learning losses will ‘map onto existing social and educational inequalities’ (p. 8). the south african numeracy chair project (sancp) aims to advance educational equity, particularly for learners in historically excluded communities and under-resourced schools. in south africa, a society with an education system with deeply entrenched inequalities (amnesty international, 2020), the gap will widen between under-resourced and well-resourced schools, and the harshest of the consequences of the school closures will be thrust on the most vulnerable learners. we have already seen this unfolding, locally and nationally (batyi, donyeli, & amner, 2020). our sancp work had since inception in 2011 used a dual resource-based and contact-based approach to school and community partnership work. this involved regular face-to-face professional development, learner mathematics clubs and family mathematics events, all accompanied by the provision of a wide range of physical research-informed resources. the sancp resources are freely available for downloading from the project website, although our partners predominantly engage with project ideas through the use of physical resources provided during mediated face-to-face interactions in professional development sessions, clubs and community events. the pandemic demanded immediate reimagination of a digitalised way of supporting mathematics learning. finding a creative strategy to access these learners in their homes became an urgent priority, and given the constraints presented by the hard lockdown regulations, the only option was to use a strategy built on digital networks. this required us to completely reimagine our work and use digital means for sharing learning resources. several dilemmas were confronted in the decision-making process of which digital modality to use, how to conceptualise what would be most useful to develop and how best to ‘distribute’ these. it required knowing and understanding some of the challenges of the living situations and limited resources of those we were supporting. this linked to considerations of what support would be easy to use and of educational value. in terms of educational value sancp resource design draws broadly on a socio-constructivist perspective on learning, which assumes that learners must actively engage with materials and ideas in order to construct knowledge and understanding. a framework of confronting dilemmas in the design of mathematics learning digital resources for use in under-served communities provides an analytic structure for this article. the questions we address include: in relation to learners’ new ‘ecology of learning’ during lockdown what digital access modality and platforms were most fit-for-purpose in sharing mathematics learning resources? what principles informed resource design and adaptation for digital distribution and use? what dilemmas were confronted in making decisions about resource design and distribution and how were these managed? description of the south african numeracy chair project covid-19 digital resource drive the sancp has been researching innovative, sustainable and practical solutions to the challenges of early numeracy education since 2011. the focus of the model that the project works to is one of blending research and development (graven & venkat, 2013). learners and teachers in historically disadvantaged public primary schools in the eastern cape are the primary focus of sancp work. all sancp numeracy intervention programmes with teachers, learners, parents and other community members are anchored by relationships built and nurtured through face-to-face events and activities. while, since 2011, we have made considerable effort to create and maintain virtual communities through our website and facebook page, and our virtual ‘members’ have expanded well beyond our eastern cape partners, our development work has mostly focused on in-person contact and sustaining communities whose membership involve physical presence and participation in the joint enterprise (wenger, 1998) of each intervention rather than managing ‘virtual’ communities. the sancp website has served largely as a repository of resources that teachers continue to access post interventions for classroom resources, and the facebook page has provided a means of sharing feedback, news and information about events. our usual way of working with teachers, learners and parents was completely halted by lockdown regulations, and our use of online platforms, in particular, needed to be immediately repurposed. in response, we initiated a project of resource development and distribution through digital means, specifically through social media. this project was implemented across the full 10-week period of stage 5 and stage 4 lockdown. it was particularly focused on our partner communities in makana and ndlambe municipalities (although the resources were available, and were used, far beyond that). in total, 22 existing sancp resources and 36 new purpose-designed primary mathematics resources were created and shared digitally from march to june 2020 on a near-daily basis. the resources included: problem-solving activities, games, puzzles and fluency practice. resource design was guided by our socio-constructivist perspective of learning which assumes that students must be actively engaged with materials to construct their own understanding and that mathematical proficiency requires developing conceptual understanding, strategic competence, procedural fluency, adaptive reasoning and a productive disposition (kilpatrick, swafford, & findell, 2001). different types of resources focused on different combinations of these strands of proficiency. for our local mathematics clubs and schools, we delivered a range of mathematics workbooks for home use but these supplementary physical provisions are not a focus of this article. our choice of platform was facebook in particular, in part because we could already reach our existing 1951 followers from the first day, and there was great potential to grow beyond that number. we explore this fully later in this article. the purpose of the facebook page thus shifted from being a platform we used to share news and information to a platform through which we could reach learners and parents in their homes and provide access to quality resources. theoretical and conceptual framework in this article, we share several dilemmas that informed our choice of digital modality, design principles and distribution. dilemmas have been used by several mathematics educators as a structuring device for post hoc reflection on design issues for a range of mathematics education interventions and practices (adler, 1998, 1999; graven, 2005). they argue that the articulation and theorisation of the dilemmas and tensions we face in the design of interventions and in our broader work is an important and necessary part of our praxis as teacher educators. dilemmas provide a useful tool for conscious reflection on challenging and complex aspects of our practice. in terms of our perspective on learning, and theoretically framing the study as noted above, our work draws largely on a socio-constructivist perspective of learning and our notion of mathematical proficiency is shaped by kilpatrick et al.’s (2001) five strands of mathematical proficiency (also evident in our south african primary curriculum documents). conceptually we draw on the work of watson (2004) in defining and delineating the ‘ecology of learning’, and on the work of greeno (1998) in reflecting on the affordances and constraints that this ecology presented in relation to intervention through digital modalities. digital modalities are here defined as the digital technologies and tools that can be used for online teaching and learning (shank, 2020). our chosen access modalities were mobile devices and the digital platform chosen was facebook. in reflecting on the design process, we use de lange’s (2015) distinction between slow and fast design, and his explanation of the role of various types of intuition. in order to structure our description of the design principles, we use the structure offered by barbosa and de oliveira’s (2013) five ‘arenas of conflict’ (p. 545): context, language, structure, distribution and levels of interaction. we use this frame in a similar manner to sullivan, knott and yang (2015) who reflected on these as dilemmas faced by task designers in the process of deciding between alternatives. within this discussion, we explain how the decisions we made worked to leverage affordances offered by the learners’ new ecology of learning in terms of digital access. watson (2004) uses the metaphor of the classroom as an ecology of participation: it is a system which offers certain kinds of interactions as possibilities (teachers talking to learners, learners talking to each other), with normal practices which are somehow regulated … and within which individuals interact in ways which themselves influence the development of the system. (p. 24) watson (2004) further explains that she defines ‘ecology’ as an environment in which: elements, human and non-human, physical and semiotic, interact with each other in ways which result in change or perpetuation of the environment and the elements and relationships within it, rather than solely of the individual elements. (p. 24) the school closures and the nature of the staggered return of grades with rotational timetabling meant that the primary site of learning for school learners shifted, and the ‘ecology of learning’ (watson, 2004) was fundamentally changed. for the initial 10 weeks of school closures the site of learning was exclusively the home. for the 5 weeks of ‘hard’ lockdown in particular, it was not possible for teachers or education development partners to distribute physical resources or learning materials to learners in their homes due to the restrictions on movement for all people. the sancp provision of support to learners (and teachers and parents) now inhabiting this new ‘ecology of learning’ required careful consideration of the affordances and constraints presented by it, particularly in relation to the use of digital modalities in the home. greeno (1998) explains that attunement to affordances and constraints, which includes understanding ‘well-coordinated patterns of participation in social practices, including … interactional conventions of communities’ (p. 9), is important for those providing resources for practices. as designers of resources and the means of their distribution, it was critical for us to understand these patterns and conventions within the ecology. we use the language of ‘affordances’ and ‘constraints’ in reflecting on this ecology, seeing affordances as ‘qualities of systems that can support interactions and therefore present possible interactions for an individual to participate in’ (greeno, 1998, p. 9) and constraints, as ‘norms, effects and relations which limit the wider possibilities of the system’ (watson, 2004, p. 23). research method and design this article emerges from our qualitative case study that post hoc reflects on (and analyses) dilemmas addressed in the digital drive to distribute resources during stage 5 and stage 4 lockdown. our methodology involved first a document review conducted prior to (and continuously throughout) the digital design project, and secondly post hoc reflection on several design dilemmas confronted in resource design in relation to the new ecology of learning context reviewed primarily in documents. document review given the sudden lockdown and the urgency of our response we focused on data that could be gathered quickly enough to inform our decisions on digital transformation of mathematics learning support. document review provided immediate sources of data that informed our planning and reflection on dilemmas confronted. we begin by describing the new ‘ecology of learning’ that learners moved into during the first 10 weeks of lockdown as gathered from a range of documents and secondly what digital modality and platforms are available in our historically disadvantaged and under-resourced eastern cape community – as available through documentary review. as a research and development project, interventions like this resource drive are not initiated without first carefully considering all available, relevant information. it is also essential to keep up to date in situations that are as unstable as was the case during the lockdown; therefore, the document review continued throughout the 10-week period. because the focus was on the provision of learning resources to the homes of learners attending schools and programmes run by our education partners, the document review focused on the makana and ndlambe municipalities in the eastern cape. every government gazette documenting the covid-19 regulations was analysed and all notices issued to schools by the dbe and the eastern cape department of education (ecdoe) that pertained to lockdown were analysed (up to 01 june 2020). all grocott’s mail (the local newspaper) articles describing the lockdown situation for the makana community were also analysed. document review continued throughout the project as government decisions were continually updated and new advice given to schools throughout 2020. for example, the initial plan was to share one existing resource per day for 21 days; however, as soon as it became clear that the lockdown and school closures would be extended, the plan shifted to designing new resources and managing them more strategically. also included in the document review were reports that allowed us insight into the affordances and constraints to learning through digital platforms that might be presented by conditions in the home, as this would become the learners’ ‘ecology of learning’. the specific reports consulted are (see full details in references): amnesty international (2020). broken and unequal: the state of education in south africa. goldstuck, a. (2020). under the skin of social media in south africa. in patricio, o. (ed.), sa social media landscape 2020: social media myths, high walls and controlled. human sciences research council (2014). state of poverty and its manifestation in the nine provinces of south africa. kemp, s. (2020). digital 2020: south africa. makana municipality. (2018). final makana municipality integrated development plan 2019–2020. ndlambe municipality. (2018). integrated development plan 2019/2020 (final review). statssa. (2012). south african population census 2011. accessed march 5, 2020 from https://wazimap.co.za/profiles/district-dc10-sarah-baartman/ statssa. (2018). community survey 2016. statssa. (2020a). general household survey 2019. the broadcast research council of south africa. (2019). the establishment survey. the statssa documents were particularly valid for analysis as most of the other documents we included all quoted statistics derived from these reports. furthermore, reliability of this data is strong and information is publicly available and accessible. the quarterly labour force survey (statssa, 2020b, 2020c) is particularly relevant and is quoted as it describes the context for south africans during lockdown. all the above-mentioned documents were reviewed and collated for answering the research questions. we acknowledge that there are several pitfalls in the use of post hoc reflections as data (shanks, 2017). cognition and intuition are notoriously difficult to capture. however, through the articulation of various dilemmas confronted and analysing the decisions made in relation to those dilemmas one is able to provide a coherent and useful account of praxis that should inform future grappling with the digital age that the pandemic has brought centre stage into education across the globe. as graven, borba, jablonka, martin and subramaniam (2021) state, the digital transition in education has recently gained speed due to the massive closure of, and disruption to, schools as a result of covid across the globe. while online learning opportunities bring the possibility for increased access to quality mathematics education resources there is enormous unevenness in systemic and individual preparedness to optimise these opportunities. this unevenness plays out across countries and within countries and there is concern that the digital divide could exacerbate inequities across gender, social class and race. through purposeful and systematic reflection on dilemmas faced working in under-resourced contexts we contribute towards important conversations and considerations for ongoing digital transformation in mathematics education post the pandemic. the design and implementation process of the sancp resource drive was carefully planned and executed. while not at the time intended to be reported as research, it is a process and product worth interrogating. in order to enhance the reliability of the claims made here and the principles that are extracted from this process and reflection, earlier drafts of the resources, personal notes made while designing the resources, artefacts produced while testing the resources and the messages to parents that were shared with each resource were used as stimuli for recall. this reflection is aimed at responding to the third question of this article: what principles informed and underpinned the design of the resources developed? in order to provide a structure to this reflection and the report on the content of the reflections resulting from it, we use barbosa and de oliveira’s (2013) five arenas of conflict that arise when making decisions during task design: context, language, structure, distribution and levels of interaction. ethical considerations the ethics in health research report (department of health, 2015) indicates that when research depends on material that is already in the public domain and no human participants are involved it does not require any ethical clearance certificate from an accredited human ethics committee. this does not mean, however, that the study is exempt from adhering to ethical research principles. in the case of this research, all documents used in the document review are in the public domain and are easily accessible. the personal reflection on the design process is that of the first author as the designer of the materials and leader of the project. the authors have no conflict of interest and no commercial interest in the success or otherwise of the resources referred to here. all material is published under a creative commons attributionnoncommercial-noderivatives 4.0 international licence, and as such is available for use and distribution by any person or organisation. attributions are clearly indicated on resources where ideas were inspired by the work of other authors and organisations. no information is included from conversations with any individual, and only comments made in the public domain on our public facebook page are mentioned, with no identifying information given in this article (though identifying information is in the public domain). results and discussion: a displaced ecology of learning in this section, we present a description of the new ‘ecology of participation’ for learners. this description is based on the findings of the document review. first the community itself is described in terms of the demographics and other socio-economic indicators. then the focus turns to understanding aspects of the ecology of participation that offer affordances and present constraints specifically related to digital use. the sancp is located in makhanda in the eastern cape – one of the poorest provinces in south africa (human sciences research council, 2014). the eastern cape has the largest number of persons and households reliant on social grants in south africa (statssa, 2020a). in 2019, the percentage of learners in the province attending no-fee schools was 76.9% and 86.4% of all learners benefitted from school feeding schemes. unemployment in the eastern cape during the period of hard lockdown was the highest of the nine provinces (statssa, 2020b), and it is the eastern cape that has been hardest hit by job losses in 2020, with employment decreasing by 13.5% between the first and third quarters of 2020 (statssa, 2020c). food security immediately became a priority concern during lockdown (maclennan, 2020), with learners unable to access their school nutrition programmes, and household income impacted by rapidly rising unemployment and the loss of all opportunities for casual work. dr imtiaz sooliman of gift of the givers was quoted as saying that he witnessed ‘the worst amount of hunger in the 28 years of the gift of the givers’ when distributing food aid in the eastern cape in july 2020 (nortier, 2020). in particular, our work is focused in certain wards of the makana and ndlambe municipalities. the school situations of the learners and teachers that we work with are well known to the sancp team. the home circumstances of the learners, however, are less understood as we have worked, even with parents and guardians, predominantly through school-based events. in planning for distribution of resources into learners’ homes, it was important to gain as clear an understanding of the home circumstances as possible. in the context of lockdown, the most accessible source of this information was from statistical publications. the communities we focused this intervention on are those in wards that feed the schools and educational programmes that we work with as sancp. the information that was sought included first the size of the target population, number of households in these target areas and number of learners in this population. this information would allow us a sense of the size of the population and a glimpse into the home to see who the other members of each household were likely to be, as these would be the people that learners would interact with as they learned. the full set of this data is provided in table 1. the table was constructed using the most up-to-date statistics available per category (eastern cape socio economic consultative council, 2017; makana municipality, 2018; ndlambe municipality, 2018; statssa, 2012, 2018, 2020a). table 1: demographics of the target communities. broader community statistics the total population that we were targeting in this intervention consisted of 128 912 individuals, of which there were 38 735 learners (< 18 years) and 38 515 households. this area included 58 public primary schools and 2 independent primary schools, 33 of which sancp has had involvement with during the past 10 years. only 22% of these schools are fee-paying schools, and most of the 58 schools are served by a school nutrition programme. a further four of the remaining schools are also supported by this programme. the lockdown context therefore undermined the food security of a staggering number of households as this nutrition was not available. as a measure of the economic status of these communities, we sought information on the percentage of adults (aged 18–64 years) who were employed. this allowed a glimpse into the possible food security of the community and generally whether the possibility would exist that there was disposable income to purchase books, learning resources, data or digital devices. we also sought information on the level of poverty in each community. poverty was defined as the proportion of the population living on r800 (±us$60) or less per month. additional information that we sought included the types of dwellings in the community and whether these dwellings had access to electricity. formal dwellings were defined as those built according to approved plans (statssa, 2020a), while informal dwellings are shacks that have been erected on land that has not been planned for housing. electricity provision was necessary to consider as the success of any digital resource drive depends on the ability to charge the devices. the percentage of adults with employment ranged from 24% to 55% across wards. the likely implication of this is that within households it is more likely than not that there is a working-age adult dependent in addition to child dependents and elderly members of the household. household disposable income could not be expected to exist, let alone stretch to purchases of learning materials. this is also reflected in the proportion of households living in poverty in this community. it varies widely across wards, with ndlambe ward 10 being as low as 26%. for the remaining areas, however, this percentage ranged from 56% to 71%. this picture of a significantly economically strained community seemed to offer only constraints; however, the high rate of electricity provision opened the possibility that digital devices may be present in the home. most dwellings were also formal, although this varied across wards. formal housing would afford learners a more conducive environment for learning than informal homes like shacks. data on household composition household composition was also an aspect to consider. rather than being in a classroom with 40 other learners and a qualified teacher, the ecology of learning for each child during hard lockdown included the other members of their households. without being able to do a survey of the households that our learners were in, the covid-19 context allowed us only to search public documentary sources available online. we sought the most recent statistical reports, particularly relying on the general household survey (statssa, 2020a) for the eastern cape statistics. for the makana and ndlambe region in particular, we relied on the most recent municipal reports (makana municipality, 2018; ndlambe municipality, 2018). unfortunately, the latter two reports were not as recent as the statssa data, but they do provide a sense of the context that could reasonably inform the decisions we made. we sought statistics on household size, generational composition, and whether learners were living with their parents or not. the approximate literacy rate and educational level attained by the adults in the community was also important to research, as younger learners in particular would require the guidance of an adult in the home to facilitate their engagement in the learning tasks. the home language of the population was also relevant to inform the design of the resources. we also needed to ascertain the level of support required for the adults who would be engaging in the learning tasks with their learners – particularly for foundation phase learners. the average household size for the homes in our community was calculated as 4 (see table 1). the general household survey (statssa, 2020a) expands this information to show the proportions of households at various sizes in the eastern cape: 22.1% of households have only 1 person, 36.8% have 2–3 people, 24.7% have 4–5 people and 16.5% have 6 or more household members. the breakdown of household composition by generations is given as: 34.5% being double generation households (learners and their parents), 18.4% are triple generation households (grandparents are part of the household) and 8.5% are skip generation households (learners living with their grandparents) (statssa, 2020a). in addition, statistics are available about whether learners live with their parents. as many as 34.6% live with neither parent and 35.4% live with just one parent (only 3.2% live with just their father). what can be hesitantly inferred from these statistics is that many homes may have only one adult with a number of learners making up the remainder of the household. double generation households are most common (34.5%), and a high percentage of households have a single parent (35.4%). this holds both affordances and constraints. for learners living in homes with other learners, learning and play can occur between learners. where an older sibling is present it is possible that the older child could assist a younger learner with understanding and using the digital resources. the educational attainment of the adults in the household is relevant here. the more educated the adults present in the household, the more likely that they are able to help their learners to work with the digital resources. the presence of an older child, or a literate adult, available to interact with the child as they work with the digital resources would make it possible for the child to work in the zone of proximal development (vygotsky, 2012). the assistance of this more knowledgeable individual would mean that the child could learn and develop rather than just be practising content that they have already mastered. functional literacy is defined as an adult who has completed grade 7 (or higher) (eastern cape socio economic consultative council [ecsecc], 2017). across the makana and ndlambe communities this varies widely from as low as 23% to as high as 97%. the literacy rate is also lower for older adults, implying that grandparents in the home are less likely to be literate. in some areas, it seems likely that a literate adult is present in the home and can assist learners in navigating digital learning resources, but for many this may not be the case. a related statistic is that 65% of adults with learners in the home never read to them and 58% of adults live in homes with zero books (south african book development council, 2016). affordances and constraints presented by this new ‘ecology of learning’ in relation to the envisioned drive to get digital learning resources into homes, the general data on the targeted community and the households within it hint at the following affordances and constraints to this project. affordances: if a device capable of receiving digital resources is in the home, there is most likely electricity to keep it charged. most housing is formal, which is a setting that is more likely to be conducive to learning as compared to informal housing. the functional literacy of adults in the home is high in some areas, meaning that an adult could assist a younger learner in understanding the activities described in the digital resources. in hard lockdown, adults were unable to attend work, making them available to help facilitate their child’s learning. most households had at least one adult. constraints: the employment rate is low in most areas, and the poverty rate is high. the food security crisis in the eastern cape during lockdown (maclennan, 2020) would be impacting these households. food security concerns could understandably override concerns for continuing education in the early stages of lockdown. the functional literacy in a number of areas is low and so it is possible that some learners do not have a literate adult to assist them in understanding the resources and how to use them. the population we hoped to reach is large both geographically and in numbers. the digital platform chosen needed to allow efficient contact with a large (preferably limitless) group. results and discussion: digital affordances and constraints in order to make the best possible decision about which digital access modality to select, and which platform would be most suitable, our document review included mining information about household access to various technologies for each of the municipal wards in which sancp works directly. this varied widely between wards, and so an average was calculated across these wards to guide our thinking and decision-making. selecting an access modality: cell phones the ecdoe (2020a) issued an instruction: learners and their parents/care-givers are requested to access the teaching and learning resources being made available on different digital platforms to ensure learners keep up to date with their schoolwork. (p. 1) these platforms included live streams and radio and television broadcasts, as well as an online repository of resources on the dbe website (https://www.education.gov.za/covid19supportpackage.aspx). all of these require certain devices and technologies to be available in the home, and in the case of the online resources, a measure of digital literacy. the ecdoe also issued instructions to principals in march 2020 to send learning materials like reading books and textbooks home with learners, as it was already thought likely that remote learning may continue for longer than initially thought. in the eastern cape most schools have learners sharing such resources in the classrooms, and thus there are not sufficient materials for each learner to take books home (amnesty international, 2020). printed learning materials could therefore not be assumed to be available to our learners, supporting our conviction that intervention was needed. hoadley (2020) describes the provincial responses to the shift to remote learning as being largely web-based, and states that ‘initial efforts at providing television and radio offerings was piecemeal, uncoordinated, poorly publicized and for the lower grades especially, unconnected to the curriculum’ (p. 15). spaull and van der berg (2020) also point out that these broadcasts only targeted grades 10–12 and early childhood development, which meant that support for grades r–9 was solely web-based. as is evident in the data in table 2, an average of only 22% of households across the makana and ndlambe communities had access to a computer in the home, which is necessary for optimal use of web-based resources. the most accessible technology was cell phones, with 86% of households having access to at least one cell phone. in addition, the prevalence of smartphones rather than ordinary cell phones is rising steadily, although there would still be a minority of these 86% of households who would have a phone that will not access the internet (the broadcast research council of south africa, 2019). televisions and radio were present in most households; however, as noted by hoadley (2020), the quality and relevance of the resources provided through those channels was poor. creating content to broadcast such learning support is also beyond what sancp was able to consider doing at the time, and these activities take time. what we needed was quick access. table 2: access to internet and other modalities for learning support in target communities. the relatively high percentage of cell phone access afforded us the opportunity to reach the widest possible ‘audience’ if we were to select this as our access modality (cabello, claro, rojas, & trucco, 2020). a constraint presented by this approach was the relatively low percentage of dwellings with internet access. most people were accessing the internet from outside the home (statssa, 2020a), and these spaces were inaccessible during lockdown. however, of those households with access to the internet, an average of 49% did so via a cell phone. at the time of planning this resource drive the percentage of mobile connections as a proportion of the total south african population was 176% (103.5 million mobile connections) and 70% of these connections were broadband (kemp, 2020), which would allow a good, stable connection to use digital learning resources. this high number however is not evenly distributed across the population with many wealthier south africans and businesses having multiple connections. being one of the poorest provinces, the eastern cape would likely have a much lower proportion of these connections than most other provinces. this said, it is an encouraging sign that access to the internet may be slightly better than the general household survey (statssa, 2020a) indicates. only 37% of the population were social media users (kemp, 2020); however, with 62% of the south african population being internet users, there is the possibility for this percentage to grow significantly (kemp, 2020). selecting a digital platform: whatsapp vs facebook having decided to focus on cell phones as the dominant access modality, a choice had to be made about the particular platform to use. cell phones do provide several constraints regarding what technologies are possible to use. many state-of-the-art learning technologies exist if one has access to high-speed internet and a computer. video-conferencing platforms like zoom enabled some teachers to reach learners during school closures, learning management systems like moodle assisted schools to distribute lessons and resources, and websites with structured learning experiences, like khan academy, were popular. none of these was feasible for efficient use with cell phones. cabello et al. (2020) explain that effective use of cell phones in learning is constrained by their small screen size, relatively slow browsing capabilities and low storage capacity. in selecting a digital platform to use it was also necessary to consider affordances from other parts of the system. in makhanda, there were several efforts to support learners’ continued learning initiated by rhodes university, among others. there were also several stories of creative initiatives by local teachers to reach their learners. many local teachers almost immediately connected with their classes by creating whatsapp groups (batyi et al., 2020) and other local organisations created whatsapp groups for early childhood development practitioners and parents in order to share learning ideas and resources. these practitioners in turn created groups with the families they worked with. whatsapp is an instant messaging platform that does not require particularly sophisticated digital literacy – in south africa it is the most used application among people with both higher and lower digital literacy (goldstuck, 2020). it also uses comparatively little data to share images, videos and audio and is thus the most economically accessible of the available options, as reflected in it being the most popular messaging app across all income groups in south africa (goldstuck, 2020). in 2020, whatsapp had 10.1 million highly active users in south africa and it was the most used application among city, town and township populations, and across all age demographics and race groups (goldstuck, 2020). teachers across the country soon formed whatsapp support groups to share resources with one another (feldtman, 2020) but soon met the group capacity imposed by whatsapp of 256 members. these groups largely migrated to using the telegram app (feldtman, 2020), which allows a wider range of file types, of larger sizes, to be shared and the group size can be as large as 200 000 members. whatsapp groups initiated by our partnership with local teachers were small enough to avoid being constrained by the 256-member limit, but it did not meet our requirements. the hope was for our digital resource drive to reach as many of the approximately 40 000 learners (statssa, 2012) in the wards that sancp works in as possible in their homes, and a group size of 256 members would soon be insufficient. telegram is also a relatively new application that has not yet been widely taken up. it is currently used by just 1% of social media and instant messaging users in south africa (goldstuck, 2020) and it was also therefore unsuitable. facebook provided the most affordances for our purposes. its highly active user base is second only to whatsapp, and as a social networking application it has the highest network usage in south africa (goldstuck, 2020). it is also of relevance to our purposes that, despite the higher data usage of the facebook app, it remains second only to whatsapp in usage across all income groups in south africa. facebook has no limitation on the number of people that can ‘follow’ a page and allows posts to be shared to other pages. our existing facebook page already had 1951 followers who would immediately gain access to the resources. additional affordances included the possibility of sharing all resources with other community facebook groups. one facebook group for makhanda residents has 39 800 members, another three groups in ndlambe each have 2060 members, 3100 members and 4300 members. given the constraint that whatsapp presents in terms of group size, and the affordance offered by facebook to reach a far larger number of homes, facebook was our chosen primary platform for this resource distribution drive. the sancp community of teachers had also already started building smaller whatsapp groups for their classes, and the possibility existed to leverage this as an affordance to reach more homes. teachers are more likely to have access to facebook and the internet, and so the possibility existed that they could save the resources and distribute them to the homes of their learners through whatsapp. the possibility of this cross-platform use informed the design of the resources and our system of managing the resources we shared. summary of constraints and affordances and the related decisions the decision-making process about which access modality to use, and which digital platform to use, is summarised in figure 1. affordances and constraints are indicated by (+) and (–) symbols, and the decision is indicated in green. figure 1: digital decision-making process including constraints and affordances. in figure 1, we include the first decision that needed to be made: how to get numeracy resources into homes. the two options were to delay our response until it became possible to distribute resources that were already purchased, or to proceed with an immediate response, which would involve extensive content development. initially the decision was to pursue the former, but as it became evident that the two-week school closure would extend to 21 days, the appropriate decision was clear – digital distribution of resources. the sancp did have a bank of resources, but these needed adaptation for these purposes. the ‘ready-to-use’ (with minor adaptation) resources were also insufficient to see us through the 21 days, if resources were posted every day as we had hoped to be able to do. when those 21 days became 10 weeks, a significant demand for more content arose. in the following section, we present a post hoc reflective description of the design process, particularly describing how the first author strived to maintain attunement to the affordances and constraints already outlined here. we will also attempt through this discussion to highlight the design principles followed and will share the tools used to achieve all this. discussion: the process and principles of digital resource design in total, there were 12 existing sancp resources that could be shared through our facebook page as is, with only minor adjustments to make them work for facebook. the project began on 26 march with a post sharing the mathsup cell phone application designed for grade r learners. it is a cell phone app, designed according to the south african curriculum and optimised to use minimal data. this sort of post fit perfectly with the type of content we envisioned sharing. our aim with the digital resource drive was to share one numeracy resource per day of lockdown. it was only as we reached 03 april 2020 that we realised that far more resources were needed and creativity was required to make the resources appealing to both parents and learners. we compared and explored resource-sharing posts from previous years and analysed the nature of those that received the most attention and positive feedback. the number of people reached in the early posts shared in this digital drive remained relatively low during the first two weeks when sharing existing resources that had not been designed for this particular purpose. this caused a need for a pause to reflect on how to increase the reach of these resources. digital resources were not reaching sufficient homes, and the homes they were reaching were largely from metropolitan areas, and not the community we were targeting. to address these concerns, plans were made to create content specifically for the purposes of sharing digitally via facebook, with capabilities of being used across device types (cell phones, tablets, computers) and across different media platforms (including whatsapp). the hope was that these resources would be attractive, mathematically conceptually sound, curriculum aligned, and so forth. figure 2 shows parts of the resources provided. figure 2: front covers of a range of the resources developed and shared. de lange (2015) writes that it is close to impossible to write about the design process in task design as it requires meta-cognitive access to mental processes, in this case from the past. it was possible to clearly map out the data used from the document review to decide on the modality and platform in focus, as these were carefully stored in computer files and notes, and the documents are still easy to gain access to online. however, the process of attempting to illuminate the design process and decisions is more challenging. it requires deep reflection and, fortunately, reflective notes made during the design process and all previous drafts of the resources have been saved as stimuli for further reflection. an explicit choice was made to focus here on literature about mathematical task design, rather than literature about mathematical resource design, as the ‘child engaging in the task at home’ was foremost in the mind of the designer during the process. we do here refer to ‘resources’, but these ‘resources’ contained minimal instructive material (if at all). rather, they presented almost exclusively tasks for learners to engage in. resource here refers to a collection of mathematical tasks. de lange (2015) makes a distinction between slow design and fast design. slow design is the ideal process when designing mathematical tasks: it is based on developmental research and generates high-quality evidence of its effectiveness and is guided by principles that inform every step of the process. this type of design relies on freedom of choice, freedom of time, freedom to explore and freedom of thought (de lange, 2015). it is not often the case that these conditions coalesce to permit this type of design process. fast design is a feature of our highly commercialised society, where products are created under time pressure, without the necessary care that would produce high-quality mathematics resources – rather it is the aesthetic appeal that is foregrounded (de lange, 2015). without a doubt, it was fast design that was needed, but rather than being a thoughtless process, much detailed consideration was given to the design of the tasks. we will reflect later on the improvements we would make, as the speed of the process meant that some compromise was necessary, but there is a process that can be reflected on and presented here. one aspect of the challenge in documenting this process is the role of intuition as a form of rapid cognition (de lange, 2015). intuition would seem to be a cognitive process that defies explanation; however, de lange (2015) refers to strategic intuition as a type of intuition. he explains: [s]trategic intuition is not a vague feeling, like ordinary intuition. strategic intuition is a clear thought. and it’s not fast, like expert intuition. it’s slow. that flash of insight you had last night might solve a problem that’s been on your mind for a month. and it doesn’t happen in familiar situations, like a tennis match. strategic intuition works in new situations. that’s when you need it most. strategic intuition can be characterized by thinking, not feeling, informed by experience, leading to a deep insight that connects seemingly unrelated knowledge to create new insight and ‘gestalts’ in unfamiliar, new, situations. strategic intuition is, in my opinion, a crucial factor in the art of educational design in mathematics and science. the problem seems to be that for obtaining strategic intuition, the crucial factor is time, especially reflective time. (pp. 295–296) strategic intuition was certainly relied on in the design process. while the lockdown regulations caused an urgency to the work, the stalling of other aspects of normal daily work and life allowed the designer the mental space to become immersed in the process, and continually reflect – each day something learned from the previous day could be acted upon to inform the next resource and posting. this process was supported by reflecting on the engagement statistics available from facebook, or comments by users, and reflecting on difficulties with technologies used to create the resources. thus, even with the need for ‘fast design’, the lockdown circumstances allowed for immersion in the project and did permit, although in a concentrated period of time, several strengths of the slow design process. the broadest guiding principle in the creation of these resources was that the majority should address the ‘numbers, operations and relationships’ part of the mathematics curriculum as it has the greatest weighting of all the content areas of the curriculum (dbe, 2011). it is the number sense developed through these activities that other topics in mathematics rely upon. learners with an ‘absence of a coherent development of number sense’ (graven & venkat, 2013, p. 16) struggle with mathematics in later grades. the ‘strategic intuition’ mentioned above was not in the absence of a guiding mathematical learning theory. as noted above, the sociocultural perspective that guides all the sancp work guided the design, as did kilpatrick et al.’s (2001) five proficiency strands. the designer aimed to address all of the five strands across the tasks. the strands are interwoven, in the same way that kilpatrick et al. describe them, but certain types of tasks foregrounded one or two strands. resources comprising problem-solving tasks were particularly aimed at encouraging the development of strategic competence, adaptive reasoning and productive disposition. there were also resources for younger learners that comprised tasks that aimed to facilitate their development of conceptual understanding of number (such as those presented in figure 3). in addition, there were tasks included in many of the resources that required practice in working with a number of similarly structured problems to promote the development of procedural fluency. all the resources, and the tasks within them, were designed with the development of ‘productive disposition’ in mind. the resources were designed to be enjoyable and attractive, with the hope that learners would choose to engage with the tasks therein. figure 3: sample of an early number concept activity requiring minimal reading. the following sections provide a more structured reflection (by the first author as the designer of the resources), using barbosa and de oliveira’s (2013) five ‘arenas of conflict’, but in a similar way to sullivan et al. (2015)’s application of this framework. these sections will be written in the first person as they comprise the post hoc reflections of the first author. this ‘arenas of conflict’ framework is applied to structure these post hoc reflections as it is closely related to the categories of ‘requirements’ generated for the resources. sullivan et al. describe the arenas of conflict as dilemmas. context as dilemma this dilemma refers to decisions made about the context of the tasks themselves, and whether one opts to use reality, or semi-reality, or a pure mathematics task (sullivan et al., 2015). in order to grant the learners using the resources a more accessible, and relatable, point of entry to the tasks, the characters were always racially diverse learners when i opted to have human characters. i also made use of anthropomorphised objects (e.g. pencils), or animals, aliens, robots, dinosaurs, etc. these were chosen for their visual appeal and their cultural neutrality. part of the reason for this was that it is very difficult to find high-quality illustrations of diverse people, and so my preference was to use attractive illustrations available of other types of characters. the majority of the contexts used were semi-real in nature. i wanted to appeal to the playfulness of learners, and their rich imaginations, in order to entice them to engage with the resources. these semi-real contexts included imagining oneself to be a robot designer and using arrays in their design, or learning number ‘tricks’ to become a magician, or being immersed in the themed resources that used gardens, the seaside, dinosaurs and space travel as the context for the mathematical tasks or even helping superhero cats to catch stars. i did not want the resources to feel ‘school-like’ as far as possible. i wanted a child to choose to investigate what was happening in the resources. there were also resources that were mathematical without links to a context. the series of ‘number concept’ resources for young learners were some of those. in these resources, numbers were presented in a variety of representations and were used in matching games, or puzzles, or number concept mats. for the older learners, in intermediate phase, there was less use of contexts and many resources contained number puzzles for them to solve. there were four arithmetic puzzle books that were developed. language as a dilemma language is a difficult dilemma to navigate when designing resources at speed within south africa’s multilingual context. english is overwhelmingly the preferred language of learning and teaching (lolt) in south africa, despite it being the native language of less than 10% of the population (robertson & graven, 2015). in the makana and ndlambe contexts, schools teach in english (a minority choose afrikaans) from grade 4 onwards, and many primary schools opt to have foundation phase learners also learn in english, despite the official policy advocating for mother tongue instruction in the early years (dbe, 2010). the demographic data in table 1 shows that the vast majority of the community is isixhosa speaking, and english is the language of the smallest group. when one adds to this the relatively low percentage of functionally literate adults in this community it is clear that language is indeed a dilemma, and this before the dense and complex nature of mathematical language is considered. in the lockdown context it was not possible to contract someone to do translations of the materials from english into isixhosa and afrikaans; however, where this was possible it was done. the number concept resources for young learners, for example, were either available in all three languages or were designed to have minimal language. figure 3 shows an example of a resource where the language required is minimal, and figure 4 shows a translated version of a task from a resource in this same series. figure 4: samples of early number concept activities that have been translated. as far as possible, resources for the foundation phase had little to no text. for the intermediate phase, when learners could be expected to have at least a basic ability to read in english, more text was incorporated, although there were always activities within the resources that did not require understanding of the text in order to interpret the task. structure as a dilemma this dilemma refers to the degree of openness of the tasks, for example whether there are multiple solutions to a problem, or whether there are a variety of methods of finding the solution, or even whether the learner themselves is required to pose the problem. because one of the overarching desires in the creation of these resources was for learners to engage deeply with the mathematics, the majority of the tasks had no prescribed method for solving the problem. there were some numeracy games that had rules, like the board game included in the dinosaur-themed resource, and such activities were generally more prescriptive, but most resources, particularly for the intermediate phase, were puzzles that were non-routine and required learners to devise their own methods. the arithmetic puzzle books are examples of this. it was also a deliberate design decision not to include introduction of algorithms in any of the resources. all the activities were at their core exploratory and promoted playful learning and urged learners to really think and devise strategies. distribution as a dilemma distribution as a dilemma refers to how to select the content of the tasks and the cognitive demand of these tasks. as explained above, the focus was to be predominantly on ‘numbers, operations and relationships’, although there were four resources that addressed ‘space and shape’ for some variation and for some creative exploration. another aspect of distribution was the grades that would be catered for. the initial plan was for foundation phase and intermediate phase resources to be shared on alternating days; however, as the project progressed it was clear that the foundation phase resources were most needed as the engagement with those posts, and the downloads of the materials, was extremely high. the foundation phase resources were regularly reaching more than 10 000 facebook users. three even reached over 35 000 users. it became clear that prioritising this level would thus be beneficial. as a result, i created intermediate phase resources that would realistically take a few days to complete (e.g. the puzzle books and the mind-bender books with olympiad questions included) so that i could focus on releasing foundation phase resources more frequently. regarding the cognitive demand of the tasks, i included a variety of levels of tasks, so that each resource could reach a range of ability levels. there were activities that learners at lower grades in the phase, or lower levels of mathematical knowledge, could engage in, as well as more cognitively challenging tasks for learners who were ready to attempt them. it should be noted that each resource is a multi-task resource, and so a range of tasks was included each time. levels of interaction as a dilemma levels of interaction as a dilemma refers to considerations of who the participants would be. in this new learning ecology that learners had moved into, this became parents, grandparents, siblings and other adult household members. from the review of the demographics, i could not assume anything about the literacy of the adults in the home, and thus had to consider that the young learner may need to engage in the tasks without the mediation of a literate adult. i designed the resources with the hopeful assumption that other household members might also engage in the activities, which is the reason for the occasional inclusion of board games or interactive possibilities, like that of the magic tricks that a child could do with someone else. the resources and tasks were as language-free as possible, although this was not always possible and there are resources that required some reading, like the two ‘math and art’ resources. these were aimed at older intermediate phase learners, however, who would more likely have been able to engage with tasks without adult assistance. i could also make no assumptions about the mathematical knowledge of the other members of the household, particularly the knowledge of how to interact with a child around these activities. to this end, the aim of the tasks was clearly stated, and there were always explanations included in the facebook posts, and on the resources, as to what level the resource was appropriate for. in addition, some of the foundation phase resources also contained more detailed explanations to the parents about how and why an activity should help their child to learn. a related issue was that the tasks had to be designed to require no resources other than the digital image. without this consideration, no interaction would be possible. because most mobile users would not be able to print the resources out and do the activities on paper, most of the tasks were easy to sketch if necessary and complete with just a pencil and paper, for example a number pyramid, using the picture on the cell phone as a reference. it was difficult to eliminate the need for paper and a pen or pencil, and so it is needed for many of the resources. where this could be avoided, however, it was. it was critical that the activities not require any more resources than that, as none could be guaranteed to be in the home. for example, where dice were needed, learners were advised to write the numbers 1 to 6 on scraps of paper and then draw one out each time they had a turn. choice as a dilemma there is another dilemma that i would like to add: that of designing the resource and task in a way that invites the child to choose to engage in it. the new ecology of learning that learners were thrust into had a critical individual absent – the teacher. in a classroom, there is a teacher managing the learning process, and directing the learners to do certain activities, and monitoring whether this is done. it is common to hear teachers bemoan the fact that so few learners do their homework. the resources that were created for this digital drive had to make the child want to look at it and want to engage with it. in the absence of the structure that school provides, the child would largely need to choose to learn. the aesthetic design of the resources is very deliberate. they incorporate bright colours, attractive characters, a clean and neat layout, and high-quality images wherever possible. this seems a superficial concern, but if the child chooses to look away rather than to examine and explore what is happening on the screen, then the resource is worthless. considerable effort was put in to ensure that the resources looked as professional as possible. this required the acquisition of many new computer skills and much patience. the effort has been worthwhile as much of the praise from parents of the resources has been for their attractiveness. if this alone encourages them to download and use it, then it was well worth that attention to detail, even while it had mostly nothing to do with the mathematics itself. resource management as a dilemma (and some technical possibilities) the final dilemma that i needed to carefully think through and refine the strategy through much trial and error was regarding how to manage the vast number of resources that were accumulating. in this section i describe the technicalities of managing this process in order to provide an example of how to manage digital resources in such a way that there is a useful repository accessible, and so that the resource can be shared on a variety of platforms and devices. each resource that was shared was available to download through a link posted either in the body of the post, or as the first comment on the post. facebook sometimes will not allow sharing of a post if its algorithms identify a link as being to a third-party site. sharing the link in the comments was possible however. every downloadable document was a pdf, so that the formatting and design remained unchanged for people opening them on different systems. the link was only useful to individuals who had a computer to view the document on, however. in order to make it accessible to everyone, every page of the resource was saved as an image, by taking a screenshot, and it was shared as part of the facebook post as a picture. this meant that even mobile users could scroll through the pictures linked to the post. this also allowed a user to save the picture onto their cell phone, and share it on other applications, like whatsapp. a technical issue that needed resolving was how to share the link, and how to store the documents so that they remain accessible at that link. i initially saved each document on google drive, but despite setting the documents up as public documents, there was repeatedly feedback from users who could not access them. i reverted to using box.com (www.app.box.com) which allowed anyone with the link to reliably download the resource. it is completely free to use and is easy to navigate. the next challenge was to shorten the url that linked to the box account, as it was extremely long and did not work well on facebook. i used another free application, bitly (www.app.bitly.com) to create shorter urls. i created a (free) account and can log in and find all of the 122 links i created during this project, thus enabling me to reuse them. in addition, it keeps a tally of the number of downloads of each file, which is a useful analytic in evaluating the effectiveness of this digital resource drive. a last tool that i found useful was a pdf compressor. because of the number of images and objects in the documents, the pdf files were extremely large and would require a large amount of data to download. the free online tool at ilovepdf.com (https://www.ilovepdf.com/compress_pdf) was the fastest and most effective of those that i tried. the pdfs were then a more reasonable size to download. a final, crucial, aspect of resource management is copyright. as the sancp we publish every resource we develop under a creative commons attribution-noncommercial-noderivatives 4.0 international licence. this allows users to copy and distribute the resources, although not for financial gain. this requires great care to be taken in the design of the resources, however, as all content must be original, or correctly attributed to the source that inspired it where relevant, or explicit permission should be granted to use it. this presented a particular challenge to the design, as any pictures that were used needed to be similarly shared under a creative commons licence, or a licence purchased for their use. the website www.freepik.com was chosen for this project as it included free images of diverse learners, and when you download any free image there is a licence issued that specifies that the image can be used to create documents to be shared. there is a premium membership option, which i opted to use so that i could get access to the larger library, but the free images are perfectly adequate and are of a good quality. feedback the digital resource drive was extremely well received and reached a number of homes beyond what we had hoped for. the facebook page grew from just 1951 followers, to over 5000 followers by the end of the 10-week period, and we received overwhelmingly positive feedback. currently (june 2021) the page has 5470 followers. a word cloud was generated by copying all of the 313 comments on the facebook posts into the online utility at www.wordclouds.com. an image is generated by calculating the frequency of each word used to determine the size of the font. it is interesting to note the prominence of words referring to the aesthetics of the resources (cute, beautiful, colours), and it is encouraging to see the words ‘fun’, ‘helpful’, ‘sharing’ and of course ‘thanks’. we did have requests for resources in isixhosa and afrikaans. this was certainly a weakness in the project. during that phase of the lockdown it was not possible to organise extensive translations. this is being addressed, however, with translations of the foundation phase and intermediate phase resources currently underway. conclusion as of 29 april 2020, distribution of teaching and learning support materials was classified as an essential service (rsa, 2020d), which enabled more direct support to be provided to homes. this, combined with the permission granted to aftercare and enrichment centres to reopen (ecdoe, 2020b) and the naming of newspapers as an essential service (rsa, 2020e), meant that we became able to distribute resources to learners in their homes through the local newspaper, grocott’s mail, and through our partner schools and aftercare centres. we were able to reformat most of the resources shared online into newspaper spreads that were delivered to homes and schools. the design principles followed in creating the digital product also served this printed product well and they have been distributed into more than 9000 homes in the makana and ndlambe municipalities – our sancp communities. while this article presents a somewhat different type of research data, it aims to capture our response as mathematics educators to a very particular moment in education in south africa, when we needed to completely reimagine how we worked to support learners in their mathematics learning (and work at speed in this process). the article makes a novel contribution to informing possible ways forward in mathematics education as the pandemic continues and in a post-pandemic world. there are many advanced learning management systems and technologies that would have allowed for a smooth transition to remote learning, as we saw happen in many of the affluent independent schools in south africa, but this is not the reality for the majority of the south african population (pournara & bowie, 2020a). what this article offers is an approach to thinking through the provision of resources that is particularly sensitive to the realities facing impoverished communities. it is by no means an ideal solution and the framing of the post hoc reflection as a navigation of dilemmas captures that through making certain choices one must accept certain challenges to arise in relation to those choices. in particular, we note that more grappling with providing online learning resources to communities who need it most is required as many aspects of our response still exclude some households. this said, the article models a way of thinking through the pressing problem of how to educate learners when the primary site of learning is displaced into homes where food security is a prominent concern, and there are few resources for learning. the pandemic is by no means over, but once the covid-19 crisis is under control, we will still have to contend with the severe suffering of many south african communities, and the vast inequalities in our education system will have widened. what this time has forced upon us is the urgency of reimagining our role as educators and as organisations that support education. there has been much creativity and innovation evident and the hopeful message would be that perhaps the pandemic is the ‘unexpected catalyst’ (pournara & bowie, 2020b, p. 6) that forces us to learn from our collective work at supporting parents, carers and teachers to ensure the continuing mathematics education of their learners through the pandemic. acknowledgements competing interests the authors declare that they have no financial or personal relationships that may have inappropriately influenced them in writing this article. authors’ contributions p.v. contributed by writing the article, reviewing the literature, collecting data and analysing data. m.g. contributed to the conceptualisation of the article, discussions on the analysis of the data and revisions made in light of the reviewer comments. funding information this work is based on the research supported by the south african research chairs initiative of the department of science and technology and the national research foundation (grant no. 74658). data availability the data that support the findings of this study are available from the corresponding author, upon reasonable request. disclaimer the views and opinions expressed in this article are those of the authors and do not necessarily reflect the official policy or position of any affiliated agency of the authors. references amnesty international. 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(1998). communities of practice: learning, meaning, and identity. new york, ny: cambridge university press. footnote note: all of the resources referred to in this article are available at www.ru.ac.za and at https://web.facebook.com/rusanc abstract introduction powerful knowledge or lived experience: the debate bridging powerful knowledge and lived experience methodology findings: tasks and teaching discussion conclusion acknowledgements references appendix 1 footnotes about the author(s) karin brodie school of education, faculty of humanities, university of the witwatersrand, johannesburg, south africa deepa gopal school of education, faculty of humanities, university of the witwatersrand, johannesburg, south africa julian moodliar school of education, faculty of humanities, university of the witwatersrand, johannesburg, south africa takalani siala school of education, faculty of humanities, university of the witwatersrand, johannesburg, south africa citation brodie, k., gopal, d., moodliar, j., & siala, t. (2021). bridging powerful knowledge and lived experience: challenges in teaching mathematics through covid-19. pythagoras, 42(1), a593. https://doi.org/10.4102/pythagoras.v42i1.593 note: teaching and learning mathematics during covid-19. original research bridging powerful knowledge and lived experience: challenges in teaching mathematics through covid-19 karin brodie, deepa gopal, julian moodliar, takalani siala received: 14 dec. 2020; accepted: 10 june 2021; published: 13 aug. 2021 copyright: © 2021. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract the coronavirus disease 2019 (covid-19) pandemic supported an investigation of ongoing challenges as to whether and how to make mathematics relevant to learners’ lifeworlds. given that covid-19 created major disruptions in all learners’ lives, we developed and taught tasks that attempted to make links between their experiences of the pandemic and disciplinary mathematical knowledge. we located our investigation in current debates about the extent to which disciplinary knowledge can be linked to learners’ out-of-school experiences. we developed and analysed two tasks about covid-19 that could support link-making and productive disciplinary engagement, and analysed one grade 10 teacher teaching these tasks. we found that linking mathematics to learners’ lifeworlds is both possible and extremely difficult in relation to task design and how the teacher mediates the tasks. in relation to task design, we argue that teachers cannot do it alone; they need to be supported by the curriculum and textbooks. in relation to mediation, we saw that teacher practices are difficult to shift, even in the best of circumstances. we articulate the complexities and nuances involved in bridging powerful knowledge and lived experience and thus contribute to debates on how to teach powerful knowledge in relation to learners’ lifeworlds. keywords: mathematics curriculum; pedagogy; relevance; powerful knowledge; learner engagement. introduction the coronavirus disease 2019 (covid-19) pandemic has created major disruptions in the lives of many south africans. a study by spaull et al. (2020) shows that three million south africans lost their jobs between february and april 2020, two million of whom were black women. the pandemic and the accompanying lockdowns have caused great hardship in south africa (sa) and around the world and exacerbated already-existing social and economic inequalities. spaull and kotze (2015) show that in poorly resourced schools, learners lag three to four grades behind their peers in better-off schools. the shutdown and concurrent loss of time in schools may have disadvantaged learners even further, particularly those who did not have access to online learning. jansen (2020) notes the unequal experiences of learners through the lockdown period and hoadley (2020) shows how much time was missed by learners in government schools, given school closures and rotational timetables when schools were open. in dealing with the coronavirus, scientists and governments have been publishing a range of information: how the virus infects people and causes damage to our bodies, how the virus spreads and how the spread can be prevented, and statistics on the demographics of those who are becoming infected, ill and dying and how these differ across countries, regions and socio-economic circumstances. much of the information has been communicated on social media and so there is also much misinformation, with severe consequences for the spread of the disease and the health of communities (barua, barua, aktar, kabir, & li, 2020). we are south african high school mathematics teachers, who studied together on a master’s programme at wits university. we have written this article together with our professor on one of our courses: studies in pedagogy. this course looks at why the school system fails the majority of poor, black learners in sa and internationally, asking why learners from marginalised communities do not achieve as well as those from better-off communities (spaull & kotze, 2015) and why many learners in all communities experience their mathematics learning as irrelevant to their lives and as boring and unenjoyable (boaler & greeno, 2000; motala, dieltiens, & sayed, 2012). the intersection of the pandemic and our course gave us an opportunity to rethink how and why we teach mathematics, particularly in relation to its relevance to learners’ lives. much has been written on the relevance of mathematics and there are contradictions and tensions relating to incorporating everyday knowledge into the curriculum. the decolonisation movement (howard & kern, 2019; le grange, 2019) argues that mathematics arises out of people’s lived experiences and can and should be taught in relation to these, particularly, but not only, to learners’ cultural traditions and social circumstances. others argue that mathematics as a discipline is somewhat removed from everyday experience and functions with its own internal rules and thus cannot be easily related to everyday lives and knowledge. our course investigated the connections and tensions between teaching disciplinary mathematical knowledge and how we as teachers might take account of who our learners are, what their lived experiences are and how we might develop rich and just pedagogies that value everyone and teach powerful knowledge (zipin, fataar, & brennan, 2015). the covid-19 pandemic gave us a unique opportunity to think about whether we could use information about the pandemic to teach mathematics and to use mathematics to develop more sound knowledge about interpreting covid-19 information among our learners. we were challenged to use the theoretical resources of the course to develop and analyse tasks that could be relevant and teach mathematics at the same time. while many others have written about making mathematics relevant, we wondered whether, given the scientific and mathematical nature of information about the pandemic, we could find new ways of connecting mathematics and learners’ lived experiences during the covid-19 pandemic. this article focuses on the following research questions: can we use mathematics to support learners’ understandings of covid-19 and, if so, how? can we use information about covid-19 to teach mathematics and, if so, how? can we use mathematics and covid-19 to develop powerful and relevant mathematics for our learners? what challenges are created in trying to link mathematics and covid-19? we begin by framing the questions in terms of the current debates. powerful knowledge or lived experience: the debate social realists (for example, young & muller, 2013) and critical sociologists (for example, zipin et al., 2015) disagree on whether powerful knowledge and everyday knowledge can be taught without diminishing either. for social realists, powerful knowledge refers to specialised knowledge, built systematically over time in a discipline such as mathematics, with agreed upon methods or criteria for determining ‘epistemic bestness’ and validity (young & muller, 2013, p. 236). powerful knowledge allows people to generalise beyond their particular experiences (young & muller, 2013) and to envisage the ‘not yet thought’ (bernstein, 2000, p. 30). the concepts of powerful knowledge are systematically related to each other and support more systematic and deeper understandings of the world, that is, powerful knowledge explains and possibly predicts what happens or might happen in reality (young & muller, 2013). powerful knowledge is dialogic in the sense that multiple voices and interpretations of scholars in a discipline contribute to this knowledge through constructing and critiquing claims and evidence (ford & forman, 2006). powerful knowledge is distinguished from ‘knowledge of the powerful’ in that the latter gains its power from those who develop and use it, while the former gains its power from what it can achieve and how it is organised. powerful knowledge is considered to be structurally distinct from context-dependent, everyday knowledge acquired from everyday discourses given the different contexts of their development and use (charlot, 2009). young (2011) argues that the primary function of schooling is to enable learners to acquire powerful knowledge because of the potential power this knowledge provides in empowering us to envisage and interpret the world beyond our everyday understandings. muller and hoadley (2019) argue that schools are the main institution where marginalised learners acquire specialised, context-independent academic knowledge and that access to powerful knowledge is a social justice issue since all learners should have epistemological access to powerful knowledge, and many, particularly from marginalised communities, can only get it at school. charlot (2009, p. 92) elaborates that to acquire powerful knowledge during schooling, learners need to move from perceiving the world as a ‘place of experience’ towards perceiving it as ‘an object of thought’, and from seeing themselves as an ‘empirical self’ and allow for the emergence of the ‘epistemic self’. in an important study, hoadley (2007) shows how working-class learners are disadvantaged when teachers are unsuccessful in linking powerful mathematics knowledge with everyday knowledge. comparing mathematics lessons in working-class and middle-class schools in cape town, hoadley notes that teachers in the working-class contexts employed mainly ‘localising strategies’ which left learners grounded in rudimentary, concrete ways of counting without moving to ‘specialising strategies’ requiring generality. conversely, middle-class learners were granted access to developing specialised knowledge of mathematics through ‘specialising strategies’ and these learners had ‘a base from which more advanced conceptual work could be done’ (hoadley, 2007, p. 693). hence, the working-class learners were left with a thin and denuded version of mathematics through valorising everyday knowledge in relation to disciplinary knowledge. this is perilous for learners from working-class contexts who are potentially denied mathematical futures since they are not granted access ‘to move, intellectually at least, beyond their local and particular circumstances’ (young, 2011, p. 15). at the same time, we know that conventional mathematics education has not supported most learners, particularly poor, black learners, to be successful in mathematics (fataar, 2012; spaull & kotze, 2015). while zipin et al. (2015) agree that powerful knowledge is an important endpoint of schooling, they argue that it is not the only endpoint and that strong distinctions between powerful knowledge and learners’ lifeworlds are counterproductive, particularly for marginalised learners (see also civil & hunter, 2015; ladson-billings, 1995). they disagree that boundary crossing between powerful and everyday knowledge is not possible, that the two knowledge forms are structurally distinct and that incorporating the everyday aspects of learners’ lives will necessarily dilute the powerful knowledge. while powerful knowledge may be defined by its structures and functions, in fact it intersects strongly with knowledge of the powerful, and powerful people continue to control powerful knowledge (zipin et al., 2015). while giving access to powerful knowledge can be seen as an element of social justice, making hard distinctions between powerful and everyday knowledge in the curriculum continues to serve the interests of the already powerful, whose cultural capital aligns with the ‘school code’ (ladson-billings, 1995; zipin et al., 2015). learners from working-class backgrounds continue to be intellectually excluded, alienated and disengaged, finding instead that their community knowledge is more useful and meaningful to them (charlot, 2009; fataar, 2012). middle-class learners are more successful at acquiring the ‘school code’ compared to working-class learners since they are enculturated into similar discourses at home (charlot, 2009; hasan, 2002). while we have presented two apparently opposing positions in the debate, we note that there is some overlap and possibilities for reconciliation, relating to how connections may be made between powerful knowledge and lived experience. in particular, young (2013) and muller and hoadley (2019) argue that while it is important for the curriculum to distinguish clearly between powerful and everyday knowledge, with powerful knowledge as a key outcome, there is space in pedagogy for teachers to make links with learners’ lived experience as long as this is done in the service of powerful knowledge. they make a strong distinction between curriculum and pedagogy. fataar (2012) notes that ‘thin’ pedagogies may produce a denuded curriculum, achieving neither powerful knowledge nor relationships with lived experience for the learners. together with fataar, we do not see such a strong distinction between curriculum and pedagogy, noting that if links are to be made, then teachers will need guidance on how to do this and a curriculum that supports it (lampen & brodie, 2020). this debate resonated with us in a number of ways. we appreciate the idea of mathematics as powerful knowledge – this is what draws us to mathematics. and yet we know that our learners tend not to see this power in mathematics, often because we do not show them how it supports stronger and deeper interpretations of the world, nor how it intersects with their lifeworlds. this may be because of the instrumental and utilitarian nature of our current schooling system, which is examination-driven and employs ‘a “thin pedagogies” approach to curriculum implementation’ (fataar, 2012, p. 62). teachers work mainly to ensure that learners pass examinations, thus emphasising the procedures of mathematics at the expense of its full power. when we try to make mathematics more relevant in class, we often find ourselves denuding the power of mathematics, as hoadley (2007) shows. we have not found many ways to show our learners what we love and find useful in mathematics. we wondered whether covid-19 would give us a new opportunity to bridge the power of mathematics with the meaning of learners’ lived experiences. bridging powerful knowledge and lived experience perhaps the best reconciliation of the two sides of the debate comes from delpit (1988). she argues that we have to teach learners both the codes of power, and respect for their own community knowledge. she argues: ‘if you are not already a participant in the culture of power, being told explicitly the rules of that culture makes acquiring power easier’ (delpit 1988, p. 282), recognising that redistributing powerful knowledge for all learners is not enough as codes for academic success will still remain implicit for working-class learners (zipin, 2015). however, we note delpit’s (1988, p. 286) concern that in redistributing cultural codes needed for academic success, learners cannot lose their cultural identities, as this would be a form of ‘cultural genocide’. this resonates with fataar’s (2012, p. 56) concern about the dangers of ‘cultural assimilation’ evident in sa’s post-apartheid schooling system. with ‘an explicit pedagogies approach’, fataar (2012, p. 58) proposes that the teacher act as a mediator in making visible the cultural codes needed for academic success while recognising that teachers need to connect the rich cultural knowledge of learners with disciplinary knowledge through their pedagogical approaches in ways that do not diminish either knowledge form. two possibilities of such connections are given by hedegaard’s (1990) ‘double move’ and engle and conant’s (2002) notion of ‘productive disciplinary engagement’. the double move refers to moving from theoretical to empirical knowledge and back again. theoretical, disciplinary knowledge is first made available to learners through important theoretical questions in the field, which learners explore with examples (in hedegaard’s case they work in life sciences). the examples help to develop the theory further, which learners can then use to see more in the examples, thus creating cycles of theoretical and empirical exploration. we wanted to see whether we could build the double move into mathematics tasks. engle and conant (2002) argue that in order to bring learners into contact with disciplinary knowledge, learners need opportunities to problematise the content, to develop authority in relation to the knowledge, to be accountable and hold others accountable to how they make the claims that they do, and they need sources and resources to support their engagement. ford and forman (2006) resonate with engle and conant’s authority and accountability when they talk about allowing learners to take on the roles of constructors and critiquers of knowledge. we were intrigued as to how these roles could be embedded in mathematics tasks. the concepts of productive disciplinary engagement (engle & conant, 2002; ford & forman, 2006) as well as the double move (hedegaard, 1990) typically require substantial class discussion, in groups or in the whole class, as well as a range of textual resources. as brodie (2007, p. 17) argues, classroom conversations ‘allow learners and the teacher to consider, question and add to each other’s thinking and to co-produce generative mathematical ideas and connections’. brodie also argues that developing conversations where there is genuine communication that supports learner engagement and thinking is difficult to achieve, particularly when discussions may take different directions from those expected by the teacher. teachers often have to make in-the-moment decisions about whether and how to follow up different learner contributions, while maintaining a focus on the key concepts they want to teach. more generally, we can take opportunities in teaching about covid-19 to ask moral and ethical questions that are not usually addressed in mathematics classrooms but can be important. for example, how do we know which sources of information can be trusted? are all sources equally valuable or powerful, especially in relation to social media? this comes back to delpit’s (1988) point about relationships between powerful knowledge and lived experience, connecting what learners experience in their everyday lives with the codes of power. we can also create opportunities for learners to experience and explore empathy, self-reflection, critical thinking and personal growth. it is important for us as teachers to teach with sensitivity and care (brodie, 2017), knowing that our learners’ experiences of the covid-19 pandemic may have been very different from each other’s and our own, and that a number of them might have experienced disruptions and even death among their families. methodology our contexts table 1 shows the variation in contexts among us, with one of us (teacher2) teaching in in a public, no-fee government school while teacher1 and teacher3 teach in private schools. table 1: our teaching contexts. our teaching contexts supported different experiences of teaching during the covid-19 lockdown. teacher2 could only rely on whatsapp to communicate with her learners, teacher1 made use of whatsapp and microsoft teams, while teacher3 had some advanced technology such as a digital pen which was used when presenting to learners in real time on google meet. teacher1’s and teacher3’s learners connected with them every day. fewer than a quarter of teacher2’s learners connected regularly, because they did not have access to devices and, if they did, could not afford data (see also jansen, 2020) and because of unstable network connectivity and regular power disruptions. it is extremely difficult to teach mathematics on whatsapp, and we felt quite despondent at the school time our learners were missing and how already existing inequalities might be exacerbated between schools. when schools re-opened, depending on the schools’ management and infrastructure, teaching and learning was very different from the past (see also hoadley, 2020). some schools opted for weekly or daily rotations for grades 8 to 11 in order to comply with standard operating procedures that are covid-19 compliant. this greatly impacted contact times with learners. some schools combined face-to-face with online learning, making teaching more demanding for teachers new to online teaching. grade 12 learners would write their final, national, high-stakes examinations on the same curriculum as in previous years, and we needed to catch up a lot of work, because of lost time during lockdown. we also needed to deal with learners’, and our, anxieties about being back at school in the midst of a pandemic, work with them on maintaining safety procedures regularly and systematically at school and at home and try to focus them on their studying during a difficult time. our own learning was disrupted after five face-to-face sessions towards the end of march 2020. fortunately, during lockdown we received free data from the university and connected on microsoft teams for our class discussions, although these were more difficult to learn from than our in-class sessions. this double disruption to our working and studying lives, together with the anxiety about a rampant disease, brought challenges and opportunities. one opportunity was to think about how we might teach about covid-19 to our learners, either online or once we got back to school. another was to think about how covid-19 might be an opportunity to bring some relevance to a mathematics curriculum dominated by rules and algorithms (boaler, 2016). data collection and analysis in this article, we analyse some ideas for mathematics tasks for learners related to covid-19. this was the central thread of the second part of our course (eight sessions) and each week we worked on different ideas in relation to covid-19 related tasks. our first set of data is therefore the tasks that we developed and analysed. we developed the tasks by looking for information in the press and on social media, and then asking questions about the information to create tasks that learners would think about, research and discuss, alone, in groups or in whole class discussions. we analysed the tasks using the analytic framework shown in table 2, developed from our theoretical framework discussed above. the process was iterative: as we analysed, we changed some aspects of the tasks, and then re-analysed. not all the analytic questions are applicable to all of the tasks but as a whole the tasks should engage learners as described in our discussion. table 2: analytic frame. towards the end of the year, we decided to try some of the tasks in one of our grade 10 classrooms. teacher3’s classroom had the best conditions and we were able to videotape the lessons. while we cannot generalise from such a small class to any other classrooms, we are able to draw out important lessons for other teachers and researchers. we used the same analytic framework to analyse the in-class enactments of the tasks (table 2), looking in particular for how what happened in class differed from the task intentions. the first author did an initial analysis of the lessons in relation to the analytic framework. after this, the other authors commented on the analysis and added some important points, while teacher3, who was the class teacher, responded to questions from the analysts and added some additional points. teacher3 had also written reflections on the lessons immediately after them and transcribed the two lessons. while we developed a set of four tasks, only two were used in the classroom, so we only discuss those two here. ethical considerations permission was granted from the research ethics committee of the university (ethics clearance number: h20/11/08). informed consent was obtained from teacher3, learners and their guardians. names of all learners have been kept anonymous to preserve the confidentiality of participants. the teacher numbers do not correspond to the order of authors, again to preserve anonymity. findings: tasks and teaching tasks we present the analysis of two tasks that can support learners to use mathematics to understand elements of the covid-19 infections in sa and also to develop mathematical knowledge based on thinking about the outbreak (see appendix). the tasks were developed from information on the covid-19 south african online portal (2020) and can be adapted for different grades and used in different ways by teachers. the images in the tasks were found in july and september 2020 and suggest ways to bring together ideas about the coronavirus and its impact, and mathematics. but we note that as things change, we would want to develop similar activities around more current data about the virus. our task questions are based on information that is presented in the images. disciplinary engagement with school content means that learners are engaged in activities that draw upon actual practices of a discipline such as constructing and critiquing knowledge (engle & conant, 2002; ford & forman, 2006). we devised questions for learners to draw on the information presented in the images about covid-19, and their knowledge, experiences and feelings about the pandemic. we hoped that the tasks and our methods of teaching would help learners’ development of mathematical concepts and we devised both mathematical and non-mathematical questions that would support the learners to problematise or deepen their ideas about covid-19. image 1 illustrates the number of deaths in different countries around the world and this gives learners a perspective on the pandemic from both a local and an international standpoint. some learners may have been directly affected by the statistics as they could have had deaths in their families due to covid-19. the numbers are given as absolute deaths rather than death rates, and we hoped that learners might raise the issue of death rates and the difficulties of comparing numbers of deaths without knowing the population of each country (which could be looked up). much of the information in the press about deaths from coivd-19 is given as absolute numbers, so this is an important question to bring up in class. question 1 can initiate a discussion about deaths from the pandemic across the world. the discussion should be relatively open at first allowing learners to explore different questions that they might ask and identify what mathematical or non-mathematical resources (engle & conant, 2002) they might use to answer their own questions. the learners’ questions can support the problematisation of everyday knowledge, for example the difference between number of deaths and death rates, and set up possibilities for using mathematical knowledge to answer these (q1.2)1. how the discussion proceeds will depend on the teacher (scott, mortimer, & aguiar, 2006) and how they make links to mathematical concepts that they want to teach (q1.1, q2.1). question 2 might help with the discussion in question 1, because it supports the learners to be more specific about what they see in the graph but, if not, it can be asked separately as an important mathematical question, supporting sense-making of the graph (q1.1, 2.1). the bar graph is presented horizontally rather than vertically, which is unusual since learners usually encounter bar graphs with vertical bars, and this might lead to some discussion (q2.2). question 3 asks an important mathematical question about percentage increase and allows learners to relate their previous disciplinary knowledge to the covid-19 numbers and to answer important questions about the numbers (q1.1). knowledge about covid-19 might also help learners to make sense of percentage increase (q2.1). question 4 asks learners which measure of central tendency is most appropriate for this data. this question should raise a number of issues relating to measures of central tendency, particularly whether and when they are appropriate in relation to a particular data set. such discussions may help to problematise the content and can serve as a catalyst for initiating a rich mathematical conversation, particularly if it is related to discussions around questions 1–3 about what the data represents (q1.2), q2.2). therefore, learners are given some authority to make judgements about the data, and are held accountable to the discipline to justify their viewpoints (q1.3, q2.3). question 5 asks learners to comment on the skewness of the data, which presents yet another opportunity to ask questions about what is represented and they are required to link their disciplinary knowledge with the covid-19 resource and take authority and accountability for their responses (q1.1, 1.2, 1.3, 2.3). the task as a whole, particularly question 4 and question 5 suggest the possibility of a double move: moving from everyday knowledge to mathematics and back again or cycling through everyday and mathematical knowledge (q3.1, 3.2). image 2 is an infographic that conveys information about covid-19 statistics in the different provinces of sa, published every day in sa. learners may have seen this in their everyday lives but not necessarily in their mathematics classrooms. this image may support learners to see the importance of mathematics in their everyday lives and may change their perception of the importance and incorporation of mathematics with lifeworld knowledge. there is some information missing, for example the numbers of tests conducted per province and the population size per province (similar to task 1). it was hoped that if learners needed these to complete the task, they would ask about them and, in the latter case, could find out what these numbers are. question 1 requires learners to think about why a province has recorded the lowest percentage of confirmed cases based on calculations they performed (q1.1, 1.2). it could raise a number of questions such as: what is a high death rate in a province, how do the death rates across provinces compare to the national death rate, and which province has been most affected by covid-19 deaths? these mathematical questions could then lead to non-mathematical questions such as: why do some provinces have higher death rates than others and might this be related to poverty or healthcare systems in the province. question 2 supports learners to talk about their perceptions about being back at school and asks learners to use statistics to support their reasoning (q1.1, q2.1). here learners would need to refer to the provincial statistics as well as the national statistics. we hoped that this question would bring up the limitations of statistics in understanding our current circumstances. question 2 also allows social issues that affected learners to be foregrounded and may allow learners to engage with the emotional and psychological effects that covid-19 has had upon their lives (q4.5). teachers will get further insight into learners’ anxieties associated with being back at school in the time of a pandemic. teachers will need to attend to basic human values of care and compassion (brodie, 2017) when listening to learners’ experiences of schooling during lockdown as there may be learners who experienced deaths in their families or learners whose family members lost jobs. this question also allows the teacher to engage with both the academic and the human project (jansen, 2016) such that teachers come to see their learners as whole people who have their own unique perceptions and experiences of schooling under lockdown. brodie (2017, p. 175) notes that attention to ‘the human and academic project are equally important in schools and in mathematics pedagogy if we want to transform our school into spaces where all can and do learn mathematics’. question 3 is not usually seen in mathematics classrooms. it asks learners to be curious and gives the learners agency to suggest what they find relevant and interesting in the image (q2.3). given question 1 in task 1, we hoped that learners would ask both mathematical and non-mathematical questions, although we expected they might need prompting to ask mathematical questions. they are encouraged to pursue or ask questions that they find interesting as a result of this image and are accountable to the teacher and fellow learners for justifying why they chose to pursue these questions. curiosity is an important attribute in a 21st century world and this attribute is typically overlooked in mathematics classrooms where learners often surrender their human agency to the agency of the discipline (boaler & greeno, 2000). question 4 and question 5 require learners to make judgements about political decisions that were taken on the basis of mathematics and the nature of the information given to the public. with question 4, statistics is used to problematise knowledge and make learners think critically about the efficacy of lockdown (q1.2). question 5 asks learners to be critical and justify whether the image is a reliable source of information (q1.2). for these questions, it will be important for the teachers to make sure that ideas such as accountability based on knowledge are made available for discussion in the classroom. as with task 1, we hoped that working through all five questions would allow the teacher to make links between covid-19 and mathematics (q3.2). we note here that research question 4 and analytic questions 4.1–4.5 relate more closely to what actually happened in the classroom and we will address them in the next section. teaching our main finding was that the learners found it difficult to link mathematical concepts with covid-19 in a sustained way through the lessons. teacher3 reflected: ‘learners tended to operate at one end of the spectrum in many instances whereby either disciplinary knowledge was favoured and everyday knowledge was made subordinate or everyday knowledge was favoured making disciplinary knowledge subordinate. … developing better links will require more skilful mediation from my part.’ our analysis shows much better linking between mathematics and everyday knowledge than seen in hoadley (2007) because the tasks and the teacher prime the learners to think mathematically; however, we do see instances where the mathematics gets lost in the everyday. we note that teacher3 is more qualified than many others, having nearly completed a master’s in education at the time of this study and was able to make important pedagogical moves such as: giving learners time to think and respond, validating learners’ responses, helping learners to articulate their ideas, and supporting learners to take authority and accountability for their thinking. however, the teacher still experienced challenges, noting that these tasks were ‘different from typical mathematics lessons whereby i present a formula and learners practise how to perform a calculation’. so even a well-qualified teacher, with expertise in the current curriculum and pedagogy, struggled to shift to making links between learners’ lifeworlds and the discipline. noting these difficulties, we also noted that the tasks and the teacher’s pedagogies did succeed in making some links between mathematics and the everyday, presenting more nuanced findings than hoadley (2007). it is in these nuances that we believe the seeds for working on more relevant curricula might lie. the teacher also expressed some dilemmas and uncertainties, which show why teaching in this way is a difficult undertaking. in order to convey the depth of what happened in the lessons, we present the tasks in the order they were taught with a narrative analysis backed up by quotes from the lessons and the teacher. task 1 for question 1 learners began by suggesting non-mathematical questions such as: how were the countries in the graph chosen? why is the sa death rate2 lower than the others when we are a poorer country? why is the united states (usa) death rate the highest? some answers were suggested: wearing or not wearing masks, and the different lockdowns in sa and the usa, one more systematic and stricter while the other was more variable. one learner made a sarcastic comment about donald trump. the teacher reworded the comment, suggesting to the student that he express his ideas in a ‘more gentle’ and ‘less-biased’ way, taking the opportunity to teach a way of working in mathematics (and other) classrooms. when the teacher asked why the questions were non-mathematical, a learner answered, ‘because you don’t have to do calculations’, suggesting a particular view of mathematics as calculations. among the mathematical questions were: why are iran and peru so close to each other given that one is war-torn and the other not (which could have been a non-mathematical question)? calculate the difference between sa and usa, to see the difference between the two extremes. calculate the difference between brazil and the usa. find the average. find the median. it is interesting to note here that the mathematical questions are expressed as instructions rather than questions and ask for a calculation, supporting the view of mathematics as calculation and suggesting a view of mathematics as instructions to do something, rather than questions to be thought about. this question worked to orient the learners to the graph and what they might see but did not immediately support any links between covid-19 and mathematics. we note that this is entirely appropriate: teaching is a long-term endeavour and not every goal is achieved with every task. we saw learners slip between actual deaths and death rate and this error was not addressed by the teacher, even though the error was expected in the task design. learners also suggested that we might find the average and median. again, the teacher could have, but did not, raise the issue about what the mean and median would show, in relation to the actual deaths and the death rate. for question 2, a learner gave an appropriate answer, noting the vertical as countries and the horizontal as number of deaths. the teacher acknowledged the answer, but did not focus on the mismatch between number of deaths and death rate, asking whether the graph seemed ‘strange’. the learners nodded and showed how it ‘should’ be oriented (with the bars vertical rather than horizontal). unfortunately, the teacher missed the opportunity to talk about how graphs appear in different forms because another learner made a comment, although the learners may have grasped this point. for question 3, the teacher wrote responses on the board and we rewrite them here: learner1: 4172/15447 = 27% learner2: 4172/15447 = 27%, 100 – 27% = 73% learner3: 4172/100 = 41.72; 15447/100 = 154.47; 154.47/41.72 = 3.7% learner4: 15447 – 4172 = 11275. 11275/15447 × 100 = 73% learner5: 15447 – 4172 = 11275. 11275/4172 = 270% learner5’s solution is correct and noting this could have led to interesting discussions about what a percentage increase higher than 100% means, and why learner4’s solution is partially correct but has a common error. also interesting is how learner2’s and learner4’s solution have the same answer with different methods and are both incorrect. the teacher decided not to discuss the solutions here, saying that the learners should go home, think about the problem and they would come back to it. the teacher reflected that: ‘three responses did not make sense to me in the moment. subsequently, i decided to probe learners who provided incorrect responses and their reasoning still appeared opaque to me. i decided to not provide finality to this question and postponed providing a solution to this question until the next day.’ when the teacher probed learner1, the learner immediately assumed that he was wrong, a typical response when learners first experience such probing. the teacher’s response is also typical for teachers who begin to allow different mathematical ideas into the classroom and become interested in methods and not only answers, particularly not only right answers. it suggests that for such strategies to work, teachers need to build ways to anticipate correct and incorrect answers and possible methods that learners might use, with different degrees of correctness. interesting in this regard are the learners’ responses to the teacher. they asked whether any of the solutions were correct, leading to the following exchange: learner2: is there a correct answer? teacher: i’m not going to answer … learner4: please … learner5: wait is there an answer that’s correct. we don’t need to know who. we just need to know someone got it. teacher: i want you guys to think. … guys, quite a few of you got different answers. i want you to compare with each other and how did different people reason about this. you all got different ideas here. on wednesday, when i see you, we’re going to discuss this question again in more detail. learner5: will you tell us the correct answer? teacher: yes, on wednesday. i want you guys to think. the exchange suggests the learners’ need for correct answers rather than to understand why particular solutions are correct. this comes from years of mathematics pedagogies where correct answers are seen to be more important than other aspects of learning mathematics such as understanding why procedures work and justifying and communicating mathematical ideas. if such pedagogies are to change, they need to support and be supported by expanded views of what mathematics is (boaler, 2016), particularly if curricula are to make links between learners’ lifeworlds and mathematics. the exchange also shows that the teacher is beginning to push learners to take authority for their ideas and to be accountable for them, taking an initial step in this direction. in summary so far, learners have engaged with the graph in mathematical and non-mathematical ways. there have not been explicit links between the mathematical and the non-mathematical but the potential for links can be seen. linking was derailed by traditional views of what counts as mathematics and the teacher’s surprise at students’ incorrect mathematical answers in question 1 and question 3. question 4 and question 5 pose mathematical questions that support stronger interpretations of the graph. the learners answered the questions mathematically and justified their answers. they eliminated mode as a measure of central tendency because no numbers are repeated and spoke about the different information that mean and median give about the graph. they returned to the question asked previously about how these countries were chosen and by implication what the measures of central tendency can show about the pandemic more generally. neither learners nor the teacher raised the issue about actual deaths and death rates referred to earlier. however, this limitation notwithstanding, in discussing the skewness of the data, the learners looked at the difference between the mean and median and thought more carefully about the information they give in relation to the graph. so, in the case of these two questions, the class was achieving links between covid-19 data and mathematical ideas about representing data, relating to questions 1.1, 1.2, 2.1, 2.2 and 3.2 in the analytic framework. although we had anticipated that the task would support the double move (hedegaard, 1990), the analysis shows that this would need to be more carefully structured into the task. the transcript also shows that the teacher pushed the learners to justify their thinking and not only their calculations: teacher: ok guys. who would like to add on to learner2? learner5: i agree. teacher: why do you agree? learner5: cause i worked out the mean and got 5 million. teacher: how did you get 5 million? none of these numbers are in millions. learner5: well i added them altogether and divided by 11. teacher: yeah. there are 11 countries. learner5: i got a wrong value. teacher: that was what i was thinking. but learner5, it wasn’t about a calculation here. learner5: it’s the average so you get a collective value of all the countries. teacher: so, you’re saying a collective value for all the countries. fair enough. and then learner2 was saying about median, do you agree with her that we could use median? here the teacher pushed learner5 for accountability for his answer and made the point that while the calculation is important, and its correctness can be estimated, in the end the actual number is not the endpoint, rather what it tells about the graph is. learner5 understood this point and made a good argument about what the mean shows about the graph. the teacher then moved on to the median. so, we see the teacher building on what was said earlier in the lesson about authority and accountability, and pushing at least one learner to achieve these, thus achieving 1.3 and 2.3 in this case. in summary then, task 1 supported the teacher and the learners to make links between some mathematical ideas and a graph of covid-19 deaths. the teacher worked hard to support link-making by the learners and was able to deal with a number of challenges as the lesson progressed. the challenges included: a view of mathematics as about calculations only, unanticipated difficulties with mathematical concepts and difficulties in making sense of these in the moment, learner dependence on correct answers and the teacher’s authority to decide on these, learners making a number of comments that may or may not be connected to the task, and how to follow up on all of these. these challenges have been documented in the literature on teaching mathematics responsively (franke et al., 2009; heaton, 2000). here we are interested in how the teacher worked with them and how others might also do so. task 2 for question 1, learners were asked to calculate the percentage of confirmed cases in the northern cape and compare this with the other provinces. two learners offered solutions: learner4 offered 56% and learner2 2.1%. the teacher asked them both to explain. learner2 added the three numbers for the northern cape and divided the number of deaths by that total. learner4 divided the number of deaths by the total positive cases in the country. learner2 may have struggled to determine an appropriate denominator because of the ambiguity in the question and the graph – should the percentage of confirmed cases be in relation to the tests conducted in the province, the total number of people in the province or, as learner4 decided, the total confirmed cases in the country? the teacher did not explicitly evaluate either answer as correct but asked what the answer of 2.1% means in relation to the size of the province in relation to other provinces, which may have been interpreted as an implicit evaluation that 2.1% is correct. faced with a choice about whether to explore the mathematics of the two very different mathematical solutions and thus the missing information on the graph, or whether to explore their meaning in relation to the current information on the graph, the teacher chose the latter. this is likely to be a choice point in many cases for a teacher trying to make links between learners’ lifeworlds and mathematics, that needs to be explored if teachers want to do this work. the teacher reflected: ‘i was surprised that learner2 could not answer the question as i predicted in my planning that this was an easy question. i then moved on to learner4 to see if she reasoned like learner2. perhaps, i could have probed learner2 more on her reasoning. maybe there could be others in the class who reasoned like learner2 about adding the three numbers.’ ideally, the teacher could explore the mathematics of the different solutions, and then link these to each other and to the graph to see whether and how they make sense and what they mean. question 2 asked the learners to relate to the image in a more personal way, as to how they felt about their own safety at home and at school. the learners spoke about their feelings and experiences initially only in terms of the everyday and the teacher accepted their contributions sensitively and tried to refocus them on the graphic by saying: ‘okay, then … i’m curious i haven’t heard from any of you … don’t any of you worry about the covid-19 pandemic, about the deaths and all of that?’ in response to the question, one learner spoke about fears of a second wave, but this was not related to the numbers on the graphic. the teacher reflected: ‘i was thinking about the infographic when i asked this question. however, learners were operating in the realm of everyday knowledge (not thinking about the infographic) and i found that it would be difficult to get them back into the realm of disciplinary knowledge with this question.’ question 3 again asked learners to generate questions, similarly to question 1 in task 1, but here it did not ask for distinctions between mathematical and non-mathematical questions. the learners’ questions varied between mathematical, which gave opportunities for linking the mathematical with the non-mathematical, and purely non-mathematical. in the first case, there were questions like: why were nearly 4 million tests conducted? what is the correlation between confirmed cases and deaths across the provinces? what is the distribution of cases and deaths across the provinces? in the latter two cases, the learners expressed the ideas in everyday language and the teacher had to recognise them as mathematical and help the learners express them mathematically, an important part of doing the linking work. a non-mathematical question was: when will covid-19 end? the teacher tried to get the learner who asked the latter question to focus on the graphic and relate her question to the numbers but it was clear that she did not want to do this, perhaps because of distress about covid-19. in question 3, the teacher did not push for accountability for the learners’ questions, for why these questions were important to ask in relation to both the epidemic and the mathematics. the teacher reflected ‘perhaps, i should have held them accountable for posing the questions they asked and asked them to give a reason why they were interested in pursuing these questions’. question 4 and question 5 followed a similar pattern. learners made non-mathematical contributions, the teacher tried to focus them on the mathematical but they stayed with everyday discussions about lockdown levels and reliability of sources. the analysis and teacher’s reflections suggest that more could have been done with question 4 to make it mathematical in class but question 5 did not lend itself to mathematical thinking, so this was an issue of task design. in summary, task 2 was less successful than task 1 in supporting links between mathematics and covid-19. while questions 1–4 could have supported such links, the teacher found it more difficult to mediate this process. the challenges that the teacher experienced in this task were focused on how to make the links, possibly because the task design supported more everyday type responses, including learners’ feelings. mathematics teachers are not used to working with these responses and so mediating such tasks is challenging. there are two additional points that we saw across the two tasks and that are important. first, the teacher encouraged the learners to use the internet to find information, both about mathematics and about covid-19. while this may be difficult in bigger classes because of monitoring what the learners are actually doing on the internet, and because many may not have access, it does bring in other sources of authority, which can be discussed in relation to authority and accountability in mathematics and beyond. second, the learners in this class participated actively and the teacher clearly had good relationships with them. in other classes it might be difficult to get learner participation, an aspect of making links that has to be worked on (scott et al., 2006). in this class, there was sarcasm from some learners, for example in the comment referred to above about donald trump, or a question about how many corpses had to be buried. the teacher dealt well with these but such comments might prove more difficult in bigger classes. while these sarcastic comments were not the reason for a focus on the everyday in this class, expecting them and working out how to deal with them so that the links and the mathematics are not derailed could be an important challenge for teachers. discussion we discuss a number of important implications that come from our analysis and that can be helpful to teachers trying to move between learners’ lifeworlds and mathematical knowledge, and to researchers studying this process. our analysis shows that making these links is both possible and difficult. we thus take a position different from both muller and hoadley (2019) and fataar (2012). we also note that suggestions for moving between these positions (delpit, 1998; engle & conant, 2002; hedegaard, 1990) are not easy to implement, and we have articulated a number of issues in relation to these. the first issue relates to task and curriculum design and we concur with jones and pepin (2016, p. 115) that task design is ‘by no means a trivial matter’. we have noted some of the task design issues in our analysis where a different formulation of the question might have led to better links being made. in particular, when relating tasks to information in the public domain, the extent to which ambiguity should be edited out is an important one. when working mathematically with real-world situations, the information is not always clear, and learners need to be able to think about what other information might be needed and what is superfluous. however, tasks that support such ambiguity may be misinterpreted and may be more difficult for the teacher to mediate. we thought carefully about the tasks and reworked them a number of times before trying them in the class, and the lessons still suggested further improvements. this finding suggests that tasks that support link-making need to be developed and tested, as conventional textbooks are. tasks that are developed by teachers can be shared and tested by others and developed into stronger tasks. ad hoc tasks developed by individual teachers are less likely to be successful. the role of tasks also points to the role of the broader curriculum in supporting link-making. as long as our curriculum focuses on performance of calculations and supports ‘thin pedagogies’ (fataar, 2012, p. 58), it will always be difficult for teachers to develop tasks relating to learners’ lifeworlds that focus on mathematical thinking and justification and that support teachers to develop learners’ authority and accountability. the second issue relates to the kinds of mathematical content that supports linking with learners’ lifeworlds. our two tasks focused on data representation, as did the two tasks that we did not analyse here. writing in the context of decolonisation, mudalay (2018) shows how difficult it was for teachers studying decolonisation to come up with tasks that related to learners’ heritage and also developed mathematical depth. while covid-19 is particularly amenable to data representation, the question remains as to how much of the mathematics curriculum can be linked with learners’ lifeworlds. while it does not have to be, and should not be, the entire curriculum, if we are to take this endeavour seriously, we cannot do it only in token ways, with one section and in a few lessons. as our analysis begins to show, the capabilities of teachers and learners to engage in this work need to be built over time. a third issue is the extent to which mathematics, as one possible way to look at the world, tells the ‘whole’ story. an aim for linking mathematics to other contexts must be to show that in complex situations, we need a range of data and methods to understand them. much of the information about the pandemic is already combined with scientific and mathematical knowledge, which makes it particularly useful for showing the use of mathematics. in their everyday lives, learners may look at graphs and newspaper articles in order to keep up to date with the trends of the coronavirus in sa without considering the deeper meaning of how the information came about or if the information is valid or not. a focus on the powerful knowledge involved in producing and interpreting the information supports learners to understand more deeply that although mathematics gives us some important ways to understand the world, it will never tell the whole story. in order to go beyond the obvious in the graphs and numbers, it is important to help learners understand that there are many other variables involved, and that we might need to find out about some of them when wanting a deeper understanding of the situation. for example, the issue of excess deaths in relation to previous years, deaths not attributed to covid-19, could have been raised in relation to task 2. applications of mathematics to the real world are seldom exact, but help us to understand our situations. our remaining points relate to how the teacher mediates the tasks. in these lessons we saw two occasions where the teacher overestimated learners’ abilities to do the mathematical calculations and was not prepared to address their unexpected errors. it might be that teachers who are used to typical mathematics lessons where they present formulae and learners practise how to perform a calculations are not aware of some of the typical errors their learners might make when performing calculations learnt in previous years. in addition, they may underestimate difficulties for learners in relating mathematical techniques learned previously to real-world situations. we saw the teacher faced with a number of choice points, which might take the class in different directions, particularly too far into either the mathematics or the everyday. these choice points will present themselves in this kind of teaching and teachers can be aware that they might happen and think in advance about how to deal with them. while it is never possible to predict everything that might happen, it may be possible to develop a set of principles as to how to bring the discussion back to linking the mathematics and the everyday, and how to make the most of mathematical and linking opportunities. finally, a crucial element of all learning, particularly mathematics learning, is that it involves emotions (frenzel, lampen, & brodie, 2019), as well as moral and ethical concerns. in particular, when linking learning to learners’ lifeworlds, emotions may come to the surface more readily. this means that teachers who do this work, need to be aware of potential emotions and be in a position to deal with them, as the teacher in this article did. conclusion the incorporation of learners’ experiences into school knowledge can have advantages and disadvantages, as charlot (2009) notes: very often an attempt is made to solve school failure by linking everything to the pupil’s daily life. this connection, however, can constitute both a support and an obstacle at the same time. (p. 91) all learners bring with them their own everyday experiences, which influence how they engage with school knowledge. as a contribution to current debates on building relationships between learners’ lifeworlds and school mathematics knowledge we have shown that this is a complex and nuanced endeavour. we articulated the complexities and illuminated some of the nuances involved through an analysis of one teacher working with two tasks related to covid-19. we have argued that linking mathematics to learners’ lifeworlds is possible and extremely difficult in relation to task design and how the teacher mediates the tasks. in relation to task design, we have argued that teachers cannot do it alone; they need to be supported by the curriculum and textbooks. we have also argued that we need to think about the full range of mathematics content in relation to applications to the real world, the extent to which the ambiguity of everyday and real-world examples can support or inhibit mathematical discussion and thinking in the classroom, and the strengths and weaknesses of mathematics as a gaze on the world. in relation to mediation, we saw that the teacher’s practices were difficult to shift. other research has shown that even in the best of circumstances, given the traditional ways of working in mathematics classrooms, changing practices is a challenge for many teachers. we saw that the teacher’s expectations of learners may overdetermine decisions made in class and that decision points as to which direction to take the class may be more difficult than usual when trying to make links. although this study was conducted in a privileged situation, we think that the questions, challenges and dilemmas raised can be applied more broadly and that this case can resonate with other, more challenging, contexts and thus can be of use to others. acknowledgements competing interests the authors have declared that no competing interest exists. authors’ contributions all authors contributed equally to this work. funding information this research received no specific grant from any funding agency in the public, commercial or not-for-profit sectors. data availability new data was created for this study. disclaimer the views and opinions expressed in this article are those of the authors and do not necessarily reflect the official policy or position of any affiliated agency of the authors. references barua, z., barua, s., aktar, s., kabir, n., & li, m. 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(2015). can social realism do social justice? debating the warrants for curriculum knowledge selection. education as change, 19(2), 9–36. https://doi.org/10.1080/16823206.2015.1085610 appendix 1 images and tasks figure 1-a1: the number of covid-19 deaths in different countries as of 13 july 2020. questions about image 1 make a list of questions that you think of when you look at this graph. divide your list into questions that are mathematical and those that are non-mathematical. provide an appropriate label for each axis. as of the 13th september 2020, south africa has reported 15 447 deaths. what was the percentage increase in deaths between 13th july 2020 and 13th september 2020? what does this tell you? which measure of central tendency do you think is most appropriate for this set of data? why? comment on the skewness of the data by comparing the mean and median. what does this tell you and why is it important? figure 2-a1: covid-19 statistics as of 13 september 2020. questions about image 2 calculate the percentage of confirmed cases in the northern cape and compare this with the other provinces. what does this suggest to you? based on these statistics (such as the confirmed cases and deaths), how do you feel being back at school and what were your experiences of schooling during lockdown? when you are looking at this image, what do you find the most interesting or revealing about this data? are there any questions that you wish to pursue or ask when you are looking at this data? when looking at information on this image, how do you feel about the current level of restrictions in south africa? do you think it was wise for restrictions to be dropped to level 1? justify your answer. do you think this information in this resource is trustworthy? justify your answer. footnotes 1. the brackets refer to the questions in the analytic framework. 2. we note that the learners’ word usage ‘death rate’ was problematic since the learners meant number of actual deaths when referring to the image. article information authors: tim dunne1 caroline long2 tracy craig3 elsie venter4 affiliations: 1department of statistical sciences, university of cape town, south africa2centre for evaluation and assessment, university of pretoria, south africa 3academic support programme for engineering in cape town, university of cape town, south africa 4independent researcher correspondence to: tracy craig postal address: aspect office, em319.1 electrical and mechanical engineering building, university of cape town, private bag x3, rondebosch 7701, south africa dates: received: 18 july 2011 accepted: 07 oct. 2012 published: 21 nov. 2012 how to cite this article: dunne, t., long, c., craig, t., & venter, e. (2012). meeting the requirements of both classroom-based and systemic assessment of mathematics proficiency: the potential of rasch measurement theory. pythagoras, 33(3), art. #19, 16 pages. http://dx.doi.org/10.4102/ pythagoras.v33i3.19 copyright notice: © 2012. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. meeting the requirements of both classroom-based and systemic assessment of mathematics proficiency: the potential of rasch measurement theory in this original research... open access • abstract • introduction    • classroom and systemic assessment       • classroom-based assessment       • systemic assessment       • reporting at an individual level       • reporting change       • inferences about systemic and classroom testing    • summation and comparison • deficit versus developmental approaches to learning and teaching • rasch measurement theory • the rasch model and its item requirements    • dichotomous items    • multiple choice items    • polytomous items • the rasch model and consequences for test design    • an illustrative example • systemic assessments and classroom intervention strategies    • complementary strategies • why rasch    • where is the catch? • discussion • conclusions • acknowledgements    • competing interests    • authors’ contributions • references • footnotes abstract top ↑ the challenges inherent in assessing mathematical proficiency depend on a number of factors, amongst which are an explicit view of what constitutes mathematical proficiency, an understanding of how children learn and the purpose and function of teaching. all of these factors impact on the choice of approach to assessment. in this article we distinguish between two broad types of assessment, classroom-based and systemic assessment. we argue that the process of assessment informed by rasch measurement theory (rmt) can potentially support the demands of both classroom-based and systemic assessment, particularly if a developmental approach to learning is adopted, and an underlying model of developing mathematical proficiency is explicit in the assessment instruments and their supporting material. an example of a mathematics instrument and its analysis which illustrates this approach, is presented. we note that the role of assessment in the 21st century is potentially powerful. this influential role can only be justified if the assessments are of high quality and can be selected to match suitable moments in learning progress and the teaching process. users of assessment data must have sufficient knowledge and insight to interpret the resulting numbers validly, and have sufficient discernment to make considered educational inferences from the data for teaching and learning responses. introduction top ↑ the assessment of mathematical proficiency is a complex task. the particular challenges inherent in this process depend on a number of factors, including the definition of what constitutes mathematical proficiency, an understanding of how children learn and the approach adopted as to the purpose and function of teaching. besides these central questions in mathematics education, there are important questions to consider about the relationship between classroom-based assessment and systemic assessment types. whilst there is potential for positive information exchange between these two types of assessment, more often there is an unnecessary conflict or simply a lack of constructive communication. classroom teachers are at times perplexed by the outcomes of systemic assessment, confused about what action to take as a result of the reported outcomes and, in the worst-case scenario, demoralised. the quest for positive information exchange demands that questions about quality at both classroom and systemic sites are addressed (see also wyatt-smith & gunn, 2009, p. 83).in this article, we differentiate explicitly between the two broad types of assessment: classroom-based and systemic (or external) assessment.1 the rationale for assessing, the demands of the stakeholders, the forms of the assessment instruments and the types of data produced can and do vary substantially. having briefly explored the differences between the two assessment types, we discuss the broad distinction between two approaches to learning and teaching, one that may be termed a developmental approach and one that may be termed a deficit approach (griffin, 2009). the particular approach adopted within a context will inevitably impact on the choice of and reasons for assessment. if systemic assessment is to be useful within the classroom the results need to be interpreted by teachers and found applicable in the classroom context. underlying this requirement of applicability is the presence of a model of developing mathematical proficiency that includes both plausible conceptual development (from the mathematical perspective) and cognitive development (from the learner perspective). a model such as envisaged here should be somewhat loosely configured and address common issues so that it does not exclude different approaches to mathematical teaching and learning (see usiskin, 2007). such a degree of coherence (from broad consensus towards a developmental model, to a working curriculum document that outlines the broad ideas, to a more specified curriculum at school level and a school programme providing more detail2) is at present a legitimate dream to work towards. also envisaged in the dream is the idea that professional development, accountability testing and formative classroom experience are integrated around core core aspects of the discipline (bennett & gitomer, 2009). the theoretical insights informing a developmental model and the elaborated assessment programme are not the immediate concern of this article. we propose merely to show how applying rasch measurement theory (rmt) may support such a project. an essential part of that support is the facility of the rasch model to yield measurement-like differences and changes. these quantities can enrich the evidence accessible from classroom-based assessment and satisfy the expectations of external stakeholders, in particular if one takes a developmental approach to learning (griffin, 2009; van wyk & andrich, 2006). an example is presented which illustrates the intervention potential of an assessment programme that adheres to rmt and within which the rasch model is applied. we advocate that this model should be seriously considered for inclusion in the approach to national systemic and external assessment programmes, in particular for mathematics. in essence, we explore the question: what model of assessment may support teaching and learning in the classroom, and in addition enable broad-based monitoring of learning progression within districts and provinces? reciprocally: how might systemic assessments not only serve their intrinsic purposes to inform decision-makers about performance levels in broad strokes, but simultaneously inform and enrich teaching and learning within the variety of classroom level challenges into which these single instruments intrude? classroom and systemic assessment the important distinctions between and commonalities within classroom-based assessment and systemic assessment types are discussed below. in addition, the complexities involved in reporting results at an individual level and monitoring change over time are noted. classroom-based assessment the teacher in the classroom is concerned with the learning processes and development of the learners in her class. successful assessment is often of a formative nature and can emerge as continuous assessment, which helps to direct learning and teaching; the summative component, recording marks for the purpose of reporting, also plays a role.3 the rationale for a teacher to run assessment exercises is to determine whole-class and, particularly, individual levels of current development, to diagnose current obstacles to learning progress, and to provide subsequent targeted scaffolding to appropriate classroom segments. in the best scenario, the forms of evidence used in classroom assessment may vary, from projects requiring extended planning to quick quizzes. such variety embraces different learning styles and different facets of mathematical proficiency and adheres to cognitive science principles (bennett & gitomer, 2009, p. 49). the stakeholders in classroom-based assessment are the teacher and the learners. the data sets produced by the classroom assessment exercises are not necessarily designed to be expressly meaningful to anyone outside the classroom, although inevitably and importantly teachers within a school community may share ideas and discuss assessments and their results. the particularity and the immediacy of a test or assessment give it currency in the classroom context and for the classroom processes, at a specific period in time. we may note that in any classroom test or assessment, the teacher is generally concerned with a current spectrum of learner skills and needs in the class, which invariably differs from the spectrum that confronts the educational decision-maker at a district or provincial level. the learners in a particular class may have test performances that are on average well above or well below the average performance associated with all learners of the corresponding grade in an entire school district or province, in the same or an equivalent test. moreover, the variation of individual test performances within any particular classroom will generally be substantially less than the overall variation in performances on the same instrument across the school district or province. systemic assessment whilst classroom assessment is generally fine-grained and topic specific, external systemic is generally broadly banded, and attempts to ‘cover the curriculum’. from the perspective of the education departments, and in some cases other stakeholders such as funders of programmes, major purposes of systemic assessment are to assess the current performance and variability within a particular cohort of learners, according to some sort of external benchmark of desired proficiency, and to monitor progress, also according to some external standards for change and performance improvements over time. overall averages (or percentage scores) and the associated pass rates (learner percentages at or above a specified pass criterion) may be deemed particular elements of interest, but their meaningfulness nonetheless has to be argued and established in a suitable robust exposition. these outcomes should be interpreted in relation to other assessment types, for example classroom-based assessment (see andrich, 2009).for systemic and external assessments, the sheer extent of the testing programmes, and the development time period and financial constraints, may impact resources and available turn-around time for testing, scoring and data capture. systemic test designers may thus be obliged to limit the types of items to multiple choice or short-answer responses, to limit testing time to (say) a single period of a school day and, in consequence, to limit the maximum number of items that can reasonably be attempted. in an ideal situation, a systemic assessment is designed to produce, from a single short dipstick event, performance data about the current health of educational systems, which is meaningful to stakeholders, district officials and state educational bodies. this body of data and its interpretation may result in decisions requiring or offering intervention or other monitoring functions. the department of basic education 2009 review claims that ‘externally set assessments are the only credible method of “demonstrating either to parents or the state whether learning is happening or not, or to what extent”’ (dada et al., 2009, p. 36, citing chisholm et al., 2000, p. 48, [emphasis added]). we contest that claim and, with andrich (2009), maintain the view that the results of external assessment must be considered in conjunction with classroom assessment, rather than alone. in fact, one may argue that to invoke only external test results to convince stakeholders whether or not learning is happening at the individual or class level, and even perhaps at the grade level in a school, amounts to dereliction of duty and is a dangerous, unethical practice. the claim (of invoking only externally set assessment) itself is unethical, however, if it does not sufficiently address the complex issues of causation that lurk within the extensive variation of student performance on the test. similar critique of inordinate emphasis on systemic tests, offered by bennett and gitomer (2009), rests primarily on two counts: firstly that systemic testing has unintended detrimental consequences for teachers, learners, administrators and policy makers, and secondly that this type of assessment generally offers limited educational value, as the assessment instrument is usually comprised largely of multiple choice or short answer questions (p. 45). a systemic assessment may in its totality give a valid overview of system-wide performance on the test instrument (through its constituent items) for the part of the subject and grade curriculum or domain which actually appears within a finite test. possibly, by astute design and professional concurrence, the test may satisfy further criteria, so as to be viewed as a valid assessment of the whole curriculum at a system-wide level. the attainment of such all-encompassing curriculum validity would, however, require a complete revision of the current systemic test design, as noted and proposed by bennett and gitomer (2009). whatever the virtues of a systemic test instrument, it simply cannot give the same level of precise inference about the performance of the individual, class or grade within a school as it does for aggregations at district or province levels. this comment applies even to the highly informative instrument4 we analyse further in this article. for that reason, any interpretation of classroom or grade performance data for a school has to be tempered with a deeper contextual understanding of those units of aggregation, for example, the particular class and the particular grade, and the school in its context and the history of its learners. reporting at an individual level a fairly recent expectation is that the results of systemic assessment be made available to parents. this new access to information may be well intentioned, but the form of the information is problematic, precisely because the data from a single and necessarily limited instrument are so fragmentary and imprecise. systemic assessment is generally not fine-grained enough to report to teachers, or parents, the results of individual learners, as if these single test performance results, ascertained from an instrument of about an hour’s duration, are on their own an adequate summative insight into a year’s progress in the classroom. even bland descriptions limited to only pass versus fail criteria for a systemic test should be supported by some vigorous and robust debate amongst curriculum specialists, and result in an explicit consensus, before such pass or fail designations of test performance outcomes are communicated. these discussions may be most productive if they occur before a test is finalised for use, and again after the tentative results are available, with explicit minutes recorded at both stages. in some systemic tests administered under the auspices of the department of education (2005), designations of performance categories are assigned to the percentage of maximum scores attained on the instruments, as in table 1. table 1: performance categories associated with percentage attained. on the basis of the systemic test score alone, a learner or parent is given a qualitative description that, however well intentioned, is simply arbitrary, invalid and possibly fraudulent, until other evidence justifies the descriptions offered. it is arguable that such descriptions are generally damaging, but especially when test design has not been informed at all by any criteria for item construction and selection that might relate to either the cut-points and the preferred 10% intervals or the adjectives chosen.table 1 indicates instead a tortured avoidance of any verbal signals that learning is in distress, and of any recognition that some children are at precarious risk in the subject. when systemic tests are designed5, there do not appear to be any explicit conditions or attempts made to warrant such achievement categorisations. their valid use would suggest explicit design and the selection and inclusion of items precisely for the vindication of such verbal descriptions. for a 40-item test, the seven performance designators seem to imply a hierarchy of items, comprising 12 simple items that all basically competent learners should have mastered, four items specifically associated with each of the five 10% interval categories, and eight items that address curriculum elements at the highest levels of cognitive demand for the particular grade cohort of the test. such a table as a criterion-referenced end-product, using cut-points and descriptions, may be a laudable goal, but we suggest it cannot be achieved in the time horizons of planning and test design currently applicable in national and provincial tests. if we are correct, then it becomes important to redesign the systemic test construction agendas and timelines to ensure that the criterion-referenced outcomes are validly constructed within the instruments. in general, current systemic tests include items that explore elements of the curriculum that warrant particular attention. for each of these elements, a test instrument will indicate how many learners exhibit the desired mastery in the associated responses. thus the instrument can validly diagnose a series of current particular needs or inadequacies. summarising the item performance of a class of moderate size in a grade will give an indication of those curriculum elements of which those learners as a group do not yet have mastery. these indications, inferred from items that have elicited evidence of low proficiency, do not however identify what factors are contributing to the performances, whether good or poor. the class item scores report states, rather than relationships or processes. they may tell us where a problem is to be found, but not why it arose and what may be necessary to address it effectively. reporting change any objective or intention to use systemic test performances to report on change between years, and possibly on trends over time, will involve an enormous amount of preparatory work to ensure the test performances for the various time periods are truly comparable. there needs to be demonstrable evidence that the associated tests are effectively equivalent. where it is not possible to use the same instrument on two separate occasions, construction of equivalence is difficult and must be undertaken rigorously. where the same instrument is used within too short a time frame, the problem of response dependence6 and appropriate targeting has to be addressed.such preparatory work will involve subject and teaching expertise, but must necessarily impact on test construction and assessment. without this work, and associated extensive piloting of all the test items or instruments in question as well as linking and equating processes, any apparent comparisons of individual test performances to measure change over time must be regarded as moot. it is safer to regard them as invalid until an equivalence relationship between performances over time or across tests has been explicitly argued and demonstrated. a major purpose of equivalent tests is to legitimate comparisons. we may wish to examine progress within an individual over time, or to contrast the competencies elicited from two distinct cohorts of learners. whilst such comparisons may and should admit and use profound qualitative insights and inferences, there is often an intention to seek numerical evidence to bolster those conclusions, and to argue their consequentiality. for that reason, inter alia, it will be of interest to obtain measurement-like outcomes of test instruments, in order to allow use of appropriate numerical differences and perhaps numerical ratios. inferences about systemic and classroom testing we argue that systemic testing is valuable as an external assessment technique at broader levels of aggregation, such as district or province, but is substantially less valuable where aggregation is narrower, such as at class and school level. a well-functioning system of external assessment would involve teachers in the development of the test instruments. it would also feed the results and analysis back into constructive professional development, intended ultimately to impact on classroom practice. in reality, the current design cycles of systemic testing and most external assessments, with or without envisaged professional development support, are too short. the cycles do not encourage adequate engagement with teachers at either the design or analysis stages. whilst engagement with teachers may not be a sufficient condition in itself to ensure subsequent effects in the classroom, it is certainly critical that assessment results make sense to teachers, and that the credibility and relevance of the outcomes are pursued. a systemic testing model proposed by bennett and gitomer (2009), provides an alternative model which avoids many of the pitfalls mentioned previously. this model includes three intersecting phases: an accountability phase, a formative phase and a professional development phase, wherein the engagement of teachers is a critical feature of the process. summation and comparison designers of any test instrument face the challenge of informing both the classroom and external stakeholders. we argue that, alongside this, traditional instruments assume validity of arithmetical functions, such as summation and comparison, which are not necessarily grounded in sound statistical theory.we note that every assessment instrument will involve the summation of item scores. the validity of adding these scores underpins all assessment practice. our current conventions of practice assume this operation is reasonable in every test, even though we may in contrast alert learners to the errors of adding apples to pears or grapes to watermelons. the unique role of rasch measurement models in confirming the admissibility of summing test item scores to obtain a test-performance indicator, and in supporting interpretations of test results, will be outlined shortly. comparison of assessment performances using numerical differences requires that there is some common scale against which the two sets of performances can be authentically captured as numbers of a common kind. then we may compare by subtraction. in effect we mimic the way we compare 23 apples with 26 pears by obtaining a distinct currency values for each individual fruit of each set, and then use additions and a subtraction. we must assure ourselves that we can discern differences by use of a common inherent unit. rasch approaches also allow evidence of change to emerge from the differences observed between two testing contexts whose comparability has been carefully constructed. the potential of the rasch model to support use of information-enriched assessment for constructive classroom intervention in order to bring about changes in learning will be described shortly. here we argue simply that educational objectives of assessing performance and monitoring for numerical evidence of change must rely on the admissibility of summing item scores and of subtracting test scores. authorities need to explicitly establish and not simply assume that the conditions for using arithmetic operations are inherently defensible parts of the assessment instruments and their processes. deficit versus developmental approaches to learning and teaching top ↑ griffin (2009) makes a distinction between deficit and developmental learning approaches. a deficit approach may ‘focus on the things that people cannot do and hence develop a “fix-it approach” to education, and thereby focus on and emphasise “cures” for learning deficits’ (p. 187). the deficit approach is common practice where systemic assessment design processes take place in a short time period within the school year, with less than optimum engagement with any teachers and schools, and constrained to the use of a single instrument for a limited extent of class time. these practices are followed by a period of data scoring and capture, an extensive analysis being performed on the data, and some form of particular aggregated data provided to the schools, many months after the assessment was designed, and of no possible diagnostic value for the same classrooms from which the data arose. invariably, the media are informed of the ‘research’ and information such as ‘x% of learners in grade z cannot handle concept y’, thereby exemplifying a deficit approach. the grade z teachers then, possibly as a result of a circular letter informing them that only x% of their learners have mastered concept y, change their teaching plans and focus an inordinate amount of energy on teaching concept y. the mathematical concept y may not singly be the problem, but may indicate a constellation of concepts that have not yet been mastered (see long, 2011; long, wendt & dunne, 2011). to focus on concept y without understanding the bigger picture may in many cases be counterproductive. certainly opting for such post hoc ‘teaching to the test’ is something of a backward move, unless of course the ‘the test is worth teaching to’ (bennett & gitomer, 2009). a developmental approach builds on and scaffolds from the existing knowledge base of individual learners. this approach, advocated by steinbring (1998), requires that a teacher be attuned to the learner’s current understanding and hence current location on a developmental path. the teacher has to be able to diagnose and analyse the various students’ current constructions of mathematical knowledge within a curriculum. then she has to compare these constructions with the mathematics knowledge required (informal assessment), and to adjust her teaching accordingly so as to facilitate the transition (steinbring, 1998). this process happens against the background of a carefully constructed sequence of learning experiences, exhibiting a suitable sequence of logical and evolving mathematical concepts and theorems that are to be learnt. the developmental approach resonates with the work of vergnaud (1988). he emphasises the important link between learners’ current intuitive knowledge and the targetted more formal knowledge, and where the teacher’s role assures scaffolding of the formal knowledge. the perceived ‘errors’ highlighted in a deficit model become the stepping stones to greater understanding and the construction of generalisable mathematical concepts.7 something of a paradigm shift is required in order to focus on a developmental trajectory which takes into account the network nature of mathematical concepts and considers that learners may learn different concepts at different rates and in different sequences. this shift may obviate a learning approach where the focus is only on those mathematical objects and skills which cannot yet be exhibited fluently. what is required is an assessment instrument which can more reliably inform teachers of the locations of learners along an intended trajectory of development. such an assessment instrument may also more reliably inform the education departments, and stakeholders such as funders, of the current learning requirements of particular cohorts of learners, at least in the associated curriculum elements, through an explicit sequential rationale. when a test is well-designed for its purpose of distinguishing between different levels of learner performance, then we may simply order individual learner performances from lowest to highest, and order test items by their observed levels of difficulty. by partitioning learner scores into a range of ordinal categories, and similarly defining ranges of item difficulties, we may ascertain associations between these groupings that suggest educationally meaningful sequences within teaching and learning. such a device is produced by the rasch measurement model and can be easily incorporated into systemic-testing design, so as to permit the provision of supplementary diagnostic information about items, inter alia for later communication after assessment results have been analysed. rasch measurement theory top ↑ we argue that the assessment opportunities provided by the application of the rasch measurement model can resolve the potential conflicts between the contrasted viewpoints discussed above: classroom based assessment and systemic type assessment, and a developmental model and a deficit model. a well-designed assessment instrument, or sets of instruments, can provide detailed information on the individual development of each learner as well as simultaneously informing external stakeholders on the educational health of an education system. the requirements of the rasch model resonate with the requirements of good educational practice.8rasch measurement theory is explained in a number of publications (andrich, 1988; rasch, 1960/1980; wilson, 2005; wright & stone, 1979, 1999). a comprehensive application of the rasch model to a mathematical area, the multiplicative conceptual field, can be found in long (2011), and an application pertaining to language assessment in griffin (2007, 2009). in this article, the purpose is merely to illustrate the application of the rasch model in one systemic test, through stipulating the requirements of the rasch measurement model and through depicting the outcomes in a form that has the potential to inform both stakeholders and teachers, and to mitigate the misunderstandings that may arise when only aggregated data is used. whilst the example test was designed for a systemic application, it exhibits features which suggest areas of improvement in a subsequent design. the choice of setting happens to be mathematical, but the methodology is not tied to any single discipline. the rasch measurement model is based on a requirement that measurement in the social sciences should aspire to the rigour that has been the hallmark of measurement in the physical sciences (wright, 1997). a great deal of qualitative and theoretical work is required in order to construct a valid measurement instrument, as in the physical sciences (such as the thermometer, ruler, scale, or clock). in the natural sciences measurement devices are designed for specific contexts. though notions such as length, mass and time have universal application, the selection of the specific instrument by which we choose to measure those characteristics is necessarily determined in part by the context in which we seek to comprehend and measure levels of extent and variation in extent. in the social sciences, including education, the first step required towards measurement-like observations is to make explicit the construct to be tested. the operationalisation of the construct as various items, indicative of various levels of proficiency, makes up a test instrument whose overall purpose is to approximate measurement of a characteristic or an ability of persons. this ability is assumed to be plausibly described by a location on a continuum, rather than merely by membership of a discrete ordered category. in designing a test instrument we are obliged to consider and specify both the construct of interest that we seek to measure, and the context or type of context within which the instrument is intended to be applicable. having identified and described the construct of interest and designed a plausible test instrument, as a collection of items selected with an educational context in mind, the next step is administering the test to the intended study group (see wright & stone, 1979). there is no requirement that the items are all of equal or equivalent difficulty. they will generally be a collection with elements at various levels of difficulty. of particular importance at this step is that the test instrument has been properly targeted to the cohort to be tested. this objective is a substantial challenge, because it involves hazarding judgements about how the overall study group will respond to the items, both individually and collectively, but doing so prior to having any corroborative information. in an educational context this challenge requires subject expertise, teaching experience and pedagogical insights into the learning journeys particular to the subject. it will preferably include cycles of engagement about suitability of items and item structures with specialists in the field, namely specialist teachers. appropriate targeting is the requirement that the instrument will be able to distinguish effectively between various levels of performance across the spectrum of learner achievement arising in a specific context of assessment. we may be particularly interested in distinguishing between overall performances at precisely those levels most frequently observed in the assessments. the adequacy of our targeting will contribute to the precision which an assessment instrument can achieve, and hence validly discern differences in ability, in the specified context. we note that targeting a test instrument, in order to maximise its prospective use for distinguishing between performances in a specified study group, is a completely different issue from using the test as an instrument to decide which learners have performances exhibiting a desired level of competence in the curriculum of the test subject. we may order a set of student performances from best to worst, regardless of what subsequent judgment we may care to make about which of them attain a pass or attain a distinction on the basis of the ordered test scores. this difference between distinguishing and deciding arises from the contrasting nature of norm-based (internal ordering) and criterion-based (external notions of pass and distinction) inferences, rasch models can admit the strengths of both norm and criterion approaches. more will be said on the matter of criterion-referencing in the illustrative example. the rasch model and its item requirements9 top ↑ the essential idea underpinning the rasch model for measuring ability by performance on a test, is that the whole test comprises a coherent set of appropriate items. each item is conceptually relevant to the purpose of the test: it consistently gives partial information about the ability which we seek to measure (justifying a possible inclusion of the item), it enriches the information provided by all the other items collectively (contradicting possible redundancy and exclusion of the item), and it is substantially free of characteristics which might obscure the information obtainable from the instrument (contributing to the precision, rather than to uncertainty, of the instrument, and being free of bias). dichotomous items in educational settings, the rasch model is a refutable hypothesis that measurement of an ability is being approximated by the test instrument outcomes in a specified context. it postulates that the ability level of a particular person can be represented by a single number βn. in its simplest form for dichotomous items (with outcomes success or failure, scored as 1 and 0), the model assumes that single numbers δi represent the difficulty levels of the items.each outcome of an interaction between a person and an item is uncertain, but has a probability governed only by these two characteristics, that is by (βn, δi). the rasch model avers that the arrays of numbers βn and δi are on the same linear scale, so that all differences between arbitrary pairs of these numbers such as (βn − δi), and hence also (βn − βm) and (δi − δj), are meaningful. through these differences we may not only assign probabilities to item outcomes, but may also measure the contrasts between ability levels of persons and the contrasts between difficulty levels of items, and offer stochastic interpretations of those contrasts. the probability of any learner answering any item correctly is a function of only the difference between the locations of ability of the specified learner and the difficulty of the particular item (βn − δi). the model demands that no other person factor or item factor or other consideration intrudes into the probability of success on the item, and that the net joint interaction effect of the person ability and item difficulty is dominated entirely by that difference. the logistic function with parameters (βn, δi), expresses the probability of a person n with ability βn responding successfully on a dichotomous item i, with two ordered categories, lower and upper designated as 0 and 1, in the equation: here p is the probability, xni is the item score variable allocated to a response of person n on dichotomous item i, the number x is an observed score value (either 0 or 1), where βn is the ability of person n and δi is the difficulty or location of item i. note that equation 1 does not require any restrictions on either of the real numbers βn or δi, but it does require that the two values can be subtracted. the function of the denominator λni in equation 1 is simply to ensure the (two) probabilities for the dichotomous item sum to 1. the relationship of item to learner is such that if a learner labelled n is at the same location on the scale as an item labelled i, then βn = δi or (βn − δi) = 0. in consequence the two probabilities for the ordered categories are equal. then substituting this zero difference for the bracketed terms into equation 1 implies that the learner of any ability level will always have a 50% chance of achieving a correct response to any dichotomous item with a difficulty level equal to his or her ability level. if an item difficulty is above the ability location of any learner, then the learner has a less than 50% chance of achieving a correct response on that item, but if the item is located lower on the scale than the person location, the learner would have a greater than 50% chance of achieving a correct response. the graph of equation 1 for a specified value of δi is obtained by setting the probability on the vertical axis, and person parameter βn on the horizontal axis (see figure 1). the result is a symmetric s-shaped ogive curve, with a midpoint at (δi, 0.5). this curve is termed the item characteristic curve. the ascending curve (from low on the left to high on the right of the figure) indicates the probability of obtaining a correct response. the descending curve (from high left to low right) gives the complementary probability of obtaining an incorrect response. figure 1: item characteristic curve and complement: probability of 0/1 responses. equation 1 suggests that if we consider the subset of all persons whose common ability is precisely βn, then each of them will always have exactly the same probability of obtaining a score one, at each and every item whose difficulty is given by a specific value δi. similarly, these persons of common ability will all have the same probability of a zero score, for all items at the specified difficulty level. moreover, this equivalence of probabilities will continue but with a revised common probability value, at any new set of items which are all at a distinct but common difficulty location δj, where δj ≠ δi.equation 1 is a stringent requirement, but it is exactly as required for a dichotomous test item to be validly considered as unbiased and equally fair to all persons who take the test. it may appear to be only one equation in this format, but each version comprises two probability statements (for the values x = 0 and x = 1).then there are 2 × n × k equations summarised within equation 1 as rasch models require that same stochastic structure for all possible n × k combinations of n persons in a study group, each interacting with each of k (dichotomous) items in a test instrument. multiple choice items items offering a multiple choice amongst a closed set of response options are handled in the same way as dichotomous items. some minor adaptations allow the analysis of test data to address the extent to which preference for the various false distractor items may exhibit patterns that vary over the ability range of the persons taking the test. polytomous items modifications of equation 1 allow the probability relationships to be extended to polytomous test items that permit maximum score categories higher than one, for partially or completely correct responses. for polytomous items we permit each item response to be recorded as an ordinal category indicated as a single number within the set 0, 1, … m, where m > 1.it is important to note that we are making an ordinal set of categories, recorded primarily by numbers. rules for allocating these number labels will be set out in a scoring memo for the polytomous item. because we assume expert construction of each item and its scoring memo, we expect that higher item scores will be associated with higher abilities βn, and conversely that lower scores will be associated with lower abilities βn. we are only saying the labelled categories 0, 1, … m are distinct and uniquely ordered. we are not saying that unit differences between the scores x and x + 1 are the same, regardless of x. we are not considering any ratios to be valid. here 2 is more than 1 but is not two times 1. likewise 3 is higher than 2 and 1, but is not 3 times 1, nor 2 plus 1. this initial ordinal structure is therefore distinct from using the category labels x as marks. but we may go on to assume the labels to be marks, and also allow addition of these marks across all items. then, for any particular item, as the marks x increase, we will expect higher total performance scores in general, and specifically, higher averaged total scores at each new higher observed label x. simultaneously, but distinctly, we also assume that higher levels of person ability βn will be associated with both higher item-score labels and marks x, for each polytomous item, and hence also with higher test total performance scores. these addition strategies are perfectly plausible and coherent, and have been common practice perhaps for many decades. but the issue of the conditions under which they can be defended as modes of obtaining objective and meaningful totals must still be addressed. the levels of person ability can range over the entire set of real numbers (-∞, ∞). a consequence of the ordering of our categories in any polytomous item is that we also expect that each such item partitions the full ability range into a sequence of (m + 1) consecutive disjoint intervals, over which the corresponding most likely item category label or score will be 0, 1, 2, … m and in that ordering. if we wish to make inferences about the relative abilities of individual persons the rasch measurement model is the only route by which to do so. all other models permit only vague general statements about the distribution of abilities for unspecified persons. software packages to perform rasch analysis through stages of model checking, diagnostic processes and estimation procedures are available on the internet, and from development laboratories. this study made particular use of the rumm2030 suite of programs. in the reported data (see table 2), the five polytomous items are represented by the average thresholds. table 2: items ordered from difficult to easy, with item location, standard error, item type, domain and item description. the rasch model and consequences for test design top ↑ good test design seeks to have every item satisfying the design criteria outlined above. what rasch methodology offers is the possibility of checking each of those item requirements, their collective functioning, and the various independence requirements. constructing a valid instrument will require some arduous tasks at item level. when the item and independence requirements are each found to be reasonably satisfied by the test item data, the astonishing power of the rasch model is harnessed. statistical theory guarantees us that under these required conditions we can not only find a valid estimate of ability for each learner, but that for any person, the sum of his or her item scores is the key element in estimating that ability, and that all other detailed information from the data is neither needed nor helpful in the estimation process. we note that this sufficiency does not imply the total performance score itself is a suitable measure of the ability, but that the person ability measure is a mathematical function involving only that person’s total score. the same statistical theory also guarantees a similar result for items: counting how many of the n persons have been assigned into each of the (m + 1) score categories of an item, that is finding that item’s score frequencies, is sufficient to obtain valid estimates for both the m thresholds of that item and for its average level of difficulty. no other information from the data is required, and no other information from the data set could possibly improve the estimation process. again this sufficiency of the (m + 1) category frequencies for the m threshold estimates does not imply the frequencies themselves are suitable measures for the thresholds, but rather that threshold estimates are simply a mathematical function involving only those frequencies, whilst the person estimates are determined by the array of total scores. these two types of simple estimation structures are extraordinary. these simplicities do not hold for any other model than the rasch measurement model. the rasch model is essentially an hypothesis that an ability is measurable, indirectly, from test instrument data in a specified context. if the observed data do not fit the requirements of the rasch model, then these measurement-like advantages, however desirable, do not arise. in consequence there is no way to coherently provide any statistical inferences relating to individual people or specific items, other than by frequency tables. any long-term intention to make statistical comparisons between or within cohorts over time is irrevocably undermined. when the data fits a rasch model, suitable transformation of the raw total scores for persons and raw frequencies of score categories of each item will enable calculation of estimates for both learner ability parameters and all item thresholds and average difficulty levels. all these estimates may then be legitimately represented and located on the same scale or linear dimension. all differences obtained from any pair of these n + m estimates have an explicit stochastic interpretation. the estimated item difficulties are calibrated to have a mean of zero10, and then the relative difficulties of the items are located accordingly. thereafter the learner proficiencies are estimated in relation to the corresponding learner performance on each of the items. figure 2 (in the illustrative example) displays a summary of item difficulty and person ability estimates in the same diagram. on the right side, all the items from the test instrument are located at their levels of relative difficulty. on the left side, all the learners are located at their individual levels of proficiency on the same vertical axis. each learner is however only shown in the figure as hidden amongst the collective contributors to the cross (×) symbols at the particular interval in which their estimates appear. note that the display gives valid insights into the test performance, but that no notions of fail, pass or distinction have been specified. figure 2: person-item map approximating person proficiency and item difficulty on a common scale. the rasch measurement model suggests an assessment system which provides statistically sound data and analysis which can inform classroom teaching as well as external stakeholders in a contextually meaningful way. we support our argument with an example drawn from recent practice in secondary school assessment. an illustrative example a test instrument (k = 40 items) was designed for the purposes of measuring learner proficiency on grade 8 mathematics. the test, as is common practice, combined several mathematical strands, such as data and probability, geometry, algebra, and number. the test was administered over a cohort of grade 8 learners (n = 49 104) in one south african province. the study data was analysed applying the rasch model, for the purposes of confirming appropriate difficulty level of the instrument as a whole for the learners and to identify and describe learner ability in relation to the test items (long & venter, 2009).the mean of all item locations is set at zero as a standard reference point in the rasch measurement model11. the item difficulties are estimated and located on the scale. the learner ability values are then estimated. the learner proficiency estimates are located on the same scale in relation to the items. for the purposes of this analysis the scale was divided into bins of equal width. the left hand side of figure 2 is a simplified histogram for the estimated ability values12. the chosen scale is the log-odds or logit scale, derived from using the logarithm of odds (the ratio ). within this scale all the parameter estimates satisfy the required measurement-like properties, and have consistent stochastic interpretations. we note that figure 2 immediately provides decision-makers with an extensive but quick diagnostic summary of which items can be correctly answered by at least half (50%) of the learners at a set of specified ability levels, and which items are correctly answered by fewer than half of the tested persons at specified ability levels. the diagram provides a label in which the item number in the test is specified, and the item content is easily obtained by reference to that label. here visual inspection of the proficiency histogram will suggest that the person (ability) mean is below the zero item mean, being located at approximately -1.0 logits. this negative location indicates that the test instrument is not appropriately targeted for the tested grade 8 group as a whole. in consequence, somewhat less than optimum information for distinguishing between performance abilities on this test is obtainable for this cohort on this test. this graphical feature of the output indicates that the test could be improved to better match the variation in the study group. the data suggest that for this study group, more items of below the current average difficulty would improve the power of the test to distinguish between proficiencies at the lower segment of the person range, where most of the study group are located. augmenting the instrument with new items in the targeted range might make the instrument appear easier in the sense of possibly improved performances for all learners who performed well enough on the new items. that artefact of apparently increased scores and likely increased percentages, necessary in seeking better power to make finer comparisons between learner performances in the mid-range, will usually require a revised view of any corresponding criterion-referenced judgments such as pass-fail or distinction-pass applicable in a revised instrument. these revisions require precisely that same expert judgment which we hope originally contributes to the design of every systemic test, and to its educational interpretation, being exercised by the inclusion of new items and the interpretation of their consequences. for learners clustered around the person mean, there are some items (below them) which are relatively easy, some items for which according to the model learners in this cluster have a 50% chance of answering correctly, but most items in the test (above them) are relatively difficult for this cluster of learners (fewer than 50% of them will answer correctly on any of the highest sets of items). table 2 presents the same items from most to least difficult vertically down a table with brief descriptions of the k = 40 items in the associated levels. the easiest items therefore address the interpretation of a bar chart (i29) and the identification of a net (i14). the items, calculating rate (i38), coordinate geometry (i40) and calculating the mean (i36) emerge as the most difficult. for ease of analysis, some equally spaced levels, also denoted as proficiency zones, have been superimposed on the person-item map (see figure 2). items i15, i13, i17 and i19 are of average difficulty and therefore aligned with the item mean set at zero (see logits -0.01 to +0.06, see also table 2). i29 and i14 are the easiest items, located at the lower end of the scale (logits -3.14, and -1.74), with i38, i40 and i36 the most difficult items, located at the top end of the vertical scale (logits 2.14−2.79). for the few learners at proficiency zones 8, 9 and above (not shown in figure 2 due to the scale chosen), there are no items which challenge their mathematical proficiency. for learners at proficiency zones 6 and 7, there are five items located at a matching level, items i39, i37, i38, i40 and i36, and for which the learners have around a 50% chance of being correct. analysis of relative locations of learner proficiency and item difficulty in separate individual construct strands (for example algebra) allows stakeholders in both classroom-based and systemic assessment to further research and provide some appropriate intervention. for example, lesson sequences may be developed which attend to the increasing algebraic demands and the associated cognitive skills proximate to current levels of interpreted ability. retrospectively, according to the model: when the amount [extent] of latent trait possessed by the candidate was equal to the amount [extent] needed to demonstrate the criterion behavior, the probability that the person could demonstrate the behavior [in this instance] was 0.50. this [criterion] was an important idea in defining a person’s [current] ability, but it was crucial to the assessment being used to improve learning, identify appropriate teaching resources and to develop current policy. (griffin, 2007, p. 90) systemic assessments and classroom intervention strategies top ↑ we now make an educational assumption. we allow that the changing proficiencies between learners mapped against the static display of item difficulty as we move up figure 2, will be very similar to the progression of proficiency on the corresponding curriculum elements particular to an individual learner. we assume that the learner is increasingly engaged in the teaching and learning classroom on tasks related to the test material and over time becomes better able to tackle items of greater difficulty up the vertical sequence. this assumption is debatable, since there is not necessarily only one pathway to mathematical sophistication in any grade. however its utility is that it allows us to interpret the static figure 2 (with item descriptions) as part of a developmental model. for each set of learners clustered at a level in figure 2, we have some idea of the types of items which the cluster can currently manage (i.e. for which they have at least 50% chance of success). we also have some idea of the types of items just some small distance above the current cluster level, and hence located in what may be called the zone of proximal development (vygotsky, 1962) for that cluster of persons: the idea of ordering criteria and locating the criterion where the probability of success for each person is 0.50 can be linked to vygotsky’s research which was driven by questions about the development of human beings and the role that formal education plays in the process. the challenge for educators was to identify students’ emerging skills and provide the right support at the right time at the right level. it was in this context that vygotsky’s construct of zone of proximal development (zpd) – the zone in which an individual is able to achieve more with assistance than he or she can manage alone – was conceptualized. (griffin, 2007, p. 90) by specifying an assumed zone of proximal development for each cluster level, the teacher uses the test information to make teaching efforts more efficient. in this structure the teacher imposes temporary clusters within the class so as to more easily divide teaching efforts and time between groups with similar current needs, as reflected by the tested subject proficiency. for example, learners located in proficiency zone 3 have four items located within a similar zone. for these learners the model probability is 0.5 or 50% for answering correctly. for learners in proficiency zone 2, these same four items will be more difficult in general. from a conceptual development perspective, we see in table 2, where the algebra items are in bold, that they are spread nicely over almost the whole range of item difficulties, and well aligned with learner proficiencies, therefore giving a fair reflection of learner proficiencies in algebra. see items i04, i027, i020, i012, i019, i015, i015, i021, i006, arranged from least difficult algebra item (logit -1.74) to most difficult algebraic fractions item (logit 0.45). the potential is there, in the case of this systemic assessment, of identifying a hierarchy of competences within algebra through which learners could be guided in the small setting of a single classroom. the hierarchy of competences evident in table 2, was derived from the responses collected from a very large sample of learners and not just from one classroom. this hierarchy could reflect increasing challenge in mastery of algebra as generally experienced by learners of that age. the development of a sequence of items, aligned with the theory of emerging proficiency in algebra, has the potential therefore to empower the researcher or professional teacher communities to structure learning opportunities in an informed manner, mapped to the needs of clusters of learners in her class whose proficiency has been mapped onto the same scale. the efficacy of the instrument depends on the theoretical work that has informed the instrument and that also informs the analysis and the inferences to be made from the analysis. but given high quality theoretical work underpinning test construction and rigour in the refining of the instrument, we propose that the application of rasch measurement theory may provide the means for meeting the needs of both the teacher or learners and the stakeholders interested in outcomes of large-scale assessment. complementary strategies the advantage of identifying and targeting current need groups, emerging even from a non-optimal systemic test as reported here, arises if the results are known quickly. in large and complex educational structures where quick turn-around from data to results at a learner level is not easily achieved, it may be useful to consider an alternative complementary assessment strategy beyond systemic testing. an external resource of a large collection of items, sufficient for several tests at any parts of the likely person ability range, along with associated already prepared diagnostic information, can be marshalled, and made available for devolved use by schools, grade leaders and teachers. there may be a need to provide facilitative scoring arrangements (e.g. electronic marking and outputs as provided for the example test in this article) so that the richness of the assessment resource feeds timeously into teaching. given suitable systemic test and scoring resources, it will then be feasible for any classroom to be focused upon its own current needs, across all the very diverse ranges of classroom proficiency and school contexts. making this option for selection and downloading of items feasible will require prior resource implications and processes. many proposed items will need to be submitted, cleared for use, piloted and, where necessary, adapted. there will be some attrition due to unsuitable proposals, and some necessity to ensure breadth of cover for the resource. all items will require grading and diagnostic ancillary information. the associated collaborations will generate teacher collegiality and contribute to professional development of classroom diagnostic skills and intervention initiatives. in this scenario, district and provincial decision-makers can usefully supplement external systemic-test results apparently signalling classrooms in current distress, with detailed analysis of the assessment initiatives and interventive strategies currently explored, or not yet explored, in those environments. thus any systemic need to address incompetence or inexperience in the classroom can be informed in part by systemic tests, and give rise to other information or information processes that will be fairer to all teachers, affirming the dedicated and competent and alerting to incompetence or neglect. why rasch top ↑ the importance of requiring data to fit rasch models is that fitting the model guarantees that scores arising from items which independently obey equation 1, may always be summed together. these person totals and category counts will always permit separate estimation of each of the n person ability parameters and each of m item difficulty parameters.only rasch models have this property of guaranteeing the summation process to obtain a valid overall test score. all other methods (whether based on so-called traditional test theory or on so-called 2-parameter and 3-parameter structures for item responses) simply assume the summation is valid, even if there is demonstrable evidence that test items scores do not behave additively. in other words, all other models for summing of test item scores into a collective indicator will only assume the internal consistency within and between item scores as an incontestable truth, whereas the rasch model allows the data to signal when such summation is dubious or false. this issue of permissible summation is not simply a mathematical nicety. it is an ethical imperative. if we claim we have an instrument that consistently accumulates scores from appropriate component parts, we are obliged to assess the extent to which both the accumulation and the behaviour of the parts are confirmed by the evidence in the data. we note that there is no requirement that the persons interacting with the items of an instrument are a random sample of any kind. the persons are simply part of the context, and not representative of any group other than themselves. we seek to make inferences about the relative abilities of any and all the persons tested. similarly, the items are not intended as a random sample from possible items. we seek to make valid inferences about the manner in which the selected items collectively discriminate between the persons who are the source the data. where is the catch? in practice the validity of the output and analysis on which figure 2 and table 2 are based, is conditional on the adequacy of the fit of the test data to the rasch model requirements. checking the requirements of the model is an extensive and difficult task, precisely because this particular model embodies all the many requirements that permit measurement-like estimates. all these requirements should be checked. it may transpire that several iterations of design, analysis and identification of problems are required, before an instrument is deemed to be satisfactory for its intended measurement purposes. the checking of the fit is sketched here so as to obviate any impression that displays like figure 2 are simply routine outputs of a test instrument and software which can be accepted without justification and analysis. the checking of model fit is the first of a set of cyclical processes, the purpose of which is to understand the data and where necessary to improve the functioning of the instrument. here we distinguish between items that fit the model, items which are under-discriminating (often when learners are simply guessing), and over-discriminating items arising from item response dependence (e.g. where a correct response on a previous item increases the probability of a correct response on a current item). a further possible violation of requirements to be considered when applying rmt is differential functioning of an item across distinct learner groups. for example, boys at an ascertained proficiency level may perform much better than girls at the same level on a particular item that involves bicycle gears. checking for these group differences is important in the interest of assuring fairness of all items for all groups. strategies for diminishing the effects of differential item functioning are to be found in the literature (andrich & hagquist, 2012; andrich & marais, 2012). the rasch model is essentially a single complex hypothesis built from several requirements about a context, about a test instrument and its constituent items, and about the way in which the context and instrument interact to produce special forms of measurement-like data. the whole purpose of the rasch model might be characterised as seeking to make valid inferences at the level of an individual person and to avoid being limited only to inferences about the patterns within a totality of persons in a given context. it is inevitable that in demanding so much more detailed utility of an instrument of any kind, there will be more stringent properties required within its construction. in addition, we will require detailed description of the contexts within which such an instrument can be validly used. here we will take care to specify all the major requirements, and indicate some of the ways in which each of those requirements may be invalidated by evidence. note that a single invalidation of any one requirement may be sufficient to send a test instrument back to a revised design stage, the beginning of a new cycle of iteration towards a data set with a validated rasch measurement model. one such context may be the mathematical abilities of learners in a specified grade in all schools of a province. a test instrument is constructed with the purpose to measure the abilities of all the learners in the context, with sufficient precision. it will be impossible for the test instrument to yield exact measures, because it is composed of discrete item scores, subject to uncertainty. however we all recognise there is a point at which non-exact measures may be subject to such high levels of uncertainty that their utility is lost. in consequence all parameter estimates should be reported with an associated standard error of measurement, or by confidence intervals, as well as by point estimates. we may note that increasing numbers of persons will imply reduced standard errors for item parameters, and increasing numbers of items will imply reduced standard errors for person parameters. the test instrument and its items are expected to explore and reflect an underlying single dimension, rather than more than one dimension. one may argue that the complexity of mathematics implies more than one dimension. detailed discussion on the topic of unidimensionality may be found in andrich (2006). here we note that unidimensionality implies all aspects of the test ‘pulling in the same direction’. undue language difficulty for example, would be an example of an unwanted dimension. on this single dimension we hypothesise that it is possible to meaningfully locate all n person abilities at particular numbers on a number line. we require that this arrangement must operate in such a way that all comparisons between person abilities would be consistently represented on the number line. we require that all k item average difficulties and all m item difficulty thresholds can be similarly organised on a single dimension, and that all comparisons of item parameters are consistently preserved. in addition, we require that the same straight line be used for both person and item arrangements, and that the two arrangements can be interwoven so that all differences of the type (βn − δi) will also be consistently preserved. further, attention must be given to any extreme scores for persons and items. no test can usefully deal with estimating abilities for persons who score either 0% or 100% correct, except when further new assumptions are justified, or when new relevant information becomes available from beyond the current data set. items on which 0% or 100% of persons are correct, tell us nothing about the distinct person abilities. these item data cannot contribute to a rasch model for distinguishing either between persons, between 0% items or between 100% items, and are therefore eliminated from the analysis. some violations of the required independence may arise only from specific persons or specific items. for each item and for each person we may calculate the corresponding item fit and person fit statistics. the values obtained for these statistics assess evidence for dependencies between item responses for any particular person, and dependencies between person responses for any particular item. the statistics identify items or persons for whom the interaction data does not conform to the required rasch expectations. after identifying anomalous persons and anomalous items, the test designers have to explore what can be learnt from those elements. for the instrument, this process may involve changing or even dropping any anomalous item(s). the wording, structure and content of the item(s) will guide this choice. in general the final form of every item should enrich the collective power of the test instrument to distinguish between various persons on the basis of their ability alone. for the specified context, finding that any particular subset of persons responds anomalously, often warrants exploring their removal from the analysis. if a person’s item responses are random or incoherent, they do not address the construct which the items are intended to embody. given that the vast majority of other learners are responding appropriately, we may eliminate the anomalous learners, precisely because their data are not contributing to an understanding of the relative difficulty of the items. in fact, including their anomalous item data will obscure the patterns in the data, and hence affect both the estimates obtained for the other learners and the estimates for the item parameters. we may eliminate such data, but must record the elimination and its rationale. this strategy still preserves a diagnostic value, for example identifying students who simply randomly guess for all or part of the instrument may have value for educational interventions. only one ability-difficulty dimension is the intended construct of interest. however, it may be the case that an instrument taps into several dimensions, all inter-related in some way. checking an instrument involves exploring if there is a suggestion that more than one dimension emerges from the data (andrich & marais, 2011). having ascertained that the data largely manifest as a single scale for the person performances and the item difficulties, we check if each of the items suitably contributes to our objective of a measurement process. this process is lengthy and detailed (andrich & marais, 2011). it is also complicated, especially when by construction we seek to have an instrument with substantive validity, and that validity requires distinct aspects of the single dimension to be included. for example, we may in a mathematics test require items that tap into algebra, arithmetic, geometry and data handling. the data should be scrutinised for violations of the homogeneity of the learner responses over any features other than ability itself. comparisons of the graphs produced by the rasch analysis software for two or more groups may assist in determining whether various explanatory variables or factors give evidence for differences between groups. specifically we may check whether or not evidence exists for suspecting any items to be under-discriminating (as when learners are guessing rather than engaging with items), or over-discriminating (as when an item requires pre-knowledge or a threshold concept). discussion top ↑ the example provided serves to illustrate the potential of an application of the rasch model to an assessment instrument should the requirements be met. the potential of such an assessment model with its subsequent analysis is dependent on the quality of the instrument, and therefore on the prior theoretical work that has preceded the development and selection of items. whilst in this example some worthwhile information is available for the stakeholders to observe, the potential for a more nuanced instrument may be envisaged. we note that the rasch model is used routinely in timss (trends in mathematics and science study) and pisa (programme for international student assessment) to scale item difficulties and proficiency scores (see wendt, bos & goy, 2011).given a well-targeted test instrument, informed by adequate theoretical investigation within the substantive discipline of the test, there is the potential for informing both the stakeholders and the educational officials. well-targeted instruments may also require some type of pilot testing or external benchmarking. as it transpired, this well-intentioned test did not match the target population very well. inferences can be explored to improve this aspect of the test instrument. nonetheless, diagnostics relevant to the teaching of the material relating directly and indirectly to the test are readily available from the design work on the construction of the test. the design work permits the explicit statements in table 2, and the ordering of items from the data, to suggest sequences of teaching and learning. it is readily conceded that further iterations with some altered or replaced items may produce revised table 2 summaries that will conceivably be mildly or radically improved in usefulness. one may ask whether the information presented in this analysis is not already known to the stakeholders and education officials. we recognise the test design as somewhat typical of assessment instruments expected by current systemic assessment programmes; they should ‘cover the curriculum’. the issues may be well known, but the problem of coherence within such a test when analysed from a developmental learning approach is less explicitly recognised. by its generality of coverage, the systemic instrument provides only scant or generic developmental information to the teacher. perhaps it is time for cycles of systemic assessment of a more focused and limited nature, for example, an instrument with a focus only on algebra where the skills and concepts may be operationalised in a set of items requiring increasingly complex and critical skills that elaborate on the key areas identified in the literature. associated specific developmental elements can be marshalled at the design stage, and modified in terms of the emerging patterns of the applied test context, to inform more specific target interventions for algebra in the classroom. conclusions top ↑ any approach to mathematics assessment almost certainly follows a predicated view of teaching and learning, which in turn rests on an understanding of the central features of mathematics. the implicit beliefs underpinning current assessment practice may benefit from debate and explicit acknowledgement of any underpinning philosophy. for example, what view of learning and what view of evidence underpins the claim that ‘external’ assessment is the only credible method of demonstrating that learning is happening in schools (dada et al., 2009)?the recommendations resulting from the department of education review (dada et al., 2009) are that continuous and broad-based assessment is limited and that external assessment at grades 3, 6 and 9 be enshrined in policy. given that this policy decision has been adopted, it is critical that the external assessments work in conjunction with classroom assessment. the relevant grade teachers, rather than being the objects of the testing policy, should be participants involved in the construction and analysis of tests. we aver that a collaborative strategy supporting regular use of formative assessments may impact more directly on their teaching, in ways that better address learner needs, and hence improve learning of the subject. in answer to the question: what model of assessment may support teaching and learning in the classroom, and in addition enable broad-based monitoring of learning progression within districts and provinces?, we advocate an approach which takes seriously the critical elements of mathematics, in the formulation of a developmental trajectory. systemic provision of a large variety of test items and their diagnostic support material, together with informed and deliberative selection by committed teachers for classroom use, with facilities for electronic data capture and/or marking, are important strategies. routine classroom tests drawn from such item bases can simultaneously support classroom innovations, whilst providing district structures with information about classroom efforts and needs. in such extended contexts, occasional systemic testing can be interpreted against a wider range of contextual information. the role of assessment in the 21st century is ‘extremely powerful’ (matters, 2009, p. 222). according to matters, this role can only be justified on condition firstly that the assessment is ‘of sufficient strength and quality to support its use’, and secondly that the ‘users of assessment data have sufficient experience and imagination to see beyond the numbers’ (p. 222). assessment against this background of theoretical rigour fulfils a requirement of the rasch measurement theory that the construct of interest be made explicit. the practical unfolding of the construct, in items that are realisations of the construct, is then formulated as a test instrument. the output from the rasch model, provided the prior requirements are met, has the potential to inform current teaching practice, to orchestrate teacher insights into the challenges of their own classrooms and initiate two-way communication between classrooms and decision-makers. acknowledgements top ↑ acknowledgement is due to the schools development unit of the university of cape town for the development of the test instrument in collaboration with caroline long and elsie venter, and to the western cape education department for commissioning the work. competing interests we declare that we have no financial or personal relationship(s) which might have inappropriately influenced our writing of this article. authors’ contributions t.d. (university of cape town) contributed to the conceptualisation of the article, and to the detailed explanations of the rasch model. he wrote extensive sections of the article. he was also involved as an advisor to the original project where the data were collected. c.l. (university of pretoria) was project leader for the original project where these data were collected. together with e.v. (independent researcher) she was responsible for the initial analysis in the original project, and independently conducted the re-analysis using rumm software for this article. she contributed to the conceptualisation of the article. she wrote the remaining elements of the article. t.c. (university of cape town) contributed to the conceptualisation of the article and thereafter assisted with critical revision of the manuscript. e.v. worked with c.l. on the pilot study analysis and the subsequent analysis of the data. references top ↑ andrich, d. (1988). rasch models for measurement. beverly hills, ca: sage publications.andrich, d. (2006). on the fractal dimension of social measurements i. perth: pearson psychometric laboratory, university of western australia. andrich, d. (2009). review of the curriculum framework for curriculum, assessment and reporting purposes in western australian schools, with particular reference to years kindergarten to year 10. perth: university of western australia. andrich, d.a., & hagquist, k. (2012). real and artificial differential item functioning. journal of educational and behavioural statistics, 37(3), 387−416. http://dx.doi.org/10.3102/1076998611411913 andrich, d., & marais, i. (2011). introductory course notes: instrument design with rasch, irt and data analysis. perth: university of western australia. pmcid:3217813 andrich, d., & marais, i. (2012). advanced course notes: instrument design with rasch, irt and data analysis. perth: university of western australia. andrich, d., sheridan, b., & luo, g. (2011). rumm2030 software and manuals. perth: university of western australia. available from http://www.rummlab.com.au/ bennett, r.e., & gitomer, g.h. (2009). transforming k-12 assessment: integrating accountability testing, formative assessment and professional development. in c. wyatt-smith, & j.j. cumming (eds.), educational assessment in the 21st century (pp. 43−62). dordrecht: springer. http://dx.doi.org/10.1007/978-1-4020-9964-9_3 black, p.j. (1998). testing: friend or foe. london: falmer press. chisholm, l., volmink, j., ndhlovu, t., potenza, e., mahomed, h., muller, et al. (2000). a south african curriculum for the 21st century. report of the review committee on curriculum 2005. pretoria: doe. available from http://www.education.gov.za/linkclick.aspx?fileticket=y%2bnxttmzkog%3d&tabid=358&mid=1301 dada, f., dipholo, t., hoadley, u., khembo, e., muller, s., & volmink, j. (2009). report of the task team for the review of the implementation of the national curriculum statement. pretoria: dbe. available from http://www.education.gov.za/linkclick.aspx?fileticket=kydmwouhvps%3d&tabid=358&mid=1261 department of education. (2005). the national protocol on assessment for schools in the general and further education and training band (grades r to 12). pretoria: doe. griffin, p. (2007). the comfort of competence and the uncertainty of assessment. studies in educational evaluation, 33, 87−99. http://dx.doi.org/10.1016/j.stueduc.2007.01.007 griffin, p. (2009). teachers’ use of assessment data. in c. wyatt-smith, & j. j. cumming (eds.), educational assessment in the 21st century: connecting theory and practice (pp. 183−208). dordrecht: springer. http://dx.doi.org/10.1007/978-1-4020-9964-9_10 long, c. (2011). mathematical, cognitive and didactic elements of the multiplicative conceptual field investigated within a rasch assessment and measurement framework. unpublished doctoral dissertation. university of cape town, cape town, south africa. available from http://web.up.ac.za/sitefiles/file/43/314/long,_m__c__(2011)__the_mulitplcative_conceptual_field_investigated_within_a_rasch_measurement_framework_.pdf long, c., & venter, e. (2009). report on the western cape grade 8 systemic assessment project. pretoria: centre for evaluation and assessment, university of pretoria. long, c., wendt, h., & dunne, t. (2011). applying rasch measurement in mathematics education research: steps towards a triangulated investigation into proficiency in the multiplicative conceptual field. educational research and evaluation, 17(5), 387−407. http://dx.doi.org/10.1080/13803611.2011.632661 matters, g. (2009). a problematic leap in the use of test data: from performance to inference. in c. wyatt-smith, & j.j. cumming (eds.), educational assessment in the 21st century: connecting theory and practice (pp. 209−225). dordrecht: springer. http://dx.doi.org/10.1007/978-1-4020-9964-9_11 rasch, g. (1960/1980). probabilistic models for some intelligence and attainment tests (expanded edition with foreword and afterword by b.d. wright). chicago, il: university of chicago press. steinbring, h. (1998). elements of epistemological knowledge for mathematics teachers. journal of mathematics teacher education, 1, 157−189. http://dx.doi.org/10.1023/a:1009984621792 thijs, a., & van den akker, j. (2009). curriculum in development. enschede: netherlands institute for curriculum development (slo). usiskin, z. (2007). would national curriculum standards with teeth benefit u.s. students and teachers? ucsmp newsletter, 37, 5−7. available from http://d75gtjwn62jkj.cloudfront.net/37.pdf van wyk, j., & andrich, d. (2006). a typology of polytomously scored items disclosed by the rasch model: implications for constructing a continuum of achievement. in d. andrich, & g. luo (eds.), report no. 2 arc linkage grant lp0454080: maintaining invariant scales in state, national and international assessments (n.p.). perth: murdoch university. vergnaud, g. (1988). multiplicative structures. in j. hiebert, & m. behr (eds.), number concepts and operations in the middle grades (pp. 141−161). hillsdale, nj: national council of teachers of mathematics. vygotsky, l.s. (1962). thought and language. cambridge, ma: mit press. http://dx.doi.org/10.1037/11193-000 wendt, h., bos, h., & goy, m. (2011). on applications of rasch models in international comparative large-scale assessments: a historical review. educational research and evaluation, 17(6), 419−446. http://dx.doi.org/10.1080/13803611.2011.634582 wilson, m. (2005). constructing measures: an item response modeling approach. london: lawrence erlbaum. wright, b.d. (1997). a history of social science measurement. educational measurement: issues and practice, 16(4), 33−45. http://dx.doi.org/10.1111/j.1745-3992.1997.tb00606.x wright, b.d., & stone, m.h. (1979). the measurement model. in b.d. wright, & m.h. stone (eds.), best test design (pp. 1−17). chicago, il: mesa press. wright, b.d., & stone, m.h. (1999). measurement essentials. wilmington, de: wide range, inc. wyatt-smith, c., & gunn, s. (2009). towards theorising assessment as critical inquiry. in c. wyatt-smith, & j.j. cumming (eds.), educational assessment in the 21st century: connecting theory and practice (pp. 83−102). dordrecht: springer. http://dx.doi.org/10.1007/978-1-4020-9964-9_5 footnotes top ↑ 1. for detailed descriptions of assessment types and a coherent framework, see black (1998).2. see thijs and van den akker (2009) for descriptions of curricula at the macro, meso and micro levels. 3. we consider the terms formative and summative assessment not as referring to discrete entities, but as depicting points on a continuum. assessment moments may have elements of both kinds. 4. we distinguish here between a highly informative instrument and an instrument which through rigorous analysis and revision may be regarded as valid and fit for purpose. 5. these divisions may be the intentions of the test designers. in practice this balance is difficult to achieve. 6. statistical techniques to resolve or account for issues of item dependence across replications of a single instrument for a particular cohort of learners are possible, and even necessary, to ensure validity of results (see andrich & marais, 2012). 7. the answers to constructed response items in a systemic test set are often found to be partly correct, thus supporting vergnaud’s (1988) notion of ‘concepts-in-action’. the transition from localised concepts-in-action to formal and generalisable concepts is the challenge of mathematics education. 8. the model was developed by georg rasch in the 1950s in order to solve an educational dilemma: that of measuring reading progress over time with different tests (rasch, 1960/1980). equating and linking of tests over time, initiated in the 1950s, are examples of the immense power of the rasch model. 9. this section may be omitted on first reading, but readers are encouraged to become familiar with the underlying mathematical logic of the rasch model. 10. there is a technical reason for setting the item mean equal to zero. a simple explanation is that there needs to be one arbitrary origin for all item difficulties because the data can only inform us about differences between item parameters in equation 1, hence differences between person and item parameters. 11. the software, rumm2030 (andrich, sheridan & luo, 2011), a programme designed to support the features and requirements of the rasch measurement model, has been applied here. 12. the terms ‘ability’ and ‘proficiency’ are both used to describe the location of persons. proficiency is the preferred term as it denotes a current state rather than an innate condition. policies for enhancing success or failure?  a glimpse into the language policy dilemma  of one bilingual african state  clement dlamini  university of the witwatersrand  dlaminicl@yahoo.co.uk  this paper is an attempt to extend the debate on language policy development, which so far  has been dominated by african  linguists and  language policy planners. little attempt has  been made in the mathematics education field to highlight the discriminatory nature of some  language  policies,  be  they  national  or  institutional  polices.  dominant  societies  have  used  language  to discriminate against other minority groups  in numerous societies all over  the  world.  most  research  studies  on  language  as  a  linguistic  capital  have  been  conducted  by  members  of  the  privileged  groups  and  the  recommendations  that  have accompanied  such  studies do not provide a practical solution as  to how  the  ‘suppressed’ minorities could be  ‘liberated’ and be able to participate in their societies in a meaningful way. while language  policies have affected immigrants in most european countries, it is a different story in africa.  in africa the people most affected by discriminatory policies are the indigenous population.  the paper presents a case study of one country in africa where the language policy has been a  big  obstacle  to  indigenous  learners’  quest  to  gain  access  to  tertiary  education.  the  paper  compares learners’ achievement in english language and mathematics. i argue in this paper  that proficiency in english language does not necessarily mean success in mathematics.   from monolingualism to bilingualism and finally to multilingualism is a journey that many african scholars have travelled, not by choice but propelled by their quest to gain access to social goods. gee (1999) uses the phrase ‘social goods’ to mean anything that constitutes a source of power, e.g. money, qualifications, social status, etc. in some contexts, african children are born monolingual, communicating only in their maternal language. they are then initiated into a second language, the language of instruction, and finally into languages of particular discourses, e. g. the language of mathematics. in all professions and disciplines there are specific languages that give a certain identity to that particular discourse. in most cases the specific language of the practice or discipline is conveyed in a natural language which should be mastered before the discipline is mastered. in most african countries masters of professional discourses have had to master the natural language in which the discourse is conveyed as well as the language of the practice in which they are engaged in. for example, in francophone and anglophone africa french and english are media of instruction in schools and second languages for a majority of learners in most former colonies of france and britain. the question of why second languages have been used as the media of instruction has been debated by many scholars over the years. some have put blame on africa’s former colonial powers and poor educational language policies that have been drawn up by the newly independent african states after attaining independence (alexander, 2001; alidou, 2001; bamgbose, 1999; bokamba, 1995). almost all previously colonized african countries have opted for the language of the colonizing power for various reasons, such as unification of different tribes in cases of multilingual countries and cost effectiveness in adopting an already established language of government, business and instruction, thus endorsing these languages as languages for development (alexander, 2001; wodak & corson, 1997). these languages have been elevated to the status of a capital and therefore have become social pythagoras, 67, 5-13 (june 2008) 5 policies for enhancing success or failure? goods that every citizen endeavours to gain access to. english language has become the linguistic capital (barwell, 2002; zevenbergen, 2000) in anglophone countries and a legitimate medium of instruction in schools. according to edwards (1994) and paulston (1999), cited in barwell (2002), political decisions determine the roles of language in education. they assert that the questions of who gets which language in school “are bound up with issues of access, power and dominance” (p. 206). this leads to what barwell (2002) refers to as ‘linguistic discrimination’ where some languages are favoured over others. this discrimination creates a situation where the dominant language becomes a form of capital (zevenbergen, 2000). who is now dominating who in independent african countries where governments are composed of people whose mother tongue is not english and yet in these countries english is the dominant language? while difficulties of learning in a second language have been documented in different studies both in the field of linguistics and the field of mathematics and science education (cummins, 2001; setati & adler, 2001; moschkovich, 1999; zevenbergen, 2000), and while there has been documented successes in bilingual education programmes (enge & chesterfield, 1996; komarek, 2000; okombo & rubgumya, 1996), there has not been any significant changes in language policies of most african states. hence, the difficulties faced by africans in accessing social goods have remained tied to language. for example, the language policies still hold the assumption that proficiency in the language of instruction at school level is an indicator of success in tertiary education. a number of african universities require or consider proficiency in the language of instruction as one of their entry requirements. this belief has also been reinforced by studies that have suggested that, in mathematics for instance, proficiency in the language of instruction is a factor in the acquisition of mathematical concepts and therefore influences the degree of achievement in mathematics (e.g. fernadez & neilsey, 1986; howie, 2002; mestre, 1988; secada, 1992; young, 1988;) that is, in cases where english is the language of learning and teaching (lolt), language proficiency has been associated with mathematics achievement. a number of researchers who have been involved in bi/multilingual research have gone beyond the issue of language proficiency and have looked at other factors that contribute to the failure or success of mathematics teaching and learning in schools. these researchers have actually looked at mathematics as a language that has its own register and have explored practices that distinguish mathematics from other disciplines and looked at ways of how the discourse practices can be used to enhance success in mathematics (e.g. barwell, 2003, in britain; gorgio & planas, 2001, in spain; moschkovich, 2002, in the united states of america; setati & adler, 2001, in south africa; zevenbergen, 2000, in australia). this paper argues for revision of language policies that have since placed indigenous citizens at a disadvantage probably because they are a barrier for access to social goods. the paper argues that policies designed through the notion of language proficiency as an indicator of success in schools and in the world of work are in fact hindering development and excluding people from participating in the development of their countries. this argument is supported by data (which is presented later) that suggest that some language policies can exclude indigenous people from accessing social goods, such as gaining access to tertiary education, to jobs, and other economic benefits they should be enjoying in their own countries. the data is presented here as an example of how one county in africa is ‘wasting’ the cream of it’s would be intellectuals through what i perceive to be unsound language policies. this country is an example of such cases in africa where the learners’ home language is not the language of instruction and where the language policy has excluded a majority of capable indigenous people from participating in the technological and scientific fields of development. acting on the premise that language proficiency is an indicator of mathematics achievement, data is presented as a point of debate as to how these studies hold in the context of african countries. an analytical glance at the effect of language proficiency on other disciplines most african countries are plagued by unemployment, diseases, high levels of poverty, the use of primitive technology, and underdevelopment among a host of other problems. although there are numerous factors that have contributed to these problems, most of them have been a result of poor national policies. most policies seem to have been adopted from their colonial masters and hence seem to be a continuation of the assimilation policies of the west. the 6 clement dlamini language-in-school policies, which are most often driven by national language policies, have proved to be the major source of underachievement of indigenous africans because of the use of a second language in teaching and learning environments. the use of a second language implies that indigenous learners first have to learn the language of instruction before they can master the discipline of the learning area. it is, therefore, logical that proficiency should play a major role in the academic achievements of learners in any learning discipline. the findings of the studies mentioned earlier do not come as a surprise. it was probably a result of these studies that influenced the design of most language policies in africa. however, a number of these studies have only focused on low achieving learners and not high achievers. another weakness of these studies is that proficiency has not been adequately discussed and defined to such an extent that it is understood in the context of the study in question. it is therefore difficult to ascertain what language competencies a learner should possess to be seen to be proficient enough in the language of instruction. at what level of schooling or of the education system does language proficiency matter? most studies that have shown a relationship between mathematics achievement and proficiency in english, for example, were conducted at the primary school level where most indigenous learners are still learning the language of instruction. at this level it is expected that learners will not be masters of a discipline that is in the language they are still learning. in this paper i argue that although lack of language proficiency may affect one’s chances of succeeding in other disciplines, it does not necessarily cause low performance in mathematics. i present an analysis of the general certificate of education (gce) examination results in swaziland which shows that high achievers in mathematics have low proficiency (assuming that the gce examinations are a measure of proficiency) in the language of instruction. i begin my argument by presenting the language policies of the country and its tertiary institutions’ admission policies and show how these policies have succeeded to strangulate the country’s development in terms of possible technological advances and curbing unemployment through the training of the cream of the product of its school system. swaziland’s national language policy and language-in-school policy although swaziland is fast becoming a heterogeneous society, it is still referred to as monolingual with siswati being the only indigenous language. sizulu is also spoken in southern swaziland. when swaziland was colonized by the british in the early 1900, english became the language of business and writing. that is, all written communication was only in english. at the time, all indigenous languages in the country had no written register. after gaining independence in 1968, the status of siswati was elevated to the status of an official language and this status has remained at that since then. although siswati and english are official languages, almost all government correspondence and communication is in english, notwithstanding the fact that more than 90% of swazis speak fluent siswati (mbatha, 2001). according to mbatha (2001) the hegemony of english is what has actually retained english as an official language, unlike in other african multilingual countries where it was viewed as a language of national unity. in swaziland it was retained as an official language because it is a language of wider communication, a lingua franca in dealing with other countries. mbatha (2001) contends that it was retained because it was viewed as a language of development, which has a large body of knowledge and also because it is a “symbol of civilized, educated people” (p. 89). the early british system of education influenced the present status of english as a language of communication in that all government communication, such as parliament bills and laws of the country are in english. whenever a need for translation arises, such translations are normally from english to siswati and not vice versa. the swaziland education system follows a transitional bilingual-education policy with a weak emphasis on siswati. transitional bilingualism is a type of bilingualism that allows the first or main language to be used as a medium of instruction and as a tool to facilitate the transition to the medium of instruction (garcia, 1997), which is english. while the national language policy suggests that siswati and english have equal status, in practice this is not the case. the transitional-education policy is somewhat contradictory to the national policy in that it elevates english to a higher status than siswati. 7 policies for enhancing success or failure? officially, siswati is the medium of instruction in grades 1 and 2 only, while english is the medium of instruction from grade 3 up to tertiary education level. however, in practice there is a strong pull towards english even in grades 1 and 2. this is because there are no guidelines for using siswati as a medium of instruction. this weak emphasis on siswati is worsened by the fact that all teaching and learning materials are written in english from grade 1 to tertiary education level. english language, as a subject, is a hurdle for failure in schools. that is, government policy states that a child who fails english language is deemed to have failed the grade level and should not proceed to the next grade. it is thus known locally as a ‘failing subject’. in school reports you would find comments like “failed because of english language”. assumptions architects of this policy had noble intentions of forcing the learner to pass english language at an early stage of the learning process so that the learner is able to face real world challenges in the world of work. however, what this policy assumes is that a child who fails english language in grade 3 cannot cope with demands of english language and other subjects in grade 4. actually, it suggests that the complexity of english language and other subjects increase with each year of study. it also suggests that the level of english of the subject in a subsequent grade is so complex that a child who passes all other subjects and fails english will not have ‘enough english’ to succeed in the following grade level. these policies are also extended to tertiary institutions. in the university of swaziland, for example, a good pass in english language is a requirement for admission into most faculties. for example, to pursue a bachelor of science (bsc) degree, a credit grade (c or better) in english language in the general certificate of education (gce) is among the entry requirements. again this requirement suggests that a bsc degree requires proficiency in english and that the level of proficiency that enabled a learner to pass other subjects in the gce is not good enough for university mathematics and science programmes. another assumption made is that english language as a subject is the only subject where learners can learn all the necessary language skills and that the english language examinations are an effective measure of proficiency. it is probably why the university of swaziland selects english language as one of the indicators of success in the bsc programmes. shocking statistics the statistical results presented here were completely unexpected considering the fact that studies have suggested that there is a strong correlation between mathematics and english language. although most of these studies were conducted in europe and america, other studies conducted in africa have also alluded to this finding (e.g. howie, 2002). my analysis has been generated from data that was drawn from an ongoing study that i am currently conducting in swaziland. the larger study is conducted in one school where the majority of learners are indigenous swazis. all indigenous swazis are second language speakers of english. the analysis that is presented in this paper is based on data that was collected even before the main study began to find out how learners in swaziland fared in english and mathematics, the two subjects that are considered in swaziland to be gateways to higher learning. the data was solicited from the examination council of swaziland (ecos). the data that was collected was the gce grades for english language and mathematics for 2006 for all the candidates of the school where the larger study was conducted, and for all high achieving mathematics learners in swaziland. this data was collected to find out how the results of the learners of the school compare with the national english language and mathematics grades. since the study was about mathematics high achieving learners, all grades of candidates in the school and in the whole country who had obtained a credit grade (see table 1) in mathematics were selected together with their corresponding english language grade. these results were analysed using the excel spreadsheet. the number of mathematics grades and their english language grades were counted and represented as shown in tables 2 and 3 below. the correlation coefficients for 2006 for benguni1 (pseudonym) school candidates and for the national candidates were calculated to show the degree of 1 benguni school has large enrolment of indigenous learners. the school is where the major study is being conducted. data from this school is used here because discourse practices and achievement patterns in this school may be similar to other typical schools where indigenous learners are a majority. furthermore, schools like benguni are a majority in swaziland. 8 clement dlamini 9 correlation between the two subjects. the formula for calculating the coefficients that is used in the excel spreadsheet is it further shows that three candidates obtained an upper a grade in mathematics. this is 10% of all the a’s in swaziland for the year 2006 (see table 3). the three candidates were among the top 1.6% candidates who obtained an upper a grade in mathematics. the other two candidates who obtained an upper a grade in mathematics and a credit grade in english language were among the only 24 candidates (1.3%) in the whole country who achieved this feat. one of these candidates was the only one who obtained a 1 grade in mathematics and 5 grade in english language in the whole country. this is a remarkable achievement by the country’s standards. ∑ ∑ −− −− −− −− = 22 )()( ))(( yyxx yyxx r where r is the pearson product-moment correlation, and x and y are scores of the two subjects while − x and are the means. this means that excel divides the sums of products of the deviation of an x value from its mean and the deviation of a y value from its mean by the product of the squared deviations. − y table 2 shows the mathematics grades and their corresponding english language grades for benguni school in 2006 and the national grades of the subjects for the same year. the number of learners who did not gain a credit in english language is calculated and then expressed as percentage of the total number of candidates. further analysis reveals the number and percentage of learners that obtained exceptional results in mathematics but did not gain a credit in english language. figures 1-3 show diagrammatic representations of the grades as shown by the plots. table 2: a comparison of the 2006 english language and mathematics grades for benguni high school candidates number of corresponding english language gradesmathematics symbols 1 2 3 4 5 6 7 8 9 totals 1 0 0 1 0 1 0 0 1 0 3 2 0 0 0 0 0 0 0 0 1 1 3 0 0 1 2 3 6 5 2 0 19 4 0 0 0 0 0 0 5 5 0 10 5 0 0 0 0 0 2 4 1 2 9 6 1 0 2 1 2 2 8 5 1 22 totals 1 0 4 3 6 10 22 14 4 64 candidates who did not gain a credit in english language = 40 (62.5%) candidates who obtained a or b in mathematics but no credit in english language = 19 (29.7%) correlation coefficient = 0.014 also shown in the table is that there is one candidate who did not gain a credit in english language. actually, this candidate gained credit grades in all the other subjects (two other upper a’s and three b’s with an aggregate2 of 17). table 2 is re-represented in figure 1. table 1 shows the gce o-level grading system. the table shows that grades 1-6 (equivalent to a-c) are categorized as a credit, 7-8 (equivalent to d-e) are a pass and 9 (equivalent to u) is a fail. the following tables show comparisons of english language and mathematics grades in the gce olevel examinations for 2006. figure 1 shows that for all the a, b and upper c grades in mathematics (square dots) all the corresponding english language grades occupied an upper y-value in the graph. the positioning of the english language grades in the graph shows that in this school the best mathematics achievers did not get an english language grade that was better than their mathematics grade. a similar analysis can be made for table 3. table 2 shows that there were 64 candidates in this school who obtained a credit grade in mathematics. table 1: the gce o-level grading system numerical symbol alphabetical symbol 1 2 upper a lower a 3 4 upper b lower b 5 6 upper c lower c 7 d 8 e 9 u ungraded 2 an aggregate is calculated by adding the best grades (including english language) from each subject group. policies for enhancing success or failure? 10 9 8 7 6 symbol 5 4 3 english maths 2 1 0 10 20 30 40 50 60 700 learners with a credit in mathematics figure 1: the 2006 english language and mathematics grades for benguni high school candidates table 3: a comparison of the gce 2006 national english language and mathematics grade number of corresponding english language symbols an analysis for the national grades of all candidates who had obtained a credit in mathematics for the years; 2001, 2002 and 2003 showed that, in 2001, out of a total of 965 learners who had gained a credit in mathematics, 604 (62.6%) did not gain a credit in english language. in 2002 and 2003, 66.6% and 63.9% respectively did not gain a credit in english language. what is of major concern is that about 20% of candidates who obtained exceptional grades (a and b) in mathematics were consistently not gaining a credit in english language in all the years for which the analysis was done. in 2001, 2002, 2003 and 2006, 17.7%, 18.7%, 16.1% and 18.6% respectively did not gain a credit in english language. the analysis presented above shows that: • one fifth of the best mathematics high achieving learners in swaziland do not gain a credit in english language. • a majority (more than 60%) of high achievers in mathematics do not gain a credit in english language table 4 shows the correlation coefficients that were calculated to find the linear relationship between the mathematics grades for high achievers in mathematics and their corresponding english language grades. the analysis shows a low positive correlation of about 0.3. it also suggests that the mathematics symbols 1 2 3 4 5 6 7 8 9 totals 1 9 4 5 3 1 2 4 2 30 2 9 8 15 9 7 12 24 9 3 96 3 20 22 45 24 29 58 94 64 16 372 4 10 6 23 14 13 25 67 44 17 219 5 13 6 25 13 17 44 100 58 31 307 6 12 15 46 37 38 94 285 215 83 825 totals 73 61 159 100 105 235 574 392 150 1849 candidates who did not gain a credit in english language = 1116 (60.35%) candidates who obtained a or b in mathematics but no credit in english language = 344 (18.6%) correlation coefficient = 0.2827 10 clement dlamini relationship between the two subjects was not affected by a change in population. for example, the doubled population of the 2006 cohort of 1849 from the 2001 cohort of 965 candidates did not introduce any significant difference in the orrelation coefficient. table 4: co ts in rs to can coefficient c rrelation coefficien tal number of different yea correlation year didates 2001 965 0.2980 2002 1217 0.2899 2003 1114 0.3422 2006 1849 0.2827 the correlation coefficient r is an index that ranges between -1 and +1. this index reflects the extent of a linear relationship. if r tends towards -1 it means there is a negative correlation between two variables. in this case that would mean candidates who obtain high grades in mathematics would tend to obtain low grades in english. if the index is zero that means there is no relationship between the variables being compared. this analysis has shown that the correlation coefficients are consistently closer to zero for the periods where data was provided, which suggests that there is a low correlation between wer th national coefficients shown in table 4. mathematics and english in swaziland. the scatter plots in figures 2 and 3 illustrate a low correlation. the correlation coefficient between the scores was calculated at 0.014, which indicates a close to zero correlation between the two subjects. this correlation coefficient is far lo an the • high achievers in mathematics may not be high achievers in english language in summarizing the findings i should point out that the data presented in tables 2 and 3 and figures 1-3 reflect a true reality of the swazi situation. sampling errors were eliminated by taking the whole population of learners in both the school and the national statistics. the data presented here suggests that high achievement in mathematics is not solely influenced by high achievement in english language. the low correlation coefficients for benguni mathematics high achievers suggest that there may be no correlation between english language and mathematics grades of swazi indigenous learners. the data has thus prompted the following conclusions: • english language is a poor predictor of future performance for learners in swaziland in the mathematics field of study 0 1 2 3 4 5 6 7 mathematics 0 1 2 3 4 5 6 7 8 9 10 english figure 2: scatter plot of mathematics and english language grades of benguni high school for 2006 figure 3: scatter plot of the gce 2006 national english language and mathematics grades 0 10 0 1 2 3 5 6 7 lish languageeng 9 8 7 6 5 4 3 2 1 4 mathematics 11 policies for enhancing success or failure? it should be noted here that although correlation coefficients measure the degree of linearity between two variables, it does not measure causality. in this case it does not indicate that high performance in mathematics causes low performance in english language. however, the trend as presented above seems to suggest that candidates in swaziland who do well in mathematics may have low proficiency in english language. conclusion the policy of higher education institutions of accepting only a credit in english language is a severe blow to the country’s development as the country is losing more than 60% of potentially capable citizens through this policy. the results discussed here probably hold for similar situations in other african countries. as mentioned earlier, the studies that associated proficiency in english with mathematics achievement were conducted in countries and in environments that were quite different from the swazi situation. the results have shown that there is need to revise or enforce language policies that are aimed at enhancing development in various countries. the school policies of failing learners because of failing a second language are counterproductive. they seem to discriminate against the indigenous populations. the data presented earlier suggests that the language-in-school policy in swaziland is successful in discriminating against the indigenous population as it discards more 60% of mathematically capable citizens yearly. the indigenous population is eliminated in the school system as they increase the repetition rates and eventually drop out of school, thus denied participation in the national development of their countries. the low correlation between english and mathematics is proof that the language policies in africa were indeed poorly planned, and that they were possibly based on empirical evidence from the west. the mere fact that a good pass in english language is an entry requirement into mathematics and science programmes in some institutions of higher learning in most anglophone states is an indication that it could have been based on an assumed high correlation between these disciplines. while this assumption may be true in some contexts, the findings in this paper suggest that it is not always true in some african contexts. the results have also shown that if the language policies of countries where the medium of instruction is in a second language are left unchecked, they will continue to discriminate against indigenous populations who will always be denied access to social goods. references alexander, n. 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"cracking the code" of mathematics classrooms: school success as a function of linguistic, social, and cultural background. in j. boaler (ed.), multiple perspectives on mathematics teaching and learning. london: ablex publishing. << /ascii85encodepages false /allowtransparency false /autopositionepsfiles true /autorotatepages /none /binding /left /calgrayprofile (dot gain 20%) /calrgbprofile (srgb iec61966-2.1) /calcmykprofile (u.s. web coated \050swop\051 v2) /srgbprofile (srgb iec61966-2.1) /cannotembedfontpolicy /warning /compatibilitylevel 1.5 /compressobjects /tags /compresspages true /convertimagestoindexed true /passthroughjpegimages true /createjobticket false /defaultrenderingintent /default /detectblends true /detectcurves 0.0000 /colorconversionstrategy /cmyk /dothumbnails false /embedallfonts true /embedopentype false /parseiccprofilesincomments true /embedjoboptions true /dscreportinglevel 0 /emitdscwarnings false /endpage -1 /imagememory 1048576 /lockdistillerparams false 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/hrv (za stvaranje adobe pdf dokumenata najpogodnijih za visokokvalitetni ispis prije tiskanja koristite ove postavke. stvoreni pdf dokumenti mogu se otvoriti acrobat i adobe reader 5.0 i kasnijim verzijama.) /hun 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can be opened with acrobat and adobe reader 5.0 and later.) >> /namespace [ (adobe) (common) (1.0) ] /othernamespaces [ << /asreaderspreads false /cropimagestoframes true /errorcontrol /warnandcontinue /flattenerignorespreadoverrides false /includeguidesgrids false /includenonprinting false /includeslug false /namespace [ (adobe) (indesign) (4.0) ] /omitplacedbitmaps false /omitplacedeps false /omitplacedpdf false /simulateoverprint /legacy >> << /addbleedmarks false /addcolorbars false /addcropmarks false /addpageinfo false /addregmarks false /convertcolors /converttocmyk /destinationprofilename () /destinationprofileselector /documentcmyk /downsample16bitimages true /flattenerpreset << /presetselector /mediumresolution >> /formelements false /generatestructure false /includebookmarks false /includehyperlinks false /includeinteractive false /includelayers false /includeprofiles false /multimediahandling /useobjectsettings /namespace [ (adobe) (creativesuite) (2.0) ] /pdfxoutputintentprofileselector /documentcmyk /preserveediting true /untaggedcmykhandling /leaveuntagged /untaggedrgbhandling /usedocumentprofile /usedocumentbleed false >> ] >> setdistillerparams << /hwresolution [2400 2400] /pagesize [612.000 792.000] >> setpagedevice abstract introduction exemplification with variation and its significance in mathematics pedagogy research methodology and data analysis sequencing, pairing, simultaneity and juxtaposition of examples the teacher and learner actions on the example set – its mediation concluding discussion acknowledgements references footnotes about the author(s) vasen pillay school of education, faculty of humanities, university of the witwatersrand, johannesburg, south africa jill adler school of education, faculty of humanities, university of the witwatersrand, johannesburg, south africa ulla runesson kempe school of education, faculty of humanities, university of the witwatersrand, johannesburg, south africa school of education and communication, jönköping university, jönköping, sweden citation pillay, v., adler, j., & runesson kempe, u. (2022). the sequencing and pairing of examples in the midst of sameness and difference: opening opportunities to learn. pythagoras, 43(1), a667. https://doi.org/10.4102/pythagoras.v43i1.667 original research the sequencing and pairing of examples in the midst of sameness and difference: opening opportunities to learn vasen pillay, jill adler, ulla runesson kempe received: 18 nov. 2021; accepted: 11 may 2022; published: 01 nov. 2022 copyright: © 2022. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract the teaching of mathematics cannot be thought of without considering the use of examples. the examples that teachers use during a lesson and how they mediate the example set is critical to what opportunities for learning are opened up during the lesson. in this article, we explore how a teacher mediates an example set with focus particularly on what is varied and what remains the same. the case that we draw on is taken from a larger learning study conducted in grade 10 mathematics classes and the lesson that is used in this article was the last lesson in the learning study cycle. we use variation theory, specifically how the sequencing, pairing and juxtaposing of examples provides learners with opportunities to discern the critical aspect of the object of learning. we analyse the teacher’s mediation of the example set on a micro level, as this enables us to illuminate and develop our argument, while simultaneously offering a detailed example of mathematics teaching. we argue that it is the systematic and deliberate structuring of variation within an example set in the midst of invariance coupled with the teacher’s mediation of both planned and learner-generated examples that is critical for opening opportunities to learn. keywords: critical aspects of object of learning; learner-generated examples; sequencing, pairing and juxtaposing examples; variation theory. introduction imagine thinking about and then teaching a mathematical concept (e.g. quadratic functions) without any appropriate examples (of quadratic functions). attempting such a task would be impossible, to say the least. the use of examples forms an integral part both of doing mathematics and in the teaching and learning of mathematics. as goldenberg and mason (2008) have argued, examples can be seen as: cultural mediating tools between learners and mathematical concepts, theorems, and techniques. they are a major means for ‘making contact’ with abstract ideas and a major means of mathematical communication, whether ‘with oneself’, or with others. examples can also provide context, while the variation in examples can help learners distinguish essential from incidental features and, if well selected, the range over which that variation is permitted. (p. 184) goldenberg and mason’s paper was part of a special issue of educational studies in mathematics dedicated to the role and use of examples (bills & watson, 2008, p. 86). research related to examples in mathematics education has evolved considerably over the past decade. for example, there was a special issue of zdm mathematics education devoted to investigations on examples in mathematical thinking and learning (antonini, presmeg, mariotti, & zaslavsky, 2011) and, more recently, a special section in the journal of mathematical behavior has focused on the roles and uses of examples in conjecturing and proving (zaslavsky & knuth, 2019). across this work, as studies attended to the use and roles of examples in mathematics teaching, research on how teachers integrate examples into their teaching remained in the background (zodik & zaslavsky, 2008). consequently, less was understood and known about how teachers use examples to provide their learners with opportunities to discern critical aspects of mathematical concepts, theorems and techniques. research on teaching and teachers’ use of examples is now gaining increasing attention (e.g. adler & pournara, 2020; al-murani, kilhamn, morgan, & watson, 2019; essien, 2021). of interest to us is that this focus on teaching, in varied ways, brings together research on example use with considerations of variation, informed by ‘variation theory’ (e.g. marton, 2015) and the illumination of structure and generality through variance amid invariance in mathematics education (e.g. watson & mason, 2006a). our article contributes to this growing research field. through a study of a selected mathematics lesson, we build an argument for the value of sequencing and pairing examples in the midst of sameness and difference in opening up opportunities to learn. to both locate the article and illuminate our theoretical orientation to studying and working on mathematics teaching, we begin with a brief introduction to variation theory and a review of literature on exemplification with variation in mathematics teaching. we then describe the context and wider research from which we have drawn the teaching case we present. in line with morris and hiebert’s (2011) call for the need of instructional products that are at an appropriate grain size and sufficiently detailed to inform teaching, we focus on and analyse a particular lesson on a micro level, as this enables us to illuminate and develop our argument, while simultaneously offering a detailed example of mathematics teaching. exemplification with variation and its significance in mathematics pedagogy the theoretical framework for this study is variation theory (marton, 2015; marton & booth, 1997; marton & pang, 2006; runesson, 2005). variation theory states that learning is a change in ways of experiencing. the way something is experienced is a function of discernment of what is to be learned, the object of learning, and particular aspects that are critical for learning something in a certain way. however, the discernment of an aspect presupposes an experience of variation of that aspect. that which is varied against a stable background is more likely to be discerned.various studies on how variation can be used to enhance learners’ learning (e.g. marton, 2015; marton & pang, 2006) indicate that how the content is handled and what aspects are made possible to discern by opening those aspects as dimensions of variation in a lesson affects what is made possible to learn. furthermore, variation theory asserts that we learn by seeing how things differ rather than how they are similar. therefore, to facilitate learning, contrasts need to precede similarities (marton, 2015). when two instances are different and compared, it is possible to notice distinguishing features among them. for instance, if the intention is to help students to discriminate between integers by finding the largest number among 3, –3, –18, these examples of numbers afford additional learning possibilities compared with the examples 6, –1, 3. in the latter it is possible to give the correct answer by just looking at the digits (1 is smaller than 3 and 6). in the former, however, where two of the digits are the same (invariant) attention is drawn to the absence or presence of the negative sign (the signs vary). furthermore, by choosing the number –18, one generally held idea from the natural numbers (18 is bigger than 3) is challenged. hence, from a variation theory perspective, to facilitate learning, the character of the variation presented and what then is possible to notice are important. in the same way, watson and mason (2006a) have argued that it is the kind of variation in the midst of what remains the same (is invariant), embedded in a set of examples presented, that is crucial for mathematics learning. through variation amid invariance, students can come to see patterns, and to see possible underlying structure or generality across the example set, both of which are central to knowing and doing mathematics. in this article, we examine a planned example set that included various representations of linear, quadratic, hyperbolic and exponential functions, as well as learner-generated examples. we focus on their pairing and sequencing through the enactment of the lesson and how these work to open opportunities for learning. following the work of watson and mason (2002) we take examples to include: anything used as raw material for intuiting relationships and inductive reasoning; illustrations of concepts and principles; contexts which illustrate or motivate a particular topic in mathematics; and particular solutions where several are possible. (p. 4) in discussing the place of examples in mathematics teaching, rowland, thwaites and huckstep (2003) distinguish between two different uses of examples. the first use is ‘inductive’ and this is described as providing (or motivating learners to provide) ‘examples of something’, where the ‘something’ is general in nature (like the quadratic function of the form: y = ax2 + bx + c). providing examples of a quadratic function (like: y = –2x2 – 5x + 3) is a particular instance of the generality. therefore, ‘we teach a (general) procedure by a (particular) performance of that procedure’ (rowland et al., 2003, p. 86). looking at the particular instance of the quadratic function used here, the value of the coefficient of x2 is significant. it is not only a key feature of the function, but, more critically, if the coefficient of x2 was set to 0, then we move out of the realm of quadratic functions and into the class of linear functions. knowing what aspects of the ‘general procedure’ could change, as indicated above, is what marton & booth (1997) called ‘dimensions of variation’. being aware of what values to use and what values not to use is usefully articulated by watson and mason (2006a) as the ‘range of permissible change’. thus, when selecting and using examples in mathematics teaching, awareness of the dimensions of variation and the range of permissible change become significant in terms of bringing the critical aspect of the object of learning into focus with the learners. as we will see in this illustrative case, curriculum sequencing also impacts possible dimensions of variation and possibilities for range of change. the second use of examples as identified by rowland et al. (2003) is what they refer to as ‘exercises’. this use is not ‘inductive’ in nature but is illustrative and intended for practice. there are additional distinctions drawn in the literature on example use. zodik and zaslavsky (2008) distinguish planned from spontaneous examples, where the latter arise in the course of the lesson. watson and mason (2005) distinguish learner-generated examples (lges) from those planned or spontaneous, but by the teacher, and the pedagogical value of eliciting lges. our focus in this article is on inductive example use. this includes the accumulated example set (planned, spontaneous and lges) during a lesson and, critically, together with the teacher’s mediation of the example set. we thus turn our attention to literature pertinent to this focus. kullberg, runesson and mårtensson (2013) have shown how the same task used by different teachers in a lesson levered different opportunities for learning. drawing on principles of variation theory, they ascribed these different affordances to the manner in which the example set emerged in their lesson in terms of sequencing, aspects that were juxtaposed and contrasted, and aspects of the content that were made explicit by the teacher. in the lesson we examine, we focus on the sequencing of the examples over the lesson, with particular attention to the pairing of examples within the sequence and their enactment and mediation by the teacher. the significance of the character of an example set in opening opportunities for learning has been reinforced by further research. both kullberg, runesson kempe and marton (2017) and al-murani et al. (2019) used principles of variation to study lessons designed and taught by teachers who themselves had been introduced to such, although in different ways. kullberg et al. studied a lesson of a teacher before and after he had participated in the processes of a learning study (pang & marton, 2003). they show that while both lessons were focused on linear equations, opportunities to learn that were made available were substantively different. in the first lesson, the patterns of variation of the examples drew attention to solving an equation. in the later lesson, attention was drawn to the meaning of equality and a solution – and thus very different critical aspects of working with equations. al-murani et al’s. study took place in the context of recent changes in primary mathematics education in the united kingdom, where lesson design has been influenced by principles of variation. they selected three lessons that were publicly available. while each lesson had a different object of learning and example set, they used principles of variation to examine what was variant and invariant, and what the teacher drew attention to. they showed how variation designed into the example sets together with the enactment of these in the lessons shaped what was made possible to learn. while these two studies had different goals, both illustrated what al-murani et al. describe as the ‘power’ of variation as an analytic tool for examining conceptual opportunities for learning in mathematics lessons. both also showed and then emphasised how what is made available to learn was a function of both the example set and what was brought into focus (i.e. mediated) by the teacher with the learners. in the lesson we examine, we couple a presentation of the example set with the actions of the teacher and learners as they engaged with the examples over the lesson. the lesson takes place in the south african context, and in a classroom context quite different from those in the studies discussed above. that there are merits to teaching with variation has been argued in the south african context (mhlolo, 2013). more recently, specific studies in south africa focused on exemplification with variation have also been reported. the recent work of adler and her colleagues draws substantively on the notion of variation as being significant for exemplification in mathematics teaching (adler & ronda, 2017; adler & venkat, 2014). their work turns the spotlight onto the deliberate and judicial use of examples by teachers to provide their learners with opportunities to discern the critical aspect of an object of learning by emphasising ‘variation amidst invariance’ (watson & mason, 2006a), ‘sameness and difference’ (marton, runesson, & tsui, 2004), and simultaneity and juxtaposition (kullberg et al., 2017). in their teacher development work, and grounded in research in the field, adler and pournara (2020) have worked with teachers on describing the object of learning. it is against this object of learning that they examined example sets that they have designed or taken from textbooks or from prescribed lesson plans to determine what is made possible to learn, paying attention to the sequencing, juxtaposition and pairing of examples. their work has not extended to how these specific uses of examples might be mediated in actual classrooms. essien (2021) has described the use of examples in teaching as a mathematical practice that has particular import in teaching and learning in multilingual classroom contexts, and thus in the preparation of teachers for this work. in so doing, his focus on mediation of the example set is on interactional practices as these are critical for learner participation in mathematics. he brings a new dimension to this field of research by illustrating how exemplification with variation theory on the one hand, and attention to meaning making as dialogic process on the other, combine to illuminate their mutual significance in opening opportunities for learning. his empirical base is mathematics being taught in teacher education in south africa, where multiple teaching practices are or could be simultaneously mediated. unlike the studies discussed above, the lesson extracts analysed were not ‘theory‑driven based on variation theory’ (essien, 2021, p. 482), that is, designed with principles of variation theory. these can nevertheless be brought to bear on the constructed example set, and, following his concern, the patterns of interaction in their enactment. like the other research, he shows how possibilities for learning lie not in an example set alone, but in classroom enactment. here too, given the focus on interactional patterns, the specificity of the selection and sequencing of the examples, and how these function amid sameness and difference across the examples are not in focus. the two research questions we thus pursue in this article are: (1) what are the sequencing and pairing of examples over the lesson, and how do these open opportunities for learning? (2) how does the teacher together with his learners act on these examples to bring the critical aspect of the object of learning in the lesson into focus? through exploring and then answering these questions we hope to contribute to the field reviewed above through illustrating the value of sequencing and pairing of examples in the midst of sameness and difference, together with how these are enacted in the lesson, and why this mediation matters. research methodology and data analysis the lesson we study in this article is drawn from a wider study that focused on teachers’ mediation of a selected object of learning through their participation in a learning study (pillay, 2013). the learning study comprised four iterations of a planned and replanned lesson across four different grade 10 classes in two schools by four teachers who taught grade 10 mathematics. a qualitative case study approach was adopted to examine the lessons in detail and in depth. all of the lesson plans were collected, and each lesson video recorded and then transcribed. in this article, we focus on one lesson by a teacher, mr nkosi (pseudonym), and specifically on his selection and sequencing of examples and their mediation. the context the two schools were located in the same township within the johannesburg city municipality, an area that remains one of the poorest urban areas in south africa. infrastructural support is limited, and a high level of unemployment endures. mathematics performance in the post-apartheid south african education system remains highly differentiated, with wide gaps in achievement across race and class divisions, and, to a lesser extent, within race and class divisions (adler & pillay, 2016). in mr nkosi’s school, only 20% of the learners passed mathematics in the 2014 national grade 12 final examinations with a mark greater than 40% (dbe, 2015). it is against this context, reflective of wider conditions in many south african secondary schools, and poor performance in mathematics across grades, that we tell the story of mr nkosi’s lesson, and illustrate how he mediated the object of learning and then what eventually emerged as the critical aspect for his class of learners. the learning study process the object of learning for the wider learning study was to enhance learners’ ability to differentiate between the linear, quadratic, hyperbolic and exponential functions across their different representations (verbal, algebraic, sets and graphical). the teachers and the researcher were aware that the object of learning as initially articulated had multiple varying aspects (class of function as well their various representations). they were nonetheless content to continue, as the lessons were targeting grade 10 learners who had ostensibly already been taught the section on algebraic functions as per the syllabus requirements. after the second iteration of the lesson, as it became clear that learners were struggling with multiple aspects, the teachers collectively decided to refine the object of learning and focus only on the algebraic representation of the functions. in a wider context, one would not expect grade 10 learners to display limited familiarity and fluency with algebraic expressions and equations. however, as the lessons unfolded, these difficulties became apparent1. indeed, it was this narrowing of the object of learning that ultimately enabled the emergence of a critical aspect that could support learners to correctly classify the algebraic representation of a function into its appropriate class of function. it was in mr nkosi’s lesson, the fourth lesson in the study (evidenced in detail in pillay, 2013), that learners were provided with opportunities to discern this critical aspect and so too the more refined object of learning. hence our choice of this lesson in the article. the data as indicated, the data included the lesson plan, and the detailed transcription of the enacted lesson. the lesson plan built on the planned example set for the wider learning study, which included examples of linear, quadratic, hyperbolic and exponential functions (pillay, 2013). to investigate the sequencing and mediation of the example set, the transcript of the lesson was chunked into episodes. a new episode was identified when the teacher or a learner introduced a new example or a different form of the algebraic representation of the example (). the examples and their sequencing are presented in table 1. table 1: overview of the lesson presented in episodes. the analytic process within each sequence, the examples used were examined on a detailed level from the point of view of variation. following watson’s and mason (2006b) argument of the importance of a systematic sequence of variation horizontally as well as vertically within the set of examples, what they call going with and across the grain (p. 4), we analysed what was constant and what varied in the examples within each episode as well as between the episodes. finally, our focus was directed to how the content was mediated by the teacher. table 1 provides the lesson overview, organised into episodes (first column) by the presentation of the example set2 (second column) we present all the examples that were used during the lesson, in the order in which they were used. this presentation enables us to illustrate mr nkosi’s deliberate sequencing and pairing of examples. in the third column we describe the teacher and learner actions in the specific episode and in relation to the example set. in the commentary in the last column for each episode we draw attention to how mr nkosi used the selected examples to draw attention to sameness and difference, thus providing learners with opportunities to discern critical aspects of a selected object of learning. following the presentation of the example set in table 1, the detailed analysis, and so findings of the study, are discussed in two sections: the first focused on the sequencing, pairing, juxtaposition and simultaneity of the examples, and the second on the teacher’s and learners’ actions and so the teacher’s mediation of the example set, and of lges. sequencing, pairing, simultaneity and juxtaposition of examples taking an overall look at the examples in table 1, we can see that the representation of the functions varied (algebraic and graphical) and the set of examples is restricted by the use of the number 2, but with varying positions in the functions. this is probably critical for the possibility to learn characteristics of a class of functions. however, what we find most interesting is the pattern of variation and invariance that unfolds in the example set when variation is introduced against a background that remains stable. as noted, following watson’s and mason (2006b) argument of the importance of analysing with and across the grain (p. 4), we examined what was constant and what varied in the examples within each episode as well as between the episodes. what was found significant and reoccurring in this lesson is, on the one hand, the juxtaposition and contrast of pairs of examples and, on the other, how the change to a new class of functions happened by keeping something invariant while varying something else. pairing and contrasting – going across the grain as can be seen in table 1, reoccurring throughout the episodes in the lesson is the pairing of examples. these chosen examples bring out differences and similarities that make it possible to notice features of classes of functions and thus to distinguish one class from the others. in the first set of episodes (1–4) only the linear function is in focus with various examples presented on the board. so, the class of function is the same while the examples vary. looking at what is different and thus what is compared within each pair of examples, the gradient varies in episode 1 (y = 2x and ), while in episode 4 the fractional form of the function is different ( and ) and in episode 3, the gradients and intercepts vary (the lges). in episode 5 a new class of functions is introduced: rational functions. what was the same in the previous episodes (the power of x = 1) is now changed in episode 5. a pair of examples, one of a linear function (), the other of a rational function (), is used to bring out a contrast between linear and rational functions. these examples are then compared; what is similar and different? by varying the position of x and 2 in the two examples (i.e., juxtaposing 2 and x with their multiplicative inverse) and varying the algebraic form of into h(x) = 2 x−1, the power of x in a linear versus a rational function is made possible to notice. in episodes 1–4 and 5–6 only one class of function was handled at a time and the pair of examples taken were examples of the same class of functions. in the next set of episodes (7–13), however, two classes of functions (exponential and quadratic) are discussed and handled simultaneously. first (episode 7) a pair of examples, one quadratic, one exponential, is juxtaposed (p(x) = x2 and g(x) = 2x). within this pair of examples, the exponent varies (exponent = 2 and exponent = x) and, subsequently, the position of the variable is also varying. after having compared these two examples, in the next moves (episodes 8–11) the class is restricted to only one class (quadratic), but with various examples of quadratic functions presented on the board. however, one of the lges is not a quadratic function (episodes 9–10). this variation was brought in by a learner, and the teacher uses and compares this example (linear function) to the other quadratic functions. again, examples are juxtaposed and contrasted. sequencing and juxtaposing – going with the grain looking at how the content is presented in terms of the sequencing of the different classes of functions, it may look like ‘taking one thing at the time’, and that the learners should master one thing before learning another. however, a closer look at differences and sameness concerning the examples when a new class of functions is introduced (i.e. what changes vertically) tells us that is not the full picture. instead, the analysis shows how the move from one class to another is carefully done using variation against a stable background. this can be seen, for instance, in the move from episode 4 to episode 5 when a new class of functions (rational functions) is introduced. what was the same (the power of x) in the previous episodes is now changed. in this switch, a pair of examples is present on the board. one of the examples (, a linear function) is the same as in episode 4. the other one is a new example (, a rational function). these examples are then compared: what is similar and different? in this way, in the move from episode 4 to episode 5, when introducing a new class of functions, one of the examples of linear functions, which the learners have demonstrated to be familiar with, is picked up and used as a link to a new class in episode 5. this pair of examples serves as a contrast between the two classes of functions. by comparing the examples, the difference in relation to the power of x is made possible to discern. in this way of contrasting examples with x1 to examples of x−1, the teacher is weaving connections (ekdahl, venkat, runesson, & askew, 2018) between episodes 1–4 and episodes 5–6 and, thus, the classes of functions. similarly, in episode 6 the teacher uses an example from the previous episode 5 () but changes the constant from positive 2 into negative 2 (). in this way, another example of a rational function (hyperbola) is presented in two different forms ( and p(x) = –2x−1). another example of ‘returning’ and weaving in examples between episodes to bring out differences and similarities, is seen in episodes 7–12. in episode 7 the examples (p(x) = x2 and g(x) = 2x) are juxtaposed and contrasted (quadratic vs exponential); thus, the variation concerns two classes of functions. in episodes 8–11 only one of them is in focus (quadratic). thus, the class of functions is the same. however, in episode 12 the pair of examples (p(x) = x2 and g(x) = 2x) used was discussed and contrasted in episode 7, hence it is picked up and contrasted once again. however, whereas the comparison of the exponent in episode 7 resulted in a focus on quadratic functions, the comparison in episode 12 led to the exponential function coming into the fore of attention. this way of returning to an example previously discussed, is similar to what happened in the move from episode 4 to episode 5. here the example () from episode 4 made up a pair together with a new example (). to conclude, there is a systematic pattern of variation and invariance in the sequence, the juxtaposition and contrasting of pairs of examples and in the moves between the different classes of functions. this we infer, from a variation theory point of view, afforded the learners opportunity to experience and discern the different powers of x and the different positions of the exponent. the teacher and learner actions on the example set – its mediation as we have argued, the example set on its own, no matter how systematically organised, also needs to be mediated. we thus turn out attention now to excerpts of teacher and learner actions. mr nkosi’s mediation of the example set it was shown in the previous sections that the example set was made up of pairs of juxtaposed examples. once the examples were introduced, mr nkosi asked learners to compare the given pair of equations. here the questions ‘what is same? what is different?’ guided the learners’ attention to critical aspects of the class of functions. the results of our analysis suggest that this comparison was made in a shift from emphasising sameness to differences and vice versa. we will use two episodes about linear functions to illustrate this. in episode 1 the two equations y = 2x and were compared. it was concluded that they were different regarding how they were read and written algebraically. so, differences between the equations were in focus. next, what they had in common was attended to. the students suggested: ‘y’, ‘x’ and ‘=’ as common features. the teacher agreed and added ‘they are all equations’ but pointed out a significant difference – the coefficients: ‘two times x’ and ‘half times x’. mr nkosi concluded: ‘so, you’ll find, as you have already indicated, that’s something in common’. he circled the unknown x in each equation, and continued: ‘you’ve got two being multiplied, and again here you‘ve got half being multiplied by x’. however, he leaves this difference and comes back to what is similar – the exponent is 1: mr nkosi if i could ask you again, what is the exponent of x in both equations? yes girl? learner one in the same way, in episode 4, by the questions he asked, mr nkosi elicited differences and sameness between the examples of equations. he initially asked the learners to read the examples and . it was then concluded that they were different regarding how they were read and written algebraically: mr nkosi no, they don’t sound the same. y is equal to half x, and y is equal to x over two. we’re saying they don’t sound the same. and hence they are not the same. so, these differences between the equations were in focus. next, what they have in common was attended to by lifting a feature that was not instantly visible in the equations – the value of x. he substituted the x-values and demonstrated that although the algebraic expressions were different, they yield the same value and were the same in that respect. he asked: ‘are they the same?’ and concluded: mr nkosi they are written differently but it is …? learners the same. mr nkosi the same. in both episodes he drew the learners’ attention to sameness which goes beyond the superficial level and irrelevant commonalities (they are written differently or sound different) and eventually ended up with that which was the critical aspect – the exponent. when in episode 1 mr nkosi asked about differences between the two examples of linear functions – how they were read and written – a contrast between visible features of the equations was made. once these differences were identified, the teacher shifted and asked about sameness and concluded that the exponent was the same (1). the learners’ answers (x, y, =) indicated that these were the features they actually discerned. they did not notice that they both were examples of linear functions. this can be interpreted as attending to superficial, not essential, features of the equations. however, these are obviously what the learners paid attention to. when being asked about commonalities, no one (audibly) answered that the class of function was in common. neither did they notice the difference between coefficients (multiplying by two vs multiplying by one half). these examples demonstrate how questions of sameness and differences supported the character of the examples in the pairs. as was described previously, the set of examples was chosen so that something changed against a stable background between and within the examples. this was mediated by the teacher’s questions and actions which afforded opportunities for the discernment of the critical aspect. following principles from variation theory, stating that learning is a matter of differentiation, we would suggest that by asking about sameness and differences and successively comparing features of the equations that are not critical and, thus, can be neglected, focus on the critical aspect of linear functions (the exponent of x) emerged and came to the fore. mr nkosi’s mediation of learner-generated examples, and so of learner actions with examples watson and mason’s (2005) idea of lges, which is essentially a process where the teacher invites learners to generate examples according to specified features, is not a typical practice in south african mathematics classrooms. they argue that learners who consistently employ example generation as an integral part of their learning strategy undergo more shifts of concept image, provide better explanations, develop broader example spaces and have a more complete understanding of the taught concept. in episodes 3, 9 and 14 we observe that mr nkosi asked his learners to generate their own examples of linear equations (episode 3), quadratic equations (episode 9) and exponential equations (episode 14). by engaging his learners in the process of generating their own examples he provided them with some opportunity to assess their own understanding and it also provided him with some insights as to whether the critical aspect had come into focus for his learners. of particular interest here is the lge q(x) = 22 + x (episode 9) as an example of a quadratic function. once it was confirmed that q(x) is an example of a linear function, the learner who generated q(x) had a follow-up question. the extract below is from the lesson transcript that deals with mr nkosi’s engagement with this follow-up question: learner can i ask a question? is it … must it always have an x to an exponent two? mr nkosi right, there is the question. must it (teacher emphasises ‘it’) always have an exponent two? learner yes. mr nkosi you got the answer? learner must x always be having an exponent two? mr nkosi must this always be an x having an exponent two? right, that’s her question. must it always be an x here? right. (erases 22 + x) we are saying x squared plus two to the exponent two (writes q(x) = x2 + 22). must there always be an x here (points to x2)? learners yes. mr nkosi you are saying? learners yes. and continuing with his attention to variation amid invariance and the critical aspect of the exponent of the variation, mr nkosi followed up further: mr nkosi yes, how about if we have it as q of x (writes q(x)) … sorry, (erases q(x)) q of r equal to r squared plus two squared (writes q(r) = r2 + 22). is this not a parabola as well? learners it’s a parabola. mr nkosi it’s a parabola. because the variable here, the highest part of the variable is still two. ok? so it doesn’t restrict us to an x. it depends on which variable we have chosen. the example generated, q(x) = 22 + x, illustrates that what has come into focus for the learner is that if one of the terms in an equation is squared, then the equation represents a quadratic function. out of focus for this learner is the critical aspect that the independent variable has to be squared in order for the equation to represent a quadratic function3. by allowing this learner to create her own example of a quadratic function mr nkosi provided opportunities for the learner to compare her understanding of the concept in focus with that which is accepted as valid. he reinforced the critical aspect being the independent variable by varying the letter used to symbolise the variable. his mediation thus provided this learner with the opportunity to develop mathematical meaning by experiencing structure and extending the range of variation which contributes to experiencing generality (watson & mason, 2002). mr nkosi’s use of lges across the lesson also provided him with a yardstick by which to measure, to some extent, the stability of his learners’ discernment of the critical aspect before varying another aspect of the object of learning and thereby moving into another class of algebraic function as is seen in the transition from episodes 3–4 to episode 5 (from linear function to rational function) and again from episodes 9–11 into episode 12 (from quadratic function to exponential function). to conclude our presentation of mr nkosi’s mediation of the example set, he consistently drew learners’ attention to what was the same and what was different within pairs of function equations, with particular attention on the exponent of the variable; by inviting lges he created opportunities for learners to not only participate and produce particular function equations, but also for him to be able to mediate where and when necessary. concluding discussion we revisit the research questions that guided our study viz. (1) what are the sequencing and pairing of examples over the lesson, and how do these open opportunities for learning?; (2) how does the teacher together with his learners act on these examples to bring the critical aspect of the object of learning in the lesson into focus? to answer these questions, we analysed mr nkosi’s lesson on a micro level focusing on both the selection of examples used during the lesson and how he mediated the example set, which includes his mediation of lges. using principles of variation theory, we examined the example set used and illustrated the value of sequencing, pairing and the juxtaposing of examples in the midst of sameness and difference to make the identified critical aspect discernible. in so doing, we further illuminated variation theory as being a practical analytic tool by which one can gaze into the teachers’ use of examples and contribute to theorising teachers’ work. looking at the selection of examples through the lens of variation theory alone was insufficient for us to determine how opportunities for the learners’ discernment of the object of learning were made possible. to gain some understanding here, we also looked at how mr nkosi mediated the example set that was created during the lesson and concluded that it is the example space together with its mediation that is of significance in providing opportunities for the learners’ discernment of the object of learning. we have shown both that and how his questions, revolving around sameness and differences, focused and helped to draw attention to the pattern of variation within and between the example sets. we thus conclude that it is the example space together with its mediation that is of significance in providing opportunities for the learners’ discernment of the object of learning. furthermore, the study gives an empirical illustration and supports watson and mason’s (2005) suggestion of the pedagogical value of eliciting lges. we have demonstrated how the teacher picked up and used a learner’s example (even if incorrect) and how these examples added to and expanded the pattern of variation. in this way our study contributes to extending previous research on variation theory designed and analysed lessons (e.g. al-murani et al., 2019; kullberg et al., 2013). this article also contributes to the literature of how teachers integrate examples into their teaching by illustrating how a teacher deliberately sequenced and paired examples that featured variation amid invariance, while explicitly bringing sameness and difference to the learners’ attention through simultaneity and juxtaposition, and contrast. this enabled learners to discern a critical aspect of an object of learning, specifically distinguishing classes of functions. furthermore, it is clear from the analysis of the lesson that it is a systematic pattern of sameness and difference between the examples that characterise their choice and mediation. this was a deliberate attempt by mr nkosi to bring the degree of each of the equations as represented by the various examples into focus for the learners. hence, we have demonstrated how a teacher can make use of principles of variation and invariance for planning the sequence of examples in the lesson and mediate the features of the examples in a way that makes them possible to discern. our analysis of mr nkosi’s lesson also contributes to the growing research in the field by describing in detail (i.e. at a micro level) how a teacher used examples to provide learners with opportunities for the discernment of an object of learning, and more specifically to research in south africa by studying a teacher’s enactment heeding the call by morris and hiebert (2011) who argue for the need of instructional products that are minute and significantly detailed to inform teaching. we also agree with essien (2021) that using examples is a mathematical practice and that doing mathematics entails exemplifying. thus, when extending this to teacher education, the use of principles of variation theory plays a central role in the mathematics teaching practice of using examples. this also resonates with adler and pournara’s (2020) framework which illustrates that teachers need to do more than just use examples; they need to use examples deliberately. we are not arguing that teaching with variation or comparing and contrasting by using examples is novel or unique. what we are arguing for is that it is not just any variation or any contrasting that matters, but a systematic and deliberate variation in the midst of invariance that is critical, or more specifically how something is changed against a stable background. in our study we have shown what variation and what sameness, the characteristics of the variation, and what is juxtaposed and compared are significant from the point of view of possibilities to learn. in this case the variation concerned the critical aspect. zooming into mr nkosi’s lesson we note that the relationship between the equation and so algebraic representation and its graphical representation of different classes of functions was brought into focus as these graphs were drawn. however, this was not simultaneously in focus, indicating that mr nkosi’s goal was to focus first on distinguishing the algebraic forms. focusing on the critical aspect that emerged during this learning study we note, mathematically speaking, that y = 2x is not linear because it is a polynomial function of the first degree. it is linear because it has a constant rate of change; y = x2 is quadratic because the rate of change of the rate of change is a constant and not that it is a polynomial function of the second degree. in terms of relating the syntax of the algebraic representation of a function to its graph, focusing on the value of the exponent of the independent variable provides some criteria for learners to able to identify the class of function represented by a given equation. it is precisely the exponent of the independent variable that emerged as a critical aspect for this group of learners as it provided them with some ‘rules’ by which to recognise the class of function given its algebraic representation. one may argue that this critical aspect is a visual cue which is insufficient since it is not grounded in mathematics. this critical aspect may thus be inadequate for the learners’ understanding of functions, and may lead to learners developing partially formed ideas and possibly what literature refers to as prototypical thinking (tall & bakar, 1992). although what emerged as the critical aspect in this study may be inadequate for learners developing a deeper understanding, it was nevertheless crucial for them, since it provided them with some resources with which to go forward in terms of their learning of mathematics. of course, we are not suggesting that their experience of learning about classes of functions remains at this level of thinking. our goal in this article was not specific to the learning and teaching of functions. it was rather to illustrate the value of sequencing and pairing of examples in the midst of sameness and difference, and the opportunities this can open for learning. it is a value, we submit, that could inform further research and practice in different mathematical domains and topics, and at different grade levels in the curriculum. implications for mathematics teacher education follow: prospective and practising teachers can themselves experience the value of deliberate exemplification as a mathematical and a mathematics teaching practice. acknowledgements competing interests the authors declare that they have no financial or personal relationships that may have inappropriately influenced them in writing this article. authors’ contributions this article is based on a larger study conducted by v.p. and therefore v.p. collected the data. v.p., j.a. and u.r.k. contributed by writing the article, reviewing the literature and analysing the data. j.a. and u.r.k. contributed to the conceptualisation of the article and guided the discussions on the analysis of the data. ethical considerations since human participants (teachers and their learners) as well as schools were involved in this study, the study adhered to ethical research principles. the relevant ethics clearance was obtained from the university general/human research ethics committee (2011ece005c). funding information this work is based on research supported by the south african research chairs initiative of the department of science and technology and the national research foundation (grant no. 71218). data availability data sharing is not applicable to this article, as no new data were created or analysed in this study. disclaimer any opinion, finding and conclusion or recommendation expressed in this material is that of the authors and the national research foundation does not accept any liability in this regard. references adler, j., & pillay, v. 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(2019). the complex interplay between examples and proving: where are we and where should we head? the journal of mathematical behavior, 53, 242–244. https://doi.org/10.1016/j.jmathb.2018.10.001 zodik, i., & zaslavsky, o. (2008). characteristics of teachers’ choice of examples in and for the mathematics classroom educational studies in mathematics, 69, 165–182. https://doi.org/10.1007/s10649-008-9140-6 footnotes 1. these difficulties do not only apply to this group of learners. each year the diagnostic reports on the grade 12 examinations, in south africa, point to learners’ difficulty in answering questions related to functions. the most recent diagnostic report based on the 2020 grade 12 mathematics paper shows that the average performance in question 4 which was based on the rational function was 55% and in question 5 which was based on the quadratic and linear functions and was 45%. in providing suggestions for improving, the examiners suggest that teachers should spend time discussing the basic concepts of functions (dbe, 2020, p. 190). 2. we do not elaborate nor mirror here the methodology used in the wider study, as this is not pertinent to the story we tell here. interested readers are referred to pillay (2013) and pillay and adler (2015) for a detailed account of the analytical tools used to analyse the lessons, and evidence of learner discernment in the lesson. 3. the grade 10 syllabus restricts work on the quadratic function to having the axis of symmetry at x = 0: vertical transformation and reflection along the x-axis. thus, in this lesson only examples of the form y = ax2 + c are explored. abstract introduction literature review research methodology results conclusion acknowledgements references about the author(s) nombuso zondo department of statistics, school of mathematics, statistics and computer science, university of kwazulu-natal, durban, south africa temesgen zewotir department of statistics, school of mathematics, statistics and computer science, university of kwazulu-natal, durban, south africa delia north department of statistics, school of mathematics, statistics and computer science, university of kwazulu-natal, durban, south africa citation zondo, n., zewotir, t., & north, d. (2020). learner performance in the 2009 to 2014 final grade 12 mathematics examination: a quantile regression approach. pythagoras, 41(1), a545. https://doi.org/10.4102/pythagoras.v41i1.545 original research learner performance in the 2009 to 2014 final grade 12 mathematics examination: a quantile regression approach nombuso zondo, temesgen zewotir, delia north received: 14 apr. 2020; accepted: 07 oct. 2020; published: 30 nov. 2020 copyright: © 2020. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract the south african education system bears evidence of fluctuations in the final grade 12 mathematics marks occurring across different learner profiles. this study reflected on the national senior certificate (nsc) mathematics results from the western cape education department for the years 2009 to 2014, the period just after the introduction of the nsc in 2008 and including the updated nsc introduced in 2014. accordingly, this study aimed to examine the learners’ performance by socio-economic school quintile and education district for the period of 2009 to 2014, for learners in the western cape. instead of the ordinary regression model, we adopted the quantile regression approach to examine the effect of school (national) quintile (nq) type and education district at different quantiles of learner performance in the mathematics examination. the results showed that there is a significant school quintile type and education district effect on learner performance in nsc mathematics examinations for learners in the western cape. in some years, there were no significant performance differences between learners from nq2 and nq4 schools in the different quantiles. similarly, learner performance differences for nq3 and nq4 schools were not significant. as we moved from 2009 to 2014, the performance difference between the lower school quintiles and the upper school quintiles narrowed, although the performance differences remained significant. these differences were smallest in 2013. this is a good sign, as it indicates that government efforts and policies, designed to narrow the historical social disparities manifested in the schools, have been somewhat successful. the identification and scrutinising of school quintile type and education district where the gap is wider will assist the government to review policies and interventions to accelerate the transformation. keywords: quantile regression; school quintile; education districts; performance; western cape; education system. introduction although the apartheid era ended in 1994, south africa still remains one of the most unequal countries globally (adjaye-gbewonyo et al., 2018; npc, 2011). this inequality has extended into its education system and has largely contributed to the unequal educational opportunities for learners from different backgrounds (graven, 2014; ogbonnaya & awuah, 2019). to redress the legacy of inequality in the education context, the schooling system is divided into government (public) schools and independent (private) schools. according to the south african schools act (no. 84 of 1996), public schools should be funded through public funds, while independent schools may apply for subsidies from their relevant provinces (dass & rinquest, 2017; franklin, 2017). for the purpose of financial allocations, government schools have been grouped according to the socio-economic status of the community within which the school is located, into five national quintiles (nq), from nq1 to nq5. each of these quintiles caters for 20% of the learners nationally, based on a ranking of the socio-economic status, which is measured by the income, unemployment rates and illiteracy within the school’s catchment area. schools in the poorest communities are classified as nq1, with those in the wealthiest communities classified as nq5 (dbe, 2006; moses, van der berg, & rich, 2017; western cape education department [wced], 2018). schools classified as nq1 to nq3 are non-fee-paying schools; these schools receive relatively more funding per learner compared to nq4 and nq5 schools (dass & rinquest, 2017; graven, 2014). however, the accuracy in the classification of these quintiles has been questioned as the classifications are based on the socio-economic status of the school’s surrounding areas rather than the status of the households of the learners that attend the school (ally & mclaren, 2016; dass & rinquest, 2017). further concerns on the classification of the quintiles have been raised and discussed in the work of hall and giese (2009), mestry and ndhlovu (2014), among others. in addition, the provinces in south africa are divided into geographic areas called education districts, as defined by the provincial member of executive council for education. the education districts provide direct vital lines of communication with schools for the provincial education departments, for the purpose of effective education management (dbe, 2018; motala, dieltiens, & sayed, 2009; sanews, 2018). generally, the education districts encompass different school quintile types and they cover a wider geographic location. notably, due to the homogeneity in the rural community and the limited number of schools within the education districts, it may seem that districts and school quintile types are similar. however, in the urban education districts, the number of schools are many, with different quintiles in the same urban education district. despite the measures taken by the south african government to address the imbalances of the past through the funding allocation system, the performance gap for learners in the different school quintile types and education districts remains a challenge. a study by zondo, zewotir and north (2021), which investigates the difficulty level and discriminatory power of the 2009 national senior certificate (nsc) mathematics examination for learners in the western cape, reported that much higher abilities are needed for learners in lower quintile schools to perform well in the mathematics examination. this study examines the mark distribution of the nsc mathematics examinations for the years 2009 to 2013, by socio-economic school quintile type and education district. furthermore, the effects of school quintile type and education district on learner performance in mathematics under the updated nsc, introduced in 2014, is investigated. the findings from 2009 to 2013 can be contrasted with those for 2014, in order to detect changes in performance. this study can potentially serve as a baseline for future analysis of learner performance in the revised mathematics curriculum. literature review mathematics is an important component of general education at school level (mullis, martin, foy, & arora, 2012; reddy, 2006). good performance in mathematics is, consequently, an important component in planning to progress from school to higher education (banerjee, 2016; baya’a, 1990; devine, fawcett, szűcs, & dowker, 2012; dbe, 2016). in south africa, an observed feature of the education system is poor performance in mathematics at school level (graven, 2014; reddy, 2006). studies on learner performance in mathematics have reported a range of factors associated with poor performance at school level. for example, makgato (2007) reported that access to resources contributed to poor performance in mathematics for learners from a selected sample of schools in pretoria, south africa. eide and showalter (1998) and, later, burnett and farkas (2009), who analysed data collected on learners from the united states, reported that some school variables and socio-economic status had a significant negative effect on learner performance in mathematics. consequently, these factors remain of global concern. south africa’s nsc, which came into effect in 2008, as a single national qualification issued across all provinces upon successfully completing grade 12, serves as a key qualification to national higher education institutions and the working environment (mahlobo, 2015; sasman, 2011). in the ongoing effort to investigate learner performance in school mathematics, various researchers have identified factors associated with learner performance (graven, 2014; howie, 2003; maree et al., 2006; mji & makgato, 2006; ogbonnaya & awuah, 2019; ramohapi, maimane, & rankhumise, 2015; spaull & kotze, 2015). south africa’s 2008–2013 indicators report on the nsc revealed that schools in the lower quintile groups accounted for the highest proportion of learners with poor performance (umalusi, 2015). for performance in mathematics in particular, the report concluded that schools in the middle quintiles were improving, while schools in the nq1 category continued to find the subject a steep challenge. in a study by reddy et al. (2012), using the trends in international mathematics and science study (timss) data, it was shown that in south africa, nq1 and nq2 schools perform at similar levels in mathematics, but the results are generally lower than nq3, nq4 and nq5 schools. other studies further reported that learners from communities with low socio-economic status developed skills more slowly and received fewer educational returns compared to their counterparts with higher socio-economic status (morgan, farkas, hillemeier, & maczuga, 2009; roscigno & ainsworth-darnell, 1999). moreover, schools in these low socio-economic communities are often under-resourced (mestry, 2014), which has a further negative effect on learner performance (aikens & barbarin, 2008). in a study by keble (2012), using data collected in selected schools in the port elizabeth area, socio-economic status and school access to resources were among the factors found to correlate with poor performance in school mathematics. in a study by longueira (2016), who looked at the effect of the school quintile type funding system on learner performance, a significant difference was reported in the marks of learners from nq3 and nq1 schools, with learners from nq3 schools performing better than their counterparts from nq1 schools. similar findings were reported for disparities in learner performance between nq2 and nq1 schools, with learners from nq1 performing the least well. in a study by legotlo, maaga, and sebego (2002) on a sample from a rural province in south africa, it was reported that the education district in which the school is located had a significant effect on learner performance. the participants further reported ineffective policies and lack of communication between schools and education district offices and also between districts and provincial offices of basic education. boateng (2014), who examined the technical efficiencies in delivering basic education for 13 education district offices in south africa, concluded that the inefficiencies potentially affect education outcomes and should be given due consideration in the reform of education policies. seemingly, school quintile type and education district have long been reliable predictors of learner performance in schools. theoretical framework the theoretical basis for the analysis of the data in this study is informed by a conceptual model developed by howie (2002; adopted from shavelson, mcdonnell, & oakes, 1987), as well as other research discussed in the literature review (boateng, 2014; graven, 2014; keble, 2012; reddy et al., 2012). the model presents factors related to learner performance in school mathematics within the south african context in terms of inputs, processes and outputs in the education system. the inputs include factors related to geospatial contexts at provincial and district levels. the processes in the education system consider the socio-economic situation within the geospatial contexts. learner performance is then used to measure outputs. accordingly, for this study, education district, school quintile type and learner performance serve as inputs, processes and outputs. the data used in this study are that of the nsc mathematics examination, made available for each year from learners in the western cape who wrote the examination at the commencement of the nsc in 2009 until 2013 and also after the commencement of the updated nsc in 2014 (grussendorff, booyse, & burroughs, 2014; ramatlapana & makonye, 2012; umalusi, 2015). objectives of the study this study reports on differences in the raw unadjusted learner marks in the nsc mathematics examination for the period of 2009 to 2014, and focuses primarily on the effects of the school quintile type and education districts for learners in the western cape on the mentioned differences of interest. the following questions are addressed in the study: what are the effects of socio-economic school quintile type on learner performance in the nsc mathematics examination for the period 2009 to 2014? what are the effects of socio-economic education district on learner performance in the nsc mathematics examination for the period 2009 to 2014? what are the differences in learner performance in the nsc mathematics examination for learners from the different socio-economic school quintile types and education districts, across the years 2009 to 2014? research methodology a quantitative research approach was used in this study to investigate performance variations among learners who wrote the nsc mathematics examination for the period 2009 to 2014, in the western cape. a regression analysis approach was used to analyse the effects of school quintile type and education district on learner performance in the examinations. regression analysis is classically used to find a relationship between a dependent variable (y) and p predictor variables x1, x2, …, xp. the ordinary regression model takes the form: ε is the error term of the model assumed to be normally distributed with mean zero and variance of σ2. the regression coefficients β1, β2, … , βp measure the change in the average value of y for a unit change of x1, x2, …, xp, while keeping the other variables fixed. the estimation of the regression coefficients β1, β2, … , βp is often performed using the ordinary least squares method which minimises the error sum of squares (hao & naiman, 2007a; zhang, 2009). however, the ordinary regression model has a number of limitations, in particular when the data are skewed (not normally distributed), multimodal or contain a high number of outliers. moreover, as noted by hao and naiman (2007b), the ordinary regression model cannot answer research questions at lower or upper values of the dependent variable (y). in other words, the ordinary regression model deals with the mean value of the dependent variable at different combinations of the predictor variables. the quantile regression alleviates such limitations of the ordinary regression model by estimating the effect of the different combinations of the predictor variables (x1, x2, …, xp) at different quantile levels of the dependent variable (y). the quantile level is the probability or the proportion of the population that is associated with a quantile and the corresponding conditional quantile of the dependent variable y is denoted by qτ(y|x). that is, the quantile level, often denoted as τ ∈ [0,1], is the value of the dependent variable below which the proportion of the conditional response population is τ (hao & naiman, 2007a; rodriguez & yao, 2017). the regression model of the dependent variable (y) on a set of independent variables x1, x2, …, xp for quantile level τ is given by: the regression coefficients β1 (τ) + β2 (τ) + … + βp (τ) are estimated by minimising the following objective function (baum, 2013): from the quantile regression approach, it may be found that the relationships between the dependent and predictor variables differ at each quantile level τ, that is, the objective function yields different regression coefficient estimates at for each quantile level τ (favero & belfiore, 2019; koenker & bassett, 1978). quantile regression generalises terms such as quartile, decile, quantile and percentile (hao & naiman, 2007b). quantile regression is a type of regression which is used for modelling conditional dependent variables and aims at estimating either the conditional median, which is a special case of quantile regression, or other quantiles of the dependent variable, which can be used to describe non-central positions of the distribution. the method of ordinary regression model estimation results in estimates that approximate the conditional mean of the dependent variable, given certain values of the predictor variables, while the quantile regression model specifies the conditional quantile function. one advantage of quantile regression, relative to ordinary regression, is that the quantile regression estimates are more robust against outliers (koenker & bassett, 1978). this means that quantile regression can provide reliable regression estimates in the presence of extreme values in the data set. the data the data used in this study include the individual raw, unadjusted learner examination marks and information on the learners’ school characteristics, the socio-economic school quintile type (nq1, nq2, nq3, nq4, nq5 and independent schools) and education district. the western cape education districts include the west coast, cape winelands, eden and central karoo, overberg, metro north, metro south, metro east and metro central (wced, 2018). the data do not include records relating to the overberg district, as they were not available in the provided data set at the time of analysis. in the years 2009 to 2013, the core content of the mathematics curriculum was examined by means of two compulsory examination papers (paper 1 and paper 2), with optional material being examined in paper 3. the data do not include records relating to learner marks in the optional paper 3 examination, as they were not available at the time of analysis; moreover, this was not a compulsory paper and consequently it was not written by all learners. the dependent variable is the unadjusted mathematics examination mark for each learner, calculated as the average mark for paper 1 and paper 2, which were written by all learners in the study, and was presented as a percentage. the predictor variables in the study were school quintile type and education district. the mark is unadjusted, in the sense that it is the raw mark that the individual learner scored for the examination, and does not include marks for the continuous assessment component. final nsc marks are (potentially) standardised by umalusi, the national standardisation body, to adjust for variations resulting from standards in the marking process and other factors in the examination process from one year to another and from one examination body to the next (umalusi, 2011). in a press release, umalusi (2011) mentioned that the standardisation of the learner marks, by subject, is a sophisticated statistical model that increases or decreases the learner marks by a proportion of their total mark in the subject. it was further stated that a high percentage of the nsc subject’s raw examination marks are accepted by umalusi and are not adjusted. for example, in 2010, raw unadjusted learner examination marks were accepted for 67.24% of the nsc subjects (umalusi, 2011). ethical consideration the data for this study were collected for a special project, run by a senior individual in the wced in south africa. special permission was obtained for the use of the anonymised data for this research study, which was personally sent to the authors by the senior individual in the wced. ethical clearance was not required for this study, as the data were properly anonymised by the wced in such a way that no identity number or examination number could be identified with either individual learner or school name. data exploration learner marks used in this study were the raw percentages a learner in the western cape obtained in the nsc mathematics examination. overall, there was an increase in the mean marks from 2009 to 2013, with a slight decrease in 2014, the year in which the updated nsc was introduced. that is, the average mathematics examination marks for learners in the western cape increased during the period 2009 to 2013; however, at the introduction of the updated nsc, the average learner marks in the mathematics examination were negatively affected (marks dropped). as shown in figure 1, the distribution of the marks varied by school quintile type. learners who attended either nq5 government schools or independent schools had considerably higher marks than learners from nq1 to nq4 schools. in addition, the spread of the marks varied by school quintile type, with learners from the lower quintile schools having a slightly more homogeneous distribution of marks, characterised by a relatively smaller range of marks, with associated smaller standard deviation. these patterns were observed across all the years from 2009 to 2014. variations in the spread of the learner marks, according to the education districts, are presented in figure 2. from figure 2, it is evident that for the period 2009 to 2010, the mark distribution was positively skewed, but not equally so, for all education districts. that is, the marks were generally more concentrated in the lower end of the distribution. similar patterns were further observed for the years 2011 and 2012 for learners who attended schools in the metro central, metro south and west coast education districts. the mark distribution for learners who attended schools in the metro east education district was spread over a wider range across the years, with the marks being slightly negatively skewed in 2013. overall, figure 1 and figure 2 indicate a skewed distribution evident in the learners’ mathematics marks for learners in the western cape. figure 1: distribution of mathematics examination marks by school quintile type 2009–2014 for learners in the western cape. figure 2: distribution of mathematics examination marks by education district 2009–2014 for learners in the western cape. results this section reports on the results from the ordinary regression and the quantile regression model techniques applied to the nsc mathematics marks. the effect of the different school quintile types and education districts on these marks was then scrutinised. the variations in learner marks in the nsc mathematics examination were examined using ordinary regression, presented along the 0.3, 0.4, 0.5, 0.6 and 0.8 quantiles. the results are presented in table 1. accordingly, the predictor variables included in the model are the school quintile type (a six-level categorical variable) and the education district (a seven-level categorical variable). for school quintile type, the nq4 category serves as a reference (baseline) category, that is, the regression parameters of the indicator variables measure the effect on mathematics performance, relative to the reference category. the selection of a reference category does not change the results; any category can be chosen to be the reference category (el-habil, 2012; schafer, 2006). similarly, for education district, cape winelands serves as the reference category. using a different reference category would not change the results as they would be relative to the reference category, that is, the model would fit equally well, producing the same likelihood (schafer, 2006). table 1a: results from the estimation of the determinants of learner marks in the national senior certificate mathematics. table 1b: results from the estimation of the determinants of learner marks in the national senior certificate mathematics. table 1c: results from the estimation of the determinants of learner marks in the national senior certificate mathematics. in examining the goodness of fit assessment results (presented in figure 3), we note that the quantile regression model fits the data adequately. from the outlier diagnostics summary, it is observed that 4.5% of the observations are greater than 1.5 times the interquartile range (iqr) above the upper quartile (i.e. q3 + 1.5 * iqr). if the distribution was normally distributed, we would have fewer than 2.5% such outliers present. accordingly, the ordinary regression model may not be the ideal model for the performance analysis. consequently, quantile regression was judged to be a better model fit for such data, since it does not make any assumptions from the distribution of the error terms in the model (rodriguez & yao, 2017). all analyses for this study were conducted using the sas 9.4 software. figure 3: diagnostic plots for the quantile regression model. the effect of socio-economic school quintile type on the performance of learners in the nsc mathematics examination for the period 2009 to 2014 in the western cape table 1 presents the results. these are presented for the ordinary regression model and for each quantile in the quantile regression results. some of the results from the ordinary regression model accord with studies in literature which analysed the final adjusted learner marks, hence they show that school quintile type is a significant determinant of learner performance. the estimated parameters represent the differences in the marks between learners in the specified school quintile types and learners in the nq4 schools when the effect of education district is controlled for. for instance, in 2009, learners from nq2 schools achieved 3.4 percentage points lower on average, compared to those from nq4 schools, while those from nq3 schools achieved 4.3 percentage points lower than those from learners in nq4 schools. the estimates for nq2 schools were marginally higher than those of nq3 schools in comparison to nq4 schools for the years 2009, 2011 and 2013. the analysis shows that for the years 2010 and 2012, learners from nq1, nq2 and nq3 schools, on average, had significantly lower marks than their nq4 counterparts, while those from nq5 and independent schools presented as having significantly higher marks. these results are consistent with findings from the literature. however, in 2013, the ordinary regression results showed that learner marks from the nq2 and nq3 schools were not significantly different from those of learners from nq4 schools. this contradicts findings in the literature, on a study using timss 2011 data, where nq4 learners were reported to perform better than those from nq2 schools (reddy et al., 2012). in 2014, the marks of learners in nq3 schools were found to be similar to those of learners in nq4 schools. in other words, there is no significant difference in the parameter estimates. quantile regression provides snapshots at different quantile levels, hence adding much more value to the relationship between the dependent variable and the predictor variables (baum, 2013). the results from the quantile regression analysis show that the effect of school quintile type and education district on learner marks in the nsc mathematics examination varied across the quantiles. while these can be read across the rows in table 1, the pattern of the effects is illustrated in figure 4 and figure 5. from table 1, the estimated parameters for school quintile type obtained by the ordinary regression model remain constant for the different school quintile types as they are based on the conditional mean of the dependent variable. in contrast, the estimated parameters, obtained using quantile regression, varied as the analysis is done at a different point of the dependent variable. for instance, for 2009, the ordinary regression model estimate for nq3 schools remained constant at –4.3, while when using quantile regression, the estimates decreased across quantile from –4.0 in quantile 0.3, to –8.0 in quantile 0.8. figure 4: parameter estimate differences between the specified school quintile type and nq4 schools across the different quantiles. figure 5: parameter estimate differences between the specified education district and the cape winelands education district across the different quantiles. from the quantile regression results presented in table 1, it is observed that in the years 2009 to 2011, learners who attended nq1, nq2 and nq3 schools had significantly lower marks than those who attended nq4 schools. this is evidenced by the negative parameter estimates, with an effect that is greater in the upper quantiles (0.6 and 0.8). the performance differences between the learners who attended nq1 and nq4 schools range from 7 to 13 percentage points, while the differences in performance between nq2 and nq4 learners range from 3 to 8 percentage points in the upper quantiles. similarly, in 2012, learners who went to nq4 schools scored significantly higher marks than their counterparts in nq1, nq2 and nq3 schools. in particular, the difference between nq3 and nq4 schools is about 4 percentage points at the median (0.5 quantile) and 6 percentage points in quantile 0.8. for the period 2013 to 2014, it is worth noting that in 2013, there was no evidence of significant performance differences between learners in nq2 and nq4 schools across all quantiles. that is, in the estimated parameters from quantile regression analysis presented in table 1 for 2013, there is no statistically significant difference in the estimates of nq2 and nq4 school quintile types, and this is true at all quantiles of interest in the study. furthermore, in 2013, the differences in mathematics marks for learners in the nq3 and nq4 schools exist only at the median (quantile 0.5), but not in the lower and upper quantiles. unlike 2009–2013, in 2014, the year in which the updated nsc was introduced, the performance difference between learners from nq4 schools from that of their nq3 counterparts is not significant in the lower end and upper end quantiles. the estimated parameters for nq2 and nq3 schools in the quantile regression analysis vary within a small range. the marks for learners in nq5 schools were significantly higher than those from nq4 schools, with these findings between the nq4 and nq5 schools being echoed in the old nsc mathematics examination results, 2009 to 2013. differences in these marks were largest in the upper quantiles. similar observations were made for learners from independent schools. the effect of education district on the performance of learners in the nsc mathematics examination for the period 2009 to 2014 in the western cape the ordinary regression model results show that the education district had a significant effect on the learner marks obtained for the years 2009 to 2014. learners who attended schools in the metro east and metro north education districts were found to have significantly higher marks than learners who attended schools in the cape winelands education district. specifically, in 2009, learners who attended schools in the metro east education district achieved 7.4 percentage points higher marks on average than those who attended schools in the cape winelands. in 2013 the marks achieved for these two education districts indicated an increase, a difference of 8.6 percentage points higher for learners in the metro east compared to those in the cape winelands. the quantile regression parameter estimates for the education district variable are presented in table 1 and figure 5. different from the school quintile type effect, it is observed that learners from the metro east and metro north education districts performed significantly better than learners from the cape winelands education district, across the different quantiles, for all the years in the study. learners who attended schools in the metro central education districts generally had marks that were statistically significantly lower than those of the cape winelands education district, for the years 2009 to 2012. in 2009, there was an estimate increase, associated with increasing quantiles. the parameter estimates for the metro central education district were negative in the lower quantiles and positive in the upper quantiles for the years 2009, 2010 and 2014. that is, the improved performance that learners from the cape winelands had over their metro central counterparts was largest in quantile 0.8. this may be due to more nq5 category schools being located in the cape winelands education district, relative to the metro central education district. in 2013, the metro east education district effects were largest in the lower end quantiles and lowest in the upper quantile, with 10 percentage points higher in quantiles 0.3 and 0.4, and 6 percentage points higher in quantile 0.8, when compared to the marks of learners in the cape winelands as presented in table 1. differences in learner performance by socio-economic school quintile type and education district across quantile and across the years 2009 to 2014 upon obtaining the parameter estimates for our predictor variables from our quantile regression investigation, we tested the equivalence of the parameters within each predictor variable, across the different quantiles. the test result indicated a rejection of the equivalence of the effect of the predictors across the different quantile levels (the p-value < 0.05). this suggests that the magnitude of impact that school quintile type has on learner performance in the nsc mathematics examination changes from one quantile to another. similar equivalence tests were done for educational district effects, and it was concluded that the effect of education district significantly varied across quantile. the test for heteroscedasticity in linear models, based on quantile regression statistics of koenker and bassett (1982), was used. we adopted cumming’s (2009) approach to test the equivalence of the school quintile type effect across the years. according to cumming, if there is no overlap in the 95% confidence intervals of the parameter estimates, then the two parameters are statistically significantly different from each other; if the two confidence intervals ‘just touch’, there is a significant effect of p of about 0.01. this is known as ‘the rule of eye’ (cumming, 2009). figure 6 presents the 95% confidence intervals for the parameter estimates of the school quintile types across the different years. accordingly, the intervals that overlap and the extent of the overlap were identified. using ‘the rule of eye’, the equality of some pairwise comparisons of the estimates across the years for the different school quintile types was rejected at a 5% level of significance. in particular, for nq1 schools, in the 0.3, 0.4 and 0.6 quantiles, it was noted that parameter estimates for the year 2010 differed significantly from those of the other years (2011–2014). the nq5 parameter estimates, from quantile regression, in the years 2009 and 2010 differed significantly across the quantiles, while for the independent schools, the estimates in the years 2009 and 2010 indicate a significant overlap across the quantiles. figure 6: parameter estimates and 95% confidence limits for the school quintile types across the years. another interesting result from the test of differences between parameter estimates across the years is that of nq3 relative to nq4 schools. for nq3 schools, there were no significant differences found for the years 2013 and 2014, across the quantiles. that is, nq3 schools were not affected by the updated nsc, which came into effect from 2014. similar observations are made for nq1 and nq2 schools, when comparing parameter estimates of 2013 and 2014. conclusion much of the existing literature has shifted to the analysis of how learner performance in mathematics influences their performance in higher institutions of learning. this is evident, for example, in the work by zewotir, north and murray (2011) and singh, granville and dika (2002). in this study we analysed learner performance in school mathematics with the aim of addressing the effects of school quintile type and education districts. we applied the quantile regression method to the nsc mathematics examination results from the wced for the period 2009 to 2014, a period during which the nsc was introduced and further revised. several methods exist to examine learner performance in mathematics. past national studies used traditional regression methods (howie, 2003; maree et al., 2006) for the analysis of learner performance, which may not be valid if certain model assumptions are not satisfied. the results in this study reported that the extent to which school quintile type and education district influenced learner performance in the nsc mathematics examination varied across the quantiles. the performance of learners from nq1, nq2 and nq3 schools generally differs from that of nq5 and independent schools. the results presented in this study are similar to those of reddy et al. (2012) who reported on the timss 2011 data set that learners from nq1 and nq2 schools performed at lower levels than those in nq4 and nq5 schools. interestingly, findings from our study reveal that in the later years (i.e. 2013 and 2014) the performance differences between learners from nq4 schools and those of nq2 and nq3 schools tend to be insignificant in the lower end and upper end quantiles. this is perhaps due to the government’s continual effort or the schools’ intervention strategies to prepare the learners for the national examination in order to narrow the historical socio-economic disparities manifested in the schools. this study had the limitation of relying on the 2009–2014 data set. however, as this was the period just after the introduction of the nsc, and included the updated nsc introduced in 2014, it is with little doubt that this study can serve as a baseline on how the performance differences between the socio-economic school quintile types and the education districts change over the years as the nsc becomes a national norm. particularly, this study sheds light on how government policies or the educators’ efforts have been somewhat successful in narrowing the historical educational inputs and systems disparities manifested in schools. in other words, as the educational inputs and systems differences are minimised among the socio-economic school quintile types, the performance differences between the schools will be insignificant. despite limitations thereof, findings from this study make an important contribution to existing literature and initiate insights and a national debate on how government can review policies and interventions to accelerate transformation between the socio-economic school quintile types and education districts where the performance gap is wide. indeed, more detailed longitudinal studies are needed to generalise to later years and to learners across the country. accordingly, one of the future directions of this study is to examine more recent patterns and trend differences among the low, intermediate and high performers, in all school quintiles in south africa. acknowledgements the authors are grateful to mr brian schreuder, superintendent-general of the western cape education department, for permission to use their data. competing interests the authors have declared that no competing interest exists. authors’ contributions all authors contributed equally to this work. funding information n.z. would also like to thank the national research foundation of south africa, teaching development grant (tdg) and universities capacity development programme (ucdp) for ongoing financial support. data availability statement data sharing is not applicable as no new data were created. disclaimer the views and opinions expressed in this article are those of the authors and do not necessarily reflect the official policy or position of any affiliated agency of the authors. references adjaye-gbewonyo, k., kawachi, i., subramanian, s.v., & avendano, m. 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(2021). the level of difficulty and discrimination power of the items of the national senior certificate mathematics examination. manuscript submitted for publication. reviewer acknowledgement pythagorashttp://www.pythagoras.org.za acknowledgement to reviewers the quality of the articles in pythagoras crucially depends on the expertise and commitment of our peer reviewers. reviewing is an important part of scholarly work, making a substantial contribution to the field. reviewers’ comments serve two purposes, guided by two inter-dependent objectives: • pythagoras wishes to publish only original papers of the highest possible quality, making a meaningful contribution to the field. reviewers advise the editor on the scholarly merits of the manuscript to help him evaluate the manuscript and to decide whether or not to publish it. reviewers are encouraged to reject a manuscript if it is scientifically flawed, merely sets out observations with no analysis, provides no new insights, or is of 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the selection of appropriate peer reviewers for manuscripts for pythagoras, we ask that you take a moment to update your electronic portfolio on www. http://pythagoras.org.za, allowing us better access to your areas of interest and expertise, in order to match reviewers with submitted manuscripts. if you would like to become a reviewer, please visit the pythagoras website and register as a reviewer. to access your details on the website, follow these steps: 1. log into pythagoras online at http://www. pythagoras.org.za 2. in your ‘user home’ select ‘edit my profile’ under the heading ‘my account’ and insert all relevant details, bio statement and reviewing interest. it is good practice as a reviewer to update your personal details regularly to ensure contact with you throughout your professional term as reviewer to pythagoras. please do not hesitate to contact me if you require assistance in performing this task. rochelle flint submissions@pythagoras. org.za tel: +27 (0)21 975 2602 fax: +27 (0)21 975 4635 page 1 of 2 no. of manuscripts processed in 2013 (outcome complete) 31 accepted without changes 0 (00.0%) accepted with minor changes (to the satisfaction of the editor)1 8 (25.8%) accepted after major revisions (re-submit, then re-review)2 4 (12.9%) rejected after review – not acceptable to be published in pythagoras3 10 (32.3%) rejected without review – not acceptable to be published in pythagoras4 9 (29%) duncan mhakure duncan samson ednei becher elmarie meyer elspeth khembo faaiz gierdien fatma aslan-tutak gail fitzsimmons gerrit stols herbert khuzwayo hugh glover hurryramsingh hurchand jacques du plessis jane watson janine hechter 1.accepted after one round of review, with ‘minor’ changes as specified by reviewers and editor. 2.accepted after two or more rounds of review, with major changes specified by reviewers and editor. 3.includes three cases where authors did not resubmit after required to make major changes. 4.all submissions undergo a preliminary review by the editor (and associate editors) to ascertain if it falls within the aims and scope of pythagoras and is of an acceptable standard. includes six cases where authors did not resubmit after extensive feedback prior to reviewing. reviewer acknowledgement pythagorashttp://www.pythagoras.org.za page 2 of 2 if you would like to become a reviewer, please visit the pythagoras website and register as a reviewer. jayaluxmi naidoo jill adler jogy alex jon star lyn webb lynette nagel lynn bowie marc north margot berger mark jacobs marthie van der walt mdutshekelwa ndlovu mercy kazima michael de villiers neil eddy nicky roberts nyna amin o. osiyemi ogbonnaya iheanachor paul webb piera biccard rajendran govender retha van niekerk sally hobden seugnet blignaut sharon mcauliffe sizwe mabizela stanley adendorff stephan du toit tim dunne vera frith wajeeh daher willy mwakapenda yael shalem yip cheung chan reviewers (continued): the potential of teacher development with geometer’s sketchpad pythagoras, 68, 15-21 (december 2008) 15 the potential of teacher development with geometer’s sketchpad gerrit stols andile mji dirk wessels university of pretoria gerrit.stols@up.ac.za tshwane university of technology mjia@tut.ac.za university of stellenbosch dcjwessels@gmail.com in this paper we document the advantages of utilising technology to enhance teachers’ instructional activities. in particular we showcase the potential and impact that the use of geometer’s sketchpad may have on the teaching and learning of geometry at school. a series of five, two-hour teacher development workshops in which geometer’s sketchpad was used were attended by 12 grade 11 and 12 teachers. the findings revealed that teachers had a better understanding of the same geometry that they initially disliked. this finding was supported by a quantitative analysis which showed a positive change in the understanding of and beliefs about geometry from when the teachers started to the end of the workshops. research has shown that teachers with different kinds of understanding of a particular topic tend to teach it differently (kinach, 2002). researchers have argued that a teacher’s level of understanding plays a major role in influencing the knowledge that learners construct (e.g. doyle, 1988; koehler & grouws, 1991). in fact, others (e.g. brown & borko, 1992) have in concurrence pointed to the importance of teachers having strong content knowledge. such content knowledge is said to give teachers the “confidence and resources to engage children at more challenging levels and undertake more adventurous learning tasks” (taylor & vinjevold, 1999, p. 161). one method that is available and which may be helpful in improving teachers’ conceptual understanding is technology. an important aspect of technology is that when used appropriately it reinforces higher cognitive skill development and complex thinking skills such as problem solving, reasoning, decision making, as well as scientific inquiry (moersch, 1999). with respect to mathematics teaching in particular, researchers have agued that technology has the potential for enhancing instruction (connell, 1998; roschelle, pea, hoadley, gordin, & means, 2000). in fact, dede (2000) indicated that technology can be used to strengthen student learning and enhance pedagogy. further, it has been pointed out that appropriate usage of technology tools may enhance teachers’ conceptual development (sanders 1998). such enhanced conceptual development should be useful to the students tutored by these teachers. to this effect, it has been argued that teachers who were comfortable with technology and possessed solid basic skills are more likely to use constructivist teaching practices (rakes, fields, & cox, 2006). related to this, these authors have also asserted: given the current emphasis on producing students with high levels of thinking skills, any tools that can encourage the use of constructivist classroom practices and encourage the development of thinking skills in students should be considered important for all teachers and students. (p. 422) recently there have also been calls for the use of technology to be incorporated into teacher education. researchers in support of the calls have even provided categories of different approaches taken by teacher trainers to bring technology into their programs (cf. garofalo, drier, harper, timmerman & shockey, 2000). the importance of technology in the teaching and learning of mathematics was also identified and stressed by the national council of teachers of mathematics (nctm) as one of the six principles of high quality mathematics education. the principle, known as the ‘technology principle’ states that technology is “…essential in teaching and learning teacher development with geometer’s sketchpad 16 mathematics; it influences the mathematics that is taught and enhances students’ learning” (nctm 2000, p. 24). in south africa, the need to incorporate technology in the classroom has been identified too. initiatives to incorporate technology into the teaching and learning context are exemplified by the support the national department of education gave to the formal launching of schoolnet sa in 1997 (cf. riordon, 2007). also, computer companies such as sahara systems have been involved in spearheading initiatives to furnish computer labs within public schools in gauteng as well as the free state, with ict infrastructure and digital equipment (network times, 2007). the importance of technology notwithstanding, it is worth noting that mathematics teachers, and not technological tools, are the key change agents needed to bring about reform in mathematics teaching. with this is in mind, we introduced geometer’s sketchpad to inservice teachers of mathematics. geometer’s sketchpad was originally designed for teaching geometry in secondary schools. it includes the classical ruler and compass constructions, as well as isometries. the program allows teachers and learners to work quickly through numerous examples (by dragging) and enables them to discover patterns, to explore and to test conjectures by constructing their own sketches. sketchpad 4 and later versions has a number of trigonometric and algebraic features, which enables teachers and learners to find relationships between symbolic and graphic representations. geometer’s sketchpad is a powerful teaching and learning medium. in fact this programme has been reported to (a) enhance mathematics teaching; (b) help with conceptual development; (c) enrich visualisation of geometry; (d) lay a foundation for analysis and deductive proof; and (e) create opportunities for creative thinking (sanders 1998). in this study, the use of the geometer’s sketchpad was based on the theoretical basis advanced by wong (1998), who argued that graphing software helps to (a) develop concepts and reinforce concepts; (b) rectify common errors; (c) check graphical solutions; (d) solve equations graphically; (e) test conjectures through problem posing; (f) become meta cognitive, to acquire information technology skills; and (g) enhance the motivation to learn. in introducing this particular programme we were conscious that it would not in itself improve the quality of education but it would help enhance the qualities identified when technology is introduced to teachers (sanders, 1998; wong, 1998). in fact, we particularly selected geometer’s sketchpad on the knowledge that the use of computers in mathematics education can be harmful when used for drill and practice, but helpful when used for exploration (wenglinsky, 1998). the main objective therefore was to examine the impact of workshops that introduced the dynamic geometry software to teachers. in particular, the study examined changes in teachers’ knowledge of and beliefs about geometry when dynamic software is introduced in the teaching and learning of geometry. method participants and procedure the participants were 15 teachers from previously disadvantaged schools in the pretoria region. all were grade 11 and 12 geometry teachers who taught mathematics in previously disadvantaged schools. participation of these individuals was at the recommendation of the local department of education. of the original 15 participants, 12 completed a series of five workshops. the three who dropped out indicated that they were not computer literate and requested to be excused. initially, a pilot study was conducted with one teacher who was not part of the final 12 participants. the aim of this pilot investigation was primarily to determine whether we could (a) monitor the change and growth in the teacher’s conceptual development and understanding of mathematics following the use of geometer’s sketchpad; (b) develop and refine training materials (teachers’ worksheets and notes) depending on feedback from the teacher; and to establish (c) logistical issues such as the duration and nature of a full basic course using geometer’s sketchpad with teachers. the pilot run for all intents and purposes proved to be extremely worthwhile. for instance, the participating teacher was complementary of the software in terms of its ease of use as well as in helping her understand geometrical theory much better. she actually indicated “i think this is a good program because you can show learners how theorems can be proved by measurement … easily dragging and changing angles…” the pilot was also useful because the teacher provided valuable feedback on the prepared manual. this allowed for some activities to be modified while one activity was removed completely because she felt it was difficult for the targeted grade levels as well as time consuming. gerrit stols, andile mji & dirk wessels 17 material a manual prepared for the workshops contained basic operational instructions for using geometer’s sketchpad, as well as problems for teachers to solve. in developing mathematics activities and materials, we followed the guidelines set by garofalo et al. (2000) for appropriate uses of technology in mathematics teaching: • introduce technology in context • address worthwhile mathematics with appropriate pedagogy • take advantage of technology • connect mathematics topics • incorporate multiple representations the manual covered most of the grade 11 and 12 geometry curricula, as well as graphs of the trigonometric functions, straight lines and parabolas. in essence the workshop manual integrated the development of computer skills (geometer’s sketchpad) and geometric discovery. typically, the starting point involved a discussion about how to conduct an investigation using geometer’s sketchpad. for example, the teachers had to draw a cyclic quadrilateral and determine some of its features. an advantage of an activity such as this is that it allowed teachers to have a hands-on experience of the operational functions of geometer’s sketchpad. also, it allowed them to create the geometric diagrams, as well as determine the veracity of geometric proofs regarding the diagrams (the cyclic quadrilateral, in this case). data collection and analysis data were drawn from a series of five two-hour workshops that were conducted by the first author over a period of three months. a mixed methods design was used to collect both qualitative and quantitative data. qualitative data was obtained through reflective journals that teachers kept and this information was augmented by one-on-one interviews conducted by the first author. here, over the duration of the workshops, teachers were encouraged to keep a reflective journal of their experiences. these reflective journals provided the researchers with documentation on a continuous cycle of enquiry. we also asked teachers to provide information about the availability of computers in general, their usage of these and where they used these. further information was obtained by asking teachers to complete a questionnaire that was intended to evaluate their beliefs about geometry as well as their feelings about what was done at each workshop. a typical question asked, for example was there is a view which suggests that geometry is the most difficult section in school mathematics, how do you feel about this? at the end of three months a summative evaluation questionnaire was once again administered. here, teachers were again asked about their views about geometry and the activities they had engaged in. a typical question in this instance was think back to when you started the workshops, how do you feel about geometry now? in addition one teacher was video-recorded during each session while working. the recordings were in order to capture all activities the teacher engaged in and to allow for a rerun of the tapes which allowed for the analysis to be as accurate as possible. quantitative data on the other hand, was obtained from paper and pencil tests involving a pre-test and a post-test. the pre-test was written before any intervention was carried out, that is before the introduction of teachers to geometer’s sketchpad. for the post-test, the same test was used after the intervention (see appendix a). the aim of the two tests was to establish whether there would be changes in teachers’ knowledge with respect to grade 11 and grade 12 mathematics topics. we measured the change as a function of the difference between the teachers’ scores in the two tests. differences were established by computing a paired-samples t-test where, if there was no statistically significant difference it would be concluded that there were no changes in teachers’ knowledge about the covered mathematics topics. results qualitative analysis with respect to the availability and usage of computers, teachers had differing self reports, as shown in the frequency distribution in table 1. the table reveals that a majority of the teachers had computers at school, for some these were functional and that they utilised word processors for their school work. only one teacher reported that he had an email address with internet access at home. table 1: computer availability and usage computer activity n % computers at school 11 92 functional computer laboratory at school 9 75 computers at home 8 67 internet at home 1 8 have an e-mail address 2 16 use word processors for school work 7 58 about their feelings with respect to teaching geometry, 10 of the 12 participants reported that teacher development with geometer’s sketchpad 18 they struggled to teach this section of mathematics. statements such as: teaching geometry is a nightmare; teaching geometry is difficult; teaching geometry is hell … were advanced by seven of the teachers. also, seven teachers blamed their learners. in this instance the teachers indicated: learners have negative attitudes; learners find it difficult; learners must learn theorems and practice geometry ... in the summative evaluation at the completion of the workshops however, attitudes changed completely and were more positive. some of the feedback revealed: i will try to open afternoon classes in the computer laboratory for my pupils so that i can give them more attention to improve their geometry; learners are more enthusiastic about mathematics. they work more problems than usual; learners will be very positive in doing geometry unlike the situation that we are having at our schools where they are negative … in fact all the teachers indicated that the workshops helped them to better understood geometry and felt more confident in teaching it. the teachers identified the fifth workshop as the most useful. in this particular workshop they had to design their own riders and solve them. this exercise by their own admission, was very meaningful in their teaching. feedback here included: i had a problem before but now i think i will approach it (geometry) with confidence; after this workshop i think i have gained confidence in teaching geometry; when i came here i had a very negative attitude as learners did not really understand geometry, but now i am positive that i will also change the learners’ attitude toward geometry. a disconcerting outcome was the view from 9 of the 12 teachers that: you need geometer’s sketchpad to teach geometry successfully. an analysis of the video recordings as well as the interviews revealed a better conceptual understanding of the problems teachers were dealing with. in fact, the recordings showed teachers solving advanced problems which on their own admission they never thought they would ever be able to tackle. the teachers were able to interpret and apply geometric principles and used these to reach generalised conclusions. for example, one teacher was very excited when he independently found the solution of the sum of the interior angles of a 100-gon. he constructed a triangle, a quadrilateral and a pentagon. he then constructed a point inside each polygon and connected this newly constructed point to all the vertices of the polygon. the teacher then concluded that the triangle contained three triangles, the quadrilateral four triangles, and the pentagon five triangles. what we found exciting was the fact that he also understood that he should subtract the newly created revolution (360°) in each case to calculate the sum of the interior angles of the polygons. his conclusion was that the sum of the interior angles of a 100-gon is “…100 triangles minus one revolution that is, 17640…” we certainly cannot paint a picture of the feelings this particular teacher displayed after solving this problem. quantitative analysis table 2 shows the means, the standard deviations and the percentage improvement of scores from the paper-and-pencil test. the table reveals that there was a positive improvement from the pre-test to the post-test in all the four questions with the percentage improvement ranging between 10,8% and 30,8%. paired-samples t-test indicated that the differences among the means were statistically significant (p < 0,05) for questions 1, 3 and 4 while the difference was not statistically significant for question 2. it was then concluded that there were changes in teachers’ knowledge about the covered mathematics topics. discussion the results reported here provide a good illustration of the potential and impact that the use of geometer’s sketchpad may have on the teaching and learning of geometry and may be mathematics at school. by their own admission, teachers indicated that they had a better understanding of the geometry that they initially disliked. they also indicated more enthusiasm for mathematics by their learners. to this effect they reported that their learners tackled more problems than it previously had been the case. these table 2: statistics from the paper-and-pencil test test question mean (sd) % improvement t p pre-test post-test q1 (6) 3,77 (2,3) 5,62 (0,6) 30,8 2,98* 0,011 q2 (6) 4,14 (2,2) 4,79 (2,0) 10,8 1,26 0,229 q3 (9) 4,46 (4,1) 6,23 (4,1) 12,7 2,83* 0,014 q4 (10) 4,46 (4,1) 6,23 (4,1) 17,7 2,21* 0,024 * p < 0,05 gerrit stols, andile mji & dirk wessels 19 qualitative findings were supported by the quantitative analysis. in this analysis, an improvement of more than 10% was reported. more than that, changes in teachers’ knowledge were found to be statistically significant. the findings of this study are consistent with arguments that technology provides an optimal medium for the application of constructivist principles to learning (murphy, 1997). to be more specific, the best improvement from preto post-test was question 1. this question was about the transformation of a trigonometric function: draw the graph of 1cos2 1 += xy on the interval [−180°; 180°]. the teachers were not allowed to use sketchpad during the preand post-tests. the statistically significant improvement of 30,8% (p < 0,05) therefore shows a growth in understanding and also indicates that sketchpad is particularly useful in developing an understanding of the transformation of functions. although there was a 10,8% (p < 0,05) improvement in question 2, it was not statistically significant. the question was to determine the equation of an altitude of a triangle (see appendix a). although the use of geometer’s sketchpad during the workshops helped the teachers to know what an altitude is, four of the teachers could still not determine the equation of the altitude. teachers’ average for question 3 in the pre-test was 78,6% and this indicates that the teachers had a high level of knowledge about the circle relationships before the intervention. however, the intervention still made a difference: table 1 shows a statistically significant improvement of 12,7% (p < 0,05) from the pre to post-test in question 3. the question concerned the relationship between the angles in a circle. the software is therefore useful for discovering relationships between angles and helped the teachers to make conjectures and to use them to answer simple questions. the purpose of question 4 was to determine if the teachers can use their knowledge to solve more complicated geometry problems. question 4 measures their ability to use these relationships to do a proof. the mean in the pre-test for question 3 was 78,6% and for question 4 is 44,6%, a difference of 17,7%. the implication is that, although the teachers knew the angle relationships of a circle, they were not able to use the knowledge effectively to solve more advanced geometric problems. in the post-test after the intervention, the average for question 3 was 91,2% and for question 4 is 62,3%, a difference of 28,9%. although the teachers’ ability to solve more advanced problems increased, the gap between knowing and applying mathematical principles also increased. this is an indication that geometer’s sketchpad is a tool for exploration but cannot be used (in the way that we used it in this study) to narrow the gap between knowledge about relationships and application of the relationships in doing proofs. an important aspect of this study was the fact that the approach of integrating the development of computer skills with geometric discovery proved successful. this is important because researchers (e.g. becker & anderson, 2000; ertmer, 1999) have shown that the attitudes and beliefs of teachers about the role of technology in the curriculum can influence how and when they use as well as integrate computers into their teaching. although sketchpad was found to promote teachers’ understanding of geometry, a major pitfall was the fact that teachers believed it was the best way to teach geometry effectively. as has been pointed out previously, teachers need to understand that technology is essential if it enhances what they know because otherwise it may also be misused and therefore be of no value to their teaching. perhaps follow-up workshops in future should include discussions of how to discover and explore geometry without the use of computers. the encouraging findings, not withstanding a limitation of this study, relates to the availability and cost of geometer’s sketchpad at schools. almost all the teachers did not have this programme in their schools. those who had the geometer’s sketchpad identified its cost as an inhibiting factor because they had used their own funds to buy it. perhaps this is an issue that education authorities should look at if a better understanding of geometry in our schools is to be envisaged. this is important because professional development activities coordinated say by the department of education could then go to a higher level. in this regard, researchers have shown that if teachers are active in professional development working side by side with their colleagues then they tend to effectively use technology with their students (frank, zhao, & borman, 2004). this study demonstrated that there is merit in teacher development with geometer’s sketchpad because it will positively change the teaching and learning of geometry in our schools. http://www.ucs.mun.ca/~emurphy/� teacher development with geometer’s sketchpad 20 acknowledgements this paper was made possible by a grant from the nrf. the statements made and views expressed are solely those of the authors. references becker, h. j., & anderson, r. e. 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(1998). graphing software: computers for mathematics instruction (cmi) project. retrieved 24 august 2007 from www.sun.ac.za/mathed/software/graphmat.pdf http://www.ucs.mun.ca/~emurphy/� gerrit stols, andile mji & dirk wessels 21 o o 140° 120° 30° d a b c appendix a question 1: draw the graph of 1cos2 1 += xy on the interval [−180°; 180°]. question 2: a(−2; 8), b(−4; −2) and c(2; 6) are the vertices of ∆abc and m is the midpoint of bc. ms ⊥ ab. calculate the equation of the line sm question 3: determine the unknowns if o is the centre of the circle: a = ……… b = ………. c = ………. d = ………. e = ………. f = ….... g = …….. h = ……. i = ……… question 4: calculate the values of x and y: (give a full explanation with reasons.) b p q f g r 1 y x 1 2 3 1 2 a(−2; 8) c(2; 6) b(−4; −2) s m y x 140° e f g 48° i 60° h word bookmarks ole_link1 ole_link2 top abstract introduction theoretical framework research design ivory college test (sara and jada) marking of ivory college test (sara and jada) arbor high test (maya) marking of maya’s test arbor high test (jono) marking of jono’s test concluding remarks acknowledgements references about the author(s) shaheeda jaffer school of education, faculty of humanities, university of cape town, cape town, south africa citation jaffer, s. (2020). evaluation and orientations to grade 10 mathematics in schools differentiated by social class. pythagoras, 41(1), a578. https://doi.org/10.4102/pythagoras.v41i1.578 original research evaluation and orientations to grade 10 mathematics in schools differentiated by social class shaheeda jaffer received: 21 sept. 2020; accepted: 08 oct. 2020; published: 17 dec. 2020 copyright: © 2020. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract in south africa, differential performance in school mathematics with respect to social class remains an enduring concern as reflected in national and international large-scale assessments. the article examines the implications of evaluation for orientations to mathematics in a school populated by learners from upper-middle-class or elite backgrounds and a school populated by learners from working-class backgrounds. the particular focus is on mathematics problems featured in tests used by two grade 10 teachers in each school and teachers’ marking of learners’ test scripts. a distinction between single-topic and multi-topic mathematics problem types is refracted through an analysis that draws on the adaptation by davis of lotman’s distinction between content orientation and expression orientation with respect to the reproduction of texts. the analysis reveals a preponderance of single-topic mathematics problems and the absence of multi-topic mathematics problems employed in the school populated by learners from working-class backgrounds and the presence of both single-topic and multi-topic problem types in the school populated by learners from middle-class or elite backgrounds. differences in the types of mathematical problems suggest differences in mathematical demand expected of learners and differences in their preparation for examinations in the two social class contexts. the selection of test problems and the marking of test scripts as instances of evaluation construct an orientation to mathematics that is expression oriented in the working-class context whereas both expression and content orientations are evident in the middle-class or elite context. the analysis provides a potential explanation for the persistent disparity in mathematics performance along social class lines in south african secondary schools. keywords: evaluation, marking, orientation to mathematics, content orientation, expression orientation, problem types, social class; computational resources. introduction narrowing the achievement gap in school mathematics in south africa with respect to social class remains a persistent social justice issue in spite of extensive curriculum reforms. difference in school mathematics performance between learners from middle-class families and learners from working-class families in south africa has to a great extent been documented in large-scale quantitative studies (e.g., reddy, van der berg, lebani, & berkowitz, 2006; spaull & kotze, 2015). we have to bear in mind though that social class and ‘race’ remain intertwined in south africa. despite ‘racial’ desegregation of more affluent schools, poorer schools remain populated largely by ‘black’ learners (spaull, 2019). thus, national assessments (e.g. matric examinations) and international assessments (e.g. third international mathematics and science study, timss) continue to reflect both a social class and ‘racial’ achievement gap in school mathematics (spaull, 2019). some smaller qualitative studies are concerned with the nature of the relation between mathematics performance and social class in an attempt to understand the underlying factors impacting on learner performance in school mathematics. most research in mathematics education in south africa (e.g. carnoy et al., 2011; eds. graven & venkat, 2017; schollar, 2008) tends to focus on learners from working-class families as part of efforts aimed at closing the social class performance gap (also see graven, 2014) since those learners constitute the majority under-performing in mathematics. there are a few exceptions, studies focusing on learners from working-class families, particularly ‘black’ learners in schools previously intended for ‘white’ learners only (e.g. feza, 2012; swanson, 2002, 2006). furthermore, there are a limited number of south african comparative studies of school mathematics in pedagogic contexts differentiated with respect to learners’ social class membership (hoadley, 2007; jaffer, 2018; jaffer & davis, 2012). thus, little is known about the pedagogic practices with regard to school mathematics in upper-middle-class or elite schools, particularly secondary schools. this article aims to contribute to our knowledge of the pedagogic practices in mathematics classrooms in a school populated by learners from upper-middle-class or elite families which is compared to a school populated by learners from working-class families. despite diversity with respect to methodology, there is convergence in the findings of local and international comparative studies concerned with school mathematics in relation to the differential distribution of knowledge across different social class settings (anyon, 1980, 1981; atweh, bleicher, & cooper, 1998; atweh & cooper, 1995; cooper & dunne, 2000; hoadley, 2005, 2007; o’halloran, 2004). the orientation to mathematics in schools populated by learners from working-class families is often described in this literature as procedural or weakly bounded from the ‘everyday’. in contrast, orientation to mathematics in schools populated by learners from middle-class families is commonly described in this literature as conceptual as opposed to procedural or as strongly bounded from the ‘everyday’. the literature thus partitions descriptions of mathematics realised in pedagogic contexts differentiated with respect to social class in terms of the academic-everyday distinction and procedural-conceptual opposition and remains relatively silent on similarities in the mathematics realised in those pedagogic contexts. this article examines the orientations to mathematics in two schools differentiated with respect to learners’ social class membership by examining how evaluation structures orientation to mathematics. in particular, the research question pursued in this article is: how does evaluation function in relation to grade 10 mathematics in two schools that differ with respect to the social class membership of their learner populations and what are the implications for learners’ orientations to mathematics? theoretical framework the general methodology underpinning the analysis of data in this article is informed by bernstein’s (2000) notion of the pedagogic device, specifically in relation to evaluation. the functioning of evaluation in the two social class pedagogic contexts is examined through the selection of test problems and the marking of learners’ test scripts. the orientations to mathematics constructed by the way in which evaluation functions in a pedagogic context is explored further through davis’s (2011) appropriation of lotman’s (1990) notions of content orientation and expression orientation. pedagogic evaluation bernstein (2000) argues that evaluation is central to pedagogy. he uses the term evaluation as opposed to assessment to signal that evaluation is a broader notion that encompasses anything that marks out criteria for the recognition and realisation of legitimate knowledge statements in a particular pedagogic situation. evaluation can be instantiated, for example, in a range of activities such as the marking of a learner’s work, teacher’s responses to learners’ responses to teacher questions, teachers’ explanations or learners’ solutions to mathematics problems (davis, 2005). pedagogy entails a relationship between two or more notional pedagogic subjects, the teacher and the learner, with the reproduction of knowledge being the knot that ties the two together, referred to as a didactic relation by chevallard (1989, p. 4). teachers and learners relate to knowledge in different ways. the teacher reproduces knowledge in order to communicate what they want learners to produce and how they should produce what counts as legitimate knowledge in a pedagogic situation. the learner always asks themselves what the teacher expects of them and how should they achieve what the teacher wants from them. the relationship between the teacher and learner is therefore essentially evaluative in that what they produce is structured by evaluation. as such, evaluation is essential for the pedagogic reproduction of knowledge. for bernstein, the centrality of evaluation is emphasised in his discussion of the pedagogic device, which serves as an analytic and descriptive resource for describing the transformation of knowledge into pedagogic communication: we can see that key to pedagogic practice is continuous evaluation. […] this is what the device is about. evaluation condenses the meaning of the whole device. we are now in a position where we can derive the whole purpose of the device. the purpose of the device is to provide a symbolic ruler of consciousness. (bernstein, 2000, p. 50) the pedagogic device, which is bernstein’s attempt to relate social structure to individual consciousness, entails three hierarchically related ‘rules’ – the distributive, recontextualising and evaluative rules (bernstein, 2000) – which together structure the production and reproduction of knowledge. the evaluative rule is derived from the recontextualising rule, which is in turn derived from the distributive rule. the distributive rule ‘mark[s] and distribute[s] who may transmit what to whom’ (bernstein, 1990, p. 158). the distributive rule, which regulates the distribution of ‘forms of knowledge, forms of consciousness and forms of practice to social groups’ (bernstein, 1990, p. 42), plays a key role in the reproduction of the social division of labour by distributing access to social goods, contributing to the reproduction of patterns of social relations. the recontextualising rule, which governs the selection of knowledge from the field of production and other discourses such as theories of learning and teaching, for the formation of pedagogic discourse (e.g. school mathematics), creates specialised pedagogic subjects (bernstein, 1996, p. 46). the evaluative rule is key in the pedagogic reproduction of knowledge. bernstein argues that pedagogic practice is characterised by the ever-present evaluative activity where evaluation distinguishes legitimate from non-legitimate knowledge statements for learners (bernstein, 1996, p. 50). note that legitimate knowledge is not necessarily knowledge that is accepted as correct in general. legitimate knowledge is that sanctioned in a specific pedagogic situation. for example, when a teacher, accepts 2 as the only solution to the equation x2 = 4 where x ∈ ℝ, then incorrect knowledge is accepted as legitimate. it is also possible that correct mathematical knowledge could be considered non-legitimate in a pedagogic situation. texts produced by teachers and learners make explicit the knowledge accepted as legitimate in a pedagogic situation and provide criteria that mark out legitimate knowledge statements from non-legitimate statements. the centrality of evaluation is underscored by bernstein (1996) when he declares that ‘evaluation condenses the meaning of the device’ (p. 50). bernstein’s claim implies that both the recontextualising rule as well as the distributive rule are entailed in the evaluative rule. researchers, however, sometimes construct their accounts of pedagogic situations in the reverse order by examining how evaluation functions in pedagogic situations to reveal the content of the recontextualising rules from which the distributive rules are derived. this article is concerned with the functioning of evaluation and its structuring effect on orientations to mathematics in pedagogic contexts. orientation to mathematics lotman’s distinction between text-oriented cultures and grammar-oriented cultures has been usefully appropriated for describing orientations to mathematics (see davis, 2011; davis & ensor, 2018; dowling, 1998; jaffer, 2010, 2018). lotman describes grammar-oriented and text-oriented societies as follows: cultures can be governed by a system of rules or by a repertoire of texts imposing models of behaviour. in the former category, texts are generated by combinations of discrete units and are judged correct or incorrect according to their conformity to the combinational rules. in the latter category, society directly generates texts, which constitute macro units from which rules can eventually be inferred, but which initially and most importantly propose models to be followed and imitated. a grammar-oriented culture [i.e. governed by a system of rules] depends on ‘handbooks’, while a text-oriented culture [i.e. governed by a repetition of model texts] depends on ‘the book’. a handbook is a code which permits further messages and texts, whereas a book is a text, generated by an as-yet-unknown rule which, once analyzed and reduced to a handbook-like form, can suggest new ways of producing further texts (eco in lotman, 1990, p. xi, italics in original). a grammar-oriented culture which is governed by a system of rules is juxtaposed with a text-oriented culture which is governed by a repetition of model texts. lotman’s categories, while useful as a heuristic for thinking about pedagogy, require adaptation for analysis of pedagogic situations. pedagogic modalities in which learners are encouraged to reproduce texts, through repetition and rehearsal, that precisely conform with texts considered as legitimate in the pedagogic context, are suggestive of lotman’s text-oriented cultures. the use of ‘model answers’ for classes of mathematics problems often used in pedagogic contexts resonates with lotman’s text orientation. his concept of grammar-oriented cultures, on the other hand, is comparable with pedagogic modalities that encourage ‘syntactical symbol manipulation and propositional descriptions of relations between mathematical objects’ (davis, 2011, p. 315). for example, a grammar orientation is in play with the recognition that the vertical column method of two-digit addition and horizontal method of two-digit addition which explicitly involves partitioning two-digit numbers into tens and units are governed by the same mathematical structure (addition over natural numbers) despite differences at the level of expression. although combinatorial rules are embodied in model texts in text-oriented cultures, the repetition and rehearsal of texts seems to suggest that individuals can produce texts without the use of combinatorial rules (see davis, 2011). davis (2011) argues that this opposition cannot apply in pedagogic situations because, following chomsky (2009), combinatorial rules are essential components of thought and language and so of mathematics, and so ought to be present in text-oriented pedagogic situations. davis (2011, pp. 316–317) adopts lotman’s distinction between expression orientation and content orientation as a more appropriate distinction for describing aspects of the realisation of mathematics in pedagogic situations, given that lotman aligns text orientation with expression orientation and grammar orientation with content orientation: lotman suggests that text-oriented societies are at the same time expression-oriented ones, while grammar-oriented societies are content-oriented. the reason for such a definition becomes clear when one considers the fact that a culture which has evolved a highly differentiated content-system has also provided expression-units corresponding to its content-units, and may therefore establish a so-called ‘grammatical’ system — this simply being a highly articulated code. on the contrary a culture which has not yet differentiated its content-units expresses (through macroscopic expressive grouping: the texts) a sort of content-nebula (eco, 1976, p. 138, italics in original). so, for davis (2011, pp. 316–317), both expression orientation and content orientation are combinatorial. an expression orientation refers to an orientation to mathematics that focuses primarily on the expressive elements required and entails a system of combinatorial rules that operates directly on the expressive elements to generate texts (davis, 2011). the ‘change sides, change signs’ rule used to solve equations is an example of combinatorial rules focusing on the expressive elements of an equation, allowing an individual to solve an equation without the notion of an equation being present. for example, the equation 2x + 1 = 5 can be solved by moving the symbols ‘+ 1’ from the left-hand side to the right-hand side, and in doing so changing to the symbols ‘– 1’. thus, combinatorial rules operate directly on the symbols. furthermore, repetition and rehearsal of texts that precisely imitate texts considered as legitimate in a pedagogic situation is suggestive of an expression orientation. for example, a teacher’s model solution to the equation (x + 1)2 = 9 which insists that the equation must be transformed into standard form (ax2 + bx + c = 0) before it can be solved is promoting an expression orientation because the teacher’s solution focuses on the form of expression. the equation (x + 1)2 = 9 can be solved by using square roots, thus bypassing the need to transform (x + 1)2 = 9 into standard form. with a content orientation, the expressive elements are secondary, functioning merely as resources for communicating mathematics (davis, 2011). operating with a content orientation implies using combinatorial rules that abide by mathematical structures. for example, an individual who produces 4 as the solution to the equation 2x + 1 = 9 as follows: 2x + 1 = 9 ⇔ 2x + 1 = 8 + 1 ⇔ 2(4) + 1 is using the notion of equality and the right cancellation theorem as computational resources situated in the field of the reals as opposed to the ‘change signs, change sides’ rule which is located outside of the field of the reals. mathematics problems previous studies show that a large proportion of mathematics lessons is spent on solving mathematics problems (see, for example, us department of education, 2003). mathematics problems are commonly used as the main vehicles for the elaboration of mathematics topics in schooling. typically, teachers use mathematics problems as worked examples to illustrate particular solution procedures, as practice exercises to provide opportunities for learners to become proficient at executing those procedures and in assessment tasks such as tests and examinations to ascertain learners’ knowledge of mathematics. mathematics problems, sometimes referred to as mathematics tasks in the literature (see eds. shimizu, kaur, huang, & clarke, 2010), entail the use of computations to solve problems. arithmetic problems, solving equations, sketching graphs or solving geometric riders are examples of mathematics problems. mathematics tasks or problems are described in different ways in the literature. boaler’s (1998) distinction between ‘open-ended’ and ‘closed’ mathematics tasks is essentially a distinction between conceptual and procedural tasks (see also kaur, 2010). the distinction between ‘realistic’ and ‘esoteric’ problems (e.g. cooper & dunne, 2000) or categorisation of mathematics tasks as ‘authentic’ or ‘real-world’ (e.g. kaiser & schwarz, 2010) can be aligned with the academic-everyday distinction. the curriculum assessment policy statement (caps) further education and training assessment guidelines stipulate the following distribution of problem types: knowledge (20%), routine procedures (35%), complex procedures (30%) and problem solving (15%) (department of basic education, 2011, p. 53). the categorisation of problem types used in this article references the topics indexed by the mathematics problems because of the larger project’s interest in the announced topic and the content associated with the topic. the announced topic refers to a name used by teachers, learners and textbooks to indicate a particular selection of mathematics contents. it should be noted that the content associated with announced topics can only be revealed when teachers and learners solve mathematics problems. chitsike (2011), davis (2013) and jaffer (2012, 2018) provide examples of studies that examine the computations employed by teachers and learners when solving mathematics problems to ascertain the content associated with topics. they reveal that the realised content in the name of a topic is not stable across pedagogic contexts and that the content realised in relation to a topic could be aligned with or diverge from the content associated with the topic from the point of view of the field of mathematics. the example related to the solving of linear equations discussed above illustrates this point. the use of the ‘change sides, change signs’ rule to solve linear equations is an example where content associated with the topic diverges from the content associated with the topic from the point of view of the field of mathematics whereas the use of the right cancellation theorem to solve a linear equation is illustrative of content that aligns with the content associated with the topic from the point of view of the field of mathematics. i distinguish between single-topic mathematics problems and multi-topic mathematics problems. a mathematics problem that indexes one announced topic is referred to as a single-topic problem, typically accompanied by problem statements such as ‘sketch the parabola’ or ‘calculate the equation of the function’. multi-topic mathematics problems involve more than one announced topic. figure 1 references five announced topics: (1) graphing parabola (problems 1.1, 1.2 and 1.3), (2) graphing a linear function (problem 1.3), (3) range of functions (problem 1.4), (4) points of intersection (problem 1.5) and (5) graphic inequalities (problem 1.6). each of the problems 1.1 to 1.6 is classified as a single-topic mathematics problem but together they constitute a multi-topic problem because problem 1 deals with more than one topic. figure 1: problem 1 of test administered by ivory college teachers. multi-topic problems encourage inter-topic connectivity because they require learners to use previously encountered computational resources such as definitions, propositions and procedures together with the newly introduced knowledge to solve mathematics problems and so connect new topics to topics dealt with previously. however, the content realised through solving mathematics problems is not the issue under discussion in this article. instead, mathematics problems serve as instances of pedagogic evaluation in that they are selected by teachers for the elaboration of mathematics topics and are used to assess learners’ acquisition of mathematics topics. social class and ‘race’ social class serves as a background contextual variable in this study and not as an explanatory category. the intention here is not to set up causal relations between social class on the one hand and orientations to mathematics on the other hand. the selection of empirical sites on the basis of the social class membership of a school’s learner population was guided by the assumption that because social class continues to be aligned with differential mathematics achievement, such differences potentially point to contrasts in the way evaluation functions. as indicated earlier ‘race’ and social class remain inextricably linked in south africa. i use the term ‘race’ in quotation marks because ‘race’ has little biological validity (yudell, roberts, desalle, & tishkoff, 2016). ‘racial’ categories were imposed on south african citizens during apartheid and are not necessarily accepted by individuals categorised as such. i use the apartheid categories because this historical legacy has shaped and continues to shape schooling in south africa. since 1994, census classifications have distinguished between ‘black african’, ‘coloured’, ‘indian or asian’ and ‘white’. i use the terms ‘african’, ‘coloured’, ‘indian’ and ‘white’ all with the first letter capitalised. the term ‘black’ is used as an encompassing term to refer to ‘africans’, ‘coloureds’, and ‘indians’. deregulation of ‘race’ as an admission requirement in all schools followed shortly after the demise of apartheid. subsequently, post-apartheid south africa has witnessed substantial transformation in the ‘racial’ demographics of school populations. ‘white’, ‘coloured’ and ‘indian’ schools have changed with respect to their ‘racial’ composition, but the learner populations of ex-department of education and training schools have to a large extent remained exclusively ‘african’ (sujee, 2004, as cited in chisholm & sujee, 2006). enrolment patterns across the south african schooling system are now determined largely by school fees. no-fee and low-fee-paying public and independent schools serve the majority of south african children – working-class and lower-working-class children who are predominantly ‘african’ and ‘coloured’ (franklin, 2017). schooling has thus become stratified with respect to social class and remains to a large extent stratified in terms of ‘race’. research design this article reports on a research study (jaffer, 2018) that is concerned with the functioning of evaluation in grade 10 mathematics classrooms differentiated with respect to social class. the study was designed as a case study (yin, 2009). although yin (2009) refers to case study as a method of research that allows for in-depth study of a small number of cases where the focus is on understanding a complex phenomenon, i do not treat case study as a specific research method. instead a ‘case’ is simply viewed as a means of describing the selection procedure of a study (brown & dowling, 1998, p. 151). the theoretical resources recruited for this article were described above. the study was designed as a comparative study of two schools that differ with respect to the social class membership of their student population. the selection of schools was based on learners’ social class membership. school fees were used as a proxy for learners’ social class membership which was confirmed through biographical information obtained from a learner questionnaire and school questionnaire completed by the principal. arbor high is a no-fee school and ivory college is a private school with school fees set at r85 000 per annum in 2012. ivory college is ‘racially’ mixed and serves learners from upper-middle-class or elite families. arbor high is populated by ‘african’ learners from working-class families. at each school, two teachers and learners in one of their grade 10 mathematics classes comprise the research participants of the study. sara and jada taught at ivory college, maya and jono at arbor high. the data collected for the research study included video-recorded mathematics lessons, curriculum and teaching resources such as textbooks, worksheets, tests designed and administered by the teachers, test scripts from the four grade 10 mathematics classes and interviews with selected learners in each class. this article focuses specifically on the mathematics problems used in a test administered by the teachers and the marking of test scripts as instances of pedagogic evaluation. the announced topics in the observed lessons of the four teachers are all related to the caps curriculum topic functions, which according to the caps pace setter is scheduled for teaching in the second term of the school year (department of basic education, 2011, p. 17). all the lessons were observed in the third term. the tests, based on the topics dealt with in observed lessons, were set and conducted by teachers at the schools. at both schools, the tests took place after the observed lessons at a time decided by each school. the tests across the two schools and within arbor high differed with respect to the announced topics. the analytic categories in relation to problem types and orientations to mathematics are independent of the announced topics. ivory college test (sara and jada) ivory college wrote a ‘common’ test, which was a test set by one of the grade 10 teachers and written by all grade 10 learners at the school. the test, referred to as common test functions 2012 (see figure 1), covered linear functions, parabolas, exponential functions and hyperbolas. the test included topics dealt with during the observed lessons as well as topics covered after the observed lessons. the test comprised four problems which are classified as multi-topic mathematics problems because they each involve more than one announced topic. ivory college learners were given considerable practice opportunities by their teachers, mostly independently of the teacher, since the tasks were either given as homework exercises or tutorials that were required to be submitted for marks. the test problems in the common test functions 2012 are very similar to the problems contained in the parabola worksheet used during the observed lessons. in addition, learners were given a tutorial on graphs which they were required to submit for marks. the tutorial covered the same topics as the common test functions 2012 and consisted of single-topic problems such as ‘sketch the graph on separate axes. show all intercepts with the coordinate axes, turning points, axes of symmetry and asymptotes’ as well as multi-topic problems such as those shown in figure 1. the graph tutorial and the parabola worksheet therefore served as preparation for the test, which in turn appeared to be intended as preparation for the examination. in addition, ivory college learners were exposed to variations in phrasing of mathematics problems. for example, problems on calculating the points of intersection of two functions were posed in different ways in the worksheet and the tutorial: (1) calculate the points of intersection of the functions f and g, (2) calculate the values for x for which f(x) = g(x), (3) solve the equation −x2 + 9 = 2x + 6 and explain how this helps find two of the points labelled on the graph or (4) calculate the coordinates of the point t. in the last problem, learners were first expected to work out that point t represents a point of intersection of two functions. mathematics problems that did not directly name the procedure required to solve the problem was another feature of the worksheet, tutorial and the common test used by ivory college teachers. learners were, therefore, expected to analyse the problem statement in order to decipher the appropriate procedure required. the selection of mathematics problems for the test, worksheet and tutorial suggests that evaluation functions in a way that attempts to move beyond recall and rehearsal of procedures for solving particular classes of mathematics problems. in other words, fostering content orientation to mathematics rather than an expression orientation. furthermore, the test, like the worksheet and tutorial, encourages inter-topic connectivity in that mathematics problems focus on more than one topic simultaneously and so require learners to select appropriate computational resources. the evaluative activity instantiated in the selection of mathematics problems for the test suggests an orientation to mathematics that is content oriented rather than expression oriented. marking of ivory college test (sara and jada) the memorandum of common test functions 2012 provides solutions to the test problem as well as the allocation of marks. the memorandum appears very specific, as suggested by the comment ‘must give co-ords’ for problem 1.5 and the details of how problem 1.3 should be marked. it is interesting that in problem 1.1, marks are allocated for factorising the quadratic equation and writing down the x-intercepts but not for setting up the equation, that is, for establishing f(x) = 0 which is central to calculating the x-intercepts of the function. the mark allocation suggests evaluative criteria that prioritise obtaining the correct values of the x-intercepts without the notion of an equation serving as a computational resource. this hypothesis is borne out by the marking of the test scripts by sara. sara mostly makes the evaluative criteria explicit to learners by correcting errors or providing evaluative commentary (see figure 2), except for one learner’s solution to problem 1.6 (discussed later). figure 2: a learner’s marked solution to problem 1.1. sara awarded full marks for problem 1.1 to two learners despite the fact that they generated the correct x-intercepts without setting up an equation (see figure 2). her marking of the learners’ solutions as correct despite the explicit absence of an equation as a computational resource suggests an orientation to mathematics that is expression oriented rather than content oriented, that is, an orientation that focuses on producing the correct expressions irrespective of the underlying mathematical structures. however, the fact that she corrected the learner’s solution indicates that she does make the evaluative criteria that the learners ought to display explicit. sara’s marking of problem 1.6 (see figure 1) which focuses on a graphical solution of the inequality f(x) < 0 (i.e. −2x2 + 2x + 4 < 0) provides further evidence of a slippage towards expression orientation. learner 1 produced ‘x < 1 and x > 2’ as the solution to problem 1.6 which was marked as correct although the correct solution to problem 1.6 is x < –1 or x > 2. her failure to correct the learner’s incorrect solution might be an oversight on her part given that her marking of learners’ test scripts was generally consistent and she indicated that learner 2’s solution is incorrect by replacing ‘and’ with ‘or’. as in the case of the marking of problem 1.1, sara awards full marks for learner 2’s incorrect solution to problem 1.6. it thus appears that the correct usage of the logical connective ‘or’ rather than ‘and’ is not deemed important. it is curious though that sara is not prepared to accept learner 3’s solution (2 < x < −1) as correct. sara writes the following on the learner’s script: ‘you can’t write this as one inequality. 2 < −1 is not true. x > −2 or x < −1’. note that sara makes a mistake when she provides the corrected solution. she stated the solution is x > −2 or x < −1. the correct solution is x > 2 or x < −1. the statement 2 < x < −1 produced by learner 3 is equivalent to the statement, x < −1 and x > 2, produced by learner 2 and learner 1. both statements imply that there is a number, x, which is simultaneously greater than 2 and less than –1 and so disrupt the order relation. however, the solution of learner 1 is marked as correct with full marks without commentary and learner 2’s solution is awarded full marks but corrected. sara’s marking of learner 1’s and learner 2’s solutions to problem 1.6 suggests that the logical connectives and order relations are not explicitly required as computational resources and indicates that content that diverges both at the level of expression and at the level of content is accepted as correct. furthermore, her marking indicates that the presence of the expressions /x < −1/ and /x > 2/ are prioritised over the solution set that satisfies the condition that f(x) < 0, suggesting an orientation to mathematics that is expression centred rather than content centred. jada’s marking was consistent across learners’ test scripts and she made the evaluative criteria explicit to learners by correcting errors or providing evaluative commentary (see figure 3). in contrast to sara, jada deducted marks for errors committed by learners. for example, she deducted one mark for not equating f(x) with 0 when solving problem 1.1. she therefore prioritises the notion of an equation as a central computational resource in solving problem 1.1. her marking of learners’ solutions stands in opposition to sara’s marking and appears to be more content oriented than expression oriented. figure 3: noa’s (p12) solution to problem 1.1. arbor high test (maya) at arbor high, each grade 10 teacher set their own mathematics test. the tests administered by the two grade 10 teachers at arbor high differed in terms of announced topics assessed. maya’s test covered the topics dealt with during the observed lessons and consisted of four test problems. problems 1, 2 and 3 entailed finding the equation of a given function provided as a sketch (see figure 4 for problems 1 and 2) and problem 4 focused on sketching the graph of the function . figure 4: extract of test on functions administered by maya. problems 1–3 are of the type ‘calculate the equation of the function’ and problem 4 of the type ‘sketch the graph of the function’, which were the problem types covered in class during the observed lessons. all the test problems are classified as single-topic problems. the test problem types are the same as the problems used in the observed lessons, but the examples differed. the test, like the problems used during the observed lessons, directly named the procedure that learners were expected to carry out. maya’s learners were not expected to analyse problems in order to select a particular procedure for solving a problem. furthermore, learners in maya’s pedagogic context were not exposed to variations in the phrasing of problem types. for example, the ‘calculate the equation of the linear function’ mathematics problems provided by maya all entailed a sketch with given intercepts. variants of the same mathematics problem could, for example, be achieved through changing the nature of the given points: two intercepts or a y-intercept and any other point on the line or any two points on the line. alternatively, the gradient of the line and another point could be provided. the absence of problems that require analysis in order to select appropriate procedures for solving the problem and the lack of variation in problem statements are suggestive of an orientation to mathematics that attempts to elicit precise responses from learners through the rehearsal of particular procedures for solving particular problems. in other words, learners are encouraged to recognise problem types and then select the correct procedure. furthermore, the test, like the problems used in the observed lessons, treated topics separately. therefore, unlike the ivory college test, the test set by maya suggests a lack of inter-topic connectivity. only sketches of functions were provided in problems 1, 2 and 3 of the test (see figure 4), as was the case for the mathematics problems presented during the observed lessons. in other words, the mathematics problems did not state what type of function was represented nor was the general equation of the function provided. learners were thus expected to determine the type of function from the sketch, suggesting that the expressive elements (the graphical expressions) serve as computational resources because it is the imagistic features of the text that learners are expected to draw on in order to determine the type of function represented. in other words, learners were expected to recognise the function type on the basis of the images (expressions) provided. the test problems and mathematics problems used in the observed lessons thus encourage an expression orientation to mathematics. marking of maya’s test the total mark for the test and mark allocation per problem were not provided to learners and the teacher’s memorandum did not show the mark allocation. from the marked scripts, it became apparent that the teacher allocated four marks per problem, bringing the total of the test to 16. in the marking of learners’ test scripts, maya at times indicated that an error was produced and deducted marks to penalise the learner. on a number of occasions though, errors produced by learners were not highlighted by the teacher and were marked as correct. an example is illustrated in figure 5 which shows learner 1’s solution to problem 1. figure 5: learner 1’s marked solution to problem 1. learner 1 identifies the sketch as representing a parabola as indicated in her choice of general formula y = ax2 + bx + q. she produces the correct value for a and although b is correct, her solution indicates that the notion of an equation, central to the computation, is absent. she produces the expression 2 – 6 = b from the expression y = –2 + b + 6, which is incorrect because y simply ‘disappears’. despite the mathematical inconsistencies produced by learner 1, the teacher awards her full marks. so, maya’s evaluation of the learner’s mathematical work reveals an orientation to mathematics that focuses on the correct expression in spite of mathematical violations. maya awards full marks to learner 2 for her solution to problem 1 despite a number of computational inconsistencies with respect to mathematics. the learner’s final equation y = −2x2 −4b + 6 is marked as correct even though it is incorrect presumably because she positioned the value of b in the ‘correct place’ in the equation. after writing down the expected general formula y = ax2 + bx + q, the learner produced the expression 6 = a(x + 0)(x + 6) which is incorrect but not pointed out by maya. later in the solution, the learner incorrectly computes the value of but her value of a is marked as correct by the teacher, presumably because the teacher assumes that the learner has merely made a computational error which is corrected later in her solution. maya’s marking of the learner’s solution suggests a strong expression orientation to mathematics since her assessment of learners’ work validates the production of the expected expressions despite divergence from the mathematics content associated with the topic. mathematics constituted in this pedagogic context is primarily a form of mathematical knowledge which diverges from mathematics at the level of expression and at the level of the content associated with the topic. the teacher’s marking of learners’ solutions suggests that her evaluation cultivates an orientation to mathematics that is expression oriented. arbor high test (jono) jono’s test (see figure 6) comprises 10 problems requiring learners to ‘find the domain and range’ of functions, two of which are linear functions (section a), three quadratic functions (section b), three hyperbolic functions (section c) and two exponential functions (section d). the test is an extract of the worksheet used during the observed lessons and resembles the worksheet both in its structure and mathematics problems. in fact, the test constitutes a selection of items from the worksheet which is evident when we compare the test to the worksheet. figure 6: extract of test on domain and range of functions administered by jono. test problem a1 corresponds with worksheet problem a5, test problem a2 with worksheet problem a1, test problems b1–b3 with worksheet problems b1–b3, test problems c1–c3 with worksheet problems c1–c3, test problem d1 with worksheet problem d1 and test problem d2 with worksheet problem d5. thus, all the problems from the test were selected from the worksheet, sometimes in the same order. all the test problems are classified as single-topic mathematics problems. jono’s learners, like maya’s learners, were not expected to analyse problems in order to select a particular procedure for solving a problem. the function type, for example linear functions or quadratic functions, was identified for the learner, thus generating a test of low complexity because learners mostly needed to recall the propositions with respect to each function type established during the observed lessons. for example, learners were expected to recall that the domain of a linear function is{x: x ∈ ℝ; −∞ < x < +∞}. note that the inclusion of –∞ < x < +∞ as part of the statement is really redundant but this was the legitimate response expected by jono. jono’s learners, like maya’s learners, were not exposed to variations in problem types. for example, mathematics problems related to domain and range of functions could be set in graphical form. in other words, learners have to deduce the domain and range from the graph of a function. the absence of problems that required analysis in order to select appropriate procedures for solving them and the lack of variation in problem statements suggest that learners were expected to rehearse and repeat particular procedures for solving particular problems, typical of an expression orientation to mathematics. furthermore, the learners had seen the test problems and worked through the problems in class. it seems that the test assesses whether learners are able to repeat the texts produced in class under test conditions. in other words, the evaluation encourages learners to reproduce texts that precisely conform with texts that are considered as legitimate in the pedagogic context, through repetition and rehearsal. the test is therefore strongly suggestive of an orientation to mathematics that is expression oriented. marking of jono’s test the test memorandum provides solutions to the test problems but how marks ought to be awarded is not made explicit. the marked tests show that two marks were allocated per test problem, half a mark each for the domain and range expressed in set builder notation and in interval notation. the memorandum provided the domain for problem a1 as {x: x ∈ ℝ; –∞ < x < ∞} in set builder notation and (–∞; ∞) for interval notation. the range for problem a1 was given as {y: y ∈ ℝ; –∞ < y < ∞} in set builder notation and for (–∞; ∞) for interval notation. the interviewed learners’ solutions to problem a1 and jono’s marking of their solutions are shown in table 1. table 1: solutions to problem a1 of jono’s learners selected to be interviewed. table 1 shows (1) solutions to problem a1 (domain and range in set builder notation and interval notation) produced by learners who were selected to be interviewed, (2) whether the solution is correct or incorrect according to the teacher’s memorandum and (3) the marks awarded by the teacher. we observe that incorrect components are treated as though they are correct. ali produces incorrect statements for the domain and range in set builder notation but is awarded full marks by the teacher. similarly, ozi and lea have three parts to the solution to problem a1 incorrect but ozi is awarded 1.5 marks and lea is awarded 1 mark. similarly, ory and zoe both produce the correct domain and range in interval notation despite the fact that both learners obtained the incorrect set builder notation. ory was awarded 1 mark and zoe was awarded 1.5 marks. jono at times corrected errors and on other occasions he neglected to identify the errors. for example, inconsistencies are evident when we compare the marking of two learners’ (ozi and ali) solutions to problem a1. the set builder notation for the domain and range in both learners’ solutions is incorrect as they violate order relations. jono corrects ozi’s domain in set builder notation by putting rings around the inequality signs to indicate that they are incorrect and writing the correct expression –∞ < x < ∞ in place of the incorrect expression –∞ > x > ∞ but does not correct ali’s solution. ali’s statements for the domain as {x/ x ∈ ℝ; –∞ > x < ∞} and range as {y/ y ∈ ℝ; –∞ > x < ∞} for the function in set builder notation are incorrect but they are not corrected by the teacher. in fact, ali makes the same mistake throughout the test but jono does not correct the error even once. the fact that ali was awarded full marks for his solution to problem a1 and the fact the jono does not correct ali’s errors suggest that jono does not require learners to use order relations as computational resources. jono’s evaluative activity as instantiated in the marking of learners’ test scripts validates content that differs from mathematics at the level of expression as well as the level of content. in addition, his evaluation of learners’ test scripts represents an extreme version of expression orientation in that as long as the expressions produced by learners resemble the correct expression according to the memorandum they are accepted as correct. concluding remarks comparing the tests across the two schools reveals differences in the types of problems set in the two schools and differences in the preparation for tests and examinations. ivory college learners’ preparation involved classwork and independent work on worksheets and tutorials that pose mathematics problems in different ways. the types of problems encountered in class and independent work as well the test included both single-topic problems as well as multi-topic problems that required learners to draw on a number of different topics. in contrast, arbor high learners were only exposed to single-topic mathematics problems in their tests which were restricted to mathematics problems encountered during the observed lessons with no variation in the way problems are phrased. the arbor high tests appear to encourage an expression orientation given the similarity of the mathematics problems used in the tests to those used during the observed lessons, with jono’s test representing an extreme case of expression orientation because the test problems were extracted from the worksheet used during the observed lessons. the tests suggest that rehearsal of model texts in the form of set solution procedures for set problem types is the primary mode of mathematics reproduction in the arbor high pedagogic contexts. the absence of multi-topic mathematics problems in the arbor high tests corresponds with the absence of multi-topic mathematics problems in the observed lessons, which indicates that topics are treated in isolation by arbor high teachers thus resulting in a lack of inter-topic connectivity. as such, arbor high learners are left to make connections between topics independently of the teacher. it could be argued that synthesis of school mathematics topics into a coherent whole is made much harder for the learners from working-class backgrounds than the learners from upper-middle-class or elite backgrounds. in addition, ivory college learners’ exposure in class and in tests to multi-topic problems which resemble examination type problems means that they appear to be better prepared with the support of their teachers for more mathematically demanding problems than their counterparts at arbor high. it appears that in the working-class contexts, learners are left to make connections to topics and tackle more complex problems on their own, that is, without instruction and support from teachers, which perhaps provides insight into why mathematics performance for the majority of secondary learners in south africa is so poor. comparing the marking of the test also reveals differences across the four pedagogic contexts. maya and jono’s marking included instances where mathematical violations were not corrected by the teacher and were accepted as correct. furthermore, their marking is inconsistent and at times learners’ solutions marked as correct did not match their memoranda. sara and jada corrected learner errors, thus making learners’ errors explicit to them. sara, however, at times made marking errors and did not deduct marks even though the solutions contained errors and on one occasion marked an incorrect solution as correct. thus, jono’s and maya’s marking encourages an expression orientation to mathematics. jono’s and maya’s marking confirms the expression orientation observed in the observed lessons (see jaffer, 2018). sara’s marking reveals aspects of expression orientation that is reminiscent of the marking in the working-class contexts although to a lesser degree. the types of problems learners are exposed to suggest an orientation that is more content centred. although not discussed here, the orientation to mathematics evident in the observed lessons taught by sara was more content oriented (see jaffer, 2018). thus, sara presents a case of a hybrid of expression orientation and content orientation. we, therefore, see differences as well as similarities in the pedagogic practices across social class contexts, disrupting entrenched narratives in the literature about stark differences between schools populated by learners from working-class backgrounds and schools populated by learners from upper-middle-class or elite backgrounds. the crucial difference, however, does appear to hinge on the absence of content orientation in the school populated by learners from working-class backgrounds and the extensive preparation and type of problems that learners in the upper-middle-class or elite context are exposed to, which perhaps provides a possible explanation for the persistent differential performance in mathematics along the lines of social class. acknowledgements this article derives from my doctoral research project. i hereby thank my supervisors, associate professor zain davis and professor emeritus paula ensor, for their guidance and support during the supervision of my phd study. i further wish to thank both supervisors for their valuable comments on a draft version of this article. competing interests i declare that i have no financial or personal relationships that may have inappropriately influenced me in writing this article. authors’ contributions i declare that i am the sole author of this article. ethical consideration ethical clearance for this research was obtained from the research ethics committee of the university of cape town in the western cape (ethical clearance number ednrec202000909). ethical issues including anonymity, confidentiality and voluntary participation were discussed with participants in the study and written consent for participation in the research was obtained from participants. funding information this work is based on research supported in part by the national research foundation of south africa under grant number 92639. the research also benefited from funding from the university’s research capacity initiative. any opinion, findings, conclusions and recommendations expressed here are those of the author and are not necessarily attributable to either of these organisations. data availability statement data sharing is not applicable to this article. disclaimer the views and opinions expressed in this article are those of the author and do not necessarily reflect the official policy or position of any affiliate agency of the author. references anyon, j. 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(2016). taking race out of human genetics. science, 351(6273), 564–565. https://doi.org/10.1126/science.aac4951 investigating a scaffold to code‐switching  as strategy in multilingual classrooms    hannatjie vorster  north‐west university (potchefstroom campus)  hannatjie.vorster@nwu.ac.za  there  is an urgent need  to  find strategies  to assist multilingual  learners who are  taught by  means of english as the language of teaching and learning (lolt) in coping with subject matter,  especially as  learners show different  levels of english proficiency. one  important strategy  is  code‐switching.  the  possibility  of  using  a  glossary,  notes  and  tests  given  in  english  and  setswana  as  scaffold  for  the  code‐switching,  was  investigated  in  a  qualitative  study.  an  intervention  was  launched  in  a  region  where  the  main  language  is  setswana.  two  schools  participated in the study, one school in a rural area and the other in a township. the teaching of  a geometry unit was undertaken in which notes, a glossary and tests in english and setswana  were used to assist the learners. the teachers used a strategy of code‐switching. interviews with  the learners and the teachers revealed that learners had a positive attitude towards the use of  both languages in written work. in contrast, the teachers had mixed feelings, especially about  the use of mathematical terminology in setswana. difficulties experienced and positive features  regarding the use of new setswana terminology are discussed.  the use of two or more languages, usually english and an indigenous language, has now become a recognised feature in multilingual classes in south africa. code-switching, as this phenomenon has become known, has however been introduced in an informal and unstructured manner. in teacher education in south africa pedagogy that includes language strategies in the teaching and learning of mathematics is still in its infancy. research carried out in the north west region in south africa (nwsa) has indicated that although oral codeswitching was used widely, no written material was made available for learners in their own language to refer to when they did homework or studied for tests (vorster, 2005). this study attempts to contribute to the search for ways to support learners in the use of their main language in combination with english to enhance conceptualisation in mathematics. the term ‘learners' main language’ refers to the indigenous language in which the learner is the most proficient. due to environmental influences, this may be a different language than the learner’s original home language. a brief discussion of literature is given to motivate why the main language of the learner is so important for conceptualisation. an overview of the current language strategies, techniques and the different language scenarios in the classroom in south africa is also crucial to a valid contribution. theoretical background the role of language in the learning of mathematics the importance of language for conceptualisation and reasoning in general and for mathematics in particular, has long been acknowledged in the research literature (gentner & goldin-meadow, 2003; pimm, 1987, 1991; usiskin, 1996; vygotsky, 1962). the following can be highlighted: when new concepts are constructed, word sense plays an important role. the “sense” of a word is understood to be all that a word arouses in our consciousness and all the different nuances of the meaning of a word in different contexts (vygotsky, 1962). mastery of word sense is important for understanding concepts. word sense is only mastered in a specific language when the proficiency of the learner has reached a certain level. according to khisty (1995), the development of proficiency in the disciplinary register within the second language can take as long as seven years. in rural areas with an english foreign language environment, it may take longer. pythagoras, 67, 33-41 (june 2008) 33 investigating a scaffold to code-switching the network theory of learning stresses the importance of connecting new knowledge into networks of existing knowledge in order for conceptualisation to occur (shunk, 1996). conceptual knowledge embedded in the learner’s main language forms important prior knowledge. this, together with word sense that is usually well developed in the main language of a learner, gives an indication that the main language plays a crucial role in facilitating the construction of learners’ concepts. usiskin (1996) stresses that oral communication, both formal and informal, is important for the learning of mathematics. by articulating their mathematical ideas and discussing it with peers, learners negotiate mathematical meaning for themselves (costello, 1991; pimm, 1987; sai, 1994). the main language of learners supplies a support system to facilitate the interaction among learners and with the teacher, while their proficiency in english is developed (khisty, 1995; setati, et al., 2002). part of learning mathematics entails gaining control over its register. it poses problems even for first language learners because of the specific structure of the mathematics register (pimm, 1991), and all the more so for second language learners. learners struggle with the specific and correct use of prepositions, the numerous synonyms in english as well as with the fact that many words have different meanings in standard english and the mathematics register (orton & frobisher, 1996; pimm, 1987). the difference in the semantic structure of english and the indigenous languages is another complicating factor. the language culture in mathematics classrooms in south africa adler (2001) identifies three different language scenarios in multilingual mathematics classes: • firstly, the urban-suburban areas where there is a strong english environment and many different main languages are found. • secondly, the additional language situation of the urban or township contexts, with less english in the environment, a strong regional language and different indigenous languages present. • thirdly, a foreign language situation where the learners mostly hear english only at school and most of the learners have the same main language. this is often found in the rural areas. the focus of this article will be on the foreign and additional language situations, as these are situations where the use of the main language as aid may be the most viable. two language strategies are currently implemented. the first strategy is to use only english in the teaching and learning of mathematics. this strategy is for example found where the teacher cannot speak the indigenous language or where many different indigenous languages are present in a class. improving the learners’ english proficiency is the main instrument to overcome situations where a varying proficiency in english complicates teaching and learning. in the meantime, teachers cope with learners’ different levels of english proficiency by using different techniques to overcome communication problems. they for instance circumvent language by using a minimum of words and concentrating on symbolic language. some teachers avoid such topics as linear programming and word problems. another technique is to use safe talk such as “ritualisation”, where questions are put to the class as a group, who then answer in a chorus (hornberger & chick, 2001). this is ‘face saving’ because learners who do not know the answers can effectively hide in the group. such a situation invariably creates an impression that everybody understands (heller & martin-jones, 2001). in the second main strategy, namely codeswitching, both the main language(s) of the learners and english are used to facilitate understanding. adler (2001) and setati (2002) contributed by describing the nature of existing code-switching practices in multilingual mathematics classrooms. the discussions on the different purposes for which code-switching is used, as described by setati (2002), give valuable insights into code-switching practices. heller and martin-jones (2001, pp. 9-11) identify “front stage” and “back stage” use of the main language. front stage describes the phenomenon where the main language is used to facilitate understanding, while “back stage” describes the situation where the main language is used only for giving instructions or in the affective domain, for example to encourage or discipline learners. some techniques that facilitate understanding are used in both the “english only” and codeswitching strategies. the technique of recasting provides the learner with examples of how to use 34 hannatjie vorster mathematical terminology and synonyms in different sentences (khisty, 1995). revoicing, where the teacher listens to the learners’ mathematical talk and revoices their expressions, leads them towards correct and more formal mathematical discourse (setati & adler, 2000). another technique is the use of the etymology and morphology of words, e.g. “sigma”, which originates from the greek letter “s” to indicate sum, or “co-interior angles” where “co” means together and “interior” inside. a more recent development is the use of multilingual mathematics dictionaries of which three are worth mentioning here. the multilingual mathematics dictionary for grade 1 to 6 (department of arts and culture, 2003) supplies a thousand existing mathematical terms in all eleven languages, mainly for teaching in primary school. the set of bilingual explanatory maths dictionaries (fricke & meyer, 2005) give only the english terminology with explanations of the terminology in both english and an indigenous language (available for isizulu, setswana, sesotho sa leboa and thsivenda). understanding concepts in mathematics and science (young et al., 2005), supplies selected mathematics and science terminology in english, xhosa, zulu and afrikaans, with a comprehensive discussion of the each term aided by illustrations. these dictionaries are important aids to code-switching and signify attempts to cope with the lack of formalised terminology in the indigenous languages. the most recent development in the field of teaching in the multilingual classrooms is the research by setati et al. (2007) on a new pedagogy for teaching and learning of mathematics in multilingual classrooms. the pedagogy focuses on a problem-centred strategy that includes a language strategy resembling the scaffolding proposed in this paper. notes were given in both english and the relevant indigenous languages, but the english mathematical terminology was not translated. the use of the language was meant to be ‘invisible’ in the sense that the focus was on the mathematics, not on the language. although the written material was made available in the learners’ indigenous languages and code-switching was taking place, attention was not drawn to use of the language. in this study however, the terminology is translated and the focus is on how learners and teachers experienced the use of two languages. these recent developments are summarised in figure 1 and the place of the proposed scaffold is indicated. figure 1: strategies, techniques and recent developments 35 investigating a scaffold to code-switching the empirical research in this qualitative study an intervention was made, involving code-switching between english and setswana. important features of the intervention were that the code-switching was supported by notes, a glossary with the main mathematical terminology of the unit, as well as tests set in both languages. the design of the geometry unit used in the intervention the intervention entailed the teaching of a geometry unit that included examples of worksheets for grade 8 learners. an english/setswana glossary of the most important terminology was composed for the learners to refer to while doing classwork and homework. learners were not required to memorise the setswana terminology, but could use the glossary as a reference in tests. setswana linguists from the translation world cc did the translation of the unit and also created new terminology where no setswana terminology existed. the translation world cc does translations of mathematics texts for various role players in the market of school mathematics. figure 2 shows some examples of the terminology in the glossary. straight lines mela e e tlhamaletseng common arm letsogo-gotlhe adjacent angles dikhutlomabapi vertically opposite angles dikhutlotsepamo-tebagano transversal molakgabaganyo horizontal line mola o o rapameng alternate angles dikhutlo-thefosano corresponding angles dikhutlo-tsamaelano co-interior angles dikhutlogare mmogo parallel lines mela e e bapileng figure 2: examples of the terminology in the glossary as the content of the notes consisted of many diagrams and short sentences, it was decided to put the sentences directly below each other, as illustrated in figure 3. measure the width of each pair of parallel lines at different places. lekanya bophara ba para nngwe le nngwe e e bapileng mo mafelong a a farologaneng. what do you find? \o bona eng? figure 3: an example of notes in two languages research procedures and results the geometry unit was used as learning material in an intervention in grade 8 classes at one rural school (school a) and one township school (school b) from the north west province in south africa. both schools were selected on the basis of convenience and feasibility. one grade 8 teacher at each school agreed to teach the geometry unit and they became partners in the research. it took the teacher in school a five weeks and the teacher in school b four weeks to teach the geometry unit. case studies were done at the two schools. it consisted of class visits with field notes and semistructured interviews with both a sample of learners from the classes where the intervention took place and the two teachers. all interviews were tape recorded and transcribed. learners’ interviews were translated from setswana into english. school a school a is a well-organised but under-resourced combined school in a rural area. there was only one grade 8 class with 33 learners, 31 of the learners’ home language was setswana and two isixhosa. four learners, namely, joseph, sanna, lina, and stephen, were selected for interviews. a range of different academic profiles were selected based on their mathematics, setswana and english marks. the fifth learner, gladys, was selected because she was one of the two learners whose home language was isixhosa. the purpose of this was to look into the position of a learner that lives in a mainly setswana environment, but whose home language is not setswana. the home language of the teacher, miriam, was setswana. she had 17 years teaching experience. she enjoyed teaching mathematics, but felt she was struggling a bit with the geometry teaching. before the intervention miriam’s language policy was to teach mathematics in english and to switch to setswana only when learners really did not understand. from the class visits during the intervention it was observed that the communication among the learners, arranged in groups, was subdued and mostly in setswana. the teacher paid a great deal of individual attention to the learners. she asked the learners individually to read the notes to her in both languages, while at her table. some of the learners read the notes in english fluently, but others could not manage well in english, while reading fluently in setswana. 36 hannatjie vorster when miriam introduced new terminology, she initially explained the setswana terminology, and also used the morphology of the terminology to explain the concepts in setswana. she then introduced the formal english terminology and also coined sentences that she often repeated e.g. “the sum of the co-interior angles is 180 degrees.” while teaching miriam habitually used setswana with code-mixing, using the english terminology as well as these coined sentences during a mainly setswana explanation. learners mostly answered questions in english, but it seemed as if english was a foreign language to many of the learners, because their communication in english would consist only of “yes” or “no” or the coined english mathematical phrases. the views of the teacher at school a before the intervention, miriam expressed the view that the learners’ english was not very good “but they are trying.” she stated that it was sometimes difficult for the learners to do their mathematics in english because they did not always understand the questions, therefore it would help them if they could learn in both languages. she expressed the view that new setswana terminology would help some learners, but for others it would be difficult “because learners are not equal.” mary was very enthusiastic about the geometry unit and expressed herself as follows after the intervention: this unit has helped us a lot. me and the learners. i called mister m. one day to come and witness them. then he was so grateful to see what the learners have done. he was very, very happy. yes, i think that the language also contributed because i used their language to explain. although miriam found the setswana terminology more difficult than she expected, she said she used the setswana terminology to explain the concepts, for example: dikuthlo-tsaemalano … dia tsamalana. that means they are the same, they are in the same position. this was also apparent during the class visits. according to miriam, the code-switching was of great value to the learners, but the setswana terminology was difficult. she was of opinion that the learners preferred the english terminology, as well as to use english in their answers to questions in class and in tests. this was true for some of the learners, as can be seen in the interviews. miriam also thought that the learners did not really use the glossary. the interviews with the learners revealed that this assumption was actually incorrect. interviews with the learners at school a a senior teacher conducted the interviews in setswana in collaboration with the researcher. the teacher gave feedback to the researcher of what the learners were saying, so that the researcher could monitor and direct further questions. learners were asked to comment on the notes, terminology, glossary, and tests given in both languages. these interviews were later transcribed, translated and interpreted. although all the learners were in favour of the notes and glossary, two views could be discerned. some learners preferred the english, but also saw the benefits of the dual use of the languages to enhance their understanding. the other learners depended more heavily on the setswana, but realised that they also need the english because they have been trained in english for a long time. the first view was expressed as follows by lina: i understand the geometry better in english, teacher. she also commented on the glossary: if you don’t understand it in english you may be able to understand it in setswana. stephen commented: i understand it in english, but in setswana i don’t understand it well. english is just fine. but also: setswana makes things easy. english should be used with setswana explanations about the tests he said: i can’t do mathematics in setswana because we have been taught the subject for quite a long time in english. the second view was expressed the clearest by joseph: i would like to be taught mathematics in english and setswana. it is necessary to have the notes and list in those languages because some learners do not understand all the english words. i was in the position to understand the teacher. 37 investigating a scaffold to code-switching furthermore, he also wanted the tests in setswana, and maintained that the setswana terminology was easier to understand. however, he had to concede that the glossary was needed: because the teacher writes things in setswana, the english words are needed. yes teacher even in setswana. gladys expressed a common view among the learners about the notes by saying: “where i did not understand, i used setswana to understand” and stated that the glossary was necessary because “you can read it in short.” sanna indicated that she sometimes knows concepts but cannot express herself: “if i don’t remember what a straight line is in english – then i can refer to setswana.” gladys’s home language was not setswana, but it was clear from the interview that her main language has become setswana. although their language at home was xhosa, she was firm in her conviction that she is more fluent in setswana. this was possibly because she lived in a setswana environment and learned setswana at school. the picture that emerged from the interviews was that all the learners felt that they benefited from these language aids. this class is an example of an ideal situation for code-switching between setswana and english with written work as scaffold. only one main language has to be considered which makes code-switching easier and notes have to be translated into only one language. school b school b is a township school that is a large wellequipped secondary school. there were five grade 8 classes with two mathematics teachers, vusi and peter. before the intervention, vusi and peter were interviewed together. however, after vusi studied the content of the geometry unit, he decided not to participate. he did not feel confident to teach the geometry unit using the setswana mathematics register. peter was willing to participate. four learners were interviewed at school b. though the intention originally was to select learners whose home language is setswana, it became clear that this would not be a true reflection of the situation at school b since the learners at this school had a mixed language profile. the learners were therefore only selected based on their academic profile in the same manner as in school a, without taking into account their home language. this resulted in a random selection of home languages. as it came about, only one learner’s home language was setswana, namely sam. thandi’s home language was isisxhosa, and thabo’s home languages were sesotho/setswana. the fourth learner, lebo’s home language was sepedi. he was transferred to school b in grade 8 from a sepedi primary school. the home language of the teacher, peter, was isizulu, as was the main language of the primary school he attended. his secondary schooling was at a setswana school and he has been teaching at school b for nine years. he felt confident about teaching mathematics. the class consisted of 56 learners. before the intervention, the classes were usually conducted mainly in english, although peter sometimes did use setswana. during the intervention, peter explained more in setswana than usually, but still used a fair amount of english and the english terminology was front staged. during class visits, it was observed that he used the morphology of the new setswana terminology as a tool when explaining concepts. he explained new concepts in setswana and then linked it to the english: it’s like the ‘alternate’ its like (the teacher) is given another chance (to explain), its almost like that, and then you say ‘refosano … tefosano’. you say ‘tefosano … dikutlho tefosano’ and then tefosano is like from the word ‘refosano’ its like ‘just for two’, you see. some of the words that they are acquainted to. learners answered questions freely and good communication was established. the learners seemed to have mastered the setswana terminology and answered questions mainly in setswana. the communication among the learners was mainly in setswana. with 56 learners in the class, there was not time to pay individual attention to all the learners. views of the teacher at school b peter expressed the opinion that his learners were not very good in english and therefore it was sometimes difficult for the learners to understand the mathematics concepts when the lolt was english. he felt that it would be easier for the learners if they could be taught in setswana only, because the concepts will be easier for them in their main language. however, he was of the opinion that if the two languages are used simultaneously, like in code-switching and with the notes in both languages, it will create confusion. 38 hannatjie vorster after the intervention, peter said that it took time to get used to the setswana mathematics register (“the lingo itself”). he was a bit ashamed to be seen teaching in setswana. learners who visited the class “wanting that and that” were amazed to hear him teaching mathematics in setswana and he had to explain why. even so, peter “enjoyed” himself. he expressed himself as follows about switching to setswana: it was something very positive, i enjoyed myself. it gave me another chance of trying to put the knowledge or the concept very clear for them. the learners participated more in the class than usual and he commented: “it makes it much easier for them to understand than in english.” he is of opinion that it was beneficial for the learners to have the setswana notes and glossary as aid, especially because some of the setswana terminology was new and the learners did not know it. interviews with the learners at school b the interviews at school b were conducted by an experienced sestwana interviewer from the university in collaboration with the researcher. the interviews were conducted mainly in setswana in the same manner, and with the same set of questions, as was done at school a. learners were asked to give comments on the notes, terminology, glossary, and tests supplied in both languages. it seemed that thandi and lebo needed the dual presentation of languages and they felt that setswana was very important. thandi was positive about all the features of the dual language presentation: the setswana helped me, i understand setswana much better than english. she was also emphatic that she needed the setswana in the tests. although lebo was sepedi, he also appreciated the dual use of language. his: “sharp – it helped me”, came over very enthusiastically. he also repeated the refrain that was heard at the rural school: “if i do not understand the english term – i refer to the setswana word list.” he liked the terminology in setswana, but for the test he preferred english. his comment “in english because i’m used to english terms” highlighted that this was a new experience for the learners and the fact that they previously used only english in written work played an important role in their views. sam and thabo had mixed feelings. although sam said: i feel much good because the usage of both english and setswana make it better to understand he did not like the setswana mathematical terminology itself. he stated that he did not need the glossary, and want the test in english only. he remarked: they are much better in english. setswana terms give me problems. thabo was more non-committal and commented on the code-switching: “it was very interesting” and on the notes and setswana terminology: “it was not a problem to have two languages.” he showed his interest in the dual language representation finally by saying that the tests should be in both languages. the learners’ different home languages did not seem to have a great impact on how much benefit they felt they had gained from the setswana aids. from the interviews, it became clear that setswana has become both thabo and thandi’s main language, possibly because they attended schools with main language setswana from grade 1. lebo, who has started in a school where the main language is setswana only in grade 8 also felt strongly that he benefited. this may be because he has encountered setswana earlier, as was established from the interview, and because sepedi is a related language. none of the learners indicated that the use of the two languages together was confusing, as peter feared. on the contrary, they found it useful. as they did not understand the mathematical setswana or the english perfectly, they could refer to both languages to make sense of the notes, and consequently maximise understanding. discussion the teachers found the setswana terminology difficult, as could be expected as they have been using english terminology during their entire careers. however, they did use the morphology of terminology to link the concepts to words and concepts that the learners knew. although some of the learners preferred the english terminology and found the setswana terminology difficult, the main 39 investigating a scaffold to code-switching impression is that the learners’ view is that they benefited from the availability of the setswana notes and the glossary. the learners differed in their views on whether setswana/english questions in the tests were necessary, but most of them welcomed it. it was significant that all the learners made it clear that they used both the english and the setswana to clarify what they did not understand in the other language. this suggests that the dual use of languages is useful so that the learners can continuously oscillate between the two, using understanding in one language to support learning in the other. the purpose of the study was not to establish whether english as lolt should be substituted by setswana. the learners’ preference for english terminology and answering questions in english is therefore not a negative feature, because in the end the aim is to facilitate the journey to the formal mathematical register of english. although the study was a qualitative study and therefore the results cannot be generalised, valuable information has been gained concerning the views and experiences of these learners and teachers. it was positively established that it benefited most learners to have these setswana language-aids available, involving code-switching, english/setswana notes, a glossary, and tests as a scaffold to enhance understanding. conclusion an evolutionary process can be seen in research on language matters in mathematics classrooms in south africa. it started with reports on informal code-switching that was initiated by the teachers themselves and continued with more structured investigations into the language journeys. dictionaries of varied structures started to appear to aid the code-switching, with the newest development being the investigations of setati’s team into developing a new pedagogy for multilingual classrooms, one of the features being the use of study material in the learners’ different main languages as well as in english. vusi’s reaction showed that some teachers were not ready to use code-switching with written work as support. however, peter and miriam showed that some teachers are ready and even enthusiastic to try this strategy. it seems that the issue of setswana terminology must not be forced. both teachers and some of the learners found the setswana terminology difficult. however, these learners’ and the teachers’ struggle with the indigenous mathematical terminology is not surprising, since both teachers and learners are accustomed to the usage of the mathematical terminology in english only. therefore, it seems that the way to go at the moment is to use support materials in the indigenous language with the mathematical terminology mainly in english. this was the language strategy in the research on a pedagogy for multilingual mathematics classes done by setati et al. (2007). on the other hand, the teachers both used the morphology of words like dikuthlomabapi and dikhutlo-thefosano in their teaching and found it useful and some learners welcomed the setswana terminology. because it is an evolutionary process, the time can be foreseen when the need for terminology in setswana may be more pressing. the use of the techniques of recasting, revoicing and facilitating conceptualisation by using the morphology and etymology of words in the main language of learners will also be more viable if such terminology is available. as a result, research into finding methods to create meaningful terminology for mathematics in setswana where such terminology does not exist, may be indicated as an area of future research. although this will be a slow process the development of mathematical registers in indigenous languages may be one of the by-products of the development of this kind of pro-active scaffold model of code-switching. references adler, j. (2001). teaching mathematics in multilingual classrooms. dordrecht: kluwer academic. costello, j. (1991). teaching and learning mathematics 11-16. london: routledge. department of arts and culture (2003). multilingual mathematics dictionary. pretoria: national language service. fricke, i., & meyer, l. (2005). bilingual explanatory maths dictionary – english. isizulu. pretoria: clever books. (also in sesotho, sesotho sa leboa, setswana and yshivenda.) gentner, d., & goldin-meadow, s. (eds.) (2003). language in mind. advances in the study of language and thought. cambridge: mit press. heller, m., & martin-jones, m. (eds.) (2001). voices of authority: education and linguistic difference. westport, conn.: ablex publishing. 40 hannatjie vorster 41 hornberger, n. h., & chick, j. k. (2001). coconstructing school safetime: safetalk practices in peruvian and south african classrooms. in m. heller & m. martin-jones (eds.), voices of authority: education and linguistic difference (pp. 31-55). westport, conn.: ablex publishing. khisty, l. l. (1995). making inequality: issues of language and meanings in mathematics teaching with hispanic students. in g. secada, e. fennema & l. adajian (eds.), new directions for equity in mathematics education (pp. 279-285). new york: cambridge university press. orton, a., & frobisher, l. (1996). insights into teaching mathematics. london: cassell. pimm, d. (1987). speaking mathematically. communication in mathematics classrooms. london: routledge. pimm, d. (1991). communicating mathematically. in k. durkin & b. shire (eds.), language in mathematical education: research and practice (pp. 18-24). philadelphia: open university press. sai, k. p. (1994). doing and talking in primary mathematics. pythagoras, 34,15-19. setati, m. (2002). language practices in intermediate multilingual mathematics classrooms. unpublished doctoral dissertation, university of the witwatersrand. setati, m., & adler, j. (2000). between languages and discourses: language practices in primary multilingual mathematics classrooms in south africa. educational studies in mathematics, 43, 243-269. setati, m., adler, j., reed, y., & bapoo, a. (2002). code switching and other language practices in mathematics, science and english language classroom in south africa. in j. adler & y. reed (eds.), challenges of teacher development: an investigation of take up in south africa (pp. 7292). pretoria: van schaik publishers. setati, m., molefe, t., duma, b., nkambule, t., mpalami, n., & langa, m. (2007). towards pedagogy for teaching mathematics in multilingual classrooms. marang symposium cd, wits centre for maths and science education, university of the witwatersrand. shunk, d. h. (1996). learning theories (2nd ed.). englewood cliffs, n.j.: prentice hall. usiskin, z. (1996). mathematics as language. in p. c. elliot, & m. j. kenney (eds.), communication in mathematics: k-12 and beyond. 1996 yearbook (pp 231-243). reston: nctm. vorster, j. a. (2005). the influence of terminology and support materials in the main language on the conceptualisation of geometry learners with limited english proficiency. unpublished master’s thesis, potchefstroom university for c.h.e. vygotsky, l. s. (1962). thought and language. (edited and translated by eugenia hanfmann and gertrude vakar.) new york: m.i.t. press. young, d., van der vlugt, j., & qanya, s., et al. (2005). understanding concepts in mathematics and science: a multilingual learning and teaching resource book in english, isixhosa, isizulu, afrikaans. cape town: maskew miller longman. << /ascii85encodepages false /allowtransparency false /autopositionepsfiles true /autorotatepages /none /binding /left /calgrayprofile (dot gain 20%) /calrgbprofile (srgb iec61966-2.1) /calcmykprofile (u.s. web coated \050swop\051 v2) /srgbprofile (srgb iec61966-2.1) /cannotembedfontpolicy /warning /compatibilitylevel 1.5 /compressobjects /tags /compresspages true /convertimagestoindexed true /passthroughjpegimages true /createjobticket false /defaultrenderingintent /default /detectblends true /detectcurves 0.0000 /colorconversionstrategy /cmyk /dothumbnails false /embedallfonts true /embedopentype false /parseiccprofilesincomments true /embedjoboptions true /dscreportinglevel 0 /emitdscwarnings false /endpage -1 /imagememory 1048576 /lockdistillerparams false /maxsubsetpct 100 /optimize true /opm 1 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/pdfxoutputintentprofileselector /documentcmyk /preserveediting true /untaggedcmykhandling /leaveuntagged /untaggedrgbhandling /usedocumentprofile /usedocumentbleed false >> ] >> setdistillerparams << /hwresolution [2400 2400] /pagesize [612.000 792.000] >> setpagedevice pyth_v41_i1_2020_contents.indd http://www.pythagoras.org.za open access table of contents i original research university students’ mental construction when learning the convergence of a series concept conilius j. chagwiza, aneshkumar maharaj, deonarain brijlall pythagoras | vol 41, no 1 | a567 | 15 december 2020 original research evaluation and orientations to grade 10 mathematics in schools differentiated by social class shaheeda jaffer pythagoras | vol 41, no 1 | a578 | 17 december 2020 original research investigating the strength of alignment between senior phase mathematics content standards and workbook activities on number patterns agnes d. qhibi, zwelithini b. dhlamini, kabelo chuene pythagoras | vol 41, no 1 | a569 | 17 december 2020 original research becoming mathematical: designing a curriculum for a mathematics club erna lampen, karin brodie pythagoras | vol 41, no 1 | a572 | 18 december 2020 original research relative difficulty of early grade compare type word problems: learning from the case of isixhosa ingrid e. mostert pythagoras | vol 41, no 1 | a538 | 21 december 2020 reviewer acknowledgement pythagoras | vol 41, no 1 | a588 | 21 december 2020 69 82 94 110 125 141 book review pedagogical narratives in mathematics education in south africa charles r. smith pythagoras | vol 41, no 1 | a573 | 25 september 2020 original research learning the function concept by exploring digital images as functions christiaan venter pythagoras | vol 41, no 1 | a524 | 31 august 2020 original research peer tutors’ views on their role in motivating learners to learn mathematics abigail k. roberts, erica d. spangenberg pythagoras | vol 41, no 1 | a520 | 28 september 2020 original research the effect of graphing calculator use on learners’ achievement and strategies in quadratic inequality problem solving levi ndlovu, mdutshekelwa ndlovu pythagoras | vol 41, no 1 | a552 | 18 november 2020 original research learner performance in the 2009 to 2014 final grade 12 mathematics examination: a quantile regression approach nombuso zondo, temesgen zewotir, delia north pythagoras | vol 41, no 1 | a545 | 30 november 2020 original research postgraduate mathematics education students’ experiences of using digital platforms for learning within the covid-19 pandemic era jayaluxmi naidoo pythagoras | vol 41, no 1 | a568 | 30 november 2020 1 4 19 32 45 58 page i of i table of contents i vol 41, no 1 (2020) issn: 1012-2346 (print) | issn: 2223-7895 (online)pythagoras mother tongue policies and mathematical  terminology in the teaching of mathematics  mercy kazima  university of malawi  mkazima@chanco.unima.mw  many countries in africa have mother tongue policies which require learners to be taught in  their  mother  tongue  for  at  least  some  of  the  years  of  primary  education.  this  paper  discusses the implementation of such policies in relation to mathematical terminology in the  teaching and learning of mathematics. i present and discuss two strategies of dealing with  mathematical terminology when teaching in the mother tongue:  the strategy of developing  mathematics  registers  in  the  local  languages  and  the  strategy  of  borrowing  from  mathematical english. i discuss the cases of nigeria and tanzania as examples of the former  and the case of malawi as an example of the latter. the discussion illustrates the strengths of  each strategy as well as potential problems. i conclude with a discussion of implications for  malawi specifically and for any african country in general.  mother tongue policies require that learners be taught in their mother tongue for some or all of the years of schooling. this is fairly manageable when the subject concerned is one that does not have its own highly specialised terminology. however, the picture becomes rather complicated when the subject is mathematics. the problem arises because the mathematical information comes in a register of the so called ‘language of wider communication’ such as english. it therefore becomes necessary to render this information into the language of the learners. in malawi this has hitherto meant rendering it into chichewa, among other local languages. however, it has been shown that chichewa is currently inadequate as a vehicle for conveying scientific information in general, and mathematical information in particular because the language lacks suitable terminology for expressing scientific and mathematical concepts and ideas (kishindo, 1987; kishindo & chiotha, 1995; kishindo & kazima, 2004). this finding is important because it shows that using chichewa as the language of instruction for mathematics needs consideration of how to handle mathematical terminology to help chichewa overcome its present limitations. mother tongue policies and issues of language of teaching and learning are often thought of as a dichotomy between the mother tongue and english (setati et al., 2008). many, including policy makers, take it to mean making a choice between teaching and learning in english or teaching and learning in the mother tongue. as setati et al. (2008) explain, debates around language and learning … create an impression that the use of the learners’ home languages for teaching and learning must necessarily exclude and be in opposition to english, and the use of english must necessarily exclude the learners’ home languages. (p. 3) setati et al. (2008) argue, however, that using home languages in the classroom does not have to be in opposition to english. the role of language in mathematics teaching and learning use of english in teaching and learning mathematics involves ordinary english (as in everyday use) and mathematical english (where words and phrases have specific meanings in mathematics) (pimm, 1987). the latter is what has been referred to as the mathematics register (halliday, 1974). the mathematics register includes words from ordinary english but having a specialised mathematics meaning, for example, ‘set’, ‘power’, ‘similar’ and ‘difference’, and also includes words like ‘polygon’, ‘isosceles’ and ‘quadrilateral’ which are borrowed from other languages (orton, 1992; pimm, 1987). pythagoras, 67, 56-63 (june 2008) 56 mercy kazima studies that have investigated learners’ understanding of a variety of mathematics words, have demonstrated that some learners do not understand many of the words that are commonly used in mathematics classrooms (for example, berenson & vidakovic, 1996; kazima, 2006; otterburn & nicholson, 1976; williams, 1995). in particular, learners have problems with words that have one meaning in mathematics and another in ordinary english. the word ‘similar’ for instance, means ‘proportional’ in mathematics yet in ordinary english it means ‘alike’. this causes confusion because what is similar in ordinary english is not necessarily similar in mathematics, and vice versa (orton, 1992). pimm (1987) and orton (1992) have given some interesting anecdotes about learners experiencing difficulties in mathematics because they do not understand the mathematical meanings of the words involved. for example, in response to the question “what is the difference between 47 and 23” one learner responded “one of the numbers is bigger than the other”. when this was not accepted, he tried “one number contains a 4 and a 7 but the other number doesn’t” (orton, 1992, p. 128). another example is that of a 13-year-old learner who counted the number of diagonals in a given figure as the number of sloping sides the figure had relative to the orientation of the page (pimm, 1987). the learner’s response to the question “if you knew the number of sides of a polygon, could you work out the number of diagonals?” was “it depends on what the shape is and which way you place it” (p. 84). the two examples show clearly that the learners did not understand the mathematical meanings of the words ‘difference’ and ‘diagonal’ and were employing the ordinary english meanings which landed them in difficulties. this is an illustration of the problems learners experience when they make errors of interpretation based on common everyday use of the words. this problem exists even for older learners as was demonstrated by monaghan (1991). he studied a-level learners in england and observed that the word ‘converges’, for example, which is commonly used in calculus courses to mean ‘approaches’ or ‘limit’, was confusing for many learners. the learners could not see how sequences of numbers could converge because ordinary use of the word is strongly linked with at least two lines ‘converging’ and eventually meeting. other mathematics words, especially those borrowed from other languages, like ‘isosceles’ and ‘quadrilateral’ do not seem to occur elsewhere outside mathematics (orton & frobisher, 1996). since the words have only one meaning, learners might be able to learn the meanings without confusion with other everyday usage in ordinary english. however, some researchers have found that although some of the words can be explained in terms of their roots and origins, the words still cause problems because often learners do not remember the meanings (berenson & vidakovic, 1996; otterburn & nicholson, 1976; williams, 1995). comprehension of mathematical word problems is another area that highlights the role of language in learning mathematics. researchers such as fasi (1999) and woo-hyung whang (1996) explored how learners understand word problems in mathematics. a common finding was that the more competent the learners were in english the better they were at comprehending word problems. in addition, many learners with low competence in english performed better on non-verbal mathematics tasks than on mathematically equivalent word problems (fasi, 1999), which suggests that language difficulties interfere with learners’ ability to solve mathematics word problems. as fasi concludes, the english in the word problems confuses and misleads many learners even when the mathematics involved is simple. other researchers (e.g. adetula, 1990; beecham, 2000; jones, 1982; berry, 1985; bunyi, 1997) also had similar results and drew the same conclusion that language presents difficulties in learners’s understanding of word problems. furthermore, many observed that when learners do not understand the word problems they often resort to ‘cue word strategy’ (adetula, 1990; jones, 1982) that is, searching for a word that will give them a hint of which arithmetic operation to carry out. for example, it has been found that the words ‘more’, ‘less’ and ‘share’ prompt learners to add, subtract, and divide, respectively, regardless of what the question is asking (adetula, 1990; jones, 1982). it is important to realise that language difficulties with word problems are not unique to second language speakers of english. research has shown that even first language speakers face difficulties with word problems (de corte & verschaffel, 1991; gibbs & orton, 1994; orton, 1992). however, one would expect that first language speakers have difficulties mainly with the mathematical english while second language 57 mother tongue policies and mathematical terminology in the teaching of mathematics speakers have difficulties in coping with the ordinary english, which they are not competent in, as well as the mathematical english. setati et. al.’s (2008) study which aims at helping learners with this problem of comprehension of word problems by using the learners’ home languages as resources in the classroom offers versions of word problems in the learners’ home languages. they conclude that this strategy improves learners’ comprehension of the word problems and so makes the mathematics accessible to all the learners because they focus on the mathematics and not the language as is the case when comprehension is a problem. an interesting observation in setati et. al.’s study is that, where one teacher translated mathematics word problems including mathematical terms into the learners’ local languages, the learners found the terms harder to understand than when presented in english. although it can be argued that it might have been the teacher’s translation that was problematic, the point is that it was not easy to provide a translation that learners could understand. as setati et al. (2008) explain the problem for learners when working with english word problems is not only with terminology but comprehension of the entire problem. this can be extended to the whole lesson to say the problem is in comprehension and communication of mathematics in the classroom, so focusing on terminology alone might not be helpful in our objectives of making mathematics accessible to learners. the discussion above clearly illustrates that language plays an important role in the teaching and learning of mathematics. we see that mathematical terms with precise meanings in mathematics but also with everyday english meanings cause confusion for many learners. on the other hand, completely new words that do not occur elsewhere in everyday english might not be confusing for learners but they do have their own potential problems. the inference can be drawn here that mathematical terms in any language which have also everyday meanings in the everyday usage of the language would cause confusion among learners. similarly completely new words that are not part of the language would have potential problems. what is a country to do when implementing mother tongue policy? how can a country deal with mathematical terminology when teaching in the mother tongue? i will discuss two different strategies; (a) developing a mathematics register and (b) borrowing from english. i will discuss three african countries: tanzania and nigeria as examples of developing a mathematics register and malawi as an example of borrowing from english the nigerian case the use of an african language as a medium of instruction has gone a long way in nigeria. metalanguage has been developed in the vernaculars for language, literature and teaching methods (emenanjo, 1990; muhammed, 1990; bamgbose, 2004). the development of meta-languages for nigerian languages is in keeping with the government’s national policy on education, which states that the medium of instruction in the first three years of primary education should be in the learner’s mother tongue. furthermore the government recognises any two of the three main languages (hausa, igbo and yoruba) as core subjects for secondary school learners (adesina, 1990). it is therefore believed that if teachers are to be produced for the use of these languages as mediums of instruction at both primary and secondary school, the teaching will extend to colleges of education and universities where teachers will be trained on how to handle the language effectively. for these reasons, the devising of mathematical terminology is considered as an important aspect of language planning. a glossary of primary mathematics has been developed in some of the languages. table 1 shows a sample of some mathematical terms in six languages: edo, efik, hausa, igbo, izon, and yoruba (bamgbose, 1986). table 1: sample of mathematical terminology in six nigerian languages set zero base sum factor point angle edo usun ihoi ezi esomu evbayagha ihe ukoko efik ebok ikpikpu besi iboroedidian fakto ntoi itun hausa tari sifili turke jumla ciduka digo kusurwa igbo ikpo efu nkweongu mgbakota facto kpom mgba izon ituu yefaa kientibi oseee diediebo tein lelei ikoki yoruba seeto ofo ipile aropo fato oju angu 58 mercy kazima in addition to the terms, it is recognised that the process of teaching the various topics involves expression in the general vocabulary of description, explication and argumentation. on the whole, these already exist in the general vocabulary of the languages, but in some cases such processes may involve technical vocabulary which has also been devised. interesting to note here is that some of the terms for example ‘besi’ (base in efik), ‘facto’ (factor in igbo) and ‘angu’ (angle in yoruba) seem to have been borrowed from mathematical english. according to bamgbose (2004) teaching and learning in these languages has so far been successful. bamgbose (2004) reports on a study that used experimental design to evaluate the effects of teaching in yoruba. the study had two main groups of learners. an experimental group which was taught in yoruba for all six years of primary education, and a control group which was taught in yoruba for the first three years and followed by english the last three years. both groups were evaluated in five subjects including mathematics. the results showed that the experimental group consistently performed better than the control group. bamgbose (2004) concludes that teaching in the learners’ language is more effective than teaching in english. by implication this claims that teaching in yoruba and the other nigerian languages, which involves use of the terms in table 1, has been successful. the tanzanian case in tanzania, issues of language were given priority in the process of nationalisation. nationalisation in education meant ‘swahilisation’ of the content as well as medium of instruction. the political and social vision underlying it was education for selfreliance. swahilization meant in practical terms, creating terminology for subjects where such did not exist. mathematical terminology has thus been developed for primary schools. this terminology has been developed on the understanding that the problem rested on ensuring that the learners understood the concepts and not the technical vocabulary, since this could be translated from english into kiswahili (abdulaziz, 1980). as a result a practical approach to the teaching of mathematics was followed. in teaching about the circle for example, learners would be involved in practical constructions; for example they would peg a piece of wood into the ground, tie a rope around it, and by using this rope draw a circle on the ground. in the process, concepts of centre, radius, diameter and circumference would be practically introduced and followed subsequently by the naming of the concepts (abdulaziz, 1980). radius for instance is called ‘nusu kipenyo’ which can be literally rendered into english as ‘half an opening’, and circle is called ‘duara’ literally meaning ‘wheel’. other mathematical terms developed in kiswahili include the ones shown in table 2. table 2: sample of mathematical terminology in kishwahili mathematical english kiswahili literal meaning fraction sehemu portion decimal sehenu za kumi portion of ten percent sehenu za mia portion of hundred positive hakika certainty negative kukana to say ‘no’ multiply nyongeza an increase remainder mabaki what remains angle pembe angle triangle pembe tatu three angles quadrilateral pembenne four angles rhombus msambamba sawa parallel and equal the swahili method has aimed at transferring the concept rather than mere translation. it is a faithful transfer from the source language into the target language of the concept conveying the term. thus a literal translation of the term has been avoided because in most cases it does not convey meaning of the term. in some cases a descriptive coinage such as ‘sehemu za mia’ has been preferred. the swahili examples illustrate how internal resources can be exploited to creatively develop a viable mathematical terminology. as can be seen, the vocabulary is already available in the language only that now the meanings have been extended to the realm of mathematics. the strength of the tanzanian strategy is that it focuses on the mathematical concept of a term and not the literal translation of the term into kiswahili. however, there is a potential problem because the literal meanings might become a source of confusion for learners. for instance the term for ‘multiply’ is ‘nyongeza’ literally meaning ‘an increase’. this could be confusing for some learners since a number can be increased by adding to it. furthermore, multiplying by zero, a negative number or a proper fraction does not increase a whole number but rather decreases it. therefore here the problem of everyday meanings interfering with mathematical meanings, as discussed earlier, might also occur. 59 mother tongue policies and mathematical terminology in the teaching of mathematics the malawian case malawi’s mother tongue policy states that all public schools should teach in the learners’ mother tongue from standard 1 to 4 – the first four years of primary education (ministry of education, 1996). there are at least two factors which led to this policy. firstly, the malawi government signed the united nations convention of learners’ rights in education which emphasised the rights of learners to be taught in their mother tongue. secondly, the malawi government also signed a memorandum at the african unity which encouraged use of african languages as mediums of instruction (chienda, 2002). local languages in malawi include chichewa, chitumbuka, chiyawo, chisena, chilomwe and others. in this paper i focus on chichewa because of its dominance as malawi’s national language. malawi’s strategy of dealing with mathematical terminology when teaching in chichewa was introduced by the malawi institute of education. this was done through textbooks written in chichewa which were introduced in malawi primary schools in 1991. before this all textbooks were in english and so mathematical terms were presented in english. important to note here is that the malawi institute of education is a government body and is the sole provider of textbooks for primary schools in malawi. the strategy is that of borrowing from english, that is, they take terms as they are in mathematical english and spell them in chichewa. table 3 gives some examples. table 3: sample of mathematical terminology in chichewa mathematical english chichewa circle seko decimal desimo factor fakitala fraction fulakishoni multiple matipo number nambala percent pelesenti quadrilateral kwadilatelo quotient koshenti rectangle rekitango square sikweya set seti triangle thirayango comparing the chichewa mathematics terms with the local language mathematical terms in the nigerian and tanzanian cases presented in table 1 and table 2 respectively, where the terms have meanings in the various languages, the chichewa terms in table 3 do not have meanings in chichewa although they are spelled in chichewa. for example, ‘triangle’ in kiswahili is ‘pembe tatu’ literally meaning ‘three angles’, and ‘set’ in yoruba is ‘seeto’ literally meaning ‘to put together’. on the other hand, the chichewa terms ‘thirayango’ and ‘seti’ for ‘triangle’ and ‘set’ respectively, do not mean anything in chichewa. the strength of this borrowing strategy is that there is no confusion with chichewa everyday use since these are not chichewa words. furthermore, the precision of meanings of the terms is not lost in interpretation. however, there are potential problems as has been observed in classrooms where english was the language of teaching and learning that learners do not always remember the meaning of english borrowed terms. similarly the malawi learners might experience the same problem with the chichewa borrowed mathematical terms. there is as yet no thorough research that has evaluated the effectiveness of this strategy for teaching and learning mathematics in malawi primary schools. however, according to a small study that explored the views of teachers and learners in the effectiveness of using these terms in teaching and learning mathematics, it was found that all the teachers in the sample said it is effective to teach mathematics using the chichewa borrowed terminology. reasons that the teachers gave for viewing this strategy as effective were mostly that their learners could easily read and write the chichewa borrowed mathematical terms (kazima, forthcoming). discussion of the three cases looking across the three cases, malawi’s strategy has the advantage of easiness. first, it is easier for the developers at malawi institute of education, since they only had to spell all the mathematics terms used in malawi schools in chichewa to present the chichewa terminologies for schools. this saved time and made the terms available for use in schools in a short time which would not be possible where a mathematics register in the language is sought. second, it is easier for teachers as they implement teaching in chichewa. the teachers do not have to learn new words for chichewa mathematical terms, since they are used to using the english words. third, it is easier for the learners. when they proceed from standard 4 to standard 5 and onwards where the language of 60 mercy kazima teaching is english, the learners do not learn new words for the mathematical terms that they have already encountered since the terms are the same. the learners only have to learn the english spellings and pronunciations of the terms in contrast to the malawi case, the nigerian and tanzanian cases of developing a mathematics register in their local languages took some time. educators had consultations with many people and discussions among themselves as they developed the mathematics registers. in tanzania for example they focused on transferring the mathematics concepts and not providing literal translations. to do this they explored their culture to find ways of presenting the mathematical concepts. this therefore demonstrated the link between mathematics in the classroom and mathematics in their culture. consequently the strategy made the mathematical concepts relevant to the learners’ everyday life. this opportunity of illustrating relevance was lost in the malawi case. however, the challenge for tanzania and nigeria is to capture precise mathematical meanings in the local language terms and for teachers to avoid everyday meanings interfering with learners’ understandings of the local language mathematical terms. malawi’s case avoids this confusion with everyday meanings but brings in its own problems of learners not remembering what the terms mean. the nigerian case is similar to malawi’s case in that the mathematical terms are one word only unlike the tanzanian case where sometimes a description in more than one word is used, for example ‘msambani sawa’ for rhombus. while single word terms might be desirable for convenience, it might not be easy to get equivalent terms in the local languages which capture the mathematical meanings. however, using a description might not necessarily capture the precise meaning of the mathematical term. for example the ‘msambani sawa’ for rhombus literally means ‘parallel and equal’ which is not a precise enough description of a rhombus since a rectangle or parallelogram can also fit the description. these are some of the dilemmas that policy makers would face in making decisions about how to deal with mathematical terminology. what does this mean for malawi? the strategy of borrowing from mathematical english has stirred discussion among mathematics educators and linguists in malawi. many have argued against the use of borrowed terminology in the malawi textbooks and in teaching. for example kishindo and kazima (2000) have argued that although the terms might have the phonological structure of chichewa, such as ‘fulakishoni’ (fraction) or ‘sikweya’ (square), they do not mean anything in chichewa. they argue further that the underlying assumption seems to be that the learners understand the english in the first place hence the free use of ‘chichewalised’ english. kishindo and kazima (2000) argue for the need to devise a mathematics register for chichewa as has been done for kiswahili in tanzania. they suggest for example that the term ‘fraction’ which is called ‘sehemu’ in kiswahili meaning ‘portion’, could be called ‘gawo’ in chichewa also meaning ‘portion’. this argument might seem reasonable but there is also an assumption made that if terms are in chichewa then learners will easily understand their meanings. this assumption might not be true because the meanings that learners should understand are the mathematical meanings and not everyday meanings of the words, and sometimes these two meanings are different. mathematical meanings are often more precise than everyday meanings. in english language it has been shown that learners find mathematical terms such as ‘similar’, ‘difference’, or ‘converge’ that have mathematical meanings different from everyday meanings difficult (monaghan, 1991; orton, 1992). therefore it can be argued that although the chichewa borrowed terms do not mean anything in chichewa, learners could learn their meanings without confusion with everyday use. other malawian researchers in this area also strongly recommend the development of a mathematics register in chichewa (for example, ; chienda, 2002; kaphesi, 2000; kaphesi, 2001; rambiki, 2004). kaphesi (2001) in particular argues that if no register is developed, then teachers might use their own chichewa interpretations of mathematical terms which could cause confusion and mathematical misconceptions among learners. there is an assumption here as well that if a term is presented in chichewa then it will not require further interpretations by the teacher. this is not necessarily correct because a chichewa term could still require interpretation and explanation by the teacher. for example calling a fraction ‘gawo’ as suggested by kishindo and kazima (2000) does not mean a teacher will not have to explain its meaning, and like anything else in teaching, the explanation is subject to the teacher’s interpretation and is open to confusion or misconceptions by the learners. 61 mother tongue policies and mathematical terminology in the teaching of mathematics it is interesting to see these arguments against the strategy of borrowing mathematical terms when according to the small survey mentioned earlier, the teachers who are the implementers seem to be happy with the strategy. the reasons the teachers gave are quite convincing especially from the point of view of the teacher whose main concern is for the child to be able to learn. the teachers all said that their learners are able to read the chichewa borrowed terms. this shows us that, for the teachers, this ability to read the terms is important for their teaching and hence a step towards effectiveness. indeed being able to read the terms makes it possible for the child to engage with the textbook. whether they understand what they read is a different matter but at least they can read the terms and the teacher can take it from there. in setati et al.’s (2008) study, it is intriguing that in some cases learners found the local language mathematical terms (as translated by their teacher) difficult to understand. indeed mathematical terms have precise meanings which are not easy to capture in one word in many african languages; often one needs a whole sentence to elaborate the precise meaning of a term. this seems to suggest that we should not try to present everything in the local languages; some mathematical terms might be best presented in their english form. this raises the question of whether it is worthwhile to engage in the huge task of trying to devise mathematics registers in chichewa and other local languages that include all the mathematical terms used in schools, as is being suggested by many interested parties in malawi. it is important to remember that it is not the name of the term that is important but the concept behind the term. for example, it is not the name ‘fulakishoni’ or ‘gawo’ that is important for learners to know but the concept of fraction. what might be a way forward for malawi is to evaluate the strategy to establish its level of effectiveness. until there is evidence of lack of effectiveness in teaching using the borrowed terminology, it would not be wise to consider alternatives. if and when there is a need to consider developing a register for malawi, again careful examination of the proposed strategy will need to be done other than copying what other countries such as tanzania or nigeria have done because what works for the other countries might not necessarily work for malawi. conclusion this paper has discussed issues of mathematical terminology in the context of implementing mother tongue policies. i have discussed borrowing from english and developing mathematics registers in the local languages as two strategies for handling mathematical terminology when teaching in the mother tongue. i have presented the case of malawi as an african example of borrowing strategy and have presented the cases of nigeria and tanzania as african examples of developing mathematics registers in the local languages. the discussion has illustrated the dilemmas that decision makers might face as they suggest ways of handling mathematical terminology when teaching in the local languages. each of the strategies discussed has strengths as well as potential problems. it is therefore advisable that whatever strategy a country decides to use for its mathematical terminology, the teachers and other implementers of the strategy be aware of the strengths and potential problems so that efforts can be made to 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(2008, april). using language as a transparent resource in the teaching and learning of mathematics in a grade 11 multilingual classroom. paper presented during the third annual symposium on teaching and learning in multilingual classrooms. university of the witwatersrand, johannesburg. williams, b. g. (1995). language in the context of mathematical education: some issues in learning and teaching mathematics in a second (or weakly known) language. unpublished master’s thesis, university of leeds. woo-hyung whang (1996). the influence of english-korean bilingualism in solving mathematics word problems. educational studies in mathematics, 30(3), 289-312. << /ascii85encodepages false /allowtransparency false /autopositionepsfiles true /autorotatepages /none /binding /left /calgrayprofile (dot gain 20%) /calrgbprofile (srgb iec61966-2.1) /calcmykprofile (u.s. web coated \050swop\051 v2) /srgbprofile (srgb iec61966-2.1) /cannotembedfontpolicy /warning /compatibilitylevel 1.5 /compressobjects /tags /compresspages true /convertimagestoindexed true /passthroughjpegimages true /createjobticket false /defaultrenderingintent /default /detectblends true /detectcurves 0.0000 /colorconversionstrategy /cmyk /dothumbnails false /embedallfonts true /embedopentype false /parseiccprofilesincomments 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can be opened with acrobat and adobe reader 5.0 and later.) >> /namespace [ (adobe) (common) (1.0) ] /othernamespaces [ << /asreaderspreads false /cropimagestoframes true /errorcontrol /warnandcontinue /flattenerignorespreadoverrides false /includeguidesgrids false /includenonprinting false /includeslug false /namespace [ (adobe) (indesign) (4.0) ] /omitplacedbitmaps false /omitplacedeps false /omitplacedpdf false /simulateoverprint /legacy >> << /addbleedmarks false /addcolorbars false /addcropmarks false /addpageinfo false /addregmarks false /convertcolors /converttocmyk /destinationprofilename () /destinationprofileselector /documentcmyk /downsample16bitimages true /flattenerpreset << /presetselector /mediumresolution >> /formelements false /generatestructure false /includebookmarks false /includehyperlinks false /includeinteractive false /includelayers false /includeprofiles false /multimediahandling /useobjectsettings /namespace [ (adobe) (creativesuite) (2.0) ] /pdfxoutputintentprofileselector /documentcmyk /preserveediting true /untaggedcmykhandling /leaveuntagged /untaggedrgbhandling /usedocumentprofile /usedocumentbleed false >> ] >> setdistillerparams << /hwresolution [2400 2400] /pagesize [612.000 792.000] >> setpagedevice introducing discussion into multilingual  mathematics classrooms: an issue of code  switching?  lyn webb and paul webb  nelson mandela metropolitan university, port elizabeth  lyn.webb@nmmu.ac.za & paul.webb@nmmu.ac.za  the  department  of  education  in  south  africa  advocates  collaborative  and  constructivist  learning; however, observations  indicate that  little discussion occurs  in most multilingual  mathematics classes. in this paper we draw on a pilot study set in the eastern cape where  teachers were introduced to the theory and practice of exploratory talk, and then tasked to  perform  an  action  research  project  on  introducing  discussion  in  their  own  multilingual  mathematics classrooms. the results of the study suggest some successes in terms of teachers  initiating exploratory  talk and highlight  the  fact  that  these successes were only achieved  where code switching between english and isixhosa formed an integral part of the process.  this study focuses on an intervention aimed at introducing teachers to the concept and practice of exploratory talk in their mathematics classrooms. the rationale for an exploratory talk approach is that it has been claimed that working in groups and talking with other learners leads to the development of mathematical reasoning (mercer & sams, 2006), and the fact that, despite learners often being seated in groups in south african classrooms, very little meaningful discussion takes place (taylor & vinjevold, 1999). the research aimed at investigating whether an intervention of this type could enable teachers to implement exploratory talk and, if so, what strategies teachers used to initiate the approach within their context of multilingual teaching and learning of mathematics. in rural areas of the eastern cape (where this study took place) almost all schools choose english as the language of learning and teaching (lolt) despite the fact that the vast majority of the teachers and learners are isixhosa main language speakers (webb & treagust, 2006). access to english in most of these schools is severely limited and there is little chance of many learners hearing english outside the school premises (alidou et al., 2006). classroom observation studies conducted in several countries in africa (benin, burkina faso, guinea-bissau, mali, mozambique, niger, south africa, togo, tanzania, ethiopia, ghana, and botswana) reveal that the use of an unfamiliar language, such as english, often results in traditional and teacher-centred teaching methods, e.g. chorus teaching, repetition, memorization and recall (alidou et al., 2006). setati and adler (2001) have also shown that when teachers use english mainly for explanation, rote learning of procedures takes place and opportunities for developing meaningful learner-centred talk and writing are limited. this situation has important implications as studies by mercer and sams (2006) and others (e.g., monaghan, 2004; rojas-drummond & fernandez, 2000; wegerif, littleton, dawes, mercer, & rowe, 2004) have shown that discourse plays a profound role in the construction of knowledge and reasoning. in their pioneering work into mathematics teaching in multilingual classrooms in south africa, both adler and setati have uncovered many ‘dilemmas and contradictions’ that teachers have to face in their dual role when teaching proficiency in english as well as mathematical concepts in the mathematics classroom (adler, 2001; setati, 1998). however, little research has been undertaken in the eastern cape where there is a widespread isixhosa/english tension in the classroom. the paucity of data in this particular setting, and adler’s (2001) call to further study the various journeys that both teachers and learners make from 26 pythagoras, 67, 26-32 (june 2008) lyn webb & paul webb informal talk in a main language to formal mathematics in another language, underpin the rationale for this study which is located within mercer’s (1995) socio-cultural framework and notions of the role of language and discourse in the development of mathematical reasoning. classroom talk classroom talk in general has been studied extensively over the past few decades and it appears that the most common kind of classroom interaction that takes place is the initiationresponse-feedback (irf) questioning cycle (sinclair & coulthard, 1975). this type of classroom talk, where the teacher asks a question, the learner makes a simple statement as an answer, and the teacher responds – usually simply to agree with the answer or reject it has been criticised as a means of keeping control of the class by the teacher rather than a pedagogic tool (webb & treagust, 2006). furthermore, this type of communication leads learners to believe that mathematical knowledge is fixed, discovered and cannot be questioned (barwell & kaiser, 2005). most researchers in the field of classroom discussion reject this form of interaction and feel that it does not even qualify to be described as ‘discussion’. unfortunately in bilingual and multilingual south african classrooms, where learners are unable to express their reasoning in english, the irf cycle abounds (taylor & vinjevold, 1999) particularly in the eastern cape (webb & treagust, 2006). the dialogic perspective of learning is based on a vygotskian viewpoint, i.e. that cognition is aided by cultural processes (vygotsky, 1978). according to barwell and kaiser (2005) dialogism is ‘practice-oriented’ and communication is seen as an ongoing process of negotiation between people and contexts. a dialogic view of learning presumes that mathematics is created in the classroom through reasoning and argumentation between teacher-and-learner and learner-and-learner (barwell & kaiser, 2005). in eastern cape classrooms, observation has shown that there is a paucity of explicit verbal argumentation between learner-and-learner. the approach used in this study emulates the strategy used by mercer (1995) towards dialogic learning, i.e., understanding and encouraging learner talk and discussion so that their reasoning becomes visible. according to mercer (1995) talk between learners can be characterised into three main types and, if they are to enable their learners to successfully engage in meaningful discussions in the classroom, teachers need to be able to differentiate between them. firstly, disputational talk is characterized by disagreements and individualized decision-making and where short assertions and counter-assertions are made. secondly, cumulative talk takes place when learners build positively but uncritically on statements and assertions made by other learners (this talk is exemplified by repetitions, confirmations and elaborations). finally, exploratory talk is actualised when learners engage critically, but constructively, with others’ ideas, and justifications and alternate hypotheses are offered with joint consensus eventually being reached. central to exploratory talk is the belief that collaborative thinking skills can be taught explicitly in order to enable both teachers and learners to understand talk as ‘thinking aloud with others’ (monaghan, 2004). this resonates with the aims of the new south african curriculum, i.e. that collaborative and constructivist measures are important for meaningful learning to take place (department of education, 2003). unfortunately exploratory talk, i.e. the type of discussion that is generally believed to best support constructivist learning, is the form of discussion least often heard in classrooms (lemke, 1990). in their study, mercer and sams (2006) attempted to raise learners’ awareness of using language as a means of thinking together; to develop learners’ abilities to use language as a tool for reasoning; and to apply language effectively in studying mathematics. the teachers in the experimental classes were trained in the introduction of exploratory talk in mathematics classrooms and were given detailed lesson plans to conduct. the learners in both their experimental and control classes were tested preand post-intervention using sats mathematics items. in the experimental classes exploratory talk ground rules were negotiated with the learners. these ground rules were that: all relevant information must be shared; all members of the group should contribute to the discussion; all opinions and ideas must be respected and considered; everyone should be asked to make his/her reasons clear; challenges and alternatives must be made explicit and negotiated; and the team must seek to reach consensus (mercer & sams, 2006). they analysed transcripts of the learner discussions focusing on effective use of exploratory talk and showed that the learners in the target classes achieved significantly better in the post-test than the learners in the control classes (mercer & sams, 2006). 27 introducing discussion into multilingual mathematics classrooms multilingual classrooms and code switching teachers have voiced the difficulties they encounter when the lolt is not the first language of either teachers or learners (setati, 1998). this is also the case of many schools in the eastern cape where the learners are not exposed to english other than in their classrooms and teachers often use code switching in order to communicate with their learners (taylor & vinjevold, 1999). in fact in many south african schools english can be considered to be more of a foreign language than an additional language (adler, 2001). the lack of developed basic interpersonal communication skills (bics) in english means that the learners have little or no framework available to them within which to develop their cognitive academic language proficiency (calp), not only orally, but also in terms of reading and writing (cummins, 1984). the difficulties that teachers experience in their classes when developing bics and then moving from using informal spoken language to formal oral and written mathematical language has been clearly recognised and emphasised by setati, adler, reed and bapoo (2002), and extensive research has been documented on code switching in south africa (adler, 2001; setati, 1998). code switching is considered to be a tool which can provide spontaneous and reactive discussion of concepts by learners and teachers in their main language (setati, 2007). according to moshkovich (2007), a misconception about code switching is that it stems from a deficit model and is used when the speaker is unable to recall suitable words or phrases in the language being spoken. she disagrees, and promotes the view that code switching should rather be seen as a complex language practice which allows for the greater use of the main language while still using the official language of learning and teaching. in the eastern cape many teachers have expressed that they feel guilty when they code switch as they consider that they are depriving learners of exposure to english. this study aimed at increasing teachers’ awareness of the benefits of code switching by immersing them in a situation where they use code switching themselves in order to solve problems. methodology this pilot study took place within the framework of a bed (hons) programme offered by the nmmu and focussed specifically on a semesterlong module which, amongst other purposes, was developed to examine language practices of 134 in-service mathematics teachers who were studying at centres in port elizabeth, king william’s town, mthatha, kokstad, east london and ngcobo. during workshops these teachers were introduced to classroom discussion (including code switching) and practised exploratory talk in groups while solving problems posed by the nonverbal and supposedly culture-free raven’s progressive matrices tests, as was done in mercer, wegerif and dawes’ (1999) study. the discussion generated by the teachers during this activity was video-taped and inspected by an english and xhosa speaker in order to investigate how the teachers engaged in discussions and which language strategies they employed when doing so. the teachers were interviewed in focus groups after this process to establish their perceptions of the efficacy of the strategy. finally, the participating teachers were tasked with conducting a mini action-research project on the development of exploratory talk in their own mathematics classrooms. the 134 action research assignments produced by the teachers were used to generate data on the type of discussion that had occurred in their classrooms and the strategies that they had used to enable their learners to participate. results the data generated by this research revealed similar results to those found in mercer and sams’ (2006) study in that teachers realised that if they provided guidance and practice in using language in order to express mathematical reasoning, learners could express themselves more effectively and that this aided them in solving mathematical problems. more specifically the data generated suggest that code switching has a positive influence on generating mathematical understandings and that developing exploratory talk is sometimes a slow process that requires guidance. vignettes from the study, discussed below, give insights into the discussion processes that took place in classrooms. code switching the video-taped data of teachers in discussion revealed that at almost all times they code switched, that is they used both english and isixhosa in a single linguistic episode (moschkovich, 2007). they used english when mentioning mathematical terms e.g. circle, rectangle, diagonal, etc. this suggests that they were more comfortable with mathematical terms in english than with the words in the mathematics register of their main language (moschkovich, 28 lyn webb & paul webb 2007). the effects of code switching are reported within the ‘vignettes’ section of this paper. developing exploratory talk at first it appeared that the teachers found it difficult to differentiate between the different types of ‘talk’. in all centres, initially, the teacher who felt s/he knew the correct answer would dominate the discussion (leading to cumulative talk). if there was disagreement, the most forceful character would prevail (disputational talk). often the facilitator had to intervene to remind the teachers that the outcome was to experience the different types of ‘talk’ and not only to find the correct answer. in most cases the teachers eventually arrived at a situation which may be described as exploratory talk, i.e. they engaged with others’ ideas and justifications and (to varying degrees) joint consensus was reached. many of the teachers reflected positively during the interviews on the understanding that they had achieved through using the tenets of exploratory talk and the ground rules based on those of mercer and sams (2006), e.g. that all information must be shared; groups must try to reach agreement; good reasons must be given for decisions; and everyone must be encouraged to speak. in discussions at all centres the teachers agreed that it would have been much more difficult to formulate solutions to the raven’s test items individually than by discussing their reasoning in groups, suggesting that they had experientially discovered the benefits of sharing their reasoning with their peers. when questioned about their own code switching and the practice of code switching in their classrooms the teachers agreed that the strategy is commonly used in their classrooms to overcome both teachers’ and learners’ lack of fluency in english, but that it is accompanied by a sense of guilt and inadequacy as they feel that they have had no official guidance or stamp of approval from the department of education for the practice. they also expressed the fear that by using code switching they are depriving their learners of opportunities associated with the use of english. many of the interviewees also noted that they found it difficult to engage in exploratory talk in problem solving groups. this they attributed to the fact that they had never before thought about discussion or the quality of their interactions. they were able to identify that their first attempts had resulted in instances of disputational and cumulative talk and they recognised that they had had to be continually reminded by the facilitator of the ground rules that they had previously discussed and accepted, particularly the practice that each participant should be free to contribute and be encouraged to do so. this bears out mercer and sam’s (2006) claim that it is necessary to provide guidance about, and practice in, using language effectively as a tool to enhance mathematical reasoning. when introducing exploratory talk into the classroom the teachers noted that their learners had to be coached constantly as practising the agreed upon ground rules did not come naturally to them (mercer & sams, 2006). classroom vignettes the action research assignments produced by the teachers were of a varied standard but in approximately 65% of cases it was possible to detect through the vignettes the type of classroom discussion promoted by the teachers (a compulsory aspect of the assignment). the following are two vignettes, as described in two different teachers’ action research assignments, which give some insight into the different types of conversations that were generated when groups of learners discussed mathematical problems in english only, and when allowed/encouraged to use their home language. the first is based on learners’ conversation when attempting to solve the following word problem in english only: there are 21 cycles in a shop. some are bicycles and some are tricycles. if there are 51 wheels altogether, how many are bicycles and how many are tricycles? lethu: do we all understand? limpho: what? lethu: what the teacher said. maria: let’s find the number of bicycles and tricycles. lethu: but how can we, if we talk? maria: this is easy, you see. limpho: what? maria: 24 bicycles and 1 tricycle. limpho: what? lethu: did we talk – or did you just tell us? bongo: yes, maria is right. i can see 24 bicycles will be having 48 wheels neh! plus three for the tricycle. maria: that’s all there is! lethu: okay. maria: do you all see? (murmurs of assent) 29 introducing discussion into multilingual mathematics classrooms of the ten learners present only four participated in the exchange, with only maria and bongo addressing the question. maria suggested an answer which was flawed and resulted in an incorrect answer as they were focussing on the number of wheels only and not the restriction that there were 21 cycles in the shop, yet the other learners agreed without engaging in the reasoning themselves. they felt they had found a feasible answer and did not try to find alternative solutions. their discussion was an example of cumulative talk, where one assertive participant’s view was ratified by the other members of the group without reasons or explanations or counter arguments. when the teacher asked the learners in this group about the low level of participation, one replied, “siyasibamba isingesi!” [“the english is restricting us!”]. the teacher commented that when the learners were engaging in talk using only english they had difficulty in expressing their ideas and reasoning, and that the few learners in the group who were proficient in english dominated the conversation. the second is a vignette of discussion that took place when a group of learners were encouraged to discuss their reasoning in isixhosa when investigating possible answers for the following problem: a family of 5 people need to cross a river. they have a raft which can carry a maximum of 100 kg. to save time they want to make as few trips as possible. jabu’s mass is 57 kg. khaba’s mass is 85 kg. linda’s mass is 38 kg. mandla’s mass is 60 kg and nandi’s mass is 35 kg. sipho: i think linda should go in the boat with jabu. lilian: why? sipho: i think it is because kaloku imass yabo ngu 93 kg is less than 100 kg. james: why don’t sithathe ukhaba kuba ngoyena unzima? [why don’t we take khaba first because he is the heaviest?] vuyo: ibuye iboat? [how will the boat return?] james: oh! i see. liyabona. vuyo: how about nandi goes with mandla then nandi comes back? bongo: what then? sinovuyo: then nandi goes with linda, then linda comes back again then goes with jabu, then comes back, then khaba gets on alone, and linda comes back and gets nandi – and then go together. ntombizondwa: perfect! just the trip! mcebisi: so can i report to the rest of the class? most of the learners in the group were prepared to express views and were prepared to offer challenges. the exploratory talk trigger words “why”, “i think” and “because” were consciously used in this short episode by a number of the participants. other vignettes were provided by teachers in their action research assignments which could be judged to represent exploratory talk, but what was noticeable was that in all of these cases the learners engaged in code switching. in none of the cases reported where english only was used did the participants evolve their discussions further than cumulative talk. discussion and implications there have been a number of studies which maintain that social activity through exploratory talk between peers in the classroom increases reasoning skills (mercer & sams, 2006; monaghan, 2004; wegerif et al., 2004). rojasdrummond and fernandez (2000) worked with english as additional language (eal) primary school children in mexico, while webb and treagust (2006) worked with isixhosa-speaking learners in science classrooms in south africa. the findings of both of these studies positively link teaching learners’ explicit language use with enhanced reasoning skills. in both studies the learners who had been taught exploratory talk principles and used code switching in classes achieved better results on tests of non-verbal reasoning than those in the control classes who had not. in this study teachers were exposed to the tenets of exploratory talk experientially and verbalised their preference for using their main language when grappling with a problem solving situation. in groups where there was a non-isixhosa speaking teacher, the discussion was in english as the teachers felt that common understanding was important. when mathematical problems were discussed the teachers code switched, using words like “parallel” and “diagonal” in english whereas their reasoning was expressed in isixhosa. they carried this positive experience through to their classrooms and realised that discussion flowed easier when they allowed their learners to use the vernacular, moreover results seemed more meaningful to the learners. both teacher-learner and learner-learner interaction increased and instances of irf questioning cycle were diminished. 30 lyn webb & paul webb it has been noted that mathematics teachers can play an important role in the development of learners’ awareness and use of language as a tool for reasoning, as well as for producing a more collaborative and inclusive classroom ethos (mercer & sams, 2006). in addition, barwell (2005) argues that policymakers and mathematics educators have stressed the difference between the precise subject language of mathematics and the more informal talk that learners use in the mathematics classrooms, and he feels that this has had a detrimental effect on inducting learners into mathematical practices. furthermore, barwell and kaiser (2005) argue that if learners can be encouraged to talk informally about their mathematical reasoning there is more chance that they will be able to develop formal mathematical discourse. the results of this study concur with this viewpoint as they indicate that when teachers do not emphasise the use of ‘english only’, informal discussion increases. without a culture of informal talk using bics, there is little possibility of developing mathematical discourse in calp and so the journey is truncated before it is even begun. one of the teachers used the metaphor of her experience of learning to drive a car. when she was concentrating on the skills of gear changing and braking she had no confidence in her ability to drive in public. it was only when the skills became transparent, or second nature to her, that she was able to shift her focus onto driving effectively. in her class she has tried to make language transparent so that the learners can focus on mathematical reasoning rather than on the language used. this has resulted in greater use of the vernacular together with code switching, and an increase in classroom discourse. in this particular teacher’s classes she has introduced a culture of argumentation through exploratory talk, which supports dialogic learning. it is hoped that continued use of these practices will result in progress from informal talk in isixhosa to the development of mathematical discourse in english. the results of the classroom research reports indicate that some teachers consider that any discussion in groups constitutes exploratory talk, in the same way that some teachers feel that seating learners in groups constitutes collaborative learning. this resonates with mercer ands sams (2006) caveat that on-going training is necessary to sensitise both teachers and learners to the nuances of the different types of talk so that the tenets of exploratory talk can be practised. as such, it seems that practical, meaningful and appropriate sociocultural mathematics cum language interventions, which include exploratory talk and code switching, need to be developed, implemented and researched if more and more learners are not to become marginalized in multilingual classrooms because of limited language acquisition. references adler, j. (2001). teaching mathematics in multilingual classrooms. dordrecht: kluwer academic publishers. alidou, h., boly, a., brock-utne, b., diallo, y., heugh, k., & wolff, h. (2006). optimizing learning and education in africa – the language factor. a stock-taking research on mother tongue and bilingual education in sub-saharan africa. gabon: unesco institute for education. barwell, r. (2005). ambiguity in the mathematics classroom. language and education, 19(2), 118-126. barwell, r., & kaiser, g. (2005). mathematics education in culturally diverse classrooms, zdm, 37(2). 61-63. cummins, j. (1984). wanted: a theoretical framework for relating language proficiency to academic achievement among bilingual students. in c. rivera (ed.), language proficiency and academic achievement. clevedon: multilingual matters. department of education, (2003). national curriculum statement grades 10-12 (general), mathematics. pretoria: government printers. lemke, j. (1990). talking science: language, learning and values. new jersey: ablex. mercer, n. (1995). the guided construction of knowledge: talk amongst teachers and learners. clevedon: multilingual matters. mercer, n., & sams, c. (2006). teaching children how to use language to solve maths problems. language and education, 20(6), 507-528. mercer, n., wegerif, r., & dawes, l. (1999). children’s talk and the development of reasoning in the classroom. british educational research journal, 25(1), 95-110. monaghan, f. (2004). thinking together – using ict to develop collaborative thinking and talk in mathematics. in o. mcnamara (ed.), proceedings of the british society for research into learning mathematics, 24(2), 69-74. moschkovich, j. (2007). using two languages when learning mathematics. educational studies in mathematics, 64(2), 121-144. rojas-drummond, s., & fernandez, m. (2000). developing exploratory talk and collective reasoning among mexican primary school children. proceedings of iii conference for sociocultural research. sao paulo, brazil. 31 introducing discussion into multilingual mathematics classrooms 32 setati, m. (1998). code switching in a senior primary class of second language learners. for the learning of mathematics, 18(1), 34-40. setati, m. (2005). teaching mathematics in a primary multilingual classroom. journal for research in mathematics education, 36(5), 447-466. setati, m. (2007). towards pedagogy for teaching mathematics in multilingual classrooms in south africa. paper presented at the second marang symposium on teaching and learning mathematics in multilingual classrooms. university of the witwatersrand, johannesburg. setati, m., & adler, j. (2001). between languages and discourses: language practices in primary mathematics classrooms in south africa. educational studies in mathematics, 43(3), 243269. setati, m., adler, j., reed, y., & bapoo, a. (2002). incomplete journeys: code switching and other language practices in mathematics, science and english language classrooms in south africa. journal of language education, 16(2), 128-149. sinclair, j., & coulthard, r. (1975). towards an analysis of discourse. oxford: oxford university press. taylor, n., & vinjevold, p, (1999). getting learning right. report of the president’s education initiative research project. johannesburg: joint education trust. vygotsky, l. (1978). mind in society: the development of higher psychological processes. cambridge, ma: harvard university press. webb, p., & treagust, d. (2006). using exploratory talk to enhance problem solving and reasoning skills in grade 7 science classrooms. research in science education, 36, 381-401. wegerif, r., littleton, k., dawes, l., mercer n., & rowe, d. 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can be opened with acrobat and adobe reader 5.0 and later.) >> /namespace [ (adobe) (common) (1.0) ] /othernamespaces [ << /asreaderspreads false /cropimagestoframes true /errorcontrol /warnandcontinue /flattenerignorespreadoverrides false /includeguidesgrids false /includenonprinting false /includeslug false /namespace [ (adobe) (indesign) (4.0) ] /omitplacedbitmaps false /omitplacedeps false /omitplacedpdf false /simulateoverprint /legacy >> << /addbleedmarks false /addcolorbars false /addcropmarks false /addpageinfo false /addregmarks false /convertcolors /converttocmyk /destinationprofilename () /destinationprofileselector /documentcmyk /downsample16bitimages true /flattenerpreset << /presetselector /mediumresolution >> /formelements false /generatestructure false /includebookmarks false /includehyperlinks false /includeinteractive false /includelayers false /includeprofiles false /multimediahandling /useobjectsettings /namespace [ (adobe) (creativesuite) (2.0) ] /pdfxoutputintentprofileselector /documentcmyk /preserveediting true /untaggedcmykhandling /leaveuntagged /untaggedrgbhandling /usedocumentprofile /usedocumentbleed false >> ] >> setdistillerparams << /hwresolution [2400 2400] /pagesize [612.000 792.000] >> setpagedevice abstract introduction understanding reflective practice theoretical framing and the six-lens framework methodology the findings concluding remarks acknowledgements references footnote about the author(s) samukeliso chikiwa department of education, faculty of education, rhodes university, makhanda, south africa mellony graven department of education, faculty of education, rhodes university, makhanda, south africa citation chikiwa, s., & graven, m. (2023). exploring the development of south african pre-service teachers’ reflective practice. pythagoras, 44(1), a678. https://doi.org/10.4102/pythagoras.v44i1.678 original research exploring the development of south african pre-service teachers’ reflective practice samukeliso chikiwa, mellony graven received: 31 jan. 2022; accepted: 14 apr. 2023; published: 28 june 2023 copyright: © 2023. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract ongoing concern about poor learner performance in mathematics has led to wide-ranging research on the subject, globally and in south africa. among the remedies identified is the reformation of pre-service teacher (pst) education programmes in a way that supports the acquisition of professional skills for pre-service teachers. developing psts’ reflective practice (rp) is a significant component of the desired reformation. our research explored psts’ rp development in the context of video-based mathematics lesson analysis. the aim was to contribute knowledge towards strengthening mathematics pst education and to report on whether increased benefits accrued from working with psts in small groups, guided by an experienced facilitator, as compared to whole-class lecturing. we draw on this extended analytic framework to compare two sets of reflections written by four selected psts based on viewing video recordings of their own teaching. one set was written in august 2018 after the psts completed three lecture sessions on rp in a mathematics methods course. the other was written in september 2019 after the four selected psts participated in three small-group, facilitator-guided sessions. the findings indicate some shifts towards higher-level reflections in the latter set, although only two of the four psts reflected at the highest level (reflectivity) following the small-group sessions. implications for pre-service mathematics teacher education and refinement of frameworks for delineating levels of reflection are discussed. contribution: the research contributes to mathematics teaching through refining and extending existing models of reflective-based practice to better analyse the shifting nature of mathematics teachers’ reflections with a view to supporting improved teaching and learning. keywords: mathematics education; reflective practice; pre-service teachers; six-lens framework; pre-service teacher education; reflections; primary school mathematics; foundation phase. introduction there is growing evidence that, despite qualifications, many teachers of primary mathematics do not have sufficient mathematical and pedagogical knowledge for teaching mathematics (e.g. venkat & spaull, 2014). this is compounded by the problem that many pre-service teachers (psts) come into their degrees with weak mathematical knowledge from their own schooling (askew et al., 2019). south african teacher education institutions are working on transforming the teaching and learning of mathematics through transforming the ways of preparing mathematics teachers to meet the pedagogical and content knowledge demands of teaching mathematics (see roberts, 2020, and the primted project). these efforts are influenced by south africa’s long history of poor learner performance in mathematics (fleisch, 2008; reddy et al., 2016, 2022; spaull & kotze, 2015). the centre for development and enterprise (cde, 2013) acknowledges that: [t]he teaching of mathematics in south african schools is amongst the worst in the world. mounting indicators on school performance and teaching reveal largely unacknowledged poor teaching of mathematics in the great majority of schools. (p. 2) developing psts’ reflective practice (rp) is increasingly being incorporated into pre-service teacher education (pte) as one of the strategies supporting transformation (cadiz, 2021). there is, however, limited understanding of how to develop psts’ rp, particularly in a context such as south africa, where psts appear to have little experience of reflecting (chikiwa & graven, 2021). our study explored the reflections of foundation phase (fp) psts as they analysed video-recorded mathematics lessons. we chose to focus on fp psts because the trajectory of poor performance in mathematics begins at this early stage in south africa (spaull & kotze, 2015). this article reports on a question from a more extensive study that focused on psts’ rp, namely: what is the nature of psts’ evolving rp when reflecting on their lessons following small-group, facilitator-led rp sessions? we thus compared the reflections that psts developed after engaging in rp in large lecture-type sessions with those they developed after being engaged in small-group sessions under the guidance of facilitators experienced in rp. we aimed to understand the opportunities and challenges of these ways of developing psts’ rp to inform pte. while we focus on developing psts’ rp in respect of the teaching of mathematics, the insights gained may extend to developing rp in other disciplines. teaching mathematics is a complex task for many teachers (ziegler & loos, 2017), hence ball, thames and phelps (2008) argue for knowledge that is specific to the teaching of mathematics. inspired by shulman’s (1987) idea of pedagogical content knowledge (pck), they developed a practice-based framework with six teacher knowledge domains that they argue are necessary for the effective teaching of mathematics which they named ‘mathematical knowledge for teaching (mkft). it is the desire of every mathematics educator to ensure that psts leave pte with adequate mkft to get started with the teaching of mathematics efficiently. developing teachers’ rp is one approach that has been found helpful in developing teachers’ mkft (karsenty et al., 2015; schwarts & karsenty, 2020). developing psts’ rp is included in pte programmes at our institution and several other south african institutions because of its perceived benefits as a basis for learning from and constantly improving one’s teaching (genor, 2005). as hutchinson and allen (1997) noted: [o]ne of the goals of teacher education is to develop each pre-service teacher into a reflective educator, one who is a life-long learner who perceives every experience as an opportunity for growth, change, and development of understanding. (p. 226) developing rp, therefore, becomes a critical element in teacher education programmes. darling-hammond (2009) claims that rp functions as a bridge between theory and practice, enabling novice teachers to link the two. liou (2001) notes that ‘critical reflection raises teachers’ awareness about teaching, enables [a] deeper understanding of variables related to teaching, and triggers positive changes in their practice’ (p. 1). while rp has become a common topic in pte research and teacher programmes worldwide, there has been less inquiry into the process of developing psts’ rp. zeichner (1994) warns that wanting ‘to prepare teachers who are reflective does not translate directly into the content of teacher education programs’ (p. 9). it is necessary to create a step-by-step process. the challenges in this process may be different for different countries with varying educational systems and cultures of critique; our research sought to contribute within the south african context of pte. this article emerges from a broader research study that explored psts developing rp through participation in sessions focused on the development of rp through video-based lesson analysis (chikiwa, 2020) discussed in the methodology section. the article seeks to explore the evolving levels of psts’ rp when reflecting on their own lessons following small-group, facilitator-led rp sessions. understanding reflective practice reflective practice is widely noted as key to professional development across various professions (for example healthcare; see norrie et al., 2012). the professional development of mathematics educators is no exception. most studies have positively reported on its impact on teaching and learning in mathematics education (chikiwa, 2020; darling-hammond, 2009; hollingsworth & clarke, 2017; karsenty et al., 2015; ward & mccotter, 2004). reflective practice enables teachers to meet the learning needs of their classes effectively. it enables teachers to adjust and respond to issues while assisting them to become aware of their underlying beliefs and assumptions about teaching and learning. the concept of rp can be traced back to 1933, when dewey first introduced what he called reflective thinking. dewey (1933) identified three attitudes that lay at the heart of rp for improved efficiency: wholeheartedness, responsibility, and open-mindedness. wholeheartedness relates to one’s commitment to working diligently and seeking every opportunity to learn something new from the present. responsibility reflects careful consideration of the consequences of one’s actions, particularly one’s preparedness to acknowledge that the decisions one makes impact the learners’ lives in ways both foreseen and unforeseen. finally, open-mindedness is the willingness to admit the possibility of error, considering various ways of looking at a particular situation or event, and being ready to change one’s viewpoint if necessary. schon (1983) applied dewey’s notion of rp to the teaching profession and defined rp as the practice by which professionals become aware of their implicit knowledge base and learn from their experience. several studies have since sought to understand rp in education, exploring its influence in teaching and learning and how to develop it (beauchamp, 2006; karsenty et al., 2015; zeichner, 1994). beauchamp (2006) carried out a systematic review of 55 articles on rp in search of a comprehensive understanding of the concept. while she could not identify a conclusive definition or meaning, she discovered that scholars agreed about rp’s significance as a tool for lifelong learning, professional development and its ability to influence effective teaching. russell (2005) notes that this absence of consensus should alert teachers and teacher educators that rp is a complex concept that they should approach with caution and preparedness. as jay and johnson (2002) put it: ‘if the concept itself seems difficult to characterize, it is even more difficult to teach’ (p. 73). the researchers found dewey’s early definition of reflection meaningful and more explicit for teacher education. dewey (1933) defines reflection as: [an] active, persistent and careful consideration of any belief or supposed form of knowledge in the light of the ground that supports it … [it] allows individuals to think critically and scientifically. (p. 9) the process of reflection is thus a systematic cycle that needs constant reiteration. reflective practice assists teachers in making links between various teaching experiences, thereby fostering progressive learning. schon (1983) distinguished between ‘reflection-in-action’, ‘reflection-on-action’ and ‘reflection-for-action’ to show how professionals solve problems associated with the practice. olteanu (2017) summarises as follows: reflection-in-action takes place during an action, and reflection-on-action takes place after an event has occurred. … reflection-for-action is thinking about future actions with the intention of improving or changing a practice. this type of reflection requires teachers to anticipate what will occur during a lesson, as well as reflect on their past experiences, before a lesson occurs (farrell 2015). the main problem teachers have in doing reflection-for-action is knowing what they should reflect on to be able to make improvements or changes in their own practice. (p. 350) schon (1983) uses the phrase ‘knowing in action’ to describe tacit knowledge that develops as teachers reflect on their practice. he likens it to riding a bike. if a person is riding a bicycle, and starts to fall, they will react in situ to regain their balance, yet may not be able to explain how when asked about it. reflective practice facilitates knowing in action and hence constitutes a basic part of teaching and learning. selmo and orsenigob (2014) concur that ‘learning from experience is enriched by reflecting on experience’ (p. 1925). it is therefore imperative that psts develop the skill to reflect on the practice of teaching, and that at least part of this development should occur within the actual practice of teaching. mewborn (1999) claims that this will create a ‘bridge across the chasm between educational theory and practice’ (p. 317). because of the many benefits associated with developing teachers’ rp, pte programmes have increasingly embraced it as a useful means of preparing teachers for effective teaching. kullman (1998) claims it leads ‘to a greater awareness among student teachers of what constitutes appropriate pedagogic practice and will lay the foundations for development, a process which will be ongoing throughout their teaching careers’ (pp. 471–472). according to shoffener (2009), exposing psts to rp during teacher preparation supports their construction of the personal knowledge they need to guide their teaching and learning decisions. she further claims that ‘by engaging in reflective thinking, pre-service teachers (psts) generate questions and, by seeking answers, support change in their educational actions, responses to criticism, and social and cultural beliefs’ (p. 145). a key challenge for pte however is that psts are mostly outside of the classroom and not themselves teaching. in this respect creative ways need to be found for psts to develop key rp skills to support their knowing in action in classroom teaching. studies conducted with psts confirm some benefits of rp in pte with the use of video-recorded lessons. for instance, hewitt et al. (2003) explored how psts reacted to scenarios shown in video-recorded lessons and confirmed that this video-stimulated reflective approach helped psts develop deeper insights into classroom practice. similarly, chen (2016) included reflection as part of the design of a teaching-learning programme for training kindergarten psts to teach mathematics. her study concludes that engaging psts in rp can assist them to gain mathematical knowledge and learn more about their own teaching methods. however, not all methods and attempts to develop rp with psts have proved successful. for example, pte efforts to involve psts in reflective journal writing have been found to be relatively ineffectual. ward and mccotter (2004) claim that most psts do badly in reflective writing assignments because they do not understand what reflection is or how to practise it. research studies have therefore raised questions about the widespread practice of writing as a means of developing rp (shoffener, 2009; ward & mccotter, 2004). thus, while there is general acknowledgement of the value of developing this practice in psts there is not general agreement on optimal ways to do this, and research tends to note that this is a relatively challenging skill to develop effectively. teacher education remains actively in search of strategies for developing psts’ rp. karsenty et al. (2015) explored video-based mathematics lesson analysis using the ‘six-lens framework’ (slf) to develop in-service teachers’ rp and mkft and reported positive results. their ongoing research continues to gather evidence, and deepen our understanding, of the effectiveness and opportunities that such a tool can provide in developing rp (see, for example, karsenty, 2018; karsenty & arcavi, 2017). while other researchers have similarly used video to stimulate reflection on practice (e.g. hollingsworth & clarke, 2017) we draw on the work of karsenty and colleagues because their slf was used as a tool to support the psts in this study in their reflections on the video. theoretical framing and the six-lens framework our research was underpinned by vygotsky’s (1978) social cultural theory of learning guided by the assumptions that social interactions play an essential role in learning; language is a fundamental tool in the learning process and learning occurs within the zone of proximal development (zpd) and is mediated by more knowledgeable others (mko). a mko can be a teacher, peer or artefact that embodies cultural or historical knowledge and functions to draw learners into their zpd (abtahi et al., 2017; graven & lerman, 2014). the zpd is described by vygotsky (1978) as: the distance between the actual developmental level as determined by independent problem solving and the level of potential development as determined through problem solving under adult guidance or in collaboration with more capable peers. (p. 86) the work of karsenty and colleagues coheres with this framework as it emphasises social learning and the importance of careful mediation in the use of video of teaching practice as an artefact for development. karsenty et al. (2015) and colleagues designed the slf in collaboration with teachers. they designed it as a tool primarily for their video-lm project (viewing, investigating and discussing environments of learning mathematics) at the weizmann institute of science in israel to enhance secondary school mathematics teachers’ reflection and mkft. the tool comprises six viewing lenses that the team considered essential for analysing and reflecting on episodes of mathematics teaching. these are mathematical and meta-mathematical ideas (mmi), teachers’ explicit and implicit goals, tasks and activities, dilemmas and decision-making (ddm), interactions with learners, and teachers’ beliefs. the lenses provide the psts with opportunities to consider concepts that can be developed under a given topic. karsenty et al. (2015) designed the mmi lens to guide the focus to the mathematical concepts and procedures the videoed teacher is advancing through the lesson. karsenty and arcavi (2017, p. 441) claim that reflecting on mmi ‘not only enables teachers to overtly explore ideas related to a certain topic; it also helps to refine, rethink and re-connect among them, and sometimes even to learn something new’. the goals lens directs the viewers to consider the explicit and implicit goals the teacher sought to achieve through the lesson. reflecting through this lens not only assists the viewing psts to understand the teachers’ actions and decisions, but also alerts them to the significance of setting meaningful goals. the tasks are the activities the teacher engages in to pursue the goals. reflecting on these exposes the psts to both content and strategies for teaching mathematics. the interactions lens focuses on how the teacher was interacting with the learners during the lesson, such as how they introduce the problems and activities, and respond to learners’ spoken and unspoken concerns. reflecting through this lens exposes psts to different ways of interacting with learners and allows them to evaluate what methods work better. the ddm lens helps viewers identify the possible dilemmas (unexpected challenges) during the lesson and what decisions the teacher makes concerning these. the teacher responds to unexpected behaviours (relating to the teaching and learning of mathematics) and learners’ responses to tasks and activities, during the lesson. lastly, the beliefs lens calls attention to the teacher’s beliefs about mathematics teaching that are displayed through how the teacher conducts the lesson. the teaching approaches chosen by the teacher for the lesson provide insight into their beliefs about the subject and how it should be taught. exposure to different teacher beliefs may help the psts develop a new conception of the subject and develop better ways of teaching it. while the framework may appear straightforward for experienced teachers, it should be remembered that psts have almost no teaching experience other than practicum and are still in the process of developing their mkft. methodology the research paradigm adopted was interpretive guided by the assumption that reality is multiple and shaped by social experiences (cohen et al. 2011). we used a qualitative case study research approach (creswell, 2014) gathering data in the form of rp written narratives. the case our article is focused on is a group of four psts participating in small-group rp sessions. phase 1 of our research included 19 out of 52 third-year psts who agreed to participate in the study. phase 2 was with only 4 of the 19 psts who were purposefully selected on the basis that they each had submitted a video recording of the mathematics lesson they taught during teaching practice as part of their course assignments. these video recordings of their own teaching provided a rich video resource for selfreflection. while five psts had submitted such video recordings, only four agreed to participate in the further sessions of phase 2 of the research. the focus of the article is on the data collected with the four psts in phase 2. these sessions with the four psts began at the end of 2018 following phase 1 of our research. in 2018, 52 psts, in their third year of bachelor of education studies at a university in south africa were taken through three sessions of analysing video-recorded mathematics lessons with the intention of developing both their mkft and their rp. the lecturer participating in our study identified the need for drawing on support tools to aid psts’ reflection on video lessons. having come across the slf of karsenty and colleagues she chose to draw on this tool as a device to support psts in focusing on a range of important aspects of video-recorded mathematics lessons as they reflected on them. since the aim of the sessions was to develop the psts’ mkft and rp, we took the opportunity to investigate how the psts fared with rp directed to mathematics education. in the whole-class rp sessions, the lecturer started by introducing the slf, and asked the psts to reflect on video-recorded mathematics lessons she perceived to be relevant to the psts’ context. she asked different groups of psts to use different lenses from the slf to discuss and report back reflections relevant to each lens. the psts were requested to write individual reflections before they could discuss as a group, then present to the whole class, leading to a class discussion. thus, psts were given opportunities to co-create knowledge and learn from each other. all 52 psts in the cohort were invited to participate in the research and all ethical protocols were followed. nineteen psts volunteered and signed consent forms for phase 1. the psts were provided anonymity and were informed they could withdraw their participation at any point. the ethics application was approved by our university education research ethics committee in 2017. elsewhere (chikiwa & graven, 2019, 2021) we have focused on the written reflections of 19 psts who participated in the broader study in term 1 and term 2 of 2018. the findings pointed to generally low levels of reflection and few shifts towards higher levels of rp. the focus of this article is on phase 2 which began towards the end of 2018 and continued in 2019 with four of these psts who had made video recordings of their own mathematics teaching practice during their teaching practicum as part of the practicum assessment. (the psts were requested by their lecturer to video-record themselves teaching any subject of choice and submit to the practicum coordinator for assessment.) in phase 2 we asked the four psts to write reflections on these videos of their own teaching using the slf they had used in lectures during 2018. they submitted these written reflections to us in the last term of 2018. in the third term of 2019 we invited the same four psts to participate in three small-group, facilitator-led rp sessions focused on selected video episodes of other teachers’ lessons as a way of scaffolding their rp skills. the video lessons used during these sessions were chosen by the facilitators for their appropriateness to fp teaching as well as their experience of these lesson videos having rich material to stimulate reflective discussion. the facilitators, being experienced in reflecting on video of mathematics lessons according to the slf, served as the mediating mkos, modelling rp focused on important mathematical aspects of teaching and learning. following these sessions (september 2019), the four psts were asked to use the slf once again to provide written reflections on the video recording of their own lesson they used previously in 2018. they completed these towards the end of the fourth term in 2019. we analysed their reflections using content analysis (stemler, 2015) and a tool we developed comprising four hierarchical levels of reflection that we adapted from existing frameworks of reflection levels. note that the slf was a pedagogical tool used by the lecturer for focusing attention on six different aspects of lessons while the framework we use for analysis relates to analysing the levels of reflection when psts reflect using the various lenses. (this is discussed further in the section that follows and is summarised in table 3.) as mentioned above, in phase 2 the four psts were taken through three reflection sessions led by experienced facilitators to support their rp development. in these sessions the facilitator probed for reflections as the psts analysed selected video recordings of other teachers’ mathematics lessons. in table 1 we share an excerpt from one of the sessions showing how the facilitator probed for deeper reflections. table 1: excerpt from a facilitator-guided reflective development session. at the start of the session the facilitator carol (who was highly experienced in the use of the slf for supporting teacher reflection on videos) provided some orienting comments that included an explicit statement about the importance of noticing mathematical over general aspects of lessons, such as: ‘our focus is on the mathematics, so we try to talk less about generic aspects such as the teacher’s voice or the teacher’s body language; these things are interesting [but] … our focus is to talk about the mathematics. so the mathematical idea, the mathematical goals, the mathematical activities, even the interactions can be about the mathematics, so the mathematics is in the centre.’ (carol) thereafter she assigned each of the psts a metaphorical lens to look through as they analysed the video-recorded mathematics lesson she had selected for the session. she played the video and allowed the psts to write reflections using the lenses they were assigned. she began the discussion by calling for more reflections through probing questions. in the process she assisted the psts by guiding them to comment on aspects of the lesson specific to each lens. in table 1 we share some examples of interactions between the psts and the facilitator. as seen in the session excerpts in table 1, the facilitator introduced the session by encouraging the psts to focus on the mathematical aspects of the lesson. as the session continued, she asked psts questions that related to each lens of the slf through which the pst was asked to analyse the lesson. the facilitator probed for deeper insights into what was happening in the lesson and why. after three such sessions focusing on different videos, we asked the four psts to provide written reflections again on the video recordings of their own teaching (which they had done a year earlier). we then analysed their written reflections using the four levels of reflection framework (flrm) and compared their first reflections on their own practice (rop1 in 2018) with their second (rop2 in 2019). below we give a brief account of how we developed our analytical framework. developing the four levels of reflection framework for analysing reflections the literature offers a wide range of useful analytical frameworks for analysing rp, but none of them quite met our needs in relation to coding and analysing written reflections of the 19 psts who participated in phase 1 of the study. we therefore had to create our own, merging and adapting existing frameworks following a detailed and wide-ranging review of existing frameworks. we began by merging elements of lee’s (2005) and muir and beswick’s (2007) levels of reflection frameworks (these resonated most strongly with the nature of the data we had) to develop a basic hierarchical rp tool for analysis with indicators. following repeated revisions to our initial framework (from ongoing analysis of samples of psts’ data over time) we finally settled on a four-level framework that merged key elements of both lee and muir and beswick’s three-level frameworks and indicators. after using samples of data in these two combined frameworks we identified the need for an additional level in our framework as well as some refinement of a set of sub-indicators across the levels. we particularly noted that ‘suggestion’ (as evidenced by our data) did not cater for reflectivity or critical reflection in its full sense as described by lee (2005) and muir and beswick (2007). we therefore created a level between their levels 2 and 3 which we named suggestions, and this shifted their third and last level to level 4 in our framework. we concurred with lee’s idea that reflectivity should go beyond identifying a classroom event and providing explanations and suggestions, to a deeper engagement with the identified event that enables the proposal of alternatives. we thus adopted lee’s level 3, reflectivity, as our highest level of reflection. (see chikiwa and graven, 2021, for the rationale for the need to extend to this four-level framework.) in table 2 we present the four levels comprising our adapted rp framework. in the third column we provide what we consider to be the key indicators for each level (adapted from the aforementioned frameworks) and in the fourth column we provide some examples from the psts’ written reflections. table 2: the four levels of reflection framework. we used the flrm that we developed to analyse our data. each statement written by the pst was broken into small chunks of single ideas to allow for coding. the chunked data was levelled against the levels of reflection in the flrm. the psts’ reflections were either general or mathematical, which led us to code each idea as such. general reflections referred to reflections that were not specific to the teaching and learning of mathematics but generally applicable to the teaching and learning of any subject, for example ‘the teacher put learners in groups of four’. a reflection was deemed to be general when the pst did not make mention of mathematical concepts, terms, symbols, numbers, or mathematical ideas. we referred to mathematical reflections as reflections that were specific to the teaching and learning of mathematics. these were evidenced by explicit mention of the mathematical concepts, terms, numbers, symbols or ideas, for example ‘the teacher wrote 39+9 on the board’, or when an implicit reference is made to mathematical terms, numbers, symbols or ideas, such as ‘the learners did this activity in groups’. if the activity that is described before this statement was mathematical, then the reflection also becomes mathematical because of the reference. following a process of repeated refinement of codes we settled on the codes in table 3 that we subsequently used for coding all the psts’ data. table 3: summary of codes developed from the analytical tool and data. we identified the codes that have an arrow with ‘e’ as indicative that the reflection was followed by an explanation. for example: ‘[the teacher] wrote the addition steps on the board md→e, to help the child that did not understand me.’ (lutho, rop1, ddm, ref 44–45) ‘a simple explanation is when the rationale given was a statement with only one idea (like the one above). the expanded explanation was a rationale with more than one idea. for example: ‘i shouldn’t have put together addition and subtraction ms→e, because learners were not really focused and couldn’t understand the subtraction part, they needed it to be done separately mee.’ (bonga rop2, ref 80–81) as in the establishment of the levels of rp and the indicators thereof, these codes were similarly developed after several rounds of coding by the authors of the article along with a third researcher who was brought in to assist in establishing whether our coding and indicators were sufficiently recognisable to others to enable consistent coding across researchers. some initial differences in coding resulted in further refining and clarification of the codes until there was general agreement between the researchers’ coding. thereafter the first author of this article coded each of the four psts’ reflections. the second author served to check agreement with the coding and small discrepancies were resolved through discussion and through reverting to the indicators and providing further clarification where necessary. below we present the comparisons we made for each of the four psts (in alphabetical order of their pseudonyms). the findings in this section we share both quantitative and qualitative data from our coding of two sets of reflections written by psts a year apart. the tables in each figure capture the quantitative descriptive statistics following the coding of all the written reflections, according to levels of reflection and according to mathematical versus general reflections. the pie charts following the tables compare the psts’ general versus mathematical reflections on each set of data. following the figures that summarise each of the four psts data we provide selected qualitative data that exemplifies and illuminates shifts from the first (rop1) to the second (rop2) written reflections. comparing bonga’s 2018 and 2019 reflections on his own practice figure 1 contains summaries of bonga’s rop1 and rop2 on the same video-recorded lesson, followed by a discussion of the comparison across the years. figure 1: comparing bonga’s 2018 and 2019 reflections on his own practice. while we see a small decrease in the total number of ideas that were coded in bonga’s rop1 (100) to the number of ideas in rop2 (92), figure 1 shows that bonga’s level 1 descriptive reflections shifted downwards from 86% to 77% and his explanation of level 2 reflections increased slightly from 14% to 16%. the decrease in the proportion of descriptions is mostly balanced by the 7% presence of mathematical suggestions in rop2 while there were no suggestions in rop1. in rop2 across the lenses, bonga supported a few more of his mathematical descriptions with explanations. excerpt 1 below provides an example of his reflections on mmi in rop1 and rop2 to illuminate the modest shift towards increased explanation. no suggestions were provided in either rop1 or rop2 for the lens of mmi. we have highlighted the explanations in both: ‘in this lesson i planned to teach learners bonds of 15 (md). i wanted learners to come up with two numbers that can be added to make up the number 15 (md). each learner would raise a hand (gd) and give the two numbers (md), explain to the class how they calculated it (md). learners had to justify their answers by explaining to the class (md→e) this was for helping them to have number sense, to help them with addition (mee) aims of this lesson was to equip my learners with ‘adding on’ instead of starting from 1 when adding (md). as learners were explaining how they arrived at making 15 with two digits (md), i helped them to start with the larger number instead of small number to do so (md).’ (bonga rop1, mmi, ref 1–10) ‘in this lesson i taught learners bonds of 15 (md→e). this was to help learners know different numbers that make up the number 15 (me) so that they can add or subtract faster without counters (mee). learners had to come up with 2 digit that would make 15 (md). this was first done as whole class (gd), learners giving the teacher numbers (md) and showing how they arrived to their answer (md). justifying their answer (md) and showing their calculations (md→e) helped learners understand their procedures (me). then we did it using a grid (md). the grid had addition on it (md→e). we used it [the grid] to look at different ways to add (me), mainly adding on (md). the second strategy was number line (md); here i introduced subtraction (md).‘ (bonga rop2, mmi, ref 1–16) through the mmi lens, which is focused on mathematical and meta-mathematical ideas, as expected, the reflections for both rop1 and rop2 are predominantly mathematical. across other lenses we see a shift towards more mathematical than general reflection ideas. the tables and pie charts in figure 1 show a significant shift in bonga’s reflections towards mathematical ideas, from 45% in rop1 to 84% in rop2. thus, in rop2 bonga focused more on mathematical descriptions and explanations across the lenses. for example, while bonga provided several similar descriptive reflections on his beliefs in relation to the lesson observed in both rop1 and rop2, the start of the rop2 reflection indicates an orientation towards increasingly foregrounding the specificity of teaching mathematics. for example, in rop1 ‘beliefs’ bonga begins his reflection with ‘any discipline can be made fun by a teacher (gd), it’s all just a time needed to be taken by the teacher (gd) and willingness to allow the children play as they learn (gd)’.on the other hand, for rop2, bonga begins with ‘maths can be done in different ways (md). it can be hard for others (md) and easy for others (md). learners have to be able to justify (gd) and show how they got to an answer (gd)’. of particular interest is that all bonga’s suggestions (l3) are mathematical rather than general. while no suggestions appear in rop1, several emerge in rop2. for example, in the task lens of rop2 bonga follows his description of learners using other learners’ fingers to count with: ‘i should have provided counters for my struggling learners’. in addition, following his reflections on each of the six lenses in rop2, bonga took the initiative to add a section, namely ‘what would i change?’. of interest is that all ideas included here are mathematical: ‘i shouldn’t have put together addition and subtraction (ms→e), because learners were not really focused (me) and couldn’t understand the subtraction part (me), they needed it to be done separately (ms→e). these two [algorithms] are already complicated for grade 1 to use on 2-digit numbers me, and putting them together was not a good idea (ms→e) because i ended up spending more time on addition and very less time on subtraction (me). many learners seemed to get confused when i wanted them to subtract (md). that was not good for the learners (md). they didn’t learn much from it (md). i should have stuck with only one (ms).’ (bonga, rop2, beliefs, ref, 80-87) in the above paragraph we see that bonga brings several l1 (descriptions), l2 (explanations), and l3 (suggestions) into dialogue with each other. when these were considered as a whole, we decided that the paragraph constituted an example of l4 reflectivity. recall that our definition of reflectivity requires that psts engage dialogically with the classroom event; it incorporates suggestions that are considered from different perspectives. we noted here that bonga brings observations, descriptions, explanations, suggestions, and justifications for these into dialogue, constituting l4 reflectivity. thus, while no single idea is coded as reflectivity, the paragraph as a whole constitutes reflectivity. for this reason, we have placed l4 as separate from the total l1, l2, and l3 ideas coded and we have put na in the table under the percentage for reflectivity. as this instance (and two cases in joy’s reflections) were the only reflections across the data sets that met the indicators for reflectivity, we did not have sufficient data to expand the sensitivity of our coding system for this level of reflection. the idea that multiple ideas together display reflectivity, rather than a single utterance or idea, suggests further consideration is needed in terms of how this l4 of the reflection framework subsumes (is constituted by) l1, l2, and l3 utterances, rather than having its own distinct utterances. the above shifts suggest that the facilitator-guided reflection sessions may have influenced bonga’s attention to focus on mathematical learning and teaching aspects of the lesson, and to extend his response to include engaging with some reflectivity, particularly in relation to engaging with suggestions and explanations (or justifications) for these suggestions. this was not however the case across psts, as we see in dumi’s reflections in figure 2. figure 2: comparing dumi’s 2018 and 2019 reflections on his own practice. comparing dumi’s 2018 and 2019 reflections on his own practice figure 2 contains summaries of dumi’s rop1 and rop2 on the same video-recorded lesson, followed by a discussion of the comparison across the years. as seen in figure 2 there was a drop in the number of ideas coded in dumi’s reflections from rop1 to rop2 (109 to 73). in rop2 dumi avoided reflecting on classroom management as he had done in rop1. while the total number of reflection ideas went down, the coding shows a proportional shift towards less description (80% to 66%) and more explanation (14% to 29%), while the suggestions remain relatively consistent across both rop1 and rop2 (at 6% in rop1 and 5% in rop2). a considerable number of his reflections in rop2 were followed by explicit explanations (see figure 2, rop2) as evidenced by his repeated use of the phrase ‘reason is that’. no paragraphs were considered at the level of reflectivity (l4). thus, the greatest shift for dumi was towards a greater proportion of explanatory reflections. to illuminate this proportional shift towards more explanation, we share his rop1 and rop2 on ‘interactions with students’. dumi’s explicit indication that he was providing reasons in rop2 (‘the reason is / for that’) was not visible in rop1. this probably suggests the impact of the facilitators’ probing for reasons behind identified events: ‘first of all, i was walking around the classroom (gd), helping learners who had questions (gd) and those who were stuck (gd). i was addressing them as a group since i grouped them (gd), but i noticed that some of them were confused (gd), i decided to sit with them one on one (gd). the time was a problem (gd→e) because, i ended up spending lot of time with some students (ge) while others were struggling (gd) and i couldn’t finish all of them to see whether they were following the activity (gd). the questions that i was asking to students, some of them were questions like: if you have 12 dominos and added 7 dominos, how many of them in total (md). i also told them that addition means putting together (md 68) and subtraction means taking away (md). some of the students had questions revolving around how to use the counters (md). some they were not familiar on how to use them [counters] (md). i also had a rule that, when i speak they should listen (gd) and when someone wants to speak must raise their hands (gd→e) so that i can give, there is order in class (ge). even though the class was a little bit chaotic (gd) when i was busy with a group of students (gd) some would make noise (gd).’ (dumi, rop1, interactions, ref 56–76) ‘firstly the teacher is explaining the lesson to the whole class (gd→e), reason [for] him to do this he wants everyone to listen and pay attention on the instructions, so he does not waste time during the lesson (gee3). he grouped them into three groups (gd) and gave each group different task (gd) and material to do (gd) and use (gd→e). reason is that he was developing different strategies for each group (ge). teacher is moving to each group (gd→e) to ensure that everyone understand the instructions and they are doing what they supposed to be doing (gee2). the teacher is asking learners during the lesson in each group questions such as: show me how did you do it? double check is the answer is right, how did you calculate it? (gd→e) reason is that he wants them to make sure that they know how they got the answer. he also wants them to self-correct themselves (gee). he kept on saying yes as his response (gd→e) to show that he is listening to the learners and he wants them to carry on (gee2). he encourages learners to do different sums (md→e) to develop different strategies of calculating and understanding the operations (gee2). teacher also instructing learners to help each other (md→e). reason for this is some learners understand it better when they get help from peers and some are developing confidence in mathematics (mee3). he encourages them by saying good to learners who are doing well (gd→e) reason for this he wants them to give their best and even if they fail, they cannot be afraid to try hard (gee2). he also instructs learners to recount when they have forgotten a number (md→e) so that they remember which number follows what (me).’ (dumi, rop2, interactions, ref 1–20) in terms of the mathematical and general foci, there was an unexpected shift away from reflecting on mathematical events towards more general events. the mathematical reflections decreased from 59% in 2018 to 41% in 2019. this could be a result of the keen interest he displayed in the lens of interaction with students, which generally motivates reflections that are more general than mathematical (see chikiwa & graven, 2022). this interest is echoed in that he began his rop2 by reflecting on ‘interactions’ instead of the mmi that is presented first in the framework and therefore commonly used first. he also wrote more reflections under this lens than the rest. thus, it may seem that the three sessions of facilitator-guided reflections did not shift dumi’s focus to mathematical reflections or increase his suggestions and reflectivity, though the move to providing more explanation could have been supported by these sessions. comparing joy’s 2018 and 2019 reflections on her own practice figure 3 contains summaries of joy’s rop1 and rop2 on the same video-recorded lesson, followed by a discussion of the comparison across the years. figure 3: comparing joy’s 2018 and 2019 reflections on her own practice. unlike bonga and dumi, there was an increase in the quantity of joy’s coded ideas from 58 in rop1 to 97 in rop2. thus, joy wrote more ideas about her teaching after the facilitator-guided sessions. joy’s reflections shifted steadily to higher levels after the series of facilitator-guided sessions. as seen in figure 3, there was a noticeable decrease in the percentage of descriptions (l1) from 76% in rop1 to 63% in rop2, and a commensurate increase in explanations (l2) from 17% to 24%. the proportion of suggestions (l3) also increased from 7% to 13%. we share an example of her reflections in rop1 and rop2 for the ‘tasks’ lens to illustrate some of the subtle shifts towards more explanation: ‘rote counting in 10s (md). mental maths activity – number plus 2 (md), measuring desks using pencils (md). writing down their measurement observation in their workbooks (md). i asked the learners leading questions (gd→e)so as to scaffold them (ge). sometimes the learners would not understand what answer i was looking for (gd). in rote counting, as mentioned before some of the learners were not counting correctly (md) or even counting at all (md). but that was difficult for me to pick up (md). by doing the mental maths activity (md→e) they got to revise on their addition skills (me). they did the measuring activity (md→e) so that they got to see how many pencils can fit into the length of their desk (me) they should have written their observation in their workbooks (gs→e). the benefits of this are they get to connect what they observe with writing it down, a form of report (ge).’ (joy, rop1, tasks, ref 29–43) ‘i introduced the task by asking questions (md→e) so that i could see how much they knew about measurement (me). the responses were accurate (gd) but only came through once i started asking leading questions (gd). this showed me that they had an idea of what measurement was (md) but did not connect the concept to their prior knowledge (md). i then demonstrated what i wanted them to do by using the board (gd) instead of using a desk and a pencil which they were also using for measuring (md). using the board may have confused some of the learners (md→e)because i just started to [measure] what may have looked like at the middle of the board which we see when two learners start measuring their desk in the middle (mee).’ (joy, rop2, tasks, ref 37–63) ‘i used the board (gd→e) because i wanted the entire class to see what i was demonstrating (ge), but in the process, i took it for granted that i was starting in the middle of the entire board (gd). a clearer demonstration could have avoided this (gs) paired up with a clearer instruction (gs). using this practical way of teaching could be beneficial to the learners (gs→e) because they can see and do what i am explaining in the abstract (gee). demonstrating how to measure would have been ideal (ms→e) because they would ideally be able to link it to daily activities of measurement (me).’ (joy, rop2, tasks, ref 37–63) ‘the potential shortfall is the form in which i communicated the instruction (gs1). the writing as consolidation of the activity (md→e) was to create a link between measuring and recording (me). however, next time i would combine the recording with the measuring (ms→e) so as to improve this part of the lesson (me). i will ask one learner to measure (ms) and the other to record (ms→e). this is because the time between them measuring and me handing out their books may be too long for others to remember what they had measured (mee).’ (joy, rop2, tasks, ref 37–63) elaborating in rop2, joy followed most of her described classroom events with a rationale or explanation. she further made several mathematical and general suggestions for improving future instruction whereas in rop1 she had only provided a single general suggestion. in rop2 she reflected with intention to improve instruction, which we see as probably a result of participation in the facilitator-guided sessions. we also found that in rop2 joy wrote two reflections at l4: gr (highlighted in the excerpt above) and mr. as mentioned in bonga’s section, l4 reflectivity constitutes a collection of reflections (description, explanation, and suggestion) that together are in dialogue with each other and at the level of paragraph. the second and third paragraphs in rop2 above point to possible instances of reflectivity. in the first instance of reflectivity, joy is having a dialogue with herself about how she used the board in a manner that hindered the learners’ conceptual understanding of measurement. she ends the dialogue with a proposal that carrying out a demonstration accurately would have helped learners to link measurement with daily activities. in the second instance of reflectivity, joy again enters self-dialogue, reflecting on the ‘less than ideal’ way she taught the learners. she provides her reason for her judgement and suggests a way forward also backed by reason. the excerpts provide some examples of the qualitative shifts in joy’s reflection between the two years. the proportion of mathematical (versus general) reflections remained relatively consistent across rop1 and rop2 at 63% to 59%. comparing lutho’s 2018 and 2019 reflections on her own practice figure 4 contains summaries of lutho’s rop1 and rop2 on the same video-recorded lesson, followed by a discussion of the comparison across the years. figure 4: comparing lutho’s 2018 and 2019 reflections on her own practice. there was a small increase in the quantity of lutho’s reflections between 2018 and 2019, from 61 to 63. in rop2 more attention was paid to some lenses while attention was removed from others. for example, lutho’s mmi reflections increased from 8 in 2018 to 14 in 2019, while her reflections on interaction halved from 12 in 2018 to 6 in 2019. as in joy’s case, we noticed steady shifts from the lower levels to higher levels of reflection. as seen in figure 4, l1 reflections decreased significantly from 90% to 75%, a shift complemented by an increase in l2 reflections from 10% to 16% and l3 reflections from 0% to 9%. there were no instances of reflectivity. we also noticed a slight shift to a more mathematical focus than general. mathematical reflections increased slightly from 57% in 2018 to 63% in 2019, while general reflections decreased from 43% to 37%. the shift in lutho’s reflections from lower to higher levels and to mathematical events of her lesson seems to point to the positive influence of facilitator-guided reflection sessions on lutho’s rp. we share her rop1 and rop2 reflections on mmi to illustrate how in rop2 she followed her described classroom events (that were similarly described in rop1) with explanations and went further to suggest what could possibly improve future instruction. ‘the teacher brought forward the money concept (md). the aim was to teach about the currency used in sa [south africa] (md). this is evident through her use of a chart (md) and the different pictures of monies (md) that she uses as manipulative materials for the children (gd→e) to acquire concrete understanding (ge). in her teaching she incorporated addition (md) when asking children how much money is needed to produce a certain amount (md).’ (lutho, rop1, mmi, ref 1–8) ‘[the lesson was on] sa money currency (md and value (md). [that is, identification (md) and recognition of currencies of different currencies (md)]. she used a chart to create a pictorial of what money looks like (md→e) for the children to see (ge). using the chart (md→e) allows them to link the pictures that are in the chart and the manipulatives that are in front of them (me). nonetheless, she could have allowed the children to explore the manipulatives themselves before she showed them the chart (ms→e) to check how much understanding of money do they already have (me). she allows the children to find the correct currency by themselves (md) after seeing the picture of that currency (md→e). this helps the children to acquire concrete understanding of what the different currencies look like (me). on the other hand all children with different intelligences are catered for (gd).’ (lutho, rop2, mmi, ref 1–14) findings and discussion across the four pre-service teachers the overview of the four psts’ reflections shows that for all psts there was an increase in the proportional percentage of ideas that went beyond l1 descriptive reflections. thus, there was a greater proportion of explanations (l2) and suggestions (l3) from rop1 to rop2 for all four of the psts, other than dumi, whose percentage of suggestions remained relatively constant (6% in rop1 and 5% in rop2). two of the psts, bonga and lutho, had not provided any suggestions in rop1 and provided several in rop2. furthermore, while no examples of l4 reflectivity reflections were found in rop1, in rop2 three examples are found in which bonga and joy each engaged with descriptions, explanations and suggestions in a dialogic way that met our definition of reflectivity. as far as the mathematical versus general balance of reflections is concerned, despite the explicit orientation in facilitator-mediated sessions towards a focus on mathematical rather than generic observations of lesson events, only bonga’s and lutho’s mathematical reflections shifted upwards. bonga’s mathematical reflections increased significantly, from 55% to 84%, while lutho had a shift from 57% to 63%. the other two psts’ proportional focus on mathematical ideas decreased slightly from rop1 to rop2, although only for dumi did this result in his focusing more on general than mathematical ideas in rop2 (59% general vs 41% mathematical). for all others, mathematical reflections continued to dominate over general reflections (59%, 63% and 84% for joy, lutho and bonga). the above suggests that the facilitator-guided reflections (and perhaps the increased experience in both practicum and studies) supported shifts towards increased explanation and suggestion for the psts in rop2. this said, descriptive (l1) reflections still dominated at between 63% and 77% of psts’ coded reflection ideas in rop2. thus, across the four psts, reflections were still predominantly at the lowest level of reflection (description) with only modest shifts for some towards a greater proportion of higher-level reflections. this suggests that while small-group, facilitator-guided reflection sessions may be helpful in supporting some psts to provide occasional suggestions and instances of reflectivity, more work is needed if we wish to shift psts’ rp towards the deeper reflectivity needed for strengthening mkft and allowing for transformation of practice. therefore, while our findings provide some support for the finding of johns (2010), who reported that facilitator-guided reflection assists novice practitioners to improve the way they reflect, the extent of this assistance appears relatively limited in the case of only a few facilitator-guided sessions. these findings of shifts across the four psts’ reflections on their own practice following three facilitator-guided sessions are similar to the limited shifts in levels of rp identified with the larger sample of 18 psts following three lecture sessions on the use of the slf for developing rp using video-recorded lesson episodes (see chikiwa & graven, 2019) limitations of the study a limitation of this research, especially in terms of the empirical contribution, is the small number of students in the sample, and the fact that the videos to which they responded in their reflections were of their own practice, while the facilitator-guided sessions featured the practice of other teachers. if the students had reflected on other teachers’ practice after facilitator-guided sessions, the results may have been different, although we do not expect this to be the case. we did not at the time deem it feasible to ask psts to do additional written tasks as they were busy with other assignments and courses. alternatively, had the facilitator-mediated sessions focused on videos of their own teaching, shifts towards increasing proportions of higher levels of reflections may have appeared. an additional limitation is that as time passed between the gathering of the data on the two sets of reflections one cannot claim that the shifts are a result of the facilitator-led session and not general experiences in their studies that might have supported strengthened rp. furthermore, we did not qualitatively code differences in various reflections within the levels of description, explanation and suggestion other than to distinguish between those that were relatively simple and those that were expanded on and connected to further explanation (as shown in table 3). further differentiation and increased sensitivity in the coding of sub-levels per level may enable researchers to track progressions in psts’ reflections that may not be visible in our coding system. further research might usefully look at developing increased levels of refinement so that progress might be tracked within each of the levels of rp over time. concluding remarks a contribution of the broader study is our adapting and developing a coding system for reflections in a way that might usefully reflect south african pst shifts in reflection. in our study the lecturer used karsenty et al.’s (2015) slf to support psts in looking at six different aspects of (looking through six different lenses at) lessons. the slf however is not an analytic framework for researching qualitatively different levels of reflection. for our analysis of psts’ reflection on lessons we thus looked at frameworks that delineated hierarchical levels of reflection. we argued in the article, with reference to our data, that the three-level frameworks of reflection of researchers in australia and the corresponding indicators for identifying reflections at each of these levels were not sufficient for our purposes. in this article we used our adapted four-level analytic framework, along with a range of sub-categories for coding rp data to analyse the shifts in psts’ rp. we found that psts, following small-group, facilitator-led sessions, provided many more suggestions. these however were at a basic level and without justification. they were thus not considered to meet the requirements of the highest level of reflection, that is, reflectivity. we thus introduced suggestion as a level between explanation and reflectivity and noted that the few instances (three) of reflectivity that we found in rop2 required consideration at the level of paragraph or at least a combination of l1, l2 and l3 ideas brought into combination with each other. thus, while coding of ideas worked for l1, l2 and l3 reflections, and these ideas could be coded into a single category (thus avoiding ideas being coded into more than one category), this coding did not work for l4 as the definition itself involves bringing ideas into dialogue with one another. considering the limitations noted above, we have only tentative recommendations that build on the insights emerging from the study. the first is that, recognising the complexity and difficulty of developing rp, pte needs to find ways to provide opportunities to model high levels of rp across multiple pst course offerings and opportunities for facilitator-led mediation of psts’ observations and reflections. tools such as the slf (karsenty et al., 2015) are useful for supporting psts in focusing on a range of aspects of lessons but are not sufficient to enable psts to reflect at higher levels. they also do not guarantee a focus on mathematical over general aspects of lessons. pre-service teachers probably need much greater exposure to opportunities for developing rp across multiple contexts (e.g. live observation of lessons, viewing video recordings of the lessons of other teachers as well as their own) and multiple course offerings, from the first year of their studies. in this way, rp may come to offer powerful support for their future teaching as a sustainable dimension of their teaching practice. acknowledgements this work is based on research supported by south african research chairs (sarch) initiative of the department of science and technology and the national research foundations (grant number, 74658). competing interests the authors declare that they have no financial or personal relationships that may have inappropriately influenced them in writing this article. authors’ contributions m.g. and s.c. collaborated on the published work. both made a substantial contribution to conceptualisation and design, acquisition of data, analysis and interpretation of data, and drafting of the manuscript. m.g. further critically revised it for important intellectual content and approved the final version to be published. ethical considerations ethical clearance to conduct this study was obtained from the rhodes university faculty of education higher degrees’ committee (no. 2017.12.08.06.) funding information the authors acknowledge national research foundation (nrf) for funding this research. data availability the data that support the findings of this study are available on request from the corresponding author, s.c. disclaimer the views and opinions expressed in this article are those 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(2017). “what is mathematics?” and why we should ask, where one should experience and learn that, and how to teach it. in proceedings of the 13th international congress on mathematical education: icme-13 (pp. 63–77). springer international publishing. footnote 1. in identifying the event as a shortfall, she is implicitly suggesting she needs to improve the way she instructs learners. abstract introduction methodology findings discussion conclusion and recommendations acknowledgements references about the author(s) george ekol mathematics education division, school of education, faculty of humanities, university of the witwatersrand, johannesburg, south africa simphiwe mlotshwa mathematics education division, school of education, faculty of humanities, university of the witwatersrand, johannesburg, south africa citation ekol, g., & mlotshwa, s. (2022). investigating the cognitive demand levels in probability and counting principles learning tasks from an online mathematics textbook. pythagoras, 43(1), a677. https://doi.org/10.4102/pythagoras.v43i1.677 original research investigating the cognitive demand levels in probability and counting principles learning tasks from an online mathematics textbook george ekol, simphiwe mlotshwa received: 22 jan. 2022; accepted: 05 july 2022; published: 16 sept. 2022 copyright: © 2022. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract this case study carried out during the 2020 coronavirus disease of 2019 (covid-19) lockdown used online data collection means to investigate the distribution of cognitive demand levels of probability and counting principles (pcp) learning tasks in a popular online grade 12 mathematics textbook, based on the pcp teachers’ rating. the teachers’ cognitive demand ratings were categorised following stein’s mathematical task framework. five mathematics teachers from four secondary schools in two provinces in south africa participated in the study by filling in an online questionnaire. we developed a rating framework named the mean cognitive demand rating (mcdr) to help us interpret the teachers’ perception of the tasks in terms of cognitive demand to the learners. data from the teachers’ ratings revealed nearly 65% of the pcp learning tasks in the online textbook were rated as high. analysis of secondary data from department of basic education diagnostic reports from 2014 to 2020, however, suggests no association between teachers’ rating of learning tasks and learner performance. contribution: this study draws attention to a long-standing underperformance in the topic of probability and suggests classroom-based study that focuses on the learners’ rating of the learning tasks themselves to understand clearly how best to support them. keywords: probability and counting principles; mean cognitive demand rating; mathematical competencies; mathematical task framework; descriptive statistical analysis; digital textbook; multiple representations; grade 12. introduction the worldwide spread of the coronavirus disease of 2019 (covid-19) in the year 2020 brought changes to the way societies live, work and study. the institutions affected by covid-19 responded by moving from physical, human-to-human interaction to virtual and online platforms. during 2020, south african schools were shut down for two and a half months before grade 7 and grade 12 pupils were permitted to return to school. during the shutdown, learning continued virtually and online for some schools, particularly the relatively well-resourced schools (mohohlwane, taylor, & shepherd, 2020). in response to the closure of schools, and in a bid to ensure that learning was taking place especially for the candidate classes, the department of basic education (dbe) issued guidelines with recommendations for children to take advantage of online learning resources to continue with schooling from home. to facilitate learning from home, some websites were zero-rated in partnership with mobile phone companies (dbe, 2020b). online learning resources include resources such as online video lessons on youtube, digital textbooks (dt), and study guides. these resources can be accessed by learners working from home on a computer or on a smartphone connected to the internet. in this study, we focus on only one online resource, namely the dt used by grade 12 mathematics learners and teachers in south africa. due to the closure of schools in 2020 due the prevalence of covid-19, we did not conduct classroom-based study involving the grade 12 learners. we noticed that this online textbook was being heavily used by both grade 12 teachers and learners and decided to investigate the teachers’ rating of learning tasks in the probability and counting chapter of the book. we were able to carry out an online survey with the grade 12 mathematics teachers who are involved in teaching probability and counting principles (pcp). digital textbook the dt approved by the dbe is freely downloadable on any mobile device such as tablets and mobile phones. the dt comprises 9 mathematics topics (sequences and series, functions, finance, trigonometry, polynomials, analytical geometry, euclidean geometry, statistics, and probability) that are taught at grade 12. the 9 topics are each written following the same format; thus, each topic begins with the revision of related concepts, followed by the content notes, a couple of worked out examples, and exercises at the end. the exercises have answers to enable learners to cross-check their solutions. in this article, we only discuss the topic of pcp. probability and counting principles probability theory is a mathematical modelling (blum et al., 2007, p. 4) of the phenomenon of chance or randomness. randomness has a specific meaning in probability and statistics (batanero et al., 1997; batanero & sanchez, 2013). suppose t is a finite probability space. we assume that the physical characteristics of an experiment in t are such that the various outcomes of the experiment have equal chance of occurring. such a probability space, where each point is assigned the same chance of outcome, is called a finite equiprobable space (moore, notz, & fligner, 2013, p. 262; spiegel, schiller, & srinivasan, 2013). however, events in t are far from being random. for example, a coin does not generate random numbers because a tossed coin obeys the laws of physics depending on the force used, angle of toss, and surface of the coin. so, why then do the results of tossing a coin look random? it is because the outcomes are extremely sensitive to the inputs, so that very small changes in the forces one applies when tossing a coin do change the outcomes, say from heads to tails and back again (moore et al., 2013, p. 262). regarding counting principles, these are techniques for determining without direct enumeration, the number of possible outcomes of a particular experiment (may, masson, & hunter, 1990, p. 189). for example, if t defined as above, has k elements, then each point in t is assigned the probability , and each event b ∈ t containing h points is assigned the probability . in other words, . we note that the formula for p(b) only applies to an equiprobable space t and not to a general space. multiple representations by multiple representations, we mean techniques of teaching pcp that include various objects such as graphs, diagrams, texts, and 3d visualisations to facilitate learners’ grasping of the underlying meaning of the concepts. context probability and fundamental counting principles are relatively new topics in the south african mathematics syllabus (zondo, zewot & north, 2020), having been introduced for the first time in grade 10 in 2012, and in grade 11 in the following year. the topic was first examined in grade 12 in 2014. since their inclusion as compulsory topics in the south african mathematics syllabus, these principles have remained a challenge to many learners, with performance remaining generally poor over the years (dbe, 2015, 2016, 2017, 2018, 2019, 2020, 2021). for example, in 2013, the average score in the national examination was 30% (dbe, 2014). although every year diagnostic reports are published stating specific concepts in pcp that learners show weakness in so that educators in schools can support them in overcoming such identified weaknesses, from 2014 to 2019, diagnostic reports strongly suggest that the same challenges faced by learners keep coming up in subsequent years. there is also a lack of research to inform the teaching and learning of pcp at the school level. for example, a database search by the first author of articles in pythagoras from 2016 to 2020 with probability as the keyword turned up only two outcomes, which were also not linked to the information on probability we were looking for. one article (murray, 2017) sought to understand how the grades obtained at school for english and mathematics affect the ‘probability’ of graduation at a university. clearly, the context of said study was different from the current one. the second article (prince & frith, 2017) discussed quantitative literacy of south african school leavers who qualify for higher education. again, this study is not related to pcp. these examples clearly confirm our assertion of little research in the topic of pcp at the secondary level. this study contributes to understanding mathematics teachers’ rating of the pcp learning tasks at grade 12 in terms of the tasks’ cognitive demand levels. it may be that the underperformance in pcp at the national certificate is contributed to by the learning tasks that learners are prepared for on for the national examinations. one of the proposals from the education authorities to try and reverse the poor performance at grade 12 is the suggestion that the teaching and learning of pcp should incorporate multiple representations of tasks. based on the mathematical task theoretical framework (stein & smith, 1998), we associate tasks with high cognitive demands with better prospects to enable learners to master pcp concepts, whereas tasks with low cognitive demand are associated with less chance of offering learners the opportunity to master pcp concepts. the objective is to understand the distribution of the cognitive demand levels of pcp learning tasks that are in the dt introduced earlier in the previous section of this study. the curriculum assessment policy statements (caps) diagnostic reports (dbe, 2018, 2020) also recommend multiple representations of concepts as a strategy for teaching pcp at grade 12. the effective learning of pcp at grade 12 requires many resources. teachers certainly play a vital role in supporting learners (fennema & franke, 1992). focus this case study investigated the grade 12 mathematics teachers’ rating of learning tasks in a pcp chapter in a popular grade 12 mathematics online textbook. teacher ratings were interpreted following stein and smith’s (1998) cognitive demand levels of learning tasks. two research questions guided the study: (1) what is grade 12 mathematics teachers’ rating of the pcp learning tasks in one popular grade 12 mathematics online textbook? (2) from the teachers’ rating of learning tasks, and from the secondary data available on grade 12 learners’ performance in pcp over the years, what can be said about the two pieces of data – might there be a link between the achievement in probability at the national level by grade 12 learners, and mean cognitive demand level of the learning tasks that the learners popularly use to prepare for the national examinations? probability knowledge for teaching the idea of probability is empirical. that is, probability describes what happens in very many trials, and we must observe many trials to pin down a probability. in this article, we use the definition based on the notion of proportion or relative frequency. relative frequency of a score is obtained by dividing the frequency of that score by the total number of scores. similarly, the probability of an experiment yielding a particular result (e.g. a coin toss yielding heads) can be defined as the number of equally likely and mutually exclusive outcomes, divided by the total number of possible, equally likely, and mutually exclusive outcomes. by equally likely, we mean that in the long run each of the possible outcomes will occur with approximately equal frequency (may et al., 1990, p. 179; moore et al., 2013, p. 260). probability knowledge for teaching (pkt) includes content knowledge of probability, and various ways of presenting this content to the learners so that learning takes place (batanero, chernoff, engel, lee, & sanchez, 2016). like mathematical knowledge for teaching, pkt can be divided into probability content knowledge (pck) and probability pedagogical content knowledge (ppck). probability content knowledge requires teachers to have specialised training in probability beyond the content covered in the high school. probability pedagogical content knowledge is knowledge about presenting the concepts to the learners so that learners easily understand them (kazima & adler, 2006; hill et al., 2008). for instance, how does a teacher present to the learners concepts such as variation and randomness, aware that many learners come into a probability class with deterministic (concrete) understanding of the world around them? take, for example, a probability experiment such as tossing a coin. in such an experiment, the outcome is not predictable every time the coin is tossed. it is, thus, not a straightforward case to generalise the probability of events arising from the experiment unless the experiment is repeated very many times. nevertheless, although an individual trial has an unpredictable outcome, there is a predictable pattern of outcomes that will be obtained over a long series of trials (moore et al., 2013, p. 260). hence, for a teacher who may be unaware of the foundation principles of probability, moving their learners from theoretical probability to the concrete results can result in misunderstanding by the learners. mathematical tasks in the mathematical task framework (mtf), a task is defined as a segment of classroom activity that is devoted to the development of a particular mathematical idea (stein & smith, 1998). a task can involve several related problems in each topic in mathematics, in this case pcp. mathematical tasks used in the classroom, or used by learners in their homework, are the foundation for their learning (doyle, 1988; stein & smith, 1998). stein and smith (1998) distinguish three phases through which tasks pass: the first phase includes tasks that are found in the instructional materials such as study guides, printed textbooks, and dts. the second phase includes tasks that are prepared by the teacher, and the third phase involves tasks that the students engage with in the classroom or at home, as reflected in figure 1. figure 1: the mathematics tasks framework. this article will limit the discussion to phase 1 of stein and smith (1998) with a focus on pcp learning tasks in a dt. learning tasks for pcp were chosen for two reasons. first, as noted earlier, pcp is a topic that learners show poor grades in at the matric level (see dbe, 2019, 2020). second, through interaction with some grade 12 mathematics teachers, we learned that the dt is widely used by learners and teachers. so, we wanted to understand the cognitive demand level of pcp tasks in the dt. this study contributes to providing research-based information on pcp at the matric level. mathematical task framework as shown in figure 1, the three phases of mathematical tasks are: (1) curriculum tasks found in the learning materials such as textbooks, and other caps-compliant learning materials, (2) tasks that teachers select and use in their classroom teaching, and (3) tasks that students implement in their day-to-day learning (stein, smith, henningsen, & silver, 2000). the three phases are interrelated. stein et al. (1998) use the mtf to classify tasks into four levels of cognitive demand, namely: (1) memorisation, (2) procedures without connections, (3) procedures with connections, and (4) doing mathematics (see table 1). according to the authors, tasks that promote memorisation and procedures without connections do not present any challenge to the learners since they do not require deep reflection to solve. tasks that involve procedures but require other information that is not obvious in the tasks are classified as procedures with connections. finally, doing mathematics is a level at which tasks are highly cognitively demanding (stein, grover, & henningsen, 1996). based on stein et al.’s (1996) classification, we associate high cognitive demand tasks with tasks that require multiple representations to solve. by ‘doing mathematics’, we mean engaging students with pcp tasks that give them the opportunity to develop their thinking and reasoning skills thus leading them to meaningful mathematical understanding (stein & smith, 1998, p. 13). table 1: cognitive demand levels used in mathematical task framework. cognitive demands of tasks the mtf is used to classify pcp tasks found in a dt in terms of either high or low cognitive demand levels. tasks that are set at a high cognitive demand level require multiple strategies to solve (stein et al., 1996). low cognitive demand tasks occupy learners with reproducing known facts. tasks promoting memorisation (level 1) and procedures without connections (level 2) require less reflection to solve and are categorised as low cognitive demand level (stein et al., 1996). for example: the probability that jabu likes tea is 0.6 and the probability that jabu likes coffee is 0.3. if the probability that jabu likes tea, coffee or both is 0.7, determine the probability that jabu likes tea and coffee. this task does not demand much more than using a formula and substituting in the respective values, then solving for the unknown. let t represent tea, and c represent coffee, p(t) is the probability that jabu likes tea and p(c) is the probability that jabu likes coffee. then, p(t∪c) = p(t) + p(c) – p(t∩c). tasks of high cognitive demand level require some thinking and reasoning to solve (stein et al., 1996). take an example adapted from moore et al. (2013): government data in country z show that 10% of adults are full-time students and that 35% of the adults are age 50 years or older. explain why we cannot conclude that because (0.10) (0.35) = 0.035, therefore about 3.5% of adults are college students aged 50 years or older. one reason is that the two events are not necessarily independent, because not all 10% of adult full-time students are above 50 years of age. moreover, it is reasonable to expect that younger adults are more likely than older adults to be college students. hence, p(college student|over 50 years) < 0.10. this example fits in level 3 or level 4 of stein et al.’s (1996) categorisation of learning tasks, namely procedures with connection, or doing mathematics. probability and counting tasks used in the study the pcp tasks used in this article are obtained from a digital grade 12 mathematics textbook. the textbook is endorsed by the dbe in south africa. the book is freely available to south african users. users are free to download and read the book on their mobile devices or print and read offline. the only restriction is for users to keep the book’s cover, title, contents, and short-codes unchanged. we chose the digital book from among other books for three reasons. first, the book is used by many grade 12 mathematics teachers and learners in south africa, so it is a popular learning resource. second, the book is freely available. third, the book covers all mathematics topics taught in grade 12 in south africa. in this article, we focus only on the topic of pcp. the pcp section is divided into eight sub-topics. for the purposes of this article, we limited our discussion to only four sub-topics, namely: the fundamental counting principles, factorial notation, tasks involving the application of counting principles, and tasks involving application of probability. we picked a total of 48 different learning tasks and asked five senior mathematics teachers at grade 12 to rate the tasks according to the four levels of cognitive demand developed by stein et al. (1996). details of the study design are contained in the methodology section. methodology design this study is a case study taking a descriptive statistical approach. this approach enables us to transform qualitative data into quantifiable form and use it to make sense of the cognitive demand levels of learning tasks in pcp. participants participants in the study are five secondary mathematics and probability teachers, pseudo-named a, b, c, d, and e to ensure anonymity. initially, seven secondary mathematics teachers (six male and one female) were contacted by email to take part in the study. a questionnaire with clear instructions was emailed to all the seven teachers to complete. however, only five teachers, all male, from four secondary schools in two provinces in south africa (gauteng and kwazulu-natal) returned the questionnaire. the five questionnaires were entered into a spreadsheet (table 2) for analysis. table 2: cognitive demand rating of 48 probability and counting principles tasks in the digital textbook by five senior mathematics teachers. data gathering process table 2 has 48 rows and 8 columns. each row represents one task taken from the dt. the first column provides the serial number of the task for identification during analysis. the second column gives the location of the task in the dt. an ‘exercise’ is a collection of tasks. for example, 10.4 (1) represents task 1 found under exercise 10.4 in the dt. it can also be observed from table 2 that exercise 10.4 has a total of seven different tasks. the remaining 47 tasks are presented in a similar format. for instance, exercise 10.5 has a total of three pcp tasks. the next five columns after column 2 are the five participant teachers who independently rated the 48 pcp tasks according to the four cognitive demand levels on a scale from 1 to 4 for each task. the last column shows the mean rating for each task. consistency of measurements the mean cognitive demand ratings (mcdr) by five senior pcp teachers from four schools in two provinces in south africa were received by email by both authors of this article. each teacher rated the 48 tasks independently of the other teachers. the first author entered the original data received from all teachers in table 2 and the second author corroborated the entries with the original submissions. table 2 was again cross-checked by the first author to ensure accuracy and consistency of measurements. in reporting the findings, we have rounded off the cognitive demand ratings of pcp tasks to the nearest digits. data analysis mean cognitive demand rating in this article, we have categorised mcdr 1 and 2 as low, and mcdr 3 and 4 as high (stein et al., 1996). an mcdr is a value (corrected to the nearest whole number) obtained from the five independent ratings of a learning task, divided by the total number of ratings. for instance, the mcdr for task number 48 in table 2, is (corrected to the nearest unit), where xi is a rating of teacher i. table 3 provides the frequency distribution of the mcdr scores obtained from table 2 and figure 2 is the corresponding chart. table 3: frequency table of the mean cognitive demand ratings of probability and counting principles learning tasks. figure 2: bar graph of the mean cognitive demand rating of probability and counting learning tasks in the digital textbook by probability and counting principles teachers. findings from table 3 and figure 2, teachers’ rating of the pcp tasks in the popular online textbook tasks reveal that 8.3% (n = 4) of the total learning tasks sampled in this study comprise facts that only require memory to solve; 12.5% (n = 6) of the tasks are procedures without connection; 64.5% (n = 31) are procedures with some connections; whereas 14.6% (n = 7) are learning tasks rated under doing mathematics, meaning, for example, tasks whose solutions are require learners to explore and understand the nature of mathematical concepts, processes, or relationships. such tasks, according to stein and smith (1998) also demand self-monitoring or self-regulation of one’s own cognitive processes. from table 3, tasks requiring memorisation and procedures without connection together account for approximately 21% of the total number of pcp tasks sampled in this study. according to stein et al.’s (1996) mtf, the above tasks are grouped under low cognitive demand level tasks which occupy learners with reproducing known facts. one hopes that these are not the kinds of task that take much of learners’ time when they prepare for pcp assessments at different school levels. however, until classroom-based studies are conducted, this remains an open question. from table 3, tasks that only need memorisation or procedures without connection are 6% more than tasks rated as doing mathematics. ‘doing mathematics’ is conceptualised as engaging students in the learning tasks (blumenfeld et al., 1991) that give them the opportunity to develop their thinking and reasoning skills (stein & smith, 1998, p. 13). an example of such tasks is: the code to a safe consists of 10 digits chosen from the digits 0 to 9. assuming that none of the digits is repeated, determine the probability of having a code with the first digit even and none of the first three digits is 0. such a task includes procedures with connection, but it also requires some reasoning skills from the learner to solve. table 3 and figure 2 clearly show that the majority (64.6%, n = 31) of the pcp learning tasks in the dt comprise procedures with connections. only about 13% of the learning tasks are rated at the highest cognitive demand level of doing mathematics. nevertheless, if tasks under procedures with connection, and tasks under doing mathematics are combined, it can be concluded that, overall, 79% of the pcp learning tasks in the dt are high-level cognitive demand tasks, and 21% are low-level cognitive demand tasks. discussion this case study focused on and investigated grade 12 senior mathematics teachers’ rating of learning tasks in a pcp chapter of a popular grade 12 online textbook. teacher ratings of tasks were interpreted following stein and smith’s (1998) cognitive demand framework. two research questions guided the study: (1) what is grade 12 mathematics teachers’ rating of the pcp learning tasks in one popular grade 12 mathematics online textbook? (2) from the teachers’ rating, might there be a link between the achievement in probability at the national level by grade 12 learners, and mean cognitive demand level of the learning tasks that the learners popularly use to prepare for the national examinations? on the first question, 65% of the learning tasks in the chapter on pcp were rated by the teachers in this sample as procedures, but with some connections to other concepts and representations, which supported learning. characteristics of such tasks include use of procedures, but after obtaining the numerical solutions, learners are expected to interpret the solutions. other examples include interpreting the concepts of probability that have been represented in a diagram such as a tree diagram. we argue concepts such as the ones in our example engage learners beyond the procedures and can help them to understand underlying concepts in the tasks (stein et al., 2000). however, findings in this study also revealed that teachers in this study rated 79% of the learning tasks in the dt as having high cognitive demand. only 21% of tasks in the dt the teachers rated as having low cognitive demand. if the teachers’ seemingly favourable rating of the learning tasks is true, the question that remains unanswered is: what explains learners’ general underperformance in the pcp topic in the grade 12 national examination? under the new caps syllabus (dbe, 2011), probability has been examined in paper 1 at grade 12 since 2014 and contributes about 18% of the total marks in paper 1 (mutara & makonye, 2014). however, since 2014, learners have performed poorly in this topic in matric examinations (see dbe, 2020). in fact, the mean percentage pass in probability is 34.7% for a period of seven years, from 2014 to 2020, respectively. in 2020 the percentage pass was 18% (see table 4), the lowest since 2014 when the topic was first examined, and the pattern does not show signs of improvement. table 4: grade 12 learners’ mean pass rate in probability (2014–2020). this leads us to the second research question, which is: what are the implications of the teacher rating of the learning tasks in the online mathematics textbook on improved performance in pcp? drawing on the secondary data from the caps document, and from our data from the teachers’ ratings of tasks, we can only offer two reflections on this question. first, the exceptionally low performance in probability in 2020 by grade 12 learners partly speaks to the learning difficulty that learners could have faced during the covid-19 closure of schools, but this observation has no direct link to our current data on teacher rating. the apparently favourable teacher rating of the online tasks is probably an indication of the confidence that teachers had (or still have) in the tasks. however, the learner performance as shown in table 4 clearly shows that there is no link between the teacher ratings of tasks and learner performance. looking at the teacher ratings and learner performance in pcp topic over the years, our study draws attention to the fact that there is a bigger problem in pcp that needs concerted effort to solve. the question of underperformance in pcp, our data have shown, cannot be fully explained by the hardships imposed on teaching during the covid-19 lockdown. the issue must be about how probability is taught, how much time is allocated to it, who teaches the topic and at what time of the curriculum calendar year it is taught as well as the resources available to both teachers and learners. all these questions remain to be followed up in future research studies. limitations to the study the study was carried out during the restrictions due to covid-19, where physical contacts were restricted as recommended by the health authorities to keep individuals and the public safe from contracting the disease. communication during the data gathering process depended mainly on email with an attached questionnaire for the teachers, and follow-up phone calls. we contacted seven pcp teachers, but in the end only five teachers returned the questionnaire. although only five teachers responded, percentage wise, it still represented a reasonable percentage considering that our initial target was seven senior teachers of probability at grade 12. we were not able to observe the teaching of pcp in the classrooms for the same reason explained above. finally, this study focused only on the learning tasks that are available to learners in the online textbook, so we missed the teaching tasks and the nuances that the teachers incorporate in their actual lessons. obviously, we also missed observing the tasks that learners implement in their learning in the classrooms (stein et al., 1996). future empirical studies should consider these uncovered areas with respect to pcp. conclusion and recommendations the study opened our eyes to the challenges in the teaching of probability that we only have been hearing about but have not investigated for ourselves. this study suggests that teachers’ rating of tasks does not count until reflected in learner output in terms of learners’ performance in the tasks. it can also be argued that learners’ rating of learning tasks should precede the teachers’ rating. in other words, teachers should rely on, and respond to, the learners’ rating of learning tasks. one direct indicator is the learner scores in the tasks that are assigned to them. we recommend empirical classroom-based studies that support teachers with different ways of teaching pcp. one possibility is writing pcp teaching support materials that complement the online materials and the caps-recommended materials, focusing on the understanding of meanings in pcp, and their applications. acknowledgements g.e. would like to thank wits university, faculty of humanities, for the small research grant for this report. competing interests the authors have declared that no competing interests exist. authors’ contributions s.m. compiled the data and wrote the first draft. g.e. added more research information on the draft, wrote and edited the final manuscript. ethical considerations this article followed all ethical standards for research without direct contact with human or animal subjects. funding information g.e. received project fund number gekl020 from the faculty of humanities research grant. data availability data sharing is not applicable as no new data were created. disclaimer the views and opinions expressed in this article are those of the authors and do not necessarily reflect the official policy or position of any affiliated agency of the authors. references batanero, c., chernoff, e.j., engel, j., lee, h.s., & sanchez, e. 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(2020). learner performance in the 2009 to 2014 final grade 12 mathematics examination: a quantile regression approach. pythagoras, 41(1), a545. https://doi.org/10.4102/pythagoras.v41i1.545 article information authors: vimolan mudaly1 deborah moore-russo2 affiliations: 1faculty of education, school of science, mathematics and technology education, university of kwazulu-natal, south africa 2department of learning and instruction, graduate school of education, university at buffalo, the state university of new york, united states correspondence to: vimolan mudaly email: mudalyv@ukzn.ac.za postal address: private bag x03, ashwood 3605, south africa dates: received: 07 apr. 2011 accepted: 08 aug. 2011 published: 15 sept. 2011 how to cite this article: mudaly, v., & moore-russo, d. (2011). south african teachers’ conceptualisations of gradient: a study of historically disadvantaged teachers in an advanced certificate in education programme. pythagoras, 32(1), art. #25, 8 pages. http://dx.doi.org/10.4102 /pythagoras.v32i1.25 copyright notice: © 2011. the authors. licensee: aosis openjournals. this work is licensed under the creative commons attribution license. issn: 1012-2346 (print) issn: 2223-7895 (online) south african teachers’ conceptualisations of gradient: a study of historically disadvantaged teachers in an advanced certificate in education programme in this original research... open access • abstract • introduction • literature review • theoretical framework    • methodology       • participants and data collection       • data analysis       • reliability and external validity       • internal validity       • ethical considerations • results • discussion    • limitations and future work    • implications for professional development • acknowledgements    • competing interests    • authors’ contributions • references • appendix abstract (back to top) this study looked at how a group of south african secondary school mathematics teachers regarded the concept of gradient (slope). results are reported from nine free-response items on a paper-and-pencil test administered to practising teachers who were pursuing qualifications to teach grades 10–12 mathematics through an advanced certificate in education (ace) programme. the findings suggest that teachers’ understanding of gradient varies greatly. a number of teachers inthe study demonstrated very little to no understanding of this important concept, whilst others demonstrated a strong understanding of gradient and were able to conceptualise it in many different ways. implications for teacher professional development are considered. introduction (back to top) adoption of a new curriculum in mathematics in south africa (department of education, 2003) brought with it the need for more appropriately qualified teachers to teach grades 10–12 mathematics. many of the mathematics teachers in the areas that serve historically disadvantaged communities are themselves members of thesame disadvantaged groups. it has been difficult for these teachers to acquire the qualifications needed to teach grades 10–12 mathematics. whilst policy changes have had less impact on private schools, the shortage of qualified mathematics teachers in public schools, particularly those in rural areas, has been labelled ‘critical’ by adler and davis (2006), who provide a succinct overview of the history of teacher education in south africa. more importantly, though, they emphasise theidea that the majority of practising teachers from historically disadvantaged groups face the challenge of having training in mathematics that constituted a three-year certification, and entered their careers with limited mathematics content knowledge. this has now created an urgent need for these teachers to become re-certificated in mathematics content knowledge. a large university in kwazulu-natal in south africa recently implemented an advanced certificate of education (ace) programme to address the shortage of qualified mathematics teachers in its region. this ace programme provides an alternate way for practising teachers, especially those serving historicallydisadvantaged communities, to obtain minimal qualifications to teach mathematics in grades 10–12. the intervention was specifically created in order to help prepare underand unqualified teachers currently teaching mathematics to meet the minimal qualifications for teaching mathematics. the ace programme offers flexible delivery at multiple open learning centres and is structured to reach some of the most remote rural and disadvantaged communities. however, little is known about the mathematical knowledge of teachers enrolled in the ace programme. is their mathematical knowledge robust enough to allow them to present the key, fundamental mathematical ideas as a ‘unified body of knowledge’ (ma, 1999, p. 122)? how deeply do these teachers understand the keyconcepts in grades 10–12 mathematics? the more that is known about how teachers in ace programmes conceptualise key concepts, the better modules in the ace programme can be adapted to provideeducational experiences that will be of maximum benefit to those teachers. it is crucial to consider teachers’ current knowledge in order to connect to and build on it, so that teachers are able to develop a strong understanding of the content represented by the concept itself, utilise various representations of the concept and recognise their particular affordances, and make curricular connections to the content both within and outside mathematics. for these reasons, this study focuses onteachers’ understanding of gradient, a key concept that cuts across many areas in mathematics, including those most commonly addressed in grades 10–12. more specifically, the study described in this article was driven by the following research question: ‘how do practising teachers, who teach historically disadvantaged students and who are pursuing qualifications to teach mathematics in grades 10–12, understand the concept of gradient?’ literature review (back to top) research provides evidence of students’ weak covariational reasoning (i.e. the ability to reason simultaneously about how one quantity changes as another relatedquantity also changes), including difficulties with the concepts of slope (or gradient, as the concept is commonly referred to in south africa) and rate of change (barr, 1981; carlson, jacobs, coe, larson & hsu, 2002; orton, 1984; stump, 2001a, 2001b; teuscher & reys, 2010). there has been particular evidence of students’ inability to make connections between various representations of these concepts. stump (2001b) found that students held varying views of gradient as an angle, aformula, rise over run or steepness, and that students did not make connections between rate of change and gradient. some studies (moore-russo, conner & rugg, 2011; stump, 1999) have helped identify and have analysed the many different ways in which the concept of gradient can be conceptualised (stanton & moore-russo, in press). moore-russo and colleagues (moore-russo et al., 2011; stanton & moore-russo, in press) have suggested 11 conceptualisations of gradient, basedon their own research and the earlier work of stump (1999, 2001a, 2001b), which are summarised in table 1. in this study the same 11 conceptualisations were used to determine south african teachers’ understanding of gradient. although very little research has investigated teachers’ understanding of gradient, coe (2007) reported that secondary mathematics teachers showed difficulty in working with average rates of change, and could not explain the use of division in the formula for gradient in the algebraic ratio conceptualisation of gradient. in her study, stump (1999) found the most common conceptualisation of gradient amongst secondary teachers to be a geometric ratio (the ratio of the rise of a linearfunction to its run). in the same study, stump also reported that teachers expressed concern about their students’ understanding of gradient; however, the teachers focused on students’ difficulties in the procedures for determining gradient rather than their conceptual notions of gradient. this prompted the suggestion that ‘… both pre-service and in-service mathematics teachers need opportunities to examine the concept of slope … [and] to construct connections amongst its variousrepresentations’ (stump, 1999, p. 142). table 1: eleven conceptualisations of gradient. theoretical framework (back to top) the mathematical knowledge for teaching framework introduced by ball, thames and phelps (2008) served as the underpinning theory for this study. shulman (1986) suggested three categories of teacher knowledge: content knowledge, pedagogical content knowledge, and curricular knowledge. elaborating on shulman’s seminal work, ball et al. (2008) divide mathematical knowledge for teaching into subject matter knowledge (smk) and pedagogical content knowledge (pck). ball et al. (2008, p. 402) have labelled smk as the ‘relatively uncharted arena of mathematical knowledge necessary for teaching that is not intertwined with the knowledge of pedagogy, students, curriculum, or other non-content domains’. their theory highlights the fundamental components of smk by further dividing itinto common content knowledge (cck) and specialised content knowledge (sck), as well as provisionally suggesting a third domain, horizon content knowledge (sometimes simply referred to as horizon knowledge). teachers’ cck provides the foundations upon which they develop their sck and horizon content knowledge. cck is outlined as the knowledge and skills held by educated adults that are essential and used in a wide variety of settings. sck represents the deep, flexible, nuanced understanding of mathematics that is uniquely related to teaching; it includes, amongst other things, knowing how concepts are represented, related, developed, and validated. it allows teachers to ‘mediatestudents’ ideas, make choices about representations of content, modify curriculum materials, and the like’ (ball & bass, 2000, p. 97). horizon content knowledge relates to understanding how mathematical knowledge is related and connected to more advanced mathematical concepts. if ace programmes are to help teachers develop their mathematical knowledge for teaching, then these programmes should be aware for teachers’ cck. the research reported in this study looked at the cck of a single concept – gradient – in a group of historically disadvantaged south african mathematics teachers enrolled in an ace programme to pursue qualifications to teach mathematics in grades 10–12. the study focused specifically on the teachers’ conceptualisations ofgradient using the 11 conceptualisations suggested by moore-russo et al. (2011) to frame the analysis of the study. methodology participants and data collection data were collected from 251 practising teachers from eight different project sites in kwazulu-natal during the summer of 2010. the data come from teachers’responses to a paper-and-pencil pre-test that was administered prior to the concept of gradient being addressed in the ace programme. all of the participant teachers were from historically disadvantaged communities, and all were teaching in schools whose student populations were historicallydisadvantaged. some were teaching grades 10–12 mathematics without appropriate qualifications; others were teaching mathematics at grade 9 and below. a third, smaller group was teaching other subject areas at the time; they were using the programme as a way to retrain to become mathematics teachers. enrolment in this ace programme had no cost implications for the teachers since funding was provided by the provincial department of education. nine of the 27 items on the pre-test were free-response and addressed the concept of gradient; only these items were used in the analysis for this study. the teachers were instructed to show all working details and to provide an explanation for how they arrived at the answer for each of the nine items. the nine items (provided inthe appendix), were purposely composed so as to elicit responses from the teachers that made use of a variety of the 11 conceptualisations of gradient (table 1). data analysis the unit of analysis was a teacher’s response to a single pre-test item. since 251 teachers answered nine items, there were exactly 2259 responses (304 of which were blank) that were analysed. data analysis began with creating task-specific rubrics for each item. each rubric was customised to address the item, but was generally based on the scoring rubric provided in table 2. the rubric for each item was based on a scale of 0 to 2. for each item the score assigned considered not only the finalanswer but also the teacher’s entire response, including all writing, equations, tables and drawings used in the explanation related to that item. reliability and external validity the test items were carefully selected after much deliberation between the researchers, and all ambiguity and unnecessary distracters were removed. the researchers ensured that the questions chosen were ones that these practising teachers would have encountered in their school learning and in their teaching. the language used was sufficiently basic so that most teachers would be able to understand the words used. the participants came from various parts of kwazulu-natal and hence we could postulate that these teachers represented a close approximation of how other teachers with similar qualifications and backgrounds would respond to the selected questions. nonetheless, no broad generalisations are being made. internal validity both members of the research team scored all teacher responses to each item independently. proportion agreement for each of the nine items was above 0.97. thecohen kappa statistic for each item was well above 0.80, which is considered ‘almost perfect’ agreement (landis & koch, 1977, p. 165). the two independent scorers reached consensus by means of discussion for each response they had coded differently. after the items were scored, the research team reviewed the data to find any patterns in teachers’ responses. ethical considerations on arrival at the lecture venue all teachers were informed by the lecturing staff that the test was intended for two purposes: firstly, it was to be used as a meansof establishing their prior knowledge so that we could measure their progress during the course of the module, and secondly, their responses were to be used in research, but their anonymity was guaranteed. all participating teachers signed a document acknowledging the fact that they were aware that their responses would be used for research purposes. table 2: general scoring rubric for teacher responses. results (back to top) teachers’ total scores on the nine items ranged from the minimum possible score of 0 to the maximum possible score of 18. figure 1 displays the distribution of thetotal scores assigned to teachers’ responses, and shows the wide distribution for the teachers’ total scores. the mean total score per teacher was 9.66 (sd 5.16). table 3 displays the frequency of scores that were assigned to teachers’ responses on each of the nine items. the mean score for all items was 1.07. those with the three highest mean scores were item 6 (1.42), item 1 (1.26) and item 4 (1.22), whilst those with the three lowest mean scores were item 3 (0.92), item 8 (0.84) and item2 (0.73). before discussing teachers’ responses to the nine items, it is necessary to outline what these items entailed. the preamble to the first three questions required theteachers to understand a contextualised problem, which stated that to take out a text-only advertisement in the newspaper cost r10.00 plus an additional r1.00 for each word. item 1 asked that the teachers should write an equation that represented the cost of the advertisement (c) in terms of the number of words (w). item 2 asked the teachers to draw a graph of the situation, and item 3 required them to write down the gradient of the graph. the rest of the items had different graphsdrawn and the teachers had to respond to particular questions based on them. although item 1, item 2 and item 3 were based on the same situation, the results for these items varied greatly: whilst 141 (or 56%) of the teachers responded with thecorrect formula for the situation in item 1, only 63 (or 25%) of the teachers were able to draw the correct corresponding graph for item 2. in fact, only six teachers scored higher on item 2 than on item 1. moreover, 27 of the teachers who drew a graph for item 2 provided graphs that were non-linear. since teachers could have used the situation – the equation (from item 1) or the graph (from item 2) – to find the gradient in item 3, it might seem reasonable to consider that more teachers would have been able to determine the correct gradient than responded correctly to either item 1 or item 2. however, only 85 (or 34%) of the teachers responded with the correct value for the gradient, and 64 teachers (26%) left this question blank (as opposed to 13 and 38 teachers who did not offer any responses for item 1 and item 2 respectively). scores for item 1 and item 2 would have been even lower had the discrete nature of the situation been taken into account in assigning scores. item 1 asked for an equation, but did not ask for the domain of the function, so responses that included the correct equation were given a score of 2, even though only one teacher included in his explanation that the domain should be limited to positive integers. teachers who drew a continuous straight-line graph that represented the equationin item 2 were also given a score of 2, since the scoring was meant to measure teachers’ understanding of slope, not their understanding of domain. only seven teachers drew discrete points in a linear manner that represented the situation with its domain limited to positive integers. figure 2 shows two examples of teacher responses for item 1 and item 2 that were scored as 1; these teachers demonstrated just a partial understanding of the concepts being tested. teachers’ responses provided evidence as to the conceptualisations of gradient that they held. when a teacher’s explanation included reasoning that related to a conceptualisation of gradient, it was noted. table 4 provides the frequency distribution for each of the 11 conceptualisations of gradient evidenced in the teachers’ responses; however, it is possible that teachers held more conceptualisations than were evidenced in their responses. the most common conceptualisations ofgradient found were parametric coefficient (for 153 teachers) and behaviour indicator (for 135 teachers), whilst the least noted were real-world situation (for 17 teachers) and physical property (for 9 teachers). the low numbers for real-world situation could be misleading since 56 of the 251 teachers responded correctly to item 1, item 2 and item 3, all of which related to a real-world situation; however, this may not be the case considering that only eight teachers considered the discretenature of the domain for the situation. table 5 displays the number of teachers who held varying amounts of conceptualisations of gradient. the mean number of conceptualisations of gradient evidencedas being held by teachers was 3.15 (sd = 2.02). as previously stated, it is possible that teachers held more conceptualisations than were evidenced in their responses. teachers’ responses also provided evidence that they held a number of misconceptions. the most common misconception (found in 29 of the teachers’ responses,and all occurring for item 5 and item 8) was that a line that goes through the origin must have a gradient of zero. the second most common misconception (which occurred in 26, or over 10% of the responses and was found exclusively on item 8), involved teachers ignoring the units marked on the graph and assigning a gradient of 2 to an increasing linear function whose gradient was 1. a related misconception occurred in 22 responses to various problems, when teachers created markingswith specific values on axes when none were given, leading them to various incorrect assumptions. a misconception that occurred in 20 responses (all on item 6) was that horizontal lines have no gradient (instead of a gradient of 0). in 19 responses, all on item 2,nonconventional labelling of units was used on the axes such that the axes were made to cross at a point other than (0, 0). fifteen teachers’ responses (on various items) confused the gradient with the y intercept in a linear equation. all other errors occurred in fewer than 15 responses. table 3: frequency distribution of scores per item. table 4: frequency distribution for teachers’ conceptualisations of gradient. table 5: frequency distribution for teachers’ varying amounts of conceptualisations of gradient. figure 1: distribution of teachers’ total scores. figure 2: examples of responses to item 1 and item 2 which were scored as 1. discussion (back to top) the findings suggest that those teachers participating in the study varied greatly in their understanding of gradient. evidence to support this comes from three findings: the wide range of total scores assigned to the teachers; the fact that on eight of the nine items, teachers’ responses were assigned scores of 0 or 2 morefrequently than scores of 1; and the wide range in the number of conceptualisations of gradient evidenced in teachers’ responses. in particular, the current study sheds light on the dire situation that faces some of the historically disadvantaged students who have teachers with little or nounderstanding of gradient. of all 2259 of the teachers’ responses, 818 (36%) received scores of 0, and only 143 of the 251 teachers were able to obtain 9 or more out of 18 points (50%) as a total score on the nine items. even though the items may not have elicited all possible conceptualisations of gradient from all teachers, it is surprising to find that 101 of the 251 teachers (40%) evidenced only two or fewer conceptualisations on all nine items. in contrast, some of the teachers displayed a strong understanding of gradient. of the 251 teachers, 42 missed two or fewer points out of the 18 possible points for the teachers’ total score. also, 34 of the teachers demonstrated that they held at least six different conceptualisations for gradient. the varied scores for item 1, item 2 and item 3 might suggest that teachers do not make the connections between various representations of functions. teachers who responded to item 1 with the correct equation and with a correct graph of that equation in item 2 often did not show evidence of relating either the equation or the graph to the situational context. an example here is the teacher who drew a graph with a positive gradient but declared that the gradient is negative (figure 3). thismay be indicative of the fact that in general teachers do not engage in the iterative process between the contextual situation and its mathematisation. results suggest that transforming from a written situation to an equation was easier for the teachers than transforming to a graph or determining the gradient. manyteachers were able to determine the equation but were not able to identify the gradient of a given situation. even though the parametric coefficient was one of the most noted conceptualisations in teachers’ responses, this suggests that it is not a conceptualisation held by all teachers. figure 3: example of a teacher’s response. limitations and future work whilst this study provides valuable data, its limitations should be noted. firstly, item selection might have impacted the types of conceptualisations evidenced by the teachers in their responses. one can only ponder how the scores might have differed if item 1, item 2 and item 3 had not been related to the same situation, or ifresponses to a linear graph with a negative slope had been included that did not have the origin as its y intercept (as did item 5). the disregarding of the units on the axes, coupled with the inappropriate and inaccurate overuse of the behaviour indicator conceptualisation, might indicate twonotions that both merit further study. the first notion is that teachers’ struggles with graphing on the cartesian plane, as noted in item 2 and item 8, impact their understanding of gradient. a large number of the misconceptions that were noted related to the axes, including their erroneously labelling and disregarding the units on the axes. the second notion is that the behaviour indicator conceptualisation is much stronger than, and possibly interferes with, other conceptualisationslike geometric ratio and trigonometric conception. this notion is supported by the large number of teachers who simply looked at the graph without using the grid that was provided on item 8. it is also supported by the responses to item 9, where teachers often looked at the graph without considering the angle and declared that the answer to the question was y = x. in general, this study provides a much-needed first look at teachers’ conceptualisations of gradient. it would be interesting to follow this study with another that includes items written to intentionally elicit conceptualisations by the teachers that were less noted in their responses in this study. perhaps the best methods to usefor such future studies would be more qualitative approaches, possibly using interviews. the idea of a key mathematical concept having multiple conceptualisations can be applied beyond this study. it would be appropriate and useful for themathematics education research community to identify other key concepts and consider their various conceptualisations in future work. implications for professional development gradient is a concept in the secondary mathematics curriculum of most countries. paradoxically, this important concept is ‘well known but not well understood’ (moore-russo et al., 2011, p. 3). this study adds to the research literature by providing additional insight into common misconceptions that seem to impact on teachers’ understanding of gradient. this has implications for the professional development of teachers that extend past the concept of gradient. teachers shouldengage in activities that require them to transform between representations of the same mathematical idea, make connections between various representations of the same idea, and consider the basic principles of the cartesian coordinate system (e.g. the orthogonal intersection of real number lines at the zero value of each, the independent nature of the scale of the two axes). one of the greatest challenges facing those involved in professional development is how to deal with teachers who enter programmes with little to no cck. whilst not the focus of this article, the methods employed in the instructional setting suggest that pre-test data can be collected to inform instructional decisions. forexample, results of the pre-test helped identify those teachers who had strong (as well as weak) understandings of gradient. the pre-test results also helped the instructional team to realise some of the common misconceptions that teachers held. during the second tutor training session, tutors were specifically instructed to concentrate on aspects that we thought were problematic. an example of what we thought could be a misconception is illustrated in figure 4: the teacher hadwritten the correct equation (c = w + 10), but drew the graph as c = w. it is perhaps the case that the teacher believed that because the gradient was equal to 1, the graph would pass through the origin. on a final note, another challenge is how to handle professional development when some teachers have weak cck whilst others seem to have strong cck. teacher educators need to develop differentiated activities in which the teachers would engage that are appropriate and challenging for both groups. to do so, teacher educators should consider the conceptualisations that most teachers hold for a particular mathematical concept, help teachers recognise that concepts can beconceptualised in various ways, challenge all teachers to develop some of their weaker conceptualisations of a concept better, and help teachers see the connections amongst and benefits related to the various conceptualisations of a concept. figure 4: an example of a teacher’s misconception. acknowledgements (back to top) we acknowledge funding from a grant from the united states agency for international development, administered through the non-governmental organisation higher education for development for research on the different modules in the ace certification programme. no specific grant was obtained for this particulararticle. competing interests we declare that we have no financial or personal relationship(s) which may have inappropriately influenced us in writing this article. authors’ contributions d.m-r. had originally conceptualised the items for the test, and v.m. made further suggestions. administering of the tests was managed by v.m., and the scoring of data was done separately then jointly by both d.m-r. and v.m. the manuscript was written with contributions from both authors, and editorial corrections were also completed by both authors. references (back to top) adler, j., & davis, z. 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(2010). slope, rate of change, and steepness: do students understand these concepts? mathematics teacher, 103(7), 519−524. appendix (back to top) appendix image contents.indd http://www.pythagoras.org.za open access table of contents original research novice mentors versus mentees: mentoring experiences in mathematics at general education and training phase ernest mahofa, stanley a. adendorff pythagoras | vol 43, no 1 | a641 | 19 august 2022 original research grade 10 teachers’ example selection, sequencing and variation during functions lessons hlamulo w. mbhiza pythagoras | vol 43, no 1 | a696 | 30 august 2022 original research the intersect of early numeracy, vocabulary, executive functions and logical reasoning in grade r hanrie s. bezuidenhout, elizabeth henning pythagoras | vol 43, no 1 | a646 | 15 september 2022 original research investigating the cognitive demand levels in probability and counting principles learning tasks from an online mathematics textbook george ekol, simphiwe mlotshwa pythagoras | vol 43, no 1 | a677 | 16 september 2022 original research the sequencing and pairing of examples in the midst of sameness and difference: opening opportunities to learn vasen pillay, jill adler, ulla runesson kempe pythagoras | vol 43, no 1 | a667 | 01 november 2022 original research prospective teachers’ cognitive engagement during virtual teaching using geogebra and desmos solomon a. tesfamicael pythagoras | vol 43, no 1 | a691 | 04 november 2022 reviewer acknowledgement pythagoras | vol 43, no 1 | a724 | 19 december 2022 64 74 85 93 101 111 126 editorial research resilience in the covid era rajendran govender pythagoras | vol 43, no 1 | a714 | 15 december 2022 review article complexities of translating mathematics tasks from english to learners’ home languages nkosinathi mpalami pythagoras | vol 43, no 1 | a560 | 18 january 2022 review article language and multilingualism in the teaching and learning of mathematics in south africa: a review of literature in pythagoras from 1994 to 2021 kathryn mclachlan, anthony a. essien pythagoras | vol 43, no 1 | a669 | 22 july 2022 review article exploring the relationship between commognition and the van hiele theory for studying problem-solving discourse in euclidean geometry education sfiso c. mahlaba, vimolan mudaly pythagoras | vol 43, no 1 | a659 | 29 july 2022 original research exploring low-tech opportunities for higher education mathematics lecturers in an emergency techno-response pedagogy antonia makina, langton kadzere pythagoras | vol 43, no 1 | a644 | 22 april 2022 original research weathering the storm: learning strategies that promote mathematical resilience vuyisile l. khumalo, surette van staden, marien a. graham pythagoras | vol 43, no 1 | a655 | 25 july 2022 original research effective communication of learning intentions and success criteria in the mathematics classroom: merlo pedagogy for senior phase south african schools lydia o. adesanya, marien a. graham pythagoras | vol 43, no 1 | a666 | 16 august 2022 1 2 8 19 30 40 51 page i of i table of contents vol 43, no 1 (2022) issn: 1012-2346 (print) | issn: 2223-7895 (online)pythagoras article information authors: percy sepeng1,2 paul webb2 affiliations: 1marang centre for mathematics and science education, university of the witwatersrand, south africa2centre for educational research, technology and innovation, nelson mandela metropolitan university, south africa correspondence to: percy sepeng postal address: private bag x3, wits 2050, south africa dates: received: 10 oct. 2011 accepted: 22 june 2012 published: 15 aug. 2012 how to cite this article: sepeng, p., & webb, p. (2012). exploring mathematical discussion in word problem-solving. pythagoras, 33(1), art. #60, 8 pages. http://dx.doi.org/10.4102/ pythagoras.v33i1.60 copyright notice: © 2012. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. exploring mathematical discussion in word problem-solving in this original research... open access • abstract • introduction    • solving word problems    • discussion in mathematics classrooms • method    • sample in this study    • research design    • design type and the intervention    • he problem-solving tasks    • coding of the tests       • overview of reliability and validity in this study       • ethical considerations       • statistical analysis of data • findings    • effect of intervention on problem solving    • practical significance of the differences    • sense-making (or realistic considerations) of word problem solving (ps tasks)    • matched-pairs t-tests • discussion    • problem solving    • sense-making • conclusion • acknowledgements    • competing interests    • authors’ contributions • references abstract top ↑ this study explored whether discussion as a teaching strategy in mathematics classrooms could have positive gains in improving learners’ problem-solving performance, as well as their ability to make sense of real wor(l)d problems. this article discusses the partial findings of a bigger study that used a pre-test intervention or post-test mixed-method design and utilised both quantitative and qualitative data. analysis of the data generated from both pre-tests and post-tests suggests that the intervention strategy significantly improved the experimental group’s problem-solving skills and sense-making performance. the statistical results illustrate that the experimental group performed significantly better than the comparison group in the post-test. the main finding of this study is that in classrooms of experimental schools in which discussion technique was successfully implemented, there was a statistically significant improvement in the learners’ competence in solving word problems. introduction top ↑ problem solving and integrated assessment are seen as the cornerstones of school mathematics and the principles and standards for school mathematics (national council of teachers of mathematics [nctm], 2000) called for mathematics instruction and assessment to focus more on conceptual understanding than on procedural knowledge or rule-driven computation (hamilton, 2004; kilpatrick, swafford & findell, 2001). major arguments for including word problems in the school mathematics curriculum have always been their potential ability to promote realistic mathematical modelling and problem-solving. solving word problems also help learners to develop the skill of knowing when and how to apply classroom mathematical knowledge as well as everyday life-knowledge when solving problems. in this article we therefore argue that there are multiple benefits and good outcomes when learners participate in mathematical discussion in the classroom. this article begins with a discussion of the current debates on using discussion as a strategy for the teaching and learning of mathematics in general. the study draws largely on the results of a research project conducted in six schools in the eastern cape province of south africa (sepeng, 2010). much of the work in this study relies both theoretically and methodologically on notions of classroom mathematics discourse and mathematical modelling. the main argument in this article is that careful use of discussion as a teaching strategy in mathematics classrooms appears to have positive gains in improving learners’ problem solving performance in word problems. the hypothesis for the study was that the introduction of mathematical discussion in the teaching and learning of word problems would improve and enhance learners’ problem solving performance and the ability to make sense of real wor(l)d problem-solving. the purpose of this article is therefore to demonstrate how the results of the study were found to support this hypothesis. solving word problems word problems have been defined differently in different studies. for the purpose of this study, the definition provided by verschaffel, greer and de corte (2000) is used. these researchers define word problems as ‘textual descriptions of situations assumed to be comprehensive to the reader, within which mathematical questions can be contextualised’ (p. ix). they also stress that word problems ‘provide, in convenient form, a possible link between the abstractions of pure mathematics and its applications to the real-world phenomena’ (p. ix). according to palm (2009) mathematical word problems include pure mathematical tasks ‘dressed up’ in a real-world situation that require students to ‘undress’ these tasks and solve them (p. 60).a further methodological issue, which socio-cultural approaches have yet to address satisfactorily, arises from the increasingly multicultural nature of mathematics classrooms. students’ interpretations of mathematics classroom interaction relate in part to their different social, cultural and linguistic backgrounds. when classroom interaction is analysed, therefore, some way of taking account of this diversity needs to be found, otherwise there is the risk that a single cultural perspective, that of the researcher, will be imposed. discursive psychology has the potential to address some of the above-mentioned issues. discussion in mathematics classrooms the use of discussion as a tool to increase reasoning has gained emphasis in classrooms worldwide, as earlier reports had predicted (yore, bisanz & hand, 2003). discussion, however, requires scaffolding and structure in order to support learning (norris & phillips, 2003). wood, williams and mcneal (2006) found variations in students’ ways of seeing and reasoning, and these were attributed to the particular differences established in classrooms early in the year pertaining to when and how to contribute to mathematical discussions and what to do as a listener. their conclusions are consistent with the findings reported by several other researchers (e.g. dekker & elshout-mohr, 2004; ding, li, piccolo & kulm, 2007; gillies & boyle, 2006; webb, nemer & ing, 2006), who also suggested that participation obligations put boundaries around the opportunities for students to share their ideas and to engage in mathematical practices. when they make a difference through classroom discourse, teachers shift students’ cognitive attention toward making sense of their mathematical experiences, rather than limiting their focus to procedural rules. in doing so, students become less engaged in solutions to problems than in the reasoning and thinking that lead to those solutions (yackel & cobb, 1996).students develop a mathematical disposition through the patterns of interaction and discourse created in the classroom and the process of ascribing meaningfulness to one another’s attempts to make sense of the world. learning about other ways to think about ideas, to reflect, and to clarify and modify thinking is fundamental to moving learning forward. carpenter, franke and levi (2003) maintain that the very nature of mathematics presupposes that students cannot learn mathematics with understanding and without engaging in discussion. however, more talk in the classrooms does not necessarily enhance student understanding. better understanding is dependent on particular pedagogical approaches, purposefully focused on developing a discourse culture that elicits clarification and produces consensus within the classroom community. a variety of situations may arise in which the outcomes are not fully realised. for example, a number of studies have reported that some students appear to thrive more than others in whole-class discussions. in their respective research, baxter, woodward and olson (2001) and ball (1993) found that highly articulate students tend to dominate classroom discussions. low academic achievers usually remain passive, and when they do participate visibly, their contributions are comparatively weaker, and their ideas sometimes muddled. nevertheless, pedagogical practices that create opportunities for students to explain their thinking and to engage fully in dialogue have been reported in research undertaken by steinberg, empson and carpenter (2004). in a study which was part of their cognitively guided instruction project, they found that classroom discussion was central to a sustained change in students’ conceptual understanding. method top ↑ sample in this study the sample consisted of grade 9 learners (n = 176) and their teachers in six township secondary schools, four of which were experimental schools (n = 107) and two of which were comparison schools (n = 69) (where no intervention took place). the six schools chosen were a convenient sample of a cluster of similar schools in port elizabeth. all the schools were functional (as opposed to dysfunctional – which is the case in many instances in south africa), had similar characteristics in their approach to teaching and learning contexts and were public and previously marginalised schools. the schools drew their learners from communities of low economic status. the two comparison schools were chosen randomly within the group identified. research design in this study we used a pre-test – intervention – post-test design. we investigated the situation in terms of the problem-solving performance of grade 9 learners with english as their second language, using a pre-test to establish how the learners solved mathematics word problems and what problems they might have mathematically. then we wanted to find out if introducing discussion into the classroom as part of the learning experience had any effect on learners’ problem-solving performances and/or strategies. a post-test, which was exactly the same as the pre-tests, investigated if there were any changes. we also used this test to find explanations or reasons for any changes in learners’ sense-making and solutions of real wor(l)d problems. in the next section we present the intervention strategy used to promote discussion in the teaching and learning of word problems. design type and the intervention in this study an intervention strategy was used to investigate the effect of discussion on the problem-solving performance of grade 9 second language learners. the intervention took place over a period of six weeks and focused on the use of writing and mathematical discussion to solve word problems. the intervention was implemented in the experimental group and its aim was to assist learners not only to identify problem situations that were problematic from a realistic point of view, but also to consider the (in)appropriateness of applying mathematical operations directly as their solutions. moreover, the teachers of the experimental schools were introduced to a typical collaborative learning context(s) where the teacher posed a task (or a problem) and the learners, after sufficient time to complete the task, engaged in a discussion of the solution methods and/or strategies that they had developed in small groups. the whole class discussion continued for, at most, 15 minutes before another short segment (three to five minutes) of group task. this cycle was repeated five to six times in a double period of about 90 minutes. during the small group interaction learners had to develop reasons to support their thinking and/or think about some relevant issue(s) or question(s) instead of solving a specific mathematics question. because of the continual emphasis on both justification and reasoning, whole classroom discussions resulted in the emergence of key concepts in wor(l)d problem solving within the contexts of realistic considerations in particular, and sense-making in general. as a consequence, the teaching approach appeared to have considerable potential for in-depth conceptual development growing out of the learners’ discursive activity. although learners had some time to explore their reasoning with one another other during the limited time allocated to small group discussion, the interruptions brought about by the teachers in the classroom discussion did not give them much time to pursue their own ideas. however, learners were expected to accept the obligation and engage in thinking about the issues at hand and in sharing their thinking within their smaller groups. as such and since the discussion inevitably focused on their reasons, learners were in a good frame of mind to compare and contrast their reasons with those of others. consequently, their thoughts and discussions formed a basis for engaging meaningfully in the subsequent classroom discussion. in some instances, concept cartoons in mathematics were used as a stimulus or trigger for discussion while learners were solving problems in their small groups. the purpose of introducing discussion was to help learners seek, share and construct knowledge when engaging in solving word problems. the discussions took the form of dialogue and talk (formal and informal) in both english and the learners’ home language (isixhosa). in promoting discussion, learners were expected to engage critically with problems and build positively on what others had said. the observations during and after the intervention were done with the aim of measuring teachers’ implementation of the strategies that they had learnt during teacher workshops. these workshops gave teachers from experimental schools the opportunity to be trained on how to get learners discussing, arguing and writing about their views and experiences when they solved mathematics word problems. the aim of promoting these strategies in their teaching was to develop and improve their approaches to the teaching and learning of word problem solving in their classrooms. these teachers were introduced to and trained in strategies to improve their pedagogical content knowledge and their ability to promote the teaching and learning of mathematics when solving word problems. the intervention also focused on the language of mathematics embedded within word problems. simple translations were provided for phrases that are often used in mathematical word problems to simplify the meaning of these problem statements. table 1 shows some examples of the translations provided. the problem-solving tasks the pre-test and post-test consisted of the following three modelling problems adapted from verschaffel, greer and van dooren (2009. learners were expected to complete these problem-solving (ps) tasks and give a written explanation of how they arrived at their answers: ps1: 100 children are transported by minibuses to a summer camp at the seaside. each minibus can hold a maximum of 8 children. how many minibuses are needed? ps2: two boys, sibusiso and vukile, are going to help so nwabo rake leaves on his plot of land. the plot is 1200 square meters. sibusiso rakes 700 square meters in four hours and vukile does 500 square meters in two hours. they get 180 rand (r) for their work. how are the boys going to divide the money so that it is fair? ps3: john’s best time to run 100 meters is 17 seconds. how long will it take him to run 1 kilometre? coding of the tests the three ps tasks were coded using a schema that was an elaboration of the classification schema developed by verschaffel, de corte and lasure (1994). the classification schema comprised fourteen categories, which were reduced to three general categories: • realistic reaction (rr): all cases where a learner either gave the (most) correct numerical solution that also took into account the real-world aspects of the problem context, as well as cases where there was a clear indication that the learner tried to take into account those real-world aspects, without giving the mathematically and situationally (most) accurate numerical answer. • no reaction (nr): all those cases where there is no indication that the learner was aware of the realistic modelling difficulty, for example, mathematically correct but situationally inaccurate and/or incorrect or inappropriate responses, computational errors, etc. this category also provides a measure of the problem-solving performance of the learner. • other reaction (or): all cases where a learner did not provide a numerical response and did not give any written comment that indicated that the learner was aware of the realistic modelling difficulty that prevented him or her from answering the problem, as well as instances where the learner generated incorrect responses with mathematical (or computational) errors. table 1: translations of word problem phrases. overview of reliability and validity in this study the primary strategy utilised in this study to ensure external validity was the provision of thick, rich and detailed descriptions so that anyone interested in transferability would have a solid framework for comparison (merriam, 1988). nixon and power (2007) point out that warranting of claims must fulfill the criteria of trustworthiness, soundness, coherence, plausibility and fruitfulness. trustworthiness refers to the quality of qualitative data collected (anastas, 2004); and in the sensof neutrality in the findings or decisions of the study (guba & lincoln, 2005).reliability is the degree to which the instrument consistently measures whatever it is measuring (ary, jacobs & razavier, 1990; best & kahn, 2003). according to silverman (1999), reliability refers to the degree of consistency with which instances are assigned to the same category by different observers or by the same observer on different occasions. neuman (2003) suggests reliability has to do with the issue of dependability. dependability of the data in this study was established by capturing the observations on a tape and video recorder, and transcribing them both manually in writing and with computer software. attempts were made to reproduce the interview scripts as accurately as possible to eliminate possible threats to the reliability of the instruments used in this study. creswell (2005) defines threats as the problems that threaten our ability to draw correct cause-and-effect inferences that arise because of the experimental procedures or the experiences of participants. ethical considerations the education, research technology and innovation committee gave prior permission to conduct this research as part of the integrated school development and improvement project offered by the centre for educational research, technology and innovation at the nelson mandela metropolitan university. after obtaining ethics clearance, the first author approached the principals and teachers of the participating schools, where their roles as participants and their rights to choose to be participants and to participate or not in the study were explained to them. both the teachers and parents (on behalf of the learners) gave their informed consent. they were assured of confidentiality and also that participation was voluntary. they were given a guarantee that they could withdraw from the study at any time and that no personal details would be disclosed. they were promised confidentiality of information collected in the schools and were assured that no portion of the data collection would be used for any purpose other than this research. learners were also assured that the test results would not influence their school marks. statistical analysis of data the quantitative statistical data generated from the pre-tests and post-tests were captured in a microsoft office excel spreadsheet and subjected to repeated measure anova techniques (using the statistica software package) which simultaneously accounted for pre-test and post-test data of the experimental and comparison groups in order to provide both descriptive and inferential statistics. anova techniques were used to determine the statistical (non)significance of the results, based on mean differences between experimental and comparison groups before and after the intervention. where necessary, the statistical technique of matched-pairs t-tests was computed to compare the mean scores of the comparison and experimental groups. cohen’s d statistics were calculated to determine whether statistically significant (p < 0.0005) pair-wise differences were practically significant. a small practical significance is noted where 0.2 < d < 0.5; a moderate practical significance is noted if 0.5 < d < 0.8; and a large practical difference is recorded if d > 0.8. expressed differently, an effect size of less than 0.2 is considered to be insignificant, an effect size between 0.2 and 0.5 is considered to be of small significance; an effect size between 0.5 and 0.8 is considered as being moderately significant, while an effect size of 0.8 and greater is considered to be highly significant. effect size as expressed by the cohen’s d statistics is defined as the difference in means divided by the pooled standard deviation and is a measure of magnitude (or significance) of the differences between the pre-test and post-test scores (gravetter & walnau, 2008). findings top ↑ the word problems in this study, which are examples of a central part of mathematics learning, can be seen as attempts to connect mathematical reasoning to learners’ everyday life experiences and/or knowledge (sepeng, 2011). in other words, these problem-solving tasks can be viewed as a manifestation of the notion that mathematics is or should be part of mundane practices in everyday life (verschaffel et al., 2009). effect of intervention on problem solving table 2 depicts a summary of the statistical analysis of learners’ word problem solving achievements as well as their realistic reactions. analysis of pre-test results indicate a statistically significant difference between the experimental and comparison groups, with the experimental group’s performance being worse than that of the comparison group (p < 0.0005). after the intervention, the experimental group’s performance was statistically significantly better (p < 0.0005) than the comparison group’s, with a mean difference (δχ) of 29.14. in table 2 a positive mean difference implies that the mean score of the experimental group was more than that of the comparison group in the post-test. practical significance of the differences as p < 0.0005 in all cases, cohen’s d was calculated in order to gauge the effect size of the practical significance of the differences in the experimental and the comparison groups. a large practical significance (d = 1.56) was noted. when the effect sizes on the rr difference between the experimental and the comparison groups were compared, a moderate practical significance (d = 0.57) was calculated (see table 2). table 3 shows the number and percentage of learners who succeeded in producing three, two, one, and zero rrs to the ps tasks. it illustrates that learners performed rather poorly in these tasks, which required not only computational skills, but realistic sense-making as well. only 1% of learners in the experimental schools produced three situationally accurate answers or reacted three times in a way that showed attention to the realistic modelling complexity of the problems. a closer look at the post-test results for the experimental schools shows a 10% improvement in the production of 2 rrs compared to a drop of 6% in the comparison schools.the mean difference (difference between the mean scores) shows [δχ = 0.47] a statistically significant (p < 0.0005) difference between the experimental and the comparison groups for the rr after the intervention. the positive mean score shows that, although the comparison group had a tendency to consider reality and sense-making when solving the word problems before the intervention (δχ = -0.02), its performance was well below the experimental group’s after the intervention. sense-making (or realistic considerations) of word problem solving (ps tasks) table 3 shows the number and percentage of learners who succeeded in producing three, two, one, and zero rrs to the ps tasks. it illustrates that learners performed rather poorly in these tasks, which required not only computational skills, but realistic sense-making as well. only 1% of learners in the experimental schools produced three situationally accurate answers or reacted three times in a way that showed attention to the realistic modelling complexity of the problems. a closer look at the post-test results for the experimental schools shows a 10% improvement in the production of 2 rrs compared to a drop of 6% in the comparison schools.the mean difference (difference between the mean scores) shows [δχ = 0.47] a statistically significant (p < 0.0005) difference between the experimental and the comparison groups for the rr after the intervention. the positive mean score shows that, although the comparison group had a tendency to consider reality and sense-making when solving the word problems before the intervention (δχ = -0.02), its performance was well below the experimental group’s after the intervention. matched-pairs t-tests table 4 shows the results of a matched-pairs t-test that was used to test whether there was a significant mean difference between experimental and comparison groups before and after the intervention (or pre-test and post-test). in addition to this, data in table 4 depict mean scores of the experimental and comparison groups for word problem solving and sense-making (rrs) in this study.at the p < 0.0005 significance level, the study gives overwhelming evidence that the problem-solving scores of the experimental group improved by 17.08 after the intervention, whilst the practical significance calculated for the experimental group is moderate. although the comparison group’s mean score was higher than that of the experimental group before the intervention (pre-test), a negative mean difference (δχ = -26.67) suggests that the comparison group not only scored well below the experimental group after the intervention, but that their scores were significantly lower than the experimental group’s. a practical non-significance was also calculated for the comparison group. despite a small practical significance found for the experimental group’s rrs, the experimental group did significantly better than the comparison group with a marginally significant improvement in sense-making scores (δχ = 0.23) in the experimental group after the intervention. table 5 provides a brief summary of findings based on learners’ solving and sense-making of word problems. table 2: experimental minus comparison groups’ scores (mean difference). table 3: number of learners producing realistic reactions. table 4: a test of a significant mean difference (pre-tests and post-tests in the experimental and comparison groups using a matched-pairs t-test). table 5: results summary. discussion top ↑ problem solving analysis of the data obtained from pre-tests and post-tests revealed that learners’ problem-solving performance in the post-tests improved over time after the intervention. statistical results illustrate that there was a statistically significant difference between the experimental and comparison groups before the intervention (pre-tests), with the comparison group performing significantly better than the experimental group. however, the experimental group performed statistically significantly better than the comparison group after the intervention. in other words, it appeared that the intervention strategy in this study (see introduction of the discussion) had a positive influence on learners’ skills in solving word problems. as a result, a large practical significance was also noted. the experimental group improved significantly from the pre-test to the post-test. sense-making the overall results of this study illustrate that learners’ performance changes dramatically when discussion is introduced into the mathematics classroom as a teaching strategy. when word problems are taught, teachers should consider carefully how to model the situation (or context), and also whether the information provided is relevant and sufficient for solving the problem (säljö, riesbeck & wyndham, 2009). the results of this study showed that before the intervention (pre-test) learners had a tendency to respond to the problems even if the information given was irrelevant to the information needed to answer the given question. it is interesting that intercultural comparison studies show similar findings (säljö et al., 2009; verschaffel et al., 2000; xin, 2009; xin, lin, zhang & yan, 2007).the statistical results revealed a significant difference between the experimental group and comparison group. the experimental group appeared to show a tendency to consider reality marginally better than the comparison group. a large significant practical difference between the experimental group and the comparison group was also noted after the intervention. the results of the study demonstrate that the introduction of discussion in the teaching and learning of word problems in mathematics not only had a positive effect on learners’ problem-solving performance, but also on their ability to consider reality when they had to solve word problems. the data generated in this study also suggest that whole-class discussion and problem-based approaches to the teaching of word problems can be applied appropriately and successfully (to certain degree) in second language teaching and learning settings, and can assist both mathematics teachers and learners to improve their knowledge of the real wor(l)d effects of mathematics problems. conclusion top ↑ in this study discussion as a strategy to improve second language learners’ word problem solving and sense-making skills was explored. the literature suggests, and initial observations appeared to confirm, that the difficulties in solving word-problems are related to effective pedagogical strategies that advance problem-based and whole-class discussion approaches to the teaching and learning of word problems in mathematics (verschaffel et al., 2000, 2009). the results of this study seem to illustrate and substantiate that what teachers do serves as a fundamental component to raising learning outcomes (douglas, 2009). in the study teachers’ pedagogical practices during the intervention resulted in lessons which involved group interaction and communication. the lessons were organised in such a way that there was some form of taking turns, where each member of a small group had to make their talk comprehensible to all (heap, 1990). in this way the lessons in the mathematics classrooms came to be dominated by cooperative learning discussions where talk within teacher-learner interactions and/or learner-to-leaner(s) interactions were of high quality.statistical analysis of variance showed statistically significant and large practically significant evidence that the introduction of discussion in the teaching and learning of word problems in mathematics increases the problem-solving and sense-making performance of second language learners. the level at which the threshold of p was set in the study was 0.0005, which meant that there was a 0.05% chance that the results were accidental. the large practical significance noted in the study implies a research result that should be viewed as important for teaching practice in mathematics classrooms. the findings of this study suggest that when discussion was introduced into the mathematics classroom, better connections between classroom mathematics and out-of-school mathematics were made and that there was better integration between the learners’ formal written mathematical language and their informal spoken mathematical language. in fact, learners not only generated more computationally correct responses, but also produced more situationally accurate and appropriate solutions to real-wor(l)d problems in the post-test (or after the intervention). although the intervention of the study targeted only a limited number of teachers and schools in port elizabeth townships, and the conclusions drawn from the study cannot be generalised, the findings provide sufficient insights from which tentative recommendations for mathematics teacher development can be drawn. analysis of the quantitative data suggests that promoting the introduction of discussion techniques in mathematics classrooms had benefits and in all probability promoted the participating learners’ problem-solving performance and significantly increased the likelihood of realistic consideration of word problem solving. however, to successfully implement such a strategy, teachers need appropriate fundamental skills and the necessary knowledge of managing and maintaining classroom discourses that allow the development of formal written mathematical language and the skills necessary for the negotiation of meaning within informal spoken mathematical language. acknowledgements top ↑ this research is supported by the centre for educational research, technology and innovation at the nelson mandela metropolitan university. any opinions, findings and conclusions or recommendations expressed in this study are those of the authors and do not necessarily reflect the views of the centre for educational research, technology and innovation. competing interests we declare that we have no institutional and/or personal or financial relationships which may have inappropriately influenced us in writing this article. authors’ contributions p.s. 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(2008). the role of vocabulary in reading comprehension: the case of secondary students learning english in singapore. relc journal, 39(1), 52–77. abstract introduction theoretical framework and literature review method findings discussion conclusion acknowledgements references appendix 1 appendix 2 about the author(s) solomon a. tesfamicael department of teacher education, faculty of social and education sciences, norwegian university of science and technology, trondheim, norway citation tesfamicael, s.a. (2022). prospective teachers’ cognitive engagement during virtual teaching using geogebra and desmos. pythagoras, 43(1), a691. https://doi.org/10.4102/pythagoras.v43i1.691 original research prospective teachers’ cognitive engagement during virtual teaching using geogebra and desmos solomon a. tesfamicael received: 17 mar. 2022; accepted: 29 aug. 2022; published: 04 nov. 2022 copyright: © 2022. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract this article focuses on teacher educators’ reflections on prospective teachers’ cognitive engagement in the teaching and learning of mathematics during the pandemic. using three interacting aspects that can foster learners’ cognitive engagement as a lens, observations and reflections from two teacher educators and anonymous screenshots of students’ work were gathered and analysed by a mathematics teacher educator. the prospective teachers’ self-regulated learning, engagement in solving tasks, and participation in productive discourse were positively surprising, showing the cognitive presence of the learners during virtual teaching. the influence of digital platforms like blackboard and digital tools like geogebra, desmos, padlet, google docs, google forms and google sheets in teaching helped us observe the learners’ cognitive engagement in real time. contribution: the geogebra class function and the desmos teacher feature helped us to observe the prospective teachers’ cognitive engagement during the online teaching. however, continuously and rapidly creating digital content on said digital platforms can be demanding for educators. keywords: cognitive engagement; prospective teachers; digital platforms; digital tools; virtual teaching. introduction the coronavirus disease 2019 (covid-19) pandemic has created challenges and opportunities for educators. on the one hand, the disruption has greatly impacted the day-to-day activities of schools, teachers, students (learners) and educators. on the other hand, the availability of different digital learning management platforms and other digital tools has allowed continuing schooling virtually, especially in those parts of the world where access to such technology is satisfactory. often, the virtual learning environment is a substitute for the in-person classroom environment, thereby creating the challenge of engaging students cognitively during virtual teaching (we will use the term virtual teaching throughout, although terms like online learning, online education, etc. could be used interchangeably in the context of our study). information and communication technology in mathematics education information and communication technology (ict) has a huge impact on education. in mathematics education, ict particularly supports mathematics teaching in many ways: composing, revising, editing, publishing, calculating, making connections, visualising data, synthesising and problem-solving (joshi, 2017). kaino (2008) summarised the four ict revolutions in connection to the history of the international commission on mathematical instruction (icme), explaining how the fourth industrial revolution or waves (according to johnston-wilder & pimm, 2005; oliver & espinosa, 2021) of ict have become the globalisation force in creating a single global community. this effect was more pronounced during the covid-19 pandemic years, starting in march 2020. nathalie sinclair claimed that we would see the return of technology in mathematics education during a virtual european society for research in mathematics education event at the beginning of february 2021 (sinclair, 2021). she reflected somewhat humorously on how people during this covid-19 era claim that ‘technology will one day rescue humanity’ (sinclair, 2021). after experiencing digital teaching fully during the last half of the spring 2021 semester, it is possible to say that such statements are understated. way before this pandemic, the national council of teachers of mathematics (nctm, 2014) suggested that technology is an essential tool for learning mathematics in this century, and all schools must ensure that all their students have access to technology. but access to technology is not uniform across the world. also, icmi has dedicated two studies, icme study 1 and icme study 17, to the integration of ict in mathematics education. this fact indicates how ict improves conceptualisation and problem-solving in mathematics education (kaino, 2008). as mishra and koehler (2006) indicated already 15 years ago, teachers and educators need to have technological pedagogical content knowledge (tpack) for using various technologies to teach, represent and facilitate knowledge creation of specific subject content. the pandemic certainly rushed the teachers’ needs for the development of tpack going forward (cheng et al., 2022). compare, for example, how reluctant workers are towards video meetings now to before the pandemic. the availability of different digital learning management platforms and other digital tools in ict has made it possible to continue schooling virtually, especially in those parts of the world where access to such technology is satisfactory. however, drijvers (2015) stated that ‘the integration of technology in mathematics education is a subtle question, and that success and failure occur at levels of learning, teaching, and research’ (p. 147). drijvers identifies factors that promote or hinder the successful integration of digital technology in mathematics education: the design, the role of the teacher, and the educational context. research questions robinson-neal (2021) recommended four research questions specific to the educational sphere. our research aligns with the third question, which states: ‘in what ways has the pandemic guided changes in classroom practice in primary, secondary, and post-secondary classrooms since instructors across the educational spectrum have had to adjust the ways they teach?’ (robinson-neal, 2021, p. 3). we had a similar experience in our teaching during this pandemic season. we have, for the first time, engaged prospective student teachers in the teaching and learning of mathematics virtually (online) almost all the time. using digital learning platforms was common before the pandemic. however, student engagement is a broad concept (attard, 2012), and studying it is problematic during this pandemic. in fact, student attendance in the virtual classroom seemed to be better than the in-person classroom attendance. but, how engaged (cognitively) the students are in such digital classrooms is not yet known in depth. in our teaching, a digital platform called blackboard and its online lecturing platform called collaborate and digital tools like geogebra, desmos, scratch, trinket.io, padlet, google docs, google forms and google sheets were used collaboratively. in this article, we focused on the use of two digital tools: geogebra and desmos. hence, we specify our research question as follows: to what extent can geogebra and desmos contribute to boosting prospective teachers’ cognitive engagement during virtual teaching? the results from this article can contribute to the kinds of instructional adjustments educators in mathematics (and other subjects) at the tertiary level have to make because of the shift to digital. the goal is to investigate if the digital tools mentioned assist in tracking the prospective teachers’ cognitive engagement during virtual teaching. in addition, we dare to share this virtual teaching and learning experience because it has become so much closer to being a norm for educational systems in the 21st century. maybe covid-19 can cause reforms across nations, and maybe it becomes a norm to do more remote, virtual, digital or blended learning at an unprecedented level, especially at the tertiary level (mulenga & marbán, 2020). theoretical framework and literature review students’ cognitive engagement engagement is difficult to study in a limited time because it is a multidimensional concept including cognitive, affective and behavioural factors (attard, 2012; fredricks et al., 2004). especially, the study of students’ mathematical engagement has a higher stake since it is considered a precursor of students’ performance in assessments and participation in science, technology, engineering and mathematics (stem) education (watt & goos, 2017). abbott (2017) describes different forms of engagement: intellectual, emotional, behavioural, physical, social and cultural. among these types, intellectual engagement points to students’ curiosity, emotional engagement refers to students’ enthusiasm, and social and physical engagements refer to working collaboratively with social and physical activities. bond et al. (2020) argued that since the concept of student engagement is complex and different definitions seem to exist, each researcher or project should state their own working definition. referring to ainley, 2012, helme and clarke (2001) pointed out that: [t]he term engagement usually refers to the extent to which a student is actively involved with the content of a learning activity, where active involvement suggests that the person acts to maintain or extend their contact with the object in order to increase their knowledge of it. (helme & clarke, 2001, p. 133) cognitive engagement includes the student’s mental effort in learning activities (hollister et al., 2022). it incorporates thoughtfulness and willingness to exert the effort necessary to comprehend complex ideas and master difficult skills (mahatmya et al., 2012). fredricks et al. (2004) discussed that cognitive engagement stems from school engagement and instruction, which stresses investment, self-regulation or being strategic. for them, ‘cognitive engagement can range from simple memorisation to the use of self-regulated learning strategies that promote deep understanding and expertise’ (fredricks et al., 2004, p. 61). appleton et al. (2006) measured student cognitive engagement in terms of control and relevance of schoolwork, future aspirations and goals, and extrinsic motivation (reschly & christenson, 2012). in this work, virtual cognitive engagement means the students engage in tasks (activities) to foster their mathematical understanding (conceptual or procedural), which demands self-regulation and collaboration with others via the online discourse. students’ cognitive engagement during online teaching actually, during the online teaching, what mattered most for us was the cognitive aspect of student engagement. this refers to active mental engagement. clarke (2002) suggested four forms of cognitive engagement: self-regulated learning, task focus, resource management and recipience. bond et al. (2020) used three indicators of student cognitive engagement: purposeful, critical thinking and self-regulation. however, helme and clarke (2001) outlined three interacting aspects that can indicate the cognitive engagement of students: individual learning (self-regulated learning), the learning environment and the tasks. this definition specifies the basis for observing, investigating and analysing prospective teachers’ cognitive engagement in this study. hence, a brief presentation of the framework is further needed. self-regulated learning self-regulated learning can be defined as an individual cognitive controller which transforms the thinking and affective aspects into emotional and motivational behaviours during the learning process (sahdan & abidin, 2017). it is one of the strategies that can help implement a student-centred learning approach in the in-person or online learning environment. since these prospective teachers were, in our experience, most likely to be alone during the virtual teaching sessions, their participation with higher-order or metacognitive components is crucial. their plan, setting goals, organising, self-monitoring and self-evaluating are critical aspects of various points during virtual learning. according to zimmerman (1990, p. 4), a self-regulated learning ‘perspective on students’ learning and achievement are not only distinctive, but it has profound implications for the way teachers should interact with students and the manner in which schools should be organised’. these implications intensify when teaching shifts suddenly to virtual. studying the self-regulation of learners during digital sessions is not an easy matter. of course, their attendance in the digital classroom, in our case, the blackboard collaborate room, shows it has increased compared to the usual in-person teaching, with close to 100% attendance most of the time. however, helme and clarke (2001) shed light on the preciseness of studies that claim time-on-task or student self-reports as indirect measures of cognitive engagement. they pointed out that self-regulated individuals actively seek learning opportunities to achieve the desired learning goals, unlike the students who use help-seeking and effort-avoiding strategies (superficial engagement). further, self-regulated students systematically use metacognitive, motivational and behavioural strategies to optimise their learning. task (activities) focus the nature of the mathematical tasks to which students are exposed determines what students learn (nctm, 1991). student learning is fostered with open-ended tasks that are not only focused on procedures but also foster conceptual understanding. tasks can be routine, simple, complex or challenging. smith and stein (1998) and other authors have developed four different levels of tasks based on cognitive demand: memorisation, procedures without connections, procedures with connections and doing mathematics (nctm, 2014). henningsen and stein (1997) emphasised the importance of engaging students in mathematical tasks that demand a cognitive process at the level of doing mathematics that engage, students in high-level mathematical thinking and reasoning. they emphasised that tasks and activities are vehicles for students’ cognitive engagement. the course we designed and provided is mostly activity and task based, favouring student-centeredness. it means that in one-day virtual teaching, there will be four to five activities or tasks that demand randomly created group work among the prospective teachers. in such types of tasks, students plan, monitor and solve the tasks independently. how well each student group engages (cognitively) may vary between groups and depend on the richness of the task provided. virtual learning environments in relation to discourse the learning environment is one of the vital elements that can contribute to promoting or restraining students’ cognitive engagement (edwards & mercer, 1987). collaborative small group and whole class interactions define an effective learning environment. computer-mediated communication (cmc) has given new opportunities for students to participate in instructional activities that demand higher cognitive engagement (abdelhafez, 2021; pavel & wysocki, 2007; richardson & newby, 2006). effective mathematics teaching facilitates discourse among students and teachers to build a shared understanding of mathematical ideas by analysing and comparing student approaches and arguments (nctm, 2014). in the course, we are involved in real-time virtual instruction on platforms like blackboard collaborate and zoom. all these platforms are equipped for whole class or small group chat spaces. in particular, the breakout rooms that are provided by the digital platforms, that is, splitting the group of students into smaller groups for group work, and making random groups when needed, foster a virtual discourse and virtual collaboration among the virtual community of inquiry (pavel & wysocki, 2007). students’ behaviour like questioning, answering teachers’ questions, exchanging ideas, justifying an argument, explaining procedures and concepts, and even expressing gestures using the platforms can be used as indicators of cognitive engagement (helme & clarke, 2001). however, it is unclear in virtual teaching how the pedagogical model developed by stein et al. (2008) for orchestrating classroom discussion using five key practices, anticipating, monitoring, selecting, sequencing and making connections between student responses, can be implemented effectively. method creswell and poth cheryl (2018) defined a case study as ‘a qualitative approach in which the investigator explores a real-life, contemporary bounded system (a case) over time, through detailed, in-depth data collection involving multiple sources of information’ (p. 96). this study uses ict to boost prospective teachers’ cognitive engagement during a second-year mathematics course focusing on mathematical content, statistics and probability. it considers the case of exploiting the provisions of digital tools like geogebra and desmos platforms for learning. almost all teaching was done virtually from january to may 2022. the shift from face-to-face to digital or virtual teaching is the phenomenon that educators are encountering. further, to define it more precisely, how student teachers are engaged cognitively during the virtual teaching on blackboard collaborate is the question about which educators wonder. how can they make sure that their students are engaged cognitively? the goal was to describe what the educators experience in light of the theoretical frameworks described above and investigate the opportunities embedded in different ict tools like geogebra and desmos. data collection and analysis primarily, the experience of instructors is considered. in the spring semester, three teacher educators were tasked with teaching a course on statistics, probability and stochastic variables for the year two student teachers in the teacher education department. one hundred student teachers were grouped into three classes. in the first half of 2021, we, for the first time, started to engage prospective teachers in the teaching and learning of programming for middle and lower secondary schools. recently, programming was introduced into the norwegian curriculum. hence, scratch and python were introduced in the course to connect concepts like combinatory statistics and probability. for objectivity, we identify ourselves as teacher educators 1, 2 and 3 (te1, te2, te3). in addition, anonymous screenshots of prospective teachers’ engagement while solving problems or tasks on the virtual platforms, like geogebra and desmos, were collected. due to the provision of the platforms, the works of all those prospective teachers who attained virtual teaching were captured and included in the study. but the data was collected during the semester randomly. the observations and reflections of te1 and te2 are used as data. te3 is the author of this article; hence, for objectivity, te3’s observations and reflections are not included. content analysis helps to evaluate the presence of words, phrases, themes, concepts or images within a piece or multiple pieces of given qualitative content data (cohen et al., 2018). the observations and reflections of te1 and te2 are analysed via content analysis. the three aspects that can help us to understand the cognitive engagement of the prospective teachers, self-regulated learning (the individual), the learner’s engagement with the tasks and their discourse on the virtual learning environment, as outlined by helme and clarke (2001), are used as a synchronising framework for the data collection and the data analysis. in addition, the documents that contain tasks and activities made by the educators using the geogebra class function and desmos teacher feature are used. the data from online sources geogebra (http://www.geogebra.org) and desmos (http://teacher.desmos.com) on the platform blackboard collaborate were collected, mostly as screenshots (see figure 1). figure 1: a task to reflect in connection to pascal’s triangle during the virtual teaching on the digital platform blackboard collaborate using the class function in geogebra. findings in fact, prospective teachers’ attendance in the virtual classroom seems to be better than in-person classroom attendance. first, the results from the class function in geogebra and the desmos teacher feature are presented. then the observations and reflections of the teacher educators are presented using the theoretical frameworks defined above in connection to students’ cognitive engagement. note that we preferred to present the data from the digital tools mentioned to grasp the teacher educators’ reflections, as they refer to these tools in their reflections. in addition, the theoretical framework presented above in connection to the students’ cognitive engagement is presented in such a way that it gives meaning, at least to the author. data from the class function in geogebra in connection to the task labelled as ‘tankeksperimentet’, roughly translated as ‘the tank experiment’ (although there is an intended pun here, where ‘tanke-eksperiment’ would translate to ‘thinking experiment’), students were given four tasks. the teacher educators could see all their students working on the four tasks in real time on the geogebra class platform. these tasks are based on the context of water pouring into a stack of tanks or buckets, supported by diagrammatic representations. task 1 asks for the water share in the tanks; task 2 asks for another row of buckets at the bottom and calculates the amounts of water in this row; task 3 substitutes water through a collection of hoses that will go from the top down so that it ends up with hoses for every possible path for the water; task 4 asks how the number of litres of water in the buckets is connected to the number of hoses. appendix 1 shows the screenshot of one class with 27 students. among those, the work from student 5 and student 2 is displayed in figure 2. the difference in the progress of the students can be seen easily. these data give insight into how the prospective teachers are engaged in solving the task, and their reasoning can also be observed almost in real time. figure 2: student 5’s and student 2’s solutions for the four tasks. another task requested students to find the different ways of colouring a figure with four squares with one square blue and one square red. this assignment is a combinatoric task, and the website link generated on geogebra is shared with the students. the screenshots of their work are provided in appendix 2, after two and four minutes. figure 3 shows the response progress of three students after two and four minutes. figure 3: screenshot of the work of students 4, 8 and 11 at two and four minutes after the task is provided to them. data collected using the desmos platform desmos, an advanced online graphing calculator, is another virtual tool implemented during our teaching. we collected data from approximately 100 students in three different groups using several tasks. figure 4 and figure 5 provide examples of the tasks and evidence of real-time student engagement during virtual teaching. in figure 4, we see one activity where students are given an interactive balance, where they are supposed to move the weights around to balance the scales – providing a foundation for understanding the concept of average through balancing. in figure 5, we see results from one of the ‘screens’ in the activity where students vote for which scale is correctly balanced. we also note that the students are automatically given names of famous scientists, providing the opportunity to use the activities as discussion points without anyone feeling they are being exposed. figure 4: data on desmos during one of the virtual classrooms showing how a group of students solved the balancing tasks to understand a set’s mean value. this example is screen 10 of 14. figure 5: data collected on desmos during one of the virtual classrooms with anonymity. observations on how students engage in solving tasks te1 reflected that the digital tools we have created to keep ourselves informed on student progress was an eye-opener. the real-time teaching platforms geogebra and desmos (including google sheets and padlet) helped us see how students are progressing and where they struggle. when constructing teaching materials, the online setting has led to new ways of doing handouts, problem-solving-based instructions, and using group work during virtual teaching. for example, we used the class mode in geogebra to see how many were on task and what they replied (see figure 6). in this mode, you can track how much of the prescribed work has been done by all the students (left) and how one particular student is responding to a task (right). figure 6: geogebra data collected showing the progress of all students’ engagement in solving a task related to making histograms (left) and one student’s engagement in estimating where the mean of a data set lies given the boxplot of that data (right). geogebra class also has the functionality to make tasks where students can respond in handwriting, which many students use, either from tablets or just drawing with the computer mouse. in desmos, we used the activity builder in teacher mode to reuse the library of desmos activities and build our own (see figure 4 and figure 5). names are anonymised in desmos so that you can use the progress through the activities as a focus for whole class discussions. desmos has an increasing library of self-paced activities to be freely used. in scratch (http://scratch.mit.edu) and python (the online python distribution of https://trinket.io/ is the one used in the course), we have created simulations in probability and statistics that students themselves can do and then comment in chats or on the microphone about how they react to them. some programs were created together, and others were prepared for the students. the use of google sheets and cloud variables in scratch has made it possible to facilitate experiments where we can collectively work out the frequencies of non-standard situations. observation of self-regulated learning te2 believed that it is important to see the students and try to read their body language when teaching. this phenomenon is a key component in understanding whether or not the students follow the lesson actively. the reading of body language is, in our experience, very difficult while you are presenting material during a virtual lesson. there are several reasons for this difficulty: (1) you have to pay attention to the screen you share, (2) the students often switch off their cameras and (3) some students don’t like to say things aloud. this difficulty might also be true in ordinary teaching but seems to be magnified when we teach digitally. because of this situation, quite a bit of time at the start of the semester was spent emphasising the importance of the students’ participation or active engagement during digital teaching. it was clear that if students wanted the best possible learning outcome, they needed to ask if they didn’t understand (like telling the instructor to go slower, faster, etc.). te1 reflected that students could appear active, but the online presence might be misleading. for instance, asking students to ‘give a thumbs up if you follow’ results in 100% affirmation, but designating them into groups means a quarter of the group disappears mysteriously, a clear effort-avoiding tactic. there also appeared to be a usual ‘gang of suspects’ that are the most active, as would be the case in face-to-face campus teaching. it could be that some of those students who disappear do so because they are put into groups with which they are not happy. when they disappear from the online platform, there is no means to call them back. or they may also use other social media platforms that are more plausible to them. observation of the learning environments using the discourse perspective the tes reflected that some students are more active than others during small group and whole class discussions. only a few use the chat features embedded in the platforms. te1 reported that the prospective teachers appreciate being put into groups randomly during the breakout rooms, leading them to collaborate with people with whom they normally would not interact. te2 reflected his preference to walk around the classroom while the students are working to observe different groups, to get a feel for their problems, to determine which group or groups should present and even to choose groups with different solutions and strategies. these goals are difficult to achieve in digital teaching. in general, some students were not turning on their cameras but participating in the discourse actively via audio. they asked their fellow students and the teacher questions whenever they are in the breakout rooms. they reflected and explained their solutions and contributed ideas and made evaluative comments. discussion in the whole online teaching period, ict utilisation is magnified: the learning platform blackboard, its zoom-like feature collaborate, and different digital tools like padlet, google drive, google sheets, excel, geogebra, desmos, trinket, scratch, python and others were used. in this work, the results are discussed based on the use of the two digital tools, geogebra and desmos, in relation to the three aspects of the students’ cognitive engagement. the affordance and limitation of geogebra and desmos in relation to tasks (activities) cognitive presence refers to the students’ questioning, reasoning, connecting, challenging and developing problem-solving techniques (lipman, 1991) during digital teaching. pavel and wysocki (2007) indicated that selecting content and supporting discourse constitutes cognitive presence in virtual teaching. the learning platforms have enabled us to design and provide tasks to students. in addition, the breakout rooms in blackboard collaborate and zoom have provided small group work where students engage in activities, solving tasks that the instructors design. there is huge potential to engage students cognitively in real time: the use of the class feature in geogebra and the desmos teacher feature. as shown in figure 6, geogebra provided the opportunity to see the engagement of the students in the given task in real time. the tool shows the progress of the students comparatively (the whole class on the left), and it also helps us to see how each individual (particular student on the right side) is engaged in solving the tasks in real time when we zoom in on specific students. tasks can be routine, simple, complex or challenging. smith and stein (1998) and other authors have developed four different levels of tasks based on cognitive demand: memorisation, procedures without connections, procedures with connections and doing mathematics (nctm, 2014). boaler and dweck (2016, p. 144) argued that ‘when mathematics tasks are opened for different ways of seeing, different methods and pathways, and different representations, everything changes’. geogebra and desmos have helped us create tasks that can foster student learning. it was possible to design tasks that demanded different levels of thinking. task 4 in figure 2 and the desmos tasks demand procedures with connections according to smith and stein’s categories. it was also possible to create open-ended tasks as well as closed ones. student learning is fostered with open-ended tasks that are not only focused on procedures but also foster conceptual understanding. however, designing and preparing digital content that can engage students in real time is not an easy task in a very limited time. it demands sufficient tpack from the educator’s or teacher’s side. whether it is the class function in geogebra, the activity builder in desmos, or creating our own simulations in scratch and python for virtual teaching, the process demands more effort than in-person teaching. in our experience, it was evident that designing new tasks and activities suitable for real-time learning demands new skills, resources and ample time if we want to engage learners cognitively during virtual teaching. the digital learning environments and tools provide for such possibilities and good development in the circumstances. the geogebra and desmos tasks showed us that there is a different pace in solving the tasks when the prospective teachers are working individually and in small groups. in virtual teaching, it was also evident that the cognitive engagement of the student teachers as a community of inquiry showed different levels of cognitive activity in relation to bloom’s (1956) taxonomy of education. some learners are at the information level, while others are as high as the evaluation level. this fact was evidenced in their engagement of learning outcomes submitted via scratch, python, geogebra and desmos. some of the students reflected that working in small groups with students who had experience with scratch and python programming from before helped them to learn faster. the affordance and limitation of geogebra and desmos in relation to self-regulated learning helme and clarke (2001) emphasised that learners’ wills and skills are crucial in order for them to be successful learners. as sahdan and abidin (2017) emphasised, self-regulated learning can be seen as a skill in learning that students can control. prospective teachers’ self-regulation during this covid-19 era of forced virtual learning is remarkable. their virtual attendance, participation in verbalising their thinking, self-monitoring, solving the tasks and engaging in the activities during virtual teaching is visible. the teacher educators utilised different technologies to foster the students’ active cognitive engagement. the use of the class function in geogebra, the desmos teacher feature, padlet and breakout rooms via blackboard collaborate and zoom are among such tools that helped us to undergo asynchronous and synchronous communications. geogebra has provided the opportunity to see the engagement of the students in the given task in real time. it helped us to see how the students are regulating their learning while they engage in solving the tasks in real time. the tool, as shown in figure 3 and figure 6, helps us see the students’ progress and not only compare but also follow everyone’s progress in solving the tasks in real time. depending on the task, it is also possible to follow up on a student’s handwriting on a certain task in real time. that shows students’ self-regulated learning in real time too. however, as reflected by all tes, the digital platforms do not allow us to see the whole activities, interactions and reactions of the students compared to in-person meetings, which, in turn, limits us in orchestrating a productive mathematical discussion as entailed by stein and smith (2008). the affordance and limitation of geogebra and desmos in relation to discourse perspective as lipman (1991) suggested, this helps the community of inquiry to engage in critical thinking and deep learning outcomes. moreover, the provision of the digital platform, synchronous and asynchronous, contributes to the community of inquiry to engage with the contents for ample amounts of time. in this context, self-regulated learning is somehow visible too. some learners are consistently active, as t1 referred to them, a usual ‘gang of suspects’. the discourse in the virtual classroom can also show cognitive engagement among the community of inquirers. both tes have reflected on their experiences and observations of the whole virtual classroom and small breakout groups provided by the platforms. as te 1 evidenced, the students appreciated being put into groups randomly, leading them to collaborate with people with whom they normally would not. the participation of most students was via audio with some with video settings opening their cameras, and some had more dynamic discourse than others. te2 echoed the limitation of the virtual platform in leading a productive classroom discourse. as teachers, we have the responsibility to develop classroom discourse that meets the criteria described by smith and stein (2008): anticipating, monitoring, selecting, sequencing and connecting. the last three phases are especially difficult to attain in real-time virtual teaching. conclusion our real-time virtual classroom observation and data taken from the virtual platforms and tools show that teaching in a virtual setting after the pandemic led to forced changes in activity style. the use of digital tools and their affordances came to the fore, and we experienced that our earlier approaches to teaching and learning would not be adequate to engage students the way we wanted. in particular, the real-time monitoring of student activity provided by desmos, geogebra, padlet, google sheets and chats was instrumental in keeping a close-to-live relationship with the student group as a whole. the introduction of programming into the norwegian curriculum made it natural to create simulations and activities as part of the teaching sequences and as ready-made programs for the students to use. the class function in geogebra and the teacher feature in desmos provided an opportunity to monitor students’ cognitive engagement. nevertheless, there are still issues that should be addressed. for example, preparing digital content and assessments that will engage the students cognitively in a short time is not an easy task at all, as stated by drijvers (2015). in our opinion, using the same content during in-person teaching would not be optimal. what works for one may not be effective for the other. hence, support might be needed, for example in the form of job-embedded professional development to boost the teachers’ knowledge as described in the tpack framework (mishra & koehler, 2006). acknowledgements the author acknowledge the contributions of rune k. and øistein g. in the preparation of this article. competing interests the author declares that they have no financial or personal relationships that may have inappropriately influenced them in writing this article. author’s contributions s.a.t. is the sole author of this article. ethical considerations this article followed all ethical standards for research without direct contact with human or animal subjects. funding information this research received no specific grant from any funding agency in the public, commercial or not-for-profit sectors. data availability data sharing is not applicable to this article as no new data were created or analysed in this study. disclaimer the views and opinions expressed in this article are those of the author and do not necessarily reflect the official policy or position of any affiliated agency of the author. references abbott, a.l. 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(1990). self-regulated learning and academic achievement: an overview. educational psychologist, 25(1), 3–17. https://doi.org/10.1207/s15326985ep2501_2 appendix 1 figure 1-a1: screenshot of the works of students using the class function in geogebra. appendix 2 figure 2-a1: screenshot of the works of students after two and four minutes using the class function in geogebra. figure 2-a2: after 4 minutes the task is delivered. abstract introduction an emergent, responsive curriculum mathematical proficiency, identities and agency the club contexts methods of analysis findings: the curriculum analysis discussion and conclusions acknowledgements references appendix 1 about the author(s) erna lampen department of curriculum studies, faculty of education, stellenbosch university, stellenbosch, south africa karin brodie school of education, faculty of humanities, university of the witwatersrand, johannesburg, south africa citation lampen, e., & brodie, k. (2020). becoming mathematical: designing a curriculum for a mathematics club. pythagoras, 41(1), a572. https://doi.org/10.4102/pythagoras.v41i1.572 original research becoming mathematical: designing a curriculum for a mathematics club erna lampen, karin brodie received: 11 aug. 2020; accepted: 19 nov. 2020; published: 18 dec. 2020 copyright: © 2020. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract mathematics clubs are becoming increasingly researched in south africa, yet what might constitute a curriculum for such clubs has not been discussed. we present and analyse a curriculum design for clubs that we set up and worked with over three years. the goals of the clubs were to foster strong mathematical proficiency, identities and agency among learners through supporting their enjoyment of mathematics as a useful way to make sense of the world. we therefore developed a curriculum where learners could become and be ‘mathematical’, through drawing on both their everyday reasoning and the mathematics that they learned in school. our analysis suggests an emergent, responsive curriculum, with tasks that can be categorised as: (1) becoming mathematical with everyday tasks, (2) becoming mathematical about mathematics, (3) being mathematical and making mathematics, and (4) mathematics. we argue that through such a curriculum, we can develop mathematical reasoning on the basis of learners’ everyday reasoning in ways that support their mathematical proficiency, identities and agency. keywords: mathematics clubs; curriculum; becoming mathematical; proficiency; identity; agency. introduction mathematics clubs are usually constituted as informal spaces, outside of school, although often based at a school or a university, where learners are encouraged to engage in mathematics as a sense-making activity, to become mathematical problem-solvers, to relate mathematics to real-world situations, and to develop increased enjoyment of mathematics (prescott & pressick-kilborn, 2015; schlosser & balzano, 2014; sherman & catapano, 2011; turner, gutierrez, & sutton, 2011). it is also hoped that clubs might improve learners’ mathematics achievement (sherman & catapano, 2011) and build on learners’ diverse experiences to support them to talk about, interact with and become more confident in mathematics (amit, fried, & abu-naja, 2007; diez-palomar, varley, & simic, 2006). south african research with clubs, mostly with primary school learners (stott & graven, 2013), works with similar goals. none of the research on clubs discusses a curriculum at the level of decisions about task selection and sequencing. we developed a number of mathematics clubs in grades 8–10 that aimed to foster strong mathematical proficiency, identities and agency among learners through supporting their enjoyment of mathematics as a useful way to make sense of the world. in particular, we wanted learners to relate mathematical reasoning in the clubs to both their everyday reasoning and the mathematics that they learn at school. given that we worked across a number of clubs, we were faced with the challenge of coordinating our activities across the clubs, while remaining responsive to the learners in individual clubs. we therefore found it important to design a club curriculum, with common goals, some common tasks and similar pedagogies across the clubs. the curriculum design principles of mathematics clubs have not been articulated and elaborated in the research, and we have developed some important principles for both task and curriculum design which can contribute to the research literature on mathematics clubs and support mathematics club leaders to reflect on and design their own curricula. the focus of this article is to present and analyse elements of our curriculum and thus to make a contribution to this gap in mathematics club research, in south africa and internationally. we write this article in the midst of and as a contribution to debates in south africa and elsewhere about the extent to which learners’ everyday knowledge, or ‘life-worlds’, should organise and permeate the mathematics curriculum (hoadley, 2018; moll, amanti, neff, & gonzalez, 1992; young & muller, 2013; zipin, fataar, & brennan, 2015). there is evidence that many teachers who try to work with everyday knowledge do not manage to relate it to mathematical knowledge, leaving learners with a denuded sense of mathematics (hoadley, 2007). at the same time, we know that many learners experience the disciplinary mathematics knowledge of the formal curriculum as alienating and unrelated to their lives and worlds (brown, brown, & bibby, 2008). drawing on young’s (young & muller, 2013) notion of powerful knowledge, mathematics is powerful because it provides a gaze on the world different from everyday knowledge, and enables us to enquire into, explain and experience the world in new ways, taking us beyond the present and the particular and enabling us to project possible worlds. it is clear that the current mathematics curriculum in most schools does not achieve such power for most learners. at the same time, drawing on learners’ everyday knowledge does not often give access to mathematics as powerful knowledge (hoadley, 2007). fataar (2012) argues that pedagogies that support social justice and learner engagement with their schooling require ‘intellectually demanding content, on the one hand, and active recognition and working with life world knowledge, on the other’ (p. 58). fataar lays a conceptual basis for such pedagogies but does not show how they might be achieved. our contribution is to suggest one such pedagogy in mathematics, which links everyday and mathematical reasoning, rather than only content. our club curriculum is premised on two assumptions. firstly, we know that much of mathematics, including current mathematical research, has its seeds in everyday experience and reasoning. euclidean geometry began with particular, concrete methods of artisans and developed into more general and abstracted work on mathematical objects (krantz & parks, 2014). research mathematicians often draw inspiration from their everyday experiences and thinking. for example, the well-known bridges of königsberg problem originated in everyday experiences and was transformed to mathematics through posing mathematical questions (gribkovskaia, halskau, & laporte, 2007). secondly, we know that learners reason powerfully in their everyday lives, particularly learners who live in stressful situations and who regularly solve daily problems of living (e.g. hemson, 2019). this reasoning must be creative and flexible to support survival, but it is most likely tacit rather than explicit and does not allow for reflection and enquiring beyond the particular. since learners who are creative outside of school often do not see mathematics as relevant to their lives, we hoped that their everyday reasoning could be utilised and extended to pose mathematical questions. we wanted to both draw on their daily reasoning as a way into mathematics and show them that mathematics can further support how they see and experience the world. a key element underlying our curriculum is linking everyday reasoning and mathematics, through reasoning in a mathematical way, or as davis (1996) calls it: the mathematical, which he distinguishes from mathematics. mathematics is the canon of concepts and activities that have been developed through ages of enquiry. the mathematical ‘is that orientation to enquiry which has allowed our mathematics to emerge. it involves noticing of sameness, pattern, and regularity amid one’s explorations. it involves comparing, ordering, creating, and naming’ (davis, 1996, pp. 92–93). for our curriculum we formulated three interrelated spaces of action and thinking: the everyday, the mathematical and the mathematics, in order to give access to mathematics as powerful knowledge. in this article we analyse our club curriculum in order to answer three main research questions: did the curriculum link learners’ everyday reasoning to mathematical reasoning, and if so how? did the curriculum support mathematical proficiency, identity and agency? if so, how? to what extent was the curriculum responsive? if so, how? we note that our analysis is of the curriculum only, and not of the ways in which it was taken up by learners. in the next two sections we present our theoretical framing of the curriculum and of our key theoretical constructs: mathematical proficiency, identity and agency. an emergent, responsive curriculum we think about curriculum as a contextualised social process (cornbleth, 1990), which embodies systemic and sociocultural influences on knowledge, teaching and learning, including ideas about what mathematical knowledge is, how it is created and what values and whose interests are inscribed in and promoted through the curriculum. our notion of curriculum thus includes and goes beyond particular curriculum content, and even beyond mathematical practices such as generalising, systematising, justifying and communicating mathematics (ball, 2003). an important premise is that it is not possible to divorce values, particularly epistemological values, from the knowledge that a curriculum seeks to engage learners with. in our case we foreground a mathematical orientation to enquiry as an epistemological value. designing a curriculum for a mathematics club may seem contradictory. mathematics clubs are usually constituted as informal spaces, outside of school, where problem solving and enjoyment are key (papanastasiou & bottiger, 2004; schlosser & balzano, 2014; sherman & catapano, 2011). the idea of a curriculum suggests a formality and prescriptiveness which might hinder these goals and we have not found many articles that carefully describe curriculum decisions taken in clubs. wallin, noren and valero (2019) argue that in swedish schools, when system-wide, but informal, mathematics clubs became more formalised, with a curriculum and planning guidelines for schools, what they term the ‘schoolification’ of after-school mathematics clubs, tensions arose for teachers and learners in the clubs as some of the original intentions of engagement and enjoyment were lost. we certainly wanted to avoid ‘schoolification’ in our clubs. however, we wanted to do more than provide a set of challenging and interesting tasks that would support learners to make sense of and enjoy mathematics. we wanted to make careful and thoughtful decisions about the sequence of tasks and about how to develop mathematical identities and agency that support a mathematical orientation to enquiry. we knew that we could not develop a curriculum with linear progression from task to task but expected that sets of tasks taken together, together with a clear set of pedagogical principles, would support our goals. so we did not start with a fully worked out plan, in the usual sense of curriculum. we started with two tasks that embodied some of our overall goals in particular ways, and worked from there, responsively, in relation to learners’ engagement with the tasks. a key element of curriculum as contextualised social process is the extent to which and ways in which the curriculum, with its attendant pedagogy, is responsive to learners (seah, andersson, bishop, & clarkson, 2016). while any enacted curriculum is interpreted by teachers and learners (remillard & heck, 2014), who does the interpreting and whose interests are served as the curriculum is enacted are dependent on the responsiveness of the teacher to the learners, and the extent to which the curriculum can shift to take account of learners’ ideas. in schools, shifts away from the official curriculum usually occur in the pacing of tasks, and in the addition, deletion or modification of tasks. bigger shifts might occur when: learners are supported to use and discuss different methods in solving problems, including those not previously seen by the teacher, allowing for learner errors as evidence of the kinds of meanings made by learners and opportunities for new learning, and supporting learners to have conversations that might change the order of the curriculum. these latter curriculum enactments are seen as more responsive to learners’ ideas in and about mathematics, and hence position learners as more mathematically agentic, as well as supporting their developing mathematical proficiency and identities. while an emergent curriculum can be more responsive to learners’ engagements, a key concern is how it achieves coherence, which is a key element of powerful knowledge. a standard mathematics curriculum coheres around systematic building of the content, although it is often not experienced as coherent by learners, who see few connections between various topics and little relevance for their own thinking. noting that the power of powerful knowledge comes from its systematicity (young & muller, 2013), we did not want to jettison the systematic building of a curriculum. however, we saw coherence and systematicity differently, in relation to content and pedagogy. our approach to content coherence was to focus on indispensable knowledge from primary school, notably multiplicative relationships between numbers, including fractions, patterned change, including linear change patterns, the properties of operations on numbers, including distribution, associativity and communativity, and spatial properties and relations between points, lines and the figures they form. we wanted to strengthen these content areas for the learners, in ways that are relevant to and supportive of high school content and reasoning, but were very clear that we were not doing additional ‘school’ teaching in the club, even though we came under intense pressure to do this, from learners, teachers, principals and parents. we found similar orientations to curriculum in the work of stott and graven (2013), based on valsiner’s zone theory. in particular they describe their use of the notion of zones of free movement (zfm) resulting from the anticipation of learners’ ‘thinking about the concept being taught, at that moment and in the future … providing a framework for cognitive activity’ (stott & graven, 2013, p. 5). we move away from the zone image with implicit borders in order to signal that everyday life experiences unfold endlessly, and high school learners are already cognitively active in their lives. we found a powerful metaphor in the double helix structure of dna (figure 1). figure 1: everyday reasoning, the mathematical and mathematics. in the metaphor of the dna helix, we aimed for learners to relate their everyday worlds and mathematics through the common, but developing, ways of reasoning that are relevant in both, and to increasingly twist and condense the spaces between these worlds. the dna image shows that these worlds can move apart and intersect, all the time being linked. this image captures how we moved systematically between the spaces of everyday reasoning, the mathematical and mathematics, with the aim to shift increasingly over time to becoming mathematical about mathematics.we will use the detailed image of structure at the bottom of the diagram to display our conceptual framework. we used particular content in the service of developing mathematical orientations to enquiry, in particular to find and use structural properties and relations between numbers, operations and geometric objects, in goal-directed tasks. we moved systematically between the spaces of everyday reasoning, the mathematical and mathematics, with the aim to shift increasingly over time to becoming mathematical about mathematics. so the coherence and systematicity of our curriculum emerged through the relationships between content and pedagogy in relation to the three important spaces for reasoning and action – the everday, the mathematical and mathematics – and through the following elements of our pedagogy: all learners can become mathematical enquirers and we have high expectations of all learners and communicate these to them. talk and discussion of ideas are encouraged and learners are shown explicitly how articulating ideas deepens understandings. errors, confusion and asking questions are explicitly supported as normal parts of learning and sense-making in mathematics. mathematical enquiry in both everyday and mathematical contexts is explicitly modelled and valued. we did not personally design all the tasks that we used in the curriculum. among the club facilitators we had many years of experience in teaching mathematics and as teacher educators, and we had many tasks that we could use. the design challenge was to select tasks, adapt and modify them to be as responsive as possible to our club learners and our goals, and to work with them in the club so that together they supported the kinds of learning we wanted. we are not claiming that we developed the most appropriate curriculum, or that there might not be other tasks that would also be useful – we certainly expect that there would be. but we do articulate a process that we believe supported a useful club curriculum for our main purposes. we are currently writing up the achievements in the clubs and can show that they did make a difference to learners’ achievement, their views of mathematics, their mathematical values and their identities. mathematical proficiency, identities and agency the conceptual framing for our article comes directly out of the conceptual framing for the design of the clubs and the curriculum. in the clubs we wanted learners to become proficient in mathematics, through building powerful mathematical knowledge and orientations to enquiry, for them to develop mathematical identities as people who choose to use mathematics to make sense of their experiences, and to support learners’ mathematical agency, in communicating and making mathematical decisions that make sense to them, and that could be justified in relation to the mathematical and the mathematics. in this article we investigate the extent to which the designed club curriculum might support these three important elements of learning mathematics. mathematical proficiency kilpatrick, swafford and findell’s (eds. 2001) well-known five strands of mathematical proficiency – conceptual understanding, procedural fluency, strategic competence, adaptive reasoning and productive disposition – are expressed as nouns and thus might be thought of as reifications, as competences that learners should display, therefore as part of mathematics. davis (1996, p. 68) argues: ‘the mathematical began to be overshadowed by the mathematics – i.e. the mode of thinking was in some way hidden by the corpus of knowledge that it spawned’. however, the strands of mathematical proficiency can also be thought of as processes and ways of thinking, to reclaim the mathematical as central to mathematics and more closely related to human experience and exploration of the world. from this perspective mathematical proficiency ‘emerges from our actions in the world and from our interactions with one another’ (davis, 1996, p. 74). it provides a focus on ‘the conversational or dialogical nature of mathematical enquiry – that is, the aspect of learning that involves an active and intersubjective questioning of the world’, rather than ‘a perception of mathematics learning as solitary and monological’ (davis, 1996, p. 94). for example, conceptual understanding is described as knowing ‘why a mathematical idea is important and the kinds of contexts in which it is useful’ and ‘connecting those ideas to what they already know’ (eds. kilpatrick et al., 2001, p. 118), and therefore requires ‘comparing, ordering, creating, and naming’, key aspects of the mathematical (davis, 1996, p. 92). similarly, the other strands, all of which are interrelated, can be seen as reflecting and producing the mathematical, as well as mathematics. more importantly for this article, we were interested in how these strands were seeded in learners’ everyday reasoning, and how the five strands of mathematical proficiency could draw on the everyday and build to the mathematical and the mathematics. because many of the learners in our clubs had not been previously successful in the predominantly numeric and algorithmic reasoning that school mathematics requires, we wanted them to experience and draw on successes in powerful enquiry in everyday spaces, which we could then relate to mathematics in school. so an important value in our clubs was the similarity between reasoning about mathematics already in the canon and reasoning in extra-mathematical situations. in both our design and our analysis, we looked for possibilities for connections between these, rather than distinctions and dichotomies between mathematical and other forms of reasoning. at the same time, we were aware that many club learners did not like mathematics as they had experienced it in school and our early experiences in the clubs suggested that tasks that looked like school mathematics tended to evoke school-type responses: diminished mathematical agency, ignoring their own reasoning, and trying to implement procedures inflexibly. we therefore chose tasks that could set the club apart from school mathematics while trying to build towards understandings of mathematics and becoming mathematical. mathematical identities mathematical identity as a research construct can be slippery (darragh, 2016; radovic, black, william, & salas, 2018). we define identity as both social and subjective (radovic et al., 2018) and as an interaction between the social and the personal, where social identity refers to learners’ positioning in communities of practice, for example the mathematics class or club, and personal identity refers to learners’ personal experiences of mathematics, including their emotions (gardee & brodie, 2019). the five strands of mathematical proficiency working together are all important in supporting learners’ personal and social identities, but in particular the strand of productive disposition and how it interacts with the other strands is crucial. for kilpatrick et al. (eds. 2001), productive disposition ‘refers to the tendency to see sense in mathematics, to perceive it as both useful and worthwhile, to believe that steady effort in learning mathematics pays off, and to see oneself as an effective learner and doer of mathematics’ (p. 131). learners’ social identities can be developed as members of a mathematical community of practice oriented towards mathematical enquiry and mathematics, and their personal identities can be developed as effective problem-solvers in and outside of mathematics. learners’ disabling beliefs about themselves as mathematical thinkers have often been developed through experiences that frame success in mathematics as related to ability rather than to effort and perseverance. we therefore combined our messages of high expectations of all learners with messages about mindset. dweck (2006) argues that people with fixed mindsets believe that intelligence is fixed, no matter how hard you work, and that success and failure can be attributed to ability rather than effort, while people with growth mindsets believe that intelligence is malleable, grows as you learn and that failure contributes to success and both can be attributed to effort and perseverance. boaler (2016) develops these notions in relation to mathematics, arguing that successful mathematics learners hold growth beliefs about themselves and about the nature of mathematics and their role in it. they ‘search for patterns and relationships and think about connections’ and they know that ‘math is a subject of growth and their role is to learn and think about new ideas’ (2016, p. 34). research has shown that learners are marginalised when success is attributed to ability, specialness or a ‘gift’, rather than challenge and perseverence, suggesting that not all learners can be successful in mathematics (eds. black, mendick, & solomon, 2009; gardee, 2019). from a growth mindset perspective, high expectations mean that it is expected that all learners can undertake and persevere with challenging mathematical tasks, with support that continues to challenge them, rather than lowers the task demands (stein, grover, & henningsen, 1996). learners’ contributions are seen as efforts towards a solution, even (or especially when) there are errors. all learners are supported to persevere to success, where success is defined as finding new directions to think about a task, explaining ideas to others and engaging in mathematical conversations, and making progress in thinking about a task, rather than getting right answers. our club pedagogy worked to consistently send these messages, both explicitly and implicitly and we have used these concepts to analyse our curriculum. mathematical agency agency can be defined in relation to different forms of intentional action or practice or in relation to choosing, consciously or unconsciously, certain actions from a range of other possible actions (gresalfi, martin, & hand, 2008; hays, 1994; noren, 2015). like identity, agency is not a dispositional attribute of a person, but emerges through social relations between persons and social contexts (biesta, priestly, & robinson, 2017). people exercise agency by transforming or reproducing their circumstances, influenced by the affordances or constraints made available to them to exercise agency in their situations. in mathematics education research, agency is sometimes discussed using andrew pickering’s (1995) distinction between disciplinary and conceptual agency in what he calls the ‘dance of agency’ (cobb, gresalfi, & hodge, 2009; gresalfi & cobb, 2006; sengupta-irving, 2016). studying the practices of mathematicians and scientists, pickering shows how mathematicians utilise disciplinary agency when utilising the current tools and methods of the discipline, and conceptual agency when developing new concepts and methods. all new mathematical ideas are constrained by current disciplinary practices, and require both mathematical autonomy and subjugation to disciplinary norms. conventional mathematics curricula constrain learners’ agency by presenting mathematics content as rules to solve mathematical problems, rendering learners dependent on teachers’ approval and validation for their mathematical decisions. so learners are completely subjugated to disciplinary norms without developing mathematical autonomy. this one-sided ‘dance’ works against mathematical proficiency and the development of robust mathematical identities. our definition for mathematical agency is based in strategic competence and adaptive reasoning, the willingness to dance with mathematics by making a plan to solve a problem and to justify the solution through argument and explanation. when confronted with problem tasks, both everyday and mathematical, strategic competence was for us a first agentic response to not knowing. adaptive reasoning as agency refers to the warrants used in argument or justification: what counts as the warrant in an everyday argument often resides in immediate and local demonstration that is ‘true for me’ and thus is difficult to argue with. in mathematical reasoning, supporting arguments go beyond local demonstration and refer to more general conceptual representations and principles, agreed to by the community. our conceptual framework is presented diagrammatically in figure 2. loosely analogous to the building blocks of the dna double helix (see the bottom of figure 1), our curriculum connects the everyday and mathematics spaces, imaged as two strings of pentagons in figure 2 (analogous to nucleotides). the mathematical space is constituted of mathematical processes, imaged as linking rectangles between the strings and analogous to the hydrogen bonds in dna that stabilise the two helixes. the role of agency, identity and proficiency is to twist the spaces ever closer around each other, a process called supercoiling in dna. figure 2: conceptual framework. the club contexts we established a number of mathematics clubs over the three years of the project. in this article we talk about the two clubs that existed for the full time of the project (the actual club time was closer to two and a half years because we did not start at the beginning of the first year). one club was in johannesburg and one in the western cape in two very different schools. the johannesburg school was a technical high school, located close to the city centre, which selected learners who could attend on the basis of their grade 8 results and where all learners continued with mathematics until grade 12 with the intention of moving on to further study in mathematics and science and technical careers. the western cape school was located in a township just outside a major town, and was not selective. most learners in this school, including all of the club learners, took mathematical literacy from grade 10. most of the learners in both schools were black (‘coloured’ in the western cape). the medium of instruction in the johannesburg school was english and in the western cape school was afrikaans. the western cape school was located in a violent area, and a number of club sessions could not take place because gang violence had happened recently. school lessons were also sometimes disrupted because of gang activity. all of the learners in the western cape school lived near to the school, whereas many of the johannesburg learners had to travel quite far, often on public transport, sometimes taking more than an hour, which impacted learners’ attendance at the club. the mathematics results at both schools were poor, with end of year averages in grade 10 being 31% and 28% at the two schools. the clubs took place once a week, after school in johannesburg and on saturday mornings in the western cape, each for about two hours. table 1 indicates the number of sessions each year in each club and the number of learners signed up for the club. the numbers indicate the maximum number of learners in any session because attendance was voluntary, although we did encourage learners to attend regularly and many did, some for all three years of the project. table 1: club sessions and learners. each club was led and managed by each author of this article, one of whom took the lead on developing the curriculum, always in consultation with the other and with other club leaders and facilitators. the johannesburg club had a second club leader, who took major responsibility in the first year, and each club had about four to six pre-service mathematics teachers and postgraduate students (experienced mathematics teachers), who helped with the club facilitation. the club facilitators were inducted into the club curriculum, including values and pedagogical principles, as outlined above, and met after each club session with the lead facilitators to reflect on and discuss how the curriculum was experienced by the learners, what the challenges were, the extent to which the tasks and pedagogy contributed to the overall curriculum goals and what the next tasks might be. regular day-long meetings between the leaders of the two clubs – three per year – contributed to the curriculum design process and coherence across the clubs. while the curricula in the two clubs followed similar design principles and used many of the same tasks, the different contexts sometimes required differences in the curriculum. this was particularly the case in the third year – when the western cape learners took mathematical literacy, although we were able to adapt some of the mathematical literacy tasks for the johannesburg learners. as we present the analysis of the club curriculum, we refer specifically to the western cape curriculum, noting that in the johannesburg club, as well as others, the process and curriculum were similar, with some differences. methods of analysis our analysis of the responsiveness of the curriculum comes mostly from planning and reflection field notes. we analysed how often our observations of learner engagement in a club session led to clear decisions to either access a different space from the previous task, or to promote specific aspects of identity, agency or proficiency with the subsequent task. our analysis consisted of three main steps: we went back to our thinking in designing the curriculum, some of which we had explicitly articulated and written down at the time and some of which was implicit and needed to be articulated and elaborated. we had written notes on each task before taking it to the club, on what mathematics we thought the task would enact, what values it might support, why we chose tasks in different non-mathematical and mathematical contexts, how we thought the learners might approach the tasks, what challenges they might experience and how the club facilitators could deal with the challenges. we also wrote reflections after each club session on how the learners approached the tasks, which parts they enjoyed, what their challenges were and how these were dealt with by the club facilitators. while we cannot make claims about how learners actually engaged without additional evidence, we did find that our reflections influenced the developing curriculum and so we use these for this purpose in the analysis. we drew on the above to systematically re-analyse each task in relation to the five strands of mathematical proficiency, and how each task supported movement between the three spaces: everyday, mathematical and mathematics. we wrote up a short memo for each task on why we chose it in relation to previous tasks and how our reflections on the club sessions influenced decisions about subsequent tasks and how we would work with them. at this point in the analysis we thought carefully about how the tasks linked with each other and reflected the goals and values of the curriculum, thus doing a curriculum analysis that went beyond the tasks. we analysed each task in relation to the ways that it might support mathematical identities and agency, again writing up memos relating each task to the preceding and subsequent tasks in relation to identity and agency, and used our initial reflections to show how we saw movement across the tasks in these key attributes. key elements that link the three spaces with each other are flexibility in using representations and moving between the general and the particular. everyday reasoning usually embodies local and immediate reasoning, with particular instances and representations. the mathematical usually embodies shared representations and reasoning across instances, while the mathematics involves shared, canonic representation and reasoning flexibly between the particular and the general, with arguments based on defined structure. in relating the mathematical to everyday reasoning, we worked with taking objects and ideas apart and putting them together again, in order to see structure, with part-whole relationships, and with building algorithmic thinking and procedural fluency outside of familiar mathematics content. in order to analyse for adaptive reasoning, we first identified whether the tasks supported appropriate justification for the learners, and in which spaces the justifications would be placed. for example, in the extra-mathematical tasks, justifications would be constrained to local, immediate reasoning and would fall completely in the everyday space, while mathematics tasks where reasoning is constrained to isolated, rote-learned claims would fall completely in the mathematics space. it became clear from our analysis that we did practically create the third space, the mathematical, by including tasks that allowed basing justifications for everyday reasoning on appropriate mathematical aspects, or basing justifications for mathematical reasoning on appropriate everyday aspects. secondly, we asked if our pedagogy utilised extra-mathematical tasks to acknowledge immediate and local reasoning supported by demonstration, and then to promote reasoning between learners to move toward the mathematical where reasoning has to take account of multiple instances and consolidate contrasting or contradicting explanations. similarly we enquired if we utilised mathematics tasks to promote reasoning that accessed the mathematical space, where imagery and analogy supported justification of the mathematics. thirdly, we reflected on whether tasks and pedagogy were successful in engaging learners in agentic ways – by promoting decision-making, hence becoming strategic in the face of problems, and stimulating sharing and argumentation. in presenting our findings below, we have included vignettes which make claims about what the learners did in the clubs. we have drawn these from our notes so they are our views of what was happening; we have included them to convey the environments that we experienced in the clubs as we interacted with learners and to ‘give life’ to our analysis. findings: the curriculum analysis our findings show that the tasks can be divided into four distinct groups (a brief description of the tasks is in appendix 1): becoming mathematical with everyday tasks (tasks 1–5) where we worked mainly between the everyday and the mathematical, except for task 2. becoming mathematical about mathematics (tasks 6–9), where we worked mainly between the mathematics and the mathematical. being mathematical and making mathematics (tasks 10–12) where we worked with all three elements, from everyday to mathematical to mathematics. mathematics (tasks 13–17), where we worked with the mathematical and the mathematics. at the beginning of the year, we worked more with the relationships between the everyday and the mathematical, in order to draw learners into the club and to make clear the distinctions and relations between club and school mathematics. towards the end of the year we were working more with the relationships between mathematics and the mathematical. this was because of the learners’ development of proficiency, identity and agency, as well as their expressed needs to focus more on schoolwork. the analysis suggests that we did develop a progression through the three spaces over time, first linking two of the three spaces (groups 1 and 2), bringing the three together (group 3), and finally working between the two mathematical spaces (group 4). in each section below we present examples of tasks from each group, with an analysis in terms of the proficiency, identity and agency that the curriculum can support. each task is presented in the context of the curriculum as a whole, so in analysing the tasks, we also analyse the curriculum. becoming mathematical with everyday tasks two tasks in this group (1 and 4) worked from everyday to mathematical reasoning and two (3 and 5) worked from mathematical to everyday reasoning, because we wanted to establish the importance of everyday reasoning from early in the curriculum. task 1 and task 2 were the only tasks that we set up before we met the learners in the club and we chose them deliberately as a pair, with the first one intended to make links between the everyday and mathematical and the second between the mathematics and the mathematical. task 1 the task to analyse the structure of a clothes peg in relation to its use was given in the first club session and the learners were unsure of what was expected of them, and obviously did not see this as a mathematics task (figure 3). however, the discussion yielded what we planned, namely analysis and reflection about the form and function of an everyday object that had resonance with the mathematical. figure 3: task 1. vignette: clothes peg. agency: the clothes peg task yielded useful information about the learners’ agency at the time. from our reflections we saw that showing their knowledge of a clothes peg by drawing was a source of anxiety, probably about being judged. the agency to claim “i don’t know!” – as a starting point for learning – seemed not to be an option. not knowing was a source of shame. yet the learners engaged in argumentation and they made drawings. we analysed the nature of the responses that we discussed in the post session reflection, and concluded that the learners’ arguments about the functions of the various shape features of the pegs had the nature of conjectures that could be supported or refuted. the nature of their arguments was analysed as an aspect of proficiency in everyday reasoning. to analyse if our pedagogy in the session promoted agency, we had to reflect on whether our engagement with the learners during the session was sensitive and responsive to their emotional experiences in the moment (frenzel at al., 2019). we knew that the learners were not used to deliberating about everyday objects in the school mathematics classroom. we also knew that if this task alienated them, they may not return to the club. we had to be open to the emotional demand on the learners of this engagement with new facilitators and a foreign kind of task. we allowed the learners to support each other emotionally by joking, teasing and playing around with the clothes peg. we acknowledged this as an act of agency rather than disruption, and observed keenly where and when the playful joking opened up opportunities to focus on the functions of the features of the peg, and so sustain enquiry. proficiency: our reflections on the clothes peg task motivated our conception that mathematical proficiency has a mirror construct in everyday spaces. in particular we conceptualised the making and reading of precise and informative diagrams as the counterpart of procedural fluency in working with everyday tasks. we conjectured on the basis of our post session field notes that the learners’ initial proficiency resided in the everyday space, where evidence was provided through demonstration and recollection of experiences. for example, the cylindrical hole through the pins of the peg supposes that washing lines or rods are cylindrical (learners clamped the peg over a pencil as evidence) and the v-shaped cut-out at the front ends of the pins have the purpose to minimise the area of the peg that is in contact with the cloth it must secure (they expressed their frustration with peg marks on their own dried clothes). their diagrams reflected varying proficiency. the learners could acknowledge that the reason for the lack of information and structure in their drawings was not because they could not draw, or that they did not know about clothes pegs, but because they did not ‘see’ certain information. playing with the peg with the purpose of enquiry helped them see the peg in a different way. their later attempts were more proficient in detail and showing structure. the clothes peg became a different object, and the learners had different eyes, at least for the moment. the learners’ diagrams of the peg allowed us to talk about an explanatory diagram as used in science and mathematics, as different from a sketch or drawing that one would produce in the arts. identity: we viewed the learners’ search for function in form in such goal-directed human-made objects as a clothes peg as becoming mathematical. the result of this search could be described as conceptual understanding of the object. analysing structure and function have underpinned mathematical enquiry through the ages (maclane, 1986). we saw productive disposition in learners’ everyday reasoning about the clothes peg as the expectation of understandable structural relations between forms and their functions, just like productive disposition in mathematics is the expectation that mathematics makes sense and that mathematics problems can be solved (eds. kilpatrick et al., 2001). task 2 the aim of the second task, four fours (https://www.youcubed.org/tasks/the-four-4s/), was to bring in mathematics content and show learners how they could reason about mathematics in similar ways as they did with the peg (figure 4). this task did not achieve our aim of promoting the mathematical and foregrounding structure and purpose, and we observed adverse effects on agency, identity and proficiency. figure 4: task 2. vignette: four fours. agency and identity: our post session notes reveal our deep concern about the effect of this task on learners’ well-being. the learners did not experiment with numbers to get a better understanding of the task, and then work systematically and relationally through the numbers, as we had expected them to. we wondered if the constraint was only lack of procedural fluency, but we agreed that their mathematical agency was constrained by the fact that they did not see a purpose for this task. they did not grasp the notion of ‘starting with the answer’. whispered talk was about meaning and purpose, such as what sum must be done, why use only four. written work showed that they nevertheless tried to make sense as they wrote simple expressions. it became clear to us that learners were not unwilling to engage, but that the problem and its meaning were beyond their comprehension. our reflections revealed that we were acutely aware of learners’ general embarrasment about their difficulties with the task, as they acted out by making fun of their peers, asked to leave the room, and delayed returning after the break (see also frenzel et al., 2019). we sensed that these learners, who were currently engaging with algebra in school, expected of themselves to have at least some proficiency with arithmetic, and given that they could not make progress, the activity was not conducive to building positive mathematical identities at this stage. our reflection on the contrast in learners’ engagement with these two tasks supported our conception that willingness to engage in strategic competence is important for agency and willingness to engage in justification can be seen as the seed for identity as a mathematical thinker. responsiveness: we understood that we would need to work hard to build their trust through our future engagement with them at the club, without compromising the principles of high expectations and of challenging their thinking. based on our reflections on how the learners interacted with the first two tasks we decided that we would need to do more work with everyday reasoning and use more tasks that moved between the everyday and the mathematical, before coming back to the mathematics. the learners’ specific difficulties with the task to express numbers by using only four 4s confirmed the importance of the design principle to focus on discerning structure and purpose. it was evident that the learners did not think about natural numbers as having additive or multiplicative structure, and that operation symbols were merely instructions to act. we spent the time on tasks 3–5 working with the structure and purpose of everyday thinking in relation to the mathematical. the facilitators’ reflections on the learners’ engagement after this group of tasks suggested that they got started on the tasks more readily, they discussed processes and solutions with each other more than they depended on the facilitators, and they were starting to write more readily on their scratchpads. after the four successful tasks in this group, we thought that we had initiated a number of important practices in our club: (1) learners showed increased agency as they were reasoning in everyday contexts and beginning to see how some of their reasoning could be seen as mathematical; (2) learners could see that this club was a different enough space from their classrooms that they could begin to take risks; and (3) learners were beginning to trust us and each other to support their thinking, rather than to deny its value. learners also asked that we help them with their mathematics revision for their mid-year examinations. we therefore imagined, with some trepidation, that we might now be ready to move into school mathematics content more explicitly and to still support what we had begun to build. becoming mathematical about mathematics in group 2, all four tasks (6–9) went from the mathematics to the mathematical. at this point in the semester, examinations were approaching and the learners asked us to help them with their schoolwork to prepare for the examinations. we responded to their expressed awareness that they did not understand their school mathematics as an indication of developing mathematical identity and agency, because they were asking for more than procedures. in task 6 and task 7, we took the opportunity to show the learners that the reasoning mathematically in everyday contexts they had done in the previous tasks can be seen as similar to reasoning mathematically in classroom mathematics contexts. we drew explicitly on seeds of strategic competence and adaptive reasoning: the fact that learners were willing to start by trying out cases, working systematically and positing reasons for their claims when they worked with the tasks in the first group. they were invited to work in the same way in their school mathematics to guess, test and improve the solutions for linear equations, before ‘doing’ the algebra that did not yet make sense to them. task 6 this task allowed us to address three fundamental issues. firstly, the meaning of the scale metaphor, often used in class. setting up an equation between expressions in mathematics is like balancing a scale in everyday life. here the objects represent weights, not quantities or physical objects (figure 5). the question ‘how many circles?’ is interpreted as ‘how many circles will balance the five blocks?’, and if not interrogated as algebraic representation, may lead to solutions such as 20y. secondly, we would use the context to emphasise equivalent action on both sides of the scale to maintain the balance, rather than the often-used ‘taking numbers over the equal sign’ in equations. thirdly, it is not always the best to start at the top and move linearly. we would emphasise strategic reasoning: start by evaluating the information in relation to the whole. figure 5: task 6. vignette: equations. identity, agency and proficiency: the learners’ evident helplessness when faced with school algebra – the reason for bad marks and anxiety about the approaching examination – suggested that we had to do identity work while becoming mathematical about mathematics. this task’s explicit invitation to think in terms of practical balancing actions supported agency. the club was lively with drawing, gesturing and argumentation, promoting learners’ identities as capable with the mathematical. we noted that the learners compared strategies with each other and had clear preferences aimed at efficiency. as with the previous extra-mathematical tasks they were not shy or withholding as they tended to be with mathematics content. the task enabled us to emphasise that algebra can be reasoned and understood and that they are capable to do so. the metaphor created opportunities for us to support becoming mathematical about school mathematics. although their algebraic proficiency was flawed and rule based, the learners had clearly met the balance metaphor before. we discussed that the objects on the scales, and hence the letter symbols they wrote for them, represent masses, that is, measurements and not physical objects. when they explained algebraic manipulation as ‘you have to take over the y and change the sign’, we challenged them with the meaning of such action on the scales. consequently they came to adopt the more mathematical description ‘take off a y on both sides’, which we supported in algebra talk as well. responsiveness: in subsequent work, we responded to their increased willingness to engage in adaptive reasoning through mediation of everyday metaphors, and we continued to relax the usual school mathematics discourse on abstract objects to emphasise processes and invite enactment. with the example ‘solve for x: 2x + 3 = 12 – x in their textbooks, we invoked the scale metaphor, for example, with ‘what number can you choose to make the left side equal to the right side?’ we held back on immediate manipulation and encouraged experimentation and guessing and testing. we discussed the structure and relationships between the components of such expressions, for example ‘on the left the result gets bigger as you choose bigger numbers for x, and on the right it gets smaller. so if one side increases and the other decreases they must be the same somewhere!’ considering how the relationship between two expressions changes as the expressions themselves take different values is at the heart of relational reasoning and allows a kind of estimation in algebra, to support knowing if one is correct. our post session notes suggest that the learners increasingly experienced that they can talk algebra in meaningful ways, not just voicing symbols, and they gradually supported their manipulation skills by making point checks and reasoning about change and relationships. before, their only recourse to verification of solutions was to ask a facilitator, or to redo the manipulation steps. being mathematical and making mathematics the group 3 tasks re-established links between everyday activity from a mathematical perspective and becoming mathematical. in contrast to tasks where the mathematics emerged from the everyday and the mathematical through our questioning and probing, we now planned explicitly to apply mathematics to extra-mathematics tasks. we worked with geometry, responding to the interest and pleasure the learners expressed when they were able to ‘make’ geometry by considering spatial relations between dots on a page (task 8). we analysed whether the tasks allowed us to refer back to aspects of proficiency in previous tasks. in relation to the mathematical, these competences included verifying results by reference to a diagram, and reading a diagram for procedural information. in these tasks we tied in the proficiency developed through tying knots and giving instructions to tie knots in task 4 and revisited the clothes peg diagram of task 1 in terms of purpose and structure. in task 10 the learners had to fold origami objects based on diagrammatic instructions, and then teach someone else to fold a figure by demonstration and giving instructions. the language of origami, such as hill folds and valley folds, had to be mastered – an aspect of procedural fluency. the activity of folding was followed with relooking at the geometric relationships between fold lines, and the learners had to identify which folds produced parallel or perpendicular lines and which lines produced right angles, specific triangles or specific quadrilaterals. we were therefore able to incorporate many of the ways of working and reasoning we introduced earlier. a new aspect that emerged and needed attention was that the origami task required learners to connect single instructions to a somewhat distant goal, which they struggled with. origami instructions are essentially algorithms that consist of smaller units. the learners struggled with the algorithmic nature of the origami and would try to get visually similar end products by ‘fudging’ results. for example, when a fold did not result in equal lengths fitting, they would cut off the excess paper. we related these non-mathematical strategies to similar fudging strategies in their number and algebra work and challenged them to develop precision and accuracy in both cases. in the planning session we conceptualised unrelated step-following in mathematics as lack of algorithmic thinking. we designed task 11 to focus on algorithms as step-by-step calculation plans towards a goal, and used an everyday situation (programming a robot to make hot chocolate) to emphasise the need for precise and exact formulation. we reflected on the algorithmic nature of the previous week’s origami instructions and followed with a task (task 12) that required of them to establish a (linear functional) calculation plan for pricing doughnuts, and reflect on their calculation plans for input and output values, and rate of change, as algorithms. agency, identity and proficiency: this third group of tasks allowed us opportunities to further discuss with the learners that we make mathematics by the questions we ask in situations. origami can seem like an art activity, but if we can see geometric shapes and relationships between lines and folds, we can use the knowledge to design new figures. for example, with the inclusion of task 10 and task 11 we could challenge them to investigate the fold lines for an origami box and then to fold a new sheet to produce two squares, one with half the area of the other. we revisited the role of writing and making diagrams as means to assist their own thinking, but also to communicate their processes and findings with others. we used action-based discourse (e.g. ‘can you fit two of these squares on the big square?’) rather than the more abstract noun-based discourse (e.g. ‘what is the area of this one?’) as invitation to imagine themselves as agents of making mathematics, rather than simply receiving mathematics instructions to execute. our reflection notes show that the learners became increasingly engaged in group discussions, and in challenging each other to explain their reasoning along with writing and gesturing. as they showed increased strategic and adaptive reasoning, as well as increased productive disposition in the tasks that related the everyday and mathematical, we wanted to establish these gains also in their engagement with school mathematics. mathematics in group 4 the tasks responded to the increased pull of school mathematics expressed by the learners. in the school context this time of the year is test time and learners again brought classroom content into the club. once again algebra was the problem. in response to the learners’ requests for help with the solution of equations, task 13 required of them to interpret number riddles, such as ‘i think of a number; it is a multiple of 7, the sum of the digits is 8 and the number is smaller than 50’. the riddles incorporated a variety of number concepts as well as logic structures such as ‘and’, ‘or’, not’ and implication. they allowed us to represent the learners’ reasoning with mathematics notation, and to establish the relation between operations on numbers, expressions as representations of numbers and equivalent expressions. we asked groups of learners to compose similar riddles for each other, using language that explicitly describes action on numbers and the consequences of such actions. this activity lent itself also to improving mathematics vocabulary, since they revised the meaning of terms like digit, remainder, multiple, twice as many, a third of, etc. responsiveness: when we judged that the learners’ discourse among themselves was mathematical in the way it described enactment and its consequences for algebraic expressions, we asked them to give such action language expressions for the equations they brought from textbooks. at this stage many learners were able to solve equations of two linear expressions by good inspection strategies, while we supported their algebraic manipulations with reference to the balancing tasks we did earlier. task 14 continued to build on these ideas in a more formal mathematics context, presenting the learners with a table of expressions, both numerical and algebraic, to be compared and circled if equivalent. expressions like 12 + 5 = 17 and 2x + 5 = 17 had to be related, and 11 + 6 = 17 had to be separated out through arguments about structure. there was no instruction to calculate, simplify or solve, so that the the focus was on the structural composition of the expressions. our planning session notes for task 15 and task 16 indicate that we wanted to maintain engagement that is mathematical, as we established the spatial-graphical relations between algebraic expressions of different functions (figure 6). for example, to justify why the graph of y = 2x is steeper than the graph of y = ½x we expected descriptive talk (e.g. if you multiply a number by 2 the answer is always bigger than if you multiply by half), rather than simply saying two is bigger than half. we further maintained the mathematical by tasks that required the learners to draw global graphs of functions in contexts, and justify their graphs by reasoning about dependence of variables, rate of change and intercepts. figure 6: task 15 and task 16. vignette: planning meeting. agency and identity: the learners’ requests for specific assistance was a clear act of mathematics agency. we experienced that the learners were aware of the difference between fluency and understanding, and they wanted to understand. although nearing their examinations, the club learners stayed with the club ethos and did not expect the club to be like school. we cautiously suggest that their identities as being capable of the mathematical were emerging, and that they expected to be able to understand. we resisted schoolification and rather than work with textbook tasks as they are, we designed tasks around the content they brought in, to continue to engage them in being mathematical as we had done previously. we noticed productive disposition when they enthusiastically took up an impromptu non-routine challenge task: write functions to explain how the height above ground level (and then the distance towards the back of the building) changes with the number of steps climbed on the stairwell next to the classroom. they organised themselves in small groups and independently carried out actual measurements. at the end of the session they presented their functions for each others’ scrutiny. discussion and conclusions in working to support access to mathematics for learners who had not experienced success in mathematics previously, we worked to build learners’ mathematical proficiency, identities and agency through an emergent and responsive curriculum. our analysis suggests a number of important principles that we developed through designing our curriculum. firstly, we worked out how to build on learners’ everyday experiences in ways that go beyond what this usually means. connecting mathematics to the everyday is often thought of in terms of content and can lead to losing the mathematics in the everyday (hoadley, 2007, 2018). our aim with this curriculum was to go beyond the everyday as content and to acknowledge that in people’s ways of being in the world, in order to manoeuvre ourselves and make sense of many everyday activities, we order, name phenomena, and notice sameness, pattern and regularity. such discerments include re-viewing objects in the world in terms of their purpose and structure. this reasoning can serve as a means to develop similar processes in mathematics – the mathematical. the seven tasks in our curriculum that worked with everyday reasoning supported learners to represent their reasoning processes diagrammatically or with physical objects, to question the validity of their own and others’ reasoning, and to see and create mathematical order in situations that might not seem mathematical. building on learners’ everyday reasoning in these tasks supported their reasoning in the tasks that were entirely in a mathematical context. secondly, we found kilpatrick et al.’s (eds. 2001) five strands of mathematical proficiency useful to conceptualise moving between the mathematical and mathematics, a purpose that they are not usually used for. we asked questions to build learners’ strategic competence and adaptive reasoning and supported a less formal and more enactive discourse to answer questions. we aimed to build learners’ productive dispositions by giving mindset messages about perseverance, that all people can do mathematics, the importance of errors, and different methods of solving problems, in mathematics and in everyday contexts. we found two limitations in kilpatrick et al.’s (eds. 2001) notion of mathematical proficiency: it did not support connections with learners’ everyday realities and reasoning and it did not give us a developmental trajectory for building the five strands. we wanted development in relation to linking the everyday with the mathematical and the mathematics. so we articulated how the five strands can be re-interpreted in relation to everyday reasoning, as we have shown in this article. we also found that working within and across the three spaces – everyday, mathematical and mathematics – supported systematic movement through a set of tasks which supported increasing linkages between the three spaces, and development within each one. given the emergent nature of the curriculum, we did not move linearly from the everyday to the mathematical and the mathematics. we cycled through these in different ways in different tasks, often working with only two, sometimes with all three, in ways that gave the learners experiences of these three spaces, in relationship with each other. the coherence of our curriculum was supported by our overarching goals that we aimed to build in each task, systematically through the three spaces. thirdly, we showed how the curriculum path through the three spaces could support the development of mathematical agency and identity. the original tasks and pedagogy showed that learners experienced anxiety and shame at having to admit to not knowing something, and at their weak mathematical skills. as facilitators we were able to see some forms of proficiency in their work, particularly in the everyday space. being responsive to their strengths and their emotions supported us to develop more tasks and ways of working that did support learners over time to feel more comfortable to try out ideas, ask questions and acknowledge that they wanted to understand and make meaning of the mathematics. so while we cannot point to a direct, linear trajectory, we have suggested that cycling through the spaces might support new forms of agency and identity. fourthly, we elaborated our notion of curriculum responsiveness. had we seen responsiveness as hearing and implementing the learners’, teachers’ and parents’ requests, we would have retaught the school curriculum and not achieved any of our goals of what can be achieved in the distinct space of the mathematics clubs. rather, we had a set of goals and values that informed our linking of the everyday, the mathematical and mathematics, and we maintained responsiveness to learners in relation to those goals – listening carefully to how they were seeing what we were trying to do. the basis for this work is the assumption that all people have the seeds of mathematical reasoning in their everyday experiences and we could build on these in order to give access to the mathematical and mathematics. but we had to listen sensitively to learners to understand where we had to change direction, and why, in relation to our goals and their experiences of the curriculum. a final question is the extent to which our curriculum might be useful to others involved in organising mathematics clubs. we have shown that it is possible to design a club curriculum without ‘schoolifying’, that is, without setting up a formal, predesigned, prescriptive curriculum, which might not be responsive to learners and might move away from the intentions of the club to promote enjoyment and sense-making of mathematics. importantly, we are not presenting this set of tasks as a design for others to follow. rather we hope that through the set of principles that we have outlined and our descriptions of how we worked to achieve them, others will find some inspiration and ways of working that will support them in building curricula for their clubs. we continued to use this method of curriculum design in our clubs in the second and third years, and found that our ‘system’ became even more emergent and responsive, and we were able to engage learners in becoming mathematical. acknowledgements competing interests the authors have declared that no competing interests exist. authors’ contributions all authors contributed equally to this work. ethical consideration ethical approval to conduct the study was obtained from the university of the witwatersrand, school of education ethics committee (subcommittee of university committee; ethical clearance number: 2016ece001s). funding information funding was received from national research foundation, south africa (ref no.: cprr150707123751; grant no.: 99098). data availability statement data sharing is not applicable to this article as no new data were created or analysed in this study. disclaimer the views and opinions expressed in this article are those of the authors and do not necessarily reflect the official policy or position of any affiliated agency of the authors. references amit, m., fried, m. n., & abu-naja, m. 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(2015). can social realism do social justice? debating the warrants for curriculum knowledge selection. education as change, 19(2), 9–36. https://doi.org/10.1080/16823206.2015.1085610 appendix 1 table 1-a1: task list and descriptions. development of in-service teachers’ concept images of the derivative 22 pythagoras, 68, 22-31 (december 2008) a case study of the development of in-service teachers’ concept images of the derivative botshiwe likwambe and iben maj christiansen faculty of education, university of kwazulu-natal, south africa likwamsh@dynalogic.co.za and christianseni@ukzn.ac.za this research focuses on the development of the concept images of the derivative concept of five students enrolled in the in-service programme advanced certificate in education (ace) at the university of kwazulu-natal, pietermaritzburg campus. for comparison, the concept images of two qualified teachers not enrolled in the programme were included. the findings show that of the five ace students who were interviewed, only one had a profound concept image in all the three layers of the derivative, with multiple representations as well as connections among representations within the layers. this one student also passed the calculus module with a distinction. the other four students had the ratio layer and graphical representation in their concept images, while the other layers and representations were pseudo-structural with very few connections. two of these students passed the calculus module, while the other two failed. all the students showed progression in their concept images, which can probably be credited to the ace calculus module. however, it is clear that even upon completion of this module, many practicing teachers have concept images of the derivative which are not encompassing all the layers and more than one or two representations. the research suggested that historically disadvantaged students experience continued inequalities in learning. the research also showed the need to expand the framework of zandieh (2000), developed to describe concept images of derivative. according to tall (1993): calculus represent the first time in which the student is faced with the limit concept, involving calculations that are no longer performed by simple arithmetic and algebra, and infinite processes that can only be carried out by indirect arguments. (p. 23) research from several countries indicate that many students rely on memorisation, struggle to translate realistic problems into calculus, and appear to prefer procedural methods rather than pursuing conceptual understanding (aspinwall, shaw, & presmeg, 1997; selden, selden, hauk, & mason, 1999; tall, 1993; white & mitchelmore, 1996). their concept images are often limited, including only some aspects of the notion of derivative (amit & vinner, 1990; bezuidenhout, 1998; orton, 1983; thompson, 1994; ubuz, 2001; zandieh, 2000). in south africa, bezuidenhout (2001) found that many mathematics lecturers have acknowledged that, while the first year students can find limits, derivatives and integrals, they lack conceptual understanding of basic calculus. his study on students from three different south african universities show that a large number of first year students have a weak understanding of basic calculus concepts and that their understanding is mainly procedural rather than conceptual. zandieh’s results (2000) show that the nine students in her study had a wide variety of representational preferences. these students had different concept images of the derivative concept and these included the basic representations that focuses on rate and slope as a way of describing the derivative. the results also show that there is no hierarchy in the layers of the concept (see botshiwe likwambe & iben christiansen 23 theoretical framework); this means that the students may learn the layers in any order. a calculus reform movement evolved in the usa during the late 1980s because of the high number of dropouts from the calculus course and poor performances by those who continued with the course (bowie, 1998). though this reform did affect south africa as well, we believe the south african situation is still very problematic given the number of unqualified teachers (cf. parker, 2004). the ace course was developed to upgrade the competencies of unor under-qualified educators, but it was our impression that the courses were not sufficiently effective on a conceptual level. it was on this basis we chose to investigate the progression of the ace students’ concept images of the derivative. studies which have been conducted in south africa on the derivative concept were mainly on one aspect of the derivative using one representation. this study involves all three layers and various representations of the derivative concept, which we will now expand on in the discussion of our theoretical framework. theoretical framework in the research, we drew on zandieh’s theoretical framework for the derivative (1997; 2000), which draws on the notion of concept images (tall & vinner, 1981; thompson, 1994), sfard’s (1991) work on the process-object duality of mathematical concepts and work on concept representations (goldin & shteingold, 2001; gravemeijer, lehrer, van oers, & verschaffel, 2003; vinner & dreyfus, 1989). zandieh’s framework was also informed by the whole community of mathematics, including researchers, teachers, students and textbooks. according to sfard (1991), a concept can be seen as a process and as an object. generally, the process aspect precedes the object aspect. in describing the process-object framework, zandieh (2000) points out that: the underlying structure of any representation of the concept of the derivative can be seen as a function whose value at any point [x] is the limit of the ratio of differences h xfhxf xf h )()( lim)(' 0 −+ = → (p.106, author’s notation) zandieh (1997, 2000) claims that the derivative has three layers: ratio, limit and function. each of these three layers can be viewed as a process and as an object. also, each of these layers can be represented graphically, verbally, paradigmatically (physically) and symbolically. the ratio layer the ratio layer is the part of the derivative that deals with the average gradient, represented in leibniz notation as x y ∆ ∆ . the ratio as a process is seen as a process of division of the numerator by the denominator, and as an object a ratio is seen as a pair of integers or as the outcome of the division process. in symbolic representation the ratio as the average gradient between two points ))(,( 00 xfx and ))(,( 00 hxfhx ++ can be represented as h xfhxf )()( 00 −+ . the ratio layer can also be represented graphically as a slope of a secant which is calculated as xindifference yindifference , and in terms of situational representation, average velocity can be represented as a ratio when one is calculating timeinchange ntdisplacemeinchange . all of the representations have both process and object aspects. the limit layer the limit as a process is seen as taking the limit and as an object it is seen as the limit value. the limit layer can be represented in several different ways. the limiting process of the gradient can be represented in leibniz notation as x y ∆ ∆ lim and in symbolic representation as h xfhxf xf h )()( lim)(' 00 00 −+ = → . note the use of x0 here, to indicate that at the limit layer, the derivative is found at a point. the limit layer can also be represented graphically. for the limiting process, one focuses at a point on a curve and keeps it fixed while choosing other points on the curve increasingly closer to the fixed point. connecting the fixed point to each of the other points in turn, a number of secants are drawn, in-service teachers’ concept images of the derivative 24 which will approach the tangent of the curve, referred to by thompson (1994) as the sliding secant. so the limiting process will be to approach the tangent to the curve at a given point. when someone understands that the gradient of the tangent is also the gradient of the curve at that point (a matter of existence and uniqueness which we will not touch upon here, but to note that it is an aspect about which zandieh’s framework remains silent), they will have grasped the fundamental assumption of the limit layer in the derivative concept. the function layer as a process, a function is described by vinner and dreyfus (1989) as a correspondence between two non-empty sets, and as an object they describe it as a set of ordered pairs. in the derivative’s function layer, the order pairs are ))(',( xfx . the function layer can be represented in many different ways. the derivative can be represented in leibniz notation as dx dy . in symbolic representation, the derivative function, that is the gradient of the curve at every point (x, f(x)), is represented as h xfhxf xf h )()( lim)(' 0 −+ = → . in situational representation, one looks at the instantaneous velocity at any point in time. pseudo-structural objects if the student does not have the internal structure as part of her/his concept image, then a pseudostructural conception occurs. according to sfard (1991), a pseudo-structural object is an intuitive understanding which does not involve an understanding of the process underlying the object. zandieh (2000) points out that the use of pseudostructural objects allows the student to formulate basic understandings of the derivative concept which differ from each other, and as a result, the students’ understanding develops from partial to a more complete understanding with only a few aspects of the concept missing expanding zandieh’s framework while zandieh acknowledges that it is in the perceived connection between the various components of the concept image that the concept exists for the individual, her framework does not allow her to identify the extent to which the students make such connections. we want to emphasise what becomes hidden in zandieh’s framework, namely that strong concept images exist in connections across layers and representations. for instance, it is desirable that learners and teachers alike will look at a graphical illustration of a derivative and interpret it symbolically or vice versa. it therefore becomes necessary to enhance the framework’s ability to capture such connections. zandieh’s framework focuses on the conceptual layers of the derivative. however, many learners and educators have procedural (distinct from process aspects of concepts) knowledge of the derivative. for instance, our data showed that some of the interviewees could find the derivative of a given function without necessarily showing concept images falling within zandieh’s framework. to capture this aspect, we found it necessary to expand the theoretical framework, so we had to include ‘instrumental understanding’ in order to include the only learning exhibited by most of the interviewees. lithner (2003) describes instrumental understanding as mastering of a rule or procedure. a student can master a rule or a procedure without any insight into the reasons that make it work, but some students can master a rule or procedure and at the same time have insight into the reasons that make it work. here, we use ‘instrumental understanding’ simply to refer to the ability to use a rule. if a student also understands the background to the rule, we will indicate this in our coding of the data. methodology we used a case study approach with the unit of analysis being the ace students at the university of kwazulu-natal, pietermaritzburg campus. this was a convenience sample. however, the students attending the ace in pietermaritzburg come from a wide range of backgrounds, though mostly historically disadvantaged. in that sense, there is no reason to believe that this group of students was markedly different in their concept knowledge from students at other higher education institutions across south africa. from the students attending the calculus module, we randomly selected students with the only criteria that there would be at least one high performing, one medium performing, and one low performing student. more than five students agreed to participate, but failed to make themselves botshiwe likwambe & iben christiansen 25 available for interviewing, despite repeated prompting, and we ended with a sample of two high performing, one medium performing, and two low performing students. while some high school classes in south africa are being taught by ‘teachers’ who have not completed grade 12 themselves, the ace students in this research had all obtained their school leaving certificates. two of them had been taught introductory calculus at high school and the other three had not. all five students completed a three year teaching diploma at different colleges of education, specialising in the teaching of mathematics. as part hereof, they were taught differential calculus. two of the students have never taught calculus since they completed their diploma. during the ace programme, they were all reintroduced to differential calculus, especially the derivative concept. the two teachers outside of the ace programme who took part in this research both have a bsc in mathematics and a higher diploma in education from the historically white university of natal. they are teaching at a historically advantaged school. megan has been teaching for more than ten years, while deanne has been teaching for two years. table 1 gives an overview of the interviewees’ background and current situation.1 here, as in the analysis, pseudonyms have been used. the first author conducted three open ended interviews with each student. matthew’s first interview was done one week before his examinations, the second interview was done just after the examinations and the last interview was done four months after the second interview. for the other four students, the first interview was done halfway into the course and the second interview at the end of the course just before the examinations. the third interview was done four months after the second interview. the two non-ace teachers were interviewed once, each, as there was no reason to assume that their concept images would change drastically over time. the interviews were recorded and transcribed. during the interviews, the students and educators were asked open-ended questions. choosing suitable questions to ask the students was a big challenge. the question, “what is a derivative?” seemed to be threatening to the students because it may be perceived as an assessment question. thus, it could have limited the possible explanations the students could have given, which is why it was only asked at the end of the interview. hence the question “what comes to your mind when i say derivative?” was used, as it also directly searches for concept images rather than concept definitions. after their answer, where necessary, the researcher followed up with deliberately vague prompts such as “what else comes to your mind?”, “tell me more about that”, “how are all the parts of the formula connected?”. examples were asked for, where she deemed it necessary or useful in obtaining more insight into their concept images. the interviewees have concept images that they have constructed, experienced and shared and which will be elicited as they interact with the researcher (cf. guba & lincoln, 1989). through interacting with their lecturer and classmates, the students have created meaning and made sense of the derivative concept. therefore, we worked from the assumption that the interviewees had concept images which did not change substantially because they were asked about them – though we recognise that some alterations of concept images are likely to occur as students are prompted to summarise or reflect on their knowledge and learning. another assumption was that what the students say during the interviews will reasonably reflect their concept table 1: the background and current situation of interviewees pseudonym of interviewee did calculus in matric? type of school where matriculated did calculus at college? number of years teaching teach matric (grade 12)? number of years teaching calculus matthew yes ex model c yes 5 yes 3 themba no village school yes 10 yes 10 nompilo yes township school yes 5 no 0 ayanda no mission school yes 6 no 0 mandisa no township school yes 10 yes 0 megan yes ex model c yes > 10 yes > 10 deanne yes ex model c yes 2 yes 1 in-service teachers’ concept images of the derivative 26 images – though we do acknowledge that it is possible that only parts of their concept images were elicited during the interviews. analysis instrument our instrument used zandieh’s instrument developed from her theoretical framework as a starting point. we added three aspects to it, reflecting the discussion in the previous section: • instrumental understanding was added as a separate category • arrows were added to reflect connections among the representations revealed in the interviews • a non-layer was added to reflect situations where interviewees’ responses could not be classified in any of the three layers. the result was the ‘matrix’ in figure 1. following zandieh, we used shaded circles to denote a complete understanding of the representation and the process involved, and unshaded circles to denote pseudo-structural understanding. instrumental understanding characterised by the interviewee not showing an understanding of reasons behind a rule is always illustrated by an un-shaded circle. the arrows from one representation to the other show any connections. next, we illustrate the use of the instrument on a number of the interviews. analysis nompilo’s case interview two botshiwe: when i say derivative what comes to your mind? nompilo: slope comes to my mind. bosthiwe: is there a situation where we can use this? nompilo: we use the derivative to find the slope of a graph. botshiwe: what else comes to your mind? nompilo: the formula of the derivative comes to my mind. botshiwe: tell me more about the formula nompilo: it is (she writes dx dy ), we use it to find the derivative. botshiwe: what else comes to your mind when i say derivative? nompilo: the chain rule and implicit differentiation. botshiwe: anything else? nompilo: nothing. placing an un-shaded circle in the slope column representation process-object layer graphical verbal paradigmatic/ physical symbolic instrumental understanding slope rate velocity difference quotient ratio limit function non-layer figure 2: nompilo’s description of the derivative representation process-object layer graphical verbal paradigmatic/ physical symbolic other instrumental understanding slope rate velocity ratio limit function non-layer figure 1: theoretical framework (source: adapted from zandieh, 2000) ? botshiwe likwambe & iben christiansen 27 and the non-layer row, the diagram (figure 2) reflects that nompilo describes the derivative as slope. if she had described the slope as a ratio, the circle would have been in the ratio row. a circle with a question mark is in the instrumental understanding column because nompilo mentions that the formula comes to her mind, but it is not clear if she is able to use the formula. interview three botshiwe: what comes to your mind when i say derivative? nompilo: slope or steepness of a graph. botshiwe: tell me more about the slope. nompilo: the slope is 12 12 xx yy − − and this is the gradient. but the gradient at a point is h xfhxf xf h )()( lim)(' 0 −+ = → . botshiwe: in the formula for gradient at a point, how are the different parts connected to each other? nompilo: )()( xfhxf −+ is change in y and h is change in x. the limit helps us to find the gradient at a point. botshiwe: is there any situation where we can use the slope in real life? nompilo: there is but i don’t remember. botshiwe: is there anything else that comes to your mind about the derivative? nompilo: i know we find the derivatives in trigonometry. botshiwe: how do you do that? nompilo: i don’t remember. botshiwe: sum up and tell me, what is a derivative? nompilo: a derivative tells us about the slope of a function. the shaded circle in the ratio row and slope column in figure 3 reflects that nompilo describes the derivative as a slope which is change in y divided by change in x, which means that she has the ratio layer represented graphically in her concept image. the un-shaded circle in the slope column and limit row indicates that this layer of her concept image is pseudo-structural; she knows it involves the limit but does not explain how. the un-shaded circle in the limit row and the difference quotient column indicates that nompilo has symbolic representation of slope in her concept image but this is again pseudo-structural. nompilo explains the calculations involved in the symbolic expression of the derivative at the ratio layer, which is why the circle in the ratio row and difference quotient is shaded. the arrows indicate that she makes connections between the graphical representation and the symbolic representation, because she describes the derivative as a slope and then gives the symbolic representation of the slope at the ratio layer. the arrows at the limit layer show that there is a connection between graphical representation and the symbolic representation, because she describes the derivative as the gradient at a point and then gives the symbolic representation of the gradient at a point. development in nompilo’s concept image while looking at the results of nompilo’s three interviews, we notice that generally the slope is the main representation in her concept image, and the ratio layer the dominant layer. while it is possible that she could have the function layer and other representations of the derivative as part of her concept image and not exhibited them during the interview, we find this unlikely given nompilo’s stated difficulties with recalling anything else of relevance. nompilo’s concept image did not seem to change during the module (from interview 1 – not shown here – to interview 3) but seems to have been strengthened after the module. representation process-object layer graphical verbal paradigmatic/ physical symbolic instrumental understanding slope rate velocity difference quotient ratio limit function non-layer figure 3: further analysis of nompilo’s view in-service teachers’ concept images of the derivative 28 deanne’s concept image deanne is a qualified educator who has been teaching for two years. botshiwe: when i say derivative what comes to your mind? deanne: slope and rate of change. botshiwe: tell me more about the slope. deanne: we use the derivative to find more about the slope. if you know the formula for the slope then you can find out the rate of change whether it’s time, it depends on what you are looking at. botshiwe: tell me more about the formula for the slope. deanne: the formula i was talking about is something like 523)( 23 −+−= xxxxf which is the formula of a graph. do you want to know the formula of finding the slope of this graph? botshiwe: yes. deanne: to find the derivative of 523)( 23 −+−= xxxxf , initially one would do it from first principles which would be finding the average gradient between two points on the graph. 523)( 23 −+−= xxxxf would be at x on the graph, then you would find another point on the graph say f(x+h). to do that you would substitute x+h into 523 23 −+− xxx which means 5)()2()(3 )( 23 −+++−+= + hxhxhx hxf to find the derivative you would then find the difference between these two and divide by h. and then we take the limit of that as h tends to zero. this is written as h xfhxf xf h )()( lim)(' 0 −+ = → . botshiwe: how are the different parts of the formula connected? deanne: )()( xfhxf −+ is the difference in y and h is the difference in x. botshiwe: how does the limit fit in? deanne: the limit fits in because we are looking at average slope between two points and we want to find the slope at a specific point. botshiwe: how would one use the slope in real life? deanne: if you were trying to model population growth you might have a formula for that population growth and you might want to find what the rate of change of population is over time. so then you would find the derivative of that formula and that would let you see at various points in time what the rate of change is when the population was increasing at that stage or decreasing. botshiwe: you mentioned rate of change. tell me more about it. deanne: ok the rate of change. what we are looking at, is the rate of change between two points and the derivative is the instantaneous rate of change. which means we are looking at how the function is changing at a specific point. botshiwe: is there anything else that comes to your mind about the derivative? deanne: looking at maximum and minimum values in a graph is one of the things that one would use the derivative for. and one does that by looking at local maxima and local minima. when the derivative is zero you will find the local maximum or local minimum in a graph because your instantaneous rate of change becomes zero as the graph reaches a maximum or minimum. botshiwe: what is a derivative? deanne: it is the instantaneous rate of change. deanne describes the derivative as slope and gives the formula of finding the slope. that is why there is a shaded circle in the ratio row and slope column in figure 4. she is able to explain the underlying idea behind the common rule of finding the derivative of a function. this shows that she has a strong ratio layer which is represented graphically in her concept image. she also describes the derivative as rate of change and explains that “it is a formula that would let you see at various points in time what the rate of change is”, so she has the function layer in her concept image. deanne gives the definition of the derivative function and representation process-object layer graphical verbal paradigmatic/ physical symbolic instrumental understanding slope rate velocity difference quotient ratio limit function non-layer figure 4: deanne’s interpretation of the derivative botshiwe likwambe & iben christiansen 29 explains the calculations that are involved. this is shown by the shaded circle in the ratio row and the difference quotient column and also the limit row and difference quotient column which shows that she has a strong symbolic representation of the ratio layer in her concept image. the shaded circle in the limit row and slope column illustrates that deanne explains how the limit is involved in finding the derivative. she also explains instantaneous rate of change, which is why there is a shaded circle in the limit row and rate of change column. her concept image of the derivative is spread across all but one representation. deanne describes the derivative as slope and gives the formula of the slope which shows that she has connections between graphical and symbolic representations at the ratio layer. she also describes the derivative as the slope at a point and explains the process using the formula. this indicates that she has connections between the graphical and symbolic representations at the limit layer. deanne has connections between the graphical and verbal representations because she mentions that if one knows the formula of the slope they can find rate of change. from these results, one can conclude that deanne has a strong concept image of the derivative which is spread across all representations except velocity which does not occur. she might have had velocity in her concept image but chosen not to mention it during the interview. the function layer is least strong in her concept image because it has one representation. these results also show that she has several connections among representations in her concept image. findings: developments in concept images of all interviewees table 2 shows the layers and representations of the derivative concept, preferred by the interviewees when they were asked, “what comes to your mind when i say derivative?” this offers an overview of the first level of analysis. all five students at some stage showed that they had instrumental understanding of the derivative concept. it is also evident from the interviews that all five students have the slope as (one of) their main representation(s) of the derivative, while the limit layer is pseudo-structural for most of the interviewees and the function layer rarely exhibited. the analysis of matthew, deanne and megan’s results show that they display all three layers. unlike zandieh’s students, it appears that these students are more likely to develop the ratio layer before the limit layer, with the function layer as the last aspect to be developed. the results of mandisa, nompilo, ayanda and themba show consistencies across representations within the layers. this means that in the ratio layer for example, if their understanding is pseudostructural in one representation, then it will be pseudo-structural in the other representations of the same layer. these four students have one or two representations of a layer. their representations do not exceed two, while matthew’s interviews show that he has more than two representations of a layer. in all the three interviews, ayanda and themba do not exhibit symbolic representations in their concept images, although ayanda is repeating the module and themba is a medium performing table 2: interviews summary interviewee interview one interview two interview three matthew representation r+s r+s+v r+v+s dominant layer ratio ratio/limit ratio/limit themba representation s r+s r+s dominant layer limit ratio limit mandisa representation none s s dominant layer ayanda representation i i s dominant layer none non-layer ratio nompilo representation i s s dominant layer none non-layer ratio megan representation s dominant layer ratio/limit deanne representation s+r dominant layer ratio/limit r= rate of change s= slope v=velocity i= instrumental understanding in-service teachers’ concept images of the derivative 30 student. mandisa, ayanda and nompilo do not have verbal representation in their concept images while themba and matthew mainly have pseudostructural verbal representation. matthew shows a steady development of the velocity representation which starts as pseudostructural in the ratio layer by the first interview. by the second interview, velocity has become more profound in the ratio layer and pseudostructural in the limit layer. the third interview shows a strong understanding of the derivative in the ratio and limit layers with velocity as a representation. themba and ayanda have a pseudo-structural understanding of the velocity representation in the non-layer during their second interviews, while mandisa and nompilo do not have velocity as their representation. these students have been through the same course and the slope has been a strong part of their concept image which has been evoked by the course. even though this is the case, the students display obvious differences in their concept images. what is needed, however, is more insight into how students’ prior knowledge and experiences influence how they interpret their experiences in a calculus class. discussion mandisa, themba, ayanda and nompilo are historically disadvantaged students while matthew is a historically advantaged student. megan and deanne are the two historically advantaged educators. from their results, we notice that mandisa, themba, ayanda and nompilo have concept images that are mainly in the ratio layer with not more than two representations. themba’s concept image is mainly pseudo-structural throughout the three interviews, while ayanda, mandisa and nompilo’s concept images become more profound at the ratio layer by the third interview. matthew has a very strong concept image which is in the ratio and limit layers and – in the third interview – with all representations present. he is able to describe the derivative as a process as well as an object. megan and deanne have concept images with three or more representations in each layer but megan’s limit layer is quite weak. the question arising from these observations is “how can the historically disadvantaged students improve their concept images?” it is problematic that the course does not seem to manage to level out the differences more. on a positive note, some learning appears to have taken place for all students in the course, but the development of the historically advantaged student’s concept image is so marked in comparison to the development of the historically disadvantaged students’ concept images that it does raise some doubts about the extent to which the course is successful in assisting these students in making sense of the material and relating it to their previous experiences. the analysis of themba and nompilo’s results show that their concept images do not differ significantly from ayanda and mandisa’s concept images, but their examination results show that they scored passing marks while ayanda and mandisa’s examination results show that they were failing. this raises questions about the extent to which the examination written by these students captured conceptual aspects of calculus or mostly instrumental competency – or if the problem is in the interviewing or what can be captured by the theoretical framework. future work will engage this, by interrogating the interplay between students, lecturer and materials. notes 1 an ‘ex-model c’ school is a school that before 1994 catered for white students, and thus was more advantaged in terms of both financial and human resources. these schools are often still better off than many of their counterparts in the townships, with their better building and equipment and more financially affluent parent body which can then contribute financially to the running of the school. thus, these schools are often more capable of attracting the best qualified teachers. many parents from historically disadvantaged background choose to send their children to ex-model c schools in the hope that the advantage will benefit them. thus, the learner body has become racially diversified, while in many cases 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(1997). the role of a formal definition in nine students’ concept image of derivative. in s. berenson, k. dawkins, m. blanton, w. coulombe, j. klob, k. norwood & l. stiff (eds.), proceedings of the 20th annual meeting of the north american chapter of the international group for the psychology of mathematics education (pp. 136-141). columbus, oh: eric clearinghouse for science, mathematics and environmental education. zandieh, m. j. (2000). a theoretical framework for analyzing student understanding of the concept of derivative. cbms issues in mathematics education, 8, 103-127. 66 p14-33 vermeulen final 14 pythagoras 66, december, 2007, pp. 14-33 does curriculum 2005 promote successful learning of elementary algebra? nelis vermeulen faculty of education and social sciences, cape peninsula university of technology email: vermeulenc@cput.ac.za this article reviews literature, previous to the development of curriculum 2005, describing possible causes and solutions for learners’ poor performance in algebra. it then analyses the revised national curriculum statement for mathematics in an attempt to determine whether it addresses these causes and suggested solutions. this analysis finds that the curriculum to a large extent does address them, but that some are either not addressed, or addressed only implicitly. consequently, curriculum 2005 may only partly promote successful learning of elementary algebra. introduction for many years now, mathematics educators have been concerned about the quality of learners’ knowledge and understanding of elementary algebra. numerous research projects have been undertaken internationally and a wealth of knowledge regarding learners’ understanding and the way in which they learn has been accumulated. based on these insights and often under the banner of “reform in algebra”, researchers have made a number of suggestions to improve the practice of elementary algebra teaching. these not only include suggested approaches in the early high school years, but also essential groundwork to be done in the primary school years. this article reviews literature previous to the development of curriculum 2005, which reflects on problems observed and reported on in learners’ knowledge and understanding of elementary algebra. it then highlights a few suggested reasons for these and briefly reports on suggestions to improve the practice of elementary algebra teaching. the article then analyses the learning outcomes as specified by the revised national curriculum statement for mathematics (department of education, 2002), and evaluates whether these outcomes and associated assessment standards correspond with the earlier mentioned suggestions for improving the practice of elementary algebra teaching. the article concludes with a number of recommendations for research and teacher training, based on lessons learnt in the usa with the implementation of the national council of teachers of mathematics (nctm) principles and standards (national council of teachers of mathematics, 2000). what is elementary algebra? if one randomly poses the question, “what is algebra?” to high school learners, the following responses are certain to emerge: algebra is “working with symbols”, “finding the unknown”, “simplifying”, “solving for x”. this extremely limited perception of algebra is hardly surprising, since “the traditional image of algebra, based on more than a century of school algebra, is one of simplifying algebraic expressions, solving equations, learning the rules for manipulating symbols” (kaput, 1999: 133). asked how they feel about algebra, learners may often provide answers such as “i don’t like it, because i don’t see why we study it”, “where are we ever going to use it?”, “it is boring”, “it is so abstract!” and “i am scared of it”. according to the revised national curriculum statement for mathematics (rncsm) “algebra is the language for investigating and communicating most of mathematics. algebra can be seen as generalised arithmetic, and can be extended to the study of functions and other relationships between variables” [italics by author] (department of education, 2002: 9). viewed as generalised arithmetic, algebra essentially refers to the following two key aspects: • the manipulations in algebra are based on exactly the same principles as those underlying calculations in arithmetic, namely the properties of operation (primarily the distributive and commutative properties). • number properties and patterns present in arithmetic can be generalised, and described in symbolic notation. bednarz, kieran and lee (1996: 3) indicate various “…conceptions (explicit and more often implicit) of algebra: the study of a language and its nelis vermeulen 15 syntax; the study of solving procedures for certain classes of problems, algebra here being conceived of not only as a tool for solving specific problems but also as a tool for expressing general solutions; the study of regularities governing numerical relations, a conception of algebra that centres on generalization and that can be widened by adding components of proof and validation; and the study of relations among quantities that vary” [italics by author]. national council of teachers of mathematics (1994) distinguishes four organising themes or conceptual organisers in algebra, namely: • functions and relations: behind the equations, tables and graphs so common to algebra is the central mathematical concept of function… functions, and the related concept of variable, give organised ways of thinking about an enormous variety of mathematical settings. • modelling: many complex phenomena can be modelled by relatively simple algebraic relationships…viewing algebraic relations in terms of the phenomena they model is an effective way of giving life to them and bringing to the study of algebra the richness of experience all students carry with them. • structure: through the efficient and compressed symbol systems of algebra, deep yet simple structures and patterns can be represented. • language and representation: algebra can be seen as a language – with its various ‘dialects’ of literal symbols, graphs, and tables. for instance, algebra can be seen as the language for generalizing arithmetic [italics by author]. (national council of teachers of mathematics, 1994: 8) booth (1986) states: “one of the important functions of algebra is to permit the concise representation of general relationships and procedures. the value of this representation, in turn, is to enable a wide range of problems, to which these relationships or procedures relate, to be more readily solved. it also allows new relationships and procedures to be derived by appropriate manipulation of the old” [italics by author]. in summary, algebra can therefore be described as: • a mathematical language that enables us to express generalisations, to investigate and describe patterns, relationships and procedures, and to derive new relationships and procedures by appropriate manipulation; • generalised arithmetic; • a study of relationships between variables; and • a tool used to solve problems inside and outside mathematics, often through modelling. elementary algebra is that subset of algebra traditionally addressed in grades 7, 8 and 9. this entails the conceptualisation, formalisation, manipulation and application of the notions variable, algebraic expression, equation and relationships between variables, namely, the function concept. problems experienced in elementary algebra most mathematics teachers are familiar with the following typical errors made by grade 8 and 9 (and even higher) learners: • conjoining or closure: 3+x = 3x or x + y = xy • overgeneralisation of the distributive property: yxyx +=+ ; 222)( yxyx +=+ or sin(x + y) = sin x + sin y • incomplete application of the distributive property: 2(x + y) = 2x + y or 62 2 64 += + x x most mathematics teachers have also observed the tendency by their learners to change an algebraic expression into an equation, as illustrated by an excerpt from an interview with a first year mathematics education student (vermeulen, 2000); i = interviewer and s = student. i: if i write there 3x+5, what does it mean to you? s: … you need to find x … you need to say 3x + 5 is equal to zero. i: hmmm… s: then you take the 5 to the other side and … numerous reports of learners’ lack of understanding and misconceptions are to be found in literature, of which only a few are highlighted below. as part of the extensive csms (concepts in secondary mathematics and science) research programme, küchemann (1981: 102) investigated close to 1,000 14-year-olds’ understanding of letter symbols in algebra. he concluded that children interpret letter symbols in the following ways, arranged here in a hierarchical way with number 6 the most appropriate interpretation of letter symbols: 1) letter evaluated: this is when a person assigns a numerical value to the letter symbol from the outset. this is done to avoid working with the does curriculum 2005 promote successful learning of elementary algebra? 16 letter. example: replacing a by 1 and b by 2, presumably because a and b are the first and second letters of the alphabet. 2) letter not used: the letter is simply ignored. example: in response to the question: “if e + f = 8, then e + f + g = …”, 34% of the respondents gave the answers 12, 9 or 15, rather than the correct 8 + g, thereby effectively ignoring the letter symbol g. 3) letter used as an object: the letter is regarded as shorthand for an object or an object in its own right. example: the familiar 2a + 3b interpreted as 2 apples and 3 bananas, or as 2 a’s and 3 b’s. other researchers refer to this phenomenon as “using letter symbols as labels”. a familiar example is the ‘studentprofessor’ problem: “write an equation using the variables s and p to represent the following statement: ‘there are six times as many students as professors at this university’. use s for the number of students and p for the number of professors.” a surprising number of people give the following equation: 6s = p, and explains it as ‘six students for every one professor’, which is similar to the comparison 10 mm = 1 cm. (clement, lockhead & monk, 1981, as cited by kinzel, 1999: 436). 4) letter used as a specific unknown: the letter symbol is regarded as a specific but unknown number. 5) letter used as a generalised number: the letter symbol is seen as representing several values rather than just one. 6) letter used as a variable: the letter symbol is seen as representing a range of unspecified values, and a systematic relationship is seen to exist between two such sets of values. the single cases indicated above are the results of learners’ lack of understanding of the meaning of the basic concepts in elementary algebra, namely, variables, algebraic expressions and equations, the purpose and underlying principles of simplification or manipulation of algebraic expressions, as well the notion of algebra as a language. numerous examples can also be quoted on learners’ lack of understanding of the function concept. causes and solutions numerous causes for children’s lack of understanding of concepts in elementary algebra have been suggested by researchers and educators, usually accompanied by suggested solutions. i focus on three categories, namely similarities and differences between arithmetic and algebra, unhelpful teaching, and learning difficulties. similarities and differences between arithmetic and algebra booth (1988), reporting on the strategies and errors in secondary mathematics (sesm) research project conducted in the united kingdom between 1980 and 1983, provides a useful framework for the classification and investigation of children’s errors in elementary algebra. during the same period, as well as subsequent to the sesm project, many other researchers arrived at similar conclusions. according to booth, many errors made by children could be traced to their ideas of aspects such as those described in the four sections below. the focus of algebraic activity and the nature of answers in arithmetic, the focus of activity traditionally is on finding particular numerical answers. in algebra, however, the aim of manipulation is not so much the finding of an “answer” in simplified form, as the replacement of one algebraic expression by another more useful, yet still equivalent, expression. “more useful” is a function of the task or activity, for example, to evaluate the expression 214)57(2 xxx −+ for any value of x would be easier if the expression is first replaced by the simpler, yet equivalent, expression 10x. on the other hand, to solve the equation 023 2 =+− xx , a more useful form of the expression on the left side would be (x – 1)(x – 2). operating in an arithmetic conceptual framework may be the reason why children are unwilling or unable to view 2n as a proper answer to the question: “what is the perimeter of a shape with n sides of which each is of length 2?” this framework may also be the cause of conjoining as in x + y = xy where children have a cognitive difficulty in accepting lack of closure. booth suggests another aspect of this problem: “not only are unclosed algebraic expressions legitimate as ‘answers’, but the expression represents both the procedure as well as the answer itself. for example, ‘n + 3’ can be both an ‘instruction’ (or procedure) statement, which states that 3 is to be added to the variable n, and an ‘answer’, which gives the result of having performed the addition.” (booth, 1988: 24). this problematic situation was termed by matz (1979) a “process-product dilemma” and later by sfard (1991) a “process-object duality”. nelis vermeulen 17 notation and convention in algebra part of the problem in children’s attempts to simplify expressions such as 2a + 5b concerns their interpretation of the operation symbol. in arithmetic, symbols such as + and = are typically interpreted in terms of actions to be performed, so that for children + means to actually perform the operation (addition) and = means to write down the answer. in algebra, however, the = sign is not the signifier of the ‘answer’ in the same sense as in arithmetic, but rather serves as an equivalence indicator. for example, the = sign in (x – 1)(x 2) = 232 +− xx simply indicates that these two expressions are equivalent. the same applies to 23 2 +− xx = (x – 1)(x 2). concatenation, that is, the juxtaposition of two symbols, is another source of error. in arithmetic, concatenation such as 32 implicitly denotes addition (by convention, 32 means 30 + 2), while in algebra concatenation such as xy by convention denotes multiplication. letters and variables although the most obvious difference between arithmetic and algebra is the use of letters in algebra, letters also appear in arithmetic, but in quite a different way, and with a completely different purpose. the letters ‘m’ and ‘c’ for instance, may be used in arithmetic to represent ‘metres’ and ‘cents’, rather than representing the number of metres or the number of cents, as in algebra. to further compound this problem, letters in algebra do sometimes serve as ‘labels’ rather than representing a (range of) unknown value(s). for example, all of us have written the area formula for a rectangle as a = l × b, where a indicates the area, l the length and b the width. even worse is the formula for the perimeter of a rectangle, when written as p = 2l + 2b. one of the most important aspects of algebra is the notion of ‘variable’. even when children do interpret letters as representing numbers, there is a strong tendency for the letter to be regarded as standing for specific or unique numbers, as noted earlier in this article (küchemann, 1981). this may well be a construct formed in children’s arithmetic conceptual framework, since in arithmetic, symbols representing quantities always do signify unique values. for example, the symbol ‘3’ represents only and exactly three. a related, but inverse, problem arising from this perception is that children often assume that different letters must represent different numerical values. for example, although they may accept that x and y in x + y can assume any value, they find it hard to accept that x and y can assume the same value. learners’ understanding of arithmetic most of the difficulties so far have been discussed from the perspective of the differences between arithmetic and algebra. however, algebra is not separate from arithmetic; indeed, in many respects algebra is generalised arithmetic. but to appreciate the generalisation of arithmetical relationships and procedures children must first be(come) explicitly aware of those relationships and procedures in the arithmetical context. in this case, the difficulties that children experience in elementary algebra are not so much difficulties in algebra itself as problems in arithmetic that remain unresolved. in reform arithmetic classrooms learners are allowed to use their own, informal computational methods, initially based on their theorems-inaction (vergnaud, 1989). the danger for algebra in this practice is as severe as in the traditional method of teaching computational procedures via the standard algorithms, since learners may never become explicitly aware of the underlying structures of these computational procedures. and these structures are exactly the same as those underlying manipulation in algebra, namely, the properties of operation, for example, the commutative and distributive property. to illustrate this, consider the following two cases. in a reform classroom a learner may calculate 32 × 6 in a number of ways; it may be very implicit, in that the child may not necessarily present it as shown here. 32 × 6 = 6 × 32 [uses the commutative property for multiplication] = 6 × (30 + 2) [decomposes 32; some learners even as 10+10+10+2] = (6 × 30) + (6 × 2) [uses the distributive property] = 180 + 12 = 192 in a traditional classroom the same calculation may be carried out as follows, using the standard algorithm for long multiplication; the learner’s reasoning appears in square brackets after each step. 32 × 6 12 [6 × 2 = 12] 18 [6 × 3 = 18; and move the digit one place to the left] 192 [2 + 0 is 2; 1 + 8 is 9; 0 + 1 = 1] does curriculum 2005 promote successful learning of elementary algebra? 18 in this procedure 32 was decomposed as 30 + 2, and 6 was multiplied with both 2 and 30 (and not 3). therefore, the distributive property was most certainly used; only, this standard procedure is so deceptively simple that a primary school learner will never realise it – unless s/he is made explicitly aware of it. and so for the learner in the reform classroom – her/his intuitive use of these properties must be gradually explicated and generalised. in algebra, these properties of operation are of crucial importance – they are exactly the principles that govern equivalence. their correct application ensures the creation of equivalent algebraic expressions. it is for this reason that a number of researchers have indicated the importance of structure. kieran (1989), amongst others, emphasises that an important part of learners’ problems in elementary algebra is their difficulty to recognise and use structure. kieran sees algebra as the formulation and manipulation of general statements about numbers, and hence hypothesises that children’s prior experience with the structure of numerical expressions in primary school should have an important effect on their ability to make sense of algebra. booth (1989) expresses the same view: …a major part of students’ difficulty in algebra stems precisely from their lack of understanding of arithmetical relations. the ability to work meaningfully in algebra, and thereby handle the notational conventions with ease, requires that students first develop a semantic understanding of arithmetic. (1989: 58) kieran (1989) states that much of school arithmetic is orientated towards “finding the answer”. this emphasis allows children to get by with informal, intuitive procedures. however, in algebra, they are required to recognise and use the structure that they have been able to avoid in arithmetic. from the above the conclusion can be drawn that structure should be a unifying concept in arithmetic and algebra. this implies that primary school learners should be given opportunities to become explicitly aware of: • the structure of numerical calculations, based on the properties of operation, as well as • the notion of equivalence, furthermore, this explicit awareness should be gradually generalised (in words and in symbolic language) to facilitate a natural transition to the use of letter symbols and the manipulation of symbolic (algebraic) expressions. unhelpful teaching orton and frobisher (1996) identify two major categories of causes of learners’ lack of understanding and errors in elementary algebra, namely, “unhelpful teaching and learning difficulties”. “unhelpful teaching” practices usually stem from the traditional approach to teaching elementary algebra, where the emphasis is on manipulation. for example, learners do not get the opportunity to understand algebra in its fullest context, as generalised arithmetic, since little attempt is made by the teacher to encourage conceptual links between arithmetic and algebra. another “unhelpful teaching” strategy is the way in which teachers introduce letter symbols and motivate manipulation with letter symbols and expressions containing them. a well-known strategy is the “fruit salad” approach where learners are taught that “3a + 4a equals 7a, because 3 apples plus 4 apples equals 7 apples. however, 3a + 4b cannot be simplified, because you cannot add 3 apples and 4 bananas – unless you want to say that it equals 7 fruit”. this perception of the meaning of letter symbols corresponds to what küchemann (1981) termed as “letters as objects’ or using the letter symbol as a label. this wrong perception of letter symbols created within learners is compounded by the use of initial letters of variable quantities, for example, h for the height, d for the distance, t for time, etc. the traditional approach to teaching elementary algebra supplies learners with the notation but does not attend to developing appropriate mental referents for the notation (kinzel, 1999: 440). to overcome this, teachers should assist learners to develop the concept of variables/-ility. this may be achieved by describing patterns and relationships in ever increasing formal language (from words to letter symbols), as well as exploring a range of input values for letter symbols in algebraic formulas or expressions. learning difficulties whereas orton and frobisher (1996) refer to “learning difficulties”, herscovics (1989) refers to “cognitive obstacles”. he states that the acquisition of new conceptual knowledge (schemata) by the learner “is strewn with cognitive obstacles”, and that cognitive obstacles should be considered normal and inherent to the learner’s construction of knowledge. from the piagetian perspective, acquisition of knowledge involves both assimilation (the integration of new knowledge into the existing cognitive structure) and accommodation (changes in the learner’s cognitive nelis vermeulen 19 structure necessitated by the acquisition of new knowledge). however, accommodation does not occur easily as existing cognitive structures are difficult to change significantly – their very existence becoming cognitive obstacles in the construction of new knowledge. several researchers, including collis (1974, as cited by herscovics, 1989: 62) have found that learners’ prior arithmetical experience can be a source of difficulty in their construction of meaning in algebra. therefore, arithmetic teaching and learning should encourage learners to construct algebraic cognitive structures right from their early primary school years. of course, initially these structures will be very simple. however, they need to be developed and expanded until the learner is introduced to algebra in the high school so that consequent learning of algebra is a process of assimilation rather than accommodation. as such, one would then speak of a natural progression from arithmetic to algebra, rather than a transition, as is currently the case. requirements of an algebra curriculum reflecting upon the research-based findings and suggestions in the previous section, it would appear that in order to promote successful learning of elementary algebra, the curriculum should incorporate a number of aspects. these are introduced in the sections below. aspect 1. enable learners to experience and appreciate algebra as generalised arithmetic the algebra curriculum should enable learners to experience and appreciate algebra as generalised arithmetic. in order to do this, it should promote the following: 1.1 learners must become explicitly aware of structure, primarily that, 1.1.1 the same principles (namely, the properties of operation) underlie computation in arithmetic and manipulation in algebra; 1.1.2 equivalence of expressions, whether numerical or algebraic, is preserved by the correct application of properties of operation, and that manipulation in algebra merely transforms expressions into more useful equivalent (and often simpler) expressions; 1.1.3 there are therefore structural similarities, but also differences, between arithmetic and algebra, and what these similarities and differences are. 1.2 learners must experience and appreciate algebra as a language used to express generalisation of patterns and procedures, thoroughly understanding the semantics and syntax of this language, including proper understanding of the meaning and function of letter symbols, algebraic expressions and equations, and manipulations performed upon them. aspect 2. enable learners to use and appreciate algebra as a means to describe relationships between variables the algebra curriculum should enable learners to use and appreciate algebra as a means to describe relationships between quantities that vary – variables – inside and outside mathematics. specific reference should be made to the following: 2.1 constructing meaning for the concept of variables/-ility; 2.2 appreciating that a relationship exists between different values of the unknown/ variable/letter symbol and the resulting values of the algebraic expression; 2.3 demonstrating the relationship between variables in several ways: verbally, using flow diagrams, tables, algebraic expressions/ formulas and graphs; 2.4 substitution into algebraic expressions and solving equations as two (inverse) processes utilised to determine values for the two variables – substitution to determine output values, and solving equations to determine input values. aspect 3. enable learners to use and appreciate algebra as a tool to solve problems the algebra curriculum should enable learners to use and appreciate algebra as a tool to solve problems, inside and outside mathematics, often through a modelling process. a spiral learning process beginning in primary school it is important to realise that the various categories and subcategories above cannot be separated in practice. they overlap extensively, supporting and reinforcing, informing and complementing one another. elementary algebra can therefore not be taught in a simple linear fashion, as was often the case with the traditional approach (letter symbols does curriculum 2005 promote successful learning of elementary algebra? 20 → algebraic expressions → solution of equations → functions and graphs). in fact, a curriculum designed to satisfy the three aspects above must commence in the early primary school years and follow a spiral approach, where these aspects are addressed in a variety of contexts while continuously generalising and formalising. curriculum 2005 the revised national curriculum statement (rncs) (department of education, 2002) can be viewed as the third attempt to formulate south africa’s outcomes-based curriculum for school grades r-9 – the foundation, intermediate and senior phases or general education and training (get). this is referred to as curriculum 2005. as was the case with the draft national curriculum statement, the rncs for mathematics is a muchsimplified version of the original curriculum documents, and contains only five learning outcomes (los), as opposed to ten in the original version. these los are: lo 1: numbers, operations and relationships lo 2: patterns, functions and algebra lo 3: space and shape (geometry) lo 4: measurement lo 5: data handling each lo is supplemented by a learning outcome focus and a phase focus for each of the three phases in get (foundation, intermediate and senior). assessment standards are provided for each grade against which learners’ attainment of the los are to be assessed. of particular interest for the purpose of this article are los 1 and 2. as stated earlier, the aspects to be addressed by the algebra curriculum – detailed above – should take place in a variety of contexts, including los 3, 4 and 5, where appropriate. however, the core of the development of learners’ knowledge and understanding of algebra will, by the very nature of algebra, take place within the first two los. in an attempt to evaluate how well this curriculum reflects perspectives towards the teaching and learning of algebra, as expressed in the three aspects on page 7, we proceed as follows: • relevant sections from the learning outcome focus of each of los 1 and 2 will be quoted from department of education (2002), followed by the focus of each lo in each of the phases (foundation, intermediate and senior). excerpted statements that address aspects 1 to 3 will be italicised and numbered by a superscript number, and these statements will form the core of the comparison between the curriculum and the requirements formulated as aspects 1 to 3. • in tabular form, the following will be listed: each of aspects 1 to 3, and their subaspects. (note: only the numbers of the relevant sub-aspects will be indicated in the tables, for example, 1.1.1, 1.1.2, etc). relevant excerpted and numbered statements from the lo focus or phase focus. (note: only the superscripted numbers of the particular excerpted statements will be indicated in the tables, e.g. 1, 2, etc). relevant assessment standards. (note: only the numbers of the relevant assessment standards will be indicated in the tables. the list of assessment standards and their corresponding numbers appear in the text adjacent to the relevant table). comments and verdicts indicating the degree to which the curriculum satisfies aspects 1 to 3 will be supplied. this will be done alternately for lo1 and lo2. learning outcome 1: number, operations and relationships the learner will be able to recognise, describe and represent numbers and their relationships, and to count, estimate, calculate and check with competence and confidence in solving problems. learning outcome focus learning outcome 1 builds the learner’s number sense, which is the foundation of further study in mathematics. it also develops the learner’s understanding of: • what different kinds of numbers mean; • how different kinds of numbers relate to each other; • the relative size of different numbers; • how different numbers can be thought about and represented in different ways; and • the effect of operating with numbers.1 essential to the development of number sense is knowledge of basic number facts, the use of efficient and accurate methods for calculation2 and measurement, and a range of strategies for estimating and checking results. nelis vermeulen 21 foundation phase focus in this phase, the number concept of the learner is developed through working with physical objects in order to count collections of objects, partition and combine quantities, skip count3 in various ways, solve contextual (word) problems, and build up and break down numbers.4 intermediate phase focus through a study of a variety of number patterns, the learner should recognise and describe properties of numbers, including identity properties, factors and multiples, and properties of operations, including commutative, associative and distributive properties.5 the purpose should be for the learner to recognise: • what the properties are; and • how they can be used to solve problems and simplify calculations.6 senior phase focus the learner should be encouraged to: • sharpen the ability to estimate and judge the reasonableness of solutions, using a variety of strategies; and • use number knowledge to develop algebraic skills.7 learning outcome 2: patterns, functions and algebra the learner will be able to recognise, describe and represent patterns and relationships, as well as to solve problems using algebraic language and skills. learning outcome focus algebra is the language for investigating and communicating most of mathematics. algebra can be seen as generalized arithmetic, and can be extended to the study of functions and other relationships between variables. a central part of this outcome is for learners to achieve efficient manipulative skills in the use of algebra. learning outcome 2 focuses on: • describing patterns and relationships8 through the use of symbolic expressions, graphs and tables.9 • identifying and analysing regularities and changes in patterns and relationships that enable learners to make predictions and solve problems. mathematical skills are developed over time. the learner should be given opportunities at every grade level to develop these skills to greater levels of sophistication.10 contexts should be selected in which the learner can use algebraic language and skills to describe patterns and relationships11 in a way that builds awareness of other learning areas, as well as human rights, social, economic, cultural, political and environmental issues. foundation phase focus in this phase, the focus of this outcome is to lay the foundation for developing algebra in the intermediate and senior phases.12 this foundation can be established by helping the learner to use physical objects and drawings to copy, extend, create and describe: • geometric patterns13 (e.g. pictures), and • numeric patterns (e.g. skip counting).14 intermediate phase focus in the intermediate phase, the study of numeric and geometric patterns is extended with a special focus on the relationships: • between terms in a sequence; and. • between the number of the term (its place in the sequence) and the term itself.15 this study of numeric and geometric patterns develops the concepts of variables, relationship and function. the understanding of these relationships by the learner will allow her or him to describe the rules generating the patterns.16 this phase has a particular focus on the use of different, yet equivalent, representations of problems and relationships – in other words: words, flow diagrams, tables and simple graphs.17 graphs are not dealt with in this learning outcome in the intermediate phase. however, the learner is given opportunities to read, interpret and draw graphs within data contexts (see learning outcome 5). senior phase focus in the senior phase, the focus of patterns, function and algebra is to formalise the rules generating patterns.18 learners should continue to: • investigate numerical and geometric patterns to establish the relationship between variables,19 and • express rules governing patterns in algebraic language or symbols.20 the learner continues to analyse situations in a variety of contexts21 in order to make sense of them, with added ability to represent and describe them in algebraic language, formulae, expressions and graphs.22 the learner should be exposed to a variety of functions and graphs to compare their does curriculum 2005 promote successful learning of elementary algebra? 22 global features, rather than to focus on the behaviour or features of particular functions or graphs.23 the learners should also study properties of algebraic expressions by manipulating them with sufficient practice to form simpler equivalent expressions24 for calculating and solving problems.25 the learners should develop an appreciation of how algebraic manipulation is useful for solving problems (and not engage in algebraic manipulation for its own sake).26 we now turn to each of the aspects 1 to 3 and their subsections individually, and compare them with the lo foci, phase foci and assessment standards to determine which of these aspects are addressed by the curriculum, and to what extent, and which are excluded or not explicitly addressed. assessment standards used in tables 1 to 12 the assessment standards used for each table (tables 1 to 12) are identified below. these appear adjacent to, or on the previous page from, the relevant table. subaspect where/how addressed in curriculum lo phase(s) numbered statements assessment standard grade(s) comments and verdict regarding 1.1.1 f, i, s 1 2 all grades in get comment: statements 1 and 2 from the lo focus are of a general nature, and are addressed in more detail in the rest of this table. 1a r f 4 1b 1; 2; 3 comment: building up and breaking down numbers pave the way towards using the distributive property, and should therefore be included in the curriculum (which it is). verdict: adequately addressed. 1c 4; 5; 6 i 5 6 1d 6 comment: properties of operations are explicitly addressed. however, teachers should note that all the assessment standards mentioned in the corresponding assessment standards column contribute towards developing and applying them, and even though their names need not be known, recognition, description and using them should be emphasised in class. teachers in reform classes should also be aware of the model describing levels of awareness of these properties (vermeulen 1995: 421422). verdict: adequately addressed. s 7 1e 7; 8 comment: the curriculum apparently encourages learners to investigate different calculating techniques, thereby evaluating explicit use of the properties of operation. this should further explicate learners’ knowledge and understanding. verdict: adequately addressed. 1.1.1 1 final verdict: 1.1.1 is adequately addressed in lo1, sometimes explicitly, but often implicitly. teachers need to be aware of assessment standards that implicitly contribute towards realising 1.1.1, as mentioned above. table 1. aspect 1: enable learners to experience and appreciate algebra as generalised arithmetic (lo1). nelis vermeulen 23 table 1 assessment standards 1a: building up and breaking down numbers to at least 10 1b: perform calculations … uses the following techniques … building up and breaking down numbers 1c: • calculates by selecting and using operations appropriate to solving problems that involve: addition and subtraction of whole numbers multiplication of at least 2/3/4 digit by 2/3 digit numbers division of at least whole 2/3/4 digit numbers by 2/3 digit numbers • uses a range of techniques to perform calculations, including building up and breaking down numbers • recognises, describes and uses the commutative, associative and distributive properties (the expectation is that learners should be able to use these properties and not necessarily know the names of the properties) 1d: multiple operations on whole numbers with or without brackets 1e: • uses a range of techniques to perform calculations, including the commutative, associative and distributive properties • recognises, describes and uses the commutative, associative and distributive properties (the expectation is that learners should be able to use these properties and not necessarily know the names of the properties). table 2 assessment standards 2a: uses conventions of algebraic notation and the commutative, associative and distributive laws to: • collect like terms • multiply or divide an algebraic expression • simplify algebraic expressions given in bracket notation 2b: uses the distributive law and manipulative skills developed in grade 8 to: • find the product of two binomials • factorise algebraic expressions subaspect where/how addressed in curriculum lo phase(s) numbered statements assessment standard grade(s) comments and verdict regarding 1.1.1 f, i, s 10 all grades in get comment: the recognition, description and use of the properties of operation in an ever-increasing level of sophistication and generalisation in arithmetic progress naturally, logically and coherently into their recognition, description and use in algebra. 2a 8 s 24 25 26 2b 9 comment: using the properties of operation to manipulate algebraic expressions should be a natural progression from using them for calculations in arithmetic. however, the curriculum does not explicitly indicate this natural, but vital, link. verdict: 1.1.1 is inadequately addressed. 1.1.1 2 final verdict: 1.1.1 is inadequately addressed in lo2. teachers need to be aware of the importance of 1.1.1 and must: • treat manipulation in algebra as a natural progression from calculation in arithmetic • emphasise the fact that the properties of operation underlie calculation in arithmetic and manipulation in algebra table 2. aspect 1 (cont.): enable learners to experience and appreciate algebra as generalised arithmetic (lo2). does curriculum 2005 promote successful learning of elementary algebra? 24 table 3 assessment standards 3a: • calculates by selecting and using operations appropriate to solving problems that involve: addition and subtraction of whole numbers multiplication of at least 2/3/4 digit by 2/3 digit numbers division of at least whole 2/3/4 digit numbers by 2/3 digit numbers • uses a range of techniques to perform calculations, including building up and breaking down numbers • recognises, describes and uses the commutative, associative and distributive properties (the expectation is that learners should be able to use these properties and not necessarily know the names of the properties) 3b: multiple operations on whole numbers with or without brackets 3c: • uses a range of techniques to perform calculations, including the commutative, associative and distributive properties • recognises, describes and uses the commutative, associative and distributive properties (the expectation is that learners should be able to use these properties and not necessarily know the names of the properties) subaspect where/how addressed in curriculum lo phase(s) numbered statements assessment standard grade(s) comments and verdict regarding 1.1.2 f, i, s 1 2 all grades in get comment: statements 1 and 2 from the lo focus are of a general nature, and are addressed in more detail in the rest of this table. f comment: equivalence need not be addressed explicitly at this level. however, where the opportunity arises, teachers should introduce it informally. 3a 4; 5; 6 i 5 6 3b 6 verdict: addressed, but very implicitly. not indicating to teachers that using properties of operations creates a new, though still equivalent (different in appearance, but equal in value) numerical expression. s 7 3c 7; 8 verdict: addressed, but very implicitly. not indicating to teachers that using properties of operations creates a new, though still equivalent (different in appearance, but equal in value) numerical expression. 1.1.2 1 final verdict: 1.1.2 is very implicitly addressed in lo1. teachers need to be aware of the notion of equivalence, and that using properties of operations conserve equivalence. unfortunately, it is addressed no better in lo2. table 3. aspect 1 (cont.): enable learners to experience and appreciate algebra as generalised arithmetic (lo1). nelis vermeulen 25 table 4 assessment standards 4a: uses conventions of algebraic notation and the commutative, associative and distributive laws to: • collect like terms • multiply or divide an algebraic expression • simplify algebraic expressions given in bracket notation 4b: uses the distributive law and manipulative skills developed in grade 8 to: • find the product of two binomials • factorise algebraic expressions subaspect where/how addressed in curriculum lo phase(s) numbered statements assessment standard grade(s) comments and verdict regarding 1.1.2 f, i, s 10 all grades in get comment: the realisation that using the properties of operations leads to equivalent numerical expressions should, through ever-increasing explication and generalisation, naturally lead to the same realisation during algebraic manipulation. 4a 8 s 24 25 26 4b 9 comment: the phase focus does mention equivalence, but this concept is not reflected in the assessment standards. verdict: equivalence is not addressed sufficiently. teachers need to know that properties of operations preserve equivalence. comment: the phase focus indicates that “algebraic manipulation is useful for solving problems” this formulation however does not explicitly enough state that the main purpose of algebraic manipulation is to create equivalent expressions which are more suitable for a specific task. neither do the assessment standards refer to this fundamental principle. verdict: algebraic manipulation as a means of constructing more convenient equivalent expressions is inadequately addressed. 1.1.2 2 final verdict: 1.1.2 is very inadequately addressed in lo2. table 4. aspect 1 (cont.): enable learners to experience and appreciate algebra as generalised arithmetic (lo2). does curriculum 2005 promote successful learning of elementary algebra? 26 table 5 assessment standards 5a: • calculates by selecting and using operations appropriate to solving problems that involve: addition and subtraction of whole numbers multiplication of at least 2/3/4 digit by 2/3 digit numbers division of at least whole 2/3/4 digit numbers by 2/3 digit numbers • uses a range of techniques to perform calculations, including building up and breaking down numbers • recognises, describes and uses the commutative, associative and distributive properties (the expectation is that learners should be able to use these properties and not necessarily know the names of the properties) 5b: multiple operations on whole numbers with or without brackets 5c: • uses a range of techniques to perform calculations, including the commutative, associative and distributive properties • recognises, describes and uses the commutative, associative and distributive properties (the expectation is that learners should be able to use these properties and not necessarily know the names of the properties) subaspect where/how addressed in curriculum lo phase(s) numbered statements assessment standard grade(s) comments and verdict regarding 1.1.3 f, i, s 1 2 all grades in get comment: statements 1 and 2 from the lo focus are of a general nature, and are addressed in more detail in the rest of this table. f comment: this aspect needs not be addressed at the foundation phase level. however, where the opportunity arises, teachers should introduce it informally 5a 4; 5; 6 i 5 6 5b 6 comment: it is important that learners develop an explicit knowledge and understanding of the structural aspects of the properties of operations, in order to transfer these to manipulation in algebra. without this explicit knowledge and understanding, learners will not be able to see the structural similarities between calculation in arithmetic and manipulation in algebra. verdict: addressed, but very implicitly. not indicating to teachers the importance of explicit knowledge of the structural aspects of properties of operations. s 7 5c 7; 8 verdict: addressed, but very implicitly. not indicating to teachers the importance of explicit knowledge of the structural aspects of properties of operations. 1.1.3 1 final verdict: 1.1.3 is very implicitly addressed in lo1. teachers need to be aware of the importance of explicit knowledge and understanding of the structural aspects of properties of operations. table 5. aspect 1 (cont.): enable learners to experience and appreciate algebra as generalised arithmetic (lo1). nelis vermeulen 27 table 6 assessment standards 6a: uses conventions of algebraic notation and the commutative, associative and distributive laws to: • collect like terms • multiply or divide an algebraic expression • simplify algebraic expressions given in bracket notation 6b: uses the distributive law and manipulative skills developed in grade 8 to: • find the product of two binomials • factorise algebraic expressions table 7 assessment standards 7a: recognises, describes and uses the commutative, associative and distributive properties (the expectation is that learners should be able to use these properties and not necessarily know the names of the properties) subaspect where/how addressed in curriculum lo phase(s) numbered statements assessment standard grade(s) comments and verdict regarding 1.1.2 f, i, s 10 all grades in get 6a 8 s 24 25 26 6b 9 comment: the phase focus merely mentions “study properties of algebraic expressions”, but it does not in any way indicate the fundamental structural similarities and differences between arithmetic (numerical) expressions and algebraic expressions. verdict: structural similarities are inadequately addressed. teachers need to know these similarities and differences. 1.1.3 2 final verdict: 1.1.3 is very inadequately addressed in lo2. table 6. aspect 1 (cont.): enable learners to experience and appreciate algebra as generalised arithmetic (lo2). subaspect where/how addressed in curriculum lo phase(s) numbered statements assessment standard grade(s) comments and verdict regarding 1.2 f i comment: it is not expected for 1.2 to be addressed in foundation or intermediate phases. s 7 7a 7; 8 comment: repeated recognition, description and use of the properties of operations should (with teacher guidance) enable learners to be explicitly aware of them. the (intuitive) need to describe them in a generalised way (words and later symbols) should arise in the senior phase. 1.2 1 final verdict: 1.2 is hardly addressed in lo 1. one would however not expect it here, but rather in lo2. table 7. aspect 1 (cont.): enable learners to experience and appreciate algebra as generalised arithmetic (lo1). does curriculum 2005 promote successful learning of elementary algebra? 28 subaspect where/how addressed in curriculum lo phase(s) numbered statements assessment standard grade(s) comments and verdict regarding 1.2 f comment: it is not expected for 1.2 to be addressed in foundation phase. i 16 8a 4 – 6 s 20 24 8b 7 8 comment: describing rules governing patterns using symbols should develop understanding of these symbols as variables and place holders for numbers. it should also develop understanding of algebraic expressions and equations as a shorthand way of expressing calculating procedures and relationships, namely, as a language. the concept of equivalence is fundamental to understanding and appreciating manipulation in algebra, and therefore the semantics and syntax of algebra as a language. 1.2 2 verdict: 1.2 is adequately addressed in lo 2. table 8. aspect 1 (cont.): enable learners to experience and appreciate algebra as generalised arithmetic (lo2). subaspects where/how addressed in curriculum lo phase(s) numbered statements assessment standard grade(s) comments and verdict regarding 2.1 – 2.4 f 3 1 9a 1; 2; 3 i 3 1 9b 4; 5; 6 s 3 1 7 9c 7; 8; 9 comment: the notion that the number of skips corresponds to a final answer, and that one could determine the final answer by using the number of skips, form the embryonic beginnings of the concept of variable, and of a relationship between variables. verdict: skip counting is implicitly addressed in 2.1 and 2.2. comment: ratio, and rate and proportion in particular, are important components of the function concept, thereby including the concept of variable. in grade 9, proportion problems should be solved within a function context, using formulae (y = kx and xy = k) and graphs. verdict: ratio, rate and proportion are adequately addressed in 2.1 to 2.4 2.1 2.2 2.3 2.4 1 final verdict: 2.1 2.4 are addressed implicitly in lo1. teachers need to be aware of the potential the indicated assessment standards have to develop the concept of variable, and utilize these in an ever-increasing generalised form, describing patterns and relationships in words, and eventually in the senior phase using flow diagrams, formulae and graphs. table 9. aspect 2: enable learners to use and appreciate algebra as a means to describe relationships between quantities that vary (namely, variables) inside and outside mathematics (lo1). nelis vermeulen 29 table 8 assessment standards 8a: • investigates and extends numeric and geometric patterns • writes number sentences to describe a problem situation • solves or completes number sentences 8b: • investigates and extends numeric and geometric patterns • describes, explains and justifies observed relationships or rules in algebra • solves number sentences/equations • uses conventions of algebraic notation and the commutative, associative and distributive laws to compare different representations of algebraic expressions involving one or more operations, selecting those which are equivalent write algebraic expressions, formulae or equations in simpler or more useful equivalent forms in context table 9 assessment standards 9a: counts forwards and backwards in ones, tens, fives, twos, etc 9b: • counts forwards and backwards in a variety of intervals • solves problems that involve ratio and rate 9c: • counts forwards and backwards in a variety of intervals • solves problems that involve ratio and rate (and proportion in grade 9) table 10 assessment standards 10a: • copies and extends simple patterns using physical objects and drawings • copies and extends simple number sequences • creates own patterns • describes observed patterns • identifies, describes and copies geometric patterns in natural and cultural artefacts 10b: • investigates and extends numeric and geometric patterns looking for relationship or rule • describes observed relationships or rules in own words • determines output values for given input values using verbal descriptions, flow diagrams tables (grade 6) • writes number sentences to describe a problem situation. • solves or completes number sentences by inspection or trial-and-improvement • determine the equivalence of different descriptions of the same relationship or rule presented verbally, in flow diagram by number sentences (grade 6) in tables (grade 6) 10c: • investigates and extends numeric and geometric patterns looking for a relationship or rule • describes, explains and justifies observed relationships in own words (or algebra – grade 8) • represents and uses relationships between variables in order to determine input and output values in a variety of ways using: verbal descriptions flow diagrams tables formulae and equations (grade 8 and 9) • solves or completes number sentences (grade 7) • solves equations by inspection, trial-andimprovement or algebraic processes (grade 8 and 9) • describes a situation by interpreting a graph of the situation, or draws a graph from the description of a situation (grade 7 and 8) • draws graphs on the cartesian plane for given equations, or determine equations from given graphs (grade 9) • determines, analyses and interprets the equivalence of different descriptions of the same relationship or rule (verbal, flow diagrams, tables, equations/expressions, graphs) does curriculum 2005 promote successful learning of elementary algebra? 30 table 11 assessment standards 11a: building up and breaking down number 11b: estimates and calculates by selecting and using operations appropriate to solving problems that involve addition, subtraction, multiplication, division, equal sharing with remainders, … 11c: • solve problems in context • solves problems that involve ratio and rate (and in grade 9, also proportion) • estimates and calculates by selecting and using operations appropriate to solving problems subaspects where/how addressed in curriculum lo phase(s) numbered statements assessment standard grade(s) comments and verdict regarding 2.1 – 2.4 f, i, s 8 10 1 – 9 comment: the lo focus explicitly states recognising and describing patterns and relationships from an early age, with increasing levels of sophistication (and thus generalisation and formalisation). f 12 13 14 10a 1; 2; 3 i 15 16 17 10b 4; 5; 66 s 18 19 20 21 22 23 10c 7; 8; 9 comment: in the foundation phase the emphasis is on the recognition and extension of patterns, while in the intermediate phase the focus shifts towards recognising the relationship between the place of a number in a sequence and its value (therefore viewing position and value as the two variables). in the senior phase these relationships are formally described using algebraic language (formulas / equations) and graphs. 2.4 is partially addressed by the assessment standard “…uses relationships between variables in order to determine input and output values … using formulae and equations (grade 8 and 9)”. however, it is not explicitly stated that substitution into formulas/algebraic expressions to find output values, and solving equations to determine input values are two inverse processes. still, the curriculum provides a solid context within which to introduce solving algebraic equations, namely as a means to find values of the input variable. 2.1 2.2 2.3 2.4 2 final verdict: 2.1 – 2.4 are thoroughly addressed in lo2. to a large extent 2.1 – 2.4 form the core of (elementary) algebra, developing within the context of patterns and relationships the rationale for algebraic notation such as letter symbols, algebraic expressions and equations, and their manipulation. table 10. aspect 2: enable learners to use and appreciate algebra as a means to describe relationships between quantities (lo2). nelis vermeulen 31 subaspect where/how addressed in curriculum lo phase(s) numbered statements assessment standard grade(s) comments and verdict regarding 3 f 4 11a 1; 2; 3 comment: young learners often intuitively calculate by breaking down numbers, a result of them modelling the (physical) problem they try to solve. verdict: although very far removed from algebra, the notion of modelling a problem using mathematics is formed, and should be encouraged, in this phase. i 6 11b 4; 5; 6 comment: solving problems implies modelling real-life situations, using mathematics to solve them. verdict: although very far removed from algebra, the notion of modelling a problem using mathematics is formed, and should be encouraged, in this phase. 1 s 7 11c 7; 8; 9 comment: solving problems imply modelling real-life situations, using mathematics to solve them. verdict: although lo1 does not refer to algebra as such, familiarity with problem solving, and therefore modelling, in a number context should pave the way for problem solving, and modelling, in a algebra context. table 11. aspect 3: enable learners to use and appreciate algebra as a tool to solve problems inside and outside mathematics, often through a modelling process (lo1). subaspect where/how addressed in curriculum lo phase(s) numbered statements assessment standard grade(s) comments and verdict regarding 3 f, i, s 8 9 verdict: the lo2 focus explicitly refers to solving problems within a patterns and relationship context. this implicitly refers to formulas/equations as a tool to solve problems. unfortunately, the latter aspect is not stated clearly enough. 2 s 19 20 21 22 23 24 25 26 12a 7; 8; 9 comment: the intention of the curriculum reflects the need for learners to use and appreciate algebra as a tool to solve problems inside and outside mathematics, often through the modelling process. verdict: 3 is adequately, however quite implicitly, addressed. table 12. aspect 3: enable learners to use and appreciate algebra as a tool to solve problems inside and outside mathematics (lo2). does curriculum 2005 promote successful learning of elementary algebra? 32 table 12 assessment standards 12a: • investigates and extends numeric and geometric patterns looking for a relationship or rule • describes, explains and justifies observed relationships in own words (or algebra – grade 8) • represents and uses relationships between variables in order to determine input and output values in a variety of ways using: verbal descriptions flow diagrams tables formulae and equations (grade 8 and 9) • solves or completes number sentences (grade 7) • solves equations by inspection, trial-andimprovement or algebraic processes (grade 8 and 9) • describes a situation by interpreting a graph of the situation, or draws a graph from the description of a situation (grade 7 and 8) • draws graphs on the cartesian plane for given equations, or determine equations from given graphs (grade 9) • determines, analyses and interprets the equivalence of different descriptions of the same relationship or rule (verbal, flow diagrams, tables, equations/expressions, graphs). conclusion it appears that the rncs (mathematics) to a fair extent succeeds in addressing current concerns and recommendations regarding the teaching and learning of (elementary) algebra, thereby fairly well aligning it with international curricula, notably that of the usa. particularly meritorious is the curriculum’s development of the notion of variables/-ility, relationships between variables, and the emphasis on increasing levels of sophistication to describe these relationships, culminating in symbolic algebraic notation (letter symbols, algebraic expressions and equations) and graphs. the curriculum designers should be congratulated with this attempt. there are however concerns and recommendations that are not explicitly addressed, as indicated in tables 1 to 12 under “verdict”. these may have been omitted either deliberately to prevent an overload of information in the curriculum document, or by mistake. whatever the case may be, teachers and teacher educators need to get the full picture, and therefore these omissions need to be addressed – either through additional documentation, or through training. another major concern is that in spite of a fairly complete curriculum statement regarding algebra, the question remains whether mathematics teachers and teacher educators are aware of the expectations and implications of this curriculum. do they read into the curriculum that algebra is not only a high school topic, but that its roots lie in and must be developed right from the foundation phase, through the intermediate and senior phases, with increasing levels of generalisation and symbolisation? recommendations based on what has been described so far, the following recommendations are made: • inset and preset programmes must address general education and training (get) teachers’ and student teachers’ beliefs about mathematics and the learning of algebra (and mathematics in general), and guide them towards a clear understanding of the nature of algebra and the learning of algebra, as intended by the curriculum. • learning programmes need to be designed to assist teachers to implement the intended algebra curriculum throughout the get phase. research needs to be conducted to inform and support these learning programmes, in particular regarding learners’ abilities to cope with the curriculum’s expectations and teachers’ abilities to implement such learning programmes. extensive research is currently being undertaken in the usa, where there is a strong drive to “algebraise” the “arithmetic curriculum” (rivera, 2006). while mathematics educators and researchers in the usa “welcome the integration of algebra into the early mathematics curriculum”, they state that “these ... do not diminish the need for research; quite the contrary, they highlight the need for a solid research base for guiding the mathematics education community along this new venture” (carraher, schliemann, brizuela & earnest, 2006). in south africa, this research field is still largely untouched. references bednarz, n., kieran, c. & lee, l. (eds.). (1996). approaches to algebra: perspectives for research and teaching. dordrecht: kluwer academic publishers. booth, l.r. (1986). difficulties in algebra. the australian mathematics teacher, 42(3), 2-4. nelis vermeulen 33 booth, l.r. (1988). children’s difficulties in beginning algebra. in a.f. coxford (ed.), the ideas of algebra, k-12 (pp 20-32). reston, nctm. booth, l.r. (1989). a question of structure. in s. wagner & c. kieran (eds.), research issues in the learning and teaching of algebra (pp 5759). reston: nctm. carraher, d.w., schliemann, a.d., brizuela, b.m. & and earnest, d. (2006). arithmetic and algebra in early mathematics education. journal for research in mathematics education, 37(2), 87-115. department of education. (2002). revised national curriculum statement grades r-9 (schools) policy. pretoria: government printer. herscovics, n. (1989). cognitive obstacles encountered in the learning of algebra. in s. wagner. & c. kieran (eds.). research issues in the learning and teaching of algebra (pp 6086). reston: nctm. kaput, j. (1999). teaching and learning a new algebra. in e. fennema & t. romberg (eds.). mathematics classrooms that promote understanding (pp 133-156). london: lawrence erlbaum. kieran, c. (1989). the early learning of algebra: a structural perspective. in s. wagner & c. kieran (eds.). research issues in the learning and teaching of algebra (pp 33-56). reston: nctm. kinzel, m.t. (1999). understanding algebraic notation from the students’ perspective. the mathematics teacher, 92(5), 436-442. küchemann, d. (1981) algebra. in k.m. hart (ed.), children’s understanding of mathematics (pp 102-110). london: john murray. matz, m. (1979). towards a process model for high school algebra errors. (working paper no. 181). cambridge: massachusetts institute of technology. national council of teachers of mathematics. (1994). a framework for constructing a vision of algebra. prepared by the algebra working group of the council of teachers of mathematics. reston, va: nctm. national council of teachers of mathematics. (2000). principles and standards for school mathematics. reston, va: nctm. orton, a. & frobisher, l. (1996). insights into teaching mathematics. london: cassell. rivera, f.d. (2006). changing the face of arithmetic: teaching children algebra. teaching children mathematics, 2(16), 206311. sfard, a. (1991). on the dual nature of mathematical conceptions: reflections on process and object as different sides of the same coin. educational studies in mathematics, 22, 1-36. vergnaud, g. (1989). multiplicative structures. in j. hiebert & m. behr (eds.). number concepts and operations in the middle grades (pp 141161). reston, virginia: nctm. vermeulen, c.f. (1995). verhoging van laerskoolleerlinge se vlak van bewustheid van die distributiewe eienskap in rekenkunde [increasing primary school learners’ level of awareness of the distributive property in arithmetic]. unpublished d ed thesis, university of stellenbosch. vermeulen, n. (2000). student teachers’ concept images of algebraic expressions. in t. nakahara & k. koyama (eds.). proceedings of the 24th conference of the international group for the psychology of mathematics education, volume 4 (pp 257-264). hiroshima: nishiki. “algebra reverses the relative importance of the factors in ordinary language. it is essentially a written language, and it endeavours to exemplify in its written structures the patterns which it is its purpose to convey. the pattern of the marks on paper is a particular instance of the pattern to be conveyed to thought. the algebraic method is our best approach to the expression of necessity, by reason of its reduction of accident to the ghostlike character of the real variable.” alfred north whitehead mathematics as a social construct pythagoras, 68, 3-14 (december 2008) 3 mathematics as a social construct: teaching mathematics in context hayley barnes and elsie venter university of pretoria hayley.barnes@up.ac.za and elsie.venter@up.ac.za why is teaching in context an important option to consider in the teaching of mathematics? what does it mean to teach mathematics from and in contexts? and what are the possible challenges associated with this practice? the aim of this paper is not to provide a comprehensive answer or solution to these questions. we attempt rather to address these questions specifically with regard to south africa and the theory of realistic mathematics education. in this article we consider a vignette of a more formal and traditional mathematics lesson and then suggest possible reasons why we need to be teaching more in context. furthermore we discuss the application of the theory of realistic mathematics education as a potential approach to facilitate teaching in context. finally we present some challenges associated with this practice. the philosophical shift that has occurred within the domain of mathematics has brought with it a wave of reform in mathematics education. the former absolutist paradigm that dominated undermined the social responsibility of mathematics in human affairs such as value, wealth and power (ernest, 1991). the shift has challenged the infallibility of mathematics and acknowledged it as a product of human inventiveness (davis & hersh, 1980) and a human activity (freudenthal, 1973), thus making it a social construct. while the reform propagating the teaching of mathematics as a social construct is a positive move, along with the introduction of mathematical literacy in many countries, how can we effectively implement this reform? this paper seeks to examine the area of teaching mathematics in context (specifically in relation to the theory of realistic mathematics education) to support this reform and the implementation thereof in school and tertiary education. a vignette of a traditional formal mathematics lesson is first simulated, followed by a discussion of mathematics as a social construct. the relevance of teaching in context is then explored while alluding to the theory of realistic mathematics education as a vehicle through which this can be done. examples of studies conducted at secondary and tertiary level are considered, concluding with some challenges of teaching in context. a lesson simulation the following vignette of a lesson will probably be familiar to many readers: today’s lesson is going to look at rounding off decimals to the nearest whole number. let us first revise the concept of rounding off. remember that rounding off helps us in estimation. last year you learnt to round off to the nearest ten, hundred and thousand. you did this by looking at which ten, hundred or thousand the number you are rounding off is closest to. let’s look at an example. write this down in your books if you have forgotten how to round off: example 1: round 63 off to the nearest ten. on the number line 63 lies between 60 and 70. it lies closer to 60 though so we round it off to 60, as the nearest ten. remember also that in this case, if the units digit is less than the number 5, we round down to the nearest ten. example 2: round 2 499 off to: a) the nearest ten b) the nearest hundred c) the nearest thousand a) let us start by looking at the units digit. it is more than 5 so we round up to the nearest ten. the answer is therefore: 2 500 mathematics as a social construct: teaching mathematics in context 4 b) now we need to look at the tens digit because we are rounding off to the nearest hundred. it is also more than 5 so we round up to the next hundred. answer: 2 500 c) for rounding off to the nearest thousand we look at the hundreds digit. it is less than 5 so we round down to the nearest thousand. answer: 2 000 now let us do one with decimals. example 3: round 4,25 off to the nearest unit or whole number. 4,25 lies on the number line between the units/whole numbers 4 and 5. it is closer to 4 though. also when rounding off the nearest whole number, we look at the tenths digit. it is less than 5 so we round down to the nearest whole number which is 4. does everyone understand? are there any questions? now let us use what we have learnt about rounding off to help us solve some problems: please solve the following: 1. 17 ÷ 4 = ? round your answer off to the nearest whole number. 2. mr farmer decides to share his 17 cows evenly between his 4 children. to avoid conflict, each child must receive the same number of cows. how many cows will each child get? if you have the class work correct, please continue with the homework which is page 23 of your textbook, numbers 1 – 8. in the approach applied in the lesson vignette above, mathematics is viewed as a ready-made system with general applicability. consequently, mathematics instruction is seen as a process of breaking up formal mathematical knowledge into learning procedures and then learning to use them accordingly. let us examine another similar problem to example 3 above: the problem: 17 people are trapped on a mountain and need to be rescued by helicopter. the helicopter can take a maximum of 4 passengers at a time, in addition to the pilot. how many trips will the helicopter need to make? this example illustrates where conventional mathematics as we perform it outside of any prescribed context, can actually support or conflict with the answer depending on the context of the problem. in solving the problem above, it is the context that must take preference over the mathematical convention of rounding off “down” to the nearest whole number when our indicator is below 5. but when we teach mathematics predominantly formally and outside of context, do our students learn to know the difference between conventional mathematics and mathematics as a tool operating within a social context? mathematics as a social construct formerly (prior to the 20th century) an absolutist view of mathematical knowledge dominated the philosophy of mathematics education. according to ernest (1991), this view accepts that mathematics consists of absolute and unchallengeable truths that can be regarded as certain knowledge based on two types of assumptions in terms of the actual mathematics (axioms and definitions) and logic (axioms, rules of inference and the formal language and its syntax). early in the twentieth century a number of antinomies and contradictions, mainly in the theory of sets and functions, were derived in mathematics (kline, 1980; kneebone, 1963; wilder, 1965 as cited in ernest, 1991), which caused a crisis within this absolutist paradigm. the certainty of mathematics and its theorems was challenged by the appearance of these contradictions (i.e., falsehoods) resulting in the development of a number of schools in the philosophy of mathematics. these schools aimed to account for the nature of mathematical knowledge and to reestablish the certainty thereof. ernest (1991) identifies the three major schools as being logicism, formalism and constructivism. without elaborating on each school, it suffices to say that the former absolutist paradigm that dominated previously, undermined the social responsibility of mathematics in human affairs such as value, wealth and power (ernest, 1991). the shift has challenged the infallibility of mathematics and acknowledged it as a product of human inventiveness (davis & hersh, 1980) and a human activity (freudenthal, 1973), thus making it a social construct. ernest (1991, p. 42) specifies the grounds for describing mathematical knowledge as a social system: hayley barnes & elsie venter 5 1. the basis of mathematical knowledge is linguistic knowledge, conventions and rules, and language is a social construction. 2. interpersonal social processes are required to turn an individual’s subjective mathematical knowledge, after publication, into accepted objective mathematical knowledge. 3. objectivity itself will be understood to be social. it therefore draws on conventionalism, in accepting that human language, rules and agreement play a central role, including the view that mathematical concepts develop and change. it also includes lakatos’ philosophical thesis that mathematical knowledge grows through conjectures and refutations. the above therefore constitutes the perspective from which the following questions are discussed in this paper: • why is teaching in context an important option to consider in the teaching of mathematics? • what does it mean to teach mathematics from and in context? • what are the possible challenges associated with this practice? the relevance of teaching in context within the scope of this paper, two main answers to the first question are proposed. the first stems directly out of the shift in mathematics education already discussed from an absolutist paradigm to a more social constructivist view. this shift to emphasizing mathematics as a social construct is certainly supported by and demonstrated in the following definition of mathematics provided in the revised national curriculum statement (rncs) by the department of education (doe, 2002, p. 4): mathematics is a human activity that involves observing, representing and investigating patterns and quantitative relationships in physical and social phenomena and between mathematical objects themselves. through this process, new mathematical ideas and insights are developed. mathematics uses its own specialised language that involves symbols and notations for describing numerical, geometric and graphical relationships. mathematical ideas and concepts build on one another to create a coherent structure. mathematics is a product of investigation by different cultures – a purposeful activity in the context of social, political and economic goals and constraints. to fully embrace the extent of this definition and to realise the goals intended by the introduction of mathematical literacy as a compulsory subject, we need to consider teaching mathematics from and in context rather than in its absolute form for the purpose of later applying in contexts. the second response to the first question lies within the results of both national as well as international studies carried out in mathematics education. in the systemic evaluation, south africa’s grade 3 and grade 6 learners performed poorly on the national average. in the 1999 and 2003 results of the trends in international mathematics and science study (timss, 2003) south africa once again was placed last out of the fifty countries that participated. although a detailed discussion and analysis of factors causing such overall poor performance is beyond the scope of this paper (see howie, 2001; 2002), it can be concluded that south african learners certainly struggle far more than the rest of the world when required to perform mathematics within a context. figures 1 shows examples of selected contextual items from timss 2003. these items were identified (as examples not as conclusive proof) according to their applicability to contexts from everyday life that one might encounter. the performance of south african grade 8 learners on the items in comparison to the international average and that of selected countries are also provided. teaching in and from context the theory of realistic mathematics education in order to examine the second question, we draw on the work currently being done by the freudenthal institute in the netherlands. they have been leaders in introducing and researching teaching mathematics in and from contexts in their theory known as realistic mathematics education (rme). realistic mathematics education has its roots in freudenthal’s interpretation of mathematics as a human activity (freudenthal, 1973; gravemeijer, 1994). to this end, freudenthal accentuated that the actual activity of doing mathematics; an activity which he propagated, should predominantly consist of organising or mathematising subject matter taken from reality. learners should therefore learn mathematics by mathematising subject matter from real contexts and their own mathematical activity rather than from the traditional view of presenting mathematics as a social construct: teaching mathematics in context 6 item 1 a garden has 14 rows. each row has 20 plants. the gardener then plants 6 more rows with 20 plants in each row. how many plants are now there altogether? answer: _________________________ item 2 a car has a fuel tank that holds 45  of fuel. the car consumes 8,5  of fuel for each 100 km driven. a trip of 350 km was started with a full tank of fuel. how much fuel remained in the tank at the end of the trip? a. 15,25  b. 16,25  c. 24,75  d. 29,75  item 3 the graph represents the distance and time of a hike taken by joshua and liam. if they both started from the same place and walked in the same direction, at what time did they meet? a. 8:00 b. 8:30 c. 9:00 d. 10:00 e. 11:00 performance on the items int. ave rsa tunisia morocco egypt netherlands nz usa item 1 61,2 17,7 48,4 38,5 36,1 88,1 76,5 78,4 item 2 26,0 18,0 18,7 18,9 20,6 37,9 23,6 24,8 item 3 62,4 19,3 14,5 47,3 81,2 72,3 79,7 figure 1: items from timss (source: timss, 2003) hayley barnes & elsie venter 7 mathematics to them as a ready-made system with general applicability (gravemeijer, 1994). these real situations can include contextual problems or mathematically authentic contexts for learners where they experience the problem presented as relevant and real. the verb mathematising or its noun mathematisation implies activities in which one engages for the purposes of generality, certainty, exactness and brevity (gravemeijer, cobb, bowers & whiteneack, as cited in rasmussen & king, 2000). through a process of progressive mathematisation, learners are given the opportunity to reinvent mathematical insights, knowledge and procedures. in doing so learners go through stages referred to in rme as horizontal and then vertical mathematisation (see figure 2). horizontal mathematisation is when learners use their informal strategies to describe and solve a contextual problem and vertical mathematisation occurs when the learners’ informal strategies lead them to solve the problem using mathematical language or to find a suitable algorithm (treffers, 1987). for example, in what we would typically refer to as a “word sum”, the process of extracting the important information required and using an informal strategy such as trial and error to solve the problem, would be the horizontal mathematising. translating the problem into mathematical language through using symbols and later progressing to selecting an algorithm such as an equation could be considered vertical mathematisation, as it involves working with the problem on different levels. the traditional formal and authoritarian approach to teaching mathematics that has dominated in south african classrooms for a number of years has not afforded learners many opportunities to make use of horizontal mathematisation. mathematics lessons are often presented in such a way that the learners are introduced to the mathematical language relevant to a particular section of work and then shown a few examples of using the correct algorithms to solve problems pertaining to the topic before being given an exercise of worksheet to complete (venter, barnes, howie, & janse van vuuren, 2004). the exercises or worksheets are usually intended to allow learners to put the algorithms they have been taught into practice and may even contain some contextual problems that require the use of these algorithms. according to the rme model depicted in figure 4, this type of approach places learners immediately in the more formal vertical mathematisation process. the danger in this is that when learners have entered that process without first having gone through a process of horizontal mathematisation, a strong possibility exists that if they forget the algorithms they were taught, they do not have a strategy in place to assist them in solving the problem. low attainers often exhibit this lack of strategy. this experience can be equated to someone being shown and told what is on the other side of a river and being expected to use what is there for their own benefit. however, they are not given or shown the bridge that assists one in crossing to the other side in order to make proper use of what is there. the horizontal mathematisation process provides this bridge (barnes, 2004). when embarking on solving a contextual problem using formal mathematical knowledge, the following steps are usually followed. first the problem needs to be translated from its contextual state into mathematical terms. available mathematical means are then drawn on in order to solve the problem, which then needs to be translated back into the original context. this process can be illustrated by the example we encountered earlier on: horizontal mathematisation ( ); vertical mathematisation ( ) figure 2: horizontal and vertical mathematisation source: adapted from gravemeijer, 1994. contextual problem mathematical language describing solving algorithm mathematics as a social construct: teaching mathematics in context 8 the problem: 17 people are trapped on a mountain and need to be rescued by helicopter. the helicopter can take a maximum of four passengers at a time, in addition to the pilot. how many trips will the helicopter need to make? translation into mathematical terms: 17 people / 4 per trip mathematical means to solve problem: 17 ÷ 4 = 4 remainder 1 or 4 1 4 or 4.25 translation back into original context: helicopter will need to make 5 trips on the other hand, in the rme problem-centred approach, the problem, rather than the use of a specific mathematical tool, is the actual aim. instead of trying to formalise the problem into mathematical terms, the learners are encouraged to describe the problem in a way that makes sense to them. this can involve using their own selfinvented symbols or pictures and identifying the central relations in the problem situation. in this way the problem is also simplified for the learner. because the symbols are meaningful for the problem-solver, further translation and interpretation of the problem is easier and using a standard procedure is not mandatory. in boxes 1 and 2 below examples of contextual problems given to grade 8 learners in a study using the theory of rme is presented (barnes, 2004). some responses from learners who took part in the study are then offered as a means to demonstrate the concept of mathematisation. the above-mentioned study took the form of a case study of 12 participants in an urban school in south africa. the learners were identified by their mathematics teachers as low attainers in mathematics. they were each part of a remediation programme in mathematics that these learners took part in while the rest of their class were enrolled for the subject of a third language. the participants were part of a mathematics intervention for two lessons per week for approximately three school terms. the intervention was designed and taught based on the principles of realistic mathematics education. in summary the theory of rme proved to be viable tool in teaching mathematics to low attainers (barnes, 2004). box 1 lesson on the cat’s pills my cat’s recent diagnosis of diabetes initiated this problem, which served as an introductory contextual problem in revisiting the concept of fractions. a discussion on what diabetes is and how it occurs in humans and cats was first embarked on with learners as an introduction (learners had little knowledge of diabetes and were not easy to convince that cats also get diabetes). learners were then presented with and asked to solve the following problem (either in groups or individually; they were given the choice): problem: my cat needs to take two types of pills and an insulin injection twice a day to control its diabetes. the cat takes half a big pill in the morning and again in the evening and a quarter of a small pill also in the morning and again in the evening. firstly, the vet has given me 17 big pills and 27 small pills to start off with, how many days will these pills last me for before i have to go back to get more? secondly, how many of each pill should i buy each month so that they last me for a whole month? about ten minutes before the end of the lesson, some learners were asked to demonstrate and explain their solutions to the class and a short whole class discussion on these explanations was held before the class was dismissed. the solutions offered by liya to the contextual problem provided in box 1 are shown in figures 3 and 4 to exemplify this (rme) process of problem solving. there are two parts to the solution as liya first tried part a (figure 3) and then realised from her answers that something was not right and then proceeded to do the solution in figure 4. it is also interesting to note how her part a solution more closely resembles the steps usually followed when using formal mathematical knowledge in an information processing approach. as these learners were probably more accustomed to using the more formal approach, and obviously had some formal knowledge in place regarding fractions, they initially often tried to go through the steps of translation into mathematical terms, searching for an adequate mathematical procedure to solve the problem and then translating it back to the context. in doing so though, it was noticed that some learners did not really have a grip on which mathematical procedure to use and even when they chose the correct one (sometimes by chance as they could not justify their decisions), they made mistakes in executing them (as can be seen in part a of liya’s solution). hayley barnes & elsie venter 9 during the course of the study (barnes, 2004) an effort was continually made to encourage learners to go through the rme approach of simplifying the contextual problem by first representing it in their own symbols and/or words and then further solving and interpreting it from there. when some of them started to do this, they found that they could more often solve the problem, using their informal strategies rather than formal procedures they were unsure of. this does not mean that they never used formal procedures or any mathematical means but that they were expected to only use them at a point in the problem-solving process when they could justify the use thereof and demonstrate an understanding of the application in that regard. box 2 you decide to start making banana bread to sell in order to earn some extra money. to start off with, you decide to make 5 loaves of banana bread. according to the recipe, each loaf requires 4 1 2 bananas. how many bananas will you need to make the 5 loaves of banana bread? show your working out in the space provided below. as can be seen in zwanela’s solution (figure 5), she correctly selected multiplication as her strategy. however, when she executed the actual multiplication, she tried to change the mixed number into an improper fraction and in doing so “lost” the denominator and got 9 instead of 9 2 , rendering her final answer incorrect. zwanela often resorted immediately to vertical mathematisation in that she searched for the “correct” formal procedure to apply. in contrast to zwanela’s more formal solution, the use of horizontal mathematisation is more evident in the solution from violet (figure 6). gravemeijer (1994) explains that by getting learners to solve a sequence of similar problems, another process is induced. the problem descriptions develop into an informal language, which is further simplified and formalised into a more formal mathematical language eventually. a similar process could be experienced in terms of the solving procedure, where solving similar kinds of problems becomes routine and actual algorithms take shape. through this learning process, formal mathematical knowledge itself can be constructed. figure 3: part a of liya’s solution to the cat pills problem figure 4: part b – liya’s second attempt at solving the cat pills problem mathematics as a social construct: teaching mathematics in context 10 rme at tertiary level international studies (doorman, 2001; streefland, 1991; treffers, 1987; vos, 2002), including studies from developing countries such as indonesia (armanto, 2002; fauzan, 2002) have shown that the rme theory is a promising direction to improve and enhance learners’ understanding in mathematics in primary and secondary schools. so how can the theory of rme be employed in tertiary education? kwon (2002) provides one example of how an rme design could be used for teaching an introductory course in differential equations at first year university level. the teaching experiment was conducted with a group of 43 students and data consisted of videotapes, field notes, copies of students’ work and records of instructional activities and decisions. materials implemented during the course were informed by rme instructional heuristic and designed to assist students to complete reinvention activities through devising their own ways of working through mathematical concepts (also known as the inquiry-oriented approach). an explicit intention of kwon’s project was to create a learning environment where students routinely offered explanations of and justifications for their explanations. a typical class period entailed students working in groups of two to four on a task presented by the instructor. cycles of whole class discussions and further group work then followed. a continuous emphasis on reasoning and whole class discussions resulted in key concepts emerging. in a later evaluation study (rasmussen, kwon, allen, marrongelle & burtch, 2004 as cited in kwon, 2005) two groups of students from four undergraduate institutions in korea and the usa were compared in terms of their understanding of central ideas and analytical methods relating to differential equations. one group was exposed to an inquiry-orientated approach based on the theory of rme. the other group, however, was taught in the traditional lecture-based manner, where a typical class consists of a review of previous work, presentation of new work and some time for students to work on their own or ask for assistance (romberg, 2001). there was no significant difference between the two groups on routine problems. however, the inquiry-orientated group did score significantly higher than the comparison (traditional lecture-based) group on the conceptual problems. figure 5: zwanela’s solution to the banana bread problem figure 6: violet’s solution to the banana bread problem hayley barnes & elsie venter 11 even after a delayed post-test a year later, kwon (2005) inferred that retention rates on procedural tasks showed no significant difference, but that retention on conceptual tasks were significantly different between the groups. once again the inquiry-orientated group scored significantly higher than the traditional group on the conceptually oriented items. in addition, kwon found that all the inquiry-oriented students out-performed the traditional group regardless of academic background or gender. this finding has important implications for the south african context when transition from school level practices to that of tertiary levels is made, especially when the quality of diverse school experiences are considered. as also demonstrated by skemp’s (1971) distinguished between relational and instructional understanding, a chasm may exist between what students are able to do and what they in fact understand. knowing what to do in a specific situation, but not necessarily understanding why it works, may limit the transfer of that procedure or skill. the increasing number of procedures that students need to commit to memory in mathematics often results in learners in secondary school and students at tertiary level becoming confused or partly remembering and trying to apply procedures they have never fully understood (daniels & anghileri, 1995). understanding on the other hand promotes remembering and enhances transfer owing to the reduced number of bits of knowledge that need to be simultaneously held in the short-term memory (hiebert & carpenter, 1992). the understanding that comes from making connections, seeing how things fit together, relating mathematics to real situations and articulating patterns and relationships also carries with it a satisfaction which can further motivate students (haylock, 1991). exposure to constructing mathematics at an early level could promote the construction of mathematics at higher levels and generating of new knowledge to the domain. compared to the more formal approach of teaching mathematics we have become accustomed to, teaching in context appears to be a vehicle through which our would-be mathematicians can express and develop themselves mathematically and thus enrich the south african community in terms of financial, economical and scientific models for living. some challenges although a necessary and positive shift, teaching in context does not come without its challenges. in a heterogeneous society such as in south africa where we have eleven official languages, the issue of language comes to the fore as a potential hurdle. research pertaining to proficiency in language as a factor in the learning of mathematics is certainly available and still currently being conducted (howie, 2002; setati, 2002). conclusive evidence of the impact of the language of learning and teaching not being in the mother-tongue is not yet available though and remains a controversial issue. when teaching in context, it is obviously important that the language of the problem be accessible to the students. in our situation visual aids, such as graphs, tables and diagrams can assist students in this regard. employing simple language that is not ambiguous is also necessary. also relating to language is the actual selection of the context from which the development of certain mathematics is to evolve. the context of the problem needs to be accessible within the framework of the student so that they are able to set about solving the problem, rather than getting lost or caught up in the context. kwon (2005) also makes the point that the tasks need to be carefully selected or designed as learning is not necessarily implied through solving a sequence of problems. this in turn calls for a high level of subject matter knowledge and understanding of the substantive and syntactic structures of the discipline of mathematics (grossman, wilson & shulman, 1990) for both designers and teachers. linking to this is the issue of material development. as bowie and frith (2006) discovered in their process of developing mathematical literacy materials, when one attempts to mathematise a context, it is necessary to have a good understanding, not only of mathematics, but also of the context. mathematical literacy teachers in south africa already therefore need to familiarise themselves with contexts such as “voting systems, mortgages, retirement funding, hiv/aids, global positioning systems and socially responsible trade (to name but a few of the contexts suggested in the current curriculum)” (bowie & frith, 2006, p.33). the learners and students themselves also need to be able to understand the context and in our diverse country we cannot, for example, assume that all learners are familiar with the notion of a formal banking system or the concept of risk and return on investments. in order to successfully implement rme, this issue of appropriate contexts will also need to be resolved. bowie and frith (2006) suggest integration of knowledge and skills across mathematics as a social construct: teaching mathematics in context 12 subjects and terrains of practice as a possible solution in addressing this issue. they stress that this integration needs to be worked in at a curriculum level in order for it to be taken seriously. but our teachers are under enough pressure in coping with the constant curriculum changes. it is our opinion that we need to start supplying teachers with materials that enable them to teach using an rme approach. at school level a number of learning materials based on the theory of rme are already available in the netherlands, the united states and the united kingdom. we could consider recontextualising these for a south african context. at tertiary level, not many such materials have been developed. however, a strong developmental research partnership between mathematicians and mathematics educators could facilitate such a process. wittmann (1998 as cited in julie, 2006) identifies the central task of mathematics education, as a field of study, as developing learning resources for productive and meaningful learning. he argues that teachers do not have the time to do this and that one of the main roles of mathematics educators is to therefore carry out the didactical analysis of subject matter in designing the resources. this is done through “theorisation and thought-experimentation which leads to hypothetical learning trajectories” (julie, 2006: 64). these hypothesised trajectories then need to be trialled, researched and further developed within real classroom situations in order to decrease the distance between the intended and implemented curriculum (julie, 2006). the rme materials mentioned above have been developed in such a manner within their respective countries (see gravemeijer, 1998; gravemeijer & cobb, 2002; treffers & goffree, 1985). a replication of a similar process in south africa would help us to re-contextualise the materials for a south african context as well as assist in capacity building where classrooms become research domains and sites of experimental implementation (julie, 2006). finally, having discussed language, contexts and material development, we cannot omit our main challenge (and source of success) in implementing a theory such as realistic mathematics education, namely teachers. curriculum developers can produce material in context to be used in mathematics classrooms, but this does not necessarily mean that teachers will implement these as intended. for teaching mathematics in context, a methodology change is required that challenges the learners to become more independent thinkers in order to become better problem-solvers as well as mathematicians. the theory of rme encourages an approach that treats each individual student of mathematics as a mathematician with the capacity to mathematise contexts into mathematical problems that can be solved (freudenthal, 1983). teachers who attempt to teach mathematics in context through a more traditional approach of giving an example and then expecting learners to practise applying the tools to a range of contextual problems have missed the point. it is therefore necessary in training mathematics teachers to teach in context, to develop the skills needed to be able to relate mathematics and context (brown & schafer, 2006) in a more problem-solving or inquiry oriented approach. an important skill that needs to be foregrounded in this regard is that of interpreting mathematical concepts and skills in relation to a context (brown & schafer, 2006). having not learnt mathematics at school or university through such an approach, we cannot assume that our mathematics teachers, even those proficient in mathematics, have mastered this skill. equipping teachers to teach in context will therefore require extensive and continued training. academic institutions willing to form partnerships with schools where mathematics educators, mathematicians and mathematics teachers could work together in developing, implementing and researching an approach of teaching in context, would go a long way to making such a venture successful. conclusion this paper has sought to examine the issue of teaching mathematics in context as a social construct, using the theory of realistic mathematics education. we are of the opinion that our current mathematical practises are still not foregrounding the shift from the absolutist paradigm to social constructivism and adequately empowering our learners. this has further implications for them as students at tertiary institutions as well as effective citizens within our democracy. while teaching in and from context is not without its challenges, especially within our diverse society, it appears to hold potential as a vehicle with which to address this problem, both at school and tertiary level. mathematics lecturers are encouraged to reflect on their own practices in the teaching of mathematics and to consider partnering with mathematics educators (from schools and tertiary institutions) to work on designing classroom experiments that engender a culture of mathematising amongst our students. hayley barnes & elsie venter 13 references armanto, d. (2002). teaching multiplication and division realistically in indonesian primary schools: a prototype of local instruction theory. unpublished doctoral dissertation. university of twente, the netherlands. barnes, h. e. (2004). a developmental case study: implementing the theory of realistic mathematics education with low attaining learners. unpublished masters dissertation. university of pretoria, south africa. bowie, l. & frith, v. (2006). concerns about the south african mathematical literacy curriculum arising from experience of materials development. pythagoras, 64, 29-36. brown, b. & schäfer, m. (2006). teacher education for mathematical literacy: a modelling approach. pythagoras, 64, 45-51. daniels, h. & anghileri, j. (1995). secondary mathematics and special educational needs. london: cassell. davis, p.j. and hersh, r. (1980). the mathematical experience. harmondsworth: penguin. department of education (2002). annual report. retrieved 21 march 2003 from http://education.pwv.gov.za/policies%20and%20 reports/2002_reports doorman, m. (2001). how to guide students? a reinvention course on modeling movement. paper presented at the netherlands and taiwan conference on common sense in mathematics education, taipei, taiwan. retrieved 22 april 2003 from www.fi.ruu.nl/en/publications.shtml ernest, p. (1991). the philosophy of mathematics education. hampshire: falmer press. fauzan, a. (2002). applying realistic mathematics education (rme) in teaching geometry in indonesian primary schools. unpublished doctoral dissertation. the netherlands, enschede: university of twente. freudenthal, h. (1973). mathematics as an educational task. dordrecht: reidel. freudenthal, h. (1983). didactical phenomenology of mathematical structures. dordrecht: reidel. gravemeijer, k., & cobb, p. (2002). designing classroom-learning environments that support mathematical learning. paper presented at the american educational research association conference in april 2001. gravemeijer, k.p.e. (1994). developing realistic mathematics education. utrecht: freudenthal institute. gravemeijer, k.p.e. (1998). developmental research as a research method. in a. sierpinska & j. kilpatrick (eds.), mathematics education as a research domain: a search for identity (book 2, pp. 277-295). dordrecht: kluwer. grossman, p.l., wilson, s.m., & shulman, l.s. (1990). teachers of substance: subject matter knowledge for teaching. in m. reynolds (ed.), knowledge base for the beginning teacher. new york: pergamon. haylock, d. (1991). teaching mathematics to low attainers, 8-12. london: paul chapman. hiebert, j., & carpenter, t.p. (1992). learning and teaching with understanding. in d. gouws (ed.), handbook for research on mathematics teaching and learning (pp. 65-97). new york: macmillan. howie, s.j. (2001). mathematics and science performance in grade 8 in south africa 1998/1999: timss-r 1999 south africa. pretoria: human sciences research council. howie, s.j. (2002). english language proficiency and contextual factors influencing mathematics achievement of secondary school pupils in south africa. unpublished doctoral dissertation. university of twente, the netherlands. julie, c. (2006). mathematical literacy: myths, further inclusions and exclusions. pythagoras, 64, 62-69. kwon, o. (2002). conceptualizing the realistic mathematics education approach in the teaching and learning of ordinary differential equations. paper presented at the 2nd international conference on the teaching of mathematics, greece. retrieved 8 december 2009 from www.eric.ed.gov/ericdocs/data/ericdocs2sql/co ntent_storage_01/0000019b/80/1a/ad/9d.pdf kwon, o. (2005). towards inquiry-oriented mathematics instruction in the university. proceedings of kaist international symposium on enhancing university mathematics teaching, may 2005, daejeon, korea. retrieved 8 december 2009 from http://mathnet.kaist.ac.kr/real/2005/8/ohnamkwo n.pdf rasmussen, c.l., & king, k.d. (2000). locating starting points in differential equations: a realistic mathematics education approach. international journal of mathematical education in science and technology, 31, 161-173. romberg, t. a. (2001). mathematical literacy: what does it mean for school mathematics. wisconsin school news, october 2001. setati, m. (2002). language practices in multilingual classrooms in south africa. unpublished doctoral dissertation. university of the witwatersrand, south africa. skemp, r. (1971). the psychology of learning mathematics. harmondsworth, uk: penguin. skemp, r. (1989). mathematics in the primary school. london: routledge. streefland, l. (1991). a paradigm of developmental research. dordrecht: kluwer academic. mathematics as a social construct: teaching mathematics in context 14 timss (2003). released items: eighth grade mathematics. retrieved 2 november 2006, from http://timss.bc.edu/pdf/t03_released_m8.pdf treffers, a. (1987). three dimensions – a model of goal and theory description in mathematics instruction. dordrecht: kluwer academic. treffers, a., & goffree, f. (1985). rational analysis of realistic mathematics education – the wiskobas program. in l. streefland (ed.), proceedings of the ninth international conference for the psychology of mathematics education, 2, 97-121. venter, e., barnes, h., howie, s.j, & jansen van vuuren, s. (2004). mpumalanga secondary science initiative – learner progress research project. pretoria: centre for evaluation and assessment. vos, p. like an ocean liner changing course: the grade 8 mathematics curriculum in the netherlands, 1995-2000. unpublished doctoral dissertation. university of twente, the netherlands. http://www.pythagoras.org.za open access page 1 of 1 reviewer acknowledgement acknowledgement to reviewers in an effort to facilitate the selection of appropriate peer reviewers for pythagoras, we ask that you take a moment to update your electronic portfolio on https://pythagoras. org.za for our files, allowing us better access to your areas of interest and expertise, in order to match reviewers with submitted manuscripts. if you would like to become a reviewer, please visit the journal website and register as a user. in order to be considered, please email submissions@ pythagoras.org.za indicating your intention to register as a reviewer for the journal. to access your 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https://pythagoras.org.za/index.php/pythagoras/user https://pythagoras.org.za/index.php/pythagoras/user mailto:publishing@aosis.co.za http://d.de generalizing the nagel line to circumscribed polygons by constructive analogy 32 pythagoras, 68, 32-40 (december 2008) generalizing the nagel line to circumscribed polygons by analogy and constructive defining michael de villiers school of science, mathematics & technology education, university of kwazulu-natal profmd@mweb.co.za this paper first discusses the genetic approach and the relevance of the history of mathematics for teaching, reasoning by analogy, and the role of constructive defining in the creation of new mathematical content. it then uses constructive defining to generate a new generalization of the nagel line of a triangle to polygons circumscribed around a circle, based on an analogy between the nagel line and the euler line of a triangle. the purpose of this paper is to heuristically present a new generalisation of the nagel line of a triangle to polygons circumscribed around a circle by making use of an interesting analogy between the nagel line and the euler line of a triangle (de villiers, 2006). i hope that by describing the invention process to perhaps contribute in some small way to teachers’ perspectives on the nature of mathematics, namely an understanding of how new mathematics is sometimes created. this may also hopefully assist and motivate teachers to attempt to recreate similar learning experiences for their students with content and concepts appropriate to their level. therefore, in order to place the invention process in a broader perspective relevant to mathematics education, i will first discuss the genetic approach, reasoning by analogy, and constructive defining, before the generalisation of the nagel line to circumscribed polygons is presented. the genetic approach like the mathematician george polya, i always wondered as a student at high school, and even to some degree at undergraduate level at university, how all the mathematics we were learning – the efficient algorithms, beautiful theorems and sometime ingenious proofs – had been discovered or invented (polya, 1945). it often seemed to me mysterious and amazing, but sometimes also capricious and intimidating, to be taught theorem after theorem, followed by carefully presented proofs, without been given any insight into how they were found, nor what may have motivated them in the first place. faced with an unrelenting barrage of mathematics presented in this way, i sometimes felt i had little choice but to succumb to dutiful memorisation and rote learning in order to pass the examinations. one of the first books that i read that seriously attempted to demystify the origins of mathematics (and that has had a life-long influence on me), was polya’s famous 1945 book how to solve it, which concentrate on the processes of mathematics, not the end-results or products. this focus on the processes of mathematics is in correspondence with what is often called the ‘genetic’ approach in mathematics education and was already propounded by the german mathematician felix klein (1924): people often think that mathematics can, or even should be, taught purely deductively, in the sense that a certain set of axioms should serve as a starting point and everything else should be derived logically from it. this approach is often justified on the authority of euclid, but does not correspond to the historical development of mathematics. ( p. 17) (the teacher) ought … to lead pupils slowly along the same path to higher ideas and finally to abstract formulations, as those that the human race in general followed from a naïve primitive state to knowledge at higher, abstract level. (p. 290) michael de villiers 33 likewise, freudenthal (1973) has criticised a deductive teaching approach, calling it a “didactical inversion” of the historical process, and that it was in fact was anti-didactical. he propagated “mathematics as an activity”: students should be given the opportunity to “re-invent” mathematics for themselves. toeplitz (1963, p. v) eloquently describes the motivation behind the genetic approach and how it may help to demystify mathematics: regarding all these basic topics in infinitesimal calculus which we teach today as canonical requisites, e.g., mean-value theorem, taylor series, the concept of convergence, the definite integral, and the differential quotient itself, the question is never raised “why so?” or “how does one arrive at them?” yet all these matters must at one time have been goals of an urgent quest, answers to burning questions, at the time, namely, when they were created. if we were to go back to the origins of these ideas, they would lose that dead appearance of cut-anddried facts and instead take on fresh and vibrant life again. however, for toeplitz it is clearly not about a complete dramatisation of the history of mathematics: the historian – the mathematical historian as well – must record all that has been, whether good or bad. i, on the contrary, want to select and utilize from mathematical history only the origins of those ideas which came to prove their value ... it is not history for its own sake in which i am interested, but the genesis at its cardinal points, of problems, facts, and proofs. (toeplitz, 1963, p. v) with reference to klein’s genetic principle, hull (1969, p. 29) similarly writes: it certainly suggests that the pattern of past mathematical discovery should be closely studied in relation to any proposed pattern of individual learning. this does not mean a slavish adherence to the historical order of development in every detail or topic; but it may well mean that the broad general lines along which the race has conducted its creative thought are also the lines along which children can most naturally learn. it is therefore encouraging that the revised national curriculum statement (rncs) (doe, 2002) acknowledges the importance of the history of mathematics by stating that the mathematics learning area should develop “an appreciation for the diverse historical, cultural and social practices of mathematics” (p. 13, bold added), and later on gives several examples like the following: grade 5 assessment standards: describes and illustrates various ways of writing numbers in different cultures (including local) throughout history. (p. 49) grade 8 assessment standards: describes and illustrates ways of measuring in different cultures throughout history (e.g. determining right angles using knotted string, leading to the theorem of pythagoras). (p. 95) however, as shown above, the rncs unfortunately seems to restrict the use and importance of the history of mathematics to only the incorporation of a few historical examples here and there. more generally, one can distinguish at least four ways in which the history of mathematics can be utilised by a teacher: 1) a complete presentation of the historical development of a concept, algorithm or theorem 2) an abbreviated presentation of the most significant moments of the historical development in chronological order 3) a presentation which includes no historical material, but is nonetheless based on an analysis of the historical development of the particular concepts, algorithms and theorems (the indirect, genetic method) 4) a presentation which is not based on the actual historical development of the particular concepts, algorithms and theorems, but attempts to simulate with the advantage of hindsight, how they might’ve been discovered and/or invented through typical mathematical processes or ways of thinking. in other words, what is suggested in 4) above, is that a detailed analysis of the history of mathematics can lead to the identification of certain general patterns and processes by which mathematical content is discovered and invented, and that these patterns and processes could then be utilized as possible teaching approaches without any direct reference to the history of the particular content being taught. so for example, it is possible to teach boolean algebra as described in de villiers (1986a), not in the actual historical order nor from the context it originally developed, but to instead begin by focussing on the modelling of generalizing the nagel line to circumscribed polygons 34 switching circuit problems, and only later on dealing with its axiomatization into a formal mathematical system of axioms, theorems and proofs (as well as its application to other areas such as logic, computer programming, biology, etc.) an approach in which learners are exposed to or engaged with the typical mathematical processes by which new content in mathematics is generally discovered, invented and organized has been called a “reconstructive” approach by human (1978:20) as follows: with this term we want to indicate that content is not directly introduced to pupils (as finished products of mathematical activity), but that the content is newly reconstructed during teaching in a typical mathematical manner by the teacher and/or the pupils. (freely translated from afrikaans) the effective implementation of a reconstructive approach therefore presupposes that teachers themselves are well acquainted and proficient in typical mathematical processes. the history of mathematics is of course not the only source by which teachers can gain useful knowledge and insight into the typical mathematical processes by which new mathematics is discovered and created. valuable insight can also be obtained by reading books on problem solving, problem posing and heuristic reasoning, and perhaps most importantly, from being mathematically active oneself, and reflecting on one’s own struggles and triumphs. it is therefore with this latter view in mind that this paper hopes to make a modestly small contribution, and to which we are now proceeding in the next sections. reasoning by analogy analogy is a kind of similarity. reasoning by analogy is often a powerful means of extending or applying mathematical results into other domains. it has often also featured in several major scientific discoveries. for example, by noticing the analogy between the movement of a projectile and the moon around the earth, newton was ultimately steered in the direction of his famous laws. similarly, kekulé is reputed to have developed the idea of a circular molecular structure for benzene from a dream of a snake swallowing its tail. when first leonardo da vinci, and then later the wright brothers, considered propellers, they respectively regarded them as “air screws” or “rotary wings”. polya (1954, p. 13) defines an analogy between two mathematical systems as follows: “two systems are analogous, if they agree in clearly definable relations of their respective parts.” for example, a circle, triangle and square in the plane are respectively analogous to a sphere, tetrahedron and cube in space. in the plane, a circle may be defined as the locus of all points equidistant from a point, but similarly in space, a sphere can be defined as the surface formed by all the points equidistant from a point. a triangle is the most elementary, finite figure in the plane bounded by straight lines, whereas a tetrahedron is the most elementary solid in space bounded by planes. lastly, congruent line segments, all perpendicular to each other, bound a square in the plane, while congruent square faces, all perpendicular to each other, bound a cube in space. sometimes an analogy can be defined explicitly, for example, when the relations are governed by the same rules or axioms. for example, the addition of numbers is analogous to the multiplication of numbers, since both are commutative and associative. in such a case where the analogy is determined by clearly defined rules, we have a duality, as the two operations can be interchanged (as long as only these laws are involved). for example, this duality between addition and multiplication extends to a fruitful analogy between arithmetic and geometric sequences to produce an interesting dual for the fibonacci sequence, involving an analogous rule tn × tn+1 = tn+2 for producing consecutive terms (see de villiers, 2000). just like the fibonacci sequence, the limit of the quotients of the logarithms of the adjacent terms of this dual sequence is also the golden ratio, i.e.: lim n →∞ logtn +1 logtn = φ . constructive defining ... the algorithmically constructive and creative definition ... models new objects out of familiar ones freudenthal (1973, p. 458) in mathematics we can distinguish between two different types of defining of concepts, namely, descriptive (a posteriori) and constructive (a priori) defining (compare de villiers, 1986b; human, 1978; krygowska, 1971). however, for the michael de villiers 35 purpose of this paper only constructive defining is relevant. constructive (a priori) defining takes places when a given definition of a concept is changed through the exclusion, generalization, specialization, replacement or addition of properties to the definition, so that a new concept is constructed in the process (see figure 1). in other words, a new concept is defined “into being”, the further properties of which can then be experimentally or logically explored. the main purpose or function of a priori defining is the production of new knowledge. t h e o r e m a b c d g h f p r o p e r t ie s o f g i v e n d e f i n i t io n v a r i a t i o n a b c e n e w d e f i n i t io n f u r t h e r e x p l o r a t i o o f p r o p e r t ie s figure 1: constructive defining a classic, historical example of constructive defining using analogy is the extension of the algebra of the real numbers by bombelli in 1569 to complex algebra with imaginary numbers, by the addition of an imaginary unit, i2 = −1, yet maintaining the same basic laws of commutativity, associativity, distributivity, closure and identities. in a letter to his sister, weil (1940, p. 339) stated: “the analogies between algebraic functions and numbers have been on the minds of all the great number theorists of the time”, and in 1946, he laid the foundations of algebraic geometry from the analogy of the theory of differentiable manifolds with some constructions from algebraic topology. another famous example is that of defining a hypercube (a four dimensional cube) by analogy from a three-dimensional cube. since a cube viewed directly from the front appears like a square within a square with corresponding vertices connected as shown in figure 2a (i.e. a 2-d representation of a 3-d object), by analogy one can consider a hypercube as a cube inside a cube (i.e. a 3-d representation of a 4-d object) as shown in figure 2b. figure 2: generalising a cube to the fourth dimension in a similar way, by using analogy we can constructively define a generalisation of the concept of a parallelogram to hexagons by defining a parallelo-hexagon as a hexagon with opposite sides equal and parallel (see figure 3). or by relaxing one or the other condition, we can generalise even further by constructively defining a parallel-hexagon as a hexagon with opposite sides parallel or a oppo-sided hexagon as a hexagon with opposite sides equal. op po -s ide d-h e x ag onpa ra lle l-he xa go n pa ra lle lo-h e x ag on pa ra lle log ram figure 3: generalising a parallelogram the analogy between the euler and nagel lines of a triangle analogy seems to have a share in all discoveries, but in some it has the lion’s share. (polya, 1954, p. 17) generalizing the nagel line to circumscribed polygons 36 a remarkable analogy between the nine-point circle and euler line on the one hand, and that of the spieker circle and nagel line on the other hand, is contrasted in table 1 and illustrated in figure 4 (coolidge,1971; de villiers, 2006; honsberger, 1995). it is insightful to compare the underlying similarity transformations implied by both results. for example, note that for the euler line, a halfturn with centre g and a scale factor of ½, maps abc onto the median triangle a′b′c′, and circumcentre o to p. but a dilation with a scale factor of 2 from centre o, maps a′b′c′ to a″b″c″, and p to h. therefore, h (the orthocentre of abc) is the circumcentre of a″b″c″. similarly, for the nagel line, a halfturn with centre g and a scale factor of ½, maps abc onto the median triangle a′b′c′, and incentre i to s. but a dilation with a scale factor of 2 from centre o, maps a′b′c′ to a″b″c″, and s to n. therefore, n (the nagel point of abc) is the incentre of a″b″c″. generalising the euler and nagel lines to quadrilaterals let us now first consider generalising the euler line to a cyclic quadrilateral. in general, given any quadrilateral abcd as shown in figure 5, the respective centroids c′, d′, a′ and b′ of triangles abd, abc, bcd and cda form a quadrilateral a′b′c′d′, similar to the original and scale factor – 1/3 (a halfturn and reduction by 1/3), with lines aa′, bb′, cc′ and dd′ concurrent in g. then this point of concurrency g (centre of similarity between abcd and a′b′c′d′) is defined as the centroid of the quadrilateral (for proofs, see de villiers, 1999; yaglom, 1968). table 1: analogy between the euler and nagel lines of a triangle the nine-point circle is the circumcircle of abc's median triangle and has radius half that of circumcircle of abc. the spieker circle is the incircle of abc's median triangle and has radius half that of incircle of abc. the circumcentre (o), centroid (g) & orthocentre (h) of any triangle abc are collinear (euler line), gh = 2go and the midpoint of oh is the centre of the nine-point circle (p) so that hp = 3 pg. the incentre (i), centroid (g) & nagel point (n) of any triangle are collinear (nagel line), gn = 2gi and the midpoint of in is the centre of the spieker circle (s) so that ns = 3 sg. figure 4: nine-point & spieker circles figure 5: centroid of quadrilateral michael de villiers 37 another result we will need is the following. for a cyclic quadrilateral, the perpendiculars from the midpoints of the sides to the opposite sides (called the maltitudes) are concurrent as shown in figure 6 (for a proof see de villiers, 1999). this point of concurrency also coincides with the common point of intersection of the four nine-point circles of triangles abc, bcd, cda and dab, and is constructively defined as the nine-point (or euler) centre of a cyclic quadrilateral by yaglom (1968, p. 24). p o a b c d figure 6: nine-point centre of a cyclic quadrilateral then as shown in figure 7, we have the interesting generalisation of the euler line, namely, that the circumcentre o, nine-point centre p, and centroid g of a cyclic quadrilateral are not only collinear, but also og = gp (for a proof, see de villiers, 1999). maintaining the same relationship as for a triangle between the circumcentre o, nine-point centre p, and orthocentre h, we can now also constructively define the ‘orthocentre’ of a cyclic quadrilateral as a point h so that op = ph, and o, p, and h are collinear. now similarly to the case for a triangle, note according to the given ratios between the points, that a dilation of -1/3 with centre g, maps abcd onto the centroid quadrilateral a′b′c′d′, and circumcentre o to o′. hence, if go′ = x, then go = 3x. but a dilation with a scale factor of 3 from centre o, maps a′b′c′d′ to a″b″c″d′′, and o′′ to h. therefore, h (which we defined as the orthocentre of a cyclic quadrilateral) is the circumcentre of a″b″c″ d′′, and we have oh = 3 × oo′ = 3 × 4x = 12x and hg = 9x = 3go (and from our definition of h in terms of p it follows that po = 6x and pg = 4x = go). on the basis of the analogy between the euler and nagel line of a triangle, and the above result for a cyclic quadrilateral, we can now constructively define the nagel line for a quadrilateral circumscribed around a circle as follows (see figure 7). for the circumscribed quadrilateral, a dilation of -1/3 with centre g, maps abcd onto the centroid quadrilateral a′b′c′d′, and incentre i to i′, and because of the half-turn, i, g and i′ are collinear. next apply a dilation with a scale factor of 3 from centre i, to map a′b′c′d′ to a″b″c″d′′, and i′′ to the point n, which we now constructively figure 7: euler line of cyclic quad and nagel line of circum quad generalizing the nagel line to circumscribed polygons 38 define as the nagel point of a circumscribed quadrilateral. then from the applied transformations we have similarly to the cyclic case that n, g and i are collinear, and ng = 3gi. if we analogously define the spieker centre s as the midpoint of ni, we also have sg = gi. generalising the euler and nagel lines to cyclic and circumscribed polygons the generalisation of both results depend on the following general theorem for the centroid of any polygon, given and proved in de villiers (1999) and yaglom (1968): “given a n-gon a1a2 a3... an (n ≥ 3)…, then the centroids of the (n-1)-gons, a1a2 a3... an −1, a2a3a4... an , etc. that subdivide it, form a n-gon a1 ' a2 ' a3 ' ...an ' similar to the original n-gon with a scale factor of − 1 n − 1 , while the centre of similarity is the centroid of the original n-gon.” using this general result, we now constructively, and in general, define the orthocentre and nagel point of a cyclic and circumscribed polygon, respectively, as the circumcentre and incentre of the n-gon a1 '' a2 '' a3 '' ...an '' obtained from the dilation of the respective centroid polygon a1 ' a2 ' a3 ' ...an ' with scale factor n – 1. as shown in figure 8, if for example for the general euler line we let go′ = x, then go = (n – 1)x, oo′ = nx and hg = n(n − 1) x − (n − 1)x = (n − 1)2 x . hence, hg go = (n − 1)2 x (n − 1)x = n − 1. it also follows that since p is the midpoint of oh that pg = n(n − 1) x 2 − (n − 1)x = (n − 1) x n 2 − 1       , and pg go = n 2 − 1, etc. in exactly the same way, it follows for the general nagel line that ng gi = n − 1 and sg go = n 2 − 1. concluding comments note that in the physical, real world context, the centroid given above is for point masses located at the vertices of cyclic and circumscribed polygons. myakishev (2006) provides a completely different generalisation of the nagel line of a circumscribed quadrilateral by considering instead the centroid of a ‘perimeter’ circumscribed quadrilateral (in other words, where all the weight is distributed along the boundary), and constructively defining a different nagel point. according to benson (2007, p. 4), apart from its usefulness in discovering new knowledge, the ability to recognise analogies is also fundamental to problem solving as it “allows the solver to connect the familiar (a previously used method, strategy, or context) to the unfamiliar (a new problem)”. problem solving research has indeed revealed that expert problem solvers in mathematics and science engaged “in metaphorical processes as they constructed problem representations, they looked for analogies between the problem at hand and other familiar situations” (silver, 1987, p. 45). an immediate research question that comes to mind is whether and how such reasoning can be developed and taught to novice learners. stefan banach (1892-1945), the founder of modern functional analysis, characterises great mathematicians by their ability to go further, and even spotting analogies between analogies: “good mathematicians see analogies. great mathematicians see analogies between analogies.” (banach, n.d.) brown and porter (n.d.) argue that one of the reasons for the current usefulness and importance of category theory, particularly in its application to scientific problems, is that it gives “an abstract mathematical setting for analogy and comparison, allowing an analysis of the process of abstracting and relating new concepts. this setting is one of the most important routes for the application of mathematics to scientific problems.” x (n-1)x n(n-1)xn(n-1)x (n-1)xx sp n i'h o' g o ig figure 8: general euler and nagel lines michael de villiers 39 it would appear that reasoning by analogy is a fundamental human ability and relates to our attempts to construct meaning and draw relationships between similar concepts or objects. from a very young age children are able to identify analogies, e.g. a lion is like a large cat, or a burrow for a rabbit is like a nest for a swallow, and as adults we often use analogies to communicate more effectively. already in ancient greek philosophy as well as the bible there appear excellent analogies – the fables of aesop and the parables of jesus draw powerful analogies with the circumstances of people, and used to convey a moral or religious message. in traditional iq and sat-tests, simple analogies like “legs are to mammals as … are to fish” have been frequently used as test-items to measure “intelligence”, “academic potential” or “reasoning ability”. use of analogy in the classroom by teachers can allow students to observe and use commonalities between different mathematical concepts, algorithms, representations, theorems, classifications, etc., thereby contributing to better integration of different components of mathematical knowledge. most teachers probably use analogies such as ‘solving inequalities is like solving equations’, or ‘balancing equations is like balancing a scale’. however, since analogies highlight commonalities, but not the essential differences, teachers ought to ensure that learners do not lose track of important underlying or fundamental distinctions. in fact, the use of some popular analogies can be dangerous as pointed out by cognitive research over the last thirty years. for example, simply stating, ‘doing algebra is just like doing arithmetic with letter symbols’, in many ways seriously trivialises the abstract meaning of letter symbols representing “variables” or “generalised numbers” (hart, 1981). even worse, is the ubiquitous use of the false or misleading analogy that 2x + 3x = 5x because 2 apples plus 3 apples is 5 apples. sadly, some teachers often still say “we can’t add 2x + 3y” because they are “unlike terms” and we can’t add apples and oranges. this completely misleading use of letter symbols representing ‘concrete objects’ has been aptly called “fruitsalad” algebra. other analogies may help learners to remember results, e.g. a ‘negative times a negative is positive since magnets of same polarity repel each other’, but provides no explanation why the result is true. in such cases, they only serve the purpose of a mnemonic (memory aid device), but provide no meaningful integration with other mathematical concepts, and no different from rote learning of “rules without reasons”. finding analogies between the properties of geometric figures on the plane and on the sphere (as well as major differences) can assist learners in developing a deeper understanding of the propositional nature of mathematics. more-over, according to lenart (2004), learning comparatively about different types of geometries can help students develop a more tolerant view toward other people with opposing views, or of different social or cultural background, or when getting into another political, economical or technical environment. given the rich, multi-lingual and cultural diversity of south africa, it seems a pity that very little research on analogical reasoning in mathematics seems to have been done locally. a study by richland, zur and holyoak (2007) found that though mathematics teachers in hong kong and japan compared to their usa counterparts used about the same number of analogies, they generally offered more cognitive support to explore the analogies more deeply and thoroughly. in particular, there was much more emphasis in the asian countries on relational reasoning, and the use of mental and visual imagery. taken altogether, these factors may well conceivably contribute to differences in performance in timms achievements for these countries (richland et al., 2007). of relevance to mathematics teachers and mathematics education researchers is a book edited by english (2004) reporting the results of a 3-year longitudinal study whose participants were children in australia and the united states. in particular, the book seeks to understand the relationship between mathematical reasoning and children’s natural tendency to create analogies. the book reports the results of empirical studies, as well as classroom discourse and case studies, which qualitatively analyse the role of discourse in the development and relationship between mathematical and analogical reasoning. generalizing the nagel line to circumscribed polygons 40 references banach, j. (n.d.). quote. retrieved 4 december 2009 from http://en.wikipedia.org/wiki/stefan_banach benson, s. (2007). problem solving by analogy/problem solving as analogy. the mathematics educator, 17(2), 2-6. retrieved 4 october 2008 from http://math.coe.uga.edu/tme/issues/v17n2/v17n2 _benson.pdf brown, r., & porter, t. (n.d.). category theory: an abstract setting for analogy and comparison. retrieved on 21 december 2008 from www.informatics.bangor.ac.uk/public/math/resear ch/ftp/cathom/05_10.pdf coolidge, j. l. (1971). a treatise on the circle and the sphere. bronx, ny: chelsea publishing company. de villiers, m. (1986a). boolean algebra at school (revised edition). stellenbosch: research unit for mathematics education at the university of stellenbosch (rumeus). de villiers, m. d. (1986b). the role of axiomatization in mathematics and mathematics teaching. stellenbosch: rumeus. retrieved on 10 december 2009 from http://mzone.mweb.co.za/residents/profmd/axiom.pdf de villiers, m. (1999). generalizations involving maltitudes. international journal of mathematics education, science and technology, 30(4), 541548. de villiers, m. (2000). a fibonacci generalization and its dual. international journal of mathematics education, science and technology, 31(3), 447477. de villiers, m. (2006). a generalization of the spieker circle and nagel line. pythagoras, 63, 3037. department of education (doe). (2002). revised national curriculum statement grades r-9 (schools), mathematics. pretoria: department of education. english, l. (ed.) (2004). mathematical and analogical reasoning of young learners. new jersey: lawrence erlbaum. hart, k. (1981). children’s understanding of mathematics: 11-16. london: john murray. honsberger, r. (1995). episodes in nineteenth & twentieth century euclidean geometry. washington, dc: the mathematical association of america. human, p. g. (1978). wiskundige werkwyses in wiskunde-onderwys. unpublished doctoral dissertation, stellenbosch university, south africa. freudenthal, h. (1973). mathematics as an educational task. dordrecht: reidel. klein, f. (1924). elementarmathematik vom höheren standpunkte aus (3rd ed.). berlin: springer. krygowska, a. z. (1971). treatment of the axiomatic method in class. in w. servais & t. varga, teaching school mathematics (pp. 124-150). london: penguin-unesco. lénárt, i. (2004). why teach plane and spherical geometry simultaneously? amesa kzn mathematics journal, 8(1), 39-48. myakishev, a. (2006). on two remarkable lines related to a quadrilateral. forum geometricorum, 6, 289-295. richland, l. e., zur, o., & holyoak, k. j. (2007). cognitive supports for analogies in the mathematics classroom. education forum, 316, 1128-1129. silver, e. (1987). foundations of cognitive theory and research for mathematics problem solving. in a. schoenfeld (ed.), cognitive science and mathematics education (pp. 33-60). new jersey: lawrence erlbaum. toeplitz, o. (1963). the calculus, a genetic approach. chicago: university press. weil, a. (1940). a letter to his sister on analogies in mathematics. retrieved 4 october 2008 from www.ams.org/notices/200503/fea-weil.pdf yaglom, i. m. (1968). geometric transformations ii. washington, dc: the mathematical association of america. abstract introduction literature overview and theoretical framework research design and methods recommendations conclusion acknowledgements references about the author(s) ernest mahofa research department, faculty of education, cape peninsula university of technology, cape town, south africa stanley a. adendorff department of mathematics (get), faculty of education, cape peninsula university of technology, cape town, south africa citation mahofa, e., & adendorff, s.a. (2022). novice mentors versus mentees: mentoring experiences in mathematics at general education and training phase. pythagoras, 43(1), a641. https://doi.org/10.4102/pythagoras.v43i1.641 original research novice mentors versus mentees: mentoring experiences in mathematics at general education and training phase ernest mahofa, stanley a. adendorff received: 19 july 2021; accepted: 04 apr. 2022; published: 19 aug. 2022 copyright: © 2022. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract in the south african educational system, student teachers are deployed to schools for practical experience, where they are monitored by lecturers from their universities. student teachers are also mentored by teachers allocated to them by the school principal. some of these mentoring teachers are themselves newly qualified and may have little or no teaching experience. this study analysed the relationship between these various role players during the teaching of mathematics in general education and training (get) phase, at secondary schools in the western cape. the theoretical framework for the study was provided by lave and wenger’s communities of practice. an ethnographic qualitative research design was used for collecting data from classroom observations and semi-structured interviews. the selected participants comprised four novice mathematics teachers, four mathematics student teachers in the get phase, two lecturers and one school principal. the purposive selection method was used to select these participants. the findings revealed that novice mentor teachers were challenged by facing (1) no or little communication and collaboration between themselves and lecturers, (2) limited cooperation between mentor and mentee in the teaching of mathematics in get phase, (3) limited mathematics content knowledge by student teachers and (4) limited mentoring skills of novice mentors. it is recommended that universities create a sound educational partnership with mentor teachers. universities should also consider the voices of novice mentor teachers in their mentoring of student teachers. keywords: communities of practice; ethnography; general education and training; mentee; mentoring; novice mentor; student teacher; teaching practice. introduction mentoring has been viewed as an important means for student teachers to gain personal and professional skills in a practical teaching and learning environment (gholam, 2018; mangope, mongwaketse, dinama, & kuyin, 2018; pennanen, heikkinen, & tynjälä, 2020; salvage, cannon, & sutters, 2015, schulleri, 2020). the mentoring process deals with many aspects, for example formal and informal interaction between mentors and mentees, sharing of pedagogical knowledge and skills, social well-being, sharing of teaching experiences gained during training at each stage of the teaching journey (duse, duse, & karkowska, 2017; kutsyuruba & godden, 2019). during mentoring, mentors and mentees share knowledge and skills gained at different stages of their professional training as they are members of a learning community (farnsworth, kleanthous, & wenger-trayner, 2016; lave & wenger, 1991; wasonga, wanzare, & dawo, 2015; wenger, mcdermott, & snyder, 2002). the process of entering the teaching profession presents students with a novel situation and major challenges. efficient and effective mentoring requires that the mentor teacher be qualified and experienced. best teaching practice supposes that experienced mentors are available to spend productive time in real classroom situations with their mentees (endeley, 2014). yet, in practice, there may be little support for mentees and mentors to establish productive relationships in teaching practice programmes, for several reasons. consequently, in some schools there may be a shortage of experienced mentors, compelling newly qualified teachers (novice mentors) to perform this function. this study sought to understand more about the experiences of novice mentors and mentees during the teaching practice programme, by allowing their voices to be heard and interventions made. the research question was: what are the experiences of mentees and novice mentor mathematics teachers at a high school at general education and training (get) phase in the western cape? research background and problem statement much research is concerned about mentoring as a one-way process whereby mentees gain from mentors (pennanen et al., 2020). this misperception prompted the researchers to conduct a study on the experiences of both mentors and mentees, so as to hear the voices of both concerning their interactions. according to the communities of practice (cop) theoretical framework, members should be committed to a domain, interact and collaborate as they share activities (sharing of mathematics knowledge) as a community of practice which had been missing in this study (lave & wenger, 1991). the lack of effective mentoring in some of the schools prompted this study. in some schools, there were no policies prescribing that mentoring should be done and how mentees and mentors should interact. therefore, this study explored the experiences of both mentors and mentees so that their voices can be heard, and stakeholders encouraged to enforce policies about mentoring programme in schools. it has been observed that most schools do not have a clear map of the mentoring programme they become involved in, while universities send student teachers to schools without getting to know the experiences of mentors and mentees during the mentoring programme. mentoring is an important aspect in the education system that enables entrant and pre-service teachers to develop professionally. however, we observed that little is known about the day-to-day experiences of both mentees and mentors during mentoring in teaching mathematics at get phase. mentoring should assist to improve the quality of learning and teaching as well as interpersonal relationships in an organisation (duse et al., 2017). however, the issue is that mentoring in schools is not properly supervised or undertaken in a professional manner. there are no policies in most of the schools to guide how mentoring should be carried out. this deficiency prompted this study. more research was done on how mentees gained knowledge and skills from their more knowledgeable mentors. however, the discussion in this study is about experiences of both mentees and mentors and what they gained from each other during the mentoring programme. purpose of the study the purpose of the study is to explore the experiences of mentees and novice mentors in the learning and teaching of mathematics at get phase at a secondary school in the western cape. the research question was: what are the experiences of mentees and novice mentor mathematics teachers at a high school at get phase in the western cape? literature overview and theoretical framework there is abundant literature regarding mentoring in the education system (bird & hudson, 2015; du plessis, 2013; endeley, 2014; fischer, & van andel, 2002; koki, 1997; leshem, 2012; smith, hayes, & shea, 2017). however, less is known about the experiences of novice mentors and mentees in the learning and teaching of mathematics in the south african context, which motivated the researchers to undertake this research. mentoring mentors should be experts with experience in the teaching field since they act as guides in the teaching career of a mentee (mckimm, jollie, & hatter, 2015; wasonga et al., 2015). in mathematics education, mentoring is important for assisting mentees to become competent to teach their subject, resulting in better teacher-learner relationships in which teachers have earned the trust of learners and so become professionally developed in mathematics education (nel & luneta, 2017; reddy, 2006, wasonga et al., 2015). moreover, nel and luneta (2017) iterated that mentoring improves mathematics content delivery and mathematics performance by learners. a mathematics mentor should be a teacher who has been trained in the knowhow of their field. the mentors should work in the same building as the teachers they are assisting, and they should be assigned a limited number of mentees at any given time. mentors need to work hand-in-hand with their mentees and make sure that the mentees are able to teach independently and gain pre-service experience (bird & hudson, 2015). the novice mentor kim and roth (2011) defined a novice teacher as a recently certified teacher from a teacher training institution. in addition, leshem (2012) defined a novice mentor as a new entrant qualified teacher from the training institution who also needs to be trained and certified to act as mentor. combining these, a novice mentor is a newly qualified and certified teacher entering the teaching profession who performs mentoring (brown & duguid, 2006). thus, a novice mentor needs further training to be a fully fledged mentor capable of conducting a programme so as to ensure that the mentee has an optimal pre-service experience. the mentee a mentee is expected to always alert the mentor about their needs during teaching practice. the interaction between the mentee and the mentor supposedly promotes a strong relationship and mutual understanding (mckimm et al., 2015). their relationship, however, might in practice be much less supportive, for instance when the mentee is given the whole of the teaching load while the mentor goes to the staffroom to do their own work (bukari & kuyini, 2014, kiggundu & nayimuli, 2009). in contrast to less support provided by a mentor, bird and hudson (2015) stated that leaving a mentee in charge of a class alone gives the mentee confidence and an ability to work without being supervised, so gaining pre-service experience. on the other hand, wasonga et al. (2015) noted that mentees gained professional development during mentoring as they engaged in practical training under the guidance of mentors. experiences by mentors and mentees according to preston, walker and ralph (2015) and gholam (2018), mentees and mentors had cordial relationships as the results showed that mentors were supportive. similarly, schulleri (2020) stated that mentors gained interpersonal skills while mentees gained classroom management skills. in corroboration, arasomwan and mashiya (2021) stated that mentees had pleasant experiences when receiving assistance required from their mentors, for instance in the use of teaching resources for effective teaching. some mentees reported that they had particular challenges in teaching learners in the medium of instruction (english) as mentors were mostly using the learners’ home languages. bukari and kayuni (2014), however, reported that some mentees were not well supported by their mentors as mentors were overloading mentees with a lot of additional responsibilities. novice mentors may feel disempowered if they are less qualified than the mentee, resulting in novice mentors feeling inferior (cranton & wright, 2008). in her study, schulleri (2020) iterated that mentors were challenged with inferiority as they felt undermined by mentees because of their lower qualifications. while most studies, such as that of salvage et al. (2015), concentrate on the mentoring experiences of mentors generally, this study focuses on novice mentors mainly. studies by salvage et al. focused on mentoring of teachers at the beginning of their teaching careers while this study focuses on mentoring novice mentors and mentees. kiggundu and nayimuli (2009) reported that mentees were faced with a negative relationship between themselves and their mentees in terms of duty overload and less opportunities to engage with learners. theoretical framework this study was informed by the cop theoretical framework as founded by lave and wenger (1991) where interaction between members with common ideas is emphasised (social learning). according to wenger et al. (2002), cop is: a group of committed people active in a common domain, with a genuine interest in each other’s expertise based on their own practice. group members combine their own interests with an open mandate from their organisations and work together in a rather informal structure. (p. 21) in this study, the cop was considered to comprise novice mentors, mentees, principals and the lecturers in the teaching practice programme as they share a common concern and work in a community (learning and teaching process). in a cop, members generate a direct connection between learning and performance. this theoretical framework was considered appropriate for this study because it recognises that in learning there is interaction of members of the group (farnsworth et al., 2016; wenger, 2000). participants start as peripheral and gradually become fully participating (mcdonald & mercieca, 2021). members of a cop possess common characteristics (farnsworth et al., 2016). they gain knowledge from each other through continuous interactions. thus, members in a cop engage with each other resulting in a collective learning environment (farnsworth et al., 2016). what makes this theoretical framework important to this study is that the mentees, mentors, lecturers and principals do share a common concern (mentoring programme) in the teaching and learning of mathematics at get phase. a cop is considered to comprise three components: the domain, community and practice (farnsworth et al., 2016, lave & wenger, 1991; smith et al., 2017; wenger, 2000, 2004, 2006). in this study, the domain was the teaching of mathematics, while the community comprised the group of participants who interacted, some being peripheral while others were fully participating. practice ‘is the body of knowledge’ (wenger, 2004, p. 4), for example skills (ways of teaching in this study) that participants shared and developed as a group. in the context of a cop, mentoring is seen as involving all members rather than mentees being told what to do by mentors. this perspective was supported by wenger (2006), smith et al. (2017), farnsworth et al. (2016) noting that in a cop, participants learn as they increase their participation. in cop there is creation and transmission of knowledge as members develop the practices and identities appropriate to that community (teaching and learning) (bettencourt, malaney, kidder, & mwangi, 2017). within it, mentor, mentee and lecturer should be able to build on trust and honesty so that they can learn from each other. this may only happen when members are able to meaningfully interact and constructively learn together. finally, seeing that cop members are experts, they ‘develop a shared repertoire of resources, experiences, stories, tools and ways of addressing recurring problems that is a shared practice’ (skalicky & west, 2006, p. 2). research design and methods an ethnographic qualitative approach was adopted in this study, with the goal of understanding the novice mentors’ and mentees’ lived experiences from their own perspective. the researchers used classroom observations and semi-structured interviews to collect data so as to generate an exhaustive understanding of novice mentors’ and mentees’ experiences in the mentoring programme (crowe et al., 2011). semi-structured interviews and classroom observations were used in recognition of the interaction between all participants as members of a group sharing ideas (farnsworth et al., 2016; wenger, 2006). the purposive selection method was used because detailed information and in-depth data was needed from particular participants (creswell, 2013). the sample comprised four novice mathematics teachers, four mathematics student teachers of get phase, one school principal and two lecturers. the selection of principals and lecturers was justified because as important stakeholders in the teaching practice process, they provided relevant data about mentoring of mentees. in terms of the cop theoretical framework, members should be share a common concern and learn from each other as well as sharing in a social learning environment (farnsworth et al., 2017; smith et al., 2017; wenger, 2006). mentors and mentees were observed when presenting their mathematics lessons as well as being interviewed to provide their experiences in the mentoring process, while the principal and lecturers were only interviewed. the latter were considered as data sources because as researchers we believed that their actions affected the mentoring process. we were responsible for the selection of participants, observing lesson presentations, interviewing, collecting, interpreting, and analysing data. we listened to the interviews and read the transcripts many times so as to acquire the essence of participants’ responses. pseudonyms were assigned to each interviewee. we took notes to develop themes from the semi-structured interviews and the classroom observations. data collected from interviews were explicated through collaizi’s (1978) phenomenological data analysis to analyse the experiences of novice mentors. the research conformed to the ethical requirements of the ethics research committee of a south african university respecting the rights to privacy, confidentiality and anonymity. participants completed consent forms before taking part in the study. participants were also informed that they had a right to pull out from their involvement in the research at any moment. the researchers obtained required permissions to undertake the study from the department of education, school, principal, university lecturers, mathematics get phase teachers and student teachers. analysis of results separated data were categorised and merged into main themes, seven of which emerged from the data about the experiences of novice mentors while mentoring student teachers: communication and collaboration between novice mentors and lecturers, guidance given to mentor teachers, mentoring skills, pedagogical competence, commitment to the mentoring process, matching of mentor and mentee, and motivation of novice mentors. pedagogical competence in terms of the subject participating mentors reported that most of the mentees lacked or had limited pedagogical and content knowledge in teaching mathematics at get phase. mentor 2 revealed that one of his mentees was trained to teach mathematical literacy, but the university insisted that the mentee teach mathematics. there was no communication stipulating which subjects and to what level the student teacher should teach during teaching practice. mentee 2 stated that: ‘i trained to teach mathematical literacy. i did not do mathematics at fet and university; it is difficult for me to factorise quadratic and trinomial algebraic expressions at grade 9 and i feel inferior in front of learners as i am prone to make mistakes.’ the above teacher’s sentiments were in line with our lesson observations in that the mentee was unable to factorise simple algebraic expressions. the mentee made many mistakes and we advised her to plan and ask for assistance from the mentor. the results of the observations are: the teacher factorised the following algebraic expression as follows: question 1: factorise completely: 4ab2c2 – 8ab3c + 12a2b2c3 divide by mentee’s explanation: at the bottom cancels with 4ab2 at the top. two c’s on top divided by one c gives one c and three b’s on top divided by two b’s at the bottom gives one b. instead, the mentee could have started with a simpler algebraic expression and used learners’ prior knowledge of finding the highest common factor (hcf) of terms and simplification of exponents. step 1: find the highest common factor (hcf) = 4ab2c the mentee lacked the knowledge of using mathematical language when factorising algebraic expressions. instead of using term ‘denominator’ he used ‘bottom’ and instead of ‘numerator’ he said ‘top’. moreover, he used the statement ‘cancelling each other’ rather than using the laws of exponents so that learners could be alerted to how particular mathematics content topics are linked to each other. when the mentee said ‘two cs on top divided by one c at the bottom gives one c and three bs on top divided by two bs at the bottom gives one b’, the inability to use proper mathematical language was of concern for the learner as mathematical language is important and is needed to form accurate mathematical concepts. in another class, mentee 2 was unable to factorise a trinomial, as illustrated below. mentee 2 stated: factorise the following, x squared minus five x plus six. you use trial and error method. look for the factors of six and write x minus three in one bracket then x minus two in another bracket as below: x2 – 5x + 6 = (x – 3)(x – 2) the mentee instructed the learners to find the factors using the trial and error method without clear explanations or demonstrations on what to do. in one of the classes we observed, the mentee introduced algebra to learners. the mentee did not move from the known to the unknown. we observed that the mentee lacked knowledge about mathematical language. these findings concurred with the findings above that the mentee had limited mathematical language and mathematical vocabulary. thus, the mentee gained experience that mathematical language and vocabulary are important aspects in learning and teaching of mathematics for learner understanding. in one of the lessons observed, mentee 4 did not use mathematical terms for instance, variables, like and unlike terms, or coefficient so that learners could get used to actively using mathematical vocabulary and so improve their understanding of mathematical terminology. these findings showed that the mentees seemed to be unfamiliar with using mathematical language and mathematical vocabulary. during our engagement with mentees after observing their lessons, we confirmed that they had limited mathematical vocabulary. thus, interaction between mentors and mentees promoted mathematical knowledge, interpersonal skills and gaining meaningful experience in teaching mathematics (farnsworth et al., 2016). teaching and learning styles in some classroom observations, we noted that the mentees used procedural discourse when teaching; they did not explain to learners why they were adding or subtracting certain terms. during the interviews, one of the mentors explained to one of the mentees that at every step in simplifying algebraic expressions, she needed to explain why it was done, so that learners could be confident in manipulating the given task. it would have been better to have advised the mentee to use questioning in a way that compelled learners to think and be challenged to arrive at the solutions themselves. during our interviews, most mentors advised the mentees to use conceptual discourse rather than procedural discourse, so that learners could provide reasons for using particular algorithms when manipulating mathematics tasks: example: cross multiplication so 4x = 2b compared to conceptual approach: these results revealed that some mentors were able to guide mentees on how to present and conduct lessons productively and effectively and involve all learners during the learning and teaching process. these results conformed to wenger and lave’s (1991) recommendation that all participants in the cop should have the opportunity to take part towards a common goal. in some observed classes the mentees used mainly a teacher-centred approach instead of learner-centred approach, not considering that learners should construct their knowledge rather than being absorbers or receivers of knowledge, as recommended by freire (1996). thus, mentees were seen as sources of knowledge that they delivered to learners. we observed that most mentees did not provide learners with a chance to think outside the box, thereby preventing them from developing into creative and critical thinkers. instead, learners acted as empty vessels that needed filling with mathematical knowledge (freire, 1996). these findings were in line with the interview results from a mentor: ‘my mentee does not give learners a chance to think. she give them the answers. spoon feeding learners is not a good strategy as mathematics need critical thinking.’ in one of the classes observed, mentee 4 favoured instrumental (rules without reasoning) rather than relational teaching and learning (skemp, 1978). here the mentee appeared to be content with learners getting the correct answers without reasoning. this showed that the mentee favoured a teacher-centred approach. the mentee concerned did not provide a platform for learners to discuss their solutions or give reasons for choosing a method suitable to them. these findings corroborated that some mentees used procedural rather than conceptual discourse. the above findings are illustrated below: mentee 4: question: simplify the following (2x – 3)2 learners’ solutions: learner a: (2x – 3)2 = 4x2 – 9 learner b: (2x – 3)2 = 4x2 + 9 learner c: (2x – 3)2 = (2x – 3)(2x – 3) = 4x2 – 6x – 6x – 9 = 4x2 + 12x – 9 the mentee provided learners with the solution and asked them to mark and write corrections without any explanations about how the question should be answered. when asked why the mentee did not clearly explain to learners, he responded that he was still behind in terms of the content that needed to be covered: teacher’s solution 1: (2x – 3)2 = 4x2 – 12x + 9 teacher’s solution 2: (2x – 3)2 = 4x2 + (2x)(−3)(2) + 9 = 4x2 – 12x + 9 no explanation was provided of how to simplify the expression, instead the mentee wrote the answer on the board for learners to copy. the mentee thus used a procedural approach. the mentee stated: ‘square 2x to get 4x2, multiply 2x by −3 to get −12x and then square 3 to get 9.’ the mentee did not explain why square 2x, why multiply 2x by –3 or why square –3. instead, the mentee could have used the method below and given learners similar questions so that they could investigate the algorithm that the teacher had used. the mentee could have used the distributive law method to make learners understand better: the errors some learners made to arrive at (2x – 3)2 = 4x2 + 9 and 4x2 – 9 were not explained by the mentee. during classroom observations, learners were not provided an opportunity to explain how they arrived at their solutions. learners need to be given an opportunity to discover their own mistakes and explain how they got their solutions. this would encourage learners to exploit and explore their mathematics understanding and so link their prior knowledge to their new understanding (slavit, & slavit, 2007). it was observed that the mentee was teaching for the sake of completing the lesson and not for learner understanding. when mentees were asked why they used a teacher-centred approach instead of a learner-centred approach, most stated that they had themselves been taught through teacher talk and had modelled their approach from their former teachers. the mentees’ mistakes and apparent lack of confidence in teaching mathematics might well result in learners being reluctant to do mathematics, as was stated by one mentee: ‘i can see that learners are not enjoying the lesson. maybe it is because i am also not confident in teaching mathematics as it is not my major subject.’ in support of this perception, one mentor stated that: ‘when a teacher is unable to explain to learners how she/he arrived at the solution, learners will doubt the teacher’s capability to teach mathematics. this is what i have observed from my mentee as learners ask my mentee to deliver the lesson alone.’ the above findings corroborate those by ball (2008) that the learners’ mathematical success is dependent on their teachers’ mathematics knowledge. thus, if teachers are not knowledgeable in mathematics the learners may also not do well as they may imitate their teachers’ negative attitude towards the learning and teaching of mathematics. furthermore, during classroom observations and interviews with both mentees and mentors, it was observed that there was a need for cooperative and collaborative learning. this learning can happen through sharing of ideas before and after a lesson. these findings corroborate those of luckenbill (2018) that lessons should be structured and presented to afford individuals with opportunities to share information through productive discussions to explore mathematics concepts. thus, when learners think mathematically, they gain skills for conceptual discourse instead of procedural discourse (setati, 2008). using google translate some mentees were able to introduce new ict skills to the mentors in their lesson presentations. one mentor stated that: ‘mentees possess a wealth of experience in ict, and we need to capitalise on this knowledge and skills they gained from their universities. i am learning new skills, which is experiencing the use of interactive whiteboard.’ these perceptions revealed that mentees had new skills that mentors gained and could improve their day-to-day teaching. thus, mentees shared new skills during the mentoring programme while mentors gained new experiences in using ict in teaching mathematics. similarly, wasonga et al. (2015) and duse et al. (2017) pointed out that beginning teachers’ training experience may influence the way they engage with their teaching. in one of the classes a mentee played a video game so that learners could engage with the content. learners were provided with questions to answer as they watched a video game about algebra. in two other classes observed, the mentees used powerpoint to present their lesson. however, there were errors in the solutions provided. these interactions showed that the mentees and mentors exchanged skills and knowledge in teaching mathematics at get phase; they interacted in a social learning environment as members of a community (farnsworth et al., 2015; wenger, 2006, wenger-trayner, & wenger-trayner, 2020). one of the mentors stated that google translate assisted her in teaching mathematics word problems: ‘this is my new experience. i have been struggling to teach word problems without knowing that there is an application which can help learners understand better and manipulate the problems correctly. i need to thank my mentee as he brought technology into my teaching resulting in experiencing new skills.’ in the example below, a mentee showed the mentor how google translate could help simplify algebraic expressions expressed in a word problem format. question: tom has a certain number of oranges, and he gave his brother half of what he has, he then gave his sister a quarter of what he gave his brother. how many oranges is tom left with? translation from english to isixhosa: utom unenani elithile leeorenji kwaye wanika umntakwabo isiqingatha soko anako, emva koko wanika udade wabo ikota yento ayinike umntakwabo. ushiye iimangile ezingaphi utom? mentor: i have experienced that you need to make sure that learners have the assumed knowledge for example, simplifying fractions. as mathematics word problems need learners to translate into isixhosa to promote better mathematics understanding, i used to crack my head thinking how i can assist these learners when they work on their own, now i got the magic, google translate can be used at anytime and anywhere so long there is internet connectivity and data to help in translating one language to another. teacher’s illustration for the question above after using google translate with learners: you need to use variables so that you can answer this question. let x be the number of the oranges tom has = x brother = sister = therefore, tom has = oranges during interviews, a mentor advised a mentee to use learner’s prior knowledge, for instance making sure one has taught all the foundation for simplifying algebraic expressions. these findings revealed that using google translate assisted the mentors with teaching word problems. one of the mentees presented a mathematics word problem to learners in powerpoint form in which he used google translate so that learners could understand the problem more clearly. we observed that learners understood better after the translation. thus, the mentors could assist learners who could not answer mathematics word problems expressed in english. these findings showed growth in professional development while working together as a learning community because participants worked together during mentoring, learning from each other as a group as noted by smith et al. (2017) that ‘in cop members work together around ideas of interest as they interact to learn together’ (p. 3). we concluded that mentees and mentors learned from each other; they developed effective communication skills, collaboration, pedagogical skills, and interpersonal skills by working as a team. the above findings supported farnsworth et al. (2016) that in a cop, members have a way of acting as they interact and share knowledge during their involvement in the group. communication and collaboration of mentors and lecturers the interview data revealed that, generally, novice mentors were unable to communicate or collaborate with lecturers during teaching practice as would be expected. some mentors reported that lecturers came to school for evaluating student teachers and left without any communication with them on the progress of the student teachers. mentor 1 stated that: ‘i do not have any document that stipulates my duties or guidelines for mentoring from the universities. i did not have any training in mentoring. i need assistance to be able to deliver my best if only i can voice my worries to the responsible authorities.’ another mentor stated that: ‘who are we in this ‘business’, these lecturers do not care about our challenges but only to send their students so that we can mentor them. we are left behind and taken as if we are not part of this programme, but we are the ones doing most of the work to assist the mentee to be a better teacher.’ this report showed that mentors needed assistance from university lecturers, but lecturers did not provide it. there was no mutual understanding or collaboration between mentors and the university lecturers. we claim that university lecturers considered novice mentor teachers as peripheral rather than being within the cop as observed by wenger (2000) and farnsworth et al. (2016) that in cop, generally new members are left in peripheral areas. as an example, lecturer 1 stated: ‘our duty is to evaluate student teachers not mentors, that is why we evaluate students and then we go to the next school because we do not have time to sit and chat to mentors.’ although mentors are not necessarily trainers or assessors, they play crucial roles in teaching practice since they spend the most time with mentees. university lecturers should deliberately make an effort to draw mentor teachers into the university mentoring programme by fostering mutual relationships. in this study, mentors, mentees and lecturers had something in common (mentoring) but did not interact and learn together. mentor 3 had this perception: ‘the sending universities should orient us about mentoring. the universities only communicate with the school management team who in turn allocate student teachers to us without any guidance.’ in collaboration with this expression, mentor 2 stated that: ‘lecturers just come and leave the school premises without any communication with me. i do not have time to share some of my challenges as these lecturers are not accepting us.’ this apparent ignorance of direct communication between novice mentors and lecturers could cause some misunderstanding during the teaching practice programme. the university appears to be saddling teachers with a mentoring burden without ensuring that these teachers are knowledgeable about the teaching practice programme. mentors could well have a divergent mindset about teaching practice, resulting in ineffective functioning of the programme. moreover, the findings showed that in teaching practice, novice mentors were initially only peripheral participants in the cop. seeing that lecturers excluded teachers from discussions of the mentoring process, they were regarded as involved in the periphery (smith et al., 2017; wenger, 2006). however, teachers were fully involved in the mentoring process as they dealt hands-on with the day-to-day activities with the mentees. therefore, they should have been considered as fully involved members, as noted by lave and wenger (1991) that ‘members that are fully involved in the activities of the community should be allowed to move from the peripheral participation into full participation’ (p. 37). as mentors and mentees acquired new knowledge and skills through practice, they should have moved to more central participation and eventually assumed a more expert role (lave & wenger, 1991), which did not happen as these teachers were isolated by lecturers. the findings that there was little or no communication between mentor teachers and the universities meant that mentees may acquire inadequate professional teaching skills during their teaching practice. finally, much teacher support from universities and other stakeholders is needed to ensure a successful teaching practice. mentoring guidance and mentoring skills some universities require that student teachers sign a learning contract with the teachers who will mentor them, but that was not apparently happening within our study since mentors were not even aware of the existence of such a contract, although that might have assisted mentors in articulating their mentoring duties properly. one of the universities supposedly specified that during teaching practice sessions, mentors were requested to complete a confidential report about the student teacher’s progress. however, there was no discussion about these reports between the lecturers and mentors. lecturers and mentors operated in different domains, in opposition to the concept of a cop. there were no policies or guidelines in place, in opposition to the recommendations of alabi (2017) that the education system should have policies in place to guide the stakeholders involved during mentoring. teachers were left on the periphery with no full participation as noted by wenger (2006). finally, there was no training offered to support mentors so that they could conduct mentoring effectively, as suggested by leshem (2012) that mentors need to be trained and certified. these findings again point to there being little or no cooperation between mentors and universities in the teaching practices programme. mentor 2 had this to say: ‘even though some student teachers signed a contract with me, this is always breached as some students just abandoned classes or absenting themselves without reporting. i do not have time to report such students on a face-to-face basis with the university officials. i think this is a great weakness or loophole or gap in the teaching practice programme.’ most of the mentors in the sample were themselves newly qualified, being new entrant teachers or in their final year, and so were evidently not knowledgeable about the mentoring process. mentor 1 articulated that: ‘i am a new teacher from university. i completed my degree last year and i do not know what is needed or expected of me when mentoring a student teacher. i am a novice teacher. i need the university and school to train me how to mentor a student teacher.’ mentor 2 stated that: ‘sometimes we just fill in forms for the sake of filling so that the student teacher could complete the course because we are not aware about the requirements of the university for mentoring a student teacher. student teachers sometimes they take advantage and do as they wish.’ in corroboration, mentor 3 said that ‘some student teachers from some of the universities did not take teaching practice seriously because there were no lecturers to evaluate them’. mentee 1 had this to say: ‘we only work on portfolios because our university have selected the students to be evaluated and if my portfolio is complete then it’s a pass for me’. during interviews we found that one of the mentors was a fourth-year student who himself needed to be mentored as he was not a qualified teacher. this student teacher was teaching alone, with no guidance and had to mentor another student teacher as well. in this case the school principal was employing fourth-year students as independent teachers before they had finished their programme, which shows a gap in the administration of teaching practice. when the principal was asked about this, he stated that there was a shortage of mentors as most experienced teachers had resigned or retired. the principal pointed out that ‘i am not aware that new entrant teachers should be qualified to be mentors. i just distribute student teachers to various departments who in turn allocate to subject teachers.’ these findings reveal that even a head of school could have limited knowledge about the mentoring process and its purpose, resulting in improper administration of teaching practice. thus, there were no policies guiding the conduct of the mentoring programme as suggested by alabi (2017) that education systems should have mentoring policies in place to guide the stakeholders involved. commitment in the mentoring process mentors reported that mentees often or regularly dodged, bunked and absented from classes, at times sitting in the staffroom or not coming to school at all. such mentees could be avoiding mentors who lacked commitment, to the extent of leaving the class to the mentee. thus, some mentors seemed to exploit the situation by abusing the mentees. mentee 4 had this to say: ‘my previous mentor was always delegating a lot of work and was not coming to school as she used me to do all her work which i am not paid for. i cannot work for someone. i decided to sit in the staffroom as i was already evaluated by my lecturer, and i even absented myself for no reason.’ this response shows that some mentees could be frustrated by a perceived lack of support from their mentors, or a feeling of being used for doing all the mentor’s duties. these results showed the lack of members having a common concern in the mentoring programme (lave & wenger, 1991; wenger-trayner & wenger-trayner, 2020). however, some mentors could be giving the mentee an opportunity to experience teaching without being supervised. mentor 4 stated that she had to leave a student with her class because she wanted to do administration duties and she had many student teachers who needed her attention. mentor 3 iterated that ‘the problem that i had was to mentor three student teachers which was a burden for me’. this picture presents mentors as overwhelmed and suggests a need for mutual understanding between mentor and mentee on how to work together in an educationally sound manner. such an understanding could foster a strong commitment to the mentoring process. mentors are expected to be committed to their duties and work well with mentees in assisting them to have a smooth transition and successful teaching practice. thus, mentors need to avoid conflicts of interest, use their time wisely and be productive in working with their mentees for a common goal in a cop (farnsworth et al., 2016; smith et al., 2017; wenger-trayner & wenger-trayner, 2020). level of motivation of mentor teachers all four mentors stated that the mentoring programme did not motivate them as it was voluntary and unpaid, resulting in them not fully participating in the programme. these findings revealed a lack of working as members of a group towards shared ideas (lave & wenger, 1991; wenger, 2006). they suggested that universities should introduce incentives to make teaching practice more efficient and their roles as mentors effective. the mentees complained that teaching practice overloads them with work for which that is not paid. as mentor 3 put it: ‘we do not get any incentives to motivate us to continue assisting mentees. we only volunteer. there is not even a special certificate or training workshops to assist us in developing good skills for mentoring.’ the lack of recognition for the roles played by mentoring teachers could lead to them undermining teaching practice and becoming reluctant to conduct an effective mentoring programme in which they participated fully as recognised members of a cop (farnsworth et al., 2016; smith et al., 2017; wenger, 2000; wenger-trayner & wenger-trayner, 2020). recommendations the universities, department of basic education (dbe) and south african council of educators (sace) should incentivise mentors by giving them developmental points through the sace continuous professional teacher development (cptd) system as well as issuing certificates during workshop programmes this idea of incentivising mentors is supported by the recommendation of sace (2012), that when teachers engage in professional development activities they should acquire points which would be added to their personal cptd points account. it is recommended that universities, school, mentors, mentees, dbe, sace and any other stakeholders collaborate to optimise the effectiveness of teaching practice. mentors should be given an opportunity to gain mentoring skills through workshops or training sessions and be awarded certificates of competence as incentives. universities should also consider the voices of novice mentor teachers, to better understand the challenges in the mentoring of student teachers. moreover, it is recommended that better mentor-mentee relationships be promoted in terms of professional development in mathematics education. universities and schools are recommended to work together so that the teachers are allocated to the learning areas they qualified for, so that better mathematics performance may be achieved by many learners. finally, the study recommends the daily use of technology by teachers so that learners understand and use learner-centred teaching approaches to help them explore mathematics and construct knowledge on their own, with the teacher being the facilitator. more research should be conducted about the experiences of mentees and mentors during teaching practice with a much larger number of participants. it is recommended that emphasis be placed more on integrating theory and practice in an effort to adequately underpin teaching theoretically. conclusion the research findings showed that mentees had limited professional competence in curriculum, subject, implementation of the curriculum and pedagogical knowledge. during their teaching practice, mentees did gain mathematical pedagogical content knowledge and skills during mentoring. the findings showed that the use of technology provided a rich lived space as learners were able to discover and work independently by visualising and hearing the content presented. the use of technology facilities like google translate was an eye opener for mentors, as they gained new skills that may reduce their challenges in teaching mathematics. there was no or little interaction (collaboration) between mentors and lecturers, which made it particularly difficult for novice mentors to do their task efficiently and effectively as mentors were invisible to lecturers who ignored them. mentoring was generally done in an informal or unplanned manner despite the expectation that schools should have well-planned or formalised mentoring programmes. the novice mentor teachers were challenged by limited collaboration with university officials, insufficient guidance on mentoring and scant motivation. other challenges for novice mentors were in respect of flawed communication, cooperation and training from the universities. schools and most importantly universities, gave inadequate attention to teacher mentoring. acknowledgements competing interests we hereby state unequivocally that we are not aware of any conflict of interest that may exist in respect of pursuing publication of our article in the pythagoras journal. authors’ contributions all the authors equally contributed to the research and writing of this article. ethical considerations ethical clearance was obtained from the ethical clearance was obtained from the faculty research ethics committee, education 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(2020). learning to make a difference: value creation in social learning spaces. vol. 1. cambridge: cambridge university press. article information authors: laurie butgereit1,2 reinhardt a. botha1 affiliations: 1institute for ict advancement, nelson mandela metropolitan university, south africa2meraka institute, council for scientific and industrial research, pretoria, south africa correspondence to: laurie butgereit email: lbutgereit@meraka.org.za postal address: po box 290, lanseria 1748, south africa dates: received: 30 sept. 2011 accepted: 02 nov. 2011 published: 25 nov. 2011 how to cite this article: butgereit, l., & botha, r.a. (2011). a model to identify mathematics topics in mxit lingo to provide tutors quick access to supporting documentation. pythagoras, 32(2), art. #59, 11 pages. http://dx.doi.org/10.4102/pythagoras.v32i2.59 copyright notice: © 2011. the authors. licensee: aosis openjournals. this work is licensed under the creative commons attribution license. issn: 1012-2346 (print) issn: 2223-7895 (online) a model to identify mathematics topics in mxit lingo to provide tutors quick access to supporting documentation in this original research... open access • abstract • introduction • research problem and objectives    • research design       • ethical considerations • the μ model    • phase 1: initial creation       • step 1: creating stemmers       • step 2: compiling mathematics vocabulary       • step 3: identifying stop words    • phase 2: on start-up of μ    • phase 3: μ processing of each message    • phase 4: μ feedback loop • model instantiation    • creating a post-stemmer    • selecting mathematics terms    • creating a pre-stemmer    • removing stop words    • processing the conversations    • correcting misspellings    • determining the topics • presenting the topic to the tutor • evaluation • discussion • conclusion • acknowledgements    • competing interests    • authors’ contributions • references • footnote abstract (back to top) dr mathtm is a mobile, online tutoring system where learners can use mxittm on their mobile phones to receive help with their mathematics homework from volunteer tutors. these conversations between learners and dr math are held in mxit lingo. mxit lingo is a heavily abbreviated, english-like language that is evolving between users of mobile phones that communicate using mxit. the dr math project has been running since january 2007 and uses volunteer tutors who are mostly university students who readily understand and use mxit lingo. however, due to the large number of simultaneous conversations that the tutors are often involved in and the diversity of topics discussed, it would often be beneficial to provide assistance regarding the mathematics topic to the tutors. this article explains how the μ model identifies the mathematics topic in the conversation. the model identifies appropriate mathematics topics in just over 75% of conversations in a corpus of conversations identified to be about mathematics topics in the school curriculum. introduction (back to top) dr mathtm is an on-going project hosted at the meraka institute1 which enables primary and secondary school learners to converse with tutors about their mathematics homework (butgereit, 2011). the learners use mxit on their mobile phones and the tutors use traditional internet-based computer workstations. mxit is a communication system which uses internet technologies over mobile phones to provide text-based communication (chigona, chigona, ngqokelela, & mpofu, 2009). whether because of the small mobile phone screen, the small keypad, or the fast pace of mxit-based conversations, an abbreviated form of english has developed when communicating using mxit, which is often called mxit lingo.although the english term ‘lingo’ is considered by some to be a slang term for a dialect of a language or the vocabulary of a specific industry or body of knowledge, the terms ‘sms lingo’, ‘net lingo’ and ‘im lingo’ are often encountered in academic literature. in this project we have therefore standardised on the term ‘mxit lingo’ to describe the specialised vocabulary, spelling, syntax and grammar used when communicating using mxit as a medium. the learners who converse with dr math usually ask their mathematics questions in this mxit lingo. some of the questions asked by learners are straightforward. for example: i wnt 2 knw hw 2 fnd th nth term tht they always ask abt other questions can be quite complicated: nadeem and jeny keep fit by skiping. nadeem cn skp 90 tyms pe min, when he starts training. each week he increase dis by 5 per min. jeny starts wt 60per min n increase dis by 10 per week. after hw many weeks wl deir number of skipz b the same the dr math business model uses volunteer tutors to answer these questions. most tutors come from universities in south africa and are typically fairly familiar with mxit lingo. however, situations do arise where tutors do not actually understand what the learners are asking; for example, the following (from now on, messages by learners will be set in bold to distinguish them from the messages from the tutors): o is the centre. e.o =squareroot of=2 and oe is perpendicular todf.fynd de sorry pls try write the question clearly i dont understand what you are asking it z a triangl n a circle which points are on the circle line e.o=root3 this article describes ongoing work on project μ. the term μ, with its pronunciation of ‘mu’, represents the phrase ‘mxit understander’. next we consider the objective of project μ and the related research problem and questions in more depth. research problem and objectives (back to top) a number of situations have occurred where tutors do not understand the questions posed in mxit lingo. such situations are becoming more common, due to a number of reasons:• increasingly, a large number of the tutors are not south african. these tutors are primarily graduate students from central africa who are studying mathematics in south africa. the mathematics knowledge of these tutors is excellent, but their home language is french. they often cannot understand what is being asked in mxit lingo. • a growing number of tutors are more ‘mature’ south africans from industry who have never used mxit or instant messaging. although their mathematics knowledge is also excellent and they are usually home language english speakers, they are not familiar with the cryptic mxit lingo. • a handful of tutors are citizens and/or residents in europe and north america. again, these tutors have excellent mathematics knowledge and english knowledge, but are often not used to some of the specific mathematics vocabulary used in south africa (such as ‘surd’ in the place of ‘radical’) and are not familiar with local words that pepper conversations (such as ‘howzit’, ‘ja’, and ‘yebo’). dr math has become extremely popular, and tutors are typically chatting with 20–30 learners concurrently. from dr math’s modest start, when only 20–30 learners were expected to participate, dr math now has over 30 000 registered learners. often a tutor needs to quickly look up some formula or definition. a good example is when tutors from an engineering background are asked questions about financial mathematics (e.g. l.b. always needs to look up the formula for compound interest). if topics can be automatically identified, supporting documentation can be timeously presented to the tutor by the platform, thereby reducing tutor response times. the research question, therefore, asked: how can mathematics topics be spotted in dr math conversations? the research objective was: to create a topic spotter which can timeously identify mathematics topics in conversations between dr math tutors and learners. research design project μ adopted a design science paradigm. baskerville (2008) argues that design science can be considered a research paradigm. design science changes the state of the world through purpose-driven development of artefacts, and thus researchers are comfortable with alternative realities. knowledge is gained through the construction and validation of artefacts (vaishnavi & kuechler, 2007, pp. 16–19).design science is characterised by having an artefact as primary output. march and smith (1995) propose four possible artefacts as outputs of design science research: constructs (which provide the vocabulary in which problems and solutions are defined and communicated); models (which use constructs to represent real-world situations); methods (which define solution processing, algorithms, and ‘best practices’); and instantiations (which are implementations of constructs, models, and methods). this article presents two artefacts: the μ model, which describes an executable process model used to identify mathematics topics, and the μ topic spotter, which is an instantiation of the μ model in the dr math tutoring platform. the interplay between model and implementation is important in the generation of new knowledge. vaishnavi and kuechler (2007, p. 12) emphasise that ‘the circumspection process is especially important in understanding design science research because it generates an understanding that could only be gained from the specific act of construction’. the μ model has been refined, based on lessons learned whilst constructing the μ topic spotter. hevner, march, park and ram (2004) established seven requirements for good design science research, which will be used at the end of this article to evaluate the research results. ethical considerations project μ uses conversations obtained from minor children. these conversations were part of the dr math project. the dr math project has received ethics clearance from the tshwane university of technology. the tshwane university of technology is not involved with the dr math project in any way and could take an objective view of the project prior to issuing ethics clearance. all conversations between learners and tutors are recorded for security, quality, and research purposes.the minor children receive daily messages from the dr math project which say: ‘never give out personal details to dr math. all conversations are recorded for security, quality and research purposes.’ tutors sign codes of conduct where they agree not to discuss any illegal activities with the learners. tutors also supply copies of identification such as copies of id books, passports, or driver’s licences. the tutors also sign informed consent documents agreeing that their tutoring conversations can be used for research purposes. whenever tutors log in to the dr math system, they receive messages which say: ‘by logging into this website, you agree that all tutoring conversations are recorded for security, quality and research purposes.’ from these messages and documents, it is clear that all participants have given their consent for their conversations to be used for research purposes. in addition, participation in the dr math project is completely voluntary, from the point of view of both the learners and the tutors. any participant could resign from the project at any time. all identities were hidden from all participants. the μ model (back to top) the μ model consists of four major phases or sections.phase 1 of the μ model consists of analysing and using initial historical data to create an initial configuration of μ. figure 1 shows that just as human infants need to encounter human language in order to learn language (barinaga, 1997), the μ model needs historical conversations and certain mathematics information in order to start processing. the μ model uses historic conversations between tutors and learners. these historic data consist of textual recordings of conversations between tutors and learners between january 2010 and july 2011. phase 1 only happens when there is a need to reconfigure the system; this may be when new trends in mxit lingo are observed and new words are identified, as feedback from phase 4 indicates in figure 1. figure 1: overview of μ model processing. the μ model must be instantiated in a computer program. every time the μ program is started, phase 2 of the μ model is executed. in phase 2 the configuration created in phase 1 is read by μ, providing it with a basis for determining the topics of conversations between tutors and learners. only minor processing of this configuration information happens at this point in time.as learners and tutors begin to converse, phase 3 of the μ model is executed. phase 3 processes the conversations according to the configuration which was read in phase 2, and attempts to determine the topics of the conversations. during phase 3 new items may be encountered which the μ model does not understand. these could be new words, new spellings, or new contractions. in the context of the dr math project, this could also include new topics in mathematics which might be added to the school curriculum. phase 4 of the μ model takes the new words, spellings, contractions or topics and adds them to the configuration of μ. it is important to the μ model that these changes can be integrated back into the model itself, as languages change and evolve over time. we now consider each of these four phases in more detail. phase 1: initial creation the initial creation phase of the μ model consisted of three steps:1. creating stemmers 2. compiling mathematics vocabulary 3. identifying stop words. step 1: creating stemmers a stemmer is a utility which removes suffixes from the ends of words, leaving just the root stem of the word. stemmers are often used in internet search engines and other information retrieval systems (hatcher & gospodnetic, 2004). stemmers take words such as factor, factoring, factorisation, and factored and remove the suffixes -or, -ing, -isation, and -ed to obtain just the root or stem of the word – fact. besides being able to stem both american english and british english, this stemmer must also be able to handle new mxit lingo suffixes. for example, the word facta is the mxit equivalent of the english word factor, illustrating the mxit lingo suffix -a which can replace the normal -er or -or english suffix. the stemmer must be able to remove multiple suffixes from words in order to handle a word such as factazashun, where the -shun suffix is the mxit lingo equivalent of the english -tion or -sion suffix. we have previously reported full details about mxit spelling conventions and mxit stemming (butgereit & botha, 2011a).stemmers can also operate at the beginning of a word. for example, the words equality and inequality only differ by the prefix of in-. another example is the word pair internal and external; the only difference between them is the prefixes attached to the beginning of the stem. the terms ‘pre-stemming’ and ‘post-stemming’ will be used to differentiate between these two types of stemming. post-stemming is absolutely critical to the μ model, and will be explained in more detail when describing the actual μ implementation. pre-stemming is also catered for in the μ model, but is not as important. throughout the rest of this document the term stemming will thus refer to post-stemming. step 2: compiling mathematics vocabulary a mathematics vocabulary must be compiled, which must contain the words and terms which are common to mathematical conversations, such as ‘parallel’, ‘factor’, ‘sum’, ‘sin’ and ‘expression’. compilation of this vocabulary could be done manually or by automatically extracting mathematical terms from pre-tagged conversations. these vocabulary lists are created in proper english format.the mathematics vocabulary needs to be classified into various topics and subtopics. this means that words such as ‘sin’, ‘cos’ and ‘tan’ need to be classified as terms in the topic of ‘trigonometry’, and words such as ‘parallel’ and ‘perpendicular’ need to classified as terms in the topic of ‘geometry’. mathematics terms can belong to more than once topic. for example, the term ‘hypotenuse’ could exist in both the ‘trigonometry’ topic and the ‘geometry’ topic. in addition, the relationship between various topics and subtopics needs to be defined. for example, the topic of ‘parabola’ could be a subtopic in the major topic of ‘algebra’. subtopics could belong to more than one major topic; for example, the subtopic ‘parabola’ could also belong to the major topic ‘graphs’. as with the compilation of the mathematics vocabulary, the determination of topics and subtopics could be done manually or in an automated manner. step 3: identifying stop words stop words are those which can be removed from a sentence without altering the major idea of the sentence. the expression ‘stop words’ is that which natural language processing practitioners use to describe these extraneous words. stop words are identified as words which have the same likelihood of occurring in documents not relevant to a topic as in those which are relevant to the topic (wilbur & sirotkin, 1992). for example, in the sentence the sin of an angle is equal to the ratio of the opposite side to the hypotenuse the words of, an, is, to, and the can be safely removed from the sentence, leaving just sin, angle, equal, ratio, opposite, side and hypotenuse.stop words for this new mxit lingo must be determined. these include words such as ‘sup’, ‘awe’, and ‘howzit’, which are common greetings in mxit lingo. as with the mathematics vocabulary and the mathematics topics, the compilation of stop words could be done manually or in an automated manner. phase 2: on start-up of μ since μ is an executable model, some initial processing must be done on the configuration files created in phase 1. figure 2 shows the aspects of processing. figure 2: start-up processing. during start-up, the stop words (which have been created in phase 1) are processed by both the pre-stemmer and post-stemmer. in addition, the mathematics terms are also processed by both stemmers. after stemming, μ then reads the unstemmed stop words, stemmed stop words, unstemmed mathematics terms, stemmed mathematics terms, and a configuration of the relationship between mathematics topics, subtopics, and the mathematics terms. this information is only read at start-up time. phase 3: μ processing of each message as the name suggests, phase 3 of the μ model is where the majority of the processing takes place. whenever a message from a learner enters the system, the steps in this phase are executed. figure 3 shows processing for both phase 3 and phase 4. all processing is assumed to belong to phase 3 unless clearly marked as phase 4. figure 3: phase 3 and phase 4 processing. as each message from a learner is received by μ, stop words are immediately removed from the conversation. this removes complete words such as ‘sup’ and ‘howzit’. the remaining words are then processed by both the pre-stemmer and post-stemmer. this stemming step changes words such as ‘factorisation’ to ‘fact’. the stemmers cater for the english and american spelling of terms, so ‘factorization’ will also be stemmed to ‘fact’.the stemmed conversation is then compared against the stemmed stop words. this is to cater for situations where the original stop word had a suffix. for example, it could be that the stop word was ‘looked’ and the conversation now being processed held the word ‘looking’. by making this comparison with the stemmed stop words, this word can also be removed as extraneous to the conversation about mathematics. at this point, there should only be stemmed words about mathematics in the remaining text. the next step is to look for unique misspellings of mathematics words which have not yet been encountered. this would cater for situations where, for example, the original mathematics term was ‘transform’ but the learner typed in ‘trasnform’ or, taking into account the fact that the stemmer has already executed, the learner may have typed in something like ‘trasnformashunz’. this attempt to find misspelled words in mxit lingo uses algorithms similar to finding normal misspelled words in word processing systems. once all the mathematics terms have been extracted or distilled from the conversation, n-gram processing is used against the mathematics terms and their relationships to determine the topic of the conversation. n-gram processing will be discussed in more detail when the model instantiation is presented. phase 4: μ feedback loop the μ model provides for a feedback loop where newly encountered spellings of words could be added to the μ configuration. in such a situation a word such as ‘trasnform’ could be added to the configuration files if it starts to appear often. model instantiation (back to top) to facilitate experimentation and to serve as proof of concept, a specific instantiation of the μ model was created. to avoid confusion, the term ‘μ model’ will be used when describing the model and the term ‘μ topic spotter’ will be used to describe the specific instantiation of that model. creating a post-stemmer a post-stemmer utility was written which catered for american english, british english, and mxit lingo. a sample routine which caters for plurals is listed below:public string singular(string word) { string stem = word; int length = word.length(); if (length > 4 && word.endswith(“ies”) ) { stem = [something] } else if ( length > 4 && word.endswith(“iez”) ) { stem = [something] } else if ( length > 3 && word.endswith(“es”) ) { stem = [something] } else if ( length > 3 && word.endswith(“ez”) ) { stem = [something] } else if ( length > 3 && word.endswith(“s”) ) { stem = [something] } else if ( length > 3 && word.endswith(“z”) ) { stem = [something] } return stem; } this sample code removes the normal english plural suffixes of -s, -es, and -ies. it also, however, removes the common mxit suffixes for plurals which are -z, -ez, and -iez. we have previously reported an in-depth discussion of the stemming facilities of the μ model (butgereit & botha, 2011a). selecting mathematics terms for the specific instantiation of the μ model for integration into the dr math tutoring platform, mathematics topics were subdivided into topics and subtopics. the topics were algebra, geometry, trigonometry, calculus, statistics, financial mathematics, number theory, logarithms, graphs, measurement, and sequences and series. for the scope of this project, three of the terms had specific definitions. the topic ‘statistics’ included probability and data handling. the term ‘number theory’ indicated the way numbers worked, including the differences between integers, natural numbers, whole numbers, real numbers, imaginary numbers, rational and irrational numbers, et cetera. the term ‘number theory’ also included concepts such as prime numbers, factoring, lowest common denominator, highest common factor, et cetera, but did not cover topics such as euler’s theorem, fermat’s theorem, waring’s problem, or riemann’s hypothesis. it referred only to concepts of how numbers work within the scope of the school syllabus. the term ‘graphs’ referred to drawing curves on a set of axes. it did not refer to the higher mathematical concept of ‘graph theory’.subtopics were also defined. these subtopics were parabolas, circles, exponents, functions, hyperbolas, lines, quadratics, solving for x, factoring expressions, simultaneous equations, inequalities, prime numbers, fractions, scientific notation, pythagoras, transformations, parallel lines with transversal, sin/cos/tan, double angles, compound interest, simple interest, effective and nominal interest, and percentages. the subtopics were classified into one or more topics. for example, the subtopic of ‘parabola’ was classified under the topic ‘algebra’ and the topic ‘graphs’. the subtopic ‘circle’ was classified under three topics: ‘geometry’, ‘algebra’, and ‘graphs’. creating a pre-stemmer pre-stemming was not as important in the specific instantiation of the μ model necessary for dr math. however, there was one prefix which needed to be specifically handled because of its high occurrence rate in conversations about mathematics.the exercise of collecting mathematics terms netted 568 common terms used in conversations about mathematics. of those 568 terms, 31 (or approximately 5%) began with the prefix in-. these words beginning with the prefix inspanned a number of mathematics topics and included the words income, increment, inequality, infinity, inflection, insolvency, instalment, integer, integral, intercept, interest, interior, interquartile, interval, and investment. no other prefix (such as con-, pri-, peror sub-) occurred with such a high percentage. special processing was done with words beginning with the prefix in-. the inwas stripped from the beginning of the word. it is important to note that the μ model has facilities to cater for any number of prefixes. in the specific case of the μ topic spotter instantiated for the dr math project, only one prefix was specifically processed. removing stop words historic data from dr math were used to ‘prime’ the μ topic spotter. a total of 17 413 conversations between tutors and learners, recorded between january 2010 and july 2011, were used as the basis of the μ topic spotter. these conversations consisted of a total of 25 715 unique words. an elementary statistical analysis of these historical data was done. of those 25 715 unique words, nearly half of them (12 969) occurred only once. these were often words which had no relation to the mathematics conversation at all but represented extra sounds a person might be making, such as laughter, kisses, anger, confusion or exasperation. for example:hahahahahahaha mwaaaaa aaaaaaarrrrrrrrrrrggggggggggghhhhhhhhh bwhahahaha heloooooooooooooo hmmmmmmmmmmm xoxoxoxoxoxoxo the stop words would be removed from the message as each message is received. considering how often it needs to be done, the removal of stop words must be a fast operation. it was therefore necessary to reduce the number of stop words. we decided that only words which had occurred more than once during the period january 2010 to july 2011 would be eligible to be stop words. this reduced the number of potential stop words to 12 746. the next step was to automatically remove any of the mathematics terms from the potential stop words, taking into account as many of the mxit spelling conventions as possible. for example, calculatd, calculate, calculated, calculater, calculates, calculatin, calculating, calculation, calculations, calculationz, calculatns, calculator, calculators, and calculatr were removed. this reduced the number of stop words by approximately 13.5% to 11 015. at this point manual intervention was necessary to remove the last vestiges of mathematics terms from the stop word list, and it was manually reduced to 10 478 words. we have previously reported our research on stop words (butgereit & botha, 2011b). processing the conversations as the conversation between learner and tutor grew, the stop words were removed. this means that a conversation which looked like:hw do i do transformation geometry ooh well what is the questions pleas explain transformation to me what grade u in? so what are the types of questions they ask you. they asked me to determine t(-1;3) under the translation (x; y) into (x; y+2) using a cartisian plane so the trasnformation is (x;y) to (x;y+2) so you put (-1;3) as the x and y. what do you get you get (3; 7) no you get (-1, 5) x remains the same but y gets 2 added to it oh, yes now i understand, so if i hv to translate t(3 ; 5) under the translation (x+4; y+2), i will get (7; 7) ? yes ohk thanks, let me do some practically peace out ok would be simply reduced to transform geomet transform grad under transl cart plan trasnform add transl under transl by removing the stop words. correcting misspellings once the stop words are removed, the remaining words are compared against the expected mathematics terms. in this example, the stem word ‘trasnform’ (which originated as the word ‘trasnformation’) is found not to match any stemmed mathematics term. the μ topic spotter works with so few words that every word is important. in the μ model n-gram processing is used to determine which mathematics term is the best possible candidate match for the word ‘trasnform’. after extensive experimentation it was determined that n-grams of length four would be used to attempt to find the best match for a stemmed word which has slipped through the stop word removal but does not match a stemmed mathematics term. n-grams are collections of n sequential letters in a word, sentence, document or file (cavnar & trenkle, 1994). the value of n can vary depending on the specific application. this means that there may be n-grams of length 2 (often called bi-grams) or of length 3 (often called tri-grams). figure 4 provides all the possible n-grams of length 4 for the word ‘transform’, where * indicates leading or trailing blanks. figure 4: n-grams of length 4 for the word transform. n-grams have been used in text classification or categorisation in many languages besides english, including arabic (khreisat, 2006) and turkish (güran, akyokuş, bayazıt & gürbüz, 2009). in addition, n-grams have been used to attempt to identify actual authors of specific segments of documents by comparing n-grams in a document where the author is known to n-grams in documents by specific authors (kešelj, peng, cercone & thomas, 2003).in this particular case the string ‘trasnform’ was also converted into n-grams of length 4 and a similarity ratio was calculated for each mathematics term. figure 5 shows the n-grams of length 4 for the misspelled word ‘trasnform’. figure 5: n-grams of length 4 for the word transform. n-gram processing calculates the similarity between two strings. the similarity is defined as the ratio of the number of common n-grams divided by the number of the union of n-grams. the calculation of the union of the n-grams can be done two different ways. as can be seen in figure 4, there are 12 n-grams of length 4 in the word ‘transform’ and, as can be seen in figure 5, there are 12 n-grams of length 4 in the word ‘trasnform’. some implementations of n-grams would calculate the union as being 24. other implementations of n-grams would calculate the union to be the union of unique n-grams. in such a case, the union would be 17. in the case of the μ topic spotter, the union was calculated from total n-grams (not unique n-grams) in terms of the misspelling corrector.table 1 shows the similarity values between two possible matches for the string ‘trasnform’. table 1: various n-gram similarities for trasnform. remembering that not all words that remain after stop word processing are, in fact, mathematics terms, it was not a simple matter of just taking the best choice for the word. a cut-off value for the n-gram similarity needed to be determined. after experimentation the value of 0.290 was used, which indicated that at least 29% of the n-grams were the same. in other words, the word with the highest similarity value which was over 0.290 was used as the properly spelled word, taking into account mxit stemming. common misspellings of mathematical terms were extracted from the historical conversations and tested by the misspelling corrector.in the case of the conversation about geometric transformations, in the set of remaining words after stop word processing the word ‘transform’ was listed three times. determining the topics once just the important words were distilled from the conversation, n-gram processing was carried out again. this second time, however, all of the important words (including the words which had spelling corrected) between the learner and tutor were converted to n-grams of length 4 and compared against the collections of various mathematics terms which had been classified into various topics and subtopics. table 2 shows the similarity values in this particular example of discussions about transformations. table 2: n-grams for topic determination. in this particular case, the subtopic ‘transformations’ with the highest similarity ratio was, in fact, the correct subtopic. however, n-gram processing is not an exact science. often, when the term with highest similarity after n-gram processing is wrong, the secondor third-ranking topic is correct. therefore, consider a case where the highest similarity is not necessarily the best match for the conversation:i ned help wif parabola graphs ok what is the formula xsqrd plus 3x plus 2 can you factor that? no u need 2 integers which mutliply up to 2 but add up to 3 wot are dey 1 n 2? yes well done the factors are (x+1) and (x+2) do you know what the roots are yes -1 n -2 as each line of the conversation was received from the learner, n-gram processing was executed. so, for example, when the first line (i ned help wif parabola graphs) was received by μ from the learner, the topic similarities were: graphs (0.141), quadratic (0.131), and parabola (0.120). when the second line (xsqrd plus 3x plus 2) was received by μ from the learner, the topic similarities were: quadratic (0.144), graphs (0.141), and parabola (0.123). it is important to point out that the n-gram processing is being done only when messages are received from the learner; however, the processing is done on the entire growing conversation including the tutor portion. that means that when the second line of the conversation from the learner is processed, the growing conversation includes the word formula from the tutor’s response (ok what is the formula). this is to ensure that the insights of the tutor are also used in providing possible help. when the third line from the learner (simply the word no) is received, the growing conversation also includes the word factor (or fact after stemming) from the tutor’s response (can you factor that?). at this stage, the topic similarities were: graphs (0.178), factoring (0.157), and quadratic (0.156). the next line received from the learner (1 n 2?) is the guess of the factors as being one and two. the growing conversation now includes the terms integers (stemmed to teg) and add. it is interesting to note that the string mutliply failed to be recognised as the word multiply. there are two reasons for this. since the spelling corrector works with stemmed words, it compares multipl and mutlipl, rather than multiply and mutliply. of the 20 common n-grams, five are shared, giving a similarity ratio of 25%. this similarity is under the cut-off of 29%. however, even with ignoring mutliply, the topic similarity values were: graphs (0.207), algebra (0.202), and measurement (0.164). when the last line was received from the learner, adding the term root to the growing list of mathematics terms, the similarity ratios were: factoring (0.234), algebra (0.222), and graphs (0.221). however, it is clear that this conversation is really not about graphs, despite the fact that the learner specifically asked about graphs. the conversation is really about factoring. for this reason the three highest-ranking topics are displayed to the tutor whenever a new line is received from the learner. presenting the topic to the tutor (back to top) when the μ topic spotter was integrated into the dr math tutoring platform, the dr math system administrator or domain expert created a list of websites or web pages which held good supporting information on each of the specific mathematics topics supported by the μ topic spotter.figure 6 shows a sample tutoring screen where the learner asked: area of a circl (note that the learner is asked by the dr math system to include the number 75, to be certain that the message comes from an actual human and thereby protect the tutors from mxit-based spam. these numbers are stripped from the messages for our analysis.) figure 6: example tutor screen. in the left column of the screen in figure 4, the three best guesses as to the topic of the mathematics conversation are displayed as links to supporting documentation. in this particular case the top three choices were circle, geometry, and trigonometry. by clicking on the link the tutor was directed to a webpage which may provide the tutor with assistance in helping the learner. evaluation (back to top) evaluation of the μ model and the μ topic spotter involved testing the topic spotter on conversations which happened after july 2011. the evaluation did not include formal user evaluation from the tutors, although some feedback was received that suggested that the μ topic spotter does identify relevant topics.for the evaluation we selected conversations from random days in august and september 2011, yielding 1399 conversations between learners and tutors. these conversations were manually reviewed and categorised into mathematical topics and subtopics by l.b. this was done prior to using the μ topic spotter on the conversations. we acknowledge that some bias may be present in this evaluation as an independent person unrelated to the project was not used to categorise the conversations in the corpus used. however, the extensive tutoring experience of l.b. in the environment provides a high degree of confidence in the correctness of the classification. nevertheless, we are embarking on a project to build a tagged and validated corpus of dr math conversations to be used by this and related projects. of the 1399 conversations, 805 did not cover any topics in the mathematics curriculum. these 805 conversations included general discussions about examinations, information about how the dr math project works, and requests for non-mathematical help, and were removed from the sample. the remaining 594 conversations varied in length as well as in mathematical content. similar to during the initial identification of mathematics terms, conversations could be assigned to more than one topic and more than one subtopic. for example, the following conversation would be classified as falling under the topics of both trigonometry and graphs: hi how can i help? hey dr maths i hv some prblms i ned 2 knw abt trig graphs coz i hv a crus prblm wif dat graph well lets start with a sin graph it starts at 0, and goes between 1 and -1 over 360 degrees the μ topic spotter then processed the 594 conversations and the topics determined by the μ topic spotter were compared with the topics which had been manually assigned. in order to be considered to be correct, at least one of the topics suggested by the μ topic spotter had to match at least one of the manually assigned topics. the results can be seen in table 3. the μ topic spotter spotted the correct topic of discussion in more than 75% of the conversations, and was thus able to supply the tutor with fast, relevant supporting documentation three-quarters of the time. table 3: results of topic-spotting tests. in addition, some characteristics of the successful and unsuccessful topic selections were generated. the average number of words and number of characters in each of the 594 conversations were calculated. the average number of lines in the conversations was ignored because the concept of a line and the length of a line varied greatly between learners using a small screen on a mobile phone and tutors using a normal internet-based workstation.as can be seen in table 4, messages which were classified correctly had 24% more words in the message and more than double the number of characters. the longer the conversation, the better the μ topic spotter is at correctly determining the topic under discussion. table 4: message sizes categorised by correct or wrong classification. discussion (back to top) this article presented the μ model that aims to identify mathematics topics in mxit conversations and discussed an instantiation, the μ topic spotter, in the dr math tutoring environment. the instantiation demonstrated the feasibility of implementing the μ model. initial evaluation of the results showed the μ topic spotter to provide appropriate supporting documentation in more than 75% of cases. considering the idiosyncratic nature of mxit lingo, this definitely represents a useful result for the purposes of the model.in this study a corpus of conversations coded by a single person was used. work has already started on providing a tagged corpus of dr math conversations for use in this and other projects. such a validated corpus will allow more formal evaluation of the model and enable several other research areas. during the implementation of the μ topic spotter several lessons were learned and observations made that will allow further refinement of the μ model and thus of the μ topic spotter. one shortcoming that was identified is grounded in the observation that mxit spelling changes often occur within a word; for example, in this article ‘trasnform’ was equated to ‘transform’, since the similarity value was just slightly above the cut-off point. however, just adapting the cut-off point introduces many new false-positives. this demonstrates the limitations of n-gram processing to cater for misspelled words. to complicate matters, transform could just as easily been spelled as ‘tr@ns4m’ in mxit lingo. currently μ model cannot equate such cases to the word transform. to cater for these strange spellings, better tokenising of mxit lingo is needed. numerals often appear in mxit words in normal conversation. for example, ‘n0t’ is a common mxit spelling of the english word ‘not’. however, in view of the fact that this implementation of the μ topic spotter was for a mathematics tutoring environment, numerals were used as word delimiters to cater for mathematical expressions. for example, learners often typed expressions such as ‘x2plus5xplus6’ without any spaces, and it was necessary for the dr math project that the word ‘plus’ could be extracted from that string of characters. as a better understanding of the mxit lingo specifically as it relates to mathematics tutoring is achieved, it may be possible to do even more domain-specific tweaking. one such suggestion is to add weighted values to the mathematical terms or, possibly, even for specific strings. for example, while searching the log files for this research one conversation was found where a learner asked about ‘that python thing’. being in a mathematics tutoring environment and not a discussion forum about snakes or programming languages, the likelihood of the question being about pythagoras theorem is high. perhaps future research could indicate whether just receiving the string ‘py’ at the beginning of a word would be sufficient to present the tutors with supporting documentation about pythagoras theorem. conclusion (back to top) the dr math project is an important on-going project with tens of thousands of learners having used the system since its inception. volunteer tutors are often swamped with questions and could use assistance in dealing with them. the μ model aimed to provide such assistance to tutors by providing timeous access to supporting documentation by automatically identifying the mathematics topic being discussed. table 5: guidelines for good design science. although there are still many opportunities for improvement, the model as it stands achieved the research objective, in that a topic spotter that can timeously identify mathematics topics in conversations between dr math tutors and learners was created. we also believe that we have met the seven requirements for good design science, as stipulated by hevner et al. (2004) and summarised in table 5. firstly, two clear, purposeful artefacts were produced, thereby meeting the basic requirement of the first guideline. guideline 2 is implicitly matched, as a relevant problem from an operational tutoring environment was selected as a problem area. providing appropriate supporting documentation in just over 75% of the conversations about mathematics clearly demonstrates the functionality and usefulness of the model, which addresses guideline 3. given the nature of mxit lingo, a model yielding a greater than 75% match can be argued to be sufficient to make a clear contribution, as required by guideline 4. the methods used in construction of the model are all well-known and accepted practice in the area, meeting the requirement for rigorous methods as stipulated by guideline 5. however, it could be argued that a corpus of dr math conversations using multiple coders would provide additional strength and rigour. guideline 6, seeing design as a search process, was met through the process of circumscription, while publication of this model contributes to the communication requirement set by guideline 7. the dr math tutoring service provides help to many learners where other assistance is not accessible. we believe that the μ model will aid tutors in helping learners more effectively, and we will therefore continue to refine the model and produce further initiatives around the model and the dr math platform. acknowledgements (back to top) we acknowledge the assistance of michelle van den heever, who holds a ba hons in applied language studies with a major in theory of second language acquisition. she reviewed many of the mxit-based conversations from a linguistic point of view and provided insight into the spelling conventions used in mxit lingo. r.b. thanks the national research foundation for partially supporting his research, while l.b. thanks the rupert family trust for the rupert gesinstigting award that she received to support her phd studies. competing interests the dr mathtm project is hosted at the meraka institute at the council for scientific and industrial research (csir). the term ‘dr math’ is a trademark of the meraka institute. l.b. is an employee of the meraka institute, csir, and, as such is eligible for various employee benefit-sharing programmes with regard to intellectual property rights. authors’ contributions l.b. is a phd student and r.b. is her supervisor. both l.b. and r.b. contributed to the conceptualisation of the research method and the resultant model in the study. l.b. did the programming and coding of conversations. l.b. wrote the manuscript with r.b. reviewing drafts thereof. references (back to top) barinaga, m. 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(1992). the automatic identification of stop words. journal of information science, 18(1), 45–55. http://dx.doi.org/10.1177/016555159201800106 footnote (back to top) 1.the meraka institute is an operating unit of the council for scientific and industrial research (csir), south africa, focused on information and communication technology. abstract introduction language and multilingualism in mathematics education different orientations to language methodological approach themes in the research discussions and concluding remarks acknowledgements references footnotes about the author(s) kathryn mclachlan mathematics education division, wits school of education, university of the witwatersrand, johannesburg, south africa anthony a. essien mathematics education division, wits school of education, university of the witwatersrand, johannesburg, south africa citation mclachlan, k., & essien, a.a. (2022). language and multilingualism in the teaching and learning of mathematics in south africa: a review of literature in pythagoras from 1994 to 2021. pythagoras, 43(1), a669. https://doi.org/10.4102/pythagoras.v43i1.669 review article language and multilingualism in the teaching and learning of mathematics in south africa: a review of literature in pythagoras from 1994 to 2021 kathryn mclachlan, anthony a. essien received: 23 nov. 2021; accepted: 11 may 2022; published: 22 july 2022 copyright: © 2022. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract this article presents a systematic review of research on language and multilingualism in mathematics education published in the south african journal pythagoras from 1994 to 2021. this time frame was chosen as the year 1994 marked the acknowledgement of 11 official languages in the new democratic south africa (including 9 indigenous languages), compared to only afrikaans and english during the apartheid era. the review considers emergent themes in the included articles and examines what the articles reveal about mathematics education in south africa. in addition to other findings, our corpus of literature indicates that research in this field of mathematics has been mostly undertaken in under-resourced schools and that research under the theme of multilingual education is at the forefront of research in south african mathematics education, while research on language policy needs more attention. research on multilingual education in our corpus of literature also reveals great awareness of the value of seeing language as a resource, as well as the benefits that accrue when learners’ home languages are taken into account in mathematics teaching and learning. the gaps in research in the field of language and multilingualism in the teaching and learning of mathematics are also noted and recommendations for future research are made. keywords: language and mathematics; multilingualism; language orientations; language-as-resource; language-responsive teaching. introduction the attention to the intertwinement of language and mathematics, that is, how language issues are imbedded in mathematics, has gained prominence in the last couple of decades. teaching and learning mathematics, although perhaps stereotypically associated with manipulating numbers, is now fundamentally seen as inextricably intertwined with language. pimm and keynes (1994) argue that ‘the teaching and learning of mathematics involves the activities of reading and writing, listening and discussing’ (p. 160, emphasis added), four activities that all rely on language. the intricate relationship between language and mathematics teaching has come to be seen as critical in mathematics education research over the past few decades especially in multilingual classrooms (see, for example, morgan, craig, schutte, & wagner, 2014). as the world has become more globalised, the challenges and opportunities of learning mathematics in multilingual spaces – that is, spaces where more than one language is present and presents a potential to be used (barwell, 2016) – have come to the fore. postcolonial countries such as south africa are examples of contexts where these challenges and opportunities intersect in a complex manner: matters of language in mathematics classrooms go beyond reading, writing, listening and discussing, to doing so in multiple languages, against a backdrop of political change and postcolonial policies. in south africa, language use in (mathematics) classrooms is intricately linked to the country’s political history. during the apartheid era, language-in-education policies served the agenda of the apartheid state and promoted the status of afrikaans and english over indigenous languages. resistance to apartheid education language policy culminated in the fatal 1976 soweto uprisings, and addressing issues of language – both within and beyond education – was a critical point in the new democratic south africa of the early 1990s (adler, 2001). south africa’s interim constitution (1993), which was enacted in 1994, saw the acknowledgement of 11 official languages in south africa, adding 9 indigenous languages to the previous official languages of afrikaans and english. this was an ideological shift towards formally recognising south africa’s multilingualism which was further evident in the final constitution of the republic of south africa (1996), and the language-in-education policy document (department of education, 1997). this language policy, still in use at the time of writing, supports multilingualism in schools. in theory, students should be taught in their first (home) language from grade 0 to grade 3 before english (or afrikaans) becomes the language of learning and teaching (lolt) from grade 4 to grade 12. however, this is not always the case, as even from grade 0 many school stakeholders prefer to do mathematics in english due to its political position as a language of prestige (see essien, 2018; setati, 2008). additionally, it has been argued that this policy promotes a system of multiple monolingualism in schools rather than the suggested multilingualism (sapire & essien, 2021). this article undertakes a systematic review of research on language and multilingualism in mathematics education as presented in the south african journal pythagoras from 1994 to 2021. the choice of this journal is due to the fact that it is the only journal in south africa that is directly and solely focused on the teaching and learning of mathematics at all levels of education. the lower bound of this time frame was selected due to its links to the momentous political transformations that occurred in 1994, and the associated changes to language and education policy. the upper bound of 2021 is the year of writing, 28 years into south africa’s democracy. this time frame allows for the analysis of research papers written during the period of active education and language (policy) changes of the 1990s, as well as subsequent years during which these changes have been implemented and have taken root. this systematic review was thus informed by the following research questions: what research has been published in pythagoras from 1994 to 2021 in the field of language and multilingualism as it concerns the teaching and learning of mathematics? what does this research reveal about language and multilingualism in mathematics education in south africa? in answering these research questions, we examine what research has been undertaken and published in pythagoras journal since south africa’s democratic dispensation in the area of language and multilingualism in mathematics education. with the country’s complex language-in-education history, we interrogate what patterns and themes are evident in the corpus of literature, how these may have changed over the 28 considered years, and what research gaps are evident. defining language is not straightforward. morgan et al. (2014) argue that in a mathematics education context, language is defined in many different ways. some definitions deal exclusively with words (spoken or written), while others include non-verbal communication, mathematical symbolism, or mathematical register (halliday, 1974). ‘language’ as a term is also used to describe natural languages used in (multilingual) classrooms. in this systematic review, however, an understanding of what constitutes language in mathematics education aligns primarily with pimm and keynes’s (1994) activities of reading, writing, listening and discussing. the current study also reviewed papers dealing with issues related to teaching and learning in multilingual classrooms, where multilingualism is stressed in the context in which teaching and learning occur. language and multilingualism in mathematics education as indicated previously, the study of language has become an active focus in (mathematics) educational research in the past few decades. as radford and barwell (2016) write, ‘language, talk, text and the production and interpretation of symbols are integral to the creation of learning, teaching and assessment, particularly in mathematics’ (p. 275). language is the medium through which mathematical ideas can be communicated and negotiated. mathematics, although imbedded in a natural language such as english, has come to be seen as a specialised language. students need to learn to acquire or appropriate a mathematical register as they learn the subject (zevenbergen, 2000) so that they can speak, hear, read and write its symbols and vocabulary with understanding. clarkson (2009) cites various models that suggest that students progress from informal language to more mathematically structured language, and ultimately to academic mathematical language as they learn mathematics. learning mathematics thus involves more than the simplistic view of working with numbers or algebra but is intricately bound to learning its language. difficulties involved in learning mathematics are compounded when this occurs in multilingual classrooms – that is, any classroom where more than one language is present, even if only one language is overtly used. increased globalisation of the last few decades has led to teachers being ‘increasingly faced with students who draw on a variety of different languages and other language practices, many of which are unfamiliar to them’ (barwell, 2016, p. 36). multilingual classrooms have been studied globally, notably in south africa by adler (2001) and setati (2005), in australia and papua new guinea by clarkson (2009, 2016), in pakistan by halai (2009), in tanzania by kajoro (2016), in the united states of america by moschkovich (1999, 2003), and in spain by gorgorió and planas (2001). common themes in this research include the use of code switching (adler, 2001; halai, 2009; setati, 2005), how to support english language learners in the mathematics classroom (clarkson, 2009; moschkovich, 1999), and the politics of language in multilingual classrooms. the politics of language has been highlighted in multiple settings – in particular in postcolonial countries – where it has been generally observed that english is seen as a preferred lolt over indigenous languages due to its association with power and prestige (adler, 2001; clarkson, 2016; halai, 2009; kajoro, 2016; setati, 2005, 2008). this leads to complexities as research has found that learning in a first language for as long as possible is most beneficial for students (king, 2003). balancing first language and english lolt is thus a contentious, political issue in language planning and policy (bamgbose, 1999; clarkson, 2016). finally, a key additional theme in the recent literature on multilingual mathematics education is that multilingualism is positioned as a resource in the classroom, and not as a problem (adler, 2001; barwell, 2018; erath, ingram, moschkovich, & prediger, 2021; gorgorió & planas, 2001; moschkovich, 1999; moschkovich & zahner, 2018; mostert & roberts, 2020). different orientations to language ruiz’s (1984) seminal work on language orientations in language planning, where he elaborates on three approaches to language planning, has become widely used and is useful in thinking of language issues in teaching and learning, the development of language policies, and what orientations or ideologies inform language policy and language practices. ruiz (1984, p. 16) defines orientation as the ‘complex of dispositions toward language and its role, and toward languages and their role in society’, and asserts that there are three orientations towards language: language-as-problem, language-as-right, and language-as-resource. in brief, the language-as-problem orientation emphasises monolingualism and the tendency to move towards the more powerful language, and also sees multilingualism as a problem to be solved (see planas & setati-phakeng, 2014; ruiz, 1984). in south africa, this orientation towards language supports the use of only english (or afrikaans) in the teaching and learning of mathematics and thus sees the presence or use of other (indigenous) languages that are present in the class as problematic. the language-as-right orientation highlights the right of an individual to use one’s own language without being discriminated against. while the language-as-right orientation is clearly articulated or enshrined in the constitution of the republic of south africa (1996), how this translates concretely into practice has been questioned by research (see, for example, planas & setati-phakeng, 2014). finally, language-as-resource sees the presence of a multiplicity of languages not from a deficit point of view as does language-as-problem but from a resource perspective – that is, as something advantageous that should be harnessed. as alstad and sopanen (2020) rightly point out, in the language-as-resource orientation, multilingualism is considered as a resource not only for the linguistically marginalised, but for everyone. in the context of south africa, this would entail seeing multilingualism as a resource not only for those whose first language is not english (lolt in most cases) but for both those who have english as a first language and those with english as an additional language. in such a situation, using the different languages present in the class to enrich the discussions becomes of utmost importance. methodological approach for this review, pythagoras was selected as the source of considered research studies. pythagoras is an open-access, peer-reviewed, accredited academic journal published by the association for mathematics education of south africa (amesa). as indicated earlier, it is the only peer-reviewed accredited journal that solely focuses on mathematics education in south africa. as our study sought to identify the extent of research on language issues and communication in mathematics education in south africa since 1994, analysing research articles from pythagoras appeared to be a useful starting point as a representative of the field. additionally, although pythagoras has published continually since 1980, only issues since 2004 are available online. content from 1980 to 2003 is thus much harder to examine and requires access to specialised libraries to retrieve. this review thus also sought to document research on language and communication from 1994 to 2003 that is less freely available to help make sure that it is not forgotten. inclusion criteria for articles in selecting articles, we included only original research whose focus aligned with pimm and keynes’s (1994) language and communication activities of reading, writing, listening and discussing (both in monolingual and multilingual studies). examples of words that appeared in literature that were selected include ‘speak’, ‘dialogue’, ‘discourse’, ‘narratives’, ‘language’, ‘multilingual’, ‘semiotics’, ‘discussion’, ‘listening’ and ‘writing’ as well as names of natural languages such as ‘isizulu’ or ‘isixhosa’, etc. two articles, namely powell (1998) whose article’s title includes the word ‘dialogue’ and mellor, clark and essien (2018) whose article’s title includes the word ‘german’, were considered but excluded from the review. this is because powell’s article discusses internal dialogue, in the sense of metacognition, while mellor et al.’s article analyses mathematics textbooks without a focus on language. as such, both articles were deemed beyond the scope of this systematic review. in total, 31 articles were included in the corpus of literature that we analysed. general overview of reviewed literature figure 1 shows the number of papers published per year. 2008 is the year with the most papers, largely influenced by the ‘special issue’ focusing on multilingualism in the teaching and learning of mathematics published that year. this special issue was prompted by a systematic review of literature between 2000 and 2007 on multilingualism in south africa – a paper that was later published in 2009 (see setati, chitera, & essien, 2009). this ‘special issue’ will be discussed in greater detail later in this article. in terms of language of writing, 30 of the research papers were written in english, with one article written in afrikaans (uys, 1999). in terms of context, 24 studies were situated in south africa and 7 studies were based in other countries. figure 1: number of papers on language and communication published in pythagoras per year. the 31 analysed papers considered a broad range of education contexts and levels, ranging from primary school to tertiary level, and including professional development situations. however, the number of articles per category varied considerably (see table 1). table 1: number of articles published per education context level. table 1 indicates that most research on language and communication in mathematics education has taken place in the high school context, with research based in tertiary institutions (both pre-service and in-service teachers) being the next most prevalent. papers published in pythagoras thus are in line with the finding described by sapire and essien (2021) that: [in] south africa, although there has been ongoing reporting on the status of the lolt in schools, there has been a dearth of research studies undertaken in early grade (grades r to 3) mathematics classes. (p. 77) among the papers analysed in this review, only one paper (mostert, 2020) considers lower primary. all the other papers on primary school are based in higher grades. it is interesting that so little research has been published (in pythagoras) on lower primary school, as this is the time where in south africa the lolt switches from mother tongue instruction to english. contrastingly, there appears to be a trend towards research being undertaken in tertiary institutions or in professional development situations. over the last 10 years, of the 14 papers published, 6 were based in tertiary institutions and 2 considered in-service teacher professional development (a combined total of 57% of published papers). only 4 papers were based on high school contexts, 1 on the primary school context, and 1 paper considered both primary and high school. it thus appears that a shift has occurred to researching situations and questions concerning prospective and practising teachers. for the corpus of literature under the review time frame, we noted no trends in publications on the basis of how well-resourced or poorly resourced the context of research is. while some papers are theoretical or have for context teacher education – and as such were not counted in terms of resource level – we noted that none of the papers on language or multilingualism published in pythagoras during our time frame had well-resourced schools as the sole context. we noted four research studies that were carried out in two or more schools wherein one was well-resourced and the other poorly resourced. seven of the papers had under-resourced schools as their research context. this, in a way, reinforces the language-as-problem orientation as one interpretation of the lack of focus on well-resourced schools could be that these schools are framed as not having language issues although they are south african multilingual schools in their own right. also of interest in our review of papers is which official languages are represented in research in south africa. our analysis revealed that 15 of the papers focused on or referenced the use of english; isizulu and isixhosa had 4 each; setswana and sesotho had 3 papers each; afrikaans had 2 and sepedi, swati, tshivenda and xitsonga had 1 paper each. we found no paper focusing on the use of isindebele in the teaching and learning of mathematics. themes in the research we examined all the papers and developed a summary spreadsheet and inductively coded the papers and then looked for themes. the themes emerged from reading the articles. our analysis of the corpus of literature in pythagoras under our time frame resulted in the identification of 11 areas of research. these themes are elaborated in table 2. table 2: emergent themes from the surveyed research outputs. table 3 provides a summary of the 11 identified areas of research in the analysed papers. we chose to categorise each paper into only one area of research. if a paper referred to more than one of the identified areas, we only categorised it into its main (most relevant) theme based on the study’s focus. for example, although tobias’s (2003) paper briefly mentions multilingual classrooms, his paper mainly examines linguistic elements of classroom mathematics. the paper was thus categorised under ‘mathematics as a language’. table 3: number of papers per area of research. two clear, predominant themes emerged based on the coding exercise: research on ‘multilingual education’, and research on ‘speaking’ in the classroom. all other themes had three or fewer papers. figure 2 shows the thematic patterns in our corpus of literature. papers of the same theme have been indicated in the same colour, while themes with only a single paper have been left in white. this analysis indicates that beyond the special issue on multilingual education in 2008, there are no time-related thematic patterns that immediately stand out. what can be seen, however, is that in the last 10 years (since 2011), four articles that link to ‘speaking’ and three articles that link to ‘discourse’ have been published. beyond this, there are no clear time-based trends. figure 2: paper topics per year. in the following section, we examine the themes as indicated in table 3. we first consider themes with three or fewer papers, before looking more specifically at the themes with greater focus, mainly ‘multilingual education’ and ‘speaking’. in some cases, we use the language orientation framework by ruiz (1984) discussed above to interpret the orientation to language evident in the body of literature. in our discussion of each article, we also indicate if the research is not based on the south african context. reading, terminology, visual communication and writing we found a total of four papers (one each) for these themes. for reading, mwale and mwakapenda (2018) explore the relationship between reading and mathematics through an investigation on the extent to which students can see mathematics in non-mathematical texts. they find that students struggle to identify mathematics in texts where they would classically not be looking for mathematical content. for terminology, atebe and schäfer (2010) examine high school students’ (in nigeria and south africa) proficiency of geometry vocabulary against a backdrop of the van hiele theory of levels of geometry understanding. they find that the participating students had low ability in basic geometry terminology, and that verbal geometry ability has a high correlation with the ability to work with visual problems based on the same terminology. also based in geometry, for visual communication, mudaly (2010) examines the role that visualisation plays in developing mathematical understanding and reasoning. the study illustrates how using visualisation tools (here, sketchpad) can quicken the process of reasoning and testing conjectures in geometry-based problem-solving. finally, for writing, us-based powell (2001) explores the use of student writing as a means by which teachers can access, examine and respond to students’ internal mathematical thinking. the diverse content of these four papers shows the ubiquitous nature of language and how it relates to mathematics teaching and learning in diverse ways. however, it is hardly surprising that there is limited attention given to issues of reading, writing and english terminology use in our extant literature. given the multilingual context in which teaching and learning are imbedded in south africa, the focus of research (as evident in table 3) has been on multilingual issues rather than language issues in relation to english use. listening, and mathematics as a language we found two papers each under the themes of listening and mathematics as a language. both papers on listening, namely breen (2004) and mhlolo and schäfer (2012), reference davis’s (1997) three listening styles of evaluative, interpretive and hermeneutic listening. however, these types of listening are used differently in both papers. for breen, listening is considered as a means of assessment. for mhlolo and schäfer, listening is regarded as an indicator of the levels of democracy present in a mathematics classroom. although breen’s article focuses on listening, the discussed research context uses written journals as a means of accessing his students’ thinking in a similar way as described by powell (2001). in terms of mathematics as a language, tobias’s (2003) paper considers the complexities of mathematical register such as vocabulary, semantic structure (how the meaning ascribed to a text points to particular mathematical operations or relationships)1, and lexical density (the ratio of content-related (mathematics-related) words to grammatical words in a given text)2. he argues that it is crucial for teachers to understand the complex linguistic aspects of the mathematics classroom so that they can support their students to navigate the language of mathematics. patkin’s (2011) paper considers the interplay of mathematical and everyday language in an israeli teacher training context. in her work, everyday language is seen as both a resource and a problem. she describes exercises undertaken using non-mathematical literature to heighten pre-service teachers’ awareness of words that carry multiple meanings within mathematical and everyday language. her work thus overlaps to a limited degree with that of mwale and mwakapenda (2018) by using written texts that would classically be deemed non-mathematical in conjunction with mathematical content. discourse and speaking our corpus of literature revealed three papers under the theme of discourse. venkat and adler (2012) discuss teacher discourse through the lens of mathematical discourse in instruction (mdi). their article examines and defines teachers’ mdi, which includes the mathematical features of teachers’ talk, actions and writing as they interact with students. the authors develop analytical language to examine how coherently a teacher moves from a stated mathematical problem towards a solution. lampen (2015) uses the concept of mdi to explore required teacher knowledge and skill to meaningfully conduct classroom discussions on the statistical mean algorithm in a connected manner. contrastingly, berger (2013) examines student discourse from the perspective of sfard’s theory of commognition. her article highlights the importance of words when discussing mathematical phenomena as indicators of (developing) student understanding. this highlights the importance of instructors carefully listening to student discourse as a means of accessing their understanding. for speaking, our analysis categorised five papers. in different ways, these articles make clear the opportunities for learning that are inherent in group dialogue, and the vital role of the discussion facilitator in this process. brodie (2007) considers classroom dialogue and how to create opportunities for learner talk and participation. she argues that the teacher-led initiation-response-feedback (irf) model does not create rich classroom discussion and proposes various alternatives to create genuine student dialogue. sepeng and webb (2012) explore the use of discussion as a teaching strategy. using carefully considered teacher-led discussion techniques, focusing on reasoning, their study indicates improvement in student problem-solving performance and ability to make sense of real-world word problems. contrastingly, daher (2012) considers palestinian pre-service teachers’ perceptions of democracy within the mathematics classroom, of which dialogue is a key element. the students considered the didactic communication acts of discussing, asking, arguing, and listening as part of democratic dialogue (classroom) practice. within the context of one professional learning community, chauraya and brodie (2018) analyse group conversations for learning opportunities. their study indicates that facilitator-led group conversations create opportunities for participating teachers to develop their knowledge. in particular, the study positions the facilitator as crucial in creating these learning opportunities. finally, also within the sphere of professional development, gierdien, smith and julie (2019) consider different ways in which university researchers and classroom teachers talk of, understand and work with the (same) teaching and learning of mathematics. they find that using ‘toolkits’ can anchor conversations and help university-based mathematics educators to bridge the gap of contrasting discursive practices. it is significant to note that talk in the mathematics classroom has enjoyed a good deal of attention in our selection of literature as creating a classroom where there is productive disciplinary engagement (engle & conant, 2002) is a key focus of research on language and mathematics in global research in recent years. semiotics and natural language our corpus of literature revealed three papers in each of the themes ‘semiotics’ and south african national languages. for semiotics, ernest’s (1998) article examines links between mathematical written signs and their meanings. this involves a complex interplay between signifiers and signifieds, with meanings existing in both private ‘maths words’ of the student and publicly between the student and the teacher or researcher. vile’s (1998) article, on the other hand, interrogates the popularity of semiotics in mathematics education at the time of writing. he argues that semiotics provides a framework to analyse meaning making that allows for systematic interpretation of classroom events for both researchers and teachers. both these studies are uk-based. finally, ubah and bansilal (2019) examine semiotic representation as part of reasoning in euclidean geometry. their study finds that some pre-service teachers struggle to move between visual and symbolic registers of representation, and that concrete manipulatives can be helpful in bridging this gap. these findings link to mudaly’s (2010) paper, with both cases suggesting that manipulatives can be helpful in developing geometric reasoning. finally, three papers discuss mathematical issues related to natural language. uys (1999) considers the difference in number structures in afrikaans and other south african languages, highlighting english, and how this can cause difficulties for afrikaans additional language learners. van laren and goba (2013), on the other hand, examine the effects of developing isizulu versions of various postgraduate certificate in education courses normally taught in english. they find that, although the isizulu courses were generally well received, difficulties were encountered with isizulu mathematics register, and having to translate english research paper content into isizulu for assignments. it additionally became apparent that, socially, isizulu instruction was seen as of a lower status than english instruction, although the course content was identical. lastly, mostert (2020) investigates isixhosa word problems in grade 1–3 classrooms. she examines the relative difficulty of different compare-type problems in isixhosa and how the isixhosa wording can affect how students experience the problem’s level of difficulty. mostert points to the importance of studying how african languages convey mathematical ideas to better understand the learning affordances of tasks. while the work of uys is orientated towards language-as-problem, van laren and goba and mostert both work with the language-as-resource orientation. the work under the theme of natural language relates to the work on multilingual education, to which we now turn. multilingual education the most prevalent theme in our corpus of literature is multilingual education. the earliest paper in this theme is rakgokong (1994) who wrote amid political change in south africa. rakgokong argues that language policy must be informed by research and have a child’s ability to perform at heart. he strongly argues against english being the sole lolt for mathematics in higher grades, particularly in constructivist classrooms where meaning is developed through discussion and negotiation. the next six articles all stem from pythagoras’s special issue on multilingual classrooms published in 2008. as indicated previously, this issue was motivated by the paucity of research on multilingual classrooms published in pythagoras as described in setati et al. (2009). in the south african context, the majority of students learn mathematics in an additional language making this topic particularly relevant and necessary to be covered in a local journal (setati & barwell, 2008). two papers in this issue consider matters of education policy. dlamini’s (2008) article considers language issues regarding university entrance in eswatini. he argues that eswatini university language policies require english proficiency for acceptance into science degrees, although english ability is not a predictor of mathematical ability. as such, he suggests that these policies discriminate against the indigenous population, and fail mathematically gifted students who struggle with english language. kazima (2008), on the other hand, investigates different ways in which countries have developed mathematical registers in indigenous languages. nigeria and tanzania have developed ‘new’ terminology while malawi has borrowed words from english. she discusses the strengths of weaknesses of these contrasting policy decisions. three articles in this issue consider matters relating to code switching. setati, molefe and langa (2008) investigate how to use indigenous languages in the mathematics classroom without compromising the development of crucial english language skills. their study indicates that having access to mathematical tasks in both english and a student’s main language is beneficial to the student. they argue that it is simplistic to think that mathematics classrooms should function in only one language. webb and webb (2008) examine code switching as a means of promoting exploratory talk. they contend that in english lolt multilingual classrooms, where english proficiency is low, students do not engage in much talk, with teachers using irf cycles. by using code switching as an element of exploratory talk, their paper describes how some teachers experienced success in achieving more student talk in their classrooms. vorster (2008) investigates how the use of multilingual material (a glossary, as well as providing test questions in english and setswana) can aid the code switching process. this study indicates that students benefit from having both languages available in their material. finally, bohlmann and pretorius (2008) investigate how in multilingual classrooms reading and language ability correlates to mathematical ability and find that mathematical performance is closely linked to reading capability. it is thus argued that poor literacy will affect mathematical learning. beyond the special issue, two final articles deal with multilingual issues. brijlall (2008) explores collaborative learning in multilingual classrooms and shows that students who worked in groups, where any language could be used for communication, performed better than students working individually. finally, ledibane, kaiser and van der walt (2018) consider the similarities in acquiring english as a second language, and mathematics ‘as a second language’ and argue that both can be acquired simultaneously. in terms of orientation towards language, like two papers in the theme of natural language, there is an orientation towards language-as-resource in all the papers under the theme of multilingual education. this is in line with international trends where, more and more, language is conceived of as a resource rather than a problem. discussions and concluding remarks the findings of this systematic review indicate that language and multilingual issues have been a regular part of pythagoras journal since 1994. in general, interest in these topics has increased substantially in the second half of this time frame (8 articles before 2008; 23 articles from 2008). this trend is mirrored in the locally based research, with six articles published before 2008 and 18 from 2008. more specifically, the extant literature also indicates that multilingual issues are at the forefront of research involving language in south african mathematics education. this is in line with the country’s multilingual context. research (for example, adler, 2001; moschkovich, 1999; setati, 2005) has since noted that teaching and learning in multilingual classrooms is complex, and various articles in this review (brijlall, 2008; setati et al., 2008; vorster, 2008; webb & webb, 2008) investigate methods for practising teachers to take on these complexities with increased success. what is surprising, however, is that beyond the special issue on multilingual classrooms which contributes six articles, only three articles have been published on multilingual education. should the special issue not have been published, multilingual education would thus be a minor theme in this review. setati et al. (2009) question the lack of research on multilingual education in the south african context (between 2000 and 2007) and posit that it may be the combined effect of the inter-disciplinary demands, necessity of multilingual research teams, and the political nature of this topic that renders it unpopular even though it is crucial. these reasons could be extended to pythagoras. it was also somewhat surprising to us to find few papers dealing with terminology in mathematics education, given that the issue of the importance of terminology as part of the mathematics register has always been a contentious issue in south africa. what also remains to be done with regard to multilingual issues is to pay some attention to translanguaging as a practice in its own right, or vis-à-vis code switching. code switching is generally seen as the use of two or more languages within a single interaction. as barwell (2016) notes, for much of the research on code switching, the analytical focus is on the languages as distinct, one from the other. what this means for research as barwell (2016) contends is that: a strong focus on distinct languages may … lead to the arbitrary separation in research and policy of multilingual mathematics classrooms in which only one language is used, from those in which two or more are used. (p. 29) this view of language as discrete and distinct is termed monoglossia and it excludes any consideration that, in reality, both teachers and learners have multilingual communicative repertoires that can be drawn upon with fluidity and flexibility. this brings us to translanguaging. garcía and wei (2014) define translanguaging as: the enaction of language practices that use different features that had previously moved independently constrained by different histories, but that now are experienced against each other in speakers’ interaction as one new whole. (p. 21, [italics in original]) rather than focus on the languages an individual speaks as distinct, one from the other, translanguaging posits that the languages with which learners and teachers come to school are fluid and flexible rather than codes between which the code switcher moves back and forth. as such, the language practices that become the focus of research on code switching for multilingual classrooms are different from what could be the focal point for research on translanguaging. another visible gap in our corpus of literature is research dealing overtly with matters of language policy in south africa. of the analysed papers, only four focus on issues of policy (dlamini, 2008; kazima, 2008; rakgokong, 1994; van laren & goba, 2003), with dlamini and kazima’s (2008) work based internationally. rakgokong’s (1994) paper, published during south africa’s stark political changes, calls for language policy of the new dispensation to be flexible and informed by education research for the child’s well-being and learning. van laren and goba’s (2013) article, on the other hand, investigates the enactment of language policy for higher education by promoting isizulu as an instruction medium. in both papers, a common theme is the lower perceived ‘status’ of education in indigenous languages compared to english, reflecting similar findings in the literature at large. matters of language policy are, of course, implicitly present in research on multilingual classrooms and bohlmann and pretorius (2008) and setati et al. (2008) both briefly speak towards policy (more directly than other analysed papers on multilingual classrooms). however, overall, it is surprising that in the south african context of political change and linguistic diversity, no papers directly focusing on language-in-education policy for grade 0–12 have been published since 1994 in pythagoras. since studies on language planning and policy implementation are widespread in south africa, it would be worthwhile to have research that focuses on the integration of studies on language policy implementation in terms of how this (research) has in turn shaped language policies and teaching practices. for example, findings of successful use of talk and materials in more than one language (setati et al., 2008; vorster, 2008) could be used as a foundation of a more flexible language policy (suggested by rakgokong, 1994) that takes into account translanguaging as a practice and how translanguaging can better serve the needs of multilingual classrooms. while it was not always easy for us to decipher clear trends in terms of the theoretical orientations in the corpus of literature, or in terms of the political orientations towards language (language-as-problem, language-as-right, language-as-resource), our review has revealed that in terms of the context of study, there was more research in under-resourced schools than in well-resourced schools. as indicated previously, this indicates an orientation towards language-as-problem. however, this is countered by the research on multilingual education in our corpus of literature, which indicates a great awareness of the value of seeing language as a resource, as well as the benefits that accrue when taking learners’ home languages into account in mathematics teaching and learning. international trends in language and multilingual issues in the teaching and learning of mathematics are shifting to language-responsive teaching of mathematics (erath & prediger, 2021; erath et al., 2021; essien, chitera, & planas, 2016; prediger, 2019; prediger & neugebauer 2021), which advocates for the development of the necessary knowledge and practices needed for the integration of ‘mathematics and language learning in a mathematics-specific way’ (prediger, 2019, p. 368). such research, which draws on the resource orientation to language, will entail, among others, empirical research that takes into account both the communicative and the epistemic functions of language as tools for thinking and knowledge acquisition (prediger, 2019). our corpus of literature also revealed that some languages are more represented in research than others. of note is that there was no research that referenced the isindebele language. while this is a gap in research, a more notable gap is the lack of research in the so-called non-official languages in south africa. while there may be 11 official languages in south africa, the reality is that south africa boasts of other languages that are not recognised as official. for example, there are the setlokwa and selobedu languages in limpopo, sepulana in mpumalanga, to mention but a few. early grade learners and teachers in these areas have to grapple with the added issue of teaching and learning in an indigenous language that is different to their home language. research into how these minority (or unofficial) language teachers and learners experience the teaching and learning of mathematics will certainly go a long way in informing the debates around language and multilingual issues in the teaching and learning of mathematics in south africa. acknowledgements competing interests the authors have declared that no competing interests exist. authors’ contributions k.m. produced the first draft. both k.m. and a.a.e. reconceptualised the data analysis and methodology after the original draft was produced. both a.a.e. and k.m. analysed the extant literature and contributed to the discussions section of the article. ethical considerations this article followed all ethical standards for research without direct contact with human or animal subjects. funding information this research received no specific grant from any funding agency in the public, commercial or not-for-profit sectors. data availability the authors confirm that the data supporting the findings of this study are available within the article. disclaimer the views and opinions expressed in this article are those of the authors. references adler, j. 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(2000). ‘cracking the code’ of mathematics classrooms: school success as a function of linguistic, social and cultural background. in j. boaler (ed.), multiple perspectives on mathematics teaching and learning (pp. 201–223). london: ablex publishing. footnotes 1. for example, drawing on other researchers, zevenbergen (2000) argues that in an additive change problem which can be formulated as 3 + 2 = x (lebo has 3 oranges, then she got 2 more. how many oranges does lebo now have?) is less complex for students than if the unknown is the first variable as in x + 2 = 5 (lebo has 2 oranges more than thabo. in total, there are 5 oranges. how many oranges does thabo have?). while both are about the equation 2 + 3 = 5, the first has a combine semantic structure, the second has a compare semantic structure. zevenbergen argues that wording a question to make the semantic structure familiar to learners may help learners better solve the problem but does not help learners in cracking the code of the mathematics register. 2. as such, a mathematics question with a higher content-grammar ratio is considered to be lexically dense and more difficult to read, which can mean that the question can be found to be more difficult for learners. article information authors: yael shalem1 ingrid sapire1 m. alejandra sorto2 affiliations: 1school of education, university of the witwatersrand, south africa2mathematics department, texas state university, united states of america correspondence to: ingrid sapire postal address: private bag 3, wits 2050, south africa dates: received: 31 jan. 2014 accepted: 06 aug. 2014 published: 04 nov. 2014 how to cite this article: shalem, y., sapire, i., & sorto, m.a. (2014). teachers’ explanations of learners’ errors in standardised mathematics assessments. pythagoras, 35(1), art. #254, 11 pages. http://dx.doi.org/10.4102/ pythagoras.v35i1.254 copyright notice: © 2014. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. teachers’ explanations of learners’ errors in standardised mathematics assessments in this original research... open access • abstract • introduction    • teachers’ knowledge of learners’ errors    • studying teachers’ explanations of learners’ errors in standardised mathematics assessment       • criterion 1: procedural understanding of the correct answer       • criterion 2: conceptual understanding of the correct answer       • criterion 3: awareness of error       • criterion 4: diagnostic reasoning of learners’ thinking in relation to error       • criterion 5: use of everyday links in explanations of error       • criterion 6: multiple explanations of error • teacher development research project    • operationalising the criteria • findings: usability of the measurement criteria • discussion: the teachers’ knowledge of error analysis • conclusion • acknowledgements    • competing interests    • authors’ contributions • references • footnotes • appendix 1 abstract top ↑ with the increased use of standardised mathematics assessments at the classroom level, teachers are encouraged, and sometimes required, to use data from these assessments to inform their practice. as a consequence, teacher educators and researchers are starting to focus on the development of analytical tools that will help them determine how teachers interpret learners’ work, in particular learners’ errors in the context of standardised and other assessments. to detect variation and associations between and within the different aspects of teacher knowledge related to mathematical error analysis, we developed an instrument with six criteria based on aspects of teachers’ knowledge related to explaining and diagnosing learners’ errors. in this study we provide evidence of the usability of the criteria by coding 572 explanations given by groups of mathematics educators (teachers and district officials) in a professional development context. the findings consist of observable trends and associations between the different criteria that describe the nature of teachers’ explanations of learners’ errors. introduction top ↑ reporting on data from standardised mathematics assessments that provide information about what learners can or can’t do is becoming a common practice in many countries. the reports set out to provide managers and teachers with reliable data, in the form of statistical averages, to be used to inform broad policy and classroom teaching. the elementary and secondary education act in the united states specifies that standardised assessment allows teachers ‘to get meaningful information about their practice, and support them in using this information to ensure that all students are getting the effective teaching they deserve’ (u.s. department of education, office of planning, evaluation and policy development, 2010, p. 15). in south africa, teachers are required ‘to interpret their own learners’ performance in national (and other) assessments’ (departments of basic education & higher education and training, 2011, p. 2) and develop better lessons on the basis of these interpretations. this requirement implies that teachers are expected to use learner data diagnostically and therefore, we argue, teachers’ involvement in error analysis of standardised and classroom assessment is no longer a professional right but a responsibility, an integral aspect of teacher knowledge. research has only recently begun to engage with the question of how to use learner data beyond that of a statistical indicator of quality, that is, beyond benchmarking for external accountability (boudett, city & murnane, 2005; cohen & hill, 2001; katz, earl & ben jaafar, 2009; katz, sutherland & earl, 2005). some attempts to examine a more balanced way between external and internal performance include shavelson, li, ruiz-primo and ayala (2002), black and wiliam (2006) and nichols, meyers and burling (2009). in south africa, reddy (2006), dempster (2006), long (2007) and dempster and zuma (2010) have each conducted small case studies on test-item profiling, arguing that this can provide useful data that can be used by teachers for formative and diagnostic purposes. notwithstanding these important contributions, there is very little research on how to analyse teacher knowledge of errors, analysis or what criteria can be used to assess teachers’ explanations of learners’ errors in standardised mathematical assessments. we hope that by analysing teachers’ explanations of learners’ errors, and through this their knowledge of error analysis, this article will advance this area of research and will also contribute to black and wiliam’s (1998) well-established argument of the positive potential impact of formative assessment.to analyse teacher knowledge of error analysis we developed an instrument with six criteria and compiled evidence of its usability as an analytical tool. this we did as part of our work with 62 mathematics teachers over a three-year period in the data informed practice improvement project (dipip, see more below). our central aim in developing the instrument was to detect variation and associations between and within the different aspects of teacher knowledge related to mathematical error analysis. with this in mind, we investigated the following research questions: 1. what is the nature of the teachers’ explanations of learners’ errors on standardised mathematical assessments? 2. what variability in the quality of the teachers’ explanations of learners’ errors can be identified using the criteria? 3. what relationship between aspects that inform the teachers’ explanations of learners’ errors are the criteria descriptors able to detect? the article proceeds as follows: in the first section we examine the idea of error analysis within the literature of teacher knowledge, focusing on its value for mathematics teaching. the next section describes the conceptual background on which we drew to develop the six criteria with a view to studying teachers’ explanations of learners’ errors. more specifically, we examine the aspects of error analysis included in three ‘domains of teacher knowledge’ (ball, hill & bass, 2005) and list the relevant criteria for studying teachers’ explanations of learners’ errors on an international standardised assessment test. in the third section we provide detail about the dipip project, explain the methodology we used to operationalise the criteria for our study of teachers’ explanations of learners’ errors and present exemplars of coding. in the last two sections, we assess the extent to which the criteria capture key error analysis aspects and use this to draw inferences about the nature of teachers’ explanations, their quality and the relationship amongst the different aspects that make up the six criteria. teachers’ knowledge of learners’ errors research in mathematics education has shown that a focus on errors, as evidence of mathematical thinking on the part of learners, helps teachers to understand learner thinking, to adjust the ways they engage with learners in the classroom situation, as well as to revise their teaching approach (adler, 2005; borasi, 1994; brodie, 2014; nesher, 1987; smith, disessa & roschelle, 1993; venkat & adler, 2012). some work has begun on developing broad classifications of learners’ errors (brodie & berger, 2010; radatz, 1979). studies on teaching dealing with learners’ errors show that teachers’ interpretive stance is essential for the process of remediation of error, without which teachers simply re-teach without engaging with the mathematical source of the error, or with its metacognitive structure (gagatsis & kyriakides, 2000; peng, 2010; peng & luo, 2009, prediger, 2010). nevertheless, there is hardly any literature that analytically examines the different aspects that make up teacher knowledge of error analysis and its relation to subject matter knowledge (smk) and to knowledge about teaching.knowledge of errors can be shown to incorporate both the substantive and syntactic dimensions of teacher subject matter knowledge. following the famous work of schwab (1978) and shulman (1986), rowland and turner (2008) propose the following definition of the substantive and syntactic dimensions of teacher subject matter knowledge: substantive knowledge encompasses the key facts, concepts, principles, structures and explanatory frameworks in a discipline, whereas syntactic knowledge concerns the rules of evidence and warrants of truth within that discipline, the nature of enquiry in the field, and how new knowledge is introduced and accepted in that community – in short, how to find out. (p. 92) this distinction is important. it suggests that subject matter knowledge includes the explanations of facts and concepts central to the discipline but also the rules of proof and evidence that a discipline community considers legitimate to use when making knowledge claims. both of these refer to aspects of teacher knowledge of error analysis. the substantive dimension foregrounds teachers’ explanations of what is erroneous and why, taking into account what a learner is expected to know, given the learner’s age and level of cognitive development. the syntactic dimension foregrounds teachers’ explanations of the process that needs to be followed to construct a truth claim, resolve a problem, get a correct solution, and so on. error analysis is an integral part of teacher knowledge, and the specific aspects informing teachers’ explanations of learners’ errors are the subject of this article. particular to the field of mathematics education, hill and ball (2009) see analysing learners’ errors as one of the four mathematical tasks of teaching ‘that recur across different curriculum materials or approaches to instruction’ (p. 70). peng and luo (2009) and peng (2010) argue that the process of error analysis includes four steps: identifying, addressing, diagnosing and correcting errors. in south africa, adler (2005) sees teachers’ knowledge of error analysis as a component of what she calls mathematics for teaching. she asks: what do teachers need to know and know how to do (mathematical problem solving) in order to deal with ranging learner responses (and so some error analysis), and in ways that produce what is usefully referred to as ‘mathematical proficiency’, a blend of conceptual understanding, procedural fluency and mathematical reasoning and problem solving skills? (adler 2005, p. 3) in this study we take up ball et al.’s (2005) idea of six domains of teacher knowledge and show the specific aspects of error analysis included in the first three domains (see next section). it is important to emphasise that underlying the attempts to classify the tasks involved in error analysis is an understanding that error analysis requires professional judgement to recognise exactly what learners do not understand, their reasoning behind the error, how that may affect their learning and which instructional practices could provide affordances (or constrain them) to address learner difficulties (shepard, 2009, p. 37). we offer the idea of ‘diagnostic reasoning’, to point to the delicate work of judgement involved in error analysis. appropriate judgement of how close or far a learner is from what is correct is core to teachers implementing appropriate assessment and feedback to learners. prediger (2010, p. 76) uses the notion of ‘diagnostic competence’ to distinguish reasoning about learners’ errors from merely grading their answers. his view is consistent with shepard’s (p. 34) work on formative assessment, in particular with the idea of using insights from learners’ work formatively to adjust instruction. making sound judgments, then, is key to formative assessment. the formative aspect lies in teachers’ explanations of learners’ errors, particularly when they are faced with misunderstandings exhibited by a large group of learners or when they need to scaffold further explanations in order to enable learners to cope with the complexity of a given task (city, elmore, fiarman & teitel, 2009). studying teachers’ explanations of learners’ errors in standardised mathematics assessment for the purpose of studying teachers’ explanations of learners’ mathematical errors we used ball’s classification of knowledge domains. we used this analysis to develop six criteria of teachers’ explanations of learners’ errors in standardised mathematics assessments within three of the domains of knowledge that define mathematics knowledge for teaching (ball et al., 2005; ball, thames & phelps, 2008; hill, ball & schilling, 2008). ball et al. (2008) explain that the first two domains elaborate the specialisation of subject-matter knowledge (common content knowledge and specialised content knowledge). the second two domains elaborate the specialisation involved in teaching mathematics from the perspective of learners, curriculum and pedagogy. these domains (knowledge of content and learners and knowledge of content and teaching) elaborate shulman’s (1986) notion of pedagogical content knowledge (pck). hill et al. (2008) argue that the first two domains are framed, primarily, by subject matter knowledge. this is very important from the perspective of examining teachers’ explanations of errors. as peng and luo (2009) argue, if teachers identify learners’ errors but interpret them with wrong mathematical knowledge, their assessment of learner performance and their plan for a teaching intervention are both meaningless. in other words, the tasks that teachers engage with in error analysis, such as sizing up the error or interpreting the source of its production, are possible because of the mathematical reasoning with which these domains of teacher knowledge equip them.in what follows we foreground key aspects of teachers’ explanations of learners’ errors relevant to each of the three domains we drew on specifically for the purpose of this analysis1. under the first domain, common content knowledge, we map aspects related to the recognition of whether a learner’s answer is correct or not. teachers need to recognise and be able to explain the crucial steps needed to get to the correct answer, the sequence of the steps and their conceptual links. because this knowledge underlies recognition of error, we include it under content knowledge. this analysis gives rise to two criteria in this domain: criterion 1: procedural understanding of the correct answer the emphasis of the criterion is on the quality of the teachers’ procedural explanations when discussing the solution to a mathematical problem. teaching mathematics involves a great deal of procedural explanation, which should be done fully and accurately for the learners to grasp and become competent in working with the procedures themselves. criterion 2: conceptual understanding of the correct answer the emphasis of the criterion is on the quality of the teachers’ conceptual links made in their explanations when discussing the solution to a mathematical problem. teaching mathematics involves conceptual explanations, which should be done with as many links as possible and in such a way that concepts can be generalised by learners and applied.the difference between procedural and conceptual understanding of the correct answer used in this study is similar to that of the categorisation of conceptual understanding and procedural fluency in the context of the strands of mathematical proficiency proposed by kilpatrick, swafford and findell (2001). conceptual understanding refers to ‘comprehension of mathematical concepts, operations, and relations’ and procedural fluency refers to ‘skill in carrying out procedure flexibly, accurately, and appropriately’ (kilpatrick et al. 2001:116). it is also in line with the difference between levels of cognitive demand, exemplified by stein, smith, henningsen and silver (2000) as ‘procedures with connections’ and ‘procedures without connections’ in the context of analysing mathematical tasks (p. 13). for example, in the text below, the explanation of the correct answer notes both the procedural and the conceptual aspects of understanding required in order to answer this question. (see table 1 for further possible levels of explanation.) table 1: domains of teacher knowledge and related error analysis criteria. question: which row contains only square numbers? (correct answer, c) explanation: 1² = 1; 2² = 4; 3² = 9; 4² = 16; 5² = 25; 6² = 36; 7² = 49; 8² = 64. therefore the row with 4, 16, 36 and 64 only has square numbers. to get this right, the learner needs to know what ‘square numbers’ mean and to be able to calculate or recognise which of the rows consists only of square numbers. (grade 8 teacher group.) the relationship between procedural and conceptual mathematics knowledge is complex and recent research insists that the two need to be seen as integrated rather than polarised when thinking about mathematical ideas (baroody, feil & johnson, 2007; long, 2005; star, 2005). notwithstanding, some mathematical problems lend themselves more to procedural explanations whilst in others the procedural and the conceptual are closely linked. there is a progression in mathematical understanding of concepts: what may be conceptual for a grade 3 learner (for example, basic addition of single digit numbers) is procedural for a grade 9 learner who will have progressed to operations at a higher level. the two criteria are thus closely aligned and yet they can be differentiated. under the second domain, specialised content knowledge, we map aspects related to mathematical knowledge required for the recognition of the nature of the error. in ball et al.’s (2008) words, the key aspect here is teachers looking for patterns in student errors, ‘sizing up whether a nonstandard approach would work in general’ (p. 400). whereas teachers’ knowledge of what counts as the explanation of the correct answer enables them to spot the error, looking for patterns in learners’ errors enables them to interpret learners’ solutions and evaluate their plausibility. knowledge of this domain enables teachers to ‘size up the source of a mathematical error’ (p. 397) and identify what mathematical steps would produce a particular error. we added the following criterion under this domain. criterion 3: awareness of error this criterion focuses on teachers’ explanations of the actual mathematical error and not on learners’ reasoning. the emphasis in the criterion is on the mathematical quality of teachers’ explanations of the actual mathematical error.under the third domain, knowledge of content and students, we map aspects related to teachers’ mathematical perspective of errors, typical of learners of different ages and social contexts in specific mathematical topics. this knowledge includes common misconceptions of specific topics (olivier, 1996) or learners’ levels of development in representing a mathematical construct (e.g. van hiele levels of geometric thinking, burger & shaughnessy, 1986). from the point of view of error analysis, this knowledge domain involves teachers explaining specific mathematical content primarily from the perspective of how learners typically learn the topic or ‘the mistakes or misconceptions that commonly arise during the process of learning the topic’ (hill et al. 2008:375). the knowledge of this domain enables teachers to explain and provide a rationale for the way the learners were reasoning when they produced the error. since it is focused on learners’ reasoning, this aspect of teacher knowledge of errors includes the ability to provide multiple explanations of the error. because contexts of learning (such as age and social background) affect understanding and because in some topics the learning develops through initial misconceptions, teachers will need to develop a repertoire of explanations, with a view to addressing differences in the classroom. we included three further criteria under this domain: criterion 4: diagnostic reasoning of learners’ thinking in relation to error the idea of teachers’ explanation of error goes beyond identifying the actual mathematical error (‘awareness of error’). the idea is to understand how teachers go beyond the mathematical error and explain the way learners were reasoning when they produced the error. the emphasis in this criterion is on the quality of the teachers’ attempt to provide a rationale for how learners were reasoning mathematically when they chose a distractor. this aspect aligns with one of the knowledge of content and students categories studied by hill et al. (2008), which they call common student errors; this refers to ‘providing explanations for errors, having a sense for what errors arise with what content, etc.’ (p. 380). criterion 5: use of everyday links in explanations of error teachers sometimes explain why learners make mathematical errors by appealing to everyday experiences that learners draw on and confuse with the mathematical context of the question. drawing on the work of walkerdine (1982), taylor (2001) cautions that:familiar contexts provide essential starting points for teaching young children to reason formally. … [but] not just any everyday example provides a suitable jumping off point for higher levels of conceptual development. (p. 3) the emphasis in this criterion is on the quality of the use of everyday knowledge in the explanation of the error, judged by the links made to the mathematical understanding that the teachers attempt to advance. for example, in the error explanation below, which is about learners’ confusion between units of measurement of capacity (between litres and millilitres) the use of ‘everyday’ enables mathematical understanding: ‘he draws on his frame of reference of how he perceives a litre to be e.g. a 1.25l of cold drink or a 1l of milk or a 2l of coke, etc.’ criterion 6: multiple explanations of error one of the challenges in the teaching of mathematics is that learners might need to hear more than one explanation of the error. this is because some explanations are more accurate or more accessible than others and errors may need to be explained in different ways for different learners. this criterion examines the teachers’ ability to offer alternative explanations of the error when they are engaging with learners’ errors, which is aligned with shulman’s (1986) aspect of pck related to ‘the ways of representing and formulating the subject that make it comprehensible to others’ (p. 9) in the context of error explanations.the set of six criteria hence span the first three of ball’s knowledge domains, providing evidence of the rich nature of error analysis activities. in the next section we explain the teacher development project from which the data for this analysis is taken and show how we operationalised the criteria. teacher development research project top ↑ the data informed practice improvement project (dipip), a teacher professional development project, was one of the first attempts in south africa to include teachers in a systematic process of interpretation of learners’ errors on a standardised mathematics test (shalem, sapire, welch, bialobrzeska & hellman, 2011). the 3-year (2007–2010) research and development programme2 included 62 mathematics teachers from grade 3–9 from a variety of johannesburg schools. schools were initially selected on the basis of their participation and results in the international competitions and assessments for schools (icas3) 2006 round of testing; later, proximity to the university campus also became a priority for practical reasons. teachers were organised into groups of three by grade level, forming eight groups of grade 3−6 teachers and six groups of grade 7–9 teachers. the groups consisted of a group leader (a mathematics specialist: staff member or postgraduate student who could contribute knowledge from outside the teaching workplace), a departmental subject advisor and two or three teachers. in this way groups were structured to include different authorities and different kinds of knowledge bases. over a period of three years, during term time, the groups met once a week, sharing ideas and learning from each other and exposing their practice to each other. six different activities were designed to provide a set of learning opportunities for the groups to reason about assessment data in the context of a professional learning community. in the project teachers mapped the icas 2006 and 2007 mathematics test items onto the curriculum, analysed learners’ errors, designed lessons, taught and reflected on their instructional practices and constructed test items. item-based statistics provided to the teachers for the analysis corresponded to 55 000 learners from gauteng who wrote the icas tests in the province. in this analysis we report on the groups’ analysis of the learners’ errors related to 332 test items.for each test item analysed by the groups, the group was requested to fill in an error analysis task template. the template was designed to guide the error analysis of learners’ choices of correct and incorrect answers. the template consisted of four parts. the first part of the template required the group to map each test item to the national curriculum expectations and grade level. the second part required the groups to anticipate learners’ achievement and comment on any particular distractor before checking the actual item achievement. the last two parts required the groups to analyse the correct answer and learners’ errors. the groups wrote up their explanations of how they thought the learners had reasoned when they selected the correct answer and each of the distractors. they were requested to write several explanations. operationalising the criteria the sample of explanations for the analysis reported on in this article related to 140 items (20 items per grade) across a range of items covering all of the content areas in the mathematics curriculum. a total of 572 texts were collected and coded (for the purpose of coding, each one of the groups’ explanations was called a ‘text’). there were 320 texts relating to the correct answer (‘answer texts’) and 252 texts relating to the most common distractor selected by learners (‘error texts’) which were analysed. the texts collected were a product of small group discussions and not of particular teachers; hence, inferences made about teachers’ explanations of learners’ errors should take this into account.the first two criteria (henceforth ‘procedural’ and ‘conceptual’) were used to analyse the answer texts. the remaining four criteria (henceforth ‘awareness’, ‘diagnostic’, ‘everyday’ and ‘multiple explanations’) were used to analyse the error texts. to capture variability in the quality of the teachers’ explanations of the correct answers and of the errors, each of the six criteria was divided into four categories: full, partial, inaccurate and not present. category descriptors were developed for the criteria (see the appendix 1 for the error analysis coding template). exemplars were developed to operationalise the criteria (shalem & sapire, 2012). the answer and error texts were entered into excel spreadsheets to facilitate the coding process. two coders were given the excel spreadsheet to record their coding decisions for each text. the coders were requested to enter one of the above four categories next to the text, according to each of the criteria’s category descriptors (see box 2 and box 3). consensus discussions between the coders were held on certain items in order to hone agreement between them. the final set of codes used in the analysis was agreed on in discussion with and through arbitration by a third expert (a member of the evaluation analysis team). the alignment between coders was 57% before the review, 71% after the review and full agreement after the arbitration. box 2: exemplars of coded error texts in relation to a number patterns item. the data was analysed quantitatively, finding observable trends and relationships evident in the sample. data was summarised for descriptive analysis. the correlation between the two criteria for the answer texts (‘procedural’ and ‘conceptual’) was calculated. similarly the correlation between two of the criteria for the error texts (‘awareness’ and ‘diagnostic’) was calculated. both were calculated using pearson’s r coefficient. exemplars of answer and error texts, the codes and the coding justification are given below. these are presented in two tables. box 1 relates to answer texts (‘procedural’ and ‘conceptual’ criteria) and box 2 relates to error texts (‘awareness’ and ‘diagnostic’ criteria). box 1: exemplars of coded answer texts in relation to a number patterns item. findings: usability of the measurement criteria top ↑ in terms of the first research question, the nature of the teachers’ explanations, we found that groups drew primarily on mathematical knowledge and less so on other possible explanations to explain the correct answer and the errors. in about 70% – 80% of all the explanation texts (that is, the answer and error texts), groups drew primarily on mathematical knowledge and much less so on other discourses. figure 1 shows that only about 15% of the answer texts and closer to 25% of the error texts did not have mathematical content (see not present). this result means that the groups did not often resort to common sense talk on learners’ errors, such as test-related explanations (e.g. the learners did not read the question well, or the learners guessed) or learner-related explanations (e.g. the question is not within the learners’ field of experience) or curriculum-related explanations (e.g. the learners have not learned this work). this finding is consistent with the following finding: despite the recommendation in the national curriculum at the time, to make links to everyday experiences when explaining mathematical concepts, 95% of the error texts included no links to learners’ everyday experiences (criterion 5) (see figure 1). only a small number of texts in the sample of error texts demonstrate teachers’ explanations that are focused on the link between an everyday phenomenon and the mathematical content of the item. these two findings are consistent with hill et al. (2008), who found in cognitive interviews with teachers responding to multiple-choice items measuring knowledge of content and students that teachers’ reasons for their choices were more often related to knowledge of learners’ errors or mathematical reasoning than test-taking skills. figure 1: percentage of texts by error analysis aspect. in terms of the second research question, variability in the quality of the teachers’ explanations, we found that the distribution of levels of quality of explanation within the domain of subject matter knowledge (‘procedural’, ‘conceptual’ and ‘awareness’) was similar across the three criteria (see figure 1). in comparison, the distribution of levels of quality of explanation within the domain of pedagogical content knowledge (‘diagnostic’, ‘everyday’ and ‘multiple’) varied across the three criteria. this suggests that in teachers’ explanations of learners’ errors matters of content knowledge are consistent in quality (most of them are partial) whilst matters of pedagogical content knowledge are inconsistent (no pattern of quality can be seen). most of the explanation texts (that is, the answer and error texts) were partial, missing crucial steps in the analysis of what mathematics is needed to answer the question. overall, across the 14 groups 50% of the answer texts on criterion 1 (‘procedural’), 42% of the answer texts on criterion 2 (‘conceptual’), 43% of the error texts on criterion 3 (‘awareness’) and 33% of the error texts on criterion 4 (‘diagnostic’) were partial. more full answer texts were found in criterion 1 (34%) than in criterion 2 (28%) (see figure 1). the evidence of teachers using predominantly partial explanations on criterion 1 is particularly interesting. partial procedural explanations of the mathematics involved in solving a particular mathematics problem may impede teachers’ capacity to identify mathematical errors, let alone to diagnose learners’ reasoning behind the errors. the extent to which the groups were able to describe learners’ reasoning behind mathematical errors (criterion 4) is an indication that the teachers struggled to think diagnostically about learners’ errors. figure 1 shows that 27% of the error texts did not attempt to explain learners’ reasoning behind the error (not present) and another 28% described learners’ reasoning without honing in on the error (inaccurate). altogether, more than 50% of the error texts demonstrated weak diagnostic reasoning. about 33% of the texts honed in on the error but the description of learner reasoning was partial. this means that in close to 90% of the error texts, groups offered no explanation, and where they did offer a mathematical explanation of learners’ reasoning it was inaccurate or partial. only 12% of the error texts were systematic and honed in on the learners’ reasoning about the error. according to ball et al. (2008), explanation of learners’ reasoning implies ‘nimbleness in thinking’ and ‘flexible thinking about meaning’. this is the criterion in which groups’ performance was the weakest and proportionally so more in the grade 3–6 group (see figure 2). the weakness in explaining learners’ reasoning is consistent with the groups’ inability to produce more than one explanation of the error (criterion 6). the four category descriptors for criterion 6 (multiple explanations) indicate a numeric count of the number of mathematically correct explanations (see the table 1−a1). the code inaccurate for criterion 6 means that one mathematically correct explanation was given. overall, 73% of the error texts provided only one mathematically feasible or convincing explanation. figure 2: procedural explanations of learners’ correct answers by grade groups. a comparison between the explanations of correct answers recorded by the two grade groups gives insight into some differences between these groups. the grade 3–6 groups were a little weaker than the grade 7–9 groups (see figure 2 and figure 3). this can be noted in the relative strength of the grade 7–9 group evidenced in higher percentages of full explanations in both criterion 1 and criterion 2 compared to the grade 3–6 group. figure 3: conceptual explanations of learners’ correct answers by grade groups. in terms of the third research question, the use of criteria enabled us to find two interesting correlations. the first correlation we found is between criterion 1 and criterion 2 (‘procedural’ and ‘conceptual’). the correlation was high (r = 0.73). this suggests that when teachers are able to provide a full explanation of the steps to be taken to arrive at a solution, their explanations also cover the conceptual links that underpin the solution and vice versa. the weaker the procedural explanations are, the weaker the conceptual explanations, and vice versa. the correlation confirms that there is interdependence between the procedural and conceptual aspects in teachers’ explanation of the mathematics that underlie a mathematical problem. the second correlation we found is between criterion 3 and criterion 4 (‘awareness’ and ‘diagnostic’). the correlation was also high (r = 0.67). this suggests that when groups demonstrate high awareness of the mathematical error (smk) they are more likely to give a better diagnosis of the learner thinking behind that error (pck). when teachers can describe the error mathematically well (smk) they are more likely to be able to delve into the cognitive process taken by the learners and describe the reasoning that led to the production of the error (pck). furthermore, in view of the finding that the teachers’ answer texts were mostly partial, we suggest that the finding that the teachers struggled to describe the mathematical way in which the learners produced the error is expected. discussion: the teachers’ knowledge of error analysis top ↑ much research in south africa suggests that teachers use more procedural and not enough conceptual explanations of mathematics and that this may explain learners’ poor performance (baroody et al., 2007; carnoy, chisholm & chilisa, 2012; long, 2005; star, 2005). this research was able to quantitatively examine the procedural and conceptual relationship and found a strong correlation. more research is needed to differentiate between strong and weak explanations of answer texts, in general and in different mathematical content areas. this is important for building the database for teachers on crucial steps in explanations, particularly in mathematical topics that underpin conceptual understanding at higher levels, such as place value. efforts in teacher education are needed to improve the quality of teachers’ procedural explanations, making sure that teachers are aware of which steps are crucial for addressing a mathematical problem and what counts as a full procedural explanation.the dominance of the partial category in all the texts, the groups’ difficulty with explaining the rationale for the ways in which the learners were reasoning and their inability, even in a group situation, to provide alternative explanations despite being requested to do so are noteworthy. these findings suggest that teachers struggle to explain the mathematical content covered by an item and particularly so when they are asked to explain it from the perspective of how learners typically learn that content. teachers seem to draw on different kinds of knowledge when explaining correct answers or errors and when providing reasons (single or multiple) behind learners’ errors. providing full procedural and conceptual explanations to correct answers and explanations of the actual mathematical error depends on teachers’ knowledge of mathematics, whilst diagnostic reasoning depends not only on mathematical knowledge but also on the degree of teachers’ familiarity with learners’ common ways of thinking when choosing an incorrect answer. that is, teachers understand when learners’ answers or reasoning are incorrect due mainly to their own understanding of the mathematics but this does not necessarily translate into an understanding of learners’ ways of thinking. evidence of the presence of different patterns in the distribution of levels for criteria grouped in the two different knowledge domains (smk and pck) highlights the difference between these domains of knowledge. error analysis aspects within the mathematical knowledge domain (criteria 1, 2 and 3) show similar patterns of distribution. this implies that these three aspects of error analysis can be interpreted and studied as a single construct. in contrast, the variation of distributions within the pck domain (criteria 4, 5 and 6) is an indication of the multidimensionality of this construct even in the specific context of error analysis. formally, the varied distribution means that providing a rationale for how learners reason mathematically is not related to the ability to provide multiple explanations of the error, nor to explaining the error by linking the rationale with everyday experiences. more development is needed for error analysis aspects in this domain of knowledge if they are going to be used as a single measurement scale. to those interested in using error analysis tasks and the proposed analytical tool to educate or develop teachers, this is an important and useful identification. conclusion top ↑ the argument of this article shows that assessment data can be used as an artefact to stimulate discussion and can provide an opportunity for teachers instead of being used against them as a naming and blaming tool. the study also shows that the six criteria and their category descriptors can be used to evaluate the variation in quality of teachers’ explanations of learners’ errors in mathematics assessments and can detect relationships between some of the aspects that inform teachers’ explanations of learners’ errors. acknowledgements top ↑ we acknowledge the funding received from the gauteng department of education (south africa), and in particular would like to thank reena rampersad and prem govender for their support of the project. we would also like to acknowledge professor karin brodie for her role in the conceptualisation of the dipip project. the views expressed in this article are those of the authors. competing interests the authors declare that they have no financial or personal relationship(s) that may have inappropriately influenced them in writing this article. authors’ contributions y.s. 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(1982). from context to text: a psychosemiotic approach to abstract thought. in m. beveridge (ed.), children thinking through language (pp. 129–155). london: edward arnold. footnotes top ↑ 1.ball et al.’s (2008) work includes six domains. the fourth domain, knowledge of content and teaching, is relevant in lesson design and teaching. having completed a round of error analysis, the teachers were tasked to plan a set of lessons around the errors identified. in their groups the teachers planned practical examples of how to engage learners’ errors in classroom situations. they then reflected on the way they engaged with learners’ error during teaching. a different set of criteria was operationalised for this domain. the curriculum mapping activity exposed the teachers in their groups to certain dimensions of the fifth and sixth domains, knowledge of curriculum and knowledge of the mathematical horizon. its analysis is reported in shalem, sapire and huntley (2013). 2.there is currently a third phase of dipip that is located in certain schools following a similar process with teacher groups in these schools. 3.icas is conducted by educational assessment australia, university of new south wales. learners from over 20 countries in asia, africa, europe, the pacific and the united states of america participate in icas each year. appendix 1 top ↑ table 1-a1: error analysis coding template. microsoft word 41-52 maharaj apos.docx pythagoras, 71, 41-52 (july 2010) 41 an apos analysis of students’ understanding  of the concept of a limit of a function    aneshkumar maharaj  school of mathematical sciences  university of kwazulu‐natal  maharaja32@ukzn.ac.za      this article reports on a study which used  the apos (action‐process‐object‐schema)  theory  framework to investigate university students’ understanding of limits of functions. the relevant  limit concepts were taught to undergraduate science students at a university in kwazulu‐natal  in  south  africa.  this  paper  reports  on  the  analysis  of  students’  responses  to  four  types  of  questions on limits of functions. the findings of this study confirmed that the limit concept is  one that students find difficult to understand, and suggests that this is possibly the result of  many  students  not  having  appropriate  mental  structures  at  the  process,  object  and  schema  levels.  in south africa grade 12 learners are exposed to an intuitive understanding of the limit of functions. this occurs in the context of the evaluation of limits of functions when finding derivatives from first principles of basic functions of the types , , , and (department of education, 2003). for this given context calculations for the limit of a function as x approaches a, create the impression that this is the same as the value of the function at a. during the past eight years, this is what my interactions with first year university mathematics students indicated. when confronted with a problem to evaluate a limit of a function, many students simply proceed to find the corresponding function value; even when the limit or this function value does not exist. this indicated that there was a need to engage with a study on students’ understanding of the concept of a limit of a function. the research questions for this study were: o how should the teaching of the concept of a limit of a function be approached? o what insights would an apos analysis of students’ understanding of the concept of a limit of a function reveal? theoretical framework this study is based on apos theory (dubinsky & mcdonald, 2001). apos theory proposes that an individual has to have appropriate mental structures to make sense of a given mathematical concept. the mental structures refer to the likely actions, processes, objects and schema required to learn the concept. research based on this theory requires that for a given concept the likely mental structures need to be detected, and then suitable learning activities should be designed to support the construction of these mental structures. asiala, et al. (1996) proposed a specific framework for research and curriculum development in undergraduate mathematics education. the framework consists of the following three components: theoretical analysis, teaching, and observations and assessment of student learning. according to asiala et al. (1996), apos theory functions according to the paradigm illustrated in figure 1. students’ understanding of the concept of a limit of a function 42 in this paradigm, theoretical analysis occurs relative to the researcher’s knowledge of the concept in question and knowledge of apos theory. this theoretical analysis helps to predict the mental structures that are required to learn the concept. for a given mathematical concept, the theoretical analysis informs the design and implementation of teaching. these are used for collection and analysis of data. the theoretical analysis guides the latter, which figure 1 indicates could lead to a modification of the initial theoretical analysis of the given mathematical concept. figure 1: paradigm: general research programme description of the apos/ace teaching apos theory and its application to teaching practice are based on two general hypotheses developed to understand the ideas of jean piaget. in some studies (see, for example, weller et al., 2003), these ideas were recast and applied to various topics in post-secondary mathematics. piaget investigated the thinking of adolescents and adults, including research mathematicians. those investigations led him to uncover common characteristics, specifically certain mental structures and mechanisms that guide concept acquisition (piaget, 1970). apos theory and its application to teaching practice are based on the following assumptions (dubinsky, 2010): o assumption on mathematical knowledge: an individual’s mathematical knowledge is his/her tendency to respond to perceived mathematical problem situations and their solutions by reflecting on them in a social context, and constructing or reconstructing mental structures to use in dealing with the situations. o hypothesis on learning: an individual does not learn mathematical concepts directly. he/she applies mental structures to make sense of a concept (piaget, 1964). learning is facilitated if the individual possesses mental structures appropriate for a given mathematical concept. if appropriate mental structures are not present, then learning the concept is almost impossible. the above imply that the goal for teaching should consist of strategies for helping students build appropriate mental structures, and guiding them to apply these structures to construct their understanding of mathematical concepts. in apos theory, the mental structures are actions, processes, objects, and schemas. in the following each of these are briefly described. then the ace teaching cycle; which constitutes the pedagogical strategies used to follow the hypothesis and the implication for teaching; is described. after these general considerations, the assumption on mathematical knowledge is focused on by making an apos analysis of the condition for the existence of a limit of a function. the result of this analysis is called a genetic decomposition. a genetic decomposition of a concept is a structured set of mental constructs which might describe how the concept can develop in the mind of an individual (asiala, et. al., 1996). so, a genetic decomposition postulates the particular actions, processes, and objects that play a role in the construction of a mental schema for dealing with a given mathematical situation. the genetic decomposition arrived at for the limit of a function concept, is indicated in the methodology section. apos theory the main mental mechanisms for building the mental structures of action, process, object, and schema are called interiorisation and encapsulation (dubinsky, 2010; weller et al., 2003). the mental structures of action, process, object, and schema constitute the acronym apos. apos theory postulates that a mathematical concept develops as one tries to transform existing physical or mental objects. the theoretical analysis design and implementation of teaching collection and analysis of data aneshkumar maharaj 43 descriptions of action, process, object and schema; given below; are based on those given by weller, arnon and dubinsky (2009). o action: a transformation is first conceived as an action, when it is a reaction to stimuli which an individual perceives as external. it requires specific teaching, and the need to perform each step of the transformation explicitly. for example, a student who requires an explicit expression to think about a limit of a function, lim , and can do little more than substitute values of close to for the variable in the expression and manipulate it, is considered to have an action understanding of a limit of a function. o process: as an individual repeats and reflects on an action, it may be interiorised into a mental process. a process is a mental structure that performs the same operation as the action, but wholly in the mind of the individual. specifically, the individual can imagine performing the transformation without having to execute each step explicitly. for example, an individual with a process understanding of the limit of a function; lim ; will construct a mental process for values of close to and think in terms of inputs, possibly unspecified, and transformations of those inputs to produce outputs. o object: if one becomes aware of a process as a totality, realises that transformations can act on that totality and can actually construct such transformations (explicitly or in one’s imagination), then we say the individual has encapsulated the process into a cognitive object. for example, for the limit of a function concept an individual may confront situations requiring him/her to apply various actions and/or processes. these could include thinking about an operation that takes two functions and produces a new function, such as in lim | | or lim . in order to operate on the one sided-limit of this new function, the process understanding must be encapsulated and converted to an object. o schema: a mathematical topic often involves many actions, processes, and objects that need to be organised and linked into a coherent framework, called a schema. it is coherent in that it provides an individual with a way of deciding, when presented with a particular mathematical situation, whether the schema applies. for example, the coherence might lie in the understanding that to determine the existence of a limit of a function, lim , the following must be considered: input values to the left and right of , the corresponding output values, and a means of transforming elements of the inputs to elements of the outputs. explanations offered by an apos analysis are limited to descriptions of the thinking which an individual might be capable. it is not asserted that such analyses describe what “really” happens in an individual’s mind, since this is probably unknowable. also, the fact that an individual possesses a certain mental structure does not mean that he or she will necessarily apply it in a given situation. this depends on other factors, for example managerial strategies, flexibility, prompts and emotional states. the main use of an apos analysis is to point to possible pedagogical strategies. data is collected to validate the analysis or to indicate that it must be reconsidered. for more details see asiala et al. (1996), and dubinsky and mcdonald (2001). the ace teaching cycle this pedagogical approach, based on apos theory and the hypothesis on learning and teaching, is a repeated cycle consisting of three components: (a) activities, (c) classroom discussion, and (e) exercises done outside of class (asiala, et. al., 1996). the activities, which form the first step of the cycle, are designed to foster the students’ development of the mental structures called for by an apos analysis. in the classroom the teacher guides the students to reflect on the activities and its relation to the mathematical concepts being studied. students do this by performing mathematical tasks. they discuss their results and listen to explanations, by fellow students or the teacher, of the mathematical meanings of what they are working on. the homework exercises are fairly standard problems. they reinforce the knowledge obtained in the activities and classroom discussions. students apply this knowledge to solve standard problems related to the topic being studied. the implementation of this approach and its effectiveness in helping students make mental constructions and learn mathematics has been reported in several research studies. a summary of early work can be found in weller et al. (2003). students’ understanding of the concept of a limit of a function 44 literature review there are many studies on students’ understanding of the concept of a limit of a function (e.g. cornu, 1992; davis & vinner, 1986; li & tall, 1993; maharajh, brijlall, & govender, 2008; monaghan, sun, & tall, 1994; tall, 1992; tall & vinner, 1981; williams, 1991). these studies indicate that students have difficulties with the concept of a limit of a function in the context of functions and continuity or series and sequences, and many of the difficulties encountered by students in dealing with other concepts; for example continuity, differentiability and integration; are related to their difficulties with limits. some researchers (cornu, 1992; sierpińska, 1987) reported that a high percentage of students have a static view of mathematics. such students can only deal with a very specific calculation that is placed before them. students with such a view will have difficulties with the limit of a function concept. the term procept is used to indicate that mathematical symbolism can ambiguously represent either a process or a concept (monaghan et al., 1994). so, the symbol lim is an example of a procept since it represents the process of getting to a specific value, or the value of the limit of the function itself. however, unlike the procepts of elementary mathematics, where an algorithm can be used to calculate the specific value of the concept, the limit value does not have a universal algorithm that works in all cases. further, the limit of a function concept is not restricted to a finite computation that gives a definitive answer. this is precisely where the distinction between an action and a process comes in. it could be argued that once a calculation involves an infinite number of steps, it could only be understood through a process conception. a commonly cited difficulty that students have in constructing a process conception of limit of a function, is their perception of a limit of a function as something that is actually never attained (cottrill, nichols, schwingendorf, thomas, & vidakovic, 1996; dubinsky, 2010). it seems that many students perform poorly because they: (a) are unable to adequately handle information given in symbolic form which represent objects [abstract entities], for example functions, and (b) lack adequate schema or frameworks, which help to organise and link different objects (maharaj, 2005). giraldo, carvalho and tall (2003) distinguish between a description of a concept, which specifies some properties of that concept and the formal concept definition. some descriptions commonly employed in the teaching of limits of functions include table of values, graphical and algebraic representations. individually these involve limitations that do not fully reflect the mathematical situation. the teaching implication is that a variety of representations should be used, and to encourage students to engage with a flexibility of mathematical conceptions of lim . participants and methodology participants and aim of module studied the participants for this study were 891 science students at a university in kwazulu-natal in 2010; about 66% of these were first year students. the students were studying a compulsory mathematics service module towards their bachelor of science degrees. major subjects for these students varied over chemistry, physics, biology, zoology and pharmacy. the aim of the module studied is to introduce students to the fundamental principles, methods, procedures and techniques of mathematics and statistics as the language of science. these students attended their lectures in one of three timetable groups. i was the lecturer to all the students. this was the context for the theoretical analysis of the limit of a function concept. theoretical analysis of the limit concept using an apos approach, genetic decomposition the theoretical analysis indicated the type of mental structures of action, process, object, and schema relevant to both the limit of a function concept and types of limit problems that the participants encountered. those mental structures were described in the section on apos theory above. the following genetic decomposition was detected. at an action level a student confronted with the limit of a function, lim , can do little more than substitute values of close to for the variable in the expression and manipulate it, and may or may not begin to see a pattern emerging. a process understanding of aneshkumar maharaj 45 the limit of a function; lim ; emerges as the student constructs a mental process for values of close to and thinks in terms of inputs, possibly unspecified, and transformations of those inputs to produce outputs. at the object level, the student sees this string as a totality and can perform mental or written actions on the one sided-limits of the given function, the process understanding is encapsulated and converted to an object, lim which may or may not exist. at the schema level the actions, processes, and objects are organised and linked into a coherent framework. this framework includes possible techniques for evaluating lim ; where could be ∞ or ∞. the ace teaching cycle this was informed by the theoretical analysis of the limit of a function concept and the types of limit problems that the students had to be exposed to. the key question for a 45 minute lecture session was: when does lim exist? activities were formulated and these were projected by use of a pc tablet. an example of an activity is given in appendix a. a reasonable time was given for students to reflect and work on each activity; they were free to discuss with other students sitting beside them and to use the prescribed textbook. while students engaged with the activities i observed how they worked, their difficulties and aspects that required further explanations. these informed my explanations; using a pc tablet; to the class. another 45 minute session was devoted to activities based on techniques for finding limits of functions, including limits at infinity. the activities and explanations incorporated use of graphical representations to answer questions on limits of functions, including limits of split-functions of the types given in symbolic notation; for types see questions 3 and 4 in appendix a, and question 1 in appendix b. activities and classroom discussions were followed by homework exercises, which students had to work on as part of their tutorial requirements. a pc tablet was used to summarise the lecture-room discussions. these summaries were available to students, on the website for the module. during the 3 hour tutorial sessions about a week later, students were in groups of about 35. in their groups they could further discuss the homework exercises with their tutors. tools for collection of data about 3 weeks after the tutorials a multiple choice questions (mcqs) test was administered to 868 students. the questions set were similar to those for the activities and homework. students were required to first work out the solutions in the space below each question and then to mark their choices on the multiple-choice-question cards. each question was given a weighting of 3 marks. students were informed that to discourage guessing, negative marking applied, i.e. -1 for each incorrect choice, which was a science faculty requirement for the use of mcqs at the university. the four questions on limits of functions are given in appendix b. note that the second question is based on the concept of continuity, which incorporates the concept of the existence of a limit of a function. the teaching for the section on continuity was similar to that outlined above, in that the ace teaching cycle was followed. the options given for each of the mcqs were constructed bearing in mind the apos levels of mental structures. analysis, findings and discussion to represent the analysis, findings and discussion for each of the four questions in a reader friendly format, the following subheadings which describe the type of question are used: o limits of split-functions represented in symbolic form o continuity application of split-functions represented in symbolic form o limits of functions not defined at o limits of rational functions at infinity under each of these subheadings the relevant test item and question analysis is given. the question analysis indicating number of student choices and the percentage (correct to one decimal place) for each of the four questions are indicated in tables 1 to 4 below. in each of these tables “*” denotes the letter of the correct answer, for example d is the correct answer for question 1. the omit index gives the number of students who did not mark any of the alternatives, and the bad index gives the number of students who marked more than one choice. students’ understanding of the concept of a limit of a function 46 limits of split-functions represented in symbolic form 1. let       3 if 6 3 if2 )( xx xx xf then lim a) is 5 b) is 3 c) is 6 d) does not exist e) is none of these table 1: question 1 analysis of student choices ( 868 a b c d* e omit index bad index 99 11,4% 81 9,3% 26 3% 444 51,2% 34 3,9% 181 20,9% 3 question 1 is based on finding the limit of a split-function f (x), as x approaches a value in the domain where the function is split. the question analysis in table 1, using the totals for choices a and b, indicates that 180 students conceptualised )(lim 3 xf x to be the same as one of the one-sided limits )(lim 3 xf x  or )(lim 3 xf x  . in the apos framework, using the genetic decomposition i arrived at, this means that those students’ mental constructions were at best at the action level. this suggests that for evaluation of limits of split-functions, approximately 21% of the students had mental constructions developed up to the action level. the numbers for choice c suggests that a total of 26 students (approximately 3%) had no idea of the basic technique for finding the limit of a split-function given in algebraic form; that is when the split-function is expressed in symbolic notation. a possible reason for 206 students selecting choices a, b or c is that they did not fully understand the concept of a splitfunction. this implies that it seems that approximately 23,7% of the students did not understand the concept of a split-function when such a function is represented in algebraic form. table 1 also indicates that 444 students marked the correct choice for question 1. in the context of my genetic decomposition this suggests that for evaluation of limits, of the type of split-functions under discussion, approximately 51,2% of the students had mental constructions developed up to the object level. table 1 indicates that 34 students chose option e. this suggests approximately 4% of the students could have made calculation errors; and if this is accepted; their mental constructions were probably functioning at the object level. so it seems that approximately 55% of the students had appropriate mental structures in place for some sort of effective schema to evaluate the limit of a split-function f (x) given in symbolic form, as x approaches a value in the domain where the function is split. table 1 also indicates that a large number of students (181, about 21%) did not indicate any choice. one of the reasons for this is that they did not have any idea of how to work out the solution of such questions and did not guess, for fear of losing marks. continuity application of split-functions represented in symbolic form 2. let          3 if1 3 if 3 152 )( 2 xkx x x xx xg the value of k that will make the function g continuous is a) -3 b) 4 c) -15 d) 2 e) none of these aneshkumar maharaj 47 table 2: question 2 analysis of student choices ( 868 a b* c d e omit index bad index 70 8,1% 229 26,4% 30 3,5% 82 9,4% 100 11,5% 352 40,6% 5 an analysis of question 2 reveals that the point of discontinuity occurs at 3. this is the crucial observation from the structure of the given split-function. the question is based on finding an unknown coefficient of one of the functions; in a split-function; which will make the entire function continuous on the interval ∞, ∞ . this requires continuity at 3, which implies that the condition lim 3 must be satisfied. the type of mental conception required here involves the formulation of lim 3 as an equation which should be treated as an object. further the successful use of this equation depends on an appropriately developed schema. this must incorporate conceptualisation of split-functions represented in symbolic form, as objects. note that in question 2, if 3 then . so options a, c and d deal with distracters which were arrived at from this structure. if this is accepted then the question analysis in table 2; using the totals for options a, c and d; indicates that 182 students (approximately 21%) possibly had mental constructions which were not even at the action level. table 2 also suggests that 229 students (approximately 26,4%) possibly had appropriately developed schema for applications on continuity of split-functions represented in symbolic form. since 100 students chose option e, this suggests approximately 11,5% of the students could have made calculation errors. if this is accepted then their mental constructions probably incorporated appropriate schema for applications on continuity of splitfunctions represented in symbolic form. so it seems that approximately 38% of the students had appropriate mental structures in place for some sort of effective schema for applications on continuity of split-functions represented in symbolic form. table 2 also indicates that a large number of students 352 (approximately 40,6%) did not indicate any choice. one of the reasons for this is that they did not have any idea of how to work out the solution of such questions and did not guess, for fear of losing marks. limits of functions not defined at 3. the following limit is equal to: lim √ 6 36 a) b) 0 c) ∞ d) e) ∞ table 3: question 3 analysis of student choices ( 868 a b c d* e omit index bad index 68 7,8% 269 31% 46 5,3% 254 29,3% 103 11,9% 124 14,3% 4 question 3 is based on the evaluation of limits of functions not defined at . the limit of the function cannot be found by finding the corresponding function value. the technique here is to express the function in factorised form √ √ √ , noting that 36 simplifies this to √ , and then finding lim √ . noting that , table 3 suggests that the 68 students (approximately 7,8%) had mental constructions which were not even at an action level. the distracters given as options b, c and e were designed for responses at a limited action level. if this is accepted then table 3 suggests that 418 students’ understanding of the concept of a limit of a function 48 students (approximately 48,2%) possibly used mental constructions at some sort of action level. the 254 students (approximately 29,3%) who marked the correct answer, conceptualised lim √ as an object. further, using this object it seems they had appropriate schema to deal with the relevant imbedded and implied information. table 3 suggests that 124 students (approximately 14,3%) did not have any idea of how to work out the solution of such questions and did not guess, for fear of losing marks. so, it seems that at least 22% of the students had mental constructions not developed to any of the levels indicated in apos theory. limits at infinity of rational functions 4. the following infinite limit is equal to: lim 3 3 8 6 10 a) b) 0 c) ∞ d) e) none of these table 4: question 4 analysis of student choices ( 868 a b c d* e omit index bad index 29 3,3% 41 4,7% 139 16% 400 46,1% 75 8,6% 182 21% 2 question 4 is based on finding the limit at infinity of rational functions; the case where the polynomials in the numerator and denominator are of the same degree. table 4 implies that the 180 students (approximately 20,7%); those who chose options b and c; had no appropriate schema to deal with finding the limits at infinity of rational functions. noting that is the reciprocal of , table 4 suggests that the 29 students (approximately 3,3%) had mental constructions not even at the action level. table 4 also implies that at least 400 students (approximately 46%) had appropriately developed schema to deal with problems based on finding the limits at infinity of rational functions; in particular the case where the polynomials in the numerator and denominator are of the same degree. the 75 students (approximately 8,6%) who chose option e, could have made calculation errors; and if this is accepted; their mental constructions probably incorporated appropriate schema for finding limits of rational functions, at infinity. so it seems that at most approximately 54,7% of the students had mental structures appropriately developed to some sort of effective schema for finding limits of rational functions, at infinity; in particular the case where the degree of the polynomial in the numerator is equal to that of the polynomial in the denominator. the above analyses indicated that the types of mcqs used basically gave information on the highest potential mental structure levels of those students, according to apos theory. if this is accepted then the findings and analyses from tables 1 to 4 are summarised in table 5. table 5 gives the highest potential levels of those students’ mental structures for the four types of questions, indicated as subheadings in italics above, into which the four mcqs were classified. for example, table 5 suggests that for question type 1 (limits of split-functions represented in symbolic form) the responses of students indicated that their mental structures were 3% at not even an action level, 21% towards a potential process level, and 54,2% towards a potential schema level. table 5 can be similarly interpreted for the other question types. the word potential is used since relevant mental structures at the action level should be in place for understanding to occur at the process level. according to apos theory, the challenge in teaching is to help such students to first attain the mental structures required for process understanding. aneshkumar maharaj 49 table 5: percentage of responses towards highest potential levels according to mental structures question type < action level action process object schema 1 3 21 54,2 2 21 37,9 3 7,8 48,2 29,3 4 3,3 2,7 54,7 for question 1 this would require the evaluation of both the one-sided-limits of the function, namely )(lim 3 xf x  and )(lim 3 xf x  , then this needs to be encapsulated to form the object ).(lim 3 xf x once students have the relevant structures for object-level understanding, then the challenge is to help such students develop some sort of effective schema, which incorporates an organisational framework for using the actions, processes and objects to evaluate limits of a function as approaches . so teaching here would require suitable teaching on the following three types of limits of functions; using different representations for )( xf ; )(lim 3 xf x  , )(lim 3 xf x  and )(lim 3 xf x . these representations should be verbal, graphical and algebraic. since this study used algebraic and graphical representations, it seems that more emphasis should be on verbal representations. this could lead to graphical representations. for example, the following three activities could accomplish these: activity 1 a function behaves in the following way near 3: as approaches 3 from the left, approaches 2. as approaches 3 from the right, approaches 1. for the above situation you are required to: 1. draw a sketch to illustrate the behaviour of near 3. 2. write the 2nd and 3rd sentences in symbolic form. 3. check that your symbolic form agrees with the sketch you drew. 4. determine with reasons if lim exists. activity 2 consider the split function 2 if 3 6 if 3 . for this function you are required to: 1. use the symbolic form to explain in your own words the behaviour of near 3. 2. use the symbolic form to draw the graph of . 3. evaluate lim and lim . 4. determine with reasons if lim exists. activity 3 1. explain in your own words, for a function , what is meant by “lim exists”. 2. explain the procedure to find the limit of lim , where is a split-function given in symbolic form. similarly for the other three question types, the interpretation of table 5 can be used to formulate activities to help students develop the relevant mental structures. students’ understanding of the concept of a limit of a function 50 conclusions useful insight into the relevant mental structures towards which teaching should focus was revealed by the apos genetic decomposition of the limit of a function concept. the findings of this study confirmed that the limit of a function concept is one that students find difficult to understand, and suggests that this is possibly the result of many students not having appropriate mental structures at the process, object and schema levels. it seems that my genetic decomposition was adequate. however, my reflections on the teaching design indicated that more time needs to be devoted to helping students develop the mental structures at the process, object and schema levels. this implies that teaching should focus on (1) verbal and graphical approaches to finding limits of functions; including split-functions in symbolic form, (2) unpacking of structures given in symbolic form, and (3) modelling possible schema. a graphical approach should facilitate the development of mental structures at the process and object levels, while a focus on symbolic structures should aid object conceptions. if schemas organise and link the relevant actions, processes and objects then this should be a part of the teaching. the impact of such a focus on teaching will require further research. references asiala, m., brown, a., de vries, d. j., dubinsky, e., mathews, d., & thomas, k. (1996). a framework for research and development in undergraduate mathematics education. research in collegiate mathematics education, 2, 1-32. cornu, b. (1992). limits. in d. tall (ed.), advanced mathematical thinking (pp. 153-166). dordrecht: kluwer academic publishers. cottrill, j., nichols, d., schwingendorf, k., thomas, k., & vidakovic, d. (1996). understanding the limit concept: beginning with a coordinated process schema. journal of mathematical behavior, 15(2), 167-192. retrieved from http://homepages.ohiodominican.edu/~cottrilj/concept-limit.pdf davis, r. b., & vinner, s. (1986). the notion of limit: some seemingly unavoidable misconception stages. journal of mathematical behavior, 5, 281-303. department of education. (2003). national curriculum statement grades 10-12 (schools): mathematics. pretoria: department of education. dubinsky, e. (2010, january). the apos theory of learning mathematics: pedagogical applications and results. paper presented at the eighteenth annual meeting of the southern african association for research in mathematics, science and technology education. durban, south africa. dubinsky, e., & mcdonald, m. a. (2001). apos: a constructivist theory of learning in undergraduate mathematics education research. in d. holton (ed.), the teaching and learning of mathematics at university level: an icmi study (pp. 275-282). dordrecht: kluwer academic publishers. retrieved from http://www.math.kent.edu/~edd/icmipaper.pdf giraldo, v., carvalho, l. m., & tall, d. o. (2003). descriptions and definitions in the teaching of elementary calculus. in n. a. pateman, b.j. dougherty, & j. zilliox (eds.), proceedings of the 27th conference of the international group for the psychology of mathematics education (vol. 2, pp. 445-452), honolulu hi: center for research and development group, university of hawaii. retrieved from http://www.warwick.ac.uk/staff/david.tall/pdfs/dot2003d-giraldo-carv-pme.pdf li, l., & tall, d. (1993). constructing different concept images of sequences and limits by programming. in i. hirabayashi, n. nohda, k. shigematsu, & f. lin (eds.), proceedings of the 17th conference of the international group for the psychology of mathematics education (vol 2, pp. 41-48). tsukuba, japan. retrieved from http://www.warwick.ac.uk/staff/david.tall/pdfs/dot1993e-lan-li-pme.pdf maharaj, a. (2005). investigating the senior certificate mathematics examination in south africa: implications for teaching. (unpublished doctoral dissertation). university of south africa, johannesburg. maharajh, n., brijlall,d., & govender, n. (2008). preservice mathematics students’ notions of the concept definition of continuity in calculus through collaborative instructional design worksheets. african journal of research in mathematics, science and technology education, 12, 93-108. monaghan, j., sun, s., & tall, d. (1994). construction of the limit concept with a computer algebra system. in j. p. ponte & j. f. matos (eds.), proceedings of the 18th conference of the international group for the psychology of mathematics education (vol. 3, pp. 279-286). lisbon: university of lisbon. retrieved from http://www.warwick.ac.uk/staff/david.tall/pdfs/dot1994c-monhn-sun-pme.pdf piaget, j. (1964). development and learning. journal of research in science teaching, 2, 176-180. aneshkumar maharaj 51 piaget, j. (1970). piaget’s theory (translated by g. cellerier and jonas langer; with the assistance of b. inhelder and h. sinclair). in p. h. mussen (ed.), carmichael's manual of child psychology, vol. 1 (3rd ed.) (pp. 703-732). new york: wiley & sons. sierpińska, a. (1987). humanities students and epistemological obstacles related to limits. educational studies in mathematics, 18, 371-397. tall, d. (1992). the transition to advanced mathematical thinking: functions, limits, infinity and proof. in d. a grouws (ed.), handbook of research on mathematics teaching and learning (pp. 495-511). new york: macmillan publishing. tall, d., & vinner, s. (1981). concept image and concept definition in mathematics with particular reference to limits and continuity. educational studies in mathematics, 12, 151-169. weller, k., clark, j., dubinsky, e., loch, s., mcdonald, m., & merkovsky, r. (2003). student performance and attitudes in courses based on apos theory and the ace teaching cycle. in a. selden, e. dubinsky, g. harel, & f. hitt (eds.), research in collegiate mathematics education v (pp. 97-131). providence, rhode island: american mathematical society. weller, k., arnon, i., & dubinsky, e. (2009). preservice teachers' understanding of the relation between a fraction or integer and its decimal expansion. canadian journal of science, mathematics and technology education, 9(1), 5-28. williams, s. r. (1991). models of limit held by college calculus students. journal for research in mathematics education, 22, 219-236. students’ understanding of the concept of a limit of a function 52 appendix a: an example of an activity appendix b: the four multiple choice questions 1. let       3 if 6 3 if2 )( xx xx xf . then lim a) is 5 b) is 3 c) is 6 d) does not exist e) is none of these 2. let          3 if1 3 if 3 152 )( 2 xkx x x xx xg the value of k that will make the function g continuous is a) -3 b) 4 c) -15 d) 2 e) none of these 3. the following limit is equal to: lim √ 6 36 a) b) 0 c) ∞ d) e) ∞ 4. the following infinite limit is equal to: lim 3 3 8 6 10 a) b) 0 c) ∞ d) e) none of these << /ascii85encodepages false /allowtransparency false /autopositionepsfiles true /autorotatepages /none /binding /left /calgrayprofile (dot gain 20%) /calrgbprofile (srgb iec61966-2.1) /calcmykprofile (u.s. web coated \050swop\051 v2) /srgbprofile (srgb iec61966-2.1) 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/hrv (za stvaranje adobe pdf dokumenata najpogodnijih za visokokvalitetni ispis prije tiskanja koristite ove postavke. stvoreni pdf dokumenti mogu se otvoriti acrobat i adobe reader 5.0 i kasnijim verzijama.) /hun 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can be opened with acrobat and adobe reader 5.0 and later.) >> /namespace [ (adobe) (common) (1.0) ] /othernamespaces [ << /asreaderspreads false /cropimagestoframes true /errorcontrol /warnandcontinue /flattenerignorespreadoverrides false /includeguidesgrids false /includenonprinting false /includeslug false /namespace [ (adobe) (indesign) (4.0) ] /omitplacedbitmaps false /omitplacedeps false /omitplacedpdf false /simulateoverprint /legacy >> << /addbleedmarks false /addcolorbars false /addcropmarks false /addpageinfo false /addregmarks false /convertcolors /converttocmyk /destinationprofilename () /destinationprofileselector /documentcmyk /downsample16bitimages true /flattenerpreset << /presetselector /mediumresolution >> /formelements false /generatestructure false /includebookmarks false /includehyperlinks false /includeinteractive false /includelayers false /includeprofiles false /multimediahandling /useobjectsettings /namespace [ (adobe) (creativesuite) (2.0) ] /pdfxoutputintentprofileselector /documentcmyk /preserveediting true /untaggedcmykhandling /leaveuntagged /untaggedrgbhandling /usedocumentprofile /usedocumentbleed false >> ] >> setdistillerparams << /hwresolution [2400 2400] /pagesize [612.000 792.000] >> setpagedevice collaborative learning in a multilingual class 52 pythagoras, 68, 52-61 (december 2008) collaborative learning in a multilingual class deonarain brijlall school of science, mathematics and technology education, university of kwazulu-natal brijlalld@ukzn.ac.za the solving of word problems dealing with fractions was investigated. two sets of learners worked in solving the same tasks on fractions. one set of learners worked collaboratively and the other group consisted of learners working independently. the selected participants consisted of two grade 8 classes at a high school in pietermaritzburg, south africa. the study was a qualitative one involving lesson observations, analysis of learner worksheets, questionnaires and interviews. the two tasks were presented in the form of word problems and the classroom comprised of multilingual learners. data yielded by these research instruments confirmed assumptions and literature claims. although it was a small scale qualitative research, interesting observations were made that could have pedagogical implications. in south africa learners have difficulty grasping key mathematical concepts. this was illustrated in the timms 2003 report, where it was suggested that ninety percent of south african learners do not possess a basic mathematical knowledge as opposed to the international average of less than thirty percent. only 0,3 percent of grade 8 learners in south africa could be classified as being able to solve non-routine problems (reddy, 2006). a major concern in south africa is that learners will not be competent enough in mathematics to advance to tertiary institutions and pursue careers in the science and technology fields. to ensure a better economy in south africa, a dramatic improvement must occur in the teaching of mathematics and more specifically, in the development of problem solving skills at school level. learners are not given enough problem solving opportunities at school to improve their problem solving abilities. the lack of problem solving skills in south africa may perhaps be a result of the way it has been taught in schools. in the past, problem solving tasks were solved individually by learners. problems presented to learners are usually abstract and foreign to them, therefore they generally acquire a dislike to problem solving tasks, believing they will be too difficult to solve. this is especially true for learners who are not achievers at mathematics (barns, 2005) or low attainers, as they are referred to in this paper. we decided therefore to gauge whether learners would achieve success in a group environment. in this regard we raised the following question: does collaborative learning in a multilingual classroom yield greater success in problem solving as compared to individual engagement? relevance of study to the south african context in the past ten years, not many papers reporting on learning in a multilingual classroom have been published locally (setati & barwell, 2008). we therefore decided to investigate the learning of mathematics by learners versed in different languages. in order to encourage mathematical problem solving, we allowed learners to discuss the understanding and solution of the word problems (presented in english) in the language the learner was comfortable with. it was found by dlamini (2008) that second language english learners obtained exceptional results in mathematics but performed poorly in english. we therefore in our investigation motivated us to let learners engage the solution of the problems in their indigenous languages. since the teacher is seen as a source of knowledge, learners constantly seek their advice and avoid tackling the problem amongst group members. for deonarain brijlall 53 true problem solving skills to develop, learners need to work independently as well as interdependently with class mates. problem solving and working together in groups are part of the critical outcomes in the revised national curriculum statement (rncs) for mathematics grades 1-9 (department of education, 2002). problem solving is continually mentioned in the learning outcomes. however, problem solving skills are considered poor in south africa (buffer & leigh, 2005; arora, 2003). if the approach to problem solving was modified and made less daunting to learners, a change in their attitude towards mathematics in general and more specifically towards problem solving may be achieved. multilingualism the diverse nature of multilingualism around the world is reflected in the wide range of multilingual classrooms in which mathematics is taught (barwell, 2005).we see a classroom as being multilingual if any of the participants (learners, teachers or others) is potentially able to draw on more than one language as they go through their work. this paper adopts this definition when it refers to a multilingual environment (when referring to the classroom in which learners with different mother tongue languages engaged in the problem solving activities). the learners draw from english, afrikaans, isizulu, sisuthu , sotho and swati. realistic mathematics education realistic mathematics education (rme) is a theory designed by the hungarian hans freudenthal (barns, 2005). he considered mathematics to be a human activity and thus believed that to learn mathematics one had to do mathematics. it focuses, as barns (2005) points out, the need to use learner’s everyday experiences in order for them to unfold mathematics by themselves and that mathematics is rather not a ‘ready-made system with general applicability’ (p. 50). with this in mind rme lines up with the rncs as the rncs calls for teaching to become more learner centred. the rncs also stresses the importance of interaction with others. rme allows for such interaction as real world problems are normally solved in a group environment, where all participants work together to solve the problem at hand, rather than solving it individually. this paper focuses on the rational number system. kilpatrick, swafford and findell (2001) note that in the usa many learners find the rational number system difficult. problem solving questions used in problem solving tasks must be familiar to the learner. the context of the question must be a reflection of the learners’ socioeconomic background in order to make it “realistic” to their personal experience. as cooper and dunne (2004) suggest, working-class learners do not experience the same background as middleclass learners, hence will not find the same questions “realistic”. by realistic it is meant that the mathematical task has been designed within a context the learner is familiar with. this is also true regarding learner’s gender, home language and cultural grouping. as a result learners’ mathematical abilities may well be underestimated when related to the context in which the question is asked. for this reason two problem tasks used in the study were related to the learners’ personal knowledge and background. learners were allowed to query the meanings of words used in the problem statement. they were also allowed to discuss these meanings in their home language when sitting with colleagues who had a similar home language. the rational number system is represented in a number of ways. the study focused on common fractions and decimal fractions. kilpatrick et al. (2001) believe that the informal knowledge learners possess in “sharing and measuring” is the foundation that rational numbers can be built upon. they also note that children struggle to link different forms of rational numbers as they battle to see how they are related. they suggest that for effective teaching to take place it would depend on how cognitively demanding the given tasks were. mathematics should be elaborated through tasks and sufficient time must be allocated to master each task. learners must also be given the opportunity to link their informal knowledge with the abstract knowledge of mathematics. kilpatrick et al. (2001) distinguish five strands of mathematical proficiency, namely: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning and productive disposition. different strategies have been suggested to attempt improving mathematics, one such strategy is collaborative learning methodology a qualitative research approach was employed in this research. this was achieved through the collection of data via observation of a lesson, analysis of learners’ worksheets, questionnaires collaborative learning in a multilingual class 54 and interviews of participants. the selected participants consisted of two grade 8 mathematics classes at a high school in pietermaritzburg, south africa. this school is a co-educational, with multilingual classrooms and mixed socio-economic backgrounds. the participants have a mixed ability in mathematics. learners were divided into two groups, the control group and the experimental group. the learners in the two classes were ranked in descending order of their average mathematics marks for the first two terms. every second learner was placed in the experimental group while the remaining learners were placed in the control group. this was to ensure learners with the same academic levels were present in both the control group and experimental group. the control group was given the activity to complete individually. this group completed the activity at the same time as the experimental group and was therefore supervised by a different teacher to that of the experimental group. the experimental group was further divided into seven groups of four and the control group comprised twenty three. learners were arranged in groups ensuring that the members in each group had an array of mathematical abilities (based on their previous performance). the researcher observed the experimental group. problem solving worksheets were used to gather information. the worksheets formed the foundation of the qualitative study and comprised of two tasks: one involved decimal fractions, the other common fractions. both tasks demanded problem solving skills and both experimental and control groups confronted these problems. in the experimental group learners were asked to initially complete the worksheet on their own and then they were placed in groups where a discussion was held amongst the group members. this was to promote participation by learners when working in a group. at the end of the discussion, each group was asked to complete a worksheet that expressed the joint views of the group. a fifty minute lesson was provided for learners to solve the problem. eight learners were selected to be interviewed. we found it difficult to interview all group members and interviewed four representatives form different groups and four from the control lot. the representatives from the groups were chosen by the members of the particular groups. the learners belonging to the control group worked individually. these learners had been linked to each spokeswoman from the experimental group. the linkage was done by choosing the control group interviewees with similar mathematical performance as their counterparts from the experimental groups. the interview was conducted in an informal manner to clarify answers given by learners in the worksheet. the interview focused on the learners’ understanding of the questions and their personal experience of either working individually or as part of a group. learners were also asked to comment on how the lesson may have been improved. diversity of learners fifty one learners completed the questionnaires. the diversity of learners is discussed under the headings of age, gender, home language, mother tongue language, medium of instruction and racial group. we have regarded home language as the most used language when communicating at home. this to us might be different from the mother tongue language which we adopt to refer to language associated to the individual via past generations. for example, the indian child in class was hindi speaking (as mother tongue), but english speaking (as home language). age table 1: age of learners age 13 14 15 16 unknown total number 10 34 5 1 1 51 table 1 shows that the age of the learners ranged from thirteen years to sixteen years old. two thirds of the learners were fourteen years old. thirteen and fourteen year old learners comprised 86 percent of the participants. gender sixty nine percent of learners were male and thirty one percent females. home language learners were asked to indicate their home language. home language is the preferred language in which the learners communicate at home. as indicated in table 2, isizulu is the dominant language with 58 percent of learners communicating in isizulu. one learner indicated afrikaans as home language. english was spoken table 2: home language of learners afrikaans english isizulu other more than 1 total 1 17 30 6 3 54 – 3 = 51 deonarain brijlall 55 by only 33 percent of learners. six learners indicated they spoke a language which was not listed in the questionnaire. these languages were xhosa (four), sisuthu (one), sotho (one) and swati (one). three learners indicated more than one home language. this demonstrates the diversity of home languages spoken by the learners with the most speaking isizulu. mother tongue language under the category mother tongue, learners were asked to indicate the language that they most identify with. the majority of learners identified with isizulu although the number decreased slightly compared to home language. english also decreased from 33 percent to 27 percent. one learner did not indicate an answer and one learner indicated two languages. seven learners selected the category “other”. the languages not indicated on the questionnaire and that were mentioned by learners were: xhosa (four), sisuthu (one), sothu (one) and swati (one). language of education all learners indicated that they are educated in english (meaning being taught in the medium of english). four learners indicated that they had been educated in two languages. the languages they indicated referred to their first additional language. what should be noted is that all learners were taught in english while only twenty-seven percent of learners indicated their mother tongue language as english and only thirty-three percent of learners indicated english as their home language. this means that the majority of learners taking part in the study were second language english speakers. it must be emphasised that despite presenting the problems in english, the participants in the group were allowed to communicate in their home language. racial group table 4: the racial grouping of learners black white coloured indian other total 34 8 7 1 1 51 table 4 depicts that a variety of racial groups were represented by the learners who participated in the research. one learner selected other and indicated he was muslim. two thirds of learners were black, while the other third comprised of white, coloured, indian and muslim learners. the questionnaire showed that all participants received their education in english. this is in contrast to most learners’ mother tongue and home language as the questionnaire reveals that most learners’ home and mother tongue language is isizulu. most participants were black males. the questionnaire does however show a diverse multicultural sample group. collaborative learning versus individual attempt for analysis purposes only those seven learners from the control group who were linked to the seven group representatives from the experimental group were used in the comparison. to obtain anonymity and distinguish between learners, the following codes were used. when referring to the group leaders, a “g” is used as opposed to individual learners where an “i” was used. a number was also shown to illustrate the group in which the learners were from or the corresponding individual learner. for example group leader 3 was coded as g3 and individual learner who was linked to group 2 was denoted as i2. g2.2 belonged to group 2 but was not the leader. the data collected from individual learners and the groups of learners was compared and analysed under the following sub-headings namely, results from problem task 1, and results from problem task 2. results from problem 1 the first problem task given to learners was: a large piece of cardboard paper is 0,01 mm thick. it is cut in half and one piece is placed on the other to make a pile. these are cut in half and all four pieces are placed in a pile. these four are cut in half and placed in a pile, and the process is continued. after the pieces have been cut and piled for the tenth time, what is the height of the pile in centimetre? the data collected from question one was separated into categories to make it easier to compare. the categories used were: (a) mathematically sound calculation with an accurate answer, (b) accurate answer with no calculation shown, (c) correct solution with incorrect conversion, (d) partially correct solution with calculation shown, (e) no understanding of the question, (f) incorrect answer with no calculation shown and (g) no answer shown. learners who obtained the correct answer and displayed a table 3: mother tongue language of learners afrikaans english isizulu other more than 1 no answer total 2 14 28 7 1 1 52 – 1 = 51 collaborative learning in a multilingual class 56 mathematically sound argument including a correct conversion fell into category “a”. category “b” was selected if a learner achieved the right answer but failed to show a method as it is unclear if the learner had a mathematically sound methodology. if a learner was able to achieve a mathematically sound answer before the conversion, and proceeded to either not convert or convert incorrectly, they were placed in group “c”. a learner who had an incorrect answer but had initially shown correct methodology was placed in category “d”. learners were placed in group “e” if their method of solving the problem had no link to the question and hence showed the learner did not have a clear understanding of the question. if only an incorrect answer was shown, they were placed in category “f”. if the question was left blank, it was placed in category “g”. table 5: categorising learners’ results for problem 1 a b c d e f g group 2 0 5 0 0 0 0 individual 0 0 0 6 0 1 0 table 5 represents the mode of the different categories for both the group leaders and the individual learners. neither the individual attempts nor the group attempts were left blank. no learner that completed the task individually was able to solve the first problem task. this is in contrast to the seven groups who were all able to solve the first problem before converting to centimetres while two of these groups converted it successfully. six out of the seven groups were able to give a mathematically sound explanation for the first question whereas this was not true for individual learners. most of the individual learners had partially correct answers. by this it is meant that learners had begun answering the question in a mathematically sound manner but had either not completed the task or alternatively had then proceeded with an incorrect procedure. most learners categorised as “d” had an answer such as learner i4 “we said that the thickness of the paper is 0,1 millimetres thick, we timed that by 10 papers which was equal to 0,1.”. here the misconception was that as the paper was folded ten times, the thickness would be equivalent to ten layers of paper. learners obtaining such an answer did not consider the number of layers increased in each time the paper was divided in half. most groups had a similar answer to that of learner g3: we changed 0,01mm into cm so it was 0,001. so each time you fold the paper you times by two because it’s two halves. so we times 0,001 by 2 and it becomes 0,002. so if you take the 0,002 and fold it again and you take the answer from the previous answer and times it by two, and then the answer of that you timed it by two until you get to the end. which the answer is 1,0243 cm. the learner converted to centimetres successfully. a correct understanding of the question was shown by the group as they realised that each time the paper was folded, double the previous answer was obtained. learners used different methods to solve the problem. g6 said a member of his group had actually done the problem in real life. diagrams and drawing were used by some learners as confirmed by g2 “yes using diagrams actually helps. i never thought so before”. group 2 and i2’s calculation of problem task 1 is shown below. as can be seen group 2’s calculation has been done in logical steps whereas i2’s calculation is not as clear. there is no connection made between his first and second line. figures 1 and 2 show the calculations of group 2 and individual learner two. figure 1: group 2’s calculation of problem 1 deonarain brijlall 57 group 2 multiplied the answer by two each time for ten times. however their conversion to centimetres was incorrect. their conversion illustrates a belief that centimetres are smaller than millimetres. i2’s calculation shows that to achieve the answer he multiplied by 10, corresponding to the second line of his calculation. however this does not explain his first line. he initially believed that as the paper was being divided by two so to must the answer. figure 3: group 1’s final answer in problem 1 group 1 went a step further in their solution and were able to grasp a connection to exponents. they achieved an answer of 0, 01 mm × 210. the group was able to conclude that instead of multiplying ten times by two, one could simply multiply by 210. what was also of significance was the fact that this group had three black members who communicated in isizulu (from observation notes of researcher). so, it seemed that the success of the collaborative work could be due to learners communicating in a common language to create meaning to ideas leading to the solution of the problem. results from problem 2 the following question was given as problem task two: four men were shipwrecked on an island. having no food, they went to work gathering pineapples. after gathering the pineapples, they were tired and all fell asleep. after some time, one of the men woke up and was very hungry so he ate 1 3 second man awoke and being hungry, ate of the pineapples – more than his proper share. he then went back to sleep. the 1 3 of the remaining pineapples and went back to sleep. the third man did the same. when the fourth man awoke, he took only his rightful share of the remaining pineapples. then there were 6 pineapples left. how many pineapples did the men gather? a similar system that was used in question one was adopted to categorise the data in question two. the categories were: (a) mathematically sound calculation with an accurate answer, (b) partly correct solution with calculation shown, (c) no understanding of the question, (d) incorrect answer with no solution, and (e) no answer. category “a” was selected if both the answer and the calculation were mathematically correct. for category “b”, learners did not achieve the correct answer but showed partial understanding in their solution. where no proper understanding of the question was shown, category “c” was selected. category “d” was selected only if an incorrect answer was shown. if no answer was present, it fell into category “e”. table 6 illustrates the learners’ results both individually and in groups for problem 2. table 6: categorising learners’ results for problem 2 a b c d e group 0 6 1 0 0 individual 0 2 2 1 2 neither individual learners nor groups were able to give the correct answer for question two. two individual learners did not attempt to write an answer and a further individual learner gave only an incorrect answer. all seven groups gave an explanation for their answers, whereas only four individual learners did the same. most groups had a similar calculation as group 4, figure 2: individual learner 2’s calculation of problem 1 collaborative learning in a multilingual class 58 g4 explains his groups answer: they told us that there were 6 pineapples left and so how many did the man gather? so i started with the 6 and i divided it by 3 because the first guy ate, i mean i divided it by a third because the first guy ate a third and the answer for that was 18. and the second guy ate a third of that. so i divided that by one third which gave me 54 and the other guy, the third guy which gave me 54. and then the other guy, the third guy, he also ate a third so i divide that by a third, which gave me 54. and then the fourth guy he got a quarter. so after the third guy i got 162 and then the last guy, i get 162 divide by a quarter. so the men gathered 648 pineapples. the group identified that three of the four men ate 1/3 of what was left and the final man ate ¼. they failed to comprehend that in order to get the answer, you had to work backwards from 6 and not start at the beginning. i6 realised this as he explains: well i didn’t finish my one because i didn’t have time. but what i did, i said a third times 6 because there was six remaining right. and then the last guy, i mean not the last guy but the third guy ate a third. that’s correct and then i timed that and i got 8 and then i did it again with the same answer and then i timed it by a third since the second guy also ate a third. and then i stopped there. as can be seen, i6 worked backwards from the third “guy” to the “second guy”. i6 did not include the pineapples the last man ate in his calculation and failed to mention that the he ate 1 4 of what was left. no learner acknowledged that the six pineapples was equal to three quarters and that one needed to find out that one quarter was equal to two pineapples. even though the group answers were incorrect, they were better reasoned over the individual learners’ answers. the group’s answers had detailed explanations. this shows that learners had at least attempted to answer the question in their groups. the groups also spent much longer on the two questions as opposed to the individual learners. individual learners were inclined to read though the question, attempt one answer and leave the question as they thought it was too difficult. two individual learners received help from each other even though they were instructed not to. as was observed the groups approach to the question was different. at least one member from each group had an initial answer for question two. however, as each learner attempted to explain their answer to the group they would realise that their answer was wrong and would try solve the problem once again. this was particularly evident in group 1 where learners kept arguing with each other. participation in groups initially the learners who worked in groups were extremely teacher dependant. they repeatedly asked the teacher for help. this occurred more frequently with second language english speaking learners. when learners were placed in groups, they became less dependent on the instructor and they communicated in the language they were comfortable with. in many instances, the researcher observed that members in a multilingual group switched between languages when communicating with different members within the group. however not all learners worked effectively in the group environment. some learners complained of others who, even though present, did not take part in the group discussion. as g2 recalls: i remember one person in my group and i don’t remember them giving any ideas. it’s just sometimes when you are working they think, okay i don’t understand and i’m just going to sit down and get the answers. figure 4: group 4’s results for problem 2 deonarain brijlall 59 this same sentiment was echoed by other learners who were unhappy working in groups. it was observed that mainly the low attaining learners did not participate in group discussions. when asked why, they responded that they either could not contribute to the discussion as they did not know what was going on, or they said that the other members in the group were not including them in the discussion and as a result they could not follow the debate. this would agree with learner g2 who suggested that if learners do not understand the work they do not participate in it. after the researcher explained to learners that to complete the task all members must participate, the low attaining learners still seemed to hold back and listen, rather than engaging in the discussion. however towards the end of the lesson, it was observed that in some circumstances the low attaining learners did contribute to their group’s discussion and were able to explain their group’s answers. learning is evident as in one particular case; a learner was troubled as he was not able to grasp the question. the learner’s group proceeded to explain the question to him. once he understood it, he was able to contribute to the discussion and became one of the main candidates that solved the problem. researcher: i told you that you had to try solve [the question] first on your own. how did you find, or feel trying to solve it on your own first? g2: i just thought that this was too hard for me researcher: really? g2: yes mam. researcher: and when you got into a group? g3: i saw that it was getting easier and i saw that i understood. and i realised that the only reason i thought it was hard was because i didn’t understand. g2 realised that not understanding what the question asked hindered his problem solving skills. i6, who completed the task individually, initially expressed he preferred working on his own but later he to changed his mind and felt that working in pairs would have been more beneficial. i3 repeatedly uttered throughout the interview that he “would have loved to be in a group”. the problem he felt with working individually was that he “couldn’t ask anyone for help”. i3 had partially answered the first question and had not attempted to answer the second one. composition of groups dissatisfaction was raised concerning group members who did not participate in their group discussion. a learner from group 4 was in a different class to those in his group and hence did not know any group members. he was not friendly towards the members in his group even though they tried to include him and he did not participate in the group discussion. g4 commented that “he totally ignored [the group] and did the whole thing by himself.” g4 also acknowledged that he received help from members of other groups, as his group had not functioned at all. in the interviews it was confirmed that learners found it difficult working in groups with people they did not know, as g2.2 comments “you might not even know that person but it will be hard just to come and talk to them and be friends with them” and as g2 commented “if i start talking to a stranger its all awkward and tense”. this suggested that learners feel more comfortable in groups where they are familiar with each other. learners may not have the confidence working with the members of the group that they had been placed in, and would have preferred being in different groups. the above two examples does not reflect badly on collaborative learning as a whole. learners in groups 1, 2, 5, 6 worked extremely well together. g2 felt that everyone participated and were helpful. he also believed that he “wouldn’t have found [the answer] on [his] own.” group 1 had two strong willed members who tried to convince each other and the rest of the group that their answer was correct. this is an indication that collaborative learning had taken place as learners were discussing various options and were debating their validity. all groups members were involved in the whole lesson even though individual members may not have been involved in the group discussion. despite learners dissatisfaction about working in groups, they expressed they do work in informal groups in class. two learners felt that they did not normally work in groups unless they were unable to understand or complete a problem. at this stage they felt that asking another learner would result in a better understanding and may result in the problem being solved. collaborative learning did take place in the lesson even though it was not necessarily in the assigned groups. learners still engaged with each other to find the solution to the problem. collaborative learning in a multilingual class 60 learners expressed that one of the major problems in the lesson was the dynamics of the group. most learners believe they achieve a greater knowledge in a group environment as opposed to working individually, but thought that if they were not placed in a good group they would rather work alone. learners felt that friendship groups would not be the best idea as they would not get any work done. all learners felt that if they were placed in different groups the outcome of the lesson may have been different. learners felt that making the groups equal with behavioural problems and academics would result in better results. discussion of data collaborative learning versus individual problem solving skills learners that were placed in groups were found to have a greater ability to solve problems than those that completed the worksheet individually. this would agree with barkley, cross and major (2005), and lyle (1996) who found that learners who worked collaboratively had a greater problem solving ability. learners working in informal groups felt that it was helpful. even learners who believed that they did not work in groups acknowledged that they only asked for help when they were not able to achieve the answer on their own. this shows that learners believe that working collaboratively is more successful than working individually when they encounter a difficult problem. the language of the worksheet was too difficult for learners as they did not initially grasp what the question was asking and hence were not able to solve the problem. the structure of the task is crucial to reduce uncertainty and ensure learners understand what is expected of them (lyle, 1996). learners had not been given questions in a similar fashion to this, but had solved problems of a similar nature from their text books. a worksheet however is seen as an important document and not as just another exercise. learners should be given similar worksheets to obtain more experience of what is expected of them. collaborative learning versus individual engagement the data shows that learners working in groups are more relaxed and are able to share valuable information with each other to arrive at the answer which has an advantage over learners who work individually. this does depend on the dynamics of the group. individual learners were more demanding on the teacher whereas learners working in groups were not as demanding. this would agree with barkley et al. (2005) who argues that collaborative learning encourages learner centred teaching. barkley et al further stresses that working in a group can improve the learner holistically. the placement of learners in groups allowed for academically strong learners to help low attaining ones. both academically strong learners and low attaining ones were advantaged by collaborative learning, as the low attaining learners were able to grasp the meaning of the question, enabling them to contribute towards the group discussion. the academically stronger learners’ would have gained conceptual understanding while explaining the question to low attaining learners, as indicated by barkley et al. more research may be done to verify this assumption. allowing learners to choose their groups may have curbed learners’ anxiety of working with those they do not know. the results show that learners found it difficult working in the groups they had been placed in especially when learners were not familiar to each other. previously learners worked more effectively in non official groups that they developed on their own. as lyle (1996) shows, familiarity with each learner in a group influences how the learners will interact, hence learners who are friends work better together. implications and recommendations implications for collaborative learning in multilingual classrooms the use of two or more languages, usually english and other indigenous languages, has become a frequent observation in multilingual classes in south africa (vorster, 2008). code-switching needs to be promoted on a larger scale in an informal or structured manner. the study has shown that collaborative learning (allowing learners to switch between languages) has a significantly greater success rate than individual engagement, although it is advised that further research be conducted to clarify this point. the role of language for conceptualisation and for mathematical problem solving skills has long been acknowledged in other researches (genter & goldin-meadow, 2003; pimm, 1987, 1991; usiskin, 1996; vygotsky, 1962). this would indicate that mathematical teachers must make use of collaborative learning in the classroom not only specifically for fractions and decimals but for all aspects of mathematics. collaborative learning promotes the interaction between low attaining deonarain brijlall 61 learners and academically stronger learners, which enables the stronger learners to help the low attaining ones improve their understanding. this invariably gives the teacher more free time to ensure that learning is taking place by ensuring learners are on task and assisting where necessary. collaborative learning also allowed the lesson to be more learner centred in line with the rncs (department of education, 2002). caution once again must be taken in the placement of learners in groups. a suggested method may include placing learners in the same group for a long period of time to allow learners to gain confidence in their group members. implications for further research further research should be conducted to verify the findings of this study. other possible research would be: • to determine if group dynamics improve problem solving skills • to determine if placement in a group for extended periods of time improves the group dynamics • learners’ difficulties in fractions and decimals • to determine if collaborative learning in multilingual classes improves both the low attaining learner and the academically strong learners simultaneously references arora, a. (2003). examining the problem-solving achievement of grade 8 students in timms 2003. timss & pirls international study centre, boston college, usa. barkley, e. f., cross, k.p., & major, c. h. (2005). collaborative learning techniques: a handbook for college faculty (1st ed.) san francisco: josseybass. barns, h. (2005). the theory of realistic mathematics education as a theoretical framework for teaching low attainders in mathematics. pythagoras, 6, 42-57. barwell, r. (2005). a framework for the comparison of pme research into multilingual mathematics education in different sociolinguistic settings. in h. l chick,. & j. vincent (eds.), proceedings of the 29 th conference of the international group for the psychology of mathematics education, 2 (pp. 145-152). melbourne: university of melbourne. buffer, a. & leigh, g. (2005). facilitating positive epistemological change through the development of multi-representational problem solving skills in physics. in r. pinto & d. couso (eds.), proceedings of the 5th international conference of the european science education research association (pp. 361363). barcelona. cooper, b. & bunne, m. (2004) constructing the ‘legitimate’ goal of a ‘realistic’ maths item: a comparison of 10-11 and 13-14-year-olds. mathematics education, (pp. 69-90). london: routledge falmer. department of education. (2002). revised national curriculum statement grades r-9 (schools): mathematics. pretoria: department of education. dlamini, c. (2008). policies for enhancing success or failure? a glimpse into the language policy dilemma of one bilingual african state. pythagoras, 67, 513. gentner, d., & goldin-meadow, s. (eds.) (2003). language in mind. advances in the study of language and thought. cambridge: mit press. kilpatrick, j., swafford, j., & findell, b. (2001). adding it up: helping children learn mathematics. washington, dc: national academy press. lyle, s. (1996). an analysis of collaborative group work in the primary school and the factors relevant to its success. language and education, 10 (1), 13-31. pimm, a. (1987). speaking mathematically, communicating in mathematics classrooms. london: routledge. pimm, d. (1991). communicating mathematically. in k. durkin & b.shire (eds.), language in mathematical education: research and practice (pp 18 – 24). philadelphia: open university press. reddy, v. (2006). mathematics and science achievement at south african schools in timss 2003. cape town: human science research council press. setati, m., & barwell, r. (2008). making mathematics accessible for multilingual learners. pythagoras, 67, 1 – 4. usiskin, z. (1996). mathematics as language. in p. c. elliot, & m. j. kenney (eds.), communication in mathematics: k-12 and beyond. 1996 yearbook (pp 231 – 243). reston: nctm. vorster, h. (2008). investigating a scaffold to codeswitching as strategy in multilingual classrooms. pythagoras, 67, 33 – 41. vygotsky, l. s. (1962). thought and language. new york: mit press. abstract introduction literature review theoretical framework methods and design results discussion conclusion acknowledgements references footnotes about the author(s) benjamin shongwe department of mathematics education, faculty of education, university of kwazulu-natal, durban, south africa citation shongwe, b. (2021). early career teacher’s approach to fraction equivalence in grade 4: a dialogic teaching perspective. pythagoras, 42(1), a623. https://doi.org/10.4102/pythagoras.v42i1.623 original research early career teacher’s approach to fraction equivalence in grade 4: a dialogic teaching perspective benjamin shongwe received: 14 apr. 2021; accepted: 30 july 2021; published: 15 nov. 2021 copyright: © 2021. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract the research presented in this article reports on the results of a case study examining the classroom practice of one early career grade 4 teacher (nox, pseudonym) as she teaches equivalent fractions. the goal was to explore the ways and extent to which her instruction reflected a dialogical teaching approach, defined as a pedagogical approach underpinned by five specific principles that can be enacted through a range of possible talk strategies to achieve sustained participation of learners and thus enhance meaningful learning. i provide a pedagogical activity to illustrate to teachers how, by instigating and developing classroom talk in the primary classes, a dialogic teaching sequence may be implemented. however, the majority of the existing literature on dialogic teaching stems from studies conducted in asian, european, and north american countries, whereas systematic research on dialogic teaching across international contexts remains limited. nox was interviewed after obtaining observational data to seek clarity on some of the observed instructional practices. analysis of transcripts using the notion of dialogic teaching as a theoretical lens revealed that there was little evidence of nox’s attempts to use talk to make learning of equivalent fractions a cumulative process. in addition, time constraint was the most significant factor in nox’s teaching of equivalent fractions: she considered the curriculum too congested. implications are drawn for evaluating dialogic teaching in primary mathematics classrooms. future, larger studies may shed light on the extent of these results and, if need be, a significant investment on initial teacher training may be necessary to underscore the value of dialogic teaching in enhancing meaningful learning of, at least, equivalent fractions. keywords: early career teacher; dialogic teaching; meaningful learning; equivalent fractions; grade 4; generalisation; pedagogical tool; learners’ ideas. introduction fractions is not only one of the most important topics in mathematics, but also one of the most multifaceted (pedersen & bjerre, 2021). an important part of this domain of mathematics is fraction equivalence. meaningful learning of fraction equivalence is crucial for learners’ success in algebra, a gatekeeper to post-school education ‘and the careers such education affords’ (kilpatrick & izsák, 2008, p. 11). learners’ understanding of the fundamental concept of fraction equivalence should reflect more than just procedural knowledge of generating equivalent fractions; they should be able to make connections among symbols, models, pictures, and context (wessel, 2020). previous research has documented that meaningful learning of fractions by learners, particularly constructing or identifying equivalent fractions and the development of equivalent sets of fractions, has been met with difficulty (aliustaoğlu, tuna, & biber, 2018; namkung, fuchs, & koziol, 2018; önal & yorulmaz, 2017; pearn, 2003). learners’ difficulties with equivalent fractions date back decades ago. cramer, post and delmas (2002) suggested that the difficulties are associated with mathematics instruction that tends to focus more on simple part-whole shading tasks, leaving little time to develop an understanding of the meaning of fractions in grade 4. the reasonable conclusion to draw from this suggestion is that attention must be on the teacher. in fact, there is broad consensus that ‘teacher quality is the single most important school variable influencing student achievement’ (organisation for economic co-operation and development [oecd], 2005, p. 2). it is for these reasons that the study focused, in the main, on a teacher’s instructional practices in the domain of equivalence. in a study conducted by jigyel and afamasaga-fuata’i (2007), it was found that most learners do not discuss ways of computing equivalent fractions unless prompted to do so. for example, putra and winsløw (2018) have pointed out that one of the main misconceptions of finding an equivalent fraction is that learners tend to add both numerator and denominator of a fraction by a positive integer (i.e., ). also, van hoof, lijnen, verschaffel and van dooren (2013) highlight the whole number bias, a term coined by ni and zhou (2005) to characterise learners’ tendency to see the numerator and denominator as separate numbers. put another way, learners tend to engage in an overextension of natural number principles (obersteiner, alibali, & marupidi, 2020). pedersen and bjerre (2021) attribute this common mistake to the fact that learners’ first encounter with quantities involves natural numbers in which every number represents a unique quantity. as a consequence, ni and zhou (2005) point out, when learners subsequently encounter rational numbers, seemingly different numbers describe the same quantity in the fraction notation. an understanding of a fraction as a number that can be written in different but equivalent forms, for example, is a prerequisite for meaningful learning of equivalent fractions (charalambous & pitta-pantazi, 2007). thus, meaningful learning of equivalent fractions is helpful in determining the number of fractions between, for example, and (jigyel & afamasaga-fuata’i, 2007). still, it leads to utilisation of number line representations of fractions (bright, behr, post, & wachsmuth, 1988). although direct teaching,1 defined here as teacher-directed instructional practice characterised by teaching of specific skills to learners, is effective in producing learning of basic mathematical skills (i.e. computation of amounts, sizes or other measurements involving addition and subtraction), dialogic teaching has been the focus of many investigations over the years because of the acknowledgement that it is of prime importance for teaching and learning in the classroom and beyond (arend & sunnen, 2015). most curricular documents have identified dialogic teaching as a desired means of teaching and learning (national council of teachers of mathematics [nctm], 2000; national governors association center for best practices [nga] & council of chief state school officers [ccsso], 2010). research studies have found that dialogic teaching engenders meaningful learning. for baroody and hume (1991), meaningful learning puts less emphasis on direct teaching and paper-and-pencil work but builds upon learners’ informal strengths rather than treating them as blank slates (tabulae rasae). however, hiebert and grouws (2007) state that ‘there is no reason to believe, based on empirical findings or theoretical arguments, that a single method of teaching is the most effective for achieving all types of learning goals’ (p. 374), alexander’s (2020) notion of dialogic teaching is particularly touted as an approach that can achieve the best educational results because of its emphasis on the active and sustained participation of learners in classroom talk. findings by mercer and sams (2006), resnick, asterhan and clarke (2015), and rojas-drummond and mercer (2004) from investigations of teachers’ interactional strategies showed that dialogic teaching indeed achieves better learning outcomes in cognitive domain (e.g. intellectual skills and verbal information) and affective domain (e.g. attitude and disposition to persevere). it is clear then that these two teaching orientations, namely direct teaching and dialogic teaching, would entail strikingly different visions of what ‘good’ teaching looks like in a mathematics classroom. in turn, they would in all likelihood have different effects on learning outcomes. as the name suggests, dialogic teaching is described as teacher-led interactions with learners – one of the two main kinds of interaction, the other being learner-learner interactions (mercer & sams, 2006). seen in this light, dialogic teaching is founded on vygotsky’s (1978) sociocultural account of learning in which the teacher – the ‘more knowledgeable other’ – provides a ‘scaffolding’ for the development of understanding of mathematical practice2 and mathematical operations, procedures, terms, and concepts. although there is some variation in the precise definition of the metaphor of ‘scaffolding’ among researchers, a key characterisation of this term is that a teacher provides a sensitive kind of help, which enables a learner engaged in a task to improve their competence and eventually achieve meaningful learning (rojas-drummond, mercer, & dabrowski, 2001). clearly, different researchers will use various terms to describe dialogic forms of teaching: pappas and varelas (2006) and wells (1999) call it dialogic inquiry, skidmore (2006) prefers to label it dialogical pedagogy, and for van der linden and renshaw (2004) it is dialogic learning. in this article, dialogic teaching is used because alexander’s (2020) principles of dialogic teaching are grounded in the teacher’s approach, in particular, their listening to a variety of learners’ ideas and chaining them into coherent lines of thinking and understanding. prior to providing a review of literature, it will be helpful to define the notions of ‘meaningful learning’ and ‘dialogic teaching’, which are particularly clearer if they are contrasted with ‘rote learning’. according to ausubel (2012), meaningful learning is a function of a learner’s disposition to relate symbolically expressed ideas (the learning task) in a nonarbitrary, and nonverbatim fashion, to already existing knowledge. in contrast, rote learning entails retention of learned material on a purely associative basis (e.g. memorising multiplication tables with no opportunities for learners to notice or to associate interrelations between 3 and 9 times tables, for example). novak (2010) points out that meaningful learning is the facilitation of the construction of valid meanings and reconstruction of misconceptions (invalid meanings) in learners’ cognitive structures. by ‘cognitive structure’ is meant knowledge organised through processing of new information for its appropriate retrieval in new contexts, as in novel mathematics problem-solving (novak, 2002). as schmidt (1993) argues, this knowledge is considered organised in a certain way, hence cognitive structure. this definition points to the connection between prior knowledge and dialogic teaching. however, it is important that i explicitly define what is meant by ‘dialogic teaching’. dialogic teaching is a term used to denote a pedagogical tool3 that harnesses the power of classroom talk to engage learners’ prior knowledge, stimulate their thinking, advance their understanding, and expand their ideas in the mathematics classroom (alexander, 2020). mercer and littleton (2007) describe dialogic teaching as ‘that in which both teachers and pupils make substantial and significant contributions and through which children’s thinking on a given idea or theme is helped to move forward’ (p. 41). in this sense, as alexander (2006) argues, dialogic teaching is structured classroom talk thus making it ‘the true foundation of learning’ (p. 9). in dialogic teaching, attention is paid to more than one point of view, more than one voice is heard in the classroom and there is ‘interanimation’ (exploration) of ideas (bakhtin, 1981; mortimer & scott, 2003). indeed, it depicts a classroom practice in which the teacher’s perspective is not prioritised; instead, learners and the teacher co-develop the object of the lesson (arend & sunnen, 2015). in short, given the complexity of educational contexts, it would be simplistic to assume that dialogic teaching would be suitable in all classrooms. the results in this article should broaden our knowledge of early career teachers’ uptake of dialogic teaching as a framework that prioritises learners’ ideas in the development of mathematical concepts. by doing so, the results should transform the links of dialogic teaching to equivalent fractions and thus improve teaching of equivalent fractions content. purpose of the present study previous research suggests that teaching approaches4 may vary across countries (aliustaoğlu et al., 2018; namkung et al., 2018; önal & yorulmaz, 2017; putra & winsløw, 2018). moreover, the majority of the existing literature on dialogic teaching stems from studies conducted in asian, european, and north american countries, whereas systematic research on dialogic teaching across international contexts remains limited. thus, dialogic teaching strategies in some countries may not be generalisable to other educational contexts. thus, we know little about how an early career teacher employs dialogic teaching to assist her grade 4 learners in the quest to acquire meaningful learning of equivalent fractions in instructional practices. the purpose of this study was to explore the dialogic teaching strategies employed by one early career grade 4 teacher. the analysis is grounded in alexander’s (2020) theoretical perspectives to broaden our understanding of dialogic teaching in classroom interactions and to transform its links to equivalent fractions. equivalent fractions, a topic that has proved to be difficult to learn, provided the impetus for this study. through a qualitative analysis of observational and interview data, the following general research question was posed: how does an early career teacher engage her learners in meaningful learning and sustained participation during the teaching of equivalent fractions? the remainder of this article is structured as follows. having provided a brief background to the study, i elaborate on the literature pertaining to dialogic teaching and exemplify the kind of strategies that may partly characterise instruction on equivalent fractions. then, i provide a framework guiding this study. next, an analysis of the results is undertaken followed by a discussion in which i interweave the literature with the findings. on the basis of the findings, i end the article with implications for classroom practice and recommendations for future research. literature review in recent years, many researchers in the field of education have shown interest in the work of robin alexander, david ausubel, and mikhail bakhtin. these scholars have conceptualised terms like ‘dialogic’ and ‘meaningful learning’, which have gained increased attention in both mathematics and science education studies on classroom interactions. the term ‘dialogic’ has gained increased attention and classroom talk has become a key topic in educational sciences (arend & sunnen, 2015). to situate the proposed study in current literature on classroom interactions, reference to the research work of these scholars is made. however, this review of literature does not pretend to provide an exhaustive overview of related research on classroom interactions; only research work related to dialogic theory will be reviewed. the basis for this approach is that these scholars’ works centre around the notion of ‘dialogic’ as an approach to investigate or theorise about classroom interactions. ausubel’s theory of meaningful learning, refined by the work of joseph novak, is anchored in piaget’s (1964) constructivist perspectives. alexander’s (2006, 2020) ‘dialogic teaching’, the focus of the proposed study, builds on the foundational works of bakhtin (1981, 2010). research on dialogic teaching teaching and learning is primarily concerned with the acquisition, retention, and use of information such as facts, propositions, principles, and vocabulary in the various disciplines (ausubel, 2000). one way in which such processes take place involves the idea of dialogic teaching. in their seminal work, sinclair and coulthard (1975) proposed what can be thought of as the archetypal form of interaction between a teacher and a learner, particularly their identification of the structural feature of discourse usually known as the initiation-response-feedback (irf) exchange. for example: teacher: what is the capital of peru? (i) student: lima. (r) teacher: yes, quite correct (f) the irf structure involves ‘closed’ questions in which there is only one answer, which is already known to the teacher. although this structure does not typify the pattern of talk in all classroom activities (learners may ask questions of teachers, or of other learners), irfs have been observed as a common feature in classrooms the world over (mercer, 1995). teachers’ relentless questioning reduces the amount and variety of the verbal contributions that their learners can make in the classroom. they argue that this behaviour must have a detrimental effect on the intellectual activity of learners. clearly, this comparison suggests that dialogic teaching is distinct from the ‘speaking and listening’ that characterises most interactions in classrooms. in a study of danish and indonesian teachers’ pedagogical approach to equivalent fractions, putra and winsløw (2018) found that danish learners tend to argue that adding the same natural number to both the numerator and denominator gives another equivalent fraction. for example, one learner incorrectly argued that . it is such learners’ difficulty with equivalent fractions that definitely points to the need to investigate the efficacy of dialogic teaching as a more effective way to teach this important concept. particular attention will be given to overcoming the difficulty learners have with the ‘equivalence’ idea, delaying the rule until there is meaningful learning of the concept of equivalent fractions (gattegno, 2010). learners should understand why a procedure works prior to using it. during interviews with two preservice teachers, shongwe (2014) found that teacher preparation programmes and beliefs strongly influence classroom talk. mercer and sams (2006) have found that teachers can act as important models for learners’ own use of language for constructing knowledge, particularly when they use probes such as ‘why’ and ‘what i heard you say was …’. the ‘why’ question evokes reasoning about why learners took a particular approach in a task and can be very useful for revealing learners’ perspectives on the task to the teacher and for stimulating their own reflection on it. research studies in both science and mathematics education have repeatedly shown that the use of dialogic teaching strategies enhances meaningful learning (alexander, 2020; mortimer & scott, 2003; rojas-drummond et al., 2001). current teaching approaches in classrooms hogan, nastasi and pressley (2000) have shown that teacher-led discussions and learner-learner discussions are two distinct processes in that the former is a more efficient means of attaining higher levels of meaning-making whereas the latter tends to be more exploratory. similarly, nathan and knuth (2003) found that, when the teacher maintained a central role, the learner-led discussions increased considerably; however, at the expense of mathematical precision. for instance, one pair of learners working on equivalent fractions in putra and winsløw’s (2018) study showed a misconception that adding the same number to both the numerator and denominator results in equivalent fractions. lamon (2020) points out that fraction instruction has traditionally focused on only one interpretation of fractions, that of part–whole comparisons, that is presented in a context that allows for whole number reasoning, after which the algorithms for symbolic operation are introduced. gee (2014) argues that classroom teaching tends to be assumed as focused only on textbook definitions of mathematical terms and procedures. for instance, defining equivalent fractions as two or more fractions that have different numerators and denominators but result in the same quantity after simplification, and unwittingly encouraging ordering of equivalent fractions by using the idea of ‘multiplicative property of one’, which i exemplify in the next section. however, geometric models offer the contexts to understand equivalent fractions (lee & lee, 2020). for learners, focusing on symbolic representation of, for example, as ‘two out of two’ does not convey the meaning that this is 2 halves which is equivalent to, for example, four quarters, until a geometric (area) model is used (figure 1). figure 1: different fractions that name the same whole (adapted from the trends in mathematics and science study). most preferably, instead, treatment of equivalent fractions using concrete models such as fraction bars to compare, for example, thirds and sixths and see that is exactly the same as provides meaningful learning of equivalent fractions. this is what is happening in most classrooms these days: teachers tend to focus on the textbook and thus ignore the value in eliciting learners’ ideas in discussions claiming that they are wasting time. in this regard, despite that lortie’s (1975) work dates decades ago, little has changed in the teachers’ daily struggles in the classroom, particularly more so for early career teachers. exemplification of dialogic teaching of equivalent fractions meaningful learning of fraction equivalence includes having an integrated knowledge, which can be displayed and articulated by means of the following five attributes: a fraction represents a quantity being measured in relation to a referent unit (emphasis mine). a fraction quantity can be represented using manipulatives or pictorially by partitioning (emphasis mine) area, collection or number-line models. equivalent fractions can be constructed from manipulatives or pictorial representations by repartitioning or chunking (emphasis mine). equivalent fractions can be constructed using symbolic notation. a fraction quantity is a member of an equivalence class in which all fraction numerals represent the same quantity (wong, 2010, p. 674). physical quantity, that is, the size or amount of a physical characteristic of an object, can be measured. for example, physical quantities encountered by young learners are: length, perimeter, area, volume, mass, etc. however, determining whether an answer about a physical quantity of an object, arrived at by using a calculator or paper-and-pencil method, is reasonable requires estimation ability. estimation is the mental process of arriving at an approximate measurement without the aid of measuring instruments. this then requires a learner to make a judgement of the size of a physical quantity relative to some specified unit. the notions of referent unit, partitioning, and chunking are at the core of the fractions domain in primary mathematics education, particularly in estimation. however, the question becomes, what do they mean? in what follows i define them in turn, paying particular focus to partitioning. the other two terms are merely defined for the purpose of completion; they are not a feature in the analysis. a referent unit is a non-standard or standard unit that can be used to estimate a quantity. for example, if area is to be measured a two-dimensional unit like a playing card can be used as a referent unit or if the length of a pencil is to be measured matchsticks can be used as referent unit. thus, the choice of an appropriate unit for measurement is a mathematical skill that is fundamental in learning to measure. partitioning means engaging in an intuitive activity that generates quantity to build knowledge about fair sharing. for example, in determining an equivalent fraction for , we can break up 289 into parts that are easy to divide (e.g. division by 24) as displayed in figure 2. the term chunking, in relation to fractions, refers to dividing a larger number that cannot be divided mentally, by repeated subtraction of the divisor and multiples of the divisor. for example, finding the answer to involves using numbers that are easy to multiply. figure 2: division with partitioning. the teaching approach adopted by teachers in relation to equivalence may manifest itself in learners’ work. kerslake (1986) and cramer and henry (2002) found that learners in the intermediate phase predominantly favour circle models over geometric and linear models in representing fractions. they also found that some learners hold misconceptions when explaining equivalence. for example, learners reasoned that is double . teaching episodes should emphasise that although the fractions being pictured are different, each shows that the same portion of the region is shaded. that is, fractions that have the same value even though they may look different are identified as equivalent fractions. therefore, equivalent fractions are fractions that are equal in value or size. figure 3 is designed to describe how dialogic teaching of equivalent fractions will look in a grade 4 class. since the objective here is to create opportunities for learners to make meaning for themselves, a paper-folding activity will lead to them ‘discovering’ a general rule (i.e. a generalisation) that the sequence of fractions arises from multiplying with the number of regions created by the folding. in this activity, learners are provided with a rectangular region, half of which is shaded. then, they are directed to fold it into two congruent parts as shown in figure 35 by the broken line through the region. this procedure is repeated over and over again, scaffolding learners’ development of the concept of equivalent fractions using prompts each time (suggested answers in brackets). this procedure is consistent with lannin, ellis, elliott and zbiek’s (2011) recommendation that teachers need to encourage their learners to connect examples, extend their ideas to new situations, and identify commonality across cases or to extend commonality beyond the domain of the original pattern from primary mathematics classes. figure 3: development of a general statement about equivalent fractions. an alternative generalisation approach, provided by putra and winsløw (2018), would be using the ‘multiplicative property of one’: for example, for nd a = 1 and b = 2: however, the ‘paper-folding’ activity is a superior method in promoting meaningful learning through dialogic teaching. as gattegno (2010) points out – using the idea of identity element for multiplication, as shown above – we can see that equivalence is concerned with a wider relationship where it is possible to replace one item by another. further, he argues, equivalence is a domain of primary mathematics that constitutes the most comprehensive relationship ‘and therefore the most useful’ (p. 129). he exemplifies this claim in everyday life, pointing out that to say that ‘he is on my left’ is equivalent to saying that ‘i am on his right’ and ‘i am taller than her’ is equivalent to ‘she is shorter than me’. theoretical framework the study reported in this article employed alexander’s (2020) notion of ‘dialogic teaching’ as a theoretical basis guiding formulation of research questions, methodology, and discussion of results. the central premise of this notion is that encouraging learners to share their thinking enables teachers to diagnose learners’ misconceptions, devise tasks to dispel misconceptions, and assess learners’ progress. dialogic teaching is based on the proposition that classroom talk must be positioned firmly at the heart of the learner’s learning and the teacher’s practice. in addition, dialogic teaching emphasises the value of the teacher’s guiding role in the development of learners’ mathematical understanding given that primary school children often lack the necessary skills to manage their joint activity (mercer & sams, 2006). thus, the concept of dialogic teaching is relevant for this study in that its foundational idea is that primary school learners need ‘guided construction of knowledge’ as they participate in group activities that offer valuable opportunities for them to construct solutions for themselves through talk, which would not be found in whole-class instruction (mercer & sams, 2006). put another way, it is through teacher’s talk that learners’ talk is prompted. dialogic teaching combines four repertoires (and their subcategories): talk for everyday life, learning talk, teaching talk, and classroom organisation. these repertoires are used flexibly, on the basis of fitness for purpose. additional to these repertoires, alexander (2020) proposes five principles: collective (the classroom is a site of joint learning and enquiry), reciprocal (participants listen to each other, share ideas and consider alternative viewpoints), supportive (participants feel able to express ideas freely, without risk of embarrassment over ‘wrong’ answers, and they help each other to reach common understandings), cumulative (participants build on their own and each other’s contributions and chain them into coherent lines of thinking and understanding), and purposeful (classroom talk, although open and dialogic, is structured with specific learning goals in view). although alexander’s notion of dialogic teaching encompasses a broad repertoire of strategies and techniques, only the following will be considered for the purpose of this study: structuring of questions so as to provoke thoughtful responses individual teacher-learner exchanges are coherent, connected lines of enquiry that do not leave learners stranded. in this article, i analysed nox’s teaching of equivalent fractions. the purpose of this article was neither to praise nor be critical of her pedagogy. similarly, i did not intend to advocate or critique the use of dialogic teaching approach in equivalent fractions. rather, the purpose of this article was to present an analysis of her instruction itself and thus offer a more precise description of what occurred in her classroom. a second purpose was to discuss why nox chose to teach equivalent fractions in the way that she did. the research questions using alexander’s (2020) perspectives as an explanatory framework in relation to dialogic teaching, the following specific research questions were posed to guide the study: how does nox structure her questions to learners so as to provoke thoughtful responses in the learning and teaching of equivalent fractions? why did nox teach equivalent fractions in the way she did? methods and design study design the study reported in this article is part of a larger project looking at the classroom practices and professional development of an early career primary mathematics teacher. a qualitative case study design was adopted. in a qualitative inquiry the researcher studies meanings constructed by participants on a phenomenon in their natural setting (denzin & lincoln, 2011). observational data as well as follow-up semi-structured interview data were collected in the spirit of naturalistic inquiry (lincoln & guba, 1985). the use of multiple data collection methods not only strengthens the validity of the findings (howitt & cramer, 2005) but also reflects an attempt to secure an in-depth understanding of nox’s instructional patterns of talk, and is of importance particularly for an individual researcher (patton, 2002). ethical considerations ethical approval was granted by the ethics in research committee of the university of kwazulu-natal. this research is part of a larger project designed to follow the experiences an early career teacher in the pedagogy of mathematics and ethical approval was granted by the ethics in research committee of at a university in south-eastern south africa with a protocol number hssrec/00001902/2020. informed consent was obtained from all participants. participants and context the empirical research reported in the present study involved data from a main project designed to follow the professional development of an early career teacher, nox (pseudonym), in a grade 4 class (nineand ten-year-olds) who were conveniently sampled to participate in the study. nox had a four-year professional teaching degree in primary mathematics. at the time of the study (2020), she had 2 years of teaching experience. she taught at a rural combined primary school in the eastern cape, south africa, which i shall call fundisanani. the school is situated in a village (rural area) and serves learners from a low socio-economic background, in an isixhosa-speaking african community with high absenteeism rate for both learners and teachers. like most rural schools in the area, it was under-resourced, had no laboratory or media centre, nor sports field. the names of all participants and schools have been changed to preserve anonymity. the context of the study was kept as normal as possible, in three ways. first, nox was observed teaching equivalent fractions in her grade 4 classroom under the general topic ‘fractions’. second, the lessons on equivalent fractions that were audio recorded were selected from the typical, prescribed primary school mathematics curriculum. third, the researcher was a spectator observer (patton, 2015) who gathered data from nox in her natural classroom setting. the class enrolment was 48 learners of multiple ability levels. the mathematics period in the class was approximately one hour long, five days per week. a typical lesson often began with the learners engaging in mental calculation of whole numbers including estimation, building up and breaking down numbers, rounding off and compensating and doubling and halving, all to be completed in 10 minutes. the curriculum was organised around units on general mathematical topics such as: numbers, operations and relationships, patterns, functions and algebra, geometry, measurement, and data handling. data collection and procedure qualitative data were collected by employing two methods: classroom observation and semi-structured interviews. the purpose of the observation was to understand how they orchestrate dialogic strategies in their context-specific settings. as already mentioned, nox was observed in her natural setting (i.e. the classroom). data were audio recorded and transcribed verbatim. an audio recorder was strapped to nox’s waist to capture all the teaching proceedings including interactions in group work and any non-verbal communication. given that the purpose of the observation was to understand how nox orchestrated dialogic strategies in her context-specific settings – that is, i was particularly interested in the teacher talk – i concentrated on recording solely nox. another reason for recording only her talk was to minimise interference with learners’ behaviour so that their talk would be natural. the classroom observation protocol was used to answer the first two research questions by documenting what nox said in telling her story on equivalence. the tool also included background information, contextual background and activities, and 10 items on a four-point likert-type scale. the items are on a continuum ranging from ‘never occurred’ to ‘very descriptive’. in this section, items on the left in the continuum of the scale are generally more ‘traditional’ and those on the right generally reflect more dialogic teaching. these items provided the operational definitions for dialogic teaching typology. the typology was meant to capture, in retrospect, the observer’s overall interpretation of the teaching approach observed in the classroom. the observation lasted 40 minutes. additionally, nox was interviewed after the lesson observations to establish reasons for the observed teaching sequences. the interview also helped in triangulation of observation data with interview data. sample questions on the interview schedule included ‘why do you ask learners to clarify their homework activities?’, ‘describe your teaching of equivalent fractions’, and ‘why do you teach equivalent fractions the way you did in the lesson?’. the interview lasted for 45 minutes. analysis the purpose of the proposed study is to investigate how practising grade 4 teachers use dialogic teaching to orchestrate classroom interaction in the context of equivalent fractions. the data were subjected to thematic analysis since this method offers ‘a more accessible form of analysis, particularly for those early in a qualitative research career’ (braun & clarke, 2006, p. 81). further, as fielden, sillence and little (2011) indicate, this analysis method allows the researcher to explore the individual experiences of participants and the meanings they attribute to them. the analysis took into account the notion that data analysis should end when it yields no further information. this analytical approach was adopted on the basis that ‘failure to reach saturation has an impact on the quality of the research conducted’ (fusch & ness, 2015, p. 1408). reaching this point added to the trustworthiness of the results. in particular, transcribed excerpts from episodes of audio recorded lessons of nox’s dialogic strategies were used to examine how she drew on dialogic teaching principles proposed by alexander (2004) to make understanding of equivalent fractions available on the social plane, at their different schools. that is, alexander’s predetermined categories were used to assess how the participating teachers make opportunities for understanding equivalent fractions available for learners in their classrooms, particularly probing learners’ answers with the ‘why do you think that?’ question. nox’s talk, rather than learner-learner interactions, was analysed. it is worth emphasising that the microphone was sensitive enough to record learner-learner talk, as well. on the whole, to investigate the two research questions underpinning this study, the analysis mainly focused on the identification of interactions using alexander’s (2020) lens to make conclusions on the orchestration of dialogic teaching by nox, noting how she handled learners’ ideas from the lens of dialogic teaching. in analysing the data obtained from the two sources – classroom observation and interview – attention was paid to whether nox’s classroom talk was truly dialogic by examining whether a question posed gave rise to a new question and the extent to which her teaching required learners to think rather than merely report someone else’s thinking (alexander, 2020; bakhtin, 2010; nystrand, 1997). the observation transcript was also coded by a doctoral student whose work was in the area of dialogic teaching. a very high inter-rater reliability – where researchers are expected to independently identify the same codes in a transcript and the codings compared for agreements – was reached (creswell, 2012). in addition to verbatim transcription of the data and the use of multiple data collection methods, a copy interview transcript and a draft of the interpreted results were sent to nox for review. she corrected two errors and provided additional information. all these measures were undertaken to improve the trustworthiness of the interpretations (creswell, 2012). collection of classroom observation data ceased when saturation was reached. data saturation is a methodological principle in qualitative research that is commonly taken to indicate that, on the basis of the data that have been analysed hitherto, further data analysis is unnecessary (fusch & ness, 2015). it refers to the point in the research process when no new information is discovered in data analysis, and this redundancy signals to researchers that data collection may cease. saturation, more broadly construed, means that a researcher can be reasonably assured that further data analysis did not necessarily add anything to the overall story of the participant (saunders et al., 2018). results to provide answers to the first research question meant analysing ways nox structured her classroom talk, especially her questions. this was also a way to search for evidence of dialogic teaching of equivalent fractions as her lesson unfolded. as already mentioned, the analysis focused solely on nox’s talk structure given the consensus that, as a teacher, she was the single most important school variable influencing learner achievement. the question remains whether her instructional practice showed traces of the principles of dialogic teaching – the lens through which the data in this study was analysed – that would have enhanced learning of equivalent fractions. to highlight the principles, we consider nox’s lesson on equivalent fractions in her grade 4 class. primary school teachers are encouraged to engage learners in mental work prior to beginning a mathematics lesson. however, interactions in this respect, though interesting, were not analysed as they fell outside the scope of the study. the results are presented in terms of the two research questions. description of nox’s lessons the first research question was formulated to assess how nox probed her learners to say more about their responses. in the classroom, nox was standing in the front of the learners facing the chalkboard. the learners were sitting at desks arranged in rows. she wrote the topic ‘equivalent fractions’ on the chalkboard. the excerpt below shows how nox introduced her lesson, proceeding from the previous lesson’s activity; perhaps some patterns of dialogic teaching could be observed. nox checked learners’ thinking on a previous activity involving different ways of representing fractions (using fractions to describe pictures). noticeable was that the work was void of context; i expected a word problem, at least. table 1 served two purposes. first, it provided the context of the classroom environment in which nox worked. learners took some time to settle down and nox had to talk at the top of her voice to address the behaviour. the learners were responding in chorus form (turn 6) until she decided to point at individual learners when seeking a response (turn 12). second, although the topic handled in this excerpt was representations of fractions, the excerpt captures nox’s commitment to collecting learners’ ideas. in turns 5 to 12, nox collected learners’ ideas and refrained from providing immediate evaluation of those ideas or responses at least from two learners before evaluating the answer and thus closing the talk with, ‘good’. this suggested that nox elicited learners’ answers or ideas through questions, which probed their point of view. table 1: routines in nox’s lessons. moving now to the gist of the lesson – that is, the focus on how she orchestrated the interactions with her learners to tell the story of equivalence – we find nox trying to engage her learners in more talk on the topic, equivalent fractions (see table 2). she used the fraction wall to introduce her learners to the concept of equivalent fractions (e.g. the idea of multiplication of numerator and denominator by the same number results in the same numerical value, except 0). she asked them to follow the line separating the two halves on the strip to see that is the same as and . table 2: nox’s instruction moves from conviction to seeking generalisation. as an early career teacher, nox deferred the same question to other learners for their ideas, using phrases such as ‘sipho thinks that half is the same as two quarters, do you agree with him?’ put another way, i sought to examine whether nox chained learners’ responses to follow coherent lines of thinking about the same idea to enhance meaningful learning of equivalent fractions. her phrasing of the question suggested that she wanted to provoke thoughtful responses and see interanimation of ideas in her classroom talk. seemingly convinced that the learning of equivalent fraction had taken place, nox proceeded to seek a general rule to check if two fractions are indeed equivalent, as shown in turn 16 (see table 3). nox’s intention to introduce abstraction here is commendable and consistent with lannin et al.’s (2011) argument that generalising is a practice that needs to permeate primary mathematics classes and beyond. she seems to believe that her learners have developed an understanding of equivalent fractions beyond the concrete level in which they used the fraction wall as a referent. however, she seems to have rushed her learners without having given them opportunities to explore with a finite number of cases involving equivalent fractions. evidence for this claim is provided by turn 20, where learner 6 asks ‘what is a? … yeah, what is the use … [of the] letters?’ learner 8 expressly voices the conclusion ‘it’s looking difficult’. in turn 21, nox seems to return to numerical fractions and . these learners’ statements and questions seem to break her attempt to sustain her probing to sustain the talk, as envisaged in the principles of dialogic teaching. table 3: learners show difficulty with making a generalisation. what is happening in the extract above is that nox seems to have involved many different learners in seeking an understanding of the rule. however, a close examination reveals that the voices she sought were on different aspects of the generalisation rule (i.e. ); learners’ ideas were collected only to confirm understanding rather than seek to chain these ideas into coherent lines of thinking about the same idea that has been made available on the social plane at a particular time of the exchange. evidence for this claim is found in turn 30 as she wraps up her lesson by fixing her focus on giving learners some ‘exercises’ to do in their bluebook. the dialogic exchanges ended abruptly as she leaves it to learners to pursue the concept of generalising equivalent fractions at ‘home’. although nox asked a question that appeared ‘dialogic’ in that it contained the ‘why’ cue, she did so from the perspective of a teacher with an authoritative stance (as she sought to maintain school mathematics position; pressing for the three principles of dialogic teaching was not explicit in the lessons observed). this need not necessarily be construed as an indictment on nox’s practice. she had to teach her learners in an environment that was not conducive to the principles of dialogic teaching; this could have limited her enactment of dialogic teaching in the classroom. learners’ responses to such a question functioned to respond to mathematics content because nox utilised cues such as ‘why …’ and ‘how do you …’ only to understand how closely the learners’ responses aligned with some school-sanctioned or predetermined disciplinary stance. figure 4 is a conceptual diagram that depicts nox’s arrangement and relationships of the key ideas in telling the story of equivalent fractions for her grade 4 class as conceived in her mind (depicted in the conceptual diagram). the basis for labelling it as ‘conceived in her mind’ arose from her agreement with it after i presented it to her during member checking to enhance the trustworthiness of the findings. that is, this diagram was discussed with nox to seek her agreement with the way i saw her teaching of the ideas encapsulated in equivalence. modifications were made; for example, because there was no episode in which she wrapped up the lesson to remind her learners of key ideas she wanted them to ‘take home’, the diagram only displays the introduction and the teaching of the content. thus, the conceptual diagram summarises the story that developed in nox’s lessons that were observed. figure 4: nox’s conceptual framework in teaching equivalent fractions. although nox mobilised learners’ ideas, she did not anchor her questions and comments in learners’ contributions. in other words, despite utilising cues such as ‘why’ and ‘do you believe’, she only truly cared about how closely the learners’ response aligned with disciplinary knowledge. this talk structure does not, in any way visible to an analyst, embody any ‘scaffolding’ intended for meaningful meaning of equivalent fractions. nox engaged her learners in a mathematical activity of talking about equivalent fractions. accordingly, she showed part of mathematical practice in her use of the ‘why’ cues to seek confirmation or further development of an idea. in her 29 turns of talk, nox asked a total of 16 questions that could be designated as ‘closed’ exchanges. during the dialogue, learner participation can be characterised as providing ‘correct’ answers intended to demonstrate to nox that her learners were indeed recalling the knowledge as transmitted in the previous lesson. nox implicitly indicated that a learner’s answer was incorrect by not recording it on the chalkboard or by ignoring it and continuing to ask other learners for the ‘answer’ (turn 4 and turn 7 of the second excerpt). worthy to mention here is that including other episodes would have served no purpose in that data saturation was reached at this point. that is, further analysis beyond this point did not yield new results. thus, further coding of observational data was no longer feasible. why did nox teach equivalence in the way that she did? dialogic teaching is an approach whose success is affected by contextual factors. hence, it was necessary to conduct an interview to understand why nox approached the teaching of equivalent fractions the way she did. school factors such as the routine of holding assembly in the morning prior to commencement of lessons, the movement of learners as they change classes, the pressure to complete the curriculum, and late arrival played a role in her teaching of equivalent fractions. although cross-national research studies have shown that a significant number of teachers exit their initial (preservice) teacher preparation programmes with inadequate knowledge of mathematics (eds. tatto, rodriguez, smith, & reckase, 2018), this factor was not investigated. asked to reflect on her teaching in the lesson, she pointed out that collecting learners’ ideas ‘makes the lesson messy; these kids get involved in arguments’. interested to understand better what she meant by ‘arguments’, i probed (see table 4). her response was: ‘they trying to outshine each other rather than answering the question’. this response pointed to the breakdown in talks meant to make the learning of equivalent fractions a cumulative process. the interpretation arising from these statements are that nox felt that allowing learners to engage in interactions among themselves is a recipe for chaos in the classroom as they begin to compete in ways that do not take their learning of equivalent fractions forward. this finding seems to corroborate nathan and knuth’s (2003) finding that although learner-learner dialogue increases participation, this takes place at the expense of mathematical precision. table 4: nox’s reflections on her lessons. during the interview, nox also raised time as a scarce resource, as captured in the following excerpt. as can be found in pedagogical exchanges in most classrooms the world over, nox clearly indicated that time limited her disposition to engage in dialogic teaching because she considered the curriculum too heavy. this suggestion brings to the fore the need for mentoring programmes or an improvement in inductive programmes for early career teachers so that they can enact reform-oriented teaching such as dialogic teaching. most probably time can be found by looking into the activities that seemed to consume teaching time. for instance, nox’s lesson was scheduled to take 40 minutes yet half of it was taken by the morning assembly lasting beyond the allocated time on the school’s timetable. there is truth in what nox is saying: she is in a daily grind in which she is facing challenges that come with being an early career teacher (lortie, 1975). put another way, she is in a routine that jackson (2004) refers to as ‘the daily grind’ of teaching in a primary classroom – where 30 or more people spend several hours each day literally side by side – and things do not always go according to expectations. another limiting factor was learners’ late arrival which disturbs the flow of teaching. nox has had to recap to bring the late learners trickling into the classroom up to speed with the ideas already entertained in the class. her efforts support the notion that teachers generally want all learners to understand the contents of their lessons (van de walle, bay-williams, lovin, & karp, 2013). her work is made more challenging by the fact that she was teaching in a rural school with very limited resources that can be used to mitigate her challenges (e.g. technology with the facility to draw the attention and interest of learners as they ‘see’ and ‘do’ the shading of area models involving equivalent fractions). discussion the purpose of this study was to explore possible instances of dialogic teaching in nox’s mathematics classroom. in particular, the exploration involved examining how she uses dialogic strategies to tell the story of equivalent fractions, that is, how different ratios can have the same value. to remind the reader, dialogical teaching, in a weaker sense, refers to classroom interactions in which multiple speakers take extended turns which take account of others’ ideas (scott, mortimer, & aguiar, 2006). the discussion of the results is organised in terms of the two research questions. searching for thoughtful responses teacher’s questions in dialogic teaching approach are structured so as to provoke thoughtful responses, which refers to questions that provoke further questions to create a coherent line of enquiry in relation to equivalent fractions. nox’s interest in collecting learners’ ideas was to give learners an opportunity to relate them to their existing knowledge, in this case knowledge of equivalent fractions. this practice was in fulfilment of ausubel’s (2012) argument that new knowledge should invoke links within the learner’s cognitive structures (i.e. the knowledge that the learner organises in ways that can be used in new contexts). the results suggested, however, that nox missed opportunities to exploit learners’ responses to provoke further questions that sustained their participation. in other words, nox-learner exchanges were disconnected in that she focused on completing the curriculum. she enacted the question-and-answer sequences, a practice whose sole purpose is to test retention of arbitrary items of given knowledge rather than to support the development of meaningful learning of equivalence. the testing of whether learners have retained the requisite knowledge is crucial for nox in accounting about learner performance. evidence for this claim is found in her interview utterances: ‘i have to remember that i have a duty to complete the curriculum’. in addition, nox-learner exchanges were not, in alexander’s (2006) terms, ‘chained into coherent lines of enquiry’ (p. 32). that is, new concepts were not connected in any substantive manner to learners’ ideas but were merely designed to be memorised, thus promoting the formation of, in ausubel’s (2012) terms, arbitrary relationships among ideas in the learners’ minds. this was evident particularly in nox’s attempt to develop the general rule. in contrast to putra and winsløw’s (2018) findings, nox seemed not to have paid attention to learners’ development of a meaningful understanding of how to make a generalisation, for example, when she referred to and why the rule works prior to applying it in their work with equivalent fractions. taking the definition of thoughtful responses into account, it is reasonable to conclude that there was little evidence that nox’s lessons reflected traces of this dialogic principle in her teaching of equivalent fractions. thus, observational evidence supported by her own words suggested that there was little evidence of meaningful learning of equivalent fractions, particularly the generalisation process. there were, however, other variables that were at work in her teaching approach. nox’s teaching practice in terms of her environment classroom observation of and interview with nox provided insight into the contextual factors that influenced her teaching of equivalent fractions. the actual time spent observing nox’s lessons was not commensurate with that allotted in the timetable. the effect of this factor was corroborated in nox’s own words. during the interview, she highlighted that ideally she would like to see her learners make sense of equivalent fractions. in other words, she indicated that she would like to see her learners build on what has been said by the previous speaker to increase the coherence of the exchanges. however, such exchanges were few and far between, because ‘if i do it, yeah … i’ll [be] behind the schedule’. the finding here is that she attributed her approach to the teaching of equivalent fractions to the limited time; the curriculum was congested to cover the prescribed content. nox pointed out that she had to resort to teaching methods that offer little in terms of meaningful learning of equivalent fractions. she acknowledged that learning equivalent fractions in arbitrary fashions undermines the connected nature of mathematics concepts. interview results sustained the inference made in the observation data that nox’s teaching approach mimicked transmission teaching. however, what emerged from the interview was that nox’s choice of teaching approach was not made with reckless disregard for meaningful learning; school environmental circumstances contributed. in sum, analysis of both classroom observation data and interview transcript call for teaching approaches that position learners’ ideas at the centre of instruction in line with current reform initiatives in curriculum documents (common core state standards initiative [ccssi], 2010; department of basic education [dbe], 2011). meaningful learning of equivalent fractions (at the grade 4 level) is important because it is a topic needed for all mathematical operations (addition, subtraction, division, and multiplication). however, personal experiences supported by research, for example davis and maher (1997), suggest that the main objectives of teachers working from the ‘transmission’ model of instruction (i.e. question-answer and listen-tell routines) have been computational and definitional, rather than pursuits of meaningful learning. teachers of young children (i.e. children in the foundation phase, namely reception year to grade 3) are often trained to teach a wide range of subjects, which leaves little room to develop mathematical knowledge (newton & newton, 2007). as a consequence, these teachers tend to avoid ‘conversational risk’ and instead focus on facts, routines, and right answers at the expense of pressing for understanding of the reasons underpinning these (carlsen, 1991). in this regard, i recommend that initial teacher training programmes make dialogic teaching an explicit outcome in their course; or it will never be learned. limitation of the study as is often the case in any research, one limitation should be borne in mind when interpreting the results of this study. learner-learner interactions were not analysed. this was not designed to discount learners’ voices; learners’ talk is a crucial part of a pedagogical event. this limitation suggests an opportunity for further research. future research may incorporate this component of classroom interactions to better understand how learners view the teaching approach adopted by early career teachers in equivalent fractions. conclusion in this article, i described how the teaching approach adopted by the focal teacher, nox, made use of talk to guide learners to think and talk about equivalent fractions. the study was framed by the concept of ‘dialogic teaching’, described here as a pedagogic approach underpinned by specific features enacted through a range of possible talk strategies. analysis of both observational and interview transcripts revealed that her teaching approach could be classified as following the irf format, despite her attempts to elicit learners’ ideas on equivalent fractions. interview responses confirmed her allegiance to this format. however, her reasons for adopting this approach were found to be reasonable, given the work environment of a daily grind, in which she practised her craft. professional development programmes can mitigate these obstacles by capacitating early career teachers with skills as they navigate their way in a teaching and learning environment such as that in which nox found herself. the complexity of teaching notwithstanding, future studies must investigate the impact of the kind of environment in which early career teachers work as they form their styles and strategies of teaching. in particular, such studies may focus on the effect of the environment on early career teachers’ ability to engage learners in meaningful learning, on a large scale (i.e. using survey methods). acknowledgements i am particularly indebted to the grade 4 early career teacher and her learners for opening their classroom and being generous with their time. competing interests the author declares that he has no financial or personal relationships that may have inappropriately influenced him in writing this article. author’s contributions the corresponding author worked the manuscript from concept right up to its conclusion. funding information this work was supported in part by the grant received from the university capacity development programme (ucdp) at ukzn. data availability the data that support the findings of this study are available from the corresponding author upon reasonable request. disclaimer any opinions, findings, and conclusions or recommendations expressed in this article are those of the author and do not necessarily reflect the views of the ucdp. references alexander, r.j. 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(2010). equivalent fractions: developing a pathway of students’ acquisition of knowledge and understanding. in b. kissane & c. hurst (eds.), proceedings of the 33rd annual conference of the mathematics education research group of australasia (pp. 673–680). fremantle: merga. footnotes 1. i follow hiebert and grouws (2007) in defining teaching as ‘classroom interactions among teachers and students around content directed toward facilitating students’ achievement of learning goals’ (p. 377). 2. by mathematical practice is meant mathematical activity involving plausible (inductive) reasoning, through which conjectures are generated, and demonstrative (deductive) reasoning that is formalised in proof (reid, 2002). 3. paraphrasing watson and mason (2005), i take ‘pedagogical tool’, in the context of mathematics education, to mean what a teacher uses as a ‘window’ into a learner’s mind, which includes items such as worksheets, textbooks, handouts, manipulatives, technology, mobile device applications, etc. 4. i join arbaugh and benbunan-fich (2006) in defining teaching approach as ‘the extent to which the instructor is the focus of the educational process and the extent to which the instructor relies upon individually or group-oriented activities’ (p. 435). 5. figure 3 is a synthesis of ideas and work adapted from marmur, yan and zazkis (2020) and pedersen and bjerre (2021). 66 p3-13 brodie final pythagoras 66, december, 2007, pp. 3-13 3 dialogue in mathematics classrooms: beyond question-andanswer methods karin brodie school of education, wits university email: karin.brodie@wits.ac.za this paper explores different kinds of interaction observed in south african mathematics classrooms in order to unpack the notion of participation in mathematics learning. it argues that conventional questionand-answer methods do not promote the kind of interaction that the new south african curriculum calls for. it presents more appropriate kinds of interactions, where teachers maintain high task demands, respond to genuine learner questions and support conversations among learners. the paper argues that combinations of different kinds of interaction are most likely to support learner participation and mathematical thinking in classrooms. introduction the new curriculum in south african schools calls for learners to participate in mathematics lessons and to express their mathematical ideas. learner talk is seen to be important because it (i) shows that learners are attending to the lesson; (ii) allows learners to express and clarify their own ideas; (iii)_enables learners to share ideas with each other; and (iv) provides teachers with information about what learners know and don’t know, and how learners are thinking and trying to make sense of ideas. teachers are encouraged to make their lessons more learner-centred by encouraging learners to contribute to the lesson. however, teachers are given very little guidance as to what such participation actually might look like. the new curriculum encourages teachers to be facilitators, although what this means, beyond “not telling” (chazan & ball, 1999), is not often elaborated (see brodie, 2003, for some ideas on this). many teachers believe that if they ask questions and learners provide answers, learners are participating in the lesson. however classroom research has identified a number of different interaction patterns, which, to varying degrees, are supportive of genuine mathematical thinking. this research shows that many questionand-answer exchanges are not helpful in developing learners’ mathematical thinking. drawing on this literature and on research with grade 10 and 11 mathematics teachers in johannesburg, i identify a number of different kinds of classroom dialogue. i show that there is a range of possibilities for learner-centred interaction in mathematics classrooms, which take us beyond the idea that teachers can encourage genuine dialogue and learner participation merely by asking questions. classroom discourse about 30 years ago sinclair and coulthard (1975) and mehan (1979) identified a key structure of classroom discourse, the initiation-responsefeedback/evaluation (irf/e) exchange structure. the teacher makes an initiation move, a learner responds, the teacher provides feedback or evaluates the learner response and then moves on to a new initiation. mehan calls this basic structure a sequence. often, the feedback/evaluation and subsequent initiation moves are combined into one turn, and sometimes the feedback/evaluation is absent or implicit. this gives rise to an extended sequence of initiation-response pairs, where the repeated initiation works to achieve the response the teacher is looking for. when this response is achieved, the teacher positively evaluates the response and the extended sequence ends. neither sinclair and coulthard (1975) nor mehan (1979) evaluated the consequences of the irf/e structure. other researchers (edwards & mercer, 1987; nystrand, gamoran, kachur & prendergast, 1997; wells, 1999) have argued that it may have both positive and negative consequences for learning. although this structure requires a learner contribution at every other turn (the response move), and therefore apparently gives learners time to talk, much research has shown that because teachers tend to ask questions to which they already know the answers (edwards & mercer, 1987) and to ‘funnel’ learners’ responses toward the answers that they want (bauersfeld, 1980), space for genuine learner contributions is limited. for this reason, some researchers suggest dialogue in mathematics classrooms: beyond question-and-answer methods 4 that a complete shift of the ire structure is necessary to achieve the goals of learner engagement and inquiry. classroom discussions should become more like conversations, with the teacher being a participant in similar ways to the learners (davis, 1997). however, there are enormous challenges involved in creating conversations in classrooms (brodie, 2007) and it can be difficult for teachers to move away from the ire exchange structure (wells, 1999). so, in trying to understand different possibilities for interaction, it is important to try to understand the benefits that the ire can afford. whether the ire has positive or negative consequences for learning will most likely depend on the nature of the elicitation and evaluation moves, which in turn influence the depth and extent of learners’ responses. initiation moves are often in the form of questions, and a number of researchers have focused on teacher questions. nystrand et al. (1997) distinguish between “test” questions and “authentic” questions. “test” questions aim to find out what learners know, and how closely their responses correspond to what the teacher requires. “authentic” questions on, the other hand, are questions which do not have pre-specified answers, which convey the teacher’s interest in what learners think, and which serve to validate learner ideas and bring them into the lesson. researchers in mathematics classrooms have identified a broader range of questions. hiebert and wearne (1993) have four categories: recall; describe strategy; generate problem; and explain underlying features. boaler and brodie (2004) have nine categories which include: getting information; probing; exploring concepts and relationships; and generating discussions. both these studies compare teachers using traditional and new curriculum materials in the united states and show that although the teachers using new curriculum materials ask a significant number of questions that require information and recall, they also ask a broader range of questions than teachers using traditional curriculum materials, and they ask more questions that require conceptual engagement from learners. while some of these questions may or may not be authentic, questions that require learners to explore meaning and relationships help distinguish between different kinds of teaching, and have positive influences on learning (boaler & brodie, 2004; hiebert & wearne, 1993). at the level of the feedback or evaluation move, researchers have shown that teachers often begin with more exploratory, higher-order questions and tasks, but that teacher and learners often collude to reduce the demands of the task, asking narrower questions (stein, grover & henningsen, 1996) and funnelling towards answers (bauersfeld, 1980). nystrand et al. (1997) distinguish high-level evaluations from the more conventional evaluations of the ire structure. high-level evaluations endorse the importance, rather than the correctness, of a learner’s response, and allow the contribution to modify or affect the course of the discussion in some way (nystrand & gamoran, 1991). nystrand and gamoran also develop the notion of uptake, which they describe as follows: many of the teacher’s questions are partly shaped by what immediately precedes them; the teacher takes the learners’ ideas seriously, and encourages and builds on them in subsequent discussion; the teacher’s next question is contingent on the learner’s idea, rather than predetermined; the teacher picks up on learner ideas, “weaving them into the fabric of an unfolding exchange”; and the learner’s ideas can change the course of the discussion. drawing on the above literature, i identify three different kinds of interaction that i observed in south african mathematics classrooms and i argue that each of these kinds of interaction can create genuine dialogue through genuine learner participation and teacher responsiveness. i begin by giving an example of the traditional questionand-answer method, and demonstrate how it might inhibit learner participation and thinking. example 1: questions and answers one of the main reasons why teacher questions are considered insufficient to ensure genuine participation and mathematical thinking is that teachers often ask very low-level questions (boaler & brodie, 2004; hiebert & wearne, 1993; jina, 2007) and they ask questions to which they already know the answers (edwards & mercer, 1987; nystrand & gamoran, 1991). edwards and mercer show that curricula that encourage learner-centred teaching can put teachers in a double-binding situation, if teachers interpret such an approach as requiring them to get the required knowledge from learners. in this situation, teachers are in the unenviable position of having to ask questions of learners to get particular knowledge from learners into the public domain. if one learner doesn’t know a particular answer, teachers then ask other learners, in the hope that someone will provide the expected response (edwards & mercer, 1987; lobato, clarke, & ellis, 2005). teachers continue to ask questions, progressively narrowing their questions karin brodie 5 until such a simple question is asked that a learner can provide it. this is the process identified as ‘funnelling’ by bauersfeld (1980). the following extract from a grade 11 classroom gives an example of funnelling. the class was working on the question: what changes and what stays the same when the graph of y = x2 shifts 4 units to the left and becomes y = (x+4)2. one learner claimed that the axes of symmetry of the two graphs are different and the teacher asked him why they are different. the learner answered that it is because the equations are different and the following exchange ensued: 27 reagile1: the equations are different. 28 teacher: the equations are different? 29 reagile: (nods head) 30 teacher: the equations are different? as long as you can have a difference in the equations, then they differ? 31 reagile: i think so. 32 teacher: okay, let’s say we have something like y equals... one was x plus four squared, what if we have another one, y equals minus x plus four squared [writes y = (x+4)2, and y = –(x+4)2 on board]. are these equations the same or different? 33 learners: they are the same. 34 teacher: they are the same? 35 reagile: ee [yes] the value of ‘a’, whether it’s negative or positive determines the shape of the graph (indicating with his hand). 36 teacher: the value of? 37 reagile: ‘a’ determines... okay, the value of ‘a’ determines the shape of the graph, so... (inaudible, indicating with his hand) 38 teacher: so the value of ‘a’ determines the shape of the graph. 39 reagile: yes. 40 teacher: so, but are you saying the equations are the same? 41 reagile: (thinks, looks uncertain) yes. 42 teacher: they are the same, if you say they are the same, you simply mean when we substitute our values of x here and here, if we say x is two here and here (points to the two equations on the board) and you simplify, you will come to the same expression, when we simplify. is that what you are saying? 1 all learner names are pseudonyms. 43 reagile: (inaudible) 44 teacher: okay, lets look at two and negative two (writes 2 and –2 on the board), are they the same? 45 reagile: no. 46 teacher: why no? 47 reagile: the other one is negative. (inaudible). 48 teacher: the other one is negative, the other one is positive. what about those equations? in this exchange the teacher challenged the learner’s argument that because the equations are different, the axes of symmetry will be different. from the exchange it is evident that the teacher wanted the learners to think beyond the particular case that they were dealing with [y = x2 and y = (x+4)2], where the equations are different, and to think about whether different equations always produce different axes of symmetry. the teacher’s question about the two equations, y = (x+4)2 and y = –(x+4)2 is a case of the teacher raising the task demands, rather than lowering them (stein, smith, henningsen, & silver, 2000). however, the teacher was so intent on getting the learners to see that different equations do not necessarily generate different axes of symmetry that he ignored reagile’s thinking and narrowed his own questions. some of the teacher’s questions indicate that he does not agree with the learner (and hence the learner is probably wrong). for example in lines 28 and 30, the teacher repeated the learner’s response as a question “the equations are different?” with a tone that suggested disagreement. similarly in lines 34 and 40, the teacher indicated disagreement that the equations on the board are the same. these are examples of a teacher asking questions to which he already knows the answers (edwards & mercer, 1987; nystrand & gamoran, 1991), and that when teachers repeat questions it usually means that the previous responses were incorrect and the teacher is looking for a different answer (edwards & mercer, 1987). an analysis of reagile’s claims in the above exchange suggests that he was arguing that the two equations essentially are the same, especially when you consider the axis of symmetry. the only difference in the two equations is the a-value, which determines the shape of the graph rather than the axis of symmetry. reagile made this argument even in the face of the teacher’s disagreement, suggesting that he was convinced of his position. although the teacher listened to reagile and repeated his contribution in line 38, he dialogue in mathematics classrooms: beyond question-and-answer methods 6 did not see it as a contribution to the more general question about differences in the axes of symmetry and so he ignored the gist of reagile’s argument. as the exchange progressed, the teacher narrowed his questions in a number of ways. by focusing attention on the features of the two equations, y = (x+4)2 and y = –(x+4)2, he removed attention from their relationship to the graphs, a move that reagile resisted by continuing to focus on the relationships between the equation and the graph. the teacher then went further by saying that they should merely consider substituting numbers into the equations. finally, he narrowed the question to whether 2 and –2 are the same or different, a question that is obvious to any grade 11 learner, and that is so simple that it completely lost its relevance and context in relation to the original task. in the exchange reagile was reduced to answering a simple question, rather than having his genuine mathematical thinking taken seriously. the above is an exchange between a very good teacher and a very strong learner that went somewhat wrong. in this exchange the teacher tried to raise the level of the task, requiring more generalization. he had the knowledge to do so and was able to put up the two equations y=(x+4)2 and y = –(x+4)2 very quickly as an example of his own thinking. in reflecting on the incident, the teacher said that he was shocked by reagile’s apparent misconception, given reagile’s strength in mathematics, which might have been a reason for his narrowing of the questions. the fact that reagile was able to articulate an argument and maintain a different position from the teacher, suggests that there is space for thinking in this class. however, in this case, the teacher was listening evaluatively and looking for a particular answer (davis, 1997), which prevented him from seeing the strengths of the learner’s thinking. the teacher eventually narrowed the task completely. the fact that even very good teachers can end up funnelling exchanges and narrowing the task suggests that examples and models of how to avoid this situation might be helpful. in the rest of this paper, i give three such examples. example 2: reversing the ire this example comes from a grade 11 lesson in which learners asked a number of questions and the teacher responded to them. in an extended exchange of about 120 turns there were 15 learner questions, averaging 1 learner question every 8 turns for about a quarter of the lesson. this is unusual, since in many classrooms learners do not ask questions regularly. extended learner questioning created a situation where the teacher was responding much of the time to learner ideas and he strategically used his responses to make teaching points. viewing the video footage of the whole sequence gives a sense of a different kind of classroom, one that is more learner-directed because learners actually initiate many of the ideas. space limitations do not allow me to show the whole sequence but i will try to convey some of the atmosphere of the dialogue by analysing a short part of the exchange here. the lesson concerned the theorem, ‘the angle subtended at the centre of a circle is twice the angle subtended at the circumference’. the learners had drawn a number of circles with angles at the centre and circumference subtended by the same arc, measured the angles and seen the relationship empirically. the teacher pushed them to articulate the relationship in appropriate mathematical terminology and they struggled, particularly with the word ‘subtend’ and with the concept that the two angles, one at the centre and one at the circumference, were subtended by the same arc. when learners could refer to particular angles by name, they could articulate the theorem, however when pushed by the teacher to articulate it more generally, without naming angles, learners struggled. the teacher spent about half the lesson on this issue, working with the ire structure, pushing learners to articulate the theorem more appropriately. in the second part of the lesson, the teacher asked a learner to put up a diagram she had drawn where a diameter subtended a ninety-degree angle at the circumference. this allowed the teacher to help the learners further to articulate the theorem and to grapple some more with the relationship between the angle at the centre and at the circumference. as they were talking about this diagram, a number of issues came up for the learners and they raised them as questions, which the teacher responded to. here is an extract from the extended exchange: 301 gavin: sir, couldn’t b be a diameter as well?2 302 teacher: couldn’t? 303 gavin: ... b be a diameter? 304 teacher: can b be a diameter? 305 learner: no, it’s not going through the centre. 2 it is not clear to the researcher which point the learner was referring to here. karin brodie 7 306 teacher: it’s not going through the centre of the circle, okay (gavin nods). david? 307 david: sir, can you say that the angle subtended from the diameter is always ninety degrees? 308 teacher: okay, david is just saying that the angle, which is subtended by the diameter is always ninety degrees. can you see there, can you see this diameter with a line over there, it subtends one angle there, this diameter, subtends one angle there (he shows how the chord subtends the different angles) so what are we basically saying here now, it’s not just arcs that subtends angles, but a chord can also subtend an angle. let’s look at that. 309 learner: are they always equal to ninety? 310 teacher: he says it’s always equal to ninety. why are they always equal to ninety? 311 david: because they’ll always be one eighty. 312 teacher: because the diameter, the angle, in other words that is formed by the diameter, is always 180. if that is always 180 then the angle at the circle like you guys rightly said will always be 90. 313 learners: aah 314 teacher: but can you see, we can see this bc as a chord, which is your diameter. so what we saying is the angle which is subtended by the diameter is always equal to ninety, is everybody getting that? 315 learners: yes. 316 teacher: where must that angle be, however? david, d’you mind being more specific? 317 david: it must be on the arc. 318 teacher: it must be? 319 david: on the arc. 320 teacher: now you are saying that any angle subtended by the diameter is always equal to? 321 david: on the circumference 322 teacher: the angle must be on the circumference of the circle okay 323 nathan: (inaudible) 324 teacher: i don’t understand what you’re saying, just say it again. 325 nathan: sir you said it must be on the chord, sir, but it is straight sir, it’s touching both sides (stretches out arms to indicate straight angle). 326 teacher: okay. no i fully agree with you nathan, it must be the diameter, you are saying that it must be subtended by the diameter, in other words this time, this time it’s the diameter that is opposite the angle, it’s the diameter that determines the size of the angle, okay? can you see that there, the same link those two points and the diameter? 327 nathan: so if it was changed sir, will the others still equal whatever? 328 teacher: what do you mean it was changed? 329 nathan: the diameter, if it was changed, say if it was seventy five or something. in the above exchange of 29 turns there are 5 learner questions. the teacher dealt with the first question (line 301) relatively quickly by getting a response from another learner that satisfied gavin. at this point, david’s hand was up and the teacher moved on to his question (line 307). the teacher responded to david’s question by revoicing it for the class – articulating it more clearly so that all could hear it and adding the weight of his voice, that it was an important question to consider (o’connor & michaels, 1996). as the teacher did this, another learner repeated the question (line 309). the teacher asked david to justify the claim in his question, which he did, with reference to the angle at the centre being 180 degrees. again, the teacher repeated david’s point and then asked another question, which related to some of the difficulties the class had been having earlier. from lines 316 to line 322, the teacher used a traditional ire sequence to get david to see that the angle should be on the circumference. then nathan came in with two questions (lines 325 and 327) which he did not articulate clearly and which the teacher spent some time trying to clarify. exchanges like this are both learnerand teacher-directed. they are learner-directed in that the learners’ questions drive the dialogue and the teacher is willing to divert from his teaching agenda somewhat to listen to and work with learners’ questions. they are teacher-directed in that the teacher is still in control; he decides to allow the learners to ask the questions and he chooses how to respond to them. in this case, the teacher had a number of ways of responding to the dialogue in mathematics classrooms: beyond question-and-answer methods 8 questions: (i) he revoiced them, so that others could contribute to answering them; (ii) he asked for justifications; (iii) he worked to clarify questions that were unclear; and (iv) he used the opportunities the questions presented to make some teaching points and reverted to traditional ire mode to do so. the first three of these responses correspond to higher-level evaluations and uptake, where the teacher shows interest in the question, takes it seriously and makes it part of the lesson (nystrand & gamoran, 1991). the fourth response, using the learners’ questions to make teaching points is characterized by a traditional ire structure, although in this case there was no funnelling. the first three responses are characterized by what can be called a reverse ire – learners initiating and the teacher responding. this reverse ire is not a complete reversal, because the teacher still teaches. he does not respond to the learner questions with answers only, as they would to him. rather he chooses from a range of teaching moves, which sometimes shift him back to a traditional ire. nevertheless, the interaction during the reversal is much more responsive and there is more real dialogue between teacher and learners. so although this exchange interrupts the traditional ire in some ways, it also maintains it. this kind of exchange goes beyond traditional question-and-answer exchanges in that there is no narrowing of the task demands, as in the previous example, and there is definitely a sense of a teacher in dialogue with his learners, while still teaching. the following example shows learners talking to each other. example 3: learner-learner dialogue this example comes from the same grade 11 classroom discussed in example 1 where the teacher funnelled his questions. in this case, the teacher supported an extended dialogue between two learners. the learners were talking about what is similar in the graphs of y = x2; y = (x+4)2 and y = (x–3)2. 149 mamokete3: oh, they are similar in... why i am saying they are similar in the y-values, we don’t have the value of q there, it shows that if it is not there, it is zero, that value of q, that is why they are the same throughout. 3 much of the talk in this lesson was in setswana. the videotapes were transcribed in the original setswana and then translated into english. 150 teacher: questions, comments? mapula? 151 mapula: which y-value, where is the y-value? for what? y-value of which point? 152 mamokete: for the turning point. 153 mapula: only? 154 mamokete: what do you mean? 155 learners: (laugh) 156 mapula: it means only they are similar. you say they are similar in yvalues, don’t you? 157 mamokete: yes. 158 mapula: so, i am asking that, you are implying that its y-value is zero? 159 mamokete: yes. 160 mapula: for the turning point? 161 mamokete: yes. 162 mapula: oh, what about there, our yvalue is not the same. 163 mamokete: the other y-value? 164 mapula: for the other points (inaudible) on this graph, that lies on the graph, the one on the graph, are they not the same? (she is pointing at the sheet) 165 mamokete: they are the same, these graphs move to left and right, so there is no way that they cannot be the same. 166 mapula: oh. 167 teacher: do you understand her question? 168 mamokete: yes. 169 teacher: what is she saying? 170 mamokete: she says i am implying that at the other points, besides the turning point, the y-value is not the same, and i said they are the same. (aganang raises hand) 171 teacher: mm. 172 aganang: but that other time you said that since there is no q, it means then that the y-value is zero, but on the other points. (she is pointing at the board) 173 mamokete: (interrupts aganang) we are talking about the turning point, i am talking about the turning point. in the exchange mapula was pushing mamokete to be specific about which points she was claiming had the same y-values. initially mamokete was referring to the turning points only; she spoke about the q value being zero in all three graphs and in line 152 she explicitly said she was talking karin brodie 9 about the turning points. however, through mapula’s questioning, particularly in line 164, she seemed to shift her view, saying that since the graphs moved horizontally, all the y-values (of corresponding points) would stay the same. this indicates a shift in her thinking, made through the conversation. however, when aganang challenged her by saying that she was contradicting an earlier point, it seemed that mamokete might shift back to her earlier position. such shifting of positions is characteristic of genuine dialogue and suggests that the learner is thinking through her ideas far more than a learner who tries to provide an answer that she thinks the teacher wants to hear. through this interaction, the girls were exploring the nature of the graphs and their relationships to the equations. they took seriously the roles of asking and answering questions to clarify each other’s thinking, and were taking up each other’s ideas in ways that teachers rarely do (nystrand et al, 1997). we can raise the question as to whether the two girls, mapula and aganang, were asking authentic questions, or whether they actually knew the answers to their questions but were pointing out contradictions and pushing their classmate to clarify her own ideas. we can’t know the definite answer to this, but from the questioners’ confidence and persistence and the actual content of their questions (for example lines 153 and 162), it seems that they did have an idea of potential answers to their questions and were pushing mamokete to think more deeply. also, the learners had worked on the task in groups the previous day, and so would have thought through the ideas and had some confidence in their own positions. in this case they were most likely modelling the way their own teacher asks questions (molefe, 2003). so while they were certainly evaluating mamokete’s ideas (see for example lines 153, 162, 172), their evaluations did not serve to narrow or funnel the task but allowed mamokete to clarify her thinking and to disagree with them (lines 165, 170). although the girls work in initiation-response turns and the exchange may look like an ire in form, their questions and evaluations play a different role from those in the traditional ire. so here we see an ire in form but not in function; the interaction is closer to a real dialogue. the teacher took only three turns in this exchange (lines 150, 167, 171), which is very unusual in mathematics classrooms. this teacher supported similar exchanges a number of times in his classroom. in this case his three turns did not make any substantial mathematical contributions but rather were directed at getting learners to talk and listen to each other. the first opened the floor for contributions, while the second and the third intervened to ensure that the learners were not miscommunicating. the teacher’s third contribution supported mamokete to repeat the point that she had learned through the conversation, which then allowed aganang to come in, suggesting a contradiction with mamokete’s earlier position. this example is probably closest to the notion of ‘facilitator’ promoted in the new south african curriculum documents. the teacher facilitated in this way a number of times during two weeks of lesson observations. at the same time, he also narrowed and funnelled his questions, as shown in the first example. this suggests that no teacher will consistently use one style of teaching and may vary between different approaches. it also raises the question of how the teacher worked in order to get the learners to be able to interact like this. this does not happen easily and requires a lot of work early in the year to establish the norms and ground rules for interacting in this way (boaler & humphreys, 2005; lampert, 2001; staples, 2004). the foregoing interaction is predominantly between two learners. the next example shows a number of learners interacting around a controversial mathematical point. example 4: whole-class dialogue this example comes from a grade 10 classroom where a learner had asked why –2 x 0 gives 0 and not –0, since a negative times a positive should give a negative. the teacher recognized this as an interesting question and allowed learners to discuss the issue: 152 teacher: eh, we can write negative zero, what d’you want to say victor? 153 victor: no, sir, zero is neutral, sir. 154 teacher: zero is neutral. what neutral? we are not driving a car here, ne? 155 victor: zero can be positive, zero can be negative, sir. 156 teacher: hah? is a zero positive or negative? 157 learners: (mutter inaudibly) both. 158 teacher: yes, grant, let’s listen to grant. come, come, let’s listen. 159 grant: sir, on a number line you won’t find a negative zero or positive zero. its just gonna be zero because it’s in between all those numbers. (inaudible) 160 teacher: so is there a difference if i write negative zero or positive zero? dialogue in mathematics classrooms: beyond question-and-answer methods 10 161 learners: no. it’s the same. 162 teacher: it’s the same thing? 163 learners: yes. 164 teacher: then, why do we write it just as zero? why don’t i write negative? 165 learner: because zero it’s…. 166 teacher: yes, fred. 167 learner: because zero, sir, it’s nothing, sir. 168 teacher: nothing. 169 fred: yes. 170 emily: sir, zero is just like x because sometimes it’s positive and sometimes it’s negative. 171 teacher: no. if i write x, if i write x, if i write x like this, is it positive x, or negative x? 172 learners: positive. 173 teacher: so if i write zero like this is it positive zero or negative zero? 174 learners: positive, negative. 175 teacher: lebo? 176 lebo: i ... positive. 177 teacher: this is positive zero, lebo? 178 lebo: ja, it is, because there’s no sign. 179 teacher: positive zero. 180 lebo: but, sir, my question still remains. 181 teacher: so, let’s, let’s write negative zero. 182 learner: ja. 183 teacher: must we write negative zero? 184 learners: no. 185 teacher: why not man? 186 lebo: why, why? 187 fred: no, zero, it’s nothing sir. you just give it a value…. 188 lebo: you can’t say... i disagree with fred, you can’t say zero is nothing. what’s the aim of writing it if it’s nothing, sir? 189 emily: zero, it’s a zero, it’s not a number (learners laughing). serious, sir. sir, you said x can represent any number but zero you can see it’s a nought sir. 190 teacher: ok, let’s get finality here. 191 lebo: if they say, if they say, zero it’s nothing. why did you, em, you give us an example like, zero times zero plus one. if zero it’s nothing, why did you say zero times zero? in the above extract of forty turns, five learners made contributions. all of these contributions were sensible and useful and many of them responded to and built on previous contributions. the learners were grappling with important mathematical ideas: is zero a number; what does the sign in front of a number indicate; and making links between their numerical and algebraic knowledge. they were also engaged in the discussion, because they found the issue interesting and had contributions to make. the role of the teacher in this discussion is interesting. i argue elsewhere that allowing for such a conversation is an important move, even though it diverted the class from the main issue they were discussing (brodie, 2007). the teacher clearly recognized the question as useful and thought it important to support a conversation around this. in contrast to the previous example, the teacher made many contributions to this conversation, almost every turn. some of his contributions commented explicitly on the mathematics, for example in line 154, when he implied that they can’t use the word neutral to talk about a number. in other cases he repeated learner contributions (lines 162, 168, 171, 177) in a “neutral” tone, allowing the learner to re-iterate their position. in others, he repeated the question under discussion (156, 160, 181, 185). in lines 181 and 185, the teacher actually suggested that they can write negative zero and, together with lebo (line 186), challenged the other learners as to why they were claiming that they could not. although the teacher remained neutral in some of his moves, in others he suggested a position, not necessarily his own position but one that would provoke and challenge the learners. he asked questions to which he knew the answers, but these questions served to elicit genuine learner thinking. none of his contributions narrowed the task, he did not funnel learners to an answer, and he took their ideas seriously, both lebo’s, whose question started the discussion, and the ideas of others who contributed (except possibly for victor’s). so even though this teacher takes almost half the turns in the discussion, learners get to express their views, hear others and build on each other’s ideas. discussion and conclusions in this paper i have built on existing literature which argues that merely engaging in questionand-answer exchanges does not necessarily allow for genuine learner participation in the lesson, nor takes learners’ mathematical thinking seriously. to illustrate this point, i used an example (example 1) of a teacher funnelling a learner through narrowing karin brodie 11 his questions, even when the learner was making a valid mathematical argument. my main argument is that there are other ways of interacting that support more genuine participation and thinking. i gave three examples (examples 2, 3 and 4) of such interaction that i have encountered in south african classrooms – and i believe that there are more waiting to be described. these three examples have a number of differences and similarities, which i will discuss below. my analysis of these examples contributes some interesting insights to the already existing literature on classroom interactions. first it shows that the ire exchange structure can be shifted, as in examples 2 and 3. the shift happened in different ways in the two examples. in example 2, the ire was partially reversed and the teacher had to respond to a number of learner questions. in example 3, learners interacted in initiationresponse turns, and their evaluations worked differently from those of a teacher, serving to push one of their peers to think more deeply. so this exchange was more like a dialogue. example 4 shows that the ire does not need to shift in order for dialogue to take place. in this case the teacher contributed in almost every other turn and many of his comments were evaluative or could be read as such by the learners, but he still managed to support a conversation. another important insight that emerges from this analysis is that even when teachers manage to shift the ire structure, they are unlikely to be able to do this all of the time. in example 2, we see how in responding to learners’ questions, the teacher switched into an ire mode in order to make some teaching points, and then switched back to a more responsive interaction pattern. it is likely that such switching between modes is appropriate as a teaching technique, both in terms of supporting participation and learning, and also in terms of expectations for teacher change. it is a far less daunting task to change some aspects of one’s practices and interaction patterns, rather than having to let go of everything. the first teacher is interesting in this regard because he is the teacher in the study who managed most to support learners to question each other. this teacher had been working on this practice for some years before i visited his classroom and was very successful in doing it. at the same time he also funnelled and narrowed his questions (as did the other two teachers). this adds to the point that teachers may be able to make some changes at some times in their teaching, but at other times may revert to more conventional methods. the processes of changing teaching practice are slow and uneven (see also slonimsky & brodie, 2007). this analysis also addresses some issues in relation to the kinds of questions that support learner participation. nystrand et al. (1997) argue that such questions should be authentic. however, in these three cases, we see a mix of authentic and non-authentic questions. in example 2, the learners’ questions were authentic, but the teacher’s were not. in example 3, the learners’ questions may or may not have been authentic. in example 4, there were very few authentic questions from the teacher, although lebo’s question that started the discussion was authentic. and so it seems that while authentic questions are part of creating genuine dialogue, they may not be sufficient, on their own. it may be the case that some authentic questions are necessary but that switching between different kinds of questions is the most helpful in supporting genuine participation. what the three examples of dialogue have in common – and what is different from the first case – is that the teachers did not narrow the task demands and did not funnel learners towards an answer. it may be that the insights of bauersfeld (1980) and stein et al. (2000) are key here, and that to increase participation and mathematical thinking, teachers need to learn to maintain the task demands. this is not an easy task as stein et al. (1996; 2000) show, because time constraints and teachers’ genuine desire to help learners serve to lower task demands. a case study of curriculum implementation in grade 10 in 2006 shows that a teacher who did select higher level tasks, as required by the new curriculum, was not able to maintain the level of the tasks during classroom interaction, and his interaction patterns served to narrow and funnel the tasks (jina, 2007; modau, 2007). this has important implications for teacher development. it might be that focusing on both the goal of selecting higher order tasks and working to maintain the level of the tasks in interaction with learners is a useful first step, which will allow teachers to focus on what to do, rather than on what not to do. finally, this study raises an important question: how did these three teachers manage to achieve genuine participation, at least for some of the time, in their lessons? this question is beyond the scope of this paper, and has been addressed somewhat by others (boaler & humphreys, 2005; chazan, 2000; lampert, 2001; staples, 2004), although in nonsouth african contexts. these studies describe how teachers do manage to create classroom dialogue in mathematics classrooms: beyond question-and-answer methods 12 cultures that support participation and learning. we need to do similar research to see how to achieve participation in the south african context. we know that such classrooms are not often seen – in south africa, the rest of africa, and beyond (brodie, lelliott, & davis, 2002; chisholm, volmink, ndhlovu, potenza, mahomed, muller, et al, 2000; cuban, 1993; sugrue, 1997; tabulawa, 1998; tatto, 1999). however, it is my view that when genuine participatory classrooms are created they can be liberating and empowering for both teacher and learners, as the above examples show. developing such classrooms is a goal that is well worth working towards. references bauersfeld, h. (1980). hidden dimensions in the so-called reality of a mathematics classroom. educational studies in mathematics, 11, 23-41. boaler, j. & brodie, k. (2004). the importance, nature and impact of teacher questions. in d.e. mcdougall & j.a. ross (eds.), proceedings of the 26th annual meeting of the north american chapter of the international group for the psychology of mathematics education (vol. 2, pp 773-781). toronto: ontario institute of studies in education/university of toronto. boaler, j. & humphreys, c. (2005). connecting mathematical ideas: middle school cases of teaching and learning. portsmouth, nh: heinemann. brodie, k. (2003). being a facilitator and a mediator in mathematics classrooms: a multidimensional task. in s. jaffer & l. burgess (eds.), proceedings of the 9th national congress of the association for mathematics education of south africa (pp 135-146). cape town: university of cape town. brodie, k. (2007). teaching with conversations: beginnings and endings. for the learning of mathematics, 27(1), 17-23. brodie, k., lelliott, t. & davis, h. (2002). forms and substance in learner-centred teaching: teachers’ take-up from an in-service programme in south africa. teaching and teacher education, 18, 541-559. chazan, d. (2000). beyond formulas in mathematics and teaching: dynamics of the high school algebra classroom. new york: teachers’ college press. chazan, d. & ball, d.l. (1999). beyond being told not to tell. for the learning of mathematics, 19(2), 2-10. chisholm, l., volmink, j., ndhlovu, t., potenza, e., mahomed, h., muller, j., et al. (2000). a south african curriculum for the twenty first century. report of the review committee on curriculum 2005. pretoria: department of education. cuban, l. (1993). how teachers taught: constancy and change in american classrooms. new york: teachers’ college press. davis, b. (1997). listening for differences: an evolving conception of mathematics teaching. journal for research in mathematics education, 28(3), 355-376. edwards, d. & mercer, n. (1987). common knowledge: the growth of understanding in the classroom. london: routledge. hiebert, j. & wearne, d. (1993). instructional tasks, classroom discourse, and students’ learning in second-grade arithmetic. american educational research journal, 30(2), 393-425. jina, z. (2007). teacher questioning and curriculum change. unpublished masters research report, university of the witwatersrand, johannesburg. lampert, m. (2001). teaching problems and the problems of teaching. new haven: yale university press. lobato, j., clarke, d. & ellis, a.b. (2005). initiating and eliciting in teaching: a reformulation of telling. journal for research in mathematics education, 36(2), 101-136. mehan, h. (1979). learning lessons: social organisation in the classroom. cambridge, ma: harvard university press. modau, a.s. (2007). a teacher’s selection and use of tasks in the old and new curriculum. unpublished masters research report, university of the witwatersrand, johannesburg. molefe, n. (2003). mathematical reasoning. unpublished honours research report, university of the witwatersrand, johannesburg. nystrand, m. & gamoran, a. (1991). student engagement: when recitation becomes conversation. in h.c. waxman & h.j. walberg (eds.), effective teaching: current research (pp 257276). berkeley, ca: mccutchan publishing corporation. nystrand, m., gamoran, a., kachur, r. & prendergast, c. (1997). opening dialogue. new york: teachers college press. o’connor, m.c. & michaels, s. (1996). shifting participant frameworks: orchestrating thinking practices in group discussion. in d. hicks (ed.), discourse, learning and schooling (pp 63-103). new york: cambridge university press. karin brodie 13 sinclair, j.m. & coulthard, r.m. (1975). towards an analysis of discourse: the english used by teachers and pupils. london: oxford university press. slonimsky, l. & brodie, k. (2007). teacher learning: development in and with social context. south african review of education, 12(1) 45-62. staples, m. (2004). developing a community of collaborative learners: reconfiguring roles, relationships, and practices in a high school mathematics classroom. unpublished phd thesis, stanford university, california. stein, m.k., grover, b.w. & henningsen, m.a. (1996). building student capacity for mathematical thinking and reasoning: an analysis of mathematical tasks used in reform classrooms. american educational research journal, 33(2), 455-488. stein, m.k., smith, m.s., henningsen, m.a. & silver, e.a. (2000). implementing standardsbased mathematics instruction: a casebook for professional development. new york: teachers college press. sugrue, c. (1997). complexities of teaching: childcentred perspectives. london: falmer press. tabulawa, r. (1998). teachers’ perspectives on classroom practice in botswana: implications for pedagogical change. international journal of qualitative studies in education, 11(2), 249268. tatto, m. (1999). improving teacher education in rural mexico: the challenges and tensions of constructivist reform. teaching and teacher education, 15, 15-35. wells, g. (1999). dialogic inquiry: toward a sociocultural practice and theory of education. cambridge: cambridge university press. “one of the big misapprehensions about mathematics that we perpetrate in our classrooms is that the teacher always seems to know the answer to any problem that is discussed. this gives students the idea that there is a book somewhere with all the right answers to all of the interesting questions, and that teachers know those answers. and if one could get hold of the book, one would have everything settled. that's so unlike the true nature of mathematics.” leon henkin article information author: craig pournara1 affiliation: 1wits school of education, university of witwatersrand, south africa correspondence to: craig pournara postal address: box 1531, pinegowrie 2013, south africa dates: received: 05 dec. 2013 accepted: 28 aug. 2014 published: 14 nov. 2014 how to cite this article: pournara, c. (2014). mathematics-for-teaching: insights from the case annuities. pythagoras, 35(1), art. #250, 12 pages. http://dx.doi.org/10.4102/ pythagoras.v35i1.250 copyright notice: © 2014. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. mathematics-for-teaching: insights from the case of annuities in this original research... open access • abstract • introduction • teachers’ mathematical knowledge for teaching • research on conceptions of annuities • a hierarchy of annuities concepts • an expanded view of the compound interest formula • two approaches to annuities    • deriving a formula for future value of annuity due using an account balance approach    • deriving a formula for future value of annuity due using an individual payment approach • dealing with a paradox in the modelling of time • different interpretations of the exponent in financial formulae • conclusion • acknowledgements    • competing interests • references • footnotes abstract top ↑ shulman’s notations of subject matter knowledge (smk) and pedagogical content knowledge (pck) have been very influential in education research on teachers’ knowledge for teaching. however, there is little empirical evidence in support of these as separate analytical constructs. furthermore, attempts to distinguish smk and pck highlight the complex and multidimensional nature of teachers’ knowledge and hence the difficulty of separating smk and pck. the author adopts the notion of mathematics-for-teaching (mft) and argues that teachers’ knowledge for teaching annuities comprises knowledge of mathematical aspects, knowledge of pedagogical aspects and contextual knowledge of finance. drawing from a larger study in which the author taught a financial mathematics course to pre-service secondary mathematics teachers, four examples of teachers’ knowledge for teaching annuities are identified, each of which illustrates how knowledge of mathematics, knowledge of pedagogy and contextual knowledge of finance are intertwined. introduction top ↑ the first time i taught annuities, it was to a group of pre-service secondary mathematics teachers. being a novice to financial mathematics i drew heavily on the content, presentation and sequencing of available texts (e.g. kitto et al., 1990; young, 1993). my attention was taken up by making sense of the mathematics and its relationship to the world of finance, so i was unable to pay much attention to the students and their learning. since that initial experience more than 10 years ago, i have taught annuities several times. each time i have become increasingly aware of the students, their interpretations of the mathematical ideas and their difficulties. this heightened awareness of the students and their learning has forced me to think more deeply about the concept of annuities and the knowledge needed to support university students and learners in schools to engage more meaningfully with annuities, going beyond definitions and calculations. ultimately this led to a study of teachers’ mathematical knowledge for teaching of financial mathematics (pournara, 2013a). in this article i elaborate four instances of mathematical knowledge for teaching annuities. i show how these involve mathematical knowledge, pedagogical knowledge and contextual knowledge of finance. i argue that this knowledge is not the substance of typical financial mathematics courses in commerce or actuarial science programmes, but that is it specialised knowledge that teachers require for teaching. in so doing, i challenge the simplistic distinctions between content knowledge and pedagogical content knowledge. the ideas presented here draw on data from a larger study of pre-service secondary mathematics teacher education in which i taught a course in financial mathematics to a group of 3rd year and 4th year bachelor of education students (pournara, 2013a). the course was specifically designed for teachers and adopted an investigative approach to studying annuities. for example, students engaged with realistic financial tasks in order to derive the annuities formulae. in analysing students’ responses to tasks, i was forced to consider more deeply my own conceptions of annuities. in the absence of an adequate literature base on conceptions of annuities, i had to consider from scratch questions like: ‘what does it take to learn annuities?’, ‘what difficulties do students experience in making sense of annuities?’ and ‘what knowledge do teachers need to teach annuities?’ teachers’ knowledge for teaching cannot be separated from the curriculum they will teach. in the south african school curriculum for mathematics (department of basic education, 2011; department of education, 2003) annuities are treated as an application of geometric series and therefore only introduced at grade 12 level, once the topic of sequences and series has been completed. consequently, there is little time for investigative approaches in the rush to complete the syllabus for the final year of schooling. furthermore, financial mathematics is a small section of the curriculum, being allocated only 7% of teaching time and approximately 6% of the marks in grade 12 national assessments. these constraints cannot be ignored in a discussion of the knowledge teachers need to teach annuities. however, my concern here is with teachers’ knowledge of annuities that will enable them to support learners to make sense of annuities in relation to the underlying mathematics and in relation to the realities of finance. in the next section i locate my perspective on teachers’ knowledge within the existing literature on teachers’ mathematical knowledge for teaching. teachers’ mathematical knowledge for teaching top ↑ the rise in interest in teachers’ subject matter knowledge can be linked to shulman’s (1986, 1987) seminal work in which he argues to re-insert knowledge of subject matter as a key component of teacher’s professional knowledge and to distinguish knowledge of the discipline from the knowledge required to transform that disciplinary knowledge for teaching. by distinguishing content knowledge (ck) or subject matter knowledge (smk) from pedagogical content knowledge (pck), he seeks to highlight two key elements of knowledge for teaching. ball, thames and phelps (2008) report that there has been unprecedented take-up of the notion of pck since the mid-eighties. for example, they count that in 2008 the abovementioned articles by shulman had been cited in more than 1200 refereed journal articles across 125 journals spread over a wide range of disciplines. however, they argue that the notion of pck has remained underdeveloped and its usefulness has been hampered because it lacks clear definition and has little empirical foundation. ruthven (2011) suggests that shulman’s taxonomy as well as the variations on the taxonomy that have followed (including ball’s work) ‘have mesmerised the field rather at the expense of the model of pedagogical reasoning that accompanied earlier accounts of the taxonomy’ (p. 86, italics mine). many in the mathematics education research community have been substantially influenced by shulman’s ideas. for example, adopting the smk-pck distinction, even (1990, 1993) elaborates a conception of smk, with particular focus on functions, and proposes seven aspects of teachers’ smk. however, huillet (2009) argues that four of even’s aspects blur the boundary between smk and pck. for example, she argues that the categories different representations and basic repertoire of key tasks and examples both relate to teaching practice and therefore inevitably involve pck as well as smk. in their attempt to extend shulman’s work, ball and her colleagues in the united states of america (ball, hill & bass, 2005; ball et al., 2008) propose additional sub-constructs within smk and pck. through a programme of extensive empirical research they have sought to provide evidence for the existence of some of these constructs and to measure teachers’ knowledge in relation to the constructs (e.g. hill, ball & schilling, 2008; hill, rowan & ball, 2005). however, to date their findings provide only limited evidence for distinguishing specialised content knowledge (sck) and knowledge of content and students (kcs). the coactiv group working in germany (e.g. krauss, baumert & blum, 2008; krauss et al., 2008) distinguished ck and pck empirically through tests of teachers’ knowledge. however, their ability to distinguish the constructs is highly dependent on their definitions of the two constructs. they define ck as advanced background knowledge of school mathematics, which goes beyond the requirements of the curriculum and enables the teacher ‘to cope with mathematically challenging situations’ (krauss, baumert & blum, 2008, p. 54). pck is defined as knowledge of explanations and representations, knowledge of students’ difficulties and errors and knowledge of multiple solutions to tasks. these definitions provide a ‘safe zone’ between ck and pck in that specialised knowledge relating to the core mathematical aspects of the curriculum is not included in either category. ball et al. (2008) argue that it is frequently difficult to distinguish clearly between sck and kcs. they cite the example of selecting tasks to test learners’ understanding of decimals, which involves both sck (drawing on the key mathematical ideas to produce a list of decimals to be ordered) and kcs (to consider examples that would give learners particular difficulty). the coactiv group avoid this blurring: whilst their definition of pck includes knowledge of learners’ difficulties, they appear to ignore knowledge of the key mathematical ideas in their definition of ck since the key ideas would not be considered advanced knowledge. given the difficulties in establishing productive yet clear boundaries between knowledge categories, and in producing empirical evidence for the existence of the proposed categories, i choose to avoid the smk-pck conundrum. i choose the term mathematics-for-teaching (mft) following adler (2005) and adler and davis (2006) to refer to teachers’ knowledge for teaching mathematics. i concur with many others in the field (e.g. ball, bass, & hill, 2004; ball et al., 2008; even, 1990, 1993; ferrini-mundy, floden, mccrory, burril & sandow, 2006; huillet, 2007, 2009) that teachers’ mathematical knowledge is complex and topic specific. i consider mft as an amalgam of smk and pck, in which fine distinctions between mathematical and pedagogical aspects are not a priority. in the context of mathematical knowledge for teaching financial mathematics, i propose three knowledge clusters: • aspects that are mainly mathematical – knowledge of essential features of a concept (even, 1990), different ways of approaching the concept (even, 1990, 1998), the relationship of the concept or topic to other areas of mathematics, applications and modelling and broader mathematical practices such as defining, conjecturing, exemplifying and proving. • aspects that are mainly pedagogical – knowledge of a basic repertoire of key tasks and examples (even, 1990), different teaching sequences and approaches (ferrini-mundy et al., 2006), explanations and learners’ conceptions. • aspects of contextual knowledge of finance – financial concepts and conventions, socioeconomic issues and financial literacy. it is this cluster of knowledge that provides insights into mft, which may not be visible when one focuses on areas such as algebra, calculus or geometry. in this article i focus on four examples of teachers’ knowledge for teaching annuities and argue that each illustrates why hard distinctions between smk and pck are unhelpful. in one sense these may each be considered as examples of sck, yet each has the learner in sight at all times, so in adler and davis’s (2006) terms they would be considered mt (primary focus on mathematics with secondary focus on teaching and pedagogy). the first example involves an expanded view of the compound interest formula, in which i argue that teachers are required to integrate knowledge of the formula, knowledge of learners’ conceptions of it and knowledge of how the formula is viewed in financial contexts beyond school. the second example deals with two different approaches to learning annuities. here i show how knowledge of mathematics, the curriculum and learners’ conceptions are intertwined. the third example concerns modelling of time in annuities scenarios and shows how knowledge of mathematical modelling, finance and students’ conceptions intersect. the final example involves different interpretations of the exponent in financial mathematics formulae. this example shows powerfully how knowledge of mathematics and learner conceptions intersect and challenges even the ‘distant definitions’ of ck and pck adopted in the coactiv work. these four issues are unlikely to be dealt with in a typical financial mathematics course in commerce or actuarial science programmes, yet they constitute important knowledge for teachers and hence exemplify specialist mathematical knowledge for teaching annuities. in the next section i provide a brief review of the limited research literature on conceptions of annuities. thereafter i propose a hierarchy of concepts related to annuities to provide a backdrop for the later discussion about elements of teachers’ knowledge for teaching annuities. research on conceptions of annuities top ↑ little research has been published on students’ conceptions of annuities at school or university level. some work has been published on approaches to teaching annuities to university students (e.g. dempsey, 2003; eddy & swanson, 1996; gardner, 2004; jalbert, jalbert & chan, 2004) but, with the exception of dempsey (2003), these are largely anecdotal. the work by hoyles, noss and their colleagues (e.g. bakker, kent, noss, hoyles & bhinder, 2006; hoyles, noss, kent & bakker, 2010) provides the only research-based evidence on thinking about annuities. as part of their study on techno-mathematical literacies in the workplace, hoyles et al. (2010) focused on two pension companies and a mortgage company in the united kingdom, all dealing with annuities-based scenarios. they identified several gaps in the knowledge of sales and service employees, including inadequate understanding of the growth of money and the notion of present value of money and of the key variables and their relationships in mortgage scenarios. they also noted a lack of appreciation of the mathematical models that underpin the documents produced by the it system, such as pension statements, and poor ability to interpret visual representations. employees were also unable to make estimates and predictions of costs for individual customers. since employees lacked this knowledge, they were unable to help customers with simple comparisons such as comparing the monthly interest rate charged on customers’ credit cards (say 1.9%) with the annual mortgage loan rate being offered (which was approximately 6% p.a.). a hierarchy of annuities concepts top ↑ given the lack of research on annuities, it has been necessary to develop a hierarchical network of links between key concepts relating to annuities in order to provide a reference point from which to undertake my own research. this network builds on a theoretical elaboration of compound interest that includes a hierarchy of interest concepts and a network of concepts related to growth factor (pournara, 2013b). the network proposed in figure 1 was developed with an eye on the requirements of the school curriculum. concepts that are higher up in the diagram build on those that are lower down. in the discussion below i use italics to identify the nodes in the diagram. figure 1: a hierarchy of annuities concepts. the notion of the time value of money underpins all aspects of annuities. time value is based on the principle that an amount of money is worth more today than the same amount of money in the future because it has potential to earn interest. the operation of compounding and its reverse, the operation of discounting, provide the mechanisms by which an amount is moved forwards or backwards in time. this is operationalised by multiplying the present value, or dividing the future value, by the unit growth factor (1 + i). (see pournara [2013b] for a discussion of growth factor and unit growth factor.) in figure 1, the operations of compounding and discounting are linked to present value and future value of single payments respectively and then to multiple payments in the four annuities scenarios. in locating multiple payments in figure 1, i begin with simple annuities where the frequency of payments corresponds with the frequency of compounding interest. here i include future value of ordinary annuity, future value of annuity due, present value of ordinary annuity and present value of annuity due, thus distinguishing payments in advance (annuity due) from payment in arrears (ordinary annuity) for both present value and future value. i include sinking funds and outstanding balance since they are included in the grade 12 curriculum. i refer here only to sinking funds that involve setting up a fund to make provision for the replacement of an asset at some time in the future, since this is the scope of sinking funds studied at school level. future value of an annuity is linked to sinking funds because it is used to determine the value of the regular payments at some point in the future when the new asset will be purchased. similarly, future value (of a single payment) is linked to sinking funds because it determines the depreciated value of the asset that will be replaced. for this reason, a link with depreciation is also indicated. this links to figure 1 in pournara (2013b). figure 1 shows how outstanding balance draws together several concepts from lower levels. the outstanding balance at some time tk can be calculated in two ways. in the retrospective method the loan is moved forward to tk, thus calculating the interest on the loan at tk as if no repayments have been made. each of the k repayments is also moved forward to tk and the outstanding balance is the difference between the future value of the loan amount and the sum of the future values of each repayment, as given by the formula: outstanding balance = in figure 1, the link from future value (of a single payment) indicates the loan and the links from future value of annuity indicate the k repayments. in the prospective method the outstanding balance at some time tk is the sum of the present values of all the repayments that have not yet been made. the outstanding payments are therefore moved back in time to tk as shown by the formula: outstanding balance = , where m is the number of outstanding payments. this is reflected by the links between present value of annuity and outstanding balance. the concepts of deferred annuity, complex annuity, escalating annuity and perpetuity are included for completeness. although deferred annuity is not specifically mentioned in the curriculum, it is possible to include it at grade 12 level since it only requires a simple adjustment for time. the other three annuity concepts are beyond the scope of the grade 12 curriculum. an expanded view of the compound interest formula top ↑ in this section i distinguish two views of the compound interest formula and argue that learners should be able to adopt either view as appropriate. knowledge of these two views is specialised knowledge for teachers and would likely not be encountered in a typical financial mathematics course, and certainly not in current school texts. i offer this as an example of mft that draws on knowledge of mathematics, curriculum and tasks, and thus the intersection of smk and pck. learners are typically introduced to the compound interest formula in grade 9 or grade 10 to determine the accumulated value of a certain amount over a period of time. whilst this is a necessary and important use of the formula, it is inadequate for developing a full grasp of annuities. i therefore propose the need for an expanded view of the compound interest formula, similar to kieran’s (1981) call for an expanded view of the equal sign. in the same way that learners must be able to view the equal sign as both a do-something operator and as an equivalence relation, so they need to view the compound interest formula from both an accumulation and an adjustment perspective as elaborated below. the accumulation view is a static view, best exemplified in questions of the form: an amount, p, is invested at a certain rate, r, compounded monthly for a certain period, n, and we want to determine a, the amount that accumulates. thus we have the general form , although this will need to be adjusted for monthly compounding. here we focus on the original amount and then on the final amount, and compare (in an additive sense) the amount by which the original has changed because this tells us how much interest has accumulated. we are only interested in the magnitude of a, not in its relative value in relation to the passage of time or corresponding changes in its buying power. the dominant message is that the principal amount accumulates interest and becomes ‘more’. the focus is thus on the nominal value of the principal amount, which is separated from time. the adjustment view focuses on the time value of money. it is a dynamic view in which we are interested in adjusting for the effects of time or, more correctly, the effects of inflation, exchange rates, and so on. so, we are concerned with the value of an amount at different points in time; we can use the compound interest formula as a mechanism to move our amount of money to different points in time along a timeline. it is therefore associated with compounding and discounting. here it may be better to refer to present value (rather than principal) and future value (rather an accumulated amount); thus, we may represent the general formula as: . from an adjustment point of view, we are less interested in the actual magnitude of the number and more concerned with its relative value at a different point in time. in real terms, the new amount may not have a higher value than the principal amount had in the past, although in nominal terms the new amount is ‘more’ because it is a larger number. but since the time value of money is linked to its buying power, the magnitude of the number must always be seen in relation to what it can buy, such as groceries or foreign currency. the accumulation view is emphasised throughout the school curriculum. typical questions require learners to solve for an unknown in the compound interest formula and the focus is generally on the difference between the initial and final amounts, even when learners are required to calculate the time period of the investment. however, when dealing with annuities, the compound interest formula needs to be viewed as an adjustment mechanism because the individual payments are being moved forwards or backwards in time to determine their contribution to a loan, outstanding balance or projected savings. unfortunately, since annuities are only introduced in grade 12, the need for an expanded view of the formula comes very late. nevertheless, based on discussions with actuaries working in the financial sector and in academia, the adjustment view is the one that they use most often and that is captured in typical statements such as ‘discounting a payment back to t0’ (i.e. the point at which a loan is taken or an annuity is purchased). this suggests that the prevalent view of the compound interest formula in school mathematics does not reflect its dominant use in the banking sector and in actuarial science. this is likely a consequence of an overemphasis on basic interest calculations for several years of schooling, an underemphasis on annuities and little attention to the notion of the time value of money. helping learners to view the compound interest formula in multiple ways is a teacher’s responsibility akin to supporting younger learners to see the equal sign as both a do-something signal and an equivalence relation. in both situations the teacher draws on knowledge of mathematics and of the ways in which the curriculum has already impacted learners’ conceptions. it is through the design and mediation of appropriate tasks that learners may be pushed to think about the formula in different ways. awareness of the emphasis on an adjustment view in the financial sector and in more advanced financial mathematics may provide additional motivation for the need to expand learners’ views of the formula. two approaches to annuities top ↑ even (1990) identifies knowledge of alternate ways of approaching a concept as an aspect of smk. in this section i describe two different ways of approaching annuities based on the pre-service teachers’ initial engagement with annuities tasks. as with different views of the compound interest formula described above, knowledge of different ways of approaching annuities is important for teachers although it is likely that others working with financial mathematics are also aware of both approaches. however, teachers require knowledge of how the two approaches differ and how they intersect, which goes beyond an awareness of their existence. textbooks typically introduce annuities problems as applications of geometric progressions. however, data from my study suggest that when students first encounter annuity-based tasks, and are left to devise their own strategies, they do not necessarily focus on each payment and make use of a geometric progression. rather, they track the account balance over time. from this observation i distinguish two different approaches to working with annuity situations. the first approach i call an account balance (ab) approach because it focuses on tracking the account balance. this approach mirrors what goes on in the bank each month (although the detail of daily interest calculations is ignored). in the case of annuity-based savings, a deposit is made, it is added to the account balance, interest accumulates and the closing balance is calculated at the end of each month. in the case of a loan, the loan is granted, then interest accumulates for the first month, a repayment is made and deducted from the capital balance; this process is repeated until the loan is repaid. this approach is easy to make sense of and, in my experience, is the approach students adopt when initially attempting to model an annuity-based scenario. the unit of analysis is account balance against time. it depends on simple iterative calculations, but when there is a departure from the perfect payment plan, all balances need to be recalculated, which makes it an inefficient and cumbersome approach. the second approach focuses on the behaviour of each individual payment over time, so i refer to it as an individual payment (ip) approach. in this approach each payment is disaggregated from the whole and its contribution to the overall balance is modelled by moving it forward (or backward) in time by means of compound interest calculations. in the case of an investment, each deposit is moved forward to the end of the investment period so one can see the contribution it makes to the final amount. in the case of a loan, each payment is moved back to the point when the loan is granted. the unit of analysis for the ip approach is individual payment against time. the ip approach does not reflect the monthly process of making payments and gaining interest. rather, it is an analytical approach that projects the growth of money into the future (or back to t0). it is mathematically more powerful than the ab approach as it draws on geometric progressions, which reflect the underlying mathematical structure of annuity-based scenarios. it is also a more efficient approach: when there is a departure from the perfect payment plan, only the changed payments need to be considered when recalculating balances. for both these reasons, it is not surprising that mathematics textbooks adopt this approach from the outset. it is important for teachers to recognise the differences in these two approaches and to be aware that learners may not initially appreciate the elegance of an ip approach, despite its prevalence in textbooks. i now move to discuss the derivation of the formula for future value of an annuity due using both approaches. in so doing, i draw on data from the study and show how the different approaches and their associated logic converge on the same algebraic formulation but with different interpretations of the algebraic forms. deriving a formula for future value of annuity due using an account balance approach as noted above, the ab approach broadly models what happens in banks on a monthly basis. in order to make sense of the real-world problem, students need to understand how this works. by doing the iterative calculations for a few months, they see the patterns modelled by the calculations. from this inductive process it is possible to derive the annuities formulae as shown in figure 2, which provides general expressions for the calculations that are done at the end of every period. in figure 2, each row (lines 1 to 6) represents a period. the middle column indicates that payment is made at the beginning of the period and that interest is gained at the end of the period. the right-hand column gives the strategically factorised expression. i call it ‘strategically’ factorised because it is possible (and tempting) to expand the expressions in the square bracket and then collect like terms, which may or may not lead to some form of factorisation. the factorised form shown below preserves the unit growth factor (1 + i), which ultimately produces the geometric series. in the middle column, the factor (1 + i) is multiplied by each term in the ‘expanding bracket’. thus, in each line the expression in the square bracket expands but the emerging pattern in exponents is easily seen. figure 2: account balance approach to generate series for future value of annuity due. obviously a strategic substitution of would substantially improve the readability of the expressions and also show more clearly the geometric progression embedded in the expanding expression. in order to move from this form to the standard formula for future value of annuity due students can do one of two things: either they must recognise the geometric progression (which may or may not include common factors), identify the first term, common ratio and then substitute appropriately into the formula for the sum of a geometric progression, or they must use the elimination method to remove all ‘middle’ terms and then complete the necessary algebraic manipulation. however, the elegance of the expressions above may not be easily visible in the struggle to derive the formula. for example, in figure 3, i show student hailey’s first attempt to derive this formula using an ab approach. figure 3: hailey’s attempt to simplify algebraic expressions for future value of annuity. hailey has compressed many lines of algebraic manipulation into this summary. she notes the common factor of , then the patterns in the highest two powers of the exponent and the constant, and she refers to the remaining terms as ‘some other stuff’. she describes this as ‘a sort of general pattern’. the structures she identifies do not lead her to the required expression and, even if they had, her approach does not reflect the thinking that will be most useful in working with annuity-based problems. rather, she needs to shift to an ip approach since the ab approach is limited in providing a useful and flexible model to deal with annuity-based scenarios. deriving a formula for future value of annuity due using an individual payment approach figure 4 shows the elegance of the ip approach for the future value of an annuity due, assuming 12 equal monthly payments. here each row (from line 1 to 6) represents the future value of a particular payment at the end of period 12. the terms generated are clearly recognisable as those of a geometric progression. figure 4: individual payment approach to generate series for future value of annuity due. whilst this method produces the same final result as the ab approach, the underlying thinking is substantially different. this approach does not model the monthly process in the bank. each payment requires one to run through the timeframe from the point at which the payment is made to the end of the investment. as with the ab approach, in order to move from this form to the standard formula for future value of annuity due, students must either apply the formula for a geometric series or use the elimination method coupled with the necessary algebraic manipulation. evidence from the study suggests that the ip approach is not intuitive (fischbein, 1999), and does not appear to be the obvious starting point for many students. it requires a substantial shift in thinking to conceive of an annuity situation in this way. however, once this shift is made, students appear to have little difficulty in adopting an ip approach. this finding thus challenges the starting point adopted by most textbooks in introducing annuities. whilst clearly indicating that there are different ways of approaching annuities, it also suggests at least two different teaching sequences and approaches. in their framework, ferrini-mundy et al. (2006) identify knowledge of different teaching sequences and approaches as an aspect that foregrounds teachers’ pedagogical knowledge. by contrast, as mentioned above, even (1990) refers to knowledge of different ways of approaching a concept. therefore, in considering the ab and ip approaches from the perspective of mft, we see again how knowledge of mathematical and pedagogical aspects intersect and are not easily separated in terms of smk and pck. dealing with a paradox in the modelling of time top ↑ the modelling of time in annuities scenarios exemplifies the intersection of mathematical and contextual aspects of mft of annuities. i discuss the importance of conventions in modelling time and show how these conventions overcome a paradox when payments are made at the end of the period. in the study, an appreciation of the paradox proved to be an important insight in order to make sense of what initially appeared to be students’ ignoring of the timeframes specified in the task. in the world of banking, an annuity payment can be made at any time of the month and interest is calculated daily and compounded monthly. (see pournara [2012], for a detailed discussion of the distinction between daily interest calculations and monthly compounding.) however, mathematical models remove this complexity by considering only two scenarios: payment in advance (annuity due), where the payment is made at the beginning of the period, and payment in arrears (ordinary annuity), where payment is made at the end of the period. furthermore, the beginning and end of the period are defined very specifically. for example, by convention a payment made at the end of a month does not gain interest in the month in which is it deposited. without this convention, we are faced with the following paradox: if payment is made at the end of the month and interest is compounded also at the end of the month, we have one process that is dependent on another and yet both processes take place simultaneously. so, given that interest is capitalised at the end of the month, if payment is not included in this interest calculation, then one could reason that the payment must be happening after the end of the month, in other words at the beginning of the next month. thus it might be argued that payment at the end of month n is really payment at the start of month n + 1. however, a different model is used for payment at the beginning of the month (annuity due) and so this interpretation is not acceptable. if, on the other hand, the payment is made before interest is calculated, then we may reason that the payment is not made at the end of the month. this gives rise to a subtle but crucial difficulty. consider the scenario in which regular monthly payments of r200 are made at an interest rate of 6% per annum compounded monthly. in table 1 the payments are made at the beginning of the month. in table 2 payments are made at the end of the month. the key difference between the tables is the position of the payment column. in table 1, payment occurs immediately after opening balance, thus modelling payment made at the beginning of the month. in table 2, payment occurs after the interest column to indicate that the payment is made at the end of the month. the column balance on which interest is calculated has been included to make explicit the amount of money in the account at the time of compounding interest. given that there are only two scenarios for modelling the timing of payments, if we want to model payment at the end of the period but place payment before balance on which interest is calculated the result is that we model payment at the beginning of the period. this is a subtle but important distinction. we may intend to model that the payment is made just before interest is calculated but the mathematical model assumes the payment has been in the account for the full month and thus gains interest for the entire period. table 1: annuity payments made at the beginning of the month. table 2: annuity payments made at the end of the month. the essential difference between the above situations hinges on whether the payment is added before interest is calculated. this translates to a time difference of a full period of interest on the latest payment, which shows that we cannot (easily) model ‘payment just before calculating interest’. there are only two options: ‘payment before’, which translates to payment at the beginning of the period, and ‘payment after’, which translates to payments at the end of the period. hence the need for the convention: a payment made at the end of the period does not gain interest in the month in which it is deposited. this is equivalent to thinking about the order as follows: a payment made at the end of the month takes place after the interest is capitalised for the month but before the beginning of the next month. whilst this statement may appear contradictory, it is nevertheless a useful way of interpreting the convention for payment at the end of a period and is thus a typical example of shulman’s (1986) description of pck as a way of ‘representing and formulating the subject that make[s] it comprehensible to others’ (p. 9). it is unlikely that students or learners will recognise and appreciate the paradox unless they are given the opportunity to model the scenario using their own ideas and approaches. similarly, if teachers only approach the teaching of financial mathematics by giving formulae and procedures then the issue may not emerge. in this case it may be argued that teachers do not need knowledge of the paradox. in the study, it was only through extended grappling with the students’ responses (before they had learned the formulae and conventions) that i came to recognise the paradox. initially i interpreted their responses through the lens of the annuities conventions and so assumed they had not paid attention to the instruction that payments were made at the end of the month. however, i was uneasy with this deficit interpretation of their reasoning because the ‘error’ was so pervasive across their responses. it was only when i recognised the paradox that i could appreciate that their models made sense to them and they were intending to model payment at end of month but their order of adding the new payment before compounding was the reverse of the convention. in this section i have shown how contextual knowledge of finance links with mathematical knowledge and knowledge of learners’ thinking in the modelling of time in financial scenarios. the modelling conventions greatly simplify the complexity of dealing with time in financial scenarios. however, students and learners who do not yet know these conventions, and thus work from an everyday understanding of banking processes, are likely to produce their own models that do not fit with convention and that may therefore be disregarded by teachers. thus teachers’ knowledge of the modelling conventions cannot be easily separated from knowledge of how someone who does not yet know the conventions might model the situation. furthermore, this needs to be accompanied by knowledge that the models do not take into account the daily workings in the world of banking. different interpretations of the exponent in financial formulae top ↑ the final example concerns the meaning of the exponent in the compound interest and annuities formulae. i argue that the exponent may be interpreted in different ways. by tracing through the derivations of the formulae, i show how this arises and i thus illustrate how a deeper consideration of the mathematics is necessary to appreciate these different interpretations. when learners are first introduced to the compound interest formula in grade 9 or grade 10, they are likely to view the exponent as the number of times the principal amount is compounded. consider the following typical question: • i invest r500 at 6% per annum compounded monthly for 4 years. how much will i have in total at the end of the 4-year period? the solution to this question involves substituting 48 for n since there are 48 compoundings over the four-year period. thus n represents number of times interest is compounded: . this interpretation of n arises from the use of the compound interest formula. when learners see the expression in the context of an annuities formula, such as , it seems reasonable to assume that n still represents the number of compounding periods, if for no other reason than the similarity in form. however, given that there are multiple payments, each gaining interest for a different length of time, there is likely to be some concern about which number of compounding periods n refers to. consider the following question: • i make payments of r500 into a savings account at the end of each month. interest is 6% per annum compounded monthly. how much will i have in total at the end of 3 years? solving this question involves substituting 36 for n in the formula for future value of an ordinary annuity since there are 36 payments in three years: . here n is treated as the number of payments. it does not represent the number of times the first payment gains interest because then since the payment does not accumulate interest in the month in which it is deposited. interviews with the pre-service teachers revealed that several of them were unsure about the meaning of n in the annuities formulae: does n represent number of compounding periods or the number of payments? i argue that the exponents may be considered to have different meanings based on the ways in which they are used in the respective formulae. i illustrate the shift in the meaning of n with reference to the formula for future value of an ordinary annuity. in tracing how the compound interest formula is initially used, and then how it is used to derive the annuities formulae, i show how students’ difficulties may emerge. consider the scenario of a single payment, p, gaining compound interest monthly at a monthly rate, i, for four months. figure 5 shows two different expressions for the future value at the end of each month. the exponent in the right-hand column represents the number of times the principal amount has been compounded. in the simplified expression the month number, subscript and exponent all have the same value for any particular month. figure 5: compounding of single payment. a similar pattern arises when working with annuities using an ab approach. consider a scenario with four equal payments, p, made at the end of each month with monthly compounding again at a monthly rate, i. figure 6 shows that the subscript is one unit larger than the highest exponent in the factorised expression. this reinforces the association of n with number of compoundings because at the end of each month interest is compounded on the balance. of course n also indicates the number of terms within the square bracket of the strategically factorised expression, but this may be less obvious than the relationship between the subscript and the highest exponent. figure 6: multiple payments using account balance approach. now consider the same annuities scenario with an ip approach. figure 7 indicates the accumulating interest on all four payments separately. each line represents the future value of a single payment at t4. all four payments are then combined to determine the future value of the annuity (fvann) at the end of month 4. this is the point at which the fourth payment is made; hence it does not accumulate any interest. figure 7: multiple payments using individual payment approach. there is a subtle shift in the meaning of the exponent across the three figures. in the compound interest calculations in figure 5, the exponent represents the number of times a payment has gained interest since it was deposited. in the ip calculations in figure 7, the exponent represents the number of times the payment will gain interest by the end of the term of the investment, in this case the end of the fourth month. whilst the exponent refers to the number of compoundings in both cases, there is a subtle difference in what it represents. there is also a difference in the ordering of the exponents. in figure 5 the exponents are ascending, whilst in figure 7 they are descending. however, the sums of the four terms are equal. in the ab approach in figure 6, the exponents represent the number of times each payment has gained interest since being deposited, as in the compound interest formula. when factorised, the resulting expression is the same as that for fvann in figure 7. however, the expression may be viewed differently. for example, learners may not consider the exponent to represent anything in particular; they may simply see it as the result of their algebraic manipulation. the shift in the interpretation of n comes when the annuities formulae are introduced. the formulae develop from geometric series, where each term represents the future value of an individual payment at some point tn. thus the expression for the future value of an ordinary annuity at tn is given by: each exponent represents the number of times a particular payment will gain interest by time tn. thus, n still represents number of compoundings. however, once the expression is manipulated to produce the formula , the meaning of n shifts. it then represents the number of payments since the annuities formula has emerged from the formula for the nth partial sum of a geometric series, where n represents the number of terms in the series and each term is associated with a payment: . ironically, even if one works from first principles to derive the annuities formulae, it may be no more convincing that n represents the number of terms. the typical elimination method for an ordinary annuity is shown in box 1. box 1: elimination method to derive formula for future value of an ordinary annuity. in lines 1 and 2 the exponents represent the number of times each payment will gain interest by time tn. lines 3 and 4 involve algebraic manipulation and the n that remains comes from the last term in line 2. thus n is still associated with compoundings. however, in line 4 the conventional formula for future value of an ordinary annuity emerges. when we use this formula, we substitute the number of payments into n, as shown in the earlier example. it is worth noting that several textbooks (e.g. basson et al., 2005; young, 1993) explicitly link n in the annuities formulae to the number of payments. yet there is no reason to accept that the meaning of n in line 4 should be any different from its meaning in lines 1, 2 and 3. the ordinary annuity scenario is a critical case here because it shows clearly that the exponent in the formula (n) is not the same as the exponent of the first payment (). in an annuity due scenario this is less clear since n indicates the value of three different elements, namely the number of payments, the number of compoundings of the first payment and the overall number of months in the chosen timeframe. a further example, involving a complex annuity, is useful to go beyond the special case of simple annuities. here we deal with a scenario in which the frequency of payments differs from the compounding frequency. consider the following example: • we make equal payments, p, every 6 months into an account where interest is compounded monthly, and the monthly interest rate is i. payments are made at the end of the month. assume the first payment is made at the end of the first month, determine how much has accumulated by the end of two years, i.e. the end of the 24th month. this scenario gives rise to the following geometric series: . if we reverse the order, we get a geometric series with first term and common ratio . summing this geometric series we get: . here we see a more complex common ratio but an exponent of 4 to indicate the four terms that are being added. it is not the case that interest is being compounded only 4 times, even for the first deposit1. this example thus reinforces that the exponent in the annuities formula represents the number of payments and not the number of compounding periods. it is likely that the struggle to accept a shift in the meaning of n is a result of modelling. if the exponent had no contextual meaning in either formula, the problem of the shift in meaning would not exist. but since the initial encounter with the exponent is in terms of the number of compounding periods, it may be disconcerting to conceive of it later as the number of payments. the derivations of the annuities formulae do not provide adequate explanation for a shift in the meaning of n. it must simply be accepted as a result of summing a geometric series. returning to the discussion of teachers’ knowledge, we see again how mathematical and pedagogical aspects are closely intertwined. the lengthy explications above show the detailed mathematical work entailed in decompressing the annuities formulae to re-explore their origins. ultimately there may be no satisfactory justification for interpreting the exponent as the number of payments when the derivation began with an interpretation of the exponent as the number of compounding periods. nevertheless, when the compound interest formula and the annuities formulae are used to solve financial mathematics problems, the exponents are interpreted differently. thus, in order to appreciate students’ and learners’ conceptions with regard to the exponent, we are obliged to consider the likely origins of these conceptions and difficulties, which leads us to reconsider the underlying mathematics and how this models the growth of payments over time. conclusion top ↑ i began this article with a discussion of shulman’s notions of smk (or ck) and pck. they are compelling ideas that have been very productive in moving the field forward with respect to research on teachers’ knowledge in general and teachers’ mathematical knowledge in particular. whilst they are widely accepted, as ball et al. (2008) note, they are defined and used in different ways. for this reason they are problematic as analytical constructs and although several attempts have been made to operationalise them (e.g. ball et al. 2008; even, 1990; krauss et al., 2008), the empirical evidence for distinguishing smk from pck is still relatively weak. in choosing the term mathematics-for-teaching to encompass an amalgam of mathematical and pedagogical knowledge, i have avoided the smk-pck distinction. furthermore, i have shown how mathematical, pedagogical and contextual knowledge of finance are intertwined in any consideration of teachers’ knowledge for teaching annuities. in so doing, i have made an initial contribution to establishing a knowledge base for teachers’ knowledge for teaching annuities. it is not simply knowledge of financial mathematics in the way that actuaries or bankers might use it. it contains elements that are fundamentally about how we come to know annuities, how we connect it with existing knowledge of other concepts, such as compound interest, and the importance of knowing the conventions of the banking world. acknowledgements top ↑ this work was supported financially by the thuthuka programme of the national research foundation (grant no: ttk2007050800004). any opinions, findings and conclusions or recommendations expressed are those of the author and the nrf does not accept any liability. competing interests i declare that i have no financial or personal relationships that may have inappropriately influenced me in writing this article. references top ↑ adler, j. 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(1993). financial mathematics: a computational approach. cape town: juta. footnotes top ↑ 1.it is also possible to determine the future value of the complex annuity by making using of an effective rate for half-yearly periods and then proceeding as with an simple annuity, since payment and compounding periods will correspond. article information authors: jayaluxmi naidoo1 reginald govender 2 affiliations: 1school of education, college of humanities, university of kwazulu-natal, south africa 2kingsway high school, kwazulu-natal, south africa correspondence to: ayaluxmi naidoo email: naidooj2@ukzn.ac.za postal address: 11 ronald place, westville 3629, south africa dates: received: 06 mar. 2014 accepted: 17 nov. 2014 published: 12 dec. 2014 how to cite this article: naidoo, j., & govender, r. (2014). exploring the use of a dynamic online software programme in the teaching and learning of trigonometric graphs. pythagoras, 35(2), art. #260, 12 pages. http://dx.doi.org/10.4102/pythagoras.v35i2.260 copyright notice: © 2014. the authors. licensee: aosis openjournals. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. exploring the use of a dynamic online software programme in the teaching and learning of trigonometric graphs in this original research... open access • abstract • introduction • technology-based tools for the teaching and learning of mathematics • dynamic technology • theoretical framing: the addie model • methodology    • ethical considerations    • data collection    • pilot phase: during the development stage    • during the evaluation stage • findings and discussion    • the worksheets    • behaviour of graphs    • interpretation of graphs    • modelling real-life contexts    • overall performance    • the semi-structured interview and observations    • easy access    • learners in control    • technology is the future    • motivation    • meaningful interaction • conclusion • acknowledgements    • competing interests    • authors’ contributions • references • appendix 1    • pencil and paper worksheet • appendix 2    • online activity worksheet    • question 1    • 1.1    • 1.2    • question 2    • question 3    • question 4 • appendix 3    • learner semi-structured interview schedule    • things to take note whilst doing interview: • footnotes abstract top ↑ trigonometry is an important section in mathematics that links algebraic, geometric and graphical reasoning. many secondary school learners are not familiar with these types of reasoning; thus, trigonometry presents a challenge for these learners. the traditional ‘chalk and talk’ method of teaching trigonometry is used by many teachers; however, this article is based on a case study that explores the innovative use of a dynamic online software program in the teaching and learning of grade 10 trigonometric graphs. the participants in this study were 25 grade 10 mathematics learners from one school in the south of durban, kwazulu-natal, south africa. data were collected by using two trigonometry worksheets, semi-structured interviews with selected learners and lesson observations. the worksheets were constructed based on the following focus areas: behaviour of graphs, interpretation of graphs, modelling and overall performance of the learner. in all focus areas, the participants’ scores improved significantly when working within the dynamic online environment. further analysis revealed the following five themes: easy access, learners in control, technology is the future, motivation and meaningful interaction. these findings would be of interest to mathematics teachers, learners, curriculum developers and mathematics teacher educators. introduction top ↑ technology is developing rapidly and captures the interest and attention of learners in contemporary society. technology-based tools transform mathematics concepts to an understandable form for learners (niess, 2006). ndlovu, wessels and de villiers (2011) explain that there is an increasing trend to integrate technology into mathematics education in many countries globally. technology-based tools for mathematics may include software (programs) and hardware (devices). this study focuses on the innovative use of the dynamic online software1 geogebra2 when teaching grade 10 trigonometric graphs. the study explores the implications of the use of this dynamic online software program when teaching trigonometric graphs in mathematics. research (anthony & walshaw, 2009; leendertz, blignaut, nieuwoudt, els & ellis, 2013; lin, 2008) has indicated that the use of technology enhances the teaching and learning of mathematics. the use of technology-based tools enriches learning, since visual data promotes and challenges explanation and justification, as opposed to the traditional methods of note taking, chalk and talk. this may add value when teaching trigonometric graphs since many learners perform poorly and struggle with trigonometric graphs (weber, 2008). additionally, many mathematics teachers face challenges when trying to effectively introduce the concept of trigonometric graphs to learners (klein & hamilton, 1997). the nature of the geogebra program is that it allows the mathematics learner the freedom to manipulate and visually notice instantaneous changes and behaviour of graphs. the geogebra program was used to create the tasks offline. once the offline dynamic file was created, it was then converted to an applet and uploaded to the internet, thus creating a dynamic online environment (govender, 2013). hence, this research study sought to answer the following question: what are the implications of using dynamic online software when teaching trigonometric graphs in grade 10 mathematics? technology-based tools for the teaching and learning of mathematics top ↑ there is a range of available technology-based tools that could be used within mathematics classrooms including scientific calculators, function graphers, spreadsheets, statistical packages, subject-specific websites, interactive whiteboards and geometry packages (niess, 2006; pierce & ball, 2009). technology-based devices, for example clickers and voting pods, may be used in collaboration with statistical packages to determine how many learners agree on a particular solution whilst working within a mathematics lesson. scientific calculators and function graphers are regularly used by learners and teachers in mathematics classrooms. in some cases these technology-based tools are allowed to be used in tests and examinations. using these tools saves time and eliminates redundant and complex calculations in mathematics. spreadsheets and statistical packages are currently used in mathematics classrooms to process raw data into meaningful information. these technology-based tools may also be used when teaching and learning specific concepts in mathematics, for example mean, mode, range, median, population, data representation and problem solving. similarly, interactive whiteboards assist the teacher, as they allow for the recording of mathematics lessons for later use or sharing of mathematics lessons in real time. these hi-tech whiteboards offer an unlimited supply of digital colour that will never run out and a list of programs to enhance the mathematics teaching process. thus, the use of technology-based tools in the teaching of mathematics ought to make a difference in a classroom. however, this also necessitates that mathematics teachers possess and exhibit sound technological pedagogical content knowledge (koh & sing, 2011). this term epitomises how a teacher integrates technology-based tools effectively within the classroom (mishra & koehler, 2006). dynamic technology top ↑ dynamic technology offers the learner the freedom to manipulate shapes and graphs as compared to the more static nature of sketches. mathematics is seen as a conceptual subject in which face-to-face contact is necessary for conveying concepts such as trigonometry (engelbrecht & harding, 2005). hence, the use of technology-based tools depends on the teacher's attitude towards these tools, since the teacher is the most important factor within the classroom. many teachers believe that technology-based tools will not enhance learning and subsequently these teachers resist their use in the classroom (niess, 2006). however, pierce and ball (2009) maintain that teachers’ opinions are bound to change as they are exposed to positive results through the use of technology-based tools within the classroom milieu. learners are required to generate as many graphs as necessary first by pen and paper and then to support these drawings by available technology-based tools (department of basic education, 2011). it is likely that learners will understand features pertaining to trigonometric graphs more quickly, through the use of dynamic online software, as compared to the traditional methods of pen, paper, chalk and talk. the integration of dynamic online software in meaningful ways may enhance the construction of knowledge in addition to the communication and dissemination of ideas in the mathematics classroom. visualisation and stimulation are the primary advantages of using dynamic online software and may be the initial step in achieving interactive learning environments that support the teaching and learning of abstract mathematics concepts (naidoo, 2012). according to robert and slavin (2009), this learning environment replaces the teacher with self-paced instructions. this promotes a learner-centred environment and allows learners to control their learning whether they are in the classroom or not. hence it is important to note that the use of dynamic online software enables the learner to become self-regulated because these programs have the capability of placing accurate solutions and processes at learners’ fingertips. thus the use of dynamic online software influences two major cornerstones in post-apartheid teaching and learning environments within south africa: learner-centred education and self-regulation. these concepts share a commonality that the learner is able to think independently and only requires the teacher's assistance when they encounter problems. the teacher acts as the facilitator; therefore, learner centredness and self-regulation may be applied and practised through the use of dynamic online technology. figure 1 illustrates the relationship between the use of dynamic online technology, learner centredness and self-regulation. as is evident this figure portrays a cyclic relationship between all three components. figure 1: the effects of dynamic online technology on teaching and learning mathematics. teaching trigonometric graphs with the aid of dynamic online software is more likely to result in an individualised learner-centred milieu within the classroom. learners are allowed to make mistakes freely without embarrassment or being looked down upon by peers when working within a dynamic online environment. this dynamic environment provides the learner with a sense of control as they do not just do mathematics but experience the mathematics. learners who experience difficulty in mathematics are likely to change their attitude to mathematics through the use of dynamic software in mathematics (pierce & ball, 2009). the environment is online and thus it is available to anyone who has internet access, permitting learning at any time and in any place such that distance does not become an obstacle to education (de villiers, 2013). furthermore, since this type of learning is not restricted to the classroom and school environment, there is a likelihood of increased and improved parental involvement. theoretical framing: the addie model top ↑ this case study adopted the analysis-design-develop-implement-evaluate (addie) model of instructional design (danks, 2011). the addie model was initially created for use by the united states armed forces in the early 1970s and entailed 19 stages or phases that were considered crucial to the advancement of educational and training programmes (hannum, 2005). now the addie model represents guidelines for building effective training and performance tools in five phases. when applied to this study the teacher is the instructional designer who would examine the delivery of teaching trigonometric graphs to enhance learning. this study has been designed according to the following five phases that collectively constitute the addie model: analysis design development implementation evaluation. the analysis stage is a process that determines what is going to be learned and for whom it is intended. the designer would need to identify the problem, goals, learner needs and existing knowledge (danks, 2011). the latter part relates to vygotsky's zone of proximal development (vygotsky, 1978), which is crucial to the teaching and learning process. the zone of proximal development can be described in terms of mathematics, as the space between what the learner knows and the mathematics they are about to learn. within the analysis stage other relevant considerations must be taken into account such as the learning environment, the boundaries, the delivery decisions and the timeline for the project (hoffman, ritchie & marshall, 2006). on completion of the analysis stage, the designing of the blueprint for the material to be implemented begins. the design stage comprises of examining and specifying how materials are to be used or learned by learners (morrison, ross & kemp, 2004). the definition of the exact content of the material needs to be established together with the design of activities and assessment. following the design stage is the development of the learning materials phase. this stage involves the creation of the course structure, content, activities and assessments. in relation to the study at this stage, the dynamic trigonometric graph content was uploaded to the web and examined to see whether or not the material operated as expected. this served as the pilot for the uploaded content and assisted in achieving validity and reliability of the uploaded activity. the implementation stage is a process of delivering the piloted and adjusted material to the learners. the delivery of the material (final product) can take on various forms such as a web page (morgan, 2011), which results in meaningful engagement with the dynamic online trigonometric graphs. it is important to note that although learners engage with the content on their own, the teacher takes on an active role. the teacher, as the developer, would need to continuously analyse, redesign and enhance the product in order for it to be delivered effectively. this would involve the teacher observing learners’ engagement with the material to see whether or not they appear enthusiastic, resistant or critical in their use of the material. lastly the evaluation stage involves measuring the effectiveness and efficiency of the material. the evaluation stage is a two-stage process: the summative and formative stages. formative evaluation encompasses the data that inform revision decisions. the summative approach evaluates the validity of the learning; validity is assessed at the completion of the project. the teacher as the designer is focused on achieving the requirements of the curriculum as stipulated by policy documents (department of basic education, 2011). this is done to ensure that quality learning takes place. thus, formative evaluation is ideal, since assessment can take place at each stage to refine and redevelop strategies for attaining predetermined goals. this allows for close inspection between progressions rather than at the end of the process. a schematic drawing of the addie model that was applied in this case study is illustrated in figure 2. as is evident in figure 2, it is essential that the use of technology-based tools do not hinder the teaching and learning of mathematics. the technology plan ought to focus on how technology-based tools may be implemented, to foster the teaching and learning process. the addie model is described as a framework for instructional design that guides the teacher to attempt material development and design in a logical way (bichelmeyer, 2005). figure 2: the addie model as used in this case study. methodology top ↑ ethical considerations ethical clearance was obtained from the participating university's research office and the department of basic education. in addition, consent was obtained from the principal, participants, parents of all participants and the governing body of the school that the participants attended. participants and parents were provided with a letter detailing the aim and methodology of the study; in addition, participants were informed of their right to withdraw from the study. all participants’ responses on both worksheets were coded and participants were allocated pseudonyms to protect their identity. participants and their parents provided permission for the video recording of interviews and observations. data collection this case study set out to explore the implications of using a dynamic online software program in the teaching and learning of trigonometric graphs in grade 10. worksheets, interviews and observations were used to collect data for this study. data collection through the use of various methods enhances the reliability and validity of the study (bertram, 2010; padayachee, boshoff, olivier & harding, 2011). pilot phase: during the development stage the worksheets (appendix 1 and appendix 2) and semi-structured interview schedule (appendix 3) were piloted with a sample of five grade 10 learners with mixed levels of mathematics ability. the learners’ mathematics ability levels for the pilot and main study were established by analysing the learner's previous test and examination scores in mathematics. piloting the research instruments assisted in assessing the validity and reliability of each instrument. there was a need to ensure that each instrument was suitable and adequate for collecting the data that was required. after minor adjustments were made to the research instruments, the main study commenced. during the evaluation stage two worksheets were answered by 25 grade 10 learners to evaluate their understanding of trigonometric graphs before and after the use of the dynamic online software. the worksheets were developed by using the geogebra software and focused on the sine and cosine functions in trigonometry only. the selected questions assessed learners’ development and understanding of trigonometric graphs. the worksheets contained four questions. question 1 and question 2 set out to assess learners’ understanding of the behaviour of trigonometric graphs. question 3 set out to examine the learners’ ability to interpret the trigonometric functions whilst examining a graphical representation. question 4 evaluated their ability to apply trigonometry to a real-life problem. learners had to answer questions using pencil and paper first (appendix 1) and then interact within the dynamic online environment to sketch graphs and complete solutions to questions in the second worksheet (appendix 2). according to the van hiele theory, the exploration of functions ought to be completed before arriving at the formulae (de villiers, 2013). this was observed as learners progressed from question 1 to question 4. the element of visualisation, which is aided by the software, plays an important role in the exploration of trigonometric graphs. it must be noted that the learners were not coached on how to answer the questions using the dynamic online environment. learners were not given repeated chances to complete the worksheets: learners completed each worksheet only once. subsequently, one-on-one interviews were conducted with five randomly selected learners after the completion of the second worksheet. a semi-structured interview schedule, which consisted of 10 items, was used. the interview schedule consisted of both closed and open-ended questions (see appendix 3). the use of open-ended questions resulted in rich qualitative data. this assisted in providing a deeper interpretation of learners’ experiences. data were also collected via observations of learners whilst they completed the worksheets. the observations were video recorded. data collected via the observation allowed the researcher to gain additional insights into each learner's methods and processes as learners completed the worksheets. findings and discussion top ↑ the worksheets twenty-five grade 10 learners of mixed ability levels participated in this study. the maximum possible score that a learner could obtain was 20 on each worksheet. one mark per correct answer was allocated: the learner would have given either had the correct (awarded 1) or incorrect answer (awarded 0). the worksheets consisted of four questions that assessed specific aspects of trigonometric graphs. learners had to first answer questions using a pencil and paper and then complete the second worksheet whilst working within a dynamic online environment. it was evident based on the analysis of the worksheets that learners’ scores improved when working within the dynamic online environment. an overall average of 91% was achieved in the online environment as compared to an overall average of 70.3% that was achieved by the same group of learners when they used pencil and paper to complete the first worksheet. it was evident that as learners interacted with the activity within a dynamic online environment, they grasped concepts and ideas more successfully. it would seem that learners could analyse and interpret the behaviour of graphs with more success as they manipulated the trigonometric graphs within the dynamic online environment. the performance of learners in the second worksheet is illustrated in figure 3. figure 3: graphical representation of the percentage of correctly answered questions in the second worksheet. the second worksheet was analysed statistically to explore learner performance in each question. a tabulated description of learner performance when working within the dynamic online environment is exhibited in table 1. table 1: learner competence when working within a dynamic online environment. behaviour of graphs the mean for this section (question 1 and question 2) was 7.72. the mean indicated that learners displayed a good understanding of the behaviour of the graphs. additionally, a median value of 8 was calculated. this was also the maximum mark for this section. the standard deviation of 0.46 indicates that the values were not widely dispersed from the mean of 7.72. the minimum value of 7.00 suggests that learners have a good understanding of the behaviour of trigonometric graphs with the aid of dynamic online technology. it is also worthy to note that learners performed better in this section as compared to the other sections. based on this evidence it would seem that as learners manipulated the trigonometric graphs within the dynamic online environment their knowledge and skills related to answering questions based on the behaviour of trigonometric graphs were enhanced. interpretation of graphs when analysing learner performance in question 3 of the worksheet, it was found that the spread from the mean of 8.72 was 1.73 (standard deviation) with a maximum value of 10 and minimum value of 6. the median is 9 with the median of the lower half (q1) being 8 and the median of the upper half (q2) being 10. this indicates that the majority of learners displayed a good understanding of the interpretation of trigonometric graphs. a minimum value of 6 and a difference of 1.28 between the mean and the total of this section indicate that learners could have done better. it is noted that this section had a pass percentage of 87% which is the second lowest when compared to the other sections. so, whilst learners could have scored higher marks in this question, it was evident that their skills and knowledge related to the interpretation of graphs were enhanced by their use of the dynamic online program. modelling real-life contexts the maximum value for question 4 was 2 with a minimum value of 0. the standard deviation was 0.54 with a mean value of 1.72. the coefficient variation was 0.31 which was the highest when compared to the other sections. it can be deduced that learners performed poorly in this section. this deduction is supported by the 86% pass in this question which was the lowest percentage achieved. overall performance when answering all questions on the worksheet, the learners displayed an overall performance of 91% with a mean of 18.16. there was a minimum outlier and minimum value of 15. despite these values, the majority of learners performed well on the worksheet whilst working within the dynamic online environment. this implies that the learners’ performance improved as a result of working within the dynamic online environment. the difference of 1.84 between the mean and the total of this section indicates that there could be room for improvement as it was 9% short of 100%. however, learners did score more than 85% in all sections. it would seem that the learner's skills, understanding and knowledge of answering questions related to trigonometric graphs was enhanced through the use of the dynamic online environment. the semi-structured interview and observations the analysis of the learner semi-structured interviews (li) was done concurrently whilst analysing data collected during the observations. all interviews were transcribed and the key ideas that emerged from the transcripts were grouped together. all key ideas were then analysed qualitatively whilst the observations were reviewed to ensure that there were no misunderstandings or miscommunications. the key ideas were then categorised according to overarching common themes. thus, the interview transcripts were analysed using thematic coding and categorisation. data collected from the semi-structured interviews and observations gave rise to the following themes: easy access, learners in control, technology is the future, motivation and meaningful interaction. easy access we live in a fast-paced technological world in which education is now at our fingertips. this is made easier with applications that run on our mobile devices such as mystery math town, abc trains and google sky maps. similarly, many mundane daily activities may be done within seconds without leaving your home. this type of lifestyle is referred to as cybersphere (hoehler, 2013). the learning environment on the internet is easily accessible for learners as it is not restricted to the school environment only. this is supported by the participants as is evident from the following excerpts: shivaan: i would go on the website, especially at home because there is internet and computer access. (li, 14 june 2013) bradley: anywhere it's available preferably at home. (li, 14 june 2013) teaching trigonometry through the use of dynamic online software is perceived as an easier mode to communicate with learners. when the teaching and learning environment is comfortable, it becomes conducive to learning. this is evident in the following excerpts: bradley: i feel more comfortable in learning how to go about the questions and processes to get the answer. (li, 14 june 2013) tahlia: i have both aspects … technical part and … tactile part … from the teacher. (li, 14 june 2013) it was noted that learners were more relaxed when working with the second worksheet whilst working in the dynamic online environment. additionally, working with mathematics within a dynamic online environment is beneficial for parents who do not have time or are not mathematically inclined to help their children with their mathematics homework. it was noted during the completion of the second worksheet that learners were confident in using the use of the dynamic online software, as is evident from this excerpt: andile: it's nice … it's really different. i feel a lot more comfortable because we are the generation that interacts a lot with technology so it's kind of normal and comfortable for us. (li, 14 june 2013) learners in control a sense of independence was noticed whilst learners completed the second worksheet. learners were engaging in their own learning process and seldom with their peers. very little assistance from the teacher was requested as opposed to when learners answered the first worksheet. learners played an active role rather than a passive one in their learning of trigonometric graphs whilst working in the dynamic online environment. this is corroborated by dylan's comment: ‘it is more easy than having a teacher speak to you and teach you in a classroom’ (li, 14 june 2013). teachers are generally seen as gatekeepers of knowledge whereas using this kind of technology allows learners to control their own learning. learners have the freedom to explore and discover independently. it was noted that as learners changed the amplitude and period (vertical and horizontal shifts) of the trigonometric graphs, they jotted down notes about the changes in the graph. it was evident that their knowledge and skills for answering questions based on trigonometric graphs were significantly enhanced by working within the dynamic online environment. following along similar lines, clements, julie, yelland and glass (2008) maintain that concepts are more likely to be lodged in long-term memory when learners make notes. thus, greater understanding takes place as learners become participators rather than spectators in the learning process. this notion is supported by the following excerpts: bradley: i find it very useful because i can be involved in the process of learning on how to do mathematics. (li, 14 june 2013) tahlia: it is a nice way to reinforce what i had learnt and i can see those visual images. … i can remember that in my mind rather the words and the written text. (li, 14 june 2013) the use of web-based learning contributes to learner-centred education since the teacher takes on the role of the facilitator. the learning experience offered through the dynamic online environment is more than just accepting some sort of given mathematical fact from the teacher. this is supported by this excerpt: shivaan: in class if you want to know something you have to ask the teacher, they would have to draw a diagram to explain to you. … this way you can do it yourself. it was much better for your understanding; you can change the amplitude. (li, 14 june 2013) through the observations, it was noted that all the learners were fully engaged in each task with very little teacher interaction or participation during the learning process. thus, self-regulation occurs when learners are given the opportunity to think about their own learning process and make meaning of this process by themselves. this is exemplified by this comment: bradley: the concepts being taught are much easier because i can get in and change … you know like the equation of the trigonometric graph … so i can see the different ways on how it changes and why it changes. (li, 14 june 2013) thus, it was evident from the data collected that the learners participating in the study were in full control as they experienced the changes in the shape and position of trigonometric graphs first hand. this first-hand experience assisted in self-assuring the participating learners of their reasoning. the learners’ reasoning skills and knowledge related to changing the values in the equations of trigonometric graphs were significantly improved. learners could now see and deduce the impact of these changes on the shape and position of the trigonometric graphs. technology is the future the very essence of integrating technology and mathematics demonstrates a progressive development in mathematics education. it moves from the unmotivated norm of pen and paper to a discipline that utilises technology-based tools within the teaching and learning process. following along similar lines andile indicated that ‘[o]bviously our world is continuously moving ahead towards technology and eventually i believe that all lessons will be taught only with technology’ (li, 14 june 2013). adding to this, walking into a class with technology according to shivaan is ‘not a daunting thing. it's a very good thing to walk into a classroom with technology as it shows development in maths’ (li, 14 june 2013). similarly, andile stated that the concepts in mathematics are more accessible at a level which we can understand. … we interact a lot with technology and this is just putting something that we learn at school in our form of communication. (li, 14 june 2013) as is evident from the excerpts from the interview transcripts, the participating learners were able to comprehend concepts being taught more easily through the use of technology-based tools. the participants were enthusiastic about using technology-based tools and they indicated that they enjoyed using these tools. the participants’ cognitive development was enhanced through the use of technology-based tools. they were of the belief that they were now exposed to new technology and how it works. this is validated by the following excerpts: tahlia: now we are living in a time where technology is all over. so if you used to it already then it won't be so foreign when you come across it later. (li, 14 june 2013) shivaan: so much technology around us and by doing this you get exposed to new types of technology. (li, 14 june 2013) as indicated earlier, there are many types of technology-based tools that may be used to support mathematical investigations and connections without any specialised training on how to use them. technology-based tools are designed to make our lifestyle easier and their emergence in mathematics is definitely useful. this implies that teachers do not need to spend as much time sketching or re-sketching a function on the board or attempt to explain a 3d phenomenon in a 2d environment. motivation it was noted through observations and the interviews that all learners described and demonstrated the completion of the activity in collaboration with the dynamic online software as an exciting and exhilarating experience. attractiveness and entertainment value are the core properties of animations, which make them a motivating form of presentation with the ability to sustain one's attention (lowe, 2001). similarly, bradley stated that: ‘i am more open to this idea of learning about maths … encourages me to go for maths’ (li, 14 june 2013). it is imperative that in teaching mathematics, learners understand what is discussed and what is being taught to them. this is even more so in south africa, since there are 11 official languages3 and thus a mathematics class will likely contain learners with diverse languages. however, technology-based lessons aid in eliminating the language barrier since the learner is exposed to visual information rather than verbal information. many learners such as bradley experienced the same difficulty with language and trigonometry. this is supported by bradley's statement: especially with like trig i can't comprehend what is being taught if it's spoken to me but were if i see it visually and be able to interact with the technology i understand it better. (li, 14 june 2013) meaningful interaction through observation, it was noted that learners did not experience a learning block. they displayed a high level of confidence and certainty during the completion of the worksheet within the dynamic online environment. this is likely due to the dynamic online environment, which offered a self-explanatory learning process aided by visualisation. the use of technology-based tools in mathematics ensures an answer that is without human error. thus, technology-based tools are ideal to use when a learner wants to check the validity of their reasoning in a mathematical problem especially when they are not in the classroom with their teacher, peers or tutors. it is important to note that learners need to initially sketch their graphs by hand. in this way, they demonstrate their skills and understanding of trigonometric graphs. however, it is just as important to expose learners to technology-based tools in mathematics. this notion is validated by andile's statement: it's just a new way of thinking about it. it's something different usually we so use by doing it by ourselves by hand and now it kind of broaden our horizons in respect to our way of thinking and feel. (li, 14 june 2013) the integration of dynamic online software assists with the visual illustration of the theoretical knowledge the learner currently possess. tahlia expressed that due to technology her knowledge about trigonometric graphs ‘has heightened because like i said you can see it …easier’ (li, 14 june 2013). it is important to note that learners may overcome language and other challenges in mathematics through the use of dynamic online software. technology-based tools create an external image that is transposed to one's current knowledge. self-discovery tasks, like the second worksheet in this study, allow learners to investigate, notice and make generalisations before the teacher could explain these to them. conclusion top ↑ this study aimed to explore the implications of the use of dynamic online software when teaching trigonometric graphs to a group of grade 10 mathematics learners. the use of technology in mathematics classrooms is not new. many technology-based tools are available and may be used in mathematics. teachers do use slides, movies, cassette players, data projectors and overhead projectors to enrich the teaching and learning process. in this study, it was apparent that the use of a dynamic online environment allowed learners to develop a well-founded and enhanced mathematical understanding of trigonometric graphs. the dynamic online environment allowed for each trigonometric graph to be visualised and decreased complicated calculations, thus allowing the learner to focus on important and critical ideas. the geogebra program allowed learners to obtain immediate feedback. this assisted in motivating and enhancing the learners’ confidence levels when working with trigonometric graphs in mathematics. furthermore, the dynamic online environment provided a platform for learner communication after school hours and assisted in overcoming issues associated with language barriers in mathematics. through actively participating in the lessons, learners achieved and demonstrated a concrete understanding of trigonometric graphs. based on the analysis of the data collected from both worksheets, it was evident that learners performed better on the second worksheet. this worksheet was completed within a dynamic online environment. it was evident that learners’ skills and knowledge related to answering questions based on trigonometric graphs were enhanced. based on learner feedback, this improved knowledge and skills were related to the learners being in control of their own learning and seeking immediate feedback and interaction, which was possible within a dynamic online environment. moreover, the use of technology-based tools in the classroom is likely to equip learners with computer skills and knowledge. due to exposure to technology-based tools, learners can become competent in their computer skills thus resulting in a confident tech-savvy generation. although dynamic online software was used, the teacher was not absent from the teaching and learning process. the teacher acted as the facilitator, which was fundamental and critical to the online learning. the teacher played an instrumental role in fostering and scaffolding learners’ mathematical understanding. despite acknowledging that the use of technology-based tools improves learning, there are a number of schools and teachers that face a variety of barriers related to issues of cost, lack of expertise, lack of the necessary infrastructure, resistance to change from traditional teaching methodologies, lack of connectivity needed to access internet services as well as lack of electricity. it is thus important that teachers and learners do not depend entirely on technology-based tools when teaching mathematics. a combination of hands-on and technology-enhanced activities advance learners’ understanding and confidence levels in mathematics. thus, allowing for a merging of contemporary and traditional teaching and learning strategies in mathematics classrooms enriches the learning process. acknowledgements top ↑ we would like to thank the university of kwazulu-natal's teaching and learning office for funding part of this study. the opinions expressed here are those of the authors and do not necessarily reflect the position, policies or endorsements of the university. competing interests no financial or personal relationship(s) have inappropriately influenced the writing of this article. authors’ contributions j.n. 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(2008). teaching trigonometric functions: lessons learnt from research. mathematics teacher, 102(2), 144–150. appendix 1 top ↑ pencil and paper worksheet trigonometric graphs top ↑ 1. graph of y = a sin bx, -180° ≤ x ≤ 180° 1.1. sketch the graph of f(x) = sin x on the graph paper provided. (5) 1.2. if you increase the value of a to 2, what do you think will happen to the shape of the graph? (1) 1.3. if you decrease the value of a to 1/2, what do you think will happen to the shape of the graph? (1) 1.4. if you increase the value of b to 2, what do you think will happen to the shape of the graph? (1) 1.5. if you decrease the value of b to 1/2, what do you think will happen to the shape of the graph? (1) 2. graph of y = a cos bx, -180° ≤ x ≤ −180° 2.1. sketch the graph of g(x) = cos x on the graph paper provided. (5) 2.2. if you increase the value of a to 3, what do you think will happen to the shape of the graph? (1) 2.3. if you decrease the value of a to 1/3, what do you think will happen to the shape of the graph? (1) 2.4. if you increase the value of b to 3, what do you think will happen to the shape of the graph? (1) 2.5. if you decrease the value of b to 1/3, what do you think will happen to the shape of the graph? (1) 3. study the graph below and answer the questions that follow. 3.1 what is the period of f(x)? (1) 3.2 write down the equation of f(x). (2) 3.3 what is the maximum value of f(x)? (1) 3.4 what is the minimum value of f(x)? (1) 3.5 which one of the following statements is correct? (write down only the correct letter.) a) f(x) is not symmetrical about any line. b) f(x) is symmetrical about the x-axis. c) f(x) is symmetrical about the y-axis. d) f(x) is symmetrical about the line y = x. (1) 4. the equation of this graph can be written as y = c + k cos κ. what are the values of c and k? (2) appendix 2 top ↑ online activity worksheet step 1: log onto your computer. step 2: go to the internet browser and enter: www.govenderrg.wikispaces.com step 3: click on the trig graphs link along the left-hand side. question 1 top ↑ answer the questions (1.1–1.4) below based on the sin and cos graph (click on the sin and cos link separately). 1.1 top ↑ set a = 1, b = 1, c = 0° and d = 0 move the slider of a so that it increases from 0 to 10. move the slider of a so that it decreases from 0 to -10. describe the shape of the graph when: a > 0 a < 0 1.2 top ↑ set a = 1, b = 1, c = 0° and d = 0 move the slider of d so that it increases from 0 to 10. move the slider of d so that it decreases from 0 to -10. describe the shape of the graph when: d > 0 d < 0 question 2 top ↑ move the sliders a, b, c, d one at a time and match the statement with the unknown. y = a · sin(b · (x ± c)) + d y = a · cos(b · (x ± c)) + d statement choose the appropriate unknown a, b, c or d 2.1 affects the period (repetitive cycle of graph) 2.2 affects the shifting of the graph (up or down) 2.3 affects the amplitude (maximum displacement from x axis) 2.4 affects the phase shift (left or right) question 3 top ↑ the following sound waves show the breathing patterns of two runners the sketch above shows the graphs f(x) = sin x + 1 and g(x) = cos x where x ∈ [0°; 360°] click on the sin vs cos graph link and set the graphs to f(x) = sin x + 1 and g(x) = cos x 3.1 write down the equation of the graph passing through a, c, e, f and g. 3.2 what is the amplitude of the graph that passes through points a, b, d, f and g? 3.3 write down the range of f(x) = sin x + 1. 3.4 write the maximum and minimum values of f(x) = sin x + 1. 3.5 write the maximum and minimum values of g(x) = cos x. 3.6 write down the coordinates of points b and g. 3.7 write the equation of the graph that undergoes a horizontal shift from its standard form. 3.8 for which value(s) x is cos x > sin x + 1? question 4 top ↑ the graph below shows one complete normal breathing cycle. the cycle consists of inhaling and exhaling. it takes place every 5 seconds. the velocity of air flow is positive when we inhale and negative when we exhale. it is measured in litres per second. if y represents velocity of air flow after x seconds, find a function of the form y = asin bx that models air flow in a normal breathing cycle. hint: period = and model the above graph using the sin graph online. appendix 3 top ↑ learner semi-structured interview schedule top ↑ do you enjoy doing mathematics? what are your thoughts about the use of technology in the maths classroom (depending on response form question 1)? is it easier to understand your maths work through the use of technology? why? what effect does the use of dynamic technology have on your mathematics understanding of concepts being taught? how do you feel about walking into a class that uses technology? with respect to question 5 does this changes your attitude towards doing mathematics? would the use of technology, such as the tool used in your maths lesson, benefit you later in life? are you comfortable using technology in maths and why? would you have understood at the level you did if the lesson had not used technology? why? would ever go back to the website to revise the concepts of trigonometric graphs? when (before test, exams, homework, etc.)? where (at home, school, travel, library, etc.)? things to take note whilst doing interview: top ↑ learner expressions and body movements when responding. emphasis in tone or speech when responding. words and questions learners had problems understanding. footnotes top ↑ fn0001. dynamic online software is used for teaching mathematics in a discovery mode. learners are encouraged to use computer software (such as geometer's sketchpad or geogebra) to construct figures that can be altered by dragging points around the computer screen whilst the underlying relationships are unchanged. fn0002. a dynamic online program that may be used for teaching mathematics. fn0003. the 11 official languages recognised are english, afrikaans and nine african languages: isizulu, isixhosa, setswana, tshivenda, xitsonga, sesotho, isindebele, siswati and sepedi. 66 p34-42 maharaj final 34 pythagoras 66, december, 2007, pp. 34-42 using a task analysis approach within a guided problem-solving model to design mathematical learning activities aneshkumar maharaj school of mathematical sciences, university of kwazulu-natal email: maharaja32@ukzn.ac.za the fet curriculum statements for mathematics advocates that knowledge integrates theory, skills and values. this paper focuses on a guided problem-solving teaching model that provides a framework to do this. a task analysis approach is used within this framework to illustrate how educators could frame mathematical questions based on the relevant mathematical content. introduction an analysis of matriculation examiners’ reports (kwazulu-natal department of education, 2000, 2001 & 2002; gauteng department of education, 2000, 2001, 2002 & 2003) indicated a number of recurring issues evident in candidates’ work that lead to poor performance: • a poor understanding of mathematical terminology and concepts; • an inability to recall and apply formulae, and poor substitution skills; • an inability to recall and apply algorithms; • a lack of manipulative skills which relate to simplifying arithmetic, algebraic and trigonometric expressions, and solving of polynomial equations of degree two and three; • an inability to answer interpretative questions; • a poor knowledge of book-work, in particular statements of and proofs of theorems or rules in the context of given diagrams. these were reflected by the candidates’ responses to the sorts of questions that are typical of those that appear in examination papers (box 1). to solve the equation in the first question a learner should be able to identify it as a quadratic equation and then recall and apply a suitable algorithm, for example, the factorisation technique for solving a quadratic equation. the second question requires an interpretation of the question together with required manipulative skills to transform the given equation to the equivalent centre-radius form, an integration of relevant knowledge from algebra and geometry supported by appropriate visualisation. these manipulative skills are required to deduce the embedded information from what is given. for the third question, a requirement is that the given information (in symbolic form) and figure be used to prove the theorem which states that if two triangles are equiangular then the corresponding sides are in proportion. this paper is based on a number of assumptions regarding the learning of mathematics: • learning mathematics requires the mastery of concepts, relationships – which include conventions such as – and algorithms, and their application. • learning proceeds from the known to the unknown. • there are networks of concepts in the brain which aid understanding, so learning should result in a gradual expansion and modification of these networks. the latter is premised on the work of hiebert and carpenter (1992) who presented a theory of understanding based on the formation and interplay of internal and external representations. • learning of a mathematical topic involves “many actions, processes, and objects that need to be organised and linked into a coherent framework, which is called a schema” (dubinsky, weller, mcdonald & brown, 2005). the authors refer to this as the apos (actions, processes, objects and schema) approach. • it is possible to plan for meaningful teaching of mathematics by simultaneously addressing the cognitive and affective domains (cangelosi, 1996), by focusing on the actions, processes, objects and schema (apos). • an apos analysis of a topic could give insight into the type of thinking that an individual might be capable of. it does not indicate what happens in an individual’s mind since this is unknowable. this paper focuses on addressing the following question, beginning with the assumptions above: how might the educator work on tasks so as to promote mathematical learning and address the recurring problematic issues indicated in the matriculation examiners’ reports? 3 aneshkumar maharaj 35 in trying to find an answer to this question, this paper will address three aspects. firstly, it discusses an interpretation of cangelosi’s problemsolving model. secondly, it illustrates how this model can be used to promote teaching and learning of mathematics, and finally, it shows how a task analysis approach can be used within this model to formulate mathematical activities. these activities are aimed at developing competencies required of mathematics educators, as implied in the fet curriculum statements for mathematics (department of education, 2003). the structure of these activities is based on work done by mason (2000). as such, the framing of questions and activities is guided by the following conjecture: if learners are both led through and invited to participate in typical mathematical activities, then they are more likely to appreciate mathematics as a discipline. this helps to focus on both the cognitive and affective domains as noted in the assumptions. it is particularly helpful for designing activities which elicit in the learner a willingness to try and appreciation of the relevant mathematical concept, during the different reasoning phases of the guided problem-solving teaching model (figure 1). the guided problem-solving model the guided problem-solving model, which is an interpretation of cangelosi’s (1996) problemsolving model, is illustrated in figure 1. this approach has three phases: 1) inductive reasoning (conceptual phase); 2) inductive and deductive reasoning (simple knowledge and knowledge of a process phase); 3) deductive reasoning (application phase). it is important to note that there is always an interplay between inductive and deductive reasoning. they occur continuously and are in a constant iterative relationship in mathematical thinking. for example, in an inductive process, there is frequently a preliminary ‘generalising’ step. a conclusion or the finalising of the inductive aspect forms the beginning of the deductive aspect. generalising at each of the different phases implies that a deductive mode of reasoning comes into play. furthermore, applications or the solving of problems requires knowledge of algorithms, and could lead to the discovery of new relationships. hence, there is an interplay between phases (see figure 1). in the inductive reasoning phase, inductive learning activities should be used to construct a concept or discover a discoverable relationship. instruction through guided inquiry could help box 1. typical examination questions box 1. typical examination questions. 1. solve for x: 523 2 =+ xx 2. determine the equation of the tangent to the circle x2 2x + y2 + 4y = 5 at the point (-2; -1). 3. refer to the figure. a b c p q r in triangles and and abc pqr a p b q c r , $ $, $ $ $ $.= = = copy the sketch in your answer book and use it to prove the theorem that states and use is to prove the theorem that states pq ab pr ac qr bc .= = using a task analysis approach within a guided problem-solving model to design mathematical learning activities 36 learners to construct a concept such as the concept of a quadratic equation or a cyclic quadrilateral, or discover a discoverable relationship, for example, a perfect square is always greater than or equal to zero. for the former, the use of an ‘examples and non-examples approach’ could be useful. the inductive and deductive reasoning phase is the intermediate phase, which focuses on the teaching of simple knowledge and algorithms. while the basis for learning meaningful mathematics is to help learners construct concepts and to discover relationships, learners also have to remember conventional names for these concepts and relationships. they should therefore be exposed, by means of a direct instruction process, to mathematical information which they are required to remember. requisite information includes the following examples. • simple knowledge: recalling specific responses (not a multi-step process) to specific stimuli. here learning activities should target the end product, namely, the correct statement of the simple knowledge. example: state the standard form of the quadratic equation. • comprehension: extracting and interpreting meaning from symbolic representations. example: what do the following mean to you? (a) x(x + 1) = 12 (b) a.b = 0 ⇒ a = 0 or b = 0. • knowledge of a process: engaging in an algorithm or multi-step process. here, learning activities should aim to help learners achieve algorithmic skills. example: outline the procedure to solve a quadratic equation by the factorisation technique. the direct instruction process for teaching of simple knowledge, comprehension and knowledge of a process should also have a number of appropriate stages. guided instructional activities could be designed to stimulate a willingness to attempt the mastery of simple knowledge, and promote comprehension and algorithmic skills. in the final phase, deductive reasoning is required. cangelosi (1996: 157) has suggested that “...deductive reasoning is the cognitive process by which people determine whether what they know about a concept or abstract relationship is applicable to some unique situation”. therefore, in this phase, the rules of logic have to be adhered to. application lessons require learners to put into practice previously developed or acquired concepts, relationships, information and algorithms. learners must determine whether or not any of these may apply to a problem. therefore, activities which stimulate learners to reason deductively should be planned in this phase. the guided problem-solving model could be useful in addressing and rectifying the recurring issues that contribute to poor performance amongst matriculation candidates. there are two simultaneous processes in this model, one an interplay, and the other a progression or drift to deduction. since the fet curriculum statements for mathematics advocate that knowledge integrates theory, skills and values, it calls for a development towards deductive reasoning. the guided problem-solving model is based on the theory of moving from the known to the inductive reasoning phase ▪ construct a concept ▪ discover a relationship ▪ willingness to try, appreciation inductive and deductive reasoning phase ▪ simple knowledge ▪ comprehension ▪ knowledge of a process deductive reasoning phase ▪ application ▪ willingness to try ▪ solve problems ▪ appreciation figure 1. a guided problem-solving teaching model. aneshkumar maharaj 37 unknown, taking into consideration how conceptual development occurs. a movement from the inductive reasoning phase towards the next two phases requires the gradual development of schemata and networks in the brain. schemata provide a mental framework for understanding and remembering information (bartlett, 1958). schemata or mental models provide meaning and organisation to experiences, and allow the individual to go beyond the given information (bruner, 1973). the nature of expertise is largely due to the possession of schemata that guide perception and problem-solving (chi, glaser & farr, 1988). if this is accepted then it seems that the facilitation of the learning process in mathematics needs to focus on the development and modification of schemata. the movement in phases within the guided problem-solving model is characterised by greater abstraction. during each of the phases, well formulated problems (or questions) should form an integral part of the design of learning activities. the educator has to design suitable learning activities to focus on the relevant aspects indicated in each of the phases. these learning activities should guide learners to attain the relevant objectives or intended outcomes for each phase. a task analysis approach, discussed below, could assist in the design of such activities. to help learners improve their understanding of mathematical concepts, activities should be designed which encourage them to explore, generalise, explain, argue and finally prove. commenting on the importance of getting learners to explain and justify their reasoning, jones (2000) indicates ... the requirement to explain and justify their reasoning requires students to make the difficult transition from a computational view to a view that conceives of mathematics as a field of intricately related structures. (2000: 58) the design of four tasks, which aim to facilitate such a transition, will be explained in this paper. task analysis approach embedded within a teaching model for mathematics the previous section led to the conclusion that a task analysis approach could be used to develop component knowledge and skills for certain problem types within the framework of the guided problem-solving model. task analysis has two useful functions (usability first, 2005): • it refers to a set of methods for decomposing tasks into subtasks to understand procedures better and provide support for those tasks. • it is useful for spotting potential errors that could arise from steps in the process which could be difficult or confusing. the second point indicates that task analysis has the potential to serve a predictive function, such as identifying possible impediments to learning. task analysis for the purpose of instructional design “... is a process of analysing and articulating the kind of learning that you expect the learners to know how to perform” (jonassen, tessmer & hannum, 1999: 3). this implies that within the framework of the fet national curriculum statements for mathematics (department of education, 2003), task analysis is useful to design baseline and diagnostic assessments. baseline assessment is used to establish what learners know or already can do, and therefore helps in the planning of activities. diagnostic assessment is used to discover the cause or causes of a learning barrier, and “therefore assists in deciding on support strategies” (department of education, 2003: 64). the performance of a task analysis on typical examination questions could be useful to educators when designing instruction. it could be used to: • determine the instructional goals and objectives, • define and describe (in detail) the tasks and sub-tasks, • specify the required type of knowledge, • select learning outcomes appropriate for instructional development, • prioritise and sequence tasks, • determine instructional activities and strategies that foster learning, • select appropriate media and learning environments, and • construct performance assessments and evaluation. jonassen et al. (1999) noted that the task analysis process has five distinct functions: • classification of tasks according to learning outcomes, • identifying or generating a list of tasks, • prioritising tasks and choosing those that are more feasible and appropriate, • identifying and describing the components of the tasks, and • defining the sequence in which instruction should occur to best facilitate learning. using a task analysis approach within a guided problem-solving model to design mathematical learning activities 38 however, for these functions to be realised, the assumption is that the educator is competent in the subject content. learner-centred activities derived from the task analysis previously built-up understanding could either hinder or promote the understanding of new ideas davis, 1992); this has implications for both learning and teaching. educators should therefore make efforts to determine whether learners have the relevant prior knowledge or abilities required for the understanding of new ideas. the task analysis and apos approach – identifying required actions (manipulative skills), processes, objects and schema – can be of benefit to educators. there are a number of possibilities that this approach opens up for professional training, including: • the design of spot tests or pre-tests, • the restructuring of questions or problems to make them more learner-centred, and • the development of summaries and strategies to aid problem-solving. • a task analysis approach could also help to pinpoint the different competencies required by mathematics educators to do these. the following sections note required competencies, and illustrate some of the possible situations in which task analysis could be useful. spot tests or pre-tests mathematics educators should have some way of determining whether or not the requisite prior knowledge, skills and abilities are really in place. this could be done by oral questioning or designing a spot test for learners to complete. spot tests could also be used to determine whether learners are able to outline the key steps in the proof of a theorem, based on the context of a given diagram. a time limit should be placed on the completion of such tests, and each test should be followed by a class discussion. spot tests or pretests could thus be useful for baseline and diagnostic assessments. task 1 suppose that the objective or specific outcome for the first lesson unit on solution of quadratic equations is: at the end of the lesson learners should be able to solve quadratic equations of the type 3 2 52x x+ = by using the factorisation technique. required for this lesson unit: 1.1 pinpoint the prior knowledge / abilities assumed. 1.2 for the prior knowledge / abilities indicated in (1.1), design a spot test that could help you to determine whether learners have these knowledge and abilities. a possible response to task 1: 1.1 prior knowledge/abilities assumed: a) ability to factorise quadratic polynomials of the types x2 4 ; x2 + 3x ; 3x2 + 2x – 5 b) knowledge and application of the zero product rule: for real numbers p and q, p.q = 0 ⇔ p = 0 or q = 0. c) ability to solve linear equations of the type: 2x + 5 = 0; 3x 2 = 0. spot test [10 minutes] 1. factorise 1.1 x2 + 3x 1.2 x2 4 1.3 3x2 + 2x 5 1.4 2x2 3x + 1 2. complete: for real numbers p and q if p . q = 0 then 2.1 what do you understand by the result given in 2. above? 3. solve for x: 3.1 x(x + 3) = 0 3.2 (x + 2) (x 2) = 0 3.3 3x2 + 2x 5 = 0 3.4 x(2x 3) = -1 4. now formulate a procedure to solve quadratic equations of the type x(ax + b) = d. box 2. example of a spot test. aneshkumar maharaj 39 an example of a spot test that may be used to pinpoint prior knowledge and abilities (1.1) is illustrated in box 2. in the context of the guided problem-solving model, this spot test targets the inductive and deductive reasoning phase. note that question 1.4 of this test is designed to formulate an algorithm (which links component parts) to solve quadratic equations. a class discussion could focus on connecting, refining and justifying steps in the algorithm. restructuring a problem – an example from analytical geometry educators should be able to work out problems based on the different sections of the syllabus. for a given problem they should be able to pinpoint the relevant prior knowledge and abilities required by learners. if the problem is deemed to be difficult for learners, educators could use this information to restructure the problem, with the aim of helping them solve it. this restructuring could include: (a) rewriting the given information clearly, (b) providing a diagram or figure to support the visualisation of the given information, and (c) formulating suitable sub-questions to enable learners solve the problem. a typical example of what would need to be done is illustrated in task 2. task 2 study the problem: determine the equation of the tangent to the circle x2 2x + y2 + 4y = 5 at the point (-2; -1). now answer the following questions based on the above problem. 2.1 solve the above problem. 2.2 write down the relevant prior knowledge/abilities required by learners to work out the problem. 2.3 restructure the above problem with the aim of helping learners to solve the problem. an example of a possible restructuring of the given problem is provided in box 3. the original problem required learners to integrate different component tasks into a solution. if the relevant component knowledge and abilities are in place, learners should also be exposed to a subsequent task such as the following: consider the problem: the straight line 0102 =−+ xy intersects with the circle with centre o (the origin) at points a and t, where a is on the x-axis. determine the co-ordinates of d if od is perpendicular to at and d lies on at. write down the key steps that lead to a solution of this problem. in the context of the guided problem-solving teaching model such activities help to make the transition to the deductive reasoning phase. deductive reasoning – an example from geometry examiners’ reports indicated that learners perform poorly in the proofs of geometry theorems. this could be as a result of the way in which these theorems were introduced to learners. for example, a theorem was stated and then the proof was given. learners also have many misconceptions and make common errors that relate to the application of the theorem’s statement. a mathematics educator could help learners by (a) pinpointing the relevant prior knowledge/ abilities required by learners to prove the theorem [first analyse the proof of the theorem], (b) designing a worksheet that helps learners arrive 1. the figure shows the circle defined by the equation x2 2x + y2 + 4y = 5. the line l is a tangent to the circle at the point m(-2;-1). 1.1 determine the co-ordinates of p, the centre of the circle. 1.2 calculate the gradient of mp. 1.3 why is mp perpendicular to the line l ? 1.4 now determine the defining equation of the tangent l. p m(-2;-1) y x l box 3. restructuring a problem – an example. using a task analysis approach within a guided problem-solving model to design mathematical learning activities 40 deductively at the statement [draw the relevant diagram, write up the given information and formulate suitable lead-on questions by using the relevant prior knowledge/abilities], (c) formalising the statement of the theorem [write the statement in the “if.....then” form and use diagrams to illustrate the “cause and effect”], and (d) giving quick and simple applications based on the pre-requisites to apply the statement of the theorem. task 3 was designed to help mathematics educators develop these competencies. task 3 the competencies stated above could be developed through the following example: 3.1 complete the following statement of a theorem: if two triangles are equiangular then their .................... 3.2 for the statement of the theorem in (3.1) above you are required to do the following: a) list the prior knowledge/abilities required by learners to prove the statement. b) design a worksheet which could enable learners to deductively arrive at the statement. 1. in ∆ abc and ∆ pqr $ $a p= $ $b q= and $ $c r= a b c k m p q r 1 2 \\ \\ / / 1.1 on ab point k is marked so that ak = pq. on ac point m is marked so that am = pr. km is joined. 1.2 prove that ∆ akm ≡ ∆ pqr. 1.3 why is $ $k q ?1 = 1.4 now show that $ $k b1 = . 1.5 deduce that km bc. 1.6 why is ak ab am ac = ? 1.7 now show that pq ab pr ac = . 1.8 point e is marked on ba such that be = qp and point f is marked on bc such that bf = qr. write down equal ratios similar to those in 1.7 by using qp, ba, qr and bc. 1.9 use the ratios from 1.7 and 1.8 to write down ratios that relate all the sides of ∆ pqr and ∆ abc. 1.10 now write down a general statement about the corresponding sides of equiangular triangles. box 4. example of a worksheet – deductive response (task 3.2b). aneshkumar maharaj 41 c) briefly outline strategies to eliminate common errors / misconceptions relating to the application of this theorem. an example of a response to (b), after the prior knowledge and abilities were detected, is provided in an example worksheet (box 4). to get learners to integrate component tasks into a solution, such an activity should be followed by a suitable spot test, for example see box 5. these activities are designed to make the transition towards the deductive reasoning phase of the guided problem-solving teaching model. develop summaries and strategies to support problem-solving summaries and strategies help to expand and improve the connections of the network in the brain. the devising of plans should be taught, as it is a form of representation, and an important part of problem-solving. to develop into successful problem-solvers learners need to spend time analysing problems and the directions that could be taken (fernandez, hadaway & wilson, 1994). this should lead to the development of frameworks (schemata) aimed at pinpointing the different phases and the processes that are necessary when solving particular types of problems. such schemata illustrate that the development of mathematical thinking is a sequence of ever more advanced transitions from operational to structural outlooks. a mathematics educator should facilitate the development of summaries and teachinglearning strategies in mathematics to help learners. these strategies could include frameworks or schemata, which involves representing, connecting and justifying procedures. task 4 has been designed as an example of what could be done to help educators develop summaries and strategies. task 4 for any one of the sections in the mathematics syllabus for grades 10 to 12, give an example of a summary or strategy that you helped your learners to develop. the following should be indicated. 4.1 the grade and topic/section. 4.2 full details of the summary or strategy. 4.3 has this summary or strategy helped your learners improve their performance in mathematics? motivate your response. an example of a strategy that may be used to solve equations is illustrated in figure 2. the schema for solving equations algebraically associates particular problem-solving procedures with each of the algebraic objects (linear, quadratic and cubic equations). such a schema requires that learners first identify the type of equation (structure) they are confronted with. this should be followed by an algorithm which outlines a particular procedure, for example, the factorisation technique for quadratic equations. such a schema can help learners to “… understand, deal with, a b c p q r in triangles abc and pqr .r̂ ĉ and q̂ b̂ ,p̂â === write down the key steps to prove the theorem that states . qr bc pq ab = box 5. example of a spot test. linear use of quadratic formula factorisation technique quadratic cubic solving equations lhs – terms in unknown rhs – constant terms ! grouping ! use of factor theorem figure 2. schema for solving equations algebraically. using a task analysis approach within a guided problem-solving model to design mathematical learning activities 42 organise, or make sense out of a perceived problem situation” (dubinsky 1991: 102), as it includes a more or less coherent collection of objects and processes. in this case, the task entails solving equations – of degree one, two or three – in one unknown. by focussing on the structural nature of such schemata, explicit descriptions of possible relationships between schemata can be detected. instruction should therefore focus on the need for, the usefulness of and the development of such schemata to aid thinking and bringing into play encoded labels for algorithms. conclusion the task analysis approach, with a focus on detecting required actions, processes, objects and schema, could be used to design process-based learner-centred activities within the framework of the guided problem-solving model. it is evident from the four tasks outlined in this paper that this approach requires competent educators who have a sound knowledge of the content to be taught. the approach can be used to design questions and activities that address the typical common errors, misconceptions and difficulties of learners as identified in the examiners’ reports. some examination questions contain complex component tasks. therefore, there is also a need to follow-up with activities that require learners to integrate component tasks. within the framework of the guided problem-solving model, such activities target the deductive reasoning phase. references bartlett, f.c. 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(2000). the student experience of mathematical proof at university level. international journal of mathematical education in science and technology, 31(1), 53-60. kwazulu-natal department of education (2000). matriculation examiners’ report for mathematics 1999. kwazulu-natal department of education (2001). matriculation examiners’ report for mathematics 2000. kwazulu-natal department of education (2002). matriculation examiners’ report for mathematics 2001. mason, j. (2000). asking mathematical questions mathematically. international journal of mathematical education in science and technology, 31(1), 97-111. usability first (2005). usability glossary: task analysis. retrieved march 9, 2005, from http://www.usabilityfirst.com/glossary/main.cgi ?function=display_term&term_id=294 66 p2 editorial pythagoras 66, december, 2007 2 thank you … from the editor my tenure as editor of pythagoras comes to an end with this issue and it would be remiss of me not to make a few comments. the table below provides an overview of the submission traffic since issue 60. it is apparent that there has been a steady increase of submissions with a corresponding increase of published papers. this points to an active mathematics education research community, and an increased commitment to research dissemination in south africa. edition number of articles received south african authors foreign authors rejected by editor and sent back for reworking reviewed by panel rejected by majority of panel accepted by majority of panel published articles 60 8 7 1 2 6 1 5 5 61 10 9 1 1 9 1 8 5 62 13 12 1 1 12 2 10 7 63 15 15 0 3 12 2 10 6 64 special edition: mathematical literacy 8 65 14 14 0 1 12 2 10 8 66 14 13 1 0 14 4 10 8 i wish to pay tribute to our reviewers who have given generously of their time in reviewing papers and providing sound and professional advice. good reviewers are fundamental for the integrity of any academic journal. it is through this process of peer review that pythagoras sustains robust and rigorous academic engagement. i also wish to thank all the authors who have submitted papers and subjected themselves to the often daunting peer review experience. i wish to encourage those whose papers were either substantially dissected or rejected not to lose heart – learn from the experience and try again! i would also like to pay tribute to our two language/formatting editors, ms mariss stevens and ms nikki köhly for their meticulous work and unfaltering support. without them pythagoras would simply not happen! my thanks also go to the amesa council for their continuous support and encouragement. appropriate dissemination (and consumption) of good quality mathematics education research in south africa is an issue of concern. pythagoras plays a crucial role in the dissemination and advocacy of good research and it is important that it remains the journal of choice for experienced and aspiring authors. i wish my successor, prof alwyn olivier from stellenbosch all the best as he takes pythagoras to new heights. marc schäfer abstract introduction literature review theoretical framing: commognitive theory research methodology findings and discussion conclusion acknowledgements references footnotes about the author(s) hlamulo w. mbhiza department of mathematics education, college of education, university of south africa, pretoria, south africa citation mbhiza, h.w. (2022). grade 10 teachers’ example selection, sequencing and variation during functions lessons. pythagoras, 43(1), a696. https://doi.org/10.4102/pythagoras.v43i1.696 original research grade 10 teachers’ example selection, sequencing and variation during functions lessons hlamulo w. mbhiza received: 02 apr. 2022; accepted: 20 june 2022; published: 30 aug. 2022 copyright: © 2022. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract examples that teachers choose and use are fundamental to what mathematics is taught and learned, and what opportunities for learning are created in mathematics classrooms. this qualitative multiple case study, using sfard’s commognitive theory, draws attention to mathematics teachers’ classroom practices during functions lessons which is unexamined in the south african context. in this article, data sets include unstructured non-participant classroom observations on functions, which were videorecorded. sfard’s commognitive theory served as an appropriate lens in interpreting and analysing teachers’ discourses and giving meaning to teachers’ classroom practices during functions lessons. the findings demonstrate that the example selection and sequences teachers used during functions lessons either constrained or enabled the development of endorsed narratives about the effect of parameters on the different families of functions. keywords: discourse; functions; examples; commognition; mathematics; teaching. introduction the concept of functions has received attention within the field of mathematics education (kabael, 2011; trigueros & martinez-planell, 2010). very few protracted studies have been conducted in south africa in respect of functions (moalosi, 2014; moeti, 2015; mudaly & mpofu, 2019; roberts, 2016), and the existing studies were conducted with learners to explore the difficulties they experience when learning the topic. felix klein in 1908 viewed functions as ‘the soul of mathematics’, and this notion has since been discussed by various researchers (hansson, 2006; mudaly & mpofu, 2019). in relation to this, sierpinska (1992, p. 32) stated that ‘functional thinking should pervade all mathematics, and at school, students should be brought up to functional thinking’. this statement resonates with eisenberg’s (1992, p. 153) iteration that developing learners’ sense of functions ‘should be one of the main goals of the school and collegiate curriculum’. while the statements focus on the learners, they indirectly address the importance of teachers’ content knowledge on functions, to ensure learners’ development of knowledge related to the topic. lloyd, beckmann, zbiek and cooney (2010) posit that functions are one of the key topics in secondary school mathematics, because of their relatedness with other topics within the mathematics curriculum such as finance and growth, algebra, and equations, as well as patterns and sequences. thus, functions can be considered a meta-discourse of algebra internationally (sfard, 2012). within the functions concept in school mathematics, both its importance and problems relating to its learning have been researched and documented in mathematics education research (moalosi, 2014; mpofu & pournara, 2018). swarthout, jones, klespis and cory (2009) posit that functions are a very important topic in the mathematics curriculum, because of the role that the topic is often seen to play as a unifying concept in mathematics. this makes it essential for learners to develop good conceptual understanding of the topic. while this is the case, moalosi’s (2014) study with grade 11 learners demonstrated that functions are a topic that learners find difficult to understand, because of the over-reliance on procedures in learning the topic. another difficulty in the learning of functions is learners’ ability to observe change between the given variables and identify the relationships between them (moeti, 2015). sierpinska (1992) also suggested that teachers should introduce functions as models of relationships drawing from real-life situations, and in turn view functions as tools for representing a system in another system. the rationale: ‘for the motivation of mathematical concepts by using concrete examples in the teaching of mathematics stems from the commonly accepted notion that, nowadays, students are interested in the study of the subject matter if they are confident in the applicability of the material they are about to learn.’ (abramovich & leonov, 2009, p. 2) the interest and confidence are influenced by the quality of a teacher because learners observe and do what they observe the interlocutor does during teaching and learning in the classroom (sfard, 2008, 2012). of concern is that research on grade 10 teachers’ discourses and approaches during functions lessons is scarce within the south african context. in the current article, i attribute the challenges concerning functions to the examples that teachers select and use during teaching. examples that teachers select and use, and their sequence are fundamental to what functional contents are taught and learned, and the opportunities for learning that are created by teachers in mathematics classrooms (pillay, 2013). accordingly, the current study intended to answer the following research questions: how do teachers select and use examples while teaching functions? how does the selection and use of examples facilitate or limit the development of learners’ knowledge of functions? my contention is that for learners to understand the different properties of the concept of functions, teachers should ensure that learners are taught to undertake the following actions: interpretation and construction of functions to help them to comprehend them. interpretation refers to ‘action by which student makes sense or gains meaning from a graph (or a portion of a graph), a functional equation, or a situation’ (leinhardt et al., 1990, p. 8). this statement includes but is also not limited to actions such as describing changes brought by the changes in the values of parameters in a graph or table of values as well as reading off the values of x and y from sketched graphs, for example determining the values of x for which f(x) = g(x). the following section focuses on the review of previous studies on the concept of functions. literature review the constitutive elements of a function earlier, anderson (1978, p. 23) stated that the constitutive elements of a function refer to the ‘raw material, a rule or a process … and an end product’. of importance to note is that not every function has a rule or process; for example, a set of ordered pairs could constitute a function. that is, merely thinking of a function as a rule or process is dangerous for learners’ conceptual development as they then easily fall into the trap of only associating formulae or equations with the existence of functions (see vinner & dreyfus, 1989). in addressing these concerns, sierpinska (1992, p. 30) asserted that the constitutive elements of a function should be viewed as ‘worlds’ and the teaching of the concept should focus on three worlds: world of changes or changing objects, world of relationships and world of rules, regularity and laws. mathematics has discourses and teachers are expected to use mathematical rules, deeds and interactions that are part of the subject’s discourse and legitimise certain forms of mathematising both orally and from learners’ written work (sfard, 2008). firstly, the world of changes entails an identification of ‘what’ is changing in given relationships and ‘how’ the change is taking place. in this sense, teachers should teach learners how to work with the idea of ‘transformation’ in functions, and pay attention to the appearance, displacement and orientation of functions (chimhande, 2013; mudaly & mpofu, 2019). for sierpinska (1992), teachers must emphasise to the learners the need to move from viewing x and y as knowns and unknowns, to conceiving them as variables and constants for meaningful understanding of functions. it is essential to note that the formation of a new function from an old function can be viewed in two ways: numerically as magnitude changes in number operations and graphically as transformations in terms of reflection, rotation, translation and enlargement. considering that this is expected of teachers, it is assumed that teachers have adequate mathematics knowledge for teaching (mkt) relating to the teaching of functions (ball, phelps, & thames, 2008). this mkt is important, especially when the understanding that teachers with a stronger knowledge base are more responsive to learners’ mathematical learning needs appropriately and effectively is seriously considered (ball et al., 2008). also, when teachers possess stronger mkt, they are more likely to make fewer language and mathematical errors during teaching and learning and select and use examples effectively to bring the mathematical concepts to the fore. secondly, teachers should teach the learners how to observe change between the given variables and identify the relationships between them. accordingly, the nature of teachers’ exemplification during functions lessons plays a crucial role in promoting or hindering learners’ understanding of the topic. in this article, a critical examination of the examples that five participating teachers selected and used while introducing functions, as well as their sequencing of such examples, enables me to unearth the effectiveness of their teaching of the topic. in addition to the above discussion, for sierpinska (1992), functions should be viewed ‘as tools of description and prediction’ (p. 32) of how variables are related to each other, making functions models of patterns in real-life phenomena. this resonate with euler’s and dirichlet’s definitions of what a function is: that functional relationships can be expressed in terms of covariation or by using a rule of correspondence between variables (bazzoni, 2015; blanton, 2008; wilkie, 2020). in describing covariational relationships, borba and confrey (1996) stated: ‘one quantity changes in a predictable or recognisable pattern, the other also changes, typically in a differing pattern. thus, if one can describe how x1 changes to x2 and how y1 changes to y2 then one has described a functional relationship between x and y.’ (p. 323) this is not sufficient to constitute a function. according to bazzoni (2015), learners need to be taught that the association between the two variables can be understood as fixed points on a cartesian plane and are usually represented by a set of ordered pairs as coined by bourbaki in the form (x; f(x)). of importance to note is that the enablement of learners’ understanding of functional relationships depends on the quality of the examples that a teacher selects and uses to guide learners towards generality about specific functions concepts (essien, 2021). thirdly, a function is considered a rule that governs the relationship between variables (sierpinska, 1992). according to van de walle (2004, p. 436), a function can be viewed as a rule ‘that uniquely defines how the first or independent variable affects the second or dependent variable’. what should be noted is that rules, patterns and laws refer to well-defined relationships, a reason for a strong link between this concept and the one discussed above. demarois and tall (1996) argued that the development of the function concept is very complex and that change, relationships and rules are not mutually exclusive pockets of knowledge about the concept. this means that the world of changes or changing objects, world of relationships and world of rules, regularity and laws discussed above are interdependent on each other and the topic should be taught likewise. for example, when learners are observing the change in the values of the independent variable, they should be able to observe how such change influences the values of the dependent variable to construe a rule that signifies a functional relationship. thus, the foregoing necessitates that as teachers teach functions to their learners, all three conceptions should be developed if enabling learners’ fluency in working with functional problems is seriously considered. in view of the above, it becomes clear that change, relationships and rules are to be seen as components of a complex association in understanding the mathematical concept of function. the set of examples teachers introduced across the different lessons can be described in terms of the following patterns of invariance and variation: in all the examples we find an equation in the form ‘y = f(x)’, together with an equation in the form ‘y = a.f(x)+q’, as stipulated in the curriculum and assessment policy statements (caps). according to resnick (1997), mathematics is ‘a science of pattern’ in which there is an emerging invariant structure when a phenomenon is undergoing variation. the following section presents the espoused theoretical framing for the current study and details how the components of sfard’s (2008) commognitive theory are used in analysing and making sense of the teachers’ teaching of functions. theoretical framing: commognitive theory sfard’s (2008) commognitive theoretical framework is a lens to analyse and interpret teachers’ communication during functions lessons, and to understand the intricacies and elements of the discourses from what is or is not endorsed by the mathematics discourse community. the commognitive theoretical framework is influenced by ludwig wittgenstein and lev vygotsky who emphasise the ‘inseparability of thought and its expression, either verbal or not’ (sfard, 2015, p. 132), which means thinking in mathematics is a well-defined form of communication, and mathematics teaching is participating in a discourse (roberts, 2016). of importance to note is that the effectiveness of teachers’ communication of mathematical contents during teaching depends on their content knowledge. participants in the mathematical discourse show their internal communication through what they say, write, draw or sketch; therefore, communication is seen in both talk and action (mudaly & mpofu, 2019). the teaching of functions requires teachers to communicate different concepts, processes and rules explicitly and effectively, which is an expression of mathematics at an intrapersonal (cognition) and interpersonal (communication) level (vygotsky, 1987). the framework helped with avoiding oversimplified views of teaching. the commognitive theory also allowed for rich descriptions and discussion of teachers’ ways of teaching functions, through its focus on the contextual, cultural, dialogical and dynamic nature of participants’ discourses in mathematics. by doing so, commognitive theory was used to account for the differences in individual teachers’ thinking and teaching methods during the lessons on functions. according to sfard (2008), mathematics is ‘autopoietic’ because it is ‘a system that contains the objects of talk along with the talk itself’ (p. 129), a feature that makes school mathematics difficult to teach and learn. thus, familiarity with ‘what the discourse is all about’ (sfard, 2008, p. 130) is needed for participation in the discourse, but paradoxically this familiarity only comes through participation by mathematics teachers and learners. even though this study focuses on teachers, the nature of classroom teaching involves learners who should actively partake in the lesson for familiarity with the mathematical object and to develop mathematical discourse. this does not take away the fact that it is the job of the teacher to ensure that every learner learns to work with functions and to create the right environment that encourages meaningful mathematics classroom discourse. an effective environment for learning functions is one in which learners are allowed and encouraged to engage in investigative processes and where teachers create opportunities to explore particular cases, create conjectures and prove them to make generalisations (moeti, 2015). the word ‘discourse’ implies the use of words and symbols in a way that is generally endorsed by members of a community (sfard, 2015, p. 45). accordingly, mathematical discourse communicates mathematical ideas that are ratified by the body of theorems, proofs and laws that govern mathematics (sfard, 2012). the unit of analysis for sfard (2008) is discourse, which is considered a special type of communication, made distinct by its repertoire of admissible actions and the way these actions are paired with re-actions. according to sfard (2008, 2012), four components can be employed to understand and describe different mathematical discourses during teaching and learning: words and their use: words specific to mathematical discourse which teachers and learners use in discourse-specific ways during mathematics teaching and learning. visual mediators: visual objects that teachers and learners operate upon during discursive activities in the classroom; examples could be graphs, tables and special symbols that are used during mathematical communication. narratives: teachers’ and learners’ utterance sequences as they speak about mathematical objects, relationships between objects and mathematical processes upon the objects, which are subject to rejection or endorsement within the mathematics community. routines: teachers’ and learners’ repetitive patterns during mathematics processes and communication about mathematical objects, for example mathematical conventions and performing calculations. in this article, i use the commognitive theory to describe discourses of functions presented by five teachers in the study, to reveal the opportunities as well as constraints for learning offered by the examples that the five teachers selected and used during teaching. the data from which this article emanates revealed that the dominant narratives were the presentations of functions as formulas, while there were limited opportunities for learners to make conjectures and engage in proof activities. the prevalent routines included the sketching of graphs from functions presented in symbolic form. the following section details the research methodology for the study, to highlight the nature of data generation i used. research methodology the empirical data in the current article consist mainly of videotaped lessons presented by five mathematics teachers at five different school sites in mpumalanga province of south africa, representing multiple cases. as reported in this article, a qualitative research approach was espoused (creswell, 2013). the qualitative approach entails ‘a systematic subjective approach used to describe life experiences and situations to give them meaning’ (burns & grove, 2003, p. 19). this approach allowed me to gain insight into teachers’ teaching practices in their uniqueness. to understand the teachers’ lived experiences, i immersed myself into the lives of the five participating teachers to explore and understand the teaching of functions as experienced by teachers. the current study used a multiple case study design. this design enabled me to understand the nature of mathematics teaching, specifically the teaching of functions within a bounded context and bounded activity (creswell, 2013). for the current study, the bounded context is schools in acornhoek and mathematics classrooms in the schools, and the bounded activity is the teaching of functions at grade 10 level. the region is classified as rural as there is dominance of residents that work on farms to sustain themselves and their families, poor transportation services and isolation from the national and provincial government offices. the study was conducted with five (5) grade 10 mathematics teachers at five (5) secondary schools in rural acornhoek, mpumalanga province of south africa, forming multiple cases. the schools and participating teachers were selected purposively, based on their participation in the wits rural teaching experience (wrte) project. also, teachers needed to possess experience and knowledge of teaching grade 10 mathematics. table 1 presents participating teachers’ biographical information. to conceal and protect teachers’ true identities, i use pseudonyms, as shown in table 1. table 1: teachers’ biographical information. the empirical data in the current study were generated by means of unstructured non-participatory classroom videotaped observations. johnson and christensen (2008, p. 206) defined observation technique to refer to ‘the watching of behavioural patterns of people in certain situations to obtain information about the phenomenon of interest’. in the context of the current study, the definition suggests that classroom observations can be used to explore and generate in-depth understanding of the nature of teachers’ classroom practices related to the teaching of functions (guthrie, 2011). the nature of my participation in the observations was non-participatory, i adopted a ‘passive, non-intrusive role’ during teaching in all the classrooms that i observed (cohen, manion, & morrison, 2011, p. 459). one way of ensuring the trustworthiness of data was through peer scrutiny of the research processes. during the study, i welcomed scrutiny of the project by peers, colleagues and academics at conferences, which allowed me to address biases and assumptions relating to my interpretations of teachers’ classroom practices during the lessons. data analysis according to nieuwenhuis (2007, pp. 99–100), ‘qualitative data analysis tends to be an ongoing and iterative process, implying that data collection, processing, analysis, and reporting are intertwined, and not necessarily a successive process’. in the current study, the analysis of observed lessons commenced during the process of data collection and units of analysis were created through ascribing codes to the teachers’ observed practices during teaching (muir & beswick, 2007). after transcription, the recorded lessons were analysed with the purpose of segmenting and distinguishing the discursive activities characterising the teachers’ respective discourses of functions. i firstly analysed each lesson for individual teachers separately, paying attention to repetitive patterns and characteristics of the use of different modalities of mathematical representations and narratives. i then compared the different lessons, searching for similarities and differences and using the identified nuances to inform and reshape my analyses of the separate lessons. i have intentionally adopted an outsider position as an attempt to view the discourses unfolding from the different teachers’ teaching in as unbiased a way as possible. in addition to the above discussion, analysing teachers’ classroom discourses while teaching the topic under study required an approach that allowed me to look within and across the different teachers’ lessons, to create a picture of the quality of each teacher’s teaching. thus, i overlaid into the tenets of the commognitive theory for both structure and generality about the teachers’ discourses during the observed lessons. i initially chunked lessons into episodes based on what activities were set and their related examples for each lesson. within the episodes, i then noted the nature of teachers’ mathematical discourse as framed by the four components of commognitive mathematical discourse. tables 2–6 depict teachers’ discourses of teaching functions across the different episodes. table 2: discourses in mafada’s teaching episodes. table 3: discourses in mutsakisi’s teaching episodes. table 4: discourses in tinyiko’s teaching episodes. table 5: discourses in zelda’s teaching episodes. table 6: discourses in jaden’s teaching episodes. findings and discussion this section addresses participating teachers’ selection of examples during algebraic function lessons, and whether and how they facilitated or constrained the learning of functions’ critical features during teaching. according to renkl (2017), it is important that teachers choose appropriate examples to facilitate and deepen the learning and understanding of the concepts and knowledge for the content. the discussion focuses specifically on how teachers worked with the examples during the lessons to help learners understand the critical features for linear functions, parabolic functions, hyperbolic functions and exponential functions. the south african caps curriculum recommends that teachers teach critical features for the different families of algebraic functions, such as the effect of different parameters, domain, range, intercepts and turning points (dbe, 2011). the curriculum further asks teachers to provide learners with opportunities to make conjectures, and prove them, to formulate generalisations, especially with the effect of different parameters for different functions. there are two categories in the teachers’ systems of variation of parameters: teachers who sequenced the examples showing the effect of one parameter while keeping the other invariant (jaden, mutsakisi and zelda), and teachers whose set of examples in the lessons simultaneously varied both parameters (mafada and tinyiko). the two categories are discussed as two sub-themes: varying parameters simultaneously and varying one parameter while keeping the other invariant. thus, intuitively we can say that always varying both a and q simultaneously does not seem as optimal in the learning process. development of parameters discourse within the function concept depends on the content teachers make available for learners to learn, the teaching approach they use to convey the notion of parameters to the learners as well as how they select and vary examples to develop learners’ thinking about the effect of different parameters on the behaviour of the functions. two of the five teachers in this study struggled to offer explanatory talk that would enable learners to make conjectures, prove them and make generalisations about the effect of the different parameters for the different families of functions they worked with in the classrooms. analysing how the teachers selected and sequenced a set of examples in each lesson enabled a view of whether and how the examples accumulate to bring the object of learning in different lessons into focus for learners, and whether there is movement to achieve generality which is one of the curriculum objectives for grade 10 level in south africa (adler & ronda, 2015; dbe, 2011). for the two sub-themes, the above statement means that teachers’ systems of examples and their sequencing reveal whether there was movement towards generality relating to the parameters of functions. this relates to curriculum statement 3 for functions which expects learners to ‘investigate the effect of a and q on the graphs defined by y = a.f(x)+q’ (dbe, 2011, p. 24). this curriculum principle envisages that teachers vary parameter a while keeping q invariant or varying parameter q while keeping a invariant to ensure that learners make conjectures, prove them and construe generalisations relating to the effect of each parameter where f(x) is defined by the following functions: and bx. the focus here is on the role of examples and how they were sequenced to enable or constrain systematic learning, as symbolic mediators to support learners’ knowledge building. varying parameters simultaneously it is discernible that in the examples that mafada and tinyiko used in their lessons, they did not use patterns in which they vary one parameter while keeping the other one invariant. table 7 illustrates the examples that the two teachers used and their sequences in selected lessons. table 7: sequences of examples in mafada’s and tinyiko’s lessons. mafada starts with y = x2 and moves on to change both a and p simultaneously in the next two examples with respect to the first example, but then the next three examples change only a with respect to the first example. then, with respect to the first example, an example is given where only p is different. then the last example is again a change in both a and p. thus, it could be said that putting examples 2 and 3 right after example 1 potentially limits the signifying of the changes that learners should observe during teaching. tinyiko’s example selection and sequencing demonstrate that she varied the examples in terms of the families of functions simultaneously and the simultaneity in the introduction of such functions limited observation opportunities for learners to observe the effect of parameters for the different functions. while mafada’s and tinyiko’s sequencing of the examples moved from simplicity to complexity, as presented in table 7, the system of variation did not however create opportunities for learners to observe what is changing as the teachers did not vary one parameter while keeping the other invariant. on this moeti (2015) states that during the teaching of quadratic functions, the sequencing of examples ‘moves from a parent function f(x) = x2 where simple example is taken to complex ones’ (p. 61). for mafada and tinyiko, the lack of an invariance-variance relationship to bring the world of changes to the fore in the example sets did not allow for systematic comparison of the different families of functions in terms of the effect of changing the values of a and q. thus, it can be said that the example sequences teachers used constrained the development of endorsed narratives about the effect of parameters on the different families of functions. according to martensson (2019, p. 7), ‘rather than telling the students the critical aspects, the teacher must structure the critical aspects in terms of variation and invariance’, to ensure that the effect of parameters is discerned and discriminated across examples. i argue that mafada’s and tinyiko’s examples across their lessons have constrained the discernment of the meaning and structure of the parameters of functions, because there was no systematicity in terms of what varies and what remains the same between two parameters. that is, the set of examples the teachers used did not demonstrate knowledge of what changes, what stays invariant and what the underlying meanings behind varying parameters a and q are. it could be said that the teachers’ use of formulas as symbolic visual mediators did not enable effective visualisation of the effect of the parameters. according to marton (2015), during teaching and learning, variation is a necessary component to enable learners to notice what they are expected to learn. the discernment of critical features related to the families of functions, particularly the effect of the parameters on the behaviour of the functions, did not occur, since there was no systematicity in terms of varying one parameter while the other remained invariant in the teachers’ lessons. i therefore argue that the variation of one parameter while the other parameter remains invariant is a precondition for learners to develop a sense of structure and meanings of the parameters, to see what is changing and what remains unchanged and the related effects on different families of functions (al-murani, kilhamn, morgan, & watson, 2019; martensson, 2019). in addition, the patterns of variation in mafada’s and tinyiko’s examples are contrary to leung’s (2012) postulation that: ‘invariants are critical features that define or generalise a phenomenon … for a major aim of mathematical activity is to separate out invariant patterns while different mathematical entities are varying, and subsequently to generalise.’ (p. 434) the ways mafada and tinyiko varied the parameters during teaching did not bring about the discernment of structure in working with the different families of functions as well as generality about the effect of the parameters a and q as per curriculum standards. i argue that using parameters simultaneously without first exploring the effect of each parameter while the other remains invariant constrains learners’ awareness of the effects of the parameters. thus, the teachers’ use of symbolic mediators in the form of algebraic equations was lacking what sfard (2008, 2012) termed ‘interpretive elaboration’, because they did not offer learners elucidations about the behaviour of given functions when some variation is introduced to the parameters of functions. this lack of interpretive elaborations and intellectual discussions with the learners about the effect of the parameters indicates that the teachers did not create a teaching and learning environment that facilitates learners’ deep understanding of functions. the following extract exemplifies the ritualistic routines in tinyiko’s teaching: 1 tinyiko: if for instance you are given y equals to x squared and i say draw a graph of that one, in other words, when i give you this there is an addition of zero (see image 57 for symbolic mediator), what is the y-intercept? 4 learners: (chorusing) zero! 5 tinyiko: the y-intercept is zero. why do you say zero? i said you can only use what? the table and the dual. by the way, how does the dual work? 7 learner: we let x be zero. 8 tinyiko: we said let x be equals to zero because you want to find the what? 9 learners: (chorusing) to find the y-intercept. 10 tinyiko: and let’s remember that the x-intercepts are also the output values. what about the y-intercepts? 12 learners: the outputs. 13 tinyiko: good, after you get all the values, all you need to do is draw the graphs. the questions ‘what is the y-intercept?’ (line 3), ‘because you want to find the what?’ (line 8), ‘i said you can only use what?’ (line 5), ‘by the way, how does the dual work?’ (line 6) and ‘what about the y-intercepts?’ (lines 10–11), all represent an elicitation technique to check whether learners gained information from the previous lessons. the conversation above serves as an example of mathematical communication where the teacher used the words y-intercept and x-intercept as if they refer to outputs and inputs based on their relatedness, which sfard called ‘saming’. i noticed though that tinyiko overlooked the idea that y-values are the output values and the x-values are the input values, but the notion of x-intercept entails a zero of a function where an input value produces an output of 0. also, using the word y-intercept to signify synonymity with output values does not explain to the learners that a y-intercept is a point where the input value is 0 on a given function, which also addresses the commognitive construct of saming. furthermore, the statement ‘the y-intercept is zero’ (line 5) reveals that intercepts are treated as a numerical value1 rather than as coordinate pairs. this cannot be left unproblematised, considering that what teachers say and do during teaching shapes learners’ development of correct mathematical word use to talk effectively about mathematical entities. according to sfard (2019, p. 1), ‘it is a common lore that teachers bear the main responsibility for what the students learn or fail to learn’, suggesting their influence regarding learners’ understanding or lack thereof for knowledge. tinyiko’s narrative in line 13 was also concerning, because the choice of words – ‘all you need to do is draw the graphs’ – illustrates the teachers’ ritualisation when working with functions, and lack of interpretive elaborations about the critical features of the different families of functions. the above discussion resonates with mason’s (2002) argument that worked-out examples might constrain learners’ ability to generalise the nature of mathematical objects and the nature and effect of the parameters on different functions, as the teachers primarily focused on ensuring that learners recognise the syntactical template of the symbolic representation for functions. from mafada’s and tinyiko’s teaching approaches, it could be argued that learners could not notice what stayed the same and what varied, resulting in learners’ inability to associate the patterns of variation with the different representations as well as the word use and narratives that go with them. al-murani et al. (2019) argued that learners’ conceptualisation about the function concept: ‘depends on discerning common and differing features among examples and experiences, generalising from these according to the scope of examples that are presented, and fusing these features into a concept.’ (p. 8) it is arguable that varying the two parameters simultaneously without first varying one while the other remains invariant makes it difficult for learners to experience the difference of their effect on the functions. varying one parameter while keeping the other invariant sieving out invariants in the parameters during the teaching of functions is an essence of experiencing the depth of the topic, and in turn developing conceptual understanding as this facilitates symbolic mediation for different functions (chimhande, 2013; moeti, 2015; sfard, 2008). table 8 depicts zelda’s, jaden’s and mutsakisi’s example sets. table 8: sequences of examples in zelda’s, jaden’s and mutsakisi’s lessons. zelda’s, jaden’s and mutsakisi’s patterns of variation in the selected lessons as presented in table 8 demonstrated systematicity in terms of varying one parameter while the other stayed invariant to guide the learners about the effect of the parameter in focus. although the degree of interpretive action differed for the three teachers, their selection and sequencing of examples demonstrated some intentionality to help learners move towards generality about the effect of the parameters on the different families of functions. for example, it can be said that zelda varied the values of parameter q in terms of both magnitude and signs for the functions y = x2, y = x2 + 1 and y = x2 – 1. zelda’s fourth example (y = –x2) was introduced to mediate learners’ thinking about the effect of changing the value of parameter a in terms of the sign and it was introduced after she was done with the effect of parameter q. similarly, jaden also varied the values of the gradient while the values of parameter q remained invariant, to bring the changes brought by changing the values of the gradient into focus. for mutsakisi, the pattern of variation also focused on varying the values of parameter q for parabolic functions y = x2, y = x2 + 2 and y = x2 + 1 while parameter a remained invariant and the two linear functions y = x and y = 2x + 3 were used as non-examples of the parabolic functions; the explanation she offered reinforced that the focus was on parabolic functions, and specifically the effect of parameter q. mutsakisi is thus varying not just a parameter, but the type of function also. accordingly, it could be argued that this can be confusing to learners trying to understand the variation in parameter q for the first time, but now having to consider two different family types simultaneously can be considered bad sequencing or variation. also, mutsakisi never used an example with a negative value for q, which is also not good practice. in addition, none of teachers used non-integer values for either a or q, which is also possibly bad for the learners’ movement towards generality. according to lo and marton (2012, p. 29), ‘the learning of an object is not possible if we cannot first discern the object from its context’. the example sequences mediated the identification of ‘what’ is changing in given relationships, ‘how’ the change is taking place as well as how the changes were linked to the different parameters, thereby guiding learners towards the endorsed narratives about the parameters of quadratic and linear functions (sfard, 2012). the following statements acted as endorsed narratives during teaching and are illustrative of how the system of variation enabled the commognitive elaborations for the three teachers: ‘m is equals to 1, m is greater than zero, in other words it means the value of m is positive … right, this has got a meaning, it has a very special impact on the graph that you are going to draw, akere [isn’t it?], to the graph that you are going to draw. [writing on the board as she continues speaking] if m is positive, the graph slants to your right, which means the graph that we are going to have will slant to your right, are we together? the graph will slant to the right because m is positive, akere, the general formula says y is equals to mx minus.’ (musakisi) ‘a determines the shape of the graph and then q is the vertical shift of the graph. i said a determines the shape akere [isn’t it?] while q is the vertical shift of the graph. now, let us check something here from our three graphs. i remember when i was introducing parabolic graphs, i drew this one and this one [writing parabolic functions y = x2 and y = –x2 in symbolic form] in the same cartesian plane and then the shape of the graphs were not the same; the other one was like this and the other one was like this [drawing sketches on the board]. we had two different shapes, mountain shape and cup shape, so we need to know when we have this and when do we have a cup shape.’ (zelda) ‘the graphs of y equals to 4x and y equals to 5x would be between the y-axis and the graph of y equals to 3x because the value is increasing. the graph of y equals to 3x is more steeper than 2x, that number 1, that number 2, that number 3 [pointing to the coefficients of x in the three functions], it is because of that number that when it is increasing, the graphs are coming closer to the x-axis. this is called the gradient; it is called the gradient of this line.’ (jaden) zelda, jaden and mutsakisi created opportunities through the system of variation and sequencing of examples to bring the idea of ‘transformation’ in functions, attention on the appearance (structure), displacement and orientation of functions, to the fore (chimhande, 2013; mpofu, 2018). conclusion in this study, the teachers who did not vary one parameter while keeping the other invariant in the examples did not engage in interpretive actions about the effects of the parameters on the different families of functions, relating to the lack of explorative routines. accordingly, this results in lack of formal word use and endorsed narratives related to quadratics and linear functions. this demonstrates that systematic variation, selection and sequencing of examples in symbolic form are the preconditions of productive communication about the behaviour of different parameters for families of functions in terms of formal words and endorsed narratives. that is, without systematic and sequential variation of parameters, teachers’ communication becomes limited to rote steps to draw graphs, and nothing is revealed to the learners about the effect of the parameters. the teachers who selected and sequenced the examples showing the variance-invariance patterns of working with parameters for different families of functions engaged in interpretive actions about the effect of the parameters; as such, the variation patterns mediated both their communication about the effect of the parameters and created opportunities for learners to learn about the notion of parameters. word use and endorsed narratives are enabled or constrained by the availability and systematicity of patterned variation or lack thereof. acknowledgements the participating teachers from the five acornhoek schools in mpumalanga are hereby acknowledged. competing interests the author declares that they have no financial or personal relationships that may have inappropriately influenced them in writing this article. authors’ contributions h.w.m. is the sole author of this article. ethical considerations before the study could commence, ethical clearance was granted by the university of the witwatersrand and access to the schools was permitted by the mpumalanga department of education (certificate number 2018ece006d). all teachers were informed of the purpose, confidentiality and voluntary nature of participation in the study before any data generation processes commenced and all participating teachers signed informed consents. i also adhered to the importance of ensuring that the identity of participants is protected, both in terms of keeping the information they provided confidential and by using pseudonyms to conceal their true identities as well as those of their respective schools. the assurances for confidentiality and anonymity in this study extended beyond protecting the teachers’ names and those of their schools to also include the avoidance of using self-identifying statements and information. funding information the financial assistance of the national institute for the humanities and social sciences (nihss), in collaboration with the south african humanities deans association (sahuda), towards this research is hereby acknowledged. opinions expressed and conclusions arrived at are those of the author and are not necessarily to be attributed to the nihss and sahuda. data availability data sharing is not applicable to this article as no new data were created or analysed in this study. disclaimer the views and opinions expressed in this article are those of the author and do not necessarily reflect the official policy or position of any affiliated agency of the author. references abramovich, s., & leonov, g. a. 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(2020). investigating students’ attention to covariation features of their constructed graphs in a figural pattern generalisation context. international journal of science and mathematics education, 18(2), 315–336. https://doi.org/10.1007/s10763-019-09955-6 footnotes 1. this observable action was frequent also across the other episodes in this lesson and in other observed lessons for the same teacher. abstract introduction emergency techno-response pedagogy problem statement and research question methodology results discussion recommendations conclusion acknowledgements references appendix 1: detailed analysis appendix 2 about the author(s) antonia makina department of curriculum and learning development, faculty of teaching and learning, university of south africa, pretoria, south africa langton kadzere midrand graduate institute, gauteng, south africa citation makina, a., & kadzere, l. (2022). exploring low-tech opportunities for higher education mathematics lecturers in an emergency techno-response pedagogy. pythagoras, 43(1), a644. https://doi.org/10.4102/pythagoras.v43i1.644 note: teaching and learning mathematics during the covid-19. original research exploring low-tech opportunities for higher education mathematics lecturers in an emergency techno-response pedagogy antonia makina, langton kadzere received: 04 aug. 2021; accepted: 05 feb. 2022; published: 22 apr. 2022 copyright: © 2022. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract the education sector, among others, was severely affected by the coronavirus disease 2019 (covid-19) pandemic. because mathematics has always been singled out as a subject that needs more verbal communication and interaction, rapid adjustments had to be made by mathematics lecturers in higher education institutions to try and facilitate normal teaching and learning remotely through emergency open distance methods. lecturers were forced to examine prevailing practices with a view to creating innovative and workable solutions to the emergency challenges without compromising the quality previously experienced during face-to-face classroom interactions. the article developed through a simple technology a conceptual framework for emergency remote teaching (ert) in an emergency techno-response pedagogy (etrp). the key was to demonstrate an innovative instructional strategy for teaching mathematics using a simple technology instead of an advanced or complicated mathematics software in the move from face-to-face to fully online teaching during a crisis. a development qualitative virtual case study was conducted that involved observing live and recorded mathematics lectures and interviewing an innovative lecturer of mathematics in the delivery of complex numbers at a graduate school in south africa. the facilitation of the lesson through a simple and inexpensive technology (microsoft onenote) guided the development of a conceptual framework for ert within an etrp. the context, input, process, and product (cipp) evaluation model was used as a theoretical framework to guide the analysis and conceptualisation of the lessons. results provided guidelines through a conceptual framework for ert that included a unique model of a lesson plan and advantages of using a simple technology in ert instead of advanced mathematical software. the article contributes to the knowledge base in planning future ert interventions. keywords: mathematics; emergency remote teaching; microsoft onenote; online learning; emergency techno-response pedagogy; lecturer. introduction the world was struck by the coronavirus disease 2019 (covid-19) pandemic in 2020, which changed the way many people operated in the business world. the education sector was no exception as the traditional way of learning, which involved learners being seated in an enclosed space while the teacher delivers the lesson, was a thing of the past. the covid-19 pandemic ‘created the largest disruption to education systems in history … affecting nearly 1.6 billion learners in more than 190 countries and all continents’ (un policy brief, 2020, p. 2). due to covid-19 regulations, which required social distancing and prevented large gatherings, the traditional way of teaching was significantly affected and not all higher education institutions had the skills to cope with remote learning needs. therefore, lecturers had to quickly find new ways of teaching that would be as effective as face-to-face classroom interactions. institutions that already gained some strength in online and open distance education before the pandemic lockdowns had a considerable advantage in offering emergency remote operations (skulmowski & rey, 2020). these advantages stemmed from institutional knowledge of online programming and the virtual operations in which many online units operated (bouchey, gratz, & kurland, 2021). however, other higher education institutions were not prepared for the advent of the pandemic, and teaching became a swim or sink endeavour. mathematics is typically easy to teach in a face-to-face environment because most mathematical issues are physically managed by the lecturer within the classroom environment (castle & mcguire, 2010; lewandowski, rosenberg, jordan parks, & siegel, 2011). for example, the pronunciations of symbols in differentiation and integration need to be verbalised for mathematical communication to be possible. real-life modelling is required to illustrate on-the-spot examples, which are not always easy to simulate. therefore, lecturers had to find alternative strategies to effectively teach in an environment similar to a face-to-face scenario, to enhance the effectiveness of lecturers’ current efforts to improve students’ success in emergency remote teaching (ert). lecturers who were skilled facilitators in face-to-face classrooms were unsure about how to implement student-centred practices in a remote context. their greatest challenge was communicating and effectively interacting with their students in the remote environment (skulmowski & rey, 2020). they also did not have financial resource support, including the availability of data and wi-fi. at the start of the pandemic, lecturers would send content using email, and audio and video recordings, but the students’ results were very unsatisfactory. students showed no interest by ignoring such communication or they submitted sub-standard work. they were clearly not coping. ultimately, lecturers who were considered excellent disseminators of knowledge were left in the deep end. therefore, the purpose of this article was to provide through a simple technology a conceptual framework for ert in an emergency techno-response pedagogy (etrp). the objective was to provide guidelines for ert in an etrp through observing a real-life scenario of a lecturer facilitation. emergency techno-response pedagogy an etrp (figure 1) is dependent on ert, which is a temporary shift from the normal modes of teaching that happens in a crisis when teaching becomes remote or distant managed (murphy, 2020). this transforms what would have otherwise been face-to-face or hybrid teaching into digital education. for many, ert happens without warning, illustrating the importance of having a quick contingency plan to solve the problems created by the crisis, either temporarily or permanently. when a crisis occurs that necessitates all universities to shut down their classroom or use blended teaching modes, ert and learning may be presented in the form of online lessons, radio, or blended learning solutions (zimmerman, 2020). online learning is a method of instruction presented over the internet and is sometimes called ‘e-learning’. while it can be defined as a way of delivering content to students over the internet using various online platforms, it is a form of education that happens at a distance rather than within a classroom setting (czerniewicz, 2020). online learning was created to leverage technology and allow students to earn degrees or attend school without having to physically attend an academic setting. figure 1: emergency techno-response pedagogy. face-to-face learning is the traditional mode of instruction, where students and lecturers attend an in-person session at the same time. the lecturer leads the class, manages the classroom, and provides different ways of motivating a diverse student audience. for example, within the teaching of mathematics, the lecturer normally presents illustrations and demonstrations of problems and problem-solving on the board while students follow along. students can participate if the lecturer poses questions or asks students to demonstrate some mathematical problem-solving on the board. therefore, in a face-to-face environment, there is live interaction between the lecturer and the students. the purpose of this article was to provide through a simple technology a conceptual framework for ert in an etrp. this article demonstrates how a mathematics lecturer attempted to simulate a face-to-face classroom setting during the pandemic using simple forms of technology. challenges in the initial emergency transition emergency remote teaching and learning at colleges and universities primarily consisted of moving face-to-face courses into a ‘virtual environment using learning management systems, real-time or recorded web conference meetings, and other internet-based tools’ (garrett, legon, fredericksen, & simunich, 2020, p. 2). many lecturers were unfamiliar with the virtual space and the available technology and software for online students. universities quickly developed plans, but they all fell into the trial-and-error space, leaving the lecturers to find innovative ways to start engaging with learners. lecturers working from home encountered problems related to a lack of computers or laptops and inadequate bandwidth, in addition to the lack of remote support structures and student tools (garrett et al., 2020). the quality of instruction therefore suffered during the emergency transition. students have always had unequal access to reliable high-speed internet and other academic technologies due to underlying inequalities in household income and regional infrastructure (ludeman & schreiber, 2020). moreover, while moving into ert, lecturers lost confidence in their ability to successfully convey their subjects to students. based on this background, the research questions were: what innovative teaching methods did lecturers develop during the covid-19 pandemic to try and achieve the same purpose and objectives that students gained in a face-to-face or blended mode of teaching, and what support services were required during the pivot to remote learning? therefore, this article provides guidelines from a conceptual framework that was derived from how a lecturer implemented a set of evidence-based ert practices to effectively communicate and interact with his mathematics students through an etrp. initial lecturer responses to ert when strict lockdown measures were implemented during the pandemic, lecturers initially responded by using the whatsapp platform to send students audio and video lectures containing information required for learning. some lecturers replaced face-to-face interaction with ‘busy work’ assignments and, as a result, their students spent more time and effort on coursework that does not typically improve their actual learning or performance in mathematics (bouchey et al., 2021). it was a counterproductive step, but the lecturers were not trained for global emergencies that require remote teaching. applications like the matlab software were difficult to master instantly in an emergency, and some lecturers used the common microsoft skype software to deliver lessons. one drawback of this platform was the limited number of participants who could use it when the pandemic struck. some lecturers used microsoft teams and zoom applications, which can accommodate more students in the class, but they could not be completely interactive, and illustrations were difficult to present. there was a need for a board to write on and present illustrations. older lecturers who were unfamiliar with the latest technological trends were stranded. and while most of the education fraternity expected this emergency to be temporary, it continued even as vaccines were distributed across the globe. this article thus analyses the instructional strategies employed by some lecturers in the delivery of ert to facilitate the development of the etrp. the focus is on teaching mathematical content in an emergency in ways that take advantage of simple and accessible technology to promote enhanced student understanding (bouchey et al., 2021). the context of mathematics students during covid-19 students were thrown into a space of uncertainty due to covid-19. suggestions to learn online were immediately presented to the students in their unprepared mental state, without regard for some students succumbing to a sense of panic. people adapt to the way they live as well as their home environment; however, if there is a crisis, new practical realities suddenly surface. this was the case for most students. for example, students were accustomed to going to school or a library at an online distance learning institution early in the morning, where there was always electricity, wi-fi, and a warm, comfortable space to sit and work. there was the additional bonus of the community of learners who were always present for emotional and spiritual support, and typical teenage engagement. during the pandemic, and based on their background of poverty, the students suddenly realised they did not have their own space to work from, and that access to electricity and wi-fi was challenging. some students realised that their cell phone was not an educationally user-friendly model (e.g. no whatsapp, low battery model, etc.) or that their cell phone bills were not within their means. therefore, in a crisis scenario caused by the pandemic, support was needed for students’ specific needs. problem statement and research question generally, mathematics teaching is better done in a face-to-face environment. however, due to covid-19, mathematics had to be taught in a remote setting that required an online platform. mathematics lecturers were not ready for the emergency online teaching, resulting in the need for a quick-thinking model that would allow those lecturers to achieve results in an emergency setting. the research question was, therefore: how can mathematics lecturers use simple technology in higher educational institutions to teach mathematics during this period of disruption caused by covid-19? therefore, this article develops a conceptual framework for ert which proposes guidelines for lecturers facing a crisis to conduct and deliver a mathematics lesson within an etrp. methodology a development qualitative virtual case study was conducted that involved observing live and recorded mathematics lectures and interviewing an innovative lecturer of mathematics in the delivery of complex numbers at a graduate school in south africa. the facilitation of the lessons through a simple and inexpensive technology (microsoft onenote) guided the development of a conceptual framework for ert within an etrp. in the first research stage, the researcher interviewed the lecturer at the graduate school on the challenges that prompted him to use microsoft onenote through zoom for the teaching of complex numbers during covid-19. in the second stage, the researcher observed each of the selected lessons at least twice, noting areas of interest from a researcher’s perspective. observations were made of similarities and differences between and among teaching methods employed in a face-to-face lesson. the participant was approached in a natural setting so that the researcher could analyse and understand the ert phenomenon during the teaching of mathematics in an etrp. interviews conducted by the researcher lasted approximately 1 hour, and were conducted via zoom web conferencing. the interviews were conducted at the beginning and at the end of the case research lessons. a semi-structured interview protocol was used to capture the strategies used by the lecturer to achieve ert of complex numbers in mathematics. as part of college algebra, a complex number, that is, the sum of a real number and an imaginary number, was chosen for the lesson to be observed. a complex number is expressed in standard form when written as a + bi where a is the real part and bi is the imaginary part. the theoretical framework in this article stufflebeam’s (2003) context, input, process, and product (cipp) evaluation model was used as a theoretical framework to systematically guide the analysis of the lecturer’s activities during the delivery of the ert during covid-19 (appendix 2). according to stufflebeam and zhang (2017), the objective of context evaluation is to assess the overall environmental readiness of a project, examine whether existing goals and priorities are attuned to needs, and assess whether proposed objectives are sufficiently responsive to assessed needs. the input evaluation component helped prescribe a responsive strategy and technological tool to best address the identified needs of ert in an etrp. the process evaluation component then monitors the strategic process and potential procedural barriers and identifies potential process adjustments. finally, the product evaluation component interprets and judges teaching and learning outcomes and interprets their merit, worth, significance, and probity (zhang et al., 2011). through the analysis and conceptualisation of the lessons using the cipp model the main strategies in the facilitation were categorised into broader strategies that led to the development of a conceptual framework for ert. permission and ethical considerations permission was sought from the lecturer who received enough information to make an informed decision regarding his participation in the study and the head of department (hod) was informed of the research that was to be conducted. appropriate steps were taken to ensure the participant was fully aware of his participation and role. the participant’s privacy and confidentiality were protected during and after the research process. ethical considerations study was approved by the lecturer participant at his home environment as it was during the covid-19 lockdown. results an interview was carried out with a recognised mathematics lecturer at a graduate school in south africa. in a qualitative case study, the researcher asked the lecturer to narrate his experience with teaching during the hard covid-19 lockdown. interview 1 when covid-19 knocked on my door, i was challenged mentally and psychologically on how i was going to deliver the subject effectively as compared to how i did it in my face-to-face classroom. until i discovered microsoft onenote i went through so many challenges so much that i decided to also find out the status of my colleagues. it seemed that we were all frustrated about what next step to take. several mathematics software (e.g. wolfram mathematica, matlab, infinite algebra 1) were recommended for the job but my colleagues and me found them very difficult. i personally didn’t have time to go through them. in addition, i wanted a support technology that did not work different from my face-to-face classroom. the hod discovered the ability of microsoft onenote in the teaching of mathematics as he had observed from one colleague from another school in cluster meeting. during the strict covid-19 conditions in level 5 where it was total lockdown with very minimum human movement, schools were totally closed. the hod called for a meeting and advised the use of microsoft onenote. each lecturer had to do a trial and error to come up with the most effective way to deliver a lesson using microsoft onenote. since we were advised by the hod to use microsoft onenote and each of the lecturers doing a trial and error, we all successfully managed to effectively deliver our lessons in our own different ways. i then sought to find out how to establish a model for the successful use of microsoft onenote as a teaching application. the success of microsoft onenote led to accept your request for an interview to observe my teaching and learning model. during an online staff meeting there was overwhelming positive feedback from the students as compared to the other different previous trials in the emergency remote teaching in the total lockdown. using the cipp model to build the conceptual framework the cipp evaluation model was useful in analysing the planning and implementation done by the lecturer in this case study for the teaching of complex numbers during the covid-19 pandemic. using the cipp model as a guide, analysis of the emergency remote mathematics teaching was made to ascertain the nature of the strategies used during the emergency period, as summarised in figure 2. figure 2: the cipp model guiding the conceptual framework. unpacking the conceptual framework for emergency remote teaching the cipp model was used to structure and analyse the five phases of the conceptual framework derived from the interviews and observations which were the context mapping, planning input evaluation, process building evaluation, the teaching and learning process evaluation and simple technology evaluation. strategies that emerged from the analysis were used to facilitate a closer examination of the ert approach to instructional delivery of complex numbers. the conceptual framework provides guidelines for teaching mathematics in an emergency response situation. the phases in the framework are analysed through lenses of the cipp model as explained below. context mapping this context evaluation phase enables the unpacking of two issues: analysing the context of the mathematics teaching space involved in ert and reflection of the etrp. during this phase, the following questions can be asked: where is my student and where am i? what does my student have and what do i have? what can my student afford and what can i afford? will my institution buy into my ideas? will this idea work and if it doesn’t what do i do? the answers that evolve here lead to the lesson planning document explained in the input evaluation. the analysis of the context requires the direct listing of the necessary resources required to carry out the reflected ideas. table 1 gives an example of the list that was given by the lecturer involved in the case study. table 1: example of requirements needed to use microsoft onenote. planning input evaluation this stage in the conceptual framework speaks to the planning document that carries out the ert in an etrp. through the interviews, lessons replay and observations a different type of planning that is not carried out in a normal face-to-face classroom was constructed (table 2). in a face-to-face classroom, lesson planning is superficial and has many assumptions. the assumptions originate from the fact that the lecturer is present in the classroom with the students. the classroom has charts, has a board, has different types of materials that can be referenced during a mathematics lesson. a lecturer can jump, shout, touch, verbally explain and re-explain or bring physical objects in order to clarify a mathematical concept. however, this is very different when teaching is done remotely, at a distance and using a simple technology. therefore, i observed the lecturer’s planning document for the lessons with great interest. the planning document illustrated in table 2 identifies the exact ‘critical mini objectives’ per each lesson. during a lesson there are main objectives. now these main objectives need to be minutely unpacked to come up with the ‘critical mini objectives’ that deal with the real person and his mathematical being. these ‘critical mini objectives’ fall into questions that ask if the students need to see, touch, smell, lift, slide, taste, etc. the ‘critical mini objectives’ enable the choice of the support technology like microsoft onenote and the tools to use within this technology in order to achieve the ‘critical mini objectives’. table 2: lesson plan for an ert in an etrp. table 2 shows an example of a lesson plan in the teaching of complex numbers in college algebra. mathematics components involve several subject matter strands, of which complex numbers were chosen for the demonstration. learning complex numbers involves several types of thinking processes, namely seeing, listening, relating, applying, and reasoning, among others. to promote students’ understanding of subject areas in mathematics, a process is followed as determined by what the student is supposed to grasp within that area of mathematics. this process guides the nature of the technology, the nature of the communication, and the interaction to be used during ert. it is against this background that the article recognises lesson planning as an important part of ert in an etrp. process building evaluation the process that the lecturer engaged here involved identifying learners’ contact details (e.g. cell numbers) and sending invitation links through whatsapp for learners to be part of the class, depending on the broadcasting software to be used. an example is given in table 3. table 3: example of the practical processes involved with the use of microsoft onenote. teaching and learning process evaluation during the phase of process evaluation, the lecturer carries out the teaching with microsoft onenote. figure 3 shows examples of the observed and replayed video lessons on complex numbers. figure 3: example of observed lessons presented through microsoft onenote and zoom, (a) lesson 1, (b) lesson 2, (c) lesson 3, (d) lesson 4, (e) lesson 5. product evaluation the advantages of using microsoft onenote through zoom are discussed based on another interview with the lecturer. the covid-19 pandemic caused significant restrictions, including minimal human movement, social distancing, and limited gatherings, among others. in this learning situation, the microsoft onenote application embedded within the microsoft office package was selected for use. the microsoft onenote application made it possible for the lecturer to effectively deliver a mathematics lesson on complex numbers in a virtual context. the main advantages of microsoft onenote, which made the delivery of the mathematics lesson possible, included: it is cost-effective and comes with the microsoft office package. it is easy to use; no intensive training is required to use the software to deliver an online mathematics lesson. it is compatible with all laptops or desktops and can be used on most cell phones. there is no significant difference between a traditional classroom blackboard in a conventional classroom, and the microsoft onenote whiteboard offered within microsoft office. the assessment results that were obtained based on the use of the microsoft onenote application proved to be even better than when teaching was presented in the face-to-face context. absenteeism was eliminated to almost zero; students were self-isolating in their homes and had nothing better to do than to join their mathematics lessons. since the use of microsoft onenote was a new experience for the students, they were excited and very happy about it. their overall motivation to learn and ask questions also increased. this could be attributed to students feeling more confident to interact in their own spaces without the negative comments that can occur during contact classroom sessions. there was some feedback from students who attended classes using microsoft onenote through zoom, stating that it was a successful alternative to strategies that might come in future. through zoom students could just click a button to raise their hand or share their written exercises. students’ classroom interactions and communication were easily managed. the other advantages of microsoft onenote are based on the personal observation and deductions of the lecturer. microsoft onenote was cost-effective: there was suddenly no transport and extra food-related costs to attending in-person classes, and parents were therefore willing to go the extra mile to avail the students of wi-fi and laptops as the lessons were being delivered remotely. in some cases, not living on campus eliminated any further costs. microsoft onenote offered a comparative face-to-face classroom environment: microsoft onenote offered several similarities compared to the traditional face-to-face classroom. it also had significant advantages over in-person classroom settings. students did not face any challenges following up on concepts being demonstrated on the microsoft onenote whiteboard and could do their immediate assessment activities using microsoft onenote. some limitations in the use of microsoft onenote: microsoft onenote, as a stand-alone vehicle, is incapable of adequately handling all facets of instructional delivery and must be offered through other online interactive applications, like zoom or teams. a few negative comments were received from the students addressing connectivity challenges, isolation, and a lack of direct student-to-student interaction. connectivity: not all students have access to high-speed internet connections. some are using internet connections from service providers (including mtn, cell c, vodacom and telkom, among others). due to their geographical locations, some do not have strong internet signals, making it difficult for them to attend and participate in the online class. isolation: most students are not used to being isolated from their lecturer and peers. they are used to the normal environment where they mingle with classmates on the campus. while this teaching method has proven to be helpful in these turbulent times, some students were complaining that it is affecting their social life. student-to-student interactions during lesson time: a few students raised concerns that it was no longer possible to chat with peers during and after class or have some short discussions on difficult mathematical concepts. this is no longer possible, yet the model works well in promoting student-teacher interactions. discussion according to the first paragraph in the interview, face-to-face lecturers who had to move to fully online teaching faced the greatest challenge. they had no technical skills and there was need to quickly adapt to the new need and environment. the use of a simple technology that did not require them to learn advanced and complicated mathematical software became key. the idea was to showcase the simple technologies among the mathematics teachers until more advanced mathematical software could be learnt. simple technologies like microsoft onenote resembled a face-to-face classroom which they were used to. the purpose of this article was to develop through a simple technology a conceptual framework for ert in an etrp. many students were unable to access a proper learner management system during the pandemic because universities and colleges were not ready for this eventuality. therefore, delivering content through microsoft onenote was a major innovative advantage. the teaching delivery method described in this article mimics the face-to-face lecture style of the ‘live’ classroom which is the students’ preferred teaching mode, as discussed during the interviews (kuhn & johnson, 2019). the etrp enabled the development of a conceptual framework that could guide the successful ert of mathematics using microsoft onenote. during the interviews, the concept of ‘emergency innovation’ was visible in the lecturer’s narratives; he chose a tool or combination of tools (support technologies) consistent with the intellectual challenges at hand. this article was initiated by the fact that universities allowed lecturers to do whatever they could in the absence of a long-term organised strategy. the conceptual framework for ert in an etrp allows higher education lecturers to act as agents for change in the teaching of mathematics or any other subject. etrp knowledge is vital as it describes the knowledge that lecturers must develop in an emergency. it also presents approaches to teaching, assessment and evaluation that can capture the complexity of a crisis in, for example, a pandemic, and its strength is its attention to the use of a simple technology instead of a complex approach to ert and learning. using the cipp evaluation model to analyse a mathematics lesson on complex numbers enabled the development of a conceptual framework for ert in an etrp. the comprehensive framework explored the complex dynamics behind the use of microsoft onenote in emergency teaching and learning. the effort of a lecturer during an emergency and outside an institutional context was analysed. it was also observed that the lecturer was forced to take some non-academic roles during the ert. these included finding out what time was most suitable for the lessons, preparing some entertainment to motivate the students, giving compassionate support to students who experienced gender-based violence, lost members of the family or had personal losses, and having empathy during the interactions. among other findings through observations, it was clear that students felt more connected with the lecturer when the lecturer gave them more non-academic support. however, these roles taken by the lecturer overloaded the lecturer with more work that went unrecognised by their respective academic institutions. the key contribution of this study is its explicit invocation of an etrp, an emergency response approach to higher education instruction that can be used during pandemics. this study draws clear connections between the responsive pedagogy and bridging the digital divide to play a more prominent role in the practice of ert. recommendations throughout the interviews, the lecturer stressed the importance of a community of lecturers to handle remote mathematics teaching during emergencies. it is therefore recommended that institutions move away from a ‘siloed’ model in which a small group of staff work exclusively to support fully online students, toward an ‘integrated’ model in which knowledge and skills regarding online teaching support are spread across the entire institution (wang, 2020, p. 7). this finding was also reflected in a research study that supported the importance of collaborating and building relationships between staff and other departments to ensure students’ full academic support could be better achieved during emergency periods (ryan & deci, 2020). discussions about the role that each lecturer was playing during covid-19 was crucial. these deliberations can provide an opportunity to build new relationships among staff members that can be shared when there is a crisis. ‘academic resilience during emergencies comes from an ethos of collectivism that is borne of mutual recognition of one another’s personhood through ubuntu (i am because you are)’ (enslin, & horstemke, 2004, p. 10). during pandemics, lecturers need to change from a pre-prepared teaching perspective to a possibility-focused perspective characterised by a self-empowered lecturer role. this goes hand in hand with kuhn and johnson’s (2019) notion of the need for a practical approach for predictive models that cater for predictive pedagogies in the future. the article provided theoretical and practical recommendations that higher education institutions and lecturers can incorporate into their e-teaching strategies during emergencies. conclusion greater attention should be paid to finding and formulating teaching strategies that use simple technologies to communicate and interact effectively with students during emergencies. this article presented a conceptual framework for ert of mathematics in terms of better preparing for future pandemics. this approach is dependent on the innovative push from lecturers and the context of the students. overall, the findings suggest the need for higher education institutions to be conscious of unforeseen situations that can suddenly appear across the globe, forcing them to quickly identify sudden gaps in student support. acknowledgements we acknowledge three lecturers from the graduate school that was used who were all willing to share their common shared experiences during covid-19. we also acknowledge the university that contributed to the final editing of the article. competing interests the authors have declared that no competing interests exist. authors’ contributions both authors contributed to the conceptualisation, methodology, formal analysis, investigation and writing of the original draft. funding information this research received no specific grant from any funding agency in the public, commercial or not-for-profit sectors. data availability the teaching audio and vision lessons were made available by participant consent to the researchers. disclaimer the views and opinions expressed in this article are those of the authors and do not necessarily reflect the official policy of any affiliated agency of the authors. references bouchey, b., gratz, e., & kurland, s. 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(2020). coronavirus and the great online-learning experiment. chronicle of higher education. 66(25), 37–47. appendix 1: detailed analysis step 1: identify the exact details of the learners (e.g. cell numbers). send invitation links through whatsapp to learners who are going to be part of the class depending on the broadcasting software to be used. on zoom you can share the meeting id and passcode to the class via a whatsapp groups. this allows them to enter your class. step 2: open microsoft onenote which is going to be the whiteboard where illustrations are going to be presented. microsoft onenote is a free software that comes with microsoft office, so to open microsoft onenote you need to go to microsoft office folder. this varies from machine to machine since in some cases it appears on the main menu. once you are on microsoft onenote open the drawing tab, which is where you find tools to use for the lesson. on the drawing tab you find board pens, board eraser and a tag for simple shapes that may be used during the session. step 3: open broadcasting software microsoft teams or zoom. both programs have a chat section where students can interact, comment or give suggestions as the lesson progresses. students have an option to click a raise hand button on both software should the need arise. this would notify the lecturer on which student might be in need of attention. students can also write their names in the chat section as a way of taking register. this allows the lecturer to monitor who is attending the class or not. step 4: share your screen on the broadcasting software and upon sharing your screen also confirm with the learners whether they can see your screen and hear you. this is when students have accessed your class or are still accessing your class using the link or meeting id and passcode that would have been supplied earlier. step 5: you can now use microsoft onenote as your whiteboard and deliver the desired lesson. students would be seeing your demonstrations on the whiteboard on their screens as well as hearing your voice. both software zoom and microsoft teams have video option where the lecturer can opt for a video where students would see the lecturer or not to be seen by students. this is normally determined by the signal strength during the lesson. appendix 2 table 1-a2: the cipp model. abstract introduction literature review conceptual framework: word order matching and static comparison models aim and objectives research design and methodology results discussion and concluding remarks acknowledgements references about the author(s) calos soneira department of pedagogy and didactics, faculty of education sciences, university of a coruña, a coruña, spain sarah bansilal mathematics and computer science education, school of education, university of kwazulu-natal, durban, south africa reginald govender mathematics and computer science education, school of education, university of kwazulu-natal, durban, south africa citation soneira, c., bansilal, s., & govender, r. (2021). insights into the reversal error from a study with south african and spanish prospective primary teachers. pythagoras, 42(1), a613. https://doi.org/10.4102/pythagoras.v42i1.613 original research insights into the reversal error from a study with south african and spanish prospective primary teachers calos soneira, sarah bansilal, reginald govender received: 01 apr. 2021; accepted: 24 aug. 2021; published: 25 nov. 2021 copyright: © 2021. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract this study, using a quantitative approach, examined spanish and south african pre-service teachers’ responses to translating word problems based on direct proportionality into equations. the participants were 79 south african and 211 spanish prospective primary school teachers who were in their second year of a bachelor of education degree. the study’s general objective was to compare the students’ proficiency in expressing direct proportionality word problems as equations, with a particular focus on the extent of the reversal error among the students’ responses. furthermore, the study sought to test the explanatory power of word order matching and the static comparison as causes of the reversal error in the two contexts. the study found that south african students had a higher proportion of correct responses across all the items. while nearly all the errors made by spanish students were reversals, the south african group barely committed reversal errors. however, a subgroup of the south african students made errors consisting of equations that do not make sense in the situation, suggesting that they had poor foundational knowledge of the multiplicative comparison relation and did not understand the functioning of the algebraic language. the study also found that the word order matching strategy has some explanatory power for the reversal error in both contexts. however, the static comparison strategy offers explanatory power only in the spanish context, suggesting that there may be a difference in curriculum and instructional approaches in the middle and secondary years of schooling, which is when equations are taught. keywords: algebra; reversal error; word problems; word order matching; static comparison; pre-service mathematics teachers. introduction over the past decades there has been much research about the learning of early algebra and algebra researchers have documented many misconceptions of students (e.g. bush & karp, 2013; kieran, 2007; knuth, alibali, hattikudur, mcneil, & stephens, 2008; swan, 2001). one error that has received much attention in algebraic research is the so-called reversal error which is exhibited by students at all levels who, when asked to express a directly proportional relationship between two quantities in terms of an equation, interchange the two variables (clement, 1982). the error gained prominence after the famous student-professor problem posed by clement (1982) and clement, lochhead and monk (1981). in this problem, students were asked to write an equation using s for students and p for professors to represent the statement ‘there are six times as many students as professors in this university’. clement et al. (1981) found that 68% of the incorrect responses were 6s = p. this error is referred to as a reversal error because of the reversal in the position of the two variables compared to their position in the correct equation 6p = s. the reversal error has been identified at different levels of study, among high school students in their initial stage of formal algebra learning (macgregor & stacey, 1993), and also among university students (clement, 1982; cohen & kanim, 2005; fisher, borchert, & bassok, 2011; gonzález-calero, arnau, & laserna-berenguer, 2015; sung-hee, phang, soo, kenney, & uhan, 2014). it is therefore important that research is conducted in this area so as to suggest measures that can be taken in the early stages of algebra learning so as to prevent the error, as well as in the later stages in order to remedy it (barbieri, miller-cotto, & booth, 2019). for teaching interventions to be effective, these need to be based on the knowledge of the specific cognitive process and contextual circumstances that trigger the misconception. in addition, the phenomenon is complex because it involves the difficulties linked to the transition from arithmetic to algebra (macgregor & stacey, 1993; malisani & spagnolo, 2009), as well as the general difficulties related to the change of representation from the natural to a formal language (gonzález-calero, berciano, & arnau, 2020). the present study looks into the cognitive processes that underly the reversal error, by exploring the extent of the reversal error and the explanatory power of the theoretical models suggested by previous research. the participants are prospective primary school teachers from two countries (south africa and spain) which represent two different educative contexts in terms of educational opportunities and resources. we take a comparative approach with the two groups (south african and spanish prospective primary school teachers). this is accomplished by means of two versions of a questionnaire in which specific changes were made to adapt each one to the idiomatic features but keeping the same core structure in both countries. by considering these two different contexts and using these specifically designed questionnaires, this study looks at the incidence of the reversal error across the two samples. therefore, as each sample is taken from a different context, the study can identify whether the reversal error is equally likely to occur in the two groups, which would suggest that the difficulty is embedded within the learning of algebra in general. in addition, if the strategies used by students differ across the groups, then this would suggest that there may be differences in curriculum approaches in the early learning of algebra, which could be explored in further studies. hence, we can gain insights into how the educative context may affect some sources that seem to prompt the reversal error, while there may be other sources that seem to be inherent to the learning of algebra. literature review much research in mathematics education has focused on systematic misconceptions that commonly occur with students from different countries and students who speak other languages. as part of the learning process of any concept, students develop misconceptions as they grapple with the concept at hand. a misconception is often revealed when a teacher is alerted by a consistent error made by a student in response to a particular situation. instructors can be alerted by a consistent error made by students, which points to a misconception. barmby (2009) believes that misconceptions are inevitable, they occur naturally and the errors that they lead to are largely predictable. some students have misconceptions about the meaning of the equal sign, which seem to persist up to tertiary level studies (booth, 1988; fyfe, matthews, & amsel, 2020). many students view the equal sign as a symbol separating two expressions, while others see it as an operational signal which indicates that the equation must be solved (kieran, 1981; matthews, rittle-johnson, mceldoon, & taylor, 2012). because students do so much computing of the answer, they often use it as a ‘write the answer’ sign, instead of seeing the sign as indicating equivalence between two expressions. there have also been many studies that have looked at students’ understanding of the variable. some studies have reported the common misconception of taking a variable as a label standing for an object instead of standing for an unknown quantity (fisher et al., 2011; gonzález-calero et al., 2015; soneira, gonzález-calero, & arnau, 2013; stacey & macgregor, 1999). other misconceptions that have been identified include students thinking that the same letter appearing at different points in a number sentence could not represent the same number (filloy, rojano, & puig, 2008), that letters can only stand for whole numbers, among many others. although rooted in arithmetic thinking, all these errors are subsequently transferred to their algebraic thinking, causing the corresponding misconceptions about the notion of equation (filloy et al., 2008; humberstone & reeve 2018). moreover, students tend to face difficulties when representing mathematical relationships by means of algebraic language (filloy et al., 2008; kieran, 2007; koedinger & nathan, 2004; stacey & macgregor, 1999). specifically, when translating word problems into equations, they tend to keep the same order of the main pieces of information as displayed in natural language although it may lead to errors (duval, 2017). martin and bassok (2005) pointed out that students often rely on translation cues to move directly from the words to writing the equation, without actually thinking about whether the solution makes sense in the setting or whether the relational statement was appropriately represented by the algebraic statement. the use of these cues can lead to correct solutions but it is often the case that students’ reliance on these translation cues leads to systematic errors such as the reversal error (landy & goldstone, 2007; lewis & mayer, 1987). another explanative model of the reversal error is that students understand the mathematical relationship, but hold misconceptions about the equal sign (e.g. clement, 1982; cohen & kanim, 2005; fisher et al., 2011; gonzález-calero et al., 2015). whatever the explanation, it is clear that the reversal error seems to be rooted in common algebra learners’ misconceptions. many studies have investigated the influence of factors such as variable symbol choice, sentence structure, semantic cues, context familiarity, among others, on students’ tendency to commit the reversal error (e.g. cohen & kanim, 2005; clement, 1982; martin & bassok, 2005; fisher et al., 2011). in terms of variable choice, some researchers (crowley, thomas, & tall, 1994; fisher, 1988; sims-knight & kaput, 1983) consider that the use of specific letters as variables may influence students when making up the equation to represent a relationship. for example, the use of the first letter of a word as the variable, such as p for the number of professors and s for the number of students, may be more likely to prompt students to make the reversal error as compared to the use of the more generic x and y symbols to represent the quantities (crowley et al., 1994; sims-knight & kaput, 1983). it was argued that the use of these first letters as symbols may prompt students to use the symbols as labels rather than as variables (crowley et al., 1994; fisher, 1988). however, research by cohen and kanim (2005) as well as soneira et al. (2018) did not find any differences in the rate of the reversal error in problems that took the variable as the first letter of the quantity and those that took the variable as a more general symbol. conceptual framework: word order matching and static comparison models basically, studies that assess the explanatory power of different students’ strategies that lead them to commit a reversal error have consisted of modifying some variable of the research instrument and then observing whether the change prompted different results in the students’ responses. in this study we investigate the explanatory power of the word order matching and static comparison strategies as well as their contextual dependence or independence. word order matching strategy the word order matching strategy involves a ‘literal, direct mapping of the words of english into the symbols of algebra’ (clement et al., 1981, p. 288). for example, in the students and professors problem, if the reversed equation 6s = p is posed, 6s would represent the clause ‘there are six times as many students’, the sign = would stand for the word ‘as’, and p for ‘professors’. students who rely on this word order strategy assumes that the order of the key words in the problem statement can be mapped directly into the order of symbols that appear in the equation. however, sometimes because of the flexibility of natural language the same multiplicative relationship can be expressed by using a syntactic structure such that the application of the word order matching leads to a reversal error, or by using a different syntactic structure such that the application of this incorrect strategy will however lead to a correct equation. we differentiate between these two versions by referring to statements ‘with syntactic obstruction’ as opposed to statements ‘without syntactic obstruction’ (soneira et al., 2018, p. 4.). statements with syntactic obstruction are those where the use of the word order matching would result in a reversal error (soneira et al., 2018, p. 4). an example of such a statement is ‘at this company, there are six times as many workers as managers’. assuming that the variable w represents the number of workers and m the number of managers, a word order matching translation would result in the equation 6w = m, which is a reversal error. in contrast, the same relationship can be expressed using a statement without syntactic obstruction (soneira et al., 2018, p. 4) as ‘in this company, the number of workers is six times the number of managers’. a word order matching translation of this latter statement will result in the equation w = 6m, which is the correct equation to represent the multiplicative comparison conveyed in both statements (with and without syntactic obstruction). students who opt for the reversal error strategy because they have been misled by the translation cues have some algorithmic understanding of equations; however, their algebraic skills may not have included a consideration of the modelling component (martin & bassok, 2005). therefore, these reversal errors may be considered as evidence of some algorithmic comprehension but rudimentary knowledge of the multiplicative comparison relation (martin & bassok, 2005). static comparison strategy research has also identified that some students may display the reversal error even in statements without syntactic obstruction (cohen & kanim, 2005; macgregor & stacey, 1993; martin & bassok, 2005; soneira et al., 2018). it can be inferred in this case that these students did not rely on the word order matching strategy because if they did so consistently, then they would not have obtained the reversal error. hence, in this case the cause of the error must be a different one, the static comparison being the most widely considered in the literature. with the static comparison strategy, the student conceives the equation as representing a correspondence or ratio between groups instead of an algebraic equivalence between quantities. in the previous student and professor example, using the equation 6s = p, the equal sign would mean that ‘six students correspond to one professor’. as asserted by clement et al. (1981): the expression 6s is used to indicate the larger group and p to indicate the smaller group. the letter s is not understood as a variable that represents the number of students but rather is treated like a label or unit attached to the number 6. the equals sign expresses a comparison or association, not a precise equivalence. (p. 288) regarding the assessment of the explanative power of the static comparison model, most studies (e.g. clement, 1982; cohen & kanim, 2005; fisher et al., 2011; martin & bossok, 2005) considered only items with quantities that had discrete magnitudes. furthermore, those that use items with continuous variables, namely landy, brookes and smount (2014), macgregor and stacey (1993), sung-hee et al. (2014) and wollman (1983), do not take into account this continuous character of the magnitude in their analysis. the only exception is the work by soneira et al. (2018), who went further to explore whether the type of magnitude used in setting up the direct proportional relationship between two quantities influenced the rate of the reversal errors in expressing the relationship. those researchers (soneira et al., 2018) considered items based on quantities that were discrete, extensive continuous and intensive continuous in magnitude and we use the same distinction in this study. the intention was to observe more closely the explanatory power of the static comparison model, which in effect builds a correspondence between two sets, for example the set of workers and the set of managers. since the sets are of different sizes, establishing the correspondence would require an understanding of the relative size between the sets. for quantities that are discrete, the student could easily set up a mental picture by taking one element of the set of managers and considering the corresponding elements of the second, for example taking a company with one manager and seeing that there will be six workers. hence, the student will infer that 6w = 1m within the static comparison model, where the sign = indicates correspondence instead of equality between the quantities represented in both sides. when magnitudes are continuous, it is harder to consider one element of a set in response to a corresponding element from the second set. for example, it is harder to consider a quantity of lemon juice or water as an element of a set. however, it may be possible to consider glasses of water or lemon juice, thus discretising that continuous quantity so as to make the mental model. in these cases where the continuous magnitudes are easier to discretise, such as water or juice, we follow soneira et al. (2018) and use the term extensive continuous to describe them. however, there are other continuous magnitude situations such as when the acceleration of a car is being compared to that of a motorbike, or when the density of two quantities are being compared, when the magnitude is harder to discretise. in such instances, we refer to these quantities as having an intensive continuous magnitude. the key point regarding intensive magnitudes is that because they are harder to discretise, it would be also harder to establish a correspondence between sets that underlies the static comparison strategy. therefore, if the static comparison has, indeed, explanative power as a model for the reversal error, a student most likely will apply it when dealing with discrete magnitudes, which in turn will lead to a higher rate of reversal errors. as this strategy is less likely to be applied when dealing with extensive continuous but discretisable magnitudes, the rate of reversal errors will be lower with this type of item. finally, it is highly improbable that the static comparison could be applied in dealing with intensive continuous magnitudes, and therefore the rate of reversal errors would be the lowest. on the other hand, if the static comparison did not have explanative power as a model for the reversal error, then there would be no differences in the rate of reversal errors depending on the type of magnitude. this is why in this study we also investigate whether the type of magnitude, whether it is discrete, extensive continuous or intensive continuous, influences the tendency to commit the reversal error. note that each model involves different cognitive processes. indeed, within the static comparison the student understands the correct ratio between quantities but falls short when trying to translate the relationship into an equation. in contrast, word order matching can be applied without comprehending the multiplicative relationship between quantities, but by incorrectly trying to replicate the verbal syntax in algebraic language. aim and objectives the purpose of this study was to consider the explanatory power of certain theoretical models as causes of the reversal error using items set within two languages: spanish and english. this study was guided by the following research objectives with respect to the samples of spanish and south african prospective primary school teachers: to examine the proficiency of the students from the two contexts in being able to express direct proportionality word problems into equations. to test the explanatory power of word order matching and the static comparison as causes of the reversal error within the two groups. research design and methodology in order to assess the explicative power and the context’s (in)dependence of the word order matching and static comparison as models for the reversal errors, we followed a quantitative approach with two samples. first, within a repeated measures design in each sample, we conduct hypothesis testing so as to assess in each context the student’s proficiency when translating verbal proportional statements into equations and the explicative power of each model for the reversal error. afterwards, as both experiments were analogous, we follow ellis (2010) and compare the effect sizes obtained in the corresponding tests in each context in order to get insight into the context’s dependence on each model’s explicative power. indeed, although the sample sizes are different, the effect sizes allow us to compare the corresponding results since the sample size is taken into account in the formulae for the effect size. in fact, the goal of comparative research is ‘to search for similarity and variation’ across different contexts or categories (given, 2008, p. 100) and in this study we look at the cognitive processes underlying the reversal error. by taking a comparative approach to the two different groups, we hope to gain deeper insights into what seems to prompt the reversal error. participants the participants were made up of two groups of prospective primary school teachers from universities in spain and south africa based on convenience sampling since the institutions were accessible to the researchers. however, the sampling criteria were taken as the educational backgrounds, admission requirements, field of study and future job qualification which were equivalent in both countries. both groups of students were in their second year of a bachelor of education (b.ed.) degree with a primary school specialisation. the sample from south africa consisted of 79 students most of whom (95%) spoke english as a second language. two-thirds of the group had attended non-fee-paying schools showing that they come from very poor socio-economic backgrounds. researchers have found that students from non-fee-paying schools in south africa generally perform less well in mathematics national and international exams when compared to their peers from more affluent schools (reddy et al., 2012; sairr, 2016). the sample from spain consisted of 211 students at a public university in their second year of study towards a b.ed. degree with a primary school specialisation. regarding their socio-economic background, they mostly belong to the low-middle and middle class and attended non-fee-paying schools. the analysis of the international assessment tests conducted in spain also conclude that students belonging to high socio-economic backgrounds obtain better academic results than their peers from poorer backgrounds (mefp, 2019, 2020). however, in spain the difference in the academic achievement between the higher and lower achievers has received much media attention in the organisation for economic co-operation and development countries (mefp, 2019). data collection instruments concerning the instrument, we adapted the questionnaire designed and already applied in quantitative research about the reversal error by soneira et al. (2018). the questionnaire consists of 12 verbal statements expressing proportionality relationships in everyday situations where six items were expressed using syntactic obstruction and six corresponding items without syntactic obstruction. furthermore, the statements involved comparisons of quantities, two of which were discrete, two extensive continuous and two intensive continuous. in the spanish university, we used the original questionnaire of soneira et al. (2018), which was written in spanish. in order to be applied in the south africa university, we translated the items into english and ensured that all the contextual situations described in the items were familiar to students in both countries. sometimes the wording and word order needed to be changed to ensure the word problem made sense and was relatable. however, the translation into english was done in a manner that preserved whether the item was expressed with or without syntactic obstruction. table 1 presents some examples of spanish statements, their direct translation to english – which could be grammatically incorrect, or at least awkward – and their modified versions, which were used in the english test. table 1: some examples of word problem statements used in spain and south africa. data collection took place across two countries, first at the university in spain and thereafter at the university in south africa. at both sites, the university learning management system was used as a medium to administer the 12 word problem statements in random order that formed the data collection instrument. each item was presented with a list of operators and numbers and with instructions to click and drag into position to create the equation (figure 1). while the user poses the equation, the program allows them to modify their attempt and once the student accepts it the equation is saved in a database and a new task is loaded. figure 1: example of a word problem appearing on the learning management system. coding of variables when coding the students’ responses, we defined three variables: ‘cor’ (correct), ‘rev’ (reversal error) and ‘other’ (other error) to code the response of each student to each problem. the variable cor takes the value 1 if the response is correct and 0 otherwise. the variable rev equals 1 if the answer corresponds to a reversal error and 0 otherwise. the variable other takes the value 1 whenever the answer is incorrect, but the error is not a reversal, otherwise it equals 0. in the case of the students and professors problem, the responses students = 6 × professors and students ÷ professors = 6 are examples of correct answers (cor = 1), 6 × students = professors and students = professors ÷ 6 are examples of reversal errors (rev = 1), while students × professors = 6 and students + 6 = professors are errors different from reversals (other = 1). thus, the variables cor and other account for the student’s proficiency and difficulties with each type of statement, and the variable rev allows for assessment of the explicative power of each model for the reversal error. statistical tests regarding the statistical tests we use, note that some choices depend on the fulfilment of certain assumptions, which in turn relies on the sample sizes and the results of other statistical tests. for the sake of clarity and to avoid repetition, in this section we describe the general procedure and in the results section we provide the statistical values that were used to check if the assumptions were fulfilled. in order to assess the explanative power of the word order matching, our assumption is that students who use the word order matching strategy are likely to respond correctly to items that are expressed without syntactic obstruction and for items expressed with syntactic obstruction, these students’ guiding strategy would prompt them to produce a reversal error. we take the factor ‘syntactic obstruction’ (our instrument is made up of six items with syntactic obstruction and six without syntactic obstruction), which has two levels: items expressed with syntactic obstruction and those that are not. given the sample sizes (79 and 211), we opted to use the paired t-test which is robust for sample sizes greater than 30 (pallant, 2010). in order to assess the explanative power of the static comparison, as explained in the conceptual framework section, our assumption is that this strategy is more or less likely to be applied depending on the type of magnitude, which will be reflected in the rate of reversal errors. hence, we take the factor ‘type of magnitude’. specifically, this factor has three levels (intensive continuous, extensive continuous, extensive discrete) which are ordered from lowest to highest according to how likely it is that the statement triggers the static comparison strategy. thus, we use a one-way within-subject factorial design. the sphericity assumption was checked by means of the mauchly’s test. if the sample distributions meet the sphericity condition, the analysis of variance test is used, otherwise we follow the protocol suggested by pardo and san martin (2010, p. 305). the latter involves applying first the multivariate approximations, specifically the pillai’s trace, the wilks’s lambda, and the hotelling’s trace. if these show statistical differences, then the factorial analysis is complete and we test pairwise comparisons by means of the paired t-test. if the multivariate approximations do not show significant differences, then we go on to calculate the statistic f with the modified degrees of freedom by the greenhouse-geisser and the huynh-feldt tests. if the latter show statistical differences, we test pairwise comparisons by means of the paired t-test. in all tests, we take a p-value of 0.05 as the decision criterion. we calculate the cohen’s delta to estimate the effect sizes of each paired t-test, and in order to provide more confidence in the results we report the corresponding p-value and the 95% confidence interval (ci) in all cases. as we conduct multiple hypothesis testing with each sample, in order to prevent type i errors we use the holm-bonferroni correction (all the reported p-values are those calculated after applying the correction). ethical considerations the ethical committees granted ethical clearance at the universities for this study. the researchers complied with all prescribed ethical measures, such as getting informed consent by the participants and keeping their anonymity. moreover, all participants were volunteers. results we first present the descriptive statistics as a general overview of the results, before going into more specific detail about the results of the statistical tests. descriptive statistics regarding the factor of syntactic obstruction, for each student and each of the variables cor, rev and other, we computed the mean in statements with syntactic obstruction, and the mean in statements without syntactic obstruction (table 2). table 2: differences depending on the syntactic obstruction. descriptive statistics. the results for the variable cor in table 2 point out that the group of south african students outperforms the spanish one regardless of the syntactic obstruction. however, the results for the variables rev and other suggest that the pattern of errors in the south african context is quite different from the spanish. in particular, regardless of the syntactic obstruction, nearly all the errors made by spanish students are reversals, while south african students barely commit reversal errors but errors consisting of equations that do not make sense in the situation stated in the problem statement, such as additive or inverse proportionality relationships. concerning the type of magnitude, we computed three scores for each student and variable. each one was the mean of the scores for the items involving discrete magnitudes, extensive continuous magnitudes and intensive continuous magnitudes (table 3). table 3: descriptive statistics by country and magnitude type. thus, the results depending on the type of magnitude (table 3) have an analogous interpretation to those depending on the syntactic obstruction. that is, south african students outperform the spaniards in all types of magnitude, and the pattern of errors was different between contexts. in the spanish case the reversal error is, by far, the predominant, while it was scarce among the south african students, whose errors consisted mainly of equations that did not make sense. taken together, the descriptive results (table 2 and table 3) suggest that among the south african students there is a subgroup who do not have rudimentary knowledge of the multiplicative comparison relation and do not understand the functioning of the algebraic language, especially in terms of expressing word problems as equations. the word order matching strategy we report next (table 4) the results of the hypothesis testing regarding the differences depending on the syntactic obstruction. table 4: contrasts for differences depending on the syntactic obstruction. we obtain (table 4) that in the south african context, there were statistically significant differences just for the variable rev, with a weak-to-moderate effect size, while in the spanish context we found statistically significant differences, with a weak effect size, for the variables cor and rev (table 3). hence, with respect to the second research objective, the word order matching has some explanatory power as a source for the reversal error in both contexts; otherwise we would not find statistical differences for the variable rev depending on the syntactic obstruction. the preceding tests also provide further insight into differences between the groups, thus addressing the first objective. indeed, in the spanish case there are differences for both the variable cor and the variable rev, and in the south african context only for rev. this is consistent with the fact that the word order matching is a model only for the reversal error and nearly all the errors in the spanish case are of this type (table 2); thus, the effect on the variables cor and rev must be similar in this context. with respect to the south african context, the fact that there are not significant differences depending on the syntactic obstruction neither for the variable cor nor the variable other would mean that, among those students who have problems with the algebraic modelling process that we identified above, the syntactic obstruction is not the main source of the error. otherwise, there would be differences depending on the syntactic obstruction. furthermore, these students did not follow, or their command of the algebraic language is so poor that they were not able to follow, the word order matching strategy. the static comparison strategy we report next the results of the hypothesis testing about the differences depending on the type of magnitude. for the sake of clarity, this is done for the variables rev, cor and other in each context, separately. concerning the variable rev, in the south african context we cannot assume sphericity (w = 0.952, p = 0.040) and neither the multivariate methods nor the methods with modified degrees of freedom showed statistical differences. in the spanish context, we cannot assume sphericity (w = 0.952, p = 0.006), but we found statistical differences by the multivariate methods. in particular, we obtained (f = 12.266, p < 0.001). therefore, we conducted all the pairwise comparisons, obtaining significant differences between all possible pairs (table 5). table 5: differences depending on the type of magnitude for the variable rev in the spanish context. therefore, in respect of the explanatory power of the static comparison as a model for the reversal error, firstly, our results point out that when taking together statements with and without syntactic obstruction, the rate of reversal errors among south african students is very low regardless of the type of magnitude (table 3). moreover, there are no differences in the variable rev depending on the type of magnitude but the effect of the syntactic obstruction was weak to moderate, stronger than in the spanish sample. these two results considered together would mean that, although south african students barely commit reversal errors, whenever this happens it is mainly due to the application of the word order matching, and not the static comparison strategy. this is also different from the spanish context, in which the statistical differences depending on the type of magnitude mean that the static comparison has some explanatory power as a source of the reversal error (table 5). regarding the variable cor, in the south african group, by mauchly’s test, sphericity can be assumed (w = 0.998, p = 0.941). thus, we apply the analysis of variance test for repeated measures, by means of which we found statistically significant differences (f = 4.426, p = 0.013). we then conducted pairwise comparisons and found statistically significant differences between intensive continuous and the extensive discrete items (t = –2.907, p = 0.005, ci = [–0.17594, –0.03592,], d = 0.327), with a weak effect size. in the spanish context, we cannot assume sphericity (w = 0.923, p < 0.001). multivariate methods showed statistical differences, so we conducted pairwise comparisons, obtaining significant differences between all possible pairs of types of magnitudes (table 6). table 6: differences depending on the type of magnitude for the variable cor in the spanish context. therefore, in relation to the second objective, in the light of table 5 and table 6, we observe that in the spanish context the results for cor and rev are consistent. moreover, given the fact that in this context nearly all the errors are reversals (table 3), this corroborates the explanatory power of the static comparison as a source of the reversal error in the spanish context. note also that in the south african sample, reversal errors were barely made, and the highest rate of correct answers was obtained for discrete magnitudes and the lowest for intensive magnitudes. with spanish students, the opposite happened (table 3). regarding this, the results about the variable other complement those about cor concerning the first objective. indeed, in the spanish group, the means of other are so low for all magnitude types (table 2) that it makes no sense to test whether there are differences between them. in the south african context, we cannot assume sphericity for the variable other (w = 0.914, p = 0.032), so we begin by applying multivariate methods. as these methods showed statistical differences, we conducted pairwise comparisons and we got statistical differences for two pairs, with a moderate effect size in the difference between intensive continuous and extensive discrete magnitudes (table 7). notice also that in the pair in which we cannot reject the null hypothesis, even after having applied the holm-bonferroni correction, the p-value is less than 0.1. table 7: differences depending on the type of magnitude for the variable other in the south african context. thus, although most errors in the south african context are not reversals, the type of magnitude still has an effect on the incidence of the errors. in addition, the performance by the south african students regarding these other kinds of errors is better with the extensive discrete magnitudes and worse with the intensive continuous ones. comparisons of quantities with extensive discrete magnitudes are less abstract, while those for intensive continuous ones are the most abstract, with the extensive continuous in the intermediate position. this means that for the south african students of our sample, the more abstract the magnitudes involved were, the more difficult they found it to express the word problem into an equation. this did not happen in the spanish sample, which contributes to the answer to the first research objective. discussion and concluding remarks overall, our results point out that the south african pre-service teachers in our sample outperform the ones in the spanish sample when expressing proportionality word problems in terms of equations. indeed, although the results are based on descriptive statistics (table 2 and table 3), it is worthwhile to note that we obtained consistent results regardless of whether we considered statements with or without syntactic obstruction and across all types of magnitude. but the difference between the groups goes beyond a mere difference in algebraic language proficiency. indeed, the type of errors were distinct between contexts. firstly, our results point out that the errors when expressing proportionality word problems in terms of equations differ between the south african and spanish pre-service teacher participants. specifically, in the spanish case nearly all the errors were reversals, while in the south african case there were other errors with a higher incidence. furthermore, even restricting our attention to the reversal error, its sources are different depending on the country. in the spanish case the word order matching and the syntactic comparison would both have some explanatory power for the reversal error, which is in line with other results about spain (gonzález-calero et al., 2015; gonzález-calero et al., 2020; soneira et al., 2018) and the united states (clement, 1982; clement et al., 1981; cohen & kanim, 2005; fisher et al., 2011; martin & bassok, 2005; wollman, 1983). in the south african context, just the word order matching and not the static comparison would have some explanative power. the role of the word order matching and the visual-spatial display of the text and the equation has been highlighted in prior research by fisher et al. (2011) and landy et al. (2014). however, these previous studies did not address the explanatory power of the static comparison. moreover, among the studies conducted in english, just those by landy et al. (2014) and sung-hee et al. (2014) consider continuous magnitudes, but even these studies do not analyse the role of the type of magnitude. the results of our study contribute to improve our understanding of translation of word problems involving multiplicative comparisons into equations in two ways. firstly, we have shown that the south african students also have the tendency to keep the structure or the ordering of meaningful components of the statements in natural language when changing the register of representation, even if this leads them to make errors. note that in this sample 90% of the students spoke isizulu as a first language; this therefore shows that isizulu-speaking students also display the tendency of the word order strategy. this tendency has been identified in studies with students who speak english (e.g. cohen & kanin, 2005; fisher et al., 2011; sung et al., 2014) and those who speak spanish (gonzález-calero et al., 2015; gonzález-calero et al., 2020; soneira et al., 2018). our study now extends this result to students who speak isizulu. regarding this, although it was 90% likely that the mother tongue of a part of the south african students was isizulu, when translating into equations from statements written in english, we observed that the word order matching had explanative power. even more, the effect size regarding the syntactic obstruction was higher in the south african (d = 0.379) than in the spanish context (d = 0.260). therefore, our results in the south african context point out that the tendency to apply the word order matching does not depend on the context but it is common for different contexts. on the one hand, this is in line with the theoretical ideas of duval (2017) or kaput (1987) about the special relevance of the natural language in the mathematical activity. on the other hand, as word order matching does not require understanding the mathematical relationship, the fact that the tendency to apply the word order matching does not depend on the context is consistent with the natural tendency to save cognitive effort (uesaka & manalo, 2012). secondly, our study points out that the static comparison may be context dependent, diminishing its explanatory power as a model for the reversal error despite being one of the classic models. indeed, we did not find evidence of the static comparison being a tendency among south african pre-service teachers; therefore, it should be context dependent. this highlights the importance of word order matching as a source of the reversal error because it is shared by groups from different populations that differ regarding other strategies when structuring equations, specifically the static comparison. it is also noteworthy that, in contrast to the students of the spanish sample, the students of the south african sample committed the highest rate of errors with intensive continuous magnitudes, and the lowest with extensive discrete magnitudes. this suggests that the more abstract the magnitude, the harder they find it to mathematise the situation. even more, when dealing with discrete magnitudes they perform clearly better than spanish participants, but the difference fades with the items that have intensive magnitudes. this is another difference between contexts. it could be due to differences in curriculum or instructional approaches at school level, although more research about the matter is needed. moreover, success in comparisons of quantities with extensive continuous magnitudes, which are easy to visualise and to reduce to discrete magnitudes by means of taking units of measurement, placed between the other two types. this could mean that some students spontaneously apply a process of discretisation whenever the magnitude’s unit of measurement allows a corporeal interpretation. for example, in the cocktail problem, it is possible to think in terms of five tablespoonfuls of tonic for each tablespoonful of gin. but, if we do so, the situation and the subsequent reasonings would be the same as with discrete magnitudes. indeed, the new magnitudes would be the number of tablespoonfuls of tonic and gin. note that this process cannot be carried out with intensive magnitudes. in this case, the quantity of substance does not correspond to the quantity of magnitude. for example, the density of the sauce can be greater than that of the water, but the quantity of water greater than that of the sauce. thus, it would be more difficult to mentally corporealise the unit of measurement, which in turn would make discretisation more difficult. soneira et al. (2018) reported that the discretisation may affect the reversal error making. in the present work we see that this phenomenon may affect the solver’s approach to the problem in a more general way, which suggests that it could be a solving strategy. even more, the discretisation process seems to be common to the south african and the spanish contexts, beyond the fact that the subsequent solving strategies differ. regarding curriculum and instructional implications, our results indicate that the teaching interventions to be conducted in order to remediate the reversal error should be different depending on the country. specifically, spanish educators should take into account both the word order matching and the static comparison strategies. on the one hand, when first introducing the algebraic language in spanish schools, the study suggests that teachers need to highlight the meaning of the equal sign as an algebraic equivalence instead of correspondence, and the fact that letters represent quantities instead of being labels for objects. this would be so because these aspects have been proposed as the reasoning underlying the use of the static comparison strategy (clement, 1982; cohen & kanim, 2005; fisher, 1988; gonzález-calero et al. 2020; soneira et al., 2018; sung-hee et al., 2014). teachers in spain need to pay attention to the tendency to keep the structure or the visual display of the sentences in natural language when expressing them as equations, as this would underlie the use of the word order matching strategy (clement, 1982; fisher et al., 2011; landy et al., 2014). for their part, teachers in south africa could focus more on preventing the word order matching. on the other hand, the answers to problems with statements presented with syntactic obstruction point out that the cause of most errors made by the south african students was not word order matching. the errors they made, equations that did not make sense in the problem situation, suggest that their mental scheme to express multiplicative relationships by means of equations is poorly developed. thus, although south african pre-service teachers’ performance was, overall, acceptable or, at least, better than that of the spanish ones, there was a small subgroup of them that seem to lack the rudiments of algebraic language. this suggests that when students are introduced to the concept of equation during the intermediate and senior phase schooling, it should be done in a manner that allows them to develop a conceptual understanding and to make meaning of the notion of equation, instead of proving rules for manipulation of equations. in terms of the study’s limitations, it would have been preferable to have a larger sample size, particularly for the south african context. given that we did not find evidence of the static comparison being a tendency in the south african context and that it means a key difference with respect to others (e.g. spain, united states), it may be convenient to increase the sample. furthermore, it would be convenient to conduct similar experiments in the south african context, both with pre-service teachers and at other educative levels. in addition, since we have detected that south african students barely commit reversal errors, but other types of errors, and that the source of the former seems not to be static comparison, further studies could be interesting. in particular, qualitative studies would be helpful to provide deeper insights into the present quantitative one. these would shed light on how students think when translating proportionality word problems into equations. it could also be of interest to identify whether the students’ specialisation phase affects the extent to which the word order matching strategy usually arises to prevent the reversal error. along similar lines, it could be interesting to identify whether south african students’ specialisation phase is related to the extent to which they face difficulties with intensive continuous magnitudes. acknowledgements competing interests the authors declare that they have no financial or personal relationships that may have inappropriately influenced the writing of this article. authors’ contributions this article was developed collaboratively by the three authors. c.s. led the conceptualisation of the article, led the experimental design, carried out the analysis of the data, and drew up the results and discussion. s.b. led in the introduction, literature review, collaborated on the analysis of the south african data, and collaborated in preparing the final version of the discussion. r.g. developed the tool, cleaned and coded the data, led the methodology, technical aspects related to standard setting, and prepared the manuscript to comply with publication standards. funding information this work was supported by spanish government through the project pgc2018-096463-b-i00 as well as the national research foundation (nrf) grant number uid 118377. data availability the data that support the findings of this study are available from the corresponding author upon reasonable request. disclaimer the views, opinions, findings and conclusions expressed in this article are those of the authors and do not necessarily reflect the official policy or position of any affiliated agency of the authors. references barbieri, c.a., miller-cotto, d., & booth, j.l. 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(1983). determining the sources of error in a translation from sentence to equation. journal for research in mathematics education, 14(3), 169–181. https://doi.org/10.2307/748380 abstract introduction research questions conceptualisation of the study methodology results discussion and conclusion acknowledgements references footnote about the author(s) vuyisile l. khumalo department of science, mathematics and technology education, faculty of education, university of pretoria, pretoria, south africa surette van staden department of science, mathematics and technology education, faculty of education, university of pretoria, pretoria, south africa marien a. graham department of science, mathematics and technology education, faculty of education, university of pretoria, pretoria, south africa citation khumalo, v.l., van staden, s., & graham, m.a. (2022). weathering the storm: learning strategies that promote mathematical resilience, pythagoras, 43(1), a655. https://doi.org/10.4102/pythagoras.v43i1.655 original research weathering the storm: learning strategies that promote mathematical resilience vuyisile l. khumalo, surette van staden, marien a. graham received: 14 sept. 2021; accepted: 17 mar. 2022; published: 25 july 2022 copyright: © 2022. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract most learner achievement studies tend to focus on identifying individual characteristics, ignoring the learning strategies that promote mathematical resilience. the focus of the study is on the assets embedded in an individual and their interplay with the environment. it is expected that resilience plays a deciding role in learners’ foreground with the potential to affect learner mathematics achievement in constrained environments positively. this study, framed within the socio-ecological perspective of resilience, explored how disadvantaged learners learn mathematics in a disadvantaged environment in the further education and training band in south africa. a total of nine (five boys and four girls) grade 12 learners learning mathematics in disadvantaged environments from johannesburg west and johannesburg central districts were purposively selected and one-on-one semi-structured interviews were conducted with them. this thematic report pays attention to the qualitative phase of a broader study that employed a sequential exploratory design that draws from the work of vygotsky, carroll and skovsmose. the framework focuses on the dynamic interactions between learners and the connection between the home and the school. accordingly, the findings revealed two interrelated themes, namely foreground and growth strategies. these themes make apparent the connection between the context and the interpretation of the context by an individual as translated into decisional processes. implications for teachers are discussed. keywords: resilience; mathematical resilience; assessment feedback; foreground; learning strategies. introduction in south africa, senior secondary learner performance remains constrained for most disadvantaged learners (christie, butler, & potterson, 2007). south africa is glaringly outperformed by much poorer countries in the southern african development community (sadc), such as mozambique and tanzania, in respect of mathematics achievement (hungi et al., 2010; taylor, 2009). poor learner performance may very well indicate learning deficits that are likely to perpetuate educational inequalities in the education system. spaull and kotze (2015) revealed that the learning gap between poorer learners (that make up about 80% of the population) and learners from affluent backgrounds (that make up about 20% of the population) is in the region of three grade levels in grade 3 and grows to four in grade 9. it is thus not surprising that learner performance in mathematics in grade 12 has been declining steadily by at least 4% from 2013 to 2015 (department of basic education, 2016). a plethora of literature shines the spotlight on factors driving poor learner mathematics achievement. prominent factors that have an impact on mathematics teaching and learning have been identified and range from the management of the teaching and learning process (bush, joubert, kiggundu, & van rooyen, 2010), language proficiency of learners, particularly english (howie, scherman, & venter, 2008), socio-economic status of learners (taylor, 2009), teacher content knowledge (van der berg, 2015) and teacher to learner factors (tachie & chireshe, 2013). to this end, more than 10 years ago, setati, chitera and essien (2009) already argued for research to explore social, interactive factors that may ensure mathematical success. through the construct of resilience, this study explores one aspect of the resilience factors that may ensure mathematical success. literature suggests that resilience is defined differently owing to the discipline within which a researcher is grounded or located. albeit not exhaustive, waxman, gray and padron (2003) list a few disciplines that use resilience: psychopathology, psychology, sociology, and anthropology, each with its own focus on aspects of resilience. to this end, ebersöhn (2017) argues that resilience is a process that involves school stakeholders, namely teachers, principals, parents, learners, and district officials who are chiefly geared towards helping teachers to teach and learners to learn. on the one hand, masten (2015) regards resilience as a dynamic concept that entails the ability to adapt to adversity. on the other hand, johnston-wilder and lee (2010) introduce the idea of mathematical resilience as ‘a positive approach that allows people to overcome affective barriers presented when learning mathematics’ (p. 1). however, windle (2011) proposed an overarching definition of resilience as ‘the process of effectively negotiating, adapting to, or managing significant sources of stress or trauma’ (p. 152). it is a definition that resonates with this study and is thus utilised as an operational definition in this study. adaptation requires changing and adjusting behaviour that is geared towards buffering the impact of risk and promoting resilience. in this regard, the study accepts, as an operational definition in this study, the suggestion by ungar, ghazinour and richter (2013) that protective processes are ‘those processes that enhance the experience of well-being among individuals who face significant adversity’ (p. 340). waxman et al. (2003) explained that learner achievement stands to benefit by tapping into resilience as a fluid construct and paying attention to those malleable factors that can make a difference. malleable factors include those factors that schools and teachers have control over and, over time, have played a critical role in enhancing and improving learner performance as evidenced in school effectiveness research reviews (purkey & smith, 1983; scheerens, luyten, steen, & de thouars, 2007). crucial in utilising resilience as a construct is the understanding that is explained by duckworth, akerman, macgregor, salter and vorhaus (2009) that resilience emanates mainly from three levels: the individual, the family, and the school. it is a stance that firmly resonates with martin-breen and anderies (2011) that understanding resilience and change ‘requires understanding the process and the dynamics within learners and between them and their environment’ (p. 13). understanding resilience at different levels invariably requires the understanding that resilience is interactive and interdependent (masten, 2015). more importantly, resilience researchers have refocused their research by directly focusing on the competence in children at risk of problematic outcomes (masten, 2014). equally, in the education sector, a move is afoot to focus and concentrate on what learners can attain and maintain, and areas where they excel (gutiérrez, 2008; martin, 2012; mkhize & nduna, 2010; setati et al., 2009) despite adversity. in so doing, the focus is on the assets embedded in an individual and their interplay with the environment that can have a profound effect in promoting positive and desirable outcomes or development. in south africa, according to ebersöhn (2016), studies on resilience are fragmented, often are regional and are mostly case studies. as observed by theron and theron (2010), resilience studies in south africa are conceptualised as the product of individual traits or protective resources and, at times, as a product of a person-context transaction. exploring how disadvantaged learners learn mathematics in a disadvantaged environment requires that this study be firmly located within the person-context transaction. it is an attempt at uncovering the processes of adaptation that learners use to learn mathematics under adverse conditions. in other words, it is uncovering how disadvantaged learners in a disadvantaged environment develop mathematical resilience. to this end, ebersöhn (2017) contends that even though schools operate in disadvantaged or harsh conditions, schools need to offer their learners valuable support systems to enhance the teaching and learning situations. in keeping with this belief, pridmore and jere (2011) utilised the concept of resilience in malawi to investigate how to disrupt the patterns of poor learner achievement in a disadvantaged environment by creating circles of support around each learner. in this instance, members of the community were used to offer encouragement for learning through peers, mentors, and youth club leaders. although the programme was able to improve the mathematics results of disadvantaged learners, improved motivation, and the capacity for independent learning (pridmore & jere, 2011), the study fell short of shedding light on how learners processed the learning experience. however, lugalia, johnston-wilder and goodall (2013) implemented an information and communication technology programme in kenya to learn algebra for all the form 1 (grade 8) learners in the school. the authors found that teachers reported progress in learners’ three mathematics examinations taken during the course of the school term as well as observable attainment-enhancing behaviours. these three examinations are set at the departmental level but marked by corresponding subject teachers at respective schools. in other words, the examinations are set externally by external examiners. to this end, the authors conclude that learner progress may be attributed to the shift that transformed teaching and learning of algebra that pays attention to the learning experience of the learners. although the study by lugalia et al. (2013) points out the collaborative interaction of elements in the context of the learner, the study was only focused on one mathematics topic in one girls-only boarding school. in south africa, mkhize and nduna (2010) argue further for interventions in disadvantaged schools that focus on enhanced cognitive development and a positive coping style that promotes resilience. increasingly, literature is observing a cohort of disadvantaged learners across different education bands that are defying all odds and developing the necessary skills to cope with general academic demands. diamond, furlong and quirk (2016), utilising grade 4 and grade 5 learner data in the united states of america, identified a group of latino learners who showed signs of improving their academic skills development in oral fluency, and verbal and non–verbal cognitive development. in south africa, taylor, van der berg, reddy and janse van rensburg (2011), using panel data set of grade 8 learners that took part in trends in international mathematics and science study (timss) 2002 and then tracked to grade 12, although surprised by the finding, also observed that disadvantaged schools with disadvantaged learners showed signs of converting their grade 8 poor learner achievement to a matric pass; in particular, one in five learners were able to convert their poor grade 8 pass to a grade 12 pass. furthermore, reddy et al. (2016), utilising the timss 2015 grade 9 results, had this to say: we can identify a number of groups, which will be able to accelerate the pace of change with the appropriate support. these groups include learners who achieved scores between 325 and 400 points in no-fee schools. this group represents the immense potential that exists amongst our learners to achieve excellence, given sufficient support. (p. 15) that said, these south african studies are thin on pointing out the collaborative interactions of elements within the context of the learner in improving their poor mathematics result. thus, this current study attempts to reveal the processes that learners undertake to develop mathematical resilience and thus improve their poor mathematics results. research questions this study forms part of a broader mixed methods design study that asked the following main question: how do disadvantaged learners learn mathematics in a disadvantaged environment? the following research sub-questions guide this report: how do learners describe their home environment in their personal aspirations and mathematics achievement? what strategies do learners use to cope with mathematics demands? conceptualisation of the study the initial study utilised a fusion from the work of vygotsky (1978), skovsmose (2012) and carroll’s model of school learning as a theoretical framework (carroll, 1963) in order to explore how disadvantaged learners successfully learn mathematics in a disadvantaged environment. however, this article reports on the contribution of skovsmose’s ideas of foregrounds to make meaning of the data. amin (2012) succinctly points out that foreground significantly differs from the psychological perspectives that foreground as an object may be distinguished from its background or as a prominent figure that can be remembered for longer than the background. according to amin, foreground is paying attention to the future. more importantly, amin stresses that foreground is intractably related to two types of developments, namely personal and national. skovsmose (2012) suggests that showing an interest in learners’ foreground is crucial for establishing meaningfulness. according to skovsmose, ‘a foreground is fragmented, partial and inconsistent constellation of bits and pieces of aspirations, hopes and frustrations’ (p. 12). thus, foregrounds are possibilities, aspirations, tendencies, obstructions, and barriers that are formed through the interpretation of the environment and of these possibilities and obstructions. foregrounds are fragmented, multidimensional, and may be promising or frightening yet collective in nature (skovsmose, 2012). a closer look at a study by hernandez-martinez and williams (2013, p. 7), particularly jenni’s narrative, illustrates the retrospective and feeding nature of foregrounds. here, jenni decides to excel in mathematics because she wants to be different from her family, which is in the medicine sector, and join the financial sector which requires mathematics achievement in school. similarly, amin (2012) utilises the concept of foregrounds to illustrate the respective and feeding nature of foregrounds and further makes the point that foregrounds are connected to personal and national development and are shaped by prevailing conditions in the context of the learner. kacerja (2011) not only confirms the feeding nature of foregrounds but also affirms that learners are likely to be more interested in mathematics if the focus is on things that have a bearing on their future. in other words, foreground is all the concerted efforts to focus one’s attention on the future. in this framework, foregrounds are posited to function in an interactive and connective way between the home and the school. it is expected that a stronger connection between success in mathematics and the learners’ foreground has the positive potential to affect learner mathematics achievement. according to valero, garcía, camelo, mancera and romero (2012), foregrounds provide a powerful alternative framework that departs from the deficit explanation of learners’ disengagement to a framework that recognises the dynamic interactions between the learners’ home and mathematics engagement. accordingly, foreground provides a framework to better understand the dynamic protective processes that promote mathematical resilience in constrained environments. methodology this research report adopts a qualitative approach framed within a pragmatist paradigm. this paradigm ‘focuses on “what works” rather than what might be considered absolutely and objectively “true” or “real”’ (weaver, 2018, p. 1286). morgan (2014) has argued that although pragmatism has been mainly associated with mixed methods research, it can be used in social research regardless of whether that research is qualitative, quantitative, or mixed methods. morgan goes on to claim that pragmatism is a new paradigm that replaces the older philosophies of knowledge that use ontology and epistemology. to capture the lived experience of mathematical learners, a one-on-one semi-structured interview was utilised for this purpose. morse (2015) revealed that a semi-structured interview is designed to elicit rich subjective responses from participants about a particular situation or phenomenon. according to morse, all the data collected are comparable as participants are asked the same questions. merriam and tisdell (2016) point out that the chief purpose of the interview is to obtain a special kind of information that can be used as an exploratory device to identify relationships or even explore deeper into the participant’s motivations and reasons for responding the way they do. data collection instruments a one-on-one semi-structured interview schedule was utilised to elicit rich data. an effective interview schedule not only contains meaningful prompts of the phenomenon of interest but also is relevant to the participants (bearman, 2019). guided by literature on resilience and mathematical resilience, the study interview schedule contained open-ended questions aimed at eliciting information on the learners’ identity, their home and community environment together with their school or classroom. table 1 provides the questions used in the interview schedule. table 1: interview questions. sample the sample consisted of nine (five boys and four girls) grade 12 learners that were purposively sampled to provide for relevance, diversity, and depth of understanding (etikan, musa, & alkassim, 2016). these are learners that attend poor schools and face poverty as a distal risk or have failed a grade, at least once, in the further education and training (fet) band. despite the adversity or threat, these learners have showed some improvement in their mathematics achievement by attaining at least 30% in mathematics performance as a minimum requirement within the fet band. schools are divided into five quintiles. quintiles 1 to 3 are classified as no-fee paying schools, while the upper quintiles represent schools that charge school fees. quintiles are determined based on how poor the community around the school is (maistry & africa, 2020). table 2 presents the spread of schools and participants across quintiles. participants were drawn from two educational districts, namely johannesburg west and johannesburg central districts. these learners were recommended for this study by their respective educators. educators came to know of this study in an information sharing meeting held once a term (three academic months) under the auspices of the johannesburg west district office and johannesburg central district office, which monitor and support schools in mathematics and other subjects. thus, three educators representing three different schools from johannesburg west district and two educators representing two schools from johannesburg central district recommended learners for this study. the researchers compared previous mathematical performance from the school learner assessment schedule. learner assessment schedules are official gauteng department of education records that track the learners’ performance per term per grade. these records are also utilised to produce the final learner report card, which determines whether a learner repeats a grade or proceeds to the next grade when all minimum requirements to proceed are met. table 2: school and gender breakdown of participants. data analysis this study utilised thematic analysis as a fitting data analysis method for exploration (onwuegbuzie & teddlie, 2003) of how disadvantaged learners in a disadvantaged environment learn mathematics in the fet phase. thematic analysis was done through atlas.ti version 8. braun and clark (2012) define thematic analysis as ‘a method for systematically identifying and offering insight into patterns of meaning (themes) across a data set’ (p. 57). the study also followed the six guiding phases to thematic analysis as espoused by braun and clark (2006, 2012). however, braun and clark (2006) also caution viewing these six phases (familiarising yourself with the data, generating initial codes, searching for themes, reviewing potential themes, defining and naming themes, producing a report) as a linear model as data analysis is a recursive process of generating initial codes, searching for themes, reviewing potential themes, defining and naming themes, resulting in producing a report. trustworthiness trustworthiness of this study was enhanced by the five trustworthiness criteria, namely credibility, transferability, dependability, researcher’s worldview, and confirmability (anney, 2014; tracy, 2010). credibility was enhanced by affording some learners the opportunity to verify the accuracy of the interview transcripts, through member checking as influenced by time and participants’ literacy levels (carlson, 2010; creswell & miller, 2000). thick descriptions of the methodology, and the study context as used in this study provide for the study transferability or generalisability (anney, 2014; carcary, 2009). anney (2014) suggests that dependability may be enhanced through an audit trail. thus, to cater for dependability, the study ensured that an audit trail, that includes the interview schedule, audio recording and transcripts, is kept safe. all the data are kept at the university of pretoria. according to creswell and miller (2000), the insider perspective is intrinsically embedded within the worldview or paradigm assumptions held by the researcher. one of the researchers attended school in a similar environment to the learners in this study and is also teaching mathematics in the fet band under similar conditions. the researcher is considered a member of the group and thus an insider. thus, the researcher provides an insider’s view and reflexivity as the researcher reflects on the cultural, social, and historical forces at play that may shape and reshape the interpretation of the data set. confirmability was ensured through a methodological description that provides the reader the opportunity, through an audit trail, to assess the extent to which the emerging constructs are grounded within the data (anney, 2014; ghafouri & ofoghi, 2016). ethical considerations ethical clearance was applied for and granted by the university research ethics committee. after the granting of the clearance certificate by the university research ethics committee, institutional permission was sought from and granted by the gauteng department of education. through school principals, school governing body permission was sought as well as parental consent. lastly, learner consent was sought for those learners over the age of 18. as for learners below 18 years, parental consent was deemed sufficient in this study. these learners’ interviews were conducted in their respective schools with either a teacher or the principal present to ensure protection. all the participants were not asked to identify their real names; thus, pseudonyms were used to protect participants in the report. results from the thematic analysis of the semi-structured interviews, two interrelated themes with respective sub-themes were constructed. these themes are foreground and growth strategy. these themes make apparent the connection between the context and the interpretation of the context by an individual as translated into decisional processes. foreground two features are important in the formation of foregrounds, the context, and the subjective interpretations of the context (skovsmose, 2012). data analysis makes visible the connections between the context as perceived by learners and their respective interpretations of the context. eight learners described their home or community environment as not conducive, noisy, ignorant, and harmful, thus they spent more time in school learning mathematics as illustrated by the following narratives:1 ‘i would say my environment is not conducive there is too much noise, there is too many activities going on, there is not a quiet place where i can sit down and study and practice maths. … i would say my, my environment is not conducive for me to learn mathematics that’s why i take time to learn at school.’ (nomsa*) ‘ayi [not] supportive.’ (pamela*) ‘it’s not helpful, iyo, iyo, like, every time even now, if i can go back home, like i will find people drinking alcohol, drug abuse, like, don’t have any, like, don’t see any inspiration there, like you don’t get any inspiration. there is noise.’ (sipho*) ‘around i-community yami abantu bastereotyped ukuthi i-maths inzima, bafuna ukwenza izinto ezi-easy, kahle bona bafuna i-easy life. [around my community the people are stereotyped to accept that mathematics is difficult and thus take the easy way out].’ (james*) ‘like the community members, they are not aware.’ (busisiwe*) even though the learners’ environment is harmful and non-supportive, teachers are expected to play a pivotal role in the formation of the learners’ foreground. teachers are not only ideally positioned to influence mathematical learners’ foreground (skovsmose, 2005), but also the teacher’s outlook is implicated in the foreground of the learner (amin, 2012). more importantly, teachers are an accessible human resource that is crucial in shaping learning opportunities and the development of mathematical resilience (gholson & martin, 2014). embedded within these narratives is the capacity of the learners to interact with the available resources (human resources) that the environment provides as they navigate their way to development of mathematical resilience (ungar, 2008). this study’s results point to teachers as a source of inspiration in the formation of the learners’ foreground, as evidenced in these narratives: ‘my teachers, when they speak about how they started their journey in mathematics and how their journeys of life started, it motivates me to say that, even though i am not at the place where are i would like to be right now in mathematics, but i can actually get where i wanna go.’ (nomsa*) ‘u-sir mahlangu* wangibuza, u-sir wangibuza ukuthi why ama-marks wakho wa-last year abheda, while la-uphefome kahle, ngamutshela ukuthi ahi, sir ukuthi bengingafundi kahle ini ini, all that stuff, mara khona manje ungibonisile ukuthi nginga yeki i-situation yase khaya ukuthi ingidifine ukuthi ngingubani, mele ngispane ngiyenzeni. [sir mahlangu* asked me why my marks are better than last year, then i told him that i was not studying and stuff like that, he then showed me that my home situation should not define me and i should just work hard.].’ (james*) ‘they [teachers] don’t want a learner who struggles in class, when you struggle, they ask you where you don’t understand and what went wrong because they seem selfish in a way cause like they tend to care about us.’ (mandla*) ‘okay, firstly, uh, as mr majola* would say that your attitude determines your altitude, meaning that i tell myself that okay, you know what vivian this is it.’ (vivian*) nonetheless, one participant, busisiwe, indicated that a former learner represented a source of inspiration: ‘he [former learner] used to study a lot last year so i sort of aspired; i wanted to be like him.’ (busisiwe*) according to skovsmose (2012), foreground entails the individual interpretation of the opportunities that the political, economic, and social context presents. these interpretations are expressed as intentions, hopes and aspirations that focus on the future. all the participants in this study expressed their desire to pursue a future career that involves mathematics. participants’ future plans included careers such as cardiologist, accountant, electrical engineer, and child psychologist: ‘well, my future plans are to study economics, so i would need a lot of mathematics especially financial mathematics.’ (nomsa*) ‘i wanna be an economist. so, economics involves statistics, so i learn that statistics is maths.’ (mandla*) ‘i plan to pursue a career in, uh, financial, uh, accounting science.’ (busisiwe*) ‘ke batla go phasa ka go bane ke batla go ba cardiologist. [i want to pass because i want to be a cardiologist.]’.’ (pamela*) ‘future plan is to go study electrical engineering.’ (sam*) ‘i want to become a child psychologist, a child and teenage psychologist.’ (vivian*) hernandez-martinez and williams (2013) revealed two major motives for doing mathematics that include the use value (with respect to pursuing a career or vocation that uses mathematics, for instance mathematics teacher) of mathematics and the exchange value (the need to do mathematics as a prerequisite for a qualification) of mathematics or providing entertainment. similarly, in this study, busisiwe and mandla, for instance, depict the use value of mathematics as they plan to pursue careers in accounting and economics which use mathematics as a base while vivian reveals the exchange value of doing mathematics. mathematical learners’ expression of their hopes is seen as a personal expression of the sense of purpose (ungar, 2008). results indicate that participants have a greater awareness that their future career choices need mathematics. it seems that such awareness is predicated on their research efforts into the requirements of their individual future careers: ‘so obvious ngaqala ngayenza i-research yami, kwatholakala ukuthi, okay ama-accountant awa, abantu abayenza i-commerce. [so obviously i started by doing my research and then i found that accountants are people who do commerce.].’ (jack*) while the participation process in mathematics learning is facilitated largely by teachers, the willingness by learners to participate in their own process of learning mathematics is equally important (gorgorió & planas, 2003). the willingness is seen through the identification of the source of change in the learners’ participation in the learning of mathematics. thus, mathematically resilient learners can identify, access, utilise and connect with competent others (masten & coatsworth, 1998). results of the semi-structured interviews indicate the varied sources of change that a mathematically resilient learner attributes their turning point to. these sources of change are a chance encounter with a teacher, the learner’s attitude, or even repeated failure. james provides an indication of a chance encounter with his mathematics teacher and says: ‘grade 11 then sir, sir there was a sir e-sikoleni, who told me ukuthi, why am i, why am i running nama chickens, while i should be flying nama eagles, so i started realising, ukuthi, okay, i can do this. [my grade 11 teacher said why am i running with chickens when i should be flying with eagles, then i started realising that i can do this.].’ (james*) nomsa attributed her turning point to her attitude towards the subject: ‘uh, i would say my attitude was my change.’ (nomsa*) repeated failure was described by jack as a source of his inspiration rather than a risk factor. jack illustrates his resilience in that he endured prolonged failure, but did not change mathematics for other subjects in school or even drop out: ‘is when like, i started like, to, to fail over and over again, like ngaqala ngafeyila for, u-grade 10 wami, ngafeyila, ngafeyila u-grade 10 kayi two, which meant ngumiriphithe kayi-three. [when i started to fail my grade 10 twice, which meant that i had to do it three times.].’ (jack*) busisiwe depicts a learner who negates the widely held stereotype that mathematics is difficult and channels her mind to stick with the subject: ‘i think most of the time we listen to what other people tell us, they always say mathematics is difficult, and then you channel your mind to that mathematics is difficult, so it got to me that, to a point where i was like, i want to see this mathematics, i want to try.’ (busisiwe*) other than that, most of the learners attributed their turning points in mathematics to an awareness that their future plans require mathematics, as evidenced by the following: ‘i think when you reach matric and grade 11, there’s, there is a realisation that you see that most of the times maths, maths is used in every university.’ (mandla*) ‘i’m in grade 12 now, i want to finish school with a higher grade in maths so that i become somebody in life.’ (sam*) ‘i told myself that if i, i don’t improve, nothing will go according to my plans or my, my goals won’t be achieved, so i just decided from that moment that i have to do the right thing, focus on my school work and forget about, like everything.’ (sipho*) ‘my turning point started here in matric, seeing that some other topics i can tackle them much more easily than others and then i just thought that, i saw actually that, okay, this is it i am going to pass maths this year other than the other years.’ (vivian*) growth strategy for a person to manage their life circumstances, a person needs a high level of forethought (as the process and beliefs that occur before efforts for learning are deployed) to wield anticipatory adaptive control. self-efficacy is an aspect of self-management that fuels the quality of success or performance (bandura & locke, 2003). a person’s self-efficacy tends to regulate human function through cognitive, motivational, affective, and decisional processes (bandura & locke, 2003). dweck (2012) has pointed out that learners with a growth mindset (the belief that intelligence is not fixed) tend to embrace challenges, persist in the face of adversity, view effort and study strategies to learn and use feedback to improve their achievement. thus, beliefs are to be accompanied by decisional processes. most participants in this study described using assessment feedback not only to reflect on their study techniques but also as an encouragement that boosts their confidence in their understanding of mathematics to get ready for their final examinations: ‘they motivate, they motivation. firstly, you fail you realize that, when you fail you feel like i have to push myself, i have disappointed myself, and then you work based on that, okay, you look at the paper, okay, where did i go wrong, where do i have to put more effort into?’ (busisiwe*) ‘ngizofuna ukubona ukuthi ngibhede kuphi and the ngilungise ama-misteki wami, so that ngizobe ready for i-finali. [when my results come back, i want to see where i went wrong, where my mistakes are so that i can get ready for the final examination.].’ (james*) ‘uh, they, they boost my confidence in maths because, maths, as i said maths is a difficult subject, so if you don’t love it, maths won’t love you back, so when i knew that i practised and the results that i get are satisfying, i will encourage, that will boost my confidence and say but you can do better because you managed to get these results whereas you worked hard for them so if you put if you keep on working harder you get more, you get more satisfactory results.’ (mandla*) ‘uh, i tend to reflect and see my mistakes take a minute to step back and reflect on what i did well and what i did right and to see if the study plan that i have uh used, is it working for me or is it not working because it would be useless to continue to using the same study skills but getting less results so i use that information to be able to see where i can improve and where i can change my study skills.’ (nomsa*) ‘i will first look at which one, which one i didn’t understand and then after then i will work on it, so that i become perfect.’ (sam*) ‘my results like they encourage me like to do more, put more work, yah effort.’ (sipho*) self-awareness is crucial for a learner to self-regulate. the learner needs to be able to deploy, in a dynamic manner, specific learning processes to develop mathematical resilience (zimmerman, 2002). data from the semi-structured interview reveal that learners rely on a variety of strategies such as their personal attitude and the identification of an entry point in their journey to develop mathematical resilience. through a positive attitude, mathematically resilient learners can stay focused on the task at hand and start with the easy tasks and progress to what is perceived to be challenging, as indicated in their comments: ‘i have a very positive attitude, even though it seems like somewhere somehow, i do doubt myself, but i remember that i do have that potential.’ (vivian*) ‘yah challenges i cope with them in … i would say an unusual way, ’cause every time uh i kinda not achieve what i wanted to achieve in mathematics i always stay positive which is very hard for people to do that.’ (nomsa*) ‘okay, if, if like i have too much work to do like yah, i get to the bottom of it like, i will not sleep, i make sure like everything is done.’ (sipha*) ‘uh, basic, basically, when, when i start revising maths, i always make sure that i drink a lot of water so that my brain will, will function with me, will work with me well with my body, so when, when i start, i start with the easy problems so that i will gain confidence.’ (mandla*) mathematically resilient learners can identify an entry point (a topic or chapter in mathematics) in mathematics that is used as leverage to understand other seemingly challenging topics in mathematics. algebra featured prominently, followed by statistics. such a stance points to a learning strategy described by zimmerman and pons (1986) as seeking and selecting information. in such an instance, learners are exercising control over their personal skills as well as the task at hand. starting with the easy topic and moving to a challenging one demonstrates confidence and they are more likely to persist doing the task to achieve the desired levels of outcomes (pintrich, 2003): ‘algebra.’ (mandla*) ‘okay, i will say algebra because like ay, it’s the like, okay, maybe i can say is the mother of mathematics, yah.’ (sipho*) ‘paper 1 [paper 1 is an algebra paper] mina ukuyenza kwami ngiyenza izinto engazaziyo, ngisebenza kakhulu entweni engiyi-understanda kakhulu. [i work mainly with a topic that i understand better].’ (james*) ‘the first chapter in mathematics, i fell in love with was stats i think because i passed.’ (busiswe*) apart from identifying an entry point to exploit, mathematically resilient learners show persistence (dweck, 2012). results from the semi-structured interview reveal that resilient mathematics learners on average spend between half an hour to three hours per day practising mathematics: ‘i spend about 30 minutes or so because i don’t wanna tire up myself and burn out.’ (mandla*) ‘sometimes boma [around] three hours, four hours.’ (sam*) ‘sometimes two hours, it can go as far as three hours.’ (busisiwe*) ‘in a day sir, day ngisebenza [i practise for] two hours.’ (james*) ‘i don’t measure my time, like i just, i just practise maths until i feel like yah, i’m enough, like it’s enough.’ (sipho*) ‘okay, uh, firstly, i practise maths for one hour, but now seeing that or i saw that, practising maths for one hour me as an individual, just doesn’t do any, any, any, any, uh. it’s not benefit able enough for me, but the more i increase the hours, like right now, i’m practising maths for two, uh, two hours and 30 minutes, so it does help me.’ (vivian*) apart from spending a considerate amount studying mathematics, mathematically resilient learners in this study have reported that they use textbooks together with study guides loaned to them by the gauteng department of education through their respective schools for the year as well as past examination papers that are freely available on the gauteng department of education website. provision of learning resources and the use thereof by learners is crucial in facilitating mathematical resilience for individual or collective needs (ungar et al., 2013): ‘i tend to use my question papers my answer series that … uh, my answer series and i use my textbook, ’cause i have noticed a trend, you would find things in a textbook, but you wouldn’t find them in a question paper and the textbook helps you relate to the question paper, so i use my textbooks and my question papers.’ (nomsa*) ‘i use study guides.’ (sam*) ‘i make sure that, i, okay, i put my textbooks just to refer if i get stuck, just to put them there on, to lay them on the table and then my questions papers as well, then yah, i just practise, practise, practise.’ (vivian*) discussion and conclusion the main purpose of this study was to explore how disadvantaged learners learn mathematics in a disadvantaged environment. analysis of the semi-structured interviews revealed two interrelated themes, namely foreground and growth strategy. foreground (forward-looking), as conceptualised by skovsmose (2012), entails the opportunities that the political, economic, and social context makes available to a person and how these opportunities are experienced and interpreted. in other words, the interpretations epitomise the processes that an individual secures for themselves, not only social resources (teachers) but also the physical resources (textbooks, study guides and past exam papers) that they need to develop mathematical resilience in the context of adversity. foregrounds are made up of obstructions that may be regarded as learning barriers that have the potential to ruin the learners’ foreground (skovsmose, 2012). through the semi-structured interviews, learners expressed their proximal risks or challenges, particularly in their respective community environments, that they must overcome in their learning of mathematics. the immediate community environment was depicted as not helpful and not supportive due to noise and ignorance. this is similar to a study conducted by neshila (2018) in namibia with mostly learners who were able to develop mathematical resilience despite coming from surroundings that are noisy with outlets such as bars. according to skovsmose (2012) foregrounds are made up of possibilities and the subjective interpretation of such possibilities. similarly, resilience located in a socio-ecological setting assumes many processes that can be taken by an individual that can lead to well-being (ungar et al., 2013). despite a noisy and harmful environment, learners in this study explicitly expressed their hopes and aspirations. data revealed a constellation of motives for doing mathematics. the constellation of motives displayed by learners in this study stands side by side with the findings revealed by a variety of studies such as kacerja (2011) and hernandez-martinez and williams (2013) when they found two major motives for doing mathematics are the use value (with respect to pursuing a career or vocation that uses mathematics, for instance, mathematics teacher) of mathematics and the exchange value (the need to do mathematics as a prerequisite for a qualification) of mathematics. emerging from the semi-structured interview data is the central role that the teacher plays in support of the formation of learners’ foreground. here, teachers are perceived by learners as a source of motivation or inspiration in support of the learners’ foreground. thus, teachers are viewed as showing an interest in the learners’ foreground. showing interest in a learner’s foreground requires that the teacher shows an understanding of the issues that confront the learner in a respectful manner (liebenberg et al., 2016; sosa & gomez, 2012). mathematics teachers are strategically positioned to take up the responsibility of assisting learners to visualise a better future (skovsmose, 2005, 2012). in other words, teachers play a pivotal role in enabling learners to construct and reconstruct their foregrounds, as foregrounds are not stable but fluid. mathematically resilient learners were found to have the ability to interpret their political, economic, and social context, which gave rise to the expression of hope and future aspiration. these expressions of hope and future aspirations mark the connection between the personal and national context (skovsmose, 2012) as experienced by the learners. douglas and strobel (2015) point out that disadvantaged learners in disadvantaged environments need to be aware of the connections between their efforts in mathematics and potential benefits that may accrue from a mathematical career in the future. as expressed by mathematically resilient learners, pursuing a career reveals a pathway to a better future. a closer look at the data from the semi-structured interviews reveals the awareness by these learners of the connections between their efforts and the possibilities of a better livelihood in the future, as revealed by snyder (2002). by extension, the data reveal the desire of learners to disentangle themselves from their disadvantaged and deleterious environment. the data from the semi-structured interviews revealed that mathematically resilient learners tend to deploy a variety of learning strategies. in particular, the analysis found that learners use assessment feedback as one of the growth strategies and can identify specific topics that are utilised to develop mathematical resilience. in this instance, such a strategy of using a variety of learning processes is referred to by zimmerman (2002) as a self-control method wherein the learner evaluates their performance and can adjust their future learning. the use of assessment feedback resonates with havnes, smith, dysthe and ludvigsen (2012) when they reveal that learners who perform better in mathematics tend to use assessment feedback to monitor their performance to identify sources of challenges. in this regime, learners interpret assessment feedback in accordance with reasonably stable beliefs concerning the subject area, the learning process, as well as the learning product. at the heart of self-evaluation is what bandura (1995) regards as physiological and emotional states utilised to judge one’s capabilities through interpretation of the learning process. identifying a topic to gain access to mathematics learning and using assessment results as a growth strategy point to the construction of meaning that may be associated with the learner’s foreground. skovsmose (2012) has pointed out that the construction of meaning in learning mathematics is profoundly related to the learners’ foreground, as foreground provides conditions for engagement. thus, the appraisal of one’s capabilities and the construction of meaning in many ways shape cognition which ultimately affects behaviours that may indicate and promote mathematical resilience (ungar, 2015). emerging from the analysis of the semi-structured interviews is the finding of the amount of time learners are prepared to spend practising mathematics. this study revealed that mathematically resilient learners tend to spend at least half an hour to two hours per day on average. learners do not shy away from setting aside a considerable amount of time in learning mathematics. setting time aside to learn mathematics depicts a commitment or persistence as found by martin and marsh (2006). skovsmose (2012) attributes such an activity to an instrumental approach. in other words, instrumentalism may provide the required commitment and the energy to focus on their mathematics learning. understanding foreground requires the acceptance of the complexities of the subjective and external factors at the political, economic, and social level (skovsmose, 2012). in this study the availability of study materials such as free textbooks and study guides as loaned to the learners by the department of education signals the interaction between the political and economic parameters as experienced by learners. in this instance, learners are loaned free study material as well as textbooks as they attend no-fee paying schools. the study found that learners use a combination of resources in their journey of mathematics. after selecting a particular topic, learners in this study indicated that they use mostly textbooks, study guides and past examination papers in preparation for their examination. this result adds to the growing debate about resources in general in the south african educational landscape, with incongruent research findings on the effects of learning resources on learner achievement. on the one hand, a study by kabi (2016) found that despite strides made by the department of basic education to address inequalities in education, learner mathematics and physical science performance has not improved but has dropped over time. on the other hand, wilson fadiji and reddy (2021), utilising the timss 2015 south african data, showed that the availability of resources had a positive relationship with mathematics achievement. the concept of foreground in this study shed light on the learners’ interpretation of the national and personal context and provided the protective processes that learners leverage to develop mathematical resilience. attending a no-fee paying school and having free access to mathematics learning material such as textbooks and study guides marks the political aspect of the foreground. the social aspect (hindrances or potential obstructions) was interpreted as unsupportive, noisy and ignorant environments, while the economic interpretation was expressed by learners’ aspirations (e.g. study electrical engineering). the use of various learning strategies (identifying a topic to leverage, use of various resources, coupled with the use of assessment feedback) represents the protective processes that mathematically resilient learners engage in to buffer the risk or threat of dropping out of mathematics. taken together, these different behaviours represent protective processes that mathematically resilient learners engage in to protect themselves against the impact of poverty. engaging in protective processes is akin to behavioural engagement as viewed through the formative work of fredricks, blumenfeld and paris (2004) that learner engagement may be behavioural, emotional, and cognitive. in this instance, spending time studying mathematics is viewed as behavioural, staying positive in the face of adversity marks the emotional behaviour and using a combination of learning strategies and resources points to the cognitive behaviour of mathematically resilient learners. a teacher that shows an interest in the learners’ foreground is a caring and respectful teacher. through showing an interest in the learners’ lives, teachers can empower (through sharing their personal journeys of life and in mathematics) their learners to develop mathematical resilience. thus, mathematics teachers are to be encouraged to share their mathematical journeys with their learners and encourage learners to identify a section or topic that can be used as a springboard to navigate mathematical demands. more importantly, teachers are urged to consider personal urgency of a learner and self-regulated learning strategies to promote mathematical resilience lest we perpetuate the current poor learner mathematics achievement by disadvantaged learners learning mathematics in a disadvantaged environment. acknowledgements competing interests the authors have declared that no competing interest exists. authors’ contributions the authors declare that they equally contributed to this article. ethical considerations ethical clearance was obtained from the university of pretoria research ethics committee sm 17/07/02. funding information this research received no specific grant from any funding agency. data availability all data collected and used are kept safe by the university of pretoria and can be made available on request. disclaimer the views and opinions expressed in this article are those of the authors and do not necessarily reflect the official policy or position of any affiliated agency of the authors. references amin, n. 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(1986). development of a structured interview for assessing student use of self-regulated learning strategies. american educational research journal, 23(4), 614–628. https://doi.org/10.3102/00028312023004614 footnote 1. pseudonyms used to protect the identity of learners and educators. microsoft word 3-12 molefe & brody.doc pythagoras, 71, 3-12 (july 2010) 3 teaching mathematics in the context  of curriculum change  nico molefe & karin brodie  school of education  university of the witwatersrand  nicholas.molefe@wits.ac.za & karin.brodie@wits.ac.za    this  paper  examines  the  practices  of  two  grade  10  mathematics  teachers  as  they  implemented  the  new  fet  curriculum.  as  we  analysed  their  practices  we  found  that  current descriptions of reform and traditional practices were inadequate to describe the  similarities and differences between  them. we  therefore developed a set of elaborated  categories, which distinguish reform‐oriented  from  traditionally‐oriented practices. we  use  these  categories  to  analyse  the  two  teachers’  practices  and  we  use  the  teachers’  practices  to  illuminate  the  categories.  we  show  that  although  some  of  the  teachers’  practices may seem similar on the surface, in fact one teacher employed these in a more  strongly reform‐oriented way than the other.  education reform in south africa has ushered in a variety of changes in relation to the teaching and learning of mathematics. these changes include teachers’ practices and how these practices influence learners’ contributions and interactions in mathematics classrooms. this paper looks at two south african teachers who teach high school mathematics and the kinds of practices that the two teachers employed in their classrooms. we use schifter’s (2001) definition of teaching practices to look at practices as being skilful, patterned regularities that occur in teachers’ classrooms. for our purposes, there are two kinds of practices: mathematical practices and teaching practices. mathematical practices include symbolising, representing, justifying and communicating mathematical ideas (rand mathematics study panel, 2003). teaching practices involve particular approaches or methods that teachers employ in their classrooms in order to teach mathematics or develop mathematical practices. so in mathematics classrooms, the two sets of practices are related. brodie (2008) argues that: “practices are simultaneously practical and more than practical as they involve particular forms of knowledge, skills and technologies to achieve the goals of the practice”; “practices are always located in historical and social contexts that give structure and meaning to the practice and situate the goals and technologies of the practice”; and “practices are always coproduced between teacher, learners and their social contexts” (brodie, 2008, p.31). internationally, curriculum developments, also called reforms, encourage teaching practices that present mathematics as a web of related concepts with different ways of representing and solving problems (national council of teachers of mathematics, 2000; new zealand ministry of education, 2007). mathematics can be explored, contested, justified, and communicated, and reform mathematics teaching develops conceptual depth, procedural flexibility, and reasoning among learners. genuine interaction among teachers and learners in mathematics classrooms is important to achieving these goals. curriculum developments in south africa over the past 15 years have encouraged the same goals for mathematics teaching (department of education, 1997, 2002, 2003)1. this is in contrast to traditional mathematics 1 this data was collected in 2006, the first year of implementation of the new fet curriculum. as we are in the process of finalising the paper for publication, the minister of basic education has indicated changes in some elements of this curriculum, in particular a move from packaging of the curriculum into outcomes and assessment standards, back to packaging in terms of topics. our paper focuses on teaching practices that engage learners with mathematics and which we believe to be important no matter how the official curriculum is packaged. teaching mathematics in the context of curriculum change 4 teaching, which has often been characterized as driven by procedures and algorithms, with very little learner engagement with the teacher, with each other and with conceptual mathematics. this distinction between traditional and reform creates a dichotomy, which can be used to argue that many teachers “fail” to achieve reform practice (lavi & shriki, 2008; nolan, 2008). however, it is more likely that when teachers, many of whom have been teaching traditionally for years, begin to implement new approaches, they develop hybrid practices, some kind of mixture between traditional and reform teaching (brodie, 2010; cuban, 1993). it is also the case that descriptions of reform practice can be coopted as descriptions of traditional practice. for example brodie (2007) notes how the “question and answer method” can be seen as allowing learner participation, because learners are given a chance to answer questions. however, if the questions are narrow and do not allow learners to think and reason, then questioning remains a traditional practice. a second example is getting learners to explain their thinking on the board. learners can write up a method and say what steps they did or they can explain the meaning and concepts behind their procedures. in both of these examples – even though what teachers and learners are doing may look the same on a superficial level, in relation to how learners actually engage with the mathematics, the two practices of questioning and learners explaining on the board can either be reform-oriented or traditionally-oriented. we use the terms reformand traditionally-oriented to indicate that practices are never purely one or the other, but can tend toward different ends of the spectrum. in this paper we analyse two teachers’ practices and in so doing we define a set of teaching practices that can be seen as either reformor traditionallyoriented, based on how they support learners to engage with mathematical ideas and reasoning. we elaborate on how each practice can be described differently, thus clarifying for teachers and researchers how to distinguish reform-oriented from traditionally-oriented practices. a number of studies have been conducted on reform teaching practices worldwide. of the studies that have been conducted so far, most were done in primary schools (ball, 1993, 1996; elbers, 2003; heaton, 2000; kazemi & stipek, 2001; lampert, 2001) with only a few in high schools (boaler, 2002a; chazan, 2000). all of these studies were conducted outside of south africa 2 . this paper looks at teacher practices in the context of south african high schools and our new curriculum. teachers, as classroom practitioners, are key to the enactment of the new curriculum. thus, research is needed not only to help teachers to understand the curriculum, but also for researchers to be in a better position to understand teachers’ strengths and challenges in enacting the curriculum in their various classrooms. theoretical framework this study draws on particular notions of mathematics, teaching and learning. kilpatrick, swafford and findell (2001) identify five strands of mathematical proficiency: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning and productive disposition. although the strands are intertwined and each is necessary for the development of the others, kilpatrick et al. argue that adaptive reasoning holds the other four strands together. adaptive reasoning refers to the capacity to think logically and includes knowledge of how to justify conclusions. it is important that learners know and understand that “answers are right because they make sense and follow a particular line of reasoning that is valid” (kilpatrick et al., 2001, p. 129), rather than merely accepting what the teacher and textbook tell them. two theories inform the notions of teaching and learning in this study, vygotsky’s (1978) socio-cultural theory and lave and wenger’s (1991) theory of situated learning. a socio-cultural perspective provides resources to understand how teachers and learners interact and mediate knowledge for each other. a situated perspective provides the notion of legitimate peripheral participation (lave & wenger, 1991), which describes how newcomers are inducted into a community of practice. situated and socio-cultural perspectives relate learning to the social situations in which learning occurs and help in identifying 2 we note that brodie (2010) has recently published a study of five south african secondary classrooms. nico molefe & karin brodie 5 practices that are developed through the learning process. both of these theories allow us to see teachers and learners as co-participants in the learning process, and both theories have informed curriculum reforms worldwide, including in south africa. bringing these notions of teaching, learning and mathematics together, we use lampert’s (1998) notion of a thinking practice to argue that teaching is a thinking practice in which teachers, through their intellectual work, focus on fostering thinking practices among their students. the notion of mathematical proficiency shows that mathematics is not only about knowledge of numbers, symbols and procedures; it is also a subject that involves justifying, communicating, thinking and reasoning. encouragement of these mathematical practices in mathematics classrooms can provide opportunities for learner development. teachers can use teaching practices to develop mathematical thinking practices and proficiency among learners. the teaching practices that teachers employ in their teaching, for example asking challenging questions, can become practices that learners use in their learning. as teachers and learners work together in a community of practice, mediating knowledge with and for each other, learners can internalise the teachers’ practices to build their own mathematical practices and reasoning (boaler, 2002b). literature review various studies have investigated both teaching and mathematical practices and how these relate to each other. these studies helped us to understand practices that teachers employ in their teaching and how these practices impact on the learning of mathematics. we used the studies to help us develop categories that we used to analyse the teachers’ practices. the literature identifies two overarching ways of seeing teaching. first, there is the initiation-responseevaluation (ire) approach (mehan, 1979), where the teacher initiates with a prompt or question and waits for learners to give their response. after the learner response has been given, the teacher then evaluates the response and continues with the next cycle of the ire structure. this approach can be constraining of learner contributions and interactions because learners only respond, usually in short phrases, to the teacher’s questions. the second approach is where the teacher engages learners to contribute with answers that they can justify through arguments that they hold in class with their teacher or among themselves. this approach is more conversational in nature (davis, 1997). elbers (2003) explains how a conversational or inquirybased approach allows for learners’ mathematical creativity to develop. in inquiry-based approaches, learners benefit from their teacher’s and from each other’s ideas. this resonates with the socio-cultural perspective’s argument that “collaboration among children in the process of being taught” is crucial to learning (davydov, 1995, p. 16). elbers (2003) bases his argument on the ideas of transforming classrooms into learning communities, wherein teachers help learners to participate in the process of knowledge construction. depending on their levels of engagement, learners are capable of making valuable contributions in class. in this way learners can be seen as co-participants with the teacher and each other in their learning of mathematics. interactions in the classroom can help individual learners to carefully observe how the teacher goes about engaging other learners with the use of practices that promote mathematical reasoning. pitting the ire against conversation again creates a dichotomy, which we want to break down. teachers who create hybrid practices will move between these and use the ire form in different ways (brodie, 2007). however, the extent and kind of interaction about mathematics in a classroom does give some indication of the reform orientation of the teaching practice and so we used this as one of the criteria in our analysis. three other important criteria were the nature of the tasks that teachers used, how they questioned and pressed learners and how they dealt with learners’ errors and misconceptions. stein, grover and henningson (1996) argue that tasks can be of low level cognitive demand, where they are not connected to mathematical meaning, or high level cognitive demand, where they are connected to mathematical meaning and concepts. it is also possible to raise or lower the level of tasks during interaction in the classroom. reform approaches tend to be characterised by high level task demands while traditional teaching mathematics in the context of curriculum change 6 teaching tends to be characterised by low level tasks (stein et al., 1996). teacher questioning can also be associated with reform or traditional approaches. kazemi and stipek (2001) use the notions of “high press” and “low press” to distinguish between approaches that teachers can use to push learners into verifying their answers with reference to mathematical concepts (high press) and approaches where teachers accept procedural explanations (low press). they argue: “high press questions encourage learners to include mathematical arguments in their explanations, while low press questions encourage procedural descriptions only” (p. 78). these high press questions can create opportunities for learners to work cooperatively or in collaboration with each other, as they will be forced to share their thinking in preparation of convincing either the teacher or their fellow learners. involving learners in the lesson allows learners’ errors or misconceptions to become visible (nesher, 1987; swan, 2001). getting learners to explain themselves creates opportunities for discussions of the errors and misconceptions that they produce. what is important is how teachers handle situations in which learners produce such errors or misconceptions. when viewed from the perspective that misconceptions make sense to learners and they therefore can inform teaching, errors and misconceptions can be accepted as a normal part of the teaching and learning process (kazemi & stipek, 2001; nesher, 1987). errors and misconceptions are signs that learners are involved in their learning, and their thinking processes are engaged, so further explanations can be encouraged from learners to understand why they made those errors. in discussing errors and misconceptions, further thinking and reasoning can be provoked and learners can develop practices of making meaningful contributions to mathematical discussions. research design this paper is based on a qualitative case study. our interest was in exploring teacher practices in the ongoing flow of mathematics lessons. we decided to work with two cases so that we could look across two different classrooms. for ethical reasons, we have used pseudonyms for the two teachers; we refer to the one teacher as mr. ronaldo, and the other as mr. thekiso. mr. ronaldo worked in a school in a formerly ‘coloured area’, which served both coloured and black learners, while mr. thekiso worked in a school in a ‘black township’, which served only black learners. neither of the schools was adequately resourced. the only equipment present in each of the two classrooms was a chalkboard. mr. thekiso’s classroom had a teacher’s table and chair but mr. ronaldo’s did not. mr. ronaldo has a secondary teachers’ diploma, and higher diploma in education where he specialised in mathematics. he also has a b.sc. honours degree in mathematics education, which included mathematics content courses and courses relating to the new curriculum in mathematics. mr thekiso has a secondary teachers’ diploma where he specialised in mathematics, and a further diploma in education, which included courses in mathematics and science content but which did not focus on the new curriculum in mathematics. both of the teachers were observed and videotaped teaching grade 10, four lessons in mr. ronaldo’s classroom and five lessons in mr. thekiso’s classroom. there were 42 learners in mr. ronaldo’s class, and 34 in mr. thekiso’s class. in the lessons, we observed how the teachers employed particular practices, how learners contributed in the lessons and engaged with mathematical practices and how learners’ thinking and reasoning were reflected in interactions with the teachers’ practices. interviews with each of the two teachers were conducted to substantiate the data analysis. these interviews were conducted after watching all the lessons and focused on what we observed in the lessons with respect to teacher practices and learner contributions. the interviews were semi-structured and helped us to understand what the teachers think about mathematics, how they teach it, why they teach in that way, and how they mediate their teaching with their learners to encourage thinking and reasoning. a semistructured interview schedule allowed us to decide about what to follow up on, and what to probe. we used interviews not to obtain substantially new information but to confirm or engage with specific events we observed during classroom teaching. we did not use an existing framework for practices. rather, we generated a framework by working through the data set and developing categories from patterns in the data. all of the lessons were watched repeatedly in order to discern common practices across the lessons that provided us with enough nico molefe & karin brodie 7 information to develop categories. the categories are both informed by the literature discussed above and grounded in the data (strauss & corbin, 1998) and provide the means to give an in-depth analysis of the data. we initially developed a long list of categories and compared them to see which of them linked to one another. we combined some and separated others. as we developed the categories we discussed whether and how they reflected reform-oriented or traditionally-oriented teaching practices. the categories are in table 1 and in the discussion below we show how these practices are used either in reform-oriented or traditionally-oriented ways. table 1: practices analytic framework categories reform orientation traditional orientation writing on the board uses the board as a public space to record learner ideas: correct or incorrect. uses the board as a public space to put up teacher’s ideas and correct responses from learners. giving classwork uses learners’ classwork as a means to see their thinking and help to make it public. uses learners’ classwork to correct incorrect ideas. giving advice to learners talks to learners about mathematics as a practice, which requires thinking. talks to learners about mathematics as a practice, which requires procedures and drill. inserting mathematical language inserts mathematical terminology to help learners express their own thinking. teaches mathematical terminology for learners to learn and reproduce. maintaining/changing the task level usually maintains higher cognitive demand tasks starts off with lower cognitive demand tasks or reduces demand of tasks in interaction with learners handling correct and incorrect responses from learners accepts all responses and tries to understand thinking behind both correct and incorrect responses. accepts correct responses and corrects or ignores incorrect responses. getting learners to explain themselves tries to access the learners’ thinking and work with it, whether correct or incorrect. expects and emphasises correct explanations from learners asking learners to repeat/reexplain pushes learners to clarify or justify their or others’ thinking. gets correct answer repeated or tests learners to see if they’ve been listening or have understood the correct answer. redirecting input from a learner to other learners redirects question to get additional ideas or clarity on current ideas under discussion. redirects incorrect answers in order to obtain correct answers recapping/summing up a section of work summarises learner and teacher ideas to show depth of discussion and build to correct mathematical ideas. summarises the correct mathematical ideas. encouraging self-evaluation by learners encourages learners to justify their own ideas and so to evaluate their correctness. encourages learners to check answers to see if they’re correct or not, and to correct them if necessary. teaching mathematics in the context of curriculum change 8 the teachers’ use of the identified practices we used the categories in table 1 to describe the practices that the two teachers employed in their classrooms. the substance of these practices showed greater or lesser alignment with practices that can be identified with traditionallyor reformoriented approaches. we discuss similarities as well as differences that we observed in the two teachers’ practices. writing on the board both teachers used the chalkboard to capture information but they did it differently. while mr. thekiso would more often than not write only correct information on the board, mr. ronaldo would write both correct as well as incorrect information on the board to engage learners. mr. ronaldo used the board to capture learners’ correct and incorrect answers for the purpose of discussion, while mr. thekiso used the board to capture learners’ correct answers and to convey correct mathematics. information written on the board puts it in the public domain and reminds the classroom community about previously discussed ideas, making it available for other members of the community to critique and give their opinions, if these practices are supported by the teacher. mr. ronaldo used the board in this way, coming back to previous ideas for more discussion. giving classwork mr. thekiso gave classwork more often during his lessons than mr. ronaldo did. he would give learners work to do, and walk around the class checking on what they were writing. thereafter, he picked up on particular points he had seen, and encouraged learners to correct mistakes or he emphasised particular points for them to remember. mr. thekiso employed this practice in traditional ways, as he preferred giving explanations to learners than supporting learners to make inputs on what they had been doing. mr. ronaldo gave classwork only once during the data collection week, and this was after realising that learners were struggling to identify perfect squares. most of the time, mr. ronaldo encouraged learners to make contributions in the lesson by participating in class discussions and debating their answers. giving classwork is important in both traditionaland reformoriented practice because it helps to make learners’ ideas accessible to the teacher. giving advice to learners teachers often talk to learners about doing mathematics, which communicates their understanding of the nature of mathematical knowledge and practices. both mr. thekiso and mr. ronaldo put emphasis on thinking in mathematics. mr. ronaldo advised his learners not to rush into giving answers, but rather give themselves time to think carefully about their answers and to justify their ideas (see below) indicating that mathematicians attempt to convince others by justifying. at times, mr. thekiso told his learners that “in mathematics we think and reason”, communicating this important idea to them. however, as will be discussed below, the extent to which he encouraged this is not always clear. inserting mathematical language/terminology both teachers used the language of mathematics to help develop mathematical ideas in the classroom. language is an important tool in mathematics that can help learners develop their reasoning as well as their communication skills. mathematics also has its own discourse and when teachers put emphasis on the importance of the correct use of mathematical language, this can help learners understand this discourse in mathematics, and eventually talk like mathematicians. mr. thekiso used different terminology for the same concepts, for example a cartesian plane and a system of axes. using different language gives learners access to a range of ways of talking about mathematics. when mr. ronaldo wanted to emphasise the word ‘difference’ in the difference of squares, which learners often ignore, he did it in an interesting way. he asked learners to factorise 22 ba  and insisted for some time that they should try to do so, thus setting up the possibility that they would understand why the difference was important. in this way, mr. ronaldo could insert the term “difference” in a way that made sense to learners. maintaining/changing the task level after working on a number of standard factorisation problems with the difference of two squares, mr. ronaldo gave learners the expression a2 + b2 to factorise. none of the learners in class was aware that the nico molefe & karin brodie 9 expression could not factorise, but mr. ronaldo wanted learners to keep on checking factors through multiplication to see if the expression could be factorised. this exercise of testing factors through multiplication helped learners think more deeply about how to justify their answers and the relationship between factorisation and multiplication. so mr. ronaldo raised the level of the standard task, which is a reform-oriented practice. in contrast, mr. thekiso tended to keep the level of the task the same or lower the demands, which is a traditionally-oriented practice. for example, when mr. thekiso asked learners to give trigonometric ratios from a triangle that had no right-angle, he immediately dropped the task when learners started giving wrong answers. he did not give them the chance to apply their thinking and see why their answers were wrong. handling correct/incorrect answers from learners mr. ronaldo dealt with both correct and incorrect answers in the same way by asking other learners whether they agreed with the response and asking for other ideas. he also encouraged learners to provide justifications of their own answers, whether correct or incorrect. working with different views from learners can help learners to see that generalisations and mathematical knowledge are built from a range of justified contributions. this is a practice that mathematicians work with and helps to bring reasoning into the mathematics classroom. mr. thekiso also redirected learners’ input and responses, though in his case it was mainly to search for the correct answer and not necessarily to press learners into justifying their responses and helping to reach more general conclusions. getting learners to explain themselves the two teachers both called learners to come and explain their answers on the board but did it in different ways. in mr. thekiso’s class, the learner at the board would usually talk to other learners or to the teacher, without much interaction between them. creating room for learners to explain themselves is a practice that goes with reform approaches, but mr. thekiso used this approach in traditionally-oriented ways. in mr. ronaldo’s lessons, the class would engage with what the learner explaining on the board was doing. the following extract shows how mr. ronaldo got learners to explain themselves. teacher: do you want to come and show us martha, (pause) come show us, come show... (pause) did you test it martha? martha: yes, sir teacher: why are you doubting yourself, (pause, as martha writes on the board), what are you doing, tell us what are you doing the last statement shows that mr. ronaldo was not only interested in the correct answer, but wanted the learners to have confidence in their thinking, as when he asked the learner (martha) to explain her answer and suggests that she should not doubt her ability to do so. asking learners to repeat or explain this was done by both teachers for purposes of maintaining the learners’ interest, or checking learners’ understanding about a particular section. the practice also helped teachers to know which learners understood the discussion, so as to make follow-ups. when one learner could not pronounce the word ‘hypotenuse’, mr. thekiso came to her rescue and helped her to do so. mr. ronaldo would use the practice of asking learners to explain themselves to push them into verbalising their thinking. the next extract shows how mr. ronaldo pushed a learner to justify her answer. teacher: no, people we are factorising this, my first question to you was, is p plus q all squared a perfect square, come with factors...yes learner: (inaudible) teacher: and ask why you say it is a perfect square, p plus q (pause) all squared, is it a perfect square in the above extract, mr. ronaldo encouraged the learner to explain another learner’s thinking by asking her to justify a correct answer. mr. ronaldo did not indicate whether the other learner’s answer was right teaching mathematics in the context of curriculum change 10 or wrong. he wanted learners to reason about the problem, particularly in the difficult context of distinguishing between p2 – q2 and (p – q)2. redirecting input redirecting input happened after a learner had given a response and his/her input was redirected to another learner. redirection of input was dealt with differently by the two teachers and happened more often in mr. ronaldo’s lessons. after learners had made inputs, mr. ronaldo would not make learners aware whether their answers were right or not. he handled both correct and incorrect answers in similar ways, redirecting these answers to other learners to encourage more mathematical ideas in the public space. mr. thekiso would indicate upfront if a learner’s response was correct or not and would only redirect if he wanted a learner to correct an incorrect contribution. redirecting input is a practice that can be linked with reform teaching, if it supports better engagement with the mathematics, which is how mr. ronaldo used it. mr. thekiso used it in traditional ways as he often gave answers or channelled learners towards the correct answer and did not make room for understanding learners’ incorrect answers. recapping or summing up a section of work this practice occurred when the teachers wanted to remind the class about what had been discussed previously, or when they wanted to check on the learners’ understanding. mr. thekiso used this practice more often to summarise work that he dealt with on that particular day. the structure of the lesson at this stage took a strictly ire format, wherein mr. thekiso would ask a question, wait for the learner’s response and either affirm or reject the response before asking the next question to repeat the cycle. mr. ronaldo did not often summarise at the end of a lesson but would sometimes leave learners with a question hanging and ask them to go home and work the answer out, such as: factorise a2 + b2. encouraging self-evaluation by learners this practice was more dominant in mr. ronaldo’s lessons when he pushed the learners to justify their answers. he had taught his learners to test their answers in order to justify them. it was in this way that the learners developed the habit of learning to justify their answers every time they put them up for public scrutiny. doing self-evaluation also built the learners’ confidence by encouraging them to boldly present their ideas to the class and allowing the class to engage them on their ideas. this practice was internalised by some learners, who challenged other learners to evaluate their answers, and at some stage even challenged their teacher to justify his answer when they did not agree with him. after learners had struggled with factorising (a + b)2 – (c – d)2, mr. ronaldo explained to them how it could be done. after he finished with his explanation, some learners were still not convinced by his explanation. they thought that the factorisation should be [(a+b) + (c–d)][(a+b) – (c+d)]. they did not accept his explanation and pushed him to check it and prove it to them. conclusion and implications the above discussion suggests that mr. ronaldo employed a number of reform-oriented practices, in particular maintaining tasks of high level cognitive demand and encouraging learners’ engagement with the mathematics, their mathematical thinking and their self-evaluation of their thinking, not only their answers. these are identified as reform-oriented practices in that they focus on learners’ developing meaning, conceptual understanding and learning to justify and communicate their mathematical thinking. mr. thekiso lowered, rather than raised, the task level, redirected input aimed at procedural fluency rather than conceptual understanding (kilpatrick et al., 2001) and supported learners to get correct answers, rather than to engage in mathematical thinking. although mr. ronaldo also worked to correct mistakes, he saw these as misconceptions that required discussion, rather than mistakes that could be easily corrected. so the analysis of differences across the two teachers’ practices suggests use of predominantly reform-oriented practices by mr. ronaldo and predominantly traditionally-oriented practices by mr. thekiso. the differences that we found between the two teachers’ practices suggest that mr. ronaldo was more “reform-oriented”, but there were still elements of traditional practices in his teaching. he did not employ reform practices all of the time (see also brodie, 2008). the reform-oriented practices that he did nico molefe & karin brodie 11 employ were related to how he involved learners in his lessons. he encouraged thinking and reasoning in learners and supported learners to justify their thinking. with regard to his traditionally-oriented practices, mr. ronaldo attributed use of these to time pressures. while we understand that time is a scarce resource for teachers, we also want to argue that it should not be a deterrent to promoting thinking and reasoning amongst learners. it can be argued that there will always be a place for traditional practices in reform-oriented teaching. for example, the ire/f structure is often thought of as characterising traditional teaching, but how it is used can lead to extended learner thinking and therefore resulting in reform-oriented teaching (brodie, 2007). we have also shown that standard descriptions of reform teaching practices, such as asking questions or getting learners to explain, need to be elaborated in order to distinguish between the substance of reform or traditionallyoriented practices. even though the two teachers had similarities in practices such as ‘writing on the board’ as well as ‘asking learners to come to the board to explain themselves’, the two teachers dealt with these practices in different ways. while mr. ronaldo would open discussions for both correct as well as incorrect answers without making learners aware which of the answers were right and which were wrong, mr. thekiso often rejected wrong answers in order to get the right answers. an important difference between the two teachers was how they handled learners’ responses. handling learners’ responses as a reform-oriented practice requires that teachers open up for more discussions from learners even if learners did not initially come up with a correct response. exchange of ideas, 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/hrv (za stvaranje adobe pdf dokumenata najpogodnijih za visokokvalitetni ispis prije tiskanja koristite ove postavke. stvoreni pdf dokumenti mogu se otvoriti acrobat i adobe reader 5.0 i kasnijim verzijama.) /hun 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can be opened with acrobat and adobe reader 5.0 and later.) >> /namespace [ (adobe) (common) (1.0) ] /othernamespaces [ << /asreaderspreads false /cropimagestoframes true /errorcontrol /warnandcontinue /flattenerignorespreadoverrides false /includeguidesgrids false /includenonprinting false /includeslug false /namespace [ (adobe) (indesign) (4.0) ] /omitplacedbitmaps false /omitplacedeps false /omitplacedpdf false /simulateoverprint /legacy >> << /addbleedmarks false /addcolorbars false /addcropmarks false /addpageinfo false /addregmarks false /convertcolors /converttocmyk /destinationprofilename () /destinationprofileselector /documentcmyk /downsample16bitimages true /flattenerpreset << /presetselector /mediumresolution >> /formelements false /generatestructure false /includebookmarks false /includehyperlinks false /includeinteractive false /includelayers false /includeprofiles false /multimediahandling /useobjectsettings /namespace [ (adobe) (creativesuite) (2.0) ] /pdfxoutputintentprofileselector /documentcmyk /preserveediting true /untaggedcmykhandling /leaveuntagged /untaggedrgbhandling /usedocumentprofile /usedocumentbleed false >> ] >> setdistillerparams << /hwresolution [2400 2400] /pagesize [612.000 792.000] >> setpagedevice relationships between mathematics and literacy:  exploring some underlying factors  carol bohlmann  elizabeth pretorius    university of cape town  university of south africa   carol.bohlmann@uct.ac.za  pretoej@unisa.ac.za  this paper focuses on grade 7 learners in two township schools where the relationships  between performance on language and reading tests in the home language and english  were investigated in relation to examination performance in mathematics. in both schools  reading ability rather than language proficiency in english emerged as a strong predictor  of mathematics achievement. the schools serve as a case study for exploring some of the  socio‐economic, teacher and classroom factors underlying differential school performance  in mathematics. because the new curriculum presupposes a highly literate environment, it  is  suggested  that  mathematics  learning  will  be  negatively  affected  if  learners  lack  adequate  reading  skills.  the  findings  suggest  that  quality  schooling  is  a  strong  determinant  of  both  reading  and  mathematical  achievement.  the  new  mathematics  curriculum  has  the  potential  to  make  a  difference  only  if  schools  improve  learners’  literacy development.   south africa’s poor performance in large national and international studies such as the trends in international mathematics and science study (timss) and the southern and eastern africa consortium for monitoring educational quality (sacmeq) is well known (see for example, department of education (doe), 2005; mothibeli, 2005; reddy, 2006). two factors impacting critically on mathematical performance are the language of learning and teaching (lolt) of mathematics, and teacher qualification. for the majority of learners in south africa, the lolt for mathematics is not the home language. although many primary schools have a home language as the lolt in the first three years of schooling, the crossover to english as medium of instruction is typically made in grade 4. there is also a pressing need for qualified teachers in the earlier phases of education. yet in the timss study, “internationally, most teachers had at least a fouryear degree qualification … (in) comparison … the south african mathematics and science teachers are among the least qualified” (reddy, 2006, p. xv). many of these large-scale numeracy studies also report similar poor performances in literacy. for example, the results of the national systemic evaluation showed grade 6 learners obtaining a national mean of 38% for literacy in the lolt (doe, 2005). in the latest progress in reading literacy study (pirls) south african grade 4 learners came last of 40 countries in the literacy assessment (pirls, 2006). in spite of poor trends in both mathematics and literacy, none of the above studies explores the relationship between numeracy and literacy in any detail. it is important to explore more closely the relationship between mathematics and literacy in a multilingual country such as south africa since learners’ proficiency in the lolt will undoubtedly affect their understanding of mathematics. this study provides a small-scale but unique perspective on the mathematics-literacy relationship by using the de facto mathematics performance of grade 7 learners in two primary schools in a south african township to see whether their mathematics performance shows any relationship to performance in independently administered literacy tests in the first language (l1) of the learners (in this case northern sotho) and in english, the lolt. in addition, the study breaks up the notion of literacy into two separate constructs, language proficiency and reading. this distinction is important for language proficiency is a necessary but not sufficient condition for reading ability. reading only develops through extensive 42 pythagoras, 67, 42-55 (june 2008) carol bohlmann & elizabeth pretorius reading, and it is reading specifically that predicts success at school for it forms the foundation for subsequent school-based learning (e.g. snow, burns, & griffin, 1998). however, the role that reading accomplishment plays in mathematics, specifically during the senior primary school years, is under-researched. two variables are constant in this study: all the learners come from the township and they all speak an african language at home. however, the two township schools they attend provide different conditions for mathematics and literacy development. there are three research questions that inform this paper: 1. how do the de facto grade 7 mathematics examination papers in the two schools reflect the five learning outcomes of the new mathematics curriculum? 2. are there significant differences in grade 7 mathematics performance, language proficiency and reading ability between the two schools? 3. are there significant relationships between grade 7 mathematics performance, language proficiency and reading ability at the two schools? answers to these questions are considered in the context of similarities and/or differences between the two schools in terms of mathematics teacher qualification, classroom-based factors and school reading practices. based on the findings we consider whether, by the end of primary school, grade 7 learners have sufficient competence in mathematics, language and reading to cope with the demands of the new curriculum and whether there really is a relationship between mathematics and literacy. in order to contextualise the study within the larger south african picture, we first identify some issues that relate to the teaching and learning of mathematics, before describing the methodology and discussing the results. the importance of language and reading in mathematics learning mathematics learning is highly dependent on literacy. all of the five mathematics learning outcomes for the senior phase highlight those activities that depend significantly on language and reading. for example, learning outcome 1 (lo 1) states “the learner will be able to recognise, describe and represent numbers and their relationships, and to count, estimate, calculate and check with competence and confidence in solving problems” [italics added] (doe, 2002, p. 61). this outcome includes activities aimed to develop learners’ understanding of how numbers relate to one another. this implies that learners are confident with expressions such as ‘less than’, ‘twice as much as’, and so on. even students at tertiary level find these concepts difficult (e.g. bohlmann, 2006). the dependence on context also implies a dependence on verbal skills. much has been written on the discourse of mathematics and its dependence on verbal skills in the lolt. it is thus not surprising that proficiency in english should feature prominently when the lolt is english. mathematics discourse generally contains items that have linguistic, cognitive and contextual dimensions (gibbs & orton, 1994). the linguistic dimension involves both the receptive level (e.g. reading) and the productive level (e.g. writing, discussing). the cognitive dimension reflects the level of complexity of the concepts and cognitive skills such as logical reasoning, critical analysis and interpretation of abstract concepts. the contextual dimension1 reflects the level of contextual support provided. difficulty with any one of these aspects has serious implications for studying mathematics. the conceptual complexity and problem-solving nature of mathematics make extensive demands on the reasoning, interpretive and strategic skills of learners, especially when these activities are done in a language that is not their primary language. many studies have investigated the role of language in mathematics (e.g. bartolini bussi, 1998; ellerton, clarkson, & clements, 2000), and they all show that poorly developed language skills (in the lolt) undermine mathematical performance. in general the mathematics register is abstract, non-redundant (prins, 1997) and conceptually dense. mathematical symbols and graphics (e.g. charts, tables and graphs) increase the conceptual density. mathematical discourse features more complex and compact relationships than ordinary discourse. furthermore, mathematics discourse is characterised by precision, requiring close attention to detail. mathematics texts are also 1 in this case the context of the problem within the text is implied, not the external context of the learner. 43 relationships between mathematics and literacy hierarchical and cumulative, such that understanding each statement is necessary for understanding subsequent statements. overlooking or misunderstanding a particular step has severe consequences for overall comprehension. the language problem remains a complex issue; it has become increasingly clear that only if students reach a sufficient level of familiarity with the use of natural language in … mathematical activities can they perform in a satisfactory way. (boero, douek, & ferrari, 2002, p. 242) however, nothing can be understood about specific mathematical concepts (e.g. the square root of a number) through natural language alone; only once students have understood the particular mathematical notion of a power are they able to comprehend roots. in other words mathematical concepts need to be acquired in the context of mathematics, irrespective of whether the acquisition occurs through reading and/or other uses of natural language (sfard, nesher, streefland, cobb, & mason, 1998). without well-developed reading and language skills learners will not be able to “develop mathematical thinking skills such as generalising, explaining, describing, observing, inferring, specializing, creating, justifying, representing, refuting and predicting” (doe, 2002, p. 63). reasoning ability, for instance, is dependent on an understanding of logical relations in language, one of many reading skills underpinning the ability to construct meaning from written material. in theory, the learning outcomes represent sound mathematical pedagogy. in practice, for the contexts to be used effectively they need to be available (with implications for resources in the school and home) and accessible (with implications for learners’ reading levels). learners need to access information and understand the context and content before they can even begin to apply any of the mathematical skills they have learnt. mathematics in the get band during the general education and training (get) band mathematics (essentially mathematical literacy, at this stage) can be described in terms of the contexts that require quantitative literacy practice, the mathematical content that is required when such activities are practiced, and the relevant reasoning and behaviour (frith & prince, 2006). we briefly consider these next. c o n t e x t s a n d c o n t e n t : steen (2001) reinforces the idea that quantitative literacy practice, unlike mathematics, is always embedded within a context. however, the practice in many mathematics classrooms in south africa still focuses on ‘chalkand-talk’ rather than engagement with content in relevant contexts. usiskin (2001) warns against the use of contrived ‘real-life’ examples. doing mathematics requires the use of authentic contexts which need to be understood as clearly as the mathematical content that is being applied. the challenge for the get band is to find contexts that are sufficiently available, accessible and relevant. van etten and smit (2005) point out that when realistic contexts are used, learners are then faced with “text consisting of a greater use of words/language … this poses a problem, since … learners’ reading skills are often under-developed” (p. 58). numeracy and literacy being mathematically literate presupposes an ability to express quantitative information coherently in verbal or visual form. kemp (1995) argues that this includes the ability to communicate clearly and fluently and to think critically and logically, skills which also underpin the revised national curriculum statement (rncs) (doe, 2002). in dealing with quantitative or mathematical ideas in context, learners should be able to interpret information presented verbally, graphically or in symbolic form, and be able to translate between these different representations. the interpretation of quantitative information and concepts is dependent on learners’ ability to comprehend and express their understanding coherently. learners must also choose the appropriate form for the expression of a quantitative idea, and produce a text that expresses that idea (e.g. an algebraic expression, table or graph). to meet these many criteria, learners need adequate preparation during the get phase. national assessments suggest that this is not yet happening. to study mathematics effectively in grades 8 and 9 and make a smooth transition to mathematics or mathematics literacy in grade 10, learners need to build up adequate language and reading skills in the lolt as well as a solid foundation in relevant mathematical concepts during their senior primary school years. this study attempts to shed some light on the mathematics-literacy interface during the final year of primary school in two different township schools. 44 carol bohlmann & elizabeth pretorius methodology broader context the two primary schools in this study are situated in a predominantly northern sotho/tswana speaking township west of pretoria. there are 26 state primary schools and one private school in the township. in the majority of these schools, schooling takes place in an african language from grades 1-3 (northern sotho, tswana, zulu, xhosa, tsonga, venda or south sotho). the switch to english as lolt is made in grade 4. thereafter the specific african languages continue to be taught as first language subjects. one of these primary schools (school b below) is involved in a long-term reading intervention programme, the aim of which is to make reading an integral part of daily school activities. the private school in the township serves as a comparison school. school b (state school) school b has just over 600 learners and 16 teachers. the school serves a socioeconomically disadvantaged community and is a quintile 12 school, i.e. a non-fee paying school, with a fixed budget provided by the department of education. northern sotho (henceforth ns) is the lolt from grade r–grade 3. english becomes the lolt from grade 4, while ns becomes a subject of instruction from grades 4–7. about half the teachers at the school have ns as their home language. in theory, this school provides additive bilingualism3, the most favourable of the various bilingual models, with initial literacy in the l1 for three years and with continued l1 support after english becomes the lolt in the fourth year. although the majority of the learners have ns as their home language, about 20% of grade 7 learners each year have different home languages (e.g. tswana, tsonga, ndebele, venda, etc.). the reading intervention project began at the school in 2005. the project’s aim is to create a culture of reading in order to improve literacy skills 2 schools are classified into 5 quintiles according to socio-economic indicators. schools in the first quintile are high poverty schools. 3 in additive bilingualism, the home language of the learners is used for initial schooling, and continues to be taught as a subject even when the change to another language as lolt is made at a later stage. and thereby also academic performance. the multilevel approach adopted emphasises resource and capacity development. a collection of ageappropriate books has been built up in english and ns in the school library, which is now fully functional and computerised. the intervention also focuses on developing instructional capacity of the teachers and supportive capacity of the parents. teachers are encouraged to create print-rich environments in their classrooms, small classroom libraries are being established, teachers attend reading workshops, and all the learners at the school are constantly encouraged to read, in both the home language and in english. (due to space constraints, interested readers are referred to pretorius & mampuru (2007) for further project details.) the mathematics teacher at the school has a 2-year senior primary teacher’s diploma and an advanced certificate in mathematics education. she has 11 years teaching experience of mathematics and physical science, life orientation and ns. the grade 7 learners have 8×35 minute mathematics periods a week, i.e. 4 hours 40 minutes of mathematics instruction per week. in order to monitor project progress, the project team assesses all the grade 7 learners at the school each year for language proficiency and reading ability in both ns and english. the language and reading data presented here come from the first two project years (n = 107 in 2005; n = 56 in 2006). mathematics is not assessed by the project; to obtain mathematics data, the final grade 7 mathematics examination papers and results were obtained for 2005 and 2006. the mathematics data thus reflect the de facto performance of the learners as assessed by the mathematics teacher. after the first year of the project, it was decided to assess grade 7 learners for language and literacy from another township school in order to extend the longitudinal data base. school m (private school) school m is a small private primary school established in the township in 1991. even though it serves the same community as the other schools in the township, many children at this school come from higher socio-economic homes, with many parents being white-collar professionals. however, out of a total of 300 learners at the school, 80 from poor homes receive full scholarships. the classrooms are well resourced and the teachers well qualified and experienced. classes are small 45 relationships between mathematics and literacy (about 25-32). reading and storybooks are an integral part of each classroom in the lower grades, and teachers have high reading expectations of learners. the grade 7 mathematics teacher has a bed (special needs) degree. she has taught mathematics for 15 years and additional mathematics for 7 years. the grade 7 learners have 6×45 minute mathematics periods a week, i.e. 4 hours 30 minutes of mathematics instruction per week. the school has a ‘straight for english’ language policy. the learners are not linguistically homogeneous but speak different african languages at home. no african languages are taught as subjects. unlike many other private schools or ex-model c schools, no learners at this school have english as l1. the school provides subtractive bilingualism, a bilingual programme often criticised for not providing support for the home languages of the learners. because literacy testing by the project team only began at this school in 2006, only data from 2006 are presented here. language and reading assessments to properly explore the mathematics-literacy relationship, the notion of literacy was broken down into a distinction between language proficiency and reading ability. language test ns and english language proficiency was operationally defined as proficiency obtained in a dictation test. because there are no standardised language tests in all official languages in south africa for different age groups, it was felt that a dictation test would tap into language proficiency in both languages without requiring the learners to actually read a text. dictation correlates “at … high levels with a vast array of other language tests” (oller, 1979, p. 58). dictation tasks and standardised language tests thus tap into similar knowledge sources but the former do so via the auditory rather than the written medium. because most language tests are written tests, they have to be read and thus they also tap into reading comprehension, resulting in covariance between language proficiency and reading comprehension. because a dictation test taps into language knowledge via auditory comprehension, it avoids this covariance trap. to ensure that the level of difficulty of the dictation test was age appropriate, the dictation passages were taken from approved textbooks currently available for grade 7 ns and english classes. for marking the dictation, spelling and punctuation were taken into account. ns words that were written conjunctively instead of disjunctively were accepted as correct, provided they were spelled correctly. reading test to make valid comparisons across languages the same reading tests were given in english and ns. on the assumption that the learners would find the ns reading test more readily comprehensible and would use their ns comprehension to facilitate understanding of the english texts, the latter were administered first, with the ns tests administered four weeks later. the time lag between testing was deemed long enough to minimise memory effects. to avoid the bias that a single type of comprehension task can engender, reading proficiency was operationally defined as proficiency obtained in a reading comprehension test where a combination of test items was used, using texts taken from existing grade 7 language textbooks. the types of questions included cloze items, identifying referents for anaphoric items, a combination of multiple choice inferential and vocabulary questions, and questions involving graphic information (finding places on a map, reading a graph, etc.). to ensure that the learners understood what was required, all the question formats were explained to the learners before they started the reading test and examples were given on the chalkboard, in the language of the comprehension test. mathematics examination the mathematics results from the final grade 7 examinations were obtained from the mathematics teachers at both schools and the results examined in relation to language proficiency and reading ability. the mathematics examinations were set and marked by the mathematics teachers at the schools. the questions were analysed according to the stated rncs learning outcomes and assessment standards for grade 7. both papers showed an attempt to cover the required learning outcomes. taking subsections into account, the paper at school b contained 18 questions, for which 40 marks were given. no time was specified. the paper at school m contained 71 questions, for a total of 120 marks, for which two hours were allocated. the formats for the examination papers at each school have remained similar for the past three or so years so can be 46 carol bohlmann & elizabeth pretorius regarded as fairly typical of mathematics assessment at the schools. see appendix 1 (school b) and appendix 2 (school m) for sample questions. the assessment standards specified for each learning outcome provide the criteria to determine achievement of learning outcomes. for lo 1 11 sets of criteria are given (broken down into a total of 26 items). for lo 2 the seven sets of criteria are further divided into a total of 16 items; for lo 3 there are 12 items; for lo 4 and lo 5 there are 15 and 19 items, representing nine and 10 sets, respectively. assessment procedures the english language and reading tests were administered first in school b, followed 3-4 weeks later by the ns tests. since no african languages are taught as a subject in school m, the ns tests were not administered to the grade 7s at this school. all tests were written during two periods allocated during school hours and administered by the project researchers. the mathematics examination, set by the relevant teachers, was written in november. results data for the different language tests and the mathematics examination were captured and analysed using spss. using the cronbach alpha model, the reliability scores for the english tests were 0,77 and 0,74 respectively, while the alpha scores for the ns tests were 0,73 and 0,75 respectively. these results are regarded as satisfactory. the first research question that was addressed was: how did the grade 7 mathematics examination paper in the two schools reflect the five learning outcomes of the new mathematics curriculum? a breakdown of the examination questions in the two schools according to the five learning outcomes is shown in table 1, followed by a description of the items in each learning outcome. table 1: learning outcomes and assessment standards no. of items assessed lo 1 n = 26 lo 2 n = 16 lo 3 n = 15 lo 4 n = 19 lo 5 n = 10 total n = 86 school b 2005 school b 2006 7 2 0 3 0 1 2 1 0 1 9 8 school m 2006 7 3 4 5 5 24 note: n is the total no. of items in the assessment standard description of items, school b: lo1: multiple operations using integers (only + and − ), associative law (involving only + and −) lo2: recognition of arithmetic sequence; description of sequence (using algebraic language); solution (using algebraic language and skill) of equations to find nth term, and to find n lo3: calculation of third angle of a triangle lo4: classification of angles and angle relationships lo5: drawing pie graphs description of items, school m: lo1: multiple operations using integers (only +, −, ×); associative law (involving only + and −); operations with common fractions (+, −, × ); comparison of decimals; equivalence of fractions; calculations with decimals (+, ×); percentages lo2: recognition of pattern; solution (using algebraic language and skill) of equations; use of relationships between variables to determine input/output values lo3: use of geometric properties of parallel and other lines to calculate angles; isosceles triangle properties; construction of triangle and quadrilateral; classification of triangle and quadrilateral lo4: classification of angles and angle relationships; calculation of length, perimeter and area, and volume; distance time and speed problems lo5: interpretation of double bar graph, and pie graph; calculation of mean and mode; calculation of range; drawing of stem-and-leaf graph mathematics assessment at school m was more rigorous than at school b, a wider variety of concepts was tested and greater use was made of placing mathematical activities within a context. how did the learners at both schools fare, faced with these differential challenges? the second research question was: are there significant differences in grade 7 mathematics performance, language proficiency and reading ability between the two schools? table 2 reflects the descriptive statistics. three patterns emerge from the results. firstly, the grade 7 learners in the private school (school m) far outperformed the learners in the township school (school b) in mathematics, despite the more challenging mathematics examination. the learners at school m also far outperformed their school b peers in english language and english reading comprehension. they also read faster. independent t-tests showed these differences to be highly significant: mathematics: t = 4,462; df 75 p < 0,0005 english language: t = 5,239; df 77 p < 0.0005 english reading: t = 8,056; df 72 p < 0,0005 47 relationships between mathematics and literacy table2: grade 7 mean performance in language proficiency, reading ability and mathematics school b 2005 n = 102 school b 2006 n = 56 school m 2006 n = 25 mean age (range of years) 13,1 (11-16) 13,6 (11-16) 13,4 (12-14) mathematics * (sd) minimum maximum percentiles 25 50 75 54,3 (18,03) 14 92 42 54 66 49,2 (16,7) 18 89 34 48 58 67,6 (18,01) 31 95 54 74 84 ns language * (sd) minimum maximum 68,6 (12,6) 1 95 67,9 (25,6) 0 98 – ns reading * (sd) minimum maximum 37,2 (17,3) 3 84 38,5 (19,6) 6 82 – english language * (sd) minimum maximum 62,7 (24,6) 0 97 56,1 (33,5) 2 100 92,4 (13,1) 45 100 english reading * (sd) minimum maximum 42,4 (18,7) 4 88 45,4 (19,9) 9 85 80,4 (11,9) 52 95 reading speed** ns english 93 106 106 131 – 169 * scores reflect percentages ** scores reflect number of words read per minute secondly, the reading performance of the grade 7 learners in school b was better in english than in ns in both years. the modest increase in reading ability in school b in both ns and english, but especially the latter, suggested that the reading intervention programme was having an effect on reading (this becomes more evident in table 5, which shows the breakdown of reading ability according to mathematics achievement). thirdly, in school b there was a difference of almost 30% between language proficiency scores and reading scores in ns. scoring well in their home language did not mean that the learners could read well in their home language. while the 2005 data in school b showed a similar but smaller gap between english language proficiency and english reading, this gap narrowed in 2006. given the differential challenges in mathematical assessment and performance at the two schools, does a relationship still obtain between mathematics and literacy? the third research question addresses this matter: are there significant relationships between grade 7 mathematics performance, language proficiency and reading ability at the two schools? pearson product moment correlations were performed, using data to correlate the results from the ns and english language and reading tests with the mathematics examination results. see table 3. table 3: correlations between mathematics, language and reading comprehension in ns and english ns language ns reading english language english reading school b 2005 mathematics exam ns reading english reading 0,50** 0,544** 0,652** – 0.792** 0,677** 0.696** 0,702** 0,792** – school b 2006 mathematics exam ns reading english reading 0,364* 0,693** 0,401** – 0,843** 0,279 0,784** 0,430** 0,843** – school m 2006 mathematics exam english reading – 0,361 0,691** 0,804** – * correlation is significant at the 0,05 level (2-tailed) ** correlation is significant at the 0,01 level (2-tailed) while all the correlations are significant (except for the relationship between mathematics and english language in school b in 2006), not all are equally robust. in both years, for both ns and english, reading ability rather than language proficiency consistently correlated more strongly with mathematics. furthermore, in both years, reading ability in english correlated more significantly with the mathematics examination score than reading ability in ns. secondly, we note the consistently strong correlations between reading ability in ns and in english: weak readers in ns were also weak readers in english and vice versa. reading ability in one language correlated more strongly with reading ability in the other language than it did with language proficiency in the same language. thirdly, there was a trend of a slightly lower correlation between ns language and ns reading than there was between english language and english reading. learners could score high in ns language but not high in ns reading (see also table 5). stepwise multiple regression analyses for each year were used to see which language and reading variables best predicted mathematical performance. significant models emerged for both years, as can be seen in table 4. 48 carol bohlmann & elizabeth pretorius table 4: multiple regression predicting mathematics* school b 2005 f = 63,776; df = 2 91; p < 0,0005 adj. r square = 0,574 predictor variables english reading english language ns language and ns reading were not significant predictors in this model beta 0,483 0,345 sig. 0,000 0,000 school b 2006 f = 37,308; df = 1 43; p < 0,0005 adj. r square = 0,452 predictor variables english reading ns language, ns reading and english language were not significant predictors in this model beta 0,682 sig. 0,000 school m 2006 f = 41,903; df = 1 23; p < 0,0005 adj. r square = 0,630 predictor variables english reading english language was not a significant predictor in this model beta 0,804 sig 0,000 * the dependent variable is mathematics. the predictor variables are ns language, ns reading, english language and english reading. in all the analyses, english reading consistently emerged as a robust predictor of performance in mathematics; ns language and reading were not significant predictors. to further explore the relationship between language proficiency, reading and mathematics, the learners’ performance on the language and reading tests were matched against their mathematics performance. the mean mathematics examination score placed learners into one of four achievement categories used by the department of education (e.g. doe, 2003), namely not achieved (0%-35%), partly achieved (36%-49%), achieved (50%-69%) and outstanding (70%-100%). language and reading performance can be seen within these four academic categories, as reflected in table 5. three main trends emerge. firstly, in both schools increased language and reading ability corresponded with category of mathematics performance. the higher the mathematics category, the higher the reading score. for example, the not achieved learners had much lower reading scores than the partially achieved learners, who in turn had lower reading scores than table 5: comparison of l1 and l2 language and reading proficiency with achievement categories in mathematics mathematics examination performance ns and english language and reading school b 2005 mean % school b 2006 mean % school m 2006 mean % 0% – 39% not achieved ns language ns reading english language english reading 39,3 15,3 22,9 18,7 50,7 21,8 30,4 32,4 – – 88,8 65,8 40% – 49% partly achieved ns language ns reading english language english reading 67,7 23,9 52,2 31,7 65 33,7 52,0 38,1 – – 95,8 72,6 50% – 69% achieved ns language ns reading english language english reading 71,7 39,7 66,2 42,1 77,0 48,3 73,2 54,8 – – 85,1 75,4 70% – 100% outstanding ns language ns reading english language english reading 77,2 54,9 86,4 67,8 88,7 58,6 85,4 70,0 – – 97,6 88,7 those in the achieved group; learners in the outstanding mathematics group were all competent readers. in school b, the outstanding learners read better than their peers in both english and ns, but especially in english. secondly, the notion of being a ‘good’ or ‘weak’ reader varied considerably and was relative to the school context. it is instructive to compare the mean reading scores of the learners who failed mathematics across the schools. the failed mathematics learners in school b could hardly read in either their l1 or the second language (l2) after seven years of primary schooling, while the failed mathematics learners in school m were, by comparison, ‘literate’ failures with a mean reading score of 65,8%. achieved mathematics learners in school b had mean reading scores that were about 10% lower than the mean reading level of the failed mathematics learners in school m. on the other hand, although lower than their private school peers, the reading scores of the outstanding learners in school b were remarkable, given their disadvantaged home and school circumstances. thirdly, we note again the large gaps between achievement in ns language and reading, as also the slightly smaller gaps in english language and english reading. learners could score high in ns language yet they struggled to read in ns. in sum, the results showed very robust relationships between mathematics and reading ability, especially reading in english despite the two different schooling contexts. 49 relationships between mathematics and literacy discussion the findings reported in this paper come from two primary schools within the same township. although one cannot generalise from two schools only, the study can be used to inform education improvement and point to future research avenues. we focus on three main trends that emerged. firstly, there were obvious differences between the quality and quantity of mathematics assessment by the teachers at the schools and learners’ concomitant performance. learners at the private school who faced the longer and more challenging examination outperformed their peers who had the shorter, less challenging mathematics examination. secondly, there were striking discrepancies between not only mathematics but also language and reading performance between school b and school m, with learners in school b generally performing poorly and learners in school m performing very well. the third salient finding relates to the consistent finding that english reading rather than english language proficiency related to mathematics examination performance. although reading in ns was not a predictor of mathematics performance, there was a consistently high correlation between ns reading and english reading. learners who comprehended texts in one language also comprehended texts in the other, and vice versa. reading levels in english also tended to be generally higher than reading levels in ns. this robust reading relationship has implications for language policy and mathematics pedagogy, as discussed below. the study is not without methodological failings; even though independent tests were not used to assess mathematics performance at the schools, the use of de facto mathematics performance results from the teachers themselves provide an authentic ‘snapshot’ of what is happening in schools and do not invalidate the findings. despite the disparities in teaching and assessment across the two schools, a robust relationship between mathematics and reading still emerged. in this study two variables were constant: all the learners came from the same township and all had an african home language, so differences in performance must be sought elsewhere. in school b conditions are similar to other disadvantaged primary schools in townships around the country. the two schools differed along several dimensions such as socio-economic status, teacher and classroom resources and practice, and language policy. these issues will be briefly visited below. poverty, literacy and mathematics it could be argued that the differences in mathematics and reading performance between the two schools are simply attributable to the poverty factor. indeed, there is plenty of evidence around the world that it is difficult to educate poor children (e.g. bradley & corwyn, 2002), because the factors associated with poverty create barriers to learning. poor children attend poorly resourced and managed schools, with large classes and fewer well qualified teachers. in the 1999 timms study ramirez (2006) found that inequalities in school resources and teacher qualifications in chile were closely tied to the socio-economic profiles of learners. poor children also have few literacy resources at home and parents with lower literacy levels. these socio-economic status (ses) related home and school factors are not conducive to creating stimulating learning environments. rather than regard poverty as a causal factor in mathematics performance it is more constructive to consider what variables might mitigate or exacerbate the effects of poverty on schooling. the human development index (hdi) for south africa was 0,684 in timss 2003 (reddy, 2006). although socio-economic circumstances impact upon learning opportunity, indonesia (hdi=0,682) and morocco (hdi=0,606) had lower hdis and higher timms positions (mathematics scores of 411 and 387, respectively, against south africa’s 264) (reddy, 2006). of the five lowest achieving countries in africa (tunisia, egypt, morocco, botswana, ghana and south africa, in that order) south africa has one of the highest gross national incomes per capita, but the lowest mean score in mathematical performance, suggesting that ses factors alone do not account for learner achievement (reddy, 2006). teacher qualification and resource management: the mathematics teachers at the schools differed in qualifications and teaching experience. there were also differences in terms of organisation and display of mathematical material. at school m the teacher was well organised and had an arch lever file for each grade 7 learner containing various mathematics activities and test results. these files were displayed prominently in the classroom and the classroom was print rich, with mathematics textbooks, dictionaries, exercise books, mathematical paraphernalia, posters, mind games and inspirational messages. the classroom was clearly a ‘mathematics classroom’. 50 carol bohlmann & elizabeth pretorius in contrast, the teacher in school b had the challenge of having to cope with much larger classes. there were no up-to-date or readily accessible records of learners’ activities, and although the classroom was swept and tidy, the classroom cupboard was disorganised. the cupboard’s broken lock may have contributed to the teacher’s disinclination to use it as an organising space for mathematics resources. the walls and notice board at the back of the classroom were bare. it did not obviously appear to be a mathematics classroom. obviously the socio-economic status of schools determines in many respects the availability of resources within the school. however, individual teachers also differ with regard to their perceptions of what constitutes good practice in their subject and to what extent they can act on their classroom environment. although school b remains a high poverty school, it has started developing a stronger reading focus. several of the teachers now try to mitigate the effects of poverty by creating print-rich environments even with their limited resources, and they include more reading activities in classroom activity. the need to create a visually rich and stimulating mathematical environment should be an integral part of mathematics teacher training, particularly in high poverty schools where learners are unlikely to be exposed to such stimuli in their homes. service delivery and ‘time on task’: based on regular classroom observations at school b over two years, some general trends have been identified. although a culture of reading is clearly starting to emerge, several aspects of daily classroom life remain challenging. lesson presentations often seem to be superficially planned and prepared. although classwork and homework appear to be more frequently given since the inception of the project, they appear to be undemanding and inconsistently managed. teachers are often absent or out of the classroom, and maximum use of time and ‘opportunity to learn’ are lost, with considerable cumulative loss of valuable time. in contrast, school m lessons start soon after the learners come in and homework is a regular feature of school life. even though learners at both schools have about 4½ hours of mathematics instruction per week, the contrast between the mathematics papers at the two schools (in terms of covering the learning outcomes) bears testimony to differences in the scope, variety of exercises, level of cognitive challenge (cf. appendix) and constructive use of classroom time. loss of ‘time on task’ was identified as a critical variable that distinguished successful from less successful schools in the grade 6 systemic evaluation (doe, 2005). in their study of teacher absence in peru, alcázar et al. (2006) also found that teachers were more often absent in poorer schools with poor infrastructures. reading and mathematics the findings suggest that during the senior primary school phase, english reading is strongly supportive of mathematics achievement. this is not surprising, given that english is the lolt. it is significant that english reading rather than the more general construct of english language proficiency seems to determine mathematics achievement. those who passed the final grade 7 mathematics examination scored higher than their peers on both ns and english reading tests. during the senior primary phase mathematics and reading are strongly related. it is a matter of concern that even after seven years of primary schooling, many learners enter high schools with exceedingly low reading levels in both their home language and in english, with concomitant low levels of mathematical proficiency. ‘catching up’ in high school, when the pedagogic focus is on content subjects and not literacy development, is unlikely. poverty per se does not determine poor reading levels; rather, it is the virtual absence of good practice on which effective reading development depends. if schools wish to improve their mathematics teaching, these findings suggest that they simultaneously give serious attention to improving reading levels. although schools cannot change the socio-economic status of the communities from which their learners come, they can make committed efforts to creating print-rich environments in the classrooms and ensure that reading is properly taught so that learners can developed appropriate literacy levels to support their mathematical development mathematics pedagogy it is clearly imprudent to generalise on the basis of examination papers from two schools. it is however cause for concern that the mathematics papers at school b were problematic. with scant assessment of three of the five learning outcomes, learners are unprepared for the continuation of these learning outcomes in grade 8. given that the practice of quantitative literacy is embedded in real-life contexts, access to resources in the home, community and schools will influence 51 relationships between mathematics and literacy the extent to which learners can find appropriate material for relevant mathematical tasks. learners’ background knowledge is also affected by their parents’ levels of education. research in the usa and chile has shown a strong correlation between the level of the parents’ education and the mathematical performance of their children (ramirez, 2006; tate & rousseau, 2002). language policy and mathematics the findings in this study revealed stark differential language and reading performances that were linked to mathematical performance in two different multilingual educational contexts. how does language policy fit into this picture? being taught in the home language confers linguistic, cognitive and affective advantages. however, the learners at school m far outperformed their peers at school b even though they were studying exclusively through the medium of english. the results from these schools do not by any means imply that subtractive bilingualism ‘works’ or that additive bilingualism ‘doesn’t work’ or that extended home language instruction won’t work. whatever the language policy at schools, it will only be as good as the quality of education that undergirds it. even if the lolt were to change to the learners’ mother tongue, the reading challenge remains – using a language in the classroom does not guarantee reading ability in that language or that reading will become a central classroom activity. reading only develops on extensive exposure to written language. the availability of books, a basic requirement for reading development, is often absent in poor schools, especially in the african languages (of the 4000 books in school b’s library, there are only 169 ns titles, most of which are storybooks for children under 10 years). the low reading levels and slow reading rates in ns in the current study point to the fact that not enough reading is being done in ns. contributing factors include classroom practices, availability of print resources in ns and the diglossic differences between spoken and written ns. ns spoken in the pretoria area (sesotho sa pretoria) is different in many respects from standard ns (sesotho sa leboa). learners do not have enough exposure to written ns to develop proficiency in using written forms of the language. the findings from the private school indicate that when conditions in a school are conducive to learning, very high language, reading and mathematics levels can be achieved, even when the lolt is not the l1 of the learners. in the special edition on education equity and quality in developing countries in the international journal of educational research, the main theme to emerge was “the centrality of school quality as a driver of …educational effectiveness” (lockheed & cueto, 2006, p. 98). conclusion although this as a small-scale study, the findings point to numerous factors contributing to mathematical effectiveness in school. within the two schools, reading ability in the lolt was a determinant of mathematical performance. across the schools, the quality of schooling seemed to be the central determinant of both reading and mathematical development in the primary school years. the new mathematics curriculum presupposes a highly literate environment. without easy access to material rich in quantitative information learners will find it difficult to find appropriate contexts in which to apply the mathematical concepts and skills they are being taught. learning will also be negatively affected if aspects of content knowledge are neglected and if general knowledge (assumed by the ‘context’) is missing. without adequate reading skills learners will be unable to properly understand mathematical topics. without adequate understanding of the topics taught learners will continue to follow rote methods of learning often used by teachers who lack confidence and knowledge. if teachers themselves are less than comfortable with the discourse of mathematics they will resort to procedural, language-free problems in their assessment tasks (as the examination questions from school b show), further undermining learners’ opportunity to grapple with these issues. without consideration of the literacy scaffolding learners require, it may well be that the new curriculum is, at present, beyond the reach of many learners, partly as a result of limited literacy. acknowledgements the ‘reading is fundamental’ project, from which this research derives, is funded by the dg murray trust and the nrf. the opinions expressed here are not necessarily those of the funders. sincere thanks to the learners and staff at both schools for participating so generously and willingly in the project. thanks are also due to the alru project team for steadfast commitment and support: sally currin, debbie mampuru, matsileng mokhwesana and nicoline wessels. 52 carol bohlmann & elizabeth pretorius 53 references alcazar, l., rogers, f. r., chaudhury, n., hammer, j., kremer, m., & muralidharan, k. 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(2005). learning material in compliance with the revised national curriculum statement: a dilemma. pythagoras, 62, 48-58. relationships between mathematics and literacy appendix 1 (school b) 54 carol bohlmann & elizabeth pretorius 55 appendix 2 (school m) << /ascii85encodepages false /allowtransparency false /autopositionepsfiles true /autorotatepages /none /binding /left /calgrayprofile (dot gain 20%) /calrgbprofile (srgb iec61966-2.1) /calcmykprofile (u.s. web coated \050swop\051 v2) /srgbprofile (srgb iec61966-2.1) /cannotembedfontpolicy /warning /compatibilitylevel 1.5 /compressobjects /tags /compresspages true /convertimagestoindexed true /passthroughjpegimages true /createjobticket false /defaultrenderingintent /default /detectblends true /detectcurves 0.0000 /colorconversionstrategy /cmyk /dothumbnails false /embedallfonts true /embedopentype false /parseiccprofilesincomments true /embedjoboptions true /dscreportinglevel 0 /emitdscwarnings false /endpage -1 /imagememory 1048576 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/hrv (za stvaranje adobe pdf dokumenata najpogodnijih za visokokvalitetni ispis prije tiskanja koristite ove postavke. stvoreni pdf dokumenti mogu se otvoriti acrobat i adobe reader 5.0 i kasnijim verzijama.) /hun 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can be opened with acrobat and adobe reader 5.0 and later.) >> /namespace [ (adobe) (common) (1.0) ] /othernamespaces [ << /asreaderspreads false /cropimagestoframes true /errorcontrol /warnandcontinue /flattenerignorespreadoverrides false /includeguidesgrids false /includenonprinting false /includeslug false /namespace [ (adobe) (indesign) (4.0) ] /omitplacedbitmaps false /omitplacedeps false /omitplacedpdf false /simulateoverprint /legacy >> << /addbleedmarks false /addcolorbars false /addcropmarks false /addpageinfo false /addregmarks false /convertcolors /converttocmyk /destinationprofilename () /destinationprofileselector /documentcmyk /downsample16bitimages true /flattenerpreset << /presetselector /mediumresolution >> /formelements false /generatestructure false /includebookmarks false /includehyperlinks false /includeinteractive false /includelayers false /includeprofiles false /multimediahandling /useobjectsettings /namespace [ (adobe) (creativesuite) (2.0) ] /pdfxoutputintentprofileselector /documentcmyk /preserveediting true /untaggedcmykhandling /leaveuntagged /untaggedrgbhandling /usedocumentprofile /usedocumentbleed false >> ] >> setdistillerparams << /hwresolution [2400 2400] /pagesize [612.000 792.000] >> setpagedevice beliefs and practices of a pre-service mathematics teacher pythagoras, 68, 41-51 (december 2008) 41 a snapshot in time: beliefs and practices of a preservice mathematics teacher through the lens of changing contexts and situations lyn webb and paul webb nelson mandela metropolitan university, port elizabeth, south africa lyn.webb@nmmu.ac.za and paul.webb@nmmu.ac.za for the last decade research on teachers’ beliefs has made a distinction between mathematics teachers’ professed and attributed beliefs (practice) and studies have either found some or no correlation between the two. in this paper we investigate the beliefs and practices of a novice teacher and conclude that inconsistency between beliefs and practices may be an observer’s perspective that is not necessarily shared by the teacher, and that the view that there is a possible disjuncture does not do justice to the complexity of the practitioner’s tasks nor to the rapidly changing contexts and situations that may occur within a single lesson. there has been extensive research in recent years on the relationships between the beliefs of mathematics teachers and their actual practice in the classroom (brodie, 2001; ensor, 1998; ernest, 1989, 1991; hoyles, 1992; lerman, 1986, 2002; pehkonen & törner, 2004; thompson, 1992; skott, 2001a, 2001b, 2004; speer, 2005; wilson & cooney, 2002). most of these studies focus on the correlation or disparity that researchers have identified between what teachers believe they should be doing in their classrooms (their espoused beliefs or professed beliefs) and what researchers infer based on observational and other data (their practice or “attributed beliefs”). some studies suggest a direct causality between beliefs and practices (schoenfeld, 1992), while other do not (hoyles, 1992). ernest (1991) claims that a lack of consistency results from institutional and contextual restraints and claims there is more consistency when the teacher engages in reflective practices. thompson (1992) summarized previous studies on beliefs by claiming that teachers’ beliefs about mathematics are enacted fairly consistently while hoyles (1992) argued that there should be no expectation of a relationship between beliefs and practices and introduced the notion of situated beliefs, i.e. that situations are co-producers of beliefs and as situations change, so do beliefs. skott (2001a) focuses on the complexities of classroom practice and maintains that consistency between beliefs and practice is a local and instantaneous phenomenon guided by critical incidents of practice. south african teachers exhibit a variety of levels of mathematical knowledge and knowledge of pedagogy; but they all appear to have difficulty in changing their teaching practice towards methods of engaging learners in a learner-centred approach (brodie, 2001). this difficulty suggests that it would be profitable to know more about the apparently complex relationships and interactions between teachers’ beliefs and practices, as well as the effect that changing classroom contexts and activities may have on their practice. rationale for the study the relationships between beliefs and practice have implications for current curriculum trends towards reform which require teachers to change their beliefs and practices in the classroom; reform that is epistemologically framed by both constructivism (von glaserveld, 1992) and sociocultural theory (vygotsky, 1986). examples of such curriculum trends are, among others, engaging learners in mathematics problem solving in context and the introduction of outcomes-based curricula in south african schools; trends which have in many ways created a new role for teachers (department of education, 2003). teachers are expected to develop and flexibly use a wide range beliefs and practices of a pre-service mathematics teacher 42 of different tasks and approaches in order to encourage learners’ active involvement in mathematical processes of experimenting, investigating, generalising and formalising, and to support their conceptual understanding and procedural competences (skott, 2001b). this study investigates and interrogates the relationships between the beliefs and practices of an individual in a cohort of further education and training (fet) band pre-service student teachers studying at the nelson mandela metropolitan university (nmmu) in port elizabeth, south africa. the rationale for using a pre-service teacher was that although many beginning teachers hold the belief that mathematics is a fixed set of rules and procedures and that learning occurs through solving problems in a step-wise fashion (phillip et al., 2007), they are generally not resistant to change and can more easily articulate their thinking in terms of the theory of what they are learning at the time than more experienced teachers who are somewhat removed from their academic experiences (vacc & bright, 1999). for the purposes of this article we have highlighted the experiences of one student as she dealt with changing methods, contexts and situations during her lessons, an approach similar to that used by skott (2001a), who also examined the beliefs and practices of one novice teacher to interrogate the roles of his school mathematics images. categorising and classifying beliefs teachers’ beliefs in terms of the nature of mathematics have been expressed in the past on a continuum ranging from an absolutist viewpoint, in which mathematical truth is unquestionable, certain and objective, to a fallibilist viewpoint, in which mathematical knowledge can be seen as a social construction and is therefore fallible, i.e., it can be revised and corrected (lerman, 1986). in turn perceptions about learning mathematics have been represented on a continuum from the mastery of skills to problem solving, while views of the teaching of mathematics have been represented by the contrasting notions of the teacher as instructor or the teacher as facilitator (lerman, 1986). ernest (1989) proposed three divisions, i.e., the instrumentalist, platonist and problem-solving views of mathematics. the instrumentalist view of mathematics is that of a set of unrelated but utilitarian rules and facts, thus reducing mathematics to the accumulation of facts, rules and skills that can be used in the pursuance of some external end. the platonist view of mathematics also sees mathematics as being a static, but unified, body of certain knowledge, and implies that mathematics is something to be discovered, not created. the problem-solving view of mathematics is that of a dynamic, continually expanding field of human creation, a cultural product, which is constantly being revised and constructed. in order to achieve methodological triangulation pehkonen and törner (2004) have used both qualitative and quantitative methods to investigate teachers’ beliefs about mathematics and their practices within their classrooms. one technique that they have used, and which has been replicated in this study, is a method of numerical and graphical selfand observer-estimation which allows the location of teacher’s beliefs and practices within three perspectives philosophically underpinned by dionne’s view of the nature of mathematics, i.e., the toolbox, system and process perspectives (pehkonen & törner, 2004). the toolbox perspective views mathematics as the mastery of skills. doing mathematics within this perspective involves calculating, using rules, following procedures and manipulating formulae. the system perspective sees mathematics as a language of logic and rigour which requires using a precise and concise language to express mathematical ideas, while the process perspective of mathematics involves a constructive process which draws from real-life experiences. dionne’s categorisations resonate strongly with ernest’s tripartite division. because many researchers question the assumption that what is revealed in responses to a questionnaire reveals a belief system (lerman, 2002; wilson & cooney, 2002) it was decided to trial the approach propounded by pehkonen and törner (2004). relationship between beliefs and practices while some researchers have suggested that beliefs are a major force in affecting teaching practice (schoenfeld, 1992; thompson, 1992), others believe that there is no direct link between beliefs and practices (hoyles, 1992). international research shows that there is often little or no correlation between teachers’ verbalization of their beliefs and their practice in the classroom (ernest, 1989; lerman, 1986; 2002). these findings give substance to the fact that, despite the emphasis in modern curricula on the fallabilist viewpoint (categorized by, amongst other factors, problem solving and the importance of understanding mathematics), most teachers teach mathematics as lyn & paul webb 43 a body of knowledge to be memorized. these international findings are mirrored by south african studies which reveal that teachers, despite professing views of teaching within the constructivist paradigm, often use traditional approaches that lead learners to see mathematics as a subject that must be memorized and which has little utility (webb, 2004). ernest (1989) feels that this incompatibility may be attributed to constraints and opportunities dictated by the social context of teaching, and that these macro schoolcontext factors are likely to cause disjuncture between teachers’ espoused and enacted beliefs. thompson (1992) cites some of the social context issues that complicate the relationship between beliefs and practices as, amongst others, the expectations held by learners, parents and colleagues, and issues of authority and control. hoyles (1992) adds that inconsistencies between beliefs and practices are accentuated when teachers are faced with an innovation, such as the contextual, problem solving approach to teaching mathematics propagated by the most recent south african national curriculum statement (department of education, 2003). she advocates that the mismatches between beliefs and practices often stem from situations, context and culture, which are the co-producers of what she calls “situated beliefs”. it is the situational, contextual and cultural nature of her situated beliefs and practices that lead her to the feeling that mismatches between beliefs expressed outside the classroom, and practices demonstrated inside the classroom, should be expected. ensor (1998) supports this view, noting that beliefs are not stable across contexts, and that differences in social situations result in multiple positioning of teachers, suggesting that beliefs are fore-grounded and backgrounded according to the context in which the person is operating at the time. this notion is supported by pehkonen and törner’s (2004) statements that beliefs are temporal and subject to continuous evaluation and change, a view echoed by lerman (2002) who maintains that changes in beliefs affect practice, and that, in turn, changes in practice affect the beliefs of the practitioner. research by skott (2001a) on mathematics teachers’ beliefs provides data which suggests that there can be simultaneous existence of multiple, and possibly conflicting, communities of practice in the course of classroom interaction. skott (2001a) maintains that interactions with students can transform beliefs and therefore it is inappropriate to describe teachers’ beliefs and practices as being inconsistent as it minimises the complexity of the teaching task. he sees communities of practice developing where the contributions of both individuals and groups become accepted in the class and become part of the mathematical discourse. within this view the role of the teacher is to sustain these individual and collective learning opportunities by adjusting his or her teaching style to each situation as it occurs. skott (2001b) maintains that micro aspects, such as classroom atmosphere and interactions between the teacher and specific groups of learners, are essential to the understanding of the social context and thus the teacher’s practices. he does not focus on the congruence or conflict between beliefs and practices, but attempts to disentangle the ways in which the multiple communities interact and frame the emergence of different strategies in teaching practice. method as suggested above, there are a number of ways of looking at beliefs on the nature of mathematics, as well as of locating teachers’ beliefs and practices in terms of this area of knowledge. what is clear though is that during the past decade, many researchers have concentrated on research conducted mostly within a cognitive framework and considered the implications of teachers’ beliefs from this perspective (speer, 2005). in contrast skott (2001a) has developed a construct of “critical incidents of practice” and barwell (2005) has concentrated on the socio-cultural contexts in which teaching and learning practices are coconstructed by participants (speer, 2005). in this paper we frame the findings generated by our study within lerman’s (1986), dionne’s (1984, as cited in pehkonen & törner, 2004) and ernest’s (1989) views of the nature of mathematics and pehkonen and törner’s (2004) technique of locating teacher’s beliefs and practices within philosophical perspectives of the nature of mathematics. in turn we use skott’s (2001a) understandings of how micro aspects of the social context of mathematics education (such as classroom climate, the teacher’s motives or the interactions between the teacher and specific learners) impinge on practices as backings and warrants for the arguments we use in terms of importance (or non-importance) of disjunctures between teachers’ beliefs and practices. our student, as part of a group of pre-service mathematics teachers, was given zollman and beliefs and practices of a pre-service mathematics teacher 44 mason’s 1992 standards belief instrument (furner, 2004, p. 56) in order to gauge her professed attitudes towards reform in teaching (appendix a). she was also given a questionnaire developed and tested by pehkonen and törner (2004) on conceptions of teaching mathematics (appendix b). because research has shown that respondents have not always responded consistently to questionnaires (wilson & cooney, 2002) these two questionnaires were used to get an indication as to whether her answers were consistent. speer (2005) warns that the discrepancy between what teachers say they believe and what is reflected in their practice may be caused by the very methods used to collect the data, or caused by a lack of shared understanding between the teachers and researchers of the terms used to describe both beliefs and practices. thus, a more visual, graphic approach was implemented where the participating pre-service student filled in a table developed by pehkonen and törner (2004) based on dionne’s perspectives concerning toolbox, system and process views of mathematics. she was also asked to distribute a total of 30 points corresponding to her estimation of her “real” teaching of mathematics and her “ideal” teaching of mathematics. the research was undertaken at the end of the student’s teaching practice so she had been teaching in classrooms for more than six months. furthermore, the student was also asked to mark a point on an equilateral triangle with “x” to indicate her “real” teaching of mathematics and to indicate her “ideal” teaching of mathematics with an “o” (appendix c). the vertices of the triangle represented the toolbox, process and system perspectives on mathematics as propounded by dionne (1984, as cited in pehkonen & törner, 2004). the student was then interviewed about her beliefs of the nature of mathematics and her views of teaching and learning mathematics in order to interrogate her responses more carefully and to ensure that she had a shared understanding of the terms used to describe beliefs and practices. as noted earlier, the student showcased in this study was one of a group of pre-service mathematics teachers who were given an opportunity to showcase their teaching practice during a videotaped lesson in the classroom. after the recorded lessons these students were reinterviewed and selected excerpts from these videotapes were used to allow them to explain the reasoning behind their actions and to enable the researcher to probe beliefs tied to specific examples of the students’ practices, as well as to attribute beliefs more accurately (speer, 2005). the data generated by the research suggested similar beliefs and teaching styles for four out of five students in the group (which can possibly be attributed to the fact that they had all been recently influenced by the input of their post graduate certificate in education (pgce) mathematics method lecturer). for the purposes of this paper we focused on our selected student, sarah, as although her beliefs showed the highest correlation in both of the questionnaires and her graphical and numerical self assessment tasks mapping her “real” and “ideal” teaching reflected the same trends in beliefs as her questionnaire responses, her professed beliefs and her classroom practice were consistent at times, but clearly disparate at others (more noticeably than the other students). this highlighting of a relatively clear-cut case is similar to the research done by skott (2001a), where he also examined the beliefs and practices of one novice teacher. results the scores achieved in zollman and masons’ (1992) standards belief questionnaire (furner, 2004) on a scale of 1= strongly disagree, 2= disagree, 3= agree, 4=strongly agree, were totalled and sarah scored 46, despite leaving out three questions relating to kindergarten and intermediate phase mathematics, which were not relevant to her as a grade 10-12 teacher. this score suggests that sarah has a tendency towards reform beliefs in teaching, encompassing beliefs that mathematics should be a meaningful, problem solving activity where active learning and solid reasoning are encouraged. during an initial interview before the teaching practice, sarah re-iterated her commitment to teaching reform i.e., that she believed in a learnercentred teaching and constructivist learning approach. however, she made some contradictory statements, for example: developing the skill of critical analysis and problem solving through mathematics is very valuable in all facets of life… free thinking is so important. was contrasted with: i think that through repetition and practice certain things become automatic without parrot-style memorization. lyn & paul webb 45 this statement suggests that, although sarah subscribes to a problem solving philosophy (process perspective), she sees repetition and mastery of skills as important in mathematics (toolbox perspective); however, she has qualified her statement which indicates that she has reflected about her stance. sarah’s response to the pehkonen and törner (2004) questionnaire also suggested that she has an innovative, learner-centred approach to mathematics teaching. although she was indecisive about the role of proofs and the role of visualization in teaching mathematics; she was positive about using varied application exercises, problem solving, developing thinking skills and stressing understanding. sarah’s scores were consistent when using both questionnaires (the standard beliefs and pehkonen and törner’s questionnaire) and they suggested a constructivist view of mathematics learning and reform teaching in both instruments. it is interesting to note that in the initial interview she was ambivalent about the role of understanding. when asked whether she felt it was more important to teach skills or emphasise learners’ understanding when teaching, she answered: neither. i know this is the easy way out, but it is true. i believe both are equally important and are used together most of the time. in the self-evaluation table sarah misread the instructions and distributed 20 points instead of 30 between the toolbox, system and process perspectives of real and ideal teaching. she gauged her “real” teaching of mathematics to be mainly toolbox (9) > system (7) > process (4). she viewed the “ideal” teaching of mathematics to be toolbox (8) = process (8) > system (4). this correlated with the ideas expressed in the interview that both skills and understanding are of equal importance. the exercise also showed that she did not feel that her “real” teaching style encouraged a constructive process. in the visual representation of her “real” teaching style, as opposed to “ideal” teaching style, sarah positioned her “real” teaching style (represented by x) midway between toolbox and system, far away from a process approach, as is shown in figure 1. her view of an “ideal” teaching style (represented by o) tended towards process, but remained midway between process and toolbox. this graphically mirrored her results in the selfevaluation table. an advantage of the graphic method is that one can draw a vector from the “real” to the “ideal” view to indicate how far from the “ideal” the teacher views his or her “real” teaching to fall. in this case sarah showed that she felt a balance between the three perspectives, leaning away from a system approach, was an ideal perspective. figure 1: sarah’s graphical representation of her “real” (x) teaching style and her “ideal” (o) teaching style on reviewing the videotape of her lesson in the classroom, two contrasting incidents occurred where she enacted reform-based teaching methods and diametrically opposed traditional teachercentred teaching methods. in the first instance sarah was showing the whole class how to find the area of a circle if given the radius. she stated the formula and wrote it on the board. she emphasized to the whole class that: all these examples are exactly the same. even if you do know, that’s fine, we’ll go over it all again. there’s only one method to do all of these because they’re all the same, just different numbers so if you get the method right you can do all of them – a, b, c, d. sarah proceeded to go through a few examples finding the area of a circle before giving them more of the same exercises to complete. she also emphasized that the answers should all be expressed in square metres: …because if you don’t you’ll be wrong and you’ll lose marks. in this incident sarah demonstrates a toolbox perspective of mathematics where mastery of skills and applying rules is emphasised. in the second incident, sarah arranged the class in pairs and emphasized that this was a co-operative beliefs and practices of a pre-service mathematics teacher 46 learning technique where discussion in pairs around solving the problem was essential. she gave them a problem and said: i am trying to get you to use all the things you have learned in the last two weeks in one sum. she emphasised the process rather than the product by stating: i want you to hand it in so we can see where you are going wrong. the problem involved a castle on a piece of circular land with a moat around it. sarah introduced a contextual element by discussing with the class the reason for a moat: in the olden days they had castles and they used to keep the baddies out with a moat. what happened when their friends came? they just put a drawbridge over it.ty the problem involved finding the area of the rectangular castle, the area of the circular land and the area of the moat. although the problem was a thinly disguised skills-based exercise, sarah stated in the interview after the lesson that she believed that she had contextualised the problem and she believed that she had engaged the learners in cooperative problem solving. this indicates that despite studying a mathematics method course, she has a superficial understanding of the nature of mathematical problem solving. in the course of a single lesson sarah had used two opposite styles of teaching with the same learners. she had expected them to learn through a repetitious approach and a collaborative approach where the learners communicated in pairs in order to solve a contextual problem. in the interview afterwards she saw no disjuncture between the two perspectives. sarah explained that in the toolbox section of her lesson she was preparing the learners for the forthcoming test that would be skills-based and in the process section she was endeavouring to introduce an element of contextual problem solving using collaborative learning. she thus divides the lesson into a conceptual section, where she believes a problem solving approach can aid cognition, and a procedural section, where traditional rote learning is her preference. conclusion and implications skott (2001b) claims that the motives for a teacher’s activities emerge in the course of classroom interactions and that previously espoused beliefs may become less significant, depending on the particular context at the time. it is apparent that sarah’s professed perspectives on mathematics were relatively constant, whether she completed a questionnaire or self-evaluation task or expressed an opinion during an interview, as she consistently expressed reform-based views. she mentioned investigations and experimentation as ideal ways of enhancing learner understanding; she claimed that learning mathematics included ways of approaching and posing problems; and she saw her role in teaching as a guide and facilitator of learning. however, her classroom practices reflected both a toolbox approach and a process approach – in lerman’s (1986) terms she demonstrated an absolutist viewpoint and a fallibilist viewpoint – at different occasions during the same lesson. she emphasised mastery of skills at one stage and a problem solving approach at another stage of the lesson. she acted as both instructor and facilitator. she demonstrated both a traditional “chalk-and-talk” style as well as a more innovative contextual, problem solving approach. her classroom practices, therefore, seemed at times to be consistent and at times to be inconsistent with her verbalised beliefs – depending on how her beliefs were situated in a particular context. sarah’s explanation that, firstly, she was carrying out instructions from the senior teacher to drill the learners for an upcoming skills test shed light on an external factor that dictated practice, as opposed to a disjuncture between her espoused beliefs and enacted practice. secondly, she endeavoured to implement a problem solving aspect into the lesson because of her innate belief in constructivist learning. her explanations validate speer’s (2005) advice that instead of focusing extensively on the differences between professed and attributed beliefs, researchers should instead focus on the explanations generated by the teachers for the reasons for their instructional decisions during specific instances of interactions with learners in the classroom. thus, it may be possible to refine existing theories concerning the link between beliefs and practices in order to understand the phenomenon better. sarah explained that although she had planned her lesson meticulously, there were many occasions where she had to “think on her feet” and had to deviate from the predetermined teaching practice. this echoes skott’s (2001b) standpoint that the motives behind teachers’ practices are not fixed by lyn & paul webb 47 previously stated beliefs, but by entities that may be transformed as a result of interactions with learners and a multiplicity of different tasks that emerge during a lesson. it is not possible, from this study, to predict whether sarah will hold her professed beliefs over time or whether her beliefs and practices are cyclical and open to change as she develops into a more experienced teacher (lerman, 2002). what is apparent, though, is that a teacher’s activity is likely to be related to the sense s/he makes of the classroom situation which will draw the focus of a certain activity at a specific moment onto a particular aim. this means that inconsistency between beliefs and practices may be an observer’s perspective that is not necessarily shared by the teacher, and that the view that there is a disjuncture may not do justice to the complexity of the teacher’s tasks (skott, 2004). we agree that the context and complexity of classroom interactions have become increasingly more demanding on novice teachers, and that this understanding has implications when trying to change, measure and understand teachers’ beliefs and practice during periods of curriculum reform, such as is currently demanded by the national curriculum statement for mathematics (department of education, 2003). university teacher training courses aim at inculcating sustainable constructivist beliefs about teaching and learning, but the question is what is being done to assist novice teachers to recognise and deal with the rapidly changing contexts and situations that may occur within a single lesson and which may challenge these beliefs as well as help them understand that what some may see as a disjuncture may simply be a product of the complexity of the practitioner’s tasks? references barwell, r. (2005). ambiguity in the mathematics classroom. language and education, 19(2), 118126. brodie, k. (2001). changing practices, changing knowledge: towards mathematics pedagogical content knowledge in south africa. pythagoras, 54, 17-25. department of education. (2003). national curriculum statement: grades 10-12 (general). mathematics. pretoria: department of education. ensor, p. (1998). teachers’ beliefs and the ‘problem’ of the social. pythagoras, 46/47, 3-7. ernest, p. (1989). the impact of beliefs on the teaching of mathematics. in mathematics teaching: the state of the art. london: falmer press. ernest, p. (1991). the philosophy of mathematics education. london: falmer press. furner, j. (2004). implementing the national council of teachers of mathematics standards: a slow process. pythagoras, 59, 46-56. hoyles, c. (1992). illuminations and reflections – teachers, methodologies and mathematics. in w. geeslin & k. graham (eds.), proceedings of the sixteenth international conference for the psychology of mathematics education, 3, 263286. durham, new hampshire, usa. lerman, s. (1986). alternative views of the nature of mathematics and their possible influence on the teaching of mathematics. unpublished doctoral dissertation, university of london, uk. lerman, s. (2002). situating research on mathematics teachers’ beliefs and on change. in g. leder, e. pehkonen & g. törner (eds.), beliefs: a hidden variable in mathematics education? (pp. 233243). dordrecht: kluwer. pehkonen, e., & törner, g. (2004). methodological considerations on investigating teachers’ beliefs of mathematics and its teaching, nordisk matematikk didadtikk, 9(1), 21-49. philipp, r., ambrose, r., lamb, l., sowder, j., schappelle, p., sowder, l., thanheiser, e. & chauvot, j. (2007). effects of early field experiences on the mathematical content knowledge and beliefs of prospective elementary school teachers: an experimental study. journal for research in mathematics education, 38(5), 438-475. schoenfeld, a. h. (1992). learning to think mathematically: problem solving, metacognition, and sense making in mathematics. in d. grouws (ed.), handbook of research on mathematics teaching and learning. new york: macmillan. skott, j. (2001a). the emerging practices of a novice teacher: the roles of his school mathematics images. journal of mathematics teacher education, 4(1), 3-28. skott, j. (2001b). why belief research raises the right question but provides the wrong type of answer. proceedings of the 3rd nordic conference on mathematics education. högskolan kristianstad. skott, j. (2004). the forced autonomy of mathematics teachers. educational studies in mathematics, 55(1-3), 227-257. speer, n. (2005). issues of method and theory in the study of mathematics teachers’ professed and attributed beliefs. educational studies in mathematics, 58(3), 361-391. beliefs and practices of a pre-service mathematics teacher 48 thompson, a.g. (1992). teachers’ beliefs and conceptions: a synthesis of the research. in d. a. grouws (ed.), handbook of research on mathematics teaching and learning (pp. 127-146). new york: macmillan. von glaserveld, e. (1992). a constructivist’s view of learning and teaching. in r. duit, f. goldberg & h. niedderer (eds.). research in physics learning: theoretical issues and empirical studies. kiel: institute for science education. vacc, n., & bright, g. (1999). elementary preservice teachers’ changing beliefs and instructional use of children’s mathematical thinking. journal of research in mathematics education, 30, 89-110. vygotsky, l. s. (1986). thought and language. cambridge, massachusetts & london: the mit press. webb, l. (2004). teachers’ understanding of the nature of mathematics. unpublished med thesis, nelson mandela metropolitan university, south africa. wilson, m., & cooney, t. (2002). mathematics teacher change and development. the role of beliefs. in g. leder, e. pehkonen, & g. törner (eds.), beliefs: a hidden variable in mathematics education? (pp. 127-148). dordrecht: kluwer. lyn & paul webb 49 appendix a: standards’ belief instrument shade in the answers that best describe your feeling about the following statements on the scantron grid provided. use the following code: directions: 1 = strongly disagree 2 = disagree 3 = agree 4 = strongly agree 1 problem solving should be a separate, distinct part of the mathematics curriculum. 1 2 3 4 2 students should share their problem-solving thinking and approaches with other students 1 2 3 4 3 mathematics can be thought of as a language that must be meaningful if students are to communicate and apply mathematics productively. 1 2 3 4 4 a major goal of mathematics instruction is to help children develop the beliefs that they have the power to control their own success in mathematics. 1 2 3 4 5 children should be encouraged to justify their solutions, thinking, and conjectures in a single way. 1 2 3 4 6 the study of mathematics should include opportunities of using mathematics in other curriculum areas. 1 2 3 4 7 the mathematics curriculum consists of several discrete strains such as computation, geometry, and measurement which can be best taught in isolation. 1 2 3 4 8 learning mathematics is a process in which students absorb information, storing it in easily retrievable fragments as a result of repeated practice and reinforcement. 1 2 3 4 9 mathematics should be thought of as a collection of concepts, skills and algorithms. 1 2 3 4 10 a demonstration of good reasoning should be regarded even more than students’ ability to find correct answers. 1 2 3 4 11 appropriate calculators should be available to all students at all times. 1 2 3 4 12 learning mathematics must be an active process. 1 2 3 4 source: zollman & mason, 1992, as cited in furner, 2004, p. 56. beliefs and practices of a pre-service mathematics teacher 50 appendix b: a questionnaire for teachers conception of teaching mathematics through the following questionnaire, we would like to get a profile of your ideas and conceptions concerning teaching mathematics. these are some statements on teaching mathematics. circle the option which best describes your opinion. 1=fully agree 2=agree 3= don’t know 4=disagree 5=fully disagree 1 1 2 3 4 5 in teaching mathematics, one should use varied exercises and applications above all else. 2 1 2 3 4 5 mathematics in school necessarily requires a concrete dimension; abstract mathematics alone is not enough. 3 1 2 3 4 5 logic is promoted in teaching mathematics, whereas creativity and originality are not stressed. 4 1 2 3 4 5 problem orientation should be the core of teaching mathematics. 5 1 2 3 4 5 in teaching mathematics, finished products take priority, not the process by which they are achieved. 6 1 2 3 4 5 doing mathematics means: working through the proofs carefully. 7 1 2 3 4 5 teaching mathematics provides an excellent opportunity to promote the development of the pupils’ thinking. 8 1 2 3 4 5 mathematics teaching is especially meant for talented pupils. 9 1 2 3 4 5 one should always make sure to visualize aspects of teaching mathematics. 10 1 2 3 4 5 indisputable formality takes priority in mathematics. 11 1 2 3 4 5 learning calculation techniques is the core of teaching mathematics. 12 1 2 3 4 5 while doing mathematics, understanding the topic is the most important idea. 13 1 2 3 4 5 in teaching mathematics, one should often realize projects without subject limits. source: pehkonen & törner, 2004, p. 45. lyn & paul webb 51 appendix c: numerical and graphical self evaluation starting point: a rough classification of mathematical views consists of the following three perspectives, which are part of every view of mathematics and the teaching of mathematics: t mathematics is a large toolbox: doing mathematics means working with figures, applying rules and procedures and using formulas. s mathematics is a formal, rigorous system: doing mathematics means providing evidence, arguing with clear and concise language and working to reach universal concepts. p mathematics is a constructive process: doing mathematics means learning to think, deriving formulas, applying reality to mathematics and working with concrete problems. question 1: distribute a total of 30 points corresponding to your estimation of factors, t, s, and p in which you value your… t s p … real teaching of mathematics … ideal teaching of mathematics for additional comments please use the reverse side of this page. question 2: acknowledge your position on the three factors mentioned above by marking points within the equilateral triangle below. x = real teaching of mathematics o = ideal teaching of mathematics for additional comments please use the reverse side of this page. thank you very much! source: pehkonen & törner, 2004, p. 46. system process toolbox abstract introduction and research gap theoretical perspective method results and discussion conclusion acknowledgements references appendix 1 footnotes about the author(s) samah g.a. elbehary curriculum and instruction department, faculty of education, tanta university, tanta, egypt citation elbehary, s.g.a. (2021). reasoning under uncertainty within the context of probability education: a case study of preservice mathematics teachers. pythagoras, 42(1), a630. https://doi.org/10.4102/pythagoras.v42i1.630 original research reasoning under uncertainty within the context of probability education: a case study of preservice mathematics teachers samah g.a. elbehary received: 22 may 2021; accepted: 31 aug. 2021; published: 17 dec. 2021 copyright: © 2021. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract interpreting phenomena under uncertainty stands as a substantial cognitive activity in our daily life. furthermore, in probability education research, there is a need for developing a unified model that involves several probabilistic conceptions. from this aspect, a central inquiry has been raised through this study: how do preservice mathematics teachers (psmts) reason under uncertainty? a multiple case study design was operated in which a purposive sample of psmts was selected to justify their reasoning in two probabilistic contexts while their responses were coded by nvivo, and corresponding categories were developed. as a result, psmts’ probabilistic reasoning was classified into mathematical (m), subjective (s), and outcome-oriented (o). besides, several biases emerged along with these modes of reasoning. while m thinkers shared equiprobability and insensitivity to prior probability, the prediction bias and the belief of allah’s willingness were yielded among s thinkers. also, the causal conception spread among o thinkers. keywords: allah’s willingness; egypt; preservice mathematics teachers; probability biases; reasoning under uncertainty. introduction and research gap interpreting phenomena under uncertainty is a substantial cognitive activity of our daily life, which is encountered constantly in everyday situations. starting from a simple question, ‘is it going to rain tomorrow?’, to a more sophisticated inquiry, ‘can a person accomplish his future goal?’. moreover, many professions such as insurance, economics, medicine, physics, and biology require making decisions under uncertainty. as batanero, chernoff, engel, lee, and sánchez (2016) declared, to function adequately in society, citizens need to overcome their deterministic reasoning and accept the existence of chance in nature. hence, this need for probabilistic reasoning has been acknowledged by educational institutions in many countries, and probability has been embedded in the official curricula at various levels, including teacher education. while the discussion of probabilistic reasoning becomes necessary for all learners at different levels of study, it is particularly crucial for the teacher education of preservice mathematics teachers (psmts), those university students who learn to teach intentionally and systematically (morris, hiebert, & spitzer, 2009). as noted in various studies, one pedagogical difficulty for teaching probability is the mathematics teachers’ lack of specific preparation in such content (ainley & monteiro, 2008; batanero, burrill, & reading, 2011; batanero, contreras, fernandes, & ojeda, 2010; franklin & mewborn, 2006; pecky & gould, 2005). furthermore, fundamental broad statistical knowledge is not adequate for teachers to effectively teach probability (batanero, godino, & roa, 2004). this situation appears obviously in the egyptian context, wherein only about 9% of all subjects during the whole duration of the four-year mathematics teachers’ preparation programme has been assigned to study statistics, including probability (elbehary, 2020). consequently, because of such limitedness and the specific characteristics of the probability subject that is not usually encountered in other mathematics areas (e.g., multifaceted view and the lack of reversibility), probability education incorporates distinct challenges for both teachers and students (batanero et al., 2016). additionally, and regarding probabilistic reasoning, it has been highlighted by stohl (2005) that ‘the success of any probability curriculum for developing students’ probabilistic reasoning depends greatly on teachers’ understanding of probability’ (p. 351). such focus on reasoning is compatible with ball, lubienski, and mewborn’s (2001) argument concerning the reasoning processes, which underpin all teaching practices. as they noted, teachers’ knowledge plays a significant role in the quality of their teaching since many activities, such as determining what students know, representing mathematical ideas, and modifying textbooks, involve reasoning and thinking. moreover, teachers’ knowledge itself that consolidates their reasoning manners has been regarded as one factor to determine students’ understanding and achievement (darling-hammond, 2000; darling-hammond & sykes, 2003; fennema & franke, 1992; mosvold & fauskanger, 2014; rivkin, hanushek, & kain, 2005; schacter & thum, 2004). because of such a relationship between teachers’ knowledge and pupils’ understanding, the limited emphasis on learning statistics and probability during teacher education may be one factor why the egyptian pupils’ achievement in data and chance remains the lowest among other mathematics areas (elbehary, 2019). the aforementioned argument strengthens teachers’ probability knowledge that tacitly includes reasoning under uncertainty to positively impact students’ learning. ‘teachers’ here denotes psmts who will teach probability to primary and lower secondary school students. yet, what is the probability knowledge that students need to acquire? the answer to such a question has been explicitly stated through various professional organisations. for example, the national council for teachers of mathematics (nctm) recommended grades 5–8 students explore situations through experimentation and simulation, construct sample spaces to determine the probabilities of various realistic phenomena, and appreciate the practice of probability knowledge in their daily life. furthermore, middle-grade students should compute probabilities for simple and compound events (nctm, 1989, 2000). these recommendations are consistent with what the american statistical association (asa) has advocated for studying theoretical probability, experimental probability and simulation processes, and hypothesis testing for college students (aliaga et al., 2005). repeatedly, about the relationship between teachers’ knowledge and students’ understanding, papaieronymou (2010) indicated teachers’ awareness to confront common probabilistic misconceptions, conduct simulations, and demonstrate probability concepts to students as crucial pedagogical content knowledge that has been highlighted by nctm, asa, the american mathematical society (ams), and the mathematical association of america (maa). acknowledging these recommendations, learning both theoretical and experimental probability is fundamental. this matches the egyptian curriculum, in which students at primary and lower-secondary grades are expected to acquire theoretical, experimental, and subjective (intuitive) interpretations of probability. on the other side, at the university level, psmts should perceive conditional probability and bayes’ theorem. chernoff and sriraman (2014, 2015) have classified the probability education research into four periods: (1) the piagetian period, which was dominated by piaget and inhelder’s (1975) investigations of people’s probabilistic reasoning. (2) the post-piagetian period, in which probabilistic reasoning was reviewed through fischbein’s (1975) research, focusing on primary and secondary intuitions, and tversky and kahneman’s (1974) investigations regarding judgmental heuristics of adults when they reason under uncertainty, in the field of psychology. (3) the contemporary research period, which witnessed a significant shift towards examining curriculum, instruction, probabilistic intuitions, and learning difficulties; it was led by a group of researchers in the mathematics education field (e.g., falk, 1986; konold, 1989, 1991). (4) the assimilation period started after 2000 while the research continues to develop models, frameworks, and theories associated with intuition and learning difficulties in probability, in line with the previous period. at this stage, the probability education research has been shifted smoothly from replicating and importing research findings of other fields (e.g., psychology) to develop its specific interpretations of results stemming from difficulties associated with teaching and learning probability, under the umbrella of mathematics education. nevertheless, recent investigations shed light on a renaissance period of psychological research in mathematics education (e.g., chernoff, 2012; chernoff & russell, 2012; chernoff & sriraman, 2015). based on this historical development of probability education research, some areas for future study have been identified; one of these concerns is developing a unified framework that models several conceptions of probability (chernoff & russell, 2014; jones, langrall, & mooney, 2007; shaughnessy, 1992). this trend of research has been recommended for further clarification, particularly regarding the witnessed contested area about the nature of probability, as the theory of probability itself has a mathematical side and a foundational or a philosophical side (chernoff & sriraman, 2015). from this aspect, a central inquiry has been raised through the current study: how do psmts reason under uncertainty? in other words, what are the characteristics of psmts’ probabilistic reasoning, which implies a cognitive activity associated with a context containing uncertainty elements (savard, 2014)? exploring psmts’ probabilistic reasoning may contribute to the existing literature by modelling their probability conceptions in one schema. theoretical perspective many previous studies showed that adults (including university students) have various conceptions about probability and relevant biases in reasoning under uncertainty (e.g., batanero & sanchez, 2005; dollard, 2011; fischbein & schnarch, 1997; kazak & pratt, 2017; konold, 1989; konold, pollatsek, well, lohmeier, & lipson, 1993; tversky & kahneman, 1974). moreover, and as noted by stohl (2005), without specific training in probability, preservice and practising teachers may employ their intuitions and beliefs. despite that, there is no further discussion that represents psmts’ probabilistic conceptions in a unified schema. to address this, the current study has acknowledged that ‘learners’ conceptions are underlined by their way of reasoning towards a certain phenomenon’ to be the essential hypothesis. alternatively, one way to identify psmts’ probabilistic conceptions is to explore how they reason under uncertainty. conception defines a mental filter to interpret a situation and make sense of it (giordan & pellaud, 2004); it helps to keep a balanced cognitive structure when learners adapt a new knowledge (piaget, 1975). conceptions can be valid in some contexts, but they cannot be generalised across all. for this reason, they preferably should not be described as misconceptions because they still work in certain circumstances (giordan, 1998). therefore, savard (2014) argued that, concerning probability knowledge, it is not reasonable to pretend that a certain conception could accurately explain a certain level of conceptual understanding. this is because classifying conceptions in a normative way does not declare the value of learners’ reasoning to understand the world. furthermore, and on the relationship between learners’ conceptions and their probabilistic reasoning, while probabilistic conceptions are rooted in different epistemologies, these epistemologies themselves are emphasised by the reasoning employed to think of a phenomenon. in this regard, konold (1989) distinguished the formal knowledge of probability and natural judgmental heuristics as two types of cognition when reasoning under uncertainty. later, savard (2014) redefined cognition under uncertainty as probabilistic reasoning versus deterministic reasoning, which has inspired the current study. in light of savard’s (2014) clarification, probabilistic reasoning considers two significant factors: variability and randomness. on one hand, variability indicates that the outcome is not fixed; it alters depending upon the probable favourable cases (i.e., theoretical probability), the frequencies (i.e., experimental probability), or some evaluation criteria (i.e., subjective probability) (canada, 2006; garfield & ben-zvi, 2005). on the other hand, randomness implies uncertainty and independence; while the former reflects that the event cannot be predicted with certainty, the latter indicates no correspondence between what happened before and the new outcome (dessart, 1995; green, 1993). in contrast, deterministic reasoning seeks correlation, using present and past information to describe an event. there is dependence between the events that might justify a result. besides, the deterministic estimation indicates an accurate prediction (briand, 2005), in which there is no uncertainty. from an educational viewpoint, probabilistic reasoning signifies a principal reason why probability is involved in the school curriculum, as the study of probability sustains the creation of probabilistic reasoning. it supports learners, formally, to structure their vague thinking about random phenomena (borovcnik & peard, 1996). additionally, because of the increasing number of events described in terms of risk, understanding the related concepts to reason under uncertainty should be investigated (martignon, 2014; pange & talbot, 2003). that is consistent with the need to overcome the individuals’ deterministic thinking and admit the presence of chance in nature (batanero et al., 2016). the above argument reflects the process of probabilistic reasoning as an expected capability that students should acquire through learning the content of probability, either at pre-university or at teacher education level. this appears clearly in forthcoming research of probability that flows from discussing informal reasoning toward clarifying fallacious reasoning (chernoff, 2012). following this trend, there is a demand for further studies that focus on fallacious reasoning, especially since many of these fallacies still account for both correct and incorrect responses. this direction emphasises the individual justification and reasoning processes rather than their typical normative answers. it also meets the renaissance period of psychological research in mathematics education, in which there is a need to investigate ‘theories about mathematics education and cognitive psychology to recognize and incorporate achievements from the other domain of research’ (gillard, dooren, schaeken, & verschaffel, 2009, p. 13). additionally, and about the current literature, few studies were reported on psmts’ knowledge and reasoning (e.g., batanero et al., 2010; dollard, 2011; estrella & olfos, 2010; ives, 2007; torres, 2014), and it recommended much more research to clarify the essential components in psmts’ preparation. for a case, within the recent contributed papers of the international conference on teaching statistics (icots 8 [2010], 9 [2014], and 10 [2018]), which is considered a platform to exchange ideas and experiences among statistics educators under the authorisation of the international association for statistic education (iase), only three papers were found. savard’s (2010) study aimed at interpreting primary school students’ probabilistic thinking in some artificial gambling situations, and it reported that those students practised deterministic reasoning to predict the outcome. besides, understanding variability was highlighted as an essential concept in thinking probabilistically. moreno and cardeñoso’s (2014) study revealed four hieratical levels of probabilistic thinking (i.e., deterministic, personalistic, uncertain, and contingency), and it confirmed a certain distance between teachers’ mental models and the standard conceptual models in probability theory. while those two papers addressed the characteristics of learners’ probabilistic thinking, the third article was provided by primi, morsanyi, and chiesi (2014) to develop a scale for measuring the basics of probabilistic reasoning ability. these studies stressed the approach of classification and assessment with less reflection on the nature of the reasoning process itself, except savard’s (2010) paper which provided a motive for the current study wherein it is not possible to pretend that a certain conception explains a level of understanding, as remarked earlier. accordingly, to contribute to the forthcoming literature, this study intended to model psmts’ reasoning under uncertainty and related conceptions, not as a hierarchical order of conceptual understanding (e.g., normative or optimal solution), but rather through emphasising the way of reasoning per se. from this aspect the current study admits two things: (1) teaching probability rarely builds upon authentic contexts and predominantly uses a theoretical approach, in which most encountered tasks in both k-12 and teacher education curricula incorporate a well-defined quantifiable sample space (e.g., tossing a coin, rolling dice); (2) the deficiency of such traditional tasks to afford an adequate foundation for learning subjective probability (azcárate, cardeñoso, & serradó, 2006; stohl, 2005). given these considerations, the current study has adapted an authentic probabilistic situation (in addition to a pedagogical activity) to explore psmts’ probabilistic reasoning (see the method section). further to this, and from a psychological viewpoint, the world of personal intuition signifies one source of success or failure in teaching since these intuitions determine whether learners accept or ignore what they learn. accordingly, kapadia and borovcnik (2010) remarked that to think probabilistically, it is time to replace heitele’s (1975) ideas with an approach that studies concepts more from a non-mathematical perspective. because of that, assessing the application of probability models to real phenomena was regarded as a crucial skill for interpreting random events (martignon, 2014), particularly in teacher education wherein the activities proposed for psmts during their preparation are generally stereotyped. it brings the concept of probability to the notion of calculating the relative frequency of occurrences of an event (musch & ehrenberg, 2002). accordingly, because teaching probability seldom depends on exploring authentic situations, employing a realistic context can cultivate analysing psmts’ probabilistic reasoning. it defines a mode of thinking associated with judgments under uncertainty and is related to real-life phenomena (falk & konold, 1992). method research design since the central goal of this study is modelling psmts’ probabilistic reasoning and related conceptions, the case study design as a form of qualitative research was employed to answer its inquiry. accordingly, the investigator can explore a bounded system (psmts at the faculty of education, tanta university, egypt) through in-depth data collection for reporting a case description and case-based themes (creswell, 2009). more precisely, the current study has utilised the multiple case study design; it considers the logic of replication, in which the inquirer replicates the procedures for each case (yin, 2003). although qualitative researchers are reluctant to generalise from one case to another because the context of cases differs, triangulating the gathered data is still necessary; it minimises bias and personal effects on the research findings (ticehurst & veal, 2000). therefore, using the multiple case study design increases the validity of the intended model, which, in this study, represents psmts’ probabilistic reasoning. as regarded by weyers, strydom, and huisamen (2008), during the data triangulation process, strong similarities could be viewed as a validation of data or conclusions. participants based on the case study research processes (creswell, 2009), the following procedures were conducted: first, a purposive sample of psmts who study the mathematics education course at the faculty of education, tanta university, egypt, during the academic year 2018/2019 was selected in light of two criteria. while the first criterion signified the participants’ convenience about time and willingness to be engaged (lopez & whitehead, 2013), prior knowledge regarding the three principal interpretations of probability (i.e., theoretical, experimental, and conditional) represented the second standard. the reason for such a criterion is clarifying psmts’ biases and conceptions that persist even under formal education. accordingly, 68 psmts were selected to participate in the current study, as they studied the three interpretations of probability in both pre-university and teacher education. in the egyptian context, learners learn theoretical and experimental probability during the primary and middle grades, while conditional probability is introduced in higher secondary school. moreover, these concepts have to be studied further as a part of a four-year preparation programme for mathematics teachers (elbehary, 2019). table 1 shows characteristics of the study sample (taken into consideration that first-year students could not participate in this study because they almost have a full schedule). table 1: the study population and sample. data collection and analysis processes secondly, two probabilistic situations (i.e., giving birth and throwing a die) were offered through a questionnaire prepared by the researcher, which should be performed within 30 minutes. this matches what hancock and algozzine (2006) noted, wherein the examined documents by a case study researcher include instruments in the form of surveys or questionnaires that often provide powerful means to collect data regarding the study questions. yet, before listing the questionnaire items, the following section describes the process of selecting the appropriate contexts within which psmts’ probabilistic reasoning can be modelled. acknowledging that psmts in this study are being prepared to teach primary and lower secondary pupils, the implemented activities of probability, provided by the national textbooks for these grades, were characterised inductively based on the context. in other words, the researcher tried to consolidate the probability tasks of similar circumstances together. consequently, seven different settings were inferred (see table 2). table 2: the probability contexts within the egyptian school curriculum. from table 2, it is apparent that the school curriculum emphasises traditional mathematical activities. hence, the activity of throwing a die was included in the study questionnaire. nonetheless, because these activities cannot provide an adequate foundation for defining subjective probability, as noted earlier, another context was utilised. for this, and in light of previously inferred probability contexts, a survey was prepared (see the appendix) and administered to the study participants. they were asked to determine the appropriateness of each context to address each probability interpretation (i.e., theoretical, experimental, and conditional), knowing that some contexts can be adapted to approach more than one concept. as a result, psmts judged that all settings could be employed to address multiple probability interpretations. nevertheless, the contexts of life expectancy, gender, and school experiences signified (1) probabilistic circumstances where the subjective interpretation strongly exists (i.e., around 25%, 22%, and 27%), and (2) balanced choices across the probability interpretations (see figure 1).1 figure 1: preservice mathematics teachers’ determination of the probability context. admitting that (1) a clear sample space, (2) easily identified chance factors, and (3) strong cultural prescription towards viewing the phenomena statistically are the criteria for judging the difficulty of the reasoning process (nisbett, krantz, jepson, & kunda, 1983), randomising devices (e.g., the activity of throwing a die) are easy to reason. nonetheless, in the social domain (e.g., gender), the sample space is often obscure, and repeatability is hard to imagine; thus, life expectancy and gender were judged as difficult contexts to reason. still, the gender context appears within the curriculum more than the life expectancy context (see table 1). as a result, the activity of giving birth was preferred to be considered in the questionnaire. therefore, to address psmts’ reasoning under uncertainty, they were asked to answer the following items: item a: knowing that there is a pregnant woman how can you determine the probability of giving birth to a girl? are there any conditions for why you have chosen the proposed ratio of probability? item b: how will you explain to your prospective learners the various strategies that one can employ to determine the probability of getting 5 in a random experiment of rolling a die one time? lastly, after the participants responded to the proposed questionnaire, their interpretations were collected and coded by nvivo software. in detail, the coding procedures were conducted as explained below at first, the obtained data from the study questionnaire were textual, with some mathematical explanations in most cases. for example, about the first item, some psmts noted that: ‘the probability of giving birth to a girl = p (g) = 1/2, since the sample space (s) = {boy, girl}, and the number of favourable outcomes = n (g) = 1’. similarly, for the second item, they responded: ‘the probability of getting five = p (5) = 1/6, because s = {1, 2, 3, 4, 5, 6} and n (5) =1’. furthermore, two rounds of coding processes were employed: (1) an inductive cycle started by investigating psmts’ responses to the first item. this is argued by linneberg and korsgaard (2019) as there is a strong tradition in qualitative research concerning developing codes directly from data rather than using theoretical or prior understandings of the researcher. (2) a deductive cycle was used to develop the inductive probabilistic reasoning categories that worked during this cycle as a lens for examining the answers to the second item (miles, huberman, & saldana, 2013). such procedures were handled case by case, starting with analysis of the responses of second-year psmts, extending to third-year psmts, and ending with fourth-year psmts, as recommended in multiple case study design. during the first cycle of coding, which is inductive, using nvivo, the following steps were followed (thomas, 2006): determine a label for each node that is a short phrase to refer to it. for example, there are no specific conditions to determine the probability, or there are some conditions like the number of sample space elements, knowing the biological or genetic state, admitting miscarriage as a possible result, knowing the previous babies’ gender, or it is a matter of allah’s will. describe the scope of each node. for example, the node of no conditions identified psmts who agreed that there are no restrictions to determine the probability, it is just a matter of mathematical calculation of the number of favourable cases (one case: girl) divided by the number of all outcomes (two: boy and girl). additionally, if psmts acknowledged some criteria to limit the probability, these criteria were categorised based on their nature (source) and focus point (i.e., determine the likelihood or approve the outcome). for a case, the assigned responses to ‘it depends on the number of sample space elements’ node have a numerical nature and probability emphasis. nevertheless, if the criterion was not mathematical (e.g., using baby sonar, observing the woman’s physical appearance), the response was committed to another distinct node. illustrate some examples of texts associated with nodes. for instance, the typical response for psmts who were assigned to the node ‘it depends on n(s)’ was: ‘if s has two elements, then p(g) = 50%; similarly, if s has three outcomes, p(g) = 33.3%’. moreover, for the non-mathematical justifications, although psmts provided some percentages to determine the probability of giving birth to a girl, they emphasised several criteria that may alter these percentages. as for the node of the ‘using baby sonar’, psmts reported 50% to describe the probability; however, they declared that knowing the results of baby sonar could change this probability. create links among several nodes. during this stage, all emerged codes were restructured into two broad categories of (1) probability-focused and (2) outcome-oriented. the probability-focused group was further branched into (1) mathematically oriented that indicates the psmts who emphasised the mathematical rules (e.g., 50% because of calculation procedures), and (2) subjectively oriented that denotes who displayed individual non-mathematical criteria; still, their focus is how to estimate the probability based on these criteria. incorporate the emerged categories into a model. following the emergence of the three groups, the principal interpretations of probability (i.e., theoretical, experimental, and conditional) were utilised as a framework in which these categories can be consolidated and theorised; that helps to model psmts’ probabilistic reasoning (see the results section). this detailed description helps ensure transparency in such a qualitative study; it reflects how the study findings are linked to the collected data (elo et al., 2014). furthermore, the second coding cycle, which was operated deductively, had intended to explore how psmts responses persisted in a different context by which study results could be further verified. thus, through implementing the backward direction, psmts’ responses to the second item (i.e., throwing a die) were analysed in light of the emerged categories from the first coding cycle. nevertheless, in a few cases, there were some discrepancies between psmts’ responses to both items. such cases were highlighted and are discussed within the results section. results and discussion to respond to the study question (i.e., how do psmts reason under uncertainty?), psmts were motivated to state their conditions and relevant criteria in which their estimation can be changed. therefore, to capture the characteristics of their probabilistic reasoning, the giving birth activity was first discussed with them in this manner: suppose a pregnant woman asks you to help her estimate the probability of giving birth to a girl? what do you think of such a situation? do you have any criteria or standards to make a judgment concerning the probability? however, the task of throwing a die didn’t need more clarification since psmts used to practise this. accordingly, the participants employed different ways of reasoning that are categorised as follows: mathematically oriented thinkers (see table 3) subjectively oriented thinkers (see table 4) outcome-oriented thinkers (see table 5). table 3: mathematically oriented thinkers (m and m* reasoning)2. table 4: subjectively oriented thinkers (s = s, s*, and s** reasoning). table 5: outcome-oriented thinkers (o, o*, and o** reasoning). characteristics of mathematically oriented thinkers [m = m and m* reasoning] the reasoning for both type m and m* (m thinkers) has a common feature of relying on theoretical probability. psmts modelled the given situation through the notion of s and the favourable outcome g for the giving birth activity. similarly, 49 psmts shaped the experiment of throwing a die using a sample s that contains six different possibilities with the favourable outcome being 5. in this regard, m thinkers (both m and m*) understood the idea of variability, wherein the result is not determined, but varies depending on the possible favourable cases (i.e., elements of s) (garfield & ben-zvi, 2005). m thinkers also maintained the equiprobable bias (lecoutre & fischbein, 1998; savard, 2014), since they judged the possibility of giving birth to a girl to be equal to a boy; furthermore, when they considered twins as a possible outcome, they supposed that the probability of giving birth to twins is as same as boy or girl.2 holding the equiprobable bias hindered m thinkers from reflecting on the base rate frequencies (i.e., the actual gender distribution). accordingly, their responses to the problem of giving birth indicated the insensitivity to the prior probability of outcomes bias (tversky & kahneman, 1974). the respondents ignored that the possibility of giving birth to a girl is slightly less than a boy as the actual gender distribution in egypt shows the ratio between boys to girls equals 1.06 at birth (nationmaster, 2021). certainly, psmts are neither required nor expected to give a specific correct percentage for the gender distribution throughout the country, but should preferably indicate the base rate frequency as a necessary factor to consider when judging under uncertainty. hence, the notion of the population should be regarded to estimate a reasonable value. for the second activity, m thinkers did not confirm the condition of equiprobability, which is required to utilise laplace theory. while the complexity of physical circumstances (e.g., air resistance, speed) illustrates why individuals cannot predict whether or not a particular outcome will occur when rolling a die (stohl, 2005), the theoretical probability considers one approach that embodies such complexities. accordingly, although psmts may think of this condition as just a premise for all chance games, there was no explicit information concerning the die regularity. such a result resembles chiesi and primi’s (2009) findings that indicate the prevalence of equiprobability and insensitivity to the prior probability biases among college students. chiesi and primi’s (2009) study participants exposed the equiprobability regardless of the actual proportion (i.e., ratio of green and blue balls). this mirrors m thinkers wherein the equiprobable bias hindered them ensuring the prior probability of giving birth to a girl based on the whole population; likewise, speculating the die regularity before determining the chance of obtaining 5. yet, m thinkers diverged between type m and m*, in that there is a slight difference between them in terms of the essence of equiprobable bias. on one hand, type m respondents thought that random events are naturally equiprobable, even when they are not (lecoutre, 1992). this is perhaps related to the representativeness heuristic: it denotes judging the likelihood of an event according to how well such event represents some aspects of the parent population or how it matches the system that generated it, which is the case of m thinkers (kustos & zelkowski, 2013; pratt & kazak, 2018). they focused on the random process, which appeared clearly in their description: ‘both situations imply random experiments, and [the] outcomes have the same chance to occur, regardless of any conditions’. in detail, for the activity of throwing a die, 13 cases were assigned to the m thinkers category. they strengthened the physical structure of the die itself, so the numbers here symbolise its various facets, and the favourable outcome of 5 means one side among six sides. although the respondents understood the meaning of numbers as a symbolisation of the die facets, they did not confirm die regularity, as noted first. one possible reason for that is the representativeness heuristic, which prevented psmts from confirming the theoretical probability assumptions (laplace theory axioms) and shaped their perception of randomness. such a link between students’ understanding of randomness and their perspectives on probability was highlighted by ives (2007). that is, m thinkers who modelled the situations through theoretical probability believed that randomness is bounded to equiprobability. as reported by batanero, green, and serrano (1998): in the classical conception of probability we would say that an object (or an event) is a random member of a given class if there is the same probability for [it as there is for] any other member of its class. (p. 115) on the other hand, type m* responses reflect a more abstract mindset; they attempted to overgeneralise applying the theoretical probability to all situations, whether these situations are realistic (e.g., giving birth) or technical (e.g., simulators). consequently, the equiprobable bias in type m* was inherited in the overgeneralization heuristic: it directed their thinking towards interpreting the giving birth problem like tossing a coin or drawing a card. accordingly, although m* thinkers admitted the limitations of s to restrict the probability of giving birth to a girl, they were reluctant to connect these mathematically stated limitations with the actual circumstances that may occur in reality. such manner of reasoning was also scrutinised in the second activity, in which 36 psmts created s with six possible outcomes (1, 2, 3, 4, 5, and 6) and accordingly determined p (5) = 1/6 (see table 3). they sharpened the assigned numbers to the die facets, in which the number 5 indicates one possible outcome among six possibilities. thus, again, m* thinkers maintained abstract mathematical reasoning. although such reasoning allowed them to acknowledge the sample space elements, they could not perceive these elements as signs of the die facets, which might or might not be symmetrical. consequently, some m* thinkers further reported that for any random experiment with six possible outcomes, every event has one possibility among them to occur. admittedly, type m* reasoning is more relevant to fischbein’s (1987) determination of secondary intuitions emerged because of formal education to replace the primary pre-existing intuitions. it forms a powerful tool that allows solving a problem without grasping all technical details (borovcnik, 2016). that causes various obstacles for the application since ‘the constituents of the modelling process are wider and also comprise – beyond mathematics – knowledge of the context as well as criteria for assessing how well models match a situation’ (borovcnik, 2016, p. 1494). characteristics of subjectively oriented thinkers [s = s, s*, and s** reasoning] interestingly, the category of subjectively oriented thinkers, noted by s and including s, s*, and s**, which matches nearly 60% (41 cases from 68) of psmts’ reasoning in the context of giving birth, did not emerge in the task of throwing a die. such disappearance pulls us back to what was discussed earlier regarding the fruitfulness of the realistic contexts in displaying subjective probability. for illustration, the die regularity, numbers assigned to its sides, or knowledge of the person who manipulates the die were not reviewed by the respondents. through such information, they can update their knowledge and further the probability when these additional information is recognised (kvatinsky & even, 2002). the common trait among s, s*, and s** reasoning is that all are rooted in the subjective interpretation, wherein psmts utilised their personal information to determine the probability of giving birth to a girl. more precisely, to explain the factors that may alter their judgment. in this regard, s thinkers stressed the variability of the outcomes: the outcomes may vary depending on what they stated as contingencies (see table 4). therefore, their responses were expressed in the ‘it depends’ form, which is relevant to bayesian reasoning that allows updating our estimation by processing new information (batanero et al., 2016; dollard, 2011; sharma, 2016). despite such commonality, s thinkers differed from both s* and s** in understanding the concept of randomness which represents a crucial element to reason under uncertainty. in type s, the respondents stated that using an ultrasound scan might change the expected probability from 50% to 100% for sure. this means that they tend to change their estimation to certainty and deny randomness, which indicates the prediction bias (briand, 2005; savard, 2008) since psmts’ prediction had the meaning of exact prediction. alike this in omitting the notion of randomness that demands independence without correlation (dessart, 1995; green, 1993), one psmt in the same category correlated the previous babies’ gender with the newborn’s gender. that is, past information provides a tool to predict the new outcome. yet, type s* and s** were not assigned to the same category. although psmts in both categories maintained the notions of randomness and variability, the nature of the reasons was quite different. while type s** criteria and stated conditions remained cognitive and practical (see table 4), in type s*, the respondents were inspired by the religious conception of allah’s will. nevertheless, emphasising this conception did not restrict them from determining a probabilistic percentage. also, s* thinkers adopted allah’s will not as a cause that influences the baby’s gender, but rather to reveal some out-of-control circumstances that may alter the predicted outcome. accordingly, as two psmts reported, the probability of giving birth to a girl equals 50%; still, we cannot certainly anticipate a baby girl because it is a matter of allah’s will. in this matter, it is worthwhile to mention that animism attribution of phenomena to god was defined in multiple studies as a personalist interpretation (amir & williams, 1999; garfield & ben-zvi, 2005; kissane & kemp, 2010; watson & moritz, 2003; watson & kelly, 2004). however, in this study, this interpretation has been highlighted to be a certain level of probabilistic reasoning that needs more explanation. in the viewpoint of chassapis and chatzivasileiou (2008), mathematics education and knowledge are culturally situated, so that they either implicitly or explicitly involve social and cultural values. accordingly, beyond the mathematical aspect of each construct, there is another aspect associated with the practice of that construct in daily life. that explains why one mathematical concept can be valued in one context and de-valued in another. in this sense, the influence of sociocultural factors on the individuals’ conceptions of probability was regarded (larose, bourque, & freiman, 2010; sharma, 2016). in chassapis and chatzivasileiou’s study, two groups from jordan (muslims) and greece (christians) were asked to attribute the unexpected event to one of chance, probability, fate, or god’s will. the study revealed interesting findings, which support the claim regarding type s* reasoning: most jordanian students assigned the unpredictability to god’s will compared to the chance for the greek students. hence, chassapis and chatzivasileiou declared that because of the characterisation of the jordanian context by the muslim doctrine that affirms god’s controllability of life, students’ responses were accompanied by the phrase ‘insha’allah’ (if god wished it). such findings mirror the case of type s* thinkers, where the respondents first determine the probability of giving birth to a girl; also, they added the term ‘insha’allah’ to reflect the limitedness of human beings in giving an exact prediction. besides, s* responses were judged to be a certain level of probabilistic reasoning and not just a personalist interpretation (or a belief). while the psmts in the present study were asked to determine the probability and to reflect the circumstances that may alter their judgment, the participants of chassapis and chatzivasileiou’s (2008) study were directed to attribute the cause of an unexpected event to one of chance, probability, fate, or god’s will. this describes why their students’ selection of ‘god’s will’ was assessed as a causality. however, s* thinkers did not operate the concept of allah’s will as a cause of the baby’s gender, but rather as one factor that may change the probability. thus, they maintained their understanding of randomness without dependence or certainty and the variability by which the outcome may vary in terms of god’s will. that is explicitly detailed by chassapis and chatzivasileiou as follows: ‘beliefs in god’s will and probabilistic thinking may be compatible in some cases leaving space to the formation of chance and probability conceptions’ (p. 204). this indicates how our religious beliefs influence conceptions of probability and probabilistic reasoning. characteristics of outcome-oriented thinkers [o = o, o*, and o** reasoning] type o, o*, and o**, denoted by o, respondents are the outcome-oriented thinkers. they tried to handle the experimental probability to answer the inquiries. although such utilisation involved several biases when psmts were asked to reason in the giving birth context, which caused the emergence of type o and o* categories, it was much better in the task of throwing a die. because of that, the sub-category of type o** thinkers, which substantially deviates from o and o*, emerged only during the analysis of psmts’ responses to the second task. in that sense, o** resembled o and o* in manipulating the experimental probability; nonetheless, o** expressed a clear understanding of such a concept without biases. these discrepancies can be further detailed as follows. first, type o and o* reasoning define psmts who focused on the favourable outcome (i.e., the exhibited outcome in the question) more than the probability. in the giving birth context, they understood the question as if it was: when would a woman give birth to a girl? (how to know, or under what circumstances?). hence, they did not discuss the event of a baby girl as a possible event, but rather as a precise event that already occurred, and we are examining its causes (i.e., why has or has not it happened?). accordingly, their responses assumed this woman is going to give birth to a girl if something appeared; similarly, if this thing did not happen, another gender is expected, in which case they attributed the phenomenon of giving birth to a girl to some causes. for instance, if the woman’s belly shape is circular, she will give birth to a girl; likewise, if her belly shape is not circular, she will not give birth to a girl. because of that, psmts’ predictions were evaluated to be right or wrong, and their responses took the form of yes-no decisions concerning whether an outcome will occur in a particular case (i.e., the next expected gender will be a girl or not) (batanero & sanchez, 2005; konold, 1989; savard, 2014). in the second activity, o and o* respondents emphasised the outcome, number 5 itself, rather than its probability. accordingly, the question was interpreted as how to get number 5: by what method, or how many trials are required to obtain number 5? consequently, as shown in table 5, eight cases explained their strategies to get a favourable outcome of 5. in detail, one psmt reported that the experimenter could control the die, which is similar to the way many experts perform in dominoes games. another remarked that number 5 would certainly appear on the first try if the die has been designed to carry number 5 at all sides. besides, two other psmts declared that ‘it is impossible to get number 5 when throwing the die once’. one of them extended his answer by stating ‘then, we have two possibilities; either (a) to increase the number of trials, or (b) to cheat the experiment and control the die by hand’. the other continued with ‘the outcome cannot be determined in prior, rather after experimenting the die’. although both responses tacitly indicated understanding the experimental probability as a long-term series of events, psmts’ intention was not assigning the probability but rather getting the number 5. also, four more psmts thought of the required number of trials at which number 5 is to appear. as they stated, it may arise at the first, second, or after six attempts. such exhibited reasoning of o and o* mirrors konold’s (1989) description of outcome-oriented thinkers. instead of specifying probability, which reflects the distribution of occurrences in a series of events, psmts predict the result of a single trial. accordingly, as detailed, instead of determining the probability of (1) giving birth to a girl or (2) getting number 5, they explained the circumstances and strategies to obtain such favourable outcomes. once again, the prediction bias emerged in o and o* responses. yet, it was wielded by s thinkers differently. although the term ‘exact’ referred to the probability that was judged by s thinkers precisely, without uncertainty (see table 4), for o and o* thinkers, it indicates the outcome. hence, while the attributed property ‘exact’ defines the probability in type s, it describes the outcome in types o and o*. expressly, in the task of throwing a die, some psmts realised the term ‘once’, meaning they cannot rely on the experimental probability unless the trails were increased. still, they exposed their alternatives, including repeating the experiment several times to obtain number 5 (not to use the frequentist approach). such an understanding of ‘once’ is operated based on the prediction bias: it oriented them towards classifying the sample space into (1) the favourable outcome and (2) all other events to satisfy their yes-no decision. for instance, if the woman›s belly shape is rounded, she will give birth to a girl (i.e., the favourable outcome); if not, any other event will occur (e.g., boy, twins, miscarriage). still, and only in the context of giving birth, there is a clear difference between type o reasoning compared to o* in terms of understanding of the notion of probability. in type o responses, psmts seem not to understand probability as quantification of our information regarding unknown phenomena. they discussed the favourable outcome of ‘girl’ not as a possible expected event but rather as if the random process had already happened, and its results became known. accordingly, they stated that if the woman had a miscarriage during the delivery process, the probability would decrease from 50% to 0 (see table 5). such reasoning neglects that after the delivery process, the situation will not be probabilistic anymore at which point there is no purpose for the prediction itself. in contrast, although type o* thinkers understood the idea of prediction, they were less conscious of the distinction between causality and conditionality, wherein distinguishing both concepts has been recognised as a crucial determinant to reason probabilistically (batanero et al., 2016). while dependence in probability characterises a bi-directional relation – if an event b is the cause of another event a, then whenever b is present, a is present too (i.e., p(a/b) =1) – the two directions included in conditional probabilities have a distinct connotation from a causal viewpoint (batanero et al., 2016; díaz, batanero, & contreras, 2010). for instance, although the conditional probability of having a baby girl to having a positive result on a sonar test is causal, the reversed direction from a positive diagnostic of sonar to having a baby girl is only indicative. consequently, type o* thinkers share the causal conception since they assumed a causal relationship at which the conditioning event b is the cause and a is the consequence (gras & totohasina, 1995; savard, 2014). interestingly, on the other side of both o and o*, the category of type o** thinkers has emerged only in psmts’ responses to the task of throwing a die. it denotes an adequate utilisation of the experimental probability, as noted at the beginning. type o** thinkers focused on the probability, which, for them, remains a posterior judgment since it is necessary to gain data (frequencies) about the outcomes for estimating the relevant probability (chernoff, 2008). accordingly, they admitted the validity of the experimental interpretation to fulfil the situation of throwing a die if and only if the experiment has been repeated many times. furthermore, their responses indicated an awareness of the law of large numbers, in which they recognised the term ‘once’. for example, two of them reported that: the probability of getting number 5 equals 1/6 through experimentations; yet, we cannot rely on one trial, rather the number of trials should be increased to make sure of the results and to get a more precise probability [i.e., theoretical probability]. although such argumentation reflects an understanding of the variability wherein the calculated percentage varies depending upon the frequencies, both psmts still reported that the probability would be 1/6. they either were keeping the theoretical probability in their mind to avoid ambiguity (stohl, 2005) or they misused probability language, so instead of reporting that the probability may approach 1/6 after a large number of identical trials, they just wrote: it will equal 1/6. additionally, nine psmts in this category did not specify any percentage, but were concerned about the ratio of number 5 frequencies to the conducted trials. according to them, manipulating the experimental approach leads to different percentages that express the likelihood of getting 5. they entirely understood the circumstances of the given situation; hence, the probability to obtain number 5, for them, depends on how many 5s will appear in a large number of identical trials. they also further explicitly explained that as much as the number of experiments increased, the experimental probability approximates the theoretical percentage (i.e., the law of large numbers). conclusion in conclusion, psmts’ reasoning under uncertainty has been modelled into three major categories: mathematically [m], subjectively [s], and outcome-oriented thinkers [o]. yet, the exposed reasoning differs based not only on the individuals’ knowledge but also on the context; that is exhibited clearly in figure 2, which represents psmts’ probabilistic reasoning in two different situations (i.e., giving birth and throwing a die). figure 2: psmts’ probabilistic reasoning manners in two different contexts. as displayed in figure 2, most psmts employed subjective reasoning when they were asked to speculate the problem of giving birth compared to the mathematical manner in the task of throwing a die. this can be interpreted in terms of two essential issues. the first issue denotes psmts’ familiarity with handling the traditional activities through manipulating the theoretical probability. nonetheless, several biases were exposed when psmts employed this interpretation. for example, equiprobability and insensitivity to the prior probability of outcomes were exhibited in the context of giving birth when psmts judged the possibility of giving birth to a girl to be equal to a boy (or twins) and dropped the gender ratio at birth in egypt. similarly, they were reluctant to confirm the die regularity before operating laplace theory in the task of throwing a die. furthermore, the grounds under such biases stayed different between m and m*, the two sub-categories of m. on one side, m thinkers were influenced by the representativeness heuristic: it oriented them to emphasise the random process and hindered verification of the required assumptions of theoretical probability. on the other side, equiprobable bias in type m* was inherited in the overgeneralisation heuristic wherein the theoretical probability was overgeneralised to both situations (e.g., realistic and technical). the second issue signifies the value of authentic contexts to display the subjective facet of the probability that implies a degree of belief in the truth of some premises. although such interpretation did not appear in the context of throwing a die, it represented the majority of psmts’ reasoning in the giving birth problem since around 60% of the participants were committed to type s thinkers and diverged among s, s*, and s**. yet, this reasoning involved some relevant biases. for instance, s thinkers shared the prediction bias that oriented them towards changing their estimation to certainty and denying randomness. accordingly, in some cases, their prediction had the meaning of exact prediction; in other cases, their responses designated a correlation in which past information provides a tool to predict the new outcome. on the other side, type s* and s** thinkers maintained the notions of randomness and variability. however, they attributed the variability of the random event to different circumstances. such circumstances varied between utilising realistic conditions (e.g., x and y chromosomes, baby sonar) for s** thinkers, compared to sharing the religious conception of allah’s will for s*. in this regard, the emergence of allah’s will notified one interesting finding that was interpreted, in this study, as a certain level of probabilistic reasoning. this judgment was grounded in the sociocultural influence on the individuals’ probability knowledge and reasoning (chassapis & chatzivasileiou, 2008; larose et al., 2010; sharma, 2016), within bishop’s (1988) perspective on mathematics education as a cultural induction at which students’ values remain an integral part of teaching and learning. in addition to m and s, another manner of reasoning emerged in psmts’ responses to both situations and was termed type o thinkers. this reasoning involved manipulating the experimental probability with some biases in type o and o* compared to a clear understanding of this concept for o** thinkers. on one hand, type o and o* responses typically mirror konold’s (1989) description of outcome-oriented thinkers. they focused on the favourable outcome more than its probability and understood the inquiry as if it was: when will this favourable outcome appear, and how to know or under what circumstances? accordingly, their predictions were evaluated as right or wrong, and their responses took the form of yes-no decisions. thus, instead of specifying probability, which reflects the distribution of occurrences in a series of events, psmts predict the result of a single trial. also, although both o and o* thinkers decided to adapt the experimental probability as a posterior judgment, they failed to perform it appropriately because of the prediction bias. this emerged once again, yet in a different manner, with type s thinkers. the exact prediction referred to the probability for s thinkers and the outcome for both o and o* thinkers. moreover, o thinkers did not fully grasp the probability that defines a quantification of our information regarding unknown phenomena. on the contrary, o* thinkers were less conscious of the difference between causality and conditionality; they assumed a causal relationship between the conditioning event and the preferable outcome (i.e., causal conception). on the other hand, besides both o and o*, o** thinkers reflected an adequate utilisation of the experimental probability. accordingly, most o** thinkers did not specify any accurate percentage; instead, they were more concerned about the ratio of frequencies to the conducted trials. they further explicitly explained that as the number of experiments increased, the experimental probability approximates the theoretical percentage (i.e., the law of large numbers). indeed, these findings provided detailed insights into psmts’ probabilistic reasoning that may work as learning trajectories, especially considering the inadequate preparation of psmts to teach probability, as noted in various studies. this acknowledges batanero et al.’s (2004) recommendation regarding the necessity to consolidate a discussion on heuristics and biases within specific courses relevant to the didactics of probability. it further matches the current renaissance period of psychology research in mathematics education, which has been tacitly handled, in this study, through discussing psmts’ knowledge of probability from a psychological perspective (i.e., probabilistic reasoning). also, it incorporated a unified framework that models several probability conceptions in one schema (i.e., m, s, and o), which may address the contested area regarding the nature of probability (i.e., mathematical and philosophical aspects). further to this, from a national perspective, these results shed light on psmts’ probability knowledge and reasoning under the umbrella of statistics education. perhaps this responded to innabi’s (2014) recommendation regarding the need to stimulate the study of statistics in the arab world, in which very little research has been conducted. besides, it may activate psmts’ awareness of their conceptions and biases that help them interpret similar biases in their prospective students’ reasoning. such a process is closely related to enhancing their knowledge of content and students (kcs), which considers significant knowledge for the psmts to acquire. finally, there are some precautions about the results’ interpretation; in other words, the study findings should not be generalised without considering certain conditions. one essential regard is the influence of the sociocultural factors in which the participants’ values, an integral part of teaching and learning, need to be considered. consequently, what can be acceptable in one context cannot be in another (e.g., allah’s will). also, regarding the study questionnaire: since the individuals’ reasoning differs depending not only on their knowledge but also on the context, the emerged categories of psmts’ probabilistic reasoning might slightly vary because of the implemented problems. accordingly, further research on psmts’ probabilistic reasoning and related conceptions is demanded (different groups, different contexts, different questions) so that the study results can be verified. acknowledgements i confirm that this work is original and not is currently under consideration for publication elsewhere. however, since this manuscript is a part of a phd research, if it has been accepted for publication, i may contact you to get permission to release it (as included in the dissertation) in the hiroshima university institutional repository. competing interests the author has declared that no competing interest exist. author’s contributions i declare that i am the sole author of this research article. ethical considerations this article followed all ethical standards for a research without direct contact with human or animal subjects. funding information this research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors. data availability as written in the acknowledgement section, the data provided in this research will be released in the hiroshima university institutional repository. disclaimer the views and opinions expressed in this article are those of the authors and do not necessarily reflect the official policy or position of any affiliated agency of the authors. references ainley, j., & monteiro, c. 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(2003). case study research: design and methods (2nd ed.). thousand oaks, ca: sage. appendix 1 the probability contexts survey the following table summarises seven various contexts in which the probability can be operated. based on your understanding of theoretical, experimental, and conditional probability, could you determine the appropriateness of each situation to approach each probability interpretation? please note that some contexts can be adapted to approach more than one concept (i.e. you may select multiple interpretations for each setting). figure 1-a1: the probability contexts survey. footnotes 1. note that the total number of the reviewed probability tasks equals 106. this is all the activities that have been raised within the lesson content of both primary and lower-secondary grades, from grade 3, when probability is first introduced, until grade 9. 2. note that: p = probability, s = sample space, g = girl, b = boy, and a = the favourable outcome. abstract introduction background of the study theoretical framework presentation and discussion of the data conclusions and recommendations for future research acknowledgements references footnote about the author(s) lydia o. adesanya department of science, mathematics and technology education, faculty of education, university of pretoria, pretoria, south africa marien a. graham department of science, mathematics and technology education, faculty of education, university of pretoria, pretoria, south africa citation adesanya, l.o., & graham, m.a. (2022). effective communication of learning intentions and success criteria in the mathematics classroom: merlo pedagogy for senior phase south african schools. pythagoras, 43(1), a666. https://doi.org/10.4102/pythagoras.v43i1.666 original research effective communication of learning intentions and success criteria in the mathematics classroom: merlo pedagogy for senior phase south african schools lydia o. adesanya, marien a. graham received: 16 nov. 2021; accepted: 20 june 2022; published: 16 aug. 2022 copyright: © 2022. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract a well-designed assessment construct is critical for improving all aspects of quality education and validating the achievement of educational reform. the global prevalence of how teachers communicate learning intentions (lis) and success criteria (sc) has been of great concern, particularly in the south african context. this study investigates how meaning equivalence reusable learning objects (merlo) pedagogy effectively transforms senior phase mathematics teachers’ daily practice in the classroom. the study adopted qualitative participatory action research to frame the evolution of teachers’ praxeologies such as teachers’ meta-didactical and didactical praxeologies, to improve teachers’ beliefs and practices to integrate merlo pedagogy as assessment activities. twelve senior phase teachers were purposively selected in gauteng, south africa. the methods used for data generation were interviews, classroom observation, document analysis, field notes and training sessions. thematic analysis was used to obtain insight into teachers’ beliefs and practice of effectively communicating lis and sc in the classroom. at the initial stage, teachers were examined with regard to their beliefs and practices of assessment practices in the classroom, which informed merlo intervention. in the second stage, teachers were asked to learn about merlo items by reading the merlo handout provided to them, participating in the workshop and sharing their opinions and views with others. in the third stage, teachers had to design merlo assessment items on their own to assess learners’ level of understanding of the mathematical concepts in senior phase. the findings revealed that the participating teachers acquired adequate knowledge and skills on merlo techniques that allowed them to structure and integrate the lesson plan of assessment activities into their mathematics classrooms. this study contributes to the body of knowledge by introducing merlo pedagogy to senior phase south african mathematical teachers as an assessment strategy. covid-19 caused some teachers to drop out of the study after the pre-merlo participation phase and, accordingly, future research suggests that more teachers be included in similar studies. keywords: assessment; merlo pedagogy; learning intentions; success criteria; mathematics classroom. introduction quality assessment practices, specifically in mathematics, is a topic that is getting a lot of recent attention in the 21st century (barana & marchisio, 2021; granberg, palm, & palmberg, 2021; see, gorard, lu, dong, & siddiqui, 2021). this study aimed to investigate how meaning equivalence reusable learning objects (merlo) pedagogy effectively transforms south african senior phase1 mathematics teachers’ daily assessment practice in the classroom. meaning equivalency is a concept that signifies shared meaning across representations: it is a polymorphous – one to many – transformation of meaning. as a pedagogical technique for teaching and assessment, merlo asks learners to sort and map significant ideas using representative target statements of specific conceptual contexts and relevant statements that may or may not share the same meaning. when merlo assessment items for various ideas are merged into a large database for a course of study, significant information about learners’ learning patterns can be obtained (etkind, kenett, & shafrir 2010; etkind, kenett, & shafrir, 2016). in a recent book chapter by etkind, prodromou and shafrir (2021), it is pointed out that merlo can be applied as a form of formative assessment (fa) and summative assessment (sa) to validate what learners know and get feedback regarding their conceptual understanding of mathematical concepts in the classroom. the merlo pedagogy could aid teachers in developing new knowledge and skills relevant to designing their lessons (etkind et al., 2021; robutti, 2015). the implementation of merlo pedagogy requires teachers to be skilful and competent to continually change their assessment practices in response to the actual requirements of their learners; updating or changing their assessment practices is critical as many studies have shown that assessment has considerable potential for enhancing learner performance (nortvedt & buchholtz, 2018; polly et al., 2017; suurtamm et al., 2016; veldhuis & van den heuvel-panhuizen, 2020). background of the study assessment practices in education have been a continuous focus for over a decade globally (clarke & luna-bazaldua, 2021). the effective use of the assessment process allows one to elicit information on what learners need to know, understand and be able to do at the end of the lesson (clarke, 2012). scholars indicate that assessment, as an integral part of classroom practices, has the potential to effectively enhance learners’ learning and performance (granberg et al., 2021; heritage & wylie, 2018). igunnu (2020) believes that assessment should produce accurate information and validate concrete learning by learners. there are two types of assessment, namely fa and sa. the former is an assessment technique used to improve learners’ performance, whereas the latter is used to evaluate learners’ performance (khechane, makara, & rambuda, 2020). south africa participates in both national (annual national assessment [ana]; van der berg, 2015) and international assessments (trends in mathematics and science study [timss]; human sciences research council [hsrc], 2020, and south and eastern africa consortium for monitoring educational quality [sacmeq]; department of basic education [dbe], 2017), and south african learners constantly perform far below the excepted standard in mathematics across all grades. authors have tried to explain this poor performance, with many of them (e.g. chavalala, 2015; outhred, 2022) linking it back to poor assessments in the schools; for example, chavalala (2015) states that the ‘low performance of south african learners in various national and international assessments can be linked to lack of quality assurance of assessment practices in schools’ (p. 7). in fact, in the ‘action plan to 2024: towards the realisation of schooling 2030’ issued by the dbe in 2020, the dbe directly links better learning outcomes to ‘more focussed assessment practices’ (dbe, 2020, p. 37). scholars have indicated that to improve education quality, teachers need to be trained, supported through professional development, and they should be willing to improve their assessment practice in terms of improving learners’ learning skills, learners’ involvement and learners’ goals and objectives (unesco institute for statistics, 2016). the latter speaks more to fa than sa (thus, the focus of this study was on fa although merlo can be used for both fa and sa), as recent research debates that sa causes teachers and learners to be overly concerned with performance rather than learning goals (ishaq, rana, & zin, 2020; karaoğlan-yilmaz, üstün, & yilmaz, 2020), whereas fa that asks probing questions helps learners to deepen their understanding (kyaruzi, strijbos, ufer, & brown, 2019) and provides ‘opportunities for further learning and conceptual development through feedback, interpretation, and dialogue between teachers and students’ (arifuddin, turmudi, & rokhmah, 2021, p. 242). kyaruzi et al. (2019), who conducted a study on mathematics fa practices in tanzania, remark that fa supports learners’ learning, which is positively related to their use of deep-level learning strategies, and knight, shum and littleton (2014) highlight the fact that fa plays a ‘crucial role in guiding a student’s epistemic beliefs’ and that fa may be the ‘disambiguation of the epistemic requirements of questions — in terms of understanding the question, its context, and the knowledge required to answer the question’ (p. 28). these preceding arguments led to our study of introducing the use of merlo items to assess the epistemic quality of what learners need to know, understand and be able to do in the mathematics classroom (hudson, henderson, & hudson, 2015). the importance of conceptual thinking skills is now recognised as a cornerstone of effective learning, understanding facts and ideas in the context of a conceptual framework (bransford, brown, & cocking, 2004), as ways of thinking that explore patterns of equivalence-of-meaning in ideas, relations, and underlying issues. several studies on merlo pedagogy development and the nature of its reflective practice have been evolved, validated, tested and implemented across different countries (australia, canada, israel, italy, russia and the netherlands) and various content areas and disciplines, including mathematics (arzarello et al., 2015; etkind et al., 2010, 2016; etkind & shafrir, 2013; persoons & di bucchianico, 2020; prodromou, 2015; robutti, carante, prodromou, & kenett, 2020a). however, merlo pedagogy used as an assessment strategy has not been developed and implemented in south africa. south africa is one of the most unequal countries in the world, not only when referring to the fact that approximately half of south africans live in poverty, that economic growth has stagnated, and the unemployment rate is almost one-third of south africans, but also in terms of its education system (francis & webster, 2019). in addition to this, south africa is performing poorly in mathematics (as mentioned earlier), and it is imperative to introduce an inexpensive and effective method into south african schools to enhance mathematics performance. therefore, this study contributes to the body of knowledge by introducing merlo pedagogy to senior phase south african mathematical teachers, which can be implemented inexpensively as an assessment strategy to promote the conceptual higher-skills thinking and understanding of mathematics in their daily practices. since the new democratic era was implemented in 1994 (dbe, 2009; kanjee & sayed, 2013), the topic of assessment has been deliberated in the south african educational system (mouton, louw, & strydom, 2013; pahad, 1999). in the past decade, curriculum changes are not an easy task to integrate and work on, as teachers are struggling to make sense of the demands placed on them in south african schools (govender, 2018; mouton et al., 2013). govender (2018), who conducted a study on teachers’ views on the curriculum changes, reported that teachers felt that the increased number of assessments increased their administrative duties. poliah (2019) asserts that assessment differs from province to province in the south african context, including regions, districts and schools. some scholars also reflect on the differences in the international external standardised testing results in timss across several years (howie, 2002; hsrc, 2011, 2020; reddy et al., 2020). these differences create doubts about the consistency of assessment. nevertheless, vandeyar and killen (2007) and poliah (2019) indicate that the dominant challenge in assessment is to find strategies that will provide an equal opportunity to all learners while allowing credible, reliable, valid and effective outcomes. furthermore, researchers have identified several obstacles associated with the application of assessment techniques in south africa (dube-xaba & makae, 2021; kanjee & sayed, 2013; poliah, 2019; vandeyar & killen, 2007; van staden & motsamai, 2017). these difficulties range from inadequate training and severe workloads to policy demands that are sometimes difficult to meet. in their critical reviews of the use of practising assessment in the teaching and learning process, scholars have observed that a well-designed assessment construct is vital for improving all aspects of quality education and validating the achievement of educational reform (adesanya & graham, 2021; sayed, kanjee, & rao, 2014). with the focus of the current article being on mathematics teaching and learning, authors have established that assessment is valuable in mathematics teaching and learning (nortvedt & buchholtz, 2018; polly et al., 2017; suurtamm et al., 2016; veldhuis & van den heuvel-panhuizen, 2020). the contribution of this study goes even further than providing south african teachers with assessment techniques that could ultimately improve learner performance. it should be noted the merlo assessment technique is something that south african teachers have not been exposed to before this study, and a recent study conducted within a south african context (warnich & lubbe, 2019) has shown that by applying innovative and alternative assessment practices, enjoyment is brought into the classroom that alleviates learner stress and enhances learner engagement. furthermore, since the adoption of assessment practice across schools in south africa, few empirical studies on learning intentions (lis) and success criteria (sc) have been conducted. this study thus investigates how merlo pedagogy as an assessment strategy can be used to understand teachers’ beliefs and practice of effectively communicating lis and sc in the south african senior phase mathematics classroom. crichton and mcdaid (2016) summarise the difference between lis and sc succinctly: ‘lis tell the learners what the intended outcome of the lesson is with regard to their learning. sc provide examples of their expected performance as a result of the lesson, ‘closing the gap’ between learners’ previous knowledge and their developing understanding.’ (p. 190) teachers’ beliefs on assessment practices in the south african context teachers’ beliefs have been a topic of great interest in mathematics didactics (lepik & pipere, 2011; ramnarain & hlatswayo, 2018). according to scholars, beliefs are the most important psychological element that should guide teacher education (grossman, 1990; holt-reynolds, 1992). teachers’ beliefs reflect how they conceptualise mathematics and its teaching and learning. it is clear that researchers, communities and policymakers around the world are interested in understanding the different ways in which teachers’ beliefs have contributed to and influenced learners’ academic performance (lepik & pipere, 2011). however, the terminologies that immersed knowledge and belief into a single construct defined teachers’ beliefs about assessment (barnes, fives, & dacey, 2015). ideally, binns and popp (2013) and ramnarain and hlatswayo (2018) emphasise the importance of teacher beliefs by asserting that it is not only a teacher’s educational experience that impacts whether a teacher would utilise a pedagogy that supports learner-centred learning, but also teachers’ beliefs, values and attitudes about knowledge and how it is acquired. scholars have indicated that pedagogical tactics are influenced by teachers’ notions about assessment strategies, teaching and learning, the nature of mathematics and classroom assessment practices (kuze & shumba, 2011; ramnarain & hlatswayo, 2018; sikko, lyngved, & pepin, 2012). scholars reveal that some of the biggest challenges to integrating assessment strategies in the mathematics classroom are teachers’ beliefs regarding effective planning and preparation, classroom management, inequity, lack of teachers’ training, lack of teaching and learning aids and materials, and lack of time (martin, mraz, & polly, 2022; panthi & belbase, 2017; schoen & lavenia, 2019). teachers’ beliefs influence their perceptions and judgment, which in turn influence their collection of pedagogy techniques and classroom behaviour (pajares, 1992). meanwhile, harwood, hansen and lotter (2006) claim that teachers’ beliefs have been found to impact teachers’ classroom practices, how they feel content should be taught, and how they believe learners learn. beliefs are thus likely to play a significant role in whether teachers aim to carry out the practice of teaching mathematics by asking questions to seek out information (crawford, 2014; karim, 2015). in a south african study, van der nest, long and engelbrecht (2018) used a qualitative approach to capture the experiences and perceptions of four grade 9 mathematics teachers who participated in a professional development programme that focussed on the use of fa activities in mathematics teaching and learning. they found that, although some of the teachers saw the potential for a deeper conceptual understanding developing in their learners using these new fa techniques introduced to them, their current focus was on their learners’ good performance in the anas, and this was taking attention away from the focus of implementing these new fa strategies meant for developing a deeper conceptual understanding of mathematics. this argument makes a direct link between south african teachers’ beliefs and learners’ mathematics performance in that, when the choice is there between creating a deeper conceptual understanding of mathematics or ‘teaching to the test’ (the anas), the focus falls on the latter, thus showing learner improvement in a sa (the anas), the results of which are released in public domains. van der nest et al. (2018) go on to say that the focus of teachers having their learners perform well in the public eye (in external tests that are not aligned with classroom teaching and learning) has a negative impact on mathematics education in south africa. an exploratory study conducted on south african teachers revealed that the achievement and failure of assessment practices are affected by teachers’ belief in the practices that they employ in the classroom (kanjee, 2020). in another south african study, it was found that teachers believe that various factors in terms of teacher workload, ineffective lesson preparation and planning, disruptive classrooms, large class sizes, and time constraints affected teachers’ inconsistent practice of quality assessment in the classroom, which they believe negatively affects learners’ mathematics performance as they believe that assessment is advantageous for identifying learners’ misconceptions in learning (adesanya & graham, 2021). more so, they also believe that assessment outcomes can be utilised to transform their teaching strategies to meet the learning intentions (lis) (adesanya & graham, 2021). however, some teachers do not use assessment outcomes effectively, and their measures to follow up learners’ performance are ineffective. these imply that teachers did not thoroughly practise assessment as they believed. therefore, for assessment practice to be effective, teachers need further training with guidelines revision, monitoring and periodic assessment (adesanya & graham, 2021; kanjee & croft, 2012; poliah, 2019; vandeyar & killen, 2007). communicating and sharing of learning intentions and success criteria with learners in mathematics classrooms regarding the role that understanding and ascertaining lis and sc plays in ensuring effective mathematics teaching, jones and edwards (2017) state: ‘learning to plan effective mathematics lessons is one of the most important capabilities you can acquire in becoming a successful teacher of mathematics. having a good lesson plan is significant for a whole host of reasons, not least in providing the structure which helps you to be confident that mathematics learning takes place during your lessons. not only does good planning result in lessons that are interesting, challenging and motivating for your students, but also good planning is closely linked to the equally demanding (but often more overt) issue of effective classroom management.’ (p. 70) graham, van staden and dzamesi (2021), who conducted a study in ghanaian mathematics classrooms, emphasise the importance of communicating lis and sc with learners and state that the lis and sc are the forces that drive the process of assessment practices in the mathematics classroom. various scholars of classroom practice concur that quality assessment requires teachers to understand, clarify, share and communicate lis and sc with their learners during the lesson (bartlett, 2015; graham et al., 2021; pryor & crossouard, 2008; wiliam, 2016). however, teachers need to carefully design a lesson plan that measures the quality of the instructional objective, which directs them to aid learners in accomplishing their learning goals. stating and clarifying lis stipulate what learners will learn during teaching (moss & brookhart, 2019). teachers need to have an action plan for what they will teach their learners in the classroom by understanding and ascertaining lis and sc in the process of teaching and learning (bennett, 2011; heritage, 2010; moss & brookhart, 2019). for instance, mathematical teachers’ pedagogical decisions about how to involve learners in higher-order conceptual thinking skills, reasoning and problem-solving have a direct impact on their learning outcomes. many learners in schools can answer simple mathematical problems, but they lack critical thinking, reasoning and problem-solving skills, especially when working on higher cognitive level open-ended questions (hoogland & tout, 2018). according to heritage (2010), the lis direct learners’ attention to what they will learn rather than the activities they will do. the teacher’s attempt to clarify and share lis with their learners which promotes them to be actively engaged in the learning process rather than passive recipients of knowledge. the lis and sc must be communicated to learners properly and in a language that they can understand. the use of easy words related to cognitive domains of learning that explain the lis and sc should be communicated to the learners so that they grasp the purpose of the class and can simply share it with their peers in the classroom (heritage, 2010). learners are so driven to learn new skills and knowledge through active engagement to prevent learning by memorisation that will not improve learners’ performance (brabeck, jeffrey & fry, 2017). based on communicating and sharing lis and sc to learners, scholars highlight that teachers should try to employ effective techniques before, during and after the fa process by ensuring that learners understand, know and communicate lis and sc with others (peers) (moss & brookhart, 2019). moss and brookhart (2019) further assert that when teachers state and communicate lis and sc with their learners appropriately, they would have a starting point to ‘plan their lesson with effective strategies that scaffold learners’ activities, act and monitors their teaching, and help their learners to become self-regulated as well as assessment-capable learners’ (p. 8). to this end, some scholars indicate that implementing effective assessment strategies during teaching and learning plays a vital role in increasing the progress and level of learners’ achievement in learning (nortvedt & buchholtz, 2018; polly et al., 2017; suurtamm et al., 2016; veldhuis & van den heuvel-panhuizen, 2020). this argument implies that if teachers do not grasp the lis and sc, it will impede the consistent practice of assessment because assessment is determined by teachers’ understanding of the aim and objectives of a content. meaning equivalence reusable learning objects pedagogy approach since the 1990s, merlo has been a pedagogy and teaching technique developed, validated and experimented with within different countries and across different content areas and disciplines (etkind, shafrir, kenett, & roytman, 2016; etkind & shafrir, 2013; etkind et al., 2010). as a pedagogical tool, merlo is appropriate for different versions of core content based on sharing the meaning across different forms of representation (arzarello et al., 2015; robutti et al., 2016, 2020a, robutti, prodromou, & aldon, 2020b). additionally, merlo is a powerful tool for problem-solving mathematical concepts known as duplication obstacles, extensive in all mathematics classrooms. generally, merlo items are made up of five statements, namely an unmarked target statement (ts) and four other statements that are developed by sharing meaning equivalence with ts and sharing surface similarity with ts (etkind et al., 2016). the four quadrants are q1, q2, q3 and q4. the relevance of the four quadrants to the ts tries to identify learners’ needs in learning, which provides teachers with the opportunities to pay much attention to how they plan their lessons and design good, effective questions for mathematics teaching. it also leads learners to choose two representations of objects that share the same mathematical meaning in the questions illustrated. q1 is about the representation that shares meaning equivalence and surface similarity with the ts. q2 is about the representations that are not similar in appearance to the ts but share meaning equivalence with the ts. q3 focuses on the representation that is similar in appearance to the ts but does not share meaning equivalence with the ts. q4 focuses on the representation that is not similar in appearance to the ts, and does not share meaning equivalence with the ts. etkind et al. (2016) recommend that q1 statements be excluded as they are extremely straightforward and ‘give away the shared meaning due to the valence match between surface similarity and meaning equivalence, a strong indicator of shared meaning between a q1 and the target statement’ (p. 318). the main ts can be written in various semiotic or symbolic systems (text, image, map, decimal, percentage fraction and others). scholars have demonstrated that designing merlo patterns requires some steps to acquire different equivalence forms of representations that share the same mathematical meaning with the ts (arzarello et al., 2015; prodromou, 2015; robutti et al., 2016, 2020a, 2020b). according to arzarello et al. (2015) and prodromou (2015), the basis for designing merlo items is to identify a close link that relates to the concept because it needs a change from old-style questions into present-day questions. furthermore, the ts was designated as an open question because ‘teachers acquired the practice of elaborating the ts (i.e., ts) as a statement, graph, or table’ (arzarello et al., 2015, p. 5). several studies have indicated that in designing merlo items, there are some difficulties in choosing items that share the same mathematical meaning with the ts (robutti et al., 2016, 2020a, 2020b). this implies that teachers must choose representation items that are linked with one another before designing merlo activities. these steps are described below: designing and preparing the statement of ts and q2 requires the facilitators to select a concept from the core content in mathematics and write down the statement as a ts, then design one or more statements that share the same meaning with the ts in different representations (i.e., tables, texts, numbers, equations diagrams) to signify q2. to design and prepare the statement for q3, the facilitators have to use concepts that appear similar (surface similarity) to the ts but do not have the same meaning as the ts. this implies that q3 does not share the same meaning as ts and q2. to design and prepare the statement for q4 is different because it does not have the equivalence of meaning and surface similarity to the ts and q2. these characteristics imply that q4 does not share the same meaning as the ts and does not appear to be the same as the ts (robutti et al., 2016, 2020a, 2020b). theoretical framework the constructivist philosophy was embraced to underpin the study as the instructional process of this study was designed under the principles of constructivism. this is because the teacher facilitates a process of learning in which learners are encouraged to be responsible and autonomous in their learning. paying attention to the growing trend of engaging practising teachers under investigation, reis-jorge (2005) proposes that the goal is to educate reflective practitioners ‘who are more acquainted with theoretical discourse and more skilful readers of research literature’ (p. 303). as a result, we regard teachers’ engagement with research literature as an activity aimed at professional development through knowledge sharing among communities in mathematics education rather than a path to complete absorption in the research endeavour. this conceptualisation led us to structure our work with the meta-didactical transposition theoretical model (arzarello et al., 2014), which was developed to describe the complex dynamic that occurs when teachers and researchers interact with one another. meta-didactical transposition meta-diactical transposition (mdt) is highly relevant to describing present actions and interactions among researchers and mathematics teachers relating to the merlo pedagogy after participation in workshop training sessions (arzarello et al., 2014; robutti, 2018). the term meta-didactical ‘refers to the fact that important issues related to the didactical transposition of knowledge are faced at a meta-level’ (robutti, 2018, p. 4). this framework was suitable for describing teachers’ praxeologies such as teachers’ meta-didactical (i.e. teacher professional development) and didactical praxeologies (i.e. mathematics praxeologies) that consist of four interrelated components of the task, technique, technology and theory (arzarello et al., 2014; robutti et al., 2020a). the task and the corresponding techniques are discussed as the practical counterpart (i.e. the praxis), while the technology and theory, in the sense of justification, are the theoretical correspondence that uphold the use of those techniques (i.e. the logo). with the purpose of the study, mdt offers an interpretative model of mathematics teachers’ praxeologies. a mathematics praxeology is made of tasks requiring teachers to use questions and actions, representing a didactical praxeology. for instance, teachers need to choose a mathematical concept (i.e. fraction) in the caps documents or a question from a test as a ts and incorporate it to design any form of representation; teachers need to repeat the same mathematical concept with a different form of graph or number line area, by using statements that have the role of q2; teachers also need to complete the items with statements that have the role of q3 or q4, which act as a distractors or guessing (i.e. the practical component). the theoretical components made by various theoretical frameworks and aspects could justify the practical components: the merlo pedagogy approach (made by the main criteria of meaning equivalence and surface similarity) and the epistemic nature of the mathematical contents, which are intertwined with pedagogical, didactical, curriculum and assessment (chevallard, 2019; shinno & yanagimoto, 2020). this framework implies that the contribution from mathematics education research dealing with learners struggling to understand mathematics concept (i.e. fractions) is closely linked with the senior phase teachers’ direct experience in the classrooms. research methodology this study was part of a larger participatory action research (par) project, that draws on the paradigms of constructivism. according to scholars, par is constructionist and knowledge is socially created (armstrong, 2019; baldwin, 2012; florian & beaton, 2018). we used par because it builds up opportunities to empower and support participants to re-think and change their practices in the education sector; it focuses on social transformation that promotes democracy and combats inequity (chevalier & buckles, 2019; kemmis, mctaggart, & nixon, 2014; riel, 2019). this larger project investigated the impact of how merlo items were used in senior phase south african mathematics classrooms for teaching and learning. twelve senior phase mathematics teachers were purposively selected from six public schools due to the participants’ uniqueness in their qualities (i.e. mathematics teachers with at least two years teaching experience of teaching mathematics) (maree & pietersen, 2019). two senior phase mathematics teachers were chosen from six public schools. the reason for adopting a purposive non-probability sampling technique in the current study was due to the qualities of the teachers’ skills and knowledge. although 12 participants were initially part of this study, due to covid-19, eight dropped out after the critical evaluation of problem identification in terms of understanding teachers’ beliefs and practice of effectively communicating of lis and sc in the south african senior phase mathematics classroom, which left only five teachers in a merlo participation programme. the south african mathematics teachers were also provided with a handout about the merlo items and their relevance in education as teaching, learning and assessment. the handout was adopted from the research papers of arzarello et al. (2015) and robutti et al. (2016). they were further provided with some examples of how to construct the different statements of a variety of merlo items. subsequently, they were asked to design a merlo item that could be used to assess learners’ understanding of the learning results across their lesson plans. during the completion of their merlo items, teachers shared their views in the training sessions; they discussed the correctness of their merlo statement with the other teachers and the researchers (i.e. the facilitators). after the completion of designing the merlo items, the teachers were required to explain the design strategies and justify the series of the steps they followed to construct the merlo items that they would communicate or share to assess their learners’ conceptual understanding of mathematical concepts. the data of this study consist of merlo items designed by south african senior phase teachers, audio-tape recordings of teachers’ face-to-face merlo presentations in the classroom, audio-tape recordings of semi-structured interviews (pre and post interviews), field notes, reflective journals and the merlo handout for intervention (the content relates to the production of merlo and the explanations and the justifications of the steps they need to follow to design merlo questions). due to the covid-19 pandemic, stricter measures were put in place by each school to keep their learners safe. ideally, we wanted to observe each teacher at least three times, but due to the situation in schools (i.e. covid-19 cases), each teacher was observed at least twice. the purpose of the first observed lesson was to allow the teachers to develop a good knowledge and understanding of using the merlo pedagogy as a form of assessment activity in their classroom. during the first lesson, which was the first cycle of the class, we focused primarily on how the teachers applied the components that constituted the mathematical praxeologies with their learners, which related to teachers’ didactical praxeologies. at the end of each lesson, feedback was given to the teachers on the areas that needed improvement. the second lesson observation aimed at checking whether there was any progressive improvement in presenting the merlo pedagogy in class. during this stage, the researchers used audio-tape recordings and written notes to assess how the teachers present and communicate merlo pedagogy as a form of assessment activity in the teaching and learning of mathematical concepts. during teachers’ didactical praxeologies (i.e. classroom implementation), the teachers integrated the knowledge acquired from the merlo pedagogy involvement into their lesson plan to guide them when teaching the concepts. by the end of the second cycle of presenting and communicating merlo pedagogy in the classroom, only one of the teachers involved the researchers in the lesson to clearly explain merlo. he said, ‘please can you explain to learners for clarity’ (sch2-fob-mt2). thematic analysis was used because it provides a way to look for patterns in the data set, connecting into meaningful categories collectively and themes that represent the study being investigated (braun & clarke, 2019; dolgobrodova, 2016). the researchers of this study listened carefully to all the audio-tape recordings and analysed the transcribed data, including coding of teacher merlo implementation in the teaching and learning of mathematical concepts. the data includes pre and post semi-structured interviews, reflective journals and merlo items designed by the five senior phase south africa mathematics teachers and merlo classroom implementation in the mathematics classroom. the analysis was based on the study of research literature in terms of teachers’ beliefs and practices of assessment which informs how merlo items were effectively presented and communicated in the mathematics classroom in the south african context (i.e. the process corresponds to classroom observation). ensuring trustworthiness trustworthiness was ensured through data collected from several sources to corroborate the facts and multiple methods. trustworthiness was ensured by member checking: participants were given copies of their transcripts to confirm the accuracy (nieuwenhuis, 2019). ethical considerations approval to conduct this study was obtained from the ethics committee of the faculty of education, university of pretoria. pseudonymns were used to project participants’ identities; for example, sch1-print-mt1 stands for a male teacher from the first school and sch1-print-ft1 stands for a female teacher from the first school. anonymity and confidentiality were ensured by not revealing any identifying information of the participants. all participants signed consent forms indicating voluntary participation. the potential participant was assured that participation in the research is entirely voluntary and that they were free to withdraw at any moment. presentation and discussion of the data theme 1: concept of assessment and sharing of learning intentions and success criteria the finding of this study informed merlo intervention to enhance effective communication of lis and sc in the mathematics classroom. the data were presented through pre interviews, teachers’ lesson plans and classroom observations. the following themes emerged. sub-theme 1.1: teachers’ beliefs on assessment practices teachers’ beliefs about assessment practices in the classroom play a vital role, as most teachers consistently believe that assessment is used to check learners’ understanding at least most of the time during teaching. one teacher (sch1-print-mt1) mentioned that it is very challenging to effectively practise assessment in the classroom because of the pressure to finish the scheme of the subject that needs to be presented within a single lesson period: ‘it will be very difficult for you to come up with the real real real assessment in class because sometimes you are pressured by accomplishing of the lesson, you come with the plan that we have, the mental maths, the presentation of the subjects, the classwork and the homework which we must just complete them within a single period.’ (sch1-print-mt1) this shortcoming of teachers’ belief to effectively practise assessment in the classroom could contribute to the inconsistency of planning, preparing and implementation as studies reviewed that the achievement and failure of assessment practice are affected by the teacher’s belief in assessment practices (barnes et al., 2015; kanjee, 2020; karim, 2015). sub-theme 1.2: communicating learning intentions and success criteria this sub-theme is concerned with teachers’ understanding of communicating the lis and sc to the learners. pre interviews and classroom observations were used to determine whether the teachers communicated the lis and sc to their learners in the mathematics classroom. from the pre interviews, only two out of the 12 teachers (sch2-mt2 and sch6-mt2) mentioned that it is used to verify whether objectives for the lesson have been reached. the lis identify what the teacher wants the learners to know at the end of the lesson. furthermore, none of the teachers mentioned the sc. according to graham et al. (2021), the sharing and clarification of lis and sc by teachers and peers at the beginning of the mathematics lesson (to achieve an understanding of the lis and sc by the learners from the beginning of the lesson) is vital for assessment to achieve an effective result. one of the teachers gave the following response: ‘assessment is a tool that you use to check whether how far the learners learned. understanding has gotten. have you reached the outcome or have you reached the objectives rather than you planned to proceed into other chapters without really checking do they understand? so, assessment helps us check whether learners understand and whether we can proceed to the next level of learning.’ (sch2-print-mt2) informing and clearly stating the lis and sc to the learners in the classroom offers a strong starting point for teachers to plan their lesson with effective teaching techniques that support learners’ activities, integrate and assess their teaching, and aid learners in becoming independent in their learning (moss & brookhart, 2019). teachers were observed in the classroom to see whether they communicated the lis and sc with the learners. interestingly, none of the teachers communicated the lis and sc clearly with the learners when teaching, which identified a gap that needed to be addressed (bartlett, 2015; heritage, 2010; wiliam, 2011). the scaffolding activities are based on temporal planning to support learners in fostering their level of understanding in learning rather than responses that provide an opportunity to reach reactions from their own learning (aydeniz, 2009; nasr, bagheri, & sadighi, 2020; sadler, 1989). teachers’ teaching plans were examined to know whether the lis and sc are clearly stated. it is interesting to note that none of the teachers planned their lesson notes; instead, they depended on the annual teaching plan (atp) provided by the dbe. from the pre interviews, one of the teachers mentioned that planning for the lesson is difficult since the learners in the classroom are not always the same; he gave the following response: ‘yeah, but it’s always, hum, can be tricky to plan them because the classes are not always the same. [laughing]. they are not always the same, and the kids are not the same, and even on that same that you think you know it; the class’s mood is not always the same. yes. most of the time, because myself, as a teacher, when i am planning a lesson, i don’t plan it down to the detail right.’ (sch2-print-mt1) another teacher stated that his planning and preparation depend on the topics that would be communicated or shared with the learners in the mathematics lesson; he gave the following response: ‘my planning and preparation is based on the topic i want to be introduce to my learners during the period of the lesson’ (sch6-print-mt2). another teacher stated that limited time, overcrowded classes and work overload impeded them not to plan their lesson; he gave the following response: ‘okay, yeah. sometimes you will find out maybe you want to assess, but because of the overcrowded classes, lack of time and overloaded work, then you cannot plan for learners.’ (sch1-print-mt2) another teacher stated that the problem they have is time to make up their lesson plan: ‘it’s time. time is one of our biggest obstacles’ (sch1-print-mt1). based on their teaching plans, teachers’ responses identified some gaps that needed to be addressed in senior phase mathematics. more so, teachers could not effectively demonstrate the lis and sc in their lesson plans and mathematics teaching. according to moss and brookhart (2019), teachers need to carefully design a lesson plan that measures the quality of the instructional objective, which directs them to aid learners in accomplishing their learning goals. it is clearly seen from pre interviews, classroom observations and teaching plans that none of the teachers wrote out the lis and sc in their teaching plans. in addition, none of the teachers communicated the lis and sc with their learners during mathematics teachings. the fact that teachers did not plan or communicate the lis and sc with the learners could be due to inadequate training or skills to consistently plan and communicate the lis and sc with their learners. theme 2: development of mathematics teachers praxeologies the theme shows the development of the south african mathematics teachers’ praxeologies when designing and implementing merlo in senior phase classrooms. the data were presented through post semi-structured interviews, reflective journals and merlo items designed by the five senior phase south african mathematics teachers and merlo classroom implementation in the mathematics classroom. the following themes emerged. sub-theme 2.1: design of merlo items of senior phase mathematics praxeology the sub-theme analyses example of merlo items designed by senior phase south african mathematics teachers. in figure 1, we show an example of merlo objects that are framed to communicate lis and sc with learners in the mathematics classrooms. the data presented, which were created by senior phase teachers, also validated their merlo item selections during the question design process. figure 1: example of merlo assessment items, on fractions. the example is designed in the mathematics content area of fractions, and the question developed was in merlo patterns. the teachers mentioned that planning the topic of fractions was a result of learners that were struggling to understand the concept of fractions. one teacher provided the following viewpoint: ‘i believe that the pedagogical tool merlo is another way of presenting new mathematical concepts to the learners because one of the challenges we face is crossing the bridge from abstract to reality. you know mathematics is one of the difficult subjects to produces contents that means real life to the learners.’ (sch1-mt1) this view was supported by moyo and machaba (2021): ‘learners’ definitions of fraction were neither complete nor precise. particularly pertinent were challenges related to the concept of equivalent fractions that include fraction elements, namely the numerator and denominator in the phase of rational number.’ (p. 1) during the merlo implementation time, the mathematical concepts of fraction were also included in their weekly teaching plan. the lesson presentation was done during regular mathematics periods, and the process is discussed in sub-theme 2.2. sub-theme 2.2: effective communication of learning intentions and sharing success criteria with the learners through mathematics praxeology this sub-theme analysed the introduction, presentation and communication of lis and sc with their learners through teachers’ mathematics praxeology. the merlo assessment items that focused on mathematics praxeology were on the topic of fractions (see figure 1). teachers introduced and presented a layout of the lesson and conveyed the lis. teachers explained and presented the content of the lesson and demonstrated the knowledge and skills obtained during the merlo lesson. teachers also explained and related the concept of fractions on the board by using the merlo pattern with an equivalent form in a different representation. teachers explained the terminologies associated with merlo, such as ts, surface similarity, meaning equivalence and quadrants (arzarello et al., 2015; prodromou, 2015; robutti et al., 2016, 2020a, 2020b). the following comment is one of the teachers’ interactions in the classroom: ‘i want to do with you the following example. but in our case, because we are following a new method, in our question we are going to call it a target statement. so, our target statement, in this case, it is .’ (sch2-ft2) the implication of this analysis was that the teachers introduce the task on the topic of fractions to their learners. the task was further linked to the use of the merlo pedagogy (technique) by following the criteria of meaning equivalence and surface similarity. afterwards, the teacher explained that q1 would not be included when solving the merlo question because they are straightforward and effortless to understand. teachers explained that when starting a merlo question, q2 would be used. teachers seemed pleased that learners were actively involved in class and understood the lesson. teachers demonstrated that as q stands for quadrant and that q2a represents the decimal fraction of 0.75. teachers demonstrated that another q2 example, as well as q3 and q4 examples, would be illustrated: ‘so here i want us to look ehm, at the board, i want us ehm, i want to call this one because this one and this one, they are written differently. this one is my opening statement which is my target statement. so, i want to call this statement 0.75; i want to call it q2a. if you look at here in your paper, there is q1; there is q2, there is q3, there is q4, right. we are not going to use q1 because q1 those are very simple questions. so, what we are doing now, we are only beginning at question q2, not only question q2 but at quadrant 2 let us call it like that. because the questions there it has, but i am very happy you people are actually following and understand. so that why i am calling this one 0.75 now because 0.75 is going to make my q2a. so, i want to make another q2, i want to make another example of q2 then i will also make an example of q3 and q4.’ the presentation of mathematics praxeology in the classroom, which is the ‘technique’, indicates that teachers further explained that the first q2 example was indicated as q2a, but another q2 example would be indicated as q2b. teachers asked learners to look at the board and identify whether the ts of the representation of has a similarity with the picture in q2b. most learners were able to identify that the representation of the ts has no surface similarity with q2b, which represents in a diagram. teachers asked learners to give a reason why they are not fractions, and the representation of q2b is in the form of a picture. another learner said that the representation of the ts is written as a fraction, and the representation of q2b is in the form of a pie chart. the teacher showed an appreciative word to learners by saying ‘very good, they are not similar’. the teacher encouraged class participation as she asked learners to identify whether the representation of the ts has the same mathematical meaning as the diagram in q2b. most of the learners answered ‘yes, they have the same mathematical meaning’. teachers appeared keen and enthusiastic about how learners were actively involved in answering the question. teachers also asked learners to represent the diagram in q2b in the form of a fraction. the majority of the learners indicated that the diagram in q2b is a representation of . teachers demonstrated on the board that the representation of the picture form and the representation of a fraction have the same mathematical meaning, but they have no surface similarity (arzarello et al., 2015; prodromou, 2015; robutti et al., 2016, 2020a, 2020b). the conversation between the teacher and the learners is given below: teacher: so am repeating q2, i am saying, i am giving you something like this because that one is q2a, but i am giving you q2b, right. teacher: again, am looking at my target statement and my q2b, i am saying, are this two similar? learner 1: no. learner 2: no. teacher: why they are not similar? learner 1: because is written in fraction and other is in a picture. learner 2: because is written in fraction and the other one is written in a pie. teacher: but are they having the same mathematical meaning? learners: yes, they have mathematical meaning. teachers: okay, i like the yes answer. so, what is the meaning of this diagram in q2b in a fraction way? learners: it’s . teacher: can you see, it means this one is written in a picture form, and this one is written as a fraction, so they have the same mathematical meaning, but the similarity does not exist. the teacher demonstrated the q3 example on the board and stated that q3 represents . learners were asked to interpret the representation of and the representation of the ts and to identify whether the representation of and the ts appeared the same. most of the learners were able to identify that the representation of and the target statement look the same (i.e. surface similarity). the majority of the learners were able to interpret that the representation of and the ts are written in an equivalent form of a fraction. the teacher also asked learners whether the representation of and the ts have the same mathematical meaning. most of the learners were able to identify that the representation of and the ts do not have the same mathematical meaning. teachers explained that the representation of and the ts do not have the same mathematical meaning because three-quarters and one-quarter are not the same (arzarello et al., 2015; prodromou, 2015; robutti et al., 2016, 2020a, 2020b). the exchanges between the teacher and the learners are given below: teacher: i am going to make another example; this one is q3. can you relate now to our open question, which we called our target statement? let us relate this two. are these two seem similar? learners: yes, they are similar. teacher: why? learners: because they are both written in fractions. teacher: are they having the same mathematical meaning? learners: no, they don’t have the same mathematical meaning? teacher: no right, because this one has one quarter and the other one has three quarters so they do not mean the same. teachers demonstrated a q4 example on the board. teachers said to learners that q4 represents 50%. learners were asked to interpret the representation of 50% in relation to the representation of the ts and to identify whether the representation of 50% and the ts have surface similarity. most of the learners were able to identify that the representation of 50% and the ts do not look the same (i.e. have no surface similarity). learner 1 said that the numbers are different. another learner said that the ts is written in a fraction form, and the representation of 50% is written as a percentage. some of the learners did not understand the merlo items but the teachers re-explained (arzarello et al., 2015; prodromou, 2015; robutti et al., 2016, 2020a, 2020b). the statements require learners to identify fractions with equal value from different representation objects. the conversation between the teacher and the learners is given below: teacher: now let us look at q4, i am writing her 50%. let’s relate our q4 here to our target statement. are these two similar? is and 50% similar? learners: no, they are not similar. teacher: why they are not similar? learner 1: because the numbers are different. learner 2: because is in fraction and 50% is in percentage. teachers also asked learners whether the representation of 50% and the ts have the same mathematical meaning. most of the learners were able to identify that the representation of 50% and the ts do not have the same mathematical meaning. teachers made it clear in an explanation that the representation of 50% and the ts do not have the same mathematical meaning and do not have surface similarity (arzarello et al., 2015; prodromou, 2015; robutti et al., 2016, 2020a, 2020b). the conversation between the teacher and the learners is given below: teacher: what about the mathematical meaning, did it carry the same meaning? learners: no, they do not have the same meaning. teacher: so both numbers do not have similarity and do not have the same meaning. this is q4 means similarity is no, the same meaning is no. so, the question that says no/no, they fall on the categories of q4. based on the teacher’s mathematics praxeology (i.e. merlo class implementation), the teacher re-explained how a merlo question should be answered. teachers demonstrated the process of relating q2, q3 and q4 to the ts (i.e. open question) by saying: ‘this is q4, q4 means similarity is no, the same meaning is no. so the question that says no/no, they fall on the categories of q4. let us go to q3; in q3, we say the similarity is there (i.e. yes) but not the same meaning (i.e. you say it’s no).’ (arzarello et al., 2015; prodromou, 2015; robutti et al., 2016, 2020a, 2020b) learners asked questions that linked to the content of the lesson in the class. teachers also politely worked around the class to check learners’ activities. this was done to allow learners to better understand what the teacher expects them to know, understand, or be able to do at the end of the lesson. this was also done to establish the sc. other teaching skills demonstrated by the teachers during the didactic delivery were learners’ involvement, classroom management, and addressing learners who are having learning difficulties and providing immediate feedback to the learners. the implication of the preceding analysis is that mathematics praxeology is made up of a task which consists of a problem that learners must solve (for example, learners might be asked to convert the representation of fractions to decimals), the technique used, and the more-or-less clear reason for applying it in the mathematics classroom. according to the mdt framework, the south african mathematics teacher’s praxeology consistently follows the guiding principles when designing merlo items. it seems that the design of merlo assessment items process follows the sequential order of ts-q2-q3-q4. teachers’ mathematics praxeology that is linked to merlo didactical praxeology provides insight into learners’ conceptual reasoning about fractions by designing an activity that focuses on identifying learners’ strengths and weaknesses. furthermore, the ts was designated as an open question because ‘teachers acquired the practice of elaborating the ts (i.e., ts) as a statement, graph, or table’ (arzarello et al., 2015, p. 5). according to arzarello et al. (2015) and prodromou (2015), the basis for designing merlo items is to identify a close link that relates to the concept because it needs a change from old-style interactions into present-day interactions. conclusions and recommendations for future research this study was part of a larger par project aimed at investigating how merlo pedagogy as an assessment strategy can be used to understand teachers’ beliefs and practice of effectively communicating of lis and sc in the south african senior phase mathematics classroom. this approach aimed at developing teachers’ adequate knowledge and skills to design merlo assessment items independently. due to the small sample size, transferability to the broader population is not possible. at the initial stage, teachers were examined with regard to their beliefs and assessment practices in the classroom, which informed merlo intervention. in the second stage, teachers were asked to learn about merlo items by reading the merlo handout provided to them, participating in the workshop, and sharing their opinions and views with others. in the third stage, teachers had to design merlo assessment items on their own to assess learners’ level of understanding of the mathematical concept in senior phase. we observed from the teachers’ mathematical praxeology that, even though teachers from different school contexts had separately worked on designing merlo assessment items, they progressively introduced and communicated similar mathematics praxeology which relates to didactical knowledge with their learners in the classroom. for instance, teachers were able to design and integrate merlo pedagogy across their lesson plan through the practical component, as well as the theoretical component. we assume that sharing ideas and experiences with others can be justified by invariants of mathematical objects, which remained unchanged after the involvement of designing merlo assessment items, which appears the same in various countries, and the design of merlo items was found in all groups because the items possess the same criteria in terms of meaning equivalence and surface similarity. these findings imply that merlo items can be employed in multiple nations as well as other course syllabi by modifying them to institutional contexts while holding on to their significant structure. the study further suggests the need for a professional learning programme based on merlo technology in grades 4–12, post-secondary institutions, and public and private contexts (etkind et al., 2016; shafrir, 2020). future research could involve having more teachers in the south african context due to the small sample size. acknowledgements competing interests the authors declare that they have no financial or personal relationships that may have inappropriately influenced them in writing this article. authors’ contributions l.o.a. is a doctoral student at the university of pretoria and conceptualised the idea and carried out the research. m.a.g. was l.o.a.’s supervisor for her doctoral degree and, as sole supervisor, was involved with all steps in the research. the themes generated in this article were discussed in detail between the authors to further ensure trustworthiness. ethical considerations ethical clearance was obtained from the university of pretoria research ethics committe sm 19/05/01. funding information m.a.g.’s research was funded by the national research foundation (nrf) [reference: csrp190415430728, grant number: 120401]. data availability the transcriptions are not made available publicly to protect the participants; this is done to ensure anonymity and confidentiality. disclaimer the views and opinions expressed in this article are those of the authors and do not necessarily reflect the official policy or position of any affiliated agency of the authors. references adesanya, l., & graham, m.a. 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(2016). leadership for teacher learning. moorabbin, australia: hawker brownlow education. footnote 1. the senior phase in the south african schooling system is grades 7 to 9; the interested reader is referred to southafricaeducation.info (2021) for a layout of the south african education structure. 66 p43-51 samson & schafer final pythagoras 66, december, 2007, pp. 43-51 43 an analysis of the influence of question design on learners’ approaches to number pattern generalisation tasks duncan samson and marc schäfer rhodes university email: d.samson@sacschool.com and m.schafer@ru.ac.za this paper reports on a study of the extent to which question design affects the solution strategies adopted by children when solving linear number pattern generalisation tasks presented in pictorial and numeric contexts. the research tool comprised a series of 22 pencil-and-paper exercises based on linear generalisation tasks set in both numeric and two-dimensional pictorial contexts. the responses to these linear generalisation questions were classified by means of stage descriptors as well as stage modifiers. the method or strategy adopted was analysed and classified into one of seven categories. in addition, a meta-analysis focused on the formula derived for the nth term in conjunction with its justification. the results of this study strongly support the notion that question design can play a critical role in influencing learners’ choice of strategy and level of attainment when solving pattern generalisation tasks. an understanding of the importance of appropriate question design has direct pedagogical application within the context of the mathematics classroom. introduction and background the connection between mathematics and the notion of pattern is prevalent at all levels of mathematical endeavour. goldin (2002: 197) describes mathematics as “the systematic description and study of pattern” while sandefur and camp (2004: 211) suggest that patterns are “the very essence of mathematics, the language in which it is expressed.” perhaps more generalised and all-encompassing, steen (1988: 616) broadly defines mathematics as “the science of patterns.” pattern, in a broad sense of the word, is by no means restricted to numeric or pictorial patterns, although this is the usual context of the word for most school mathematics syllabi. working with number patterns or number sequences in the classroom offers valuable opportunities for recognising, describing, extending and creating patterns (hargreaves, threlfall, frobisher & shorrocks-taylor, 1999: 67). it has been suggested that these processes have considerable value as a precursor to formal algebra (english & warren, 1998). searching for patterns is also an important strategy for mathematical problem-solving (stacey, 1989: 147). furthermore, in their seminal paper on an organising principle for mathematics curricula, cuoco, goldenberg and mark (1996) identify the search for pattern as a critical habit of mind. the study of pattern has become an integral component across all grades of the south african school mathematics curriculum (department of education, 2002; department of education, 2003b). in the intermediate phase (grades 4-6) the importance of number pattern activities is in “laying the foundation for the study of formal algebra in the senior phase while at the same time developing important mathematical thinking skills” (department of education, 2003a: 37). number pattern activities in the senior phase (grades 7-9) are essentially an extension of the intermediate phase. however, in grades 8 and 9 there is an expectation that learners “use algebra and algebraic processes in their description of these patterns” (department of education, 2003a: 39). within the further education and training (fet) band (grades 10-12) learners will “solve problems related to arithmetic, geometric and other sequences and series” as well as “explore real-life and purely mathematical number patterns and problems which develop the ability to generalise, justify and prove” (department of education, 2003b: 12). there are a variety of different number patterns which fall under the above framework, including: linear or arithmetic sequences, quadratic sequences, power sequences, geometric sequences, and fibonacci-type sequences. while number patterns can be explored purely numerically – namely, in terms of patterns presented as a sequence of numerical symbols – implicit in the requirement that learners be able to “provide explanations and justifications and attempt to prove conjectures” (department of education, 2003b: 18) is the condition that at least some of the an analysis of the influence of question design on learners’ approaches to number pattern generalisation tasks 44 pattern questions be set in non-numeric or pictorial contexts. there are numerous pictorial and practical contexts in which pattern questions can be set, among the most obvious being dot patterns, tiling patterns, matchstick patterns as well as twoand three-dimensional building block patterns. such pattern tasks usually require some form of generalisation of the pattern, usually in terms of algebraic symbols. it can be argued that setting pattern questions within a pictorial context should allow for greater scope in terms of learner problem-solving strategies, since a pictorial representation can readily be reduced to a purely numeric equivalent, provided the pictorial context has been meaningfully understood. however, although pattern problems presented in a pictorial and/or practical context have the potential to widen the scope of solution strategies for some learners, it can be argued that for others this may well create additional complications. an understanding of how question design of such pattern generalisation tasks is likely to influence the approach adopted by children would greatly assist teachers in terms of their choice of such activities. it is within this context that this paper finds both impetus and import. theoretical framework while embracing the basic tenets of constructivism, central to the broader study is the fundamental notion that constructivism is a descriptive rather than prescriptive philosophy (towers & davis, 2002: 314). built onto this philosophy is the firm belief in the use of both language and notation systems/representations as important mediators in the process of knowledge construction – both in terms of their contribution to the organisation of the thinking process itself, as well as the cyclical nature of reflection (kaput, 1991). the role of visualisation is also a central focus of the broader study. while generalisation problems presented in a pictorial or practical context have the potential to widen the scope of solution strategies for some learners, it is acknowledged that this may well create additional complications for others (orton, orton & roper, 1999). the types of generalisation activities considered purposefully in this paper include those presented in pictorial contexts, thus allowing for a possible connection to a referential context that has the potential to aid and enhance the generalisation process. within the context of the broader study, justification is seen to play a central role in communication of mathematical understanding. learners’ justifications of their generalisations are seen to provide “…a window to view their understanding of the general nature of their rules” (lannin, 2005: 251). methodology and data generation this paper focuses on more quantitative aspects within the broader study, which is based on a qualitative investigation framed within an interpretive paradigm. it attempts to interrogate learners’ responses to various linear generalisation tasks from both a technical as well as strategic viewpoint. more specifically, the research question under consideration is: to what extent does question design influence (a) strategy choice, (b) stage progress, (c) contextual connectivity – the extent to which the justification of the general term makes reference to the pictorial context – and (d) the diversity of expressions for the general term? the essential character underpinning the data analysis is the treatment of all responses, particularly those that are unexpected or idiosyncratic, with a genuine interest in understanding their character and origins – a firm conviction that “the constructions of others … have integrity and sensibility within another’s framework” (confrey, 1990: 108). a case study methodological strategy was adopted and an appropriate group of research participants was identified – the members of a mixed gender, high ability grade 9 class of 24 learners at an independent school in grahamstown, south africa. over a period of three months, the 24 research participants each completed a series of 22 pencil-and-paper exercises based on linear generalisation tasks set in both numeric and two 4 ; 7 ; 10 ; 13 ; . . . figure 1. number sequences. position 1 2 3 4 . . . number 3 5 7 9 . . . figure 2. tabular form. duncan samson and marc schäfer 45 dimensional pictorial contexts. for each pattern, participants were required to provide numerical values for the ‘next’, 10th and 50th terms (stages 1, 2 and 3), as well as a written articulation of their reasoning at each stage. participants were also asked to provide an algebraic expression for the nth term (stage 4), and provide a justification for their expression. the structure of the six different question design formats was guided by insights gleaned from the literature review (see, for example, stacey, 1989; orton, 1997). these six formats are summarised below, where the dependent variable refers to the numerical value of the term itself, while the independent variable refers to the position of the term in the sequence. • questions 1-5: a single pictorial term in which the underlying structure is unambiguous. both dependent and independent variable mentioned in the context of the picture. • questions 6–10: two non-consecutive pictorial terms. both dependent and independent variable mentioned in the context of the picture. • questions 11–13: three consecutive purely pictorial terms. • questions 14–16: three consecutive pictorial terms with numerical value of dependent variable indicated. • questions 17–19: three consecutive purely numeric terms (dependent variable indicated). • questions 20–22: three consecutive purely numeric terms in table format (dependent and independent variables indicated). the responses to the various linear generalisation questions were classified by means of stage descriptors as well as stage modifiers. the method or strategy adopted for determining each of the ‘next’, 10th and 50th terms was carefully analysed and classified into one of seven categories. in addition, a separate framework was used to characterise each learner’s justification of the nth term in terms of the extent to which the justification was linked to the pictorial context. numeric patterns were presented as a simple sequence of numbers (figure 1) as well as in tabular form (figure 2). pictorial patterns were presented using three consecutive terms (figure 3), two non-consecutive terms (figure 4) or one single term (figure 5). the use of single terms was restricted to cases where a single pictorial term provides an unambiguous explanation of the 3 houses2 houses1 house figure 3. three consecutive terms 7 squares require 22 matches 3 squares require 10 matches figure 4. two non-consecutive terms the diagram shows a fence containing 5 upright poles and 12 horizontal rails figure 5. a single pictorial term an analysis of the influence of question design on learners’ approaches to number pattern generalisation tasks 46 underlying structure. the literature review undertaken to inform the broader study suggested that linear sequences would be most appropriate in terms of eliciting rich data at all levels of the pattern generalisation process. accordingly, 22 linear/arithmetic sequences of the type cax ± )0( ≠c were chosen. the choice of sequences with non-zero constant terms was a purposeful attempt at ensuring that choice of an inappropriate strategy would not produce a spurious yet numerically correct answer. the 22 sequences were split between pictorial and non-pictorial contexts. data analysis (a) stage classification for each of the 22 questions, participants were asked to provide numeric values for the ‘next’, 10th and 50th terms, as well as an algebraic representation for the nth term. using the nomenclature of stacey (1989: 150), the 10th and 50th terms represent “near generalisation” and “far generalisation” tasks respectively. determin-ing the 10th term thus represents a task which can be accomplished by means of step-by-step counting or drawing, while determining the 50th term represents a task which goes beyond reasonable practical limits of such a step-by-step approach. the nth term denotes an algebraic generalisation of the pattern. stage descriptors and stage modifiers were used to classify the responses to the various linear generalisation questions. a similar model was used to that employed by orton and orton (1996; 1999). the various stage descriptors can be summarised as follows: • stage 0: no progress • stage 1: ‘next’ term correctly provided • stage 2: ‘next’ and 10th terms correctly provided • stage 3: ‘next’, 10th and 50th terms correctly provided • stage 4: ‘next’, 10th, 50th and nth terms correctively provided. the above scheme is not intended as a hierarchical classification system, but rather as a qualitative framework for analysis. thus, since it is possible for a learner to correctly determine the 50th term despite having incorrectly determined the 10th term (for example), stage modifiers were used to cover all possibilities. the use of both stage descriptors as well as stage modifiers thus allowed for both a quantitative as well as qualitative description of the level of attainment of each participant for each pattern generalisation task. (b) strategy classification the method or strategy adopted for determining each of the ‘next’, 10th and 50th terms was carefully analysed and classified into one of seven categories, namely: counting, chunking, difference product, explicit, whole-object uncorrected, wholeobject corrected, and the nature of numerical terms. the counting method (or method of successive addition) represents a recursive approach whereby subsequent terms are determined by successively adding the identified constant difference to previous terms. the explicit method refers to a strategy where a general formula is first derived for the nth term and the desired term is then calculated directly from the general formula by using the independent variable, namely, the position of the term. provided the general term has been correctly formulated, the explicit method will yield any number of algebraically equivalent expressions for the nth term. (c) justification classification in each question, learners were asked to justify their general formula, that is, to explain why their formula for the nth term works. an important aspect of the justification process was an analysis of the extent to which learners used the pictorial scenario as a referential context for the use of a generic example in their justification of the general term. to this end, responses were rated in terms of whether or not the justification was specifically linked to the pictorial – rather than numerical – context, using a contextual connectivity rating (ccr). only those questions that had a pictorial element (questions 1–16) were rated. scores of 1, ½ or 0 were awarded depending on the extent to which the pictorial context featured in the justification. results, analysis and discussion each of the 22 pattern generalisation tasks used in the broader study fell into one of the six different question design formats. analysis of the influence of question design on (a) strategy choice, (b) stage progress, (c) contextual connectivity, and (d) the diversity of expressions for the general term is discussed in detail below. duncan samson and marc schäfer 47 (a) influence of question design on strategy macgregor and stacey (1993) cite one of the main causes of difficulty in formulating algebraic rules as being learners’ tendency to focus on the recursive patterns of one variable rather than the relationship linking the two variables. similar observations have been made by other researchers, for example, orton (1997). this part of the analysis focuses on the extent to which question design either attracts or discourages a recursive approach. the counting strategy (recursive approach) was used in one of two different modes, either (a) on its own as sole strategy, or (b) in combination with an explicit strategy. table 1 shows the percentage of total responses using a counting strategy (as sole strategy) for stages 1, 2 and 3. the value under the “total” column indicates the number of responses using the counting strategy as a percentage of the total responses (using any strategy) for stages 1, 2 and 3 combined. the rationale behind considering only those responses that used counting as the sole strategy was the fact that when counting and explicit strategies were used in combination, the counting strategy was used simply to check the answer derived from the explicit strategy, and was thus not critical to a correct response at that stage. table 1 reveals some interesting trends. there is a dramatic drop in the number of learners using the counting strategy when two non-consecutive pictorial terms are used instead of one single pictorial term. there could be two possible reasons for this. firstly, a single pictorial term may not be a sufficient scaffold to enable some learners to derive a general expression. a second diagram, physically drawn by the learner, may have been necessary in order to see the general structure underlying the pictorial context. thus, using a counting strategy at stage 1 may have been a necessary prerequisite to moving to an explicit strategy at stage 2. secondly, questions that incorporated two non-consecutive pictorial terms tended to have slightly bigger physical structures compared to the single term scenario, and drawing the next diagram in such a case may have been considered impractical by some learners. there is a dramatic increase in the number of learners using the counting strategy when three consecutive purely pictorial terms are used instead of two non-consecutive pictorial terms. this increase is even more pronounced when the three consecutive terms are accompanied by an indication of the dependent variable. the initial increase could be a result of two possibilities. firstly, the fact that the three consecutive pictorial terms are the first three terms in the sequence, the physical structures of the pictorial representations are a little less complex than in the case of the two non-consecutive terms. this may have encouraged learners simply to draw the next term rather than looking for an explicit strategy. secondly, because the three consecutive terms give a physical stage questions next (stage 1) 10th (stage 2) 50th (stage 3) total 1–5: single pictorial term; dependent and independent variable mentioned. 36.7 % 2.5 % 0.0 % 13.1 % 6–10: two non-consecutive pictorial terms; dependent and independent variable mentioned. 17.5 % 0.0 % 0.0 % 5.8 % 11–13: three consecutive purely pictorial terms. 38. 9 % 2.8 % 0.0 % 13.9 % 14–16: three consecutive pictorial terms; numerical value of dependent variable indicated. 47.2 % 2.8 % 0.0 % 16.6 % 17–19: three consecutive purely numeric terms; dependent variable indicated. 51.4 % 8.3 % 0.0 % 19.9 % 20–22: three consecutive purely numeric terms in table format; dependent and independent variables indicated. 40.3 % 1.4 % 0.0 % 13.9 % table 1. percentage of total responses using counting as sole strategy. an analysis of the influence of question design on learners’ approaches to number pattern generalisation tasks 48 representation of growth, learners may have been drawn to the recursive nature of the pattern and simply added the common difference to the third term in order to obtain a numerical value for the next term. this seemed to be slightly more often the case than simply drawing the next term and counting the number of elements. the even greater increase when the three consecutive terms are accompanied by an indication of the dependent variable can be explained in terms of the common difference having been made somewhat more explicit by the inclusion of the dependent variable. learners were thus drawn particularly towards a recursive strategy. the simple presentation of three consecutive purely numeric terms resulted in the highest proportion of learners opting for the recursive strategy. just over 51% of all responses at stage 1 made use of the counting strategy in the three questions (17–19) presented in this format. furthermore, just over 8% of the responses at stage 2 also made use of the counting strategy, far more than in any other question design. once again, the common difference becomes immediately clear from the given terms, and learners seem to have been drawn towards this, and used a recursive approach as a result. interestingly, when the three consecutive numeric terms are put into table format, which necessarily includes the independent variable, there is a slight drop in the tendency to pattern recursively. one can only surmise that the explicit presence of both dependent and independent variables assisted some learners in seeing a general relation between the two and hence being more inclined to use an explicit strategy over a recursive approach. the above observations lend support to the findings of hershkowitz, dreyfus, ben-zvi, friedlander, hadas and resnick (2002), that the presentation of consecutive terms encourages recursion, while terms presented non-consecutively tend to encourage generalisation by means of the independent variable. hershkowitz et al. (ibid.) also found that the use of a pictorial context, particularly if non-consecutive terms were presented, tends to encourage explicit generalisations. (b) influence of question design on stage progress table 2 shows the average total stage attainment (tsa) values for each of the six different question designs. the average tsa values are indicative of the level of attainment/progress made by the research participants as a whole. the tsa value was calculated for each individual question by awarding 1 point for a correct stage 1 response, 2 points for a correct stage 2 response, 3 points for a correct stage 3 response, and 4 points for a correct stage 4 response. the highest obtainable score for a single question is thus 10 (1+2+3+4) for a learner who correctly answered all four stages. although the majority of the average tsa values lie fairly close to one another, of interest are the highest and lowest values, which are well distanced from the rest of the cluster. the highest level of attainment (average tsa = 9.32) was achieved on those questions presented purely numerically, in tabular format. the explicit presence of both the dependent and independent variable, along with the fact that the terms were consecutive and hence made the common difference easier to recognise, all seem to have allowed for greater overall attainment. this finds resonance with a study by english and warren (1998) where students found it easier to generalise, both verbally and symbolically, when patterns were presented in tabular form as opposed to pictorial form. the lowest level of attainment (average tsa = 7.97) was achieved on those questions presented as three consecutive purely pictorial terms. in these questions, no mention was made of either the dependent or independent variable. this is an interesting observation when taken in conjunction with the adopted strategy. question designs that make use of (a) three consecutive purely pictorial terms (questions 11–13), and (b) three consecutive purely numeric terms in tabular format (questions 20–22) show almost identical values for the percentage of total responses using an explicit strategy (85.2% vs. 84.7%). however, there is a questions average tsa 1 – 5 8.91 6 – 10 8.98 11 – 13 7.97 14 – 16 8.68 17 – 19 8.75 20 – 22 9.32 table 2. average tsa per question type. duncan samson and marc schäfer 49 marked difference in level of attainment in these two question types (7.97 for the former, 9.32 for the latter). this adds weight to the notion that a pictorial representation is only of benefit if the underlying structure can be clearly seen. despite the fact that learners made almost equal use of an explicit strategy in the two question types, the lower level of success in the purely pictorial context would seem to suggest the use of explicit strategies based on misinterpretation of the general structure inherent in the pictorial context. thus, while a purely pictorial context may be useful to some learners, to others it may well create complications. a contextualised indication of both the dependent and independent variable (for example: for 2 squares you will need 7 matchsticks), in conjunction with the pictorial representation (questions 1–5 and 6–10) seemed to be most successful in alleviating this problem. (c) influence of question design on contextual connectivity table 3 shows the average contextual connectivity rating (ccr) for each of the four different question designs that were based on a pictorial context (questions 1–16). the ccr is indicative of the extent to which the justification of the general term makes reference to the pictorial context. the results shown in table 3 reveal a fascinating trend. the effect of presenting consecutive terms (questions 11–13 and 14–16) seems to have a big influence on moving learners’ nt justifications away from the referential context (the pictorial representation) toward a more numerically based argument. this effect is even more pronounced in those questions (14–16) where the pictorial context is presented in conjunction with values for the dependent variable. the most likely explanation for this observation is that consecutive terms attract attention to the common difference, hence away from the underlying general structure inherent in the pictorial context, and thus to a more numeric approach to extracting and justifying the general formula for nt . there is also a slight decrease in the average ccr value when moving from questions involving a single pictorial term (questions 1-5) to those making use of two non-consecutive pictorial terms (questions 6-10). it is worth keeping in mind that both these question types make contextualised reference to both the dependent and independent variables. thus, the slight decrease can probably be ascribed to the presence of more numeric points of reference. (d) influence of question design on diversity of expressions for nt table 4 shows the average number of nt variations per question type. this gives an indication of the diversity of responses in formulating a general algebraic expression for the nth term. only correct stage 4 (i.e. nt ) responses have been considered. the dramatic drop in the number of correct nt variations for those questions incorporating purely numeric terms is both expected and understandable, since the lack of a referential (pictorial) context severely limits the scope of readily identifiable variations in nt . without a pictorial frame of reference, expressions for nt can only be derived from purely numeric considerations, the resulting expressions usually taking the form dna )1( −+ or )( dadn −+ , or those deriving fortuitously from a guess-and-check approach. questions average ccr 1 – 5 0.86 6 – 10 0.80 11 – 13 0.64 14 – 16 0.40 17 – 19 20 – 22 table 3. average ccr per question type. questions average number of correct nt variations 1 – 5 3.6 6 – 10 6.2 11 – 13 4.7 14 – 16 6.0 17 – 19 2.7 20 – 22 2.3 table 4. average number of nt variations per question type. an analysis of the influence of question design on learners’ approaches to number pattern generalisation tasks 50 the increase in the number of correct nt variations when moving from questions involving a single pictorial term (questions 1–5) to those making use of two non-consecutive pictorial terms (questions 6–10) can probably be ascribed to learners’ enhanced appreciation of the underlying general structure inherent in the pictorial context as a result of the additional term. the same argument could be applied when moving from two pictorial terms (questions 6–10) to three pictorial terms (questions 11–13 and 14–16). the value of 4.7 (questions 11–13) is thus somewhat anomalous, and is probably a result of the specific questions chosen for that particular design type. responses to stage 4 in question 13 gave rise to seven different nt variations, while question 11 and question 12 had only 4 and 3 respectively. it is worth bearing in mind that some pictorial designs yield fewer accessible (easily identifiable) expressions for nt , and this is likely to have been the case in this situation. comparing the number of correct nt variations per question type with average ccr values should be treated with extreme caution. there is no reason to assume that a high ccr value implies a high diversity of nt variations. the ccr value relates to the contextualisation of the justification for the nth term, but the justification itself is not necessarily an indication of the approach used to derive the algebraic expression for nt . it is thus hardly surprising that there is little correlation between the average ccr values per question type and the average number of nt variations per question type. conclusion this paper is based on a broader study in which a case study approach was adopted as a methodological strategy. although the emphasis of a case study is to optimise understanding of the specific case under scrutiny rather than generalisation beyond that case, a case study can nonetheless be a useful small step towards a larger generalisation, or an increasingly refined generalisation (stake, 1994 & 1995; cohen & manion, 1994). thus, although any general trends or patterns observed are only relevant to the group of 24 research participants who took part in the study, such “generalisations” could be broadened or increasingly refined by future research involving further samples from the larger population. learners’ responses gave evidence of the complex interplay between the number pattern itself, the nature of the question design and the specific numeric/pictorial context chosen. choice of strategy, level of stage progression, contextual connectivity, and the diversity of nt expressions are a manifestation of this interwoven complexity in conjunction with the diverse cognitive skills of each individual learner. there is thus a high degree of interconnectedness, and correlations between different aspects should be treated with due circumspection. the emphasis of the national curriculum statement (ncs) on investigation as a pedagogical approach to number pattern generalisation tasks, as well as its requirement that learners be able to investigate number patterns and hence “make conjectures and generalisations” as well as “provide explanations and justifications and attempt to prove conjectures” (department of education, 2003b: 18), has important pedagogical implications for classroom practitioners. an understanding of how question design of such pattern generalisation tasks is likely to influence the approach adopted by children would greatly assist teachers in terms of their choice of such activities. it is within this pedagogical context that this paper finds practical significance. the results highlighted in this paper give strong support to the notion that question design can play a key role in influencing which strategies are adopted by learners when solving pattern generalisation tasks, in both pictorial and purely numeric contexts. this observation is central to the theme of the broader study, and the notion that different contexts – numeric versus pictorial – will resonate differently with different learners. while a pictorial context may be helpful to some learners, for others it may simply create additional complications. it would be interesting to repeat this study with other high ability groups of learners, possibly with an augmented selection of patterning questions. this would serve to broaden and/or increasingly refine any localised “generalisations” identified in this paper. in addition, it would add further insight into the complex interplay between the number pattern, the nature of the question design and the specific numeric/pictorial context chosen. references cohen, l. & manion, l. 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(2002). structuring occasions. educational studies in mathematics, 49, 313-340. “the riddle does not exist. if a question can be put at all, then it can also be answered.” ludwig wittgenstein abstract introduction theoretical background and literature review research material and methods data analysis results limitations of the study conclusion and recommendation acknowledgements references about the author(s) kakoma luneta department of childhood education, faculty of education, university of johannesburg, johannesburg, south africa mekonnen y. legesse department of mathematics college of natural and computational sciences, haramaya university, ethiopia citation luneta, k., & legesse, m.y. (2023). discourse-based mathematics instruction on grade 11 learners’ mathematical proficiency in algebra topics, pythagoras, 44(1), a686. https://doi.org/10.4102/pythagoras.v44i1.686 original research discourse-based mathematics instruction on grade 11 learners’ mathematical proficiency in algebra topics kakoma luneta, mekonnen y. legesse received: 14 mar. 2022; accepted: 28 apr. 2023; published: 07 july 2023 copyright: © 2023. the author(s). licensee: aosis. this is an open access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abstract school algebra serves as the language of mathematics and a foundational subject for learning advanced mathematics courses. this makes developing learners’ proficiency in algebra the most desirable instructional goal of school mathematics. despite having such importance emphasis, however, studies indicate that the vast majority of learners are characterised by inadequate mathematics proficiency levels in general and in the algebra syllabus topics in particular. consequently, this quasi-experimental study attempted to investigate the efficacy of using discourse-based instruction as an instructional approach to developing proficiency in algebra unit topics. one hundred and six (n = 106) grade 11 learners participated in the study and were randomly grouped into an experimental group (n = 52) and a control group (n = 54). using a test instrument that consisted of 24 rasch-validated items, both pre-test and post-test data were collected from both groups under similar conditions. the mann-whitney u statistical analysis of the pre-test data revealed no significant difference between the control and experimental groups. the mann-whitney u analysis performed on the post-test data demonstrated that the experimental group scored significantly higher in the post-test scores when compared to the control group after the intervention. the study findings provided evidence of the efficacy of discourse-based instruction over teacher-centred instruction for developing learners’ algebra proficiency. contribution: the study has contributed to the conceptual and practical understanding of how discourse-based instruction can be used to concretise learners’ proficiency in basic algebra. keywords: classroom discourse; discourse-based approach; mathematical proficiency; experimental study; teaching method. introduction this manuscript reports the findings of a doctoral study (legesse, 2022) that investigated the effectiveness of discourse-based instruction of algebra and function syllabus topics. in this dynamically changing world, mathematics plays an essential role in scientific investigation, technological progress, and every walk of life (suh & seshaiyer, 2013). mathematics educators and researchers advocate the teaching of mathematics for developing mathematically proficient citizens (suh, 2007; suh & seshaiyer, 2013). for a learner to be mathematically proficient in a domain of mathematics, the learner must demonstrate conceptual understanding (comprehension of mathematical concepts, operations, and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently, and appropriately), strategic competence (the ability to formulate, represent, and solve mathematical problems), adaptive reasoning (the capacity for logical thought, reflection, explanation, and justification), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile), coupled with a belief in diligence and one’s efficacy (kilpatrick et al., 2001, p. 116). like many other countries around the world, ethiopia has envisioned the teaching and learning of mathematics to enable all learners to be mathematically competent (ethiopian ministry of education [moe], 2010a). despite this vision, the findings of the ethiopian national assessment and examination agency (enaea, 2010) showed that the majority of grade 10 and grade 12 learners scored below the average passing mark of 50% in the mathematics assessment examination. the assessment results further revealed that inadequate mathematical proficiency across different content domains of school mathematics has remained a persistent problem for the vast majority of learners (legesse et al., 2020). to alleviate this problem, research literature in mathematics education recommends different forms of instructional approach for enabling learners to develop mathematical understanding, problem-solving, and thinking (bennett, 2014; bradford, 2007; cross, 2009; manouchehri & enderson, 1999; national council of teachers of mathematics [nctm], 1991, 2014). among the recommendations was the use of classroom discourse orchestrated around mathematical tasks that foster learners’ understanding of mathematical ideas, communication, and problem-solving skills (bennett, 2014; cross, 2009; legesse et al., 2020; nctm, 1991; smith & stein, 2011; walshaw & anthony, 2008). moreover, the literature suggests the engagement of learners in mathematical discursive practices of explaining, justifying, listening to, sharing, comparing, evaluating, and interpreting each other’s mathematical ideas and reasons and constructing convincing arguments (bennett, 2014; rumsey & langrall, 2016; smith, 2018; smith & stein, 2011) to enhance their mathematical proficiency and achievement results (anthony & hunter, 2017; bennett, 2014; bradford, 2007; smith, 2018). however, evaluation studies of the effectiveness of discourse-based mathematics teaching and learning on targeted academic outcomes in different cultural contexts are scant (bradford, 2007; rumsey & langrall, 2016). in particular, there is limited practical effort in using mathematical discourse as a teaching strategy in ethiopian school classrooms. in an ethiopian context, poor performance in mathematics is often attributed to teacher-dominated classroom teaching practices (dhoj & verspoor, 2013). deressa (2004) characterised teacher-dominated instruction as being associated with poor social interactions and communication between teachers and learners and among learners over the learning content. one approach to improving learners’ mathematics proficiency learning outcomes could be exploring the potential of research-informed teaching approaches, namely the discourse-based approach. consequently, the purpose of this study was to investigate the effectiveness of discourse-based instruction of algebra and function topics on grade 11 learners’ mathematical proficiency compared to traditional lecture-based instruction. school algebra serves as the language of mathematics and a foundational subject for learning higher mathematics, science subjects, and engineering courses (grønmo, 2018). improving the classroom instruction of algebra enables learners to pursue learning advanced mathematics; hence, proficiency in algebra plays an important role in college and university mathematics courses (grønmo, 2018; kilpatrick et al., 2001). the present study, therefore, attempted to determine if discourse-based instruction enables learners to enhance their proficiency in algebra unit topics based on the ethiopian grade 11 mathematics syllabus. the research question was: are there differences in mathematical proficiency scores between grade 11 learners who were taught algebra unit topics using discourse-based instruction and those learners who were taught the same content using the traditional lecture method before and after the treatment? this study used the term discourse to refer to dialogic talk that promotes the teaching and learning of mathematics through engaging in sociocultural practices, such as questioning, reasoning, listening, sharing, explaining, and justifying (alexander, 2008; brown & hirst, 2007; steele, 2001). a discourse counts as mathematical if it is about mathematical objects and involves the use of symbols, notations, representations, and definitions (moschkovich, 2007; sfard, 2008). mathematical discourse includes the communication of definitions, rules, procedures, ideas, theorems, and proofs, verbal explanations of problem-solving strategies, and justification of reasons (shilo & kramarski, 2018). theoretical background and literature review the present study is guided by a sociocultural perspective. a sociocultural perspective underlines the role of dialogic talk in the construction of knowledge in a social setting (alexander, 2008). through engaging in dialogic talk that promotes social interactions, learners develop mathematical understanding and construct meanings by explaining how mathematical tasks can be accomplished or how procedures work, and by challenging each other’s ideas, comparing different solution strategies, sharing ideas and reasoning (bradford, 2007; gravemeijer & cobb, 2006; hiebert & wearne, 1993; lampert, 1990; steele, 2002). mathematical discourse as a vehicle for learning promotes the construction of mathematical understanding by helping learners to clarify and organize their thoughts, facilitating personal and collective sense-making, supporting building connections between representations and multiple strategies, enabling learners to use others as a resource of ideas to challenge and broaden understanding, and helping learners learn the mathematical language. (anthony & hunter, 2017, p. 101) hoyles (1985) describes mathematical understanding as the ability to form a view of the mathematical idea, step back and reflect upon it, use it appropriately and flexibly, communicate it effectively to another, reflect on another’s perspective of the idea, and incorporate another’s perspective into one’s own [schema] or challenge and logically reject this alternative view. (p. 212) this also involves asking clarification questions, sharing ideas and reasoning, agreeing and disagreeing with each other’s ideas, and respecting and listening to each other’s ideas to promote the construction of mathematical knowledge (chapin et al., 2003). engaging learners in collaborative classroom discourse that incorporates mathematical discursive activities such as agreeing and disagreeing with others’ ideas, explaining their reasoning and thinking, and discussing and comparing solution methods promotes the development of the understanding of mathematical topics (anthony & hunter, 2017; kazemi, 2008). to create a classroom environment that invites all learners to participate, the teacher should build confidence in learners that everyone can contribute to the classroom lesson by informing all learners to respect each other’s responses, share, and exchange their ideas, listen to each other’s responses, appreciate making mistakes as an opportunity for learning, and to question each other (cobb, 1994; maguire & neill, 2006; yackel & cobb, 1996). sociomathematical norms refer to what counts as an acceptable mathematical explanation and justification of the ‘why’ and ‘how’ aspects of teachers’ and learners’ activities that are specific to the mathematical discussion (cobb, 1994; maguire & neill, 2006; stephan, 2014; yackel & cobb, 1996). the teacher and learners may set criteria for deciding what is regarded as an acceptable explanation, or a different solution method (stephan, 2014). the teacher is responsible for establishing a ‘psychological safety’ learning environment in which learners feel free to respectfully express their thinking and reasoning during group discussions of mathematical ideas (cobb et al., 1990). the teacher should create a classroom culture in which learners are expected to explain and justify their solution strategies and reasoning (cobb et al., 1990). regardless of the method of instruction, sociomathematical norms are present in all classrooms; however, what counts as an acceptable mathematical explanation and solution in traditional classrooms may not be a mathematically elegant and sophisticated explanation and solution in learner-centred classrooms (stephan, 2014; yackel & cobb, 1996). for instance, just describing the solution procedure for solving a quadratic equation might be counted as an acceptable explanation in traditional classrooms (stephan, 2014). setting the use of accurate mathematical terms, symbols, representations, syntax, rules, and notations as sociomathematical norms can help learners develop mathematical language proficiency (pourdavood & wachira, 2015; summers, 2012). questions that can be used to establish sociomathematical norms include: how can you prove that your answer is right? can you prove it in more than one way? how is your solution strategy different from that of another learner’s solution strategy? do you agree or disagree with another learner’s solution? why? why does strategy a work? why does strategy b not work? (kazemi, 2008, p. 414). this study conceived discourse-based mathematics instruction as the teaching and learning of mathematics through engaging in dialogic talk orchestrated around carefully designed tasks in which the topics to be learned are embedded. discourse-based instruction can be characterised by the engagement of learners in dialogue-elicited tasks wherein the learner-teacher, learner-learner, and learner-group interactions are anchored on discourse practices of challenging each other’s ideas, sharing ideas, agreeing and disagreeing with solution strategies and ideas, comparing solution procedures, and explaining problem-solving strategies (legesse et al., 2020, 2021). in such classroom learning environments, mathematics learning is more than ‘appending’ new knowledge to existing knowledge; it involves the reconstruction of understanding and building a web of interconnected conceptual understanding that fosters the transfer of learning to new contexts (hiebert et al., 1997). some evaluation studies indicate that discourse-oriented forms of instruction positively influenced mathematics learning outcomes (bradford, 2007; cross, 2009; legesse et al., 2020; sepeng & webb, 2012; smith, 2018). bradford (2007) found that discourse-oriented instruction in pre-algebra classes helped low-achieving learners improve their mathematics achievement and problem-solving skills. a quasi-experimental study by legesse et al. (2020) examined the effects of discourse-based mathematics instruction on grade 11 learners’ acquisition of conceptual and procedural knowledge of probability and statistics topics in an ethiopian secondary school. legesse et al. found that discourse-based instruction of probability and statistics increased learners’ knowledge of concepts and procedures. moreover, the discussion-based teaching strategy improved the experimental group of grade 9 learners’ word problem-solving performance when compared to the control group taught with the traditional lecture method (sepeng & webb, 2012). research material and methods the present study employed a quasi-experimental design with a pre-test and post-test control group (shadish et al., 2002). a quasi-experimental design with a control group is an appropriate method of inquiry when the primary intent of the study is to evaluate the effectiveness of an instructional intervention (goodwin, 2009). such a quasi-experimental design is also appropriate when the randomisation of each participant is impossible for practical reasons (gall et al., 2003). thus, a quasi-experimental study with a pre-test and post-test control group established a cause-effect relationship between discourse-based instruction and proficiency in algebra unit topics. participant selection this study was conducted with grade 11 learners in a randomly selected public school in bahir dar, ethiopia. the school principal provided permission to conduct the study with the intended groups of learners. the study lasted for about 10 class weeks. during each class week, there were five lessons and each lesson was 42 minutes. in collaboration with the head of the mathematics department at the participating school, two comparable grade 11 mathematics teachers, in terms of professional and academic qualifications, who were willing to be involved in the study were recruited. the learner population in the participating school has similar ethnicity and socioeconomic status. all grade 11 learners enrolled in the natural science stream were allocated into 14 sections. after this, each section was assigned a unique natural number. fourteen equal-size pieces of paper were prepared and each piece was numbered 1 through 14. then each piece was folded and placed in a carton. after shaking the box and mixing the pieces well, an independent person randomly picked two pieces of paper one after the other without replacement, representing two sections of grade 11 learners. the two sections were assigned to either the control group (n = 54) or the experimental group (n = 52). one hundred and six grade 11 learners (n = 106) participated in this study. after being briefed about the study, all learners in the selected classes offered their verbal consent for participation. names and details of participants were kept confidential while analysing the data and reporting the results. based on prior studies in a similar context (e.g. legesse et al., 2020), there was an anticipation that the data would violate the assumptions for the parametric tests. accordingly, the mann-whitney u test was chosen for analysing and comparing differences in the pre-test and post-test scores between the control and experimental groups. using the evidence from the literature (e.g. happ et al., 2019; wmwssp, n.d.), a sample size calculation based on the expected medium effect size (r = 0.349) for the mann-whitney u test (mangiafico, 2016), power (1–β):0.80, and two-sided type i error (α: 0.05) produced an estimated total sample size of 112 participants. this power analysis indicated that the sample size of 106 was sufficient to detect a statistically significant difference between the control and experimental groups on the continuous dependent variable. locating the control and experimental groups in the same school controls differences in the physical conditions (fraenkel et al., 2012) and avoids possible threats due to differential selection (gall et al., 2003). on the other hand, it might cause treatment contamination. danga and korb (2014) examined the effect of treatment diffusion using an educational experimental design in a nigerian secondary school where learners had maximum opportunities to exchange information and talk about their classroom teaching and teach each other. the study found that placing the control group and the experimental group in the same school did not result in a potential treatment diffusion to affect learners in the control group (danga & korb, 2014). in this study, the authors made efforts to prevent or reduce potential treatment diffusion (shadish et al., 2002) between the control group and the experimental group by making the groups blind to one another and by taking the groups to different classes in physically separated buildings (rhoads, 2011). under such circumstances, it was most unlikely there would be information leakage about the nature of treatment between the control and experimental groups. implementation of discourse-based mathematics instruction two comparable mathematics teachers (both male) in terms of academic profiles and professional qualifications were both chosen from the grade 11 classes involved in the study. after recruitment, the two teachers received two different training sessions of four hours each on how to design and implement discourse-based lessons. the training involved the design of discourse-elicited tasks, crafting different questioning strategies, facilitation strategies of learners’ classroom discourse, and setting up and maintaining social and sociomathematical norms. during the training session, the teachers were maintained blind to the hypotheses of the study (rhoads, 2011). the training was offered in such a way that the teachers did not disclose information to their learners about the new method of teaching. after the training, using a coin toss, the teachers were assigned to teach either the experimental group or the control group. both the control and experimental groups were taught the same unit topics outlined in the first two chapters of the grade 11 mathematics syllabus (moe, 2009). the unit topics include relation and function, inverse relation, graphs of relation and inverse relation, even and odd functions, one-to-one and onto functions, absolute value and signum functions, inverse functions and their graphs, simplification of rational expressions, arithmetic operations on rational expressions, rational functions and their graphs, solving rational equations and inequalities, and some applications of rational equations and inequalities as word problems. the experimenter teacher was provided with a lesson plan format, prototype lessons, and an intervention guide, that assist in discourse-based lesson preparation and implementation while the other teacher was strictly informed not to implement discourse-based instruction in the control group but to use the traditional lecture method. the classroom discourse in the experimental class was structured in individual work, small-group discourse, and whole-class discourse. the experimental group underwent discourse-based mathematics instruction where learners actively engaged with mathematics learning through explaining, justifying, conjecturing, comparing, sharing, and questioning. learners in the experimental group were grouped into heterogeneous small groups (webb, 1991) of four or five learners. the experimenter teacher was responsible for establishing classroom rules that equally entertain and respect all learners’ ideas and contributions (bennett, 2014) and creating discourse-elicited mathematical tasks (lampert, 1990). the teacher briefed his learners on how to respectfully listen to and share ideas, communicate, agree or disagree with each other’s ideas, work in groups, and maintain classroom norms (bennett, 2014; legesse et al., 2021). the experimenter teacher followed the implementation process illustrated in table 1. table 1: the process of implementing discourse-based mathematics instruction (legesse et al., 2020). as outlined in table 1, the design and implementation of discourse-based instruction goes through planning and creating of tasks that embed the learning topics in the stages of task presentation: individual work (each learner thinks about the given task), small-group discussion, whole-class discussion, and reflection. the task implementation requires the active involvement of learners in individual work (give learners the task and allow each learner to think about it individually for some minutes), small-group discourse (encourage learners to work together in small groups to talk about the task, discuss, and listen to each other’s ideas), whole-class discourse (allow learners to present and share their ideas and strategies), and reflection (allow learners to reflect on the task through open-ended questions, such as ‘what solution strategies did you find very useful for solving rational equations?’ ‘what challenges have you faced in today’s lesson?’ ‘what concepts have you understood well and what concepts do still you want to understand?’). the teacher is responsible for establishing classroom rules that equally entertain and respect all learners’ ideas and contributions (bennett, 2014) and creating discourse-elicited mathematical tasks (lampert, 1990). learners should respectfully challenge, listen to, and share each others’ ideas, work in groups, and be accountable for their learning (bennett, 2014). by applying the five practices model – anticipating, monitoring, selecting, sequencing, and connecting (smith & stein, 2011, p. 8) – the experimenter teacher (a) facilitated and guided learners’ discourse of conversations about the given tasks. the experimental group of learners engaged in discursive activities that include generating examples and non-examples, agreeing and disagreeing, comparing equation-solving strategies, exploring different simplification strategies for rational expressions, explaining rational equation-solving strategies, describing relations using multiple representations, and evaluating mathematical statements. from the leading author’s classroom observations, a sample lesson on identifying functions graphically in the experimental class is presented below. planning for instruction: the teacher articulated the goal of the lesson and asked learners to open their mathematics textbooks. then the teacher wrote the task on the chalkboard. task choice or design: the teacher read and presented the task shown in figure 1. the questions asked were ‘which of the following graphs represent functions?’ and ‘which graph does not represent a function? explain your answer.’ individual work: the teacher allowed learners to think about the task independently for some minutes. small group discourse: the teacher encouraged learners to discuss and exchange their ideas in small groups for about six minutes. after ensuring all groups completed the task, the teacher asked the class to share their ideas and explanations. whole-class discourse: the teacher selected a learner from a group. the learner answered ‘the graphs in (a) and (b) represent functions’. the teacher posed the question ‘how did you know that?’ the same learner verbally explained that ‘if we draw a vertical line, it crosses exactly at one point. so, the graphs represent functions by a vertical line test’. the teacher asked the class ‘do you all agree with this explanation?’ the class approved the explanation by saying loudly ‘yes’. the teacher asked, ‘what about the graphs in (c) and (d)?’. some other selected learners answered that the graphs in (c) and (d) do not represent a function. the teacher asked the class ‘who would like to explain why these graphs do not represent functions?’ a learner explained to the class by saying ‘a vertical line crosses the graphs at two points’. the teacher asked ‘do you agree?’ the class agreed with the learners’ explanations. the teacher said ‘but what does it mean when the vertical line crosses the graph at two points?’ a learner explained by saying that ‘two numbers in the domain are related to the same number in the range’. figure 1: a discourse-elicited task presented to the experimental class: by a vertical line test, blocks (a) and (b) represent functions, whereas blocks (c) and (d) do not. the control group was taught the same unit topics with the traditional lecture method and attended the same class hours per week in similar shifts as the experimental group. the duration of each lesson was 42 minutes. the traditional lecture method can be characterised as the ‘chalk and talk’ method where the teacher mainly dominates the classroom talk, and the majority of learners are passive listeners to the teacher’s lecture with a lack of opportunities for challenging ideas, or discussing with peers to construct an understanding of concepts (rosenthal, 1995). the lesson observations showed that the classroom instruction in the control group was dominated by the teacher’s demonstration of rules and procedures for arithmetic operations of rational expressions, simplification strategies, equation solving, and graphing procedures for rational functions. learners often engaged with procedural and computation tasks that did not provide opportunities for learners to be actively engaged in discourse practices of explaining, justifying, questioning and answering, agreeing and disagreeing, and comparing solution strategies. learners copied notes and worked on examples and were passive listeners to what was explained by the teacher. findings from the observations were consistent with the traditional teacher-centred forms of teaching practices (deressa, 2004; lampert, 1990). in an ethiopian context, the traditional teaching method was found to manifest poor social interactions and communication between teacher and learners and among learners over the learning content (deressa, 2004). the implementation of the tasks is based on teacher-led demonstration where learners are accustomed to rehearsing facts and mimicking formulas, solution procedures, and rules to do similar tasks (hsu, 2013). the instruction focuses on teaching procedures and computational skills. the teacher-dominated way of teaching mathematics restricts opportunities for learners to explain, share, and discuss their ideas and thinking (hsu, 2013). for instance, the teaching of simplification of rational expressions in a teacher-dominated classroom focuses on enabling learners to memorise and practise procedural rules by assigning a set of routine tasks (kooloos et al., 2019). data collection instrument the variables involved in the present study were methods of teaching as the independent variable and mathematical proficiency scores as the dependent variable. test scores as quantitative and classroom observations as qualitative data were collected. the total test score (total sum of scores in each strand) was used as a measure of participants’ proficiency in algebra. the maximum possible score assumed was 48 while the minimum was 0. during the progression of the experiment, the researcher as a neutral observer made unannounced classroom observations twice per week in both groups. accordingly, the researcher observed and documented teachers’ actions, learners’ participation, and classroom activities. anchoring to the characterisations of the four strands of mathematical proficiency (see table 2), the authors developed 24 test items from reviewing different literature sources including national mathematics examination papers. the constructs to be measured were operationalised through the test items that covered a variety of algebra and function topics outlined in the grade 11 mathematics syllabus (moe, 2009). more specifically, the rasch partial credit model (wilson, 2005) was applied to validate the test items using winsteps software (version 4.4.2). the rasch analysis provided psychometric evidence of item reliability (0.98), person reliability (0.79), item separation (7.66), and person separation (1.97), item fit, item difficulty, item polarity, local independence, and unidimensionality that made up the test instrument. measures of item difficulty were found to be within the range from –1.96 logits to 1.92 logits; which is located inside the normal range from –3 logits to 3 logits (timofte & siminiciuc, 2018). items with difficulty measures below –2.0 or above 2.0 are described as easy and hard items (de ayala, 2009). table 2: characterisations of the four strands of mathematics proficiency (legesse, 2022). before the start of the experiment, pre-test data were collected using rasch-validated test items (see table 3) on the dependent variable from both groups. the time permitted to complete the test was 2 hours. after the completion of the implementation phase, using the same test instrument, post-test data were collected from both groups under similar conditions. the same classroom teachers and two doctoral students invigilated the administration of the pre test and post test. the researcher attended the invigilation processes as a neutral observer. learners’ test performance was marked using an item-specific scoring rubric (0/1/2). an initial format of the scoring rubric was created after analysing each item for its consistency with the content and characterisations of the proficiency strands. it was then reviewed by doctoral students and senior mathematics teachers. the modified version of the rubric was used for scoring learners’ test performance. table 3: sample test items (legesse, 2022). data analysis results statistical data analyses were performed using the statistical package for the social sciences (spss, version 23). the normality of the test data was examined by considering the absolute sizes of the z-ratios for skewness and kurtosis (orcan, 2020). the z-ratio is determined by dividing the value of skewness and kurtosis by their standard errors (orcan, 2020). for a small sample size, when the absolute sizes of the z-ratios for skewness and kurtosis are greater than 1.96, the data would be non-normally distributed (orcan, 2020). as shown in table 4, the absolute size of the z ratio for the skewness of pre-test scores in the experimental group and that of post-test scores in the control group are greater than 1.96. the values of skewness and kurtosis indicated that the pre-test scores and post-test scores were non-normally distributed for each sample (orcan, 2020). results of the shapiro-wilk test supported that the raw scores on the dependent variable failed to meet the normality assumption for each sample. table 4: descriptive statistics for pre-test scores and post-test scores. davison (1999) recommended non-parametric tests when the violation of the assumptions for parametric tests becomes evident. consequently, the authors performed mann-whitney u test analyses on rank-converted pre-test and post-test raw scores (feys, 2016; pallant, 2016). pre-test and post-test data analysis to determine if there was a significant difference in the pre-test scores between the control group and the experimental group (see table 5) before the treatment, the pre-test scores were analysed using the mann-whitney u test. table 5: normality test for pre-test scores and post-test scores. the result of the mann-whitney u test indicated no statistically significant difference in ranked pre-test scores between the control group and the experimental group (u = 1281, z = –0.786, p = 0.432) (see table 6), which showed that both groups were comparable at the start of the experiment. table 6: results of the mann-whitney u test for the pre-test scores. the result of the mann-whitney test on the post-test scores showed that the ranked post-test scores in the experimental group were significantly higher compared to the ranked post-test scores in the control group (u = 1068.50, z = –2.124, p = 0.034) after the intervention (see table 7), which revealed that the experimental group scored statistically higher in the post-test scores than the control group. table 7: results of the mann-whitney u test on the post-test scores. calculation of the effect size for the mann-whitney u test (r = 0.21) using the formula , where z refers to the absolute standardised test statistic z and n is the total number of participants (pallant, 2016; tomczak & tomczak, 2014), showed that the degree to which the experimental group had post-test scores with higher ranks than the control group was small (mangiafico, 2016). this value of effect size might suggest that learners should be exposed to discourse-based instruction for a relatively longer period of intervention for having a bigger outcome difference. description of discourse types observed in the experimental class the analysis of discourse types focused on capturing and describing the types of discourse practices enacted in the experimental class. using the discourse observation protocol (weaver et al., 2005), learners’ actions and teachers’ actions were observed for the potential discursive indicators during classroom instruction. a total of eight classroom lesson observations were conducted during the implementation phase, apart from the first and last weeks of the experiment period. themes of potential discursive indicators were identified by reviewing the literature (e.g. moschkovich, 2007; weaver et al., 2005). the discourse of conversation utterances made between learners and teacher and among learners along the segments of each observed lesson were coded and categorised for the following discourse types: sharing of ideas, solution procedures and strategies (s), conceptual explanation (e), exemplifying (ex), questioning (q), justification (j), comparison and contrast (cc), agreeing and disagreeing (ad), and generalising (g) (weaver et al., 2005). the discourse types extracted from the observed lessons are summarised and presented in table 8. table 8: discourse types extracted from lessons observed in the experimental class with discursive activities (legesse, 2022). descriptive analysis of classroom instruction in the control group a total of six classroom observations were made for the control group. the classroom instruction predominantly focused on the teacher’s demonstration of rules and procedures for arithmetic operations of rational expressions, simplification strategies, rational equation solving, and graphing procedures for rational functions. learners often engaged with procedural and computation tasks that did not provide opportunities for learners to foster the construction of mathematical ideas and concepts through engaging in discourse practices of explaining, justifying, questioning and answering, agreeing and disagreeing, and comparing solutions strategies. descriptive analysis of the observed lessons revealed that learners in the control group lacked opportunities to compare and explain different solution strategies and graphing procedures and to express their thoughts in the process of simplifying rational expressions and solving equations and inequalities. the teacher often interrogated learners’ understanding by gathering information about their mastery of the rules, definitions, and algorithms, using questions such as ‘what did you recall about yesterday’s lesson?’, ‘what is the definition of relation?’, and ‘what are the graphing procedures for rational functions?’. for instance, the teaching of simplification of rational expressions focuses on enabling learners to memorise and practise procedural rules by assigning a set of routine tasks (kooloos et al., 2019). such procedural-oriented mathematics teaching restricts opportunities for learners to explain, share, and discuss their ideas and thinking (hsu, 2013). classroom observations in the control group revealed that the instruction was characterised by the absence of questions that probed learners’ justification and reflection. the teacher questioning strategy mainly manifested a characteristic of teacher asking-learner salient-teacher answering for definitions, procedures, and rules. the nature of classroom discourse was dominated by the unidirectional flow of ideas from the teacher to learners, less frequent interactions among learners and between the teacher and learners, asking closed questions, lack of opportunities for learners to conjecture, discuss, and share their ideas and solution strategies. the teacher was the sole responsible person to approve learners’ responses and to state rules and procedures to be followed without soliciting learners’ thinking (lampert, 1990). limitations of the study unwanted researcher-related bias and teacher-related variables such as teaching experience, teaching style, difference in their views about mathematics and its teaching, lack of discourse-oriented learning materials, and relatively large class size might influence the implementation of discourse-based teaching as was witnessed by bradford (2007). the study did not provide enough information on which strands of proficiency either group might perform better relative to the other one. a separate analysis of groups’ scores in easy and difficult items was also interesting. furthermore, the study did not examine the interaction of the mathematical tasks, questions, and discourse types to influence learners’ learning and this limitation was also observed in other similar studies (hiebert & wearne, 1993). after the completion of the experiment, the researcher planned to treat the control group with discourse-based instruction. however, about two months after the completion of the experiment, it was impossible to expose the control group to the treatment in a similar manner to the experimental group because schools were closed due to the covid-19 pandemic. to circumvent this unexpected situation, the control group learners received classroom activities that were enacted in the experimental class in the form of handouts and group assignments via their mathematics teacher after schools reopened. conclusion and recommendation the study examined the effectiveness of discourse-based mathematics instruction on grade 11 learners’ proficiency in the syllabus topics of algebra and function. results of the data analyses showed that learners who engaged in discourse-based instruction of algebra topics demonstrated better performance in mathematics proficiency than those who were taught the same syllabus topics through teacher-centred instruction. it appeared that discourse-based instruction was more effective in developing learners’ proficiency than the teacher-centred method for teaching mathematics. that is, learners in the experimental group gained a better understanding of mathematical concepts and procedures than those learners in the control group. this was most likely the result of their regular engagement in the discourse-based learning activities that promote individual thinking and discussions of mathematical ideas where learners compared solution methods, explained and posed questions for clarification, and shared and listened to one another’s ideas and reasoning. the findings in this study were consistent with theoretical and empirical literature (e.g. bradford, 2007; cross, 2009; franke et al., 2015; legesse et al., 2020; sepeng & webb, 2012; smith, 2018; stein, 2007). for instance, a sociocultural learning perspective contends that a classroom environment that creates a learning platform for social interactions and cultural contexts fosters the construction of knowledge and understanding. overall, the study findings provided innovative ideas on how school mathematics should be taught in ethiopian classroom contexts and that assist mathematics teachers in being aware of the potential benefits of discourse-based mathematics teaching for enhanced conceptual learning outcomes. based on these findings, the study recommended discourse-based instruction for teaching mathematics at upper secondary and university levels. it proposed discourse-based education as a component of mathematics teachers’ professional development programme. for these findings to be highly informative, replication studies should be conducted across different grade levels and other mathematical content domains with similar or mixed research designs. acknowledgements this manuscript is based on a doctoral thesis submitted to the university of johannesburg for which the university provided funding. competing interests the authors declare that they have no financial or personal relationships that may have inappropriately influenced them in writing this article. authors’ contributions k.l., conceived the research and conceptualised the research design and methodology as well as the initial analysis of the data. m.y.l., collected the data and developed the data analysis tools, finalised the analysis of the data, and the initial writing of the manuscript. ethical considerations ethical clearance to conduct this study was obtained from the university of johannessburg faculty of education research ethics committee (no. sem 2-2019-031). funding information this research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors. data availability data sharing is not applicable to this article as the participants did not authorise the authors to share the data collected and analysed in this study. disclaimer the opinions, findings, conclusions, and recommendations communicated in this manuscript are those of the authors and do not necessarily reflect the views of other persons or any institution. references alexander, r. 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(2006). socio-mathematical norms, argumentation, and autonomy in mathematics. journal for research in mathematics education, 27(4), 458–477. using language as a transparent resource  in the teaching and learning of mathematics  in a grade 11 multilingual classroom  mamokgethi setati  terence molefe  mampho langa  university of south africa  setatrm@unisa.ac.za  fons luminus high school  molefebt@yahoo.com  oprah winfrey leadership  academy for girls  mampho.langa@owla.co.za  in this paper, we draw on a study conducted in grade 11 classrooms to explore how the  learners’  home  languages  can  be  used  for  teaching  and  learning  mathematics  in  multilingual classrooms in south africa. this report is part of a wider study that is still in  progress. based on an analysis of data collected through lesson observations in a grade 11  class and learner interviews we argue for the deliberate, proactive and strategic use of the  learners’  home  languages  as  a  transparent  resource  in  the  teaching  and  learning  of  mathematics  in  multilingual  classrooms.  such  use  of  the  languages  will  ensure  that  learners gain access to mathematical knowledge without losing access to english, which  many parents, teachers and  learners presently see as a necessary condition for gaining  access to social goods such as higher education and employment.   what does it mean to teach or learn mathematics in a language that is not your home, first or main language? this is the situation in the majority of classrooms in south africa. in these classrooms the language of learning and teaching (lolt) is english – one of the eleven official languages; however, neither the teacher nor the learners have english as their main, home or first language. research shows that teachers and learners in these classrooms prefer that english be used as the lolt (setati, in press). in addition to this, anecdotal evidence shows that a majority of learners in these classrooms are not motivated to study mathematics, and they are doing it either because they have to or because they know that it is needed for any higher education study related to science, technology or commerce. what strategies are appropriate for use in these classrooms? embedded in this question are issues about language and learning and also about motivating learners’ interest in mathematics. in this paper we draw on a wider study exploring relevant pedagogies for teaching and learning mathematics in multilingual classrooms. we specifically focus on data collected in one grade 11 classroom to explore how the learner’s home languages can be used as a transparent resource in the teaching and learning of mathematics in multilingual classrooms. we begin the paper with a review of research on teaching and learning mathematics in multilingual classrooms. through this we expose three prevalent dichotomies in research on teaching and learning mathematics in multilingual classrooms. first, is the dichotomy between using english as lolt as opposed to using the learners’ home language(s) as lolt. second, is the dichotomy about drawing on socio-political perspectives when analysing interactions in multilingual mathematics classrooms as opposed to drawing on cognitive perspectives. the third dichotomy is about gaining access to mathematical knowledge as opposed to access to english. we then discuss the theory that informed the analysis we present in the paper. this discussion provides a theoretical background for the argument for the deliberate, proactive and strategic use of the learners’ home languages as a transparent resource in the teaching and learning of mathematics. 14 pythagoras, 67, 14-25 (june 2008) mamokgethi setati, terence molefe & mampho langa teaching and learning mathematics in multilingual classrooms there is a continuing debate in south african education and the public domain regarding which language should be used for teaching mathematics in multilingual classrooms in which children are still learning english. this debate is due to the fact that learners in many of these classrooms are not yet fully fluent in english which is the lolt in their classrooms. while the south african language in education policy (liep) encourages multilingualism (department of education, 1997) and research supports the use of the learners’ home languages (moschkovich, 2002), a recent analysis of the learners’ and teachers’ language choices for teaching and learning mathematics shows a preference for english (setati, in press). in this analysis setati argues that the desire to gain access to social goods (e.g. jobs, higher education) overrides the multilingual teachers’ and learners’ concern for epistemological access. the view with many teachers and learners is that english is an international language and in south africa it is important for higher education, jobs and subsequently a better life. given this background, we argue that it is crucial to explore ways of drawing on the learners’ home languages for teaching and learning mathematics in multilingual classrooms without denying them access to english. previous research argues that the learners’ home languages are a resource for mathematics learning (adler, 2001; moschkovich, 1999; setati, 2005; setati & adler, 2000). the challenge, however, is that in a context such as south africa, where the hegemony of english is so prevalent, regarding the learners’ home languages as a resource tends to be seen as a threat to multilingual learners’ development of fluency in english. as sachs (1994) pointed out, in south africa “all language rights are rights against english” (p. 1). hence our argument that for the use of the learners’ home languages in the teaching and learning of mathematics in multilingual classrooms to be successful it must ensure that the learners gain epistemological access without losing access to english. granville et al. (1998) present a similar idea in relation to the south african language in education policy, where they argue for english without g(u)ilt. what is new about our argument is the different orientation we bring by focusing on learning and teaching rather than policy. debates around language and learning in south africa tend to create a dichotomy between learning in english and learning in the home languages. they create an impression that the use of the learners’ home languages for teaching and learning must necessarily exclude and be in opposition to english, and the use of english must necessarily exclude the learners’ home languages. in an article entitled ‘why don't kids learn maths and science successfully?’, sarah howie of the university of pretoria is quoted as saying “the most significant factor in learning science and maths isn’t whether the learners are rich or poor. it’s whether they are fluent in english” (science in africa, 2003). this she said drawing on her analysis of south africa’s poor performance in the third international mathematics and science study of 1995 (see also howie 2003, 2004). in the same article howie makes an impassioned call on south africa to choose only one language for teaching and learning mathematics in multilingual classrooms: let’s stop sitting on the fence and make a hard decision. we must either shore up the mother tongue teaching of maths and sciences, or switch completely to english if we want to succeed. our argument in this paper is that in a multilingual country such as south africa the choices are not as simplistic as howie suggests. our argument is informed by a holistic view of multilingual learners (different from howie’s monolingual view), in our view multilingual learners have a unique and specific language configuration and therefore they should not be considered as the sum of two or more complete or incomplete monolinguals. to explain this different but complete language system in multilinguals, grosjean (1985) uses an analogy from the domain of athletics: the high hurdler blends two types of competencies: that of high jumping and that of sprinting. when compared individually with the sprinter or the high jumper, the hurdler meets neither level of competence, and yet when taken as a whole, the hurdler is an athlete in his or her own right. no expert in track and field would ever compare a high hurdler to a sprinter or to a high jumper, even though the former blends certain characteristics of the latter two. in many ways the bilingual is like the high hurdler. (p. 471) 15 using language as a transparent resource the use of the learners’ home languages as a transparent resource that we are exploring in this paper is informed by this holistic view of multilingual learners. we accept that the idea of drawing on the learners’ home languages during teaching is not necessarily new. the use of codeswitching as a learning and teaching resource in bilingual and multilingual mathematics classrooms has been the focus of research in the recent past (e.g. adendorff, 1993; adler, 1998, 2001; arthur, 1994; khisty, 1995; merritt, cleghorn, abagi, & bunyi, 1992; moschkovich, 1996, 1999; ncedo, peires, & morar, 2002; setati, 1998; setati, 2005; setati & adler, 2000). these studies have argued for the use of the learners’ home languages in teaching and learning mathematics, as a support needed while learners continue to develop proficiency in the lolt at the same time as learning mathematics. all of these studies seem to be in agreement that to facilitate multilingual learners’ participation and success in mathematics teachers should recognise their home languages as legitimate languages of mathematical communication. the practical manifestation of the use of the learners’ home languages in these studies is through code-switching, mainly to provide explanation to learners in their home languages. in all of these studies code-switching is presented as spontaneous and reactive, the learner’s home languages are only used in oral communication and never in written texts. what we are advocating in this paper is the deliberate, strategic and proactive use of the learners’ home languages. this strategy recognizes the fact that learners want access to english and thus while we draw on the learners’ home languages and foreground the quality of the mathematics tasks used during teaching, we also ensure that english is still available to learners and they can continue to develop fluency in it. research into the complex relationship between bilingualism/multilingualism and mathematics teaching and learning argues that bilingualism/multilingualism per se does not impede mathematics learning (clarkson, 1991; dawe, 1983; stephens, waywood, clarke, & izard, 1993; zepp, 1989). this field of research has been criticised because of its cognitive orientation and its inevitable deficit model of the bilingual learner (martin-jones & romaine, 1986; frederickson & cline, 1990; both in baker, 1993). learner performance (and by implication, mathematical achievement) is determined by a complex set of inter-related factors. poor performance by multilingual learners thus cannot be attributed to the learners’ limited language proficiencies in isolation from the wider social, cultural and political factors that infuse schooling. in our view, this past research informed by a cognitive perspective presents an implicit argument in support of the maintenance of learners’ home languages, and of the potential benefits of learners using their home language(s) as a resource in their mathematics learning. multilingualism is the norm in many south african classrooms, rather than the exception. hence the need for south african mathematics education not only to treat the multilingual learner as the norm but also to view his or her facility across languages as a resource rather than a problem (baker, 1993). through our work in this study we have come to recognise that separating cognitive matters from the socio-political issues relating to language and power when exploring the use of language(s) for teaching and learning mathematics in multilingual classrooms is not productive. while we accept that cognitively oriented research does not deal with the socio-political issues relating to the context in which teaching and learning takes place, we acknowledge that it is useful in helping us attend to issues relating to the quality of the mathematics and its teaching and learning in multilingual classrooms. in this study we are thus moving against dichotomies, not only of language choices but also of theoretical perspectives. theoretical underpinnings this study is broadly informed by an understanding of language as “a transparent resource” (lave & wenger, 1991). while the notion of transparency as used by lave and wenger is not usually applied to language as a resource nor to learning in school, it is illuminating of language use in multilingual classrooms (see also adler, 2001). lave and wenger (1991) argue that access to a practice relates to the dual visibility and invisibility of its resources: invisibility is in the form of unproblematic interpretation and integration into activity, and visibility is in the form of extended access to information. this is not a dichotomous distinction, since these two crucial characteristics are in a complex interplay, their relation being one of both conflict and synergy. (p. 103) for language in the classroom to be useful it must be both visible and invisible: visible so that it is clearly seen and understood by all; and invisible in 16 mamokgethi setati, terence molefe & mampho langa 17 that when interacting with written texts and discussing mathematics, this use of language should not distract the learners’ attention from the mathematical task under discussion but facilitate their mathematics learning. this idea is similar to the use of technology in mathematics learning. the technology needs to be visible so that the learners can notice and use it. however it also needs to be simultaneously invisible so that the learners’ attention is focussed on the mathematics problem that they are trying to solve. like technology, language needs to be a transparent resource. as lave and wenger argue the idea of the visibility and invisibility of a resource is not a dichotomous distinction, it is not about whether to focus on language or mathematics, it is about recognising that the two are intertwined and are constantly in complex interplay. we found lave and wenger’s concept of transparency useful in conceptualising a strategy for using language in teaching and learning mathematics in multilingual classrooms. multilingual classrooms are characterised by complex multiple teaching demands: the learners’ limited proficiency in the language of learning and teaching (english); the challenge to develop the learners’ mathematical proficiency as well as the presence of multiple languages. the strategy we are exploring is guided by two main principles, which are informed by the theoretical assumptions elaborated in the discussion above. first, it is the deliberate use of the learners’ home languages. we emphasise the word deliberate because with this strategy the use of the learners’ home languages is deliberate, proactive and strategic and not spontaneous and reactive as it happens with codeswitching. second, is that through the selection of real world interesting and challenging mathematical tasks, learners would develop a different orientation towards mathematics than they had and would be more motivated to study and use it (gutstein, 2003). many learners in multilingual classrooms in south africa have what gutstein (2003) describes as “the typical and well documented disposition with which most mathematics teachers are familiar – mathematics as a rote-learned, decontextualised series of rules and procedures to memorise, regurgitate and not understand” (p. 46). in this study we selected high cognitive demand tasks (stein, smith, henningsen, & silver, 2000), that present real world problems that the learners can find interesting and useful to engage with. the study the study presented in this paper was undertaken in a grade 11 class taught by terence, the second author. the data we are presenting here was collected in terence’s classroom in a multilingual high school in soweto, johannesburg. there were 36 learners in his class and they had the following home languages: setswana, xitsonga, isizulu and tshivenda. each of the learners was able to communicate in at least four languages and they were learning english as a subject at second language level as well as their respective home languages as subjects at first language level. terence is multilingual and fluent in eight languages1, which includes all the home languages of his learners as well as english. his home language is setswana. at the time of the study terence had been teaching mathematics at secondary school level for 15 years. data was collected through lesson observations and individual learner interviews. lessons were observed and video recorded for four consecutive days. at the end of the four days four learners from different home language groups were interviewed by mampho, the third author and a former teacher in the school who was not present during lesson observations. the interview focused on their reflections and views about the lessons. on the next page is the task that the learners were working on during the lessons observed and analysed in this paper. this task was translated into the four home languages of the learners in the class. during the lessons learners were organised into seven home language groups: two setswana groups, two tshivenda groups, two isizulu groups and one xitsonga group. six of the groups had five learners and one group had six learners and they were given tasks in two language versions (english and their home language). learners were explicitly made aware of the two language versions of the task and encouraged to communicate in any language including their home languages at any stage during the lessons. all the lessons and learner interviews were video recorded and then transcribed. to analyse the video-recording and the transcribed data we looked 1 this kind of multilingualism is not unusual. given the integration of different ethnic groups, a majorty of african teachers (indeed african people in general) in the gauteng province are multilingual and can communicate in at least four languages. using language as a transparent resource cost of electricity the brahm park electricity department charges r40,00 monthly service fees then an additional 20c per kilowatt-hour (kwh). a kilowatt-hour is the amount of electricity used in one hour at a constant power of one kilowatt. 1. the estimated monthly electricity consumption of a family home is 560 kwh. predict what the monthly account would be for electricity. 2. three people live in a townhouse. their monthly electricity account is approximately r180,00. how many kilowatt-hours per month do they usually use? 3. in winter the average electricity consumption increases by 20%, what would the monthly bills be for the family home in (1) above and for the townhouse? 4. in your opinion, what may be the reason for the increase in the average electricity consumption in (3) above? 5. determine a formula to assist the electricity department to calculate the monthly electricity bill for any household. state clearly what your variables represent and the units used. 6. a) complete the following table showing the cost of electricity in rand for differing amounts of electricity used: consumption (kwh) 0 100 200 300 400 500 600 700 800 900 cost (in rand) b) draw a graph on the set of axes below to illustrate the cost of different units of electricity at the rate charged by the brahm park electricity department. electricity costs 0 50 100 150 200 250 300 0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 consumption (kwh) co st (r an d) after careful consideration, the electricity department decided to alter their costing structure. they decide that there will no longer be a monthly service fee of r40,00 but now each kilowatt-hour will cost 25c. 7. what would be the new monthly electricity accounts for the family home and the townhouse? 8. a) complete the following table showing the cost of electricity in rand for differing amounts of electricity used using the new costing structure: consumption (kwh) 0 100 200 300 400 500 600 700 800 900 cost (in rand) b) draw a graph on the same set of axes in question 6 b) to illustrate the cost of electricity for different units of electricity using the new costing structure. 9. do both the family home and the townhouse benefit from this new costing structure? explain. 10. if people using the electricity had the option of choosing either of the two costing structures, which would you recommend? clearly explain your answer using tables you have completed and graphs drawn in questions 6 a) and 6 b) and 8 a) and 8 b) above. malati materials (30-04-2005) 18 mamokgethi setati, terence molefe & mampho langa for presences (what was visible) and absences (what was invisible) in what the learners were talking about. in the lesson observation data, what was most visible was learners’ attempts to find the solutions to the questions in the task without much focus to the language. there was only one incident during the lessons observed, which we discuss in more detail below, where language became visible but not simultaneously invisible in one of the groups. all the lessons and learner interviews were video recorded and then transcribed. to analyse the video-recording and the transcribed data we looked for presences (what was visible) and absences (what was invisible) in what the learners were talking about. in the lesson observation data, what was most visible was learners’ attempts to find the solutions to the questions in the task without much focus to the language. there was only one incident during the lessons observed, which we discuss in more detail below, where language became visible but not simultaneously invisible in one of the groups. language was, however, constantly visible for terence, the teacher. for instance when asking learners to read he would specify which language they should read in. in the section that follows we discuss, with evidence from lesson observation data, incidents during the lessons when language was transparent (visible and invisible) and also when it was visible but not simultaneously invisible (i.e. not transparent). when language was visible and invisible in our analysis we found that when language was transparent learners’ interactions were conceptual – learners’ interactions were focused not only on what the solution is but also why it is correct. the two extracts we are analysing in this section happened in the tshivenda group and are typical of how language functioned as a transparent resource during interactions between terence and the learners and also between the learners themselves. both extracts are taken from the first lesson at the time when the learners were beginning their work on the task and they needed to understand the following statement in the problem: the brahm park electricity department charges r40,00 monthly service fees then an additional 20c per kilowatt-hour (kwh). a kilowatt-hour is the amount of electricity used in one hour at a constant power of one kilowatt the extract below shows the interaction between terence and the learners in the group. here terence is working with them on the two charges mentioned in the problem, the r40 monthly service fee and the additional 20c per kilowatt-hour. 1. terence: forty rhanda heyi, vhoibadhala when [when is the forty rand paid?] 2. sipho: in a month. 3. terence: twenty cents yone [what about the twenty cents?] 4. given: twenty cents yo ediwa. [twenty cents is added.] 5. terence: why i ediwa [why is it added?] 6. learners: (silent). 7. terence: vhoi edela mini? twenty cents vhoi edela mini [why is it added? why is twenty cents added?] 8. learners: (inaudible). 9. terence: okay, if you use electricity ukho bhadala forty rand? [okay, if you use electricity will you pay forty rand?] 10. learners: yes meneer [sir] 11. terence: if unga shumisanga electricity ukho bhadala forty rand [if you did not consume electricity, will you pay forty rand?] 12. sipho: no, no no … 13. given: haena, whether ushumisile ore haushumisanga, ukhobhadala forty rhanda [no, whether you have consumed electricity or not, you pay the forty rand.] 14. terence: whether ushumisile ore haushumisanga? [whether consumed or not?] 15. sipho: eya, yes, it is a must. 16. terence: it is a must? one very noticeable thing about the extract above is the fact that it is in a mixture of english and the learners’ home language (tshivenda). this as indicated earlier was typical of interactions during the lessons observed in this class. the unproblematic move between tshivenda and english without explicit negotiation between interactants is an indication that language is functioning as a transparent resource. whilst it is visible it is also invisible enough to be used without distracting attention from the task. this invisibility of language as a mediating tool allows focus on and thus supports visibility of the mathematics the learners are discussing (lave & wenger, 1991). at the same time, the visibility of language (i.e. tasks given in two languages) is necessary for allowing its unproblematic invisible use. in the extract above, the learners are struggling to understand the phrase “an additional 20c per kilowatt-hour (kwh)”. while they understand that everyone who has electricity is supposed to pay the r40 monthly cost and also that 20c is added, they seem to be having difficulty in understanding why the 20c is added. in utterances 5 to 7 terence asks 19 using language as a transparent resource them why 20c was added. seeing that they are not able to answer his question, he moves back to asking them about the r40 in utterance 9. by doing this terence is separating the r40 from the 20c so that the learners can see that while everyone who has electricity is required to pay r40, as to how much they pay thereafter depends on the amount of kilowatt hours they used and 1 kwh costs 20c. the extract above ends with terence in utterance 16 having established with the learners that the r40 payment is a mandatory service fee for everyone who has electricity. in the extract below the learners carry on with the discussion (on their own) about when and why 20c is added. 17. given: hei, nayo … ar … (giggles) … so forty rhanda hi monthly cost ne, then ba yieda nga twenty cents kha kilowatt for one hour. then after that, angado shumisa …, baibidza mini? heyi … ndoshumisa one kilowatt nga twenty cents kha one hour [hey, this question … ar …(giggles) … so forty rand is the monthly cost, then they add twenty cents per kilowatt-hour. …, they use…, what do they call it? hey … they use one kilowatt-hour for twenty cents.] 18. sipho: eya [yes] 19. given: boyieda, maybe boshumisa twenty cents nga one hour [they add it, maybe they use twenty cents per hour.] 20. sipho: eya, yantha [yes, one hour] 21. given: iba … [it becomes…] 22. given and sipho (together): forty rand twenty cents. 23. sipho: yes, vhoibadela monthly, ngangwedzi ya hona. yo fhelela, yes. sesiyaqubheka. [yes, they pay it monthly, each month. it is complete, yes. we continue.] the transparent use of language continues in the above extract. the learners do not focus on what language is used for what; they are focused on communicating their understanding. this transparency of language enables conceptual interactions between the learners and the teacher and also among learners themselves. using their home language, tshivenda, as a legitimate language of interaction together with english made it possible for them to understand that in this case 1 kwh costs 20c. the learners are not concerned about the correctness of their grammar in tshivenda and in english, they are more focused on gaining an understanding of the problem and having both language versions serves as a resource that they can draw as and when they need to. in the next section we focus on the incident during lesson one, when language was not a transparent resource. when language was visible but not simultaneously invisible as indicated earlier when a resource is too visible it distracts attention from the subject matter. the extract below indicates learners in the tshivenda group interacting with each other on the answer to questions 3 and 4 in the task sheet, which stated as follows: 3. in winter the average electricity consumption increases by 20%, what would the monthly bills be for the family home in (1) above and for the townhouse? 4. in your opinion, what may be the reason for the increase in the average electricity consumption in (3) above? in the extract they are trying to attend to what the questions are asking them to do but they are also dealing with the issue of language what language to give the answer in. the extract suggests that the struggle with questions in this instance is worsened by their struggle first with understanding what the problem is asking them to do and second with finding the tshivenda words to use. 1. given: di ya benefita. hapfa neh, kha summer, kha winter vha badhela seven hundred neh, kha townhouse. then kha…kha mudi kha winter, kha botshibadhela one-fifty two, kha botshibadhela vhugai, one-forty. so vha benefitha ngavhugai? the amount… [they benefit. here, in summer… in winter they pay seven hundred for the townhouse. then the household in winter they pay one-fifty two, and in summer how much do they pay, one-forty. so, how much do they benefit? the amount…] 2. patience: (interrupts given) eya, mara i think vha budzisa huri vha kho inkhriza ngavhugai. apfa vhobadhela one-fifty two and apfa summer vha kho vhadela one-forty. [yes but i think what is being asked is by how much will it increase. here they pay one-fifty two and here in summer they pay one-forty.] 3. given: ndikhongwala nga tshivenda zwino. [i am now going to write in tshivenda.] 4. sipho: eya, ngwala nga tshivenda ngwananga [yes, write in tshivenda baby!] 5. given: (writes in tasksheet) ndikhoneta nga english. ritshi… kana ndimini u…u...u… [i am tired of english. we say… by the way what is to…to…to…] 6. sipho: khezwo! ngwala nga english [aha! write in english.] 7. given: hae, kana ndimini u… [no, by the way what is to…] 8. patience: kwitani [to do what?] 9. sipho: inkhriza [to increase?] 20 mamokgethi setati, terence molefe & mampho langa 10. given: kutanga… kutanganisa, tshitangadzisa mudi kha…kha summer na mudi kha winter, ritshi mini? mudi… hu tanganisa ndimini ngatshivenda [to add… to add, we add the household in…in summer and in winter, what do we say? household… what is to ‘tanganisa’ (add) in tshivenda?] 11. patience: ndingutanganisa [is to ‘tanganisa’ (add).] during this interaction, language was made visible by given in utterance 3 when she indicated that she is going to write the answer in tshivenda. until then language had been a transparent resource in the group as their focus was on providing answers to the questions in the task. by making language visible given moved the learners’ attention from the mathematical content of the task to the language. as we can see, from utterance 5, given indicates that she is now going to write the answer in tshivenda because she is “tired of english”. this was responded to by sipho in a seemingly sarcastic manner in utterance 4 by saying “yes, write in tshivenda baby!” it is the inclusion of the word “baby” which suggests sarcasm in sipho’s tone. in utterance 6 sipho responds to given’s struggle for a tshivenda word for increase by saying “aha! write in english”. the inclusion of the word “aha!” in this case suggests that sipho knew that given would not be able to write the answer in tshivenda. it is important to remember that these learners study their home languages as subjects at first language level. however, doing mathematics, which includes reading-writing-speaking-listening mathematically is something that they have never done in their home languages but only in english despite their limited fluency in it. so the challenge that given is experiencing here is about doing mathematics in tshivenda only, a constraint that she imposed on herself as terence did not force them to give their solutions in their home languages only. according to terence’s instructions their answer would have still been accepted if it were given in english or in a mixture of english and tshivenda. in utterance 9 sipho gives the transliterated word for increase, which is inkhriza, however given does not accept it as he is looking for a pure tshivenda term for increase. in utterance 11 patience offers “tanganisa”, this means ‘add’ and not ‘increase’. the word for ‘increase’ in tshivenda is ‘huengedza’, which literally means ‘to make more’. it is evident from these learners’ interactions in the above extract that even if learners know the answer, if they do not have the words to express it then their answer may never be known. it needs to be noted that while these learners wanted to respond to the question in tshivenda only, it was neither necessary nor required. with the invisibility of language, it is possible that these learners may not have felt the need to respond in one language only. they could have used both tshivenda and english to write their answer. by insisting on using only tshivenda, language was no longer an invisible resource but an obstacle they needed to overcome by for example, finding the pure tshivenda term for increase to use in their response to the question. it is important to note what precipitated this visibility of language in the extract above. in utterance 5 given indicated that she is “tired of english”. but why is given tired of english? why does she feel empowered to express her feelings against english now when they are supposed to be solving a problem? we raise these questions here not to answer them but to signal our view that the use of language as a transparent resource is not necessarily achieved overnight especially in multilingual classrooms such as in south africa where the political role of language both historically and in the current practices cannot be ignored. in the section that follows we explore language as a transparent resource in the learners’ reflections on the lessons. language as a transparent resource in the learners’ reflections on the lessons as indicated earlier, reflective individual interviews were conducted with learners after the first week of lessons. terence selected four learners from different home language groups for interviewing so as to get their reflections on the lessons observed. the interviews were semistructured and were conducted by mampho langa (the third author) who used to be the head of the mathematics department in the school. we decided that mampho conduct the interviews because we thought that the learners would be free to talk frankly about their experiences of the lessons. in our view mampho was the best person to conduct the interview also because she was not part of the team that collected the lesson observation data. furthermore the learners were not only familiar with her but also with her position as head of the mathematics department and thus comfortable about talking with her about their mathematics learning. learners selected languages they wanted to be interviewed in. the interviews were video recorded and then transcribed. 21 using language as a transparent resource in analysing the learner interviews we again looked at what was visible, i.e. presences (what the learners were talking about) and what was invisible, i.e. absences (what they were not talking about). we expected that the two main changes that we introduced (language and nature of the task) would be most visible in what the learners talk about during the interviews. in the extracts below from interviews with individual learners the interviewer asked them the same open question about what was happening in their class: mampho: i understand this week you had visitors in your class, what was happening? sindiswa: er…, we were learning a lesson in which we can calculate electricity er …. amount … er … the way in which the electricity department can calculate the amount of electricity unit per household. nhlanhla: we were learning about how to calculate …er…er… kilowatts of the electricity, how do we … like … how can we calculate them and when … at …, besifunda mem ukuthi ugesi udleka kakhulu nini. [we were learning about when there is high electricity consumption.] colbert: er …we were just solving for electricity, kilowatt per hour, for comparing if they are using card or the meter, which is both, i think are the same. sipho: er, the visitors they were doing research. gošho gore ba sheba gore bothata … bothata ba rona bo mo kae, ka … ka … maths, then they found out that er… ba bang ha ba understende dilanguage, like english so, then ha ba botsa karabo then they can’t find the answer. so mr molefe then decided to … to … make it in … in english and vernacular language to … to …, for us to understand. three of the learners above point to the mathematical task that they were working on during the lesson thus suggesting that is what they found as central to what was happening during the lesson. as explained earlier, the strategy we are exploring in this class centres around two principles: (1) the deliberate use of the learners’ home languages, and (2) the selection of interesting and challenging mathematics tasks. given these principles we had thought the use of the learners’ home languages in the tasks given would be the most prominent thing for the learners to notice. we thought so because language is often referred to as one of the main constraints in teaching and learning mathematics in multilingual classrooms. what is emerging in the extracts above is that for the learners the context of the task, cost of electricity, was more prominent. given our expectation that the learners would point to language as most prominent about what was different about the lessons, mampho probed further as below: mampho: but what was so special about the lessons? sindiswa: it does not include those maths … maths. it is not different, but those words used in maths didn’t occur, didn’t occur but we weren’t using them. … er … ‘simplifying’, ‘finding the formulas’, ‘similarities’, … nhlanhla: hayi, no mem, ku-different… okokuqala mem, ilokhuza, la sidila ngama-calculations awemali, manje ku-maths asisebenzi ngemali. [no mam, it is different. firstly mam, we were working with money and usually in maths we do not work with money.] colbert: iya, basenzele in order to … ukuthi ibe simple and easy to us, because most of people, uyabona, aba-understendi like i … like i-card ne meter. abanye bathi i-meter is … i-price yakhona i-much uyabona, i-card iless i-price yakhona, that’s why uyabona. so, abantu abana-knowledge, uyabona, bakhuluma just for the sake of it. so, i think for us, because we have learnt something, both are the same. [yes, you see they made it easy for us, because most people do not understand, like card or using a meter. some say when using the card you pay less than when using the meter, you see. so people do not have knowledge out there, they just talk for the sake of it. so i think, for us we have learnt something, both are the same.] sipho: gošho gore ba sheba gore bothata … bothata ba rona bo mo kae, ka … ka … maths, then they found out that er… ba bang ha ba understende dilanguage, like english so, then ha ba botsa karabo then they can’t find the answer. so mr molefe then decided to … to … make it in … in english and vernacular language to … to …, for us to understand. [they were looking at the problem… where our problem is, with… with… maths, then they found that er … some of us do not understand languages like english, so when they ask for the answer we can’t find it. so mr. molefe then decided to… to … make it in english and vernacular language to … to… for us to understand.] in responding to mampho’s question above, both sindiswa and sipho point to language. sindiswa points to how the task differs from the textbook tasks that they are used to. sindiswa says of the observed lesson, that the absence of many of the 22 mamokgethi setati, terence molefe & mampho langa terms usually associated with the mathematics classroom was significant, even though the essence of the lesson and activity remained unchanged. it is evident that for sindiswa language played a clear role in the “feel” of the lesson. this is echoed by sipho, whose response and choice of language is very interesting. what stood out for him about the lessons was what the introduction of learners’ home languages allowed learners to do. it changed the dimensions of the interaction, increased participation and intervened at the level of meaning. noticeably, he does not say “ha ba botsa karabo then they don’t know the answer” [“when they ask for the answer they don’t know the answer”], he says “ha ba botsa karabo then they can’t find the answer” [“when they ask for the answer they can’t find the answer”]. in sipho’s analysis, the learners may have the answer in one language, but their inability to find it in another language (english) has direct effects on their participation and performance. on the face of it, sindiswa and sipho address different aspects of the changed lesson. however, both highlight the manner in which the use of language (or the absence of certain kinds of language) can either enable comprehension or constrain learning. both see the actual mathematical activity as unchanged. for sindiswa, when “difficult” words are minimised, then learners and teachers can get on with the usual business of mathematics, focusing on the task and allowing learners to experience mathematics differently and more fluently. nhlanhla and colbert point to the nature of the task. for nhlanhla what stood out the most is the fact that in mathematics they usually do not deal with calculations involving money and so these lessons were special because they involved money calculations. this resonates with colbert’s focus on the value of the task beyond the lesson. for him it was about clarifying a real life situation that he never understood – the fact that the cost for electricity will ultimately be the same in both costing structures. what colbert is referring to is his learning about two different costing systems for electricity as described in the problem. in his view both options end up costing the same. while colbert’s analysis of the task is mathematically incorrect, it is clear that the context of the task presented a real life problem that, as he says, people in his neighbourhood have been arguing about. looking at the graphs below illustrating the cost of electricity for the two options, it is clear that they intersect at the point (800, 200), which means that if electricity consumption is more than 800 kwh then the cost of electricity will be cheaper when using the first costing structure. cost of electricity for the two options 0 50 100 150 200 250 0 100 200 300 400 500 600 700 800 900 consumption (kwh) c o st (r an d s) given the learners’ seeming reluctance to talk about the fact that their home languages were used, mampho asked them a direct question about the way in which their home languages were used in the task: mampho: i understand that the tasks that were given were written in both your home language and english. tell me about that. sindiswa: it was fine. it was just the same. it was the same as doing it in english, because i understand both languages. nhlanhla: i think mem leyo kusinikeza amaphepha o itwo kuya nceda mem, ngoba, like mina, kukhona amanye ama-questions bengingawaunderstandi, i-home language iyakhona ukusiza ukuthi ngiwa understande. [mam i think that one of giving us tasks in two languages is very helpful mam, because like there are questions that i did not understand and my home language helped me understand.] colbert: iya, i think is a good idea, uyabona, ngoba iyenza ukuthi … iyenze izinto zibe simple, ngoba if singa-understendi ngeenglish, sicheka ku … our languages, aba simple bese siyakhomphera. [yes, i think it is a good idea, you see it makes things simple because if we do not understand in english we check in our home languages and it is simple because we can compare.] sipho: iya, i did understand in english and vernac. i did benefit. from the learners’ responses above it is clear that none of them experienced the use of their home languages as a distracter or constraint. in fact nhlanhla and colbert explained that having their home language versions was helpful. the silences and presences in the learner interviews are interesting. we find it interesting that the 23 using language as a transparent resource interviewer had to explicitly raise the issue of language for the learners to talk about it. this to us suggests the transparency of language as a resource. while the home languages were visible in the sense that the learners were for the first time given written text during the mathematics lessons in their home languages, they are also invisible in that they are not distracting the learners’ attention from the mathematics tasks they are doing. the learners are not focusing on the languages but on the mathematics of the task. as lave and wenger (1991) argue, for a resource to be useful it needs to be both visible and invisible. in their view the invisibility is in the form of unproblematic interpretation and integration (of the artefact – in this case the translated versions of the task) and visibility in the form of extended access to information. while the unusual use of the learners’ home languages in the task can be noticed and used, when invisible, it did not distract learners from the task. the learners were at liberty to choose which language version they wanted to refer to at any time. this, we argue, contributes to the relevance of the strategy we are exploring. the learners are given an opportunity to draw on the linguistic resources they have, and at the same time the presence of english assures them of the fact that they are not loosing access to the language of power, which they so much want to gain access to. conclusion mathematics education begins in language, it advances and stumbles because of language, and its outcomes are often assessed in language. (durkin, 1991, p 3) the above quotation captures the important role of language as a resource in the teaching and learning of mathematics. while it is a resource that can help advance mathematics learning, it can also be a stumbling block for successful learning. the major challenge in multilingual classrooms in south africa is the fact that while the power of english is unavoidable, many learners do not have the level of fluency that enables them to engage in mathematical tasks set in english. in this paper we have explored a strategy for using language as a transparent resource in the teaching and learning mathematics in multilingual classrooms. this strategy is guided by two main principles – the deliberate, proactive and strategic use of the learners’ main languages and the selection of real life, interesting and high cognitive demand mathematics tasks. our analysis shows that with this strategy language becomes a visible but invisible resource in the sense that while learners can draw on different languages at any time they want, language is also invisible because it does not disturb their focus on the mathematics. we argue that our proposed strategy recognizes the political nature of english and thus while it draws on the learners’ home languages, it does not present them as being in opposition to english but as working together with english to make mathematics more accessible t the learners. through our exploration of this strategy in this study, we became more and more aware of the challenge of translating tasks into multiple languages. translation is never a straight-forward enterprise, it is complex. as multilingual speakers of languages from different conceptual worlds we know from experience of living in language, what monolinguals know theoretically from training, that much loss and distortion of meaning can occur in translation. yet translation is part of how meaning is transferred, made and re-negotiated; therefore, this aspect of linguistic activity remains an important consideration. this is why the deliberate, proactive and strategic use of the learners’ home languages pedagogically is so important. much remains to be done! acknowledgements the work presented in this paper was supported by a grant from the national research foundation (ttk2007051500040). any ideas expressed are, however, those of the authors and therefore the national research foundation does not accept any liability. we are grateful to the learners in terence’s class as well as the school management for agreeing to participate in this study. references adendorff, r. 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/pdfxoutputintentprofileselector /documentcmyk /preserveediting true /untaggedcmykhandling /leaveuntagged /untaggedrgbhandling /usedocumentprofile /usedocumentbleed false >> ] >> setdistillerparams << /hwresolution [2400 2400] /pagesize [612.000 792.000] >> setpagedevice